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1. [generated] kernel theorem Acc.inv (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:acc-inv
   0 dependencies 0 dependents
2. Affirming the consequent is a valid inference refuted · search-certificate · F:affirming-the-consequent C:fallacyC:converseC:countermodel +1
   0 dependencies 0 dependents
3. The 1x1 determinant over an abstract commutative ring is its single entry, and it matches Rat.det symbolically proved · kernel-lean · F:alg-comm-ring-det-one
   0 dependencies 0 dependents
4. In an abstract group, a left inverse equals a right inverse proved · kernel-lean · F:alg-group-inv-unique
   1 dependencies 3 dependents
5. In an abstract monoid, the identity element is unique proved · kernel-lean · F:alg-monoid-ident-unique
   1 dependencies 0 dependents
6. In an abstract ordered ring, multiplying a nonneg-vs-le pair on the left preserves the order proved · kernel-lean · F:alg-mul-le-mul-of-nonneg-left
   1 dependencies 0 dependents
7. In an abstract group, the operation cancels on the left proved · kernel-lean · F:alg-mul-left-cancel
   1 dependencies 1 dependents
8. In an abstract ring, x*(-1)=-x proved · kernel-lean · F:alg-mul-neg-one
   3 dependencies 0 dependents
9. In an abstract group, the inverse of the inverse is the original element proved · kernel-lean · F:alg-neg-neg
   2 dependencies 0 dependents
10. In an abstract ordered ring, the generic numeral builder distributes over addition proved · kernel-lean · F:alg-ofnat-add
    1 dependencies 2 dependents
11. In an abstract ordered ring with 0<=1, the generic numeral builder is monotone proved · kernel-lean · F:alg-ofnat-le-ofnat-of-le
    2 dependencies 1 dependents
12. In an abstract monoid, npow distributes over addition of exponents proved · kernel-lean · F:alg-pow-add
    1 dependencies 0 dependents
13. In an abstract ring, any element times zero is zero proved · kernel-lean · F:alg-ring-mul-zero
    2 dependencies 2 dependents
14. In an abstract ring, a-a=0, defined generically proved · kernel-lean · F:alg-sub-self
    2 dependencies 0 dependents
15. In an abstract Setoid-flavored group, a fixed left factor can be cancelled proved · kernel-lean · F:algs-add-left-cancel
    0 dependencies 1 dependents
16. A linear combination of succ n terms splits as the combination without index i plus the i-th term proved · kernel-lean · F:algs-exchange-lincomb-removeat-insertat
    1 dependencies 0 dependents
17. Over a constructive field carried as a setoid, a scalar apart from zero cancels proved · kernel-lean · F:algs-field-mul-left-cancel
    1 dependencies 3 dependents
18. Deleting an index from a family and putting its value back rebuilds the family, at every index proved · kernel-lean · F:algs-index-insertat-removeat
    0 dependencies 1 dependents
19. In an abstract Setoid-flavored group, the inverse of the inverse is the original element (up to equiv) proved · kernel-lean · F:algs-inv-inv
    1 dependencies 1 dependents
20. In an abstract Setoid-flavored group, a left inverse equals a right inverse (up to equiv) proved · kernel-lean · F:algs-inv-unique
    1 dependencies 2 dependents
21. In an abstract Setoid-flavored ring, an element times negative one is equivalent to its negation proved · kernel-lean · F:algs-mul-neg-one
    1 dependencies 1 dependents
22. In an abstract Setoid-flavored ring, any element times zero is equivalent to zero proved · kernel-lean · F:algs-mul-zero
    1 dependencies 5 dependents
23. In an abstract Setoid-flavored ring, double negation is equivalent to the identity proved · kernel-lean · F:algs-neg-neg
    1 dependencies 2 dependents
24. In an abstract Setoid-flavored ordered ring, the generic numeral builder distributes over addition (up to equiv) proved · kernel-lean · F:algs-ofnat-add
    1 dependencies 1 dependents
25. In an abstract Setoid-flavored ordered ring with 0<=1, the generic numeral builder is monotone proved · kernel-lean · F:algs-ofnat-le-ofnat-of-le
    2 dependencies 0 dependents
26. The Bernoulli parameter is its mean, over an arbitrary setoid ordered ring and with no hypothesis on the parameter proved · kernel-lean · F:algs-orderedring-bernoulli-expectation
    2 dependencies 2 dependents
27. The Bernoulli weights are a distribution, discharged from a construction rather than assumed, in every setoid ordered ring proved · kernel-lean · F:algs-orderedring-bernoulli-is-distribution
    3 dependencies 1 dependents
28. The Bernoulli weights are nonnegative at EVERY index, not only inside the two-outcome range proved · kernel-lean · F:algs-orderedring-bernoulli-mass-nonneg
    0 dependencies 1 dependents
29. The Bernoulli variance is q(1-q) in a ring with NO commutativity, because the identity needs commutativity exactly once and it is derivable proved · kernel-lean · F:algs-orderedring-bernoulli-variance
    4 dependencies 1 dependents
30. Markov's inequality holds in every setoid ordered ring, with no strict order and no positivity hypothesis proved · kernel-lean · F:algs-orderedring-markov-inequality
    0 dependencies 1 dependents
31. Negating the RIGHT factor is a separate theorem from negating the left one, in a ring without commutative multiplication proved · kernel-lean · F:algs-orderedring-mul-neg
    1 dependencies 1 dependents
32. Independent events are uncorrelated, over an arbitrary setoid ordered ring proved · kernel-lean · F:algs-orderedring-uncorrelated-of-independent
    0 dependencies 0 dependents
33. The setoid ordered ring's additive unit law is right-handed, so the left one is a theorem proved · kernel-lean · F:algs-orderedring-zero-add
    0 dependencies 3 dependents
34. An antidiagonal walk whose cell family vanishes off the first column collapses to its head cell proved · kernel-lean · F:algs-poly-antidiagfrom-head
    2 dependencies 1 dependents
35. The antidiagonal walk sums to the same value when the cell family is transposed proved · kernel-lean · F:algs-poly-antidiagfrom-rev
    1 dependencies 1 dependents
36. The polynomial ring over an abstract Setoid-flavored commutative ring is a commutative ring proved · kernel-lean · F:algs-poly-comm-ring
    3 dependencies 0 dependents
37. Polynomial multiplication over an abstract Setoid-flavored commutative ring is left-distributive proved · kernel-lean · F:algs-poly-distrib-l
    1 dependencies 4 dependents
38. The polynomial ring over an abstract Setoid-flavored commutative ring is a module over it proved · kernel-lean · F:algs-poly-module
    1 dependencies 1 dependents
39. Polynomial multiplication over an abstract Setoid-flavored commutative ring is associative proved · kernel-lean · F:algs-poly-mul-assoc
    1 dependencies 1 dependents
40. Polynomial multiplication over an abstract Setoid-flavored commutative ring is commutative proved · kernel-lean · F:algs-poly-mul-comm
    1 dependencies 1 dependents
41. In an abstract Setoid-flavored ring, an element minus itself is equivalent to zero proved · kernel-lean · F:algs-sub-self
    1 dependencies 1 dependents
42. The kernel of a homomorphism of abstract Setoid-flavored groups is a subgroup proved · kernel-lean · F:algs-subgroup-ker
    2 dependencies 0 dependents
43. Invariance of basis number at zero: a space with an empty basis has no basis of any other length proved · kernel-lean · F:algs-vectorspace-basis-zero-unique
    1 dependencies 0 dependents
44. In a vector space over a constructive field, a scalar apart from zero can be solved for proved · kernel-lean · F:algs-vectorspace-solve-smul
    1 dependencies 2 dependents
45. The alternating binomial row sum vanishes proved · cas-certificate · F:alternating-binomial-row-sum-zero
    0 dependencies 0 dependents
46. The Apery numbers satisfy Apery's second-order recurrence proved · cas-certificate · F:apery-numbers-recurrence
    0 dependencies 0 dependents
47. No barber shaves exactly those who do not shave themselves proved · smt-clausal · F:barber-no-such-barber C:barber-paradoxC:russells-paradoxC:self-reference +2
    0 dependencies 0 dependents
48. The BCNF repair of the street/city/zip schema rejoins exactly and cannot enforce its own dependency; two other splits lose information proved · search-certificate · F:bcnf-decomposition-lossless-not-dependency-preserving
    1 dependencies 0 dependents
49. The binomial row sum is a power of two proved · cas-certificate · F:binomial-row-sum-two-power
    0 dependencies 0 dependents
50. Boolean conjunction is commutative proved · imported-kernel-lean · F:bool-and-comm C:commutativityC:boolean-algebra
    0 dependencies 0 dependents
51. A 13-gate NAND-only circuit computes a 1-bit full adder, checked by exhaustive replay of all 8 input rows proved · cas-certificate · F:cas-boolean-circuit-nand-only-full-adder
    0 dependencies 0 dependents
52. character tables over exact cyclotomic fields, checked by orthogonality, degrees and Galois consistency computed · cas-certificate · F:cas-character-tables-verified-over-cyclotomic-fields
    0 dependencies 0 dependents
53. The CAS's (x+1)(x-1) = x^2-1 is kernel-reconstructed at a FREE x, and the CAS-refuted variant is rejected by the same kernel proved · cas-certificate · F:cas-difference-of-squares-free-x-kernel-checked
    0 dependencies 0 dependents
54. the CAS enclosure layer computes proof-carrying rational enclosures of real constants and definite integrals computed · cas-certificate · F:cas-enclosure-layer-certified-real-number-enclosures
    0 dependencies 0 dependents
55. For p(x)=x^3-6x on [-3,2], the maximum is interior -- kernel-reconstructed, not just CAS-asserted proved · cas-certificate · F:cas-evt-endpoint-exclusion-cubic-kernel-checked
    0 dependencies 0 dependents
56. For p(x)=x^3-6x, the derivative p'=3x^2-6 changes sign on (-2,-1) -- kernel-reconstructed, not just CAS-asserted proved · cas-certificate · F:cas-extremum-deriv-sign-bracket-kernel-checked
    0 dependencies 0 dependents
57. Extreme Value Theorem for p(x)=x^3-6x on [-3,2]: axeyum-cas names the interior argmax -sqrt(2) exactly proved · cas-certificate · F:cas-extremum-irrational-argmax
    0 dependencies 0 dependents
58. x^400 + x^5 + x^3 + x^2 + 1 is irreducible over GF(2), by a Rabin certificate checked twice with disjoint arithmetic proved · cas-certificate · F:cas-gf2-degree-400-trinomial-irreducible
    0 dependencies 0 dependents
59. No trinomial x^8+x^k+1 is irreducible over GF(2), established by a shard-manifest Exhausted row whose exhaustion is re-run rather than trusted proved · cas-certificate · F:cas-gf2-degree-8-trinomial-exhaustion-rederived
    0 dependencies 0 dependents
60. Karatsuba's 3-multiplication algorithm for degree-2 GF(2) polynomial products replays exactly against all 12 target coefficients proved · cas-certificate · F:cas-gf2-tensor-karatsuba-degree-2-rank-three
    0 dependencies 0 dependents
61. Gosper's algorithm records WHICH of its acceptance modes admitted an antidifference, and correctly reports the geometric identity for k*2^k as weaker than the full telescoping test it uses for k proved · cas-certificate · F:cas-gosper-acceptance-mode-distinguishes-geometric-from-telescoping
    0 dependencies 0 dependents
62. The polynomial system x+y=3, xy=5, x^2+y^2=1 is inconsistent, witnessed by an explicit cofactor identity summing to the constant 1 proved · cas-certificate · F:cas-groebner-cofactor-unit-ideal-witness
    0 dependencies 0 dependents
63. Inverse function theorem for p(x)=x^5+x on [0,2]: axeyum-cas names p^-1(3) exactly as a degree-5 algebraic number proved · cas-certificate · F:cas-inverse-quintic-degree-five-witness
    0 dependencies 0 dependents
64. x^3-2 has exactly one real root in (1,2), named exactly by the CAS's IVT certificate proved · cas-certificate · F:cas-ivt-cbrt2-in-1-2
    0 dependencies 0 dependents
65. The sign bracket of x^4-2 on (1,2) is kernel-reconstructed -- the one-degree-higher cost-curve point proved · cas-certificate · F:cas-ivt-degree4-sign-bracket-kernel-checked-cost-curve
    0 dependencies 0 dependents
66. The sign bracket of x^3-2 on (1,2) is kernel-reconstructed, not just CAS-internal proved · cas-certificate · F:cas-ivt-sign-bracket-cbrt2-kernel-checked
    0 dependencies 0 dependents
67. The integral of 3x^2+2x is x^3+x^2, and the answer proves itself: differentiating it back and zero-testing against the integrand certifies exact equality proved · cas-certificate · F:cas-lib-integrate-polynomial-certified
    0 dependencies 0 dependents
68. finite matrix groups over F_p and over Q: orders, isomorphism types, and a decided finiteness question computed · cas-certificate · F:cas-matrix-groups-orders-and-isomorphisms
    0 dependencies 0 dependents
69. Mean Value Theorem for p(x)=x^3 on [0,3]: axeyum-cas names the witness c = sqrt(3) exactly proved · cas-certificate · F:cas-mvt-cubic-witness-sqrt3
    0 dependencies 0 dependents
70. For p(x)=x^3 on [0,3], the secant endpoints p(3)=27 and p(0)=0 are kernel-reconstructed, not just CAS-asserted proved · cas-certificate · F:cas-mvt-secant-endpoints-kernel-checked
    0 dependencies 0 dependents
71. axeyum-cas's compositeness certificate is independently re-checked across five composites, including the smallest Carmichael number proved · cas-certificate · F:cas-ntheory-compositeness-certificate
    0 dependencies 0 dependents
72. axeyum-cas's CRT certificate (both the solution and the inconsistency direction) is independently re-checked across six systems, and every one of them is re-derived inside the kernel proved · cas-certificate · F:cas-ntheory-crt-certificate
    0 dependencies 0 dependents
73. axeyum-cas's prime factorization certificate is independently re-checked across eight instances, including negatives and units proved · cas-certificate · F:cas-ntheory-factorization-certificate
    1 dependencies 0 dependents
74. axeyum-cas's Pratt certificate conditions are re-derived inside the kernel for every prime below the measured unary-numeral ceiling proved · cas-certificate · F:cas-ntheory-pratt-certificate-kernel-reconstructed
    1 dependencies 0 dependents
75. 2^89-1 is prime: axeyum-cas's Pratt certificate is independently re-checked proved · cas-certificate · F:cas-ntheory-pratt-primality-mersenne89
    0 dependencies 2 dependents
76. The mixed_general_case partial-fraction certificate's coefficient-matching identity is kernel-reconstructed over Rat, needing a NEW Rational-coefficient poly x poly bridge that did not exist before this fact proved · cas-certificate · F:cas-partial-fractions-mixed-general-case-kernel-checked
    1 dependencies 0 dependents
77. Partial-fraction decomposition of (x+1)/((x-1)^2(x^2+1)): axeyum-cas certifies all four rungs in one general construction proved · cas-certificate · F:cas-partial-fractions-mixed-general-case
    0 dependencies 1 dependents
78. The real root of x^5 - x - 1 is irrational: axeyum-cas decides it and certifies the answer by the rational root theorem proved · cas-certificate · F:cas-quintic-real-root-is-irrational
    0 dependencies 0 dependents
79. The Horowitz-Ostrogradsky split of x/(x-1)^2 has rational part -1/(x-1), verified by an exact polynomial identity plus a divisibility guard neither of which suffices alone proved · cas-certificate · F:cas-ratint-horowitz-x-over-x-minus-one-squared
    0 dependencies 0 dependents
80. The Smith normal form of a concrete singular-looking 3x3 integer matrix is diag(2,6,12), with the invariant-factor chain verified proved · cas-certificate · F:cas-smith-normal-form-two-six-twelve
    0 dependencies 0 dependents
81. A cubic planar field is certified globally exponentially stable with an exactly rational decay rate 1/26 proved · cas-certificate · F:cas-sos-damped-rotation-lyapunov
    0 dependencies 0 dependents
82. An unbounded-horizon unreachability result for a cubic planar system, from one barrier and two Positivstellensatz multipliers proved · cas-certificate · F:cas-sos-energy-barrier-unreachability
    0 dependencies 0 dependents
83. The Motzkin form is nonnegative and is not a sum of squares, certified both ways in exact rationals proved · cas-certificate · F:cas-sos-motzkin-psd-not-sos
    0 dependencies 0 dependents
84. Taylor's theorem with Lagrange remainder for p(x)=x^4 at n=1: axeyum-cas names the witness xi = sqrt(2/3) exactly proved · cas-certificate · F:cas-taylor-quartic-lagrange-witness
    0 dependencies 0 dependents
85. For p(x)=x^4 at a=0,n=1,b=2, the Taylor remainder's left-hand side p(2)-T_1(2)=16 is kernel-reconstructed, not just CAS-asserted proved · cas-certificate · F:cas-taylor-remainder-lhs-kernel-checked
    0 dependencies 0 dependents
86. Cassini's identity restated through det2: det2 applied to the ofInt-cast Fibonacci entries equals ofInt((-1)^(n+1)) proved · kernel-lean · F:cassini-as-determinant-of-a-matrix-power
    4 dependencies 0 dependents
87. Cassini's identity: fib(n+2)*fib(n) - fib(n+1)^2 = (-1)^(n+1), over the constructed integers proved · kernel-lean · F:cassini-identity-over-constructed-integers
    16 dependencies 1 dependents
88. The forgetful projection from groups to monoids is a functor proved · kernel-lean · F:cats-forget-grp-mon
    1 dependencies 0 dependents
89. The category of groups exists, with morphisms bundled as a subtype proved · kernel-lean · F:cats-grp-bundled-hom
    1 dependencies 2 dependents
90. The product of two groups is a categorical product in CatS.grp, and it costs no congruence bookkeeping proved · kernel-lean · F:cats-grp-product
    2 dependencies 0 dependents
91. Two initial objects of a setoid-enriched category are isomorphic proved · kernel-lean · F:cats-initial-unique
    0 dependencies 6 dependents
92. The product's universal property in a setoid-enriched category needs three conjuncts, not one proved · kernel-lean · F:cats-is-product
    1 dependencies 2 dependents
93. Functors between setoid-enriched categories compose proved · kernel-lean · F:cats-isfunctor-comp
    1 dependencies 0 dependents
94. Between two deloopings of a monoid map a natural transformation is exactly an intertwiner proved · kernel-lean · F:cats-isnat-ofmonoid
    1 dependencies 0 dependents
95. The natural numbers are an initial object of the category of pointed unary algebras proved · kernel-lean · F:cats-nat-initial-object
    2 dependencies 0 dependents
96. Two products of the same pair are isomorphic, by the isomorphism the universal property names proved · kernel-lean · F:cats-product-unique-upto-iso
    2 dependencies 0 dependents
97. The Cauchy-Schwarz inequality over the constructed reals, in squared form, with the dot product a genuine inner product proved · kernel-lean · F:cauchy-schwarz-over-constructed-plane
    21 dependencies 4 dependents
98. The Chu-Vandermonde convolution satisfies a first-order recurrence in p proved · cas-certificate · F:chu-vandermonde-convolution-recurrence
    0 dependencies 1 dependents
99. Chu-Vandermonde convolution, closed form at symbolic parameters proved · cas-certificate · F:chu-vandermonde-convolution
    1 dependencies 0 dependents
100. A circle is exactly the conic with A = C and B = 0 proved · kernel-lean · F:circle-is-the-a-eq-c-b-zero-conic
     15 dependencies 0 dependents
101. Collatz conjecture conjectured · route not assigned · F:collatz-reaches-one C:collatz-conjectureC:collatz-sequenceC:parity
     0 dependencies 0 dependents
102. [generated] kernel theorem Complex.abs_add_le (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-add-le
     32 dependencies 8 dependents
103. [generated] kernel theorem Complex.abs_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-congr
     2 dependencies 13 dependents
104. [generated] kernel theorem Complex.abs_I (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-i
     12 dependencies 1 dependents
105. [generated] kernel theorem Complex.abs_im_le (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-im-le
     12 dependencies 0 dependents
106. [generated] kernel theorem Complex.abs_le_add_abs_sub (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-le-add-abs-sub
     12 dependencies 0 dependents
107. [generated] kernel theorem Complex.abs_mul_le_of_bounds (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-mul-le-of-bounds
     8 dependencies 4 dependents
108. [generated] kernel theorem Complex.abs_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-mul
     5 dependencies 2 dependents
109. [generated] kernel theorem Complex.abs_neg (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-neg
     14 dependencies 2 dependents
110. [generated] kernel theorem Complex.abs_nonneg (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-nonneg
     4 dependencies 4 dependents
111. [generated] kernel theorem Complex.abs_ofReal (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-ofreal
     10 dependencies 2 dependents
112. [generated] kernel theorem Complex.abs_one (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-one
     7 dependencies 0 dependents
113. [generated] kernel theorem Complex.abs_re_le (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-re-le
     11 dependencies 0 dependents
114. [generated] kernel theorem Complex.abs_sub_le (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-sub-le
     5 dependencies 0 dependents
115. [generated] kernel theorem Complex.abs_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-abs-zero
     11 dependencies 3 dependents
116. [generated] kernel theorem Complex.add_assoc (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-add-assoc
     5 dependencies 4 dependents
117. Addition on the constructed complex numbers is commutative proved · kernel-lean · F:complex-add-comm C:commutativity
     7 dependencies 1 dependents
118. Addition on the constructed complex numbers respects Complex.Equiv on both arguments proved · kernel-lean · F:complex-add-congr
     1 dependencies 22 dependents
119. [generated] kernel theorem Complex.add_div (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-add-div
     12 dependencies 1 dependents
120. Every constructed complex number has an additive inverse proved · kernel-lean · F:complex-add-neg
     6 dependencies 3 dependents
121. [generated] kernel theorem Complex.add_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-add-pow
     37 dependencies 0 dependents
122. Zero is a right additive identity on the constructed complex numbers proved · kernel-lean · F:complex-add-zero
     5 dependencies 9 dependents
123. No relation on the constructed complex numbers satisfies seven of the Real package's ordered-ring laws proved · kernel-lean · F:complex-admits-no-compatible-order C:complex-numbersC:ordered-field
     5 dependencies 0 dependents
124. [generated] kernel theorem Complex.apart_irrefl (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-apart-irrefl
     15 dependencies 0 dependents
125. [generated] kernel theorem Complex.apart_of_normSq_pos (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-apart-of-normsq-pos
     10 dependencies 0 dependents
126. [generated] kernel theorem Complex.apart_symm (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-apart-symm
     16 dependencies 0 dependents
127. [generated] kernel theorem Complex.bounded_on_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-bounded-on-add
     8 dependencies 0 dependents
128. [generated] kernel theorem Complex.bounded_on_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-bounded-on-mul
     8 dependencies 0 dependents
129. [generated] kernel theorem Complex.bounded_on_unfold (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-bounded-on-unfold
     0 dependencies 0 dependents
130. Complex conjugation is additive proved · kernel-lean · F:complex-conj-add C:complex-conjugate
     9 dependencies 0 dependents
131. [generated] kernel theorem Complex.conj_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-congr
     1 dependencies 0 dependents
132. [generated] kernel theorem Complex.conj_conj (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-conj
     9 dependencies 0 dependents
133. [generated] kernel theorem Complex.conj_div (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-div
     6 dependencies 0 dependents
134. [generated] kernel theorem Complex.conj_I (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-i
     7 dependencies 0 dependents
135. [generated] kernel theorem Complex.conj_inv (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-inv
     16 dependencies 1 dependents
136. Complex conjugation is multiplicative proved · kernel-lean · F:complex-conj-mul C:complex-conjugate
     13 dependencies 2 dependents
137. [generated] kernel theorem Complex.conj_ofReal (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-ofreal
     7 dependencies 0 dependents
138. [generated] kernel theorem Complex.conj_one (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-one
     7 dependencies 1 dependents
139. [generated] kernel theorem Complex.conj_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-pow
     5 dependencies 0 dependents
140. [generated] kernel theorem Complex.conj_sub (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-sub
     9 dependencies 0 dependents
141. [generated] kernel theorem Complex.conj_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-conj-zero
     7 dependencies 0 dependents
142. [generated] kernel theorem Complex.div_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-div-congr
     2 dependencies 0 dependents
143. [generated] kernel theorem Complex.div_mul_cancel (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-div-mul-cancel
     6 dependencies 0 dependents
144. [generated] kernel theorem Complex.div_self (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-div-self
     1 dependencies 1 dependents
145. [generated] kernel theorem Complex.eq_conj_iff_real (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-eq-conj-iff-real
     20 dependencies 0 dependents
146. [generated] kernel theorem Complex.eq_zero_of_normSq_eq_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-eq-zero-of-normsq-eq-zero
     12 dependencies 1 dependents
147. [generated] kernel theorem Complex.Equiv.not_zero_I (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-equiv-not-zero-i
     1 dependencies 0 dependents
148. [generated] kernel theorem Complex.Equiv.not_zero_one (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-equiv-not-zero-one
     1 dependencies 0 dependents
149. CReal.Equiv-lifted equivalence on the constructed complex numbers is reflexive proved · kernel-lean · F:complex-equiv-refl
     1 dependencies 38 dependents
150. Complex.Equiv is symmetric proved · kernel-lean · F:complex-equiv-symm
     1 dependencies 19 dependents
151. Complex.Equiv is transitive proved · kernel-lean · F:complex-equiv-trans
     1 dependencies 40 dependents
152. factorQuotient has the expected degree bound: polyDegreeLt (factorQuotient c a n) n proved · kernel-lean · F:complex-factorquotient-degreelt
     2 dependencies 0 dependents
153. factorQuotient's recursive relationship across a degree increment proved · kernel-lean · F:complex-factorquotient-succ-eq
     12 dependencies 0 dependents
154. factorQuotient: the COMPUTED synthetic-division quotient, never extracted by Exists-elimination proved · kernel-lean · F:complex-factorquotient
     0 dependencies 0 dependents
155. The complex finite geometric series as a quotient proved · kernel-lean · F:complex-geom-series-div C:geometric-series
     9 dependencies 0 dependents
156. [generated] kernel theorem Complex.geom_sum_eq_zero_of_root_of_unity (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-geom-sum-eq-zero-of-root-of-unity
     24 dependencies 0 dependents
157. [generated] kernel theorem Complex.hasDerivative_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-add
     28 dependencies 1 dependents
158. [generated] kernel theorem Complex.hasDerivative_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-congr
     9 dependencies 1 dependents
159. [generated] kernel theorem Complex.hasDerivative_const (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-const
     19 dependencies 1 dependents
160. [generated] kernel theorem Complex.hasDerivative_id (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-id
     22 dependencies 2 dependents
161. [generated] kernel theorem Complex.hasDerivative_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-mul
     36 dependencies 1 dependents
162. [generated] kernel theorem Complex.hasDerivative_neg (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-neg
     17 dependencies 1 dependents
163. [generated] kernel theorem Complex.hasDerivative_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivative-pow
     20 dependencies 1 dependents
164. [generated] kernel theorem Complex.HasDerivativeOn.spec (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-hasderivativeon-spec
     0 dependencies 5 dependents
165. [generated] kernel theorem Complex.holomorphic_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-add
     2 dependencies 0 dependents
166. [generated] kernel theorem Complex.holomorphic_const (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-const
     1 dependencies 0 dependents
167. [generated] kernel theorem Complex.holomorphic_id (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-id
     1 dependencies 0 dependents
168. [generated] kernel theorem Complex.holomorphic_neg (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-neg
     2 dependencies 0 dependents
169. [generated] kernel theorem Complex.holomorphic_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-pow
     1 dependencies 0 dependents
170. [generated] kernel theorem Complex.holomorphic_spec (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-holomorphic-spec
     0 dependencies 2 dependents
171. hornerFromTop's diagonal equals polyEval: the sum-level bridge proved · kernel-lean · F:complex-hornerfromtop-diag-eq-polyeval
     24 dependencies 0 dependents
172. hornerFromTop's recursive step: hornerFromTop c a (m+1) (j+1) = hornerFromTop c a m j + a^(j+1)*c(m+1) proved · kernel-lean · F:complex-hornerfromtop-succsucc
     0 dependencies 2 dependents
173. hornerFromTop's base equation on the inner recursion: hornerFromTop c a (m+1) 0 = c(m+1) proved · kernel-lean · F:complex-hornerfromtop-succzero
     0 dependencies 0 dependents
174. hornerFromTop's base equation on the outer recursion: hornerFromTop c a 0 j = c 0 proved · kernel-lean · F:complex-hornerfromtop-zero
     0 dependencies 1 dependents
175. hornerFromTop: the synthetic-division partial-sum accumulator, built by nested Nat.rec proved · kernel-lean · F:complex-hornerfromtop
     0 dependencies 0 dependents
176. [generated] kernel theorem Complex.I_is_fourth_root (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-i-is-fourth-root
     16 dependencies 0 dependents
177. The imaginary unit squares to negative one proved · kernel-lean · F:complex-i-sq C:imaginary-unit
     12 dependencies 1 dependents
178. [generated] kernel theorem Complex.im_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-im-congr
     0 dependencies 0 dependents
179. [generated] kernel theorem Complex.inDisc_I_of_two_sided (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-indisc-i-of-two-sided
     2 dependencies 0 dependents
180. [generated] kernel theorem Complex.inDisc_I_offset (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-indisc-i-offset
     17 dependencies 1 dependents
181. [generated] kernel theorem Complex.inDisc_of_two_sided (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-indisc-of-two-sided
     2 dependencies 0 dependents
182. [generated] kernel theorem Complex.inDisc_ofReal_offset (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-indisc-ofreal-offset
     12 dependencies 1 dependents
183. [generated] kernel theorem Complex.inv_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-inv-congr
     4 dependencies 1 dependents
184. A witnessed multiplicative inverse cancels on the constructed complex numbers proved · kernel-lean · F:complex-inv-mul-cancel
     3 dependencies 2 dependents
185. [generated] kernel theorem Complex.inv_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-inv-mul
     8 dependencies 0 dependents
186. Multiplication distributes over addition on the constructed complex numbers proved · kernel-lean · F:complex-left-distrib C:distributive-property
     11 dependencies 4 dependents
187. [generated] kernel theorem Complex.mul_apart_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-mul-apart-zero
     12 dependencies 1 dependents
188. Multiplication on the constructed complex numbers is associative proved · kernel-lean · F:complex-mul-assoc C:associativity
     14 dependencies 5 dependents
189. Multiplication on the constructed complex numbers is commutative proved · kernel-lean · F:complex-mul-comm C:commutativity
     12 dependencies 8 dependents
190. Multiplication on the constructed complex numbers respects Complex.Equiv on both arguments proved · kernel-lean · F:complex-mul-congr
     3 dependencies 18 dependents
191. A complex number times its own conjugate is its squared modulus proved · kernel-lean · F:complex-mul-conj C:complex-modulus
     15 dependencies 0 dependents
192. [generated] kernel theorem Complex.mul_div_assoc (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-mul-div-assoc
     1 dependencies 1 dependents
193. [generated] kernel theorem Complex.mul_eq_zero_not_both_apart_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-mul-eq-zero-not-both-apart-zero
     13 dependencies 0 dependents
194. A constructed complex number with positive squared modulus has a multiplicative inverse proved · kernel-lean · F:complex-mul-inv-cancel C:multiplicative-inverse
     15 dependencies 3 dependents
195. One is a right multiplicative identity on the constructed complex numbers proved · kernel-lean · F:complex-mul-one
     11 dependencies 8 dependents
196. The complex finite geometric-sum identity proved · kernel-lean · F:complex-mul-sub-one-geom C:geometric-series
     22 dependencies 2 dependents
197. [generated] kernel theorem Complex.mul_sumRange (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-mul-sumrange
     5 dependencies 5 dependents
198. Zero annihilates multiplication on the constructed complex numbers proved · kernel-lean · F:complex-mul-zero
     10 dependencies 5 dependents
199. [generated] kernel theorem Complex.neg_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-neg-congr
     1 dependencies 2 dependents
200. [generated] kernel theorem Complex.neg_div (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-neg-div
     13 dependencies 1 dependents
201. [generated] kernel theorem Complex.normSq_add_le (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-add-le
     7 dependencies 1 dependents
202. [generated] kernel theorem Complex.normSq_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-add
     15 dependencies 1 dependents
203. [generated] kernel theorem Complex.normSq_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-congr
     2 dependencies 2 dependents
204. [generated] kernel theorem Complex.normSq_conj (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-conj
     13 dependencies 3 dependents
205. [generated] kernel theorem Complex.normSq_eq_zero_iff (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-eq-zero-iff
     2 dependencies 0 dependents
206. [generated] kernel theorem Complex.normSq_eq_zero_of_eq_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-eq-zero-of-eq-zero
     9 dependencies 2 dependents
207. [generated] kernel theorem Complex.normSq_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-mul
     15 dependencies 4 dependents
208. [generated] kernel theorem Complex.normSq_nonneg (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-nonneg
     5 dependencies 3 dependents
209. [generated] kernel theorem Complex.normSq_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-normsq-pow
     10 dependencies 0 dependents
210. [generated] kernel theorem Complex.ofNat_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofnat-add
     6 dependencies 2 dependents
211. [generated] kernel theorem Complex.ofNat_eq_cast (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofnat-eq-cast
     8 dependencies 0 dependents
212. [generated] kernel theorem Complex.ofNat_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofnat-mul
     8 dependencies 0 dependents
213. [generated] kernel theorem Complex.ofNat_succ (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofnat-succ
     0 dependencies 0 dependents
214. [generated] kernel theorem Complex.ofNat_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofnat-zero
     0 dependencies 0 dependents
215. [generated] kernel theorem Complex.ofReal_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofreal-add
     6 dependencies 1 dependents
216. [generated] kernel theorem Complex.ofReal_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ofreal-mul
     11 dependencies 0 dependents
217. [generated] kernel theorem Complex.one_is_root_of_unity (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-one-is-root-of-unity
     4 dependencies 1 dependents
218. polyAdd is pointwise coefficient-function addition proved · kernel-lean · F:complex-polyadd
     0 dependencies 0 dependents
219. polyDegreeLt is closed under polyAdd at a shared bound proved · kernel-lean · F:complex-polydegreelt-polyadd
     4 dependencies 0 dependents
220. polyDegreeLt is subadditive under polyMul (the Cauchy-product convolution) proved · kernel-lean · F:complex-polydegreelt-polymul
     28 dependencies 0 dependents
221. polyDegreeLt is closed under polyScale proved · kernel-lean · F:complex-polydegreelt-polyscale
     4 dependencies 0 dependents
222. polyDegreeLt is a PROPOSITION, deliberately not a computed degree proved · kernel-lean · F:complex-polydegreelt
     0 dependencies 0 dependents
223. polyEval is additive over polyAdd, up to Complex.Equiv proved · kernel-lean · F:complex-polyeval-polyadd
     16 dependencies 0 dependents
224. polyEval is multiplicative over polyMul -- ONLY under degree bounds proved · kernel-lean · F:complex-polyeval-polymul
     44 dependencies 0 dependents
225. polyEval is linear over polyScale, up to Complex.Equiv proved · kernel-lean · F:complex-polyeval-polyscale
     19 dependencies 0 dependents
226. polyEval's recursive unfolding: one more Horner step proved · kernel-lean · F:complex-polyeval-succ
     0 dependencies 1 dependents
227. polyEval at length 0 is the zero polynomial's value proved · kernel-lean · F:complex-polyeval-zero
     0 dependencies 1 dependents
228. polyEval is the total Horner evaluation of a truncated polynomial proved · kernel-lean · F:complex-polyeval
     0 dependencies 0 dependents
229. polyMul is the total antidiagonal (Cauchy-product) convolution proved · kernel-lean · F:complex-polymul
     0 dependencies 0 dependents
230. polyScale is uniform coefficient-function scaling by a constant proved · kernel-lean · F:complex-polyscale
     0 dependencies 0 dependents
231. [generated] kernel theorem Complex.pos_bound_conj (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-pos-bound-conj
     4 dependencies 2 dependents
232. [generated] kernel theorem Complex.pow_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-pow-add
     6 dependencies 2 dependents
233. [generated] kernel theorem Complex.pow_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-pow-mul
     4 dependencies 1 dependents
234. [generated] kernel theorem Complex.pow_succ (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-pow-succ
     0 dependencies 1 dependents
235. [generated] kernel theorem Complex.pow_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-pow-zero
     0 dependencies 1 dependents
236. [generated] kernel theorem Complex.ptolemy_identity (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ptolemy-identity
     15 dependencies 1 dependents
237. [generated] kernel theorem Complex.ptolemy_inequality_sq (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-ptolemy-inequality-sq
     6 dependencies 0 dependents
238. [generated] kernel theorem Complex.re_add_im (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-re-add-im
     12 dependencies 0 dependents
239. [generated] kernel theorem Complex.re_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-re-congr
     0 dependencies 0 dependents
240. The complex numbers are constructible in this kernel at zero trusted declarations, as a pair setoid over the constructed reals proved · kernel-lean · F:complex-ring-constructed-axiom-free C:complex-numbersC:real-numbers
     15 dependencies 1 dependents
241. [generated] kernel theorem Complex.root_of_unity_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-root-of-unity-mul
     19 dependencies 0 dependents
242. [generated] kernel theorem Complex.root_of_unity_pow (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-root-of-unity-pow
     7 dependencies 0 dependents
243. [generated] kernel theorem Complex.sub_div (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sub-div
     5 dependencies 0 dependents
244. [generated] kernel theorem Complex.sumRange_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-add
     11 dependencies 5 dependents
245. [generated] kernel theorem Complex.sumRange_congr_lt (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-congr-lt
     5 dependencies 4 dependents
246. [generated] kernel theorem Complex.sumRange_congr (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-congr
     3 dependencies 7 dependents
247. [generated] kernel theorem Complex.sumRange_diagonal (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-diagonal
     16 dependencies 1 dependents
248. [generated] kernel theorem Complex.sumRange_mul_double (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-mul-double
     4 dependencies 2 dependents
249. [generated] kernel theorem Complex.sumRange_mul_eq_diag_add_corner (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-mul-eq-diag-add-corner
     3 dependencies 1 dependents
250. [generated] kernel theorem Complex.sumRange_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-mul
     4 dependencies 1 dependents
251. [generated] kernel theorem Complex.sumRange_rect_eq_diag_add_corner (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-rect-eq-diag-add-corner
     10 dependencies 2 dependents
252. [generated] kernel theorem Complex.sumRange_shiftFront (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-shiftfront
     5 dependencies 1 dependents
253. [generated] kernel theorem Complex.sumRange_split (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-split
     6 dependencies 2 dependents
254. [generated] kernel theorem Complex.sumRange_succ (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-succ
     0 dependencies 0 dependents
255. [generated] kernel theorem Complex.sumRange_swap (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-swap
     6 dependencies 0 dependents
256. [generated] kernel theorem Complex.sumRange_zero (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-sumrange-zero
     0 dependencies 0 dependents
257. [generated] kernel theorem Complex.uniformlyContinuous_add (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuous-add
     22 dependencies 0 dependents
258. [generated] kernel theorem Complex.uniformlyContinuous_const (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuous-const
     12 dependencies 0 dependents
259. [generated] kernel theorem Complex.uniformlyContinuous_id (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuous-id
     0 dependencies 1 dependents
260. [generated] kernel theorem Complex.uniformlyContinuous_mul (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuous-mul
     36 dependencies 0 dependents
261. [generated] kernel theorem Complex.uniformlyContinuous_of_hasDerivative (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuous-of-hasderivative
     36 dependencies 0 dependents
262. [generated] kernel theorem Complex.UniformlyContinuousOn.spec (complex prelude, axiom-free, prose not curated) proved · kernel-lean · F:complex-uniformlycontinuouson-spec
     0 dependencies 3 dependents
263. [generated] kernel theorem congrFun' (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:congrfun
     0 dependencies 0 dependents
264. The conic discriminant is invariant under a rotation of the plane proved · kernel-lean · F:conic-discriminant-invariant-under-rotation
     6 dependencies 0 dependents
265. Six conjunctive-query containment questions decided by homomorphism and by counterexample database, agreeing across three independent routes proved · search-certificate · F:conjunctive-query-containment-homomorphism-certified
     0 dependencies 0 dependents
266. The continuum hypothesis is independent of ZFC open · route not assigned · F:continuum-hypothesis-independent C:continuum-hypothesisC:independence-of-axiomsC:zfc-intuition +2
     0 dependencies 0 dependents
267. A conditional is equivalent to its contrapositive proved · smt-term-level · F:contraposition C:contrapositiveC:logical-equivalenceC:converse
     0 dependencies 1 dependents
268. The probabilistic Cauchy-Schwarz inequality over the constructed rationals: cov(X,Y)^2 <= var(X)*var(Y), in squared form proved · kernel-lean · F:covariance-sq-le-variance-mul-over-constructed-rationals
     8 dependencies 0 dependents
269. [generated] kernel theorem CPoint.apollonius_from_stewart (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-apollonius-from-stewart
     14 dependencies 0 dependents
270. [generated] kernel theorem CPoint.apollonius_median (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-apollonius-median
     20 dependencies 0 dependents
271. [generated] kernel theorem CPoint.area_zero_of_collinear (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-area-zero-of-collinear
     15 dependencies 0 dependents
272. Cauchy-Schwarz on the constructed plane, in its UNSQUARED form proved · kernel-lean · F:cpoint-cauchy-schwarz-unsquared
     5 dependencies 2 dependents
273. The Cauchy-Schwarz inequality proved · kernel-lean · F:cpoint-cauchy-schwarz C:cauchy-schwarz-inequality
     21 dependencies 2 dependents
274. [generated] kernel theorem CPoint.centroid_distSq (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-centroid-distsq
     22 dependencies 0 dependents
275. [generated] kernel theorem CPoint.centroid_median (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-centroid-median
     12 dependencies 0 dependents
276. [generated] kernel theorem CPoint.centroid_ratio (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-centroid-ratio
     19 dependencies 0 dependents
277. Ceva's theorem, sufficiency direction proved · kernel-lean · F:cpoint-ceva-concurrent-of-ratio-product C:cevas-theorem
     16 dependencies 0 dependents
278. [generated] kernel theorem CPoint.ceva_ratio_product_of_concurrent (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-ceva-ratio-product-of-concurrent
     17 dependencies 0 dependents
279. [generated] kernel theorem CPoint.cevian_pair_meet (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cevian-pair-meet
     16 dependencies 0 dependents
280. [generated] kernel theorem CPoint.circumcentre_difference_dots (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-difference-dots
     12 dependencies 1 dependents
281. [generated] kernel theorem CPoint.circumcentre_identity (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-identity
     8 dependencies 1 dependents
282. [generated] kernel theorem CPoint.circumcentre_on_perp_bisectors (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-on-perp-bisectors
     1 dependencies 1 dependents
283. [generated] kernel theorem CPoint.circumcentre_orthocentre_construction (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-orthocentre-construction
     16 dependencies 0 dependents
284. [generated] kernel theorem CPoint.circumcentre_third_distance (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-third-distance
     10 dependencies 3 dependents
285. [generated] kernel theorem CPoint.circumcentre_unique (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-circumcentre-unique
     16 dependencies 0 dependents
286. [generated] kernel theorem CPoint.collinear_of_area_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-collinear-of-area-zero
     16 dependencies 0 dependents
287. [generated] kernel theorem CPoint.cross_annihilates_difference (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cross-annihilates-difference
     16 dependencies 1 dependents
288. [generated] kernel theorem CPoint.cross_self_left (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cross-self-left
     8 dependencies 0 dependents
289. [generated] kernel theorem CPoint.cross_self_right (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cross-self-right
     7 dependencies 0 dependents
290. [generated] kernel theorem CPoint.cross_swap_bc (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cross-swap-bc
     14 dependencies 0 dependents
291. [generated] kernel theorem CPoint.cross_translate (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-cross-translate
     15 dependencies 0 dependents
292. Squared distance on the constructed plane is symmetric proved · kernel-lean · F:cpoint-distsq-comm
     13 dependencies 6 dependents
293. Squared distance on the constructed plane respects CPoint.Equiv on both arguments proved · kernel-lean · F:cpoint-distsq-congr
     3 dependencies 1 dependents
294. [generated] kernel theorem CPoint.distSq_double_sum_bound (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-distsq-double-sum-bound
     17 dependencies 0 dependents
295. [generated] kernel theorem CPoint.distSq_eq_zero_iff (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-distsq-eq-zero-iff
     2 dependencies 0 dependents
296. [generated] kernel theorem CPoint.distSq_eq_zero_of_equiv (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-distsq-eq-zero-of-equiv
     5 dependencies 1 dependents
297. [generated] kernel theorem CPoint.distSq_self_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-distsq-self-zero
     7 dependencies 1 dependents
298. [generated] kernel theorem CPoint.distSq_triangle_sq_bound (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-distsq-triangle-sq-bound
     18 dependencies 0 dependents
299. The dot product on the constructed plane distributes over addition in its left argument proved · kernel-lean · F:cpoint-dot-add-left
     8 dependencies 5 dependents
300. The dot product on the constructed plane distributes over addition in its right argument proved · kernel-lean · F:cpoint-dot-add-right
     7 dependencies 7 dependents
301. The dot product on the constructed plane is commutative proved · kernel-lean · F:cpoint-dot-comm C:dot-product
     2 dependencies 12 dependents
302. The dot product on the constructed plane respects CPoint.Equiv on both arguments proved · kernel-lean · F:cpoint-dot-congr
     2 dependencies 21 dependents
303. Negating the left argument of the constructed-plane dot product negates the result proved · kernel-lean · F:cpoint-dot-neg-left
     14 dependencies 9 dependents
304. Polarization identity for the sum of two points proved · kernel-lean · F:cpoint-dot-self-add C:polarization-identity
     7 dependencies 10 dependents
305. [generated] kernel theorem CPoint.dot_self_add3 (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-dot-self-add3
     5 dependencies 1 dependents
306. A point's dot product with itself is non-negative proved · kernel-lean · F:cpoint-dot-self-nonneg C:non-negativity
     6 dependencies 2 dependents
307. Polarization identity for the difference of two points proved · kernel-lean · F:cpoint-dot-self-sub C:polarization-identity
     13 dependencies 10 dependents
308. [generated] kernel theorem CPoint.dot_self_zero_iff (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-dot-self-zero-iff
     2 dependencies 0 dependents
309. [generated] kernel theorem CPoint.dot_self_zero_of_eq_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-dot-self-zero-of-eq-zero
     5 dependencies 2 dependents
310. The dot product on the constructed plane distributes over subtraction in its left argument proved · kernel-lean · F:cpoint-dot-sub-left
     14 dependencies 4 dependents
311. The dot product on the constructed plane distributes over subtraction in its right argument proved · kernel-lean · F:cpoint-dot-sub-right
     12 dependencies 5 dependents
312. [generated] kernel theorem CPoint.eq_zero_of_distSq_eq_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-eq-zero-of-distsq-eq-zero
     9 dependencies 1 dependents
313. [generated] kernel theorem CPoint.eq_zero_of_dot_self_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-eq-zero-of-dot-self-zero
     5 dependencies 2 dependents
314. The centroid vector identity anchoring the Euler line proved · kernel-lean · F:cpoint-euler-line C:eulers-line
     15 dependencies 1 dependents
315. [generated] kernel theorem CPoint.euler_quadrilateral (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-euler-quadrilateral
     15 dependencies 0 dependents
316. [generated] kernel theorem CPoint.heron_sixteen_area_sq (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-heron-sixteen-area-sq
     15 dependencies 0 dependents
317. Lagrange's identity in two dimensions proved · kernel-lean · F:cpoint-lagrange-identity C:lagranges-identity
     16 dependencies 0 dependents
318. The squared distance from a point to a linear interpolation of two other points, expanded proved · kernel-lean · F:cpoint-lerp-dist-sq
     17 dependencies 2 dependents
319. Linear interpolation at the scalar one-half is the midpoint, on the constructed plane proved · kernel-lean · F:cpoint-lerp-half-is-midpoint
     16 dependencies 1 dependents
320. [generated] kernel theorem CPoint.lerp_one (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-lerp-one
     10 dependencies 0 dependents
321. [generated] kernel theorem CPoint.lerp_zero (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-lerp-zero
     6 dependencies 0 dependents
322. [generated] kernel theorem CPoint.medial_triangle_cross_quarter (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-medial-triangle-cross-quarter
     15 dependencies 0 dependents
323. [generated] kernel theorem CPoint.menelaus_collinear_of_ratio_product (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-menelaus-collinear-of-ratio-product
     15 dependencies 0 dependents
324. [generated] kernel theorem CPoint.midpoint_dist_sq_quarter (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-midpoint-dist-sq-quarter
     18 dependencies 1 dependents
325. [generated] kernel theorem CPoint.nine_point_centre_equidistant (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-nine-point-centre-equidistant
     8 dependencies 0 dependents
326. [generated] kernel theorem CPoint.nine_point_centre_on_euler_line (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-nine-point-centre-on-euler-line
     14 dependencies 0 dependents
327. [generated] kernel theorem CPoint.nine_point_radius_ab (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-nine-point-radius-ab
     15 dependencies 1 dependents
328. [generated] kernel theorem CPoint.nine_point_radius_bc (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-nine-point-radius-bc
     15 dependencies 1 dependents
329. [generated] kernel theorem CPoint.orthocentre_identity (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-orthocentre-identity
     14 dependencies 1 dependents
330. [generated] kernel theorem CPoint.orthocentre_third_altitude (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-orthocentre-third-altitude
     7 dependencies 0 dependents
331. [generated] kernel theorem CPoint.parallelogram_diagonals_bisect (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-parallelogram-diagonals-bisect
     9 dependencies 0 dependents
332. The parallelogram law proved · kernel-lean · F:cpoint-parallelogram-law C:parallelogram-law
     15 dependencies 0 dependents
333. [generated] kernel theorem CPoint.parallelogram_opposite_sides_eq (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-parallelogram-opposite-sides-eq
     13 dependencies 0 dependents
334. [generated] kernel theorem CPoint.perp_bisector_iff_dot (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-perp-bisector-iff-dot
     22 dependencies 1 dependents
335. [generated] kernel theorem CPoint.perp_bisector_midpoint (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-perp-bisector-midpoint
     20 dependencies 0 dependents
336. [generated] kernel theorem CPoint.power_difference_linear (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-power-difference-linear
     22 dependencies 0 dependents
337. [generated] kernel theorem CPoint.power_of_centre (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-power-of-centre
     6 dependencies 0 dependents
338. [generated] kernel theorem CPoint.power_zero_iff_on_circle (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-power-zero-iff-on-circle
     8 dependencies 1 dependents
339. The Pythagorean theorem restated in squared distance proved · kernel-lean · F:cpoint-pythagoras-distsq C:pythagorean-theorem
     1 dependencies 0 dependents
340. The Pythagorean theorem proved · kernel-lean · F:cpoint-pythagoras C:pythagorean-theorem
     14 dependencies 1 dependents
341. [generated] kernel theorem CPoint.radical_axis_iff_dot (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-radical-axis-iff-dot
     21 dependencies 1 dependents
342. [generated] kernel theorem CPoint.Scalar.centroid_self (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-centroid-self
     12 dependencies 0 dependents
343. [generated] kernel theorem CPoint.Scalar.midpoint_comm (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-midpoint-comm
     3 dependencies 0 dependents
344. [generated] kernel theorem CPoint.Scalar.midpoint_diag_core (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-midpoint-diag-core
     6 dependencies 1 dependents
345. [generated] kernel theorem CPoint.Scalar.midpoint_self (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-midpoint-self
     11 dependencies 0 dependents
346. [generated] kernel theorem CPoint.Scalar.midpoint_vector_swap (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-midpoint-vector-swap
     10 dependencies 2 dependents
347. One minus one-half is one-half, for the constructed-plane scalar one-half proved · kernel-lean · F:cpoint-scalar-one-sub-inv2
     13 dependencies 1 dependents
348. [generated] kernel theorem CPoint.Scalar.sum_of_midpoints_perm (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-sum-of-midpoints-perm
     6 dependencies 1 dependents
349. [generated] kernel theorem CPoint.Scalar.sum_perm (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-scalar-sum-perm
     6 dependencies 2 dependents
350. The constructed-plane scalar three is positive, with an explicit precision-0 bound witness proved · kernel-lean · F:cpoint-scalar-threeposbound
     5 dependencies 5 dependents
351. The constructed-plane scalar two is positive, with an explicit precision-0 bound witness proved · kernel-lean · F:cpoint-scalar-twoposbound
     3 dependencies 16 dependents
352. Stewart's theorem specialised to the median proved · kernel-lean · F:cpoint-stewart-median C:stewarts-theorem
     9 dependencies 1 dependents
353. Stewart's theorem proved · kernel-lean · F:cpoint-stewart C:stewarts-theorem
     19 dependencies 1 dependents
354. [generated] kernel theorem CPoint.thales_converse (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-thales-converse
     22 dependencies 0 dependents
355. Thales' theorem (angle in a semicircle is a right angle) proved · kernel-lean · F:cpoint-thales C:thales-theorem
     22 dependencies 0 dependents
356. [generated] kernel theorem CPoint.two_circles_meet_on_radical_axis (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-two-circles-meet-on-radical-axis
     7 dependencies 0 dependents
357. [generated] kernel theorem CPoint.varignon_diagonals_bisect (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-varignon-diagonals-bisect
     1 dependencies 0 dependents
358. [generated] kernel theorem CPoint.varignon_vector_parallel (cpoint prelude, axiom-free, prose not curated) proved · kernel-lean · F:cpoint-varignon-vector-parallel
     1 dependencies 0 dependents
359. Cramer's rule, forward direction: a solution of a 2x2 rational linear system with nonzero determinant must be the quotient of determinants proved · kernel-lean · F:cramer-rule-forward-direction-over-constructed-rationals
     10 dependencies 0 dependents
360. [generated] kernel theorem CReal.abs_add_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-add-le
     14 dependencies 18 dependents
361. CReal.abs_cases_of_order_decision: the absolute value of a real is equivalent to it or to its negation, on an explicit order-decision hypothesis proved · kernel-lean · F:creal-abs-cases-of-order-decision
     6 dependencies 0 dependents
362. Absolute value on the constructed reals respects CReal.Equiv proved · kernel-lean · F:creal-abs-congr
     2 dependencies 32 dependents
363. [generated] kernel theorem CReal.abs_diff_le_of_deriv_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-diff-le-of-deriv-bound
     27 dependencies 1 dependents
364. [generated] kernel theorem CReal.abs_diff_le_of_small_image (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-diff-le-of-small-image
     25 dependencies 1 dependents
365. [generated] kernel theorem CReal.abs_diff_sub_le_of_deriv_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-diff-sub-le-of-deriv-bound
     11 dependencies 1 dependents
366. [generated] kernel theorem CReal.abs_le_of_two_sided (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-le-of-two-sided
     13 dependencies 10 dependents
367. A two-sided bound implies a bound on the absolute value, on the constructed reals proved · kernel-lean · F:creal-abs-le
     1 dependencies 61 dependents
368. A product of constructed reals is bounded by the product of bounds on its factors' absolute values proved · kernel-lean · F:creal-abs-mul-le-of-bounds
     28 dependencies 17 dependents
369. [generated] kernel theorem CReal.abs_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-nonneg
     5 dependencies 17 dependents
370. [generated] kernel theorem CReal.abs_pow_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-pow-le
     14 dependencies 1 dependents
371. [generated] kernel theorem CReal.abs_sumRange_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-abs-sumrange-le
     16 dependencies 4 dependents
372. Addition on the constructed reals is associative proved · kernel-lean · F:creal-add-assoc C:associativity
     14 dependencies 220 dependents
373. Addition on the constructed reals is commutative proved · kernel-lean · F:creal-add-comm C:commutativity
     2 dependencies 262 dependents
374. Addition on the constructed reals respects CReal.Equiv proved · kernel-lean · F:creal-add-congr C:equivalence-relation
     4 dependencies 283 dependents
375. Addition preserves order on the constructed reals proved · kernel-lean · F:creal-add-le-add
     4 dependencies 107 dependents
376. [generated] kernel theorem CReal.add_lt_add_of_le_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-add-lt-add-of-le-of-lt
     4 dependencies 5 dependents
377. Every constructed real has an additive inverse under CReal.Equiv proved · kernel-lean · F:creal-add-neg C:additive-inverse
     2 dependencies 243 dependents
378. Addition on the constructed reals is right-cancellative proved · kernel-lean · F:creal-add-right-cancel
     7 dependencies 36 dependents
379. Zero is a right additive identity on the constructed reals proved · kernel-lean · F:creal-add-zero C:zero
     9 dependencies 271 dependents
380. Alternating series: the bracketing inequality E(n) <= E(n+j) <= E(n+j+1) for every offset j proved · kernel-lean · F:creal-alternatingbracket
     19 dependencies 1 dependents
381. The alternating-series bracket, dual form: every partial sum sits at or below the odd bracket proved · kernel-lean · F:creal-alternatingbracketupper
     19 dependencies 1 dependents
382. Alternating series: the even partial sum is <= the next (odd-length) partial sum proved · kernel-lean · F:creal-alternatingeleo
     11 dependencies 0 dependents
383. The alternating-series lower bracket, closed against the actual limit proved · kernel-lean · F:creal-alternatinglowerbound
     9 dependencies 2 dependents
384. The alternating-series upper bracket, closed against the actual limit proved · kernel-lean · F:creal-alternatingupperbound
     20 dependencies 2 dependents
385. [generated] kernel theorem CReal.antiderivative_abs_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-antiderivative-abs-le
     7 dependencies 0 dependents
386. [generated] kernel theorem CReal.antitone_of_nonpos_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-antitone-of-nonpos-deriv
     12 dependencies 0 dependents
387. [generated] kernel theorem CReal.apart_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-apart-congr
     1 dependencies 0 dependents
388. [generated] kernel theorem CReal.apart_cotrans (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-apart-cotrans
     1 dependencies 0 dependents
389. [generated] kernel theorem CReal.apart_irrefl (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-apart-irrefl
     1 dependencies 0 dependents
390. CReal.apart_of_not_equiv_of_order_decision: Markov's principle for the reals -- non-equivalence yields apartness -- on an explicit order-decision hypothesis proved · kernel-lean · F:creal-apart-of-not-equiv-of-order-decision
     1 dependencies 0 dependents
391. [generated] kernel theorem CReal.apart_symm (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-apart-symm
     0 dependencies 0 dependents
392. [generated] kernel theorem CReal.apart_zero_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-apart-zero-one
     1 dependencies 0 dependents
393. The constructed reals are Archimedean proved · kernel-lean · F:creal-archimedean C:archimedean-property
     7 dependencies 3 dependents
394. A constructed real's sequence stays within its own regularity bound proved · kernel-lean · F:creal-bound-within
     13 dependencies 24 dependents
395. [generated] kernel theorem CReal.bounded_of_uniformly_continuous (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bounded-of-uniformly-continuous
     59 dependencies 6 dependents
396. [generated] kernel theorem CReal.bounded_on_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bounded-on-add
     19 dependencies 0 dependents
397. [generated] kernel theorem CReal.bounded_on_id_unit (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bounded-on-id-unit
     9 dependencies 1 dependents
398. [generated] kernel theorem CReal.bounded_on_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bounded-on-mul
     8 dependencies 0 dependents
399. [generated] kernel theorem CReal.bounded_on_unfold (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bounded-on-unfold
     0 dependencies 6 dependents
400. [generated] kernel theorem CReal.bucketClampLower (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketclamplower
     13 dependencies 4 dependents
401. [generated] kernel theorem CReal.bucketClampUpper (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketclampupper
     10 dependencies 4 dependents
402. [generated] kernel theorem CReal.bucketClose (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketclose
     40 dependencies 1 dependents
403. [generated] kernel theorem CReal.bucketIndexBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketindexbound
     19 dependencies 1 dependents
404. [generated] kernel theorem CReal.bucketIndexFloorLower (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketindexfloorlower
     14 dependencies 5 dependents
405. [generated] kernel theorem CReal.bucketIndexFloorUpper (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-bucketindexfloorupper
     15 dependencies 4 dependents
406. [generated] kernel theorem CReal.cauchy_of_abs_diff_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cauchy-of-abs-diff-le
     15 dependencies 2 dependents
407. A Cauchy witness transports across a pointwise-Equiv sequence proved · kernel-lean · F:creal-cauchyofpointwiseequiv
     7 dependencies 2 dependents
408. [generated] kernel theorem CReal.clamp_id (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-clamp-id
     9 dependencies 2 dependents
409. [generated] kernel theorem CReal.clamp_mono (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-clamp-mono
     2 dependencies 1 dependents
410. A raw rational Cauchy bound at TWO DIFFERENT sample indices lifts to a CReal closeness bound proved · kernel-lean · F:creal-close-within-of-within-indexed
     20 dependencies 0 dependents
411. A raw rational Cauchy bound on two CReal samples lifts to a CReal closeness bound proved · kernel-lean · F:creal-close-within-of-within
     20 dependencies 1 dependents
412. Uniform continuity on [a,b] gives Equiv-congruence, but ONLY inside [a,b] proved · kernel-lean · F:creal-congrofuniformlycontinuous
     11 dependencies 3 dependents
413. [generated] kernel theorem CReal.constant_of_zero_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-constant-of-zero-deriv
     15 dependencies 1 dependents
414. [generated] kernel theorem CReal.continuous_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-continuous-add
     1 dependencies 0 dependents
415. [generated] kernel theorem CReal.continuous_comp (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-continuous-comp
     0 dependencies 0 dependents
416. [generated] kernel theorem CReal.continuous_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-continuous-const
     1 dependencies 0 dependents
417. [generated] kernel theorem CReal.continuous_id (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-continuous-id
     0 dependencies 0 dependents
418. [generated] kernel theorem CReal.continuous_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-continuous-mul
     1 dependencies 0 dependents
419. [generated] kernel theorem CReal.converges_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-add
     14 dependencies 5 dependents
420. [generated] kernel theorem CReal.converges_bounded (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-bounded
     12 dependencies 1 dependents
421. [generated] kernel theorem CReal.converges_cauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-cauchy
     8 dependencies 4 dependents
422. [generated] kernel theorem CReal.converges_comp_eventually (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-comp-eventually
     29 dependencies 1 dependents
423. [generated] kernel theorem CReal.converges_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-le
     12 dependencies 2 dependents
424. [generated] kernel theorem CReal.converges_lower_bound_shift (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-lower-bound-shift
     16 dependencies 5 dependents
425. [generated] kernel theorem CReal.converges_lower_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-lower-bound
     7 dependencies 3 dependents
426. [generated] kernel theorem CReal.converges_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-mul
     21 dependencies 4 dependents
427. [generated] kernel theorem CReal.converges_neg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-neg
     3 dependencies 2 dependents
428. [generated] kernel theorem CReal.converges_of_cauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-of-cauchy
     7 dependencies 7 dependents
429. [generated] kernel theorem CReal.converges_of_close (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-of-close
     3 dependencies 6 dependents
430. [generated] kernel theorem CReal.converges_of_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-of-const
     3 dependencies 5 dependents
431. [generated] kernel theorem CReal.converges_of_equiv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-of-equiv
     0 dependencies 1 dependents
432. [generated] kernel theorem CReal.converges_of_scaled_cauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-of-scaled-cauchy
     7 dependencies 1 dependents
433. [generated] kernel theorem CReal.converges_squeeze (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-squeeze
     10 dependencies 0 dependents
434. [generated] kernel theorem CReal.converges_sub (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-sub
     2 dependencies 1 dependents
435. A convergent sequence of constructed reals has a unique limit proved · kernel-lean · F:creal-converges-unique C:limit-of-a-sequence
     7 dependencies 5 dependents
436. [generated] kernel theorem CReal.converges_upper_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-converges-upper-bound
     9 dependencies 7 dependents
437. [generated] kernel theorem CReal.cosDominant16Over25CauchyBody (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosdominant16over25cauchybody
     22 dependencies 0 dependents
438. [generated] kernel theorem CReal.cosFn_one_equiv_cosOne (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfn-one-equiv-cosone
     41 dependencies 0 dependents
439. [generated] kernel theorem CReal.cosFnPartialHasDerivative (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnpartialhasderivative
     12 dependencies 1 dependents
440. [generated] kernel theorem CReal.cosFnTerm_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnterm-congr
     1 dependencies 2 dependents
441. [generated] kernel theorem CReal.cosFnTermAbsLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfntermabsle
     18 dependencies 1 dependents
442. [generated] kernel theorem CReal.cosFnTermAbsLeWide (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfntermabslewide
     21 dependencies 1 dependents
443. [generated] kernel theorem CReal.cosFnTermDerivCoeff (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfntermderivcoeff
     17 dependencies 1 dependents
444. [generated] kernel theorem CReal.cosFnTermHasDerivative (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfntermhasderivative
     27 dependencies 1 dependents
445. [generated] kernel theorem CReal.cosFnUniformConverges (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnuniformconverges
     27 dependencies 1 dependents
446. [generated] kernel theorem CReal.cosFnWideHasDerivative (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnwidehasderivative
     6 dependencies 0 dependents
447. [generated] kernel theorem CReal.cosFnWideUniformConverges (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnwideuniformconverges
     40 dependencies 2 dependents
448. [generated] kernel theorem CReal.cosFnWideUniformlyContinuous (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosfnwideuniformlycontinuous
     9 dependencies 0 dependents
449. cos(1)'s even-count partial sums are a lower bracket proved · kernel-lean · F:creal-cosone-alternating-lower
     6 dependencies 1 dependents
450. cos(1)'s odd-count partial sums are an upper bracket proved · kernel-lean · F:creal-cosone-alternating-upper
     6 dependencies 1 dependents
451. cos(1) <= 1/0! = 1, the exact real upper bound on cos(1) proved · kernel-lean · F:creal-cosone-le-exp-term-zero
     8 dependencies 0 dependents
452. cosOne is at most 4 -- a loose, uniform bound, not a sharp estimate proved · kernel-lean · F:creal-cosone-le-four
     32 dependencies 0 dependents
453. cos(1) is nonnegative: 0 <= cos(1) proved · kernel-lean · F:creal-cosone-nonneg
     1 dependencies 0 dependents
454. cos(1) is constructed as a total, axiom-free CReal value proved · kernel-lean · F:creal-cosone
     0 dependencies 0 dependents
455. The cosine-at-1 partial sums converge to cosOne proved · kernel-lean · F:creal-cosoneconverges
     27 dependencies 5 dependents
456. cosSeriesPartial is the n-th partial sum defining cosOne proved · kernel-lean · F:creal-cosseriespartial
     0 dependencies 0 dependents
457. [generated] kernel theorem CReal.cosSeriesPartialIsPowerSeries (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-cosseriespartialispowerseries
     7 dependencies 0 dependents
458. cosTerm is the n-th signed term of the cosine-at-1 Taylor series proved · kernel-lean · F:creal-costerm
     0 dependencies 0 dependents
459. Each cosine series term is bounded by e's own reused geometric dominant proved · kernel-lean · F:creal-costermabsledominant
     31 dependencies 4 dependents
460. crossingClose: f(sampled crossing point) is close to f(c) -- STATED with two UNDISCHARGED hypotheses proved · kernel-lean · F:creal-crossingclose
     16 dependencies 0 dependents
461. The cross-width crossing bound, with both domain hypotheses discharged by construction proved · kernel-lean · F:creal-crossingcloseclamped
     22 dependencies 1 dependents
462. crossingIndex is a total, axiom-free bucket index for a level crossing proved · kernel-lean · F:creal-crossingindex
     0 dependencies 0 dependents
463. crossingLower: a slack lower bound at the sampled crossing bucket proved · kernel-lean · F:creal-crossinglower
     27 dependencies 1 dependents
464. The crossing sample point never falls below its own base point a proved · kernel-lean · F:creal-crossingsamplegea
     9 dependencies 1 dependents
465. crossingSampleLower: the sampled-point form of the slack lower bound proved · kernel-lean · F:creal-crossingsamplelower
     11 dependencies 2 dependents
466. crossingSampleUpper: the sampled-point form of the slack upper bound proved · kernel-lean · F:creal-crossingsampleupper
     14 dependencies 2 dependents
467. crossingUpper: a slack upper bound at the sampled crossing bucket proved · kernel-lean · F:creal-crossingupper
     25 dependencies 1 dependents
468. [generated] kernel theorem CReal.density (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-density
     2 dependencies 0 dependents
469. [generated] kernel theorem CReal.diff_le_of_strict_mono_magnitude (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-diff-le-of-strict-mono-magnitude
     18 dependencies 1 dependents
470. e is the limit of its own defining exponential-series partial sums proved · kernel-lean · F:creal-e-converges
     27 dependencies 4 dependents
471. e <= 4, one uniform bound holding at every index proved · kernel-lean · F:creal-e-le-four
     30 dependencies 0 dependents
472. e <= 3, via a genuine {0, 1, k+2} case split at the mathematical kink proved · kernel-lean · F:creal-e-le-three
     39 dependencies 0 dependents
473. Euler's number e is constructed as a total, axiom-free CReal value proved · kernel-lean · F:creal-e
     0 dependencies 0 dependents
474. [generated] kernel theorem CReal.eq_zero_of_add_eq_zero_of_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-eq-zero-of-add-eq-zero-of-nonneg
     8 dependencies 1 dependents
475. [generated] kernel theorem CReal.eq_zero_of_mul_self_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-eq-zero-of-mul-self-zero
     36 dependencies 5 dependents
476. [generated] kernel theorem CReal.Equiv.not_zero_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-equiv-not-zero-one
     1 dependencies 3 dependents
477. A uniform bound on the pointwise gap builds CReal.Equiv proved · kernel-lean · F:creal-equiv-of-bounded C:equivalence-relation
     10 dependencies 8 dependents
478. [generated] kernel theorem CReal.equiv_of_le_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-equiv-of-le-le
     2 dependencies 23 dependents
479. Pointwise-equal sequences build CReal.Equiv-equivalent reals proved · kernel-lean · F:creal-equiv-of-pointwise C:equivalence-relation
     3 dependencies 15 dependents
480. CReal.Equiv is reflexive proved · kernel-lean · F:creal-equiv-refl C:equivalence-relation
     3 dependencies 377 dependents
481. CReal.Equiv is symmetric proved · kernel-lean · F:creal-equiv-symm C:equivalence-relation
     2 dependencies 325 dependents
482. CReal.Equiv is transitive proved · kernel-lean · F:creal-equiv-trans C:equivalence-relationC:transitivity
     10 dependencies 344 dependents
483. [generated] kernel theorem CReal.equiv_zero_of_rate (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-equiv-zero-of-rate
     15 dependencies 3 dependents
484. [generated] kernel theorem CReal.equiv_zero_of_small (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-equiv-zero-of-small
     16 dependencies 7 dependents
485. [generated] kernel theorem CReal.equivAbsDiffLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-equivabsdiffle
     16 dependencies 2 dependents
486. CReal.evt_approx_max: the Extreme Value Theorem's honest row 1 -- an approximate maximum proved · kernel-lean · F:creal-evt-approx-max
     4 dependencies 0 dependents
487. CReal.evt_attained_max_decides_sign: an attained maximiser for a linear family decides the sign of an arbitrary real proved · kernel-lean · F:creal-evt-attained-max-decides-sign
     23 dependencies 0 dependents
488. [generated] kernel theorem CReal.exp_dominant_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-exp-dominant-nonneg
     9 dependencies 1 dependents
489. [generated] kernel theorem CReal.exp_term_abs_le_dominant (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-exp-term-abs-le-dominant
     9 dependencies 7 dependents
490. [generated] kernel theorem CReal.exp_term_le_dominant (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-exp-term-le-dominant
     15 dependencies 3 dependents
491. [generated] kernel theorem CReal.exp_term_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-exp-term-nonneg
     7 dependencies 6 dependents
492. The dominating geometric partial sums 2*sum(1/2)^i form a Cauchy sequence proved · kernel-lean · F:creal-expdominantcauchy
     7 dependencies 1 dependents
493. [generated] kernel theorem CReal.expFn_one_equiv_e (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expfn-one-equiv-e
     41 dependencies 0 dependents
494. [generated] kernel theorem CReal.expFnTermAbsLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expfntermabsle
     18 dependencies 1 dependents
495. [generated] kernel theorem CReal.expFnUniformConverges (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expfnuniformconverges
     27 dependencies 1 dependents
496. [generated] kernel theorem CReal.expOfModulus_le_trueExpOfModulus (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expofmodulus-le-trueexpofmodulus
     3 dependencies 0 dependents
497. [generated] kernel theorem CReal.expSeriesPartialConverges (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expseriespartialconverges
     3 dependencies 0 dependents
498. [generated] kernel theorem CReal.expSeriesPartialIsPowerSeries (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-expseriespartialispowerseries
     7 dependencies 0 dependents
499. expTerm is antitone: 1/(n+1)! <= 1/n! proved · kernel-lean · F:creal-expterm-antitone
     14 dependencies 4 dependents
500. 1/n! is dominated by the geometric term 2/2^n proved · kernel-lean · F:creal-expterm-le-geom
     16 dependencies 1 dependents
501. expTerm(1) = 1 exactly, as a literal CReal equality proved · kernel-lean · F:creal-expterm-one-eq-one
     2 dependencies 0 dependents
502. expTerm(0) = 1 exactly, as a literal CReal equality proved · kernel-lean · F:creal-expterm-zero-eq-one
     2 dependencies 0 dependents
503. [generated] kernel theorem CReal.expTermSuccScale (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-exptermsuccscale
     13 dependencies 1 dependents
504. [generated] kernel theorem CReal.fermat_interiorExtremum (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-fermat-interiorextremum
     38 dependencies 1 dependents
505. The constructed reals are a field, and the inverse must be existential because a functional one is undefinable proved · kernel-lean · F:creal-field-setoid
     7 dependencies 0 dependents
506. [generated] kernel theorem CReal.fineBlockSum_close (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-fineblocksum-close
     24 dependencies 1 dependents
507. [generated] kernel theorem CReal.fineSample_close (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-finesample-close
     38 dependencies 1 dependents
508. [generated] kernel theorem CReal.fineSample_in_bounds (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-finesample-in-bounds
     34 dependencies 2 dependents
509. [generated] kernel theorem CReal.geom_pair_within (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-pair-within
     6 dependencies 2 dependents
510. [generated] kernel theorem CReal.geom_sum_bounded (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-sum-bounded
     12 dependencies 0 dependents
511. [generated] kernel theorem CReal.geom_tail_bounded_div (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-tail-bounded-div
     12 dependencies 1 dependents
512. [generated] kernel theorem CReal.geom_tail_bounded (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-tail-bounded
     12 dependencies 1 dependents
513. [generated] kernel theorem CReal.geom_tail_within_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-tail-within-le
     2 dependencies 1 dependents
514. [generated] kernel theorem CReal.geom_tail_within (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geom-tail-within
     21 dependencies 1 dependents
515. The base-1/2 geometric partial sums form a Cauchy sequence proved · kernel-lean · F:creal-geomcauchy
     5 dependencies 1 dependents
516. [generated] kernel theorem CReal.geomCauchyBody16Over25 (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomcauchybody16over25
     20 dependencies 0 dependents
517. [generated] kernel theorem CReal.geomCauchyBodyOfGap (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomcauchybodyofgap
     5 dependencies 1 dependents
518. The geometric series sum x^n is Cauchy for any ratio 0<=x<1 proved · kernel-lean · F:creal-geomcauchyoflt
     6 dependencies 1 dependents
519. The ordered-pair Cauchy bound for the general-ratio geometric series proved · kernel-lean · F:creal-geomcauchyofltordered
     13 dependencies 2 dependents
520. [generated] kernel theorem CReal.geomCauchyOrdered16Over25 (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomcauchyordered16over25
     20 dependencies 4 dependents
521. [generated] kernel theorem CReal.geomCauchyOrderedHalf (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomcauchyorderedhalf
     33 dependencies 6 dependents
522. [generated] kernel theorem CReal.geomCauchyOrderedOfGap (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomcauchyorderedofgap
     3 dependencies 2 dependents
523. [generated] kernel theorem CReal.geomHalfInvLeafBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomhalfinvleafbound
     33 dependencies 1 dependents
524. A constant multiple of a general-ratio geometric series is Cauchy proved · kernel-lean · F:creal-geomscaledcauchyoflt
     7 dependencies 3 dependents
525. [generated] kernel theorem CReal.geomYBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomybound
     13 dependencies 1 dependents
526. [generated] kernel theorem CReal.geomYBoundRaw (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-geomyboundraw
     12 dependencies 1 dependents
527. [generated] kernel theorem CReal.hasDerivative_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-add
     31 dependencies 3 dependents
528. Fundamental theorem of calculus, part I, for a uniformly continuous integrand proved · kernel-lean · F:creal-hasderivative-antiderivative-of-uc
     2 dependencies 0 dependents
529. [generated] kernel theorem CReal.hasDerivative_antiderivative (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-antiderivative
     46 dependencies 2 dependents
530. [generated] kernel theorem CReal.hasDerivative_chain_id_sq (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-chain-id-sq
     8 dependencies 0 dependents
531. [generated] kernel theorem CReal.hasDerivative_chain (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-chain
     44 dependencies 1 dependents
532. [generated] kernel theorem CReal.hasDerivative_closeOfEquiv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-closeofequiv
     25 dependencies 3 dependents
533. [generated] kernel theorem CReal.hasDerivative_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-congr
     8 dependencies 6 dependents
534. [generated] kernel theorem CReal.hasDerivative_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-const
     17 dependencies 3 dependents
535. [generated] kernel theorem CReal.hasDerivative_cube (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-cube
     4 dependencies 0 dependents
536. [generated] kernel theorem CReal.hasDerivative_id (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-id
     17 dependencies 4 dependents
537. [generated] kernel theorem CReal.hasDerivative_integral_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-integral-const
     24 dependencies 0 dependents
538. The product rule holds for derivatives of constructed-real functions proved · kernel-lean · F:creal-hasderivative-mul C:product-rule
     41 dependencies 3 dependents
539. [generated] kernel theorem CReal.hasDerivative_neg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-neg
     20 dependencies 5 dependents
540. [generated] kernel theorem CReal.hasDerivative_pow_two (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-pow-two
     7 dependencies 0 dependents
541. [generated] kernel theorem CReal.hasDerivative_pow (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-pow
     17 dependencies 1 dependents
542. [generated] kernel theorem CReal.hasDerivative_smul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-smul
     21 dependencies 2 dependents
543. [generated] kernel theorem CReal.hasDerivative_sq (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-sq
     27 dependencies 3 dependents
544. [generated] kernel theorem CReal.hasDerivative_sub (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-sub
     2 dependencies 6 dependents
545. [generated] kernel theorem CReal.hasDerivative_uniform_limit (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-uniform-limit
     38 dependencies 1 dependents
546. [generated] kernel theorem CReal.hasDerivative_unique (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-hasderivative-unique
     48 dependencies 0 dependents
547. The defining modulus-of-continuity property of CReal.HasDerivativeOn proved · kernel-lean · F:creal-hasderivativeon-spec
     0 dependencies 13 dependents
548. [generated] kernel theorem CReal.integral_abs_le_of_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-abs-le-of-bound
     22 dependencies 3 dependents
549. [generated] kernel theorem CReal.integral_abs_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-abs-le
     23 dependencies 1 dependents
550. The integral of a sum is the sum of the integrals proved · kernel-lean · F:creal-integral-add
     5 dependencies 2 dependents
551. [generated] kernel theorem CReal.integral_by_parts (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-by-parts
     13 dependencies 0 dependents
552. The integral of a constant function is base times height proved · kernel-lean · F:creal-integral-const
     5 dependencies 4 dependents
553. CReal.integral is the limit of its own defining Riemann-sum sequence proved · kernel-lean · F:creal-integral-converges
     2 dependencies 8 dependents
554. Fundamental theorem of calculus, part II (evaluation rule), for a uniformly continuous integrand proved · kernel-lean · F:creal-integral-eq-antideriv-diff-of-uc
     2 dependencies 0 dependents
555. [generated] kernel theorem CReal.integral_eq_antideriv_diff (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-eq-antideriv-diff
     29 dependencies 2 dependents
556. Order passes to the integral proved · kernel-lean · F:creal-integral-le
     21 dependencies 2 dependents
557. A constant factor pulls out of the integral proved · kernel-lean · F:creal-integral-scale
     27 dependencies 0 dependents
558. [generated] kernel theorem CReal.integral_split (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-split
     39 dependencies 1 dependents
559. [generated] kernel theorem CReal.integral_sub_linear_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integral-sub-linear-le
     19 dependencies 1 dependents
560. CReal.integral does not depend on which uniform-continuity witness is supplied proved · kernel-lean · F:creal-integral-witness-independent
     4 dependencies 0 dependents
561. The constructed Riemann integral exists as a total, axiom-free CReal function proved · kernel-lean · F:creal-integral
     0 dependencies 0 dependents
562. [generated] kernel theorem CReal.integralEndpointClose (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integralendpointclose
     73 dependencies 1 dependents
563. [generated] kernel theorem CReal.integralSplitAnywhere (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integralsplitanywhere
     39 dependencies 2 dependents
564. [generated] kernel theorem CReal.integralSplitArbitrary (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-integralsplitarbitrary
     59 dependencies 1 dependents
565. [generated] kernel theorem CReal.inv_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-inv-congr
     8 dependencies 3 dependents
566. [generated] kernel theorem CReal.inv_index_irrelevant (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-inv-index-irrelevant
     2 dependencies 0 dependents
567. [generated] kernel theorem CReal.inv_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-inv-nonneg
     18 dependencies 4 dependents
568. [generated] kernel theorem CReal.inverse_lipschitz_of_pos_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-inverse-lipschitz-of-pos-deriv
     22 dependencies 0 dependents
569. [generated] kernel theorem CReal.invLeOfPosBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-invleofposbound
     14 dependencies 2 dependents
570. CReal.ivt_approx: the constructive approximate Intermediate Value Theorem -- roots to any requested accuracy proved · kernel-lean · F:creal-ivt-approx
     38 dependencies 0 dependents
571. CReal.ivt_bisect_approx: ivt_approx's accuracy bound restated at a named, computable point proved · kernel-lean · F:creal-ivt-bisect-approx
     38 dependencies 2 dependents
572. CReal.ivt_bisect_cauchy_bound: the named bisection points at any two accuracies are close, as a real-valued bound proved · kernel-lean · F:creal-ivt-bisect-cauchy-bound
     11 dependencies 1 dependents
573. CReal.ivt_bisect_cauchy: the named bisection points, under a uniformly positive derivative, form a genuine Cauchy sequence proved · kernel-lean · F:creal-ivt-bisect-cauchy
     2 dependencies 1 dependents
574. CReal.ivt_bisect_invariant: the computable bisection bracket satisfies the same six-part invariant as the existential one proved · kernel-lean · F:creal-ivt-bisect-invariant
     37 dependencies 1 dependents
575. CReal.ivt_exact_root_at: the exact-root theorem generalized to an arbitrary target value proved · kernel-lean · F:creal-ivt-exact-root-at
     16 dependencies 0 dependents
576. IVT's ADR-0603 row 2: an exact root of the plateau family decides the sign of an arbitrary real proved · kernel-lean · F:creal-ivt-exact-root-decides-sign
     18 dependencies 0 dependents
577. CReal.ivt_exact_root: an EXACT root, at the price of a uniformly positive derivative on the whole interval proved · kernel-lean · F:creal-ivt-exact-root
     33 dependencies 1 dependents
578. CReal.ivt_iter: n-fold bisection, bracket width shrinking geometrically as (1/2)^n proved · kernel-lean · F:creal-ivt-iter
     9 dependencies 1 dependents
579. CReal.ivt_step: one bisection step that halves the bracket without ever deciding an exact sign proved · kernel-lean · F:creal-ivt-step
     27 dependencies 1 dependents
580. A constructed real is at most its own absolute value proved · kernel-lean · F:creal-le-abs-self
     1 dependencies 43 dependents
581. [generated] kernel theorem CReal.le_add_of_abs_sub_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-add-of-abs-sub-le
     13 dependencies 2 dependents
582. Adding a nonnegative rational to a constructed real does not decrease it proved · kernel-lean · F:creal-le-add-of-nonneg
     12 dependencies 6 dependents
583. The order on the constructed reals respects CReal.Equiv on both sides proved · kernel-lean · F:creal-le-congr
     3 dependencies 176 dependents
584. The max of two constructed reals is at least the left one proved · kernel-lean · F:creal-le-max-left
     5 dependencies 12 dependents
585. The max of two constructed reals is at least the right one proved · kernel-lean · F:creal-le-max-right
     5 dependencies 10 dependents
586. [generated] kernel theorem CReal.le_min (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-min
     1 dependencies 9 dependents
587. CReal.Equiv-equivalent constructed reals are ordered by <= proved · kernel-lean · F:creal-le-of-equiv
     0 dependencies 46 dependents
588. [generated] kernel theorem CReal.le_of_forall_le_add_rate (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-of-forall-le-add-rate
     14 dependencies 3 dependents
589. [generated] kernel theorem CReal.le_of_forall_le_add_small (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-of-forall-le-add-small
     14 dependencies 4 dependents
590. [generated] kernel theorem CReal.le_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-of-lt
     3 dependencies 33 dependents
591. [generated] kernel theorem CReal.le_of_mul_le_mul_left (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-of-mul-le-mul-left
     11 dependencies 5 dependents
592. [generated] kernel theorem CReal.le_of_sq_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-le-of-sq-le
     3 dependencies 3 dependents
593. The order on the constructed reals is reflexive proved · kernel-lean · F:creal-le-refl
     2 dependencies 126 dependents
594. CReal.le_total_of_order_decision: totality of the order on the constructed reals, on an explicit order-decision hypothesis proved · kernel-lean · F:creal-le-total-of-order-decision
     1 dependencies 1 dependents
595. The order on the constructed reals is transitive proved · kernel-lean · F:creal-le-trans C:order-relationC:transitivity
     7 dependencies 149 dependents
596. Multiplication distributes over addition on the constructed reals proved · kernel-lean · F:creal-left-distrib C:distributivity
     17 dependencies 136 dependents
597. [generated] kernel theorem CReal.limit_dist (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-limit-dist
     13 dependencies 0 dependents
598. [generated] kernel theorem CReal.limitSeq_regular (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-limitseq-regular
     8 dependencies 0 dependents
599. [generated] kernel theorem CReal.lipschitz_of_deriv_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lipschitz-of-deriv-bound
     26 dependencies 1 dependents
600. [generated] kernel theorem CReal.lt_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-congr
     5 dependencies 14 dependents
601. [generated] kernel theorem CReal.lt_cotrans (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-cotrans
     20 dependencies 8 dependents
602. [generated] kernel theorem CReal.lt_irrefl (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-irrefl
     17 dependencies 7 dependents
603. [generated] kernel theorem CReal.lt_of_le_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-of-le-of-lt
     3 dependencies 3 dependents
604. [generated] kernel theorem CReal.lt_of_lt_of_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-of-lt-of-le
     1 dependencies 5 dependents
605. [generated] kernel theorem CReal.lt_trans (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-lt-trans
     2 dependencies 2 dependents
606. CReal.lub_decides_em: a supremum for an arbitrary inhabited bounded set decides an arbitrary proposition proved · kernel-lean · F:creal-lub-decides-em
     8 dependencies 0 dependents
607. [generated] kernel theorem CReal.mag_bound_le_sumRange_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mag-bound-le-sumrange-of-lt
     14 dependencies 0 dependents
608. Constructed-real max respects CReal.Equiv in both arguments proved · kernel-lean · F:creal-max-congr
     4 dependencies 5 dependents
609. max is the least upper bound among its two arguments' common upper bounds proved · kernel-lean · F:creal-max-le
     1 dependencies 14 dependents
610. [generated] kernel theorem CReal.max_mono_right (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-max-mono-right
     4 dependencies 1 dependents
611. [generated] kernel theorem CReal.max_sub_min (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-max-sub-min
     24 dependencies 1 dependents
612. [generated] kernel theorem CReal.maxRange_mono (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-mono
     2 dependencies 1 dependents
613. [generated] kernel theorem CReal.maxRange_self_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-self-le
     2 dependencies 1 dependents
614. [generated] kernel theorem CReal.maxRange_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-succ
     0 dependencies 0 dependents
615. [generated] kernel theorem CReal.maxRange_transport (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-transport
     6 dependencies 1 dependents
616. [generated] kernel theorem CReal.maxRange_ub (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-ub
     3 dependencies 1 dependents
617. [generated] kernel theorem CReal.maxRange_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-maxrange-zero
     0 dependencies 0 dependents
618. [generated] kernel theorem CReal.mesh_count_width (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mesh-count-width
     13 dependencies 12 dependents
619. [generated] kernel theorem CReal.mesh_le_of_ge (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mesh-le-of-ge
     24 dependencies 3 dependents
620. [generated] kernel theorem CReal.meshLevelCount_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-meshlevelcount-succ
     0 dependencies 0 dependents
621. [generated] kernel theorem CReal.meshLevelCount_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-meshlevelcount-zero
     0 dependencies 0 dependents
622. [generated] kernel theorem CReal.meshMax_mono (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-meshmax-mono
     2 dependencies 0 dependents
623. [generated] kernel theorem CReal.meshMax_step_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-meshmax-step-le
     22 dependencies 1 dependents
624. [generated] kernel theorem CReal.meshReciprocalMul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-meshreciprocalmul
     6 dependencies 3 dependents
625. A scaled Archimedean mesh bound: (m+1) * (b-a) * 1/n3 <= 1/n2 given a large enough mesh count proved · kernel-lean · F:creal-meshscaledleofge
     10 dependencies 0 dependents
626. [generated] kernel theorem CReal.min_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-min-congr
     4 dependencies 0 dependents
627. [generated] kernel theorem CReal.min_le_left (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-min-le-left
     5 dependencies 8 dependents
628. [generated] kernel theorem CReal.min_le_right (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-min-le-right
     5 dependencies 10 dependents
629. [generated] kernel theorem CReal.min_mono_left (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-min-mono-left
     4 dependencies 1 dependents
630. [generated] kernel theorem CReal.monotone_of_le_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-monotone-of-le-succ
     2 dependencies 3 dependents
631. [generated] kernel theorem CReal.monotone_of_nonneg_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-monotone-of-nonneg-deriv
     48 dependencies 3 dependents
632. Multiplication on the constructed reals is associative proved · kernel-lean · F:creal-mul-assoc C:associativity
     16 dependencies 108 dependents
633. Multiplication on the constructed reals is commutative proved · kernel-lean · F:creal-mul-comm C:commutativity
     4 dependencies 190 dependents
634. Multiplication on the constructed reals respects CReal.Equiv proved · kernel-lean · F:creal-mul-congr C:equivalence-relation
     15 dependencies 206 dependents
635. A constructed real bounded away from zero has a multiplicative inverse proved · kernel-lean · F:creal-mul-inv-cancel C:multiplicative-inverse
     30 dependencies 35 dependents
636. Constructed-real order is preserved by a nonnegative left multiplier proved · kernel-lean · F:creal-mul-le-mul-of-nonneg-left
     14 dependencies 61 dependents
637. The product of two nonnegative constructed reals is nonnegative proved · kernel-lean · F:creal-mul-nonneg
     17 dependencies 37 dependents
638. One is a right multiplicative identity on the constructed reals proved · kernel-lean · F:creal-mul-one
     8 dependencies 136 dependents
639. [generated] kernel theorem CReal.mul_pos (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-pos
     14 dependencies 2 dependents
640. [generated] kernel theorem CReal.mul_riemannSum (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-riemannsum
     5 dependencies 1 dependents
641. [generated] kernel theorem CReal.mul_self_abs (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-self-abs
     16 dependencies 9 dependents
642. [generated] kernel theorem CReal.mul_self_sqrt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-self-sqrt
     51 dependencies 7 dependents
643. [generated] kernel theorem CReal.mul_sub_one_geom_tail (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-sub-one-geom-tail
     15 dependencies 1 dependents
644. The finite geometric series partial-sum identity over the constructed reals proved · kernel-lean · F:creal-mul-sub-one-geom C:geometric-series
     14 dependencies 2 dependents
645. [generated] kernel theorem CReal.mul_sumRange (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mul-sumrange
     5 dependencies 16 dependents
646. Zero annihilates multiplication on the constructed reals proved · kernel-lean · F:creal-mul-zero C:zero
     2 dependencies 116 dependents
647. A power-series-shaped congruence, DERIVED MECHANICALLY by the setoid congruence deriver rather than hand-built proved · kernel-lean · F:creal-mulpowcongr
     3 dependencies 2 dependents
648. [generated] kernel theorem CReal.mvt_interiorExtremum (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-mvt-interiorextremum
     21 dependencies 0 dependents
649. [generated] kernel theorem CReal.natDivSuccStepLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-natdivsuccsteple
     6 dependencies 1 dependents
650. [generated] kernel theorem CReal.natSqrtLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-natsqrtle
     1 dependencies 1 dependents
651. [generated] kernel theorem CReal.natSqrtLt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-natsqrtlt
     1 dependencies 1 dependents
652. [generated] kernel theorem CReal.natSqrtSpec (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-natsqrtspec
     14 dependencies 2 dependents
653. Negation on the constructed reals respects CReal.Equiv proved · kernel-lean · F:creal-neg-congr C:equivalence-relation
     3 dependencies 148 dependents
654. Negative 4 is at most cosOne -- the matching loose lower bound proved · kernel-lean · F:creal-neg-four-le-cosone
     34 dependencies 0 dependents
655. The negation of a constructed real is at most its absolute value proved · kernel-lean · F:creal-neg-le-abs
     1 dependencies 39 dependents
656. [generated] kernel theorem CReal.neg_le_neg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-neg-le-neg
     1 dependencies 45 dependents
657. Negating both factors leaves a product of constructed reals unchanged up to equivalence proved · kernel-lean · F:creal-neg-mul-neg
     5 dependencies 8 dependents
658. [generated] kernel theorem CReal.neg_sub_swap (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-neg-sub-swap
     8 dependencies 13 dependents
659. (-1)^(2k) = 1 for every k, by plain induction (no parity case split) proved · kernel-lean · F:creal-negonepowdouble
     7 dependencies 3 dependents
660. [generated] kernel theorem CReal.not_apart_one_of_pow_succ_eq_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-apart-one-of-pow-succ-eq-one
     7 dependencies 0 dependents
661. [generated] kernel theorem CReal.not_equiv_abs_neg_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-equiv-abs-neg-one
     4 dependencies 0 dependents
662. [generated] kernel theorem CReal.not_equiv_mul_one_one_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-equiv-mul-one-one-zero
     5 dependencies 0 dependents
663. [generated] kernel theorem CReal.not_equiv_of_apart (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-equiv-of-apart
     4 dependencies 0 dependents
664. [generated] kernel theorem CReal.not_le_one_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-le-one-zero
     3 dependencies 1 dependents
665. [generated] kernel theorem CReal.not_le_zero_neg_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-not-le-zero-neg-one
     8 dependencies 2 dependents
666. [generated] kernel theorem CReal.ofNat_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofnat-add
     3 dependencies 5 dependents
667. [generated] kernel theorem CReal.ofNat_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofnat-le
     3 dependencies 9 dependents
668. [generated] kernel theorem CReal.ofNat_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofnat-mul
     3 dependencies 3 dependents
669. The rational-to-constructed-real embedding is additive proved · kernel-lean · F:creal-ofrat-add
     1 dependencies 57 dependents
670. The rational-to-constructed-real embedding is order-preserving proved · kernel-lean · F:creal-ofrat-le
     5 dependencies 102 dependents
671. The rational-to-constructed-real embedding is multiplicative proved · kernel-lean · F:creal-ofrat-mul
     1 dependencies 51 dependents
672. [generated] kernel theorem CReal.ofRat_neg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofrat-neg
     1 dependencies 7 dependents
673. [generated] kernel theorem CReal.ofRat_pos (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofrat-pos
     4 dependencies 9 dependents
674. [generated] kernel theorem CReal.ofRat_pow (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofrat-pow
     4 dependencies 2 dependents
675. [generated] kernel theorem CReal.ofRat_sub (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ofrat-sub
     1 dependencies 2 dependents
676. [generated] kernel theorem CReal.one_le_pow_of_one_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-one-le-pow-of-one-le
     10 dependencies 0 dependents
677. [generated] kernel theorem CReal.one_pow (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-one-pow
     3 dependencies 2 dependents
678. [generated] kernel theorem CReal.order_reflect_of_pos_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-order-reflect-of-pos-deriv
     3 dependencies 0 dependents
679. pi <= 4, from the same geometric domination the construction already needed proved · kernel-lean · F:creal-pi-le-four
     4 dependencies 0 dependents
680. pi is constructed as a total, axiom-free CReal value, with no root and no intermediate value theorem proved · kernel-lean · F:creal-pi
     0 dependencies 3 dependents
681. polyAdd: total coefficientwise sum of two polynomial coefficient functions proved · kernel-lean · F:creal-polyadd
     0 dependencies 0 dependents
682. polyDegreeLt is closed under polyAdd at the same bound proved · kernel-lean · F:creal-polydegreelt-polyadd
     4 dependencies 0 dependents
683. polyDegreeLt is closed under polyScale proved · kernel-lean · F:creal-polydegreelt-polyscale
     4 dependencies 0 dependents
684. polyDegreeLt: the PROPOSITION that a coefficient function vanishes past a bound, not a computed degree proved · kernel-lean · F:creal-polydegreelt
     0 dependencies 0 dependents
685. polyEval is additive: evaluating a coefficientwise sum equals the sum of the evaluations proved · kernel-lean · F:creal-polyeval-polyadd
     7 dependencies 0 dependents
686. polyEval is scalar-linear: evaluating a scaled polynomial equals scaling the evaluation proved · kernel-lean · F:creal-polyeval-polyscale
     6 dependencies 0 dependents
687. polyEval's recursive step: evaluating one more coefficient adds one more term proved · kernel-lean · F:creal-polyeval-succ
     0 dependencies 0 dependents
688. polyEval at degree bound 0 is zero (the empty sum) proved · kernel-lean · F:creal-polyeval-zero
     0 dependencies 0 dependents
689. polyEval: total Horner-style evaluation of a coefficient function over CReal proved · kernel-lean · F:creal-polyeval
     0 dependencies 0 dependents
690. polyScale: total scalar multiplication of a polynomial coefficient function proved · kernel-lean · F:creal-polyscale
     0 dependencies 0 dependents
691. [generated] kernel theorem CReal.pos_bound_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pos-bound-of-lt
     9 dependencies 8 dependents
692. [generated] kernel theorem CReal.pos_of_pos_bound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pos-of-pos-bound
     7 dependencies 0 dependents
693. [generated] kernel theorem CReal.pow_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-add
     6 dependencies 4 dependents
694. [generated] kernel theorem CReal.pow_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-congr
     2 dependencies 4 dependents
695. [generated] kernel theorem CReal.pow_half_le_natDivSucc (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-half-le-natdivsucc
     28 dependencies 4 dependents
696. [generated] kernel theorem CReal.pow_le_natDivSucc_of_gap (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-natdivsucc-of-gap
     36 dependencies 1 dependents
697. [generated] kernel theorem CReal.pow_le_natDivSucc_of_lt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-natdivsucc-of-lt
     39 dependencies 1 dependents
698. [generated] kernel theorem CReal.pow_le_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-one
     8 dependencies 6 dependents
699. [generated] kernel theorem CReal.pow_le_pow_of_base_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-pow-of-base-le
     6 dependencies 3 dependents
700. [generated] kernel theorem CReal.pow_le_pow_of_le_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-pow-of-le-one
     5 dependencies 1 dependents
701. [generated] kernel theorem CReal.pow_le_pow_of_one_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-le-pow-of-one-le
     8 dependencies 0 dependents
702. [generated] kernel theorem CReal.pow_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-nonneg
     3 dependencies 27 dependents
703. [generated] kernel theorem CReal.pow_pos (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-pos
     2 dependencies 0 dependents
704. [generated] kernel theorem CReal.pow_succ_gt_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-succ-gt-one
     13 dependencies 1 dependents
705. [generated] kernel theorem CReal.pow_succ_lt_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-succ-lt-one
     11 dependencies 1 dependents
706. [generated] kernel theorem CReal.pow_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-succ
     0 dependencies 0 dependents
707. [generated] kernel theorem CReal.pow_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-pow-zero
     0 dependencies 0 dependents
708. [generated] kernel theorem CReal.powerSeriesCauchyWithinRadius (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-powerseriescauchywithinradius
     3 dependencies 0 dependents
709. [generated] kernel theorem CReal.powerSeriesConvergesWithinRadius (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-powerseriesconvergeswithinradius
     4 dependencies 0 dependents
710. A power-series term is dominated by M*r^j whenever the coefficients are bounded by M and 0<=x<=r proved · kernel-lean · F:creal-powerseriesterm-abs-le
     13 dependencies 1 dependents
711. The power-series term function respects CReal.Equiv in its evaluation point proved · kernel-lean · F:creal-powerseriesterm-congr
     3 dependencies 2 dependents
712. The j-th power-series term, coefficients times x to the j, is a total constructed function proved · kernel-lean · F:creal-powerseriesterm
     0 dependencies 0 dependents
713. [generated] kernel theorem CReal.powerSeriesTermRadiusBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-powerseriestermradiusbound
     14 dependencies 2 dependents
714. A power series with bounded coefficients converges uniformly on [0,r] once the dominating geometric series is Cauchy proved · kernel-lean · F:creal-powerseriesuniformconvergeson
     16 dependencies 0 dependents
715. [generated] kernel theorem CReal.powMulDistrib (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-powmuldistrib
     7 dependencies 3 dependents
716. [generated] kernel theorem CReal.rat_approx_lower (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-rat-approx-lower
     11 dependencies 3 dependents
717. [generated] kernel theorem CReal.rat_approx_upper (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-rat-approx-upper
     11 dependencies 3 dependents
718. [generated] kernel theorem CReal.ratIndexRatioLeOne (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ratindexratioleone
     3 dependencies 2 dependents
719. [generated] kernel theorem CReal.ratioDecayBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ratiodecaybound
     11 dependencies 1 dependents
720. [generated] kernel theorem CReal.ratSqLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ratsqle
     14 dependencies 8 dependents
721. [generated] kernel theorem CReal.ratSqSandwich (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ratsqsandwich
     5 dependencies 1 dependents
722. [generated] kernel theorem CReal.ratUnitEqOne (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-ratuniteqone
     1 dependencies 27 dependents
723. [generated] kernel theorem CReal.reblockBlock_eq_fineBlockSum (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-reblockblock-eq-fineblocksum
     29 dependencies 1 dependents
724. [generated] kernel theorem CReal.regular_of_kregular (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-regular-of-kregular
     1 dependencies 5 dependents
725. [generated] kernel theorem CReal.regular_of_scaled_cauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-regular-of-scaled-cauchy
     5 dependencies 3 dependents
726. Every constructed real's representing sequence is regular proved · kernel-lean · F:creal-regular
     0 dependencies 50 dependents
727. The crossing-close bound restated at an ordinary Riemann-sum sample point proved · kernel-lean · F:creal-riemannsamplecrossingclose
     1 dependencies 0 dependents
728. [generated] kernel theorem CReal.riemannSum_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-add
     7 dependencies 1 dependents
729. Riemann sums over a fixed interval form a Cauchy sequence as the mesh refines proved · kernel-lean · F:creal-riemannsum-cauchy
     18 dependencies 6 dependents
730. [generated] kernel theorem CReal.riemannSum_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-const
     19 dependencies 1 dependents
731. A sufficiently fine fixed-mesh Riemann sum lies within an explicit, accuracy-derived distance of the integral proved · kernel-lean · F:creal-riemannsum-integral-close
     4 dependencies 0 dependents
732. [generated] kernel theorem CReal.riemannSum_le_on (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-le-on
     14 dependencies 1 dependents
733. [generated] kernel theorem CReal.riemannSum_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-le
     11 dependencies 0 dependents
734. [generated] kernel theorem CReal.riemannSum_reblock_close (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-reblock-close
     18 dependencies 1 dependents
735. [generated] kernel theorem CReal.riemannSum_sample_in_bounds (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-sample-in-bounds
     27 dependencies 10 dependents
736. A shared mid-anchor sample index turns two independent closeness bounds into one, with the index left free proved · kernel-lean · F:creal-riemannsum-sharedaccuracyclose-at
     5 dependencies 0 dependents
737. [generated] kernel theorem CReal.riemannSum_sharedAccuracyClose (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsum-sharedaccuracyclose
     11 dependencies 3 dependents
738. The exact mesh-point split identity discharged from a uniform-continuity witness alone proved · kernel-lean · F:creal-riemannsum-split-exact-of-uc
     31 dependencies 1 dependents
739. Riemann-sum interval splitting is an EXACT identity at a mesh-aligned split point proved · kernel-lean · F:creal-riemannsum-split-exact
     16 dependencies 0 dependents
740. Scaling both halves of a mesh split by the same factor never moves the split point proved · kernel-lean · F:creal-riemannsum-split-scale-invariant
     19 dependencies 1 dependents
741. [generated] kernel theorem CReal.riemannSumAddCauchyCross (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumaddcauchycross
     27 dependencies 1 dependents
742. [generated] kernel theorem CReal.riemannSumDeepCauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumdeepcauchy
     12 dependencies 1 dependents
743. [generated] kernel theorem CReal.riemannSumDeepCauchyCross (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumdeepcauchycross
     12 dependencies 1 dependents
744. [generated] kernel theorem CReal.riemannSumDeepCauchyCrossFolded (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumdeepcauchycrossfolded
     13 dependencies 1 dependents
745. [generated] kernel theorem CReal.riemannSumDeepCauchyFolded (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumdeepcauchyfolded
     13 dependencies 1 dependents
746. [generated] kernel theorem CReal.riemannSumTotalEpsLe (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-riemannsumtotalepsle
     23 dependencies 7 dependents
747. [generated] kernel theorem CReal.rolle_interiorExtremum (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-rolle-interiorextremum
     12 dependencies 1 dependents
748. [generated] kernel theorem CReal.sampleLowerBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-samplelowerbound
     9 dependencies 6 dependents
749. [generated] kernel theorem CReal.samplePoint_reblock (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-samplepoint-reblock
     15 dependencies 0 dependents
750. [generated] kernel theorem CReal.sampleUpperBound (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sampleupperbound
     9 dependencies 7 dependents
751. [generated] kernel theorem CReal.scale_cancel_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-scale-cancel-le
     15 dependencies 2 dependents
752. Closeness at an arbitrary shared index implies closeness at the canonical Cauchy index proved · kernel-lean · F:creal-sharedindextocanonical
     3 dependencies 8 dependents
753. [generated] kernel theorem CReal.sinDominant16Over25CauchyBody (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sindominant16over25cauchybody
     22 dependencies 0 dependents
754. [generated] kernel theorem CReal.sinFnTerm_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sinfnterm-congr
     1 dependencies 1 dependents
755. [generated] kernel theorem CReal.sinFnTermAbsLeWide (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sinfntermabslewide
     21 dependencies 1 dependents
756. [generated] kernel theorem CReal.sinFnUniformConverges (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sinfnuniformconverges
     43 dependencies 2 dependents
757. [generated] kernel theorem CReal.sinFnUniformlyContinuous (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sinfnuniformlycontinuous
     9 dependencies 0 dependents
758. sin(1)'s even-count partial sums are a lower bracket proved · kernel-lean · F:creal-sinone-alternating-lower
     6 dependencies 1 dependents
759. sin(1)'s odd-count partial sums are an upper bracket proved · kernel-lean · F:creal-sinone-alternating-upper
     6 dependencies 1 dependents
760. sin(1) <= 1/1! = 1, the exact real upper bound on sin(1) proved · kernel-lean · F:creal-sinone-le-exp-term-one
     8 dependencies 0 dependents
761. sin(1) is nonnegative: 0 <= sin(1) proved · kernel-lean · F:creal-sinone-nonneg
     1 dependencies 0 dependents
762. sin(1) is constructed as a total, axiom-free CReal value proved · kernel-lean · F:creal-sinone
     0 dependencies 0 dependents
763. sin(1)'s partial sums converge to the constructed value sinOne proved · kernel-lean · F:creal-sinoneconverges
     27 dependencies 2 dependents
764. sinSeriesPartial: the partial sums of the sin(1) series proved · kernel-lean · F:creal-sinseriespartial
     0 dependencies 0 dependents
765. sinTerm: the k-th term of the sin(1) alternating series proved · kernel-lean · F:creal-sinterm
     0 dependencies 0 dependents
766. sinTerm is dominated by the shared geometric bound expDominant proved · kernel-lean · F:creal-sintermabsledominant
     32 dependencies 1 dependents
767. [generated] kernel theorem CReal.speedup_close (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-speedup-close
     1 dependencies 6 dependents
768. [generated] kernel theorem CReal.splitPointApprox (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-splitpointapprox
     61 dependencies 1 dependents
769. Squares of constructed reals are nonnegative proved · kernel-lean · F:creal-sq-nonneg
     5 dependencies 14 dependents
770. [generated] kernel theorem CReal.sqrt_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-congr
     42 dependencies 9 dependents
771. [generated] kernel theorem CReal.sqrt_le_sqrt (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-le-sqrt
     39 dependencies 4 dependents
772. [generated] kernel theorem CReal.sqrt_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-mul
     11 dependencies 3 dependents
773. [generated] kernel theorem CReal.sqrt_nonneg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-nonneg
     4 dependencies 5 dependents
774. [generated] kernel theorem CReal.sqrt_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-one
     28 dependencies 2 dependents
775. [generated] kernel theorem CReal.sqrt_sq (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-sq
     53 dependencies 7 dependents
776. [generated] kernel theorem CReal.sqrt_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrt-zero
     17 dependencies 1 dependents
777. [generated] kernel theorem CReal.sqrtApproxKRegular (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrtapproxkregular
     40 dependencies 0 dependents
778. [generated] kernel theorem CReal.sqrtApproxSqBracket (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sqrtapproxsqbracket
     22 dependencies 7 dependents
779. [generated] kernel theorem CReal.strict_antitone_of_neg_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-strict-antitone-of-neg-deriv
     15 dependencies 0 dependents
780. [generated] kernel theorem CReal.strict_injective_of_pos_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-strict-injective-of-pos-deriv
     1 dependencies 0 dependents
781. [generated] kernel theorem CReal.strict_mono_comp (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-strict-mono-comp
     2 dependencies 0 dependents
782. [generated] kernel theorem CReal.strict_mono_magnitude (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-strict-mono-magnitude
     48 dependencies 2 dependents
783. [generated] kernel theorem CReal.strict_mono_of_pos_deriv (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-strict-mono-of-pos-deriv
     51 dependencies 3 dependents
784. [generated] kernel theorem CReal.subdivisionPoint_in_bounds (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-subdivisionpoint-in-bounds
     26 dependencies 4 dependents
785. [generated] kernel theorem CReal.sumRange_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-add
     7 dependencies 5 dependents
786. [generated] kernel theorem CReal.sumRange_cauchy_dominated_ordered_normalized (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-cauchy-dominated-ordered-normalized
     12 dependencies 6 dependents
787. [generated] kernel theorem CReal.sumRange_cauchy_dominated_ordered (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-cauchy-dominated-ordered
     7 dependencies 1 dependents
788. Absolute convergence implies convergence, Cauchy form proved · kernel-lean · F:creal-sumrange-cauchy-of-abs-cauchy
     2 dependencies 1 dependents
789. Dominated convergence, Cauchy form: a series bounded by a Cauchy series is Cauchy proved · kernel-lean · F:creal-sumrange-cauchy-of-dominated
     5 dependencies 5 dependents
790. The comparison test for series of nonnegative terms proved · kernel-lean · F:creal-sumrange-comparisontest
     12 dependencies 0 dependents
791. [generated] kernel theorem CReal.sumRange_congr (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-congr
     3 dependencies 11 dependents
792. [generated] kernel theorem CReal.sumRange_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-const
     13 dependencies 6 dependents
793. Absolute convergence implies convergence, Converges form proved · kernel-lean · F:creal-sumrange-converges-of-abs-converges
     3 dependencies 0 dependents
794. Dominated convergence: a series bounded by a Cauchy series converges proved · kernel-lean · F:creal-sumrange-converges-of-dominated
     2 dependencies 2 dependents
795. [generated] kernel theorem CReal.sumRange_double (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-double
     4 dependencies 0 dependents
796. [generated] kernel theorem CReal.sumRange_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-le
     4 dependencies 11 dependents
797. [generated] kernel theorem CReal.sumRange_mono_outer (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-mono-outer
     6 dependencies 3 dependents
798. [generated] kernel theorem CReal.sumRange_pow_half_closed_form (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-pow-half-closed-form
     24 dependencies 3 dependents
799. [generated] kernel theorem CReal.sumRange_reblock (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-reblock
     4 dependencies 0 dependents
800. [generated] kernel theorem CReal.sumRange_seq_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-seq-succ
     0 dependencies 0 dependents
801. [generated] kernel theorem CReal.sumRange_seq_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-seq-zero
     0 dependencies 0 dependents
802. [generated] kernel theorem CReal.sumRange_split (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-split
     6 dependencies 7 dependents
803. [generated] kernel theorem CReal.sumRange_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-succ
     0 dependencies 0 dependents
804. [generated] kernel theorem CReal.sumRange_tail_cauchy_within (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-tail-cauchy-within
     3 dependencies 1 dependents
805. [generated] kernel theorem CReal.sumRange_tail_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-tail-le
     14 dependencies 1 dependents
806. [generated] kernel theorem CReal.sumRange_tail_within_cauchy (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-tail-within-cauchy
     6 dependencies 1 dependents
807. [generated] kernel theorem CReal.sumRange_tail_within_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-tail-within-le
     2 dependencies 0 dependents
808. [generated] kernel theorem CReal.sumRange_tail_within (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-tail-within
     7 dependencies 2 dependents
809. [generated] kernel theorem CReal.sumRange_telescope_ge (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-telescope-ge
     4 dependencies 3 dependents
810. [generated] kernel theorem CReal.sumRange_telescope_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-telescope-le
     4 dependencies 0 dependents
811. A finite telescoping sum of constructed-real differences collapses to its endpoints proved · kernel-lean · F:creal-sumrange-telescope C:telescoping-sum
     8 dependencies 2 dependents
812. [generated] kernel theorem CReal.sumRange_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-sumrange-zero
     0 dependencies 0 dependents
813. The ratio test: a geometrically-decaying-ratio series is Cauchy proved · kernel-lean · F:creal-sumrangeratiotest
     3 dependencies 0 dependents
814. CReal.supOn_approx_lub: the approximate least-upper-bound law -- supOn is approached to any requested accuracy proved · kernel-lean · F:creal-supon-approx-lub
     18 dependencies 1 dependents
815. CReal.supOn_ub: the upper-bound law -- supOn dominates F at every point of [a, b] proved · kernel-lean · F:creal-supon-ub
     45 dependencies 1 dependents
816. CReal.supOn exists as a total, axiom-free CReal function -- the supremum value of a uniformly continuous function on [a, b] proved · kernel-lean · F:creal-supon
     0 dependencies 3 dependents
817. 3 <= pi, four terms, and every rational chosen to keep the formed Nats small proved · kernel-lean · F:creal-three-le-pi
     15 dependencies 0 dependents
818. CReal.trichotomy_of_order_decision: trichotomy on the constructed reals, on an explicit order-decision hypothesis proved · kernel-lean · F:creal-trichotomy-of-order-decision
     1 dependencies 1 dependents
819. [generated] kernel theorem CReal.trueExpOfModulus_mono (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-trueexpofmodulus-mono
     2 dependencies 0 dependents
820. [generated] kernel theorem CReal.trueExpOfModulus_step_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-trueexpofmodulus-step-le
     1 dependencies 1 dependents
821. [generated] kernel theorem CReal.trueExpOfModulus_succ (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-trueexpofmodulus-succ
     0 dependencies 0 dependents
822. [generated] kernel theorem CReal.trueExpOfModulus_zero (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-trueexpofmodulus-zero
     0 dependencies 0 dependents
823. 2 <= e, by an EVENTUAL argument proved · kernel-lean · F:creal-two-le-e
     8 dependencies 0 dependents
824. 2 <= pi, the cheapest honest lower bound proved · kernel-lean · F:creal-two-le-pi
     13 dependencies 0 dependents
825. [generated] kernel theorem CReal.two_sided_of_abs_sub_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-two-sided-of-abs-sub-le
     14 dependencies 4 dependents
826. Uniform convergence is closed under termwise addition proved · kernel-lean · F:creal-uniform-converges-add
     18 dependencies 0 dependents
827. x*(1/2)^n converges uniformly to the zero function on [0,1] proved · kernel-lean · F:creal-uniform-converges-geom-half
     20 dependencies 0 dependents
828. The identity sequence converges uniformly to the identity function on any [a,b] proved · kernel-lean · F:creal-uniform-converges-id
     16 dependencies 0 dependents
829. A uniform limit of uniformly continuous functions is uniformly continuous proved · kernel-lean · F:creal-uniform-limit-uniformly-continuous
     19 dependencies 2 dependents
830. [generated] kernel theorem CReal.uniformConvergesNeg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformconvergesneg
     15 dependencies 1 dependents
831. [generated] kernel theorem CReal.UniformConvergesOn.spec (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformconvergeson-spec
     0 dependencies 7 dependents
832. UniformConvergesOn: a one-constructor TYPE (not Prop) carrying the uniform rate as data proved · kernel-lean · F:creal-uniformconvergeson
     0 dependencies 0 dependents
833. [generated] kernel theorem CReal.uniformConvergesShift (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformconvergesshift
     4 dependencies 1 dependents
834. [generated] kernel theorem CReal.uniformly_continuous_add (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-add
     23 dependencies 5 dependents
835. [generated] kernel theorem CReal.uniformly_continuous_const (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-const
     12 dependencies 9 dependents
836. [generated] kernel theorem CReal.uniformly_continuous_id (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-id
     0 dependencies 8 dependents
837. [generated] kernel theorem CReal.uniformly_continuous_mul (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-mul
     40 dependencies 5 dependents
838. [generated] kernel theorem CReal.uniformly_continuous_neg (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-neg
     15 dependencies 1 dependents
839. [generated] kernel theorem CReal.uniformly_continuous_poly_example (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-poly-example
     5 dependencies 0 dependents
840. [generated] kernel theorem CReal.uniformly_continuous_sq (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-sq
     2 dependencies 1 dependents
841. [generated] kernel theorem CReal.uniformly_continuous_sub (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-uniformly-continuous-sub
     2 dependencies 2 dependents
842. Uniform continuity on an interval restricts to any sub-interval with the SAME modulus proved · kernel-lean · F:creal-uniformlycontinuouson-restrict
     2 dependencies 4 dependents
843. The defining modulus-of-continuity property of CReal.UniformlyContinuousOn proved · kernel-lean · F:creal-uniformlycontinuouson-spec
     0 dependencies 19 dependents
844. The Weierstrass M-test: a uniform series bound gives uniform convergence proved · kernel-lean · F:creal-weierstrassmtest
     40 dependencies 5 dependents
845. [generated] kernel theorem CReal.within_of_two_sided_le (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-within-of-two-sided-le
     3 dependencies 2 dependents
846. [generated] kernel theorem CReal.zero_lt_one (creal prelude, axiom-free, prose not curated) proved · kernel-lean · F:creal-zero-lt-one
     4 dependencies 22 dependents
847. The cross binomial row sum equals a central binomial coefficient proved · cas-certificate · F:cross-binomial-row-sum
     0 dependencies 0 dependents
848. De Morgan's laws for conjunction and disjunction proved · smt-term-level · F:de-morgan-laws C:de-morgans-lawsC:logical-equivalenceC:negation
     0 dependencies 0 dependents
849. [generated] kernel theorem demorgan_or_not_and (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:demorgan-or-not-and
     3 dependencies 0 dependents
850. Multiplicativity of the 2x2 determinant over the constructed rationals: det2(AB) = det2(A)*det2(B) proved · kernel-lean · F:determinant-multiplicative-over-constructed-rationals
     12 dependencies 0 dependents
851. Double negation elimination proved · smt-term-level · F:double-negation-elimination C:negationC:proof-by-contradictionC:non-constructive-proof +1
     1 dependencies 0 dependents
852. [generated] kernel theorem Eq.symm (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:eq-symm
     0 dependencies 13 dependents
853. Euclid I.20 on the constructed Euclidean plane, in its UNSQUARED form proved · kernel-lean · F:euclidean-plane-triangle-inequality-unsquared
     7 dependencies 0 dependents
854. Mathlib's Extreme Value Theorem, imported (labeled scaffolding, not ours) proved · imported-kernel-lean · F:evt-mathlib-import-compact-exists-is-max-on
     0 dependencies 0 dependents
855. A contradiction entails everything proved · smt-term-level · F:ex-falso-quodlibet C:contradiction-logicalC:proof-by-contradictionC:consistency
     0 dependencies 0 dependents
856. Excluded middle is not derivable in intuitionistic propositional logic proved · kernel-lean · F:excluded-middle-not-intuitionistic C:intuitionismC:excluded-middleC:non-constructive-proof +2
     2 dependencies 1 dependents
857. Law of excluded middle proved · smt-term-level · F:excluded-middle C:excluded-middleC:intuitionismC:truth-value
     0 dependencies 2 dependents
858. Exportation: the propositional form of the deduction theorem proved · smt-term-level · F:exportation C:deductionC:implicationC:assumption +1
     0 dependencies 0 dependents
859. A Farkas refutation closes over the constructed reals resting on zero carrier axioms, where the same refutation over the axiomatized AxReal package rests on 30 proved · kernel-lean · F:farkas-refutation-over-constructed-reals C:real-numbersC:linear-programming-duality
     2 dependencies 1 dependents
860. Fermat's Last Theorem open · route not assigned · F:fermat-last-theorem C:fermats-last-theorem
     0 dependencies 0 dependents
861. Fermat's little theorem over the axiom-free naturals: a^p is congruent to a modulo any prime p proved · kernel-lean · F:fermat-little-theorem-over-constructed-naturals
     7 dependencies 2 dependents
862. The product of two finite complex partial sums equals the diagonal-grouped convolution plus a corner term, which the naive Cauchy identity omits proved · kernel-lean · F:finite-cauchy-product-over-complex-equals-triangle-plus-corner
     4 dependencies 0 dependents
863. A square double sum over the naturals splits exactly into the antidiagonal triangle plus the corner, which is the honest form of the finite Cauchy product proved · kernel-lean · F:finite-double-sum-rectangle-eq-triangle-plus-corner
     6 dependencies 1 dependents
864. The subsets of an initial segment are exhaustively enumerable by a natural-number code, and a bounded search over that enumeration decides a membership-congruent property for every subset proved · kernel-lean · F:finset-subset-search-reflection
     1 dependencies 2 dependents
865. A pairing of the naturals whose inverse is a plain structural recursion on the code proved · kernel-lean · F:fo-code-pairing-round-trip
     0 dependencies 6 dependents
866. Completeness of the first-order natural deduction calculus (OPEN in this ledger) open · route not assigned · F:fo-completeness-henkin
     4 dependencies 0 dependents
867. Falsehood is not derivable from the empty context in the first-order calculus proved · kernel-lean · F:fo-consistency
     1 dependencies 2 dependents
868. The diagonal identity: diag of a formula's code is the code of that formula applied to its own numeral proved · kernel-lean · F:fo-diagonal-identity
     1 dependencies 1 dependents
869. The diagonal lemma for the first-order calculus (OPEN; the arithmetization half and the Leibniz rule are landed, representability in Q is not) open · route not assigned · F:fo-diagonal-lemma
     6 dependencies 1 dependents
870. Goedel's first incompleteness theorem, in the hypothesis-carrying form (OPEN in this ledger) open · route not assigned · F:fo-first-incompleteness
     4 dependencies 0 dependents
871. The Goedel numbering of first-order formulas is injective proved · kernel-lean · F:fo-formula-code-injective
     2 dependencies 4 dependents
872. The fuel-additive decoder round trip for first-order formulas proved · kernel-lean · F:fo-formula-decode-code
     4 dependencies 3 dependents
873. The SELF-fuelled decoder inverts the numbering (OPEN, and NOT reachable from the size bound; the fuel-additive round trip IS proved) open · route not assigned · F:fo-formula-decoder
     5 dependencies 4 dependents
874. Robinson's Q is consistent, by exhibiting a model proved · kernel-lean · F:fo-q-consistency
     2 dependencies 0 dependents
875. Q proves every true numeral addition and every true numeral multiplication proved · kernel-lean · F:fo-q-numeral-arithmetic
     1 dependencies 1 dependents
876. The pairing function has a defining formula in Q, and Q proves its graph at every numeral triple proved · kernel-lean · F:fo-q-pair-representable
     3 dependencies 0 dependents
877. Robinson's Q as a first-order context, and the natural numbers are a model of it proved · kernel-lean · F:fo-q-robinson-context
     1 dependencies 4 dependents
878. Q with the order axioms is consistent, by exhibiting a model proved · kernel-lean · F:fo-qle-consistency
     2 dependencies 0 dependents
879. Robinson's Q extended by six order axioms has ℕ as a model proved · kernel-lean · F:fo-qle-order-context
     1 dependencies 1 dependents
880. Soundness of first-order natural deduction with respect to Tarski satisfaction proved · kernel-lean · F:fo-soundness
     1 dependencies 6 dependents
881. Substitution on Goedel codes agrees with substitution on first-order syntax proved · kernel-lean · F:fo-subst-code-commutes
     2 dependencies 2 dependents
882. The substitution lemma for first-order satisfaction proved · kernel-lean · F:fo-substitution-lemma
     0 dependencies 2 dependents
883. The Goedel numbering of first-order terms is injective proved · kernel-lean · F:fo-term-code-injective
     1 dependencies 3 dependents
884. The fuel-additive decoder round trip for first-order terms proved · kernel-lean · F:fo-term-decode-code
     3 dependencies 3 dependents
885. Validity in first-order logic is undecidable open · route not assigned · F:fol-validity-undecidable C:halting-problem-logicC:undecidable-statementC:computable-function +3
     0 dependencies 0 dependents
886. binary16 addition under roundNearestTiesToEven is monotone in its first argument proved · smt-clausal · F:fp16-add-monotone-rne
     0 dependencies 0 dependents
887. Narrowing binary16 to bfloat16 and back is the identity refuted · search-certificate · F:fp16-bf16-roundtrip-not-identity
     0 dependencies 0 dependents
888. In binary16 under roundNearestTiesToEven, x+x and 2*x are the same value proved · smt-clausal · F:fp16-doubling-add-equals-mul-two
     0 dependencies 0 dependents
889. Widening binary16 to binary32 and narrowing back is the identity proved · smt-clausal · F:fp16-fp32-roundtrip-identity
     0 dependencies 0 dependents
890. In binary32 under roundNearestTiesToEven, x+x and 2*x are the same value proved · smt-clausal · F:fp32-doubling-add-equals-mul-two
     0 dependencies 0 dependents
891. fp8 E5M2 addition under roundNearestTiesToEven is monotone in its first argument proved · smt-clausal · F:fp8-add-monotone-rne
     0 dependencies 0 dependents
892. fp8 E5M2 addition under roundNearestTiesToEven is associative refuted · search-certificate · F:fp8-add-not-associative
     0 dependencies 0 dependents
893. The Franel numbers satisfy a second-order recurrence proved · cas-certificate · F:franel-numbers-recurrence
     0 dependencies 0 dependents
894. The rational coordinate plane satisfies Playfair's parallel axiom proved · kernel-lean · F:geo-affine-model-rational-plane
     1 dependencies 1 dependents
895. The real coordinate plane satisfies Playfair's parallel axiom, with parallelism stated positively proved · kernel-lean · F:geo-affine-model-real-plane
     2 dependencies 0 dependents
896. Two distinct lines meet in at most one point, in any incidence structure proved · kernel-lean · F:geo-distinct-lines-meet-once
     1 dependencies 0 dependents
897. Hilbert's three incidence axioms are consistent, witnessed by the rational coordinate plane proved · kernel-lean · F:geo-incidence-model-rational-plane
     0 dependencies 3 dependents
898. The real coordinate plane is a model of Hilbert's incidence axioms, with apartness as a witnessed positive bound proved · kernel-lean · F:geo-incidence-model-real-plane
     1 dependencies 1 dependents
899. the centroid-divides-medians certificate's two cofactor identities (3P.x and 3P.y) are kernel-reconstructed over Rat, needing both the fractional cast and the Rational poly x poly bridge proved · cas-certificate · F:geometry-centroid-divides-medians-kernel-checked
     1 dependencies 0 dependents
900. the medians of a non-degenerate triangle meet at (A+B+C)/3 proved · cas-certificate · F:geometry-centroid-divides-medians
     0 dependencies 1 dependents
901. the polar line of a point on a conic is the tangent line there proved · cas-certificate · F:geometry-conic-polar-is-tangent
     0 dependencies 0 dependents
902. Desargues' theorem in the affine plane: perspective from a point implies perspective from a line proved · cas-certificate · F:geometry-desargues-affine-perspective
     0 dependencies 0 dependents
903. Euler's line: the circumcentre, the centroid and the orthocentre of a non-degenerate triangle are collinear proved · cas-certificate · F:geometry-euler-line
     0 dependencies 0 dependents
904. Leibniz's centroid formula over the constructed reals: |PA|^2+|PB|^2+|PC|^2 = 3|PG|^2 + (|GA|^2+|GB|^2+|GC|^2) in CPoint proved · kernel-lean · F:geometry-leibniz-centroid-over-constructed-reals
     1 dependencies 0 dependents
905. The medians-concurrent certificate's cofactor identity is kernel-reconstructed with FRACTIONAL generator coefficients -- the first CAS->kernel bridge to embed a genuinely non-integer Rat literal proved · cas-certificate · F:geometry-medians-cofactor-identity-kernel-checked
     2 dependencies 0 dependents
906. the medians of a triangle are concurrent proved · cas-certificate · F:geometry-medians-concurrent
     0 dependencies 1 dependents
907. the altitudes of a triangle are concurrent proved · cas-certificate · F:geometry-orthocentre-altitudes-concurrent
     0 dependencies 1 dependents
908. The orthocentre certificate's cofactor identity is kernel-reconstructed at symbolic coordinates -- the first multivariate CAS->kernel bridge proved · cas-certificate · F:geometry-orthocentre-cofactor-identity-kernel-checked
     1 dependencies 2 dependents
909. Pappus's hexagon theorem: the three cross intersections are collinear, and ONE non-degeneracy condition suffices proved · cas-certificate · F:geometry-pappus-hexagon
     0 dependencies 0 dependents
910. the parallelogram-diagonals-bisect certificate's two cofactor identities (midpoint x and y agreement) are kernel-reconstructed over Rat, needing both the fractional cast and the Rational poly x poly bridge proved · cas-certificate · F:geometry-parallelogram-diagonals-bisect-kernel-checked
     1 dependencies 0 dependents
911. the diagonals of a non-flat parallelogram bisect each other proved · cas-certificate · F:geometry-parallelogram-diagonals-bisect
     0 dependencies 1 dependents
912. Pascal's theorem on a parabola: the three diagonal points of an inscribed hexagon are collinear proved · cas-certificate · F:geometry-pascal-parabola-hexagon
     0 dependencies 0 dependents
913. The rhombus certificate's cofactor identity is kernel-reconstructed with POLYNOMIAL cofactors -- the first CAS->kernel bridge that multiplies two non-constant polynomials proved · cas-certificate · F:geometry-rhombus-cofactor-identity-kernel-checked
     2 dependencies 0 dependents
914. the diagonals of a non-flat rhombus are perpendicular proved · cas-certificate · F:geometry-rhombus-diagonals-perpendicular
     0 dependencies 1 dependents
915. Simson's line: the feet of the perpendiculars from a concyclic point are collinear, and the minimal condition set depends on the FIELD rather than on the budget proved · cas-certificate · F:geometry-simson-line
     0 dependencies 0 dependents
916. Stewart's theorem over the constructed reals, in squared-parametric form: the cevian relation in CPoint proved · kernel-lean · F:geometry-stewart-over-constructed-reals
     19 dependencies 1 dependents
917. the medians of a tetrahedron meet at the centroid (A+B+C+D)/4, given that the tetrahedron is non-degenerate proved · cas-certificate · F:geometry-tetrahedron-medians-concurrent
     0 dependencies 0 dependents
918. the perpendicular bisector planes of a tetrahedron's edges are concurrent proved · cas-certificate · F:geometry-tetrahedron-perpendicular-bisectors-concurrent
     0 dependencies 0 dependents
919. The Thales certificate's cofactor identity is kernel-reconstructed at symbolic coordinates -- and it is disclosed here as REFL-SHAPED, not a genuine combination proved · cas-certificate · F:geometry-thales-cofactor-identity-kernel-checked
     1 dependencies 0 dependents
920. Thales' theorem: an angle inscribed in a semicircle is right proved · cas-certificate · F:geometry-thales-right-angle-in-semicircle
     0 dependencies 1 dependents
921. Varignon's theorem: the midpoint quadrilateral is a parallelogram proved · cas-certificate · F:geometry-varignon-midpoint-parallelogram
     0 dependencies 0 dependents
922. Varignon's theorem over the constructed reals: the midpoint quadrilateral is a parallelogram in CPoint proved · kernel-lean · F:geometry-varignon-over-constructed-reals
     1 dependencies 0 dependents
923. Half-degree shape classification under binary polynomial composition proved · cas-certificate · F:gf2-composition-shape-classification
     0 dependencies 1 dependents
924. A shaped degree-eight source has a two-step degree-eight composition chain refuted · cas-certificate · F:gf2-degree-eight-octuple-two-step-chain
     1 dependencies 0 dependents
925. Binary irreducible monomial composition criterion and iteration proved · cas-certificate · F:gf2-general-monomial-composition-criterion
     0 dependencies 0 dependents
926. Degree-seven joint Witt shifted-trace closed form proved · cas-certificate · F:gf2-witt-shifted-degree-seven-closed-form
     0 dependencies 0 dependents
927. Godel's first incompleteness theorem open · route not assigned · F:godel-first-incompleteness C:godel-incompletenessC:incompleteness-theoremC:completeness-logical +4
     0 dependencies 0 dependents
928. Strong Goldbach conjecture conjectured · route not assigned · F:goldbach-strong C:goldbach-conjectureC:prime-number
     0 dependencies 0 dependents
929. Hall's condition on a one-element index set produces a system of distinct representatives, with the representative computed from the count rather than chosen proved · kernel-lean · F:hall-base-case-singleton
     3 dependencies 2 dependents
930. Hall's condition survives on the complement of a critical subset proved · kernel-lean · F:hall-critical-subset-split
     5 dependencies 2 dependents
931. Splitting a finite set of naturals at a subset splits its count exactly, so a nonempty proper part and its complement are both strictly smaller proved · kernel-lean · F:hall-descent-measure
     4 dependencies 1 dependents
932. Hall's marriage theorem: a finite family of finite sets has a system of distinct representatives if and only if every sub-family covers at least its own size proved · kernel-lean · F:hall-marriage-theorem
     10 dependencies 0 dependents
933. Two systems of distinct representatives whose images are disjoint glue into one on the union of their index sets, with no disjointness required of the index sets themselves proved · kernel-lean · F:hall-matching-union-disjoint-images
     2 dependencies 2 dependents
934. Strict slack lets one value be deleted from every neighbourhood proved · kernel-lean · F:hall-noncritical-single-value-deletion
     6 dependencies 1 dependents
935. Deleting a fixed set from every member of an indexed family commutes with taking the family's union, so the union loses at most that set's own size proved · kernel-lean · F:hall-union-under-family-deletion
     3 dependencies 4 dependents
936. The 3-element Gödel/Łukasiewicz Heyting chain refutes excluded middle proved · kernel-lean · F:heyting-3-chain-refutes-excluded-middle C:intuitionismC:excluded-middleC:non-constructive-proof
     0 dependencies 1 dependents
937. Addition on the integers is associative proved · kernel-lean · F:int-add-assoc C:associativityC:integers
     9 dependencies 59 dependents
938. Addition on the integers is commutative proved · kernel-lean · F:int-add-comm C:commutativityC:integers
     2 dependencies 87 dependents
939. The order on the integers is compatible with addition proved · kernel-lean · F:int-add-le-add C:ordered-ringC:integers
     4 dependencies 19 dependents
940. A strict integer inequality survives addition of a non-strict one proved · kernel-lean · F:int-add-lt-add-of-le-of-lt C:ordered-ringC:integers
     6 dependencies 5 dependents
941. [generated] kernel theorem Int.add_modEq_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-add-modeq-left
     2 dependencies 2 dependents
942. [generated] kernel theorem Int.add_modEq_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-add-modeq-right
     2 dependencies 0 dependents
943. [generated] kernel theorem Int.add_neg_cancel_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-add-neg-cancel-right
     4 dependencies 26 dependents
944. Every integer has an additive inverse proved · kernel-lean · F:int-add-neg C:additive-inverseC:integers
     1 dependencies 50 dependents
945. Adding zero to an integer is the identity proved · kernel-lean · F:int-add-zero
     0 dependencies 81 dependents
946. The Brahmagupta-Fibonacci identity over the constructed integers proved · kernel-lean · F:int-brahmagupta-fibonacci
     12 dependencies 3 dependents
947. The constructed Int is THE integers: every generated aperiodic Z-structure is in structure-preserving bijection with it proved · kernel-lean · F:int-categoricity C:integersC:peano-axioms
     7 dependencies 1 dependents
948. [generated] kernel theorem Int.Characterization.cases (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-cases
     0 dependencies 0 dependents
949. [generated] kernel theorem Int.Characterization.categorical_at_int (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-categorical-at-int
     11 dependencies 0 dependents
950. [generated] kernel theorem Int.Characterization.categorical (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-categorical
     5 dependencies 1 dependents
951. [generated] kernel theorem Int.Characterization.cross (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-cross
     2 dependencies 2 dependents
952. [generated] kernel theorem Int.Characterization.discrete_everywhere (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-discrete-everywhere
     7 dependencies 0 dependents
953. [generated] kernel theorem Int.Characterization.discrete (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-discrete
     1 dependencies 0 dependents
954. [generated] kernel theorem Int.Characterization.induction (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-induction
     0 dependencies 2 dependents
955. The constructed Int is initial among Z-structures proved · kernel-lean · F:int-characterization-initial
     4 dependencies 0 dependents
956. [generated] kernel theorem Int.Characterization.injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-injective
     5 dependencies 2 dependents
957. [generated] kernel theorem Int.Characterization.iso (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iso
     3 dependencies 0 dependents
958. [generated] kernel theorem Int.Characterization.iter_at_int (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-at-int
     0 dependencies 1 dependents
959. [generated] kernel theorem Int.Characterization.iter_down_injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-down-injective
     3 dependencies 1 dependents
960. [generated] kernel theorem Int.Characterization.iter_pred (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-pred
     2 dependencies 5 dependents
961. [generated] kernel theorem Int.Characterization.iter_succ (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-succ
     0 dependencies 5 dependents
962. [generated] kernel theorem Int.Characterization.iter_up_injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-up-injective
     1 dependencies 1 dependents
963. [generated] kernel theorem Int.Characterization.iter_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-iter-zero
     0 dependencies 3 dependents
964. [generated] kernel theorem Int.Characterization.le_total (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-le-total
     1 dependencies 0 dependents
965. [generated] kernel theorem Int.Characterization.of_nat_or_neg (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-of-nat-or-neg
     0 dependencies 0 dependents
966. [generated] kernel theorem Int.Characterization.rec_unique (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-rec-unique
     1 dependencies 2 dependents
967. [generated] kernel theorem Int.Characterization.shift (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-shift
     0 dependencies 2 dependents
968. [generated] kernel theorem Int.Characterization.surjective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-surjective
     2 dependencies 2 dependents
969. [generated] kernel theorem Int.Characterization.up_injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-up-injective
     0 dependencies 1 dependents
970. [generated] kernel theorem Int.Characterization.zero_lt_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-zero-lt-one
     1 dependencies 0 dependents
971. [generated] kernel theorem Int.Characterization.zero_ne_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-characterization-zero-ne-one
     2 dependencies 0 dependents
972. The constructed Int is a discretely ordered ring generated by 1, with unique maps out proved · kernel-lean · F:int-characterization C:integersC:ordered-ring
     2 dependencies 1 dependents
973. [generated] kernel theorem Int.coprime_of_bezout_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-coprime-of-bezout-one
     7 dependencies 5 dependents
974. [generated] kernel theorem Int.coprimeFactorialOfLtPrime (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-coprimefactorialofltprime
     2 dependencies 1 dependents
975. [generated] kernel theorem Int.crt_exists (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-crt-exists
     13 dependencies 0 dependents
976. [generated] kernel theorem Int.crt_unique (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-crt-unique
     9 dependencies 0 dependents
977. The termination certificate of the two-squares descent, read off the factorisation proved · kernel-lean · F:int-descent-multiplier-bounds
     8 dependencies 1 dependents
978. [generated] kernel theorem Int.dvd_add (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-add
     1 dependencies 7 dependents
979. [generated] kernel theorem Int.dvd_factorial_of_le (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-factorial-of-le
     8 dependencies 1 dependents
980. [generated] kernel theorem Int.dvd_gcd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-gcd
     4 dependencies 2 dependents
981. [generated] kernel theorem Int.dvd_mul_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-mul-left
     1 dependencies 4 dependents
982. [generated] kernel theorem Int.dvd_mul_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-mul-right
     0 dependencies 7 dependents
983. [generated] kernel theorem Int.dvd_of_nat_abs_dvd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-of-nat-abs-dvd
     2 dependencies 14 dependents
984. [generated] kernel theorem Int.dvd_refl (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-refl
     1 dependencies 7 dependents
985. [generated] kernel theorem Int.dvd_trans (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-dvd-trans
     1 dependencies 11 dependents
986. [generated] kernel theorem Int.ediv_add_emod (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-ediv-add-emod
     11 dependencies 38 dependents
987. [generated] kernel theorem Int.ediv_emod_unique (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-ediv-emod-unique
     18 dependencies 16 dependents
988. [generated] kernel theorem Int.emod_eq_zero_iff_dvd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-emod-eq-zero-iff-dvd
     7 dependencies 4 dependents
989. [generated] kernel theorem Int.emod_lt_of_pos (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-emod-lt-of-pos
     10 dependencies 19 dependents
990. [generated] kernel theorem Int.emod_neg (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-emod-neg
     0 dependencies 2 dependents
991. [generated] kernel theorem Int.emod_nonneg (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-emod-nonneg
     6 dependencies 20 dependents
992. Equality of integers is decidable proved · kernel-lean · F:int-equality-is-decidable C:decidabilityC:integers
     3 dependencies 9 dependents
993. [generated] kernel theorem Int.euclid_infinitude (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euclid-infinitude
     1 dependencies 0 dependents
994. [generated] kernel theorem Int.euclid_lemma (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euclid-lemma
     4 dependencies 4 dependents
995. Euclidean division exists for a negative dividend proved · kernel-lean · F:int-euclid-neg-succ
     14 dependencies 1 dependents
996. Euclidean division exists for a nonnegative dividend proved · kernel-lean · F:int-euclid-of-nat
     3 dependencies 1 dependents
997. Euclidean decomposition over the integers is derived, not assumed proved · kernel-lean · F:int-euclidean-decomposition C:euclidean-division
     6 dependencies 1 dependents
998. [generated] kernel theorem Int.euler_criterion_neg_one_imp_not_residue (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-criterion-neg-one-imp-not-residue
     11 dependencies 1 dependents
999. [generated] kernel theorem Int.euler_criterion_pm_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-criterion-pm-one
     33 dependencies 0 dependents
1000. [generated] kernel theorem Int.euler_criterion_residue_imp_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-criterion-residue-imp-one
      29 dependencies 1 dependents
1001. Euler's totient theorem over the constructed integers (Int.euler_totient_theorem) proved · kernel-lean · F:int-euler-totient-theorem
      30 dependencies 2 dependents
1002. [generated] kernel theorem Int.euler_unit_coprime_iff (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-unit-coprime-iff
      22 dependencies 1 dependents
1003. [generated] kernel theorem Int.euler_unit_coprime (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-unit-coprime
      21 dependencies 1 dependents
1004. [generated] kernel theorem Int.euler_unit_injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-unit-injective
      9 dependencies 1 dependents
1005. [generated] kernel theorem Int.euler_unit_perm_injective (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-unit-perm-injective
      5 dependencies 1 dependents
1006. [generated] kernel theorem Int.euler_unit_perm_maps_into (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-euler-unit-perm-maps-into
      4 dependencies 1 dependents
1007. Int.even_iff_nat_abs_even : forall n, Iff (Odd n) (Nat.Odd (natAbs n)) proved · kernel-lean · F:int-even-iff-nat-abs-even
      1 dependencies 0 dependents
1008. Int.Even n := Nat.Even (natAbs n) -- magnitude alone decides parity proved · kernel-lean · F:int-even
      0 dependencies 1 dependents
1009. Every residue class modulo a positive integer has a representative in the symmetric band proved · kernel-lean · F:int-exists-centered-representative
      15 dependencies 2 dependents
1010. [generated] kernel theorem Int.factorial_interior_modeq_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorial-interior-modeq-one
      36 dependencies 2 dependents
1011. [generated] kernel theorem Int.factorial_pos (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorial-pos
      5 dependencies 0 dependents
1012. [generated] kernel theorem Int.factorial_sq_modeq_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorial-sq-modeq-one
      29 dependencies 0 dependents
1013. [generated] kernel theorem Int.factorial_succ (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorial-succ
      0 dependencies 0 dependents
1014. [generated] kernel theorem Int.factorial_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorial-zero
      0 dependencies 0 dependents
1015. [generated] kernel theorem Int.factorialEqOfNatFactorial (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-factorialeqofnatfactorial
      2 dependencies 1 dependents
1016. Fermat's theorem on sums of two squares proved · kernel-lean · F:int-fermat-two-squares
      14 dependencies 1 dependents
1017. Int.fib_rec -- the Fibonacci recurrence at EVERY integer index proved · kernel-lean · F:int-fib-rec
      14 dependencies 3 dependents
1018. The first supplementary law of quadratic reciprocity, non-residue half, over the constructed integers proved · kernel-lean · F:int-firstsupplementarylawnotresidue
      23 dependencies 0 dependents
1019. The first supplementary law of quadratic reciprocity, residue half, over the constructed integers proved · kernel-lean · F:int-firstsupplementarylawresidue
      3 dependencies 1 dependents
1020. [generated] kernel theorem Int.gauss_lemma (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gauss-lemma
      8 dependencies 6 dependents
1021. Gauss's lemma (quadratic residues), over the constructed integers proved · kernel-lean · F:int-gausslemmasigncount
      17 dependencies 2 dependents
1022. [generated] kernel theorem Int.gaussSignProdEqPowNegOneOfCount (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gausssignprodeqpownegoneofcount
      1 dependencies 1 dependents
1023. Gauss's-lemma per-term congruence: a*k = (sign) * gaussFold(pp,a,k) mod pp proved · kernel-lean · F:int-gausstermmodeq
      7 dependencies 1 dependents
1024. gcd is commutative on the integers proved · kernel-lean · F:int-gcd-comm C:greatest-common-divisor
      10 dependencies 1 dependents
1025. [generated] kernel theorem Int.gcd_dvd_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gcd-dvd-left
      2 dependencies 11 dependents
1026. [generated] kernel theorem Int.gcd_dvd_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gcd-dvd-right
      2 dependencies 11 dependents
1027. [generated] kernel theorem Int.gcd_eq_gcd_ab (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gcd-eq-gcd-ab
      10 dependencies 4 dependents
1028. [generated] kernel theorem Int.gcd_one_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gcd-one-right
      2 dependencies 0 dependents
1029. [generated] kernel theorem Int.gcd_zero_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-gcd-zero-right
      11 dependencies 2 dependents
1030. [generated] kernel theorem Int.int_isCommRing (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-int-iscommring
      8 dependencies 0 dependents
1031. [generated] kernel theorem Int.is_quadratic_residue_mul (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-is-quadratic-residue-mul
      3 dependencies 0 dependents
1032. [generated] kernel theorem Int.is_quadratic_residue_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-is-quadratic-residue-one
      2 dependencies 0 dependents
1033. Quadratic-residue-hood respects congruence, over the constructed integers proved · kernel-lean · F:int-isquadraticresidue-of-modeq
      1 dependencies 1 dependents
1034. [generated] kernel theorem Int.le_antisymm (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-le-antisymm
      4 dependencies 5 dependents
1035. A non-strict integer inequality is a nonnegative natural offset proved · kernel-lean · F:int-le-dest
      5 dependencies 5 dependents
1036. [generated] kernel theorem Int.le_of_lt (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-le-of-lt
      1 dependencies 11 dependents
1037. Adding a natural number does not decrease an integer proved · kernel-lean · F:int-le-of-nat-add
      5 dependencies 6 dependents
1038. The integer order is reflexive proved · kernel-lean · F:int-le-refl
      0 dependencies 28 dependents
1039. The integer order is total proved · kernel-lean · F:int-le-total
      1 dependencies 14 dependents
1040. The integer order is transitive proved · kernel-lean · F:int-le-trans
      1 dependencies 11 dependents
1041. Multiplication distributes over addition on the integers proved · kernel-lean · F:int-left-distrib C:distributivityC:integers
      8 dependencies 40 dependents
1042. Euler's criterion for the Legendre symbol defined by Gauss's count proved · kernel-lean · F:int-legendre-sym-modeq-pow
      1 dependencies 0 dependents
1043. A strict integer inequality is a positive natural offset proved · kernel-lean · F:int-lt-dest
      6 dependencies 6 dependents
1044. [generated] kernel theorem Int.lt_irrefl (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-lt-irrefl
      1 dependencies 35 dependents
1045. [generated] kernel theorem Int.lt_of_le_of_lt (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-lt-of-le-of-lt
      2 dependencies 3 dependents
1046. [generated] kernel theorem Int.lt_of_le_of_ne (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-lt-of-le-of-ne
      1 dependencies 11 dependents
1047. [generated] kernel theorem Int.lt_of_lt_of_le (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-lt-of-lt-of-le
      2 dependencies 6 dependents
1048. Adding a positive natural number strictly increases an integer proved · kernel-lean · F:int-lt-of-nat-add
      7 dependencies 4 dependents
1049. [generated] kernel theorem Int.lt_trans (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-lt-trans
      2 dependencies 2 dependents
1050. [generated] kernel theorem Int.mod_modEq (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mod-modeq
      3 dependencies 1 dependents
1051. [generated] kernel theorem Int.modEq_add_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-add-left
      2 dependencies 7 dependents
1052. [generated] kernel theorem Int.modEq_add_mul_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-add-mul-left
      14 dependencies 18 dependents
1053. [generated] kernel theorem Int.modEq_add_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-add-right
      13 dependencies 9 dependents
1054. [generated] kernel theorem Int.modEq_cancel (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-cancel
      6 dependencies 5 dependents
1055. [generated] kernel theorem Int.modEq_iff_dvd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-iff-dvd
      11 dependencies 24 dependents
1056. [generated] kernel theorem Int.modEq_inverse_exists (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-inverse-exists
      5 dependencies 3 dependents
1057. [generated] kernel theorem Int.modEq_inverse_unique (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-inverse-unique
      10 dependencies 4 dependents
1058. [generated] kernel theorem Int.modEq_mul_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-mul-left
      6 dependencies 13 dependents
1059. [generated] kernel theorem Int.modEq_mul_right (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-mul-right
      2 dependencies 4 dependents
1060. [generated] kernel theorem Int.modEq_mul (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-mul
      3 dependencies 15 dependents
1061. [generated] kernel theorem Int.modEq_neg_modulus (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-neg-modulus
      1 dependencies 1 dependents
1062. [generated] kernel theorem Int.modEq_of_nat_modEq (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-of-nat-modeq
      6 dependencies 2 dependents
1063. [generated] kernel theorem Int.modEq_of_neg_modulus (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-of-neg-modulus
      1 dependencies 0 dependents
1064. [generated] kernel theorem Int.modEq_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-one
      4 dependencies 0 dependents
1065. [generated] kernel theorem Int.modEq_pow (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-pow
      2 dependencies 2 dependents
1066. [generated] kernel theorem Int.modEq_prodRange_lt (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-prodrange-lt
      4 dependencies 3 dependents
1067. [generated] kernel theorem Int.modEq_prodRange (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-prodrange
      2 dependencies 1 dependents
1068. Congruence mod n is reflexive on the integers proved · kernel-lean · F:int-modeq-refl C:modular-arithmeticC:congruence
      0 dependencies 14 dependents
1069. [generated] kernel theorem Int.modEq_sub (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modeq-sub
      6 dependencies 0 dependents
1070. A signed finite sum reduces modulo n term by term, with no positivity hypothesis proved · kernel-lean · F:int-modeq-sumrange
      4 dependencies 0 dependents
1071. Congruence mod n is symmetric on the integers proved · kernel-lean · F:int-modeq-symm C:modular-arithmeticC:congruence
      1 dependencies 36 dependents
1072. Congruence mod n is transitive on the integers proved · kernel-lean · F:int-modeq-trans C:modular-arithmeticC:congruence
      1 dependencies 30 dependents
1073. [generated] kernel theorem Int.modulus_modEq_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-modulus-modeq-zero
      3 dependencies 1 dependents
1074. Multiplication on the integers is associative proved · kernel-lean · F:int-mul-assoc C:associativityC:integers
      5 dependencies 69 dependents
1075. Multiplication on the integers is commutative proved · kernel-lean · F:int-mul-comm C:commutativityC:integers
      1 dependencies 90 dependents
1076. [generated] kernel theorem Int.mul_eq_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mul-eq-zero
      3 dependencies 5 dependents
1077. [generated] kernel theorem Int.mul_inv_of_pow (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mul-inv-of-pow
      6 dependencies 7 dependents
1078. [generated] kernel theorem Int.mul_le_mul_of_nonneg_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mul-le-mul-of-nonneg-left
      6 dependencies 8 dependents
1079. Product of two negative-represented integers, at the representation level proved · kernel-lean · F:int-mul-neg-of-nat-neg-succ
      1 dependencies 2 dependents
1080. Product of a negOfNat integer and a nonnegative integer, at the representation level proved · kernel-lean · F:int-mul-neg-of-nat-of-nat
      1 dependencies 3 dependents
1081. Product of a negSucc integer and a negOfNat integer, at the representation level proved · kernel-lean · F:int-mul-neg-succ-neg-of-nat
      0 dependencies 4 dependents
1082. Multiplication distributes over integer negation on the right proved · kernel-lean · F:int-mul-neg
      3 dependencies 18 dependents
1083. The product of two nonnegative integers is nonnegative proved · kernel-lean · F:int-mul-nonneg
      1 dependencies 12 dependents
1084. Product of a nonnegative integer and a negOfNat integer, at the representation level proved · kernel-lean · F:int-mul-of-nat-neg-of-nat
      0 dependencies 2 dependents
1085. Multiplying an integer by one is the identity proved · kernel-lean · F:int-mul-one
      1 dependencies 55 dependents
1086. [generated] kernel theorem Int.mul_pos (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mul-pos
      3 dependencies 5 dependents
1087. [generated] kernel theorem Int.mul_sub (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-mul-sub
      2 dependencies 10 dependents
1088. Multiplying an integer by zero gives zero proved · kernel-lean · F:int-mul-zero
      0 dependencies 35 dependents
1089. Every unit mod n has a multiplicative order (Int.order_exists) proved · kernel-lean · F:int-multiplicative-order-exists
      8 dependencies 0 dependents
1090. [generated] kernel theorem Int.nat_abs_dvd_nat_abs_of_dvd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-nat-abs-dvd-nat-abs-of-dvd
      1 dependencies 20 dependents
1091. [generated] kernel theorem Int.nat_abs_mul (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-nat-abs-mul
      1 dependencies 16 dependents
1092. [generated] kernel theorem Int.nat_abs_neg_of_nat (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-nat-abs-neg-of-nat
      0 dependencies 4 dependents
1093. [generated] kernel theorem Int.nat_abs_neg (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-nat-abs-neg
      1 dependencies 3 dependents
1094. [generated] kernel theorem Int.nat_abs_pow (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-nat-abs-pow
      2 dependencies 0 dependents
1095. Negation distributes over addition on the constructed integers proved · kernel-lean · F:int-neg-add
      2 dependencies 0 dependents
1096. Sum of two negOfNat integers, at the representation level proved · kernel-lean · F:int-neg-of-nat-add-neg-of-nat
      4 dependencies 2 dependents
1097. Sum of a negOfNat integer and a nonnegative integer reduces to subNatNat proved · kernel-lean · F:int-neg-of-nat-add-of-nat
      2 dependencies 3 dependents
1098. Multiplying by negative one is negation proved · kernel-lean · F:int-neg-one-mul
      3 dependencies 26 dependents
1099. Adding a negSucc integer into a normalized difference, at the representation level proved · kernel-lean · F:int-neg-succ-add-sub-nat-nat
      2 dependencies 2 dependents
1100. Multiplying a negSucc integer into a normalized difference, at the representation level proved · kernel-lean · F:int-neg-succ-mul-sub-nat-nat
      4 dependencies 1 dependents
1101. [generated] kernel theorem Int.negOfNat_add_subNatNat (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-negofnat-add-subnatnat
      9 dependencies 1 dependents
1102. No integer lies strictly between zero and one proved · kernel-lean · F:int-no-integer-strictly-between-zero-and-one C:discretenessC:integers
      4 dependencies 2 dependents
1103. Int.odd_iff_nat_abs_odd : forall n, Iff (Odd n) (Nat.Odd (natAbs n)) proved · kernel-lean · F:int-odd-iff-nat-abs-odd
      1 dependencies 0 dependents
1104. Int.Odd n := Nat.Odd (natAbs n) -- magnitude alone decides parity proved · kernel-lean · F:int-odd
      0 dependencies 3 dependents
1105. Sum of a nonnegative integer and a negOfNat integer reduces to subNatNat proved · kernel-lean · F:int-of-nat-add-neg-of-nat
      1 dependencies 5 dependents
1106. Adding a nonnegative integer into a normalized difference, at the representation level proved · kernel-lean · F:int-of-nat-add-sub-nat-nat
      7 dependencies 3 dependents
1107. Multiplying a nonnegative integer into a normalized difference, at the representation level proved · kernel-lean · F:int-of-nat-mul-sub-nat-nat
      4 dependencies 1 dependents
1108. [generated] kernel theorem Int.of_nat_nat_abs_of_nonneg (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-of-nat-nat-abs-of-nonneg
      0 dependencies 19 dependents
1109. [generated] kernel theorem Int.of_nat_pow (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-of-nat-pow
      0 dependencies 1 dependents
1110. Multiplying an integer by one on the left is the identity proved · kernel-lean · F:int-one-mul
      2 dependencies 23 dependents
1111. The multiplicative order divides the totient (Int.order_dvd_totient) proved · kernel-lean · F:int-order-dvd-totient
      1 dependencies 0 dependents
1112. [generated] kernel theorem Int.pow_add (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-pow-add
      3 dependencies 4 dependents
1113. a^m is 1 mod n exactly when the order divides m (Int.pow_modeq_one_iff_order_dvd) proved · kernel-lean · F:int-pow-modeq-one-iff-order-dvd
      0 dependencies 1 dependents
1114. [generated] kernel theorem Int.pow_mul (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-pow-mul
      1 dependencies 0 dependents
1115. [generated] kernel theorem Int.pow_neg_one_of_even (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-pow-neg-one-of-even
      8 dependencies 3 dependents
1116. [generated] kernel theorem Int.pow_neg_one_of_odd (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-pow-neg-one-of-odd
      8 dependencies 2 dependents
1117. [generated] kernel theorem Int.pow_prime_sub_one_modeq_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-pow-prime-sub-one-modeq-one
      12 dependencies 3 dependents
1118. The successor case of integer exponentiation proved · kernel-lean · F:int-pow-succ
      0 dependencies 9 dependents
1119. The base case of integer exponentiation proved · kernel-lean · F:int-pow-zero
      0 dependencies 2 dependents
1120. The powers of a primitive root are pairwise incongruent (Int.primitive_root_pow_injective) proved · kernel-lean · F:int-primitive-root-pow-injective
      3 dependencies 0 dependents
1121. [generated] kernel theorem Int.prod_range_pairing_collapse (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prod-range-pairing-collapse
      25 dependencies 1 dependents
1122. The generalized distributive law: a product of sums expands into a sum over the function space proved · kernel-lean · F:int-prod-range-sum-range-expand
      2 dependencies 0 dependents
1123. [generated] kernel theorem Int.prodRange_congr_lt (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-congr-lt
      2 dependencies 4 dependents
1124. [generated] kernel theorem Int.prodRange_congr (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-congr
      0 dependencies 3 dependents
1125. [generated] kernel theorem Int.prodRange_constPow (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-constpow
      0 dependencies 1 dependents
1126. [generated] kernel theorem Int.prodRange_mul (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-mul
      3 dependencies 4 dependents
1127. [generated] kernel theorem Int.prodRange_permute (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-permute
      19 dependencies 4 dependents
1128. [generated] kernel theorem Int.prodRange_scaledIndexEqPowMulFactorial (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-scaledindexeqpowmulfactorial
      2 dependencies 2 dependents
1129. [generated] kernel theorem Int.prodRange_shiftFront (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-shiftfront
      3 dependencies 3 dependents
1130. A finite product over the constructed integers splits at a symbolic point proved · kernel-lean · F:int-prodrange-split
      2 dependencies 1 dependents
1131. [generated] kernel theorem Int.prodRange_succ (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-succ
      0 dependencies 0 dependents
1132. [generated] kernel theorem Int.prodRange_swap_adjacent (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-swap-adjacent
      9 dependencies 1 dependents
1133. [generated] kernel theorem Int.prodRange_swap (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-swap
      18 dependencies 3 dependents
1134. [generated] kernel theorem Int.prodRange_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrange-zero
      0 dependencies 0 dependents
1135. [generated] kernel theorem Int.prodRangeIf_const_eq_pow_count (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-const-eq-pow-count
      1 dependencies 2 dependents
1136. [generated] kernel theorem Int.prodRangeIf_coprime (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-coprime
      16 dependencies 1 dependents
1137. [generated] kernel theorem Int.prodRangeIf_factor_const_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-factor-const-left
      3 dependencies 1 dependents
1138. [generated] kernel theorem Int.prodRangeIf_modeq (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-modeq
      2 dependencies 1 dependents
1139. [generated] kernel theorem Int.prodRangeIf_permute (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-permute
      2 dependencies 1 dependents
1140. [generated] kernel theorem Int.prodRangeIf_succ (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-succ
      0 dependencies 0 dependents
1141. [generated] kernel theorem Int.prodRangeIf_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-prodrangeif-zero
      0 dependencies 0 dependents
1142. The law of quadratic reciprocity proved · kernel-lean · F:int-quadratic-reciprocity
      6 dependencies 0 dependents
1143. The second supplementary law of quadratic reciprocity, over the constructed integers proved · kernel-lean · F:int-secondsupplementarylaw
      10 dependencies 0 dependents
1144. [generated] kernel theorem Int.self_inverse_mod_prime (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-self-inverse-mod-prime
      18 dependencies 2 dependents
1145. The strict decrease of the two-squares descent measure proved · kernel-lean · F:int-sq-add-sq-lt-sq-of-bounds
      11 dependencies 3 dependents
1146. Squaring is monotone on a symmetric band over the integers proved · kernel-lean · F:int-sq-le-sq-of-neg-le-of-le
      7 dependencies 1 dependents
1147. Every integer square is nonnegative proved · kernel-lean · F:int-sq-nonneg C:integers
      2 dependencies 5 dependents
1148. A normalized difference with a common left addend reduces to the remainder proved · kernel-lean · F:int-sub-nat-nat-add-left
      2 dependencies 6 dependents
1149. Adding a negSucc integer to a normalized difference, at the representation level proved · kernel-lean · F:int-sub-nat-nat-add-neg-succ
      6 dependencies 3 dependents
1150. Adding a nonnegative integer to a normalized difference, at the representation level proved · kernel-lean · F:int-sub-nat-nat-add-of-nat
      4 dependencies 2 dependents
1151. A normalized difference with a common right addend reduces to the negated remainder proved · kernel-lean · F:int-sub-nat-nat-add-right
      2 dependencies 9 dependents
1152. The integer borrow has exactly two outcomes, and each is witnessed by a natural proved · kernel-lean · F:int-sub-nat-nat-elim C:integersC:proof-by-cases
      6 dependencies 11 dependents
1153. The normalized integer difference is invariant under a common shift proved · kernel-lean · F:int-sub-nat-nat-shift C:integersC:natural-numbers
      2 dependencies 3 dependents
1154. A normalized difference is unchanged by adding one to both sides proved · kernel-lean · F:int-sub-nat-nat-succ-succ
      1 dependencies 2 dependents
1155. [generated] kernel theorem Int.subNatNat_add_add_left (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-subnatnat-add-add-left
      3 dependencies 4 dependents
1156. [generated] kernel theorem Int.subNatNat_zero (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-subnatnat-zero
      2 dependencies 3 dependents
1157. The function-space sum peels its first index into an ordinary range sum proved · kernel-lean · F:int-sum-maps-succ
      0 dependencies 0 dependents
1158. A finite sum indexed by a function space, with no aggregate type proved · kernel-lean · F:int-sum-maps
      0 dependencies 0 dependents
1159. Sums of two squares are closed under multiplication (over the constructed integers) proved · kernel-lean · F:int-sum-of-two-squares-multiplicative
      1 dependencies 2 dependents
1160. The signed finite sum is additive in the summand proved · kernel-lean · F:int-sumrange-add
      3 dependencies 1 dependents
1161. Pointwise-equal families have equal signed finite sums proved · kernel-lean · F:int-sumrange-congr
      0 dependencies 1 dependents
1162. Negation pulls out of a signed finite sum proved · kernel-lean · F:int-sumrange-neg
      2 dependencies 1 dependents
1163. A signed finite sum of coerced naturals is the coerced natural sum proved · kernel-lean · F:int-sumrange-ofnat
      0 dependencies 0 dependents
1164. Subtraction inside a signed finite sum over the constructed integers proved · kernel-lean · F:int-sumrange-sub
      2 dependencies 0 dependents
1165. The signed finite sum adds the fresh term on the right proved · kernel-lean · F:int-sumrange-succ
      0 dependencies 0 dependents
1166. The empty signed finite sum over the constructed integers is zero proved · kernel-lean · F:int-sumrange-zero
      0 dependencies 0 dependents
1167. Int.induction_on -- two-sided induction over the constructed integers proved · kernel-lean · F:int-two-sided-induction
      0 dependencies 1 dependents
1168. The full sum-of-two-squares characterisation for composite n (OPEN, and deliberately out of scope for the lane that opened it) open · route not assigned · F:int-two-squares-composite-characterisation
      3 dependencies 0 dependents
1169. A vanishing descent multiplier forces m to divide p proved · kernel-lean · F:int-two-squares-degenerate-divides
      3 dependencies 2 dependents
1170. Both cross terms of the conjugate Brahmagupta grouping vanish modulo the multiplier proved · kernel-lean · F:int-two-squares-descent-cross-terms
      5 dependencies 3 dependents
1171. The entry point of Fermat's two-squares descent proved · kernel-lean · F:int-two-squares-descent-entry
      22 dependencies 1 dependents
1172. The reusable descent step for sums of two squares proved · kernel-lean · F:int-two-squares-descent-step
      8 dependencies 3 dependents
1173. Euler's descent: a small multiple of a prime that is a sum of two squares descends to the prime proved · kernel-lean · F:int-two-squares-descent
      15 dependencies 1 dependents
1174. An integer congruent to 3 modulo 4 is not a sum of two squares proved · kernel-lean · F:int-two-squares-mod-four-refutation
      7 dependencies 1 dependents
1175. The descent's next multiplier exists proved · kernel-lean · F:int-two-squares-next-multiplier
      3 dependencies 2 dependents
1176. [generated] kernel theorem Int.wilson_converse (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-wilson-converse
      19 dependencies 1 dependents
1177. [generated] kernel theorem Int.wilson_iff (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-wilson-iff
      2 dependencies 0 dependents
1178. [generated] kernel theorem Int.wilson (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-wilson
      22 dependencies 2 dependents
1179. Wilson's theorem split at the half: (p-1)! congruent to m! times ((-1)^m times m!), over the constructed integers proved · kernel-lean · F:int-wilsonhalfsplit
      30 dependencies 1 dependents
1180. [generated] kernel theorem Int.zero_lt_one (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-zero-lt-one
      0 dependencies 17 dependents
1181. [generated] kernel theorem Int.zero_subNatNat (integer prelude, axiom-free, prose not curated) proved · kernel-lean · F:int-zero-subnatnat
      0 dependencies 2 dependents
1182. Integrand congruence holds in every integration space, from monotonicity alone proved · kernel-lean · F:intspace-integral-congr
      0 dependencies 3 dependents
1183. The L1 distance on [a,b] is the Riemann integral of the absolute difference, definitionally proved · kernel-lean · F:intspace-l1-interval-distance-is-the-integral-of-the-absolute-difference
      1 dependencies 0 dependents
1184. An integrated pointwise distance on an integration space is a metric on the bundled carrier proved · kernel-lean · F:intspace-l1-is-a-metric-space
      1 dependencies 1 dependents
1185. Measure is derived from the integral, and its two bounds hold in every integration space proved · kernel-lean · F:intspace-measure-derived-from-integral
      1 dependencies 0 dependents
1186. Monotone convergence of integrals, with the classical principle as an explicit hypothesis proved · kernel-lean · F:intspace-monotone-convergence-on-a-hypothesis
      1 dependencies 0 dependents
1187. The rational finite expectation is an integral in the finite integration space, across the embedding of the rationals in the reals proved · kernel-lean · F:intspace-rat-expectation-integral
      0 dependencies 0 dependents
1188. Mathlib's Intermediate Value Theorem, imported (labeled scaffolding, not ours) proved · imported-kernel-lean · F:ivt-mathlib-import-intermediate-value-icc
      0 dependencies 0 dependents
1189. The law of cosines over the constructed reals, in the classical c^2 = a^2 + b^2 - 2ab cos C form proved · kernel-lean · F:law-of-cosines-over-constructed-plane
      22 dependencies 0 dependents
1190. The law of sines over the constructed reals, as |U x V| = ||U|| ||V|| sin, with the cross product identified with the existing triangle determinant proved · kernel-lean · F:law-of-sines-over-constructed-plane
      22 dependencies 0 dependents
1191. Official Lean's own kernel accepts the 73 constructed-real declarations it will not take as theorems, once they are exported as definitions open · route not assigned · F:lean-kernel-accepts-the-non-prop-residue-of-the-constructed-real-carrier
      1 dependencies 0 dependents
1192. Official Lean's own kernel accepts every representable declaration of the constructed-real carrier, and refuses the rest by a rule of its own proved · kernel-lean · F:lean-kernel-accepts-the-whole-constructed-real-carrier
      0 dependencies 1 dependents
1193. A Lean query module over the constructed reals shrinks 257x when the shared development is imported rather than inlined proved · kernel-lean · F:lean-query-module-shrinks-by-a-shared-import lean-module-exportconstructed-reals
      0 dependencies 0 dependents
1194. Append is associative over a computed List proved · kernel-lean · F:list-append-assoc
      0 dependencies 1 dependents
1195. Appending nil on the right is the identity for List.append proved · kernel-lean · F:list-append-nil
      0 dependencies 1 dependents
1196. Count distributes additively over append proved · kernel-lean · F:list-count-append
      2 dependencies 2 dependents
1197. A list and its reversal have the same per-element counts proved · kernel-lean · F:list-count-reverse
      1 dependencies 1 dependents
1198. A list's per-element count agrees with its computed multiset's count proved · kernel-lean · F:list-count-to-multiset
      4 dependencies 0 dependents
1199. Right fold distributes over append proved · kernel-lean · F:list-foldr-append
      0 dependencies 1 dependents
1200. The length of an append is the sum of the two lengths proved · kernel-lean · F:list-length-append
      2 dependencies 1 dependents
1201. Mapping a function over a list does not change its length proved · kernel-lean · F:list-length-map
      0 dependencies 0 dependents
1202. Reversing a list does not change its length proved · kernel-lean · F:list-length-reverse
      1 dependencies 0 dependents
1203. The empty list is a left identity for append proved · imported-kernel-lean · F:list-nil-append C:identity-element
      0 dependencies 0 dependents
1204. List.Perm is preserved by swapping an append's operands proved · kernel-lean · F:list-perm-append-comm
      3 dependencies 0 dependents
1205. List.Perm is reflexive proved · kernel-lean · F:list-perm-refl
      1 dependencies 0 dependents
1206. A list is a List.Perm of its own reversal proved · kernel-lean · F:list-perm-reverse
      2 dependencies 0 dependents
1207. List.Perm is symmetric proved · kernel-lean · F:list-perm-symm
      2 dependencies 0 dependents
1208. Reversing an append swaps and reverses the operands proved · kernel-lean · F:list-reverse-append
      2 dependencies 1 dependents
1209. Reversing a list twice recovers the original proved · kernel-lean · F:list-reverse-reverse
      1 dependencies 0 dependents
1210. Summing an appended list Nat is the sum of the two sums proved · kernel-lean · F:list-sum-append
      2 dependencies 0 dependents
1211. Fourteen rows of a 90-row outbound load plan are an irreducible infeasible subsystem proved · search-certificate · F:loadplan-hazmat-iis
      0 dependencies 0 dependents
1212. Conjunction: left projection proved · kernel-lean · F:logic-and-left
      0 dependencies 22 dependents
1213. Conjunction: right projection proved · kernel-lean · F:logic-and-right
      0 dependencies 21 dependents
1214. Boolean false is not true proved · kernel-lean · F:logic-bool-false-ne-true
      0 dependencies 9 dependents
1215. Boolean true is not false proved · kernel-lean · F:logic-bool-true-ne-false
      2 dependencies 5 dependents
1216. A decidable proposition's decision procedure returns true iff the proposition holds proved · kernel-lean · F:logic-decidable-decide-eq-true-iff
      1 dependencies 0 dependents
1217. Excluded middle for decidable propositions proved · kernel-lean · F:logic-decidable-em
      0 dependencies 0 dependents
1218. A decidable proposition's negation holds when its decision procedure says false proved · kernel-lean · F:logic-decidable-of-decide-eq-false
      1 dependencies 0 dependents
1219. A decidable proposition holds when its decision procedure says true proved · kernel-lean · F:logic-decidable-of-decide-eq-true
      1 dependencies 1 dependents
1220. De Morgan: not-p and not-q implies not-(p or q) proved · kernel-lean · F:logic-demorgan-not-or-converse
      3 dependencies 0 dependents
1221. De Morgan: not-(p or q) implies not-p and not-q proved · kernel-lean · F:logic-demorgan-not-or
      0 dependencies 0 dependents
1222. Double-negation elimination from excluded middle proved · kernel-lean · F:logic-dne-of-em
      1 dependencies 0 dependents
1223. Excluded middle from double-negation elimination proved · kernel-lean · F:logic-em-of-dne
      1 dependencies 0 dependents
1224. Excluded middle from Peirce's law proved · kernel-lean · F:logic-em-of-peirce
      0 dependencies 0 dependents
1225. Biconditional: forward direction proved · kernel-lean · F:logic-iff-mp
      0 dependencies 17 dependents
1226. Biconditional: backward direction proved · kernel-lean · F:logic-iff-mpr
      0 dependencies 19 dependents
1227. Modus tollens (kernel route) proved · kernel-lean · F:logic-modus-tollens
      0 dependencies 0 dependents
1228. Law of non-contradiction (kernel route) proved · kernel-lean · F:logic-noncontradiction
      2 dependencies 0 dependents
1229. Double negation distributes over conjunction proved · kernel-lean · F:logic-not-not-and
      0 dependencies 0 dependents
1230. Excluded middle is irrefutable proved · kernel-lean · F:logic-not-not-em
      0 dependencies 1 dependents
1231. Double-negation introduction proved · kernel-lean · F:logic-not-not-intro
      0 dependencies 2 dependents
1232. Triple negation collapses to single negation proved · kernel-lean · F:logic-not-not-not
      1 dependencies 0 dependents
1233. Disjunction elimination (case analysis) proved · kernel-lean · F:logic-or-elim
      0 dependencies 19 dependents
1234. Disjunctive syllogism (left) proved · kernel-lean · F:logic-or-resolve-left
      0 dependencies 1 dependents
1235. Peirce's law from excluded middle proved · kernel-lean · F:logic-peirce-of-em
      1 dependencies 0 dependents
1236. Ceva's and Menelaus' theorems over the constructed reals, in a division-free ratio parametrisation, with Ceva proved in both directions proved · kernel-lean · F:menelaus-and-ceva-over-constructed-plane
      15 dependencies 0 dependents
1237. A closed interval of the constructive reals is Bishop-compact proved · kernel-lean · F:metric-bishop-compactness-of-a-closed-real-interval
      1 dependencies 1 dependents
1238. The Extreme Value Theorem, in its constructive approximate form, over an arbitrary metric space proved · kernel-lean · F:metric-extreme-value-theorem-over-a-metric-space
      1 dependencies 1 dependents
1239. The real interval's Extreme Value Theorem is an instance of the metric-space one: one statement, two independent proofs proved · kernel-lean · F:metric-interval-evt-is-an-instance-of-the-metric-evt
      2 dependencies 0 dependents
1240. Completeness transfers to the product of two metric spaces proved · kernel-lean · F:metric-product-completeness-transfer
      0 dependencies 0 dependents
1241. Completeness of the real line, as an instance of completeness for an arbitrary metric space proved · kernel-lean · F:metric-space-completeness-of-the-real-line
      3 dependencies 3 dependents
1242. The four-point inequality, proved once for every metric space proved · kernel-lean · F:metric-space-four-point-inequality
      0 dependencies 0 dependents
1243. Uniform continuity on a real interval implies continuity as a notion of the metric layer proved · kernel-lean · F:metric-uniform-continuity-on-the-reals-is-a-metric-notion
      1 dependencies 0 dependents
1244. Mathlib v4.30 source proposition Int.add_assoc proved · kernel-lean · F:ml430-int-add-assoc-749cb0ff
      7 dependencies 1 dependents
1245. Mathlib v4.30 source proposition Int.add_comm proved · kernel-lean · F:ml430-int-add-comm-c5722728
      2 dependencies 1 dependents
1246. Mathlib v4.30 source proposition Int.add_ediv_of_dvd_left open · route not assigned · F:ml430-int-add-ediv-of-dvd-left-52ee6c5c
      0 dependencies 0 dependents
1247. Mathlib v4.30 source proposition Int.add_ediv_of_dvd_right open · route not assigned · F:ml430-int-add-ediv-of-dvd-right-3ead15d8
      0 dependencies 0 dependents
1248. Mathlib v4.30 source proposition Int.add_emod open · route not assigned · F:ml430-int-add-emod-b5735756
      0 dependencies 0 dependents
1249. Mathlib v4.30 source proposition Int.add_emod_emod open · route not assigned · F:ml430-int-add-emod-emod-87d0ffc8
      0 dependencies 0 dependents
1250. Mathlib v4.30 source proposition Int.add_emod_eq_add_emod_left open · route not assigned · F:ml430-int-add-emod-eq-add-emod-left-71885891
      0 dependencies 0 dependents
1251. Mathlib v4.30 source proposition Int.add_emod_eq_add_emod_right open · route not assigned · F:ml430-int-add-emod-eq-add-emod-right-c2cf373e
      0 dependencies 0 dependents
1252. Mathlib v4.30 source proposition Int.add_emod_left open · route not assigned · F:ml430-int-add-emod-left-dd3f807c
      0 dependencies 0 dependents
1253. Mathlib v4.30 source proposition Int.add_emod_right open · route not assigned · F:ml430-int-add-emod-right-4345648e
      0 dependencies 0 dependents
1254. Mathlib v4.30 source proposition Int.add_le_add proved · kernel-lean · F:ml430-int-add-le-add-a76ad5ce
      4 dependencies 0 dependents
1255. Mathlib v4.30 source proposition Int.add_le_add_iff_left proved · kernel-lean · F:ml430-int-add-le-add-iff-left-83bf44fa
      6 dependencies 1 dependents
1256. Mathlib v4.30 source proposition Int.add_le_add_iff_right proved · kernel-lean · F:ml430-int-add-le-add-iff-right-995cbaa5
      3 dependencies 5 dependents
1257. Mathlib v4.30 source proposition Int.add_le_add_left proved · kernel-lean · F:ml430-int-add-le-add-left-4a766697
      2 dependencies 8 dependents
1258. Mathlib v4.30 source proposition Int.add_le_add_right proved · kernel-lean · F:ml430-int-add-le-add-right-0c9ed495
      2 dependencies 5 dependents
1259. Mathlib v4.30 source proposition Int.add_le_add_three proved · kernel-lean · F:ml430-int-add-le-add-three-ade2c97e
      9 dependencies 0 dependents
1260. Mathlib v4.30 source proposition Int.add_le_iff_le_sub proved · kernel-lean · F:ml430-int-add-le-iff-le-sub-66d51eea
      7 dependencies 0 dependents
1261. Mathlib v4.30 source proposition Int.add_le_of_le_neg_add proved · kernel-lean · F:ml430-int-add-le-of-le-neg-add-6481d6ea
      11 dependencies 0 dependents
1262. Mathlib v4.30 source proposition Int.add_le_of_le_sub_left proved · kernel-lean · F:ml430-int-add-le-of-le-sub-left-b5bfaca2
      11 dependencies 0 dependents
1263. Mathlib v4.30 source proposition Int.add_le_of_le_sub_right proved · kernel-lean · F:ml430-int-add-le-of-le-sub-right-6cf92559
      11 dependencies 0 dependents
1264. Mathlib v4.30 source proposition Int.add_left_cancel proved · kernel-lean · F:ml430-int-add-left-cancel-eca3a10d
      6 dependencies 1 dependents
1265. Mathlib v4.30 source proposition Int.add_left_comm proved · kernel-lean · F:ml430-int-add-left-comm-519fffa6
      3 dependencies 0 dependents
1266. Mathlib v4.30 source proposition Int.add_left_inj proved · kernel-lean · F:ml430-int-add-left-inj-befcca48
      2 dependencies 0 dependents
1267. Mathlib v4.30 source proposition Int.add_left_neg proved · kernel-lean · F:ml430-int-add-left-neg-c1bf80e1
      2 dependencies 6 dependents
1268. Mathlib v4.30 source proposition Int.add_modEq_left proved · kernel-lean · F:ml430-int-add-modeq-left-ee732b5b
      2 dependencies 0 dependents
1269. Mathlib v4.30 source proposition Int.add_modEq_right proved · kernel-lean · F:ml430-int-add-modeq-right-e58108ee
      2 dependencies 0 dependents
1270. Mathlib v4.30 source proposition Int.add_mul proved · kernel-lean · F:ml430-int-add-mul-66aa025b
      2 dependencies 4 dependents
1271. Mathlib v4.30 source proposition Int.add_mul_ediv_left open · route not assigned · F:ml430-int-add-mul-ediv-left-3f1150d1
      0 dependencies 0 dependents
1272. Mathlib v4.30 source proposition Int.add_mul_ediv_right open · route not assigned · F:ml430-int-add-mul-ediv-right-edd6fbac
      0 dependencies 0 dependents
1273. Mathlib v4.30 source proposition Int.add_neg_cancel_left proved · kernel-lean · F:ml430-int-add-neg-cancel-left-580f2be4
      4 dependencies 0 dependents
1274. Mathlib v4.30 source proposition Int.add_neg_cancel_right proved · kernel-lean · F:ml430-int-add-neg-cancel-right-98f97323
      4 dependencies 0 dependents
1275. Mathlib v4.30 source proposition Int.add_neg_eq_sub proved · kernel-lean · F:ml430-int-add-neg-eq-sub-32f34729
      0 dependencies 2 dependents
1276. Mathlib v4.30 source proposition Int.add_one_ediv_two_mul_two_of_odd proved · kernel-lean · F:ml430-int-add-one-ediv-two-mul-two-of-odd-3c9ef32f
      4 dependencies 0 dependents
1277. Mathlib v4.30 source proposition Int.add_one_le_of_not_le open · route not assigned · F:ml430-int-add-one-le-of-not-le-fd7eda8b
      0 dependencies 0 dependents
1278. Mathlib v4.30 source proposition Int.add_two_le_iff_lt_of_even_sub open · route not assigned · F:ml430-int-add-two-le-iff-lt-of-even-sub-5f816323
      0 dependencies 0 dependents
1279. Mathlib v4.30 source proposition Int.div_dvd_iff_dvd_mul open · route not assigned · F:ml430-int-div-dvd-iff-dvd-mul-a8c3badc
      0 dependencies 0 dependents
1280. Mathlib v4.30 source proposition Int.div_le_div_iff_of_dvd_of_neg_of_neg open · route not assigned · F:ml430-int-div-le-div-iff-of-dvd-of-neg-of-neg-2a406b84
      0 dependencies 0 dependents
1281. Mathlib v4.30 source proposition Int.div_le_div_iff_of_dvd_of_neg_of_pos open · route not assigned · F:ml430-int-div-le-div-iff-of-dvd-of-neg-of-pos-9c8d8a09
      0 dependencies 0 dependents
1282. Mathlib v4.30 source proposition Int.div_le_div_iff_of_dvd_of_pos_of_neg open · route not assigned · F:ml430-int-div-le-div-iff-of-dvd-of-pos-of-neg-9de4290e
      0 dependencies 0 dependents
1283. Mathlib v4.30 source proposition Int.div_le_div_iff_of_dvd_of_pos_of_pos open · route not assigned · F:ml430-int-div-le-div-iff-of-dvd-of-pos-of-pos-ae84c83e
      0 dependencies 0 dependents
1284. Mathlib v4.30 source proposition Int.div_le_iff_of_dvd_of_neg open · route not assigned · F:ml430-int-div-le-iff-of-dvd-of-neg-56f5d86e
      0 dependencies 0 dependents
1285. Mathlib v4.30 source proposition Int.div_le_iff_of_dvd_of_pos open · route not assigned · F:ml430-int-div-le-iff-of-dvd-of-pos-d6e9563a
      0 dependencies 0 dependents
1286. Mathlib v4.30 source proposition Int.div_lt_div_iff_of_dvd_of_neg_of_neg open · route not assigned · F:ml430-int-div-lt-div-iff-of-dvd-of-neg-of-neg-aaf3dbfe
      0 dependencies 0 dependents
1287. Mathlib v4.30 source proposition Int.div_lt_div_iff_of_dvd_of_neg_of_pos open · route not assigned · F:ml430-int-div-lt-div-iff-of-dvd-of-neg-of-pos-f131fc0e
      0 dependencies 0 dependents
1288. Mathlib v4.30 source proposition Int.div_lt_div_iff_of_dvd_of_pos open · route not assigned · F:ml430-int-div-lt-div-iff-of-dvd-of-pos-a3d09d53
      0 dependencies 0 dependents
1289. Mathlib v4.30 source proposition Int.dvd_coe_gcd proved · kernel-lean · F:ml430-int-dvd-coe-gcd-6bda035e
      3 dependencies 0 dependents
1290. Mathlib v4.30 source proposition Int.dvd_coe_gcd_iff proved · kernel-lean · F:ml430-int-dvd-coe-gcd-iff-143eb02f
      4 dependencies 0 dependents
1291. Mathlib v4.30 source proposition Int.dvd_emod_add_of_dvd_add open · route not assigned · F:ml430-int-dvd-emod-add-of-dvd-add-3d3b6f29
      0 dependencies 0 dependents
1292. Mathlib v4.30 source proposition Int.dvd_gcd proved · kernel-lean · F:ml430-int-dvd-gcd-aebc8aa1
      2 dependencies 0 dependents
1293. Mathlib v4.30 source proposition Int.dvd_gcd_iff proved · kernel-lean · F:ml430-int-dvd-gcd-iff-5c30733d
      6 dependencies 0 dependents
1294. Mathlib v4.30 source proposition Int.dvd_gcd_mul_gcd_iff_dvd_mul proved · kernel-lean · F:ml430-int-dvd-gcd-mul-gcd-iff-dvd-mul-8ea752a5
      5 dependencies 0 dependents
1295. Mathlib v4.30 source proposition Int.dvd_gcd_mul_iff_dvd_mul proved · kernel-lean · F:ml430-int-dvd-gcd-mul-iff-dvd-mul-12f61b99
      4 dependencies 0 dependents
1296. Mathlib v4.30 source proposition Int.dvd_mul proved · kernel-lean · F:ml430-int-dvd-mul-3a7b94cd
      18 dependencies 0 dependents
1297. Mathlib v4.30 source proposition Int.dvd_mul_emod_add_of_dvd_mul_add open · route not assigned · F:ml430-int-dvd-mul-emod-add-of-dvd-mul-add-8a4f82e3
      0 dependencies 0 dependents
1298. Mathlib v4.30 source proposition Int.dvd_mul_gcd_iff_dvd_mul proved · kernel-lean · F:ml430-int-dvd-mul-gcd-iff-dvd-mul-22d6488e
      5 dependencies 1 dependents
1299. Mathlib v4.30 source proposition Int.dvd_natCast open · route not assigned · F:ml430-int-dvd-natcast-84ee4207
      0 dependencies 0 dependents
1300. Mathlib v4.30 source proposition Int.dvd_of_dvd_mul_left_of_gcd_one proved · kernel-lean · F:ml430-int-dvd-of-dvd-mul-left-of-gcd-one-649e349b
      3 dependencies 1 dependents
1301. Mathlib v4.30 source proposition Int.dvd_of_dvd_mul_right_of_gcd_one proved · kernel-lean · F:ml430-int-dvd-of-dvd-mul-right-of-gcd-one-77817ff0
      2 dependencies 0 dependents
1302. Mathlib v4.30 source proposition Int.dvd_of_mul_dvd open · route not assigned · F:ml430-int-dvd-of-mul-dvd-c09d49d1
      0 dependencies 0 dependents
1303. Mathlib v4.30 source proposition Int.ediv_gcd_ne_zero_if_ne_zero_right proved · kernel-lean · F:ml430-int-ediv-gcd-ne-zero-if-ne-zero-right-e1bda815
      5 dependencies 0 dependents
1304. Mathlib v4.30 source proposition Int.ediv_gcd_ne_zero_of_ne_zero_left proved · kernel-lean · F:ml430-int-ediv-gcd-ne-zero-of-ne-zero-left-2778814c
      5 dependencies 0 dependents
1305. Mathlib v4.30 source proposition Int.ediv_two_mul_two_add_one_of_odd proved · kernel-lean · F:ml430-int-ediv-two-mul-two-add-one-of-odd-a7ec30d7
      3 dependencies 0 dependents
1306. Mathlib v4.30 source proposition Int.ediv_two_mul_two_of_even proved · kernel-lean · F:ml430-int-ediv-two-mul-two-of-even-0095e2a6
      4 dependencies 0 dependents
1307. Mathlib v4.30 source proposition Int.emod_eq_sub_self_emod open · route not assigned · F:ml430-int-emod-eq-sub-self-emod-8ceed103
      0 dependencies 0 dependents
1308. Mathlib v4.30 source proposition Int.emod_two_ne_one proved · kernel-lean · F:ml430-int-emod-two-ne-one-5b930333
      2 dependencies 0 dependents
1309. Mathlib v4.30 source proposition Int.emod_two_ne_zero proved · kernel-lean · F:ml430-int-emod-two-ne-zero-d07d008f
      2 dependencies 0 dependents
1310. Mathlib v4.30 source proposition Int.eq_natCast_toNat open · route not assigned · F:ml430-int-eq-natcast-tonat-0270a7b2
      0 dependencies 0 dependents
1311. Mathlib v4.30 source proposition Int.eq_of_mod_eq_of_natAbs_sub_lt_natAbs open · route not assigned · F:ml430-int-eq-of-mod-eq-of-natabs-sub-lt-natabs-67e7052a
      0 dependencies 0 dependents
1312. Mathlib v4.30 source proposition Int.eq_of_mul_eq_one open · route not assigned · F:ml430-int-eq-of-mul-eq-one-0e6b8782
      0 dependencies 0 dependents
1313. Mathlib v4.30 source proposition Int.eq_one_or_neg_one_of_mul_eq_neg_one open · route not assigned · F:ml430-int-eq-one-or-neg-one-of-mul-eq-neg-one-b8df542e
      0 dependencies 0 dependents
1314. Mathlib v4.30 source proposition Int.eq_one_or_neg_one_of_mul_eq_neg_one' open · route not assigned · F:ml430-int-eq-one-or-neg-one-of-mul-eq-neg-one-be9181e7
      0 dependencies 0 dependents
1315. Mathlib v4.30 source proposition Int.eq_one_or_neg_one_of_mul_eq_one open · route not assigned · F:ml430-int-eq-one-or-neg-one-of-mul-eq-one-79096e09
      0 dependencies 0 dependents
1316. Mathlib v4.30 source proposition Int.eq_one_or_neg_one_of_mul_eq_one' open · route not assigned · F:ml430-int-eq-one-or-neg-one-of-mul-eq-one-9cbac9dc
      0 dependencies 0 dependents
1317. Mathlib v4.30 source proposition Int.eq_zero_of_dvd_of_nonneg_of_lt open · route not assigned · F:ml430-int-eq-zero-of-dvd-of-nonneg-of-lt-87fe63a0
      0 dependencies 0 dependents
1318. Mathlib v4.30 source proposition Int.eq_zero_or_eq_zero_of_mul_eq_zero proved · kernel-lean · F:ml430-int-eq-zero-or-eq-zero-of-mul-eq-zero-da702fe4
      3 dependencies 0 dependents
1319. Mathlib v4.30 source proposition Int.even_add proved · kernel-lean · F:ml430-int-even-add-3c4536e3
      7 dependencies 0 dependents
1320. Mathlib v4.30 source proposition Int.even_add' proved · kernel-lean · F:ml430-int-even-add-bc8e1394
      7 dependencies 0 dependents
1321. Mathlib v4.30 source proposition Int.even_add_one proved · kernel-lean · F:ml430-int-even-add-one-af33da18
      8 dependencies 0 dependents
1322. Mathlib v4.30 source proposition Int.exists_gcd_one' proved · kernel-lean · F:ml430-int-exists-gcd-one-657db3e2
      7 dependencies 0 dependents
1323. Mathlib v4.30 source proposition Int.exists_gcd_one proved · kernel-lean · F:ml430-int-exists-gcd-one-d8820780
      7 dependencies 0 dependents
1324. Mathlib v4.30 source proposition Int.exists_greatest_of_bdd open · route not assigned · F:ml430-int-exists-greatest-of-bdd-540c90cf
      0 dependencies 0 dependents
1325. Mathlib v4.30 source proposition Int.exists_least_of_bdd open · route not assigned · F:ml430-int-exists-least-of-bdd-5b2bde8b
      0 dependencies 0 dependents
1326. Mathlib v4.30 source proposition Int.fib_add proved · kernel-lean · F:ml430-int-fib-add-181b6a2c
      9 dependencies 2 dependents
1327. Mathlib v4.30 source proposition Int.fib_add_one proved · kernel-lean · F:ml430-int-fib-add-one-33f1b748
      1 dependencies 0 dependents
1328. Mathlib v4.30 source proposition Int.fib_add_two proved · kernel-lean · F:ml430-int-fib-add-two-739358dd
      2 dependencies 2 dependents
1329. Mathlib v4.30 source proposition Int.fib_dvd proved · kernel-lean · F:ml430-int-fib-dvd-ffb3c5c1
      0 dependencies 0 dependents
1330. Mathlib v4.30 source proposition Int.fib_eq_fib_add_two_sub_fib_add_one proved · kernel-lean · F:ml430-int-fib-eq-fib-add-two-sub-fib-add-one-0dab3f6d
      1 dependencies 0 dependents
1331. Mathlib v4.30 source proposition Int.fib_eq_zero proved · kernel-lean · F:ml430-int-fib-eq-zero-8193c7cb
      0 dependencies 0 dependents
1332. Mathlib v4.30 source proposition Int.fib_gcd proved · kernel-lean · F:ml430-int-fib-gcd-3a8bfdec
      1 dependencies 0 dependents
1333. Mathlib v4.30 source proposition Int.fib_natCast proved · kernel-lean · F:ml430-int-fib-natcast-d5886be4
      0 dependencies 1 dependents
1334. Mathlib v4.30 source proposition Int.fib_neg proved · kernel-lean · F:ml430-int-fib-neg-b4021d37
      0 dependencies 1 dependents
1335. Mathlib v4.30 source proposition Int.fib_neg_one proved · kernel-lean · F:ml430-int-fib-neg-one-107bdfc6
      0 dependencies 0 dependents
1336. Mathlib v4.30 source proposition Int.fib_neg_two proved · kernel-lean · F:ml430-int-fib-neg-two-379ecfa0
      0 dependencies 0 dependents
1337. Mathlib v4.30 source proposition Int.fib_of_nonneg proved · kernel-lean · F:ml430-int-fib-of-nonneg-438018c5
      0 dependencies 0 dependents
1338. Mathlib v4.30 source proposition Int.fib_of_odd proved · kernel-lean · F:ml430-int-fib-of-odd-66560495
      10 dependencies 1 dependents
1339. Mathlib v4.30 source proposition Int.fib_one proved · kernel-lean · F:ml430-int-fib-one-df6c44d8
      0 dependencies 0 dependents
1340. Mathlib v4.30 source proposition Int.fib_two proved · kernel-lean · F:ml430-int-fib-two-33d76423
      0 dependencies 0 dependents
1341. Mathlib v4.30 source proposition Int.fib_two_mul proved · kernel-lean · F:ml430-int-fib-two-mul-0e70f3dd
      9 dependencies 1 dependents
1342. Mathlib v4.30 source proposition Int.fib_two_mul_add_one_eq_natFib_natAbs proved · kernel-lean · F:ml430-int-fib-two-mul-add-one-eq-natfib-natabs-61a8342b
      4 dependencies 0 dependents
1343. Mathlib v4.30 source proposition Int.fib_two_mul_add_one_pos proved · kernel-lean · F:ml430-int-fib-two-mul-add-one-pos-8977f65f
      10 dependencies 0 dependents
1344. Mathlib v4.30 source proposition Int.fib_two_mul_add_two proved · kernel-lean · F:ml430-int-fib-two-mul-add-two-0ba4a948
      9 dependencies 0 dependents
1345. Mathlib v4.30 source proposition Int.fib_zero proved · kernel-lean · F:ml430-int-fib-zero-439bfeee
      0 dependencies 0 dependents
1346. Mathlib v4.30 source proposition Int.gcd_div proved · kernel-lean · F:ml430-int-gcd-div-5e01872f
      26 dependencies 0 dependents
1347. Mathlib v4.30 source proposition Int.gcd_div_gcd_div_gcd proved · kernel-lean · F:ml430-int-gcd-div-gcd-div-gcd-2db608dc
      16 dependencies 3 dependents
1348. Mathlib v4.30 source proposition Int.gcd_dvd_iff proved · kernel-lean · F:ml430-int-gcd-dvd-iff-66fa03b3
      9 dependencies 0 dependents
1349. Mathlib v4.30 source proposition Int.gcd_emod open · route not assigned · F:ml430-int-gcd-emod-66050e90
      0 dependencies 0 dependents
1350. Mathlib v4.30 source proposition Int.gcd_eq_gcd_ab proved · kernel-lean · F:ml430-int-gcd-eq-gcd-ab-63005aef
      4 dependencies 4 dependents
1351. Mathlib v4.30 source proposition Int.gcd_eq_one_of_gcd_mul_right_eq_one_left proved · kernel-lean · F:ml430-int-gcd-eq-one-of-gcd-mul-right-eq-one-left-8533eb82
      3 dependencies 0 dependents
1352. Mathlib v4.30 source proposition Int.gcd_eq_one_of_gcd_mul_right_eq_one_right proved · kernel-lean · F:ml430-int-gcd-eq-one-of-gcd-mul-right-eq-one-right-a9b19222
      4 dependencies 0 dependents
1353. Mathlib v4.30 source proposition Int.gcd_fib proved · kernel-lean · F:ml430-int-gcd-fib-73bdafc2
      2 dependencies 1 dependents
1354. Mathlib v4.30 source proposition Int.gcd_greatest proved · kernel-lean · F:ml430-int-gcd-greatest-5b31c5fe
      11 dependencies 0 dependents
1355. Mathlib v4.30 source proposition Int.gcd_ne_one_iff_gcd_mul_right_ne_one proved · kernel-lean · F:ml430-int-gcd-ne-one-iff-gcd-mul-right-ne-one-ae6099bd
      3 dependencies 0 dependents
1356. Mathlib v4.30 source proposition Int.induct_rcc_left open · route not assigned · F:ml430-int-induct-rcc-left-b1583f11
      0 dependencies 0 dependents
1357. Mathlib v4.30 source proposition Int.induct_rcc_right open · route not assigned · F:ml430-int-induct-rcc-right-5d3f1a22
      0 dependencies 0 dependents
1358. Mathlib v4.30 source proposition Int.induct_rco_left open · route not assigned · F:ml430-int-induct-rco-left-44aea79a
      0 dependencies 0 dependents
1359. Mathlib v4.30 source proposition Int.induct_rco_right open · route not assigned · F:ml430-int-induct-rco-right-f62a54c5
      0 dependencies 0 dependents
1360. Mathlib v4.30 source proposition Int.induct_roc_left open · route not assigned · F:ml430-int-induct-roc-left-ba1f9783
      0 dependencies 0 dependents
1361. Mathlib v4.30 source proposition Int.induct_roc_right open · route not assigned · F:ml430-int-induct-roc-right-0f595642
      0 dependencies 0 dependents
1362. Mathlib v4.30 source proposition Int.induct_roo_left open · route not assigned · F:ml430-int-induct-roo-left-c3e749b7
      0 dependencies 0 dependents
1363. Mathlib v4.30 source proposition Int.induct_roo_right open · route not assigned · F:ml430-int-induct-roo-right-5c8bce01
      0 dependencies 0 dependents
1364. Mathlib v4.30 source proposition Int.le.elim proved · kernel-lean · F:ml430-int-le-elim-efa70bfa
      1 dependencies 0 dependents
1365. Mathlib v4.30 source proposition Int.le_of_ofNat_le_ofNat proved · kernel-lean · F:ml430-int-le-of-ofnat-le-ofnat-483e0f18
      0 dependencies 0 dependents
1366. Mathlib v4.30 source proposition Int.le_sub_one_of_not_le open · route not assigned · F:ml430-int-le-sub-one-of-not-le-fc32b89d
      0 dependencies 0 dependents
1367. Mathlib v4.30 source proposition Int.lt.elim proved · kernel-lean · F:ml430-int-lt-elim-35b9d0f8
      1 dependencies 0 dependents
1368. Mathlib v4.30 source proposition Int.lt_of_ofNat_lt_ofNat proved · kernel-lean · F:ml430-int-lt-of-ofnat-lt-ofnat-968e5a43
      0 dependencies 2 dependents
1369. Mathlib v4.30 source proposition Int.lt_of_sum_four_squares_eq_mul open · route not assigned · F:ml430-int-lt-of-sum-four-squares-eq-mul-449dddb6
      0 dependencies 0 dependents
1370. Mathlib v4.30 source proposition Int.lt_of_toNat_lt open · route not assigned · F:ml430-int-lt-of-tonat-lt-0b04114f
      0 dependencies 0 dependents
1371. Mathlib v4.30 source proposition Int.lt_or_eq_of_le open · route not assigned · F:ml430-int-lt-or-eq-of-le-28b5d4e8
      0 dependencies 0 dependents
1372. Mathlib v4.30 source proposition Int.lt_toNat open · route not assigned · F:ml430-int-lt-tonat-18a10163
      0 dependencies 0 dependents
1373. Mathlib v4.30 source proposition Int.mod_modEq proved · kernel-lean · F:ml430-int-mod-modeq-6bec7847
      3 dependencies 0 dependents
1374. Mathlib v4.30 source proposition Int.ModEq.add proved · kernel-lean · F:ml430-int-modeq-add-b805125a
      3 dependencies 2 dependents
1375. Mathlib v4.30 source proposition Int.ModEq.add_left proved · kernel-lean · F:ml430-int-modeq-add-left-6e17c69a
      2 dependencies 0 dependents
1376. Mathlib v4.30 source proposition Int.ModEq.add_left_cancel' proved · kernel-lean · F:ml430-int-modeq-add-left-cancel-062ad5fe
      5 dependencies 3 dependents
1377. Mathlib v4.30 source proposition Int.ModEq.add_left_cancel proved · kernel-lean · F:ml430-int-modeq-add-left-cancel-c1adde5a
      4 dependencies 0 dependents
1378. Mathlib v4.30 source proposition Int.ModEq.add_right_cancel proved · kernel-lean · F:ml430-int-modeq-add-right-cancel-d7366811
      4 dependencies 0 dependents
1379. Mathlib v4.30 source proposition Int.ModEq.add_right_cancel' proved · kernel-lean · F:ml430-int-modeq-add-right-cancel-f74acb64
      2 dependencies 1 dependents
1380. Mathlib v4.30 source proposition Int.ModEq.add_right proved · kernel-lean · F:ml430-int-modeq-add-right-fa8f7abe
      10 dependencies 0 dependents
1381. Mathlib v4.30 source proposition Int.ModEq.cancel_left_div_gcd proved · kernel-lean · F:ml430-int-modeq-cancel-left-div-gcd-b2d407e8
      24 dependencies 1 dependents
1382. Mathlib v4.30 source proposition Int.ModEq.cancel_right_div_gcd proved · kernel-lean · F:ml430-int-modeq-cancel-right-div-gcd-00cd73fa
      2 dependencies 0 dependents
1383. Mathlib v4.30 source proposition Int.modEq_comm proved · kernel-lean · F:ml430-int-modeq-comm-1e4bcc07
      1 dependencies 0 dependents
1384. Mathlib v4.30 source proposition Int.ModEq.dvd proved · kernel-lean · F:ml430-int-modeq-dvd-1ce6b46e
      7 dependencies 0 dependents
1385. Mathlib v4.30 source proposition Int.ModEq.dvd_iff proved · kernel-lean · F:ml430-int-modeq-dvd-iff-b7ffeff8
      11 dependencies 0 dependents
1386. Mathlib v4.30 source proposition Int.ModEq.eq proved · kernel-lean · F:ml430-int-modeq-eq-696e85f7
      0 dependencies 0 dependents
1387. Mathlib v4.30 source proposition Int.ModEq.mul proved · kernel-lean · F:ml430-int-modeq-mul-6736aa2e
      14 dependencies 2 dependents
1388. Mathlib v4.30 source proposition Int.modEq_neg proved · kernel-lean · F:ml430-int-modeq-neg-d6ff57b6
      7 dependencies 2 dependents
1389. Mathlib v4.30 source proposition Int.ModEq.neg proved · kernel-lean · F:ml430-int-modeq-neg-f649f6c5
      7 dependencies 1 dependents
1390. Mathlib v4.30 source proposition Int.ModEq.of_dvd proved · kernel-lean · F:ml430-int-modeq-of-dvd-b9c41fce
      10 dependencies 2 dependents
1391. Mathlib v4.30 source proposition Int.ModEq.of_mul_left proved · kernel-lean · F:ml430-int-modeq-of-mul-left-c4ccd51e
      2 dependencies 1 dependents
1392. Mathlib v4.30 source proposition Int.ModEq.of_mul_right proved · kernel-lean · F:ml430-int-modeq-of-mul-right-c92b7bf0
      3 dependencies 0 dependents
1393. Mathlib v4.30 source proposition Int.modEq_one proved · kernel-lean · F:ml430-int-modeq-one-01d9de39
      3 dependencies 0 dependents
1394. Mathlib v4.30 source proposition Int.ModEq.refl proved · kernel-lean · F:ml430-int-modeq-refl-30e15520
      0 dependencies 0 dependents
1395. Mathlib v4.30 source proposition Int.modEq_sub proved · kernel-lean · F:ml430-int-modeq-sub-3148f130
      7 dependencies 0 dependents
1396. Mathlib v4.30 source proposition Int.ModEq.symm proved · kernel-lean · F:ml430-int-modeq-symm-984a6e67
      0 dependencies 3 dependents
1397. Mathlib v4.30 source proposition Int.ModEq.trans proved · kernel-lean · F:ml430-int-modeq-trans-6d7863e0
      0 dependencies 1 dependents
1398. Mathlib v4.30 source proposition Int.modulus_modEq_zero proved · kernel-lean · F:ml430-int-modulus-modeq-zero-5b57a898
      3 dependencies 0 dependents
1399. Mathlib v4.30 source proposition Int.mul_ediv_le_mul_ediv_assoc open · route not assigned · F:ml430-int-mul-ediv-le-mul-ediv-assoc-114d9ea9
      0 dependencies 0 dependents
1400. Mathlib v4.30 source proposition Int.mul_eq_neg_one_iff_eq_one_or_neg_one open · route not assigned · F:ml430-int-mul-eq-neg-one-iff-eq-one-or-neg-one-baa79d49
      0 dependencies 0 dependents
1401. Mathlib v4.30 source proposition Int.mul_eq_one_iff_eq_one_or_neg_one open · route not assigned · F:ml430-int-mul-eq-one-iff-eq-one-or-neg-one-073ece31
      0 dependencies 0 dependents
1402. Mathlib v4.30 source proposition Int.mul_le_mul_of_le_of_le_of_nonneg_of_nonneg open · route not assigned · F:ml430-int-mul-le-mul-of-le-of-le-of-nonneg-of-nonneg-5d7e65e5
      0 dependencies 0 dependents
1403. Mathlib v4.30 source proposition Int.mul_le_mul_of_le_of_le_of_nonneg_of_nonpos open · route not assigned · F:ml430-int-mul-le-mul-of-le-of-le-of-nonneg-of-nonpos-7e223ee0
      0 dependencies 0 dependents
1404. Mathlib v4.30 source proposition Int.mul_le_mul_of_le_of_le_of_nonpos_of_nonneg open · route not assigned · F:ml430-int-mul-le-mul-of-le-of-le-of-nonpos-of-nonneg-a449a2a6
      0 dependencies 0 dependents
1405. Mathlib v4.30 source proposition Int.mul_le_mul_of_le_of_le_of_nonpos_of_nonpos open · route not assigned · F:ml430-int-mul-le-mul-of-le-of-le-of-nonpos-of-nonpos-7980025b
      0 dependencies 0 dependents
1406. Mathlib v4.30 source proposition Int.mul_le_mul_of_natAbs_le open · route not assigned · F:ml430-int-mul-le-mul-of-natabs-le-e87cd800
      0 dependencies 0 dependents
1407. Mathlib v4.30 source proposition Int.mul_neg_iff proved · kernel-lean · F:ml430-int-mul-neg-iff-79c18955
      18 dependencies 0 dependents
1408. Mathlib v4.30 source proposition Int.mul_nonneg_iff proved · kernel-lean · F:ml430-int-mul-nonneg-iff-4a7185c1
      8 dependencies 0 dependents
1409. Mathlib v4.30 source proposition Int.mul_nonneg_of_nonneg_or_nonpos proved · kernel-lean · F:ml430-int-mul-nonneg-of-nonneg-or-nonpos-79da88c9
      8 dependencies 1 dependents
1410. Mathlib v4.30 source proposition Int.mul_nonpos_iff proved · kernel-lean · F:ml430-int-mul-nonpos-iff-4da0d0b9
      15 dependencies 0 dependents
1411. Mathlib v4.30 source proposition Int.mul_pos_iff proved · kernel-lean · F:ml430-int-mul-pos-iff-e4bfa31d
      17 dependencies 0 dependents
1412. Mathlib v4.30 source proposition Int.natAbs_coe_sub_coe_le_of_le proved · kernel-lean · F:ml430-int-natabs-coe-sub-coe-le-of-le-d2800d86
      5 dependencies 0 dependents
1413. Mathlib v4.30 source proposition Int.natAbs_coe_sub_coe_lt_of_lt proved · kernel-lean · F:ml430-int-natabs-coe-sub-coe-lt-of-lt-e0566dd0
      5 dependencies 0 dependents
1414. Mathlib v4.30 source proposition Int.natAbs_emod_two proved · kernel-lean · F:ml430-int-natabs-emod-two-18514063
      3 dependencies 0 dependents
1415. Mathlib v4.30 source proposition Int.natAbs_eq_iff_mul_self_eq proved · kernel-lean · F:ml430-int-natabs-eq-iff-mul-self-eq-f1c49fdf
      0 dependencies 0 dependents
1416. Mathlib v4.30 source proposition Int.natAbs_inj_of_nonneg_of_nonneg proved · kernel-lean · F:ml430-int-natabs-inj-of-nonneg-of-nonneg-db3f2d3d
      0 dependencies 0 dependents
1417. Mathlib v4.30 source proposition Int.natAbs_inj_of_nonneg_of_nonpos proved · kernel-lean · F:ml430-int-natabs-inj-of-nonneg-of-nonpos-b5d96f53
      2 dependencies 0 dependents
1418. Mathlib v4.30 source proposition Int.natAbs_inj_of_nonpos_of_nonneg proved · kernel-lean · F:ml430-int-natabs-inj-of-nonpos-of-nonneg-ecdb334a
      2 dependencies 0 dependents
1419. Mathlib v4.30 source proposition Int.natAbs_inj_of_nonpos_of_nonpos proved · kernel-lean · F:ml430-int-natabs-inj-of-nonpos-of-nonpos-87475bd8
      3 dependencies 0 dependents
1420. Mathlib v4.30 source proposition Int.natAbs_le_iff_mul_self_le proved · kernel-lean · F:ml430-int-natabs-le-iff-mul-self-le-20242c1d
      0 dependencies 0 dependents
1421. Mathlib v4.30 source proposition Int.natAbs_le_of_dvd_ne_zero open · route not assigned · F:ml430-int-natabs-le-of-dvd-ne-zero-a8fcd923
      0 dependencies 0 dependents
1422. Mathlib v4.30 source proposition Int.natAbs_lt_iff_mul_self_lt proved · kernel-lean · F:ml430-int-natabs-lt-iff-mul-self-lt-5f19cd1f
      0 dependencies 0 dependents
1423. Mathlib v4.30 source proposition Int.natCast_dvd open · route not assigned · F:ml430-int-natcast-dvd-cb62f648
      0 dependencies 0 dependents
1424. Mathlib v4.30 source proposition Int.natCast_dvd_ofNat open · route not assigned · F:ml430-int-natcast-dvd-ofnat-8d8307a2
      0 dependencies 0 dependents
1425. Mathlib v4.30 source proposition Int.natCast_ediv open · route not assigned · F:ml430-int-natcast-ediv-2b4e8324
      0 dependencies 0 dependents
1426. Mathlib v4.30 source proposition Int.natCast_inj open · route not assigned · F:ml430-int-natcast-inj-9dc3b9c2
      0 dependencies 0 dependents
1427. Mathlib v4.30 source proposition Int.natCast_le_zero open · route not assigned · F:ml430-int-natcast-le-zero-bfce410a
      0 dependencies 0 dependents
1428. Mathlib v4.30 source proposition Int.ne_zero_of_gcd proved · kernel-lean · F:ml430-int-ne-zero-of-gcd-f71f00df
      2 dependencies 0 dependents
1429. Mathlib v4.30 source proposition Int.neg_modEq_neg proved · kernel-lean · F:ml430-int-neg-modeq-neg-30d98479
      6 dependencies 0 dependents
1430. Mathlib v4.30 source proposition Int.negSucc_ediv_negSucc open · route not assigned · F:ml430-int-negsucc-ediv-negsucc-74a87fde
      0 dependencies 0 dependents
1431. Mathlib v4.30 source proposition Int.negSucc_ediv_ofNat_succ open · route not assigned · F:ml430-int-negsucc-ediv-ofnat-succ-27c87d32
      0 dependencies 0 dependents
1432. Mathlib v4.30 source proposition Int.not_le_eq open · route not assigned · F:ml430-int-not-le-eq-a1aa34a4
      0 dependencies 0 dependents
1433. Mathlib v4.30 source proposition Int.not_lt_eq open · route not assigned · F:ml430-int-not-lt-eq-120ae6c2
      0 dependencies 0 dependents
1434. Mathlib v4.30 source proposition Int.not_prime_of_int_mul proved · kernel-lean · F:ml430-int-not-prime-of-int-mul-e3060f5d
      9 dependencies 0 dependents
1435. Mathlib v4.30 source proposition Int.Odd.of_mul_left proved · kernel-lean · F:ml430-int-odd-of-mul-left-b580971e
      2 dependencies 0 dependents
1436. Mathlib v4.30 source proposition Int.Odd.of_mul_right proved · kernel-lean · F:ml430-int-odd-of-mul-right-d6d1fc1d
      2 dependencies 0 dependents
1437. Mathlib v4.30 source proposition Int.ofNat_eq_coe open · route not assigned · F:ml430-int-ofnat-eq-coe-f82fca8a
      0 dependencies 0 dependents
1438. Mathlib v4.30 source proposition Int.ofNat_eq_natCast open · route not assigned · F:ml430-int-ofnat-eq-natcast-001cc3e2
      0 dependencies 0 dependents
1439. Mathlib v4.30 source proposition Int.ofNat_one open · route not assigned · F:ml430-int-ofnat-one-8ef5badc
      0 dependencies 0 dependents
1440. Mathlib v4.30 source proposition Int.ofNat_two open · route not assigned · F:ml430-int-ofnat-two-20e97f9e
      0 dependencies 0 dependents
1441. Mathlib v4.30 source proposition Int.ofNat_zero open · route not assigned · F:ml430-int-ofnat-zero-0d364f22
      0 dependencies 0 dependents
1442. Mathlib v4.30 source proposition Int.Prime.dvd_mul' proved · kernel-lean · F:ml430-int-prime-dvd-mul-23b73e69
      3 dependencies 0 dependents
1443. Mathlib v4.30 source proposition Int.Prime.dvd_mul proved · kernel-lean · F:ml430-int-prime-dvd-mul-90351ba0
      3 dependencies 0 dependents
1444. Mathlib v4.30 source proposition Int.sq_ne_two_mod_four open · route not assigned · F:ml430-int-sq-ne-two-mod-four-3d69c9b6
      0 dependencies 0 dependents
1445. Mathlib v4.30 source proposition Int.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul proved · kernel-lean · F:ml430-int-succ-dvd-or-succ-dvd-of-succ-sum-dvd-mul-435a4948
      13 dependencies 0 dependents
1446. Mathlib v4.30 source proposition Int.two_le_iff_pos_of_even open · route not assigned · F:ml430-int-two-le-iff-pos-of-even-64eb33a6
      0 dependencies 0 dependents
1447. Outcome-blind mutation of Nat.fib_eq_zero open · route not assigned · F:ml430-mutation-1432b2277cf2cc26c1d11cd6
      0 dependencies 0 dependents
1448. Outcome-blind mutation of Nat.sqrt_le_self open · route not assigned · F:ml430-mutation-2086302b3a338591b3179871
      0 dependencies 0 dependents
1449. Outcome-blind mutation of Int.ne_zero_of_gcd open · route not assigned · F:ml430-mutation-48fe130e2b8eadb6f626b66f
      0 dependencies 0 dependents
1450. Outcome-blind mutation of Nat.Prime.pred_pos open · route not assigned · F:ml430-mutation-5179f333b8333ecff8adc223
      0 dependencies 0 dependents
1451. Outcome-blind mutation of Nat.factorial_ne_zero open · route not assigned · F:ml430-mutation-7afa5ec620720a1501bf349d
      0 dependencies 0 dependents
1452. Outcome-blind mutation of Nat.lor_comm open · route not assigned · F:ml430-mutation-a6dd1759bce60d820292e107
      0 dependencies 0 dependents
1453. Outcome-blind mutation of Int.fib_eq_zero open · route not assigned · F:ml430-mutation-aabb80b1f89f0c5847364692
      0 dependencies 0 dependents
1454. Outcome-blind mutation of Int.ModEq.symm open · route not assigned · F:ml430-mutation-aca37b68d3cdf06f0127def9
      0 dependencies 0 dependents
1455. Outcome-blind mutation of Nat.not_coprime_zero_zero open · route not assigned · F:ml430-mutation-c20db9b4c60b816ce738bdf2
      0 dependencies 0 dependents
1456. Outcome-blind mutation of Nat.ModEq.symm open · route not assigned · F:ml430-mutation-c86940b52af8159ca9b381d6
      0 dependencies 0 dependents
1457. Outcome-blind mutation of Nat.log_le_self open · route not assigned · F:ml430-mutation-e8583599cfae2d40cefae3f0
      0 dependencies 0 dependents
1458. Outcome-blind mutation of Nat.choose_self open · route not assigned · F:ml430-mutation-edb05acf07d9ef3f9f8232fc
      0 dependencies 0 dependents
1459. Mathlib v4.30 source proposition Nat.abundant_iff_not_perfect_and_not_deficient proved · kernel-lean · F:ml430-nat-abundant-iff-not-perfect-and-not-deficient-9763e268
      5 dependencies 0 dependents
1460. Mathlib v4.30 source proposition Nat.Abundant.mul_left proved · kernel-lean · F:ml430-nat-abundant-mul-left-4de4fbe7
      5 dependencies 0 dependents
1461. Mathlib v4.30 source proposition Nat.Abundant.of_dvd proved · kernel-lean · F:ml430-nat-abundant-of-dvd-686548ce
      6 dependencies 0 dependents
1462. Mathlib v4.30 source proposition Nat.abundant_twelve proved · kernel-lean · F:ml430-nat-abundant-twelve-24ce1ba6
      1 dependencies 0 dependents
1463. Mathlib v4.30 source proposition Nat.add_add_add_comm proved · kernel-lean · F:ml430-nat-add-add-add-comm-74d2c151
      2 dependencies 2 dependents
1464. Mathlib v4.30 source proposition Nat.add_assoc proved · kernel-lean · F:ml430-nat-add-assoc-8c87a1f1
      0 dependencies 81 dependents
1465. Mathlib v4.30 source proposition Nat.add_choose proved · kernel-lean · F:ml430-nat-add-choose-eb49fa11
      8 dependencies 0 dependents
1466. Mathlib v4.30 source proposition Nat.add_choose_mul_factorial_mul_factorial proved · kernel-lean · F:ml430-nat-add-choose-mul-factorial-mul-factorial-26ba01ef
      4 dependencies 1 dependents
1467. Mathlib v4.30 source proposition Nat.add_comm proved · kernel-lean · F:ml430-nat-add-comm-56a2d614
      2 dependencies 176 dependents
1468. Mathlib v4.30 source proposition Nat.add_descFactorial_eq_ascFactorial proved · kernel-lean · F:ml430-nat-add-descfactorial-eq-ascfactorial-5faac784
      0 dependencies 1 dependents
1469. Mathlib v4.30 source proposition Nat.add_div_left proved · kernel-lean · F:ml430-nat-add-div-left-1b15b2b2
      7 dependencies 0 dependents
1470. Mathlib v4.30 source proposition Nat.add_div_of_dvd_add_add_one proved · kernel-lean · F:ml430-nat-add-div-of-dvd-add-add-one-f17dffc0
      22 dependencies 0 dependents
1471. Mathlib v4.30 source proposition Nat.add_div_right proved · kernel-lean · F:ml430-nat-add-div-right-4b60b393
      5 dependencies 1 dependents
1472. Mathlib v4.30 source proposition Nat.add_eq proved · kernel-lean · F:ml430-nat-add-eq-ab0eab69
      0 dependencies 0 dependents
1473. Mathlib v4.30 source proposition Nat.add_eq_left proved · kernel-lean · F:ml430-nat-add-eq-left-8e12789f
      2 dependencies 2 dependents
1474. Mathlib v4.30 source proposition Nat.add_eq_max_iff proved · kernel-lean · F:ml430-nat-add-eq-max-iff-39576e6d
      6 dependencies 0 dependents
1475. Mathlib v4.30 source proposition Nat.add_eq_min_iff proved · kernel-lean · F:ml430-nat-add-eq-min-iff-e60cf432
      5 dependencies 0 dependents
1476. Mathlib v4.30 source proposition Nat.add_eq_one_iff proved · kernel-lean · F:ml430-nat-add-eq-one-iff-f8463abc
      6 dependencies 0 dependents
1477. Mathlib v4.30 source proposition Nat.add_eq_right proved · kernel-lean · F:ml430-nat-add-eq-right-9067eb1a
      2 dependencies 2 dependents
1478. Mathlib v4.30 source proposition Nat.add_eq_three_iff proved · kernel-lean · F:ml430-nat-add-eq-three-iff-799a0a8f
      6 dependencies 0 dependents
1479. Mathlib v4.30 source proposition Nat.add_eq_two_iff proved · kernel-lean · F:ml430-nat-add-eq-two-iff-25385c65
      6 dependencies 0 dependents
1480. Mathlib v4.30 source proposition Nat.add_eq_zero proved · kernel-lean · F:ml430-nat-add-eq-zero-64233539
      1 dependencies 0 dependents
1481. Mathlib v4.30 source proposition Nat.add_factorial_le_factorial_add proved · kernel-lean · F:ml430-nat-add-factorial-le-factorial-add-b0400cf6
      9 dependencies 1 dependents
1482. Mathlib v4.30 source proposition Nat.add_factorial_lt_factorial_add proved · kernel-lean · F:ml430-nat-add-factorial-lt-factorial-add-7501a8c8
      16 dependencies 1 dependents
1483. Mathlib v4.30 source proposition Nat.add_factorial_succ_le_factorial_add_succ proved · kernel-lean · F:ml430-nat-add-factorial-succ-le-factorial-add-succ-e8145feb
      3 dependencies 0 dependents
1484. Mathlib v4.30 source proposition Nat.add_factorial_succ_lt_factorial_add_succ proved · kernel-lean · F:ml430-nat-add-factorial-succ-lt-factorial-add-succ-ec0fa8d3
      3 dependencies 0 dependents
1485. Mathlib v4.30 source proposition Nat.add_le_pair open · route not assigned · F:ml430-nat-add-le-pair-4af701ca
      0 dependencies 0 dependents
1486. Mathlib v4.30 source proposition Nat.add_max_add_left proved · kernel-lean · F:ml430-nat-add-max-add-left-37eb9f8d
      2 dependencies 0 dependents
1487. Mathlib v4.30 source proposition Nat.add_max_add_right proved · kernel-lean · F:ml430-nat-add-max-add-right-178bc311
      2 dependencies 0 dependents
1488. Mathlib v4.30 source proposition Nat.add_min_add_left proved · kernel-lean · F:ml430-nat-add-min-add-left-9728864e
      2 dependencies 0 dependents
1489. Mathlib v4.30 source proposition Nat.add_min_add_right proved · kernel-lean · F:ml430-nat-add-min-add-right-b483207e
      2 dependencies 0 dependents
1490. Mathlib v4.30 source proposition Nat.add_mod_left proved · kernel-lean · F:ml430-nat-add-mod-left-6b337077
      9 dependencies 0 dependents
1491. Mathlib v4.30 source proposition Nat.add_mod_right proved · kernel-lean · F:ml430-nat-add-mod-right-c047c67a
      7 dependencies 0 dependents
1492. Mathlib v4.30 source proposition Nat.add_modEq_left proved · kernel-lean · F:ml430-nat-add-modeq-left-e3b1fba9
      0 dependencies 0 dependents
1493. Mathlib v4.30 source proposition Nat.add_modEq_right proved · kernel-lean · F:ml430-nat-add-modeq-right-e2f11f21
      0 dependencies 0 dependents
1494. Mathlib v4.30 source proposition Nat.add_mul_div_left proved · kernel-lean · F:ml430-nat-add-mul-div-left-e20827dd
      4 dependencies 2 dependents
1495. Mathlib v4.30 source proposition Nat.add_mul_div_right proved · kernel-lean · F:ml430-nat-add-mul-div-right-44a689e4
      5 dependencies 1 dependents
1496. Mathlib v4.30 source proposition Nat.add_mul_lt_mul_of_lt_of_lt open · route not assigned · F:ml430-nat-add-mul-lt-mul-of-lt-of-lt-4d1d3fbc
      0 dependencies 0 dependents
1497. Mathlib v4.30 source proposition Nat.add_mul_mod_self_left proved · kernel-lean · F:ml430-nat-add-mul-mod-self-left-108b5fe0
      7 dependencies 1 dependents
1498. Mathlib v4.30 source proposition Nat.add_mul_mod_self_right proved · kernel-lean · F:ml430-nat-add-mul-mod-self-right-ac5b3624
      8 dependencies 0 dependents
1499. Mathlib v4.30 source proposition Nat.add_one_lt_of_even proved · kernel-lean · F:ml430-nat-add-one-lt-of-even-3464b374
      1 dependencies 0 dependents
1500. Mathlib v4.30 source proposition Nat.add_one_mul_choose_eq proved · kernel-lean · F:ml430-nat-add-one-mul-choose-eq-d364de16
      2 dependencies 0 dependents
1501. Mathlib v4.30 source proposition Nat.add_pos_right proved · kernel-lean · F:ml430-nat-add-pos-right-e43374dc
      3 dependencies 2 dependents
1502. Mathlib v4.30 source proposition Nat.and_assoc proved · kernel-lean · F:ml430-nat-and-assoc-273b60d8
      5 dependencies 0 dependents
1503. Mathlib v4.30 source proposition Nat.and_comm proved · kernel-lean · F:ml430-nat-and-comm-7525d05a
      3 dependencies 2 dependents
1504. Mathlib v4.30 source proposition Nat.and_div_two proved · kernel-lean · F:ml430-nat-and-div-two-1a2f7c33
      27 dependencies 0 dependents
1505. Mathlib v4.30 source proposition Nat.and_le_left proved · kernel-lean · F:ml430-nat-and-le-left-6d04acb7
      0 dependencies 3 dependents
1506. Mathlib v4.30 source proposition Nat.and_le_right proved · kernel-lean · F:ml430-nat-and-le-right-a3f80076
      2 dependencies 0 dependents
1507. Mathlib v4.30 source proposition Nat.and_mod_two_eq_one proved · kernel-lean · F:ml430-nat-and-mod-two-eq-one-3e873792
      9 dependencies 0 dependents
1508. Mathlib v4.30 source proposition Nat.and_one_is_mod proved · kernel-lean · F:ml430-nat-and-one-is-mod-d861e96b
      6 dependencies 0 dependents
1509. Mathlib v4.30 source proposition Nat.and_or_distrib_left proved · kernel-lean · F:ml430-nat-and-or-distrib-left-fe131f64
      9 dependencies 0 dependents
1510. Mathlib v4.30 source proposition Nat.and_or_distrib_right proved · kernel-lean · F:ml430-nat-and-or-distrib-right-0daaa284
      9 dependencies 0 dependents
1511. Mathlib v4.30 source proposition Nat.and_self proved · kernel-lean · F:ml430-nat-and-self-06a84ccc
      0 dependencies 0 dependents
1512. Mathlib v4.30 source proposition Nat.ascFactorial_eq_div proved · kernel-lean · F:ml430-nat-ascfactorial-eq-div-87d768e8
      9 dependencies 0 dependents
1513. Mathlib v4.30 source proposition Nat.ascFactorial_zero proved · kernel-lean · F:ml430-nat-ascfactorial-zero-fd183202
      0 dependencies 1 dependents
1514. Mathlib v4.30 source proposition Nat.avg_comm open · route not assigned · F:ml430-nat-avg-comm-7c5dd07e
      0 dependencies 0 dependents
1515. Mathlib v4.30 source proposition Nat.avg_le_left open · route not assigned · F:ml430-nat-avg-le-left-4a112778
      0 dependencies 0 dependents
1516. Mathlib v4.30 source proposition Nat.avg_le_right open · route not assigned · F:ml430-nat-avg-le-right-9fa159da
      0 dependencies 0 dependents
1517. Mathlib v4.30 source proposition Nat.avg_lt_left open · route not assigned · F:ml430-nat-avg-lt-left-14dbe48e
      0 dependencies 0 dependents
1518. Mathlib v4.30 source proposition Nat.avg_lt_right open · route not assigned · F:ml430-nat-avg-lt-right-e1a68d05
      0 dependencies 0 dependents
1519. Mathlib v4.30 source proposition Nat.base_induction proved · kernel-lean · F:ml430-nat-base-induction-83561d4c
      14 dependencies 0 dependents
1520. Mathlib v4.30 source proposition Nat.bertrand open · route not assigned · F:ml430-nat-bertrand-d35d0477
      0 dependencies 0 dependents
1521. Mathlib v4.30 source proposition Nat.bit_add proved · kernel-lean · F:ml430-nat-bit-add-b1edcc8e
      2 dependencies 0 dependents
1522. Mathlib v4.30 source proposition Nat.bit_add' proved · kernel-lean · F:ml430-nat-bit-add-d487108f
      2 dependencies 0 dependents
1523. Mathlib v4.30 source proposition Nat.bit_div_two open · route not assigned · F:ml430-nat-bit-div-two-d74e7898
      0 dependencies 0 dependents
1524. Mathlib v4.30 source proposition Nat.bit_eq_zero_iff open · route not assigned · F:ml430-nat-bit-eq-zero-iff-6b701e2b
      0 dependencies 0 dependents
1525. Mathlib v4.30 source proposition Nat.bit_false proved · kernel-lean · F:ml430-nat-bit-false-98b0bf2a
      0 dependencies 0 dependents
1526. Mathlib v4.30 source proposition Nat.bit_false_apply proved · kernel-lean · F:ml430-nat-bit-false-apply-5962146d
      0 dependencies 0 dependents
1527. Mathlib v4.30 source proposition Nat.bit_false_zero proved · kernel-lean · F:ml430-nat-bit-false-zero-d996adbf
      0 dependencies 0 dependents
1528. Mathlib v4.30 source proposition Nat.bit_le proved · kernel-lean · F:ml430-nat-bit-le-40743fa3
      2 dependencies 0 dependents
1529. Mathlib v4.30 source proposition Nat.bit_lt_bit proved · kernel-lean · F:ml430-nat-bit-lt-bit-295453cc
      7 dependencies 0 dependents
1530. Mathlib v4.30 source proposition Nat.bit_mod_two_eq_one_iff open · route not assigned · F:ml430-nat-bit-mod-two-eq-one-iff-d9b00bec
      0 dependencies 0 dependents
1531. Mathlib v4.30 source proposition Nat.bit_mod_two_eq_zero_iff open · route not assigned · F:ml430-nat-bit-mod-two-eq-zero-iff-b69a9790
      0 dependencies 0 dependents
1532. Mathlib v4.30 source proposition Nat.bit_ne_zero proved · kernel-lean · F:ml430-nat-bit-ne-zero-181bd65c
      7 dependencies 0 dependents
1533. Mathlib v4.30 source proposition Nat.bit_ne_zero_iff open · route not assigned · F:ml430-nat-bit-ne-zero-iff-d811128e
      0 dependencies 0 dependents
1534. Mathlib v4.30 source proposition Nat.bit_true proved · kernel-lean · F:ml430-nat-bit-true-2456e237
      0 dependencies 0 dependents
1535. Mathlib v4.30 source proposition Nat.bit_true_apply proved · kernel-lean · F:ml430-nat-bit-true-apply-02338ebc
      0 dependencies 0 dependents
1536. Mathlib v4.30 source proposition Nat.bitwise_bit' proved · kernel-lean · F:ml430-nat-bitwise-bit-4c4b28a8
      0 dependencies 1 dependents
1537. Mathlib v4.30 source proposition Nat.bitwise_comm proved · kernel-lean · F:ml430-nat-bitwise-comm-1a273bae
      1 dependencies 2 dependents
1538. Mathlib v4.30 source proposition Nat.bitwise_swap proved · kernel-lean · F:ml430-nat-bitwise-swap-7175e90e
      1 dependencies 1 dependents
1539. Mathlib v4.30 source proposition Nat.bitwise_zero open · route not assigned · F:ml430-nat-bitwise-zero-7c0e3f82
      0 dependencies 0 dependents
1540. Mathlib v4.30 source proposition Nat.ble_eq_true_of_le proved · kernel-lean · F:ml430-nat-ble-eq-true-of-le-5ce4ac2e
      2 dependencies 31 dependents
1541. Mathlib v4.30 source proposition Nat.ble_self_eq_true proved · kernel-lean · F:ml430-nat-ble-self-eq-true-839df126
      0 dependencies 2 dependents
1542. Mathlib v4.30 source proposition Nat.ble_succ_eq_true proved · kernel-lean · F:ml430-nat-ble-succ-eq-true-000a69f4
      0 dependencies 3 dependents
1543. Mathlib v4.30 source proposition Nat.case_strong_induction_on open · route not assigned · F:ml430-nat-case-strong-induction-on-0ec18022
      0 dependencies 0 dependents
1544. Mathlib v4.30 source proposition Nat.cauchy_induction' open · route not assigned · F:ml430-nat-cauchy-induction-64736fbc
      0 dependencies 0 dependents
1545. Mathlib v4.30 source proposition Nat.cauchy_induction open · route not assigned · F:ml430-nat-cauchy-induction-fc316053
      0 dependencies 0 dependents
1546. Mathlib v4.30 source proposition Nat.cauchy_induction_mul open · route not assigned · F:ml430-nat-cauchy-induction-mul-7c7491ec
      0 dependencies 0 dependents
1547. Mathlib v4.30 source proposition Nat.cauchy_induction_two_mul open · route not assigned · F:ml430-nat-cauchy-induction-two-mul-d6a1fcd8
      0 dependencies 0 dependents
1548. Mathlib v4.30 source proposition Nat.ceilRoot_eq_zero open · route not assigned · F:ml430-nat-ceilroot-eq-zero-a9d23c47
      0 dependencies 0 dependents
1549. Mathlib v4.30 source proposition Nat.ceilRoot_ne_zero open · route not assigned · F:ml430-nat-ceilroot-ne-zero-11a86167
      0 dependencies 0 dependents
1550. Mathlib v4.30 source proposition Nat.ceilRoot_one_left open · route not assigned · F:ml430-nat-ceilroot-one-left-fffb7237
      0 dependencies 0 dependents
1551. Mathlib v4.30 source proposition Nat.ceilRoot_one_right open · route not assigned · F:ml430-nat-ceilroot-one-right-e08e200c
      0 dependencies 0 dependents
1552. Mathlib v4.30 source proposition Nat.ceilRoot_pow_self open · route not assigned · F:ml430-nat-ceilroot-pow-self-19e66788
      0 dependencies 0 dependents
1553. Mathlib v4.30 source proposition Nat.ceilRoot_zero_left open · route not assigned · F:ml430-nat-ceilroot-zero-left-c1253d24
      0 dependencies 0 dependents
1554. Mathlib v4.30 source proposition Nat.ceilRoot_zero_right open · route not assigned · F:ml430-nat-ceilroot-zero-right-697fdf9e
      0 dependencies 0 dependents
1555. Mathlib v4.30 source proposition Nat.choose_eq_zero_of_lt proved · kernel-lean · F:ml430-nat-choose-eq-zero-of-lt-92ebab29
      7 dependencies 1 dependents
1556. Mathlib v4.30 source proposition Nat.choose_le_add proved · kernel-lean · F:ml430-nat-choose-le-add-9c463139
      2 dependencies 1 dependents
1557. Mathlib v4.30 source proposition Nat.choose_le_choose proved · kernel-lean · F:ml430-nat-choose-le-choose-907b5042
      4 dependencies 2 dependents
1558. Mathlib v4.30 source proposition Nat.choose_le_descFactorial open · route not assigned · F:ml430-nat-choose-le-descfactorial-67e8cc84
      0 dependencies 0 dependents
1559. Mathlib v4.30 source proposition Nat.choose_le_pow open · route not assigned · F:ml430-nat-choose-le-pow-75c38848
      0 dependencies 0 dependents
1560. Mathlib v4.30 source proposition Nat.choose_le_succ proved · kernel-lean · F:ml430-nat-choose-le-succ-62ae968b
      4 dependencies 1 dependents
1561. Mathlib v4.30 source proposition Nat.choose_le_two_pow open · route not assigned · F:ml430-nat-choose-le-two-pow-ef6f7dcd
      0 dependencies 0 dependents
1562. Mathlib v4.30 source proposition Nat.choose_lt_descFactorial open · route not assigned · F:ml430-nat-choose-lt-descfactorial-add66985
      0 dependencies 0 dependents
1563. Mathlib v4.30 source proposition Nat.choose_lt_pow open · route not assigned · F:ml430-nat-choose-lt-pow-e5cc4f38
      0 dependencies 0 dependents
1564. Mathlib v4.30 source proposition Nat.choose_lt_two_pow open · route not assigned · F:ml430-nat-choose-lt-two-pow-233c6d67
      0 dependencies 0 dependents
1565. Mathlib v4.30 source proposition Nat.choose_middle_le_pow open · route not assigned · F:ml430-nat-choose-middle-le-pow-de15c8ab
      0 dependencies 0 dependents
1566. Mathlib v4.30 source proposition Nat.choose_mono proved · kernel-lean · F:ml430-nat-choose-mono-a1af9c18
      1 dependencies 0 dependents
1567. Mathlib v4.30 source proposition Nat.choose_ne_zero proved · kernel-lean · F:ml430-nat-choose-ne-zero-49c3d3cb
      7 dependencies 0 dependents
1568. Mathlib v4.30 source proposition Nat.choose_one_right proved · kernel-lean · F:ml430-nat-choose-one-right-7eda8e39
      6 dependencies 2 dependents
1569. Mathlib v4.30 source proposition Nat.choose_self proved · kernel-lean · F:ml430-nat-choose-self-25bb9fb8
      0 dependencies 0 dependents
1570. Mathlib v4.30 source proposition Nat.choose_succ_self proved · kernel-lean · F:ml430-nat-choose-succ-self-e396f6c2
      1 dependencies 0 dependents
1571. Mathlib v4.30 source proposition Nat.choose_succ_succ proved · kernel-lean · F:ml430-nat-choose-succ-succ-671856b6
      0 dependencies 0 dependents
1572. Mathlib v4.30 source proposition Nat.choose_symm_add proved · kernel-lean · F:ml430-nat-choose-symm-add-e4b68161
      1 dependencies 0 dependents
1573. Mathlib v4.30 source proposition Nat.choose_symm_of_eq_add proved · kernel-lean · F:ml430-nat-choose-symm-of-eq-add-9b5f9a20
      3 dependencies 1 dependents
1574. Mathlib v4.30 source proposition Nat.choose_zero_right proved · kernel-lean · F:ml430-nat-choose-zero-right-1ed2802a
      0 dependencies 1 dependents
1575. Mathlib v4.30 source proposition Nat.choose_zero_succ proved · kernel-lean · F:ml430-nat-choose-zero-succ-62c6520b
      0 dependencies 0 dependents
1576. Mathlib v4.30 source proposition Nat.clog_anti_left proved · kernel-lean · F:ml430-nat-clog-anti-left-d72bd6cd
      2 dependencies 0 dependents
1577. Mathlib v4.30 source proposition Nat.clog_antitone_left proved · kernel-lean · F:ml430-nat-clog-antitone-left-44a87771
      1 dependencies 2 dependents
1578. Mathlib v4.30 source proposition Nat.clog_eq_one proved · kernel-lean · F:ml430-nat-clog-eq-one-f7834503
      10 dependencies 0 dependents
1579. Mathlib v4.30 source proposition Nat.clog_mono proved · kernel-lean · F:ml430-nat-clog-mono-74b44081
      4 dependencies 0 dependents
1580. Mathlib v4.30 source proposition Nat.clog_mono_right proved · kernel-lean · F:ml430-nat-clog-mono-right-8d87a410
      0 dependencies 4 dependents
1581. Mathlib v4.30 source proposition Nat.clog_monotone proved · kernel-lean · F:ml430-nat-clog-monotone-48fe50c6
      1 dependencies 0 dependents
1582. Mathlib v4.30 source proposition Nat.clog_of_left_le_one proved · kernel-lean · F:ml430-nat-clog-of-left-le-one-a6640cee
      7 dependencies 0 dependents
1583. Mathlib v4.30 source proposition Nat.clog_of_right_le_one proved · kernel-lean · F:ml430-nat-clog-of-right-le-one-8f11b62f
      7 dependencies 0 dependents
1584. Mathlib v4.30 source proposition Nat.clog_one_left proved · kernel-lean · F:ml430-nat-clog-one-left-b496af12
      0 dependencies 1 dependents
1585. Mathlib v4.30 source proposition Nat.clog_one_right proved · kernel-lean · F:ml430-nat-clog-one-right-1ce3d52f
      0 dependencies 1 dependents
1586. Mathlib v4.30 source proposition Nat.clog_pos proved · kernel-lean · F:ml430-nat-clog-pos-00852cb8
      3 dependencies 0 dependents
1587. Mathlib v4.30 source proposition Nat.clog_zero_left proved · kernel-lean · F:ml430-nat-clog-zero-left-1c61a5bf
      0 dependencies 1 dependents
1588. Mathlib v4.30 source proposition Nat.clog_zero_right proved · kernel-lean · F:ml430-nat-clog-zero-right-d42d47b1
      0 dependencies 1 dependents
1589. Mathlib v4.30 source proposition Nat.coprime_add_self_left proved · kernel-lean · F:ml430-nat-coprime-add-self-left-5e93448c
      2 dependencies 1 dependents
1590. Mathlib v4.30 source proposition Nat.coprime_add_self_right proved · kernel-lean · F:ml430-nat-coprime-add-self-right-c3ed0f45 C:greatest-common-divisor
      12 dependencies 2 dependents
1591. Mathlib v4.30 source proposition Nat.Coprime.coprime_div_left proved · kernel-lean · F:ml430-nat-coprime-coprime-div-left-6f7082bd
      10 dependencies 0 dependents
1592. Mathlib v4.30 source proposition Nat.Coprime.coprime_div_right proved · kernel-lean · F:ml430-nat-coprime-coprime-div-right-7a8ce438
      10 dependencies 0 dependents
1593. Mathlib v4.30 source proposition Nat.Coprime.coprime_dvd_left proved · kernel-lean · F:ml430-nat-coprime-coprime-dvd-left-2ce391d2
      1 dependencies 0 dependents
1594. Mathlib v4.30 source proposition Nat.Coprime.coprime_dvd_right proved · kernel-lean · F:ml430-nat-coprime-coprime-dvd-right-4a2670ae
      1 dependencies 0 dependents
1595. Mathlib v4.30 source proposition Nat.Coprime.coprime_mul_left proved · kernel-lean · F:ml430-nat-coprime-coprime-mul-left-fb5bd11a
      4 dependencies 0 dependents
1596. Mathlib v4.30 source proposition Nat.Coprime.coprime_mul_left_right proved · kernel-lean · F:ml430-nat-coprime-coprime-mul-left-right-910d7d8f
      4 dependencies 0 dependents
1597. Mathlib v4.30 source proposition Nat.Coprime.coprime_mul_right proved · kernel-lean · F:ml430-nat-coprime-coprime-mul-right-70e4e946
      3 dependencies 0 dependents
1598. Mathlib v4.30 source proposition Nat.Coprime.coprime_mul_right_right proved · kernel-lean · F:ml430-nat-coprime-coprime-mul-right-right-9599ecd3
      3 dependencies 0 dependents
1599. Mathlib v4.30 source proposition Nat.Coprime.dvd_mul_left proved · kernel-lean · F:ml430-nat-coprime-dvd-mul-left-e799d04c
      4 dependencies 0 dependents
1600. Mathlib v4.30 source proposition Nat.Coprime.dvd_mul_right proved · kernel-lean · F:ml430-nat-coprime-dvd-mul-right-7cd1c3c8
      4 dependencies 0 dependents
1601. Mathlib v4.30 source proposition Nat.Coprime.dvd_of_dvd_mul_left proved · kernel-lean · F:ml430-nat-coprime-dvd-of-dvd-mul-left-b0608cb9
      1 dependencies 0 dependents
1602. Mathlib v4.30 source proposition Nat.Coprime.dvd_of_dvd_mul_right proved · kernel-lean · F:ml430-nat-coprime-dvd-of-dvd-mul-right-efc3a4ec
      2 dependencies 0 dependents
1603. Mathlib v4.30 source proposition Nat.Coprime.eq_of_mul_eq_zero proved · kernel-lean · F:ml430-nat-coprime-eq-of-mul-eq-zero-a2026bd5
      2 dependencies 0 dependents
1604. Mathlib v4.30 source proposition Nat.coprime_factorizationLCMLeft_factorizationLCMRight open · route not assigned · F:ml430-nat-coprime-factorizationlcmleft-factorizationlcmright-e7db70ce
      0 dependencies 0 dependents
1605. Mathlib v4.30 source proposition Nat.coprime_fermatNumber_fermatNumber proved · kernel-lean · F:ml430-nat-coprime-fermatnumber-fermatnumber-161e79c7
      23 dependencies 0 dependents
1606. Mathlib v4.30 source proposition Nat.coprime_iff_isRelPrime proved · kernel-lean · F:ml430-nat-coprime-iff-isrelprime-0c08eb25 C:coprime-integers
      5 dependencies 0 dependents
1607. Mathlib v4.30 source proposition Nat.Coprime.lcm_eq_mul proved · kernel-lean · F:ml430-nat-coprime-lcm-eq-mul-edf52888
      2 dependencies 1 dependents
1608. Mathlib v4.30 source proposition Nat.Coprime.mul_add_mul_ne_mul proved · kernel-lean · F:ml430-nat-coprime-mul-add-mul-ne-mul-51b56f70
      21 dependencies 0 dependents
1609. Mathlib v4.30 source proposition Nat.Coprime.odd_of_left proved · kernel-lean · F:ml430-nat-coprime-odd-of-left-ed80ab44
      1 dependencies 2 dependents
1610. Mathlib v4.30 source proposition Nat.Coprime.odd_of_right proved · kernel-lean · F:ml430-nat-coprime-odd-of-right-8dc1decc
      1 dependencies 0 dependents
1611. Mathlib v4.30 source proposition Nat.Coprime.of_dvd proved · kernel-lean · F:ml430-nat-coprime-of-dvd-18fcd09f
      2 dependencies 0 dependents
1612. Mathlib v4.30 source proposition Nat.coprime_of_dvd' proved · kernel-lean · F:ml430-nat-coprime-of-dvd-6f652673
      19 dependencies 0 dependents
1613. Mathlib v4.30 source proposition Nat.Coprime.of_dvd_left proved · kernel-lean · F:ml430-nat-coprime-of-dvd-left-b0e2aa94 C:greatest-common-divisorC:divisibility
      6 dependencies 5 dependents
1614. Mathlib v4.30 source proposition Nat.Coprime.of_dvd_right proved · kernel-lean · F:ml430-nat-coprime-of-dvd-right-a640bd56 C:greatest-common-divisorC:divisibility
      6 dependencies 7 dependents
1615. Mathlib v4.30 source proposition Nat.coprime_of_lt_minFac open · route not assigned · F:ml430-nat-coprime-of-lt-minfac-0f79bdba
      0 dependencies 0 dependents
1616. Mathlib v4.30 source proposition Nat.coprime_of_lt_prime proved · kernel-lean · F:ml430-nat-coprime-of-lt-prime-1978a919
      6 dependencies 4 dependents
1617. Mathlib v4.30 source proposition Nat.coprime_one_left_iff proved · kernel-lean · F:ml430-nat-coprime-one-left-iff-45945e80
      2 dependencies 4 dependents
1618. Mathlib v4.30 source proposition Nat.coprime_one_right_iff proved · kernel-lean · F:ml430-nat-coprime-one-right-iff-42fed4ce
      2 dependencies 2 dependents
1619. Mathlib v4.30 source proposition Nat.coprime_or_dvd_of_prime proved · kernel-lean · F:ml430-nat-coprime-or-dvd-of-prime-65f47114
      4 dependencies 6 dependents
1620. Mathlib v4.30 source proposition Nat.coprime_primes proved · kernel-lean · F:ml430-nat-coprime-primes-5769049f
      6 dependencies 1 dependents
1621. Mathlib v4.30 source proposition Nat.coprime_self_add_left proved · kernel-lean · F:ml430-nat-coprime-self-add-left-51351fa1
      3 dependencies 0 dependents
1622. Mathlib v4.30 source proposition Nat.coprime_self_add_right proved · kernel-lean · F:ml430-nat-coprime-self-add-right-966e5434
      3 dependencies 1 dependents
1623. Mathlib v4.30 source proposition Nat.Coprime.symmetric proved · kernel-lean · F:ml430-nat-coprime-symmetric-9b5cfa12
      6 dependencies 10 dependents
1624. Mathlib v4.30 source proposition Nat.coprime_two_left proved · kernel-lean · F:ml430-nat-coprime-two-left-1b47e7c4
      10 dependencies 4 dependents
1625. Mathlib v4.30 source proposition Nat.coprime_two_right proved · kernel-lean · F:ml430-nat-coprime-two-right-7c5a1850
      2 dependencies 1 dependents
1626. Mathlib v4.30 source proposition Nat.deficient_iff_not_abundant_and_not_perfect proved · kernel-lean · F:ml430-nat-deficient-iff-not-abundant-and-not-perfect-18bbe30a
      5 dependencies 0 dependents
1627. Mathlib v4.30 source proposition Nat.deficient_one proved · kernel-lean · F:ml430-nat-deficient-one-75f44529
      0 dependencies 0 dependents
1628. Mathlib v4.30 source proposition Nat.descFactorial_le proved · kernel-lean · F:ml430-nat-descfactorial-le-2b8cc09a
      2 dependencies 0 dependents
1629. Mathlib v4.30 source proposition Nat.descFactorial_of_lt proved · kernel-lean · F:ml430-nat-descfactorial-of-lt-fbcf5d26
      7 dependencies 0 dependents
1630. Mathlib v4.30 source proposition Nat.descFactorial_one proved · kernel-lean · F:ml430-nat-descfactorial-one-d4856d4a
      1 dependencies 0 dependents
1631. Mathlib v4.30 source proposition Nat.descFactorial_self proved · kernel-lean · F:ml430-nat-descfactorial-self-899fc0e0
      2 dependencies 0 dependents
1632. Mathlib v4.30 source proposition Nat.descFactorial_zero proved · kernel-lean · F:ml430-nat-descfactorial-zero-966b01df
      0 dependencies 0 dependents
1633. Mathlib v4.30 source proposition Nat.diag_induction open · route not assigned · F:ml430-nat-diag-induction-12c954e8
      0 dependencies 0 dependents
1634. Mathlib v4.30 source proposition Nat.dist_add_add_left proved · kernel-lean · F:ml430-nat-dist-add-add-left-92fa4403
      0 dependencies 1 dependents
1635. Mathlib v4.30 source proposition Nat.dist_add_add_right proved · kernel-lean · F:ml430-nat-dist-add-add-right-6e5d8bbb
      2 dependencies 0 dependents
1636. Mathlib v4.30 source proposition Nat.dist_comm proved · kernel-lean · F:ml430-nat-dist-comm-1fa29a04
      1 dependencies 2 dependents
1637. Mathlib v4.30 source proposition Nat.dist_eq_intro proved · kernel-lean · F:ml430-nat-dist-eq-intro-294b44ad
      8 dependencies 0 dependents
1638. Mathlib v4.30 source proposition Nat.dist_eq_zero proved · kernel-lean · F:ml430-nat-dist-eq-zero-5ae5b706
      1 dependencies 0 dependents
1639. Mathlib v4.30 source proposition Nat.dist_mul_left proved · kernel-lean · F:ml430-nat-dist-mul-left-92624d63
      2 dependencies 1 dependents
1640. Mathlib v4.30 source proposition Nat.dist_mul_right proved · kernel-lean · F:ml430-nat-dist-mul-right-d4e0c33d
      2 dependencies 0 dependents
1641. Mathlib v4.30 source proposition Nat.dist_pos_of_ne proved · kernel-lean · F:ml430-nat-dist-pos-of-ne-00f5e22f
      7 dependencies 0 dependents
1642. Mathlib v4.30 source proposition Nat.dist_self proved · kernel-lean · F:ml430-nat-dist-self-0cfa5426
      2 dependencies 1 dependents
1643. Mathlib v4.30 source proposition Nat.dist.triangle_inequality proved · kernel-lean · F:ml430-nat-dist-triangle-inequality-b35e82d3
      12 dependencies 0 dependents
1644. Mathlib v4.30 source proposition Nat.div_dvd_div_left proved · kernel-lean · F:ml430-nat-div-dvd-div-left-b56f6f7c
      10 dependencies 0 dependents
1645. Mathlib v4.30 source proposition Nat.div_gcd_pos_of_pos_left proved · kernel-lean · F:ml430-nat-div-gcd-pos-of-pos-left-dd878a3f
      5 dependencies 0 dependents
1646. Mathlib v4.30 source proposition Nat.div_gcd_pos_of_pos_right proved · kernel-lean · F:ml430-nat-div-gcd-pos-of-pos-right-8d26808c
      5 dependencies 0 dependents
1647. Mathlib v4.30 source proposition Nat.div_lt_of_lt_mul proved · kernel-lean · F:ml430-nat-div-lt-of-lt-mul-818dc4c7
      4 dependencies 4 dependents
1648. Mathlib v4.30 source proposition Nat.div_lt_self' open · route not assigned · F:ml430-nat-div-lt-self-4c19ff1c
      0 dependencies 0 dependents
1649. Mathlib v4.30 source proposition Nat.div_mul_cancel proved · kernel-lean · F:ml430-nat-div-mul-cancel-99799a00
      6 dependencies 2 dependents
1650. Mathlib v4.30 source proposition Nat.div_right_comm open · route not assigned · F:ml430-nat-div-right-comm-ab79d5f8
      0 dependencies 0 dependents
1651. Mathlib v4.30 source proposition Nat.div_two_mul_two_add_one_of_odd proved · kernel-lean · F:ml430-nat-div-two-mul-two-add-one-of-odd-9e3e8b82
      3 dependencies 0 dependents
1652. Mathlib v4.30 source proposition Nat.div_two_mul_two_of_even proved · kernel-lean · F:ml430-nat-div-two-mul-two-of-even-9ccc5340
      4 dependencies 0 dependents
1653. Mathlib v4.30 source proposition Nat.divMaxPow_base_mul open · route not assigned · F:ml430-nat-divmaxpow-base-mul-63c19e29
      0 dependencies 0 dependents
1654. Mathlib v4.30 source proposition Nat.divMaxPow_one_left open · route not assigned · F:ml430-nat-divmaxpow-one-left-2e311f8c
      0 dependencies 0 dependents
1655. Mathlib v4.30 source proposition Nat.divMaxPow_one_right open · route not assigned · F:ml430-nat-divmaxpow-one-right-4696cbec
      0 dependencies 0 dependents
1656. Mathlib v4.30 source proposition Nat.divMaxPow_self open · route not assigned · F:ml430-nat-divmaxpow-self-19761ae0
      0 dependencies 0 dependents
1657. Mathlib v4.30 source proposition Nat.divMaxPow_zero_left open · route not assigned · F:ml430-nat-divmaxpow-zero-left-8f1e9599
      0 dependencies 0 dependents
1658. Mathlib v4.30 source proposition Nat.divMaxPow_zero_right open · route not assigned · F:ml430-nat-divmaxpow-zero-right-b1dfd200
      0 dependencies 0 dependents
1659. Mathlib v4.30 source proposition Nat.dvd_add proved · kernel-lean · F:ml430-nat-dvd-add-0c5bcc91
      1 dependencies 11 dependents
1660. Mathlib v4.30 source proposition Nat.dvd_add_iff_left proved · kernel-lean · F:ml430-nat-dvd-add-iff-left-332cbe04
      2 dependencies 1 dependents
1661. Mathlib v4.30 source proposition Nat.dvd_add_iff_right proved · kernel-lean · F:ml430-nat-dvd-add-iff-right-bf79c0cd
      3 dependencies 6 dependents
1662. Mathlib v4.30 source proposition Nat.dvd_add_self_left open · route not assigned · F:ml430-nat-dvd-add-self-left-5568ca79
      0 dependencies 0 dependents
1663. Mathlib v4.30 source proposition Nat.dvd_add_self_right open · route not assigned · F:ml430-nat-dvd-add-self-right-5220f882
      0 dependencies 0 dependents
1664. Mathlib v4.30 source proposition Nat.dvd_antisymm proved · kernel-lean · F:ml430-nat-dvd-antisymm-507f9026
      5 dependencies 8 dependents
1665. Mathlib v4.30 source proposition Nat.dvd_ceilRoot_pow open · route not assigned · F:ml430-nat-dvd-ceilroot-pow-80ce502a
      0 dependencies 0 dependents
1666. Mathlib v4.30 source proposition Nat.Dvd.dvd.nat_lcm_left proved · kernel-lean · F:ml430-nat-dvd-dvd-nat-lcm-left-6143311e
      2 dependencies 0 dependents
1667. Mathlib v4.30 source proposition Nat.Dvd.dvd.nat_lcm_right proved · kernel-lean · F:ml430-nat-dvd-dvd-nat-lcm-right-d05db50b
      2 dependencies 0 dependents
1668. Mathlib v4.30 source proposition Nat.dvd_gcd proved · kernel-lean · F:ml430-nat-dvd-gcd-e5184fc5
      6 dependencies 24 dependents
1669. Mathlib v4.30 source proposition Nat.dvd_gcd_iff proved · kernel-lean · F:ml430-nat-dvd-gcd-iff-b8485987
      4 dependencies 3 dependents
1670. Mathlib v4.30 source proposition Nat.dvd_gcd_mul_gcd_iff_dvd_mul proved · kernel-lean · F:ml430-nat-dvd-gcd-mul-gcd-iff-dvd-mul-07fec722
      3 dependencies 0 dependents
1671. Mathlib v4.30 source proposition Nat.dvd_gcd_mul_iff_dvd_mul proved · kernel-lean · F:ml430-nat-dvd-gcd-mul-iff-dvd-mul-0afe640a
      2 dependencies 3 dependents
1672. Mathlib v4.30 source proposition Nat.dvd_iff_mod_eq_zero proved · kernel-lean · F:ml430-nat-dvd-iff-mod-eq-zero-d795bfff
      4 dependencies 1 dependents
1673. Mathlib v4.30 source proposition Nat.dvd_iff_prime_pow_dvd_dvd open · route not assigned · F:ml430-nat-dvd-iff-prime-pow-dvd-dvd-4458751e
      0 dependencies 0 dependents
1674. Mathlib v4.30 source proposition Nat.dvd_lcm_left proved · kernel-lean · F:ml430-nat-dvd-lcm-left-c83bcebc
      13 dependencies 8 dependents
1675. Mathlib v4.30 source proposition Nat.dvd_lcm_of_dvd_left proved · kernel-lean · F:ml430-nat-dvd-lcm-of-dvd-left-141a64bb
      3 dependencies 0 dependents
1676. Mathlib v4.30 source proposition Nat.dvd_lcm_of_dvd_right proved · kernel-lean · F:ml430-nat-dvd-lcm-of-dvd-right-61a50fc3
      3 dependencies 0 dependents
1677. Mathlib v4.30 source proposition Nat.dvd_lcm_right proved · kernel-lean · F:ml430-nat-dvd-lcm-right-18ab8e2f
      12 dependencies 8 dependents
1678. Mathlib v4.30 source proposition Nat.dvd_left_iff_eq open · route not assigned · F:ml430-nat-dvd-left-iff-eq-b9d7bb6d
      0 dependencies 0 dependents
1679. Mathlib v4.30 source proposition Nat.dvd_mod_iff proved · kernel-lean · F:ml430-nat-dvd-mod-iff-2d082f10
      2 dependencies 0 dependents
1680. Mathlib v4.30 source proposition Nat.dvd_mul proved · kernel-lean · F:ml430-nat-dvd-mul-ebd102e2
      15 dependencies 0 dependents
1681. Mathlib v4.30 source proposition Nat.dvd_mul_gcd_iff_dvd_mul proved · kernel-lean · F:ml430-nat-dvd-mul-gcd-iff-dvd-mul-f9517e6b
      3 dependencies 1 dependents
1682. Mathlib v4.30 source proposition Nat.dvd_mul_left proved · kernel-lean · F:ml430-nat-dvd-mul-left-a1a8a4b8
      2 dependencies 3 dependents
1683. Mathlib v4.30 source proposition Nat.dvd_mul_left_of_dvd proved · kernel-lean · F:ml430-nat-dvd-mul-left-of-dvd-200e20a4
      2 dependencies 2 dependents
1684. Mathlib v4.30 source proposition Nat.dvd_mul_right proved · kernel-lean · F:ml430-nat-dvd-mul-right-a87a83c4
      0 dependencies 43 dependents
1685. Mathlib v4.30 source proposition Nat.dvd_of_forall_prime_mul_dvd proved · kernel-lean · F:ml430-nat-dvd-of-forall-prime-mul-dvd-5898723b C:divisibilityC:prime-number
      19 dependencies 0 dependents
1686. Mathlib v4.30 source proposition Nat.dvd_of_lcm_left_dvd proved · kernel-lean · F:ml430-nat-dvd-of-lcm-left-dvd-d6b2407c
      3 dependencies 0 dependents
1687. Mathlib v4.30 source proposition Nat.dvd_of_lcm_right_dvd proved · kernel-lean · F:ml430-nat-dvd-of-lcm-right-dvd-61bd1a60
      3 dependencies 0 dependents
1688. Mathlib v4.30 source proposition Nat.dvd_pow_iff_ceilRoot_dvd open · route not assigned · F:ml430-nat-dvd-pow-iff-ceilroot-dvd-f00c5188
      0 dependencies 0 dependents
1689. Mathlib v4.30 source proposition Nat.dvd_right_iff_eq open · route not assigned · F:ml430-nat-dvd-right-iff-eq-4ce70bc6
      0 dependencies 0 dependents
1690. Mathlib v4.30 source proposition Nat.dvd_two_of_totient_le_one proved · kernel-lean · F:ml430-nat-dvd-two-of-totient-le-one-3642bf31
      17 dependencies 0 dependents
1691. Mathlib v4.30 source proposition Nat.eq_of_beq_eq_true proved · kernel-lean · F:ml430-nat-eq-of-beq-eq-true-6ec35ee4
      0 dependencies 28 dependents
1692. Mathlib v4.30 source proposition Nat.eq_or_eq_of_totient_eq_totient proved · kernel-lean · F:ml430-nat-eq-or-eq-of-totient-eq-totient-d4d154c7 C:eulers-totient-functionC:divisibility
      13 dependencies 0 dependents
1693. Mathlib v4.30 source proposition Nat.eq_zero_of_gcd_eq_zero_left proved · kernel-lean · F:ml430-nat-eq-zero-of-gcd-eq-zero-left-72cc4246
      2 dependencies 1 dependents
1694. Mathlib v4.30 source proposition Nat.eq_zero_of_gcd_eq_zero_right proved · kernel-lean · F:ml430-nat-eq-zero-of-gcd-eq-zero-right-24054a86
      2 dependencies 0 dependents
1695. Mathlib v4.30 source proposition Nat.eq_zero_of_lcm_eq_zero proved · kernel-lean · F:ml430-nat-eq-zero-of-lcm-eq-zero-d09b7af7
      3 dependencies 0 dependents
1696. Mathlib v4.30 source proposition Nat.euler_four_squares open · route not assigned · F:ml430-nat-euler-four-squares-21d8c900
      0 dependencies 0 dependents
1697. Mathlib v4.30 source proposition Nat.even_add proved · kernel-lean · F:ml430-nat-even-add-31386639
      2 dependencies 0 dependents
1698. Mathlib v4.30 source proposition Nat.even_add' proved · kernel-lean · F:ml430-nat-even-add-39e3bc07
      2 dependencies 0 dependents
1699. Mathlib v4.30 source proposition Nat.even_add_one proved · kernel-lean · F:ml430-nat-even-add-one-15b5cb18
      3 dependencies 1 dependents
1700. Mathlib v4.30 source proposition Nat.even_div proved · kernel-lean · F:ml430-nat-even-div-395c6b5e
      3 dependencies 0 dependents
1701. Mathlib v4.30 source proposition Nat.even_iff proved · kernel-lean · F:ml430-nat-even-iff-024826e9
      7 dependencies 14 dependents
1702. Mathlib v4.30 source proposition Nat.even_xor proved · kernel-lean · F:ml430-nat-even-xor-78a39432
      0 dependencies 0 dependents
1703. Mathlib v4.30 source proposition Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le open · route not assigned · F:ml430-nat-exists-add-mul-eq-of-gcd-dvd-of-mul-pred-le-e4a87d78
      0 dependencies 0 dependents
1704. Mathlib v4.30 source proposition Nat.exists_eq_pow_mul_and_not_dvd open · route not assigned · F:ml430-nat-exists-eq-pow-mul-and-not-dvd-ea40a197
      0 dependencies 0 dependents
1705. Mathlib v4.30 source proposition Nat.exists_eq_pow_of_exponent_coprime_of_pow_eq_pow open · route not assigned · F:ml430-nat-exists-eq-pow-of-exponent-coprime-of-pow-eq-pow-17408247
      0 dependencies 0 dependents
1706. Mathlib v4.30 source proposition Nat.exists_eq_pow_of_pow_eq_pow open · route not assigned · F:ml430-nat-exists-eq-pow-of-pow-eq-pow-127883b7
      0 dependencies 0 dependents
1707. Mathlib v4.30 source proposition Nat.exists_eq_two_pow_mul_odd open · route not assigned · F:ml430-nat-exists-eq-two-pow-mul-odd-e7310d1c
      0 dependencies 0 dependents
1708. Mathlib v4.30 source proposition Nat.exists_mul_mod_eq_gcd proved · kernel-lean · F:ml430-nat-exists-mul-mod-eq-gcd-8bf9ec7e
      15 dependencies 0 dependents
1709. Mathlib v4.30 source proposition Nat.exists_mul_self open · route not assigned · F:ml430-nat-exists-mul-self-e73ca9fa
      1 dependencies 0 dependents
1710. Mathlib v4.30 source proposition Nat.exists_not_and_succ_of_not_zero_of_exists open · route not assigned · F:ml430-nat-exists-not-and-succ-of-not-zero-of-exists-07952ae8
      0 dependencies 0 dependents
1711. Mathlib v4.30 source proposition Nat.exists_pow_eq_iff' open · route not assigned · F:ml430-nat-exists-pow-eq-iff-b8198130
      0 dependencies 0 dependents
1712. Mathlib v4.30 source proposition Nat.exists_pow_eq_iff open · route not assigned · F:ml430-nat-exists-pow-eq-iff-f581ca30
      0 dependencies 0 dependents
1713. Mathlib v4.30 source proposition Nat.exists_prime_lt_and_le_two_mul open · route not assigned · F:ml430-nat-exists-prime-lt-and-le-two-mul-a80e6056
      0 dependencies 0 dependents
1714. Mathlib v4.30 source proposition Nat.exists_prime_lt_and_le_two_mul_eventually open · route not assigned · F:ml430-nat-exists-prime-lt-and-le-two-mul-eventually-5743fe5b
      0 dependencies 0 dependents
1715. Mathlib v4.30 source proposition Nat.exists_prime_lt_and_le_two_mul_succ open · route not assigned · F:ml430-nat-exists-prime-lt-and-le-two-mul-succ-0c2eec3c
      0 dependencies 0 dependents
1716. Mathlib v4.30 source proposition Nat.factorial_dvd_ascFactorial proved · kernel-lean · F:ml430-nat-factorial-dvd-ascfactorial-44a4e641
      7 dependencies 0 dependents
1717. Mathlib v4.30 source proposition Nat.factorial_dvd_descFactorial proved · kernel-lean · F:ml430-nat-factorial-dvd-descfactorial-bbf6124f
      2 dependencies 0 dependents
1718. Mathlib v4.30 source proposition Nat.factorial_dvd_factorial proved · kernel-lean · F:ml430-nat-factorial-dvd-factorial-e9d14845
      6 dependencies 1 dependents
1719. Mathlib v4.30 source proposition Nat.factorial_le proved · kernel-lean · F:ml430-nat-factorial-le-d0f4a912
      4 dependencies 2 dependents
1720. Mathlib v4.30 source proposition Nat.factorial_lt_of_lt proved · kernel-lean · F:ml430-nat-factorial-lt-of-lt-d6c2125d
      7 dependencies 1 dependents
1721. Mathlib v4.30 source proposition Nat.factorial_ne_zero proved · kernel-lean · F:ml430-nat-factorial-ne-zero-5fc0b0a1
      3 dependencies 0 dependents
1722. Mathlib v4.30 source proposition Nat.factorial_pos proved · kernel-lean · F:ml430-nat-factorial-pos-f1dd2405
      0 dependencies 3 dependents
1723. Mathlib v4.30 source proposition Nat.factorizationLCMLeft_dvd_left open · route not assigned · F:ml430-nat-factorizationlcmleft-dvd-left-324f9e3c
      0 dependencies 0 dependents
1724. Mathlib v4.30 source proposition Nat.factorizationLCMLeft_mul_factorizationLCMRight open · route not assigned · F:ml430-nat-factorizationlcmleft-mul-factorizationlcmright-0689efd5
      0 dependencies 0 dependents
1725. Mathlib v4.30 source proposition Nat.factorizationLCMLeft_pos open · route not assigned · F:ml430-nat-factorizationlcmleft-pos-820998f6
      0 dependencies 0 dependents
1726. Mathlib v4.30 source proposition Nat.factorizationLCMLeft_zero_left open · route not assigned · F:ml430-nat-factorizationlcmleft-zero-left-ea02654b
      0 dependencies 0 dependents
1727. Mathlib v4.30 source proposition Nat.factorizationLCMLeft_zero_right open · route not assigned · F:ml430-nat-factorizationlcmleft-zero-right-e1f61ad0
      0 dependencies 0 dependents
1728. Mathlib v4.30 source proposition Nat.factorizationLCMRight_dvd_right open · route not assigned · F:ml430-nat-factorizationlcmright-dvd-right-2460f9fd
      0 dependencies 0 dependents
1729. Mathlib v4.30 source proposition Nat.factorizationLCMRight_pos open · route not assigned · F:ml430-nat-factorizationlcmright-pos-dc3f4b60
      0 dependencies 0 dependents
1730. Mathlib v4.30 source proposition Nat.factorizationLCMRight_zero_right open · route not assigned · F:ml430-nat-factorizationlcmright-zero-right-3f1474ee
      0 dependencies 0 dependents
1731. Mathlib v4.30 source proposition Nat.factorizationLCRight_zero_left open · route not assigned · F:ml430-nat-factorizationlcright-zero-left-ef4c6211
      0 dependencies 0 dependents
1732. Mathlib v4.30 source proposition Nat.fastFib_eq open · route not assigned · F:ml430-nat-fastfib-eq-cde11774
      0 dependencies 0 dependents
1733. Mathlib v4.30 source proposition Nat.fermat_primeFactors_one_lt open · route not assigned · F:ml430-nat-fermat-primefactors-one-lt-58343c6f
      0 dependencies 0 dependents
1734. Mathlib v4.30 source proposition Nat.fermatNumber_mono proved · kernel-lean · F:ml430-nat-fermatnumber-mono-b051cee6
      4 dependencies 0 dependents
1735. Mathlib v4.30 source proposition Nat.fermatNumber_ne_one proved · kernel-lean · F:ml430-nat-fermatnumber-ne-one-91232d67
      5 dependencies 0 dependents
1736. Mathlib v4.30 source proposition Nat.fermatNumber_one proved · kernel-lean · F:ml430-nat-fermatnumber-one-b1b0798f
      0 dependencies 0 dependents
1737. Mathlib v4.30 source proposition Nat.fermatNumber_strictMono proved · kernel-lean · F:ml430-nat-fermatnumber-strictmono-acbcb8c6
      4 dependencies 0 dependents
1738. Mathlib v4.30 source proposition Nat.fermatNumber_two proved · kernel-lean · F:ml430-nat-fermatnumber-two-3aa3bfc4
      0 dependencies 0 dependents
1739. Mathlib v4.30 source proposition Nat.fermatNumber_zero proved · kernel-lean · F:ml430-nat-fermatnumber-zero-ca7aac67
      0 dependencies 0 dependents
1740. Mathlib v4.30 source proposition Nat.fib_add proved · kernel-lean · F:ml430-nat-fib-add-81ea2485
      9 dependencies 0 dependents
1741. Mathlib v4.30 source proposition Nat.fib_add_two proved · kernel-lean · F:ml430-nat-fib-add-two-b86e0c82
      1 dependencies 11 dependents
1742. Mathlib v4.30 source proposition Nat.fib_add_two_strictMono proved · kernel-lean · F:ml430-nat-fib-add-two-strictmono-c1e86d4d
      20 dependencies 1 dependents
1743. Mathlib v4.30 source proposition Nat.fib_coprime_fib_succ proved · kernel-lean · F:ml430-nat-fib-coprime-fib-succ-162fc738
      1 dependencies 1 dependents
1744. Mathlib v4.30 source proposition Nat.fib_dvd proved · kernel-lean · F:ml430-nat-fib-dvd-f80f3de1
      1 dependencies 0 dependents
1745. Mathlib v4.30 source proposition Nat.fib_eq_zero proved · kernel-lean · F:ml430-nat-fib-eq-zero-61879073
      0 dependencies 0 dependents
1746. Mathlib v4.30 source proposition Nat.fib_gcd proved · kernel-lean · F:ml430-nat-fib-gcd-d1d98407
      0 dependencies 2 dependents
1747. Mathlib v4.30 source proposition Nat.fib_le_fib_succ proved · kernel-lean · F:ml430-nat-fib-le-fib-succ-d1ef4a3d
      4 dependencies 2 dependents
1748. Mathlib v4.30 source proposition Nat.fib_lt_fib proved · kernel-lean · F:ml430-nat-fib-lt-fib-3582b881
      6 dependencies 1 dependents
1749. Mathlib v4.30 source proposition Nat.fib_lt_fib_succ proved · kernel-lean · F:ml430-nat-fib-lt-fib-succ-b4305b68
      2 dependencies 0 dependents
1750. Mathlib v4.30 source proposition Nat.fib_mono proved · kernel-lean · F:ml430-nat-fib-mono-cc6afe09
      2 dependencies 1 dependents
1751. Mathlib v4.30 source proposition Nat.fib_one proved · kernel-lean · F:ml430-nat-fib-one-02785c52
      0 dependencies 0 dependents
1752. Mathlib v4.30 source proposition Nat.fib_pos proved · kernel-lean · F:ml430-nat-fib-pos-9e67bd8e
      0 dependencies 0 dependents
1753. Mathlib v4.30 source proposition Nat.fib_strictMonoOn proved · kernel-lean · F:ml430-nat-fib-strictmonoon-905810a9
      7 dependencies 1 dependents
1754. Mathlib v4.30 source proposition Nat.fib_two proved · kernel-lean · F:ml430-nat-fib-two-2f3715f3
      0 dependencies 0 dependents
1755. Mathlib v4.30 source proposition Nat.findGreatest_eq open · route not assigned · F:ml430-nat-findgreatest-eq-87c06f0f
      0 dependencies 0 dependents
1756. Mathlib v4.30 source proposition Nat.findGreatest_eq_iff open · route not assigned · F:ml430-nat-findgreatest-eq-iff-426770ce
      0 dependencies 0 dependents
1757. Mathlib v4.30 source proposition Nat.findGreatest_eq_zero_iff open · route not assigned · F:ml430-nat-findgreatest-eq-zero-iff-3498d512
      0 dependencies 0 dependents
1758. Mathlib v4.30 source proposition Nat.findGreatest_is_greatest open · route not assigned · F:ml430-nat-findgreatest-is-greatest-27ee7d92
      0 dependencies 0 dependents
1759. Mathlib v4.30 source proposition Nat.findGreatest_le open · route not assigned · F:ml430-nat-findgreatest-le-8e41e6aa
      0 dependencies 0 dependents
1760. Mathlib v4.30 source proposition Nat.findGreatest_mono open · route not assigned · F:ml430-nat-findgreatest-mono-2566201e
      0 dependencies 0 dependents
1761. Mathlib v4.30 source proposition Nat.findGreatest_mono_left open · route not assigned · F:ml430-nat-findgreatest-mono-left-065903af
      0 dependencies 0 dependents
1762. Mathlib v4.30 source proposition Nat.findGreatest_mono_right open · route not assigned · F:ml430-nat-findgreatest-mono-right-8d67c0d3
      0 dependencies 0 dependents
1763. Mathlib v4.30 source proposition Nat.findGreatest_of_ne_zero open · route not assigned · F:ml430-nat-findgreatest-of-ne-zero-04670c87
      0 dependencies 0 dependents
1764. Mathlib v4.30 source proposition Nat.findGreatest_of_not open · route not assigned · F:ml430-nat-findgreatest-of-not-0229e05d
      0 dependencies 0 dependents
1765. Mathlib v4.30 source proposition Nat.floorRoot_eq_zero open · route not assigned · F:ml430-nat-floorroot-eq-zero-a3be4438
      0 dependencies 0 dependents
1766. Mathlib v4.30 source proposition Nat.forall_ne_zero_iff open · route not assigned · F:ml430-nat-forall-ne-zero-iff-37819136
      0 dependencies 0 dependents
1767. Mathlib v4.30 source proposition Nat.gcd_dvd_mul proved · kernel-lean · F:ml430-nat-gcd-dvd-mul-81cb13df
      2 dependencies 0 dependents
1768. Mathlib v4.30 source proposition Nat.gcd_fib_add_self proved · kernel-lean · F:ml430-nat-gcd-fib-add-self-5a92d5e3
      1 dependencies 0 dependents
1769. Mathlib v4.30 source proposition Nat.gcd_greatest proved · kernel-lean · F:ml430-nat-gcd-greatest-0a04214a
      0 dependencies 0 dependents
1770. Mathlib v4.30 source proposition Nat.gcd_le_mul proved · kernel-lean · F:ml430-nat-gcd-le-mul-7e3800f7
      4 dependencies 0 dependents
1771. Mathlib v4.30 source proposition Nat.gcd_mul_lcm proved · kernel-lean · F:ml430-nat-gcd-mul-lcm-b7217ace
      9 dependencies 6 dependents
1772. Mathlib v4.30 source proposition Nat.induct_rcc_left open · route not assigned · F:ml430-nat-induct-rcc-left-4c2df6c6
      0 dependencies 0 dependents
1773. Mathlib v4.30 source proposition Nat.induct_rcc_right open · route not assigned · F:ml430-nat-induct-rcc-right-daac46fe
      0 dependencies 0 dependents
1774. Mathlib v4.30 source proposition Nat.land_assoc proved · kernel-lean · F:ml430-nat-land-assoc-ad4775b8
      1 dependencies 1 dependents
1775. Mathlib v4.30 source proposition Nat.land_bit proved · kernel-lean · F:ml430-nat-land-bit-b9ab7475
      0 dependencies 1 dependents
1776. Mathlib v4.30 source proposition Nat.land_comm proved · kernel-lean · F:ml430-nat-land-comm-7e6ad72e
      1 dependencies 1 dependents
1777. Mathlib v4.30 source proposition Nat.lcm_assoc proved · kernel-lean · F:ml430-nat-lcm-assoc-cb00bb43
      9 dependencies 0 dependents
1778. Mathlib v4.30 source proposition Nat.lcm_comm proved · kernel-lean · F:ml430-nat-lcm-comm-d5f8aae0
      5 dependencies 0 dependents
1779. Mathlib v4.30 source proposition Nat.lcm_div proved · kernel-lean · F:ml430-nat-lcm-div-eb5d8892
      18 dependencies 0 dependents
1780. Mathlib v4.30 source proposition Nat.lcm_dvd proved · kernel-lean · F:ml430-nat-lcm-dvd-07899eea
      12 dependencies 6 dependents
1781. Mathlib v4.30 source proposition Nat.lcmUpto_dvd_factorial open · route not assigned · F:ml430-nat-lcmupto-dvd-factorial-713c6bc6
      0 dependencies 0 dependents
1782. Mathlib v4.30 source proposition Nat.lcmUpto_ne_zero open · route not assigned · F:ml430-nat-lcmupto-ne-zero-418388ac
      0 dependencies 0 dependents
1783. Mathlib v4.30 source proposition Nat.lcmUpto_pos open · route not assigned · F:ml430-nat-lcmupto-pos-e4917519
      0 dependencies 0 dependents
1784. Mathlib v4.30 source proposition Nat.ldiff_bit proved · kernel-lean · F:ml430-nat-ldiff-bit-6be49bb8
      0 dependencies 0 dependents
1785. Mathlib v4.30 source proposition Nat.le_add_one_of_avg_eq_left open · route not assigned · F:ml430-nat-le-add-one-of-avg-eq-left-210e3373
      0 dependencies 0 dependents
1786. Mathlib v4.30 source proposition Nat.le_add_one_of_avg_eq_right open · route not assigned · F:ml430-nat-le-add-one-of-avg-eq-right-58e5e606
      0 dependencies 0 dependents
1787. Mathlib v4.30 source proposition Nat.le_antisymm proved · kernel-lean · F:ml430-nat-le-antisymm-79dccead
      2 dependencies 53 dependents
1788. Mathlib v4.30 source proposition Nat.le_avg_left open · route not assigned · F:ml430-nat-le-avg-left-5067bf7f
      0 dependencies 0 dependents
1789. Mathlib v4.30 source proposition Nat.le_avg_right open · route not assigned · F:ml430-nat-le-avg-right-3a424d9e
      0 dependencies 0 dependents
1790. Mathlib v4.30 source proposition Nat.le_div_two_iff_mul_two_le open · route not assigned · F:ml430-nat-le-div-two-iff-mul-two-le-28da1d50
      0 dependencies 0 dependents
1791. Mathlib v4.30 source proposition Nat.le_fib_add_one proved · kernel-lean · F:ml430-nat-le-fib-add-one-5284f0bf
      10 dependencies 0 dependents
1792. Mathlib v4.30 source proposition Nat.le_fib_self proved · kernel-lean · F:ml430-nat-le-fib-self-0cbccb4d
      11 dependencies 1 dependents
1793. Mathlib v4.30 source proposition Nat.le_induction open · route not assigned · F:ml430-nat-le-induction-2f088ac3
      0 dependencies 0 dependents
1794. Mathlib v4.30 source proposition Nat.le_max_left proved · kernel-lean · F:ml430-nat-le-max-left-685a3331
      2 dependencies 0 dependents
1795. Mathlib v4.30 source proposition Nat.le_max_right proved · kernel-lean · F:ml430-nat-le-max-right-3cd92fc9
      2 dependencies 0 dependents
1796. Mathlib v4.30 source proposition Nat.le_min proved · kernel-lean · F:ml430-nat-le-min-69904590
      2 dependencies 1 dependents
1797. Mathlib v4.30 source proposition Nat.le_min_of_le_of_le proved · kernel-lean · F:ml430-nat-le-min-of-le-of-le-407ecd4b
      1 dependencies 1 dependents
1798. Mathlib v4.30 source proposition Nat.le_nth_of_lt_nth_succ open · route not assigned · F:ml430-nat-le-nth-of-lt-nth-succ-6d69133f
      0 dependencies 0 dependents
1799. Mathlib v4.30 source proposition Nat.le_nthRoot_iff open · route not assigned · F:ml430-nat-le-nthroot-iff-82e243be
      0 dependencies 0 dependents
1800. Mathlib v4.30 source proposition Nat.le_of_ble_eq_true proved · kernel-lean · F:ml430-nat-le-of-ble-eq-true-646f4e10
      2 dependencies 22 dependents
1801. Mathlib v4.30 source proposition Nat.le_of_lt_succ proved · kernel-lean · F:ml430-nat-le-of-lt-succ-120bd6db
      9 dependencies 52 dependents
1802. Mathlib v4.30 source proposition Nat.le_of_succ_le_succ proved · kernel-lean · F:ml430-nat-le-of-succ-le-succ-a180a72c
      1 dependencies 73 dependents
1803. Mathlib v4.30 source proposition Nat.le_refl proved · kernel-lean · F:ml430-nat-le-refl-fd7d9e15
      6 dependencies 16 dependents
1804. Mathlib v4.30 source proposition Nat.le_sqrt open · route not assigned · F:ml430-nat-le-sqrt-e6996680
      2 dependencies 3 dependents
1805. Mathlib v4.30 source proposition Nat.le_sqrt_of_eq_mul open · route not assigned · F:ml430-nat-le-sqrt-of-eq-mul-503c5afe
      1 dependencies 0 dependents
1806. Mathlib v4.30 source proposition Nat.le_succ_iff_eq_or_le open · route not assigned · F:ml430-nat-le-succ-iff-eq-or-le-0cd6331b
      0 dependencies 0 dependents
1807. Mathlib v4.30 source proposition Nat.le_three_of_sqrt_eq_one open · route not assigned · F:ml430-nat-le-three-of-sqrt-eq-one-0c48a868
      1 dependencies 0 dependents
1808. Mathlib v4.30 source proposition Nat.le_zero_eq open · route not assigned · F:ml430-nat-le-zero-eq-336b502d
      0 dependencies 0 dependents
1809. Mathlib v4.30 source proposition Nat.log_anti_left proved · kernel-lean · F:ml430-nat-log-anti-left-b72490ec
      2 dependencies 0 dependents
1810. Mathlib v4.30 source proposition Nat.log_antitone_left proved · kernel-lean · F:ml430-nat-log-antitone-left-20d1326c
      1 dependencies 1 dependents
1811. Mathlib v4.30 source proposition Nat.log_div_mul_self proved · kernel-lean · F:ml430-nat-log-div-mul-self-04282351
      24 dependencies 0 dependents
1812. Mathlib v4.30 source proposition Nat.log_eq_one_iff' proved · kernel-lean · F:ml430-nat-log-eq-one-iff-63d772fb
      20 dependencies 0 dependents
1813. Mathlib v4.30 source proposition Nat.log_eq_one_iff proved · kernel-lean · F:ml430-nat-log-eq-one-iff-a89f8bab
      16 dependencies 0 dependents
1814. Mathlib v4.30 source proposition Nat.log_eq_zero_iff proved · kernel-lean · F:ml430-nat-log-eq-zero-iff-819cea74
      3 dependencies 0 dependents
1815. Mathlib v4.30 source proposition Nat.log_le_clog proved · kernel-lean · F:ml430-nat-log-le-clog-ac8ab2d4
      2 dependencies 0 dependents
1816. Mathlib v4.30 source proposition Nat.log_le_self proved · kernel-lean · F:ml430-nat-log-le-self-da387172
      3 dependencies 0 dependents
1817. Mathlib v4.30 source proposition Nat.log_lt_self proved · kernel-lean · F:ml430-nat-log-lt-self-529f89fa
      0 dependencies 1 dependents
1818. Mathlib v4.30 source proposition Nat.log_mono_right proved · kernel-lean · F:ml430-nat-log-mono-right-b8939fee
      0 dependencies 2 dependents
1819. Mathlib v4.30 source proposition Nat.log_monotone proved · kernel-lean · F:ml430-nat-log-monotone-52fad774
      1 dependencies 0 dependents
1820. Mathlib v4.30 source proposition Nat.log_of_lt proved · kernel-lean · F:ml430-nat-log-of-lt-89eaf42e
      1 dependencies 4 dependents
1821. Mathlib v4.30 source proposition Nat.log_one_left proved · kernel-lean · F:ml430-nat-log-one-left-73efc119
      0 dependencies 0 dependents
1822. Mathlib v4.30 source proposition Nat.log_one_right proved · kernel-lean · F:ml430-nat-log-one-right-282332ef
      0 dependencies 0 dependents
1823. Mathlib v4.30 source proposition Nat.log_zero_left proved · kernel-lean · F:ml430-nat-log-zero-left-9ec8541e
      0 dependencies 0 dependents
1824. Mathlib v4.30 source proposition Nat.log_zero_right proved · kernel-lean · F:ml430-nat-log-zero-right-8ea186db
      0 dependencies 3 dependents
1825. Mathlib v4.30 source proposition Nat.log2_eq_log_two proved · kernel-lean · F:ml430-nat-log2-eq-log-two-28085932
      1 dependencies 0 dependents
1826. Mathlib v4.30 source proposition Nat.lor_assoc proved · kernel-lean · F:ml430-nat-lor-assoc-82c4d0fd
      1 dependencies 0 dependents
1827. Mathlib v4.30 source proposition Nat.lor_bit proved · kernel-lean · F:ml430-nat-lor-bit-a2f98c7c
      0 dependencies 1 dependents
1828. Mathlib v4.30 source proposition Nat.lor_comm proved · kernel-lean · F:ml430-nat-lor-comm-2666d7ef
      1 dependencies 0 dependents
1829. Mathlib v4.30 source proposition Nat.lt_min proved · kernel-lean · F:ml430-nat-lt-min-1a793099
      1 dependencies 0 dependents
1830. Mathlib v4.30 source proposition Nat.lt_of_mul_lt_mul_left proved · kernel-lean · F:ml430-nat-lt-of-mul-lt-mul-left-234e8530
      4 dependencies 2 dependents
1831. Mathlib v4.30 source proposition Nat.lt_of_mul_lt_mul_right proved · kernel-lean · F:ml430-nat-lt-of-mul-lt-mul-right-54c1120b
      5 dependencies 1 dependents
1832. Mathlib v4.30 source proposition Nat.lt_of_testBit open · route not assigned · F:ml430-nat-lt-of-testbit-72f64ab8
      0 dependencies 0 dependents
1833. Mathlib v4.30 source proposition Nat.lt_pow_nthRoot_add_one open · route not assigned · F:ml430-nat-lt-pow-nthroot-add-one-7442ac90
      0 dependencies 0 dependents
1834. Mathlib v4.30 source proposition Nat.lt_succ_sqrt open · route not assigned · F:ml430-nat-lt-succ-sqrt-39389df2
      0 dependencies 1 dependents
1835. Mathlib v4.30 source proposition Nat.lt_xor_cases proved · kernel-lean · F:ml430-nat-lt-xor-cases-c43a1e85
      10 dependencies 0 dependents
1836. Mathlib v4.30 source proposition Nat.max_comm proved · kernel-lean · F:ml430-nat-max-comm-a9a3642b
      1 dependencies 0 dependents
1837. Mathlib v4.30 source proposition Nat.mod_lcm proved · kernel-lean · F:ml430-nat-mod-lcm-ee6bdd41 C:least-common-multipleC:modular-arithmetic
      14 dependencies 0 dependents
1838. Mathlib v4.30 source proposition Nat.mod_modEq proved · kernel-lean · F:ml430-nat-mod-modeq-436e4c10
      0 dependencies 0 dependents
1839. Mathlib v4.30 source proposition Nat.mod_mul proved · kernel-lean · F:ml430-nat-mod-mul-beaccbad
      20 dependencies 2 dependents
1840. Mathlib v4.30 source proposition Nat.mod_mul_left_div_self proved · kernel-lean · F:ml430-nat-mod-mul-left-div-self-0aca6c6e
      25 dependencies 0 dependents
1841. Mathlib v4.30 source proposition Nat.mod_mul_left_mod proved · kernel-lean · F:ml430-nat-mod-mul-left-mod-9b785abc
      13 dependencies 0 dependents
1842. Mathlib v4.30 source proposition Nat.mod_mul_right_div_self proved · kernel-lean · F:ml430-nat-mod-mul-right-div-self-900e0b01
      24 dependencies 1 dependents
1843. Mathlib v4.30 source proposition Nat.mod_mul_right_mod proved · kernel-lean · F:ml430-nat-mod-mul-right-mod-a481eff8
      12 dependencies 0 dependents
1844. Mathlib v4.30 source proposition Nat.ModEq.add proved · kernel-lean · F:ml430-nat-modeq-add-1561afa8
      3 dependencies 2 dependents
1845. Mathlib v4.30 source proposition Nat.ModEq.add_iff_left proved · kernel-lean · F:ml430-nat-modeq-add-iff-left-b719aac5
      2 dependencies 0 dependents
1846. Mathlib v4.30 source proposition Nat.ModEq.add_iff_right proved · kernel-lean · F:ml430-nat-modeq-add-iff-right-84daa45f
      2 dependencies 0 dependents
1847. Mathlib v4.30 source proposition Nat.ModEq.add_le_of_lt proved · kernel-lean · F:ml430-nat-modeq-add-le-of-lt-c774015b
      7 dependencies 0 dependents
1848. Mathlib v4.30 source proposition Nat.ModEq.add_left_cancel' proved · kernel-lean · F:ml430-nat-modeq-add-left-cancel-e5287cf6
      0 dependencies 0 dependents
1849. Mathlib v4.30 source proposition Nat.ModEq.add_left_cancel proved · kernel-lean · F:ml430-nat-modeq-add-left-cancel-fb96581c
      6 dependencies 1 dependents
1850. Mathlib v4.30 source proposition Nat.ModEq.add_left proved · kernel-lean · F:ml430-nat-modeq-add-left-e83f0700
      0 dependencies 0 dependents
1851. Mathlib v4.30 source proposition Nat.ModEq.add_right proved · kernel-lean · F:ml430-nat-modeq-add-right-8e2ca0cc
      0 dependencies 0 dependents
1852. Mathlib v4.30 source proposition Nat.ModEq.add_right_cancel' proved · kernel-lean · F:ml430-nat-modeq-add-right-cancel-e871facf
      0 dependencies 0 dependents
1853. Mathlib v4.30 source proposition Nat.ModEq.add_right_cancel proved · kernel-lean · F:ml430-nat-modeq-add-right-cancel-f0ab48e4
      6 dependencies 1 dependents
1854. Mathlib v4.30 source proposition Nat.ModEq.cancel_left_div_gcd proved · kernel-lean · F:ml430-nat-modeq-cancel-left-div-gcd-57ef8287
      9 dependencies 2 dependents
1855. Mathlib v4.30 source proposition Nat.ModEq.cancel_left_div_gcd' proved · kernel-lean · F:ml430-nat-modeq-cancel-left-div-gcd-cfca1225
      4 dependencies 0 dependents
1856. Mathlib v4.30 source proposition Nat.ModEq.cancel_left_of_coprime proved · kernel-lean · F:ml430-nat-modeq-cancel-left-of-coprime-f89af373
      1 dependencies 0 dependents
1857. Mathlib v4.30 source proposition Nat.ModEq.cancel_right_div_gcd proved · kernel-lean · F:ml430-nat-modeq-cancel-right-div-gcd-22a4f40d
      2 dependencies 0 dependents
1858. Mathlib v4.30 source proposition Nat.ModEq.comm proved · kernel-lean · F:ml430-nat-modeq-comm-24b71e7a
      1 dependencies 0 dependents
1859. Mathlib v4.30 source proposition Nat.ModEq.dvd_iff proved · kernel-lean · F:ml430-nat-modeq-dvd-iff-8f130450
      3 dependencies 1 dependents
1860. Mathlib v4.30 source proposition Nat.ModEq.gcd_eq proved · kernel-lean · F:ml430-nat-modeq-gcd-eq-5167ff4f
      14 dependencies 1 dependents
1861. Mathlib v4.30 source proposition Nat.ModEq.of_dvd proved · kernel-lean · F:ml430-nat-modeq-of-dvd-d75cc374
      0 dependencies 1 dependents
1862. Mathlib v4.30 source proposition Nat.ModEq.of_mul_left proved · kernel-lean · F:ml430-nat-modeq-of-mul-left-88d20bca
      0 dependencies 1 dependents
1863. Mathlib v4.30 source proposition Nat.ModEq.of_mul_right proved · kernel-lean · F:ml430-nat-modeq-of-mul-right-43078e1c
      1 dependencies 0 dependents
1864. Mathlib v4.30 source proposition Nat.modEq_one proved · kernel-lean · F:ml430-nat-modeq-one-516d46e8
      0 dependencies 0 dependents
1865. Mathlib v4.30 source proposition Nat.ModEq.refl proved · kernel-lean · F:ml430-nat-modeq-refl-d870c8f5
      0 dependencies 0 dependents
1866. Mathlib v4.30 source proposition Nat.ModEq.symm proved · kernel-lean · F:ml430-nat-modeq-symm-0a3d4d18
      0 dependencies 2 dependents
1867. Mathlib v4.30 source proposition Nat.ModEq.trans proved · kernel-lean · F:ml430-nat-modeq-trans-ef9d1c46
      0 dependencies 1 dependents
1868. Mathlib v4.30 source proposition Nat.modulus_modEq_zero proved · kernel-lean · F:ml430-nat-modulus-modeq-zero-fd9af096
      0 dependencies 0 dependents
1869. Mathlib v4.30 source proposition Nat.monotone_primeCounting open · route not assigned · F:ml430-nat-monotone-primecounting-8c35da98
      0 dependencies 0 dependents
1870. Mathlib v4.30 source proposition Nat.monotone_primeCounting' open · route not assigned · F:ml430-nat-monotone-primecounting-9ecb2222
      0 dependencies 0 dependents
1871. Mathlib v4.30 source proposition Nat.mul_add_lt_mul_of_lt_of_lt open · route not assigned · F:ml430-nat-mul-add-lt-mul-of-lt-of-lt-3567e87f
      0 dependencies 0 dependents
1872. Mathlib v4.30 source proposition Nat.mul_lt_mul_left proved · kernel-lean · F:ml430-nat-mul-lt-mul-left-af33301e
      4 dependencies 4 dependents
1873. Mathlib v4.30 source proposition Nat.mul_lt_mul_right proved · kernel-lean · F:ml430-nat-mul-lt-mul-right-de5b6046
      6 dependencies 0 dependents
1874. Mathlib v4.30 source proposition Nat.multichoose_one open · route not assigned · F:ml430-nat-multichoose-one-b210386a
      1 dependencies 0 dependents
1875. Mathlib v4.30 source proposition Nat.multichoose_one_right open · route not assigned · F:ml430-nat-multichoose-one-right-7755072d
      1 dependencies 0 dependents
1876. Mathlib v4.30 source proposition Nat.multichoose_zero_right open · route not assigned · F:ml430-nat-multichoose-zero-right-6ef827c8
      0 dependencies 2 dependents
1877. Mathlib v4.30 source proposition Nat.nat_repr_len_aux open · route not assigned · F:ml430-nat-nat-repr-len-aux-e53093d1
      0 dependencies 0 dependents
1878. Mathlib v4.30 source proposition Nat.not_coprime_zero_zero proved · kernel-lean · F:ml430-nat-not-coprime-zero-zero-6c4e8dd8
      2 dependencies 0 dependents
1879. Mathlib v4.30 source proposition Nat.not_exists_sq open · route not assigned · F:ml430-nat-not-exists-sq-3678b0c0
      0 dependencies 0 dependents
1880. Mathlib v4.30 source proposition Nat.not_prime_of_dvd_of_ne proved · kernel-lean · F:ml430-nat-not-prime-of-dvd-of-ne-4ff592c0
      0 dependencies 2 dependents
1881. Mathlib v4.30 source proposition Nat.not_two_dvd_bit1 open · route not assigned · F:ml430-nat-not-two-dvd-bit1-d5b4bf00
      0 dependencies 0 dependents
1882. Mathlib v4.30 source proposition Nat.nth_add open · route not assigned · F:ml430-nat-nth-add-a9dfaba9
      0 dependencies 0 dependents
1883. Mathlib v4.30 source proposition Nat.nth_add_one open · route not assigned · F:ml430-nat-nth-add-one-3fc46c88
      0 dependencies 0 dependents
1884. Mathlib v4.30 source proposition Nat.nth_eq_zero_mono open · route not assigned · F:ml430-nat-nth-eq-zero-mono-2dbbfdd5
      0 dependencies 0 dependents
1885. Mathlib v4.30 source proposition Nat.nth_false open · route not assigned · F:ml430-nat-nth-false-969b255e
      0 dependencies 0 dependents
1886. Mathlib v4.30 source proposition Nat.nth_mem_anti open · route not assigned · F:ml430-nat-nth-mem-anti-db30d674
      0 dependencies 0 dependents
1887. Mathlib v4.30 source proposition Nat.nth_mem_of_ne_zero open · route not assigned · F:ml430-nat-nth-mem-of-ne-zero-c1d86fab
      0 dependencies 0 dependents
1888. Mathlib v4.30 source proposition Nat.nth_ne_zero_anti open · route not assigned · F:ml430-nat-nth-ne-zero-anti-aab2a2dc
      0 dependencies 0 dependents
1889. Mathlib v4.30 source proposition Nat.nth_of_forall open · route not assigned · F:ml430-nat-nth-of-forall-eef6b587
      0 dependencies 0 dependents
1890. Mathlib v4.30 source proposition Nat.nth_true open · route not assigned · F:ml430-nat-nth-true-8ba7e1e1
      0 dependencies 0 dependents
1891. Mathlib v4.30 source proposition Nat.nthRoot_eq_of_le_of_lt open · route not assigned · F:ml430-nat-nthroot-eq-of-le-of-lt-55ab78a4
      0 dependencies 0 dependents
1892. Mathlib v4.30 source proposition Nat.nthRoot_lt_iff open · route not assigned · F:ml430-nat-nthroot-lt-iff-a71b6995
      0 dependencies 0 dependents
1893. Mathlib v4.30 source proposition Nat.nthRoot.lt_pow_go_succ_aux open · route not assigned · F:ml430-nat-nthroot-lt-pow-go-succ-aux-7ebd25d8
      0 dependencies 0 dependents
1894. Mathlib v4.30 source proposition Nat.nthRoot_one_right open · route not assigned · F:ml430-nat-nthroot-one-right-9744d156
      0 dependencies 0 dependents
1895. Mathlib v4.30 source proposition Nat.nthRoot_pow open · route not assigned · F:ml430-nat-nthroot-pow-02612a4b
      0 dependencies 0 dependents
1896. Mathlib v4.30 source proposition Nat.nthRoot_zero_left open · route not assigned · F:ml430-nat-nthroot-zero-left-8560aafb
      0 dependencies 0 dependents
1897. Mathlib v4.30 source proposition Nat.odd_fermatNumber proved · kernel-lean · F:ml430-nat-odd-fermatnumber-251041a5
      5 dependencies 0 dependents
1898. Mathlib v4.30 source proposition Nat.Odd.of_mul_left proved · kernel-lean · F:ml430-nat-odd-of-mul-left-2c6c2553
      0 dependencies 1 dependents
1899. Mathlib v4.30 source proposition Nat.Odd.of_mul_right proved · kernel-lean · F:ml430-nat-odd-of-mul-right-fe6d20ff
      2 dependencies 0 dependents
1900. Mathlib v4.30 source proposition Nat.odd_totient_iff proved · kernel-lean · F:ml430-nat-odd-totient-iff-b6a6596f
      2 dependencies 0 dependents
1901. Mathlib v4.30 source proposition Nat.odd_totient_iff_eq_one proved · kernel-lean · F:ml430-nat-odd-totient-iff-eq-one-d0491d84
      9 dependencies 1 dependents
1902. Mathlib v4.30 source proposition Nat.one_ascFactorial proved · kernel-lean · F:ml430-nat-one-ascfactorial-8bacb017
      0 dependencies 0 dependents
1903. Mathlib v4.30 source proposition Nat.pow_of_pow_add_prime proved · kernel-lean · F:ml430-nat-pow-of-pow-add-prime-ab61d0d3
      13 dependencies 0 dependents
1904. Mathlib v4.30 source proposition Nat.Prime.coprime_choose_of_lt open · route not assigned · F:ml430-nat-prime-coprime-choose-of-lt-3ccff0b4
      0 dependencies 0 dependents
1905. Mathlib v4.30 source proposition Nat.Prime.coprime_descFactorial_of_lt_of_le proved · kernel-lean · F:ml430-nat-prime-coprime-descfactorial-of-lt-of-le-716dffc3
      14 dependencies 0 dependents
1906. Mathlib v4.30 source proposition Nat.Prime.coprime_factorial_of_lt proved · kernel-lean · F:ml430-nat-prime-coprime-factorial-of-lt-2dbea201
      9 dependencies 2 dependents
1907. Mathlib v4.30 source proposition Nat.Prime.coprime_iff_not_dvd proved · kernel-lean · F:ml430-nat-prime-coprime-iff-not-dvd-a85d46a4
      5 dependencies 1 dependents
1908. Mathlib v4.30 source proposition Nat.Prime.coprime_pow_of_not_dvd proved · kernel-lean · F:ml430-nat-prime-coprime-pow-of-not-dvd-2752b17f
      3 dependencies 0 dependents
1909. Mathlib v4.30 source proposition Nat.Prime.deficient proved · kernel-lean · F:ml430-nat-prime-deficient-89e0badf
      7 dependencies 2 dependents
1910. Mathlib v4.30 source proposition Nat.Prime.deficient_pow open · route not assigned · F:ml430-nat-prime-deficient-pow-9c5e1fef
      0 dependencies 0 dependents
1911. Mathlib v4.30 source proposition Nat.Prime.dvd_choose_add open · route not assigned · F:ml430-nat-prime-dvd-choose-add-235cc461
      0 dependencies 0 dependents
1912. Mathlib v4.30 source proposition Nat.Prime.dvd_choose_self open · route not assigned · F:ml430-nat-prime-dvd-choose-self-b81c2438
      0 dependencies 0 dependents
1913. Mathlib v4.30 source proposition Nat.Prime.dvd_factorial proved · kernel-lean · F:ml430-nat-prime-dvd-factorial-5ace903f
      7 dependencies 0 dependents
1914. Mathlib v4.30 source proposition Nat.Prime.dvd_iff_eq proved · kernel-lean · F:ml430-nat-prime-dvd-iff-eq-b7446896
      1 dependencies 0 dependents
1915. Mathlib v4.30 source proposition Nat.Prime.dvd_iff_not_coprime proved · kernel-lean · F:ml430-nat-prime-dvd-iff-not-coprime-77854741 C:prime-numberC:greatest-common-divisor
      12 dependencies 3 dependents
1916. Mathlib v4.30 source proposition Nat.Prime.dvd_lcm proved · kernel-lean · F:ml430-nat-prime-dvd-lcm-237d267c
      10 dependencies 1 dependents
1917. Mathlib v4.30 source proposition Nat.Prime.dvd_mul proved · kernel-lean · F:ml430-nat-prime-dvd-mul-59446847
      3 dependencies 0 dependents
1918. Mathlib v4.30 source proposition Nat.Prime.dvd_mul_of_dvd_ne proved · kernel-lean · F:ml430-nat-prime-dvd-mul-of-dvd-ne-6c253439
      2 dependencies 0 dependents
1919. Mathlib v4.30 source proposition Nat.Prime.dvd_of_dvd_pow proved · kernel-lean · F:ml430-nat-prime-dvd-of-dvd-pow-e76f834a
      10 dependencies 1 dependents
1920. Mathlib v4.30 source proposition Nat.Prime.dvd_or_dvd proved · kernel-lean · F:ml430-nat-prime-dvd-or-dvd-4ae88221
      15 dependencies 8 dependents
1921. Mathlib v4.30 source proposition Nat.Prime.dvd_or_dvd_of_dvd_lcm proved · kernel-lean · F:ml430-nat-prime-dvd-or-dvd-of-dvd-lcm-58280948
      1 dependencies 0 dependents
1922. Mathlib v4.30 source proposition Nat.Prime.eq_one_of_pow proved · kernel-lean · F:ml430-nat-prime-eq-one-of-pow-846d2949
      9 dependencies 0 dependents
1923. Mathlib v4.30 source proposition Nat.Prime.eq_one_or_self_of_dvd proved · kernel-lean · F:ml430-nat-prime-eq-one-or-self-of-dvd-48de0abb
      0 dependencies 1 dependents
1924. Mathlib v4.30 source proposition Nat.Prime.eq_two_or_odd' proved · kernel-lean · F:ml430-nat-prime-eq-two-or-odd-25691fc9
      1 dependencies 0 dependents
1925. Mathlib v4.30 source proposition Nat.Prime.eq_two_or_odd proved · kernel-lean · F:ml430-nat-prime-eq-two-or-odd-44a91651
      1 dependencies 0 dependents
1926. Mathlib v4.30 source proposition Nat.Prime.even_iff proved · kernel-lean · F:ml430-nat-prime-even-iff-d068ec82
      15 dependencies 3 dependents
1927. Mathlib v4.30 source proposition Nat.Prime.five_le_of_ne_two_of_ne_three proved · kernel-lean · F:ml430-nat-prime-five-le-of-ne-two-of-ne-three-c069e786
      15 dependencies 0 dependents
1928. Mathlib v4.30 source proposition Nat.Prime.mod_two_eq_one_iff_ne_two proved · kernel-lean · F:ml430-nat-prime-mod-two-eq-one-iff-ne-two-25c35e73
      3 dependencies 0 dependents
1929. Mathlib v4.30 source proposition Nat.Prime.mul_eq_prime_sq_iff proved · kernel-lean · F:ml430-nat-prime-mul-eq-prime-sq-iff-d3fd2e31
      11 dependencies 0 dependents
1930. Mathlib v4.30 source proposition Nat.Prime.ne_one proved · kernel-lean · F:ml430-nat-prime-ne-one-5b7f8845
      1 dependencies 0 dependents
1931. Mathlib v4.30 source proposition Nat.Prime.ne_zero proved · kernel-lean · F:ml430-nat-prime-ne-zero-b4e8b4d5
      3 dependencies 1 dependents
1932. Mathlib v4.30 source proposition Nat.Prime.not_abundant proved · kernel-lean · F:ml430-nat-prime-not-abundant-d2558ed6
      4 dependencies 0 dependents
1933. Mathlib v4.30 source proposition Nat.Prime.not_coprime_iff_dvd proved · kernel-lean · F:ml430-nat-prime-not-coprime-iff-dvd-c83110ca
      16 dependencies 0 dependents
1934. Mathlib v4.30 source proposition Nat.Prime.not_dvd_mul proved · kernel-lean · F:ml430-nat-prime-not-dvd-mul-cb3a915e
      2 dependencies 0 dependents
1935. Mathlib v4.30 source proposition Nat.Prime.not_dvd_one proved · kernel-lean · F:ml430-nat-prime-not-dvd-one-be7e780f
      1 dependencies 2 dependents
1936. Mathlib v4.30 source proposition Nat.Prime.not_perfect proved · kernel-lean · F:ml430-nat-prime-not-perfect-15c1235d
      2 dependencies 0 dependents
1937. Mathlib v4.30 source proposition Nat.Prime.not_prime_pow' proved · kernel-lean · F:ml430-nat-prime-not-prime-pow-5f14afc6
      9 dependencies 0 dependents
1938. Mathlib v4.30 source proposition Nat.Prime.not_prime_pow proved · kernel-lean · F:ml430-nat-prime-not-prime-pow-d6480abf
      9 dependencies 0 dependents
1939. Mathlib v4.30 source proposition Nat.Prime.odd_of_ne_two proved · kernel-lean · F:ml430-nat-prime-odd-of-ne-two-91e1195f
      12 dependencies 0 dependents
1940. Mathlib v4.30 source proposition Nat.Prime.one_le proved · kernel-lean · F:ml430-nat-prime-one-le-03eb3095
      2 dependencies 2 dependents
1941. Mathlib v4.30 source proposition Nat.Prime.one_lt proved · kernel-lean · F:ml430-nat-prime-one-lt-bc90cfc7
      0 dependencies 1 dependents
1942. Mathlib v4.30 source proposition Nat.Prime.pos proved · kernel-lean · F:ml430-nat-prime-pos-85eeeeea
      2 dependencies 0 dependents
1943. Mathlib v4.30 source proposition Nat.Prime.pred_pos proved · kernel-lean · F:ml430-nat-prime-pred-pos-4e67ac4c
      4 dependencies 1 dependents
1944. Mathlib v4.30 source proposition Nat.Prime.sum_four_squares open · route not assigned · F:ml430-nat-prime-sum-four-squares-a60fd297
      0 dependencies 0 dependents
1945. Mathlib v4.30 source proposition Nat.primeCounting'_add_le open · route not assigned · F:ml430-nat-primecounting-add-le-1ce4f43d
      0 dependencies 0 dependents
1946. Mathlib v4.30 source proposition Nat.primeCounting_add_le open · route not assigned · F:ml430-nat-primecounting-add-le-3fef2bb6
      0 dependencies 0 dependents
1947. Mathlib v4.30 source proposition Nat.primeCounting'_eq_zero_iff open · route not assigned · F:ml430-nat-primecounting-eq-zero-iff-b4a33465
      0 dependencies 0 dependents
1948. Mathlib v4.30 source proposition Nat.Primrec.add open · route not assigned · F:ml430-nat-primrec-add-ea539e24
      0 dependencies 0 dependents
1949. Mathlib v4.30 source proposition Nat.Primrec.casesOn' open · route not assigned · F:ml430-nat-primrec-caseson-68977e97
      0 dependencies 0 dependents
1950. Mathlib v4.30 source proposition Nat.Primrec.casesOn1 open · route not assigned · F:ml430-nat-primrec-caseson1-f533b9fe
      0 dependencies 0 dependents
1951. Mathlib v4.30 source proposition Nat.Primrec.const open · route not assigned · F:ml430-nat-primrec-const-a120c9a1
      0 dependencies 0 dependents
1952. Mathlib v4.30 source proposition Nat.Primrec.mul open · route not assigned · F:ml430-nat-primrec-mul-2e55be0c
      0 dependencies 0 dependents
1953. Mathlib v4.30 source proposition Nat.Primrec.of_eq open · route not assigned · F:ml430-nat-primrec-of-eq-d42f5250
      0 dependencies 0 dependents
1954. Mathlib v4.30 source proposition Nat.Primrec.pow open · route not assigned · F:ml430-nat-primrec-pow-cf3df8b0
      0 dependencies 0 dependents
1955. Mathlib v4.30 source proposition Nat.Primrec.prec1 open · route not assigned · F:ml430-nat-primrec-prec1-61e68b20
      0 dependencies 0 dependents
1956. Mathlib v4.30 source proposition Nat.Primrec.pred open · route not assigned · F:ml430-nat-primrec-pred-57bf43f2
      0 dependencies 0 dependents
1957. Mathlib v4.30 source proposition Nat.Primrec.swap open · route not assigned · F:ml430-nat-primrec-swap-eab7e860
      0 dependencies 0 dependents
1958. Mathlib v4.30 source proposition Nat.self_le_factorial proved · kernel-lean · F:ml430-nat-self-le-factorial-cfdffc69
      7 dependencies 0 dependents
1959. Mathlib v4.30 source proposition Nat.Simproc.le_add_le open · route not assigned · F:ml430-nat-simproc-le-add-le-bc6f7c5c
      0 dependencies 0 dependents
1960. Mathlib v4.30 source proposition Nat.size_bit proved · kernel-lean · F:ml430-nat-size-bit-c601dbf0
      21 dependencies 0 dependents
1961. Mathlib v4.30 source proposition Nat.size_eq_zero proved · kernel-lean · F:ml430-nat-size-eq-zero-020ad98c
      5 dependencies 0 dependents
1962. Mathlib v4.30 source proposition Nat.size_le_size proved · kernel-lean · F:ml430-nat-size-le-size-c4b98f53
      0 dependencies 0 dependents
1963. Mathlib v4.30 source proposition Nat.size_one proved · kernel-lean · F:ml430-nat-size-one-e23e5f71
      0 dependencies 0 dependents
1964. Mathlib v4.30 source proposition Nat.sq_add_sq_mul open · route not assigned · F:ml430-nat-sq-add-sq-mul-7964e174
      0 dependencies 0 dependents
1965. Mathlib v4.30 source proposition Nat.sqrt_eq open · route not assigned · F:ml430-nat-sqrt-eq-79ae8eae
      0 dependencies 1 dependents
1966. Mathlib v4.30 source proposition Nat.sqrt_eq' open · route not assigned · F:ml430-nat-sqrt-eq-c036815b
      0 dependencies 0 dependents
1967. Mathlib v4.30 source proposition Nat.sqrt_eq_zero open · route not assigned · F:ml430-nat-sqrt-eq-zero-53666a3b
      1 dependencies 0 dependents
1968. Mathlib v4.30 source proposition Nat.sqrt_le open · route not assigned · F:ml430-nat-sqrt-le-7918582b
      0 dependencies 3 dependents
1969. Mathlib v4.30 source proposition Nat.sqrt_le_self open · route not assigned · F:ml430-nat-sqrt-le-self-1ed5eb85
      1 dependencies 0 dependents
1970. Mathlib v4.30 source proposition Nat.sqrt_le_sqrt open · route not assigned · F:ml430-nat-sqrt-le-sqrt-6e2bfc47
      2 dependencies 0 dependents
1971. Mathlib v4.30 source proposition Nat.sqrt_lt open · route not assigned · F:ml430-nat-sqrt-lt-4909537f
      0 dependencies 4 dependents
1972. Mathlib v4.30 source proposition Nat.sqrt_lt_self open · route not assigned · F:ml430-nat-sqrt-lt-self-ff7a155a
      1 dependencies 0 dependents
1973. Mathlib v4.30 source proposition Nat.sqrt_pos open · route not assigned · F:ml430-nat-sqrt-pos-f75e5114
      1 dependencies 0 dependents
1974. Mathlib v4.30 source proposition Nat.sqrt_succ_le_succ_sqrt open · route not assigned · F:ml430-nat-sqrt-succ-le-succ-sqrt-6b041183
      1 dependencies 0 dependents
1975. Mathlib v4.30 source proposition Nat.Squarefree.ext_iff open · route not assigned · F:ml430-nat-squarefree-ext-iff-7218327d
      0 dependencies 0 dependents
1976. Mathlib v4.30 source proposition Nat.stirlingFirst_eq_zero_of_lt proved · kernel-lean · F:ml430-nat-stirlingfirst-eq-zero-of-lt-6f46764f
      4 dependencies 1 dependents
1977. Mathlib v4.30 source proposition Nat.stirlingFirst_one_right proved · kernel-lean · F:ml430-nat-stirlingfirst-one-right-84dfc371
      2 dependencies 0 dependents
1978. Mathlib v4.30 source proposition Nat.stirlingFirst_self proved · kernel-lean · F:ml430-nat-stirlingfirst-self-4d06a0eb
      2 dependencies 1 dependents
1979. Mathlib v4.30 source proposition Nat.stirlingFirst_succ_self_left proved · kernel-lean · F:ml430-nat-stirlingfirst-succ-self-left-135bbfbf
      4 dependencies 0 dependents
1980. Mathlib v4.30 source proposition Nat.stirlingFirst_succ_succ proved · kernel-lean · F:ml430-nat-stirlingfirst-succ-succ-61c94738
      0 dependencies 0 dependents
1981. Mathlib v4.30 source proposition Nat.stirlingFirst_succ_zero proved · kernel-lean · F:ml430-nat-stirlingfirst-succ-zero-a58c6f3c
      0 dependencies 0 dependents
1982. Mathlib v4.30 source proposition Nat.stirlingFirst_zero proved · kernel-lean · F:ml430-nat-stirlingfirst-zero-ae5f4939
      0 dependencies 0 dependents
1983. Mathlib v4.30 source proposition Nat.stirlingFirst_zero_succ proved · kernel-lean · F:ml430-nat-stirlingfirst-zero-succ-d25889f3
      0 dependencies 0 dependents
1984. Mathlib v4.30 source proposition Nat.stirlingSecond_eq_zero_of_lt proved · kernel-lean · F:ml430-nat-stirlingsecond-eq-zero-of-lt-f3caf8bd
      4 dependencies 0 dependents
1985. Mathlib v4.30 source proposition Nat.stirlingSecond_one_right proved · kernel-lean · F:ml430-nat-stirlingsecond-one-right-ef2ad447
      0 dependencies 0 dependents
1986. Mathlib v4.30 source proposition Nat.succ_pred_prime proved · kernel-lean · F:ml430-nat-succ-pred-prime-4feb123f
      2 dependencies 1 dependents
1987. Mathlib v4.30 source proposition Nat.sum_four_squares open · route not assigned · F:ml430-nat-sum-four-squares-5cfacf06
      0 dependencies 0 dependents
1988. Mathlib v4.30 source proposition Nat.testBit_eq_inth open · route not assigned · F:ml430-nat-testbit-eq-inth-ffa07392
      0 dependencies 0 dependents
1989. Mathlib v4.30 source proposition Nat.testBit_land open · route not assigned · F:ml430-nat-testbit-land-dfef7ca4
      0 dependencies 0 dependents
1990. Mathlib v4.30 source proposition Nat.testBit_ldiff open · route not assigned · F:ml430-nat-testbit-ldiff-16f94162
      0 dependencies 0 dependents
1991. Mathlib v4.30 source proposition Nat.testBit_lor open · route not assigned · F:ml430-nat-testbit-lor-7644e067
      0 dependencies 0 dependents
1992. Mathlib v4.30 source proposition Nat.totient_coprime_totient_iff proved · kernel-lean · F:ml430-nat-totient-coprime-totient-iff-3932cf83
      18 dependencies 0 dependents
1993. Mathlib v4.30 source proposition Nat.totient_dvd_of_dvd proved · kernel-lean · F:ml430-nat-totient-dvd-of-dvd-9622e44a C:eulers-totient-functionC:divisibility
      1 dependencies 1 dependents
1994. Mathlib v4.30 source proposition Nat.totient_eq_one_iff proved · kernel-lean · F:ml430-nat-totient-eq-one-iff-68d883a0
      16 dependencies 3 dependents
1995. Mathlib v4.30 source proposition Nat.totient_eq_zero proved · kernel-lean · F:ml430-nat-totient-eq-zero-3be161d6
      2 dependencies 4 dependents
1996. Mathlib v4.30 source proposition Nat.totient_even proved · kernel-lean · F:ml430-nat-totient-even-28e0415f
      31 dependencies 2 dependents
1997. Mathlib v4.30 source proposition Nat.totient_gcd_mul_totient_mul proved · kernel-lean · F:ml430-nat-totient-gcd-mul-totient-mul-2e1d13c7 C:eulers-totient-functionC:divisibility
      2 dependencies 0 dependents
1998. Mathlib v4.30 source proposition Nat.zero_ascFactorial proved · kernel-lean · F:ml430-nat-zero-ascfactorial-af4fcdca
      0 dependencies 0 dependents
1999. Mathlib v4.30 source proposition Nat.zero_of_testBit_eq_false open · route not assigned · F:ml430-nat-zero-of-testbit-eq-false-e244c9a1
      0 dependencies 0 dependents
2000. Modus ponens is a valid inference proved · smt-term-level · F:modus-ponens-valid C:modus-ponensC:inference-ruleC:valid-argument
      0 dependencies 1 dependents
2001. Modus tollens is a valid inference proved · smt-term-level · F:modus-tollens-valid C:modus-tollensC:inference-ruleC:contrapositive
      2 dependencies 0 dependents
2002. NAND defines negation, conjunction and disjunction proved · smt-term-level · F:nand-functional-completeness C:nandC:logical-connectiveC:logical-equivalence
      0 dependencies 0 dependents
2003. Addition on the naturals is associative proved · kernel-lean · F:nat-add-assoc C:associativityC:addition
      0 dependencies 60 dependents
2004. Addition on the naturals is commutative proved · kernel-lean · F:nat-add-comm C:commutativity
      2 dependencies 117 dependents
2005. [generated] kernel theorem Nat.add_le_add_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-le-add-left
      0 dependencies 48 dependents
2006. [generated] kernel theorem Nat.add_le_add_right (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-le-add-right
      3 dependencies 45 dependents
2007. [generated] kernel theorem Nat.add_left_cancel (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-left-cancel
      3 dependencies 11 dependents
2008. [generated] kernel theorem Nat.add_lt_add_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-lt-add-left
      8 dependencies 23 dependents
2009. [generated] kernel theorem Nat.add_pow_modeq_prime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-pow-modeq-prime
      22 dependencies 2 dependents
2010. [generated] kernel theorem Nat.add_pow_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-pow-one
      4 dependencies 0 dependents
2011. [generated] kernel theorem Nat.add_pow_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-pow-zero
      0 dependencies 0 dependents
2012. [generated] kernel theorem Nat.add_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-pow
      23 dependencies 2 dependents
2013. Addition cancels on the right proved · kernel-lean · F:nat-add-right-cancel C:additionC:inverse-operation
      1 dependencies 9 dependents
2014. [generated] kernel theorem Nat.add_right_comm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-right-comm
      4 dependencies 39 dependents
2015. Subtraction undoes addition on the naturals proved · kernel-lean · F:nat-add-sub-cancel-left C:inverse-operationC:subtractionC:addition
      5 dependencies 13 dependents
2016. [generated] kernel theorem Nat.add_sub_cancel_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-sub-cancel-of-le
      3 dependencies 5 dependents
2017. [generated] kernel theorem Nat.add_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-add-succ
      0 dependencies 23 dependents
2018. Zero is a right identity for addition on the naturals proved · kernel-lean · F:nat-add-zero C:identity-elementC:zeroC:addition
      0 dependencies 47 dependents
2019. The alternating sum of a constant over the subsets of a NON-EMPTY ground set vanishes, and in Nat's graded form that is add_comm proved · kernel-lean · F:nat-alternating-subset-sum-vanishes
      3 dependencies 0 dependents
2020. ascFactorial(n, 1) = n proved · kernel-lean · F:nat-asc-factorial-one
      3 dependencies 0 dependents
2021. ascFactorial(n, k+1) = (n+k) * ascFactorial(n, k) proved · kernel-lean · F:nat-asc-factorial-succ
      1 dependencies 1 dependents
2022. ascFactorial(n, 0) = 1 proved · kernel-lean · F:nat-asc-factorial-zero
      0 dependencies 3 dependents
2023. [generated] kernel theorem Nat.beq_eq_false_of_ne (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-beq-eq-false-of-ne
      2 dependencies 16 dependents
2024. [generated] kernel theorem Nat.beq_eq_true_iff (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-beq-eq-true-iff
      3 dependencies 0 dependents
2025. [generated] kernel theorem Nat.beq_eq_true_of_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-beq-eq-true-of-eq
      1 dependencies 22 dependents
2026. [generated] kernel theorem Nat.beq_refl (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-beq-refl
      0 dependencies 2 dependents
2027. [generated] kernel theorem Nat.bezout_of_scaled (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-bezout-of-scaled
      4 dependencies 1 dependents
2028. [generated] kernel theorem Nat.bijective_of_injective_on (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-bijective-of-injective-on
      1 dependencies 0 dependents
2029. [generated] kernel theorem Nat.bijective_on_comp (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-bijective-on-comp
      1 dependencies 1 dependents
2030. [generated] kernel theorem Nat.bijective_on_perm_inverse (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-bijective-on-perm-inverse
      9 dependencies 1 dependents
2031. bit-halving recursion is independent of the fuel, given two sufficient fuels proved · kernel-lean · F:nat-binary-rec-fuel-irrelevance
      6 dependencies 1 dependents
2032. the recursive equation for bit-halving recursion at a successor proved · kernel-lean · F:nat-binary-rec-succ
      1 dependencies 0 dependents
2033. bit(false, n) <= bit(true, n) proved · kernel-lean · F:nat-bit-false-le-bit-true
      2 dependencies 0 dependents
2034. bit(false, n) = 2n proved · kernel-lean · F:nat-bit-false
      0 dependencies 1 dependents
2035. 0 < bit(true, n) proved · kernel-lean · F:nat-bit-true-pos
      3 dependencies 0 dependents
2036. bit(true, n) = 2n + 1 proved · kernel-lean · F:nat-bit-true
      0 dependencies 2 dependents
2037. bitwise and_fn 3 5 = land 3 5 proved · kernel-lean · F:nat-bitwise-and-eq-land-three-five
      0 dependencies 1 dependents
2038. bitwise and_fn m n = land m n, for all m and n proved · kernel-lean · F:nat-bitwise-and-eq-land
      2 dependencies 0 dependents
2039. bitwise combines two bit-appended naturals bit-by-bit, given no leading-zero ambiguity proved · kernel-lean · F:nat-bitwise-bit C:bitwise-operations
      4 dependencies 0 dependents
2040. bitwise is commutative for a commutative combinator proved · kernel-lean · F:nat-bitwise-comm C:bitwise-operations
      3 dependencies 0 dependents
2041. bitwise or_fn 3 5 = lor 3 5 proved · kernel-lean · F:nat-bitwise-or-eq-lor-three-five
      0 dependencies 1 dependents
2042. bitwise or_fn m n = lor m n, for all m and n proved · kernel-lean · F:nat-bitwise-or-eq-lor
      2 dependencies 0 dependents
2043. bitwise commutes with swapping the combinator's argument order proved · kernel-lean · F:nat-bitwise-swap C:bitwise-operations
      3 dependencies 0 dependents
2044. bitwise xor_fn 3 5 = 6 proved · kernel-lean · F:nat-bitwise-xor-three-five
      0 dependencies 0 dependents
2045. bitwise f 0 n = if f false true then n else 0 proved · kernel-lean · F:nat-bitwise-zero-left
      0 dependencies 0 dependents
2046. bitwise f m 0 = if f true false then m else 0 proved · kernel-lean · F:nat-bitwise-zero-right
      0 dependencies 1 dependents
2047. The boolean order test is false above its bound proved · kernel-lean · F:nat-ble-eq-false-of-lt
      4 dependencies 8 dependents
2048. [generated] kernel theorem Nat.ble_eq_true_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-ble-eq-true-of-le
      4 dependencies 11 dependents
2049. Two distinct naturals are compared one way or the other, never both proved · kernel-lean · F:nat-ble-select-add-of-ne
      6 dependencies 1 dependents
2050. [generated] kernel theorem Nat.ble_self_eq_true (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-ble-self-eq-true
      0 dependencies 1 dependents
2051. [generated] kernel theorem Nat.ble_succ_eq_true (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-ble-succ-eq-true
      0 dependencies 1 dependents
2052. [generated] kernel theorem Nat.cantor_diagonal_neg (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-cantor-diagonal-neg
      1 dependencies 0 dependents
2053. [generated] kernel theorem Nat.cantor_diagonal (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-cantor-diagonal
      2 dependencies 1 dependents
2054. [generated] kernel theorem Nat.cantor_no_fixed_point (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-cantor-no-fixed-point
      0 dependencies 0 dependents
2055. [generated] kernel theorem Nat.catalan_mul_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-catalan-mul-succ
      9 dependencies 0 dependents
2056. [generated] kernel theorem Nat.choose_add_convolution (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-choose-add-convolution
      20 dependencies 1 dependents
2057. A binomial coefficient is bounded by the corresponding power of two proved · kernel-lean · F:nat-choose-le-two-pow C:binomial-coefficientC:counting
      3 dependencies 0 dependents
2058. C(a, a) = 1 proved · kernel-lean · F:nat-choose-self C:binomial-coefficient
      2 dependencies 5 dependents
2059. A binomial coefficient above the diagonal is zero proved · kernel-lean · F:nat-choose-succ-self-eq-zero C:binomial-coefficient
      2 dependencies 3 dependents
2060. Pascal's rule for binomial coefficients proved · kernel-lean · F:nat-choose-succ-succ C:binomial-coefficient
      0 dependencies 10 dependents
2061. Binomial coefficients are symmetric proved · kernel-lean · F:nat-choose-symm C:binomial-coefficientC:counting
      17 dependencies 3 dependents
2062. The odd central binomial coefficient is at most 4^m proved · kernel-lean · F:nat-choose-two-mul-succ-le-four-pow
      0 dependencies 0 dependents
2063. Choosing zero elements has exactly one way proved · kernel-lean · F:nat-choose-zero-right C:binomial-coefficientC:counting
      0 dependencies 9 dependents
2064. The ceiling logarithm in base one is zero proved · kernel-lean · F:nat-clog-one-left
      0 dependencies 0 dependents
2065. The ceiling logarithm at n = 1 is zero proved · kernel-lean · F:nat-clog-one-right
      0 dependencies 0 dependents
2066. The ceiling logarithm in base zero is zero proved · kernel-lean · F:nat-clog-zero-left
      0 dependencies 0 dependents
2067. The ceiling logarithm at n = 0 is zero proved · kernel-lean · F:nat-clog-zero-right
      0 dependencies 0 dependents
2068. [generated] kernel theorem Nat.comp_assoc (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-comp-assoc
      0 dependencies 1 dependents
2069. [generated] kernel theorem Nat.conjugate_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-conjugate-injective
      0 dependencies 2 dependents
2070. [generated] kernel theorem Nat.conjugate_maps_into (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-conjugate-maps-into
      0 dependencies 2 dependents
2071. [generated] kernel theorem Nat.coprime_fib_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-coprime-fib-succ
      10 dependencies 0 dependents
2072. [generated] kernel theorem Nat.coprime_lcm_eq_mul (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-coprime-lcm-eq-mul
      3 dependencies 1 dependents
2073. [generated] kernel theorem Nat.coprime_mul_dvd (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-coprime-mul-dvd
      4 dependencies 2 dependents
2074. A Bezout witness of 1 certifies coprimality proved · kernel-lean · F:nat-coprime-of-bezout-one C:greatest-common-divisorC:coprimeC:bezout-identity
      17 dependencies 1 dependents
2075. a number below minFac(n) and not zero is coprime to n proved · kernel-lean · F:nat-coprime-of-lt-minfac C:coprime-numbers
      16 dependencies 0 dependents
2076. [generated] kernel theorem Nat.coprime_of_lt_prime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-coprime-of-lt-prime
      5 dependencies 4 dependents
2077. The cross-bound counting law where the inverse is total: four hypotheses and no injectivity proved · kernel-lean · F:nat-countrange-bij-of-inverse
      1 dependencies 0 dependents
2078. The cross-bound counting law: a constructive bijection between two selected sets equates their counts proved · kernel-lean · F:nat-countrange-bij
      15 dependencies 3 dependents
2079. [generated] kernel theorem Nat.countRange_compl (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-compl
      4 dependencies 2 dependents
2080. [generated] kernel theorem Nat.countRange_congr (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-congr
      0 dependencies 5 dependents
2081. [generated] kernel theorem Nat.countRange_eq_pred_of_only_zero_false (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-eq-pred-of-only-zero-false
      3 dependencies 1 dependents
2082. Counting a predicate IS summing its selector, definitionally proved · kernel-lean · F:nat-countrange-eq-sumrange
      0 dependencies 3 dependents
2083. A countRange over a predicate false everywhere below the bound is zero proved · kernel-lean · F:nat-countrange-eq-zero-of-all-false
      4 dependencies 4 dependents
2084. Cardinality of an injection, at the level of bare counts proved · kernel-lean · F:nat-countrange-le-of-injon
      13 dependencies 2 dependents
2085. [generated] kernel theorem Nat.countRange_le_of_subset (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-le-of-subset
      4 dependencies 1 dependents
2086. [generated] kernel theorem Nat.countRange_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-le
      4 dependencies 2 dependents
2087. The floor-counting lemma in executable form proved · kernel-lean · F:nat-countrange-mul-succ-le-eq-floor
      2 dependencies 1 dependents
2088. The floor-counting lemma, with the quotient emitted rather than projected proved · kernel-lean · F:nat-countrange-mul-succ-le-eq-min
      8 dependencies 1 dependents
2089. [generated] kernel theorem Nat.countRange_split (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-split
      3 dependencies 3 dependents
2090. Counting the indices below a threshold saturates at the minimum proved · kernel-lean · F:nat-countrange-succ-le-eq-min
      7 dependencies 1 dependents
2091. [generated] kernel theorem Nat.countRange_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-succ
      0 dependencies 5 dependents
2092. [generated] kernel theorem Nat.countRange_union_add_inter (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-union-add-inter
      5 dependencies 2 dependents
2093. [generated] kernel theorem Nat.countRange_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-countrange-zero
      0 dependencies 7 dependents
2094. The rectangle partition at complementary predicates, with no hypothesis proved · kernel-lean · F:nat-countrectangle-partition-compl
      1 dependencies 0 dependents
2095. A rectangle of lattice points, partitioned by two complementary predicates, with no set of lattice points proved · kernel-lean · F:nat-countrectangle-partition
      7 dependencies 2 dependents
2096. The CRT residue-pairing self-map is injective on a block of coprime dimensions proved · kernel-lean · F:nat-crt-self-map-injective-on C:chinese-remainder-theoremC:eulers-totient-function
      8 dependencies 1 dependents
2097. [generated] kernel theorem Nat.crt_unique (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-crt-unique
      13 dependencies 2 dependents
2098. n < k implies descFactorial(n, k) = 0 proved · kernel-lean · F:nat-desc-factorial-of-lt
      8 dependencies 0 dependents
2099. descFactorial(n, 1) = n proved · kernel-lean · F:nat-desc-factorial-one
      3 dependencies 0 dependents
2100. descFactorial(n, k+1) = (n-k) * descFactorial(n, k) proved · kernel-lean · F:nat-desc-factorial-succ
      1 dependencies 3 dependents
2101. descFactorial(n, 0) = 1 proved · kernel-lean · F:nat-desc-factorial-zero
      0 dependencies 2 dependents
2102. The Dirichlet convolution of two arithmetic functions is commutative, and its proof is the divisor reindexing plus the cofactor law proved · kernel-lean · F:nat-dirichlet-convolution-commutes
      3 dependencies 0 dependents
2103. [generated] kernel theorem Nat.div_mod_add_multiple (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-add-multiple
      2 dependencies 9 dependents
2104. [generated] kernel theorem Nat.div_mod_bounds (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-bounds
      2 dependencies 6 dependents
2105. [generated] kernel theorem Nat.div_mod_exact_exists (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-exact-exists
      1 dependencies 1 dependents
2106. The computed quotient and remainder satisfy the division specification proved · kernel-lean · F:nat-div-mod-exec C:division-algorithmC:divisionC:remainder
      8 dependencies 54 dependents
2107. Division with remainder always exists for a positive divisor proved · kernel-lean · F:nat-div-mod-exists C:division-algorithmC:divisionC:remainder
      1 dependencies 5 dependents
2108. [generated] kernel theorem Nat.div_mod_lt_mul_iff (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-lt-mul-iff
      7 dependencies 11 dependents
2109. [generated] kernel theorem Nat.div_mod_mul_le_iff (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-mul-le-iff
      7 dependencies 2 dependents
2110. [generated] kernel theorem Nat.div_mod_remainder_eq_of_mod_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-remainder-eq-of-mod-eq
      2 dependencies 2 dependents
2111. [generated] kernel theorem Nat.div_mod_remainder_eq_zero_iff_dvd (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-remainder-eq-zero-iff-dvd
      4 dependencies 5 dependents
2112. [generated] kernel theorem Nat.div_mod_same_remainder_mod_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mod-same-remainder-mod-eq
      3 dependencies 1 dependents
2113. The quotient and remainder of a division are unique proved · kernel-lean · F:nat-div-mod-unique C:division-algorithmC:remainder
      9 dependencies 28 dependents
2114. [generated] kernel theorem Nat.div_mul_cancel_of_dvd (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-mul-cancel-of-dvd
      5 dependencies 17 dependents
2115. At the Eisenstein shape the row floor never exceeds the rectangle's height proved · kernel-lean · F:nat-div-mul-succ-le-of-le
      17 dependencies 1 dependents
2116. [generated] kernel theorem Nat.div_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-succ
      0 dependencies 0 dependents
2117. [generated] kernel theorem Nat.div_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-div-zero
      0 dependencies 5 dependents
2118. A sum over the divisors of a positive n is unchanged by the substitution d -> n/d, because the map that fixes the non-divisors is a genuine involution proved · kernel-lean · F:nat-divisor-aggregate-reindex
      3 dependencies 1 dependents
2119. [generated] kernel theorem Nat.dvd_add_iff_right (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-add-iff-right
      4 dependencies 4 dependents
2120. Divisibility cancels out of a sum on the right proved · kernel-lean · F:nat-dvd-add-right-cancel-of-pos C:divisibilityC:addition
      7 dependencies 8 dependents
2121. A common divisor of two numbers divides their sum proved · kernel-lean · F:nat-dvd-add C:divisibilityC:factorC:multiple
      1 dependencies 11 dependents
2122. [generated] kernel theorem Nat.dvd_antisymm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-antisymm
      6 dependencies 8 dependents
2123. A positive natural up to n divides n! proved · kernel-lean · F:nat-dvd-factorial-of-le C:factorialC:divisibility
      9 dependencies 2 dependents
2124. The gcd is exactly the common divisors' upper bound proved · kernel-lean · F:nat-dvd-gcd-iff C:greatest-common-divisorC:divisibilityC:coprime
      5 dependencies 1 dependents
2125. A common divisor divides the gcd proved · kernel-lean · F:nat-dvd-gcd C:greatest-common-divisorC:divisibility
      6 dependencies 16 dependents
2126. [generated] kernel theorem Nat.dvd_lcm_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-lcm-left
      13 dependencies 6 dependents
2127. [generated] kernel theorem Nat.dvd_lcm_right (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-lcm-right
      12 dependencies 6 dependents
2128. A divisor of the modulus decides divisibility of a value from its remainder proved · kernel-lean · F:nat-dvd-mod-iff C:divisibilityC:remainderC:modular-arithmetic
      4 dependencies 4 dependents
2129. Divisibility survives multiplying the dividend proved · kernel-lean · F:nat-dvd-mul-right-of-dvd C:divisibilityC:multiplication
      3 dependencies 18 dependents
2130. A factor divides its product proved · kernel-lean · F:nat-dvd-mul C:divisibilityC:factorC:multiplication
      0 dependencies 28 dependents
2131. [generated] kernel theorem Nat.dvd_of_mod_eq_zero_of_pos (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-of-mod-eq-zero-of-pos
      6 dependencies 1 dependents
2132. Every factor below the bound divides a bounded product proved · kernel-lean · F:nat-dvd-prodrange-of-lt
      7 dependencies 1 dependents
2133. Divisibility is reflexive proved · kernel-lean · F:nat-dvd-refl C:divisibility
      1 dependencies 20 dependents
2134. [generated] kernel theorem Nat.dvd_sumRange_of_forall_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-sumrange-of-forall-lt
      4 dependencies 1 dependents
2135. Divisibility is transitive proved · kernel-lean · F:nat-dvd-trans C:divisibilityC:transitivity
      1 dependencies 25 dependents
2136. [generated] kernel theorem Nat.dvd_two_pow_classify (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-two-pow-classify
      15 dependencies 3 dependents
2137. [generated] kernel theorem Nat.dvd_two_pow_mul_classify (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-two-pow-mul-classify
      15 dependencies 0 dependents
2138. [generated] kernel theorem Nat.dvd_two_pow_succ_iff_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-two-pow-succ-iff-of-le
      10 dependencies 1 dependents
2139. [generated] kernel theorem Nat.dvd_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-dvd-zero
      1 dependencies 11 dependents
2140. Eisenstein's counting identity proved · kernel-lean · F:nat-eisenstein-count-identity
      7 dependencies 1 dependents
2141. Eisenstein's floor-sum identity without the min proved · kernel-lean · F:nat-eisenstein-floor-sum-min-free
      14 dependencies 1 dependents
2142. Eisenstein's lattice rectangle, counted two ways, is a sum of floors proved · kernel-lean · F:nat-eisenstein-floor-sum
      6 dependencies 1 dependents
2143. Eisenstein's lemma as a congruence mod 2 proved · kernel-lean · F:nat-eisenstein-lemma-modeq
      6 dependencies 0 dependents
2144. Eisenstein's lemma: the Gauss count and the floor sum have the same parity proved · kernel-lean · F:nat-eisenstein-lemma
      12 dependencies 2 dependents
2145. Nat.em_implies_lpo: excluded middle yields the limited principle of omniscience over the naturals proved · kernel-lean · F:nat-em-implies-lpo
      1 dependencies 4 dependents
2146. [generated] kernel theorem Nat.eq_equivalence_on (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-eq-equivalence-on
      0 dependencies 0 dependents
2147. [generated] kernel theorem Nat.eq_of_beq_eq_true (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-eq-of-beq-eq-true
      0 dependencies 17 dependents
2148. two naturals with the same bits at every index are equal proved · kernel-lean · F:nat-eq-of-testbit-eq
      23 dependencies 5 dependents
2149. Only 1 divides 1 proved · kernel-lean · F:nat-eq-one-of-dvd-one C:divisibility
      5 dependencies 18 dependents
2150. [generated] kernel theorem Nat.eqOn_refl (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-eqon-refl
      0 dependencies 0 dependents
2151. [generated] kernel theorem Nat.eqOn_symm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-eqon-symm
      0 dependencies 0 dependents
2152. [generated] kernel theorem Nat.eqOn_trans (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-eqon-trans
      0 dependencies 0 dependents
2153. Euclid's lemma: a prime dividing a product divides a factor proved · kernel-lean · F:nat-euclid-lemma C:fundamental-theorem-of-arithmeticC:prime-numberC:divisibility
      20 dependencies 8 dependents
2154. [generated] kernel theorem Nat.even_of_even_sq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-even-of-even-sq
      9 dependencies 1 dependents
2155. Every Nat is exactly n = 2*(n/2) or n = 2*(n/2)+1, with a COMPUTED half proved · kernel-lean · F:nat-even-or-odd
      16 dependencies 4 dependents
2156. Even (xor m n) iff Even m iff Even n proved · kernel-lean · F:nat-even-xor C:bitwise-xor
      12 dependencies 0 dependents
2157. a nonzero natural has a highest set bit, with every bit above it zero proved · kernel-lean · F:nat-exists-most-significant-bit
      1 dependencies 1 dependents
2158. Every natural number at least 2 has a prime divisor proved · kernel-lean · F:nat-exists-prime-dvd C:prime-numberC:divisibilityC:infinitude-of-primes
      11 dependencies 5 dependents
2159. [generated] kernel theorem Nat.exists_prime_factorization (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-exists-prime-factorization
      19 dependencies 1 dependents
2160. There is no largest prime proved · kernel-lean · F:nat-exists-prime-gt C:infinitude-of-primesC:prime-number
      11 dependencies 2 dependents
2161. An exact divisibility exponent is unique proved · kernel-lean · F:nat-exponent-unique-of-exact-dvd
      5 dependencies 1 dependents
2162. [generated] kernel theorem Nat.factorial_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-factorial-succ
      0 dependencies 8 dependents
2163. [generated] kernel theorem Nat.factorial_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-factorial-zero
      0 dependencies 2 dependents
2164. Every element of the computed prime factorization is prime proved · kernel-lean · F:nat-factorization-prime
      0 dependencies 1 dependents
2165. [generated] kernel theorem Nat.fib_add_two (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-fib-add-two
      2 dependencies 9 dependents
2166. [generated] kernel theorem Nat.fib_add (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-fib-add
      11 dependencies 0 dependents
2167. [generated] kernel theorem Nat.fib_pos_of_pos (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-fib-pos-of-pos
      3 dependencies 2 dependents
2168. [generated] kernel theorem Nat.Fin.isLt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-fin-islt
      0 dependencies 0 dependents
2169. [generated] kernel theorem Nat.Fin.val_mk (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-fin-val-mk
      0 dependencies 0 dependents
2170. The bounded universal search is congruent in its predicate proved · kernel-lean · F:nat-finset-all-below-congr
      1 dependencies 1 dependents
2171. A false bounded loop yields an index at which the predicate fails proved · kernel-lean · F:nat-finset-all-below-false-witness
      3 dependencies 3 dependents
2172. A bounded Boolean universal reflects back to a pointwise fact proved · kernel-lean · F:nat-finset-all-below-true-at
      4 dependencies 7 dependents
2173. Counting a finite set over a longer range than its own bound changes nothing proved · kernel-lean · F:nat-finset-card-eq-count-range-add
      4 dependencies 4 dependents
2174. Cardinality of an injection between two finite sets proved · kernel-lean · F:nat-finset-card-le-of-injon
      2 dependencies 2 dependents
2175. Cardinality is monotone under decided inclusion proved · kernel-lean · F:nat-finset-card-le-of-subset
      5 dependencies 0 dependents
2176. Euler's totient is a finite-set cardinality proved · kernel-lean · F:nat-finset-card-totatives
      1 dependencies 0 dependents
2177. Inclusion-exclusion for a computed finite set proved · kernel-lean · F:nat-finset-card-union-add-card-inter
      5 dependencies 1 dependents
2178. The pigeonhole principle over a finite set, with an explicit colliding pair proved · kernel-lean · F:nat-finset-exists-collision
      7 dependencies 0 dependents
2179. A finite set of naturals with a positive cardinality has a member, and the member is computed by a bounded search rather than chosen proved · kernel-lean · F:nat-finset-exists-memb-of-card-pos
      5 dependencies 2 dependents
2180. Membership implies the index is below the bound proved · kernel-lean · F:nat-finset-lt-bound-of-memb
      2 dependencies 6 dependents
2181. The pigeonhole principle over a finite set, in refutation form proved · kernel-lean · F:nat-finset-pigeonhole
      3 dependencies 1 dependents
2182. Sums agree when the decided membership does proved · kernel-lean · F:nat-finset-sum-congr-of-beq
      5 dependencies 0 dependents
2183. Summing over a finite set past its own bound changes nothing proved · kernel-lean · F:nat-finset-sum-eq-sum-range-if-add
      5 dependencies 2 dependents
2184. A sum over a disjoint union splits proved · kernel-lean · F:nat-finset-sum-union-disjoint
      8 dependencies 1 dependents
2185. The signed Gauss fold preserves the sum over [1, m] proved · kernel-lean · F:nat-gauss-fold-sumrange-eq
      5 dependencies 1 dependents
2186. [generated] kernel theorem Nat.gauss_lemma (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-gauss-lemma
      18 dependencies 11 dependents
2187. The two Gauss counting exponents and n*m have the same parity proved · kernel-lean · F:nat-gausscount-sum-even
      7 dependencies 2 dependents
2188. The two Gauss counting exponents sum to n*m modulo 2 proved · kernel-lean · F:nat-gausscount-sum-modeq
      7 dependencies 0 dependents
2189. Bezout's identity holds for the natural gcd proved · kernel-lean · F:nat-gcd-bezout C:bezout-identityC:greatest-common-divisor
      18 dependencies 7 dependents
2190. [generated] kernel theorem Nat.gcd_cofactors_coprime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-gcd-cofactors-coprime
      3 dependencies 3 dependents
2191. The natural gcd divides its first argument proved · kernel-lean · F:nat-gcd-dvd-left C:greatest-common-divisorC:divisibility
      1 dependencies 44 dependents
2192. The natural gcd divides its second argument proved · kernel-lean · F:nat-gcd-dvd-right C:greatest-common-divisorC:divisibility
      1 dependencies 40 dependents
2193. The gcd divides both of its arguments proved · kernel-lean · F:nat-gcd-dvd C:greatest-common-divisorC:divisibility
      8 dependencies 2 dependents
2194. gcd times lcm recovers the product proved · kernel-lean · F:nat-gcd-mul-lcm C:greatest-common-divisorC:least-common-multiple
      9 dependencies 3 dependents
2195. The Euclidean algorithm's descent step is correct proved · kernel-lean · F:nat-gcd-succ C:euclidean-algorithmC:greatest-common-divisorC:remainder
      4 dependencies 4 dependents
2196. gcd(0, a) = a proved · kernel-lean · F:nat-gcd-zero-left C:greatest-common-divisorC:zero
      4 dependencies 13 dependents
2197. [generated] kernel theorem Nat.group_identity_unique (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-group-identity-unique
      0 dependencies 0 dependents
2198. [generated] kernel theorem Nat.group_inverse_unique (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-group-inverse-unique
      0 dependencies 0 dependents
2199. [generated] kernel theorem Nat.group_left_cancel (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-group-left-cancel
      0 dependencies 0 dependents
2200. The parity of the rounded-up half is decided by m mod 4 proved · kernel-lean · F:nat-half-ceil-parity
      4 dependencies 1 dependents
2201. General inclusion-exclusion for a finite family of decidable subsets, as two Nat sums rather than one signed one, and the two-set case is the pre-existing lemma verbatim proved · kernel-lean · F:nat-inclusion-exclusion-general
      11 dependencies 0 dependents
2202. [generated] kernel theorem Nat.injective_on_comp (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-injective-on-comp
      0 dependencies 2 dependents
2203. [generated] kernel theorem Nat.injective_on_imp_surjective_on (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-injective-on-imp-surjective-on
      19 dependencies 6 dependents
2204. A self-map of a finite initial segment is either injective there or has an explicit duplicate pair, constructively proved · kernel-lean · F:nat-injective-on-or-duplicate
      7 dependencies 1 dependents
2205. [generated] kernel theorem Nat.injective_on_p_imp_surjective_on_p (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-injective-on-p-imp-surjective-on-p
      2 dependencies 0 dependents
2206. [generated] kernel theorem Nat.inverseIndex_fixed_point (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-fixed-point
      29 dependencies 1 dependents
2207. [generated] kernel theorem Nat.inverseIndex_fixes_last (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-fixes-last
      27 dependencies 1 dependents
2208. [generated] kernel theorem Nat.inverseIndex_fixes_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-fixes-zero
      15 dependencies 1 dependents
2209. [generated] kernel theorem Nat.inverseIndex_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-injective
      27 dependencies 1 dependents
2210. [generated] kernel theorem Nat.inverseIndex_interior_fixed_point_free (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-interior-fixed-point-free
      6 dependencies 1 dependents
2211. [generated] kernel theorem Nat.inverseIndex_involutive (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-involutive
      26 dependencies 1 dependents
2212. [generated] kernel theorem Nat.inverseIndex_maps_into (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-inverseindex-maps-into
      15 dependencies 2 dependents
2213. A prime has exactly two divisors, so the Bool primality predicate answers true proved · kernel-lean · F:nat-is-prime-eq-true-of-prime
      16 dependencies 1 dependents
2214. land is associative proved · kernel-lean · F:nat-land-assoc C:bitwise-and
      6 dependencies 0 dependents
2215. landAux fuel-irrelevance: any sufficient fuel agrees with the canonical one proved · kernel-lean · F:nat-land-aux-eq-land-of-le
      0 dependencies 2 dependents
2216. land decodes through a bit-appended argument proved · kernel-lean · F:nat-land-bit C:bitwise-and
      7 dependencies 0 dependents
2217. land is commutative proved · kernel-lean · F:nat-land-comm C:bitwise-and
      3 dependencies 1 dependents
2218. land(1, 1) = 1 proved · kernel-lean · F:nat-land-one-one
      0 dependencies 0 dependents
2219. land(3, 5) = 1 proved · kernel-lean · F:nat-land-three-five
      0 dependencies 0 dependents
2220. land(0, n) = 0 proved · kernel-lean · F:nat-land-zero-left
      0 dependencies 4 dependents
2221. land(m, 0) = 0 proved · kernel-lean · F:nat-land-zero-right
      1 dependencies 3 dependents
2222. [generated] kernel theorem Nat.lcm_comm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lcm-comm
      8 dependencies 0 dependents
2223. [generated] kernel theorem Nat.lcm_dvd (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lcm-dvd
      13 dependencies 5 dependents
2224. [generated] kernel theorem Nat.lcm_zero_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lcm-zero-left
      2 dependencies 7 dependents
2225. ldiff decodes through a bit-appended argument proved · kernel-lean · F:nat-ldiff-bit C:bitwise-and
      5 dependencies 0 dependents
2226. ldiff(5, 3) = 4 proved · kernel-lean · F:nat-ldiff-five-three
      1 dependencies 0 dependents
2227. ldiff(3, 5) = 2 proved · kernel-lean · F:nat-ldiff-three-five
      0 dependencies 1 dependents
2228. ldiff(0, n) = 0 proved · kernel-lean · F:nat-ldiff-zero-left
      0 dependencies 1 dependents
2229. ldiff(m, 0) = m proved · kernel-lean · F:nat-ldiff-zero-right
      1 dependencies 1 dependents
2230. n is <= n plus anything proved · kernel-lean · F:nat-le-add-right C:order-relation
      0 dependencies 120 dependents
2231. <= on the naturals is antisymmetric proved · kernel-lean · F:nat-le-antisymm C:order-relation
      3 dependencies 31 dependents
2232. <= destructs into an additive witness proved · kernel-lean · F:nat-le-dest C:order-relationC:addition
      0 dependencies 27 dependents
2233. [generated] kernel theorem Nat.le_intro (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-intro
      1 dependencies 1 dependents
2234. [generated] kernel theorem Nat.le_of_add_le_add_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-add-le-add-left
      5 dependencies 5 dependents
2235. [generated] kernel theorem Nat.le_of_add_le_add_right (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-add-le-add-right
      3 dependencies 28 dependents
2236. [generated] kernel theorem Nat.le_of_ble_eq_true (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-ble-eq-true
      2 dependencies 11 dependents
2237. A divisor of a positive natural does not exceed it proved · kernel-lean · F:nat-le-of-dvd C:divisibilityC:order-relation
      3 dependencies 32 dependents
2238. [generated] kernel theorem Nat.le_of_lt_add_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-lt-add-one
      10 dependencies 0 dependents
2239. [generated] kernel theorem Nat.le_of_lt_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-lt-succ
      10 dependencies 30 dependents
2240. [generated] kernel theorem Nat.le_of_mul_le_mul_left_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-mul-le-mul-left-succ
      6 dependencies 1 dependents
2241. [generated] kernel theorem Nat.le_of_mul_le_mul_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-of-mul-le-mul-left
      4 dependencies 2 dependents
2242. <= cancels a shared successor proved · kernel-lean · F:nat-le-of-succ-le-succ C:order-relation
      1 dependencies 58 dependents
2243. The order on the naturals is reflexive proved · imported-kernel-lean · F:nat-le-refl C:order-relationC:natural-numbers
      0 dependencies 4 dependents
2244. [generated] kernel theorem Nat.le_succ_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-succ-of-le
      7 dependencies 8 dependents
2245. <= is preserved by successor on both sides proved · kernel-lean · F:nat-le-succ-succ C:order-relation
      0 dependencies 157 dependents
2246. Every natural number is below its successor proved · imported-kernel-lean · F:nat-le-succ C:order-relationC:natural-numbers
      0 dependencies 4 dependents
2247. [generated] kernel theorem Nat.le_sumRange_of_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-le-sumrange-of-lt
      9 dependencies 1 dependents
2248. <= on the naturals is total proved · kernel-lean · F:nat-le-total C:order-relation
      2 dependencies 57 dependents
2249. <= on the naturals is transitive proved · kernel-lean · F:nat-le-trans C:order-relationC:transitivity
      0 dependencies 215 dependents
2250. The bounded least-divisor search finds a minimal divisor or exhausts its bound proved · kernel-lean · F:nat-least-divisor-search C:divisibilityC:factor
      11 dependencies 1 dependents
2251. The least residues reconcile with the folded ones proved · kernel-lean · F:nat-leastresidue-sumrange-reconcile
      13 dependencies 1 dependents
2252. Multiplication distributes over addition on the left proved · kernel-lean · F:nat-left-distrib C:distributivityC:multiplicationC:addition
      2 dependencies 34 dependents
2253. Nat.lnp_decidable: the least-number principle holds constructively for a Bool-valued predicate over the naturals proved · kernel-lean · F:nat-lnp-decidable
      1 dependencies 2 dependents
2254. Nat.lnp_unrestricted_implies_em: the unrestricted least-number principle over the naturals yields excluded middle for every proposition proved · kernel-lean · F:nat-lnp-unrestricted-implies-em
      3 dependencies 2 dependents
2255. Nat.lnp_unrestricted_implies_lpo: the unrestricted least-number principle yields the limited principle of omniscience proved · kernel-lean · F:nat-lnp-unrestricted-implies-lpo
      2 dependencies 0 dependents
2256. The floor logarithm of n never exceeds n proved · kernel-lean · F:nat-log-le-self
      1 dependencies 0 dependents
2257. Below its own base, a number has floor logarithm zero proved · kernel-lean · F:nat-log-of-lt
      1 dependencies 1 dependents
2258. The floor logarithm in base one is zero proved · kernel-lean · F:nat-log-one-left
      0 dependencies 0 dependents
2259. The floor logarithm of one is zero, in every base proved · kernel-lean · F:nat-log-one-right
      0 dependencies 0 dependents
2260. The floor logarithm in base zero is zero proved · kernel-lean · F:nat-log-zero-left
      0 dependencies 0 dependents
2261. The floor logarithm of zero is zero, in every base proved · kernel-lean · F:nat-log-zero-right
      0 dependencies 0 dependents
2262. The fuel is always an upper bound on the fuel-recursive floor logarithm proved · kernel-lean · F:nat-logaux-le-fuel
      3 dependencies 2 dependents
2263. lor is associative proved · kernel-lean · F:nat-lor-assoc C:bitwise-or
      4 dependencies 0 dependents
2264. lor decodes through a bit-appended argument proved · kernel-lean · F:nat-lor-bit C:bitwise-or
      5 dependencies 0 dependents
2265. lor is commutative proved · kernel-lean · F:nat-lor-comm C:bitwise-or
      3 dependencies 1 dependents
2266. lor(3, 5) = 7 proved · kernel-lean · F:nat-lor-three-five
      0 dependencies 0 dependents
2267. lor(0, n) = n proved · kernel-lean · F:nat-lor-zero-left
      0 dependencies 2 dependents
2268. lor(m, 0) = m proved · kernel-lean · F:nat-lor-zero-right
      1 dependencies 2 dependents
2269. Nat.lpo_implies_llpo: the limited principle of omniscience yields the lesser limited principle proved · kernel-lean · F:nat-lpo-implies-llpo
      1 dependencies 0 dependents
2270. Nat.lpo_implies_markov: the limited principle of omniscience yields Markov's principle proved · kernel-lean · F:nat-lpo-implies-markov
      1 dependencies 1 dependents
2271. Nat.lpo_implies_wlpo: the limited principle of omniscience yields its weak form proved · kernel-lean · F:nat-lpo-implies-wlpo
      2 dependencies 1 dependents
2272. [generated] kernel theorem Nat.lt_add_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-add-one
      6 dependencies 1 dependents
2273. < on the naturals is irreflexive proved · kernel-lean · F:nat-lt-irrefl C:order-relation
      3 dependencies 84 dependents
2274. The false side of the Nat ble/le bridge, in its strict form proved · kernel-lean · F:nat-lt-of-ble-eq-false
      2 dependencies 3 dependents
2275. [generated] kernel theorem Nat.lt_of_le_of_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-of-le-of-lt
      9 dependencies 33 dependents
2276. [generated] kernel theorem Nat.lt_of_lt_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-of-lt-of-le
      8 dependencies 95 dependents
2277. [generated] kernel theorem Nat.lt_of_not_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-of-not-le
      3 dependencies 1 dependents
2278. bit-i disagreement plus agreement above i forces the order proved · kernel-lean · F:nat-lt-of-testbit
      21 dependencies 1 dependents
2279. <= splits into < or = proved · kernel-lean · F:nat-lt-or-eq-of-le C:order-relation
      1 dependencies 91 dependents
2280. [generated] kernel theorem Nat.lt_or_ge (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-or-ge
      3 dependencies 42 dependents
2281. [generated] kernel theorem Nat.lt_pow_size (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-pow-size
      1 dependencies 2 dependents
2282. [generated] kernel theorem Nat.lt_succ_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-succ-of-le
      8 dependencies 9 dependents
2283. [generated] kernel theorem Nat.lt_succ_self (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-lt-succ-self
      6 dependencies 52 dependents
2284. The least prime factor divides its argument proved · kernel-lean · F:nat-min-fac-dvd
      6 dependencies 2 dependents
2285. A prime is its own least prime factor proved · kernel-lean · F:nat-min-fac-eq-self-of-prime
      3 dependencies 0 dependents
2286. The least factor of a number at least 2 is prime proved · kernel-lean · F:nat-min-fac-prime
      12 dependencies 0 dependents
2287. The least prime factor of a number at least 2 is at least 2 proved · kernel-lean · F:nat-min-fac-two-le
      6 dependencies 2 dependents
2288. [generated] kernel theorem Nat.mod_eq_add_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-add-left
      2 dependencies 7 dependents
2289. Congruence mod m is preserved by adding on the right proved · kernel-lean · F:nat-mod-eq-add-right C:congruenceC:modular-arithmeticC:addition
      3 dependencies 8 dependents
2290. [generated] kernel theorem Nat.mod_eq_add (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-add
      3 dependencies 0 dependents
2291. [generated] kernel theorem Nat.mod_eq_cancel (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-cancel
      15 dependencies 2 dependents
2292. [generated] kernel theorem Nat.mod_eq_iff_div_mod_remainder_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-iff-div-mod-remainder-eq
      2 dependencies 3 dependents
2293. Nat.mod_eq_iff_mod_eq: the existential balanced-witness congruence agrees with the executable Nat.mod comparison proved · kernel-lean · F:nat-mod-eq-iff-mod-eq C:modular-arithmetic
      3 dependencies 1 dependents
2294. Congruence mod m is preserved by multiplying on the left proved · kernel-lean · F:nat-mod-eq-mul-left C:congruenceC:modular-arithmeticC:multiplication
      3 dependencies 3 dependents
2295. Congruence mod m is preserved by multiplying on the right proved · kernel-lean · F:nat-mod-eq-mul-right C:congruenceC:modular-arithmeticC:multiplication
      2 dependencies 3 dependents
2296. Congruences may be multiplied proved · kernel-lean · F:nat-mod-eq-mul C:congruenceC:modular-arithmeticC:clock-arithmetic
      3 dependencies 1 dependents
2297. Congruence mod m is reflexive proved · kernel-lean · F:nat-mod-eq-refl C:congruenceC:modular-arithmetic
      0 dependencies 5 dependents
2298. [generated] kernel theorem Nat.mod_eq_self_of_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-self-of-lt
      7 dependencies 7 dependents
2299. [generated] kernel theorem Nat.mod_eq_symm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-eq-symm
      0 dependencies 8 dependents
2300. Congruence mod m is transitive proved · kernel-lean · F:nat-mod-eq-trans C:congruenceC:modular-arithmeticC:transitivity
      5 dependencies 11 dependents
2301. Congruence to zero characterizes divisibility proved · kernel-lean · F:nat-mod-eq-zero-iff-dvd C:divisibilityC:modular-arithmetic
      7 dependencies 2 dependents
2302. A divisor's dividend is congruent to zero modulo it proved · kernel-lean · F:nat-mod-eq-zero-of-dvd C:divisibilityC:congruenceC:modular-arithmetic
      3 dependencies 2 dependents
2303. The remainder is smaller than a positive modulus proved · kernel-lean · F:nat-mod-lt C:division-algorithmC:euclidean-division
      2 dependencies 23 dependents
2304. [generated] kernel theorem Nat.mod_self (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-self
      4 dependencies 1 dependents
2305. [generated] kernel theorem Nat.mod_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-succ
      0 dependencies 0 dependents
2306. n mod 2 = 0 or n mod 2 = 1 proved · kernel-lean · F:nat-mod-two-eq-zero-or-one
      3 dependencies 7 dependents
2307. [generated] kernel theorem Nat.mod_two_mul_split (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-two-mul-split
      16 dependencies 1 dependents
2308. [generated] kernel theorem Nat.mod_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mod-zero
      0 dependencies 8 dependents
2309. [generated] kernel theorem Nat.modAdd_isGroup (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-modadd-isgroup
      20 dependencies 0 dependents
2310. Congruence modulo m is an equivalence relation proved · kernel-lean · F:nat-modeq-equivalence-on C:modular-arithmeticC:congruenceC:equivalence-relation
      3 dependencies 0 dependents
2311. [generated] kernel theorem Nat.modEq_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-modeq-pow
      2 dependencies 1 dependents
2312. The Moebius function lands on a carrier without negatives as a pair of Nat-valued halves whose sum is its absolute value and whose product is zero proved · kernel-lean · F:nat-moebius-as-a-graded-pair
      1 dependencies 0 dependents
2313. [generated] kernel theorem Nat.monotone_of_le_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-monotone-of-le-succ
      1 dependencies 2 dependents
2314. Multiplication on the naturals is associative proved · kernel-lean · F:nat-mul-assoc C:associativityC:multiplication
      1 dependencies 56 dependents
2315. Multiplication on the naturals is commutative proved · kernel-lean · F:nat-mul-comm C:commutativityC:multiplication
      2 dependencies 109 dependents
2316. Divisibility multiplies proved · kernel-lean · F:nat-mul-dvd-mul
      2 dependencies 0 dependents
2317. [generated] kernel theorem Nat.mul_eq_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-eq-zero
      3 dependencies 8 dependents
2318. Multiplication by a fixed left factor preserves <= proved · kernel-lean · F:nat-mul-le-mul-left C:order-relationC:multiplication
      2 dependencies 39 dependents
2319. [generated] kernel theorem Nat.mul_left_cancel_of_pos (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-left-cancel-of-pos
      3 dependencies 32 dependents
2320. No lattice point sits on the line through a coprime pair proved · kernel-lean · F:nat-mul-ne-mul-of-coprime-of-lt
      4 dependencies 1 dependents
2321. One is a right identity for multiplication on the naturals proved · kernel-lean · F:nat-mul-one C:identity-elementC:multiplication
      1 dependencies 91 dependents
2322. [generated] kernel theorem Nat.mul_sub_left_distrib_total (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-sub-left-distrib-total
      5 dependencies 4 dependents
2323. [generated] kernel theorem Nat.mul_sub_left_distrib (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-sub-left-distrib
      6 dependencies 2 dependents
2324. [generated] kernel theorem Nat.mul_succ_add_lt_of_le_of_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-succ-add-lt-of-le-of-lt
      6 dependencies 2 dependents
2325. The 1-based side condition the rectangle partition consumes proved · kernel-lean · F:nat-mul-succ-ne-mul-succ-of-coprime
      3 dependencies 1 dependents
2326. [generated] kernel theorem Nat.mul_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-succ
      0 dependencies 16 dependents
2327. The division algorithm, summed over a range proved · kernel-lean · F:nat-mul-sumrange-div-add-leastresidue
      4 dependencies 1 dependents
2328. [generated] kernel theorem Nat.mul_sumRange_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-sumrange-pow
      3 dependencies 0 dependents
2329. [generated] kernel theorem Nat.mul_sumRange (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-mul-sumrange
      1 dependencies 4 dependents
2330. Every natural times zero is zero proved · kernel-lean · F:nat-mul-zero C:multiplicationC:zero
      0 dependencies 35 dependents
2331. multichoose(n, 1) = n proved · kernel-lean · F:nat-multichoose-one-right
      1 dependencies 0 dependents
2332. multichoose(1, k) = 1 proved · kernel-lean · F:nat-multichoose-one
      3 dependencies 0 dependents
2333. multichoose(n, 0) = 1 proved · kernel-lean · F:nat-multichoose-zero-right
      1 dependencies 0 dependents
2334. A divisor cannot carry more copies of a prime than the multiset does proved · kernel-lean · F:nat-multiset-count-le-of-dvd-prod
      9 dependencies 0 dependents
2335. A multiset of primes carries exactly its recorded multiplicity proved · kernel-lean · F:nat-multiset-not-pow-succ-count-dvd-prod
      10 dependencies 2 dependents
2336. A multiset's product is divisible by each element to its multiplicity proved · kernel-lean · F:nat-multiset-pow-count-dvd-prod
      4 dependencies 2 dependents
2337. Uniqueness of prime factorization, as multiplicity agreement proved · kernel-lean · F:nat-multiset-prime-factorization-unique
      7 dependencies 1 dependents
2338. The product of a multiset sum is the product of the parts proved · kernel-lean · F:nat-multiset-prod-add
      6 dependencies 2 dependents
2339. Every predicate-selected sub-product of a multiset divides the whole product proved · kernel-lean · F:nat-multiset-prod-sel-dvd-prod
      5 dependencies 0 dependents
2340. Two selections of a prime-supported multiset with equal selected products agree on the support proved · kernel-lean · F:nat-multiset-prod-sel-injective
      3 dependencies 0 dependents
2341. The product of a singleton multiset is its element proved · kernel-lean · F:nat-multiset-prod-singleton
      5 dependencies 0 dependents
2342. [generated] kernel theorem Nat.ne_of_beq_eq_false (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-ne-of-beq-eq-false
      1 dependencies 15 dependents
2343. [generated] kernel theorem Nat.no_rational_sqrt_two (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-no-rational-sqrt-two
      18 dependencies 0 dependents
2344. [generated] kernel theorem Nat.not_dvd_one_add_mul_of_two_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-not-dvd-one-add-mul-of-two-le
      8 dependencies 0 dependents
2345. No natural of size at least 2 divides 1 proved · kernel-lean · F:nat-not-dvd-one-of-two-le C:divisibility
      8 dependencies 9 dependents
2346. [generated] kernel theorem Nat.not_le_of_not_ble_eq_true (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-not-le-of-not-ble-eq-true
      2 dependencies 1 dependents
2347. No natural number is less than zero proved · kernel-lean · F:nat-not-lt-zero C:order-relationC:zero
      1 dependencies 29 dependents
2348. Nat.not_prime_of_pow_mod_ne: Fermat's little theorem run backwards -- a computable compositeness certificate (ADR-0603 row 1) proved · kernel-lean · F:nat-not-prime-of-pow-mod-ne C:prime-numberC:modular-arithmeticC:fermats-little-theorem
      4 dependencies 0 dependents
2349. [generated] kernel theorem Nat.not_succ_le_self (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-not-succ-le-self
      1 dependencies 19 dependents
2350. No successor is <= zero proved · kernel-lean · F:nat-not-succ-le-zero C:order-relation
      0 dependencies 51 dependents
2351. The factorial of a natural number is always at least 1 proved · kernel-lean · F:nat-one-le-factorial C:factorialC:counting
      4 dependencies 15 dependents
2352. [generated] kernel theorem Nat.one_le_left_of_mul (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-one-le-left-of-mul
      4 dependencies 1 dependents
2353. A product of two naturals each >= 1 is itself >= 1 proved · kernel-lean · F:nat-one-le-mul
      6 dependencies 54 dependents
2354. A divisor of a positive natural is itself positive proved · kernel-lean · F:nat-one-le-of-dvd-pos C:divisibilityC:order-relation
      1 dependencies 28 dependents
2355. [generated] kernel theorem Nat.one_le_right_of_mul (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-one-le-right-of-mul
      4 dependencies 2 dependents
2356. 1 is a left identity for multiplication proved · kernel-lean · F:nat-one-mul C:multiplicationC:identity-element
      3 dependencies 75 dependents
2357. [generated] kernel theorem Nat.one_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-one-pow
      1 dependencies 7 dependents
2358. [generated] kernel theorem Nat.Peano.categorical (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-categorical
      2 dependencies 0 dependents
2359. The constructed Nat is THE natural numbers, up to unique isomorphism proved · kernel-lean · F:nat-peano-categoricity C:peano-axiomsC:natural-numbers
      2 dependencies 1 dependents
2360. [generated] kernel theorem Nat.Peano.induction (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-induction
      0 dependencies 0 dependents
2361. The constructed Nat is initial among pointed unary algebras proved · kernel-lean · F:nat-peano-initial C:natural-numbers
      3 dependencies 0 dependents
2362. [generated] kernel theorem Nat.Peano.injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-injective
      0 dependencies 2 dependents
2363. [generated] kernel theorem Nat.Peano.iter_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-iter-succ
      0 dependencies 1 dependents
2364. [generated] kernel theorem Nat.Peano.iter_unique (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-iter-unique
      0 dependencies 1 dependents
2365. [generated] kernel theorem Nat.Peano.iter_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-iter-zero
      0 dependencies 1 dependents
2366. [generated] kernel theorem Nat.Peano.succ_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-succ-injective
      1 dependencies 1 dependents
2367. [generated] kernel theorem Nat.Peano.surjective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-surjective
      0 dependencies 2 dependents
2368. [generated] kernel theorem Nat.Peano.zero_ne_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-peano-zero-ne-succ
      0 dependencies 4 dependents
2369. [generated] kernel theorem Nat.permInverse_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-perminverse-left
      14 dependencies 2 dependents
2370. [generated] kernel theorem Nat.permInverse_right (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-perminverse-right
      8 dependencies 2 dependents
2371. [generated] kernel theorem Nat.pigeonhole (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pigeonhole
      3 dependencies 0 dependents
2372. The first index law: powers add over a product proved · kernel-lean · F:nat-pow-add C:index-lawsC:exponentC:power
      4 dependencies 7 dependents
2373. A power divides a higher power of the same base proved · kernel-lean · F:nat-pow-dvd-pow-of-le
      3 dependencies 2 dependents
2374. [generated] kernel theorem Nat.pow_half_split (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-half-split
      18 dependencies 2 dependents
2375. [generated] kernel theorem Nat.pow_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-injective
      4 dependencies 2 dependents
2376. [generated] kernel theorem Nat.pow_lt_pow_of_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-lt-pow-of-lt
      7 dependencies 3 dependents
2377. [generated] kernel theorem Nat.pow_lt_pow_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-lt-pow-succ
      6 dependencies 3 dependents
2378. [generated] kernel theorem Nat.pow_mul_prime_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-mul-prime-injective
      3 dependencies 0 dependents
2379. [generated] kernel theorem Nat.pow_pos (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-pos
      7 dependencies 13 dependents
2380. Fermat's little theorem for the naturals proved · kernel-lean · F:nat-pow-prime-modeq-self C:prime-numberC:modular-arithmeticC:fermats-little-theorem
      8 dependencies 2 dependents
2381. [generated] kernel theorem Nat.pow_sq_aux_eq_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-sq-aux-eq-pow
      24 dependencies 1 dependents
2382. [generated] kernel theorem Nat.pow_sq_eq_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-sq-eq-pow
      1 dependencies 2 dependents
2383. [generated] kernel theorem Nat.pow_sq_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-sq-succ
      2 dependencies 0 dependents
2384. [generated] kernel theorem Nat.pow_sq_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-sq-zero
      2 dependencies 0 dependents
2385. [generated] kernel theorem Nat.pow_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-succ
      0 dependencies 17 dependents
2386. [generated] kernel theorem Nat.pow_two_ne_pow_two_mul_prime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-two-ne-pow-two-mul-prime
      10 dependencies 0 dependents
2387. [generated] kernel theorem Nat.pow_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow-zero
      0 dependencies 13 dependents
2388. [generated] kernel theorem Nat.pow2_geom_sum (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pow2-geom-sum
      5 dependencies 1 dependents
2389. [generated] kernel theorem Nat.pred_le_pred (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pred-le-pred
      2 dependencies 4 dependents
2390. [generated] kernel theorem Nat.pred_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pred-le
      0 dependencies 5 dependents
2391. [generated] kernel theorem Nat.pred_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pred-succ
      0 dependencies 0 dependents
2392. [generated] kernel theorem Nat.pred_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-pred-zero
      0 dependencies 0 dependents
2393. The inclusive prime-counting function is monotone proved · kernel-lean · F:nat-prime-counting-mono
      2 dependencies 0 dependents
2394. The prime-counting function is monotone proved · kernel-lean · F:nat-prime-counting-prime-mono
      0 dependencies 2 dependents
2395. The prime-counting function is unbounded proved · kernel-lean · F:nat-prime-counting-prime-unbounded
      6 dependencies 0 dependents
2396. A prime divides the interior binomial coefficients of its own row proved · kernel-lean · F:nat-prime-dvd-choose C:prime-numberC:binomial-coefficientC:divisibility
      12 dependencies 1 dependents
2397. A prime power passes through a factor the prime does not divide proved · kernel-lean · F:nat-prime-pow-dvd-of-dvd-mul-of-not-dvd
      11 dependencies 1 dependents
2398. The primorial exists in this kernel as a computable fold over minFac proved · kernel-lean · F:nat-primorial-construction
      0 dependencies 0 dependents
2399. The computed prime factorization multiplies back to its argument proved · kernel-lean · F:nat-prod-factorization
      1 dependencies 1 dependents
2400. Extending a product past its support changes nothing proved · kernel-lean · F:nat-prod-range-add-of-one-above
      2 dependencies 1 dependents
2401. prodRange respects a pointwise identity of its function proved · kernel-lean · F:nat-prod-range-congr
      0 dependencies 1 dependents
2402. A product of ones is one proved · kernel-lean · F:nat-prod-range-eq-one-of-below
      1 dependencies 1 dependents
2403. A product over a pointwise product splits into two products proved · kernel-lean · F:nat-prod-range-mul
      2 dependencies 1 dependents
2404. [generated] kernel theorem Nat.prodRange_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-prodrange-succ
      0 dependencies 1 dependents
2405. [generated] kernel theorem Nat.prodRange_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-prodrange-zero
      0 dependencies 0 dependents
2406. [generated] kernel theorem Nat.prodRangeIf_congr_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-prodrangeif-congr-lt
      2 dependencies 1 dependents
2407. [generated] kernel theorem Nat.prodRangeIf_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-prodrangeif-succ
      0 dependencies 1 dependents
2408. [generated] kernel theorem Nat.prodRangeIf_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-prodrangeif-zero
      0 dependencies 0 dependents
2409. [generated] kernel theorem Nat.restrict_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-restrict-injective
      11 dependencies 2 dependents
2410. [generated] kernel theorem Nat.restrict_maps_into (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-restrict-maps-into
      13 dependencies 2 dependents
2411. [generated] kernel theorem Nat.restrict_pair_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-restrict-pair-injective
      21 dependencies 0 dependents
2412. [generated] kernel theorem Nat.restrict_pair_maps_into (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-restrict-pair-maps-into
      23 dependencies 0 dependents
2413. Multiplication distributes over addition on the right proved · kernel-lean · F:nat-right-distrib C:distributivityC:multiplicationC:addition
      2 dependencies 12 dependents
2414. Nat.RM.self_halting_not_decidable: no total decider is correct about its own self-referential halting proved · kernel-lean · F:nat-rm-self-halting-not-decidable
      2 dependencies 0 dependents
2415. [generated] kernel theorem Nat.setCompl_inter (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setcompl-inter
      0 dependencies 0 dependents
2416. [generated] kernel theorem Nat.setCompl_involutive (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setcompl-involutive
      0 dependencies 0 dependents
2417. [generated] kernel theorem Nat.setCompl_union (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setcompl-union
      0 dependencies 0 dependents
2418. [generated] kernel theorem Nat.setDiff_eq_inter_compl (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setdiff-eq-inter-compl
      0 dependencies 0 dependents
2419. [generated] kernel theorem Nat.setInter_absorb (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setinter-absorb
      0 dependencies 0 dependents
2420. [generated] kernel theorem Nat.setInter_assoc (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setinter-assoc
      0 dependencies 0 dependents
2421. [generated] kernel theorem Nat.setInter_comm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setinter-comm
      0 dependencies 0 dependents
2422. [generated] kernel theorem Nat.setInter_idem (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setinter-idem
      0 dependencies 0 dependents
2423. [generated] kernel theorem Nat.setInter_union_distrib (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setinter-union-distrib
      0 dependencies 0 dependents
2424. [generated] kernel theorem Nat.setUnion_absorb (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setunion-absorb
      0 dependencies 0 dependents
2425. [generated] kernel theorem Nat.setUnion_assoc (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setunion-assoc
      0 dependencies 0 dependents
2426. [generated] kernel theorem Nat.setUnion_comm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setunion-comm
      0 dependencies 0 dependents
2427. [generated] kernel theorem Nat.setUnion_idem (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setunion-idem
      0 dependencies 0 dependents
2428. [generated] kernel theorem Nat.setUnion_inter_distrib (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-setunion-inter-distrib
      0 dependencies 0 dependents
2429. [generated] kernel theorem Nat.size_aux_lt_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-size-aux-lt-pow
      26 dependencies 1 dependents
2430. [generated] kernel theorem Nat.size_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-size-zero
      0 dependencies 1 dependents
2431. sqrt(1) = 1 proved · kernel-lean · F:nat-sqrt-one
      0 dependencies 0 dependents
2432. sqrt(0) = 0 proved · kernel-lean · F:nat-sqrt-zero
      0 dependencies 0 dependents
2433. Course-of-values recursion on the naturals, as a named kernel principle with its unfolding equation proved · kernel-lean · F:nat-strong-induction
      1 dependencies 3 dependents
2434. Subtraction undoes addition when the order holds proved · kernel-lean · F:nat-sub-add-cancel C:subtractionC:additionC:inverse-operation
      4 dependencies 39 dependents
2435. [generated] kernel theorem Nat.sub_eq_zero_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-eq-zero-of-le
      1 dependencies 5 dependents
2436. [generated] kernel theorem Nat.sub_le_iff_le_add (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-le-iff-le-add
      10 dependencies 1 dependents
2437. [generated] kernel theorem Nat.sub_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-le
      2 dependencies 4 dependents
2438. [generated] kernel theorem Nat.sub_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-lt
      4 dependencies 4 dependents
2439. Every natural minus itself is zero proved · kernel-lean · F:nat-sub-self C:subtractionC:zero
      1 dependencies 11 dependents
2440. Subtracting a truncated difference recovers the subtrahend, over the constructed naturals proved · kernel-lean · F:nat-sub-sub-self
      2 dependencies 1 dependents
2441. [generated] kernel theorem Nat.sub_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-succ
      0 dependencies 2 dependents
2442. [generated] kernel theorem Nat.sub_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sub-zero
      0 dependencies 4 dependents
2443. [generated] kernel theorem Nat.subset_antisymm (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-subset-antisymm
      0 dependencies 0 dependents
2444. [generated] kernel theorem Nat.subset_inter_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-subset-inter-left
      0 dependencies 0 dependents
2445. [generated] kernel theorem Nat.subset_refl (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-subset-refl
      0 dependencies 0 dependents
2446. A sum indexed by the subsets of [0,n+1) splits into the subsets that omit n and the subsets that contain it, and over a predicate-valued subset that split is Eq.refl proved · kernel-lean · F:nat-subset-sum-split-law
      3 dependencies 2 dependents
2447. [generated] kernel theorem Nat.subset_trans (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-subset-trans
      0 dependencies 0 dependents
2448. [generated] kernel theorem Nat.subset_union_left (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-subset-union-left
      0 dependencies 0 dependents
2449. The alternating sum of a constant over a non-empty masked ground set vanishes proved · kernel-lean · F:nat-subsets-sum-sel-on-const-of-mem
      7 dependencies 0 dependents
2450. The mask-restricted subset fold visits two-to-the-masked-count subsets proved · kernel-lean · F:nat-subsets-sum-subsets-on-card
      6 dependencies 0 dependents
2451. Nat succ_add proved · kernel-lean · F:nat-succ-add
      0 dependencies 67 dependents
2452. [generated] kernel theorem Nat.succ_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-injective
      0 dependencies 22 dependents
2453. [generated] kernel theorem Nat.succ_le_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-le-succ
      8 dependencies 22 dependents
2454. [generated] kernel theorem Nat.succ_mul_choose_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-mul-choose-eq
      15 dependencies 3 dependents
2455. (a+1) * b = a*b + b proved · kernel-lean · F:nat-succ-mul C:multiplicationC:addition
      1 dependencies 44 dependents
2456. [generated] kernel theorem Nat.succ_ne_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-ne-zero
      0 dependencies 25 dependents
2457. [generated] kernel theorem Nat.succ_pred_of_pos (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-pred-of-pos
      1 dependencies 54 dependents
2458. [generated] kernel theorem Nat.succ_sub_of_le (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-sub-of-le
      2 dependencies 4 dependents
2459. [generated] kernel theorem Nat.succ_sub_succ_eq_sub (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-succ-sub-succ-eq-sub
      1 dependencies 0 dependents
2460. Subtraction is unaffected by adding one to both sides proved · kernel-lean · F:nat-succ-sub-succ C:subtraction
      0 dependencies 10 dependents
2461. [generated] kernel theorem Nat.sum_choose_row (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sum-choose-row
      4 dependencies 1 dependents
2462. [generated] kernel theorem Nat.sum_choose_sq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sum-choose-sq
      5 dependencies 0 dependents
2463. [generated] kernel theorem Nat.sum_fib (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sum-fib
      6 dependencies 0 dependents
2464. [generated] kernel theorem Nat.sum_testBit_eq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sum-testbit-eq
      3 dependencies 1 dependents
2465. [generated] kernel theorem Nat.sum_testBit_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sum-testbit-lt
      16 dependencies 3 dependents
2466. [generated] kernel theorem Nat.sumDivisors_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumdivisors-one
      0 dependencies 0 dependents
2467. [generated] kernel theorem Nat.sumDivisors_prime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumdivisors-prime
      18 dependencies 1 dependents
2468. [generated] kernel theorem Nat.sumDivisors_two_pow_eq_geom_sum (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumdivisors-two-pow-eq-geom-sum
      27 dependencies 1 dependents
2469. [generated] kernel theorem Nat.sumDivisors_two_pow (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumdivisors-two-pow
      2 dependencies 0 dependents
2470. [generated] kernel theorem Nat.sumRange_add (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-add
      4 dependencies 11 dependents
2471. [generated] kernel theorem Nat.sumRange_congr_lt (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-congr-lt
      2 dependencies 12 dependents
2472. [generated] kernel theorem Nat.sumRange_congr (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-congr
      0 dependencies 13 dependents
2473. A finite sum of a constant, over the constructed naturals proved · kernel-lean · F:nat-sumrange-const
      0 dependencies 3 dependents
2474. [generated] kernel theorem Nat.sumRange_diagonal (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-diagonal
      10 dependencies 2 dependents
2475. Summing over a range is invariant under an injective self-map proved · kernel-lean · F:nat-sumrange-permute
      12 dependencies 2 dependents
2476. Two families differing at one index have sums differing by that much proved · kernel-lean · F:nat-sumrange-point-change
      8 dependencies 1 dependents
2477. [generated] kernel theorem Nat.sumRange_rect_eq_diag_add_corner (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-rect-eq-diag-add-corner
      6 dependencies 0 dependents
2478. [generated] kernel theorem Nat.sumRange_shiftFront (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-shiftfront
      3 dependencies 5 dependents
2479. [generated] kernel theorem Nat.sumRange_split (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-split
      3 dependencies 5 dependents
2480. [generated] kernel theorem Nat.sumRange_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-succ
      0 dependencies 5 dependents
2481. Fubini for finite sums over the constructed naturals proved · kernel-lean · F:nat-sumrange-swap
      4 dependencies 1 dependents
2482. [generated] kernel theorem Nat.sumRange_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-sumrange-zero
      0 dependencies 2 dependents
2483. A range sum splits into its selected and unselected parts proved · kernel-lean · F:nat-sumrangeif-compl
      5 dependencies 0 dependents
2484. Conditional sums agree when predicate and summand agree below the bound proved · kernel-lean · F:nat-sumrangeif-congr-lt
      2 dependencies 2 dependents
2485. The conditional sum's successor equation proved · kernel-lean · F:nat-sumrangeif-succ
      0 dependencies 0 dependents
2486. The empty conditional sum is zero proved · kernel-lean · F:nat-sumrangeif-zero
      0 dependencies 0 dependents
2487. [generated] kernel theorem Nat.symmetric_group_isGroupOnFn (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-symmetric-group-isgrouponfn
      5 dependencies 0 dependents
2488. every bit at or above a value's own magnitude bound is zero proved · kernel-lean · F:nat-testbit-eq-zero-of-lt
      14 dependencies 1 dependents
2489. each bit of a bitwise AND is the product of the two operand bits proved · kernel-lean · F:nat-testbit-land
      34 dependencies 3 dependents
2490. [generated] kernel theorem Nat.testBit_le_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-testbit-le-one
      5 dependencies 5 dependents
2491. each bit of a bitwise OR is the max of the two operand bits proved · kernel-lean · F:nat-testbit-lor
      29 dependencies 2 dependents
2492. [generated] kernel theorem Nat.testBit_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-testbit-succ
      0 dependencies 2 dependents
2493. a bit of xor m n is the per-bit XOR of the corresponding bits of m and n proved · kernel-lean · F:nat-testbit-xor
      29 dependencies 7 dependents
2494. [generated] kernel theorem Nat.testBit_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-testbit-zero
      0 dependencies 0 dependents
2495. The prime step: multiplying by a prime preserves totient divisibility proved · kernel-lean · F:nat-totient-dvd-totient-mul-prime C:eulers-totient-functionC:prime-number
      4 dependencies 1 dependents
2496. Euler's totient is multiplicative on coprime arguments proved · kernel-lean · F:nat-totient-mul-of-coprime C:eulers-totient-functionC:chinese-remainder-theoremC:multiplicative-function
      16 dependencies 4 dependents
2497. Multiplying a modulus by one of its own divisors scales the totient exactly proved · kernel-lean · F:nat-totient-mul-of-dvd C:eulers-totient-functionC:divisibility
      6 dependencies 3 dependents
2498. Euler's totient at a prime power proved · kernel-lean · F:nat-totient-prime-pow C:eulers-totient-functionC:prime-number
      9 dependencies 0 dependents
2499. [generated] kernel theorem Nat.totient_prime (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-totient-prime
      9 dependencies 1 dependents
2500. [generated] kernel theorem Nat.transposition_injective (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-transposition-injective
      1 dependencies 2 dependents
2501. [generated] kernel theorem Nat.transposition_involutive (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-transposition-involutive
      12 dependencies 1 dependents
2502. [generated] kernel theorem Nat.transposition_maps_into (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-transposition-maps-into
      2 dependencies 2 dependents
2503. [generated] kernel theorem Nat.two_le_succ_or_eq_one (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-two-le-succ-or-eq-one
      2 dependencies 12 dependents
2504. [generated] kernel theorem Nat.union_eq_right_of_subset (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-union-eq-right-of-subset
      0 dependencies 0 dependents
2505. [generated] kernel theorem Nat.valuation_at_two_mul_sq (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-valuation-at-two-mul-sq
      11 dependencies 0 dependents
2506. Nat.wlpo_and_markov_imply_lpo: LPO factors exactly as the weak principle plus Markov's principle proved · kernel-lean · F:nat-wlpo-and-markov-imply-lpo
      2 dependencies 0 dependents
2507. xor is associative proved · kernel-lean · F:nat-xor-assoc
      2 dependencies 1 dependents
2508. xor a b != 0 iff a != b proved · kernel-lean · F:nat-xor-ne-zero-iff
      5 dependencies 1 dependents
2509. xor a (xor a b) = b proved · kernel-lean · F:nat-xor-xor-cancel-left
      4 dependencies 2 dependents
2510. xor(xor(a, b), b) = a proved · kernel-lean · F:nat-xor-xor-cancel-right
      1 dependencies 1 dependents
2511. Nat zero_add proved · kernel-lean · F:nat-zero-add
      0 dependencies 96 dependents
2512. A positive choice from zero is zero proved · kernel-lean · F:nat-zero-choose-succ C:binomial-coefficient
      0 dependencies 4 dependents
2513. [generated] kernel theorem Nat.zero_div (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-zero-div
      0 dependencies 8 dependents
2514. Zero is a lower bound for every natural number proved · kernel-lean · F:nat-zero-le C:order-relation
      0 dependencies 216 dependents
2515. [generated] kernel theorem Nat.zero_lt_of_ne_zero (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-zero-lt-of-ne-zero
      2 dependencies 13 dependents
2516. [generated] kernel theorem Nat.zero_lt_succ (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-zero-lt-succ
      9 dependencies 34 dependents
2517. [generated] kernel theorem Nat.zero_mod (nat prelude, axiom-free, prose not curated) proved · kernel-lean · F:nat-zero-mod
      0 dependencies 6 dependents
2518. Zero is a left absorbing element for multiplication on the naturals proved · kernel-lean · F:nat-zero-mul C:zero
      0 dependencies 55 dependents
2519. a natural number with every bit zero is zero proved · kernel-lean · F:nat-zero-of-testbit-eq-zero
      3 dependencies 1 dependents
2520. The rational 1/(j+1) is antitone in j, the lemma two independent constructive-analysis developments had stopped at proved · kernel-lean · F:natdivsucc-antitone-over-constructed-rationals
      10 dependencies 27 dependents
2521. No integer squares to minus one proved · smt-term-level · F:no-integer-square-is-minus-one
      0 dependencies 0 dependents
2522. No proposition is equivalent to its own negation proved · smt-term-level · F:no-self-negating-proposition C:liar-paradoxC:russells-paradoxC:self-reference +1
      0 dependencies 0 dependents
2523. [generated] kernel theorem not_not_imp (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:not-not-imp
      0 dependencies 0 dependents
2524. [generated] kernel theorem not_not_not_intro (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:not-not-not-intro
      1 dependencies 0 dependents
2525. Two QF_NRA refutation certificates reconstruct to a kernel-checked Lean False over the CONSTRUCTED reals proved · kernel-lean · F:nra-refutations-reconstruct-over-constructed-reals
      0 dependencies 0 dependents
2526. [generated] kernel theorem Or.resolve_right (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:or-resolve-right
      0 dependencies 1 dependents
2527. A reconstructed Farkas refutation holds in every ordered commutative ring, and rests on no axiom proved · kernel-lean · F:ordered-ring-farkas-refutation C:ordered-ringC:linear-programming-duality
      0 dependencies 1 dependents
2528. The ordered-ring interface telescope is byte-identical over Real and over the axiom-free Int development proved · kernel-lean · F:ordered-ring-interface-is-the-same-over-the-axiom-free-integers
      2 dependencies 0 dependents
2529. An order-line schema has exactly two candidate keys, is not in BCNF, and is not in 3NF -- with every subset of the attributes examined proved · search-certificate · F:orders-candidate-keys-and-normal-forms
      1 dependencies 0 dependents
2530. Two implied and two unimplied functional dependencies on a committed order-line schema, each with a replayable certificate proved · search-certificate · F:orders-fd-implication-certified
      0 dependencies 2 dependents
2531. The focus-directrix property of the parabola, stated without a square root proved · kernel-lean · F:parabola-focus-directrix-without-square-roots
      15 dependencies 0 dependents
2532. Peirce's law proved · smt-term-level · F:peirce-law C:implicationC:if-then-statementC:intuitionism +1
      0 dependencies 0 dependents
2533. Isometries of the constructed plane: the monoid, translations, rotations and reflections as instances, and a constructive refutation that doubling is one proved · kernel-lean · F:plane-isometries-and-the-doubling-non-example
      23 dependencies 0 dependents
2534. A product over an initial segment collapses to 1 mod p when a fixed-point-free involution pairs each factor with its inverse proved · kernel-lean · F:prodrange-collapses-under-a-fixed-point-free-involution
      20 dependencies 1 dependents
2535. Excluded middle for propositions, as Lean proves it proved · imported-kernel-lean · F:prop-excluded-middle-classical C:excluded-middleC:intuitionism
      0 dependencies 0 dependents
2536. sin^2 + cos^2 = 1 for the angle between two vectors of the constructed plane, proved by algebra with no trigonometry in it proved · kernel-lean · F:pythagorean-identity-for-the-plane-angle
      22 dependencies 0 dependents
2537. QF_NIA single-variable polynomial UNSAT carries an independently re-derivable refutation proved · smt-term-level · F:qf-nia-univariate-unsat-is-certified
      0 dependencies 0 dependents
2538. Negation exchanges the two quantifiers proved · smt-term-level · F:quantifier-negation-duality C:quantified-negationC:universal-quantifierC:existential-quantifier +1
      0 dependencies 0 dependents
2539. The two-colour Rado number of x - y = z is 5, as a kernel theorem proved · kernel-lean · F:rado-r2-schur-two C:rado-number
      0 dependencies 0 dependents
2540. The four-colour Rado number of 5(x-y) = 3z is 625 computed · search-certificate · F:rado-r4-a5-b3 C:rado-number
      0 dependencies 0 dependents
2541. The four-colour Rado number of 5(x-y) = 4z is 741 computed · search-certificate · F:rado-r4-a5-b4 C:rado-number
      0 dependencies 0 dependents
2542. The two-colour Ramsey number R(3,3) is 6, as a kernel theorem proved · kernel-lean · F:ramsey-r33-six
      0 dependencies 0 dependents
2543. [generated] kernel theorem Rat.abs_add (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-add
      5 dependencies 0 dependents
2544. [generated] kernel theorem Rat.abs_le_of_le_of_neg_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-le-of-le-of-neg-le
      3 dependencies 0 dependents
2545. [generated] kernel theorem Rat.abs_mul (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-mul
      19 dependencies 0 dependents
2546. [generated] kernel theorem Rat.abs_neg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-neg
      5 dependencies 1 dependents
2547. [generated] kernel theorem Rat.abs_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-nonneg
      1 dependencies 0 dependents
2548. [generated] kernel theorem Rat.abs_sub_comm (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-sub-comm
      2 dependencies 0 dependents
2549. [generated] kernel theorem Rat.abs_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-abs-zero
      5 dependencies 0 dependents
2550. Addition on the rationals is associative proved · kernel-lean · F:rat-add-assoc C:associativity
      10 dependencies 84 dependents
2551. Addition on the rationals is commutative proved · kernel-lean · F:rat-add-comm C:commutativity
      5 dependencies 84 dependents
2552. The cross-multiplication numerator of a rational sum proved · kernel-lean · F:rat-add-cross
      3 dependencies 5 dependents
2553. Addition preserves the rational order in both arguments proved · kernel-lean · F:rat-add-le-add C:order-relation
      9 dependencies 106 dependents
2554. Rational order: adding a le and a strict lt gives a strict lt proved · kernel-lean · F:rat-add-lt-add-of-le-of-lt
      10 dependencies 3 dependents
2555. The arithmetic core of Gaussian elimination's clearing step over the rationals proved · kernel-lean · F:rat-add-neg-div-mul-cancel
      6 dependencies 1 dependents
2556. Rational addition renormalises and negation is an additive inverse proved · kernel-lean · F:rat-add-neg-inverse C:rational-arithmetic
      1 dependencies 1 dependents
2557. Every rational has an additive inverse proved · kernel-lean · F:rat-add-neg C:additive-inverse
      8 dependencies 36 dependents
2558. The sum of two nonnegative rationals is nonnegative proved · kernel-lean · F:rat-add-nonneg
      2 dependencies 13 dependents
2559. [generated] kernel theorem Rat.add_sub_cancel_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-add-sub-cancel-left
      4 dependencies 2 dependents
2560. Zero is a right identity for rational addition proved · kernel-lean · F:rat-add-zero C:identity-element
      6 dependencies 90 dependents
2561. [generated] kernel theorem Rat.bernoulli_harmonic_bound (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-bernoulli-harmonic-bound
      11 dependencies 1 dependents
2562. [generated] kernel theorem Rat.bernoulli_law_of_large_numbers (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-bernoulli-law-of-large-numbers
      20 dependencies 0 dependents
2563. [generated] kernel theorem Rat.bernoulli (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-bernoulli
      11 dependencies 0 dependents
2564. Chebyshev's bound for a sum of uncorrelated identically-distributed trials mentions no variance at all proved · kernel-lean · F:rat-binomial-chebyshev
      2 dependencies 0 dependents
2565. The mean of a sum of m identically-distributed trials is m times the common mean, in two lines over the existing shelf proved · kernel-lean · F:rat-binomial-expectation
      9 dependencies 0 dependents
2566. The variance of a sum of m identically-distributed uncorrelated trials is m*q(1-q), and it is at the rationals because the record has no mulComm proved · kernel-lean · F:rat-binomial-variance
      9 dependencies 1 dependents
2567. [generated] kernel theorem Rat.ble_eq_true_of_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-ble-eq-true-of-le
      3 dependencies 4 dependents
2568. [generated] kernel theorem Rat.ble_refl (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-ble-refl
      2 dependencies 0 dependents
2569. [generated] kernel theorem Rat.ble_total (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-ble-total
      3 dependencies 0 dependents
2570. [generated] kernel theorem Rat.ble_trans (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-ble-trans
      3 dependencies 0 dependents
2571. A two-sided bound survives rational addition proved · kernel-lean · F:rat-bounds-add C:triangle-inequality
      2 dependencies 38 dependents
2572. A two-sided bound survives rational multiplication proved · kernel-lean · F:rat-bounds-mul C:triangle-inequality
      10 dependencies 14 dependents
2573. Negating a two-sided bound keeps it a bound proved · kernel-lean · F:rat-bounds-neg C:triangle-inequality
      2 dependencies 38 dependents
2574. A rational is sandwiched by plus-or-minus the natDivSucc built from |numerator| proved · kernel-lean · F:rat-bounds-num
      15 dependencies 1 dependents
2575. Chebyshev's inequality over the constructed rationals proved · kernel-lean · F:rat-chebyshev-inequality C:chebyshev-inequality
      3 dependencies 3 dependents
2576. Chebyshev's inequality for the sample mean of pairwise-uncorrelated random variables proved · kernel-lean · F:rat-chebyshev-samplemean-uncorrelated
      3 dependencies 1 dependents
2577. Gaussian elimination's clearing sweep leaves the pivot row and everything above it untouched proved · kernel-lean · F:rat-clear-below-off
      1 dependencies 4 dependents
2578. Gaussian elimination's clearing sweep preserves a column already zero from the pivot row down proved · kernel-lean · F:rat-clear-below-preserves-zero
      1 dependencies 2 dependents
2579. A whole pivot step leaves every row above the cursor untouched proved · kernel-lean · F:rat-clear-below-row-swap-off
      4 dependencies 1 dependents
2580. Gaussian elimination's clearing sweep zeroes the pivot column below the pivot row proved · kernel-lean · F:rat-clear-below-zero
      7 dependencies 1 dependents
2581. [generated] kernel theorem Rat.covariance_add_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-covariance-add-right
      4 dependencies 1 dependents
2582. Covariance is symmetric proved · kernel-lean · F:rat-covariance-comm
      2 dependencies 4 dependents
2583. [generated] kernel theorem Rat.covariance_smul_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-covariance-smul-left
      5 dependencies 1 dependents
2584. Covariance squared is bounded by the product of variances, given a positive variance proved · kernel-lean · F:rat-covariance-sq-le-variance-mul-of-pos
      19 dependencies 2 dependents
2585. Covariance squared is bounded by the product of variances when both variances are zero proved · kernel-lean · F:rat-covariance-sq-le-variance-mul-of-zero-zero
      19 dependencies 2 dependents
2586. Cauchy-Schwarz for covariance: cov(X,Y)^2 <= var(X)*var(Y) proved · kernel-lean · F:rat-covariance-sq-le-variance-mul C:cauchy-schwarz-inequality
      8 dependencies 0 dependents
2587. [generated] kernel theorem Rat.covariance_sumVars_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-covariance-sumvars-left
      11 dependencies 2 dependents
2588. [generated] kernel theorem Rat.covariance_sumVars (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-covariance-sumvars
      3 dependencies 0 dependents
2589. Cramer's rule pins down the first unknown of a nonsingular 2x2 rational linear system proved · kernel-lean · F:rat-cramer-two-unique-x C:cramers-rule
      10 dependencies 0 dependents
2590. [generated] kernel theorem Rat.cramer_two_unique_y (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-cramer-two-unique-y
      10 dependencies 0 dependents
2591. [generated] kernel theorem Rat.cramer2_solves (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-cramer2-solves
      10 dependencies 0 dependents
2592. [generated] kernel theorem Rat.cross_of_eq (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-cross-of-eq
      0 dependencies 1 dependents
2593. A rational's denominator is always positive proved · kernel-lean · F:rat-den-pos C:rational-numbers
      0 dependencies 34 dependents
2594. Rat.det_alternating: the determinant vanishes when two distinct rows agree, at symbolic dimension proved · kernel-lean · F:rat-det-alternating
      6 dependencies 2 dependents
2595. Rat.det_col_expansion: cofactor expansion along the first COLUMN, at symbolic dimension proved · kernel-lean · F:rat-det-col-expansion
      7 dependencies 1 dependents
2596. Rat.det_congr: the general-n determinant respects pointwise equality of matrices proved · kernel-lean · F:rat-det-congr-pointwise
      2 dependencies 4 dependents
2597. Rat.det_eq_det2: the general-n determinant agrees with the fixed 2x2 one, symbolically in the matrix proved · kernel-lean · F:rat-det-general-n-eq-det2
      4 dependencies 5 dependents
2598. Rat.det_eq_det3: the general-n determinant agrees with the fixed 3x3 one, symbolically in the matrix proved · kernel-lean · F:rat-det-general-n-eq-det3
      6 dependencies 0 dependents
2599. Rat.det_eval_example: the general-n determinant recursion computes, at a non-symmetric 3x3, to an independently derived value proved · kernel-lean · F:rat-det-general-n-evaluates
      0 dependencies 0 dependents
2600. Rat.det_matId: the determinant of the identity matrix is 1 at a symbolic dimension proved · kernel-lean · F:rat-det-mat-id-general-n
      6 dependencies 1 dependents
2601. The Cauchy-Binet expansion over the constructed rationals: a product matrix determinant as a sum over the function space of index maps proved · kernel-lean · F:rat-det-mat-mul-expand
      14 dependencies 1 dependents
2602. Determinant multiplicativity over the constructed rationals at a symbolic dimension: det (A*B) n = det A n * det B n proved · kernel-lean · F:rat-det-mat-mul
      5 dependencies 0 dependents
2603. Rat.det_minor_col_comm: the two orders of deleting a pair of columns give the same determinant proved · kernel-lean · F:rat-det-minor-col-comm
      2 dependencies 0 dependents
2604. Rat.det_row_expansion: cofactor expansion along a GENERAL row, at symbolic dimension proved · kernel-lean · F:rat-det-row-expansion
      9 dependencies 1 dependents
2605. The selection lemma's injective half over the constructed rationals proved · kernel-lean · F:rat-det-row-selection-injective
      19 dependencies 1 dependents
2606. The selection lemma over the constructed rationals: reindexing a matrix's rows by a self-map of [0,n) multiplies its determinant by the permutation matrix's proved · kernel-lean · F:rat-det-row-selection
      10 dependencies 1 dependents
2607. Rat.det_transpose: the determinant is invariant under transpose, at symbolic dimension proved · kernel-lean · F:rat-det-transpose
      3 dependencies 0 dependents
2608. [generated] kernel theorem Rat.det2_eq_zero_of_lin_dep (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det2-eq-zero-of-lin-dep
      8 dependencies 0 dependents
2609. The 2x2 identity matrix over ℚ has determinant 1 proved · kernel-lean · F:rat-det2-id C:determinant
      4 dependencies 0 dependents
2610. [generated] kernel theorem Rat.det2_row_add (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det2-row-add
      6 dependencies 0 dependents
2611. [generated] kernel theorem Rat.det2_scale_row (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det2-scale-row
      3 dependencies 0 dependents
2612. Swapping the two rows of a 2x2 determinant negates it proved · kernel-lean · F:rat-det2-swap-rows
      2 dependencies 1 dependents
2613. [generated] kernel theorem Rat.det3_cofactor_row1 (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-cofactor-row1
      0 dependencies 1 dependents
2614. [generated] kernel theorem Rat.det3_example_diagonal (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-example-diagonal
      1 dependencies 0 dependents
2615. [generated] kernel theorem Rat.det3_example_generic (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-example-generic
      1 dependencies 0 dependents
2616. [generated] kernel theorem Rat.det3_example_singular (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-example-singular
      1 dependencies 0 dependents
2617. [generated] kernel theorem Rat.det3_id (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-id
      5 dependencies 0 dependents
2618. [generated] kernel theorem Rat.det3_ofInt (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-ofint
      4 dependencies 3 dependents
2619. [generated] kernel theorem Rat.det3_scale_row (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-det3-scale-row
      3 dependencies 0 dependents
2620. The dot product is additive in its first argument proved · kernel-lean · F:rat-dotn-add-left
      3 dependencies 1 dependents
2621. Cauchy-Schwarz for the finite rational dot product proved · kernel-lean · F:rat-dotn-cauchy-schwarz C:cauchy-schwarz-inequality
      28 dependencies 0 dependents
2622. The rational dot product is symmetric proved · kernel-lean · F:rat-dotn-comm C:inner-product
      2 dependencies 1 dependents
2623. The dot product of a vector with itself is nonnegative proved · kernel-lean · F:rat-dotn-self-nonneg
      2 dependencies 1 dependents
2624. The dot product is homogeneous (scalar multiplication) in its first argument proved · kernel-lean · F:rat-dotn-smul-left
      3 dependencies 1 dependents
2625. [generated] kernel theorem Rat.dotN_succ (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-dotn-succ
      0 dependencies 1 dependents
2626. [generated] kernel theorem Rat.dotN_two (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-dotn-two
      3 dependencies 0 dependents
2627. [generated] kernel theorem Rat.dotN_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-dotn-zero
      0 dependencies 1 dependents
2628. Two zero rows in a row pass the echelon step test proved · kernel-lean · F:rat-echelon-step-ok-both-cols
      3 dependencies 1 dependents
2629. A leading entry that moved strictly right passes the echelon step test proved · kernel-lean · F:rat-echelon-step-ok-of-lt
      2 dependencies 3 dependents
2630. Cross-multiplication equality determines rational equality proved · kernel-lean · F:rat-eq-of-cross
      10 dependencies 14 dependents
2631. The decided rational zero test implies the propositional one: isZeroB x = true entails x = 0 proved · kernel-lean · F:rat-eq-zero-of-is-zero-b
      2 dependencies 0 dependents
2632. A rational with numerator zero is zero proved · kernel-lean · F:rat-eq-zero-of-num-zero
      3 dependencies 4 dependents
2633. [generated] kernel theorem Rat.eta (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-eta
      2 dependencies 1 dependents
2634. [generated] kernel theorem Rat.expectation_add (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-add
      3 dependencies 6 dependents
2635. [generated] kernel theorem Rat.expectation_const (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-const
      2 dependencies 2 dependents
2636. [generated] kernel theorem Rat.expectation_indicator_le_one (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-indicator-le-one
      5 dependencies 0 dependents
2637. [generated] kernel theorem Rat.expectation_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-le
      2 dependencies 1 dependents
2638. The expectation of a nonnegative random variable is nonnegative proved · kernel-lean · F:rat-expectation-nonneg
      2 dependencies 2 dependents
2639. [generated] kernel theorem Rat.expectation_smul (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-smul
      3 dependencies 5 dependents
2640. The one cross term the fourth power of a sum does not kill proved · kernel-lean · F:rat-expectation-sq-sumvars-mul-sq
      16 dependencies 1 dependents
2641. [generated] kernel theorem Rat.expectation_sumVars (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-expectation-sumvars
      8 dependencies 3 dependents
2642. [generated] kernel theorem Rat.ext (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-ext
      4 dependencies 1 dependents
2643. The rationals are a constructive field whose apartness is tight proved · kernel-lean · F:rat-field-setoid-tight
      3 dependencies 1 dependents
2644. The squared-means AM-GM over the constructed rationals, proved by a producer-found sum-of-squares certificate proved · kernel-lean · F:rat-four-mul-le-sq-add
      14 dependencies 0 dependents
2645. The fourth-moment tail bound is the concentration bound this carrier can state; Hoeffding is blocked on the joint law, not on the exponential proved · kernel-lean · F:rat-fourth-moment-inequality
      4 dependencies 1 dependents
2646. The fourth central moment of a sum of four-wise uncorrelated centred variables proved · kernel-lean · F:rat-fourth-moment-sumvars-le
      28 dependencies 1 dependents
2647. A 1/m^2 concentration tail, the first Hoeffding-class rate this carrier can state proved · kernel-lean · F:rat-fourth-moment-tail-sumvars
      8 dependencies 0 dependents
2648. [generated] kernel theorem Rat.gcd_one_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-gcd-one-right
      2 dependencies 0 dependents
2649. [generated] kernel theorem Rat.indicator_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-indicator-le
      3 dependencies 1 dependents
2650. [generated] kernel theorem Rat.indicator_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-indicator-nonneg
      3 dependencies 0 dependents
2651. [generated] kernel theorem Rat.int_le_natAbs (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-le-natabs
      0 dependencies 1 dependents
2652. Integer order cancels a common positive multiplier on the right proved · kernel-lean · F:rat-int-le-of-mul-le-mul-right
      7 dependencies 28 dependents
2653. [generated] kernel theorem Rat.int_le_or_lt (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-le-or-lt
      4 dependencies 1 dependents
2654. [generated] kernel theorem Rat.int_lt_of_mul_lt_mul_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-lt-of-mul-lt-mul-right
      6 dependencies 7 dependents
2655. Integer order is preserved by multiplying by a natural on the right proved · kernel-lean · F:rat-int-mul-le-mul-right
      3 dependencies 18 dependents
2656. [generated] kernel theorem Rat.int_mul_lt_mul_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-mul-lt-mul-right
      5 dependencies 7 dependents
2657. Integer multiplication cancels a positive natural factor proved · kernel-lean · F:rat-int-mul-right-cancel
      8 dependencies 12 dependents
2658. [generated] kernel theorem Rat.int_neg_natAbs_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-neg-natabs-le
      0 dependencies 0 dependents
2659. [generated] kernel theorem Rat.int_nonneg_of_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-nonneg-of-nonneg
      2 dependencies 5 dependents
2660. [generated] kernel theorem Rat.int_of_nat_pos (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-of-nat-pos
      0 dependencies 1 dependents
2661. [generated] kernel theorem Rat.int_one_le_of_pos (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-one-le-of-pos
      5 dependencies 1 dependents
2662. [generated] kernel theorem Rat.int_pos_of_pos (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-int-pos-of-pos
      2 dependencies 2 dependents
2663. Right distributivity of integer multiplication over addition proved · kernel-lean · F:rat-int-right-distrib
      3 dependencies 7 dependents
2664. A natural coerced into the integers is nonnegative proved · kernel-lean · F:rat-int-zero-le-of-nat
      1 dependencies 18 dependents
2665. Zero times an integer is zero proved · kernel-lean · F:rat-int-zero-mul
      2 dependencies 21 dependents
2666. Rational inversion reverses order on positives proved · kernel-lean · F:rat-inv-le-of-pos-le
      9 dependencies 1 dependents
2667. The reciprocal of 1/(n+1) is n+1 proved · kernel-lean · F:rat-inv-natdivsucc
      10 dependencies 4 dependents
2668. The reciprocal of a positive rational is positive proved · kernel-lean · F:rat-inv-pos
      8 dependencies 6 dependents
2669. [generated] kernel theorem Rat.inv_sub_inv (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-inv-sub-inv
      5 dependencies 0 dependents
2670. [generated] kernel theorem Rat.inv2_bottom_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-inv2-bottom-left
      6 dependencies 1 dependents
2671. [generated] kernel theorem Rat.inv2_bottom_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-inv2-bottom-right
      7 dependencies 1 dependents
2672. [generated] kernel theorem Rat.inv2_top_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-inv2-top-left
      7 dependencies 1 dependents
2673. [generated] kernel theorem Rat.inv2_top_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-inv2-top-right
      6 dependencies 1 dependents
2674. The row-echelon Bool predicate is exactly the adjacent-pair condition proved · kernel-lean · F:rat-is-echelon-of-pairs
      4 dependencies 2 dependents
2675. The decided pivot-column test and the pivot-row search are the same scan proved · kernel-lean · F:rat-is-pivot-col-b-eq-ble
      0 dependencies 3 dependents
2676. Rat.laplaceSummand_rowI: one function on the square is BOTH cofactor parametrisations proved · kernel-lean · F:rat-laplace-summand-row-i
      3 dependencies 1 dependents
2677. [generated] kernel theorem Rat.le_abs_self (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-abs-self
      1 dependencies 2 dependents
2678. Rational order is antisymmetric proved · kernel-lean · F:rat-le-antisymm
      2 dependencies 11 dependents
2679. [generated] kernel theorem Rat.le_max_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-max-left
      2 dependencies 12 dependents
2680. [generated] kernel theorem Rat.le_max_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-max-right
      2 dependencies 11 dependents
2681. [generated] kernel theorem Rat.le_min (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-min
      1 dependencies 1 dependents
2682. [generated] kernel theorem Rat.le_of_abs_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-of-abs-le
      2 dependencies 0 dependents
2683. [generated] kernel theorem Rat.le_of_ble_eq_true (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-of-ble-eq-true
      7 dependencies 7 dependents
2684. The Archimedean property of the constructed rationals proved · kernel-lean · F:rat-le-of-le-add-natdivsucc C:archimedean-property
      10 dependencies 9 dependents
2685. A strict rational order implies the non-strict one proved · kernel-lean · F:rat-le-of-lt
      1 dependencies 38 dependents
2686. [generated] kernel theorem Rat.le_of_sub_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-le-of-sub-le
      6 dependencies 32 dependents
2687. Rational order is total: le or the reverse strict order holds proved · kernel-lean · F:rat-le-or-lt
      1 dependencies 17 dependents
2688. The rational order is reflexive proved · kernel-lean · F:rat-le-refl C:order-relation
      1 dependencies 108 dependents
2689. The rational order is total proved · kernel-lean · F:rat-le-total C:total-order
      1 dependencies 2 dependents
2690. The rational order is transitive proved · kernel-lean · F:rat-le-trans C:transitivity
      6 dependencies 102 dependents
2691. The leading-index scan reads nothing but its own row proved · kernel-lean · F:rat-leading-index-congr-row
      1 dependencies 1 dependents
2692. A zero row's leading index is the column count proved · kernel-lean · F:rat-leading-index-eq-cols-of-zero-row
      3 dependencies 1 dependents
2693. A row zero left of a nonzero entry has that entry's column as its leading index proved · kernel-lean · F:rat-leading-index-eq-of-first-nonzero
      3 dependencies 5 dependents
2694. A pivot column recovers its own index from the row the search found proved · kernel-lean · F:rat-leading-index-pivot-row-of-col
      2 dependencies 2 dependents
2695. In echelon form a nonzero row's leading index beats every row above it proved · kernel-lean · F:rat-leading-index-strict-below
      6 dependencies 1 dependents
2696. Multiplication distributes over addition on the rationals proved · kernel-lean · F:rat-left-distrib C:distributivity
      11 dependencies 51 dependents
2697. The abstract linear combination over the rationals IS the concrete rational finite sum, definitionally proved · kernel-lean · F:rat-lincomb-eq-sumrange
      2 dependencies 0 dependents
2698. The strict rational order is irreflexive proved · kernel-lean · F:rat-lt-irrefl
      1 dependencies 14 dependents
2699. Passing the echelon step test with the lower row inside the width forces a strict increase proved · kernel-lean · F:rat-lt-of-echelon-step-ok
      5 dependencies 1 dependents
2700. Rational order: le followed by lt gives lt proved · kernel-lean · F:rat-lt-of-le-of-lt
      7 dependencies 7 dependents
2701. Rational order: strict lt followed by le gives strict lt proved · kernel-lean · F:rat-lt-of-lt-of-le
      7 dependencies 8 dependents
2702. [generated] kernel theorem Rat.lt_of_not_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-lt-of-not-le
      3 dependencies 0 dependents
2703. [generated] kernel theorem Rat.lt_of_sq_lt (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-lt-of-sq-lt
      6 dependencies 1 dependents
2704. [generated] kernel theorem Rat.lt_trans (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-lt-trans
      6 dependencies 0 dependents
2705. The rational order is trichotomous proved · kernel-lean · F:rat-lt-trichotomy C:total-order
      2 dependencies 5 dependents
2706. Markov's inequality for the constructed rationals proved · kernel-lean · F:rat-markov-constructed
      2 dependencies 2 dependents
2707. [generated] kernel theorem Rat.markov_inequality (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-markov-inequality
      3 dependencies 1 dependents
2708. Rat.matSkip_comm: deleting index b then a hits the same injection as deleting a then succ b, when a <= b proved · kernel-lean · F:rat-mat-skip-comm
      1 dependencies 2 dependents
2709. The 2x2 adjugate inverse over the constructed rationals satisfies A^-1 * A = I at every entry, stated through the general matMul/matId encoding proved · kernel-lean · F:rat-matinv2-matmul-top-left
      6 dependencies 1 dependents
2710. Matrix multiplication over the constructed rationals is associative at symbolic dimension, stated pointwise proved · kernel-lean · F:rat-matmul-assoc
      5 dependencies 1 dependents
2711. The rational identity matrix is a left unit for matrix multiplication, on its range proved · kernel-lean · F:rat-matmul-id-left
      3 dependencies 1 dependents
2712. The rational identity matrix is a right unit for matrix multiplication, on its range proved · kernel-lean · F:rat-matmul-id-right
      3 dependencies 1 dependents
2713. The 2x2 adjugate inverse over the constructed rationals satisfies A * A^-1 = I at every entry -- the direction matrix.rs never proved -- stated through the general matMul/matId encoding proved · kernel-lean · F:rat-matmul-matinv2-top-left
      10 dependencies 0 dependents
2714. Matrix transpose over the constructed rationals reverses a product at symbolic dimension, stated pointwise -- (AB)^T = B^T A^T proved · kernel-lean · F:rat-mattranspose-mul
      2 dependencies 0 dependents
2715. Matrix transpose over the constructed rationals is an involution at symbolic dimension, stated pointwise proved · kernel-lean · F:rat-mattranspose-transpose
      0 dependencies 0 dependents
2716. [generated] kernel theorem Rat.max_cases (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-max-cases
      7 dependencies 5 dependents
2717. [generated] kernel theorem Rat.max_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-max-le
      1 dependencies 11 dependents
2718. [generated] kernel theorem Rat.min_cases (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-min-cases
      7 dependencies 6 dependents
2719. [generated] kernel theorem Rat.min_le_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-min-le-left
      2 dependencies 3 dependents
2720. [generated] kernel theorem Rat.min_le_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-min-le-right
      2 dependencies 3 dependents
2721. [generated] kernel theorem Rat.mk_congr (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mk-congr
      0 dependencies 1 dependents
2722. Three-variable AM-GM over the constructed rationals, whose certificate FORCES a denominator-clearing scale proved · kernel-lean · F:rat-mul-add-le-sq-add-sq-three
      21 dependencies 0 dependents
2723. [generated] kernel theorem Rat.mul_adj2_bottom_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-adj2-bottom-left
      3 dependencies 1 dependents
2724. [generated] kernel theorem Rat.mul_adj2_bottom_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-adj2-bottom-right
      3 dependencies 2 dependents
2725. [generated] kernel theorem Rat.mul_adj2_top_left (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-adj2-top-left
      1 dependencies 2 dependents
2726. [generated] kernel theorem Rat.mul_adj2_top_right (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-adj2-top-right
      3 dependencies 1 dependents
2727. Multiplication on the rationals is associative proved · kernel-lean · F:rat-mul-assoc C:associativity
      7 dependencies 39 dependents
2728. Multiplication on the rationals is commutative proved · kernel-lean · F:rat-mul-comm C:commutativity
      5 dependencies 90 dependents
2729. The cross-multiplication numerator of a rational product proved · kernel-lean · F:rat-mul-cross
      3 dependencies 9 dependents
2730. [generated] kernel theorem Rat.mul_eq_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-eq-zero
      10 dependencies 0 dependents
2731. Every nonzero rational has a multiplicative inverse proved · kernel-lean · F:rat-mul-inv-cancel-of-ne-zero C:multiplicative-inverse
      3 dependencies 12 dependents
2732. A negative rational times its inverse is one proved · kernel-lean · F:rat-mul-inv-cancel-of-neg
      16 dependencies 1 dependents
2733. A positive rational times its inverse is one proved · kernel-lean · F:rat-mul-inv-cancel
      13 dependencies 14 dependents
2734. A rational quotient minus one equals the difference over the divisor proved · kernel-lean · F:rat-mul-inv-sub-one
      2 dependencies 1 dependents
2735. Rational order is preserved by a nonnegative left multiplier proved · kernel-lean · F:rat-mul-le-mul-of-nonneg-left
      11 dependencies 20 dependents
2736. Rational order is preserved by a nonnegative right multiplier proved · kernel-lean · F:rat-mul-le-mul-of-nonneg-right
      2 dependencies 11 dependents
2737. [generated] kernel theorem Rat.mul_left_cancel_of_ne_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-left-cancel-of-ne-zero
      4 dependencies 2 dependents
2738. Multiplication distributes over rational negation on the right proved · kernel-lean · F:rat-mul-neg
      4 dependencies 40 dependents
2739. [generated] kernel theorem Rat.mul_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-mul-nonneg
      9 dependencies 7 dependents
2740. One is a right identity for rational multiplication proved · kernel-lean · F:rat-mul-one C:identity-element
      4 dependencies 46 dependents
2741. The product of two positive rationals is positive proved · kernel-lean · F:rat-mul-pos
      11 dependencies 5 dependents
2742. Rational multiplication renormalises proved · kernel-lean · F:rat-mul-renormalises C:rational-arithmetic
      1 dependencies 2 dependents
2743. A difference of rational products splits along each factor's error proved · kernel-lean · F:rat-mul-sub-mul C:distributivity
      7 dependencies 6 dependents
2744. A scalar factors out of a rational sumRange proved · kernel-lean · F:rat-mul-sumrange C:summation
      2 dependencies 9 dependents
2745. Zero absorbs rational multiplication proved · kernel-lean · F:rat-mul-zero C:zero
      7 dependencies 29 dependents
2746. [generated] kernel theorem Rat.nat_abs_mul_of_nat (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-nat-abs-mul-of-nat
      1 dependencies 1 dependents
2747. [generated] kernel theorem Rat.nat_div_cross (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-nat-div-cross
      3 dependencies 1 dependents
2748. [generated] kernel theorem Rat.nat_dvd_antisymm_pos (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-nat-dvd-antisymm-pos
      3 dependencies 1 dependents
2749. [generated] kernel theorem Rat.nat_gauss (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-nat-gauss
      15 dependencies 1 dependents
2750. Bishop's natural sampling indices are closed under composition proved · kernel-lean · F:rat-nat-index-compose C:index-arithmetic
      5 dependencies 5 dependents
2751. A commuted-index algebraic identity used to re-index natural sums proved · kernel-lean · F:rat-nat-index-symm
      3 dependencies 3 dependents
2752. [generated] kernel theorem Rat.nat_mul_right_cancel (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-nat-mul-right-cancel
      2 dependencies 1 dependents
2753. Rational division by a fixed successor denominator adds numerators proved · kernel-lean · F:rat-natdivsucc-add C:rational-arithmetic
      7 dependencies 124 dependents
2754. [generated] kernel theorem Rat.natDivSucc_antitone (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-natdivsucc-antitone
      10 dependencies 22 dependents
2755. A doubled natDivSucc numerator and denominator halve to the same value proved · kernel-lean · F:rat-natdivsucc-halve C:rational-arithmetic
      5 dependencies 46 dependents
2756. natDivSucc is monotone in its numerator proved · kernel-lean · F:rat-natdivsucc-le-add-left
      5 dependencies 36 dependents
2757. 1/(n+1) is at most 1 proved · kernel-lean · F:rat-natdivsucc-le-one
      5 dependencies 5 dependents
2758. A deeper Bishop sampling index reads back to a coarser bound proved · kernel-lean · F:rat-natdivsucc-le-scaled C:rational-arithmetic
      7 dependencies 24 dependents
2759. natDivSucc built from a positive rational's denominator undershoots it proved · kernel-lean · F:rat-natdivsucc-lt-of-pos
      13 dependencies 3 dependents
2760. Scaling a natDivSucc bound by a whole number stays a single natDivSucc proved · kernel-lean · F:rat-natdivsucc-mul C:rational-arithmetic
      6 dependencies 58 dependents
2761. natDivSucc is positive when its numerator is at least one proved · kernel-lean · F:rat-natdivsucc-pos
      9 dependencies 17 dependents
2762. Scaling numerator and denominator of natDivSucc by the same successor leaves it unchanged proved · kernel-lean · F:rat-natdivsucc-scale
      5 dependencies 48 dependents
2763. A rational the decided test calls nonzero is nonzero propositionally proved · kernel-lean · F:rat-ne-zero-of-is-zero-b-false
      0 dependencies 0 dependents
2764. A rational's negation added on the left cancels it proved · kernel-lean · F:rat-neg-add-cancel
      2 dependencies 19 dependents
2765. Negation distributes over rational addition proved · kernel-lean · F:rat-neg-add
      5 dependencies 17 dependents
2766. If two rationals sum to zero, the first is the negation of the second proved · kernel-lean · F:rat-neg-eq-of-add-eq-zero
      4 dependencies 5 dependents
2767. [generated] kernel theorem Rat.neg_le_abs (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-neg-le-abs
      1 dependencies 2 dependents
2768. Negation reverses the rational order proved · kernel-lean · F:rat-neg-le-neg C:order-relation
      6 dependencies 68 dependents
2769. [generated] kernel theorem Rat.neg_le_of_abs_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-neg-le-of-abs-le
      4 dependencies 0 dependents
2770. [generated] kernel theorem Rat.neg_mul_le_of_bounds (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-neg-mul-le-of-bounds
      11 dependencies 1 dependents
2771. Multiplication distributes over rational negation on the left proved · kernel-lean · F:rat-neg-mul
      2 dependencies 26 dependents
2772. Double negation cancels on the rationals proved · kernel-lean · F:rat-neg-neg C:additive-inverse
      2 dependencies 35 dependents
2773. Negating a nonnegative rational gives a nonpositive one proved · kernel-lean · F:rat-neg-nonpos-of-nonneg C:non-negativity
      2 dependencies 16 dependents
2774. Negating a rational difference swaps its operands proved · kernel-lean · F:rat-neg-sub C:additive-inverse
      3 dependencies 48 dependents
2775. The negation of zero is zero, for the constructed rationals proved · kernel-lean · F:rat-neg-zero
      2 dependencies 17 dependents
2776. A rational with nonnegative numerator is nonnegative proved · kernel-lean · F:rat-nonneg-of-int-nonneg
      2 dependencies 17 dependents
2777. Adding two normalized rationals equals normalizing the cross-multiplied sum proved · kernel-lean · F:rat-normalize-add-normalize
      7 dependencies 13 dependents
2778. Rat.normalize respects cross-multiplication equality proved · kernel-lean · F:rat-normalize-congr C:congruence
      6 dependencies 22 dependents
2779. A normalized rational's cross-multiplication identity against its inputs proved · kernel-lean · F:rat-normalize-cross
      5 dependencies 37 dependents
2780. Multiplying two normalized rationals equals normalizing the cross-multiplied product proved · kernel-lean · F:rat-normalize-mul-normalize
      6 dependencies 18 dependents
2781. The rational smart constructor normalises proved · kernel-lean · F:rat-normalize-reduces C:rational-numbers
      1 dependencies 2 dependents
2782. [generated] kernel theorem Rat.not_zero_le_neg_of_nat (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-not-zero-le-neg-of-nat
      1 dependencies 1 dependents
2783. A rational matrix with no rows has every column free proved · kernel-lean · F:rat-nullity-zero-rows
      2 dependencies 0 dependents
2784. [generated] kernel theorem Rat.of_nat_inj (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-of-nat-inj
      0 dependencies 1 dependents
2785. Coercing integers into the rationals is additive proved · kernel-lean · F:rat-ofint-add
      3 dependencies 2 dependents
2786. Coercing integers into the rationals is multiplicative proved · kernel-lean · F:rat-ofint-mul
      2 dependencies 2 dependents
2787. Coercing integers into the rationals commutes with negation proved · kernel-lean · F:rat-ofint-neg
      0 dependencies 2 dependents
2788. [generated] kernel theorem Rat.one_ne_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-one-ne-zero
      2 dependencies 1 dependents
2789. The echelon predicate can be read back as the adjacent-pair condition proved · kernel-lean · F:rat-pairs-of-is-echelon
      9 dependencies 0 dependents
2790. The pivot-row scan answers the first row with the given leading index proved · kernel-lean · F:rat-pivot-row-of-col-eq-of-first
      12 dependencies 0 dependents
2791. A pivot column of a rational matrix is led by a row that exists proved · kernel-lean · F:rat-pivot-row-of-col-lt-rows
      2 dependencies 3 dependents
2792. An exhausted pivot scan means the column is zero at every row it passed proved · kernel-lean · F:rat-pivot-search-column-zero
      5 dependencies 1 dependents
2793. A pivot found in range is at or below where the search started proved · kernel-lean · F:rat-pivot-search-ge-start
      4 dependencies 1 dependents
2794. Gaussian elimination's pivot search never returns an out-of-range row proved · kernel-lean · F:rat-pivot-search-le-rows
      0 dependencies 4 dependents
2795. A pivot found in range by Gaussian elimination is nonzero proved · kernel-lean · F:rat-pivot-search-ne-zero
      1 dependencies 4 dependents
2796. Row-echelon form implies the pivot section proved · kernel-lean · F:rat-pivot-section-of-is-echelon
      4 dependencies 3 dependents
2797. Evaluating the pointwise sum of two rational polynomials adds their values proved · kernel-lean · F:rat-polyeval-add C:polynomial-evaluation
      3 dependencies 0 dependents
2798. [generated] kernel theorem Rat.polyEval_deg1 (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-polyeval-deg1
      5 dependencies 1 dependents
2799. [generated] kernel theorem Rat.polyEval_smul (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-polyeval-smul
      3 dependencies 0 dependents
2800. Rational polynomial evaluation unfolds one degree at a time proved · kernel-lean · F:rat-polyeval-succ C:polynomial-evaluation
      0 dependencies 1 dependents
2801. [generated] kernel theorem Rat.polyEval_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-polyeval-zero
      0 dependencies 1 dependents
2802. [generated] kernel theorem Rat.pow_add (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-add
      3 dependencies 1 dependents
2803. [generated] kernel theorem Rat.pow_natDivSucc_two (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-natdivsucc-two
      5 dependencies 1 dependents
2804. [generated] kernel theorem Rat.pow_one (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-one
      3 dependencies 1 dependents
2805. [generated] kernel theorem Rat.pow_sub_add (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-sub-add
      2 dependencies 0 dependents
2806. [generated] kernel theorem Rat.pow_succ (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-succ
      0 dependencies 3 dependents
2807. [generated] kernel theorem Rat.pow_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-pow-zero
      0 dependencies 1 dependents
2808. [generated] kernel theorem Rat.prob_complement (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-prob-complement
      2 dependencies 0 dependents
2809. [generated] kernel theorem Rat.prob_le_one (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-prob-le-one
      12 dependencies 0 dependents
2810. The column-form rank of a rational matrix is at most its column count proved · kernel-lean · F:rat-rank-cols-le-cols
      2 dependencies 1 dependents
2811. The column rank is at most the row rank, with no hypothesis proved · kernel-lean · F:rat-rank-cols-le-rank
      5 dependencies 0 dependents
2812. The row rank and the column rank of a rational matrix agree, given one property of the echelon form proved · kernel-lean · F:rat-rank-eq-rank-cols-of-pivot-section
      7 dependencies 4 dependents
2813. Row rank equals column rank over the rationals proved · kernel-lean · F:rat-rank-eq-rank-cols
      2 dependencies 2 dependents
2814. The rank of a rational matrix is at most its column count, given one property of the echelon form proved · kernel-lean · F:rat-rank-le-cols-of-pivot-section
      2 dependencies 1 dependents
2815. The rank of a rational matrix is at most its column count proved · kernel-lean · F:rat-rank-le-cols
      3 dependencies 0 dependents
2816. The rank of a rational matrix is at most its row count proved · kernel-lean · F:rat-rank-le-rows
      1 dependencies 1 dependents
2817. Rank-nullity in the row form over the rationals, given one property of the echelon form proved · kernel-lean · F:rat-rank-nullity-rows-of-pivot-section
      2 dependencies 1 dependents
2818. Rank-nullity over the rationals, in the row form proved · kernel-lean · F:rat-rank-nullity-rows
      3 dependencies 0 dependents
2819. Rank-nullity over the rationals, in column form proved · kernel-lean · F:rat-rank-nullity
      2 dependencies 4 dependents
2820. A rational matrix with no columns has rank zero proved · kernel-lean · F:rat-rank-zero-cols
      3 dependencies 0 dependents
2821. [generated] kernel theorem Rat.rat_isField (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-rat-isfield
      10 dependencies 1 dependents
2822. [generated] kernel theorem Rat.rat_isOrderedField (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-rat-isorderedfield
      4 dependencies 0 dependents
2823. [generated] kernel theorem Rat.reduced (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-reduced
      0 dependencies 3 dependents
2824. Rational multiplication distributes over addition on the right proved · kernel-lean · F:rat-right-distrib C:distributivity
      2 dependencies 26 dependents
2825. Adding a multiple of one row to another is inverted by adding the negated multiple, provided the rows differ proved · kernel-lean · F:rat-row-add-mul-inverse
      5 dependencies 0 dependents
2826. Gaussian elimination lands in row-echelon form proved · kernel-lean · F:rat-row-echelon-is-echelon
      26 dependencies 1 dependents
2827. Scaling a row of a rational matrix by a nonzero factor is inverted by scaling by its reciprocal proved · kernel-lean · F:rat-row-scale-inverse
      5 dependencies 0 dependents
2828. Swapping two rows of a rational matrix twice restores it, with no side condition proved · kernel-lean · F:rat-row-swap-involutive
      1 dependencies 0 dependents
2829. The pivot swap preserves a column already zero from the pivot row down proved · kernel-lean · F:rat-row-swap-preserves-zero-range
      4 dependencies 1 dependents
2830. Rat.normalize reconstructs a rational from its own numerator and denominator proved · kernel-lean · F:rat-self-normalize C:rational-numbers
      3 dependencies 20 dependents
2831. A rational square is never negative proved · kernel-lean · F:rat-sq-nonneg C:non-negativity
      9 dependencies 15 dependents
2832. [generated] kernel theorem Rat.sq_sub_sq (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sq-sub-sq
      3 dependencies 0 dependents
2833. The error of a rational sum is the sum of the errors proved · kernel-lean · F:rat-sub-add-add C:subtraction
      3 dependencies 14 dependents
2834. Rational subtraction telescopes proved · kernel-lean · F:rat-sub-add-sub C:telescoping-sum
      3 dependencies 42 dependents
2835. Rearranging a rational order inequality across subtraction proved · kernel-lean · F:rat-sub-le-of-le
      6 dependencies 29 dependents
2836. [generated] kernel theorem Rat.sub_max_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sub-max-le
      8 dependencies 4 dependents
2837. [generated] kernel theorem Rat.sub_min_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sub-min-le
      6 dependencies 2 dependents
2838. Right distributivity of rational multiplication over subtraction proved · kernel-lean · F:rat-sub-mul
      4 dependencies 10 dependents
2839. Subtracting two negations flips the subtraction proved · kernel-lean · F:rat-sub-neg-sub
      2 dependencies 3 dependents
2840. A rational minus itself is zero proved · kernel-lean · F:rat-sub-self
      1 dependencies 15 dependents
2841. Rat.sumRange_head_of_tail_zero: a finite sum whose tail vanishes equals its first summand proved · kernel-lean · F:rat-sum-range-head-of-tail-zero
      2 dependencies 1 dependents
2842. Rat.sumRange_matSkip: summing along the injection that misses j, plus f j, is the whole sum proved · kernel-lean · F:rat-sum-range-mat-skip
      5 dependencies 1 dependents
2843. Rat.sumRange_peel_head: a finite sum is its first summand plus the shifted sum of the rest proved · kernel-lean · F:rat-sum-range-peel-head
      2 dependencies 2 dependents
2844. Summing over a range distributes over pointwise addition of rational sequences proved · kernel-lean · F:rat-sumrange-add C:summation
      2 dependencies 6 dependents
2845. [generated] kernel theorem Rat.sumRange_congr_lt (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-congr-lt
      2 dependencies 7 dependents
2846. sumRange respects pointwise-equal summands proved · kernel-lean · F:rat-sumrange-congr
      0 dependencies 31 dependents
2847. [generated] kernel theorem Rat.sumRange_diagonal (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-diagonal
      9 dependencies 1 dependents
2848. [generated] kernel theorem Rat.sumRange_eq_zero_of_lt (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-eq-zero-of-lt
      3 dependencies 6 dependents
2849. [generated] kernel theorem Rat.sumRange_le (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-le
      4 dependencies 3 dependents
2850. [generated] kernel theorem Rat.sumRange_mul_double (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-mul-double
      3 dependencies 1 dependents
2851. [generated] kernel theorem Rat.sumRange_mul_eq_diag_add_corner (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-mul-eq-diag-add-corner
      2 dependencies 0 dependents
2852. [generated] kernel theorem Rat.sumRange_mul (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-mul
      3 dependencies 1 dependents
2853. [generated] kernel theorem Rat.sumRange_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-nonneg
      4 dependencies 4 dependents
2854. [generated] kernel theorem Rat.sumRange_rect_eq_diag_add_corner (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-rect-eq-diag-add-corner
      6 dependencies 1 dependents
2855. [generated] kernel theorem Rat.sumRange_split (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-split
      2 dependencies 1 dependents
2856. Summing over a range unfolds one step at the top proved · kernel-lean · F:rat-sumrange-succ C:summation
      0 dependencies 3 dependents
2857. [generated] kernel theorem Rat.sumRange_swap (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-swap
      2 dependencies 3 dependents
2858. [generated] kernel theorem Rat.sumRange_zero (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-sumrange-zero
      0 dependencies 3 dependents
2859. [generated] kernel theorem Rat.taylor_deg1 (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-taylor-deg1
      4 dependencies 0 dependents
2860. Two-variable AM-GM over the constructed rationals, proved by a producer-found sum-of-squares certificate proved · kernel-lean · F:rat-two-mul-le-sq-add-sq
      14 dependencies 0 dependents
2861. [generated] kernel theorem Rat.uniform_is_distribution (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-uniform-is-distribution
      10 dependencies 0 dependents
2862. Rat.unskip_matSkip: unskip p is a left inverse of the injection matSkip p, unconditionally proved · kernel-lean · F:rat-unskip-mat-skip
      0 dependencies 0 dependents
2863. [generated] kernel theorem Rat.variance_add_eq (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-add-eq
      8 dependencies 2 dependents
2864. [generated] kernel theorem Rat.variance_add_of_uncorrelated (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-add-of-uncorrelated
      2 dependencies 1 dependents
2865. [generated] kernel theorem Rat.variance_eq (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-eq
      13 dependencies 3 dependents
2866. [generated] kernel theorem Rat.variance_indicator_le_quarter (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-indicator-le-quarter
      16 dependencies 1 dependents
2867. [generated] kernel theorem Rat.variance_indicator (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-indicator
      6 dependencies 1 dependents
2868. Variance of a rational-valued random variable is nonnegative proved · kernel-lean · F:rat-variance-nonneg C:variance
      2 dependencies 3 dependents
2869. [generated] kernel theorem Rat.variance_sampleMean_uncorrelated (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-samplemean-uncorrelated
      2 dependencies 0 dependents
2870. [generated] kernel theorem Rat.variance_scaled_add_nonneg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-scaled-add-nonneg
      4 dependencies 2 dependents
2871. [generated] kernel theorem Rat.variance_scaled_mean (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-scaled-mean
      1 dependencies 2 dependents
2872. [generated] kernel theorem Rat.variance_smul (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-smul
      6 dependencies 2 dependents
2873. [generated] kernel theorem Rat.variance_sumVars (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-variance-sumvars
      14 dependencies 3 dependents
2874. The weak law of large numbers over the constructed rationals proved · kernel-lean · F:rat-weak-law-of-large-numbers C:law-of-large-numbers
      1 dependencies 1 dependents
2875. Zero is a left identity for rational addition proved · kernel-lean · F:rat-zero-add C:identity-element
      2 dependencies 38 dependents
2876. [generated] kernel theorem Rat.zero_le_max_neg (rat prelude, axiom-free, prose not curated) proved · kernel-lean · F:rat-zero-le-max-neg
      6 dependencies 2 dependents
2877. A natural-indexed rational division is never negative proved · kernel-lean · F:rat-zero-le-natdivsucc C:non-negativity
      7 dependencies 135 dependents
2878. Zero is strictly less than one, for the constructed rationals proved · kernel-lean · F:rat-zero-lt-one
      1 dependencies 11 dependents
2879. ℚ is a field: Rat.inv is proved to invert, at zero trusted declarations proved · kernel-lean · F:rationals-are-a-field-axiom-free C:rational-numbersC:ordered-field
      0 dependencies 1 dependents
2880. The 30 AxReal axioms are satisfiable: a Bishop setoid over the constructed rationals models all 22 laws at zero trusted declarations proved · kernel-lean · F:real-axioms-modelled-by-constructed-setoid C:real-numbersC:cauchy-sequence
      3 dependencies 4 dependents
2881. The constructed reals have a multiplicative inverse whose modulus is an explicit natural, and it is a function on the reals rather than on representatives proved · kernel-lean · F:real-inverse-is-built-and-well-defined C:real-numbersC:constructive-analysis
      1 dependencies 0 dependents
2882. No function on all of the constructed reals is a multiplicative inverse, and the modulus that would make one possible cannot be extracted from positivity proved · kernel-lean · F:real-inverse-is-partial-and-its-modulus-is-data C:real-numbersC:constructive-analysis
      1 dependencies 1 dependents
2883. The constructed reals carry max, min and a total absolute value, built with no index shift and no decision procedure proved · kernel-lean · F:real-lattice-is-constructed-axiom-free C:real-numbersC:constructive-analysisC:order-theory
      1 dependencies 0 dependents
2884. The binary propositional resolution rule is sound proved · smt-term-level · F:resolution-rule-sound C:inference-ruleC:soundnessC:satisfiability +1
      0 dependencies 0 dependents
2885. Cauchy-Schwarz in R^n at SYMBOLIC dimension, in its unsquared norm form proved · kernel-lean · F:rn-cauchy-schwarz-symbolic-dimension
      3 dependencies 1 dependents
2886. The Euclidean plane is the n = 2 case of R^n, not a parallel construction proved · kernel-lean · F:rn-cpoint-agreement
      1 dependencies 0 dependents
2887. R^n is a constructive metric space, for every n at once proved · kernel-lean · F:rn-is-a-metric-space-at-every-dimension
      3 dependencies 1 dependents
2888. Five rows of a 102-row ICU night roster are an irreducible infeasible subsystem proved · search-certificate · F:roster-icu-night-iis
      0 dependencies 0 dependents
2889. A five-constraint critical chain against a delivery deadline, refuted in the Lean kernel proved · kernel-lean · F:schedule-critical-chain-infeasible
      0 dependencies 1 dependents
2890. Five rows of a 60-row project schedule are an irreducible infeasible subsystem proved · search-certificate · F:schedule-deadline-iis
      1 dependencies 0 dependents
2891. No shipped reconstruction route BUILDS the AxReal axiom package: the trusted surface is declared but never reached proved · kernel-lean · F:shipped-front-door-reaches-no-real-axiom
      1 dependencies 0 dependents
2892. The shipped LRA/SOS front door reconstructs over the constructed reals, and the refutation it returns rests on zero carrier axioms proved · kernel-lean · F:shipped-front-door-refutes-over-constructed-reals C:real-numbersC:linear-programming-duality
      2 dependencies 1 dependents
2893. The optimal sorting network on 3 channels has exactly 3 comparators proved · smt-clausal · F:sorting-network-optimal-size-n3 C:sorting-algorithm
      0 dependencies 0 dependents
2894. The optimal sorting network on 4 channels has exactly 5 comparators proved · smt-clausal · F:sorting-network-optimal-size-n4 C:sorting-algorithm
      0 dependencies 0 dependents
2895. The optimal sorting network on 5 channels has exactly 9 comparators proved · smt-clausal · F:sorting-network-optimal-size-n5 C:sorting-algorithm
      0 dependencies 0 dependents
2896. The optimal sorting network on 6 channels has exactly 12 comparators proved · smt-clausal · F:sorting-network-optimal-size-n6 C:sorting-algorithm
      0 dependencies 0 dependents
2897. The sum of squared binomial coefficients is the central binomial coefficient proved · cas-certificate · F:squared-binomial-row-sum-central
      0 dependencies 0 dependents
2898. [generated] kernel theorem axeyum.string.2.all_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-all-append
      3 dependencies 1 dependents
2899. [generated] kernel theorem axeyum.string.2.all_iff_All (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-all-iff-all
      4 dependencies 0 dependents
2900. [generated] kernel theorem axeyum.string.2.all_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-all-nil
      0 dependencies 1 dependents
2901. [generated] kernel theorem axeyum.string.2.all_of_isPrefix (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-all-of-isprefix
      2 dependencies 0 dependents
2902. [generated] kernel theorem axeyum.string.2.append_assoc (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-append-assoc
      0 dependencies 3 dependents
2903. [generated] kernel theorem axeyum.string.2.append_eq_foldr (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-append-eq-foldr
      0 dependencies 0 dependents
2904. [generated] kernel theorem axeyum.string.2.append_left_cancel (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-append-left-cancel
      0 dependencies 0 dependents
2905. [generated] kernel theorem axeyum.string.2.append_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-append-nil
      0 dependencies 6 dependents
2906. [generated] kernel theorem axeyum.string.2.at_cons_zero (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-at-cons-zero
      0 dependencies 0 dependents
2907. [generated] kernel theorem axeyum.string.2.at_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-at-nil
      0 dependencies 0 dependents
2908. [generated] kernel theorem axeyum.string.2.char_beq_eq_true_of_eq (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-char-beq-eq-true-of-eq
      1 dependencies 0 dependents
2909. [generated] kernel theorem axeyum.string.2.char_beq_refl (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-char-beq-refl
      0 dependencies 3 dependents
2910. [generated] kernel theorem axeyum.string.2.char_eq_of_beq_eq_true (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-char-eq-of-beq-eq-true
      0 dependencies 2 dependents
2911. [generated] kernel theorem axeyum.string.2.cons_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-cons-append
      0 dependencies 0 dependents
2912. [generated] kernel theorem axeyum.string.2.contains_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-nil
      1 dependencies 0 dependents
2913. [generated] kernel theorem axeyum.string.2.contains_of_containsBool_eq_true (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-of-containsbool-eq-true
      2 dependencies 0 dependents
2914. [generated] kernel theorem axeyum.string.2.contains_of_isPrefix (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-of-isprefix
      1 dependencies 1 dependents
2915. [generated] kernel theorem axeyum.string.2.contains_of_isSuffix (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-of-issuffix
      1 dependencies 0 dependents
2916. [generated] kernel theorem axeyum.string.2.contains_refl (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-refl
      2 dependencies 0 dependents
2917. [generated] kernel theorem axeyum.string.2.contains_substr (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-contains-substr
      2 dependencies 0 dependents
2918. [generated] kernel theorem axeyum.string.2.drop_succ_cons (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-drop-succ-cons
      0 dependencies 0 dependents
2919. [generated] kernel theorem axeyum.string.2.drop_succ_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-drop-succ-nil
      0 dependencies 0 dependents
2920. [generated] kernel theorem axeyum.string.2.drop_zero (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-drop-zero
      0 dependencies 0 dependents
2921. [generated] kernel theorem axeyum.string.2.foldr_cons (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-foldr-cons
      0 dependencies 0 dependents
2922. [generated] kernel theorem axeyum.string.2.foldr_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-foldr-nil
      0 dependencies 0 dependents
2923. [generated] kernel theorem axeyum.string.2.isPrefix_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-append
      0 dependencies 0 dependents
2924. [generated] kernel theorem axeyum.string.2.isPrefix_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-nil
      1 dependencies 1 dependents
2925. [generated] kernel theorem axeyum.string.2.isPrefix_of_isPrefixBool_eq_true (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-of-isprefixbool-eq-true
      2 dependencies 2 dependents
2926. [generated] kernel theorem axeyum.string.2.isPrefix_refl (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-refl
      1 dependencies 0 dependents
2927. [generated] kernel theorem axeyum.string.2.isPrefix_take (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-take
      1 dependencies 0 dependents
2928. [generated] kernel theorem axeyum.string.2.isPrefix_trans (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefix-trans
      1 dependencies 0 dependents
2929. [generated] kernel theorem axeyum.string.2.isPrefixBool_eq_true_of_isPrefix (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefixbool-eq-true-of-isprefix
      1 dependencies 1 dependents
2930. [generated] kernel theorem axeyum.string.2.isPrefixBool_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-isprefixbool-nil
      0 dependencies 0 dependents
2931. [generated] kernel theorem axeyum.string.2.isSuffix_drop (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-drop
      1 dependencies 0 dependents
2932. [generated] kernel theorem axeyum.string.2.isSuffix_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-nil
      1 dependencies 0 dependents
2933. [generated] kernel theorem axeyum.string.2.isSuffix_of_isSuffixBool_eq_true (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-of-issuffixbool-eq-true
      2 dependencies 0 dependents
2934. [generated] kernel theorem axeyum.string.2.isSuffix_refl (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-refl
      1 dependencies 0 dependents
2935. [generated] kernel theorem axeyum.string.2.isSuffix_reverse_mp (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-reverse-mp
      1 dependencies 1 dependents
2936. [generated] kernel theorem axeyum.string.2.isSuffix_reverse_mpr (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffix-reverse-mpr
      2 dependencies 1 dependents
2937. [generated] kernel theorem axeyum.string.2.isSuffixBool_eq_true_of_isSuffix (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-issuffixbool-eq-true-of-issuffix
      2 dependencies 0 dependents
2938. [generated] kernel theorem axeyum.string.2.length_cons (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-length-cons
      0 dependencies 0 dependents
2939. [generated] kernel theorem axeyum.string.2.length_eq_foldr (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-length-eq-foldr
      0 dependencies 0 dependents
2940. [generated] kernel theorem axeyum.string.2.length_map (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-length-map
      0 dependencies 0 dependents
2941. [generated] kernel theorem axeyum.string.2.length_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-length-nil
      0 dependencies 0 dependents
2942. [generated] kernel theorem axeyum.string.2.map_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-append
      0 dependencies 1 dependents
2943. [generated] kernel theorem axeyum.string.2.map_cons (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-cons
      0 dependencies 0 dependents
2944. [generated] kernel theorem axeyum.string.2.map_id (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-id
      0 dependencies 0 dependents
2945. [generated] kernel theorem axeyum.string.2.map_map (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-map
      0 dependencies 0 dependents
2946. [generated] kernel theorem axeyum.string.2.map_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-nil
      0 dependencies 0 dependents
2947. [generated] kernel theorem axeyum.string.2.map_reverse (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-map-reverse
      1 dependencies 0 dependents
2948. [generated] kernel theorem axeyum.string.2.nil_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-nil-append
      0 dependencies 5 dependents
2949. [generated] kernel theorem axeyum.string.2.reverse_append (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-reverse-append
      2 dependencies 3 dependents
2950. [generated] kernel theorem axeyum.string.2.reverse_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-reverse-nil
      0 dependencies 0 dependents
2951. [generated] kernel theorem axeyum.string.2.reverse_reverse (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-reverse-reverse
      1 dependencies 1 dependents
2952. [generated] kernel theorem axeyum.string.2.str_beq_eq_true_of_eq (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-str-beq-eq-true-of-eq
      1 dependencies 0 dependents
2953. [generated] kernel theorem axeyum.string.2.str_beq_refl (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-str-beq-refl
      1 dependencies 1 dependents
2954. [generated] kernel theorem axeyum.string.2.str_eq_of_beq_eq_true (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-str-eq-of-beq-eq-true
      1 dependencies 0 dependents
2955. [generated] kernel theorem axeyum.string.2.substr_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-substr-nil
      0 dependencies 0 dependents
2956. [generated] kernel theorem axeyum.string.2.substr_zero_len (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-substr-zero-len
      0 dependencies 0 dependents
2957. [generated] kernel theorem axeyum.string.2.substr_zero_zero (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-substr-zero-zero
      0 dependencies 0 dependents
2958. [generated] kernel theorem axeyum.string.2.take_append_drop (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-take-append-drop
      1 dependencies 3 dependents
2959. [generated] kernel theorem axeyum.string.2.take_succ_cons (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-take-succ-cons
      0 dependencies 0 dependents
2960. [generated] kernel theorem axeyum.string.2.take_succ_nil (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-take-succ-nil
      0 dependencies 0 dependents
2961. [generated] kernel theorem axeyum.string.2.take_zero (string prelude, axiom-free, prose not curated) proved · kernel-lean · F:string-take-zero
      0 dependencies 0 dependents
2962. Only one direction of the frame law is an axiom; the converse is free proved · kernel-lean · F:top-frame-converse-distributive-law-is-a-theorem
      0 dependencies 0 dependents
2963. The frame distributive law, as an equivalence, for every frame proved · kernel-lean · F:top-frame-distributive-law-over-countable-joins
      0 dependencies 0 dependents
2964. Density of the rationals IS the covering property of the open-ball frame proved · kernel-lean · F:top-open-ball-frame-covers-the-real-line
      0 dependencies 1 dependents
2965. Two balls with strictly separated brackets share no point of the real line proved · kernel-lean · F:top-separated-balls-share-no-real-point
      1 dependencies 0 dependents
2966. The Tseitin clauses for an AND gate define the gate proved · smt-term-level · F:tseitin-and-gate C:satisfiabilityC:logical-equivalenceC:formal-proof-checking
      0 dependencies 0 dependents
2967. Twin prime conjecture conjectured · route not assigned · F:twin-prime-unbounded C:twin-prime-conjectureC:prime-numberC:prime-gap
      0 dependencies 0 dependents
2968. The k-weighted binomial row sum proved · cas-certificate · F:weighted-binomial-row-sum
      0 dependencies 0 dependents
2969. [generated] kernel theorem WellFounded.fix_eq (logic prelude, axiom-free, prose not curated) proved · kernel-lean · F:wellfounded-fix-eq
      0 dependencies 3 dependents
2970. Wilson's theorem: for prime p, (p-1)! is congruent to -1 modulo p, proved constructively with an empty axiom footprint proved · kernel-lean · F:wilson-theorem-over-constructed-integers
      21 dependencies 0 dependents
2971. Exclusive-or is associative proved · smt-term-level · F:xor-associative C:exclusive-orC:associativity-abstractC:logical-connective
      0 dependencies 0 dependents