The rational identity matrix is a right unit for matrix multiplication, on its range

Identifier: F:rat-matmul-id-right
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a matrix A over the constructed rationals (a function Nat -> Nat -> Rat), a dimension n, and indices i and j WITH j < n, the (i, j) entry of A * matId equals A i j. The bound is on the COLUMN index j here, where the left unit law (F:rat-matmul-id-left) bounds the row index i: the summation runs over the shared inner index, so which of the two outer indices has to lie in range depends on which side the identity sits. Stated pointwise; `funext` is absent from this kernel so an Eq between matrices is not available.

Formal statement

theorem Rat.matMul_id_right : ((x0 : ((x0 : AxNat) -> ((x1 : AxNat) -> Rat))) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : AxNat.lt x3 x1) -> Eq.{1} Rat (Rat.matMul x0 Rat.matId x1 x2 x3) (x0 x2 x3))))))

Dependencies

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3 direct dependencies 1 direct dependents

Evidence

kernel-Rat.matMul_id_right

Kind: kernel-term
Status: checked

Supports: Rat.matMul_id_right is admitted by the trusted kernel gate with the type recorded in formal.statement. It shares Rat.sumRange_delta with the left unit law and is the shorter of the two: the delta's hypothesis wants `t != j`, which is exactly the shape Rat.matId_off_diag takes on this side, so no ne_symm is needed and the tail closes with mul_one without a mul_comm.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Rat.matMul_id_right 2>/dev/null | grep -cE '^Rat\.matMul_id_right[[:space:]]'

Evidence notes

Verified BOTH directions on this tree. Real name: prints 1, exit 0. A fabricated name prints 0 and exits 1. Pass ONE name per invocation -- this tool consumes only its FIRST name argument and silently ignores the rest. `--release` is MANDATORY (debug SIGABRTs on a stack overflow). `[[:space:]]` and NOT `\t`.

evaluation-right-unit

Kind: instance-pin
Status: checked

Supports: A * matId agrees with A at every in-range cell, computed rather than inferred from the left unit law: `rat_mat_mul_id_left_needs_its_bound` evaluates BOTH sides -- matId * A and A * matId -- at all four cells of A = [[1,2],[3,4]] with n = 2, and `rat_mat_mul_id_laws_hold_at_a_concrete_instance` applies this theorem with its bound discharged by Nat.lt_succ_self (`Lt 1 2`), deliberately NOT reusing the `Lt 0 2` proof the left law takes, since the two laws bound different indices.

Checker command

test "$(scripts/cargo-serialized.sh test -p axeyum-lean-kernel --lib rat_prelude::rat_prelude_tests::rat_mat_mul_id_laws_hold_at_a_concrete_instance -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Evidence notes

That test also carries its own non-vacuity check: the inferred statement is asserted NOT defeq to the false equation `2 = 3`, so the positive comparison against `2 = 2` can fail. Anchored on the exact `1 passed; 0 failed` count so an unmatched filter cannot read as success.

footprint-Rat.matMul_id_right

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Rat prelude's trusted surface is empty, so no theorem in it can rest on anything asserted.

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free rat

Evidence notes

Re-measured on this tree: `rat: axiom=0 opaque=0 quotient=0 total_trusted=0`, exit 0. Covers Axiom, Opaque AND Quotient.

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane rat-matrix-layer, Rat.matMul_id_right in crates/axeyum-lean-kernel/src/rat_prelude/matrix_n.rs",
  "source": "classical"
}