feat: add symplectic matrices have determinant 1 eval problem#283
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feat: add symplectic matrices have determinant 1 eval problem#283kim-em wants to merge 1 commit into
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This PR adds the fact that every symplectic matrix has determinant 1 (§39 of Knill's "Some Fundamental Theorems in Mathematics", listed as an open TODO in `Mathlib/LinearAlgebra/SymplecticGroup.lean`). Mathlib has `Matrix.symplecticGroup` and proves `IsUnit (det A)` via `symplectic_det` but not `det A = 1`; the sign requires the Pfaffian argument and is not formalized in any other proof assistant we found. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds the symplectic-matrix
det = 1identity as a new lean-eval challenge problem — §39 of Oliver Knill's Some Fundamental Theorems in Mathematics.For any commutative ring
Rand2n × 2nsymplectic matrixAoverR(i.e.A * J * Aᵀ = JwithJ = [[0, -I], [I, 0]]),det A = 1. Taking determinants ofAᵀ J A = Jonly forces(det A)² = 1; the sign requires the Pfaffian argument.mathlib has
Matrix.symplecticGroupand provessymplectic_det(IsUnit (det A)) — but the determinant-equals-one claim is an explicit TODO at the head ofMathlib/LinearAlgebra/SymplecticGroup.lean. A search found no formalization of the identity in any other proof assistant.🤖 Prepared with Claude Code