Add Euler's criterion and quadratic reciprocity for Zmod - #307
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Factoring.v comes from Andres Erbsen's WIP ZmodSqrt branch (andres-erbsen/stdlib@5bd6d0db, theories/ZArith/Factoring.v). It defines the p-adic valuation `val`, the prime-power factorization `ppfactor`, the sorted prime factorization `factor` and Euler's `totient` on positive numbers, all computable, and proves the fundamental theorem of arithmetic together with the induction principles `factor_ind` and `ppfactor_ind`. The two unfinished scratch lemmas of the WIP (`in_ppfactor`, `val_ext`) are left out. Zdivisibility.v gains `Z.coprime_mul_l_iff`, `Z.coprime_mul_r_iff`, `Z.coprime_pow_l_iff`, `Z.coprime_pow_r_iff` and `Z.coprime_prime_prime`, which the same WIP used. The generic list lemmas it needed go to their natural homes: `Forall_repeat` and `repeat_inj` in List.v; `Sorted_repeat`, `StronglySorted_app`, `NoDup_StronglySorted`, `HdRel_map`, `Sorted_map`, `StronglySorted_Permutation_unique` and `Sorted_Permutation_unique` in Sorted.v (which now requires Permutation); `fold_right_Permutation` in Permutation.v. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01YJbiRBkJtM7x6PvYV6hmEm
QuadraticReciprocity.v is the fully proved part of Andres Erbsen's WIP ZmodSqrt branch (andres-erbsen/stdlib@5bd6d0db, theories/Zmod/ZmodSqrt.v): the Chinese-remainder decomposition of `Zmod.elements` and `Zstar.elements` for coprime moduli, the product of all units modulo a prime (`prod_elements_prime`), Euler's criterion for `Zstar` and `Zmod`, and the law of quadratic reciprocity for odd primes (`quadratic_reciprocity'`). The square-root algorithms and their partially proved correctness lemmas are not included. The generic lemmas it needs move to their homes: `filter_filter`, `negb_existsb`, `existsb_as_filter` and the `list_prod` lemmas in List.v, `NoDup_list_prod` in Finite.v, `Permutation_partition` in Permutation.v, `Z.mod_prod_mod_factor_l/r` in Zdiv.v and `Z.coprime_comm` in Zdivisibility.v. The WIP's `Permutation_filter` already exists on master as a `Proper` instance. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01YJbiRBkJtM7x6PvYV6hmEm
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Closing: the author of the development asked for this to go to mit-plv/fiat-crypto instead of stdlib for now. Sorry for the noise. |
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Depends on #306 (stacked on its branch); review the last commit only.
Adds
theories/Zmod/QuadraticReciprocity.v: the Chinese-remainder decomposition ofZmod.elements/Zstar.elementsfor coprime moduli, the product of all units modulo a prime, Euler's criterion forZstarandZmod, and the law of quadratic reciprocity for odd primes (quadratic_reciprocity':Z.smodulo (q^((p-1)/2)) p * Z.smodulo (p^((q-1)/2)) q = (-1)^((q-1)/2*((p-1)/2))). This is the fully proved part of Andres Erbsen's WIP ZmodSqrt branch (andres-erbsen/stdlib@5bd6d0db); the square-root algorithms are left for a later PR.The generic lemmas it needs go to
List.v,Finite.v,Permutation.v,Zdiv.vandZdivisibility.v.Written by Claude Fable 5.1.
🤖 Generated with Claude Code
https://claude.ai/code/session_01YJbiRBkJtM7x6PvYV6hmEm