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Merge pull request #26 from math-comp/ssrZ
Add the `ssrZ` library
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‎Make‎

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theories/ssrZ.v
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theories/zify_ssreflect.v
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theories/zify_algebra.v
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theories/zify.v

‎README.md‎

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This small library enables the use of the Micromega tactics for goals stated
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with the definitions of the Mathematical Components library by extending the
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zify tactic.
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This small library enables the use of the Micromega arithmetic solvers of Coq
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for goals stated with the definitions of the Mathematical Components library
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by extending the zify tactic.
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## Meta
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```
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## File contents
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- `zify_ssreflect.v`: Z-ification instances for the `coq-mathcomp-ssreflect`
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library
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- `zify_algebra.v`: Z-ification instances for the `coq-mathcomp-algebra`
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library
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- `zify.v`: re-exports all the Z-ification instances
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- `ssrZ.v`: provides a minimal facility for reasoning about `Z` and relating
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`Z` and `int`

‎coq-mathcomp-zify.opam‎

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synopsis: "Micromega tactics for Mathematical Components"
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description: """
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This small library enables the use of the Micromega tactics for goals stated
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with the definitions of the Mathematical Components library by extending the
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zify tactic."""
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This small library enables the use of the Micromega arithmetic solvers of Coq
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for goals stated with the definitions of the Mathematical Components library
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by extending the zify tactic."""
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build: [make "-j%{jobs}%" ]
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install: [make "install"]

‎meta.yml‎

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Micromega tactics for Mathematical Components
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description: |-
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This small library enables the use of the Micromega tactics for goals stated
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with the definitions of the Mathematical Components library by extending the
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zify tactic.
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This small library enables the use of the Micromega arithmetic solvers of Coq
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for goals stated with the definitions of the Mathematical Components library
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by extending the zify tactic.
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authors:
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- name: Kazuhiko Sakaguchi
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[MathComp](https://math-comp.github.io) 1.12.0 or later
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namespace: mathcomp.zify
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documentation: |-
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## File contents
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- `zify_ssreflect.v`: Z-ification instances for the `coq-mathcomp-ssreflect`
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library
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- `zify_algebra.v`: Z-ification instances for the `coq-mathcomp-algebra`
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library
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- `zify.v`: re-exports all the Z-ification instances
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- `ssrZ.v`: provides a minimal facility for reasoning about `Z` and relating
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`Z` and `int`
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---

‎theories/ssrZ.v‎

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From Coq Require Import ZArith ZifyClasses Zify ZifyInst ZifyBool.
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From Coq Require Export Lia.
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From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.
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From mathcomp Require Import div choice fintype tuple finfun bigop finset prime.
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From mathcomp Require Import order binomial ssralg countalg ssrnum ssrint.
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From mathcomp Require Import zify_ssreflect.
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Set Implicit Arguments.
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Unset Strict Implicit.
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Unset Printing Implicit Defensive.
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Import Order.Theory GRing.Theory Num.Theory.
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Definition int_of_Z (n : Z) : int :=
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match n with
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| Z0 => Posz 0
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| Zpos p => Posz (Pos.to_nat p)
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| Zneg p => Negz (Pos.to_nat p).-1
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end.
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Definition Z_of_int (n : int) : Z :=
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match n with
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| Posz n => Z.of_nat n
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| Negz n' => Z.opp (Z.of_nat (n' + 1))
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end.
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Lemma int_of_ZK : cancel int_of_Z Z_of_int.
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Proof. by case=> //= p; lia. Qed.
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Lemma Z_of_intK : cancel Z_of_int int_of_Z.
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Proof.
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case=> [[|n]|n] //=.
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congr Posz; lia.
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rewrite addnC /=; congr Negz; lia.
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Qed.
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Module ZInstances.
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Implicit Types (m n : Z).
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Fact eqZP : Equality.axiom Z.eqb.
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Proof. by move=> x y; apply: (iffP idP); lia. Qed.
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Canonical Z_eqType := EqType Z (EqMixin eqZP).
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Canonical Z_choiceType := ChoiceType Z (CanChoiceMixin int_of_ZK).
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Canonical Z_countType := CountType Z (CanCountMixin int_of_ZK).
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Definition Z_zmodMixin :=
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ZmodMixin Zplus_assoc Zplus_comm Zplus_0_l Zplus_opp_l.
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Canonical Z_zmodType := ZmodType Z Z_zmodMixin.
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Definition Z_ringMixin :=
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RingMixin
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Zmult_assoc Zmult_1_l Zmult_1_r Zmult_plus_distr_l Zmult_plus_distr_r isT.
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Canonical Z_ringType := RingType Z Z_ringMixin.
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Canonical Z_comRingType := ComRingType Z Zmult_comm.
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Definition unitZ := [qualify a n : Z | (n == Z.pos xH) || (n == Z.neg xH)].
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Definition invZ n := n.
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Fact mulVZ : {in unitZ, left_inverse 1%R invZ *%R}.
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Proof. by move=> n /pred2P[] ->. Qed.
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Fact unitZPl m n : (n * m = 1)%R -> m \is a unitZ.
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Proof. case: m n => [|[m|m|]|[m|m|]] [|n|n] //= []; lia. Qed.
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Fact invZ_out : {in [predC unitZ], invZ =1 id}.
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Proof. exact. Qed.
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Fact idomain_axiomZ m n : (m * n = 0)%R -> (m == 0%R) || (n == 0%R).
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Proof. by case: m n => [|m|m] []. Qed.
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Canonical Z_unitRingType :=
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UnitRingType Z (ComUnitRingMixin mulVZ unitZPl invZ_out).
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Canonical Z_comUnitRing := [comUnitRingType of Z].
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Canonical Z_idomainType := IdomainType Z idomain_axiomZ.
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Canonical Z_countZmodType := [countZmodType of Z].
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Canonical Z_countRingType := [countRingType of Z].
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Canonical Z_countComRingType := [countComRingType of Z].
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Canonical Z_countUnitRingType := [countUnitRingType of Z].
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Canonical Z_countComUnitRingType := [countComUnitRingType of Z].
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Canonical Z_countIdomainType := [countIdomainType of Z].
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Fact leZ_add m n : Z.leb 0 m -> Z.leb 0 n -> Z.leb 0 (m + n). Proof. lia. Qed.
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Fact leZ_mul m n : Z.leb 0 m -> Z.leb 0 n -> Z.leb 0 (m * n). Proof. lia. Qed.
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Fact leZ_anti m : Z.leb 0 m -> Z.leb m 0 -> m = Z0. Proof. lia. Qed.
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Fact subZ_ge0 m n : Z.leb 0 (n - m)%R = Z.leb m n.
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Proof. by rewrite /GRing.add /GRing.opp /=; lia. Qed.
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Fact leZ_total m n : Z.leb m n || Z.leb n m. Proof. lia. Qed.
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Fact normZN m : Z.abs (- m) = Z.abs m. Proof. lia. Qed.
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Fact geZ0_norm m : Z.leb 0 m -> Z.abs m = m. Proof. lia. Qed.
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Fact ltZ_def m n : (Z.ltb m n) = (n != m) && (Z.leb m n).
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Proof. by rewrite eqE /=; lia. Qed.
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Definition Mixin : realLeMixin [idomainType of Z] :=
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RealLeMixin
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leZ_add leZ_mul leZ_anti subZ_ge0 (leZ_total 0) normZN geZ0_norm ltZ_def.
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Canonical Z_porderType := POrderType ring_display Z Mixin.
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Canonical Z_latticeType := LatticeType Z Mixin.
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Canonical Z_distrLatticeType := DistrLatticeType Z Mixin.
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Canonical Z_orderType := OrderType Z leZ_total.
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Canonical Z_numDomainType := NumDomainType Z Mixin.
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Canonical Z_normedZmodType := NormedZmodType Z Z Mixin.
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Canonical Z_realDomainType := [realDomainType of Z].
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Fact Z_of_intE (n : int) : Z_of_int n = (n%:~R)%R.
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Proof.
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have Hnat (m : nat) : Z.of_nat m = (m%:R)%R.
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by elim: m => // m; rewrite Nat2Z.inj_succ -Z.add_1_l mulrS => ->.
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case: n => n; rewrite /intmul /=; first exact: Hnat.
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by congr Z.opp; rewrite Nat2Z.inj_add /= mulrSr Hnat.
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Qed.
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Fact Z_of_int_is_additive : additive Z_of_int.
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Proof. by move=> m n; rewrite !Z_of_intE raddfB. Qed.
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Canonical Z_of_int_additive := Additive Z_of_int_is_additive.
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Fact int_of_Z_is_additive : additive int_of_Z.
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Proof. exact: can2_additive Z_of_intK int_of_ZK. Qed.
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Canonical int_of_Z_additive := Additive int_of_Z_is_additive.
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Fact Z_of_int_is_multiplicative : multiplicative Z_of_int.
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Proof. by split => // n m; rewrite !Z_of_intE rmorphM. Qed.
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Canonical Z_of_int_rmorphism := AddRMorphism Z_of_int_is_multiplicative.
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Fact int_of_Z_is_multiplicative : multiplicative int_of_Z.
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Proof. exact: can2_rmorphism Z_of_intK int_of_ZK. Qed.
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Canonical int_of_Z_rmorphism := AddRMorphism int_of_Z_is_multiplicative.
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End ZInstances.
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Canonical ZInstances.Z_eqType.
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Canonical ZInstances.Z_choiceType.
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Canonical ZInstances.Z_countType.
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Canonical ZInstances.Z_zmodType.
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Canonical ZInstances.Z_ringType.
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Canonical ZInstances.Z_comRingType.
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Canonical ZInstances.Z_unitRingType.
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Canonical ZInstances.Z_comUnitRing.
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Canonical ZInstances.Z_idomainType.
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Canonical ZInstances.Z_countZmodType.
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Canonical ZInstances.Z_countRingType.
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Canonical ZInstances.Z_countComRingType.
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Canonical ZInstances.Z_countUnitRingType.
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Canonical ZInstances.Z_countComUnitRingType.
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Canonical ZInstances.Z_countIdomainType.
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Canonical ZInstances.Z_porderType.
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Canonical ZInstances.Z_latticeType.
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Canonical ZInstances.Z_distrLatticeType.
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Canonical ZInstances.Z_orderType.
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Canonical ZInstances.Z_numDomainType.
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Canonical ZInstances.Z_normedZmodType.
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Canonical ZInstances.Z_realDomainType.
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Canonical ZInstances.Z_of_int_additive.
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Canonical ZInstances.int_of_Z_additive.
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Canonical ZInstances.Z_of_int_rmorphism.
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Canonical ZInstances.int_of_Z_rmorphism.

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