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Better examples about reduction modulo p (#36)
* Better examples about reduction modulo p --------- Co-authored-by: Assia Mahboubi <[email protected]>
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| Original file line number | Diff line number | Diff line change |
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| (*****************************************************************************) | ||
| (* * Trocq *) | ||
| (* _______ * Copyright (C) 2023 Inria & MERCE *) | ||
| (* |__ __| * (Mitsubishi Electric R&D Centre Europe) *) | ||
| (* | |_ __ ___ ___ __ _ * Cyril Cohen <[email protected]> *) | ||
| (* | | '__/ _ \ / __/ _` | * Enzo Crance <[email protected]> *) | ||
| (* | | | | (_) | (_| (_| | * Assia Mahboubi <[email protected]> *) | ||
| (* |_|_| \___/ \___\__, | ************************************************) | ||
| (* | | * This file is distributed under the terms of *) | ||
| (* |_| * GNU Lesser General Public License Version 3 *) | ||
| (* * see LICENSE file for the text of the license *) | ||
| (*****************************************************************************) | ||
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| From Coq Require Import ssreflect. | ||
| From Trocq Require Import Trocq. | ||
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| Set Universe Polymorphism. | ||
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| Declare Scope int_scope. | ||
| Delimit Scope int_scope with int. | ||
| Delimit Scope int_scope with ℤ. | ||
| Local Open Scope int_scope. | ||
| Declare Scope Zmod9_scope. | ||
| Delimit Scope Zmod9_scope with Zmod9. | ||
| Local Open Scope Zmod9_scope. | ||
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| Definition unop_param {X X'} RX {Y Y'} RY | ||
| (f : X -> Y) (g : X' -> Y') := | ||
| forall x x', RX x x' -> RY (f x) (g x'). | ||
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| Definition binop_param {X X'} RX {Y Y'} RY {Z Z'} RZ | ||
| (f : X -> Y -> Z) (g : X' -> Y' -> Z') := | ||
| forall x x', RX x x' -> forall y y', RY y y' -> RZ (f x y) (g x' y'). | ||
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| (*** | ||
| We setup an axiomatic context in order not to develop | ||
| arithmetic modulo in Coq/HoTT. | ||
| **) | ||
| Axiom (int@{i} : Type@{i}) (zero : int) (add : int -> int -> int) | ||
| (mul : int -> int -> int) (one : int) | ||
| (mod3 : int -> int). | ||
| Axiom (addC : forall m n, add m n = add n m). | ||
| Axiom (Zmod9 : Type) (zerop : Zmod9) (addp : Zmod9 -> Zmod9 -> Zmod9) | ||
| (mulp : Zmod9 -> Zmod9 -> Zmod9) (onep : Zmod9). | ||
| Axiom (modp : int -> Zmod9) (reprp : Zmod9 -> int) | ||
| (reprpK : forall x, modp (reprp x) = x) | ||
| (modp3 : Zmod9 -> Zmod9). | ||
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| Definition eqmodp (x y : int) := modp x = modp y. | ||
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| (* for now translations need the support of a global reference: *) | ||
| Definition eq_Zmod9 (x y : Zmod9) := (x = y). | ||
| Arguments eq_Zmod9 /. | ||
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| Notation "0" := zero : int_scope. | ||
| Notation "0" := zerop : Zmod9_scope. | ||
| Notation "1" := one : int_scope. | ||
| Notation "1" := onep : Zmod9_scope. | ||
| Notation "x + y" := (add x%int y%int) : int_scope. | ||
| Notation "x + y" := (addp x%Zmod9 y%Zmod9) : Zmod9_scope. | ||
| Notation "x * y" := (mul x%int y%int) : int_scope. | ||
| Notation "x * y" := (mulp x%Zmod9 y%Zmod9) : Zmod9_scope. | ||
| Notation not A := (A -> Empty). | ||
| Notation "m ^ 2" := (m * m)%int (at level 2) : int_scope. | ||
| Notation "m ^ 2" := (m * m)%Zmod9 (at level 2) : Zmod9_scope. | ||
| Notation "m ³" := (m * m * m)%int (at level 2) : int_scope. | ||
| Notation "m ³" := (m * m * m)%Zmod9 (at level 2) : Zmod9_scope. | ||
| Notation "m % 3" := (mod3 m)%int (at level 2) : int_scope. | ||
| Notation "m % 3" := (modp3 m)%Zmod9 (at level 2) : Zmod9_scope. | ||
| Notation "x ≡ y" := (eqmodp x%int y%int) | ||
| (format "x ≡ y", at level 70) : int_scope. | ||
| Notation "x ≢ y" := (not (eqmodp x%int y%int)) | ||
| (format "x ≢ y", at level 70) : int_scope. | ||
| Notation "x ≠ y" := (not (x = y)). | ||
| Notation "ℤ/9ℤ" := Zmod9. | ||
| Notation ℤ := int. | ||
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| Definition Rp := SplitSurj.toParam (SplitSurj.Build reprpK). | ||
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| Axiom Rzero : Rp zero zerop. | ||
| Axiom Rone : Rp one onep. | ||
| Variable Rmod3 : unop_param Rp Rp mod3 modp3. | ||
| Variable Radd : binop_param Rp Rp Rp add addp. | ||
| Variable Rmul : binop_param Rp Rp Rp mul mulp. | ||
| Variable Reqmodp01 : forall (m : int) (x : Zmod9), Rp m x -> | ||
| forall n y, Rp n y -> Param01.Rel (eqmodp m n) (eq_Zmod9 x y). | ||
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| Trocq Use Rp Rmul Rzero Rone Radd Rmod3 Param10_paths Reqmodp01. | ||
| Trocq Use Param01_sum Param01_Empty Param10_Empty. | ||
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| Lemma flt3_step : forall (m n p : ℤ), | ||
| m * n * p % 3 ≢ 0 -> (m³ + n³)%ℤ ≠ p³%ℤ. | ||
| Proof. | ||
| trocq=> /=. | ||
| Admitted. |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,133 @@ | ||
| (*****************************************************************************) | ||
| (* * Trocq *) | ||
| (* _______ * Copyright (C) 2023 Inria & MERCE *) | ||
| (* |__ __| * (Mitsubishi Electric R&D Centre Europe) *) | ||
| (* | |_ __ ___ ___ __ _ * Cyril Cohen <[email protected]> *) | ||
| (* | | '__/ _ \ / __/ _` | * Enzo Crance <[email protected]> *) | ||
| (* | | | | (_) | (_| (_| | * Assia Mahboubi <[email protected]> *) | ||
| (* |_|_| \___/ \___\__, | ************************************************) | ||
| (* | | * This file is distributed under the terms of *) | ||
| (* |_| * GNU Lesser General Public License Version 3 *) | ||
| (* * see LICENSE file for the text of the license *) | ||
| (*****************************************************************************) | ||
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| From Coq Require Import ssreflect. | ||
| From HoTT Require Import HoTT. | ||
| Require Import Hierarchy. | ||
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| Set Universe Polymorphism. | ||
| Unset Universe Minimization ToSet. | ||
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| Print sum. | ||
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| Inductive sumR | ||
| A A' (AR : A -> A' -> Type) B B' (BR : B -> B' -> Type) : A + B -> A' + B' -> Type := | ||
| | inlR a a' (aR : AR a a') : sumR A A' AR B B' BR (inl a) (inl a') | ||
| | inrR b b' (bR : BR b b') : sumR A A' AR B B' BR (inr b) (inr b'). | ||
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| (* *) | ||
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| Definition sum_map | ||
| (A A' : Type) (AR : Param10.Rel A A') (B B' : Type) (BR : Param10.Rel B B') : | ||
| A + B -> A' + B' := | ||
| fun p => | ||
| match p with | ||
| | inl a => inl (map AR a) | ||
| | inr b => inr (map BR b) | ||
| end. | ||
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| Definition inl_inj {A B} {a1 a2 : A} : | ||
| @inl A B a1 = inl a2 -> a1 = a2 := | ||
| fun e => | ||
| match e in @paths _ _ (inl a1) return _ = a1 with | ||
| | @idpath _ _ => @idpath _ a1 | ||
| end. | ||
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| Definition inr_inj {A B} {b1 b2 : B} : | ||
| @inr A B b1 = inr b2 -> b1 = b2 := | ||
| fun e => | ||
| match e in @paths _ _ (inr b1) return _ = b1 with | ||
| | @idpath _ _ => @idpath _ b1 | ||
| end. | ||
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| Definition inl_inr {A B a b} : @inl A B a = inr b -> False. | ||
| Proof. discriminate. Defined. | ||
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| Definition inr_inl {A B b a} : @inr A B b = inl a -> False. | ||
| Proof. discriminate. Defined. | ||
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| Definition sum_map_in_R | ||
| (A A' : Type) (AR : Param2a0.Rel A A') (B B' : Type) (BR : Param2a0.Rel B B') : | ||
| forall p p', sum_map A A' AR B B' BR p = p' -> sumR A A' AR B B' BR p p'. | ||
| Proof. | ||
| case=> [a|b] [a'|b']; do ?discriminate. | ||
| - by move=> /inl_inj<-; constructor; apply: map_in_R. | ||
| - by move=> /inr_inj<-; constructor; apply: map_in_R. | ||
| Defined. | ||
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| (* *) | ||
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| Definition sum_R_in_map | ||
| (A A' : Type) (AR : Param2b0.Rel A A') (B B' : Type) (BR : Param2b0.Rel B B') : | ||
| forall p p', sumR A A' AR B B' BR p p' -> sum_map A A' AR B B' BR p = p'. | ||
| Proof. | ||
| by move=> _ _ [a a' aR|b b' bR]/=; apply: ap; apply: R_in_map. | ||
| Defined. | ||
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| Definition sum_R_in_mapK | ||
| (A A' : Type) (AR : Param40.Rel A A') (B B' : Type) (BR : Param40.Rel B B') : | ||
| forall p p' (r : sumR A A' AR B B' BR p p'), | ||
| sum_map_in_R A A' AR B B' BR p p' (sum_R_in_map A A' AR B B' BR p p' r) = r. | ||
| Proof. | ||
| move=> _ _ [a a' aR|b b' bR]/=; rewrite /internal_paths_rew. | ||
| Admitted. | ||
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| Definition Map0_sum A A' (AR : Param00.Rel A A') B B' (BR : Param00.Rel B B') : | ||
| Map0.Has (sumR A A' AR B B' BR). | ||
| Proof. constructor. Defined. | ||
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| Definition Map1_sum A A' (AR : Param10.Rel A A') B B' (BR : Param10.Rel B B') : | ||
| Map1.Has (sumR A A' AR B B' BR). | ||
| Proof. constructor. exact (sum_map A A' AR B B' BR). Defined. | ||
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| Definition Map2a_sum A A' (AR : Param2a0.Rel A A') B B' (BR : Param2a0.Rel B B') : | ||
| Map2a.Has (sumR A A' AR B B' BR). | ||
| Proof. | ||
| unshelve econstructor. | ||
| - exact (sum_map A A' AR B B' BR). | ||
| - exact (sum_map_in_R A A' AR B B' BR). | ||
| Defined. | ||
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| Definition Map2b_sum A A' (AR : Param2b0.Rel A A') B B' (BR : Param2b0.Rel B B') : | ||
| Map2b.Has (sumR A A' AR B B' BR). | ||
| Proof. | ||
| unshelve econstructor. | ||
| - exact (sum_map A A' AR B B' BR). | ||
| - exact (sum_R_in_map A A' AR B B' BR). | ||
| Defined. | ||
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| Definition Map3_sum A A' (AR : Param30.Rel A A') B B' (BR : Param30.Rel B B') : | ||
| Map3.Has (sumR A A' AR B B' BR). | ||
| Proof. | ||
| unshelve econstructor. | ||
| - exact (sum_map A A' AR B B' BR). | ||
| - exact (sum_map_in_R A A' AR B B' BR). | ||
| - exact (sum_R_in_map A A' AR B B' BR). | ||
| Defined. | ||
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| Definition Map4_sum A A' (AR : Param40.Rel A A') B B' (BR : Param40.Rel B B') : | ||
| Map4.Has (sumR A A' AR B B' BR). | ||
| Proof. | ||
| unshelve econstructor. | ||
| - exact (sum_map A A' AR B B' BR). | ||
| - exact (sum_map_in_R A A' AR B B' BR). | ||
| - exact (sum_R_in_map A A' AR B B' BR). | ||
| - exact (sum_R_in_mapK A A' AR B B' BR). | ||
| Defined. | ||
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| Definition Param01_sum A A' (AR : Param01.Rel A A') B B' (BR : Param01.Rel B B') : | ||
| Param01.Rel (A + B) (A' + B'). | ||
| exists (sumR A A' AR B B' BR). | ||
| - exact: Map0_sum. | ||
| - admit. | ||
| Admitted. |
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