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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import ZArith_base.
Require Import Zpow_def.
Require Import BinInt.
Require Import BinNat.
Require Import Ndiv_def Zdiv_def.
Require Import Setoid.
Require Export Ring2.
Require Export Ring2_polynom.
Import List.
Set Implicit Arguments.
(* An object to return when an expression is not recognized as a constant *)
Definition NotConstant := false.
(** Z is a ring and a setoid*)
Lemma Zsth : Setoid_Theory Z (@eq Z).
constructor;red;intros;subst;trivial.
Qed.
Definition Zr : Ring Z.
apply (@Build_Ring Z Z0 (Zpos xH) Zplus Zmult Zminus Zopp (@eq Z));
try apply Zsth; try (red; red; intros; rewrite H; reflexivity).
exact Zplus_comm. exact Zplus_assoc.
exact Zmult_1_l. exact Zmult_1_r. exact Zmult_assoc.
exact Zmult_plus_distr_l. intros; apply Zmult_plus_distr_r. trivial. exact Zminus_diag.
Defined.
(** Two generic morphisms from Z to (abrbitrary) rings, *)
(**second one is more convenient for proofs but they are ext. equal*)
Section ZMORPHISM.
Variable R : Type.
Variable Rr: Ring R.
Existing Instance Rr.
Ltac rrefl := gen_reflexivity Rr.
(*
Print HintDb typeclass_instances.
Print Scopes.
*)
Fixpoint gen_phiPOS1 (p:positive) : R :=
match p with
| xH => 1
| xO p => (1 + 1) * (gen_phiPOS1 p)
| xI p => 1 + ((1 + 1) * (gen_phiPOS1 p))
end.
Fixpoint gen_phiPOS (p:positive) : R :=
match p with
| xH => 1
| xO xH => (1 + 1)
| xO p => (1 + 1) * (gen_phiPOS p)
| xI xH => 1 + (1 +1)
| xI p => 1 + ((1 + 1) * (gen_phiPOS p))
end.
Definition gen_phiZ1 z :=
match z with
| Zpos p => gen_phiPOS1 p
| Z0 => 0
| Zneg p => -(gen_phiPOS1 p)
end.
Definition gen_phiZ z :=
match z with
| Zpos p => gen_phiPOS p
| Z0 => 0
| Zneg p => -(gen_phiPOS p)
end.
Notation "[ x ]" := (gen_phiZ x).
Definition get_signZ z :=
match z with
| Zneg p => Some (Zpos p)
| _ => None
end.
Ltac norm := gen_ring_rewrite.
Ltac add_push := gen_add_push.
Ltac rsimpl := simpl; set_ring_notations.
Lemma same_gen : forall x, gen_phiPOS1 x == gen_phiPOS x.
Proof.
induction x;rsimpl.
ring_rewrite IHx. destruct x;simpl;norm.
ring_rewrite IHx;destruct x;simpl;norm.
gen_reflexivity.
Qed.
Lemma ARgen_phiPOS_Psucc : forall x,
gen_phiPOS1 (Psucc x) == 1 + (gen_phiPOS1 x).
Proof.
induction x;rsimpl;norm.
ring_rewrite IHx ;norm.
add_push 1;gen_reflexivity.
Qed.
Lemma ARgen_phiPOS_add : forall x y,
gen_phiPOS1 (x + y) == (gen_phiPOS1 x) + (gen_phiPOS1 y).
Proof.
induction x;destruct y;simpl;norm.
ring_rewrite Pplus_carry_spec.
ring_rewrite ARgen_phiPOS_Psucc.
ring_rewrite IHx;norm.
add_push (gen_phiPOS1 y);add_push 1;gen_reflexivity.
ring_rewrite IHx;norm;add_push (gen_phiPOS1 y);gen_reflexivity.
ring_rewrite ARgen_phiPOS_Psucc;norm;add_push 1;gen_reflexivity.
ring_rewrite IHx;norm;add_push(gen_phiPOS1 y); add_push 1;gen_reflexivity.
ring_rewrite IHx;norm;add_push(gen_phiPOS1 y);gen_reflexivity.
add_push 1;gen_reflexivity.
ring_rewrite ARgen_phiPOS_Psucc;norm;add_push 1;gen_reflexivity.
Qed.
Lemma ARgen_phiPOS_mult :
forall x y, gen_phiPOS1 (x * y) == gen_phiPOS1 x * gen_phiPOS1 y.
Proof.
induction x;intros;simpl;norm.
ring_rewrite ARgen_phiPOS_add;simpl;ring_rewrite IHx;norm.
ring_rewrite IHx;gen_reflexivity.
Qed.
(*morphisms are extensionaly equal*)
Lemma same_genZ : forall x, [x] == gen_phiZ1 x.
Proof.
destruct x;rsimpl; try ring_rewrite same_gen; gen_reflexivity.
Qed.
Lemma gen_Zeqb_ok : forall x y,
Zeq_bool x y = true -> [x] == [y].
Proof.
intros x y H.
assert (H1 := Zeq_bool_eq x y H);unfold IDphi in H1.
ring_rewrite H1;gen_reflexivity.
Qed.
Lemma gen_phiZ1_add_pos_neg : forall x y,
gen_phiZ1
match (x ?= y)%positive with
| Eq => Z0
| Lt => Zneg (y - x)
| Gt => Zpos (x - y)
end
== gen_phiPOS1 x + -gen_phiPOS1 y.
Proof.
intros x y.
assert (H0 := Pminus_mask_Gt x y). unfold Pos.gt in H0.
assert (H1 := Pminus_mask_Gt y x). unfold Pos.gt in H1.
rewrite Pos.compare_antisym in H1.
destruct (Pos.compare_spec x y) as [H|H|H].
subst. ring_rewrite ring_opp_def;gen_reflexivity.
destruct H1 as [h [Heq1 [Heq2 Hor]]];trivial.
unfold Pminus; ring_rewrite Heq1;rewrite <- Heq2.
ring_rewrite ARgen_phiPOS_add;simpl;norm.
ring_rewrite ring_opp_def;norm.
destruct H0 as [h [Heq1 [Heq2 Hor]]];trivial.
unfold Pminus; rewrite Heq1;rewrite <- Heq2.
ring_rewrite ARgen_phiPOS_add;simpl;norm.
set_ring_notations. add_push (gen_phiPOS1 h). ring_rewrite ring_opp_def ; norm.
Qed.
Lemma match_compOpp : forall x (B:Type) (be bl bg:B),
match CompOpp x with Eq => be | Lt => bl | Gt => bg end
= match x with Eq => be | Lt => bg | Gt => bl end.
Proof. destruct x;simpl;intros;trivial. Qed.
Lemma gen_phiZ_add : forall x y, [x + y] == [x] + [y].
Proof.
intros x y; repeat ring_rewrite same_genZ; generalize x y;clear x y.
induction x;destruct y;simpl;norm.
apply ARgen_phiPOS_add.
apply gen_phiZ1_add_pos_neg.
replace Eq with (CompOpp Eq);trivial.
rewrite Pos.compare_antisym;simpl.
ring_rewrite match_compOpp.
ring_rewrite ring_add_comm.
apply gen_phiZ1_add_pos_neg.
ring_rewrite ARgen_phiPOS_add; norm.
Qed.
Lemma gen_phiZ_opp : forall x, [- x] == - [x].
Proof.
intros x. repeat ring_rewrite same_genZ. generalize x ;clear x.
induction x;simpl;norm.
ring_rewrite ring_opp_opp. gen_reflexivity.
Qed.
Lemma gen_phiZ_mul : forall x y, [x * y] == [x] * [y].
Proof.
intros x y;repeat ring_rewrite same_genZ.
destruct x;destruct y;simpl;norm;
ring_rewrite ARgen_phiPOS_mult;try (norm;fail).
ring_rewrite ring_opp_opp ;gen_reflexivity.
Qed.
Lemma gen_phiZ_ext : forall x y : Z, x = y -> [x] == [y].
Proof. intros;subst;gen_reflexivity. Qed.
(*proof that [.] satisfies morphism specifications*)
Lemma gen_phiZ_morph : @Ring_morphism Z R Zr Rr.
apply (Build_Ring_morphism Zr Rr gen_phiZ);simpl;try gen_reflexivity.
apply gen_phiZ_add. intros. ring_rewrite ring_sub_def.
replace (x-y)%Z with (x + (-y))%Z. ring_rewrite gen_phiZ_add.
ring_rewrite gen_phiZ_opp. gen_reflexivity.
reflexivity.
apply gen_phiZ_mul. apply gen_phiZ_opp. apply gen_phiZ_ext.
Defined.
End ZMORPHISM.
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