diff options
Diffstat (limited to 'plugins/ring/Ring_abstract.v')
| -rw-r--r-- | plugins/ring/Ring_abstract.v | 94 |
1 files changed, 44 insertions, 50 deletions
diff --git a/plugins/ring/Ring_abstract.v b/plugins/ring/Ring_abstract.v index e6e2dda9..a00b7bcd 100644 --- a/plugins/ring/Ring_abstract.v +++ b/plugins/ring/Ring_abstract.v @@ -1,19 +1,15 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(* $Id: Ring_abstract.v 14641 2011-11-06 11:59:10Z herbelin $ *) - Require Import LegacyRing_theory. Require Import Quote. Require Import Ring_normalize. -Unset Boxed Definitions. - Section abstract_semi_rings. Inductive aspolynomial : Type := @@ -141,14 +137,13 @@ Hint Resolve (SR_plus_zero_right2 T). Hint Resolve (SR_mult_one_right T). Hint Resolve (SR_mult_one_right2 T). (*Hint Resolve (SR_plus_reg_right T).*) -Hint Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) +Hint Resolve eq_refl eq_sym eq_trans. Hint Immediate T. Remark iacs_aux_ok : forall (x:A) (s:abstract_sum), iacs_aux x s = Aplus x (interp_acs s). Proof. - simple induction s; simpl in |- *; intros. + simple induction s; simpl; intros. trivial. reflexivity. Qed. @@ -163,8 +158,8 @@ Lemma abstract_varlist_insert_ok : simple induction s. trivial. - simpl in |- *; intros. - elim (varlist_lt l v); simpl in |- *. + simpl; intros. + elim (varlist_lt l v); simpl. eauto. rewrite iacs_aux_ok. rewrite H; auto. @@ -182,13 +177,13 @@ Proof. auto. - simpl in |- *; elim (varlist_lt v v0); simpl in |- *. + simpl; elim (varlist_lt v v0); simpl. repeat rewrite iacs_aux_ok. - rewrite H; simpl in |- *; auto. + rewrite H; simpl; auto. simpl in H0. repeat rewrite iacs_aux_ok. - rewrite H0. simpl in |- *; auto. + rewrite H0. simpl; auto. Qed. Lemma abstract_sum_scalar_ok : @@ -197,9 +192,9 @@ Lemma abstract_sum_scalar_ok : Amult (interp_vl Amult Aone Azero vm l) (interp_acs s). Proof. simple induction s. - simpl in |- *; eauto. + simpl; eauto. - simpl in |- *; intros. + simpl; intros. rewrite iacs_aux_ok. rewrite abstract_varlist_insert_ok. rewrite H. @@ -213,22 +208,22 @@ Lemma abstract_sum_prod_ok : Proof. simple induction x. - intros; simpl in |- *; eauto. + intros; simpl; eauto. destruct y as [| v0 a0]; intros. - simpl in |- *; rewrite H; eauto. + simpl; rewrite H; eauto. - unfold abstract_sum_prod in |- *; fold abstract_sum_prod in |- *. + unfold abstract_sum_prod; fold abstract_sum_prod. rewrite abstract_sum_merge_ok. rewrite abstract_sum_scalar_ok. - rewrite H; simpl in |- *; auto. + rewrite H; simpl; auto. Qed. Theorem aspolynomial_normalize_ok : forall x:aspolynomial, interp_asp x = interp_acs (aspolynomial_normalize x). Proof. - simple induction x; simpl in |- *; intros; trivial. + simple induction x; simpl; intros; trivial. rewrite abstract_sum_merge_ok. rewrite H; rewrite H0; eauto. rewrite abstract_sum_prod_ok. @@ -450,14 +445,13 @@ Hint Resolve (Th_plus_zero_right2 T). Hint Resolve (Th_mult_one_right T). Hint Resolve (Th_mult_one_right2 T). (*Hint Resolve (Th_plus_reg_right T).*) -Hint Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) +Hint Resolve eq_refl eq_sym eq_trans. Hint Immediate T. Lemma isacs_aux_ok : forall (x:A) (s:signed_sum), isacs_aux x s = Aplus x (interp_sacs s). Proof. - simple induction s; simpl in |- *; intros. + simple induction s; simpl; intros. trivial. reflexivity. reflexivity. @@ -466,15 +460,15 @@ Qed. Hint Extern 10 (_ = _ :>A) => rewrite isacs_aux_ok: core. Ltac solve1 v v0 H H0 := - simpl in |- *; elim (varlist_lt v v0); simpl in |- *; rewrite isacs_aux_ok; - [ rewrite H; simpl in |- *; auto | simpl in H0; rewrite H0; auto ]. + simpl; elim (varlist_lt v v0); simpl; rewrite isacs_aux_ok; + [ rewrite H; simpl; auto | simpl in H0; rewrite H0; auto ]. Lemma signed_sum_merge_ok : forall x y:signed_sum, interp_sacs (signed_sum_merge x y) = Aplus (interp_sacs x) (interp_sacs y). simple induction x. - intro; simpl in |- *; auto. + intro; simpl; auto. simple induction y; intros. @@ -482,8 +476,8 @@ Lemma signed_sum_merge_ok : solve1 v v0 H H0. - simpl in |- *; generalize (varlist_eq_prop v v0). - elim (varlist_eq v v0); simpl in |- *. + simpl; generalize (varlist_eq_prop v v0). + elim (varlist_eq v v0); simpl. intro Heq; rewrite (Heq I). rewrite H. @@ -503,8 +497,8 @@ Lemma signed_sum_merge_ok : auto. - simpl in |- *; generalize (varlist_eq_prop v v0). - elim (varlist_eq v v0); simpl in |- *. + simpl; generalize (varlist_eq_prop v v0). + elim (varlist_eq v v0); simpl. intro Heq; rewrite (Heq I). rewrite H. @@ -522,7 +516,7 @@ Lemma signed_sum_merge_ok : Qed. Ltac solve2 l v H := - elim (varlist_lt l v); simpl in |- *; rewrite isacs_aux_ok; + elim (varlist_lt l v); simpl; rewrite isacs_aux_ok; [ auto | rewrite H; auto ]. Lemma plus_varlist_insert_ok : @@ -534,12 +528,12 @@ Proof. simple induction s. trivial. - simpl in |- *; intros. + simpl; intros. solve2 l v H. - simpl in |- *; intros. + simpl; intros. generalize (varlist_eq_prop l v). - elim (varlist_eq l v); simpl in |- *. + elim (varlist_eq l v); simpl. intro Heq; rewrite (Heq I). repeat rewrite isacs_aux_ok. @@ -561,9 +555,9 @@ Proof. simple induction s. trivial. - simpl in |- *; intros. + simpl; intros. generalize (varlist_eq_prop l v). - elim (varlist_eq l v); simpl in |- *. + elim (varlist_eq l v); simpl. intro Heq; rewrite (Heq I). repeat rewrite isacs_aux_ok. @@ -574,10 +568,10 @@ Proof. rewrite (Th_opp_def T). auto. - simpl in |- *; intros. + simpl; intros. solve2 l v H. - simpl in |- *; intros; solve2 l v H. + simpl; intros; solve2 l v H. Qed. @@ -585,9 +579,9 @@ Lemma signed_sum_opp_ok : forall s:signed_sum, interp_sacs (signed_sum_opp s) = Aopp (interp_sacs s). Proof. - simple induction s; simpl in |- *; intros. + simple induction s; simpl; intros. - symmetry in |- *; apply (Th_opp_zero T). + symmetry ; apply (Th_opp_zero T). repeat rewrite isacs_aux_ok. rewrite H. @@ -611,14 +605,14 @@ Proof. simple induction s. trivial. - simpl in |- *; intros. + simpl; intros. rewrite plus_varlist_insert_ok. rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). repeat rewrite isacs_aux_ok. rewrite H. auto. - simpl in |- *; intros. + simpl; intros. rewrite minus_varlist_insert_ok. repeat rewrite isacs_aux_ok. rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). @@ -635,11 +629,11 @@ Lemma minus_sum_scalar_ok : Aopp (Amult (interp_vl Amult Aone Azero vm l) (interp_sacs s)). Proof. - simple induction s; simpl in |- *; intros. + simple induction s; simpl; intros. - rewrite (Th_mult_zero_right T); symmetry in |- *; apply (Th_opp_zero T). + rewrite (Th_mult_zero_right T); symmetry ; apply (Th_opp_zero T). - simpl in |- *; intros. + simpl; intros. rewrite minus_varlist_insert_ok. rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). repeat rewrite isacs_aux_ok. @@ -648,7 +642,7 @@ Proof. rewrite (Th_plus_opp_opp T). reflexivity. - simpl in |- *; intros. + simpl; intros. rewrite plus_varlist_insert_ok. repeat rewrite isacs_aux_ok. rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). @@ -668,16 +662,16 @@ Proof. simple induction x. - simpl in |- *; eauto 1. + simpl; eauto 1. - intros; simpl in |- *. + intros; simpl. rewrite signed_sum_merge_ok. rewrite plus_sum_scalar_ok. repeat rewrite isacs_aux_ok. rewrite H. auto. - intros; simpl in |- *. + intros; simpl. repeat rewrite isacs_aux_ok. rewrite signed_sum_merge_ok. rewrite minus_sum_scalar_ok. @@ -691,7 +685,7 @@ Qed. Theorem apolynomial_normalize_ok : forall p:apolynomial, interp_sacs (apolynomial_normalize p) = interp_ap p. Proof. - simple induction p; simpl in |- *; auto 1. + simple induction p; simpl; auto 1. intros. rewrite signed_sum_merge_ok. rewrite H; rewrite H0; reflexivity. |