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diff --git a/plugins/romega/ReflOmegaCore.v b/plugins/romega/ReflOmegaCore.v new file mode 100644 index 00000000..c82abfc8 --- /dev/null +++ b/plugins/romega/ReflOmegaCore.v @@ -0,0 +1,3216 @@ +(* -*- coding: utf-8 -*- *) +(************************************************************************* + + PROJET RNRT Calife - 2001 + Author: Pierre Crégut - France Télécom R&D + Licence du projet : LGPL version 2.1 + + *************************************************************************) + +Require Import List Bool Sumbool EqNat Setoid Ring_theory Decidable ZArith_base. +Delimit Scope Int_scope with I. + +(* Abstract Integers. *) + +Module Type Int. + + Parameter int : Set. + + Parameter zero : int. + Parameter one : int. + Parameter plus : int -> int -> int. + Parameter opp : int -> int. + Parameter minus : int -> int -> int. + Parameter mult : int -> int -> int. + + Notation "0" := zero : Int_scope. + Notation "1" := one : Int_scope. + Infix "+" := plus : Int_scope. + Infix "-" := minus : Int_scope. + Infix "*" := mult : Int_scope. + Notation "- x" := (opp x) : Int_scope. + + Open Scope Int_scope. + + (* First, int is a ring: *) + Axiom ring : @ring_theory int 0 1 plus mult minus opp (@eq int). + + (* int should also be ordered: *) + + Parameter le : int -> int -> Prop. + Parameter lt : int -> int -> Prop. + Parameter ge : int -> int -> Prop. + Parameter gt : int -> int -> Prop. + Notation "x <= y" := (le x y): Int_scope. + Notation "x < y" := (lt x y) : Int_scope. + Notation "x >= y" := (ge x y) : Int_scope. + Notation "x > y" := (gt x y): Int_scope. + Axiom le_lt_iff : forall i j, (i<=j) <-> ~(j<i). + Axiom ge_le_iff : forall i j, (i>=j) <-> (j<=i). + Axiom gt_lt_iff : forall i j, (i>j) <-> (j<i). + + (* Basic properties of this order *) + Axiom lt_trans : forall i j k, i<j -> j<k -> i<k. + Axiom lt_not_eq : forall i j, i<j -> i<>j. + + (* Compatibilities *) + Axiom lt_0_1 : 0<1. + Axiom plus_le_compat : forall i j k l, i<=j -> k<=l -> i+k<=j+l. + Axiom opp_le_compat : forall i j, i<=j -> (-j)<=(-i). + Axiom mult_lt_compat_l : + forall i j k, 0 < k -> i < j -> k*i<k*j. + + (* We should have a way to decide the equality and the order*) + Parameter compare : int -> int -> comparison. + Infix "?=" := compare (at level 70, no associativity) : Int_scope. + Axiom compare_Eq : forall i j, compare i j = Eq <-> i=j. + Axiom compare_Lt : forall i j, compare i j = Lt <-> i<j. + Axiom compare_Gt : forall i j, compare i j = Gt <-> i>j. + + (* Up to here, these requirements could be fulfilled + by any totally ordered ring. Let's now be int-specific: *) + Axiom le_lt_int : forall x y, x<y <-> x<=y+-(1). + + (* Btw, lt_0_1 could be deduced from this last axiom *) + +End Int. + + + +(* Of course, Z is a model for our abstract int *) + +Module Z_as_Int <: Int. + + Open Scope Z_scope. + + Definition int := Z. + Definition zero := 0. + Definition one := 1. + Definition plus := Zplus. + Definition opp := Zopp. + Definition minus := Zminus. + Definition mult := Zmult. + + Lemma ring : @ring_theory int zero one plus mult minus opp (@eq int). + Proof. + constructor. + exact Zplus_0_l. + exact Zplus_comm. + exact Zplus_assoc. + exact Zmult_1_l. + exact Zmult_comm. + exact Zmult_assoc. + exact Zmult_plus_distr_l. + unfold minus, Zminus; auto. + exact Zplus_opp_r. + Qed. + + Definition le := Zle. + Definition lt := Zlt. + Definition ge := Zge. + Definition gt := Zgt. + Lemma le_lt_iff : forall i j, (i<=j) <-> ~(j<i). + Proof. + split; intros. + apply Zle_not_lt; auto. + rewrite <- Zge_iff_le. + apply Znot_lt_ge; auto. + Qed. + Definition ge_le_iff := Zge_iff_le. + Definition gt_lt_iff := Zgt_iff_lt. + + Definition lt_trans := Zlt_trans. + Definition lt_not_eq := Zlt_not_eq. + + Definition lt_0_1 := Zlt_0_1. + Definition plus_le_compat := Zplus_le_compat. + Definition mult_lt_compat_l := Zmult_lt_compat_l. + Lemma opp_le_compat : forall i j, i<=j -> (-j)<=(-i). + Proof. + unfold Zle; intros; rewrite <- Zcompare_opp; auto. + Qed. + + Definition compare := Zcompare. + Definition compare_Eq := Zcompare_Eq_iff_eq. + Lemma compare_Lt : forall i j, compare i j = Lt <-> i<j. + Proof. intros; unfold compare, Zlt; intuition. Qed. + Lemma compare_Gt : forall i j, compare i j = Gt <-> i>j. + Proof. intros; unfold compare, Zgt; intuition. Qed. + + Lemma le_lt_int : forall x y, x<y <-> x<=y+-(1). + Proof. + intros; split; intros. + generalize (Zlt_left _ _ H); simpl; intros. + apply Zle_left_rev; auto. + apply Zlt_0_minus_lt. + generalize (Zplus_le_lt_compat x (y+-1) (-x) (-x+1) H). + rewrite Zplus_opp_r. + rewrite <-Zplus_assoc. + rewrite (Zplus_permute (-1)). + simpl in *. + rewrite Zplus_0_r. + intro H'; apply H'. + replace (-x+1) with (Zsucc (-x)); auto. + apply Zlt_succ. + Qed. + +End Z_as_Int. + + + + +Module IntProperties (I:Int). + Import I. + + (* Primo, some consequences of being a ring theory... *) + + Definition two := 1+1. + Notation "2" := two : Int_scope. + + (* Aliases for properties packed in the ring record. *) + + Definition plus_assoc := ring.(Radd_assoc). + Definition plus_comm := ring.(Radd_comm). + Definition plus_0_l := ring.(Radd_0_l). + Definition mult_assoc := ring.(Rmul_assoc). + Definition mult_comm := ring.(Rmul_comm). + Definition mult_1_l := ring.(Rmul_1_l). + Definition mult_plus_distr_r := ring.(Rdistr_l). + Definition opp_def := ring.(Ropp_def). + Definition minus_def := ring.(Rsub_def). + + Opaque plus_assoc plus_comm plus_0_l mult_assoc mult_comm mult_1_l + mult_plus_distr_r opp_def minus_def. + + (* More facts about plus *) + + Lemma plus_0_r : forall x, x+0 = x. + Proof. intros; rewrite plus_comm; apply plus_0_l. Qed. + + Lemma plus_0_r_reverse : forall x, x = x+0. + Proof. intros; symmetry; apply plus_0_r. Qed. + + Lemma plus_assoc_reverse : forall x y z, x+y+z = x+(y+z). + Proof. intros; symmetry; apply plus_assoc. Qed. + + Lemma plus_permute : forall x y z, x+(y+z) = y+(x+z). + Proof. intros; do 2 rewrite plus_assoc; f_equal; apply plus_comm. Qed. + + Lemma plus_reg_l : forall x y z, x+y = x+z -> y = z. + Proof. + intros. + rewrite (plus_0_r_reverse y), (plus_0_r_reverse z), <-(opp_def x). + now rewrite plus_permute, plus_assoc, H, <- plus_assoc, plus_permute. + Qed. + + (* More facts about mult *) + + Lemma mult_assoc_reverse : forall x y z, x*y*z = x*(y*z). + Proof. intros; symmetry; apply mult_assoc. Qed. + + Lemma mult_plus_distr_l : forall x y z, x*(y+z)=x*y+x*z. + Proof. + intros. + rewrite (mult_comm x (y+z)), (mult_comm x y), (mult_comm x z). + apply mult_plus_distr_r. + Qed. + + Lemma mult_0_l : forall x, 0*x = 0. + Proof. + intros. + generalize (mult_plus_distr_r 0 1 x). + rewrite plus_0_l, mult_1_l, plus_comm; intros. + apply plus_reg_l with x. + rewrite <- H. + apply plus_0_r_reverse. + Qed. + + + (* More facts about opp *) + + Definition plus_opp_r := opp_def. + + Lemma plus_opp_l : forall x, -x + x = 0. + Proof. intros; now rewrite plus_comm, opp_def. Qed. + + Lemma mult_opp_comm : forall x y, - x * y = x * - y. + Proof. + intros. + apply plus_reg_l with (x*y). + rewrite <- mult_plus_distr_l, <- mult_plus_distr_r. + now rewrite opp_def, opp_def, mult_0_l, mult_comm, mult_0_l. + Qed. + + Lemma opp_eq_mult_neg_1 : forall x, -x = x * -(1). + Proof. + intros; now rewrite mult_comm, mult_opp_comm, mult_1_l. + Qed. + + Lemma opp_involutive : forall x, -(-x) = x. + Proof. + intros. + apply plus_reg_l with (-x). + now rewrite opp_def, plus_comm, opp_def. + Qed. + + Lemma opp_plus_distr : forall x y, -(x+y) = -x + -y. + Proof. + intros. + apply plus_reg_l with (x+y). + rewrite opp_def. + rewrite plus_permute. + do 2 rewrite plus_assoc. + now rewrite (plus_comm (-x)), opp_def, plus_0_l, opp_def. + Qed. + + Lemma opp_mult_distr_r : forall x y, -(x*y) = x * -y. + Proof. + intros. + rewrite <- mult_opp_comm. + apply plus_reg_l with (x*y). + now rewrite opp_def, <-mult_plus_distr_r, opp_def, mult_0_l. + Qed. + + Lemma egal_left : forall n m, n=m -> n+-m = 0. + Proof. intros; subst; apply opp_def. Qed. + + Lemma ne_left_2 : forall x y : int, x<>y -> 0<>(x + - y). + Proof. + intros; contradict H. + apply (plus_reg_l (-y)). + now rewrite plus_opp_l, plus_comm, H. + Qed. + + (* Special lemmas for factorisation. *) + + Lemma red_factor0 : forall n, n = n*1. + Proof. symmetry; rewrite mult_comm; apply mult_1_l. Qed. + + Lemma red_factor1 : forall n, n+n = n*2. + Proof. + intros; unfold two. + now rewrite mult_comm, mult_plus_distr_r, mult_1_l. + Qed. + + Lemma red_factor2 : forall n m, n + n*m = n * (1+m). + Proof. + intros; rewrite mult_plus_distr_l. + f_equal; now rewrite mult_comm, mult_1_l. + Qed. + + Lemma red_factor3 : forall n m, n*m + n = n*(1+m). + Proof. intros; now rewrite plus_comm, red_factor2. Qed. + + Lemma red_factor4 : forall n m p, n*m + n*p = n*(m+p). + Proof. + intros; now rewrite mult_plus_distr_l. + Qed. + + Lemma red_factor5 : forall n m , n * 0 + m = m. + Proof. intros; now rewrite mult_comm, mult_0_l, plus_0_l. Qed. + + Definition red_factor6 := plus_0_r_reverse. + + + (* Specialized distributivities *) + + Hint Rewrite mult_plus_distr_l mult_plus_distr_r mult_assoc : int. + Hint Rewrite <- plus_assoc : int. + + Lemma OMEGA10 : + forall v c1 c2 l1 l2 k1 k2 : int, + (v * c1 + l1) * k1 + (v * c2 + l2) * k2 = + v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2). + Proof. + intros; autorewrite with int; f_equal; now rewrite plus_permute. + Qed. + + Lemma OMEGA11 : + forall v1 c1 l1 l2 k1 : int, + (v1 * c1 + l1) * k1 + l2 = v1 * (c1 * k1) + (l1 * k1 + l2). + Proof. + intros; now autorewrite with int. + Qed. + + Lemma OMEGA12 : + forall v2 c2 l1 l2 k2 : int, + l1 + (v2 * c2 + l2) * k2 = v2 * (c2 * k2) + (l1 + l2 * k2). + Proof. + intros; autorewrite with int; now rewrite plus_permute. + Qed. + + Lemma OMEGA13 : + forall v l1 l2 x : int, + v * -x + l1 + (v * x + l2) = l1 + l2. + Proof. + intros; autorewrite with int. + rewrite plus_permute; f_equal. + rewrite plus_assoc. + now rewrite <- mult_plus_distr_l, plus_opp_l, mult_comm, mult_0_l, plus_0_l. + Qed. + + Lemma OMEGA15 : + forall v c1 c2 l1 l2 k2 : int, + v * c1 + l1 + (v * c2 + l2) * k2 = v * (c1 + c2 * k2) + (l1 + l2 * k2). + Proof. + intros; autorewrite with int; f_equal; now rewrite plus_permute. + Qed. + + Lemma OMEGA16 : forall v c l k : int, (v * c + l) * k = v * (c * k) + l * k. + Proof. + intros; now autorewrite with int. + Qed. + + Lemma sum1 : forall a b c d : int, 0 = a -> 0 = b -> 0 = a * c + b * d. + Proof. + intros; elim H; elim H0; simpl in |- *; auto. + now rewrite mult_0_l, mult_0_l, plus_0_l. + Qed. + + + (* Secondo, some results about order (and equality) *) + + Lemma lt_irrefl : forall n, ~ n<n. + Proof. + intros n H. + elim (lt_not_eq _ _ H); auto. + Qed. + + Lemma lt_antisym : forall n m, n<m -> m<n -> False. + Proof. + intros; elim (lt_irrefl _ (lt_trans _ _ _ H H0)); auto. + Qed. + + Lemma lt_le_weak : forall n m, n<m -> n<=m. + Proof. + intros; rewrite le_lt_iff; intro H'; eapply lt_antisym; eauto. + Qed. + + Lemma le_refl : forall n, n<=n. + Proof. + intros; rewrite le_lt_iff; apply lt_irrefl; auto. + Qed. + + Lemma le_antisym : forall n m, n<=m -> m<=n -> n=m. + Proof. + intros n m; do 2 rewrite le_lt_iff; intros. + rewrite <- compare_Lt in H0. + rewrite <- gt_lt_iff, <- compare_Gt in H. + rewrite <- compare_Eq. + destruct compare; intuition. + Qed. + + Lemma lt_eq_lt_dec : forall n m, { n<m }+{ n=m }+{ m<n }. + Proof. + intros. + generalize (compare_Lt n m)(compare_Eq n m)(compare_Gt n m). + destruct compare; [ left; right | left; left | right ]; intuition. + rewrite gt_lt_iff in H1; intuition. + Qed. + + Lemma lt_dec : forall n m: int, { n<m } + { ~n<m }. + Proof. + intros. + generalize (compare_Lt n m)(compare_Eq n m)(compare_Gt n m). + destruct compare; [ right | left | right ]; intuition discriminate. + Qed. + + Lemma lt_le_iff : forall n m, (n<m) <-> ~(m<=n). + Proof. + intros. + rewrite le_lt_iff. + destruct (lt_dec n m); intuition. + Qed. + + Lemma le_dec : forall n m: int, { n<=m } + { ~n<=m }. + Proof. + intros; destruct (lt_dec m n); [right|left]; rewrite le_lt_iff; intuition. + Qed. + + Lemma le_lt_dec : forall n m, { n<=m } + { m<n }. + Proof. + intros; destruct (le_dec n m); [left|right]; auto; now rewrite lt_le_iff. + Qed. + + + Definition beq i j := match compare i j with Eq => true | _ => false end. + + Lemma beq_iff : forall i j, beq i j = true <-> i=j. + Proof. + intros; unfold beq; generalize (compare_Eq i j). + destruct compare; intuition discriminate. + Qed. + + Lemma beq_true : forall i j, beq i j = true -> i=j. + Proof. + intros. + rewrite <- beq_iff; auto. + Qed. + + Lemma beq_false : forall i j, beq i j = false -> i<>j. + Proof. + intros. + intro H'. + rewrite <- beq_iff in H'; rewrite H' in H; discriminate. + Qed. + + Lemma eq_dec : forall n m:int, { n=m } + { n<>m }. + Proof. + intros; generalize (beq_iff n m); destruct beq; [left|right]; intuition. + Qed. + + Definition bgt i j := match compare i j with Gt => true | _ => false end. + + Lemma bgt_iff : forall i j, bgt i j = true <-> i>j. + Proof. + intros; unfold bgt; generalize (compare_Gt i j). + destruct compare; intuition discriminate. + Qed. + + Lemma bgt_true : forall i j, bgt i j = true -> i>j. + Proof. intros; now rewrite <- bgt_iff. Qed. + + Lemma bgt_false : forall i j, bgt i j = false -> i<=j. + Proof. + intros. + rewrite le_lt_iff, <-gt_lt_iff, <-bgt_iff; intro H'; now rewrite H' in H. + Qed. + + Lemma le_is_lt_or_eq : forall n m, n<=m -> { n<m } + { n=m }. + Proof. + intros. + destruct (eq_dec n m) as [H'|H']. + right; intuition. + left; rewrite lt_le_iff. + contradict H'. + apply le_antisym; auto. + Qed. + + Lemma le_neq_lt : forall n m, n<=m -> n<>m -> n<m. + Proof. + intros. + destruct (le_is_lt_or_eq _ _ H); intuition. + Qed. + + Lemma le_trans : forall n m p, n<=m -> m<=p -> n<=p. + Proof. + intros n m p; do 3 rewrite le_lt_iff; intros A B C. + destruct (lt_eq_lt_dec p m) as [[H|H]|H]; subst; auto. + generalize (lt_trans _ _ _ H C); intuition. + Qed. + + (* order and operations *) + + Lemma le_0_neg : forall n, 0 <= n <-> -n <= 0. + Proof. + intros. + pattern 0 at 2; rewrite <- (mult_0_l (-(1))). + rewrite <- opp_eq_mult_neg_1. + split; intros. + apply opp_le_compat; auto. + rewrite <-(opp_involutive 0), <-(opp_involutive n). + apply opp_le_compat; auto. + Qed. + + Lemma le_0_neg' : forall n, n <= 0 <-> 0 <= -n. + Proof. + intros; rewrite le_0_neg, opp_involutive; intuition. + Qed. + + Lemma plus_le_reg_r : forall n m p, n + p <= m + p -> n <= m. + Proof. + intros. + replace n with ((n+p)+-p). + replace m with ((m+p)+-p). + apply plus_le_compat; auto. + apply le_refl. + now rewrite <- plus_assoc, opp_def, plus_0_r. + now rewrite <- plus_assoc, opp_def, plus_0_r. + Qed. + + Lemma plus_le_lt_compat : forall n m p q, n<=m -> p<q -> n+p<m+q. + Proof. + intros. + apply le_neq_lt. + apply plus_le_compat; auto. + apply lt_le_weak; auto. + rewrite lt_le_iff in H0. + contradict H0. + apply plus_le_reg_r with m. + rewrite (plus_comm q m), <-H0, (plus_comm p m). + apply plus_le_compat; auto. + apply le_refl; auto. + Qed. + + Lemma plus_lt_compat : forall n m p q, n<m -> p<q -> n+p<m+q. + Proof. + intros. + apply plus_le_lt_compat; auto. + apply lt_le_weak; auto. + Qed. + + Lemma opp_lt_compat : forall n m, n<m -> -m < -n. + Proof. + intros n m; do 2 rewrite lt_le_iff; intros H; contradict H. + rewrite <-(opp_involutive m), <-(opp_involutive n). + apply opp_le_compat; auto. + Qed. + + Lemma lt_0_neg : forall n, 0 < n <-> -n < 0. + Proof. + intros. + pattern 0 at 2; rewrite <- (mult_0_l (-(1))). + rewrite <- opp_eq_mult_neg_1. + split; intros. + apply opp_lt_compat; auto. + rewrite <-(opp_involutive 0), <-(opp_involutive n). + apply opp_lt_compat; auto. + Qed. + + Lemma lt_0_neg' : forall n, n < 0 <-> 0 < -n. + Proof. + intros; rewrite lt_0_neg, opp_involutive; intuition. + Qed. + + Lemma mult_lt_0_compat : forall n m, 0 < n -> 0 < m -> 0 < n*m. + Proof. + intros. + rewrite <- (mult_0_l n), mult_comm. + apply mult_lt_compat_l; auto. + Qed. + + Lemma mult_integral : forall n m, n * m = 0 -> n = 0 \/ m = 0. + Proof. + intros. + destruct (lt_eq_lt_dec n 0) as [[Hn|Hn]|Hn]; auto; + destruct (lt_eq_lt_dec m 0) as [[Hm|Hm]|Hm]; auto; exfalso. + + rewrite lt_0_neg' in Hn. + rewrite lt_0_neg' in Hm. + generalize (mult_lt_0_compat _ _ Hn Hm). + rewrite <- opp_mult_distr_r, mult_comm, <- opp_mult_distr_r, opp_involutive. + rewrite mult_comm, H. + exact (lt_irrefl 0). + + rewrite lt_0_neg' in Hn. + generalize (mult_lt_0_compat _ _ Hn Hm). + rewrite mult_comm, <- opp_mult_distr_r, mult_comm. + rewrite H. + rewrite opp_eq_mult_neg_1, mult_0_l. + exact (lt_irrefl 0). + + rewrite lt_0_neg' in Hm. + generalize (mult_lt_0_compat _ _ Hn Hm). + rewrite <- opp_mult_distr_r. + rewrite H. + rewrite opp_eq_mult_neg_1, mult_0_l. + exact (lt_irrefl 0). + + generalize (mult_lt_0_compat _ _ Hn Hm). + rewrite H. + exact (lt_irrefl 0). + Qed. + + Lemma mult_le_compat : + forall i j k l, i<=j -> k<=l -> 0<=i -> 0<=k -> i*k<=j*l. + Proof. + intros. + destruct (le_is_lt_or_eq _ _ H1). + + apply le_trans with (i*l). + destruct (le_is_lt_or_eq _ _ H0); [ | subst; apply le_refl]. + apply lt_le_weak. + apply mult_lt_compat_l; auto. + + generalize (le_trans _ _ _ H2 H0); clear H0 H1 H2; intros. + rewrite (mult_comm i), (mult_comm j). + destruct (le_is_lt_or_eq _ _ H0); + [ | subst; do 2 rewrite mult_0_l; apply le_refl]. + destruct (le_is_lt_or_eq _ _ H); + [ | subst; apply le_refl]. + apply lt_le_weak. + apply mult_lt_compat_l; auto. + + subst i. + rewrite mult_0_l. + generalize (le_trans _ _ _ H2 H0); clear H0 H1 H2; intros. + destruct (le_is_lt_or_eq _ _ H); + [ | subst; rewrite mult_0_l; apply le_refl]. + destruct (le_is_lt_or_eq _ _ H0); + [ | subst; rewrite mult_comm, mult_0_l; apply le_refl]. + apply lt_le_weak. + apply mult_lt_0_compat; auto. + Qed. + + Lemma sum5 : + forall a b c d : int, c <> 0 -> 0 <> a -> 0 = b -> 0 <> a * c + b * d. + Proof. + intros. + subst b; rewrite mult_0_l, plus_0_r. + contradict H. + symmetry in H; destruct (mult_integral _ _ H); congruence. + Qed. + + Lemma one_neq_zero : 1 <> 0. + Proof. + red; intro. + symmetry in H. + apply (lt_not_eq 0 1); auto. + apply lt_0_1. + Qed. + + Lemma minus_one_neq_zero : -(1) <> 0. + Proof. + apply lt_not_eq. + rewrite <- lt_0_neg. + apply lt_0_1. + Qed. + + Lemma le_left : forall n m, n <= m -> 0 <= m + - n. + Proof. + intros. + rewrite <- (opp_def m). + apply plus_le_compat. + apply le_refl. + apply opp_le_compat; auto. + Qed. + + Lemma OMEGA2 : forall x y, 0 <= x -> 0 <= y -> 0 <= x + y. + Proof. + intros. + replace 0 with (0+0). + apply plus_le_compat; auto. + rewrite plus_0_l; auto. + Qed. + + Lemma OMEGA8 : forall x y, 0 <= x -> 0 <= y -> x = - y -> x = 0. + Proof. + intros. + assert (y=-x). + subst x; symmetry; apply opp_involutive. + clear H1; subst y. + destruct (eq_dec 0 x) as [H'|H']; auto. + assert (H'':=le_neq_lt _ _ H H'). + generalize (plus_le_lt_compat _ _ _ _ H0 H''). + rewrite plus_opp_l, plus_0_l. + intros. + elim (lt_not_eq _ _ H1); auto. + Qed. + + Lemma sum2 : + forall a b c d : int, 0 <= d -> 0 = a -> 0 <= b -> 0 <= a * c + b * d. + Proof. + intros. + subst a; rewrite mult_0_l, plus_0_l. + rewrite <- (mult_0_l 0). + apply mult_le_compat; auto; apply le_refl. + Qed. + + Lemma sum3 : + forall a b c d : int, + 0 <= c -> 0 <= d -> 0 <= a -> 0 <= b -> 0 <= a * c + b * d. + Proof. + intros. + rewrite <- (plus_0_l 0). + apply plus_le_compat; auto. + rewrite <- (mult_0_l 0). + apply mult_le_compat; auto; apply le_refl. + rewrite <- (mult_0_l 0). + apply mult_le_compat; auto; apply le_refl. + Qed. + + Lemma sum4 : forall k : int, k>0 -> 0 <= k. + Proof. + intros k; rewrite gt_lt_iff; apply lt_le_weak. + Qed. + + (* Lemmas specific to integers (they use lt_le_int) *) + + Lemma lt_left : forall n m, n < m -> 0 <= m + -(1) + - n. + Proof. + intros; apply le_left. + now rewrite <- le_lt_int. + Qed. + + Lemma lt_left_inv : forall x y, 0 <= y + -(1) + - x -> x < y. + Proof. + intros. + generalize (plus_le_compat _ _ _ _ H (le_refl x)); clear H. + now rewrite plus_0_l, <-plus_assoc, plus_opp_l, plus_0_r, le_lt_int. + Qed. + + Lemma OMEGA4 : forall x y z, x > 0 -> y > x -> z * y + x <> 0. + Proof. + intros. + intro H'. + rewrite gt_lt_iff in H,H0. + destruct (lt_eq_lt_dec z 0) as [[G|G]|G]. + + rewrite lt_0_neg' in G. + generalize (plus_le_lt_compat _ _ _ _ (le_refl (z*y)) H0). + rewrite H'. + pattern y at 2; rewrite <-(mult_1_l y), <-mult_plus_distr_r. + intros. + rewrite le_lt_int in G. + rewrite <- opp_plus_distr in G. + assert (0 < y) by (apply lt_trans with x; auto). + generalize (mult_le_compat _ _ _ _ G (lt_le_weak _ _ H2) (le_refl 0) (le_refl 0)). + rewrite mult_0_l, mult_comm, <- opp_mult_distr_r, mult_comm, <-le_0_neg', le_lt_iff. + intuition. + + subst; rewrite mult_0_l, plus_0_l in H'; subst. + apply (lt_not_eq _ _ H); auto. + + apply (lt_not_eq 0 (z*y+x)); auto. + rewrite <- (plus_0_l 0). + apply plus_lt_compat; auto. + apply mult_lt_0_compat; auto. + apply lt_trans with x; auto. + Qed. + + Lemma OMEGA19 : forall x, x<>0 -> 0 <= x + -(1) \/ 0 <= x * -(1) + -(1). + Proof. + intros. + do 2 rewrite <- le_lt_int. + rewrite <- opp_eq_mult_neg_1. + destruct (lt_eq_lt_dec 0 x) as [[H'|H']|H']. + auto. + congruence. + right. + rewrite <-(mult_0_l (-(1))), <-(opp_eq_mult_neg_1 0). + apply opp_lt_compat; auto. + Qed. + + Lemma mult_le_approx : + forall n m p, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m. + Proof. + intros n m p. + do 2 rewrite gt_lt_iff. + do 2 rewrite le_lt_iff; intros. + contradict H1. + rewrite lt_0_neg' in H1. + rewrite lt_0_neg'. + rewrite opp_plus_distr. + rewrite mult_comm, opp_mult_distr_r. + rewrite le_lt_int. + rewrite <- plus_assoc, (plus_comm (-p)), plus_assoc. + apply lt_left. + rewrite le_lt_int. + rewrite le_lt_int in H0. + apply le_trans with (n+-(1)); auto. + apply plus_le_compat; [ | apply le_refl ]. + rewrite le_lt_int in H1. + generalize (mult_le_compat _ _ _ _ (lt_le_weak _ _ H) H1 (le_refl 0) (le_refl 0)). + rewrite mult_0_l. + rewrite mult_plus_distr_l. + rewrite <- opp_eq_mult_neg_1. + intros. + generalize (plus_le_compat _ _ _ _ (le_refl n) H2). + now rewrite plus_permute, opp_def, plus_0_r, plus_0_r. + Qed. + + (* Some decidabilities *) + + Lemma dec_eq : forall i j:int, decidable (i=j). + Proof. + red; intros; destruct (eq_dec i j); auto. + Qed. + + Lemma dec_ne : forall i j:int, decidable (i<>j). + Proof. + red; intros; destruct (eq_dec i j); auto. + Qed. + + Lemma dec_le : forall i j:int, decidable (i<=j). + Proof. + red; intros; destruct (le_dec i j); auto. + Qed. + + Lemma dec_lt : forall i j:int, decidable (i<j). + Proof. + red; intros; destruct (lt_dec i j); auto. + Qed. + + Lemma dec_ge : forall i j:int, decidable (i>=j). + Proof. + red; intros; rewrite ge_le_iff; destruct (le_dec j i); auto. + Qed. + + Lemma dec_gt : forall i j:int, decidable (i>j). + Proof. + red; intros; rewrite gt_lt_iff; destruct (lt_dec j i); auto. + Qed. + +End IntProperties. + + + + +Module IntOmega (I:Int). +Import I. +Module IP:=IntProperties(I). +Import IP. + +(* \subsubsection{Definition of reified integer expressions} + Terms are either: + \begin{itemize} + \item integers [Tint] + \item variables [Tvar] + \item operation over integers (addition, product, opposite, subtraction) + The last two are translated in additions and products. *) + +Inductive term : Set := + | Tint : int -> term + | Tplus : term -> term -> term + | Tmult : term -> term -> term + | Tminus : term -> term -> term + | Topp : term -> term + | Tvar : nat -> term. + +Delimit Scope romega_scope with term. +Arguments Scope Tint [Int_scope]. +Arguments Scope Tplus [romega_scope romega_scope]. +Arguments Scope Tmult [romega_scope romega_scope]. +Arguments Scope Tminus [romega_scope romega_scope]. +Arguments Scope Topp [romega_scope romega_scope]. + +Infix "+" := Tplus : romega_scope. +Infix "*" := Tmult : romega_scope. +Infix "-" := Tminus : romega_scope. +Notation "- x" := (Topp x) : romega_scope. +Notation "[ x ]" := (Tvar x) (at level 0) : romega_scope. + +(* \subsubsection{Definition of reified goals} *) + +(* Very restricted definition of handled predicates that should be extended + to cover a wider set of operations. + Taking care of negations and disequations require solving more than a + goal in parallel. This is a major improvement over previous versions. *) + +Inductive proposition : Set := + | EqTerm : term -> term -> proposition (* equality between terms *) + | LeqTerm : term -> term -> proposition (* less or equal on terms *) + | TrueTerm : proposition (* true *) + | FalseTerm : proposition (* false *) + | Tnot : proposition -> proposition (* negation *) + | GeqTerm : term -> term -> proposition + | GtTerm : term -> term -> proposition + | LtTerm : term -> term -> proposition + | NeqTerm : term -> term -> proposition + | Tor : proposition -> proposition -> proposition + | Tand : proposition -> proposition -> proposition + | Timp : proposition -> proposition -> proposition + | Tprop : nat -> proposition. + +(* Definition of goals as a list of hypothesis *) +Notation hyps := (list proposition). + +(* Definition of lists of subgoals (set of open goals) *) +Notation lhyps := (list hyps). + +(* a single goal packed in a subgoal list *) +Notation singleton := (fun a : hyps => a :: nil). + +(* an absurd goal *) +Definition absurd := FalseTerm :: nil. + +(* \subsubsection{Traces for merging equations} + This inductive type describes how the monomial of two equations should be + merged when the equations are added. + + For [F_equal], both equations have the same head variable and coefficient + must be added, furthermore if coefficients are opposite, [F_cancel] should + be used to collapse the term. [F_left] and [F_right] indicate which monomial + should be put first in the result *) + +Inductive t_fusion : Set := + | F_equal : t_fusion + | F_cancel : t_fusion + | F_left : t_fusion + | F_right : t_fusion. + +(* \subsubsection{Rewriting steps to normalize terms} *) +Inductive step : Set := + (* apply the rewriting steps to both subterms of an operation *) + | C_DO_BOTH : step -> step -> step + (* apply the rewriting step to the first branch *) + | C_LEFT : step -> step + (* apply the rewriting step to the second branch *) + | C_RIGHT : step -> step + (* apply two steps consecutively to a term *) + | C_SEQ : step -> step -> step + (* empty step *) + | C_NOP : step + (* the following operations correspond to actual rewriting *) + | C_OPP_PLUS : step + | C_OPP_OPP : step + | C_OPP_MULT_R : step + | C_OPP_ONE : step + (* This is a special step that reduces the term (computation) *) + | C_REDUCE : step + | C_MULT_PLUS_DISTR : step + | C_MULT_OPP_LEFT : step + | C_MULT_ASSOC_R : step + | C_PLUS_ASSOC_R : step + | C_PLUS_ASSOC_L : step + | C_PLUS_PERMUTE : step + | C_PLUS_COMM : step + | C_RED0 : step + | C_RED1 : step + | C_RED2 : step + | C_RED3 : step + | C_RED4 : step + | C_RED5 : step + | C_RED6 : step + | C_MULT_ASSOC_REDUCED : step + | C_MINUS : step + | C_MULT_COMM : step. + +(* \subsubsection{Omega steps} *) +(* The following inductive type describes steps as they can be found in + the trace coming from the decision procedure Omega. *) + +Inductive t_omega : Set := + (* n = 0 and n!= 0 *) + | O_CONSTANT_NOT_NUL : nat -> t_omega + | O_CONSTANT_NEG : nat -> t_omega + (* division and approximation of an equation *) + | O_DIV_APPROX : int -> int -> term -> nat -> t_omega -> nat -> t_omega + (* no solution because no exact division *) + | O_NOT_EXACT_DIVIDE : int -> int -> term -> nat -> nat -> t_omega + (* exact division *) + | O_EXACT_DIVIDE : int -> term -> nat -> t_omega -> nat -> t_omega + | O_SUM : int -> nat -> int -> nat -> list t_fusion -> t_omega -> t_omega + | O_CONTRADICTION : nat -> nat -> nat -> t_omega + | O_MERGE_EQ : nat -> nat -> nat -> t_omega -> t_omega + | O_SPLIT_INEQ : nat -> nat -> t_omega -> t_omega -> t_omega + | O_CONSTANT_NUL : nat -> t_omega + | O_NEGATE_CONTRADICT : nat -> nat -> t_omega + | O_NEGATE_CONTRADICT_INV : nat -> nat -> nat -> t_omega + | O_STATE : int -> step -> nat -> nat -> t_omega -> t_omega. + +(* \subsubsection{Rules for normalizing the hypothesis} *) +(* These rules indicate how to normalize useful propositions + of each useful hypothesis before the decomposition of hypothesis. + The rules include the inversion phase for negation removal. *) + +Inductive p_step : Set := + | P_LEFT : p_step -> p_step + | P_RIGHT : p_step -> p_step + | P_INVERT : step -> p_step + | P_STEP : step -> p_step + | P_NOP : p_step. + +(* List of normalizations to perform : with a constructor of type + [p_step] allowing to visit both left and right branches, we would be + able to restrict to only one normalization by hypothesis. + And since all hypothesis are useful (otherwise they wouldn't be included), + we would be able to replace [h_step] by a simple list. *) + +Inductive h_step : Set := + pair_step : nat -> p_step -> h_step. + +(* \subsubsection{Rules for decomposing the hypothesis} *) +(* This type allows to navigate in the logical constructors that + form the predicats of the hypothesis in order to decompose them. + This allows in particular to extract one hypothesis from a + conjonction with possibly the right level of negations. *) + +Inductive direction : Set := + | D_left : direction + | D_right : direction + | D_mono : direction. + +(* This type allows to extract useful components from hypothesis, either + hypothesis generated by splitting a disjonction, or equations. + The last constructor indicates how to solve the obtained system + via the use of the trace type of Omega [t_omega] *) + +Inductive e_step : Set := + | E_SPLIT : nat -> list direction -> e_step -> e_step -> e_step + | E_EXTRACT : nat -> list direction -> e_step -> e_step + | E_SOLVE : t_omega -> e_step. + +(* \subsection{Efficient decidable equality} *) +(* For each reified data-type, we define an efficient equality test. + It is not the one produced by [Decide Equality]. + + Then we prove two theorem allowing to eliminate such equalities : + \begin{verbatim} + (t1,t2: typ) (eq_typ t1 t2) = true -> t1 = t2. + (t1,t2: typ) (eq_typ t1 t2) = false -> ~ t1 = t2. + \end{verbatim} *) + +(* \subsubsection{Reified terms} *) + +Open Scope romega_scope. + +Fixpoint eq_term (t1 t2 : term) {struct t2} : bool := + match t1, t2 with + | Tint st1, Tint st2 => beq st1 st2 + | (st11 + st12), (st21 + st22) => eq_term st11 st21 && eq_term st12 st22 + | (st11 * st12), (st21 * st22) => eq_term st11 st21 && eq_term st12 st22 + | (st11 - st12), (st21 - st22) => eq_term st11 st21 && eq_term st12 st22 + | (- st1), (- st2) => eq_term st1 st2 + | [st1], [st2] => beq_nat st1 st2 + | _, _ => false + end. + +Close Scope romega_scope. + +Theorem eq_term_true : forall t1 t2 : term, eq_term t1 t2 = true -> t1 = t2. +Proof. + simple induction t1; intros until t2; case t2; simpl in *; + try (intros; discriminate; fail); + [ intros; elim beq_true with (1 := H); trivial + | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5; + elim H with (1 := H4); elim H0 with (1 := H5); + trivial + | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5; + elim H with (1 := H4); elim H0 with (1 := H5); + trivial + | intros t21 t22 H3; elim andb_prop with (1 := H3); intros H4 H5; + elim H with (1 := H4); elim H0 with (1 := H5); + trivial + | intros t21 H3; elim H with (1 := H3); trivial + | intros; elim beq_nat_true with (1 := H); trivial ]. +Qed. + +Ltac trivial_case := unfold not in |- *; intros; discriminate. + +Theorem eq_term_false : + forall t1 t2 : term, eq_term t1 t2 = false -> t1 <> t2. +Proof. + simple induction t1; + [ intros z t2; case t2; try trivial_case; simpl in |- *; unfold not in |- *; + intros; elim beq_false with (1 := H); simplify_eq H0; + auto + | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *; + intros t21 t22 H3; unfold not in |- *; intro H4; + elim andb_false_elim with (1 := H3); intros H5; + [ elim H1 with (1 := H5); simplify_eq H4; auto + | elim H2 with (1 := H5); simplify_eq H4; auto ] + | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *; + intros t21 t22 H3; unfold not in |- *; intro H4; + elim andb_false_elim with (1 := H3); intros H5; + [ elim H1 with (1 := H5); simplify_eq H4; auto + | elim H2 with (1 := H5); simplify_eq H4; auto ] + | intros t11 H1 t12 H2 t2; case t2; try trivial_case; simpl in |- *; + intros t21 t22 H3; unfold not in |- *; intro H4; + elim andb_false_elim with (1 := H3); intros H5; + [ elim H1 with (1 := H5); simplify_eq H4; auto + | elim H2 with (1 := H5); simplify_eq H4; auto ] + | intros t11 H1 t2; case t2; try trivial_case; simpl in |- *; intros t21 H3; + unfold not in |- *; intro H4; elim H1 with (1 := H3); + simplify_eq H4; auto + | intros n t2; case t2; try trivial_case; simpl in |- *; unfold not in |- *; + intros; elim beq_nat_false with (1 := H); simplify_eq H0; + auto ]. +Qed. + +(* \subsubsection{Tactiques pour éliminer ces tests} + + Si on se contente de faire un [Case (eq_typ t1 t2)] on perd + totalement dans chaque branche le fait que [t1=t2] ou [~t1=t2]. + + Initialement, les développements avaient été réalisés avec les + tests rendus par [Decide Equality], c'est à dire un test rendant + des termes du type [{t1=t2}+{~t1=t2}]. Faire une élimination sur un + tel test préserve bien l'information voulue mais calculatoirement de + telles fonctions sont trop lentes. *) + +(* Les tactiques définies si après se comportent exactement comme si on + avait utilisé le test précédent et fait une elimination dessus. *) + +Ltac elim_eq_term t1 t2 := + pattern (eq_term t1 t2) in |- *; apply bool_eq_ind; intro Aux; + [ generalize (eq_term_true t1 t2 Aux); clear Aux + | generalize (eq_term_false t1 t2 Aux); clear Aux ]. + +Ltac elim_beq t1 t2 := + pattern (beq t1 t2) in |- *; apply bool_eq_ind; intro Aux; + [ generalize (beq_true t1 t2 Aux); clear Aux + | generalize (beq_false t1 t2 Aux); clear Aux ]. + +Ltac elim_bgt t1 t2 := + pattern (bgt t1 t2) in |- *; apply bool_eq_ind; intro Aux; + [ generalize (bgt_true t1 t2 Aux); clear Aux + | generalize (bgt_false t1 t2 Aux); clear Aux ]. + + +(* \subsection{Interprétations} + \subsubsection{Interprétation des termes dans Z} *) + +Fixpoint interp_term (env : list int) (t : term) {struct t} : int := + match t with + | Tint x => x + | (t1 + t2)%term => interp_term env t1 + interp_term env t2 + | (t1 * t2)%term => interp_term env t1 * interp_term env t2 + | (t1 - t2)%term => interp_term env t1 - interp_term env t2 + | (- t)%term => - interp_term env t + | [n]%term => nth n env 0 + end. + +(* \subsubsection{Interprétation des prédicats} *) + +Fixpoint interp_proposition (envp : list Prop) (env : list int) + (p : proposition) {struct p} : Prop := + match p with + | EqTerm t1 t2 => interp_term env t1 = interp_term env t2 + | LeqTerm t1 t2 => interp_term env t1 <= interp_term env t2 + | TrueTerm => True + | FalseTerm => False + | Tnot p' => ~ interp_proposition envp env p' + | GeqTerm t1 t2 => interp_term env t1 >= interp_term env t2 + | GtTerm t1 t2 => interp_term env t1 > interp_term env t2 + | LtTerm t1 t2 => interp_term env t1 < interp_term env t2 + | NeqTerm t1 t2 => (interp_term env t1)<>(interp_term env t2) + | Tor p1 p2 => + interp_proposition envp env p1 \/ interp_proposition envp env p2 + | Tand p1 p2 => + interp_proposition envp env p1 /\ interp_proposition envp env p2 + | Timp p1 p2 => + interp_proposition envp env p1 -> interp_proposition envp env p2 + | Tprop n => nth n envp True + end. + +(* \subsubsection{Inteprétation des listes d'hypothèses} + \paragraph{Sous forme de conjonction} + Interprétation sous forme d'une conjonction d'hypothèses plus faciles + à manipuler individuellement *) + +Fixpoint interp_hyps (envp : list Prop) (env : list int) + (l : hyps) {struct l} : Prop := + match l with + | nil => True + | p' :: l' => interp_proposition envp env p' /\ interp_hyps envp env l' + end. + +(* \paragraph{sous forme de but} + C'est cette interpétation que l'on utilise sur le but (car on utilise + [Generalize] et qu'une conjonction est forcément lourde (répétition des + types dans les conjonctions intermédiaires) *) + +Fixpoint interp_goal_concl (c : proposition) (envp : list Prop) + (env : list int) (l : hyps) {struct l} : Prop := + match l with + | nil => interp_proposition envp env c + | p' :: l' => + interp_proposition envp env p' -> interp_goal_concl c envp env l' + end. + +Notation interp_goal := (interp_goal_concl FalseTerm). + +(* Les théorèmes qui suivent assurent la correspondance entre les deux + interprétations. *) + +Theorem goal_to_hyps : + forall (envp : list Prop) (env : list int) (l : hyps), + (interp_hyps envp env l -> False) -> interp_goal envp env l. +Proof. + simple induction l; + [ simpl in |- *; auto + | simpl in |- *; intros a l1 H1 H2 H3; apply H1; intro H4; apply H2; auto ]. +Qed. + +Theorem hyps_to_goal : + forall (envp : list Prop) (env : list int) (l : hyps), + interp_goal envp env l -> interp_hyps envp env l -> False. +Proof. + simple induction l; simpl in |- *; [ auto | intros; apply H; elim H1; auto ]. +Qed. + +(* \subsection{Manipulations sur les hypothèses} *) + +(* \subsubsection{Définitions de base de stabilité pour la réflexion} *) +(* Une opération laisse un terme stable si l'égalité est préservée *) +Definition term_stable (f : term -> term) := + forall (e : list int) (t : term), interp_term e t = interp_term e (f t). + +(* Une opération est valide sur une hypothèse, si l'hypothèse implique le + résultat de l'opération. \emph{Attention : cela ne concerne que des + opérations sur les hypothèses et non sur les buts (contravariance)}. + On définit la validité pour une opération prenant une ou deux propositions + en argument (cela suffit pour omega). *) + +Definition valid1 (f : proposition -> proposition) := + forall (ep : list Prop) (e : list int) (p1 : proposition), + interp_proposition ep e p1 -> interp_proposition ep e (f p1). + +Definition valid2 (f : proposition -> proposition -> proposition) := + forall (ep : list Prop) (e : list int) (p1 p2 : proposition), + interp_proposition ep e p1 -> + interp_proposition ep e p2 -> interp_proposition ep e (f p1 p2). + +(* Dans cette notion de validité, la fonction prend directement une + liste de propositions et rend une nouvelle liste de proposition. + On reste contravariant *) + +Definition valid_hyps (f : hyps -> hyps) := + forall (ep : list Prop) (e : list int) (lp : hyps), + interp_hyps ep e lp -> interp_hyps ep e (f lp). + +(* Enfin ce théorème élimine la contravariance et nous ramène à une + opération sur les buts *) + +Theorem valid_goal : + forall (ep : list Prop) (env : list int) (l : hyps) (a : hyps -> hyps), + valid_hyps a -> interp_goal ep env (a l) -> interp_goal ep env l. +Proof. + intros; simpl in |- *; apply goal_to_hyps; intro H1; + apply (hyps_to_goal ep env (a l) H0); apply H; assumption. +Qed. + +(* \subsubsection{Généralisation a des listes de buts (disjonctions)} *) + + +Fixpoint interp_list_hyps (envp : list Prop) (env : list int) + (l : lhyps) {struct l} : Prop := + match l with + | nil => False + | h :: l' => interp_hyps envp env h \/ interp_list_hyps envp env l' + end. + +Fixpoint interp_list_goal (envp : list Prop) (env : list int) + (l : lhyps) {struct l} : Prop := + match l with + | nil => True + | h :: l' => interp_goal envp env h /\ interp_list_goal envp env l' + end. + +Theorem list_goal_to_hyps : + forall (envp : list Prop) (env : list int) (l : lhyps), + (interp_list_hyps envp env l -> False) -> interp_list_goal envp env l. +Proof. + simple induction l; simpl in |- *; + [ auto + | intros h1 l1 H H1; split; + [ apply goal_to_hyps; intro H2; apply H1; auto + | apply H; intro H2; apply H1; auto ] ]. +Qed. + +Theorem list_hyps_to_goal : + forall (envp : list Prop) (env : list int) (l : lhyps), + interp_list_goal envp env l -> interp_list_hyps envp env l -> False. +Proof. + simple induction l; simpl in |- *; + [ auto + | intros h1 l1 H (H1, H2) H3; elim H3; intro H4; + [ apply hyps_to_goal with (1 := H1); assumption | auto ] ]. +Qed. + +Definition valid_list_hyps (f : hyps -> lhyps) := + forall (ep : list Prop) (e : list int) (lp : hyps), + interp_hyps ep e lp -> interp_list_hyps ep e (f lp). + +Definition valid_list_goal (f : hyps -> lhyps) := + forall (ep : list Prop) (e : list int) (lp : hyps), + interp_list_goal ep e (f lp) -> interp_goal ep e lp. + +Theorem goal_valid : + forall f : hyps -> lhyps, valid_list_hyps f -> valid_list_goal f. +Proof. + unfold valid_list_goal in |- *; intros f H ep e lp H1; apply goal_to_hyps; + intro H2; apply list_hyps_to_goal with (1 := H1); + apply (H ep e lp); assumption. +Qed. + +Theorem append_valid : + forall (ep : list Prop) (e : list int) (l1 l2 : lhyps), + interp_list_hyps ep e l1 \/ interp_list_hyps ep e l2 -> + interp_list_hyps ep e (l1 ++ l2). +Proof. + intros ep e; simple induction l1; + [ simpl in |- *; intros l2 [H| H]; [ contradiction | trivial ] + | simpl in |- *; intros h1 t1 HR l2 [[H| H]| H]; + [ auto + | right; apply (HR l2); left; trivial + | right; apply (HR l2); right; trivial ] ]. + +Qed. + +(* \subsubsection{Opérateurs valides sur les hypothèses} *) + +(* Extraire une hypothèse de la liste *) +Definition nth_hyps (n : nat) (l : hyps) := nth n l TrueTerm. + +Theorem nth_valid : + forall (ep : list Prop) (e : list int) (i : nat) (l : hyps), + interp_hyps ep e l -> interp_proposition ep e (nth_hyps i l). +Proof. + unfold nth_hyps in |- *; simple induction i; + [ simple induction l; simpl in |- *; [ auto | intros; elim H0; auto ] + | intros n H; simple induction l; + [ simpl in |- *; trivial + | intros; simpl in |- *; apply H; elim H1; auto ] ]. +Qed. + +(* Appliquer une opération (valide) sur deux hypothèses extraites de + la liste et ajouter le résultat à la liste. *) +Definition apply_oper_2 (i j : nat) + (f : proposition -> proposition -> proposition) (l : hyps) := + f (nth_hyps i l) (nth_hyps j l) :: l. + +Theorem apply_oper_2_valid : + forall (i j : nat) (f : proposition -> proposition -> proposition), + valid2 f -> valid_hyps (apply_oper_2 i j f). +Proof. + intros i j f Hf; unfold apply_oper_2, valid_hyps in |- *; simpl in |- *; + intros lp Hlp; split; [ apply Hf; apply nth_valid; assumption | assumption ]. +Qed. + +(* Modifier une hypothèse par application d'une opération valide *) + +Fixpoint apply_oper_1 (i : nat) (f : proposition -> proposition) + (l : hyps) {struct i} : hyps := + match l with + | nil => nil (A:=proposition) + | p :: l' => + match i with + | O => f p :: l' + | S j => p :: apply_oper_1 j f l' + end + end. + +Theorem apply_oper_1_valid : + forall (i : nat) (f : proposition -> proposition), + valid1 f -> valid_hyps (apply_oper_1 i f). +Proof. + unfold valid_hyps in |- *; intros i f Hf ep e; elim i; + [ intro lp; case lp; + [ simpl in |- *; trivial + | simpl in |- *; intros p l' (H1, H2); split; + [ apply Hf with (1 := H1) | assumption ] ] + | intros n Hrec lp; case lp; + [ simpl in |- *; auto + | simpl in |- *; intros p l' (H1, H2); split; + [ assumption | apply Hrec; assumption ] ] ]. +Qed. + +(* \subsubsection{Manipulations de termes} *) +(* Les fonctions suivantes permettent d'appliquer une fonction de + réécriture sur un sous terme du terme principal. Avec la composition, + cela permet de construire des réécritures complexes proches des + tactiques de conversion *) + +Definition apply_left (f : term -> term) (t : term) := + match t with + | (x + y)%term => (f x + y)%term + | (x * y)%term => (f x * y)%term + | (- x)%term => (- f x)%term + | x => x + end. + +Definition apply_right (f : term -> term) (t : term) := + match t with + | (x + y)%term => (x + f y)%term + | (x * y)%term => (x * f y)%term + | x => x + end. + +Definition apply_both (f g : term -> term) (t : term) := + match t with + | (x + y)%term => (f x + g y)%term + | (x * y)%term => (f x * g y)%term + | x => x + end. + +(* Les théorèmes suivants montrent la stabilité (conditionnée) des + fonctions. *) + +Theorem apply_left_stable : + forall f : term -> term, term_stable f -> term_stable (apply_left f). +Proof. + unfold term_stable in |- *; intros f H e t; case t; auto; simpl in |- *; + intros; elim H; trivial. +Qed. + +Theorem apply_right_stable : + forall f : term -> term, term_stable f -> term_stable (apply_right f). +Proof. + unfold term_stable in |- *; intros f H e t; case t; auto; simpl in |- *; + intros t0 t1; elim H; trivial. +Qed. + +Theorem apply_both_stable : + forall f g : term -> term, + term_stable f -> term_stable g -> term_stable (apply_both f g). +Proof. + unfold term_stable in |- *; intros f g H1 H2 e t; case t; auto; simpl in |- *; + intros t0 t1; elim H1; elim H2; trivial. +Qed. + +Theorem compose_term_stable : + forall f g : term -> term, + term_stable f -> term_stable g -> term_stable (fun t : term => f (g t)). +Proof. + unfold term_stable in |- *; intros f g Hf Hg e t; elim Hf; apply Hg. +Qed. + +(* \subsection{Les règles de réécriture} *) +(* Chacune des règles de réécriture est accompagnée par sa preuve de + stabilité. Toutes ces preuves ont la même forme : il faut analyser + suivant la forme du terme (élimination de chaque Case). On a besoin d'une + élimination uniquement dans les cas d'utilisation d'égalité décidable. + + Cette tactique itère la décomposition des Case. Elle est + constituée de deux fonctions s'appelant mutuellement : + \begin{itemize} + \item une fonction d'enrobage qui lance la recherche sur le but, + \item une fonction récursive qui décompose ce but. Quand elle a trouvé un + Case, elle l'élimine. + \end{itemize} + Les motifs sur les cas sont très imparfaits et dans certains cas, il + semble que cela ne marche pas. On aimerait plutot un motif de la + forme [ Case (?1 :: T) of _ end ] permettant de s'assurer que l'on + utilise le bon type. + + Chaque élimination introduit correctement exactement le nombre d'hypothèses + nécessaires et conserve dans le cas d'une égalité la connaissance du + résultat du test en faisant la réécriture. Pour un test de comparaison, + on conserve simplement le résultat. + + Cette fonction déborde très largement la résolution des réécritures + simples et fait une bonne partie des preuves des pas de Omega. +*) + +(* \subsubsection{La tactique pour prouver la stabilité} *) + +Ltac loop t := + match t with + (* Global *) + | (?X1 = ?X2) => loop X1 || loop X2 + | (_ -> ?X1) => loop X1 + (* Interpretations *) + | (interp_hyps _ _ ?X1) => loop X1 + | (interp_list_hyps _ _ ?X1) => loop X1 + | (interp_proposition _ _ ?X1) => loop X1 + | (interp_term _ ?X1) => loop X1 + (* Propositions *) + | (EqTerm ?X1 ?X2) => loop X1 || loop X2 + | (LeqTerm ?X1 ?X2) => loop X1 || loop X2 + (* Termes *) + | (?X1 + ?X2)%term => loop X1 || loop X2 + | (?X1 - ?X2)%term => loop X1 || loop X2 + | (?X1 * ?X2)%term => loop X1 || loop X2 + | (- ?X1)%term => loop X1 + | (Tint ?X1) => loop X1 + (* Eliminations *) + | match ?X1 with + | EqTerm x x0 => _ + | LeqTerm x x0 => _ + | TrueTerm => _ + | FalseTerm => _ + | Tnot x => _ + | GeqTerm x x0 => _ + | GtTerm x x0 => _ + | LtTerm x x0 => _ + | NeqTerm x x0 => _ + | Tor x x0 => _ + | Tand x x0 => _ + | Timp x x0 => _ + | Tprop x => _ + end => destruct X1; auto; Simplify + | match ?X1 with + | Tint x => _ + | (x + x0)%term => _ + | (x * x0)%term => _ + | (x - x0)%term => _ + | (- x)%term => _ + | [x]%term => _ + end => destruct X1; auto; Simplify + | (if beq ?X1 ?X2 then _ else _) => + let H := fresh "H" in + elim_beq X1 X2; intro H; try (rewrite H in *; clear H); + simpl in |- *; auto; Simplify + | (if bgt ?X1 ?X2 then _ else _) => + let H := fresh "H" in + elim_bgt X1 X2; intro H; simpl in |- *; auto; Simplify + | (if eq_term ?X1 ?X2 then _ else _) => + let H := fresh "H" in + elim_eq_term X1 X2; intro H; try (rewrite H in *; clear H); + simpl in |- *; auto; Simplify + | (if _ && _ then _ else _) => rewrite andb_if; Simplify + | (if negb _ then _ else _) => rewrite negb_if; Simplify + | _ => fail + end + +with Simplify := match goal with + | |- ?X1 => try loop X1 + | _ => idtac + end. + +Ltac prove_stable x th := + match constr:x with + | ?X1 => + unfold term_stable, X1 in |- *; intros; Simplify; simpl in |- *; + apply th + end. + +(* \subsubsection{Les règles elle mêmes} *) +Definition Tplus_assoc_l (t : term) := + match t with + | (n + (m + p))%term => (n + m + p)%term + | _ => t + end. + +Theorem Tplus_assoc_l_stable : term_stable Tplus_assoc_l. +Proof. + prove_stable Tplus_assoc_l (ring.(Radd_assoc)). +Qed. + +Definition Tplus_assoc_r (t : term) := + match t with + | (n + m + p)%term => (n + (m + p))%term + | _ => t + end. + +Theorem Tplus_assoc_r_stable : term_stable Tplus_assoc_r. +Proof. + prove_stable Tplus_assoc_r plus_assoc_reverse. +Qed. + +Definition Tmult_assoc_r (t : term) := + match t with + | (n * m * p)%term => (n * (m * p))%term + | _ => t + end. + +Theorem Tmult_assoc_r_stable : term_stable Tmult_assoc_r. +Proof. + prove_stable Tmult_assoc_r mult_assoc_reverse. +Qed. + +Definition Tplus_permute (t : term) := + match t with + | (n + (m + p))%term => (m + (n + p))%term + | _ => t + end. + +Theorem Tplus_permute_stable : term_stable Tplus_permute. +Proof. + prove_stable Tplus_permute plus_permute. +Qed. + +Definition Tplus_comm (t : term) := + match t with + | (x + y)%term => (y + x)%term + | _ => t + end. + +Theorem Tplus_comm_stable : term_stable Tplus_comm. +Proof. + prove_stable Tplus_comm plus_comm. +Qed. + +Definition Tmult_comm (t : term) := + match t with + | (x * y)%term => (y * x)%term + | _ => t + end. + +Theorem Tmult_comm_stable : term_stable Tmult_comm. +Proof. + prove_stable Tmult_comm mult_comm. +Qed. + +Definition T_OMEGA10 (t : term) := + match t with + | ((v * Tint c1 + l1) * Tint k1 + (v' * Tint c2 + l2) * Tint k2)%term => + if eq_term v v' + then (v * Tint (c1 * k1 + c2 * k2)%I + (l1 * Tint k1 + l2 * Tint k2))%term + else t + | _ => t + end. + +Theorem T_OMEGA10_stable : term_stable T_OMEGA10. +Proof. + prove_stable T_OMEGA10 OMEGA10. +Qed. + +Definition T_OMEGA11 (t : term) := + match t with + | ((v1 * Tint c1 + l1) * Tint k1 + l2)%term => + (v1 * Tint (c1 * k1) + (l1 * Tint k1 + l2))%term + | _ => t + end. + +Theorem T_OMEGA11_stable : term_stable T_OMEGA11. +Proof. + prove_stable T_OMEGA11 OMEGA11. +Qed. + +Definition T_OMEGA12 (t : term) := + match t with + | (l1 + (v2 * Tint c2 + l2) * Tint k2)%term => + (v2 * Tint (c2 * k2) + (l1 + l2 * Tint k2))%term + | _ => t + end. + +Theorem T_OMEGA12_stable : term_stable T_OMEGA12. +Proof. + prove_stable T_OMEGA12 OMEGA12. +Qed. + +Definition T_OMEGA13 (t : term) := + match t with + | (v * Tint x + l1 + (v' * Tint x' + l2))%term => + if eq_term v v' && beq x (-x') + then (l1+l2)%term + else t + | _ => t + end. + +Theorem T_OMEGA13_stable : term_stable T_OMEGA13. +Proof. + unfold term_stable, T_OMEGA13 in |- *; intros; Simplify; simpl in |- *; + apply OMEGA13. +Qed. + +Definition T_OMEGA15 (t : term) := + match t with + | (v * Tint c1 + l1 + (v' * Tint c2 + l2) * Tint k2)%term => + if eq_term v v' + then (v * Tint (c1 + c2 * k2)%I + (l1 + l2 * Tint k2))%term + else t + | _ => t + end. + +Theorem T_OMEGA15_stable : term_stable T_OMEGA15. +Proof. + prove_stable T_OMEGA15 OMEGA15. +Qed. + +Definition T_OMEGA16 (t : term) := + match t with + | ((v * Tint c + l) * Tint k)%term => (v * Tint (c * k) + l * Tint k)%term + | _ => t + end. + + +Theorem T_OMEGA16_stable : term_stable T_OMEGA16. +Proof. + prove_stable T_OMEGA16 OMEGA16. +Qed. + +Definition Tred_factor5 (t : term) := + match t with + | (x * Tint c + y)%term => if beq c 0 then y else t + | _ => t + end. + +Theorem Tred_factor5_stable : term_stable Tred_factor5. +Proof. + prove_stable Tred_factor5 red_factor5. +Qed. + +Definition Topp_plus (t : term) := + match t with + | (- (x + y))%term => (- x + - y)%term + | _ => t + end. + +Theorem Topp_plus_stable : term_stable Topp_plus. +Proof. + prove_stable Topp_plus opp_plus_distr. +Qed. + + +Definition Topp_opp (t : term) := + match t with + | (- - x)%term => x + | _ => t + end. + +Theorem Topp_opp_stable : term_stable Topp_opp. +Proof. + prove_stable Topp_opp opp_involutive. +Qed. + +Definition Topp_mult_r (t : term) := + match t with + | (- (x * Tint k))%term => (x * Tint (- k))%term + | _ => t + end. + +Theorem Topp_mult_r_stable : term_stable Topp_mult_r. +Proof. + prove_stable Topp_mult_r opp_mult_distr_r. +Qed. + +Definition Topp_one (t : term) := + match t with + | (- x)%term => (x * Tint (-(1)))%term + | _ => t + end. + +Theorem Topp_one_stable : term_stable Topp_one. +Proof. + prove_stable Topp_one opp_eq_mult_neg_1. +Qed. + +Definition Tmult_plus_distr (t : term) := + match t with + | ((n + m) * p)%term => (n * p + m * p)%term + | _ => t + end. + +Theorem Tmult_plus_distr_stable : term_stable Tmult_plus_distr. +Proof. + prove_stable Tmult_plus_distr mult_plus_distr_r. +Qed. + +Definition Tmult_opp_left (t : term) := + match t with + | (- x * Tint y)%term => (x * Tint (- y))%term + | _ => t + end. + +Theorem Tmult_opp_left_stable : term_stable Tmult_opp_left. +Proof. + prove_stable Tmult_opp_left mult_opp_comm. +Qed. + +Definition Tmult_assoc_reduced (t : term) := + match t with + | (n * Tint m * Tint p)%term => (n * Tint (m * p))%term + | _ => t + end. + +Theorem Tmult_assoc_reduced_stable : term_stable Tmult_assoc_reduced. +Proof. + prove_stable Tmult_assoc_reduced mult_assoc_reverse. +Qed. + +Definition Tred_factor0 (t : term) := (t * Tint 1)%term. + +Theorem Tred_factor0_stable : term_stable Tred_factor0. +Proof. + prove_stable Tred_factor0 red_factor0. +Qed. + +Definition Tred_factor1 (t : term) := + match t with + | (x + y)%term => + if eq_term x y + then (x * Tint 2)%term + else t + | _ => t + end. + +Theorem Tred_factor1_stable : term_stable Tred_factor1. +Proof. + prove_stable Tred_factor1 red_factor1. +Qed. + +Definition Tred_factor2 (t : term) := + match t with + | (x + y * Tint k)%term => + if eq_term x y + then (x * Tint (1 + k))%term + else t + | _ => t + end. + +Theorem Tred_factor2_stable : term_stable Tred_factor2. +Proof. + prove_stable Tred_factor2 red_factor2. +Qed. + +Definition Tred_factor3 (t : term) := + match t with + | (x * Tint k + y)%term => + if eq_term x y + then (x * Tint (1 + k))%term + else t + | _ => t + end. + +Theorem Tred_factor3_stable : term_stable Tred_factor3. +Proof. + prove_stable Tred_factor3 red_factor3. +Qed. + + +Definition Tred_factor4 (t : term) := + match t with + | (x * Tint k1 + y * Tint k2)%term => + if eq_term x y + then (x * Tint (k1 + k2))%term + else t + | _ => t + end. + +Theorem Tred_factor4_stable : term_stable Tred_factor4. +Proof. + prove_stable Tred_factor4 red_factor4. +Qed. + +Definition Tred_factor6 (t : term) := (t + Tint 0)%term. + +Theorem Tred_factor6_stable : term_stable Tred_factor6. +Proof. + prove_stable Tred_factor6 red_factor6. +Qed. + +Definition Tminus_def (t : term) := + match t with + | (x - y)%term => (x + - y)%term + | _ => t + end. + +Theorem Tminus_def_stable : term_stable Tminus_def. +Proof. + prove_stable Tminus_def minus_def. +Qed. + +(* \subsection{Fonctions de réécriture complexes} *) + +(* \subsubsection{Fonction de réduction} *) +(* Cette fonction réduit un terme dont la forme normale est un entier. Il + suffit pour cela d'échanger le constructeur [Tint] avec les opérateurs + réifiés. La réduction est ``gratuite''. *) + +Fixpoint reduce (t : term) : term := + match t with + | (x + y)%term => + match reduce x with + | Tint x' => + match reduce y with + | Tint y' => Tint (x' + y') + | y' => (Tint x' + y')%term + end + | x' => (x' + reduce y)%term + end + | (x * y)%term => + match reduce x with + | Tint x' => + match reduce y with + | Tint y' => Tint (x' * y') + | y' => (Tint x' * y')%term + end + | x' => (x' * reduce y)%term + end + | (x - y)%term => + match reduce x with + | Tint x' => + match reduce y with + | Tint y' => Tint (x' - y') + | y' => (Tint x' - y')%term + end + | x' => (x' - reduce y)%term + end + | (- x)%term => + match reduce x with + | Tint x' => Tint (- x') + | x' => (- x')%term + end + | _ => t + end. + +Theorem reduce_stable : term_stable reduce. +Proof. + unfold term_stable in |- *; intros e t; elim t; auto; + try + (intros t0 H0 t1 H1; simpl in |- *; rewrite H0; rewrite H1; + (case (reduce t0); + [ intro z0; case (reduce t1); intros; auto + | intros; auto + | intros; auto + | intros; auto + | intros; auto + | intros; auto ])); intros t0 H0; simpl in |- *; + rewrite H0; case (reduce t0); intros; auto. +Qed. + +(* \subsubsection{Fusions} + \paragraph{Fusion de deux équations} *) +(* On donne une somme de deux équations qui sont supposées normalisées. + Cette fonction prend une trace de fusion en argument et transforme + le terme en une équation normalisée. C'est une version très simplifiée + du moteur de réécriture [rewrite]. *) + +Fixpoint fusion (trace : list t_fusion) (t : term) {struct trace} : term := + match trace with + | nil => reduce t + | step :: trace' => + match step with + | F_equal => apply_right (fusion trace') (T_OMEGA10 t) + | F_cancel => fusion trace' (Tred_factor5 (T_OMEGA10 t)) + | F_left => apply_right (fusion trace') (T_OMEGA11 t) + | F_right => apply_right (fusion trace') (T_OMEGA12 t) + end + end. + +Theorem fusion_stable : forall t : list t_fusion, term_stable (fusion t). +Proof. + simple induction t; simpl in |- *; + [ exact reduce_stable + | intros stp l H; case stp; + [ apply compose_term_stable; + [ apply apply_right_stable; assumption | exact T_OMEGA10_stable ] + | unfold term_stable in |- *; intros e t1; rewrite T_OMEGA10_stable; + rewrite Tred_factor5_stable; apply H + | apply compose_term_stable; + [ apply apply_right_stable; assumption | exact T_OMEGA11_stable ] + | apply compose_term_stable; + [ apply apply_right_stable; assumption | exact T_OMEGA12_stable ] ] ]. +Qed. + +(* \paragraph{Fusion de deux équations dont une sans coefficient} *) + +Definition fusion_right (trace : list t_fusion) (t : term) : term := + match trace with + | nil => reduce t (* Il faut mettre un compute *) + | step :: trace' => + match step with + | F_equal => apply_right (fusion trace') (T_OMEGA15 t) + | F_cancel => fusion trace' (Tred_factor5 (T_OMEGA15 t)) + | F_left => apply_right (fusion trace') (Tplus_assoc_r t) + | F_right => apply_right (fusion trace') (T_OMEGA12 t) + end + end. + +(* \paragraph{Fusion avec annihilation} *) +(* Normalement le résultat est une constante *) + +Fixpoint fusion_cancel (trace : nat) (t : term) {struct trace} : term := + match trace with + | O => reduce t + | S trace' => fusion_cancel trace' (T_OMEGA13 t) + end. + +Theorem fusion_cancel_stable : forall t : nat, term_stable (fusion_cancel t). +Proof. + unfold term_stable, fusion_cancel in |- *; intros trace e; elim trace; + [ exact (reduce_stable e) + | intros n H t; elim H; exact (T_OMEGA13_stable e t) ]. +Qed. + +(* \subsubsection{Opérations affines sur une équation} *) +(* \paragraph{Multiplication scalaire et somme d'une constante} *) + +Fixpoint scalar_norm_add (trace : nat) (t : term) {struct trace} : term := + match trace with + | O => reduce t + | S trace' => apply_right (scalar_norm_add trace') (T_OMEGA11 t) + end. + +Theorem scalar_norm_add_stable : + forall t : nat, term_stable (scalar_norm_add t). +Proof. + unfold term_stable, scalar_norm_add in |- *; intros trace; elim trace; + [ exact reduce_stable + | intros n H e t; elim apply_right_stable; + [ exact (T_OMEGA11_stable e t) | exact H ] ]. +Qed. + +(* \paragraph{Multiplication scalaire} *) +Fixpoint scalar_norm (trace : nat) (t : term) {struct trace} : term := + match trace with + | O => reduce t + | S trace' => apply_right (scalar_norm trace') (T_OMEGA16 t) + end. + +Theorem scalar_norm_stable : forall t : nat, term_stable (scalar_norm t). +Proof. + unfold term_stable, scalar_norm in |- *; intros trace; elim trace; + [ exact reduce_stable + | intros n H e t; elim apply_right_stable; + [ exact (T_OMEGA16_stable e t) | exact H ] ]. +Qed. + +(* \paragraph{Somme d'une constante} *) +Fixpoint add_norm (trace : nat) (t : term) {struct trace} : term := + match trace with + | O => reduce t + | S trace' => apply_right (add_norm trace') (Tplus_assoc_r t) + end. + +Theorem add_norm_stable : forall t : nat, term_stable (add_norm t). +Proof. + unfold term_stable, add_norm in |- *; intros trace; elim trace; + [ exact reduce_stable + | intros n H e t; elim apply_right_stable; + [ exact (Tplus_assoc_r_stable e t) | exact H ] ]. +Qed. + +(* \subsection{La fonction de normalisation des termes (moteur de réécriture)} *) + + +Fixpoint rewrite (s : step) : term -> term := + match s with + | C_DO_BOTH s1 s2 => apply_both (rewrite s1) (rewrite s2) + | C_LEFT s => apply_left (rewrite s) + | C_RIGHT s => apply_right (rewrite s) + | C_SEQ s1 s2 => fun t : term => rewrite s2 (rewrite s1 t) + | C_NOP => fun t : term => t + | C_OPP_PLUS => Topp_plus + | C_OPP_OPP => Topp_opp + | C_OPP_MULT_R => Topp_mult_r + | C_OPP_ONE => Topp_one + | C_REDUCE => reduce + | C_MULT_PLUS_DISTR => Tmult_plus_distr + | C_MULT_OPP_LEFT => Tmult_opp_left + | C_MULT_ASSOC_R => Tmult_assoc_r + | C_PLUS_ASSOC_R => Tplus_assoc_r + | C_PLUS_ASSOC_L => Tplus_assoc_l + | C_PLUS_PERMUTE => Tplus_permute + | C_PLUS_COMM => Tplus_comm + | C_RED0 => Tred_factor0 + | C_RED1 => Tred_factor1 + | C_RED2 => Tred_factor2 + | C_RED3 => Tred_factor3 + | C_RED4 => Tred_factor4 + | C_RED5 => Tred_factor5 + | C_RED6 => Tred_factor6 + | C_MULT_ASSOC_REDUCED => Tmult_assoc_reduced + | C_MINUS => Tminus_def + | C_MULT_COMM => Tmult_comm + end. + +Theorem rewrite_stable : forall s : step, term_stable (rewrite s). +Proof. + simple induction s; simpl in |- *; + [ intros; apply apply_both_stable; auto + | intros; apply apply_left_stable; auto + | intros; apply apply_right_stable; auto + | unfold term_stable in |- *; intros; elim H0; apply H + | unfold term_stable in |- *; auto + | exact Topp_plus_stable + | exact Topp_opp_stable + | exact Topp_mult_r_stable + | exact Topp_one_stable + | exact reduce_stable + | exact Tmult_plus_distr_stable + | exact Tmult_opp_left_stable + | exact Tmult_assoc_r_stable + | exact Tplus_assoc_r_stable + | exact Tplus_assoc_l_stable + | exact Tplus_permute_stable + | exact Tplus_comm_stable + | exact Tred_factor0_stable + | exact Tred_factor1_stable + | exact Tred_factor2_stable + | exact Tred_factor3_stable + | exact Tred_factor4_stable + | exact Tred_factor5_stable + | exact Tred_factor6_stable + | exact Tmult_assoc_reduced_stable + | exact Tminus_def_stable + | exact Tmult_comm_stable ]. +Qed. + +(* \subsection{tactiques de résolution d'un but omega normalisé} + Trace de la procédure +\subsubsection{Tactiques générant une contradiction} +\paragraph{[O_CONSTANT_NOT_NUL]} *) + +Definition constant_not_nul (i : nat) (h : hyps) := + match nth_hyps i h with + | EqTerm (Tint Nul) (Tint n) => + if beq n Nul then h else absurd + | _ => h + end. + +Theorem constant_not_nul_valid : + forall i : nat, valid_hyps (constant_not_nul i). +Proof. + unfold valid_hyps, constant_not_nul in |- *; intros; + generalize (nth_valid ep e i lp); Simplify; simpl in |- *. + + elim_beq i1 i0; auto; simpl in |- *; intros H1 H2; + elim H1; symmetry in |- *; auto. +Qed. + +(* \paragraph{[O_CONSTANT_NEG]} *) + +Definition constant_neg (i : nat) (h : hyps) := + match nth_hyps i h with + | LeqTerm (Tint Nul) (Tint Neg) => + if bgt Nul Neg then absurd else h + | _ => h + end. + +Theorem constant_neg_valid : forall i : nat, valid_hyps (constant_neg i). +Proof. + unfold valid_hyps, constant_neg in |- *; intros; + generalize (nth_valid ep e i lp); Simplify; simpl in |- *. + rewrite gt_lt_iff in H0; rewrite le_lt_iff; intuition. +Qed. + +(* \paragraph{[NOT_EXACT_DIVIDE]} *) +Definition not_exact_divide (k1 k2 : int) (body : term) + (t i : nat) (l : hyps) := + match nth_hyps i l with + | EqTerm (Tint Nul) b => + if beq Nul 0 && + eq_term (scalar_norm_add t (body * Tint k1 + Tint k2)%term) b && + bgt k2 0 && + bgt k1 k2 + then absurd + else l + | _ => l + end. + +Theorem not_exact_divide_valid : + forall (k1 k2 : int) (body : term) (t i : nat), + valid_hyps (not_exact_divide k1 k2 body t i). +Proof. + unfold valid_hyps, not_exact_divide in |- *; intros; + generalize (nth_valid ep e i lp); Simplify. + rewrite (scalar_norm_add_stable t e), <-H1. + do 2 rewrite <- scalar_norm_add_stable; simpl in *; intros. + absurd (interp_term e body * k1 + k2 = 0); + [ now apply OMEGA4 | symmetry; auto ]. +Qed. + +(* \paragraph{[O_CONTRADICTION]} *) + +Definition contradiction (t i j : nat) (l : hyps) := + match nth_hyps i l with + | LeqTerm (Tint Nul) b1 => + match nth_hyps j l with + | LeqTerm (Tint Nul') b2 => + match fusion_cancel t (b1 + b2)%term with + | Tint k => if beq Nul 0 && beq Nul' 0 && bgt 0 k + then absurd + else l + | _ => l + end + | _ => l + end + | _ => l + end. + +Theorem contradiction_valid : + forall t i j : nat, valid_hyps (contradiction t i j). +Proof. + unfold valid_hyps, contradiction in |- *; intros t i j ep e l H; + generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H); + case (nth_hyps i l); auto; intros t1 t2; case t1; + auto; case (nth_hyps j l); + auto; intros t3 t4; case t3; auto; + simpl in |- *; intros z z' H1 H2; + generalize (refl_equal (interp_term e (fusion_cancel t (t2 + t4)%term))); + pattern (fusion_cancel t (t2 + t4)%term) at 2 3 in |- *; + case (fusion_cancel t (t2 + t4)%term); simpl in |- *; + auto; intro k; elim (fusion_cancel_stable t); simpl in |- *. + Simplify; intro H3. + generalize (OMEGA2 _ _ H2 H1); rewrite H3. + rewrite gt_lt_iff in H0; rewrite le_lt_iff; intuition. +Qed. + +(* \paragraph{[O_NEGATE_CONTRADICT]} *) + +Definition negate_contradict (i1 i2 : nat) (h : hyps) := + match nth_hyps i1 h with + | EqTerm (Tint Nul) b1 => + match nth_hyps i2 h with + | NeqTerm (Tint Nul') b2 => + if beq Nul 0 && beq Nul' 0 && eq_term b1 b2 + then absurd + else h + | _ => h + end + | NeqTerm (Tint Nul) b1 => + match nth_hyps i2 h with + | EqTerm (Tint Nul') b2 => + if beq Nul 0 && beq Nul' 0 && eq_term b1 b2 + then absurd + else h + | _ => h + end + | _ => h + end. + +Definition negate_contradict_inv (t i1 i2 : nat) (h : hyps) := + match nth_hyps i1 h with + | EqTerm (Tint Nul) b1 => + match nth_hyps i2 h with + | NeqTerm (Tint Nul') b2 => + if beq Nul 0 && beq Nul' 0 && + eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term) + then absurd + else h + | _ => h + end + | NeqTerm (Tint Nul) b1 => + match nth_hyps i2 h with + | EqTerm (Tint Nul') b2 => + if beq Nul 0 && beq Nul' 0 && + eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term) + then absurd + else h + | _ => h + end + | _ => h + end. + +Theorem negate_contradict_valid : + forall i j : nat, valid_hyps (negate_contradict i j). +Proof. + unfold valid_hyps, negate_contradict in |- *; intros i j ep e l H; + generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H); + case (nth_hyps i l); auto; intros t1 t2; case t1; + auto; intros z; auto; case (nth_hyps j l); + auto; intros t3 t4; case t3; auto; intros z'; + auto; simpl in |- *; intros H1 H2; Simplify. +Qed. + +Theorem negate_contradict_inv_valid : + forall t i j : nat, valid_hyps (negate_contradict_inv t i j). +Proof. + unfold valid_hyps, negate_contradict_inv in |- *; intros t i j ep e l H; + generalize (nth_valid _ _ i _ H); generalize (nth_valid _ _ j _ H); + case (nth_hyps i l); auto; intros t1 t2; case t1; + auto; intros z; auto; case (nth_hyps j l); + auto; intros t3 t4; case t3; auto; intros z'; + auto; simpl in |- *; intros H1 H2; Simplify; + [ + rewrite <- scalar_norm_stable in H2; simpl in *; + elim (mult_integral (interp_term e t4) (-(1))); intuition; + elim minus_one_neq_zero; auto + | + elim H2; clear H2; + rewrite <- scalar_norm_stable; simpl in *; + now rewrite <- H1, mult_0_l + ]. +Qed. + +(* \subsubsection{Tactiques générant une nouvelle équation} *) +(* \paragraph{[O_SUM]} + C'est une oper2 valide mais elle traite plusieurs cas à la fois (suivant + les opérateurs de comparaison des deux arguments) d'où une + preuve un peu compliquée. On utilise quelques lemmes qui sont des + généralisations des théorèmes utilisés par OMEGA. *) + +Definition sum (k1 k2 : int) (trace : list t_fusion) + (prop1 prop2 : proposition) := + match prop1 with + | EqTerm (Tint Null) b1 => + match prop2 with + | EqTerm (Tint Null') b2 => + if beq Null 0 && beq Null' 0 + then EqTerm (Tint 0) (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term) + else TrueTerm + | LeqTerm (Tint Null') b2 => + if beq Null 0 && beq Null' 0 && bgt k2 0 + then LeqTerm (Tint 0) + (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term) + else TrueTerm + | _ => TrueTerm + end + | LeqTerm (Tint Null) b1 => + if beq Null 0 && bgt k1 0 + then match prop2 with + | EqTerm (Tint Null') b2 => + if beq Null' 0 then + LeqTerm (Tint 0) + (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term) + else TrueTerm + | LeqTerm (Tint Null') b2 => + if beq Null' 0 && bgt k2 0 + then LeqTerm (Tint 0) + (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term) + else TrueTerm + | _ => TrueTerm + end + else TrueTerm + | NeqTerm (Tint Null) b1 => + match prop2 with + | EqTerm (Tint Null') b2 => + if beq Null 0 && beq Null' 0 && (negb (beq k1 0)) + then NeqTerm (Tint 0) + (fusion trace (b1 * Tint k1 + b2 * Tint k2)%term) + else TrueTerm + | _ => TrueTerm + end + | _ => TrueTerm + end. + + +Theorem sum_valid : + forall (k1 k2 : int) (t : list t_fusion), valid2 (sum k1 k2 t). +Proof. + unfold valid2 in |- *; intros k1 k2 t ep e p1 p2; unfold sum in |- *; + Simplify; simpl in |- *; auto; try elim (fusion_stable t); + simpl in |- *; intros; + [ apply sum1; assumption + | apply sum2; try assumption; apply sum4; assumption + | rewrite plus_comm; apply sum2; try assumption; apply sum4; assumption + | apply sum3; try assumption; apply sum4; assumption + | apply sum5; auto ]. +Qed. + +(* \paragraph{[O_EXACT_DIVIDE]} + c'est une oper1 valide mais on préfère une substitution a ce point la *) + +Definition exact_divide (k : int) (body : term) (t : nat) + (prop : proposition) := + match prop with + | EqTerm (Tint Null) b => + if beq Null 0 && + eq_term (scalar_norm t (body * Tint k)%term) b && + negb (beq k 0) + then EqTerm (Tint 0) body + else TrueTerm + | NeqTerm (Tint Null) b => + if beq Null 0 && + eq_term (scalar_norm t (body * Tint k)%term) b && + negb (beq k 0) + then NeqTerm (Tint 0) body + else TrueTerm + | _ => TrueTerm + end. + +Theorem exact_divide_valid : + forall (k : int) (t : term) (n : nat), valid1 (exact_divide k t n). +Proof. + unfold valid1, exact_divide in |- *; intros k1 k2 t ep e p1; + Simplify; simpl; auto; subst; + rewrite <- scalar_norm_stable; simpl; intros; + [ destruct (mult_integral _ _ (sym_eq H0)); intuition + | contradict H0; rewrite <- H0, mult_0_l; auto + ]. +Qed. + + +(* \paragraph{[O_DIV_APPROX]} + La preuve reprend le schéma de la précédente mais on + est sur une opération de type valid1 et non sur une opération terminale. *) + +Definition divide_and_approx (k1 k2 : int) (body : term) + (t : nat) (prop : proposition) := + match prop with + | LeqTerm (Tint Null) b => + if beq Null 0 && + eq_term (scalar_norm_add t (body * Tint k1 + Tint k2)%term) b && + bgt k1 0 && + bgt k1 k2 + then LeqTerm (Tint 0) body + else prop + | _ => prop + end. + +Theorem divide_and_approx_valid : + forall (k1 k2 : int) (body : term) (t : nat), + valid1 (divide_and_approx k1 k2 body t). +Proof. + unfold valid1, divide_and_approx in |- *; intros k1 k2 body t ep e p1; + Simplify; simpl; auto; subst; + elim (scalar_norm_add_stable t e); simpl in |- *. + intro H2; apply mult_le_approx with (3 := H2); assumption. +Qed. + +(* \paragraph{[MERGE_EQ]} *) + +Definition merge_eq (t : nat) (prop1 prop2 : proposition) := + match prop1 with + | LeqTerm (Tint Null) b1 => + match prop2 with + | LeqTerm (Tint Null') b2 => + if beq Null 0 && beq Null' 0 && + eq_term b1 (scalar_norm t (b2 * Tint (-(1)))%term) + then EqTerm (Tint 0) b1 + else TrueTerm + | _ => TrueTerm + end + | _ => TrueTerm + end. + +Theorem merge_eq_valid : forall n : nat, valid2 (merge_eq n). +Proof. + unfold valid2, merge_eq in |- *; intros n ep e p1 p2; Simplify; simpl in |- *; + auto; elim (scalar_norm_stable n e); simpl in |- *; + intros; symmetry in |- *; apply OMEGA8 with (2 := H0); + [ assumption | elim opp_eq_mult_neg_1; trivial ]. +Qed. + + + +(* \paragraph{[O_CONSTANT_NUL]} *) + +Definition constant_nul (i : nat) (h : hyps) := + match nth_hyps i h with + | NeqTerm (Tint Null) (Tint Null') => + if beq Null Null' then absurd else h + | _ => h + end. + +Theorem constant_nul_valid : forall i : nat, valid_hyps (constant_nul i). +Proof. + unfold valid_hyps, constant_nul in |- *; intros; + generalize (nth_valid ep e i lp); Simplify; simpl in |- *; + intro H1; absurd (0 = 0); intuition. +Qed. + +(* \paragraph{[O_STATE]} *) + +Definition state (m : int) (s : step) (prop1 prop2 : proposition) := + match prop1 with + | EqTerm (Tint Null) b1 => + match prop2 with + | EqTerm b2 b3 => + if beq Null 0 + then EqTerm (Tint 0) (rewrite s (b1 + (- b3 + b2) * Tint m)%term) + else TrueTerm + | _ => TrueTerm + end + | _ => TrueTerm + end. + +Theorem state_valid : forall (m : int) (s : step), valid2 (state m s). +Proof. + unfold valid2 in |- *; intros m s ep e p1 p2; unfold state in |- *; Simplify; + simpl in |- *; auto; elim (rewrite_stable s e); simpl in |- *; + intros H1 H2; elim H1. + now rewrite H2, plus_opp_l, plus_0_l, mult_0_l. +Qed. + +(* \subsubsection{Tactiques générant plusieurs but} + \paragraph{[O_SPLIT_INEQ]} + La seule pour le moment (tant que la normalisation n'est pas réfléchie). *) + +Definition split_ineq (i t : nat) (f1 f2 : hyps -> lhyps) + (l : hyps) := + match nth_hyps i l with + | NeqTerm (Tint Null) b1 => + if beq Null 0 then + f1 (LeqTerm (Tint 0) (add_norm t (b1 + Tint (-(1)))%term) :: l) ++ + f2 + (LeqTerm (Tint 0) + (scalar_norm_add t (b1 * Tint (-(1)) + Tint (-(1)))%term) :: l) + else l :: nil + | _ => l :: nil + end. + +Theorem split_ineq_valid : + forall (i t : nat) (f1 f2 : hyps -> lhyps), + valid_list_hyps f1 -> + valid_list_hyps f2 -> valid_list_hyps (split_ineq i t f1 f2). +Proof. + unfold valid_list_hyps, split_ineq in |- *; intros i t f1 f2 H1 H2 ep e lp H; + generalize (nth_valid _ _ i _ H); case (nth_hyps i lp); + simpl in |- *; auto; intros t1 t2; case t1; simpl in |- *; + auto; intros z; simpl in |- *; auto; intro H3. + Simplify. + apply append_valid; elim (OMEGA19 (interp_term e t2)); + [ intro H4; left; apply H1; simpl in |- *; elim (add_norm_stable t); + simpl in |- *; auto + | intro H4; right; apply H2; simpl in |- *; elim (scalar_norm_add_stable t); + simpl in |- *; auto + | generalize H3; unfold not in |- *; intros E1 E2; apply E1; + symmetry in |- *; trivial ]. +Qed. + + +(* \subsection{La fonction de rejeu de la trace} *) + +Fixpoint execute_omega (t : t_omega) (l : hyps) {struct t} : lhyps := + match t with + | O_CONSTANT_NOT_NUL n => singleton (constant_not_nul n l) + | O_CONSTANT_NEG n => singleton (constant_neg n l) + | O_DIV_APPROX k1 k2 body t cont n => + execute_omega cont (apply_oper_1 n (divide_and_approx k1 k2 body t) l) + | O_NOT_EXACT_DIVIDE k1 k2 body t i => + singleton (not_exact_divide k1 k2 body t i l) + | O_EXACT_DIVIDE k body t cont n => + execute_omega cont (apply_oper_1 n (exact_divide k body t) l) + | O_SUM k1 i1 k2 i2 t cont => + execute_omega cont (apply_oper_2 i1 i2 (sum k1 k2 t) l) + | O_CONTRADICTION t i j => singleton (contradiction t i j l) + | O_MERGE_EQ t i1 i2 cont => + execute_omega cont (apply_oper_2 i1 i2 (merge_eq t) l) + | O_SPLIT_INEQ t i cont1 cont2 => + split_ineq i t (execute_omega cont1) (execute_omega cont2) l + | O_CONSTANT_NUL i => singleton (constant_nul i l) + | O_NEGATE_CONTRADICT i j => singleton (negate_contradict i j l) + | O_NEGATE_CONTRADICT_INV t i j => + singleton (negate_contradict_inv t i j l) + | O_STATE m s i1 i2 cont => + execute_omega cont (apply_oper_2 i1 i2 (state m s) l) + end. + +Theorem omega_valid : forall t : t_omega, valid_list_hyps (execute_omega t). +Proof. + simple induction t; simpl in |- *; + [ unfold valid_list_hyps in |- *; simpl in |- *; intros; left; + apply (constant_not_nul_valid n ep e lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros; left; + apply (constant_neg_valid n ep e lp H) + | unfold valid_list_hyps, valid_hyps in |- *; + intros k1 k2 body n t' Ht' m ep e lp H; apply Ht'; + apply + (apply_oper_1_valid m (divide_and_approx k1 k2 body n) + (divide_and_approx_valid k1 k2 body n) ep e lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros; left; + apply (not_exact_divide_valid i i0 t0 n n0 ep e lp H) + | unfold valid_list_hyps, valid_hyps in |- *; + intros k body n t' Ht' m ep e lp H; apply Ht'; + apply + (apply_oper_1_valid m (exact_divide k body n) + (exact_divide_valid k body n) ep e lp H) + | unfold valid_list_hyps, valid_hyps in |- *; + intros k1 i1 k2 i2 trace t' Ht' ep e lp H; apply Ht'; + apply + (apply_oper_2_valid i1 i2 (sum k1 k2 trace) (sum_valid k1 k2 trace) ep e + lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros; left; + apply (contradiction_valid n n0 n1 ep e lp H) + | unfold valid_list_hyps, valid_hyps in |- *; + intros trace i1 i2 t' Ht' ep e lp H; apply Ht'; + apply + (apply_oper_2_valid i1 i2 (merge_eq trace) (merge_eq_valid trace) ep e + lp H) + | intros t' i k1 H1 k2 H2; unfold valid_list_hyps in |- *; simpl in |- *; + intros ep e lp H; + apply + (split_ineq_valid i t' (execute_omega k1) (execute_omega k2) H1 H2 ep e + lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros i ep e lp H; left; + apply (constant_nul_valid i ep e lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros i j ep e lp H; left; + apply (negate_contradict_valid i j ep e lp H) + | unfold valid_list_hyps in |- *; simpl in |- *; intros n i j ep e lp H; + left; apply (negate_contradict_inv_valid n i j ep e lp H) + | unfold valid_list_hyps, valid_hyps in |- *; + intros m s i1 i2 t' Ht' ep e lp H; apply Ht'; + apply (apply_oper_2_valid i1 i2 (state m s) (state_valid m s) ep e lp H) ]. +Qed. + + +(* \subsection{Les opérations globales sur le but} + \subsubsection{Normalisation} *) + +Definition move_right (s : step) (p : proposition) := + match p with + | EqTerm t1 t2 => EqTerm (Tint 0) (rewrite s (t1 + - t2)%term) + | LeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t2 + - t1)%term) + | GeqTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t1 + - t2)%term) + | LtTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t2 + Tint (-(1)) + - t1)%term) + | GtTerm t1 t2 => LeqTerm (Tint 0) (rewrite s (t1 + Tint (-(1)) + - t2)%term) + | NeqTerm t1 t2 => NeqTerm (Tint 0) (rewrite s (t1 + - t2)%term) + | p => p + end. + +Theorem move_right_valid : forall s : step, valid1 (move_right s). +Proof. + unfold valid1, move_right in |- *; intros s ep e p; Simplify; simpl in |- *; + elim (rewrite_stable s e); simpl in |- *; + [ symmetry in |- *; apply egal_left; assumption + | intro; apply le_left; assumption + | intro; apply le_left; rewrite <- ge_le_iff; assumption + | intro; apply lt_left; rewrite <- gt_lt_iff; assumption + | intro; apply lt_left; assumption + | intro; apply ne_left_2; assumption ]. +Qed. + +Definition do_normalize (i : nat) (s : step) := apply_oper_1 i (move_right s). + +Theorem do_normalize_valid : + forall (i : nat) (s : step), valid_hyps (do_normalize i s). +Proof. + intros; unfold do_normalize in |- *; apply apply_oper_1_valid; + apply move_right_valid. +Qed. + +Fixpoint do_normalize_list (l : list step) (i : nat) + (h : hyps) {struct l} : hyps := + match l with + | s :: l' => do_normalize_list l' (S i) (do_normalize i s h) + | nil => h + end. + +Theorem do_normalize_list_valid : + forall (l : list step) (i : nat), valid_hyps (do_normalize_list l i). +Proof. + simple induction l; simpl in |- *; unfold valid_hyps in |- *; + [ auto + | intros a l' Hl' i ep e lp H; unfold valid_hyps in Hl'; apply Hl'; + apply (do_normalize_valid i a ep e lp); assumption ]. +Qed. + +Theorem normalize_goal : + forall (s : list step) (ep : list Prop) (env : list int) (l : hyps), + interp_goal ep env (do_normalize_list s 0 l) -> interp_goal ep env l. +Proof. + intros; apply valid_goal with (2 := H); apply do_normalize_list_valid. +Qed. + +(* \subsubsection{Exécution de la trace} *) + +Theorem execute_goal : + forall (t : t_omega) (ep : list Prop) (env : list int) (l : hyps), + interp_list_goal ep env (execute_omega t l) -> interp_goal ep env l. +Proof. + intros; apply (goal_valid (execute_omega t) (omega_valid t) ep env l H). +Qed. + + +Theorem append_goal : + forall (ep : list Prop) (e : list int) (l1 l2 : lhyps), + interp_list_goal ep e l1 /\ interp_list_goal ep e l2 -> + interp_list_goal ep e (l1 ++ l2). +Proof. + intros ep e; simple induction l1; + [ simpl in |- *; intros l2 (H1, H2); assumption + | simpl in |- *; intros h1 t1 HR l2 ((H1, H2), H3); split; auto ]. +Qed. + +(* A simple decidability checker : if the proposition belongs to the + simple grammar describe below then it is decidable. Proof is by + induction and uses well known theorem about arithmetic and propositional + calculus *) + +Fixpoint decidability (p : proposition) : bool := + match p with + | EqTerm _ _ => true + | LeqTerm _ _ => true + | GeqTerm _ _ => true + | GtTerm _ _ => true + | LtTerm _ _ => true + | NeqTerm _ _ => true + | FalseTerm => true + | TrueTerm => true + | Tnot t => decidability t + | Tand t1 t2 => decidability t1 && decidability t2 + | Timp t1 t2 => decidability t1 && decidability t2 + | Tor t1 t2 => decidability t1 && decidability t2 + | Tprop _ => false + end. + +Theorem decidable_correct : + forall (ep : list Prop) (e : list int) (p : proposition), + decidability p = true -> decidable (interp_proposition ep e p). +Proof. + simple induction p; simpl in |- *; intros; + [ apply dec_eq + | apply dec_le + | left; auto + | right; unfold not in |- *; auto + | apply dec_not; auto + | apply dec_ge + | apply dec_gt + | apply dec_lt + | apply dec_ne + | apply dec_or; elim andb_prop with (1 := H1); auto + | apply dec_and; elim andb_prop with (1 := H1); auto + | apply dec_imp; elim andb_prop with (1 := H1); auto + | discriminate H ]. +Qed. + +(* An interpretation function for a complete goal with an explicit + conclusion. We use an intermediate fixpoint. *) + +Fixpoint interp_full_goal (envp : list Prop) (env : list int) + (c : proposition) (l : hyps) {struct l} : Prop := + match l with + | nil => interp_proposition envp env c + | p' :: l' => + interp_proposition envp env p' -> interp_full_goal envp env c l' + end. + +Definition interp_full (ep : list Prop) (e : list int) + (lc : hyps * proposition) : Prop := + match lc with + | (l, c) => interp_full_goal ep e c l + end. + +(* Relates the interpretation of a complete goal with the interpretation + of its hypothesis and conclusion *) + +Theorem interp_full_false : + forall (ep : list Prop) (e : list int) (l : hyps) (c : proposition), + (interp_hyps ep e l -> interp_proposition ep e c) -> interp_full ep e (l, c). +Proof. + simple induction l; unfold interp_full in |- *; simpl in |- *; + [ auto | intros a l1 H1 c H2 H3; apply H1; auto ]. +Qed. + +(* Push the conclusion in the list of hypothesis using a double negation + If the decidability cannot be "proven", then just forget about the + conclusion (equivalent of replacing it with false) *) + +Definition to_contradict (lc : hyps * proposition) := + match lc with + | (l, c) => if decidability c then Tnot c :: l else l + end. + +(* The previous operation is valid in the sense that the new list of + hypothesis implies the original goal *) + +Theorem to_contradict_valid : + forall (ep : list Prop) (e : list int) (lc : hyps * proposition), + interp_goal ep e (to_contradict lc) -> interp_full ep e lc. +Proof. + intros ep e lc; case lc; intros l c; simpl in |- *; + pattern (decidability c) in |- *; apply bool_eq_ind; + [ simpl in |- *; intros H H1; apply interp_full_false; intros H2; + apply not_not; + [ apply decidable_correct; assumption + | unfold not at 1 in |- *; intro H3; apply hyps_to_goal with (2 := H2); + auto ] + | intros H1 H2; apply interp_full_false; intro H3; + elim hyps_to_goal with (1 := H2); assumption ]. +Qed. + +(* [map_cons x l] adds [x] at the head of each list in [l] (which is a list + of lists *) + +Fixpoint map_cons (A : Set) (x : A) (l : list (list A)) {struct l} : + list (list A) := + match l with + | nil => nil + | l :: ll => (x :: l) :: map_cons A x ll + end. + +(* This function breaks up a list of hypothesis in a list of simpler + list of hypothesis that together implie the original one. The goal + of all this is to transform the goal in a list of solvable problems. + Note that : + - we need a way to drive the analysis as some hypotheis may not + require a split. + - this procedure must be perfectly mimicked by the ML part otherwise + hypothesis will get desynchronised and this will be a mess. + *) + +Fixpoint destructure_hyps (nn : nat) (ll : hyps) {struct nn} : lhyps := + match nn with + | O => ll :: nil + | S n => + match ll with + | nil => nil :: nil + | Tor p1 p2 :: l => + destructure_hyps n (p1 :: l) ++ destructure_hyps n (p2 :: l) + | Tand p1 p2 :: l => destructure_hyps n (p1 :: p2 :: l) + | Timp p1 p2 :: l => + if decidability p1 + then + destructure_hyps n (Tnot p1 :: l) ++ destructure_hyps n (p2 :: l) + else map_cons _ (Timp p1 p2) (destructure_hyps n l) + | Tnot p :: l => + match p with + | Tnot p1 => + if decidability p1 + then destructure_hyps n (p1 :: l) + else map_cons _ (Tnot (Tnot p1)) (destructure_hyps n l) + | Tor p1 p2 => destructure_hyps n (Tnot p1 :: Tnot p2 :: l) + | Tand p1 p2 => + if decidability p1 + then + destructure_hyps n (Tnot p1 :: l) ++ + destructure_hyps n (Tnot p2 :: l) + else map_cons _ (Tnot p) (destructure_hyps n l) + | _ => map_cons _ (Tnot p) (destructure_hyps n l) + end + | x :: l => map_cons _ x (destructure_hyps n l) + end + end. + +Theorem map_cons_val : + forall (ep : list Prop) (e : list int) (p : proposition) (l : lhyps), + interp_proposition ep e p -> + interp_list_hyps ep e l -> interp_list_hyps ep e (map_cons _ p l). +Proof. + simple induction l; simpl in |- *; [ auto | intros; elim H1; intro H2; auto ]. +Qed. + +Hint Resolve map_cons_val append_valid decidable_correct. + +Theorem destructure_hyps_valid : + forall n : nat, valid_list_hyps (destructure_hyps n). +Proof. + simple induction n; + [ unfold valid_list_hyps in |- *; simpl in |- *; auto + | unfold valid_list_hyps at 2 in |- *; intros n1 H ep e lp; case lp; + [ simpl in |- *; auto + | intros p l; case p; + try + (simpl in |- *; intros; apply map_cons_val; simpl in |- *; elim H0; + auto); + [ intro p'; case p'; + try + (simpl in |- *; intros; apply map_cons_val; simpl in |- *; elim H0; + auto); + [ simpl in |- *; intros p1 (H1, H2); + pattern (decidability p1) in |- *; apply bool_eq_ind; + intro H3; + [ apply H; simpl in |- *; split; + [ apply not_not; auto | assumption ] + | auto ] + | simpl in |- *; intros p1 p2 (H1, H2); apply H; simpl in |- *; + elim not_or with (1 := H1); auto + | simpl in |- *; intros p1 p2 (H1, H2); + pattern (decidability p1) in |- *; apply bool_eq_ind; + intro H3; + [ apply append_valid; elim not_and with (2 := H1); + [ intro; left; apply H; simpl in |- *; auto + | intro; right; apply H; simpl in |- *; auto + | auto ] + | auto ] ] + | simpl in |- *; intros p1 p2 (H1, H2); apply append_valid; + (elim H1; intro H3; simpl in |- *; [ left | right ]); + apply H; simpl in |- *; auto + | simpl in |- *; intros; apply H; simpl in |- *; tauto + | simpl in |- *; intros p1 p2 (H1, H2); + pattern (decidability p1) in |- *; apply bool_eq_ind; + intro H3; + [ apply append_valid; elim imp_simp with (2 := H1); + [ intro H4; left; simpl in |- *; apply H; simpl in |- *; auto + | intro H4; right; simpl in |- *; apply H; simpl in |- *; auto + | auto ] + | auto ] ] ] ]. +Qed. + +Definition prop_stable (f : proposition -> proposition) := + forall (ep : list Prop) (e : list int) (p : proposition), + interp_proposition ep e p <-> interp_proposition ep e (f p). + +Definition p_apply_left (f : proposition -> proposition) + (p : proposition) := + match p with + | Timp x y => Timp (f x) y + | Tor x y => Tor (f x) y + | Tand x y => Tand (f x) y + | Tnot x => Tnot (f x) + | x => x + end. + +Theorem p_apply_left_stable : + forall f : proposition -> proposition, + prop_stable f -> prop_stable (p_apply_left f). +Proof. + unfold prop_stable in |- *; intros f H ep e p; split; + (case p; simpl in |- *; auto; intros p1; elim (H ep e p1); tauto). +Qed. + +Definition p_apply_right (f : proposition -> proposition) + (p : proposition) := + match p with + | Timp x y => Timp x (f y) + | Tor x y => Tor x (f y) + | Tand x y => Tand x (f y) + | Tnot x => Tnot (f x) + | x => x + end. + +Theorem p_apply_right_stable : + forall f : proposition -> proposition, + prop_stable f -> prop_stable (p_apply_right f). +Proof. + unfold prop_stable in |- *; intros f H ep e p; split; + (case p; simpl in |- *; auto; + [ intros p1; elim (H ep e p1); tauto + | intros p1 p2; elim (H ep e p2); tauto + | intros p1 p2; elim (H ep e p2); tauto + | intros p1 p2; elim (H ep e p2); tauto ]). +Qed. + +Definition p_invert (f : proposition -> proposition) + (p : proposition) := + match p with + | EqTerm x y => Tnot (f (NeqTerm x y)) + | LeqTerm x y => Tnot (f (GtTerm x y)) + | GeqTerm x y => Tnot (f (LtTerm x y)) + | GtTerm x y => Tnot (f (LeqTerm x y)) + | LtTerm x y => Tnot (f (GeqTerm x y)) + | NeqTerm x y => Tnot (f (EqTerm x y)) + | x => x + end. + +Theorem p_invert_stable : + forall f : proposition -> proposition, + prop_stable f -> prop_stable (p_invert f). +Proof. + unfold prop_stable in |- *; intros f H ep e p; split; + (case p; simpl in |- *; auto; + [ intros t1 t2; elim (H ep e (NeqTerm t1 t2)); simpl in |- *; + generalize (dec_eq (interp_term e t1) (interp_term e t2)); + unfold decidable in |- *; tauto + | intros t1 t2; elim (H ep e (GtTerm t1 t2)); simpl in |- *; + generalize (dec_gt (interp_term e t1) (interp_term e t2)); + unfold decidable in |- *; rewrite le_lt_iff, <- gt_lt_iff; tauto + | intros t1 t2; elim (H ep e (LtTerm t1 t2)); simpl in |- *; + generalize (dec_lt (interp_term e t1) (interp_term e t2)); + unfold decidable in |- *; rewrite ge_le_iff, le_lt_iff; tauto + | intros t1 t2; elim (H ep e (LeqTerm t1 t2)); simpl in |- *; + generalize (dec_gt (interp_term e t1) (interp_term e t2)); + unfold decidable in |- *; repeat rewrite le_lt_iff; + repeat rewrite gt_lt_iff; tauto + | intros t1 t2; elim (H ep e (GeqTerm t1 t2)); simpl in |- *; + generalize (dec_lt (interp_term e t1) (interp_term e t2)); + unfold decidable in |- *; repeat rewrite ge_le_iff; + repeat rewrite le_lt_iff; tauto + | intros t1 t2; elim (H ep e (EqTerm t1 t2)); simpl in |- *; + generalize (dec_eq (interp_term e t1) (interp_term e t2)); + unfold decidable; tauto ]). +Qed. + +Theorem move_right_stable : forall s : step, prop_stable (move_right s). +Proof. + unfold move_right, prop_stable in |- *; intros s ep e p; split; + [ Simplify; simpl in |- *; elim (rewrite_stable s e); simpl in |- *; + [ symmetry in |- *; apply egal_left; assumption + | intro; apply le_left; assumption + | intro; apply le_left; rewrite <- ge_le_iff; assumption + | intro; apply lt_left; rewrite <- gt_lt_iff; assumption + | intro; apply lt_left; assumption + | intro; apply ne_left_2; assumption ] + | case p; simpl in |- *; intros; auto; generalize H; elim (rewrite_stable s); + simpl in |- *; intro H1; + [ rewrite (plus_0_r_reverse (interp_term e t0)); rewrite H1; + rewrite plus_permute; rewrite plus_opp_r; + rewrite plus_0_r; trivial + | apply (fun a b => plus_le_reg_r a b (- interp_term e t)); + rewrite plus_opp_r; assumption + | rewrite ge_le_iff; + apply (fun a b => plus_le_reg_r a b (- interp_term e t0)); + rewrite plus_opp_r; assumption + | rewrite gt_lt_iff; apply lt_left_inv; assumption + | apply lt_left_inv; assumption + | unfold not in |- *; intro H2; apply H1; + rewrite H2; rewrite plus_opp_r; trivial ] ]. +Qed. + + +Fixpoint p_rewrite (s : p_step) : proposition -> proposition := + match s with + | P_LEFT s => p_apply_left (p_rewrite s) + | P_RIGHT s => p_apply_right (p_rewrite s) + | P_STEP s => move_right s + | P_INVERT s => p_invert (move_right s) + | P_NOP => fun p : proposition => p + end. + +Theorem p_rewrite_stable : forall s : p_step, prop_stable (p_rewrite s). +Proof. + simple induction s; simpl in |- *; + [ intros; apply p_apply_left_stable; trivial + | intros; apply p_apply_right_stable; trivial + | intros; apply p_invert_stable; apply move_right_stable + | apply move_right_stable + | unfold prop_stable in |- *; simpl in |- *; intros; split; auto ]. +Qed. + +Fixpoint normalize_hyps (l : list h_step) (lh : hyps) {struct l} : hyps := + match l with + | nil => lh + | pair_step i s :: r => normalize_hyps r (apply_oper_1 i (p_rewrite s) lh) + end. + +Theorem normalize_hyps_valid : + forall l : list h_step, valid_hyps (normalize_hyps l). +Proof. + simple induction l; unfold valid_hyps in |- *; simpl in |- *; + [ auto + | intros n_s r; case n_s; intros n s H ep e lp H1; apply H; + apply apply_oper_1_valid; + [ unfold valid1 in |- *; intros ep1 e1 p1 H2; + elim (p_rewrite_stable s ep1 e1 p1); auto + | assumption ] ]. +Qed. + +Theorem normalize_hyps_goal : + forall (s : list h_step) (ep : list Prop) (env : list int) (l : hyps), + interp_goal ep env (normalize_hyps s l) -> interp_goal ep env l. +Proof. + intros; apply valid_goal with (2 := H); apply normalize_hyps_valid. +Qed. + +Fixpoint extract_hyp_pos (s : list direction) (p : proposition) {struct s} : + proposition := + match s with + | D_left :: l => + match p with + | Tand x y => extract_hyp_pos l x + | _ => p + end + | D_right :: l => + match p with + | Tand x y => extract_hyp_pos l y + | _ => p + end + | D_mono :: l => match p with + | Tnot x => extract_hyp_neg l x + | _ => p + end + | _ => p + end + + with extract_hyp_neg (s : list direction) (p : proposition) {struct s} : + proposition := + match s with + | D_left :: l => + match p with + | Tor x y => extract_hyp_neg l x + | Timp x y => if decidability x then extract_hyp_pos l x else Tnot p + | _ => Tnot p + end + | D_right :: l => + match p with + | Tor x y => extract_hyp_neg l y + | Timp x y => extract_hyp_neg l y + | _ => Tnot p + end + | D_mono :: l => + match p with + | Tnot x => if decidability x then extract_hyp_pos l x else Tnot p + | _ => Tnot p + end + | _ => + match p with + | Tnot x => if decidability x then x else Tnot p + | _ => Tnot p + end + end. + +Definition co_valid1 (f : proposition -> proposition) := + forall (ep : list Prop) (e : list int) (p1 : proposition), + interp_proposition ep e (Tnot p1) -> interp_proposition ep e (f p1). + +Theorem extract_valid : + forall s : list direction, + valid1 (extract_hyp_pos s) /\ co_valid1 (extract_hyp_neg s). +Proof. + unfold valid1, co_valid1 in |- *; simple induction s; + [ split; + [ simpl in |- *; auto + | intros ep e p1; case p1; simpl in |- *; auto; intro p; + pattern (decidability p) in |- *; apply bool_eq_ind; + [ intro H; generalize (decidable_correct ep e p H); + unfold decidable in |- *; tauto + | simpl in |- *; auto ] ] + | intros a s' (H1, H2); simpl in H2; split; intros ep e p; case a; auto; + case p; auto; simpl in |- *; intros; + (apply H1; tauto) || + (apply H2; tauto) || + (pattern (decidability p0) in |- *; apply bool_eq_ind; + [ intro H3; generalize (decidable_correct ep e p0 H3); + unfold decidable in |- *; intro H4; apply H1; + tauto + | intro; tauto ]) ]. +Qed. + +Fixpoint decompose_solve (s : e_step) (h : hyps) {struct s} : lhyps := + match s with + | E_SPLIT i dl s1 s2 => + match extract_hyp_pos dl (nth_hyps i h) with + | Tor x y => decompose_solve s1 (x :: h) ++ decompose_solve s2 (y :: h) + | Tnot (Tand x y) => + if decidability x + then + decompose_solve s1 (Tnot x :: h) ++ + decompose_solve s2 (Tnot y :: h) + else h :: nil + | Timp x y => + if decidability x then + decompose_solve s1 (Tnot x :: h) ++ decompose_solve s2 (y :: h) + else h::nil + | _ => h :: nil + end + | E_EXTRACT i dl s1 => + decompose_solve s1 (extract_hyp_pos dl (nth_hyps i h) :: h) + | E_SOLVE t => execute_omega t h + end. + +Theorem decompose_solve_valid : + forall s : e_step, valid_list_goal (decompose_solve s). +Proof. + intro s; apply goal_valid; unfold valid_list_hyps in |- *; elim s; + simpl in |- *; intros; + [ cut (interp_proposition ep e1 (extract_hyp_pos l (nth_hyps n lp))); + [ case (extract_hyp_pos l (nth_hyps n lp)); simpl in |- *; auto; + [ intro p; case p; simpl in |- *; auto; intros p1 p2 H2; + pattern (decidability p1) in |- *; apply bool_eq_ind; + [ intro H3; generalize (decidable_correct ep e1 p1 H3); intro H4; + apply append_valid; elim H4; intro H5; + [ right; apply H0; simpl in |- *; tauto + | left; apply H; simpl in |- *; tauto ] + | simpl in |- *; auto ] + | intros p1 p2 H2; apply append_valid; simpl in |- *; elim H2; + [ intros H3; left; apply H; simpl in |- *; auto + | intros H3; right; apply H0; simpl in |- *; auto ] + | intros p1 p2 H2; + pattern (decidability p1) in |- *; apply bool_eq_ind; + [ intro H3; generalize (decidable_correct ep e1 p1 H3); intro H4; + apply append_valid; elim H4; intro H5; + [ right; apply H0; simpl in |- *; tauto + | left; apply H; simpl in |- *; tauto ] + | simpl in |- *; auto ] ] + | elim (extract_valid l); intros H2 H3; apply H2; apply nth_valid; auto ] + | intros; apply H; simpl in |- *; split; + [ elim (extract_valid l); intros H2 H3; apply H2; apply nth_valid; auto + | auto ] + | apply omega_valid with (1 := H) ]. +Qed. + +(* \subsection{La dernière étape qui élimine tous les séquents inutiles} *) + +Definition valid_lhyps (f : lhyps -> lhyps) := + forall (ep : list Prop) (e : list int) (lp : lhyps), + interp_list_hyps ep e lp -> interp_list_hyps ep e (f lp). + +Fixpoint reduce_lhyps (lp : lhyps) : lhyps := + match lp with + | (FalseTerm :: nil) :: lp' => reduce_lhyps lp' + | x :: lp' => x :: reduce_lhyps lp' + | nil => nil (A:=hyps) + end. + +Theorem reduce_lhyps_valid : valid_lhyps reduce_lhyps. +Proof. + unfold valid_lhyps in |- *; intros ep e lp; elim lp; + [ simpl in |- *; auto + | intros a l HR; elim a; + [ simpl in |- *; tauto + | intros a1 l1; case l1; case a1; simpl in |- *; try tauto ] ]. +Qed. + +Theorem do_reduce_lhyps : + forall (envp : list Prop) (env : list int) (l : lhyps), + interp_list_goal envp env (reduce_lhyps l) -> interp_list_goal envp env l. +Proof. + intros envp env l H; apply list_goal_to_hyps; intro H1; + apply list_hyps_to_goal with (1 := H); apply reduce_lhyps_valid; + assumption. +Qed. + +Definition concl_to_hyp (p : proposition) := + if decidability p then Tnot p else TrueTerm. + +Definition do_concl_to_hyp : + forall (envp : list Prop) (env : list int) (c : proposition) (l : hyps), + interp_goal envp env (concl_to_hyp c :: l) -> + interp_goal_concl c envp env l. +Proof. + simpl in |- *; intros envp env c l; induction l as [| a l Hrecl]; + [ simpl in |- *; unfold concl_to_hyp in |- *; + pattern (decidability c) in |- *; apply bool_eq_ind; + [ intro H; generalize (decidable_correct envp env c H); + unfold decidable in |- *; simpl in |- *; tauto + | simpl in |- *; intros H1 H2; elim H2; trivial ] + | simpl in |- *; tauto ]. +Qed. + +Definition omega_tactic (t1 : e_step) (t2 : list h_step) + (c : proposition) (l : hyps) := + reduce_lhyps (decompose_solve t1 (normalize_hyps t2 (concl_to_hyp c :: l))). + +Theorem do_omega : + forall (t1 : e_step) (t2 : list h_step) (envp : list Prop) + (env : list int) (c : proposition) (l : hyps), + interp_list_goal envp env (omega_tactic t1 t2 c l) -> + interp_goal_concl c envp env l. +Proof. + unfold omega_tactic in |- *; intros; apply do_concl_to_hyp; + apply (normalize_hyps_goal t2); apply (decompose_solve_valid t1); + apply do_reduce_lhyps; assumption. +Qed. + +End IntOmega. + +(* For now, the above modular construction is instanciated on Z, + in order to retrieve the initial ROmega. *) + +Module ZOmega := IntOmega(Z_as_Int). |