summaryrefslogtreecommitdiff
path: root/plugins/romega/ReflOmegaCore.v
diff options
context:
space:
mode:
Diffstat (limited to 'plugins/romega/ReflOmegaCore.v')
-rw-r--r--plugins/romega/ReflOmegaCore.v3216
1 files changed, 3216 insertions, 0 deletions
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).