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(* Ancien bug signale par Laurent Thery sur la condition de garde *)
Require Import Bool.
Require Import ZArith.
Definition rNat := positive.
Inductive rBoolOp : Set :=
| rAnd : rBoolOp
| rEq : rBoolOp.
Definition rlt (a b : rNat) : Prop :=
(a ?= b)%positive Datatypes.Eq = Datatypes.Lt.
Definition rltDec : forall m n : rNat, {rlt m n} + {rlt n m \/ m = n}.
intros n m; generalize (nat_of_P_lt_Lt_compare_morphism n m);
generalize (nat_of_P_gt_Gt_compare_morphism n m);
generalize (Pcompare_Eq_eq n m); case ((n ?= m)%positive Datatypes.Eq).
intros H' H'0 H'1; right; right; auto.
intros H' H'0 H'1; left; unfold rlt in |- *.
apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
intros H' H'0 H'1; right; left; unfold rlt in |- *.
apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
apply H'0; auto.
Defined.
Definition rmax : rNat -> rNat -> rNat.
intros n m; case (rltDec n m); intros Rlt0.
exact m.
exact n.
Defined.
Inductive rExpr : Set :=
| rV : rNat -> rExpr
| rN : rExpr -> rExpr
| rNode : rBoolOp -> rExpr -> rExpr -> rExpr.
Fixpoint maxVar (e : rExpr) : rNat :=
match e with
| rV n => n
| rN p => maxVar p
| rNode n p q => rmax (maxVar p) (maxVar q)
end.
(* Check bug #1491 *)
Require Import Streams.
Definition decomp (s:Stream nat) : Stream nat :=
match s with Cons _ s => s end.
CoFixpoint bx0 : Stream nat := Cons 0 bx1
with bx1 : Stream nat := Cons 1 bx0.
Lemma bx0bx : decomp bx0 = bx1.
simpl. (* used to return bx0 in V8.1 and before instead of bx1 *)
reflexivity.
Qed.
(* Check mutually inductive statements *)
Require Import ZArith_base Omega.
Open Scope Z_scope.
Inductive even: Z -> Prop :=
| even_base: even 0
| even_succ: forall n, odd (n - 1) -> even n
with odd: Z -> Prop :=
| odd_succ: forall n, even (n - 1) -> odd n.
Lemma even_pos_odd_pos: forall n, even n -> n >= 0
with odd_pos_even_pos : forall n, odd n -> n >= 1.
Proof.
intros.
destruct H.
omega.
apply odd_pos_even_pos in H.
omega.
intros.
destruct H.
apply even_pos_odd_pos in H.
omega.
Qed.
CoInductive a : Prop := acons : b -> a
with b : Prop := bcons : a -> b.
Lemma a1 : a
with b1 : b.
Proof.
apply acons.
assumption.
apply bcons.
assumption.
Qed.
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