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(* Check that dependent rewrite applies on arbitrary terms *)
Inductive listn : nat -> Set :=
| niln : listn 0
| consn : forall n : nat, nat -> listn n -> listn (S n).
Axiom
ax :
forall (n n' : nat) (l : listn (n + n')) (l' : listn (n' + n)),
existS _ (n + n') l = existS _ (n' + n) l'.
Lemma lem :
forall (n n' : nat) (l : listn (n + n')) (l' : listn (n' + n)),
n + n' = n' + n /\ existT _ (n + n') l = existT _ (n' + n) l'.
Proof.
intros n n' l l'.
dependent rewrite (ax n n' l l').
split; reflexivity.
Qed.
(* Used to raise an anomaly instead of an error in 8.1 *)
(* Submitted by Y. Makarov *)
Parameter N : Set.
Parameter E : N -> N -> Prop.
Axiom e : forall (A : Set) (EA : A -> A -> Prop) (a : A), EA a a.
Theorem th : forall x : N, E x x.
intro x. try rewrite e.
Abort.
(* Behavior of rewrite wrt conversion *)
Require Import Arith.
Goal forall n, 0 + n = n -> True.
intros n H.
rewrite plus_0_l in H.
Abort.
(* Rewrite dependent proofs from left-to-right *)
Lemma l1 :
forall x y (H:x = y:>nat) (P:forall x y, x=y -> Type), P x y H -> P x y H.
intros x y H P H0.
rewrite H.
rewrite H in H0.
assumption.
Qed.
(* Rewrite dependent proofs from right-to-left *)
Lemma l2 :
forall x y (H:x = y:>nat) (P:forall x y, x=y -> Type), P x y H -> P x y H.
intros x y H P H0.
rewrite <- H.
rewrite <- H in H0.
assumption.
Qed.
(* Check rewriting dependent proofs with non-symmetric equalities *)
Lemma l3:forall x (H:eq_true x) (P:forall x, eq_true x -> Type), P x H -> P x H.
intros x H P H0.
rewrite H.
rewrite H in H0.
assumption.
Qed.
(* Dependent rewrite *)
Require Import JMeq.
Goal forall A B (a:A) (b:B), JMeq a b -> JMeq b a -> True.
inversion 1; (* Goal is now [JMeq a a -> True] *) dependent rewrite H3.
Undo.
intros; inversion H; dependent rewrite H4 in H0.
Undo.
intros; inversion H; dependent rewrite <- H4 in H0.
Abort.
(* Test conversion between terms with evars that both occur in K-redexes and
are elsewhere solvable.
This is quite an artificial example, but it used to work in 8.2.
Since rewrite supports conversion on terms without metas, it
was successively unifying (id 0 ?y) and 0 where ?y was not a
meta but, because coming from a "_", an evar.
After commit r12440 which unified the treatment of metas and
evars, it stopped to work. Chung-Kil Hur's Heq package used
this feature. Solved in r13...
*)
Parameter g : nat -> nat -> nat.
Definition K (x y:nat) := x.
Goal (forall y, g y (K 0 y) = 0) -> g 0 0 = 0.
intros.
rewrite (H _).
reflexivity.
Qed.
Goal (forall y, g (K 0 y) y = 0) -> g 0 0 = 0.
intros.
rewrite (H _).
reflexivity.
Qed.
(* Example of rewriting of a degenerated pattern using the right-most
argument of the goal. This is sometimes used in contribs, even if
ad hoc. Here, we have the extra requirement that checking types
needs delta-conversion *)
Axiom s : forall (A B : Type) (p : A * B), p = (fst p, snd p).
Definition P := (nat * nat)%type.
Goal forall x:P, x = x.
intros. rewrite s.
Abort.
(* Test second-order unification and failure of pattern-unification *)
Goal forall (P: forall Y, Y -> Prop) Y a, Y = nat -> (True -> P Y a) -> False.
intros.
(* The next line used to succeed between June and November 2011 *)
(* causing ill-typed rewriting *)
Fail rewrite H in H0.
Abort.
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