(* 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)),
    existT _ (n + n') l = existT _ (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 Nat.add_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.

(* Test subst in the presence of a dependent let-in *)
(* Was not working prior to May 2014 *)

Goal forall x y, x=y+0 -> let z := x+1 in x+1=y -> z=z -> z=x.
intros.
subst x. (* was failing *)
subst z.
rewrite H0.
auto with arith.
Qed.

(* Check that evars are instantiated when the term to rewrite is
   closed, like in the case it is open *)

Goal exists x, S 0 = 0 -> S x = 0.
eexists. intro H.
rewrite H.
reflexivity.
Abort.

(* Check that rewriting within evars still work (was broken in 8.5beta1) *)

Goal forall (a: unit) (H: a = tt), exists x y:nat, x = y.
intros; eexists; eexists.
rewrite H.
Undo.
subst.
Abort.

(* Check that iterated rewriting does not rewrite in the side conditions *)
(* Example from Sigurd Schneider, extracted from contrib containers *)

Lemma EQ
  : forall (e e' : nat), True -> e = e'.
Admitted.

Lemma test (v1 v2 v3: nat) (v' : v1 = v2) : v2 = v1.
Proof.
  rewrite <- (EQ v1 v2) in *.
  exact v'.
  (* There should be only two side conditions *)
  exact I.
  exact I.
Qed.
