(*
Copyright (c) 2014-2015, Verdi Team
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:

1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*)

Ltac subst_max :=
  repeat match goal with
           | [ H : ?X = _ |- _ ]  => subst X
           | [H : _ = ?X |- _] => subst X
         end.

Ltac inv H := inversion H; subst_max.
Ltac invc H := inv H; clear H.
Ltac invcs H := invc H; simpl in *.

Ltac break_if :=
  match goal with
    | [ |- context [ if ?X then _ else _ ] ] =>
      match type of X with
        | sumbool _ _ => destruct X
        | _ => destruct X eqn:?
      end
    | [ H : context [ if ?X then _ else _ ] |- _] =>
      match type of X with
        | sumbool _ _ => destruct X
        | _ => destruct X eqn:?
      end
  end.

Ltac break_match_hyp :=
  match goal with
    | [ H : context [ match ?X with _ => _ end ] |- _] =>
      match type of X with
        | sumbool _ _ => destruct X
        | _ => destruct X eqn:?
      end
  end.

Ltac break_match_goal :=
  match goal with
    | [ |- context [ match ?X with _ => _ end ] ] =>
      match type of X with
        | sumbool _ _ => destruct X
        | _ => destruct X eqn:?
      end
  end.

Ltac break_match := break_match_goal || break_match_hyp.


Ltac break_exists :=
  repeat match goal with
           | [H : exists _, _ |- _ ] => destruct H
         end.

Ltac break_exists_exists :=
  repeat match goal with
           | H:exists _, _ |- _ =>
             let x := fresh "x" in
             destruct H as [x]; exists x
         end.

Ltac break_and :=
  repeat match goal with
           | [H : _ /\ _ |- _ ] => destruct H
         end.

Ltac solve_by_inversion' tac :=
  match goal with
    | [H : _ |- _] => solve [inv H; tac]
  end.

Ltac solve_by_inversion := solve_by_inversion' auto.

Ltac apply_fun f H:=
  match type of H with
    | ?X = ?Y => assert (f X = f Y)
  end.

Ltac conclude H tac :=
  (let H' := fresh in
   match type of H with
     | ?P -> _ => assert P as H' by (tac)
   end; specialize (H H'); clear H').

Ltac concludes :=
  match goal with
    | [ H : ?P -> _ |- _ ] => conclude H auto
  end.

Ltac forward H :=
  let H' := fresh in
   match type of H with
     | ?P -> _ => assert P as H'
   end.

Ltac forwards :=
  match goal with
    | [ H : ?P -> _ |- _ ] => forward H
  end.

Ltac find_contradiction :=
  match goal with
    | [ H : ?X = _, H' : ?X = _ |- _ ] => rewrite H in H'; solve_by_inversion
  end.

Ltac find_rewrite :=
  match goal with
    | [ H : ?X _ _ _ _ = _, H' : ?X _ _ _ _ = _ |- _ ] => rewrite H in H'
    | [ H : ?X = _, H' : ?X = _ |- _ ] => rewrite H in H'
    | [ H : ?X = _, H' : context [ ?X ] |- _ ] => rewrite H in H'
    | [ H : ?X = _ |- context [ ?X ] ] => rewrite H
  end.

Ltac find_rewrite_lem lem :=
  match goal with
    | [ H : _ |- _ ] =>
      rewrite lem in H; [idtac]
  end.

Ltac find_rewrite_lem_by lem t :=
  match goal with
    | [ H : _ |- _ ] =>
      rewrite lem in H by t
  end.

Ltac find_erewrite_lem lem :=
  match goal with
    | [ H : _ |- _] => erewrite lem in H by eauto
  end.

Ltac find_reverse_rewrite :=
  match goal with
    | [ H : _ = ?X _ _ _ _, H' : ?X _ _ _ _ = _ |- _ ] => rewrite <- H in H'
    | [ H : _ = ?X, H' : context [ ?X ] |- _ ] => rewrite <- H in H'
    | [ H : _ = ?X |- context [ ?X ] ] => rewrite <- H
  end.

Ltac find_inversion :=
  match goal with
    | [ H : ?X _ _ _ _ _ _ = ?X _ _ _ _ _ _ |- _ ] => invc H
    | [ H : ?X _ _ _ _ _ = ?X _ _ _ _ _ |- _ ] => invc H
    | [ H : ?X _ _ _ _ = ?X _ _ _ _ |- _ ] => invc H
    | [ H : ?X _ _ _ = ?X _ _ _ |- _ ] => invc H
    | [ H : ?X _ _ = ?X _ _ |- _ ] => invc H
    | [ H : ?X _ = ?X _ |- _ ] => invc H
  end.

Ltac prove_eq :=
  match goal with
    | [ H : ?X ?x1 ?x2 ?x3 = ?X ?y1 ?y2 ?y3 |- _ ] =>
      assert (x1 = y1) by congruence;
        assert (x2 = y2) by congruence;
        assert (x3 = y3) by congruence;
        clear H
    | [ H : ?X ?x1 ?x2 = ?X ?y1 ?y2 |- _ ] =>
      assert (x1 = y1) by congruence;
        assert (x2 = y2) by congruence;
        clear H
    | [ H : ?X ?x1 = ?X ?y1 |- _ ] =>
      assert (x1 = y1) by congruence;
        clear H
  end.

Ltac tuple_inversion :=
  match goal with
    | [ H : (_, _, _, _) = (_, _, _, _) |- _ ] => invc H
    | [ H : (_, _, _) = (_, _, _) |- _ ] => invc H
    | [ H : (_, _) = (_, _) |- _ ] => invc H
  end.

Ltac f_apply H f :=
  match type of H with
    | ?X = ?Y =>
      assert (f X = f Y) by (rewrite H; auto)
  end.

Ltac break_let :=
  match goal with
    | [ H : context [ (let (_,_) := ?X in _) ] |- _ ] => destruct X eqn:?
    | [ |- context [ (let (_,_) := ?X in _) ] ] => destruct X eqn:?
  end.

Ltac break_or_hyp :=
  match goal with
    | [ H : _ \/ _ |- _ ] => invc H
  end.

Ltac copy_apply lem H :=
  let x := fresh in
  pose proof H as x;
    apply lem in x.

Ltac copy_eapply lem H :=
  let x := fresh in
  pose proof H as x;
    eapply lem in x.

Ltac conclude_using tac :=
  match goal with
    | [ H : ?P -> _ |- _ ] => conclude H tac
  end.

Ltac find_higher_order_rewrite :=
  match goal with
    | [ H : _ = _ |- _ ] => rewrite H in *
    | [ H : forall _, _ = _ |- _ ] => rewrite H in *
    | [ H : forall _ _, _ = _ |- _ ] => rewrite H in *
  end.

Ltac find_reverse_higher_order_rewrite :=
  match goal with
    | [ H : _ = _ |- _ ] => rewrite <- H in *
    | [ H : forall _, _ = _ |- _ ] => rewrite <- H in *
    | [ H : forall _ _, _ = _ |- _ ] => rewrite <- H in *
  end.

Ltac clean :=
  match goal with
    | [ H : ?X = ?X |- _ ] => clear H
  end.

Ltac find_apply_hyp_goal :=
  match goal with
    | [ H : _ |- _ ] => solve [apply H]
  end.

Ltac find_copy_apply_lem_hyp lem :=
  match goal with
    | [ H : _ |- _ ] => copy_apply lem H
  end.

Ltac find_apply_hyp_hyp :=
  match goal with
    | [ H : forall _, _ -> _,
        H' : _ |- _ ] =>
      apply H in H'; [idtac]
    | [ H : _ -> _ , H' : _ |- _ ] =>
      apply H in H'; auto; [idtac]
  end.

Ltac find_copy_apply_hyp_hyp :=
  match goal with
    | [ H : forall _, _ -> _,
        H' : _ |- _ ] =>
      copy_apply H H'; [idtac]
    | [ H : _ -> _ , H' : _ |- _ ] =>
      copy_apply H H'; auto; [idtac]
  end.

Ltac find_apply_lem_hyp lem :=
  match goal with
    | [ H : _ |- _ ] => apply lem in H
  end.

Ltac find_eapply_lem_hyp lem :=
  match goal with
    | [ H : _ |- _ ] => eapply lem in H
  end.

Ltac insterU H :=
  match type of H with
    | forall _ : ?T, _ =>
      let x := fresh "x" in
      evar (x : T);
      let x' := (eval unfold x in x) in
        clear x; specialize (H x')
  end.

Ltac find_insterU :=
  match goal with
    | [ H : forall _, _ |- _ ] => insterU H
  end.

Ltac eapply_prop P :=
  match goal with
    | H : P _ |- _ =>
      eapply H
  end.

Ltac isVar t :=
    match goal with
      | v : _ |- _ =>
        match t with
          | v => idtac
        end
    end.

Ltac remGen t :=
  let x := fresh in
  let H := fresh in
  remember t as x eqn:H;
    generalize dependent H.

Ltac remGenIfNotVar t := first [isVar t| remGen t].

Ltac rememberNonVars H :=
  match type of H with
    | _ ?a ?b ?c ?d ?e =>
      remGenIfNotVar a;
      remGenIfNotVar b;
      remGenIfNotVar c;
      remGenIfNotVar d;
      remGenIfNotVar e
    | _ ?a ?b ?c ?d =>
      remGenIfNotVar a;
      remGenIfNotVar b;
      remGenIfNotVar c;
      remGenIfNotVar d
    | _ ?a ?b ?c =>
      remGenIfNotVar a;
      remGenIfNotVar b;
      remGenIfNotVar c
    | _ ?a ?b =>
      remGenIfNotVar a;
      remGenIfNotVar b
    | _ ?a =>
      remGenIfNotVar a
  end.

Ltac generalizeEverythingElse H :=
  repeat match goal with
           | [ x : ?T |- _ ] =>
             first [
                 match H with
                   | x => fail 2
                 end |
                 match type of H with
                   | context [x] => fail 2
                 end |
                 revert x]
         end.

Ltac prep_induction H :=
  rememberNonVars H;
  generalizeEverythingElse H.

Ltac econcludes :=
  match goal with
    | [ H : ?P -> _ |- _ ] => conclude H eauto
  end.

Ltac find_copy_eapply_lem_hyp lem :=
  match goal with
    | [ H : _ |- _ ] => copy_eapply lem H
  end.

Ltac apply_prop_hyp P Q :=
  match goal with
  | [ H : context [ P ], H' : context [ Q ] |- _ ] =>
    apply H in H'
  end.


Ltac eapply_prop_hyp P Q :=
  match goal with
  | [ H : context [ P ], H' : context [ Q ] |- _ ] =>
    eapply H in H'
  end.

Ltac copy_eapply_prop_hyp P Q :=
  match goal with
    | [ H : context [ P ], H' : context [ Q ] |- _ ] =>
      copy_eapply H H'
  end.

Ltac find_false :=
  match goal with
    | H : _ -> False |- _ => exfalso; apply H
  end.

Ltac injc H :=
  injection H; clear H; intro; subst_max.

Ltac find_injection :=
  match goal with
    | [ H : ?X _ _ _ _ _ _ = ?X _ _ _ _ _ _ |- _ ] => injc H
    | [ H : ?X _ _ _ _ _ = ?X _ _ _ _ _ |- _ ] => injc H
    | [ H : ?X _ _ _ _ = ?X _ _ _ _ |- _ ] => injc H
    | [ H : ?X _ _ _ = ?X _ _ _ |- _ ] => injc H
    | [ H : ?X _ _ = ?X _ _ |- _ ] => injc H
    | [ H : ?X _ = ?X _ |- _ ] => injc H
  end.

Ltac aggresive_rewrite_goal :=
  match goal with H : _ |- _ => rewrite H end.

Ltac break_exists_name x :=
  match goal with
  | [ H : exists _, _ |- _ ] => destruct H as [x H]
  end.