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(**************************************************************************)
(* *)
(* SMTCoq *)
(* Copyright (C) 2011 - 2019 *)
(* *)
(* See file "AUTHORS" for the list of authors *)
(* *)
(* This file is distributed under the terms of the CeCILL-C licence *)
(* *)
(**************************************************************************)
Require Import Bool ZArith.
Require Import State.
(** Handling quantifiers with veriT **)
(* verit silently transforms an <implb a b> into a <or (not a) b> when
instantiating a quantified theorem with <implb> *)
Lemma impl_split a b:
implb a b = true -> orb (negb a) b = true.
Proof.
intro H.
destruct a; destruct b; trivial.
(* alternatively we could do <now verit_base H.> but it forces us to have veriT
installed when we compile SMTCoq. *)
Qed.
Hint Resolve impl_split : smtcoq_core.
(* verit silently transforms an <implb (a || b) c> into a <or (not a) c>
or into a <or (not b) c> when instantiating such a quantified theorem *)
Lemma impl_or_split_right a b c:
implb (a || b) c = true -> negb b || c = true.
Proof.
intro H.
destruct a; destruct c; intuition.
Qed.
Lemma impl_or_split_left a b c:
implb (a || b) c = true -> negb a || c = true.
Proof.
intro H.
destruct a; destruct c; intuition.
Qed.
(* verit considers equality modulo its symmetry, so we have to recover the
right direction in the instances of the theorems *)
Definition hidden_eq (a b : Z) := (a =? b)%Z.
Ltac all_rew :=
repeat match goal with
| [ |- context [ (?A =? ?B)%Z]] =>
change (A =? B)%Z with (hidden_eq A B)
end;
repeat match goal with
| [ |- context [ hidden_eq ?A ?B] ] =>
replace (hidden_eq A B) with (B =? A)%Z;
[ | now rewrite Z.eqb_sym]
end.
(* An automatic tactic that takes into account all those transformations *)
Ltac vauto :=
try (let H := fresh "H" in
intro H; try (all_rew; apply H);
match goal with
| [ |- is_true (negb ?A || ?B) ] =>
try (eapply impl_or_split_right; apply H);
eapply impl_or_split_left; apply H
end;
apply H);
auto with smtcoq_core.
(*
Local Variables:
coq-load-path: ((rec "." "SMTCoq"))
End:
*)
|
