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(*  The Why3 Verification Platform   /   The Why3 Development Team  *)
(*  Copyright 2010-2023 --  Inria - CNRS - Paris-Saclay University  *)
(*                                                                  *)
(*  This software is distributed under the terms of the GNU Lesser  *)
(*  General Public License version 2.1, with the special exception  *)
(*  on linking described in file LICENSE.                           *)
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(* This file is generated by Why3's Coq-realize driver *)
(* Beware! Only edit allowed sections below    *)
Require Import BuiltIn.
Require BuiltIn.
Require HighOrd.

Require Import ClassicalEpsilon.

(* Why3 assumption *)
Definition map (a:Type) (b:Type) := a -> b.

Global Instance map_WhyType : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, WhyType (map a b).
Proof.
intros.
repeat split.
exact (fun _ => why_inhabitant).
intros x y.
apply excluded_middle_informative.
Qed.

(* Why3 goal *)
Definition set {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b} :
  (a -> b) -> a -> b -> a -> b.
Proof.
intros m x y.
intros x'.
destruct (why_decidable_eq x x') as [H|H].
exact y.
exact (m x').
Defined.

(* Why3 goal *)
Lemma set'def {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b} :
  forall (f:a -> b) (x:a) (v:b) (y:a),
  ((y = x) -> ((set f x v y) = v)) /\ (~ (y = x) -> ((set f x v y) = (f y))).
Proof.
intros f x v y.
unfold set.
case why_decidable_eq.
intros <-.
split ; try easy ; intros H ; now elim H. (* TODO: replace by easy after 8.4 *)
intros H.
split ; intros H'.
now elim H.
easy.
Qed.

