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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Morphism instances for propositional connectives.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
91405 Orsay, France *)
Require Import Coq.Classes.Morphisms.
Require Import Coq.Program.Basics.
Require Import Coq.Program.Tactics.
(** Standard instances for [not], [iff] and [impl]. *)
(** Logical negation. *)
Program Instance not_impl_morphism :
Morphism (impl --> impl) not.
Program Instance not_iff_morphism :
Morphism (iff ++> iff) not.
(** Logical conjunction. *)
Program Instance and_impl_morphism :
Morphism (impl ==> impl ==> impl) and.
Program Instance and_iff_morphism :
Morphism (iff ==> iff ==> iff) and.
(** Logical disjunction. *)
Program Instance or_impl_morphism :
Morphism (impl ==> impl ==> impl) or.
Program Instance or_iff_morphism :
Morphism (iff ==> iff ==> iff) or.
(** Logical implication [impl] is a morphism for logical equivalence. *)
Program Instance iff_iff_iff_impl_morphism : Morphism (iff ==> iff ==> iff) impl.
(** Morphisms for quantifiers *)
Program Instance ex_iff_morphism {A : Type} : Morphism (pointwise_relation A iff ==> iff) (@ex A).
Next Obligation.
Proof.
unfold pointwise_relation in H.
split ; intros.
destruct H0 as [x₁ H₁].
exists x₁. rewrite H in H₁. assumption.
destruct H0 as [x₁ H₁].
exists x₁. rewrite H. assumption.
Qed.
Program Instance ex_impl_morphism {A : Type} :
Morphism (pointwise_relation A impl ==> impl) (@ex A).
Next Obligation.
Proof.
unfold pointwise_relation in H.
exists H0. apply H. assumption.
Qed.
Program Instance ex_inverse_impl_morphism {A : Type} :
Morphism (pointwise_relation A (inverse impl) ==> inverse impl) (@ex A).
Next Obligation.
Proof.
unfold pointwise_relation in H.
exists H0. apply H. assumption.
Qed.
Program Instance all_iff_morphism {A : Type} :
Morphism (pointwise_relation A iff ==> iff) (@all A).
Next Obligation.
Proof.
unfold pointwise_relation, all in *.
intuition ; specialize (H x0) ; intuition.
Qed.
Program Instance all_impl_morphism {A : Type} :
Morphism (pointwise_relation A impl ==> impl) (@all A).
Next Obligation.
Proof.
unfold pointwise_relation, all in *.
intuition ; specialize (H x0) ; intuition.
Qed.
Program Instance all_inverse_impl_morphism {A : Type} :
Morphism (pointwise_relation A (inverse impl) ==> inverse impl) (@all A).
Next Obligation.
Proof.
unfold pointwise_relation, all in *.
intuition ; specialize (H x0) ; intuition.
Qed.
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