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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 *)
(************************************************************************)
(*i $Id: Logic.v 8642 2006-03-17 10:09:02Z notin $ i*)
Set Implicit Arguments.
Require Import Notations.
(** *** Propositional connectives *)
(** [True] is the always true proposition *)
Inductive True : Prop :=
I : True.
(** [False] is the always false proposition *)
Inductive False : Prop :=.
(** [not A], written [~A], is the negation of [A] *)
Definition not (A:Prop) := A -> False.
Notation "~ x" := (not x) : type_scope.
Hint Unfold not: core.
(** [and A B], written [A /\ B], is the conjunction of [A] and [B]
[conj p q] is a proof of [A /\ B] as soon as
[p] is a proof of [A] and [q] a proof of [B]
[proj1] and [proj2] are first and second projections of a conjunction *)
Inductive and (A B:Prop) : Prop :=
conj : A -> B -> A /\ B
where "A /\ B" := (and A B) : type_scope.
Section Conjunction.
Variables A B : Prop.
Theorem proj1 : A /\ B -> A.
Proof.
destruct 1; trivial.
Qed.
Theorem proj2 : A /\ B -> B.
Proof.
destruct 1; trivial.
Qed.
End Conjunction.
(** [or A B], written [A \/ B], is the disjunction of [A] and [B] *)
Inductive or (A B:Prop) : Prop :=
| or_introl : A -> A \/ B
| or_intror : B -> A \/ B
where "A \/ B" := (or A B) : type_scope.
(** [iff A B], written [A <-> B], expresses the equivalence of [A] and [B] *)
Definition iff (A B:Prop) := (A -> B) /\ (B -> A).
Notation "A <-> B" := (iff A B) : type_scope.
Section Equivalence.
Theorem iff_refl : forall A:Prop, A <-> A.
Proof.
split; auto.
Qed.
Theorem iff_trans : forall A B C:Prop, (A <-> B) -> (B <-> C) -> (A <-> C).
Proof.
intros A B C [H1 H2] [H3 H4]; split; auto.
Qed.
Theorem iff_sym : forall A B:Prop, (A <-> B) -> (B <-> A).
Proof.
intros A B [H1 H2]; split; auto.
Qed.
End Equivalence.
(** [(IF_then_else P Q R)], written [IF P then Q else R] denotes
either [P] and [Q], or [~P] and [Q] *)
Definition IF_then_else (P Q R:Prop) := P /\ Q \/ ~ P /\ R.
Notation "'IF' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3)
(at level 200, right associativity) : type_scope.
(** *** First-order quantifiers *)
(** [ex P], or simply [exists x, P x], or also [exists x:A, P x],
expresses the existence of an [x] of some type [A] in [Set] which
satisfies the predicate [P]. This is existential quantification.
[ex2 P Q], or simply [exists2 x, P x & Q x], or also
[exists2 x:A, P x & Q x], expresses the existence of an [x] of
type [A] which satisfies both predicates [P] and [Q].
Universal quantification is primitively written [forall x:A, Q]. By
symmetry with existential quantification, the construction [all P]
is provided too.
*)
(* Remark: [exists x, Q] denotes [ex (fun x => Q)] so that [exists x,
P x] is in fact equivalent to [ex (fun x => P x)] which may be not
convertible to [ex P] if [P] is not itself an abstraction *)
Inductive ex (A:Type) (P:A -> Prop) : Prop :=
ex_intro : forall x:A, P x -> ex (A:=A) P.
Inductive ex2 (A:Type) (P Q:A -> Prop) : Prop :=
ex_intro2 : forall x:A, P x -> Q x -> ex2 (A:=A) P Q.
Definition all (A:Type) (P:A -> Prop) := forall x:A, P x.
(* Rule order is important to give printing priority to fully typed exists *)
Notation "'exists' x , p" := (ex (fun x => p))
(at level 200, x ident, right associativity) : type_scope.
Notation "'exists' x : t , p" := (ex (fun x:t => p))
(at level 200, x ident, right associativity,
format "'[' 'exists' '/ ' x : t , '/ ' p ']'")
: type_scope.
Notation "'exists2' x , p & q" := (ex2 (fun x => p) (fun x => q))
(at level 200, x ident, p at level 200, right associativity) : type_scope.
Notation "'exists2' x : t , p & q" := (ex2 (fun x:t => p) (fun x:t => q))
(at level 200, x ident, t at level 200, p at level 200, right associativity,
format "'[' 'exists2' '/ ' x : t , '/ ' '[' p & '/' q ']' ']'")
: type_scope.
(** Derived rules for universal quantification *)
Section universal_quantification.
Variable A : Type.
Variable P : A -> Prop.
Theorem inst : forall x:A, all (fun x => P x) -> P x.
Proof.
unfold all in |- *; auto.
Qed.
Theorem gen : forall (B:Prop) (f:forall y:A, B -> P y), B -> all P.
Proof.
red in |- *; auto.
Qed.
End universal_quantification.
(** *** Equality *)
(** [eq x y], or simply [x=y] expresses the equality of [x] and
[y]. Both [x] and [y] must belong to the same type [A].
The definition is inductive and states the reflexivity of the equality.
The others properties (symmetry, transitivity, replacement of
equals by equals) are proved below. The type of [x] and [y] can be
made explicit using the notation [x = y :> A]. This is Leibniz equality
as it expresses that [x] and [y] are equal iff every property on
[A] which is true of [x] is also true of [y] *)
Inductive eq (A:Type) (x:A) : A -> Prop :=
refl_equal : x = x :>A
where "x = y :> A" := (@eq A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Notation "x <> y :> T" := (~ x = y :>T) : type_scope.
Notation "x <> y" := (x <> y :>_) : type_scope.
Implicit Arguments eq_ind [A].
Implicit Arguments eq_rec [A].
Implicit Arguments eq_rect [A].
Hint Resolve I conj or_introl or_intror refl_equal: core v62.
Hint Resolve ex_intro ex_intro2: core v62.
Section Logic_lemmas.
Theorem absurd : forall A C:Prop, A -> ~ A -> C.
Proof.
unfold not in |- *; intros A C h1 h2.
destruct (h2 h1).
Qed.
Section equality.
Variables A B : Type.
Variable f : A -> B.
Variables x y z : A.
Theorem sym_eq : x = y -> y = x.
Proof.
destruct 1; trivial.
Defined.
Opaque sym_eq.
Theorem trans_eq : x = y -> y = z -> x = z.
Proof.
destruct 2; trivial.
Defined.
Opaque trans_eq.
Theorem f_equal : x = y -> f x = f y.
Proof.
destruct 1; trivial.
Defined.
Opaque f_equal.
Theorem sym_not_eq : x <> y -> y <> x.
Proof.
red in |- *; intros h1 h2; apply h1; destruct h2; trivial.
Qed.
Definition sym_equal := sym_eq.
Definition sym_not_equal := sym_not_eq.
Definition trans_equal := trans_eq.
End equality.
Definition eq_ind_r :
forall (A:Type) (x:A) (P:A -> Prop), P x -> forall y:A, y = x -> P y.
intros A x P H y H0; elim sym_eq with (1 := H0); assumption.
Defined.
Definition eq_rec_r :
forall (A:Type) (x:A) (P:A -> Set), P x -> forall y:A, y = x -> P y.
intros A x P H y H0; elim sym_eq with (1 := H0); assumption.
Defined.
Definition eq_rect_r :
forall (A:Type) (x:A) (P:A -> Type), P x -> forall y:A, y = x -> P y.
intros A x P H y H0; elim sym_eq with (1 := H0); assumption.
Defined.
End Logic_lemmas.
Theorem f_equal2 :
forall (A1 A2 B:Type) (f:A1 -> A2 -> B) (x1 y1:A1)
(x2 y2:A2), x1 = y1 -> x2 = y2 -> f x1 x2 = f y1 y2.
Proof.
destruct 1; destruct 1; reflexivity.
Qed.
Theorem f_equal3 :
forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) (x1 y1:A1)
(x2 y2:A2) (x3 y3:A3),
x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.
Proof.
destruct 1; destruct 1; destruct 1; reflexivity.
Qed.
Theorem f_equal4 :
forall (A1 A2 A3 A4 B:Type) (f:A1 -> A2 -> A3 -> A4 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4),
x1 = y1 -> x2 = y2 -> x3 = y3 -> x4 = y4 -> f x1 x2 x3 x4 = f y1 y2 y3 y4.
Proof.
destruct 1; destruct 1; destruct 1; destruct 1; reflexivity.
Qed.
Theorem f_equal5 :
forall (A1 A2 A3 A4 A5 B:Type) (f:A1 -> A2 -> A3 -> A4 -> A5 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4) (x5 y5:A5),
x1 = y1 ->
x2 = y2 ->
x3 = y3 -> x4 = y4 -> x5 = y5 -> f x1 x2 x3 x4 x5 = f y1 y2 y3 y4 y5.
Proof.
destruct 1; destruct 1; destruct 1; destruct 1; destruct 1; reflexivity.
Qed.
Hint Immediate sym_eq sym_not_eq: core v62.
(** Other notations *)
Notation "'exists' ! x , P" :=
(exists x', (fun x => P) x' /\ forall x'', (fun x => P) x'' -> x' = x'')
(at level 200, x ident, right associativity,
format "'[' 'exists' ! '/ ' x , '/ ' P ']'") : type_scope.
Notation "'exists' ! x : A , P" :=
(exists x' : A, (fun x => P) x' /\ forall x'':A, (fun x => P) x'' -> x' = x'')
(at level 200, x ident, right associativity,
format "'[' 'exists' ! '/ ' x : A , '/ ' P ']'") : type_scope.
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