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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
Require Datatypes.
(* [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.
Hints Unfold not : core.
Section Conjunction.
(* [and A B], written [A /\ B], is the conjunction of [A] and [B] *)
(* [conj A B p q], written [<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 -> (and A B).
Variables A,B : Prop.
Theorem proj1 : (and A B) -> A.
Proof.
Induction 1; Trivial.
Qed.
Theorem proj2 : (and A B) -> B.
Proof.
Induction 1; Trivial.
Qed.
End Conjunction.
Section Disjunction.
(* [or A B], written [A \/ B], is the disjunction of [A] and [B] *)
Inductive or [A,B:Prop] : Prop :=
or_introl : A -> (or A B)
| or_intror : B -> (or A B).
End Disjunction.
Section Equivalence.
(* [iff A B], written [A <-> B], expresses the equivalence of [A] and [B] *)
Definition iff := [P,Q:Prop] (and (P->Q) (Q->P)).
Theorem iff_refl : (a:Prop) (iff a a).
Proof.
Split; Auto.
Qed.
Theorem iff_trans : (a,b,c:Prop) (iff a b) -> (iff b c) -> (iff a c).
Proof.
Intros a b c (H1,H2) (H3,H4); Split; Auto.
Qed.
Theorem iff_sym : (a,b:Prop) (iff a b) -> (iff b a).
Proof.
Intros a b (H1,H2); Split; Auto.
Qed.
End Equivalence.
(* [(IF P Q R)], or more suggestively [(either P and_then Q or_else R)],
denotes either [P] and [Q], or [~P] and [Q] *)
Definition IF := [P,Q,R:Prop] (or (and P Q) (and (not P) R)).
Section First_order_quantifiers.
(* [(ex A P)], or simply [(EX x | P(x))], expresses the existence of an
[x] of type [A] which satisfies the predicate [P] ([A] is of type
[Set]). This is existential quantification. *)
(* [(ex2 A P Q)], or simply [(EX x | P(x) & Q(x))], expresses the
existence of an [x] of type [A] which satisfies both the predicates
[P] and [Q] *)
(* Universal quantification (especially first-order one) is normally
written [(x:A)(P x)]. For duality with existential quantification, the
construction [(all A P)], or simply [(All P)], is provided as an
abbreviation of [(x:A)(P x)] *)
Inductive ex [A:Set;P:A->Prop] : Prop
:= ex_intro : (x:A)(P x)->(ex A P).
Inductive ex2 [A:Set;P,Q:A->Prop] : Prop
:= ex_intro2 : (x:A)(P x)->(Q x)->(ex2 A P Q).
Definition all := [A:Set][P:A->Prop](x:A)(P x).
End First_order_quantifiers.
Section Equality.
(* [(eq A x y)], or simply [x=y], expresses the (Leibniz') 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) are proved below *)
Inductive eq [A:Set;x:A] : A->Prop
:= refl_equal : (eq A x x).
End Equality.
Hints Resolve I conj or_introl or_intror refl_equal : core v62.
Hints Resolve ex_intro ex_intro2 : core v62.
Section Logic_lemmas.
Theorem absurd : (A:Prop)(C:Prop) A -> (not A) -> C.
Proof.
Unfold not; Intros A C h1 h2.
Elim (h2 h1).
Qed.
Section equality.
Variable A,B : Set.
Variable f : A->B.
Variable x,y,z : A.
Theorem sym_eq : (eq ? x y) -> (eq ? y x).
Proof.
Induction 1; Trivial.
Defined.
Opaque sym_eq.
Theorem trans_eq : (eq ? x y) -> (eq ? y z) -> (eq ? x z).
Proof.
Induction 2; Trivial.
Defined.
Opaque trans_eq.
Theorem f_equal : (eq ? x y) -> (eq ? (f x) (f y)).
Proof.
Induction 1; Trivial.
Defined.
Opaque f_equal.
Theorem sym_not_eq : (not (eq ? x y)) -> (not (eq ? y x)).
Proof.
Red; Intros h1 h2; Apply h1; Elim h2; Trivial.
Qed.
Definition sym_equal := sym_eq.
Definition sym_not_equal := sym_not_eq.
Definition trans_equal := trans_eq.
End equality.
Theorem eq_rect: (A:Set)(x:A)(P:A->Type)(P x)->(y:A)(eq ? x y)->(P y).
Proof.
Intros.
Cut (identity A x y).
NewDestruct 1; Auto.
Elim H; Auto.
Qed.
Definition eq_ind_r : (A:Set)(x:A)(P:A->Prop)(P x)->(y:A)(eq ? y x)->(P y).
Intros A x P H y H0; Case sym_eq with 1:= H0; Trivial.
Defined.
Definition eq_rec_r : (A:Set)(x:A)(P:A->Set)(P x)->(y:A)(eq ? y x)->(P y).
Intros A x P H y H0; Case sym_eq with 1:= H0; Trivial.
Defined.
Definition eq_rect_r : (A:Set)(x:A)(P:A->Type)(P x)->(y:A)(eq ? y x)->(P y).
Intros A x P H y H0; Elim sym_eq with 1:= H0; Trivial.
Defined.
End Logic_lemmas.
Theorem f_equal2 : (A1,A2,B:Set)(f:A1->A2->B)(x1,y1:A1)(x2,y2:A2)
(eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? (f x1 x2) (f y1 y2)).
Proof.
Induction 1; Induction 1; Trivial.
Qed.
Theorem f_equal3 : (A1,A2,A3,B:Set)(f:A1->A2->A3->B)(x1,y1:A1)(x2,y2:A2)
(x3,y3:A3)(eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3)
-> (eq ? (f x1 x2 x3) (f y1 y2 y3)).
Proof.
Induction 1; Induction 1; Induction 1; Trivial.
Qed.
Theorem f_equal4 : (A1,A2,A3,A4,B:Set)(f:A1->A2->A3->A4->B)
(x1,y1:A1)(x2,y2:A2)(x3,y3:A3)(x4,y4:A4)
(eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3) -> (eq ? x4 y4)
-> (eq ? (f x1 x2 x3 x4) (f y1 y2 y3 y4)).
Proof.
Induction 1; Induction 1; Induction 1; Induction 1; Trivial.
Qed.
Theorem f_equal5 : (A1,A2,A3,A4,A5,B:Set)(f:A1->A2->A3->A4->A5->B)
(x1,y1:A1)(x2,y2:A2)(x3,y3:A3)(x4,y4:A4)(x5,y5:A5)
(eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3) -> (eq ? x4 y4) -> (eq ? x5 y5)
-> (eq ? (f x1 x2 x3 x4 x5) (f y1 y2 y3 y4 y5)).
Proof.
Induction 1; Induction 1; Induction 1; Induction 1; Induction 1; Trivial.
Qed.
Hints Immediate sym_eq sym_not_eq : core v62.
Syntactic Definition Ex := ex | 1.
Syntactic Definition Ex2 := ex2 | 1.
Syntactic Definition All := all | 1.
|
