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.
Inductive and (A B:Prop) : Prop :=
conj : A -> B -> A /\ B
where "A /\ B" := (and A B) : type_scope.
Section Conjunction.
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
|
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) : type_scope.
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.
Notation "'exists' x , p" := (ex (fun x => p))
(at level 200, x ident) : type_scope.
Notation "'exists' x : t , p" := (ex (fun x:t => p))
(at level 200, x ident, 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) : 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,
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 (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. The type of x and y can be made explicit
using the notation x = y :> A
|
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.