Library Coq.Init.Logic


Set Implicit Arguments.

Require Export Notations.

Notation "A -> B" := (forall (_ : A), B) : type_scope.

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.


Create the "core" hint database, and set its transparent state for variables and constants explicitly.


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.

  Theorem proj2 : A /\ B -> B.

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.

Theorem iff_trans : forall A B C:Prop, (A <-> B) -> (B <-> C) -> (A <-> C).

Theorem iff_sym : forall A B:Prop, (A <-> B) -> (B <-> A).

End Equivalence.

Hint Unfold iff: extcore.

Backward direction of the equivalences above does not need assumptions

Theorem and_iff_compat_l : forall A B C : Prop,
  (B <-> C) -> (A /\ B <-> A /\ C).

Theorem and_iff_compat_r : forall A B C : Prop,
  (B <-> C) -> (B /\ A <-> C /\ A).

Theorem or_iff_compat_l : forall A B C : Prop,
  (B <-> C) -> (A \/ B <-> A \/ C).

Theorem or_iff_compat_r : forall A B C : Prop,
  (B <-> C) -> (B \/ A <-> C \/ A).

Theorem imp_iff_compat_l : forall A B C : Prop,
  (B <-> C) -> ((A -> B) <-> (A -> C)).

Theorem imp_iff_compat_r : forall A B C : Prop,
  (B <-> C) -> ((B -> A) <-> (C -> A)).

Theorem not_iff_compat : forall A B : Prop,
  (A <-> B) -> (~ A <-> ~B).

Some equivalences

Theorem neg_false : forall A : Prop, ~ A <-> (A <-> False).

Theorem and_cancel_l : forall A B C : Prop,
  (B -> A) -> (C -> A) -> ((A /\ B <-> A /\ C) <-> (B <-> C)).

Theorem and_cancel_r : forall A B C : Prop,
  (B -> A) -> (C -> A) -> ((B /\ A <-> C /\ A) <-> (B <-> C)).

Theorem and_comm : forall A B : Prop, A /\ B <-> B /\ A.

Theorem and_assoc : forall A B C : Prop, (A /\ B) /\ C <-> A /\ B /\ C.

Theorem or_cancel_l : forall A B C : Prop,
  (B -> ~ A) -> (C -> ~ A) -> ((A \/ B <-> A \/ C) <-> (B <-> C)).
Theorem or_cancel_r : forall A B C : Prop,
  (B -> ~ A) -> (C -> ~ A) -> ((B \/ A <-> C \/ A) <-> (B <-> C)).
Theorem or_comm : forall A B : Prop, (A \/ B) <-> (B \/ A).

Theorem or_assoc : forall A B C : Prop, (A \/ B) \/ C <-> A \/ B \/ C.
Lemma iff_and : forall A B : Prop, (A <-> B) -> (A -> B) /\ (B -> A).

Lemma iff_to_and : forall A B : Prop, (A <-> B) <-> (A -> B) /\ (B -> A).

(IF_then_else P Q R), written IF P then Q else R denotes either P and Q, or ~P and R

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.

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 .. y , p" := (ex (fun x => .. (ex (fun y => p)) ..))
  (at level 200, x binder, right associativity,
   format "'[' 'exists' '/ ' x .. y , '/ ' 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 : A , p & q" := (ex2 (A:=A) (fun x => p) (fun x => q))
  (at level 200, x ident, A at level 200, p at level 200, right associativity,
    format "'[' 'exists2' '/ ' x : A , '/ ' '[' p & '/' q ']' ']'")
  : type_scope.

Notation "'exists2' ' x , p & q" := (ex2 (fun x => p) (fun x => q))
  (at level 200, x strict pattern, p at level 200, right associativity) : type_scope.
Notation "'exists2' ' x : A , p & q" := (ex2 (A:=A) (fun x => p) (fun x => q))
  (at level 200, x strict pattern, A at level 200, p at level 200, right associativity,
    format "'[' 'exists2' '/ ' ' x : A , '/ ' '[' 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.

  Theorem gen : forall (B:Prop) (f:forall y:A, B -> P y), B -> all P.

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 :=
    eq_refl : 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.



Hint Resolve I conj or_introl or_intror : core.
Hint Resolve eq_refl: core.
Hint Resolve ex_intro ex_intro2: core.


Section Logic_lemmas.

  Theorem absurd : forall A C:Prop, A -> ~ A -> C.

  Section equality.
    Variables A B : Type.
    Variable f : A -> B.
    Variables x y z : A.

    Theorem eq_sym : x = y -> y = x.


    Theorem eq_trans : x = y -> y = z -> x = z.


    Theorem eq_trans_r : x = y -> z = y -> x = z.

    Theorem f_equal : x = y -> f x = f y.


    Theorem not_eq_sym : x <> y -> y <> x.

  End equality.

  Definition eq_sind_r :
    forall (A:Type) (x:A) (P:A -> SProp), P x -> forall y:A, y = x -> P y.

  Definition eq_ind_r :
    forall (A:Type) (x:A) (P:A -> Prop), P x -> forall y:A, y = x -> P y.
  Defined.


  Definition eq_rec_r :
    forall (A:Type) (x:A) (P:A -> Set), P x -> forall y:A, y = x -> P y.
  Defined.

  Definition eq_rect_r :
    forall (A:Type) (x:A) (P:A -> Type), P x -> forall y:A, y = x -> P y.
  Defined.
End Logic_lemmas.

Module EqNotations.
  Notation "'rew' H 'in' H'" := (eq_rect _ _ H' _ H)
    (at level 10, H' at level 10,
     format "'[' 'rew' H in '/' H' ']'").
  Notation "'rew' [ P ] H 'in' H'" := (eq_rect _ P H' _ H)
    (at level 10, H' at level 10,
     format "'[' 'rew' [ P ] '/ ' H in '/' H' ']'").
  Notation "'rew' <- H 'in' H'" := (eq_rect_r _ H' H)
    (at level 10, H' at level 10,
     format "'[' 'rew' <- H in '/' H' ']'").
  Notation "'rew' <- [ P ] H 'in' H'" := (eq_rect_r P H' H)
    (at level 10, H' at level 10,
     format "'[' 'rew' <- [ P ] '/ ' H in '/' H' ']'").
  Notation "'rew' -> H 'in' H'" := (eq_rect _ _ H' _ H)
    (at level 10, H' at level 10, only parsing).
  Notation "'rew' -> [ P ] H 'in' H'" := (eq_rect _ P H' _ H)
    (at level 10, H' at level 10, only parsing).

End EqNotations.

Import EqNotations.

Section equality_dep.
  Variable A : Type.
  Variable B : A -> Type.
  Variable f : forall x, B x.
  Variables x y : A.

  Theorem f_equal_dep : forall (H: x = y), rew H in f x = f y.

End equality_dep.

Section equality_dep2.

  Variable A A' : Type.
  Variable B : A -> Type.
  Variable B' : A' -> Type.
  Variable f : A -> A'.
  Variable g : forall a:A, B a -> B' (f a).
  Variables x y : A.

  Lemma f_equal_dep2 : forall {A A' B B'} (f : A -> A') (g : forall a:A, B a -> B' (f a))
  {x1 x2 : A} {y1 : B x1} {y2 : B x2} (H : x1 = x2),
  rew H in y1 = y2 -> rew f_equal f H in g x1 y1 = g x2 y2.

End equality_dep2.

Lemma rew_opp_r : forall A (P:A->Type) (x y:A) (H:x=y) (a:P y), rew H in rew <- H in a = a.

Lemma rew_opp_l : forall A (P:A->Type) (x y:A) (H:x=y) (a:P x), rew <- H in rew H in a = a.

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.


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.

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.

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.

Theorem f_equal_compose : forall A B C (a b:A) (f:A->B) (g:B->C) (e:a=b),
  f_equal g (f_equal f e) = f_equal (fun a => g (f a)) e.

The groupoid structure of equality

Theorem eq_trans_refl_l : forall A (x y:A) (e:x=y), eq_trans eq_refl e = e.

Theorem eq_trans_refl_r : forall A (x y:A) (e:x=y), eq_trans e eq_refl = e.

Theorem eq_sym_involutive : forall A (x y:A) (e:x=y), eq_sym (eq_sym e) = e.

Theorem eq_trans_sym_inv_l : forall A (x y:A) (e:x=y), eq_trans (eq_sym e) e = eq_refl.

Theorem eq_trans_sym_inv_r : forall A (x y:A) (e:x=y), eq_trans e (eq_sym e) = eq_refl.

Theorem eq_trans_assoc : forall A (x y z t:A) (e:x=y) (e':y=z) (e'':z=t),
  eq_trans e (eq_trans e' e'') = eq_trans (eq_trans e e') e''.

Theorem rew_map : forall A B (P:B->Type) (f:A->B) x1 x2 (H:x1=x2) (y:P (f x1)),
  rew [fun x => P (f x)] H in y = rew f_equal f H in y.

Theorem eq_trans_map : forall {A B} {x1 x2 x3:A} {y1:B x1} {y2:B x2} {y3:B x3},
  forall (H1:x1=x2) (H2:x2=x3) (H1': rew H1 in y1 = y2) (H2': rew H2 in y2 = y3),
  rew eq_trans H1 H2 in y1 = y3.

Lemma map_subst : forall {A} {P Q:A->Type} (f : forall x, P x -> Q x) {x y} (H:x=y) (z:P x),
  rew H in f x z = f y (rew H in z).

Lemma map_subst_map : forall {A B} {P:A->Type} {Q:B->Type} (f:A->B) (g : forall x, P x -> Q (f x)),
  forall {x y} (H:x=y) (z:P x), rew f_equal f H in g x z = g y (rew H in z).

Lemma rew_swap : forall A (P:A->Type) x1 x2 (H:x1=x2) (y1:P x1) (y2:P x2), rew H in y1 = y2 -> y1 = rew <- H in y2.

Lemma rew_compose : forall A (P:A->Type) x1 x2 x3 (H1:x1=x2) (H2:x2=x3) (y:P x1),
  rew H2 in rew H1 in y = rew (eq_trans H1 H2) in y.

Extra properties of equality

Theorem eq_id_comm_l : forall A (f:A->A) (Hf:forall a, a = f a), forall a, f_equal f (Hf a) = Hf (f a).

Theorem eq_id_comm_r : forall A (f:A->A) (Hf:forall a, f a = a), forall a, f_equal f (Hf a) = Hf (f a).

Lemma eq_refl_map_distr : forall A B x (f:A->B), f_equal f (eq_refl x) = eq_refl (f x).

Lemma eq_trans_map_distr : forall A B x y z (f:A->B) (e:x=y) (e':y=z), f_equal f (eq_trans e e') = eq_trans (f_equal f e) (f_equal f e').

Lemma eq_sym_map_distr : forall A B (x y:A) (f:A->B) (e:x=y), eq_sym (f_equal f e) = f_equal f (eq_sym e).

Lemma eq_trans_sym_distr : forall A (x y z:A) (e:x=y) (e':y=z), eq_sym (eq_trans e e') = eq_trans (eq_sym e') (eq_sym e).

Lemma eq_trans_rew_distr : forall A (P:A -> Type) (x y z:A) (e:x=y) (e':y=z) (k:P x),
    rew (eq_trans e e') in k = rew e' in rew e in k.

Lemma rew_const : forall A P (x y:A) (e:x=y) (k:P),
    rew [fun _ => P] e in k = k.


Notation sym_eq := eq_sym (only parsing).
Notation trans_eq := eq_trans (only parsing).
Notation sym_not_eq := not_eq_sym (only parsing).

Notation refl_equal := eq_refl (only parsing).
Notation sym_equal := eq_sym (only parsing).
Notation trans_equal := eq_trans (only parsing).
Notation sym_not_equal := not_eq_sym (only parsing).

Hint Immediate eq_sym not_eq_sym: core.

Basic definitions about relations and properties

Definition subrelation (A B : Type) (R R' : A->B->Prop) :=
  forall x y, R x y -> R' x y.

Definition unique (A : Type) (P : A->Prop) (x:A) :=
  P x /\ forall (x':A), P x' -> x=x'.

Definition uniqueness (A:Type) (P:A->Prop) := forall x y, P x -> P y -> x = y.

Unique existence

Notation "'exists' ! x .. y , p" :=
  (ex (unique (fun x => .. (ex (unique (fun y => p))) ..)))
  (at level 200, x binder, right associativity,
   format "'[' 'exists' ! '/ ' x .. y , '/ ' p ']'")
  : type_scope.

Lemma unique_existence : forall (A:Type) (P:A->Prop),
  ((exists x, P x) /\ uniqueness P) <-> (exists! x, P x).

Lemma forall_exists_unique_domain_coincide :
  forall A (P:A->Prop), (exists! x, P x) ->
  forall Q:A->Prop, (forall x, P x -> Q x) <-> (exists x, P x /\ Q x).

Lemma forall_exists_coincide_unique_domain :
  forall A (P:A->Prop),
  (forall Q:A->Prop, (forall x, P x -> Q x) <-> (exists x, P x /\ Q x))
  -> (exists! x, P x).

Being inhabited

The predicate inhabited can be used in different contexts. If A is thought as a type, inhabited A states that A is inhabited. If A is thought as a computationally relevant proposition, then inhabited A weakens A so as to hide its computational meaning. The so-weakened proof remains computationally relevant but only in a propositional context.

Inductive inhabited (A:Type) : Prop := inhabits : A -> inhabited A.

Hint Resolve inhabits: core.

Lemma exists_inhabited : forall (A:Type) (P:A->Prop),
  (exists x, P x) -> inhabited A.

Lemma inhabited_covariant (A B : Type) : (A -> B) -> inhabited A -> inhabited B.

Declaration of stepl and stepr for eq and iff

Lemma eq_stepl : forall (A : Type) (x y z : A), x = y -> x = z -> z = y.

Declare Left Step eq_stepl.
Declare Right Step eq_trans.

Lemma iff_stepl : forall A B C : Prop, (A <-> B) -> (A <-> C) -> (C <-> B).

Declare Left Step iff_stepl.
Declare Right Step iff_trans.


Equality for ex
Section ex.
  Local Unset Implicit Arguments.
  Definition eq_ex_uncurried {A : Type} (P : A -> Prop) {u1 v1 : A} {u2 : P u1} {v2 : P v1}
             (pq : exists p : u1 = v1, rew p in u2 = v2)
  : ex_intro P u1 u2 = ex_intro P v1 v2.

  Definition eq_ex {A : Type} {P : A -> Prop} (u1 v1 : A) (u2 : P u1) (v2 : P v1)
             (p : u1 = v1) (q : rew p in u2 = v2)
  : ex_intro P u1 u2 = ex_intro P v1 v2
    := eq_ex_uncurried P (ex_intro _ p q).

  Definition eq_ex_hprop {A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q)
             (u1 v1 : A) (u2 : P u1) (v2 : P v1)
             (p : u1 = v1)
    : ex_intro P u1 u2 = ex_intro P v1 v2
    := eq_ex u1 v1 u2 v2 p (P_hprop _ _ _).

  Lemma rew_ex {A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : exists p, Q x p) {y} (H : x = y)
  : rew [fun a => exists p, Q a p] H in u
    = match u with
        | ex_intro _ u1 u2
          => ex_intro
               (Q y)
               (rew H in u1)
               (rew dependent H in u2)
      end.
End ex.

Equality for ex2
Section ex2.
  Local Unset Implicit Arguments.

  Definition eq_ex2_uncurried {A : Type} (P Q : A -> Prop) {u1 v1 : A}
             {u2 : P u1} {v2 : P v1}
             {u3 : Q u1} {v3 : Q v1}
             (pq : exists2 p : u1 = v1, rew p in u2 = v2 & rew p in u3 = v3)
  : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3.

  Definition eq_ex2 {A : Type} {P Q : A -> Prop}
             (u1 v1 : A)
             (u2 : P u1) (v2 : P v1)
             (u3 : Q u1) (v3 : Q v1)
             (p : u1 = v1) (q : rew p in u2 = v2) (r : rew p in u3 = v3)
  : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3
    := eq_ex2_uncurried P Q (ex_intro2 _ _ p q r).

  Definition eq_ex2_hprop {A} {P Q : A -> Prop}
             (P_hprop : forall (x : A) (p q : P x), p = q)
             (Q_hprop : forall (x : A) (p q : Q x), p = q)
             (u1 v1 : A) (u2 : P u1) (v2 : P v1) (u3 : Q u1) (v3 : Q v1)
             (p : u1 = v1)
    : ex_intro2 P Q u1 u2 u3 = ex_intro2 P Q v1 v2 v3
    := eq_ex2 u1 v1 u2 v2 u3 v3 p (P_hprop _ _ _) (Q_hprop _ _ _).

  Lemma rew_ex2 {A x} {P : A -> Type}
        (Q : forall a, P a -> Prop)
        (R : forall a, P a -> Prop)
        (u : exists2 p, Q x p & R x p) {y} (H : x = y)
  : rew [fun a => exists2 p, Q a p & R a p] H in u
    = match u with
        | ex_intro2 _ _ u1 u2 u3
          => ex_intro2
               (Q y)
               (R y)
               (rew H in u1)
               (rew dependent H in u2)
               (rew dependent H in u3)
      end.
End ex2.