feat(library/logic): add basic definitions

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/logic.lean

@@ -1,2 +1,77 @@
-definition [inline] Bool : Type.{0} := Type.{1}
+definition Bool [inline] := Type.{0}
 
+inductive false : Bool :=
+-- No constructors
+
+theorem false_elim (c : Bool) (H : false)
+:= @false_rec c H
+
+inductive true : Bool :=
+| trivial : true
+
+definition not (a : Bool) := a → false
+precedence `¬`:40
+notation `¬` a := not a
+
+theorem not_intro {a : Bool} (H : a → false) : ¬ a
+:= H
+
+theorem not_elim {a : Bool} (H1 : ¬ a) (H2 : a) : false
+:= H1 H2
+
+theorem absurd {a : Bool} (H1 : a) (H2 : ¬ a) : false
+:= H2 H1
+
+theorem mt {a b : Bool} (H1 : a → b) (H2 : ¬ b) : ¬ a
+:= λ Ha : a, absurd (H1 Ha) H2
+
+theorem contrapos {a b : Bool} (H : a → b) : ¬ b → ¬ a
+:= λ Hnb : ¬ b, mt H Hnb
+
+theorem absurd_elim {a : Bool} (b : Bool) (H1 : a) (H2 : ¬ a) : b
+:= false_elim b (absurd H1 H2)
+
+inductive and (a b : Bool) : Bool :=
+| and_intro : a → b → and a b
+
+infixr `/\` 35 := and
+infixr `∧`  35 := and
+
+theorem and_elim_left {a b : Bool} (H : a ∧ b) : a
+:= and_rec (λ a b, a) H
+
+theorem and_elim_right {a b : Bool} (H : a ∧ b) : b
+:= and_rec (λ a b, b) H
+
+inductive or (a b : Bool) : Bool :=
+| or_intro_left  : a → or a b
+| or_intro_right : b → or a b
+
+infixr `\/` 30 := or
+infixr `∨` 30 := or
+
+theorem or_elim (a b c : Bool) (H1 : a ∨ b) (H2 : a → c) (H3 : b → c) : c
+:= or_rec H2 H3 H1
+
+inductive eq {A : Type} (a : A) : A → Bool :=
+| eq_intro : eq A a a  -- TODO: use elaborator in inductive_cmd module, we should not need to type A here
+
+infix `=` 50 := eq
+
+theorem refl {A : Type} (a : A) : a = a
+:= @(@eq_intro A) a -- TODO: fix '@', we should not need to use two '@'
+
+theorem subst {A : Type} {a b : A} {P : A → Bool} (H1 : a = b) (H2 : P a) : P b
+:= eq_rec H2 H1
+
+theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c
+:= subst H2 H1
+
+theorem symm {A : Type} {a b : A} (H : a = b) : b = a
+:= subst H (refl a)
+
+-- theorem congr1 {A B : Type} {f g : A → B} (H : f = g) (a : A) : f a = g a
+-- := subst H (refl (f a)) -- TODO: check unifier does not work in this case
+
+theorem congr2 {A B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b
+:= subst H (refl (f a))