121 lines
3.3 KiB
Text
121 lines
3.3 KiB
Text
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
|
||
|
||
notation `assume` binders `,` r:(scoped f, f) := r
|
||
notation `take` binders `,` r:(scoped f, f) := r
|
||
|
||
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
|
||
:= assume Ha : a, absurd (H1 Ha) H2
|
||
|
||
theorem contrapos {a b : Bool} (H : a → b) : ¬ b → ¬ a
|
||
:= assume 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 :=
|
||
| refl : eq a a
|
||
|
||
infix `=` 50 := eq
|
||
|
||
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))
|
||
|
||
theorem congr2 {A B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b
|
||
:= subst H (refl (f a))
|
||
|
||
definition cast {A B : Type} (H : A = B) (a : A) : B
|
||
:= eq_rec a H
|
||
|
||
-- TODO(Leo): check why unifier needs 'help' in the following theorem
|
||
theorem cast_refl.{l} {A : Type.{l}} (a : A) : @cast.{l} A A (refl A) a = a
|
||
:= refl (@cast.{l} A A (refl A) a)
|
||
|
||
definition iff (a b : Bool) := (a → b) ∧ (b → a)
|
||
infix `↔` 50 := iff
|
||
|
||
theorem iff_intro {a b : Bool} (H1 : a → b) (H2 : b → a) : a ↔ b
|
||
:= and_intro H1 H2
|
||
|
||
theorem iff_elim {a b c : Bool} (H1 : (a → b) → (b → a) → c) (H2 : a ↔ b) : c
|
||
:= and_rec H1 H2
|
||
|
||
theorem iff_elim_left {a b : Bool} (H : a ↔ b) : a → b
|
||
:= iff_elim (assume H1 H2, H1) H
|
||
|
||
theorem iff_elim_right {a b : Bool} (H : a ↔ b) : b → a
|
||
:= iff_elim (assume H1 H2, H2) H
|
||
|
||
inductive Exists {A : Type} (P : A → Bool) : Bool :=
|
||
| exists_intro : ∀ (a : A), P a → Exists P
|
||
|
||
notation `∃` binders `,` r:(scoped P, Exists P) := r
|
||
|
||
theorem exists_elim {A : Type} {P : A → Bool} {B : Bool} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B) : B
|
||
:= Exists_rec H2 H1
|
||
|
||
definition inhabited (A : Type) := ∃ x : A, true
|
||
|
||
theorem inhabited_intro {A : Type} (a : A) : inhabited A
|
||
:= exists_intro a trivial
|
||
|
||
theorem inhabited_elim {A : Type} {B : Bool} (H1 : inhabited A) (H2 : A → B) : B
|
||
:= exists_elim H1 (λ (a : A) (H : true), H2 a)
|
||
|
||
theorem inhabited_Bool : inhabited Bool
|
||
:= inhabited_intro true
|
||
|
||
theorem inhabited_fun (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B)
|
||
:= inhabited_elim H (take (b : B), inhabited_intro (λ a : A, b))
|