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