69 lines
1.5 KiB
Text
69 lines
1.5 KiB
Text
definition Prop [inline] := Type.{0}
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definition false : Prop := ∀x : Prop, x
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check false
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theorem false_elim (C : Prop) (H : false) : C
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:= H C
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definition eq {A : Type} (a b : A)
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:= ∀ P : A → Prop, P a → P b
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check eq
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infix `=`:50 := eq
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theorem refl {A : Type} (a : A) : a = a
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:= λ P H, H
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definition true : Prop
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:= false = false
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theorem trivial : true
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:= refl false
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theorem subst {A : Type} {P : A -> Prop} {a b : A} (H1 : a = b) (H2 : P a) : P b
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:= H1 _ H2
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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 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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inductive nat : Type :=
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zero : nat,
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succ : nat → nat
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namespace nat end nat open nat
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print "using strict implicit arguments"
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abbreviation symmetric {A : Type} (R : A → A → Prop) := ∀ ⦃a b⦄, R a b → R b a
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check symmetric
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variable p : nat → nat → Prop
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check symmetric p
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axiom H1 : symmetric p
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axiom H2 : p zero (succ zero)
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check H1
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check H1 H2
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print "------------"
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print "using implicit arguments"
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abbreviation symmetric2 {A : Type} (R : A → A → Prop) := ∀ {a b}, R a b → R b a
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check symmetric2
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check symmetric2 p
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axiom H3 : symmetric2 p
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axiom H4 : p zero (succ zero)
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check H3
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check H3 H4
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print "-----------------"
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print "using strict implicit arguments (ASCII notation)"
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abbreviation symmetric3 {A : Type} (R : A → A → Prop) := ∀ {{a b}}, R a b → R b a
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check symmetric3
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check symmetric3 p
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axiom H5 : symmetric3 p
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axiom H6 : p zero (succ zero)
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check H5
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check H5 H6
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