2014-07-22 16:43:18 +00:00
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abbreviation Prop : Type.{1} := Type.{0}
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2014-06-14 15:01:52 +00:00
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section
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parameter A : Type
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2014-07-22 16:43:18 +00:00
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definition eq (a b : A) : Prop
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:= ∀P : A → Prop, P a → P b
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2014-06-14 15:01:52 +00:00
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2014-07-22 16:43:18 +00:00
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theorem subst (P : A → Prop) (a b : A) (H1 : eq a b) (H2 : P a) : P b
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2014-06-14 15:01:52 +00:00
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:= H1 P H2
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theorem refl (a : A) : eq a a
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2014-07-22 16:43:18 +00:00
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:= λ (P : A → Prop) (H : P a), H
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2014-06-14 15:01:52 +00:00
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theorem symm (a b : A) (H : eq a b) : eq b a
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:= subst (λ x : A, eq x a) a b H (refl a)
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theorem trans (a b c : A) (H1 : eq a b) (H2 : eq b c) : eq a c
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:= subst (λ x : A, eq a x) b c H2 H1
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end
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check subst.{1}
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check refl.{1}
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check symm.{1}
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check trans.{1}
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