28 lines
967 B
Text
28 lines
967 B
Text
infix:50 " ≅ " => HEq
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theorem ex1 {α : Sort u} {a b : α} (h : a ≅ b) : a = b :=
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match h with
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| HEq.refl _ => rfl
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def ex2 {α : Sort u2} {a : α} {motive : {β : Sort u2} → β → Sort u1} (m : motive a) {β : Sort u2} {b : β} (h : a ≅ b) : motive b :=
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match h, m with
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| HEq.refl _, m => m
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def ex3 {α : Sort u} {a : α} {p : α → Sort v} {b : α} (h₁ : a ≅ b) (h₂ : p a) : p b :=
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match h₁, h₂ with
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| HEq.refl _, h₂ => h₂
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theorem ex4 {α β : Sort u} {a : α} {b : β} (h : a ≅ b) : b ≅ a :=
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match h with
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| HEq.refl _ => HEq.refl _
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theorem ex5 {α : Sort u} {a a' : α} (h : a = a') : a ≅ a' :=
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match h with
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| rfl => HEq.refl _
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theorem ex6 {α β : Sort u} (h : α = β) (a : α) : cast h a ≅ a :=
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match h with
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| rfl => HEq.refl _
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theorem ex7 {α β σ : Sort u} {a : α} {b : β} {c : σ} (h₁ : a ≅ b) (h₂ : b ≅ c) : a ≅ c :=
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match h₁, h₂ with
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| HEq.refl _, HEq.refl _ => HEq.refl _
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