84 lines
2 KiB
Text
84 lines
2 KiB
Text
new_frontend
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def f (x : Nat) : Nat :=
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match x with
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| 30 => 31
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| y+1 => y
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| 0 => 10
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#eval f 20
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#eval f 0
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#eval f 30
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universes u
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theorem ex1 {α : Sort u} {a b : α} (h : a ≅ b) : a = b :=
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match α, a, b, h with
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| _, _, _, HEq.refl _ => rfl
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theorem ex2 {α : Sort u} {a b : α} (h : a ≅ b) : a = b :=
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match a, b, h with
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| _, _, HEq.refl _ => rfl
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theorem ex3 {α : Sort u} {a b : α} (h : a ≅ b) : a = b :=
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match b, h with
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| _, HEq.refl _ => rfl
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theorem ex4 {α β : Sort u} {b : β} {a a' : α} (h₁ : a = a') (h₂ : a' ≅ b) : a ≅ b :=
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match β, a', b, h₁, h₂ with
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| _, _, _, rfl, HEq.refl _ => HEq.refl _
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theorem ex5 {α β : Sort u} {b : β} {a a' : α} (h₁ : a = a') (h₂ : a' ≅ b) : a ≅ b :=
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match a', h₁, h₂ with
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| _, rfl, h₂ => h₂
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theorem ex6 {α β : Sort u} {b : β} {a a' : α} (h₁ : a = a') (h₂ : a' ≅ b) : a ≅ b :=
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begin
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subst h₁;
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assumption
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end
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theorem ex7 (a : Bool) (p q : Prop) (h₁ : a = true → p) (h₂ : a = false → q) : p ∨ q :=
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match h:a with
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| true => Or.inl $ h₁ h
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| false => Or.inr $ h₂ h
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def head {α} (xs : List α) (h : xs = [] → False) : α :=
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match he:xs with
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| [] => False.elim $ h he
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| x::_ => x
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variables {α : Type u} {p : α → Prop}
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theorem ex8 {a1 a2 : {x // p x}} (h : a1.val = a2.val) : a1 = a2 :=
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match a1, a2, h with
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| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
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universes v
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variables {β : α → Type v}
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theorem ex9 {p₁ p₂ : Sigma (fun a => β a)} (h₁ : p₁.1 = p₂.1) (h : p₁.2 ≅ p₂.2) : p₁ = p₂ :=
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match p₁, p₂, h₁, h with
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| ⟨_, _⟩, ⟨_, _⟩, rfl, HEq.refl _ => rfl
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inductive F : Nat → Type
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| z : {n : Nat} → F (n+1)
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| s : {n : Nat} → F n → F (n+1)
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def f0 {α : Sort u} (x : F 0) : α :=
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nomatch x
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def f1 {α : Sort u} (x : F 0 × Bool) : α :=
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nomatch x
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def f2 {α : Sort u} (x : Sum (F 0) (F 0)) : α :=
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nomatch x
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def f3 {α : Sort u} (x : Bool × F 0) : α :=
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nomatch x
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def f4 (x : Sum (F 0 × Bool) Nat) : Nat :=
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match x with
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| Sum.inr x => x
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#eval f4 $ Sum.inr 100
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