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