98 lines
2.4 KiB
Text
98 lines
2.4 KiB
Text
theorem tst0 {p q : Prop } (h : p ∨ q) : q ∨ p :=
|
||
by {
|
||
induction h;
|
||
{ apply Or.inr; assumption };
|
||
{ apply Or.inl; assumption }
|
||
}
|
||
|
||
theorem tst0' {p q : Prop } (h : p ∨ q) : q ∨ p := by
|
||
induction h
|
||
focus
|
||
apply Or.inr
|
||
assumption
|
||
focus
|
||
apply Or.inl
|
||
assumption
|
||
|
||
theorem tst1 {p q : Prop } (h : p ∨ q) : q ∨ p := by
|
||
induction h with
|
||
| inr h2 => exact Or.inl h2
|
||
| inl h1 => exact Or.inr h1
|
||
|
||
theorem tst6 {p q : Prop } (h : p ∨ q) : q ∨ p :=
|
||
by {
|
||
cases h with
|
||
| inr h2 => exact Or.inl h2
|
||
| inl h1 => exact Or.inr h1
|
||
}
|
||
|
||
theorem tst7 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] :=
|
||
by {
|
||
induction xs with
|
||
| nil => exact rfl
|
||
| cons z zs ih => exact absurd rfl (h z zs)
|
||
}
|
||
|
||
theorem tst8 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] := by {
|
||
induction xs;
|
||
exact rfl;
|
||
exact absurd rfl $ h _ _
|
||
}
|
||
|
||
theorem tst9 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] := by
|
||
cases xs with
|
||
| nil => exact rfl
|
||
| cons z zs => exact absurd rfl (h z zs)
|
||
|
||
theorem tst10 {p q : Prop } (h₁ : p ↔ q) (h₂ : p) : q := by
|
||
induction h₁ with
|
||
| intro h _ => exact h h₂
|
||
|
||
def Iff2 (m p q : Prop) := p ↔ q
|
||
|
||
theorem tst11 {p q r : Prop } (h₁ : Iff2 r p q) (h₂ : p) : q := by
|
||
induction h₁ using Iff.rec with
|
||
| intro h _ => exact h h₂
|
||
|
||
theorem tst12 {p q : Prop } (h₁ : p ∨ q) (h₂ : p ↔ q) (h₃ : p) : q := by
|
||
fail_if_success induction h₁ using Iff.casesOn
|
||
induction h₂ using Iff.casesOn with
|
||
| intro h _ =>
|
||
exact h h₃
|
||
|
||
inductive Tree
|
||
| leaf₁
|
||
| leaf₂
|
||
| node : Tree → Tree → Tree
|
||
|
||
def Tree.isLeaf₁ : Tree → Bool
|
||
| leaf₁ => true
|
||
| _ => false
|
||
|
||
theorem tst13 (x : Tree) (h : x = Tree.leaf₁) : x.isLeaf₁ = true := by
|
||
cases x with
|
||
| leaf₁ => rfl
|
||
| _ => injection h
|
||
|
||
theorem tst14 (x : Tree) (h : x = Tree.leaf₁) : x.isLeaf₁ = true := by
|
||
induction x with
|
||
| leaf₁ => rfl
|
||
| _ => injection h
|
||
|
||
inductive Vec (α : Type) : Nat → Type
|
||
| nil : Vec α 0
|
||
| cons : (a : α) → {n : Nat} → (as : Vec α n) → Vec α (n+1)
|
||
|
||
def getHeads {α β} {n} (xs : Vec α (n+1)) (ys : Vec β (n+1)) : α × β := by
|
||
cases xs
|
||
cases ys
|
||
apply Prod.mk
|
||
repeat
|
||
trace_state
|
||
assumption
|
||
done
|
||
|
||
theorem ex1 (n m o : Nat) : n = m + 0 → m = o → m = o := by
|
||
intro (h₁ : n = m) h₂
|
||
rw [← h₁, ← h₂]
|
||
assumption
|