new_frontend

@[recursor 4]
def Or.elim2 {p q r : Prop} (major : p ∨ q) (left : p → r) (right : q → r) : r :=
Or.elim major left right

theorem tst0 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
  induction h;
  { apply Or.inr; assumption };
  { 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 tst2 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by induction h using elim2 with
   | left _  => apply Or.inr; assumption
   | right _ => apply Or.inl; assumption

theorem tst3 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
  induction h using elim2 with
  | right h => exact Or.inl h
  | left h  => exact Or.inr h
}

theorem tst4 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
  induction h using elim2 with
  | right h => ?myright
  | left h  => ?myleft;
  case myleft  => exact Or.inr h;
  case myright => exact Or.inl h;
}

theorem tst5 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
  induction h using elim2 with
  | right h => _
  | left h  => refine Or.inr ?_; exact h;
  case right => exact Or.inl h
}

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 using Iff.elim with
   | _ 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.elim with
   | _ h _ => exact h h₂

theorem tst12 {p q : Prop } (h₁ : p ∨ q) (h₂ : p ↔ q) (h₃ : p) : q :=
by {
  failIfSuccess induction h₁ using Iff.elim;
  induction h₂ using Iff.elim with
  | _ h _ => exact h h₃
}