91 lines
2.3 KiB
Text
91 lines
2.3 KiB
Text
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
|
||
}
|