English
Induction principle for closures for predicates with two arguments:
Русский
Индуктивный принцип для замыкания предикатов с двумя аргументами.
LaTeX
$$closure_induction₂: {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop} → (mem_mem : ∀ (x) (hx : x ∈ s) (y) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy)) → (zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _)) → (add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz) → (add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz)) → (mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz) → (mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz)) {x y : R} (hx : x ∈ closure s) (hy : y ∈ closure s) : p x y hx hy$$
Lean4
/-- An induction principle for closure membership for predicates with two arguments. -/
@[elab_as_elim]
theorem closure_induction₂ {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop}
(mem_mem : ∀ (x) (hx : x ∈ s) (y) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy))
(zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
(add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
(add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
(mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
(mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz)) {x y : R}
(hx : x ∈ closure s) (hy : y ∈ closure s) : p x y hx hy := by
induction hy using closure_induction with
| mem z hz =>
induction hx using closure_induction with
| mem _ h => exact mem_mem _ h _ hz
| zero => exact zero_left _ _
| mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
| zero => exact zero_right x hx
| mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
