English
A refinement of cons_of_le with strengthened parameters; still preserves IsChain under GT.
Русский
Улучшенная версия cons_of_le; цепь сохраняется под GT.
LaTeX
$$$IsChain\\, GT\\, (a :: m) \\ leftarrow (IsChain\\, GT\\, (a :: as)) \\land (IsChain\\, GT\\, m) \\land (m \\le as) $$$
Lean4
/-- Given a chain from `a` to `b`, and a predicate true at `a`, if `r x y → p x → p y` then
the predicate is true everywhere in the chain.
That is, we can propagate the predicate down the chain.
-/
theorem cons_induction (p : α → Prop) (l : List α) (h : IsChain r (a :: l)) (carries : ∀ ⦃x y : α⦄, r x y → p x → p y)
(initial : p a) : ∀ i ∈ l, p i := fun _ hi => h.induction _ _ carries (fun _ => initial) _ (mem_cons_of_mem _ hi)
