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[Merged by Bors] - feat(Order/WellFounded): the minimum of f '' s is f applied to the minimum of s #38992

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[Merged by Bors] - feat(Order/WellFounded): the minimum of f '' s is f applied to the minimum of s #38992

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feat(Order/WellFounded): the minimum of `f '' s` is `f` applied to th…
SnirBroshi May 6, 2026
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SnirBroshi May 7, 2026
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8 changes: 8 additions & 0 deletions Mathlib/Order/WellFounded.lean
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Expand Up @@ -154,6 +154,14 @@ theorem isWellOrder_iff_exists_not_lt_and_eq_or_gt :
· grind [wellFounded_iff_has_min]
· grind [h {a, b} <| by simp]

/-- The minimum of `f '' s` is `f` applied to the minimum of `s`. -/
theorem min_image {r : β → β → Prop} [Std.Trichotomous r] (wf : WellFounded r) (f : α → β)
{s : Set α} (hne : s.Nonempty) :
wf.min (f '' s) (hne.image f) = f (wf.onFun (f := f) |>.min s hne) := by
apply min_eq_of_forall_not_lt wf <| Set.mem_image_of_mem f <| min_mem wf.onFun s hne
rintro _ ⟨a, has, rfl⟩
exact wf.onFun.not_lt_min s has

theorem not_rel_apply_succ [h : IsWellFounded α r] (f : ℕ → α) : ∃ n, ¬ r (f (n + 1)) (f n) := by
by_contra! hf
exact (wellFounded_iff_isEmpty_descending_chain.1 h.wf).elim ⟨f, hf⟩
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