feat(Order/WellFounded): characterize WellFoundedLT and IsWellOrder in terms of nonempty sets#37353
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…r` in terms of nonempty sets
PR summary 00fca21215Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Komyyy
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Apr 6, 2026
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| simpa [lt_iff_le_not_ge] using hm x hxs | ||
| · have ⟨⟨m, hms⟩, hm⟩ := h s hs | ||
| refine ⟨m, hms, fun x hxs ↦ ?_⟩ | ||
| simpa [lt_iff_le_not_ge] using @hm ⟨x, hxs⟩ |
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Minimal is more appropriate here:
Suggested change
| simpa [lt_iff_le_not_ge] using @hm ⟨x, hxs⟩ | |
| theorem wellFoundedLT_iff_exists_minimal [Preorder α] : | |
| WellFoundedLT α ↔ ∀ s : Set α, s.Nonempty → ∃ m, Minimal (· ∈ s) m := by |
| simpa [lt_iff_le_not_ge] using hm x hxs | ||
| · have ⟨⟨m, hms⟩, hm⟩ := h s hs | ||
| refine ⟨m, hms, fun x hxs ↦ ?_⟩ | ||
| simpa [lt_iff_le_not_ge] using @hm ⟨x, hxs⟩ |
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Now you can prove more elegantly:
Suggested change
| simpa [lt_iff_le_not_ge] using @hm ⟨x, hxs⟩ | |
| simp only [WellFoundedLT, isWellFounded_iff, wellFounded_iff_has_min, not_lt_iff_le_imp_ge, | |
| Minimal] |
| refine ⟨m, hms, fun x hxs ↦ ?_⟩ | ||
| simpa [lt_iff_le_not_ge] using @hm ⟨x, hxs⟩ | ||
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| theorem isWellOrder_iff_exists_not_lt_and_eq_or_lt : |
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Better name:
Suggested change
| theorem isWellOrder_iff_exists_not_lt_and_eq_or_lt : | |
| theorem isWellOrder_iff_exists_not_lt_and_eq_or_gt : |
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| theorem isWellOrder_iff_exists_not_lt_and_eq_or_lt : | ||
| IsWellOrder α r ↔ ∀ s : Set α, s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬r x m ∧ (m = x ∨ r m x) := by | ||
| refine ⟨fun h s hs ↦ ?_, fun h ↦ @IsWellOrder.mk α r ⟨?_⟩ ⟨fun a b ↦ ?_⟩⟩ |
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Structure notation works well here:
Suggested change
| refine ⟨fun h s hs ↦ ?_, fun h ↦ @IsWellOrder.mk α r ⟨?_⟩ ⟨fun a b ↦ ?_⟩⟩ | |
| refine ⟨fun h s hs ↦ ?_, fun h ↦ { wf := ?_, trichotomous a b := ?_} ⟩ |
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Both of these use the existing characterization of
WellFounded.