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refactor: rename SN.isNormalizable to SN.normalizable #736

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Merged
chenson2018 merged 2 commits into from
Aug 9, 2026
Merged

refactor: rename SN.isNormalizable to SN.normalizable #736

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5fdc1f8
feat: Add lemma SN.to_WN: SN implies Normalizable
lengyijun Jul 21, 2026
e3c603f
reprove SN.isNormalizable
lengyijun Aug 9, 2026
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19 changes: 9 additions & 10 deletions Cslib/Foundations/Relation/Confluence.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -187,6 +187,14 @@ lemma SN.onFun_of_image {r : β → β → Prop} {f : α → β} (hx : SN r (f x

lemma SN.of_normal (hx : Normal r x) : SN r x := SN.intro fun y hy => (hx ⟨y, hy⟩).elim

theorem SN.normalizable (hx : SN r x) : Normalizable r x := by
induction hx with | intro x h ih =>
by_cases hy: (∃ y, r x y)
· obtain ⟨y, hy⟩ := hy
obtain ⟨z, hz, hnormal⟩ := ih y hy
exact ⟨z, .trans (.single hy) hz, hnormal⟩
· exists x

lemma Terminating.apply (hr : Terminating r) (x : α) : SN r x := WellFounded.apply hr x

lemma Terminating.iff_forall_sn : Terminating r ↔ ∀ x, SN r x :=
Expand Down Expand Up @@ -220,17 +228,8 @@ lemma Terminating.subtype_sn (r : α → α → Prop) :
Terminating (α := {x // SN r x}) (fun a b => r a b) :=
iff_forall_sn.mpr fun x => x.property.onFun_of_image

theorem SN.isNormalizable (hx : SN r x) : Normalizable r x := by
-- restrict to the subtype where all elements are `SN`, so `flip r` is well-founded
obtain ⟨⟨y, hsn⟩, hred : ReflTransGen r x y, hnorm⟩ :=
(Terminating.subtype_sn r).has_min
(s := Subtype.val ⁻¹' ({y | ReflTransGen r x y})) ⟨⟨x, hx⟩, ReflTransGen.refl⟩
use y, hred
intro ⟨z, hyz⟩
exact hnorm ⟨z, hsn.of_rel hyz⟩ (.tail hred hyz) hyz

theorem Terminating.isNormalizing (hr : Terminating r) : Normalizing r :=
fun x => (hr.apply x).isNormalizable
fun x => (hr.apply x).normalizable

theorem Terminating.isConfluent_iff_all_unique_Normal (ht : Terminating r) :
Confluent r ↔ ∀ a : α, ∃! n : α, ReflTransGen r a n ∧ Normal r n := by
Expand Down
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