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feat: prove the distributivity of forall over OmegaSequence.flatten #532

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chenson2018 merged 2 commits into from
Apr 30, 2026
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feat: prove the distributivity of forall over OmegaSequence.flatten #532

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a968c05
feat: prove the distributivity of forall over OmegaSequence.flatten
ctchou Apr 29, 2026
bb44b1a
incorporate Chris Henson's comments
ctchou Apr 30, 2026
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11 changes: 10 additions & 1 deletion Cslib/Foundations/Data/OmegaSequence/Flatten.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -150,7 +150,7 @@ theorem flatten_drop [Inhabited α]
(flatten_take_drop h_ls n).2

/-- `ls n` is the segement from position `ls.cumLen n` to position `ls.cumLen (n + 1) - 1`
of ls.flatten` -/
of `ls.flatten` -/
@[simp, scoped grind =]
theorem extract_flatten [Inhabited α] {ls : ωSequence (List α)} (h_ls : ∀ k, (ls k).length > 0)
(n : ℕ) : ls.flatten.extract (ls.cumLen n) (ls.cumLen (n + 1)) = ls n := by
Expand All @@ -159,6 +159,15 @@ theorem extract_flatten [Inhabited α] {ls : ωSequence (List α)} (h_ls : ∀ k
have h_take := flatten_take h_ls' 1
grind [extract_eq_drop_take]

/-- Distributivity of "forall" over `flatten`. -/
theorem forall_flatten_iff [Inhabited α] {ls : ωSequence (List α)} (h_ls : ∀ k, (ls k).length > 0)
(p : α → Prop) : (∀ n, p (ls.flatten n)) ↔ ∀ k, (ls k).Forall p := by
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ctchou marked this conversation as resolved.
constructor
· simp only [List.forall_iff_forall_mem, List.forall_mem_iff_getElem, ← extract_flatten h_ls]
grind
· have := segment_upper_bound (cumLen_strictMono h_ls)
grind [List.forall_iff_forall_mem, flatten_def]

/-- Given an ω-sequence `s` and a function `f : ℕ → ℕ`, `s.toSegs f` is the ω-sequence
whose `n`-th element is the list `s.extract (f n) (f (n + 1))`. In all its uses, the
function `f` will always be assumed to be strictly monotonic with `f 0 = 0`. -/
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