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[Merged by Bors] - feat: two easy lemmas about List.Lex #6395

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[Merged by Bors] - feat: two easy lemmas about List.Lex #6395

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feat: two easy lemmas about List.Lex
dagurtomas Aug 6, 2023
dd8167b
golf
eric-wieser Aug 6, 2023
df9daf6
Update Mathlib/Data/List/Lex.lean
dagurtomas Aug 6, 2023
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13 changes: 13 additions & 0 deletions Mathlib/Data/List/Lex.lean
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Expand Up @@ -61,6 +61,19 @@ theorem not_nil_right (r : α → α → Prop) (l : List α) : ¬Lex r l [] :=
fun.
#align list.lex.not_nil_right List.Lex.not_nil_right

theorem nil_left_or_nil_eq {r : α → α → Prop} (l : List α) : List.Lex r [] l ∨ [] = l := by
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Can you add a version assuming IsRefl r that says List.Lex r [] l?

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@dagurtomas dagurtomas Aug 6, 2023 •
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That result is not true. List.Lex is not reflexive even though r is.

Indeed, we have: List.Lex.not_nil_right : ¬List.Lex r l [], in particular ¬List.Lex r [] []

induction l with
| nil => exact Or.inr rfl
| cons a as _ => exact Or.inl (by apply nil)

@[simp]
theorem singleton_iff {r : α → α → Prop} (a b : α) : List.Lex r [a] [b] ↔ r a b := by
refine' ⟨fun h ↦ _, List.Lex.rel⟩
by_contra h'
cases h
· apply not_nil_right r []; assumption
· apply h'; assumption

instance isOrderConnected (r : α → α → Prop) [IsOrderConnected α r] [IsTrichotomous α r] :
IsOrderConnected (List α) (Lex r) where
conn := aux where
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