[Merged by Bors] - feat: two easy lemmas about List.Lex #6395
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dagurtomas
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Mathlib/Data/List/Lex.lean
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| @@ -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 | |||
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| 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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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 [] []
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eric-wieser
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eric-wieser
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Mathlib/Data/List/Lex.lean
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| refine' ⟨fun h ↦ _, List.Lex.rel⟩ | ||
| by_contra h' | ||
| cases h | ||
| · apply not_nil_right r []; assumption | ||
| · apply h'; assumption | ||
| theorem singleton_iff {r : α → α → Prop} (a b : α) : List.Lex r [a] [b] ↔ r a b := | ||
| ⟨fun | rel h => h, List.Lex.rel⟩ |
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Proves that `[]` is the "bottom element" for `List.Lex r` for any relation `r` and that `List.Lex r [a] [b]` iff `r a b` Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Proves that `[]` is the "bottom element" for `List.Lex r` for any relation `r` and that `List.Lex r [a] [b]` iff `r a b` Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Nobeling's theorem: the Z-module of locally constant maps from a profinite set to the integers is free. - [x] depends on: #6360 - [x] depends on: #6373 - [x] depends on: #6395 - [x] depends on: #6396 - [x] depends on: #6432 - [x] depends on: #6520 - [x] depends on: #6578 - [x] depends on: #6589 - [x] depends on: #6722 - [x] depends on: #7829 - [x] depends on: #7895 - [x] depends on: #7896 - [x] depends on: #7897 - [x] depends on: #7899
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Nobeling's theorem: the Z-module of locally constant maps from a profinite set to the integers is free. - [x] depends on: #6360 - [x] depends on: #6373 - [x] depends on: #6395 - [x] depends on: #6396 - [x] depends on: #6432 - [x] depends on: #6520 - [x] depends on: #6578 - [x] depends on: #6589 - [x] depends on: #6722 - [x] depends on: #7829 - [x] depends on: #7895 - [x] depends on: #7896 - [x] depends on: #7897 - [x] depends on: #7899
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Proves that
[]is the "bottom element" forList.Lex rfor any relationrand thatList.Lex r [a] [b]iffr a b