372 lines
11 KiB
Text
372 lines
11 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Data.Set.Finite
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import Mathlib.Data.Finset.Sort
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import Mathlib.Logic.Equiv.Fin
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/-!
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# Index notation for a type
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In this file we will define an index of a Lorentz tensor as a
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string satisfying certain properties.
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For example, the string `ᵘ¹²` is an index of a real Lorentz tensors.
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The first character `ᵘ` specifies the color of the index, and the subsequent characters
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`¹²` specify the id of the index.
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Strings of indices e.g. `ᵘ¹²ᵤ₄₃`` are defined elsewhere.
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-/
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open Lean
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open String
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/-- The class defining index notation on a type `X`.
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Normally `X` will be taken as the type of colors of a `TensorStructure`. -/
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class IndexNotation (X : Type) where
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/-- The list of characters describing the index notation e.g.
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`{'ᵘ', 'ᵤ'}` for real tensors. -/
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charList : Finset Char
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/-- An equivalence between `X` (colors of indices) and `charList`.
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This takes every color of index to its notation character. -/
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notaEquiv : X ≃ charList
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namespace IndexNotation
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variable (X : Type) [IndexNotation X]
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variable [Fintype X] [DecidableEq X]
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/-!
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## Lists of characters forming an index
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Here we define `listCharIndex` and properties thereof.
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-/
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/-- The map taking a color to its notation character. -/
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def nota {X : Type} [IndexNotation X] (x : X) : Char :=
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(IndexNotation.notaEquiv).toFun x
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/-- A character is a `notation character` if it is in `charList`. -/
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def isNotationChar (c : Char) : Bool :=
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if c ∈ charList X then true else false
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/-- A character is a numeric superscript if it is e.g. `⁰`, `¹`, etc. -/
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def isNumericSupscript (c : Char) : Bool :=
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c = '¹' ∨ c = '²' ∨ c = '³' ∨ c = '⁴' ∨ c = '⁵' ∨ c = '⁶' ∨ c = '⁷' ∨ c = '⁸' ∨ c = '⁹' ∨ c = '⁰'
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/-- Given a character `f` which is a notation character, this is true if `c`
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is a subscript when `f` is a subscript or `c` is a superscript when `f` is a
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superscript. -/
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def IsIndexId (f : Char) (c : Char) : Bool :=
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(isSubScriptAlnum f ∧ isNumericSubscript c) ∨
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(¬ isSubScriptAlnum f ∧ isNumericSupscript c)
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/-- The proposition for a list of characters to be the tail of an index
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e.g. `['¹', '⁷', ...]` -/
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def listCharIndexTail (f : Char) (l : List Char) : Prop :=
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l ≠ [] ∧ List.all l (fun c => IsIndexId f c)
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instance : Decidable (listCharIndexTail f l) := instDecidableAnd
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/-- The proposition for a list of characters to be the characters of an index
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e.g. `['ᵘ', '¹', '⁷', ...]` -/
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def listCharIndex (l : List Char) : Prop :=
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if h : l = [] then True
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else
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let sfst := l.head h
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if ¬ isNotationChar X sfst then False
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else
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listCharIndexTail sfst l.tail
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/-- An auxillary rewrite lemma to prove that `listCharIndex` is decidable. -/
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lemma listCharIndex_iff (l : List Char) : listCharIndex X l
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↔ (if h : l = [] then True else
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let sfst := l.head h
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if ¬ isNotationChar X sfst then False
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else listCharIndexTail sfst l.tail) := by
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rw [listCharIndex]
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instance : Decidable (listCharIndex X l) :=
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@decidable_of_decidable_of_iff _ _
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(@instDecidableDite _ _ _ _ _ <|
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fun _ => @instDecidableDite _ _ _ _ _ <|
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fun _ => instDecidableListCharIndexTail)
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(listCharIndex_iff X l).symm
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/-!
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## The definition of an index and its properties
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-/
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/-- An index is a non-empty string satisfying the condtion `listCharIndex`,
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e.g. `ᵘ¹²` or `ᵤ₄₃` etc. -/
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def Index : Type := {s : String // listCharIndex X s.toList ∧ s.toList ≠ []}
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instance : DecidableEq (Index X) := Subtype.instDecidableEq
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namespace Index
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variable {X : Type} [IndexNotation X] [Fintype X] [DecidableEq X]
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/-- Creats an index from a non-empty list of characters satisfying `listCharIndex`. -/
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def ofCharList (l : List Char) (h : listCharIndex X l ∧ l ≠ []) : Index X := ⟨l.asString, h⟩
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instance : ToString (Index X) := ⟨fun i => i.val⟩
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/-- Gets the first character in an index e.g. `ᵘ` as an element of `charList X`. -/
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def head (s : Index X) : charList X :=
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⟨s.val.toList.head (s.prop.2), by
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have h := s.prop.1
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have h2 := s.prop.2
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simp [listCharIndex] at h
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simp_all only [toList, ne_eq, Bool.not_eq_true, ↓reduceDIte]
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simpa [isNotationChar] using h.1⟩
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/-- The color associated to an index. -/
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def toColor (s : Index X) : X := (IndexNotation.notaEquiv).invFun s.head
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/-- A map from super and subscript numerical characters to the natural numbers,
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returning `0` on all other characters. -/
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def charToNat (c : Char) : Nat :=
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match c with
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| '₀' => 0
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| '₁' => 1
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| '₂' => 2
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| '₃' => 3
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| '₄' => 4
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| '₅' => 5
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| '₆' => 6
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| '₇' => 7
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| '₈' => 8
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| '₉' => 9
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| '⁰' => 0
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| '¹' => 1
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| '²' => 2
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| '³' => 3
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| '⁴' => 4
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| '⁵' => 5
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| '⁶' => 6
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| '⁷' => 7
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| '⁸' => 8
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| '⁹' => 9
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| _ => 0
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/-- The numerical characters associated with an index. -/
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def tail (s : Index X) : List Char := s.val.toList.tail
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/-- The natural numbers assocaited with an index. -/
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def tailNat (s : Index X) : List Nat := s.tail.map charToNat
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/-- The id of an index, as a natural number. -/
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def id (s : Index X) : Nat := s.tailNat.foldl (fun a b => 10 * a + b) 0
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end Index
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/-!
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## List of indices
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-/
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/-- The type of lists of indices. -/
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structure IndexList where
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val : List (Index X)
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namespace IndexList
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variable {X : Type} [IndexNotation X] [Fintype X] [DecidableEq X]
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variable (l : IndexList X)
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/-- The number of indices in an index list. -/
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def length : ℕ := l.val.length
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lemma ext (h : l.val = l2.val) : l = l2 := by
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cases l
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cases l2
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simp_all
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/-- The map of from `Fin s.numIndices` into colors associated to an index list. -/
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def colorMap : Fin l.length → X :=
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fun i => (l.val.get i).toColor
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/-- The map of from `Fin s.numIndices` into the natural numbers associated to an index list. -/
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def idMap : Fin l.length → Nat :=
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fun i => (l.val.get i).id
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lemma idMap_cast {l1 l2 : IndexList X} (h : l1 = l2) (i : Fin l1.length) :
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l1.idMap i = l2.idMap (Fin.cast (by rw [h]) i) := by
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subst h
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rfl
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/-- Given a list of indices a subset of `Fin l.numIndices × Index X`
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of pairs of positions in `l` and the corresponding item in `l`. -/
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def toPosSet (l : IndexList X) : Set (Fin l.length × Index X) :=
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{(i, l.val.get i) | i : Fin l.length}
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/-- Equivalence between `toPosSet` and `Fin l.numIndices`. -/
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def toPosSetEquiv (l : IndexList X) : l.toPosSet ≃ Fin l.length where
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toFun := fun x => x.1.1
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invFun := fun x => ⟨(x, l.val.get x), by simp [toPosSet]⟩
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left_inv x := by
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have hx := x.prop
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simp [toPosSet] at hx
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simp only [List.get_eq_getElem]
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obtain ⟨i, hi⟩ := hx
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have hi2 : i = x.1.1 := by
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obtain ⟨val, property⟩ := x
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obtain ⟨fst, snd⟩ := val
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simp_all only [Prod.mk.injEq]
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subst hi2
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simp_all only [Subtype.coe_eta]
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right_inv := by
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intro x
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rfl
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lemma toPosSet_is_finite (l : IndexList X) : l.toPosSet.Finite :=
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Finite.intro l.toPosSetEquiv
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instance : Fintype l.toPosSet where
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elems := Finset.map l.toPosSetEquiv.symm.toEmbedding Finset.univ
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complete := by
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intro x
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simp_all only [Finset.mem_map_equiv, Equiv.symm_symm, Finset.mem_univ]
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/-- Given a list of indices a finite set of `Fin l.length × Index X`
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of pairs of positions in `l` and the corresponding item in `l`. -/
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def toPosFinset (l : IndexList X) : Finset (Fin l.length × Index X) :=
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l.toPosSet.toFinset
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/-- The construction of a list of indices from a map
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from `Fin n` to `Index X`. -/
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def fromFinMap {n : ℕ} (f : Fin n → Index X) : IndexList X where
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val := (Fin.list n).map f
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@[simp]
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lemma fromFinMap_numIndices {n : ℕ} (f : Fin n → Index X) :
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(fromFinMap f).length = n := by
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simp [fromFinMap, length]
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/-!
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## Appending index lists.
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-/
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section append
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variable {X : Type} [IndexNotation X] [Fintype X] [DecidableEq X]
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variable (l l2 l3 : IndexList X)
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instance : HAppend (IndexList X) (IndexList X) (IndexList X) where
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hAppend := fun l l2 => {val := l.val ++ l2.val}
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@[simp]
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lemma append_length : (l ++ l2).length = l.length + l2.length := by
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simp [IndexList.length]
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exact List.length_append l.val l2.val
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lemma append_assoc : l ++ l2 ++ l3 = l ++ (l2 ++ l3) := by
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apply ext
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change l.val ++ l2.val ++ l3.val = l.val ++ (l2.val ++ l3.val)
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exact List.append_assoc l.val l2.val l3.val
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def appendEquiv {l l2 : IndexList X} : Fin l.length ⊕ Fin l2.length ≃ Fin (l ++ l2).length :=
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finSumFinEquiv.trans (Fin.castOrderIso (List.length_append _ _).symm).toEquiv
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def appendInl : Fin l.length ↪ Fin (l ++ l2).length where
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toFun := appendEquiv ∘ Sum.inl
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inj' := by
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intro i j h
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simp [Function.comp] at h
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exact h
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def appendInr : Fin l2.length ↪ Fin (l ++ l2).length where
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toFun := appendEquiv ∘ Sum.inr
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inj' := by
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intro i j h
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simp [Function.comp] at h
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exact h
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@[simp]
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lemma appendInl_appendEquiv :
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(l.appendInl l2).trans appendEquiv.symm.toEmbedding =
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{toFun := Sum.inl, inj' := Sum.inl_injective} := by
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ext i
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simp [appendInl]
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@[simp]
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lemma appendInr_appendEquiv :
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(l.appendInr l2).trans appendEquiv.symm.toEmbedding =
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{toFun := Sum.inr, inj' := Sum.inr_injective} := by
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ext i
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simp [appendInr]
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@[simp]
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lemma append_val {l l2 : IndexList X} : (l ++ l2).val = l.val ++ l2.val := by
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rfl
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@[simp]
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lemma idMap_append_inl {l l2 : IndexList X} (i : Fin l.length) :
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(l ++ l2).idMap (appendEquiv (Sum.inl i)) = l.idMap i := by
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simp [appendEquiv, idMap]
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rw [List.getElem_append_left]
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rfl
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@[simp]
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lemma idMap_append_inr {l l2 : IndexList X} (i : Fin l2.length) :
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(l ++ l2).idMap (appendEquiv (Sum.inr i)) = l2.idMap i := by
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simp [appendEquiv, idMap, IndexList.length]
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rw [List.getElem_append_right]
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simp
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omega
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omega
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@[simp]
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lemma colorMap_append_inl {l l2 : IndexList X} (i : Fin l.length) :
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(l ++ l2).colorMap (appendEquiv (Sum.inl i)) = l.colorMap i := by
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simp [appendEquiv, colorMap, IndexList.length]
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rw [List.getElem_append_left]
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@[simp]
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lemma colorMap_append_inl' :
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(l ++ l2).colorMap ∘ appendEquiv ∘ Sum.inl = l.colorMap := by
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funext i
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simp
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@[simp]
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lemma colorMap_append_inr {l l2 : IndexList X} (i : Fin l2.length) :
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(l ++ l2).colorMap (appendEquiv (Sum.inr i)) = l2.colorMap i := by
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simp [appendEquiv, colorMap, IndexList.length]
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rw [List.getElem_append_right]
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simp
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omega
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omega
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@[simp]
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lemma colorMap_append_inr' :
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(l ++ l2).colorMap ∘ appendEquiv ∘ Sum.inr = l2.colorMap := by
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funext i
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simp
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lemma colorMap_sumELim (l1 l2 : IndexList X) :
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Sum.elim l1.colorMap l2.colorMap =
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(l1 ++ l2).colorMap ∘ appendEquiv := by
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funext x
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match x with
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| Sum.inl i => simp
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| Sum.inr i => simp
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end append
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end IndexList
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end IndexNotation
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