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defined StrictSeries, copied API from JordanHolder
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CommAlg/StrictSeries.lean
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CommAlg/StrictSeries.lean
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/-
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Most of this file is Copyright (c) 2021 Chris Hughes. All rights reserved.
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Released under Apache 2.0 license as described in the Mathlib file LICENSE.
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Authors: Chris Hughes
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-/
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import Mathlib.Order.Lattice
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import Mathlib.Data.List.Sort
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import Mathlib.Logic.Equiv.Fin
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import Mathlib.Logic.Equiv.Functor
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import Mathlib.Data.Fintype.Card
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import Mathlib.Order.Monotone.Basic
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structure StrictSeries (X : Type u) [LT X] : Type u where
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length : ℕ
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toFun : Fin (length + 1) → X
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step' : ∀ i : Fin length, (toFun (Fin.castSucc i)) < (toFun (Fin.succ i))
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--StrictMono toFun
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namespace StrictSeries
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section FinLemmas
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-- TODO: move these to `VecNotation` and rename them to better describe their statement
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variable {α : Type _} {m n : ℕ} (a : Fin m.succ → α) (b : Fin n.succ → α)
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theorem append_castAdd_aux (i : Fin m) :
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Matrix.vecAppend (Nat.add_succ _ _).symm (a ∘ Fin.castSucc) b
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(Fin.castSucc <| Fin.castAdd n i) =
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a (Fin.castSucc i) := by
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cases i
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simp [Matrix.vecAppend_eq_ite, *]
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#align composition_series.append_cast_add_aux StrictSeries.append_castAdd_aux
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theorem append_succ_castAdd_aux (i : Fin m) (h : a (Fin.last _) = b 0) :
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Matrix.vecAppend (Nat.add_succ _ _).symm (a ∘ Fin.castSucc) b (Fin.castAdd n i).succ =
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a i.succ := by
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cases' i with i hi
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simp only [Matrix.vecAppend_eq_ite, hi, Fin.succ_mk, Function.comp_apply, Fin.castSucc_mk,
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Fin.val_mk, Fin.castAdd_mk]
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split_ifs with h_1
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· rfl
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· have : i + 1 = m := le_antisymm hi (le_of_not_gt h_1)
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calc
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b ⟨i + 1 - m, by simp [this]⟩ = b 0 := congr_arg b (by simp [Fin.ext_iff, this])
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_ = a (Fin.last _) := h.symm
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_ = _ := congr_arg a (by simp [Fin.ext_iff, this])
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#align composition_series.append_succ_cast_add_aux StrictSeries.append_succ_castAdd_aux
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theorem append_natAdd_aux (i : Fin n) :
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Matrix.vecAppend (Nat.add_succ _ _).symm (a ∘ Fin.castSucc) b
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(Fin.castSucc <| Fin.natAdd m i) =
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b (Fin.castSucc i) := by
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cases i
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simp only [Matrix.vecAppend_eq_ite, Nat.not_lt_zero, Fin.natAdd_mk, add_lt_iff_neg_left,
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add_tsub_cancel_left, dif_neg, Fin.castSucc_mk, not_false_iff, Fin.val_mk]
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#align composition_series.append_nat_add_aux StrictSeries.append_natAdd_aux
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theorem append_succ_natAdd_aux (i : Fin n) :
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Matrix.vecAppend (Nat.add_succ _ _).symm (a ∘ Fin.castSucc) b (Fin.natAdd m i).succ =
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b i.succ := by
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cases' i with i hi
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simp only [Matrix.vecAppend_eq_ite, add_assoc, Nat.not_lt_zero, Fin.natAdd_mk,
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add_lt_iff_neg_left, add_tsub_cancel_left, Fin.succ_mk, dif_neg, not_false_iff, Fin.val_mk]
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#align composition_series.append_succ_nat_add_aux StrictSeries.append_succ_natAdd_aux
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end FinLemmas
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section LT
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variable {X : Type u} [LT X]
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instance coeFun : CoeFun (StrictSeries X) fun x => Fin (x.length + 1) → X where
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coe := StrictSeries.toFun
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instance inhabited [Inhabited X] : Inhabited (StrictSeries X) :=
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⟨{ length := 0
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toFun := default
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step' := fun x => x.elim0 }⟩
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theorem step (s : StrictSeries X) :
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∀ i : Fin s.length, (s (Fin.castSucc i)) < (s (Fin.succ i)) :=
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s.step'
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theorem coeFn_mk (length : ℕ) (toFun step) :
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(@StrictSeries.mk X _ length toFun step : Fin length.succ → X) = toFun :=
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rfl
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theorem lt_succ (s : StrictSeries X) (i : Fin s.length) :
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s (Fin.castSucc i) < s (Fin.succ i) :=
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(s.step _)
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instance membership : Membership X (StrictSeries X) :=
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⟨fun x s => x ∈ Set.range s⟩
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theorem mem_def {x : X} {s : StrictSeries X} : x ∈ s ↔ x ∈ Set.range s :=
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Iff.rfl
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/-- The ordered `List X` of elements of a `StrictSeries X`. -/
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def toList (s : StrictSeries X) : List X :=
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List.ofFn s
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/-- Two `StrictSeries` are equal if they are the same length and
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have the same `i`th element for every `i` -/
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theorem ext_fun {s₁ s₂ : StrictSeries X} (hl : s₁.length = s₂.length)
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(h : ∀ i, s₁ i = s₂ (Fin.cast (congr_arg Nat.succ hl) i)) : s₁ = s₂ := by
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cases s₁; cases s₂
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-- Porting note: `dsimp at *` doesn't work. Why?
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dsimp at hl h
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subst hl
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simpa [Function.funext_iff] using h
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@[simp]
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theorem length_toList (s : StrictSeries X) : s.toList.length = s.length + 1 := by
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rw [toList, List.length_ofFn]
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theorem toList_ne_nil (s : StrictSeries X) : s.toList ≠ [] := by
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rw [← List.length_pos_iff_ne_nil, length_toList]; exact Nat.succ_pos _
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theorem chain'_toList (s : StrictSeries X) : List.Chain' (· < ·) s.toList :=
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List.chain'_iff_get.2
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(by
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intro i hi
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simp only [toList, List.get_ofFn]
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rw [length_toList] at hi
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exact s.step ⟨i, hi⟩)
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@[simp]
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theorem mem_toList {s : StrictSeries X} {x : X} : x ∈ s.toList ↔ x ∈ s := by
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rw [toList, List.mem_ofFn, mem_def]
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/-- Make a `StrictSeries X` from the ordered list of its elements. -/
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def ofList (l : List X) (hl : l ≠ []) (hc : List.Chain' (· < ·) l) : StrictSeries X
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where
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length := l.length - 1
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toFun i :=
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l.nthLe i
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(by
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conv_rhs => rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt (List.length_pos_of_ne_nil hl))]
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exact i.2)
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step' := fun ⟨i, hi⟩ => List.chain'_iff_get.1 hc i hi
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theorem length_ofList (l : List X) (hl : l ≠ []) (hc : List.Chain' (· < ·) l) :
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(ofList l hl hc).length = l.length - 1 :=
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rfl
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theorem ofList_toList (s : StrictSeries X) :
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ofList s.toList s.toList_ne_nil s.chain'_toList = s := by
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refine' ext_fun _ _
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· rw [length_ofList, length_toList, Nat.succ_sub_one]
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· rintro ⟨i, hi⟩
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simp [ofList, toList, -List.ofFn_succ]
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@[simp]
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theorem ofList_toList' (s : StrictSeries X) :
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ofList s.toList s.toList_ne_nil s.chain'_toList = s :=
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ofList_toList s
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@[simp]
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theorem toList_ofList (l : List X) (hl : l ≠ []) (hc : List.Chain' (· < ·) l) :
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toList (ofList l hl hc) = l := by
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refine' List.ext_get _ _
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· rw [length_toList, length_ofList,
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tsub_add_cancel_of_le (Nat.succ_le_of_lt <| List.length_pos_of_ne_nil hl)]
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· intro i hi hi'
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dsimp [ofList, toList]
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rw [List.get_ofFn]
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rfl
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/-- The last element of a `StrictSeries` -/
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def top (s : StrictSeries X) : X :=
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s (Fin.last _)
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theorem top_mem (s : StrictSeries X) : s.top ∈ s :=
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mem_def.2 (Set.mem_range.2 ⟨Fin.last _, rfl⟩)
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/-- The first element of a `StrictSeries` -/
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def bot (s : StrictSeries X) : X :=
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s 0
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theorem bot_mem (s : StrictSeries X) : s.bot ∈ s :=
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mem_def.2 (Set.mem_range.2 ⟨0, rfl⟩)
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/-- Remove the largest element from a `StrictSeries`. If the toFun `s`
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has length zero, then `s.eraseTop = s` -/
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@[simps]
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def eraseTop (s : StrictSeries X) : StrictSeries X
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where
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length := s.length - 1
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toFun i := s ⟨i, lt_of_lt_of_le i.2 (Nat.succ_le_succ tsub_le_self)⟩
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step' i := by
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have := s.step ⟨i, lt_of_lt_of_le i.2 tsub_le_self⟩
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cases i
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exact this
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theorem top_eraseTop (s : StrictSeries X) :
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s.eraseTop.top = s ⟨s.length - 1, lt_of_le_of_lt tsub_le_self (Nat.lt_succ_self _)⟩ :=
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show s _ = s _ from
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congr_arg s
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(by
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ext
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simp only [eraseTop_length, Fin.val_last, Fin.coe_castSucc, Fin.coe_ofNat_eq_mod,
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Fin.val_mk])
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@[simp]
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theorem bot_eraseTop (s : StrictSeries X) : s.eraseTop.bot = s.bot :=
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rfl
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/-- Append two composition toFun `s₁` and `s₂` such that
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the least element of `s₁` is the maximum element of `s₂`. -/
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@[simps length]
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def append (s₁ s₂ : StrictSeries X) (h : s₁.top = s₂.bot) : StrictSeries X where
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length := s₁.length + s₂.length
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toFun := Matrix.vecAppend (Nat.add_succ s₁.length s₂.length).symm (s₁ ∘ Fin.castSucc) s₂
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step' i := by
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refine' Fin.addCases _ _ i
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· intro i
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rw [append_succ_castAdd_aux _ _ _ h, append_castAdd_aux]
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exact s₁.step i
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· intro i
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rw [append_natAdd_aux, append_succ_natAdd_aux]
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exact s₂.step i
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theorem coe_append (s₁ s₂ : StrictSeries X) (h) :
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⇑(s₁.append s₂ h) = Matrix.vecAppend (Nat.add_succ _ _).symm (s₁ ∘ Fin.castSucc) s₂ :=
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rfl
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@[simp]
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theorem append_castAdd {s₁ s₂ : StrictSeries X} (h : s₁.top = s₂.bot) (i : Fin s₁.length) :
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append s₁ s₂ h (Fin.castSucc <| Fin.castAdd s₂.length i) = s₁ (Fin.castSucc i) := by
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rw [coe_append, append_castAdd_aux _ _ i]
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@[simp]
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theorem append_succ_castAdd {s₁ s₂ : StrictSeries X} (h : s₁.top = s₂.bot)
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(i : Fin s₁.length) : append s₁ s₂ h (Fin.castAdd s₂.length i).succ = s₁ i.succ := by
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rw [coe_append, append_succ_castAdd_aux _ _ _ h]
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@[simp]
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theorem append_natAdd {s₁ s₂ : StrictSeries X} (h : s₁.top = s₂.bot) (i : Fin s₂.length) :
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append s₁ s₂ h (Fin.castSucc <| Fin.natAdd s₁.length i) = s₂ (Fin.castSucc i) := by
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rw [coe_append, append_natAdd_aux _ _ i]
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@[simp]
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theorem append_succ_natAdd {s₁ s₂ : StrictSeries X} (h : s₁.top = s₂.bot) (i : Fin s₂.length) :
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append s₁ s₂ h (Fin.natAdd s₁.length i).succ = s₂ i.succ := by
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rw [coe_append, append_succ_natAdd_aux _ _ i]
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/-- Add an element to the top of a `StrictSeries` -/
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@[simps length]
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def snoc (s : StrictSeries X) (x : X) (hsat : s.top < x) : StrictSeries X where
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length := s.length + 1
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toFun := Fin.snoc s x
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step' i := by
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refine' Fin.lastCases _ _ i
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· rwa [Fin.snoc_castSucc, Fin.succ_last, Fin.snoc_last, ← top]
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· intro i
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rw [Fin.snoc_castSucc, ← Fin.castSucc_fin_succ, Fin.snoc_castSucc]
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exact s.step _
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#align composition_series.snoc StrictSeries.snoc
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@[simp]
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theorem top_snoc (s : StrictSeries X) (x : X) (hsat : s.top < x) :
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(snoc s x hsat).top = x :=
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Fin.snoc_last (α := fun _ => X) _ _
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#align composition_series.top_snoc StrictSeries.top_snoc
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@[simp]
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theorem snoc_last (s : StrictSeries X) (x : X) (hsat : s.top < x) :
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snoc s x hsat (Fin.last (s.length + 1)) = x :=
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Fin.snoc_last (α := fun _ => X) _ _
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#align composition_series.snoc_last StrictSeries.snoc_last
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@[simp]
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theorem snoc_castSucc (s : StrictSeries X) (x : X) (hsat : s.top < x)
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(i : Fin (s.length + 1)) : snoc s x hsat (Fin.castSucc i) = s i :=
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Fin.snoc_castSucc (α := fun _ => X) _ _ _
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#align composition_series.snoc_cast_succ StrictSeries.snoc_castSucc
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@[simp]
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theorem bot_snoc (s : StrictSeries X) (x : X) (hsat : s.top < x) :
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(snoc s x hsat).bot = s.bot := by rw [bot, bot, ← snoc_castSucc s _ _ 0, Fin.castSucc_zero]
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#align composition_series.bot_snoc StrictSeries.bot_snoc
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theorem mem_snoc {s : StrictSeries X} {x y : X} {hsat : s.top < x} :
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y ∈ snoc s x hsat ↔ y ∈ s ∨ y = x := by
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simp only [snoc, mem_def]
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constructor
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· rintro ⟨i, rfl⟩
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refine' Fin.lastCases _ (fun i => _) i
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· right
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simp
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· left
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simp
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· intro h
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rcases h with (⟨i, rfl⟩ | rfl)
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· use Fin.castSucc i
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simp
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· use Fin.last _
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simp
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#align composition_series.mem_snoc StrictSeries.mem_snoc
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end LT
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section Preorder
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variable {X : Type _} [Preorder X]
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protected theorem strictMono (s : StrictSeries X) : StrictMono s :=
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Fin.strictMono_iff_lt_succ.2 s.lt_succ
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protected theorem injective (s : StrictSeries X) : Function.Injective s :=
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s.strictMono.injective
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@[simp]
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protected theorem inj (s : StrictSeries X) {i j : Fin s.length.succ} : s i = s j ↔ i = j :=
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s.injective.eq_iff
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theorem total {s : StrictSeries X} {x y : X} (hx : x ∈ s) (hy : y ∈ s) : x ≤ y ∨ y ≤ x := by
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rcases Set.mem_range.1 hx with ⟨i, rfl⟩
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rcases Set.mem_range.1 hy with ⟨j, rfl⟩
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rw [s.strictMono.le_iff_le, s.strictMono.le_iff_le]
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exact le_total i j
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theorem toList_injective : Function.Injective (@StrictSeries.toList X _) :=
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fun s₁ s₂ (h : List.ofFn s₁ = List.ofFn s₂) => by
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have h₁ : s₁.length = s₂.length :=
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Nat.succ_injective
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((List.length_ofFn s₁).symm.trans <| (congr_arg List.length h).trans <| List.length_ofFn s₂)
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have h₂ : ∀ i : Fin s₁.length.succ, s₁ i = s₂ (Fin.cast (congr_arg Nat.succ h₁) i) :=
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congr_fun <| List.ofFn_injective <| h.trans <| List.ofFn_congr (congr_arg Nat.succ h₁).symm _
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cases s₁
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cases s₂
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dsimp at h h₁ h₂
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subst h₁
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simp only [mk.injEq, heq_eq_eq, true_and]
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simp only [Fin.cast_refl] at h₂
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exact funext h₂
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theorem toList_sorted (s : StrictSeries X) : s.toList.Sorted (· < ·) :=
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List.pairwise_iff_get.2 fun i j h => by
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dsimp [toList]
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rw [List.get_ofFn, List.get_ofFn]
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exact s.strictMono h
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theorem toList_nodup (s : StrictSeries X) : s.toList.Nodup :=
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s.toList_sorted.nodup
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/-- Two `StrictSeries` on a preorder are equal if they have the same elements. See also `ext_fun`. -/
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@[ext]
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theorem ext {s₁ s₂ : StrictSeries X} (h : ∀ x, x ∈ s₁ ↔ x ∈ s₂) : s₁ = s₂ :=
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toList_injective <|
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List.eq_of_perm_of_sorted
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(by
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classical
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exact List.perm_of_nodup_nodup_toFinset_eq s₁.toList_nodup s₂.toList_nodup
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(Finset.ext <| by simp [*]))
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s₁.toList_sorted s₂.toList_sorted
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@[simp]
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theorem le_top {s : StrictSeries X} (i : Fin (s.length + 1)) : s i ≤ s.top :=
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s.strictMono.monotone (Fin.le_last _)
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theorem le_top_of_mem {s : StrictSeries X} {x : X} (hx : x ∈ s) : x ≤ s.top :=
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let ⟨_i, hi⟩ := Set.mem_range.2 hx
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hi ▸ le_top _
|
||||
|
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@[simp]
|
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theorem bot_le {s : StrictSeries X} (i : Fin (s.length + 1)) : s.bot ≤ s i :=
|
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s.strictMono.monotone (Fin.zero_le _)
|
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|
||||
theorem bot_le_of_mem {s : StrictSeries X} {x : X} (hx : x ∈ s) : s.bot ≤ x :=
|
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let ⟨_i, hi⟩ := Set.mem_range.2 hx
|
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hi ▸ bot_le _
|
||||
|
||||
-- TODO this should be in section LT
|
||||
theorem length_pos_of_mem_ne {s : StrictSeries X} {x y : X} (hx : x ∈ s) (hy : y ∈ s)
|
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(hxy : x ≠ y) : 0 < s.length :=
|
||||
let ⟨i, hi⟩ := hx
|
||||
let ⟨j, hj⟩ := hy
|
||||
have hij : i ≠ j := mt s.inj.2 fun h => hxy (hi ▸ hj ▸ h)
|
||||
hij.lt_or_lt.elim
|
||||
(fun hij => lt_of_le_of_lt (zero_le (i : ℕ)) (lt_of_lt_of_le hij (Nat.le_of_lt_succ j.2)))
|
||||
fun hji => lt_of_le_of_lt (zero_le (j : ℕ)) (lt_of_lt_of_le hji (Nat.le_of_lt_succ i.2))
|
||||
|
||||
-- TODO this should be in section LT
|
||||
theorem forall_mem_eq_of_length_eq_zero {s : StrictSeries X} (hs : s.length = 0) {x y}
|
||||
(hx : x ∈ s) (hy : y ∈ s) : x = y :=
|
||||
by_contradiction fun hxy => pos_iff_ne_zero.1 (length_pos_of_mem_ne hx hy hxy) hs
|
||||
|
||||
theorem eraseTop_top_le (s : StrictSeries X) : s.eraseTop.top ≤ s.top := by
|
||||
simp [eraseTop, top, s.strictMono.le_iff_le, Fin.le_iff_val_le_val, tsub_le_self]
|
||||
|
||||
-- TODO this should be in section LT
|
||||
theorem mem_eraseTop_of_ne_of_mem {s : StrictSeries X} {x : X} (hx : x ≠ s.top) (hxs : x ∈ s) :
|
||||
x ∈ s.eraseTop := by
|
||||
rcases hxs with ⟨i, rfl⟩
|
||||
have hi : (i : ℕ) < (s.length - 1).succ := by
|
||||
conv_rhs => rw [← Nat.succ_sub (length_pos_of_mem_ne ⟨i, rfl⟩ s.top_mem hx), Nat.succ_sub_one]
|
||||
exact lt_of_le_of_ne (Nat.le_of_lt_succ i.2) (by simpa [top, s.inj, Fin.ext_iff] using hx)
|
||||
refine' ⟨Fin.castSucc i, _⟩
|
||||
simp [Fin.ext_iff, Nat.mod_eq_of_lt hi]
|
||||
|
||||
theorem mem_eraseTop {s : StrictSeries X} {x : X} (h : 0 < s.length) :
|
||||
x ∈ s.eraseTop ↔ x ≠ s.top ∧ x ∈ s := by
|
||||
simp only [mem_def]
|
||||
dsimp only [eraseTop]
|
||||
constructor
|
||||
· rintro ⟨i, rfl⟩
|
||||
have hi : (i : ℕ) < s.length := by
|
||||
conv_rhs => rw [← Nat.succ_sub_one s.length, Nat.succ_sub h]
|
||||
exact i.2
|
||||
simp [top, Fin.ext_iff, ne_of_lt hi, -Set.mem_range, Set.mem_range_self]
|
||||
· intro h
|
||||
exact mem_eraseTop_of_ne_of_mem h.1 h.2
|
||||
|
||||
theorem lt_top_of_mem_eraseTop {s : StrictSeries X} {x : X} (h : 0 < s.length)
|
||||
(hx : x ∈ s.eraseTop) : x < s.top := by
|
||||
rw [mem_eraseTop h] at hx
|
||||
let ⟨i, hi⟩ := Set.mem_range.2 hx.2
|
||||
rw [←hi]
|
||||
apply s.strictMono
|
||||
apply lt_of_le_of_ne i.le_last
|
||||
intro hc
|
||||
exact ((hc ▸ hi).symm ▸ hx).1 rfl
|
||||
--hi ▸ le_top _
|
||||
-- lt_of_le_of_ne (le_top_of_mem ((mem_eraseTop h).1 hx).2) ((mem_eraseTop h).1 hx).1
|
||||
-- #align composition_series.lt_top_of_mem_erase_top StrictSeries.lt_top_of_mem_eraseTop
|
||||
|
||||
theorem isMaximal_eraseTop_top {s : StrictSeries X} (h : 0 < s.length) :
|
||||
s.eraseTop.top < s.top := lt_top_of_mem_eraseTop h (top_mem _)
|
||||
-- have : s.length - 1 + 1 = s.length := by
|
||||
-- conv_rhs => rw [← Nat.succ_sub_one s.length]; rw [Nat.succ_sub h]
|
||||
-- rw [top_eraseTop, top]
|
||||
-- convert s.step ⟨s.length - 1, Nat.sub_lt h zero_lt_one⟩; ext; simp [this]
|
||||
-- #align composition_series.is_maximal_erase_top_top StrictSeries.isMaximal_eraseTop_top
|
||||
|
||||
-- TODO should be in LT
|
||||
theorem eq_snoc_eraseTop {s : StrictSeries X} (h : 0 < s.length) :
|
||||
s = snoc (eraseTop s) s.top (isMaximal_eraseTop_top h) := by
|
||||
ext x
|
||||
simp [mem_snoc, mem_eraseTop h]
|
||||
by_cases h : x = s.top <;> simp [*, s.top_mem]
|
||||
|
||||
-- TODO should be in LT
|
||||
@[simp]
|
||||
theorem snoc_eraseTop_top {s : StrictSeries X} (h : s.eraseTop.top < s.top) :
|
||||
s.eraseTop.snoc s.top h = s :=
|
||||
have h : 0 < s.length :=
|
||||
Nat.pos_of_ne_zero
|
||||
(by
|
||||
intro hs
|
||||
refine' ne_of_gt h _
|
||||
simp [top, Fin.ext_iff, hs])
|
||||
(eq_snoc_eraseTop h).symm
|
||||
|
||||
-- section `Equivalent` doesn't apply here
|
||||
|
||||
theorem length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero {s₁ s₂ : StrictSeries X}
|
||||
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top) (hs₁ : s₁.length = 0) : s₂.length = 0 := by
|
||||
have : s₁.bot = s₁.top := congr_arg s₁ (Fin.ext (by simp [hs₁]))
|
||||
have : Fin.last s₂.length = (0 : Fin s₂.length.succ) :=
|
||||
s₂.injective (hb.symm.trans (this.trans ht)).symm
|
||||
-- Porting note: Was `simpa [Fin.ext_iff]`.
|
||||
rw [Fin.ext_iff] at this
|
||||
simpa
|
||||
|
||||
theorem length_pos_of_bot_eq_bot_of_top_eq_top_of_length_pos {s₁ s₂ : StrictSeries X}
|
||||
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top) : 0 < s₁.length → 0 < s₂.length :=
|
||||
not_imp_not.1
|
||||
(by
|
||||
simp only [pos_iff_ne_zero, Ne.def, not_iff_not, Classical.not_not]
|
||||
exact length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb.symm ht.symm)
|
||||
|
||||
theorem eq_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero {s₁ s₂ : StrictSeries X}
|
||||
(hb : s₁.bot = s₂.bot) (ht : s₁.top = s₂.top) (hs₁0 : s₁.length = 0) : s₁ = s₂ := by
|
||||
have : ∀ x, x ∈ s₁ ↔ x = s₁.top := fun x =>
|
||||
⟨fun hx => forall_mem_eq_of_length_eq_zero hs₁0 hx s₁.top_mem, fun hx => hx.symm ▸ s₁.top_mem⟩
|
||||
have : ∀ x, x ∈ s₂ ↔ x = s₂.top := fun x =>
|
||||
⟨fun hx =>
|
||||
forall_mem_eq_of_length_eq_zero
|
||||
(length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero hb ht hs₁0) hx s₂.top_mem,
|
||||
fun hx => hx.symm ▸ s₂.top_mem⟩
|
||||
ext
|
||||
simp [*]
|
||||
|
||||
end Preorder
|
||||
|
||||
end StrictSeries
|
||||
Loading…
Reference in a new issue