95 lines
4.2 KiB
Text
95 lines
4.2 KiB
Text
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/-
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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 HepLean.Mathematics.List.InsertIdx
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/-!
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# List lemmas
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-/
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namespace HepLean.List
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open Fin
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open HepLean
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variable {n : Nat}
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lemma insertionSortMin_lt_length_succ {α : Type} (r : α → α → Prop) [DecidableRel r]
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(i : α) (l : List α) :
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insertionSortMinPos r i l < (insertionSortDropMinPos r i l).length.succ := by
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rw [insertionSortMinPos]
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simp only [List.length_cons, List.insertionSort.eq_2, insertionSortDropMinPos,
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Nat.succ_eq_add_one]
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rw [eraseIdx_length']
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simp
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/-- Given a list `i :: l` the left-most minimial position `a` of `i :: l` wrt `r`
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as an element of `Fin (insertionSortDropMinPos r i l).length.succ`. -/
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def insertionSortMinPosFin {α : Type} (r : α → α → Prop) [DecidableRel r] (i : α) (l : List α) :
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Fin (insertionSortDropMinPos r i l).length.succ :=
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⟨insertionSortMinPos r i l, insertionSortMin_lt_length_succ r i l⟩
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lemma insertionSortMin_lt_mem_insertionSortDropMinPos {α : Type} (r : α → α → Prop) [DecidableRel r]
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[IsTotal α r] [IsTrans α r] (a : α) (l : List α)
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(i : Fin (insertionSortDropMinPos r a l).length) :
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r (insertionSortMin r a l) ((insertionSortDropMinPos r a l)[i]) := by
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let l1 := List.insertionSort r (a :: l)
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have hl1 : l1.Sorted r := by exact List.sorted_insertionSort r (a :: l)
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simp only [l1] at hl1
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rw [insertionSort_eq_insertionSortMin_cons r a l] at hl1
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simp only [List.sorted_cons, List.mem_insertionSort] at hl1
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apply hl1.1 ((insertionSortDropMinPos r a l)[i])
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simp
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lemma insertionSortMinPos_insertionSortEquiv {α : Type} (r : α → α → Prop) [DecidableRel r]
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(a : α) (l : List α) :
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insertionSortEquiv r (a ::l) (insertionSortMinPos r a l) =
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⟨0, by simp [List.orderedInsert_length]⟩ := by
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rw [insertionSortMinPos]
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exact
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Equiv.apply_symm_apply (insertionSortEquiv r (a :: l)) ⟨0, insertionSortMinPos.proof_1 r a l⟩
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lemma insertionSortEquiv_gt_zero_of_ne_insertionSortMinPos {α : Type} (r : α → α → Prop)
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[DecidableRel r] (a : α) (l : List α) (k : Fin (a :: l).length)
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(hk : k ≠ insertionSortMinPos r a l) :
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⟨0, by simp [List.orderedInsert_length]⟩ < insertionSortEquiv r (a :: l) k := by
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by_contra hn
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simp only [List.insertionSort.eq_2, List.length_cons, not_lt] at hn
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refine hk ((Equiv.apply_eq_iff_eq_symm_apply (insertionSortEquiv r (a :: l))).mp ?_)
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simp_all only [List.length_cons, ne_eq, Fin.le_def, nonpos_iff_eq_zero, List.insertionSort.eq_2]
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simp only [Fin.ext_iff]
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omega
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lemma insertionSortMin_lt_mem_insertionSortDropMinPos_of_lt {α : Type} (r : α → α → Prop)
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[DecidableRel r] (a : α) (l : List α)
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(i : Fin (insertionSortDropMinPos r a l).length)
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(h : (insertionSortMinPosFin r a l).succAbove i < insertionSortMinPosFin r a l) :
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¬ r ((insertionSortDropMinPos r a l)[i]) (insertionSortMin r a l) := by
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simp only [Fin.getElem_fin, insertionSortMin, List.get_eq_getElem, List.length_cons]
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have h1 : (insertionSortDropMinPos r a l)[i] =
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(a :: l).get (finCongr (eraseIdx_length_succ (a :: l) (insertionSortMinPos r a l))
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((insertionSortMinPosFin r a l).succAbove i)) := by
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trans (insertionSortDropMinPos r a l).get i
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simp only [Fin.getElem_fin, List.get_eq_getElem]
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simp only [insertionSortDropMinPos, List.length_cons, Nat.succ_eq_add_one,
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finCongr_apply, Fin.coe_cast]
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rw [eraseIdx_get]
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simp only [List.length_cons, Function.comp_apply, List.get_eq_getElem, Fin.coe_cast]
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rfl
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erw [h1]
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simp only [List.length_cons, Nat.succ_eq_add_one, List.get_eq_getElem,
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Fin.coe_cast]
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apply insertionSortEquiv_order
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simpa using h
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simp only [List.insertionSort.eq_2, List.length_cons, finCongr_apply]
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apply lt_of_eq_of_lt (insertionSortMinPos_insertionSortEquiv r a l)
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simp only [List.insertionSort.eq_2]
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apply insertionSortEquiv_gt_zero_of_ne_insertionSortMinPos r a l
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simp only [List.length_cons, ne_eq, Fin.ext_iff, Fin.coe_cast]
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have hl : (insertionSortMinPos r a l).val = (insertionSortMinPosFin r a l).val := by
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rfl
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simp only [hl, Nat.succ_eq_add_one, Fin.val_eq_val, ne_eq]
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exact Fin.succAbove_ne (insertionSortMinPosFin r a l) i
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end HepLean.List
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