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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.Probability.MatroidProphet.Main
import Mathlib.Data.Fintype.EquivFin
namespace OAI
/-!
# Timestamp representations of finite insertion orders
The marked-pivot proof uses injective natural-number timestamps, whereas the
manuscript speaks about positions in a finite interleaving. We prove that
timestamps can be normalized to a permutation of the finite position set
without changing any prefix, first-spanning pivot, or recorded mark vector.
No bound on gaps between timestamps is imposed.
-/
namespace MatroidProphet.Pivots
open Set Finset
variable {α β : Type*} [Fintype β]
/-- The position of an occurrence is the number of its strict predecessors. -/
noncomputable def orderPosition (time : β → ℕ) (o : β) : ℕ := by
classical
exact (Finset.univ.filter (fun p => time p < time o)).card
lemma orderPosition_mono (time : β → ℕ) {o p : β} (h : time o ≤ time p) :
orderPosition time o ≤ orderPosition time p := by
classical
apply Finset.card_le_card
intro a ha
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
lt_of_lt_of_le (Finset.mem_filter.mp ha).2 h⟩
lemma orderPosition_strict (time : β → ℕ) {o p : β} (h : time o < time p) :
orderPosition time o < orderPosition time p := by
classical
apply Finset.card_lt_card
apply Finset.ssubset_iff_subset_ne.mpr
constructor
· intro a ha
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
lt_trans (Finset.mem_filter.mp ha).2 h⟩
· intro heq
have ho : o ∈ Finset.univ.filter (fun a => time a < time p) := by simp [h]
rw [← heq] at ho
simp at ho
lemma orderPosition_lt_iff (time : β → ℕ) (o p : β) :
orderPosition time o < orderPosition time p ↔ time o < time p := by
refine ⟨?_, orderPosition_strict time⟩
intro h
by_contra hn
exact (not_lt_of_ge (orderPosition_mono time (Nat.le_of_not_gt hn))) h
lemma orderPosition_le_iff (time : β → ℕ) (o p : β) :
orderPosition time o ≤ orderPosition time p ↔ time o ≤ time p := by
simp only [← not_lt, orderPosition_lt_iff]
lemma orderPosition_lt_card (time : β → ℕ) (o : β) :
orderPosition time o < Fintype.card β := by
classical
change (Finset.univ.filter (fun p => time p < time o)).card < Fintype.card β
rw [← Finset.card_univ]
apply Finset.card_lt_card
apply Finset.ssubset_iff_subset_ne.mpr
refine ⟨Finset.filter_subset _ _, ?_⟩
intro heq
have ho : o ∈ (Finset.univ : Finset β) := Finset.mem_univ _
rw [← heq] at ho
simp at ho
/-- The normalized timestamp lies in the actual finite position set. -/
noncomputable def finiteOrderPosition (time : β → ℕ) (o : β) : Fin (Fintype.card β) :=
⟨orderPosition time o, orderPosition_lt_card time o⟩
lemma finiteOrderPosition_injective (time : β → ℕ) (htime : Function.Injective time) :
Function.Injective (finiteOrderPosition time) := by
intro o p heq
apply htime
have hval := congrArg Fin.val heq
change orderPosition time o = orderPosition time p at hval
apply Nat.le_antisymm
· exact (orderPosition_le_iff time o p).mp hval.le
· exact (orderPosition_le_iff time p o).mp hval.ge
lemma finiteOrderPosition_bijective (time : β → ℕ) (htime : Function.Injective time) :
Function.Bijective (finiteOrderPosition time) := by
rw [Fintype.bijective_iff_injective_and_card]
exact ⟨finiteOrderPosition_injective time htime, by simp⟩
omit [Fintype β] in
/-- Only the relative occurrence order, not the timestamp magnitudes, matters. -/
lemma isPivot_iff_of_same_order (M : Matroid α) (label : β → α)
(time time' : β → ℕ) (horder : ∀ o p, time o < time p ↔ time' o < time' p)
(e : α) (o : β) : IsPivot M label time e o ↔ IsPivot M label time' e o := by
have hbefore : {p | time p < time o} = {p | time' p < time' o} := by
ext p
exact horder p o
have hthrough : {p | time p ≤ time o} = {p | time' p ≤ time' o} := by
ext p
simp only [mem_ofPred_eq, ← not_lt, horder o p]
simp only [IsPivot, hbefore, hthrough]
lemma isPivot_orderPosition_iff (M : Matroid α) (label : β → α)
(time : β → ℕ) (e : α) (o : β) :
IsPivot M label (orderPosition time) e o ↔ IsPivot M label time e o :=
isPivot_iff_of_same_order M label (orderPosition time) time
(orderPosition_lt_iff time) e o
lemma recordsPivots_orderPosition_iff {r q n : ℕ} (M : Matroid α)
(label : Occurrence r q → α) (test : Fin n → α) (time : Occurrence r q → ℕ)
(oldMark : Fin r → Bool) (movableMark : Fin q → Bool) (t : Finset (Fin n)) :
RecordsPivots M label test (orderPosition time) oldMark movableMark t ↔
RecordsPivots M label test time oldMark movableMark t := by
simp only [RecordsPivots, isPivot_orderPosition_iff]
lemma oldOrdered_orderPosition_iff {r q : ℕ} (time : Occurrence r q → ℕ) :
OldOrdered (orderPosition time) ↔ OldOrdered time := by
simp only [OldOrdered, orderPosition_lt_iff]
/-- The counted family is exactly the family obtained from genuine finite
interleavings. Arbitrary natural-number gaps neither add nor remove patterns. -/
lemma mem_markedPivotPatterns_iff_finite_order [Fintype α] {r q n : ℕ}
(M : Matroid α) (b : ℕ → α) (a : Fin q → α) (test : Fin n → α)
(oldMark : Fin r → Bool) (t : Finset (Fin n)) :
t ∈ markedPivotPatterns M b a test oldMark ↔
∃ position : Occurrence r q ≃ Fin (Fintype.card (Occurrence r q)),
OldOrdered (fun o => (position o).val) ∧
∃ movableMark : Fin q → Bool,
RecordsPivots M (occurrenceLabel b a) test (fun o => (position o).val)
oldMark movableMark t := by
classical
constructor
· intro ht
obtain ⟨time, horder, htime, movableMark, hrecords⟩ := (Finset.mem_filter.mp ht).2
let position := Equiv.ofBijective (finiteOrderPosition time)
(finiteOrderPosition_bijective time htime)
refine ⟨position, ?_, movableMark, ?_⟩
· change OldOrdered (orderPosition time)
exact (oldOrdered_orderPosition_iff time).mpr horder
· change RecordsPivots M (occurrenceLabel b a) test (orderPosition time)
oldMark movableMark t
exact (recordsPivots_orderPosition_iff M (occurrenceLabel b a) test time
oldMark movableMark t).mpr hrecords
· rintro ⟨position, horder, movableMark, hrecords⟩
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, (fun o => (position o).val),
horder, (fun o p h => position.injective (Fin.ext h)), movableMark, hrecords⟩
end MatroidProphet.Pivots
end OAI
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