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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 | import OAI.Probability.MatroidProphet.Pivots.Interleavings
import Mathlib.Data.Fintype.Fin
namespace OAI
namespace MatroidProphet
namespace Pivots
open Finset
variable {r q : ℕ}
def oldBefore (time : Occurrence r q → ℕ) (t : ℕ) : ℕ :=
(Finset.univ.filter (fun i : Fin r => time (Sum.inl i) < t)).card
def movableBefore (time : Occurrence r q → ℕ) (t : ℕ) : ℕ :=
(Finset.univ.filter (fun i : Fin q => time (Sum.inr i) < t)).card
lemma oldBefore_mono (time : Occurrence r q → ℕ) : Monotone (oldBefore time) := by
intro u v huv
apply Finset.card_le_card
intro i hi
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
lt_of_lt_of_le (Finset.mem_filter.mp hi).2 huv⟩
lemma movableBefore_mono (time : Occurrence r q → ℕ) : Monotone (movableBefore time) := by
intro u v huv
apply Finset.card_le_card
intro i hi
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,
lt_of_lt_of_le (Finset.mem_filter.mp hi).2 huv⟩
lemma oldBefore_lt_after (time : Occurrence r q → ℕ) (i : Fin r) {t : ℕ}
(hit : time (Sum.inl i) < t) :
oldBefore time (time (Sum.inl i)) < oldBefore time t := by
apply Finset.card_lt_card
apply Finset.ssubset_iff_subset_ne.mpr
constructor
· intro j hj
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, lt_trans (Finset.mem_filter.mp hj).2 hit⟩
· intro heq
have hi : i ∈ Finset.univ.filter (fun j : Fin r => time (Sum.inl j) < t) := by simp [hit]
rw [← heq] at hi
simp at hi
lemma movableBefore_lt_after (time : Occurrence r q → ℕ) (i : Fin q) {t : ℕ}
(hit : time (Sum.inr i) < t) :
movableBefore time (time (Sum.inr i)) < movableBefore time t := by
apply Finset.card_lt_card
apply Finset.ssubset_iff_subset_ne.mpr
constructor
· intro j hj
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, lt_trans (Finset.mem_filter.mp hj).2 hit⟩
· intro heq
have hi : i ∈ Finset.univ.filter (fun j : Fin q => time (Sum.inr j) < t) := by simp [hit]
rw [← heq] at hi
simp at hi
lemma movableBefore_self_lt (time : Occurrence r q → ℕ) (i : Fin q) :
movableBefore time (time (Sum.inr i)) < q := by
have hcard : (Finset.univ.filter (fun j : Fin q => time (Sum.inr j) < time (Sum.inr i))).card <
(Finset.univ : Finset (Fin q)).card := by
apply Finset.card_lt_card
apply Finset.ssubset_iff_subset_ne.mpr
refine ⟨Finset.filter_subset _ _, ?_⟩
intro heq
have hi : i ∈ (Finset.univ : Finset (Fin q)) := Finset.mem_univ _
rw [← heq] at hi
simp at hi
simpa [movableBefore] using hcard
lemma old_time_lt_iff (time : Occurrence r q → ℕ) (horder : OldOrdered time)
(i j : Fin r) : time (Sum.inl i) < time (Sum.inl j) ↔ i < j := by
refine ⟨fun hij => ?_, fun hij => horder i j hij⟩
by_contra h
rcases lt_or_eq_of_le (le_of_not_gt h) with hji | rfl
· exact Nat.lt_asymm (horder j i hji) hij
· exact Nat.lt_irrefl _ hij
lemma oldBefore_self (time : Occurrence r q → ℕ) (horder : OldOrdered time) (i : Fin r) :
oldBefore time (time (Sum.inl i)) = i.val := by
unfold oldBefore
have heq : Finset.univ.filter (fun j : Fin r => time (Sum.inl j) < time (Sum.inl i)) =
Finset.Iio i := by
ext j
simp [old_time_lt_iff time horder]
rw [heq, Fin.card_Iio]
def scoreExponent (s : ℕ) (time : Occurrence r q → ℕ) : Occurrence r q → ℕ
| Sum.inl i => (s + 1) * (i.val + 1)
| Sum.inr i => (s + 1) * oldBefore time (time (Sum.inr i)) +
movableBefore time (time (Sum.inr i)) + 1
@[simp] lemma scoreExponent_old (s : ℕ) (time : Occurrence r q → ℕ) (i : Fin r) :
scoreExponent s time (Sum.inl i) = (s + 1) * (i.val + 1) := rfl
lemma scoreExponent_preserves_order (s : ℕ) (hqs : q ≤ s)
(time : Occurrence r q → ℕ) (horder : OldOrdered time)
{o p : Occurrence r q} (hop : time o < time p) :
scoreExponent s time o < scoreExponent s time p := by
cases o with
| inl i =>
cases p with
| inl j =>
have hij := (old_time_lt_iff time horder i j).mp hop
exact Nat.mul_lt_mul_of_pos_left (Nat.add_lt_add_right hij 1) (by omega)
| inr j =>
have hcnt := oldBefore_lt_after time i hop
rw [oldBefore_self time horder i] at hcnt
have hmul := Nat.mul_le_mul_left (s + 1) (Nat.succ_le_of_lt hcnt)
simp only [Nat.succ_eq_add_one] at hmul
simp only [scoreExponent]
omega
| inr i =>
have hbound := lt_of_lt_of_le (movableBefore_self_lt time i) hqs
have hmono := oldBefore_mono time (Nat.le_of_lt hop)
cases p with
| inl j =>
rw [oldBefore_self time horder j] at hmono
have hmul := Nat.mul_le_mul_left (s + 1) hmono
simp only [scoreExponent, Nat.mul_add, Nat.mul_one]
omega
| inr j =>
have hstrict := movableBefore_lt_after time i hop
rcases Nat.eq_or_lt_of_le hmono with heq | hlt
· simp only [scoreExponent]
rw [heq]
omega
· have hmul := Nat.mul_le_mul_left (s + 1) (Nat.succ_le_of_lt hlt)
simp only [Nat.succ_eq_add_one, Nat.mul_add, Nat.mul_one] at hmul
simp only [scoreExponent]
omega
lemma scoreExponent_injective (s : ℕ) (hqs : q ≤ s)
(time : Occurrence r q → ℕ) (horder : OldOrdered time) (htime : Function.Injective time) :
Function.Injective (scoreExponent s time) := by
intro o p heq
apply htime
apply le_antisymm
· by_contra h
have hlt := scoreExponent_preserves_order s hqs time horder (Nat.lt_of_not_ge h)
omega
· by_contra h
have hlt := scoreExponent_preserves_order s hqs time horder (Nat.lt_of_not_ge h)
omega
end Pivots
end MatroidProphet
end OAI
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