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e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.SelectorDefinitions
import Lean.Elab.Tactic.Omega
namespace OAI
/-!
# Delaying one moved coordinate by one color
When a moved coordinate has final color `i`, replacing it by `i.succ` removes
exactly that coordinate from the prefixSet at cut `i`. Every other cut stays
unchanged, and total movement decreases by one.
-/
noncomputable section
open scoped BigOperators
namespace TalagrandThreshold.Selector
variable {α : Type*} [Fintype α] [DecidableEq α] {s : ℕ}
/-- Raise the final color of `x` to the color immediately after cut `i`. -/
def delayColor (z : Coloring α s) (x : α) (i : Fin s) : Coloring α s :=
Function.update z x i.succ
omit [Fintype α] in
@[simp] theorem delayColor_apply_self (z : Coloring α s) (x : α) (i : Fin s) :
delayColor z x i x = i.succ := by
simp [delayColor]
omit [Fintype α] in
theorem delayColor_apply_of_ne (z : Coloring α s) (x : α) (i : Fin s)
(y : α) (hy : y ≠ x) : delayColor z x i y = z y := by
simp [delayColor, hy]
omit [Fintype α] in
theorem earlier_delayColor (a z : Coloring α s) (x : α) (i : Fin s)
(hx : (z x).val = i.val) (hmove : (z x).val < (a x).val)
(hearlier : Earlier a z) : Earlier a (delayColor z x i) := by
intro y
by_cases hy : y = x
· subst y
rw [delayColor_apply_self]
change i.val + 1 ≤ (a x).val
omega
· rw [delayColor_apply_of_ne z x i y hy]
exact hearlier y
/-- Delaying a coordinate cannot introduce a newly moved vertex. -/
theorem changes_delayColor_subset (a z : Coloring α s) (x : α) (i : Fin s)
(hx : (z x).val = i.val) : changes a (delayColor z x i) ⊆ changes a z := by
intro y hy
have hlt : delayColor z x i y < a y := (Finset.mem_filter.mp hy).2
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ y, ?_⟩
by_cases hyx : y = x
· subst y
rw [delayColor_apply_self] at hlt
change i.val + 1 < (a x).val at hlt
change (z x).val < (a x).val
omega
· rw [delayColor_apply_of_ne z x i y hyx] at hlt
exact hlt
theorem prefix_delayColor_self (z : Coloring α s) (x : α) (i : Fin s)
(_hx : (z x).val = i.val) :
prefixSet (delayColor z x i) i = (prefixSet z i).erase x := by
ext y
by_cases hy : y = x
· subst y
simp only [prefixSet, Finset.mem_filter, Finset.mem_univ, true_and,
Finset.mem_erase, delayColor_apply_self]
simp
· constructor
· intro hnew
have hle : (delayColor z x i y).val ≤ i.val :=
(Finset.mem_filter.mp hnew).2
rw [delayColor_apply_of_ne z x i y hy] at hle
exact Finset.mem_erase.mpr
⟨hy, Finset.mem_filter.mpr ⟨Finset.mem_univ y, hle⟩⟩
· intro hold
have hmem : y ∈ prefixSet z i := (Finset.mem_erase.mp hold).2
have hle : (z y).val ≤ i.val := (Finset.mem_filter.mp hmem).2
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ y, ?_⟩
rw [delayColor_apply_of_ne z x i y hy]
exact hle
theorem prefix_delayColor_of_ne (z : Coloring α s) (x : α) (i j : Fin s)
(hx : (z x).val = i.val) (hji : j ≠ i) :
prefixSet (delayColor z x i) j = prefixSet z j := by
ext y
by_cases hy : y = x
· subst y
have hvals : j.val ≠ i.val := fun h => hji (Fin.ext h)
simp only [prefixSet, Finset.mem_filter, Finset.mem_univ, true_and,
delayColor_apply_self]
change i.val + 1 ≤ j.val ↔ (z x).val ≤ j.val
omega
· simp only [prefixSet, Finset.mem_filter, Finset.mem_univ, true_and]
rw [delayColor_apply_of_ne z x i y hy]
/-- Exactly one crossing disappears at the affected cut. -/
theorem crossings_delayColor_self (a z : Coloring α s) (x : α) (i : Fin s)
(hx : (z x).val = i.val) (hmove : (z x).val < (a x).val) :
crossings a (delayColor z x i) i + 1 = crossings a z i := by
unfold crossings
rw [prefix_delayColor_self z x i hx, Finset.erase_sdiff_comm]
apply Finset.card_erase_add_one
simp only [Finset.mem_sdiff, prefixSet, Finset.mem_filter, Finset.mem_univ, true_and]
constructor <;> omega
theorem crossings_delayColor_of_ne (a z : Coloring α s) (x : α) (i j : Fin s)
(hx : (z x).val = i.val) (hji : j ≠ i) :
crossings a (delayColor z x i) j = crossings a z j := by
unfold crossings
rw [prefix_delayColor_of_ne z x i j hx hji]
/-- The delay removes precisely one unit of total movement. -/
theorem movement_delayColor_add_one (a z : Coloring α s) (x : α) (i : Fin s)
(hx : (z x).val = i.val) (hmove : (z x).val < (a x).val) :
movement a (delayColor z x i) + 1 = movement a z := by
let f : α → ℕ := fun y => (a y).val - (z y).val
let g : α → ℕ := fun y => (a y).val - (delayColor z x i y).val
have hrest : (∑ y ∈ Finset.univ.erase x, g y) =
∑ y ∈ Finset.univ.erase x, f y := by
apply Finset.sum_congr rfl
intro y hy
have hne : y ≠ x := (Finset.mem_erase.mp hy).1
dsimp only [g, f]
rw [delayColor_apply_of_ne z x i y hne]
have hpoint : g x + 1 = f x := by
dsimp only [g, f]
rw [delayColor_apply_self]
change (a x).val - (i.val + 1) + 1 = (a x).val - (z x).val
omega
change (∑ y, g y) + 1 = ∑ y, f y
rw [← Finset.sum_erase_add Finset.univ g (Finset.mem_univ x),
← Finset.sum_erase_add Finset.univ f (Finset.mem_univ x), hrest]
omega
end TalagrandThreshold.Selector
end
end OAI
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