| 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} |
|
|
| / |
| 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 |
|
|
| / |
| 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] |
|
|
| / |
| 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] |
|
|
| / |
| 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 |
|
|
