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fcab081 | 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 | import OAI.Combinatorics.HadwigerCounterexample.FiniteProjectionCenters
namespace OAI
/-! # Conditional Flags -/
noncomputable section
namespace HadwigerCounterexample
open Finset Real
open scoped BigOperators
variable {A B C X Y : Type} [Fintype A] [Fintype B] [Fintype C] [Fintype X] [Fintype Y]
variable {w : Wgt X}
def stained (w : Wgt X) (f : X→Y) (i : Y) : Wgt X := by
classical exact if mapW w f i≠0 then onW w (fun x => f x=i) else w
theorem stained_law (f : X→Y) (hw : IsLaw w) (i : Y) : IsLaw (stained w f i) := by
unfold stained; split
next h => exact on_law hw.nz (lt_of_le_of_ne ((mapW_law hw f).nz _) (Ne.symm h))
next => exact hw
theorem stain_rebuild (f : X→Y) (P : Y→Prop) (hw : IsLaw w) :
subW w (P∘f) = mixW (subW (mapW w f) P) (stained w f) := by
classical
have hh := disintegrate hw.nz f
funext i
have hi (y) :
mapW w f y * stained w f y i=
mapW w f y * onW w (fun x => f x=y) i := by
by_cases h : mapW w f y=0 <;> simp [stained,h]
have he := congr_fun hh i
unfold subW mixW Function.comp
unfold mixW at he
rw [he,avg,Finset.sum_mul]
unfold avg at *
apply sum_congr rfl
intro j _
rw [mul_right_comm _ (ind _), hi]
by_cases hx : f i=j
· rw [hx]
simp only [onW_val,ind_zero hx,mul_zero,zero_div,zero_mul]
theorem stain_inputs (f : X→Y) (hw : IsLaw w) :
w=mixW (mapW w f) (stained w f) := by
have ht := stain_rebuild f (fun _ => True) hw
unfold subW at ht
simpa only [Function.comp_apply,ind_true,mul_one] using ht
variable {μ : Wgt A}
def colored (μ : Wgt A) (w : Wgt X) (v : A→ X→ Y) :=
semi μ (fun i => mapW w (v i))
def coloredP (w : Wgt X) (v : A→ X→Y) (K : A→X→B) :=
fun i : A×Y => mapW (stained w (v i.1) i.2) (K i.1)
def flaggedW (μ : Wgt A) (w : Wgt X) (v : A→X→Y) (K : A→X→B) (P : A→Y→Prop) :=
mixW μ (fun i => mapW (subW w (P i ∘ v i)) (K i))
theorem semi_total (μ : Wgt A) (v : A → Wgt Y) :
tot (semi μ v)=avg μ (fun i=>tot (v i)) := by
simpa only [avg_const,mul_one] using semi_avg μ v (fun i=>1)
theorem colored_ultra (hμ : IsLaw μ) (hw : IsLaw w) (K : A→X→B) (v : A→X→Y) :
Ultra (colored μ w v) (coloredP w v K) := by
have ht (i) := mapW_law hw (v i)
refine ⟨⟨fun x => mul_nonneg (hμ.nz _) ((ht _).nz _),?_⟩,
fun i => mapW_law (stained_law _ hw _) (K i.1)⟩
unfold colored; rw [semi_total]
simp_rw [(ht _).total]
exact hμ.avg_const _
theorem flag_catalog (K : A→ X→B) (v : A→ X→ Y) (P : A→ Y→ Prop)
(hw : IsLaw w) (hμ : IsLaw μ) :
catalog (colored μ w v) (coloredP w v K)
(flaggedW μ w v K P) := by
classical
let W := colored μ w v
have hh := (colored_ultra hμ hw K v).law
refine ⟨subW W (fun i => P i.1 i.2),subW_nonneg hh.nz _,subW_le hh.nz _,?_⟩
unfold flaggedW
simp_rw [stain_rebuild (v _) (P _) hw,map_mix]
funext x
simp only [mixW,coloredP,colored,W,avg,subW,semi,Finset.mul_sum,Fintype.sum_prod_type]
exact sum_congr rfl fun i _ => sum_congr rfl fun j _ => by ring
theorem colored_budget [Nonempty Y] (K : A→ X→B) (v : A→ X→ Y) (f : B→C)
(hw : IsLaw w) (hμ : IsLaw μ) :
budget (colored μ w v) (coloredP w v K) f ≤
budget μ (fun i => mapW w (K i)) f + Real.log (num Y) := by
unfold budget colored coloredP
rw [semi_avg,← hμ.avg_const (Real.log _),← avg_add]
apply avg_mono hμ.nz
intro i
let u := mapW w (v i)
have he : IsLaw u := mapW_law hw (v i)
let q (j : Y) := mapW (mapW (stained w (v i) j) (K i)) f
have hq (j) : IsLaw (q j) := mapW_law (mapW_law (stained_law _ hw _) _) _
have ht : mapW (mapW w (K i)) f=mixW u q := by
calc mapW _ _ = mapW (mapW (mixW u (stained w (v i))) (K i)) f :=
congrArg (fun z => mapW (mapW z (K i)) f) (stain_inputs (v i) hw)
_ = _ := by rw [map_mix,map_mix]
change avg u (fun j => tension (q j) _)≤ tension _ _+_
rw [tension_mix,add_comm _ (log _),ht]
exact add_le_add_left (mix_charge he hq) _
end HadwigerCounterexample
end
end OAI
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