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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Combinatorics.HadwigerCounterexample.FiniteConditioning
namespace OAI
/-! # Conditional Product Laws -/
noncomputable section
namespace HadwigerCounterexample
open scoped BigOperators
variable {A B I : Type} [Fintype A] [Fintype B]
theorem subLaw_pair {w : Wgt A} {v : Wgt B} (hw : SubLaw w) (hv : SubLaw v) :
SubLaw (pairW w v) := by
refine ⟨fun x => mul_nonneg (hw.nz _) (hv.nz _),?_⟩
rw [pairW_tot]
exact (mul_le_mul hw.total hv.total (tot_nonneg hv.nz) (by norm_num)).trans_eq (one_mul _)
theorem pair_unif :
unif (α:=A×B) = pairW unif unif := by
funext x; simp [pairW,unif_apply,mul_comm]
theorem unifOn_slice (p : A × B → Prop) (x : A) (y : B) :
unifOn p (x,y) =
mapW (unifOn p) Prod.fst x * unifOn (fun y => p (x,y)) y := by
classical
have hm : mapW (unifOn p) Prod.fst x =
nbr (fun y => p (x,y)) / nbr p := by
simp only [mapW,pr,avg,Fintype.sum_prod_type,unifOn_val, nbr,tot]
rw [Finset.sum_eq_single x]
· simp [ind_true,div_eq_mul_inv,Finset.sum_mul]
· intro b _ hb
simp [ind,hb]
· simp
rw [hm,unifOn_val,unifOn_val]
by_cases h : p (x,y)
· have hh {A : Type} [Fintype A] (p : A → Prop) (x : A) (hx : p x) : 0 < nbr p := by
have hp := weight_le_tot (fun y => ind_nonneg (p y)) x
exact lt_of_lt_of_le (ind_one hx ▸ zero_lt_one) hp
rw [ind_one h]
have hv := hh _ _ h
have hw := hh (fun y => p (x,y)) _ h
field_simp
· rw [ind_zero h]
ring
theorem unifOn_and (p q : A → Prop) :
onW (unifOn p) q = unifOn (fun x => p x ∧ q x) := by
classical
have hp : pr (unifOn p) q = nbr (fun x => p x ∧ q x) / nbr p := by
simp only [pr,avg,unifOn_val,div_eq_mul_inv,ind_and,nbr,tot,Finset.sum_mul]
refine Finset.sum_congr rfl fun i _ => ?_
ring
funext x
rw [onW_val,hp,unifOn_val,unifOn_val,ind_and]
by_cases h : p x
· by_cases g : q x
· have hl (p : A → Prop) (h : p x) : 0 < nbr p :=
lt_of_lt_of_le (ind_one h ▸ zero_lt_one) (weight_le_tot (fun y => ind_nonneg (p y)) x)
have hh := hl p h
have hg := hl (fun x => p x ∧ q x) ⟨h,g⟩
field_simp
· rw [ind_zero g]
ring
· rw [ind_zero h]
ring
theorem unifOn_and_prod (p : A → Prop) (q : B → Prop) :
unifOn (fun x : A × B => p x.1 ∧ q x.2) = pairW (unifOn p) (unifOn q) := by
have h :
nbr (fun x : A × B => p x.1 ∧ q x.2) = nbr p * nbr q := by
change tot _ = _ * _
simp_rw [ind_and]
exact pairW_tot (fun x => ind (p x)) (fun y => ind (q y))
funext x
simp only [pairW,unifOn_val,h,ind_and]
ring_nf
theorem pr_cap {w v : Wgt A} {c : ℝ} (h : ∀ x, w x ≤ c*v x) (p : A → Prop) :
pr w p ≤ c*pr v p := calc
_ ≤ pr (fun x => c*v x) p := pr_mono_left (w:=w) (v:=fun x => c*v x) h _
_ = _ := by simp only [pr,avg,Finset.mul_sum,mul_assoc]
def semi (w : Wgt A) (k : A → Wgt B) : Wgt (A × B) := fun x => w x.1 * k x.1 x.2
theorem semi_avg (w : Wgt A) (k : A → Wgt B) (f : A × B → ℝ) :
avg (semi w k) f = avg w (fun x => avg (k x) (fun y => f (x,y))) := by
simp only [avg,semi,Fintype.sum_prod_type,Finset.mul_sum,mul_assoc]
theorem pair_semi_avg (w w' : Wgt A) (k k' : A → Wgt B) (f : (A × B) × (A × B) → ℝ) :
avg (pairW (semi w k) (semi w' k')) f =
avg (pairW w w') (fun x =>
avg (pairW (k x.1) (k' x.2)) (fun y => f ((x.1,y.1), (x.2,y.2)))) := by
simp only [pairW_avg, semi_avg]
refine avg_congr fun x _ => ?_
exact avg_avg _ _ _
theorem pair_semi_pr (w w' : Wgt A) (k k' : A → Wgt B) (p : A × A → Prop) (q : B × B → Prop) :
pr (pairW (semi w k) (semi w' k')) (fun x => p (x.1.1,x.2.1) ∧ q (x.1.2,x.2.2)) =
avg (pairW w w') (fun x => ind (p x) * pr (pairW (k x.1) (k' x.2)) q) := by
unfold pr
rw [pair_semi_avg]
simp_rw [ind_and]
refine avg_congr fun x _ => ?_
exact avg_mul' _ _ _
variable [Fintype I] [DecidableEq I]
/-- Reassociate two arrays of independent columns into componentwise independent pairs. -/
theorem prodW_pair_pr (w v : I → Wgt A) (p : I → A × A → Prop) :
pr (pairW (prodW w) (prodW v)) (fun x => ∀ i, p i (x.1 i,x.2 i)) =
∏ i, pr (pairW (w i) (v i)) (p i) := by
simp_rw [pr,ind_all,pairW_avg]
have hx (x : I → A) :
avg (prodW v) (fun y => ∏ i, ind (p i (x i,y i))) =
∏ i, avg (v i) (fun y => ind (p i (x i,y))) :=
prodW_avg _ (fun i y => ind (p i (x i,y)))
simp_rw [hx]
exact prodW_avg w (fun i x => avg (v i) (fun y => ind (p i (x,y))))
end HadwigerCounterexample
end
end OAI
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