File size: 7,353 Bytes
d5b7741 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import OAI.Combinatorics.ContingencyTables.Sampling.SmallGraphProfiles
import OAI.Combinatorics.ContingencyTables.Transport.ContextChildVariance
namespace OAI
/-! # Context leaf vertices are actual balanced or single-defect states -/
namespace ContingencyTables.SmallGraphProfiles
open scoped BigOperators Classical
open SmallContextCoordinates SmallContextEnumeration SmallContextFibres
open PhysicalCompletionFibres LargePairEnumeration BalancedFamily BalancedBox
open BalancedEnumeration IntegerWeightedTransport FirstPaperProfiles ExcessContextProfiles FactorialSignatures
universe u
lemma pair_total_removeOne {C : Type*} [DecidableEq C]
(D : C×Bool → ℕ) (c a : C) (b : Bool) (hd : 0 < D (c,b)) :
removeOne D (c,b) (a,false)+removeOne D (c,b) (a,true)=
D (a,false)+D (a,true)-(if a=c then 1 else 0) := by
by_cases ha : a=c
· subst a
cases b <;> simp only [removeOne,Prod.mk.injEq,Bool.false_eq_true,
Bool.true_eq_false,↓reduceIte,and_false,Nat.sub_zero] at hd ⊢ <;> omega
· simp only [removeOne,Prod.mk.injEq,ha,false_and,↓reduceIte,Nat.sub_zero]
variable {I J : Type u} [LinearOrder I] [LinearOrder J]
omit [LinearOrder I] [LinearOrder J] in
lemma balanced_of_pair_totals (small : I → J → Bool) (B : I → J → ℕ) (U : ℕ)
(hB : ∀ i j, small i j=true → B i j=U) (z : SmallProfile small)
(hpair : ∀ a : Cells small, z (a,false)+z (a,true)=U) : Balanced small B z := by
intro i j
cases hij : small i j
· simp only [smallView,smallQ,hij,Bool.false_eq_true,↓reduceDIte]
· have hh := hpair ⟨(i,j),hij⟩
simp only [smallView,smallQ,hij,dite_eq_left,hB i j hij]
omega
lemma pattern_after_one_removal (small : I → J → Bool) (B : I → J → ℕ) (U : ℕ)
(hU : 1 ≤ U) (hB : ∀ i j, small i j=true → B i j=U)
(s : I×J) (hs : small s.1 s.2=true) (D : SmallProfile small)
(hpair : ∀ a : Cells small, D (a,false)+D (a,true)=if a=⟨s,hs⟩ then U+1 else U)
(i : SmallView (fun a : I×J => small a.1 a.2)) (hi : 0 < D i)
(hz : ∀ a, removeOne D i a ≤ U) : Pattern small B (removeOne D i) := by
rcases i with ⟨c,b⟩
by_cases hc : c=⟨s,hs⟩
· subst c
left
apply balanced_of_pair_totals small B U hB
intro a
have hh := pair_total_removeOne D ⟨s,hs⟩ a b hi
rw [hpair a] at hh
by_cases ha : a=⟨s,hs⟩
· rw [ite_eq_left ha,ite_eq_left ha] at hh
omega
· rw [ite_eq_right ha,ite_eq_right ha] at hh
omega
· right
have hc' : c.val≠s := fun he => hc (Subtype.ext he)
refine ⟨c.val,s,c.property,hs,?_⟩
apply defect_of_pair_totals small B U hU hB c.val s c.property hs hc' _ hz
intro a
have hh := pair_total_removeOne D c a b hi
rw [hpair a] at hh
have hp : removeOne D (c,b) (a,false)+removeOne D (c,b) (a,true)=
if a=c then U-1 else if a=⟨s,hs⟩ then U+1 else U := by
by_cases hac : a=c
· have has : a≠⟨s,hs⟩ := fun he => hc (hac.symm.trans he)
rw [ite_eq_right has,ite_eq_left hac] at hh
rw [ite_eq_left hac]
exact hh
· rw [ite_eq_right hac] at hh ⊢
by_cases has : a=⟨s,hs⟩
· rw [ite_eq_left has] at hh ⊢
omega
· rw [ite_eq_right has] at hh ⊢
omega
simpa only [Subtype.ext_iff] using hp
lemma assemble_bounded (small : I → J → Bool) (s : I×J) (hs : small s.1 s.2=true)
(σ : I×J → ℕ) (U : ℕ) (hσ : ∀ a : Earlier small s, σ a.val.val ≤ U)
{n : ℕ} (e : Fin n ≃ Later small s)
(v : Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ)
(hv : InBox (bounds (fun _ : Bool => U) (widths (fun _ : Fin n => U))) v) :
∀ a, assemble small s hs σ U e v a ≤ U := by
rw [bounds_constant] at hv
intro a
obtain ⟨w,rfl⟩ := (contextEquiv small s hs e U).surjective a
rw [assemble_apply]
cases w with
| inl p =>
rcases p with ⟨p,b⟩
cases b
· exact hσ p
· exact Nat.sub_le _ _
| inr w => exact hv w
lemma base_pair_total (small : I → J → Bool) (s : I×J) (hs : small s.1 s.2=true)
(σ : I×J → ℕ) (U l : ℕ) (hlU : l ≤ U)
(hσ : ∀ a : Earlier small s, σ a.val.val ≤ U) {n : ℕ} (e : Fin n ≃ Later small s)
(q : Assignments (widths (fun _ : Fin n => U))) (a : Cells small) :
assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (a,false)+
assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (a,true)=
if a=⟨s,hs⟩ then U+1 else U := by
obtain ⟨v,rfl⟩ := (cellSplitEquiv small s hs).surjective a
rcases v with p|(_|t)
· have hps : p.val≠⟨s,hs⟩ := by
intro he
have h := p.property
rw [he] at h
exact h.ne rfl
change assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (p.val,false)+
assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (p.val,true)=
if p.val=⟨s,hs⟩ then U+1 else U
simp only [assemble_prefix,Bool.false_eq_true,↓reduceIte,hps]
have hb := hσ p
omega
· change assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (⟨s,hs⟩,false)+
assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (⟨s,hs⟩,true)=
if (⟨s,hs⟩ : Cells small)=⟨s,hs⟩ then U+1 else U
rw [ite_eq_left rfl]
simp only [assemble_special,embedding_root,rootDisplay,Bool.false_eq_true,↓reduceIte]
omega
· have hts : t.val≠⟨s,hs⟩ := by
intro he
have h := t.property
rw [he] at h
exact h.ne rfl
change assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (t.val,false)+
assemble small s hs σ U e (embedding _ (rootDisplay U l) q) (t.val,true)=
if t.val=⟨s,hs⟩ then U+1 else U
have hf := assemble_later small s hs σ U e (embedding _ (rootDisplay U l) q) (e.symm t) false
have ht := assemble_later small s hs σ U e (embedding _ (rootDisplay U l) q) (e.symm t) true
simp only [Equiv.apply_symm_apply] at hf ht
rw [hf,ht,ite_eq_right hts]
exact embedding_pair_total (fun _ : Fin n => U) (rootDisplay U l) q (e.symm t)
/-- Every box-valid transport leaf has exactly an allowed physical
small-graph pattern. No multi-defect vertex enters the local energy. -/
theorem leaf_pattern (small : I → J → Bool) (B : I → J → ℕ)
(s : I×J) (hs : small s.1 s.2=true) (σ : I×J → ℕ) (U l : ℕ)
(hU : 1 ≤ U) (hlU : l ≤ U) (hB : ∀ i j, small i j=true → B i j=U)
(hσ : ∀ a : Earlier small s, σ a.val.val ≤ U) {n : ℕ} (e : Fin n ≃ Later small s)
(x : Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ)
(hx : x ∈ leafVertices _ (rootDisplay U l))
(hbox : InBox (bounds (fun _ : Bool => U) (widths (fun _ : Fin n => U))) x) :
Pattern small B (assemble small s hs σ U e x) := by
obtain ⟨q,_,hx⟩ := Finset.mem_biUnion.mp hx
obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp hx
have hi' := (Finset.mem_filter.mp hi).2
have hb := assemble_bounded small s hs σ U hσ e _ hbox
rw [assemble_removeOne] at hb ⊢
apply pattern_after_one_removal small B U hU hB s hs
(assemble small s hs σ U e (embedding _ (rootDisplay U l) q))
(base_pair_total small s hs σ U l hlU hσ e q)
(contextEquiv small s hs e U (.inr i)) _ hb
rw [assemble_apply]
exact hi'
end ContingencyTables.SmallGraphProfiles
end OAI
|