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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 | import OAI.Combinatorics.ContingencyTables.Rooted.RootedCut
import OAI.Combinatorics.ContingencyTables.Rooted.ForestAffineCoefficients
namespace OAI
/-! # Cut coefficients for an explicitly selected forest root
The same signed cut calculation applies to the executed root selector.
In particular, coefficient size and the maximum-forest capacity bound
do not depend on a classical choice of cut representative.
-/
namespace ContingencyTables.RootedAffine
open scoped BigOperators Classical
open CompanionFactorialApplication RootedCompletion RootedCut GraphCompletionCounts
open ForestCapacity ForestCut MaximumForest
variable {I J : Type*} [Fintype I] [Fintype J]
noncomputable def coefficient (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (a : I×J) : ℝ :=
if A.small a.1 a.2=false ∧ ¬A.T.Adj (.inl a.1) (.inr a.2)
then -sign A.T S t a/sign A.T S t t.val else 0
lemma coefficient_abs_le_one (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (a : I×J) : |coefficient A S t a|≤1 := by
unfold coefficient
split_ifs
· rw [abs_div,abs_neg,sign_abs_self A.T S A.hT,div_one]
exact sign_abs_le_one A.T S t a
· norm_num
lemma capacity_le_of_coefficient_ne_zero (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (a : I×J) (ha : coefficient A S t a≠0) :
A.b a.1 a.2≤A.b t.val.1 t.val.2 := by
have hf : A.small a.1 a.2=false ∧ ¬A.T.Adj (.inl a.1) (.inr a.2) := by
by_contra hn
exact ha (by simp only [coefficient,hn,ite_false])
have hcross : treeCut A.T S t (.inl a.1)≠treeCut A.T S t (.inr a.2) := by
intro he
apply ha
simp [coefficient,hf,sign,he]
have h := crossing_free_edge_le (cellGraph A.small) A.T (capacity A.b)
A.hTG A.hT A.hmax (s(Sum.inl t.val.1,Sum.inr t.val.2))
(by simpa using t.property)
(away A.T (.inl t.val.1) (.inr t.val.2) (S.root (.inl t.val.1)))
(show (cellGraph A.small).Adj (.inl a.1) (.inr a.2) from hf.1) hf.2 hcross
simpa only [capacity_cell] using h
theorem normalized_coefficient_bound (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (a : I×J) (B : ℝ) (hB : 0<B) :
|coefficient A S t a*((A.b a.1 a.2:ℝ)+1)/B/(A.b t.val.1 t.val.2:ℝ)|≤2/B := by
have hb : (1:ℝ)≤A.b t.val.1 t.val.2 := by
exact_mod_cast (A.hlarge _ _ (A.hTG t.property)).2
by_cases ha : coefficient A S t a=0
· simp only [ha,zero_mul,zero_div,abs_zero]
positivity
have hcap : (A.b a.1 a.2:ℝ)≤A.b t.val.1 t.val.2 := by
exact_mod_cast capacity_le_of_coefficient_ne_zero A S t a ha
rw [abs_div,abs_div,abs_mul,abs_of_pos hB,
abs_of_pos (by linarith : (0:ℝ)<A.b t.val.1 t.val.2),
abs_of_nonneg (by positivity : (0:ℝ)≤(A.b a.1 a.2:ℝ)+1)]
apply (div_le_iff₀ (by linarith : (0:ℝ)<A.b t.val.1 t.val.2)).mpr
rw [div_le_iff₀ hB]
have hh := mul_le_mul_of_nonneg_right (coefficient_abs_le_one A S t a)
(by positivity : (0:ℝ)≤(A.b a.1 a.2:ℝ)+1)
field_simp [hB.ne']
nlinarith
noncomputable def smallContribution (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (x q : I → J → ℝ) : ℝ :=
∑ a : I×J,if A.small a.1 a.2 then
indicator (treeCut A.T S t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)-
indicator (treeCut A.T S t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0
noncomputable def offset (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (x q : I → J → ℝ) : ℝ :=
A.X t.val.1 t.val.2-smallContribution A S t x q/sign A.T S t t.val-
∑ a : I×J,coefficient A S t a*(A.X a.1 a.2:ℝ)
noncomputable def affineTree (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (x q : I → J → ℝ) (z : I×J → ℝ) : ℝ :=
AffinePenalty.affineValue (offset A S t x q) (coefficient A S t) z
theorem affine_of_nonroot_equations (A : Instance I J) (S : Selection A.T)
(t : TreeCell A.T) (x q z : I → J → ℝ)
(hr : ∀ i,Sum.inl i≠S.root (.inl i) →
rowDifference A.small x z (fun i j => (A.X i j:ℝ)) i=0)
(hc : ∀ j,Sum.inr j≠S.root (.inr j) →
columnDifference A.small q z (fun i j => (A.X i j:ℝ)) j=0) :
z t.val.1 t.val.2=affineTree A S t x q (fun a => z a.1 a.2) := by
let f : I×J → ℝ := fun a => cutTerm A.small
(fun i => treeCut A.T S t (.inl i)) (fun j => treeCut A.T S t (.inr j))
x q z (fun i j => (A.X i j:ℝ)) a.1 a.2
have hsum : (∑ a,f a)=0 := by
rw [Fintype.sum_prod_type]
exact cut_sum_zero A.small _ _ x q z _
(fun i hi => hr i (treeCut_nonroot A.T S A.hT t (.inl i) hi))
(fun j hj => hc j (treeCut_nonroot A.T S A.hT t (.inr j) hj))
have hterm (a : I×J) : f a=
(if a=t.val then sign A.T S t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)+
(if A.small a.1 a.2 then
indicator (treeCut A.T S t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)-
indicator (treeCut A.T S t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0)-
sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2) := by
by_cases he : a=t.val
· subst a
have hs : A.small t.val.1 t.val.2=false := A.hTG t.property
simp only [f,cutTerm,hs,Bool.false_eq_true,ite_false,ite_true,coefficient,t.property,
not_true_eq_false,and_false,sign,mul_zero,zero_mul,add_zero,sub_zero]
· by_cases hs : A.small a.1 a.2=true
· simp [f,cutTerm,hs,he,coefficient]
· have hs' : A.small a.1 a.2=false := Bool.eq_false_iff.mpr hs
by_cases ht : A.T.Adj (.inl a.1) (.inr a.2)
· have hz := other_tree_sign_zero A.T S t a ht he
simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht,
not_true_eq_false,and_false,zero_add,mul_zero,zero_mul,sub_zero]
change sign A.T S t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))=0
rw [hz,zero_mul]
· simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht,
not_false_eq_true,and_self,ite_true,zero_add,zero_sub]
change sign A.T S t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))=
-(sign A.T S t t.val*(-sign A.T S t a/sign A.T S t t.val)*
(z a.1 a.2-A.X a.1 a.2))
field_simp [sign_self_ne_zero A.T S A.hT t]
simp_rw [hterm] at hsum
rw [Finset.sum_sub_distrib,Finset.sum_add_distrib] at hsum
have hsingle : (∑ a : I×J,
if a=t.val then sign A.T S t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)=
sign A.T S t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2) := by simp
rw [hsingle] at hsum
change sign A.T S t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2)+
smallContribution A S t x q-
(∑ a : I×J,sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2))=0 at hsum
have hprod : (∑ a : I×J,sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2))=
sign A.T S t t.val*((∑ a : I×J,coefficient A S t a*z a.1 a.2)-
∑ a : I×J,coefficient A S t a*(A.X a.1 a.2:ℝ)) := by
rw [mul_sub,Finset.mul_sum,Finset.mul_sum,←Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro a _
ring
rw [hprod] at hsum
unfold affineTree AffinePenalty.affineValue offset
apply mul_left_cancel₀ (sign_self_ne_zero A.T S A.hT t)
rw [mul_add,mul_sub,mul_sub,←mul_div_assoc,
mul_div_cancel_left₀ _ (sign_self_ne_zero A.T S A.hT t)]
nlinarith [hsum]
end ContingencyTables.RootedAffine
end OAI
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