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e5a97c6 | 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 | import OAI.Combinatorics.PerfectMatching.DyadicEntropy
namespace OAI
section
namespace BinaryMatching
noncomputable section
open Finset
variable {n : ℕ}
lemma sum_pairs (f : Fin n → Fin n → ℝ) :
(∑ e : Pair n, f e.val.1 e.val.2)=∑ i,∑ j,if i < j then f i j else 0 := by
let s : Finset (Fin n × Fin n) := univ.filter (fun ij => ij.1 < ij.2)
have hs : ∀ ij : Fin n × Fin n,ij∈s ↔ ij.1 < ij.2 := by simp [s]
rw [←sum_subtype s hs (fun ij => f ij.1 ij.2)]
rw [show s=univ.filter (fun ij => ij.1 < ij.2) from rfl,sum_filter,Fintype.sum_prod_type]
lemma sum_skip (g : Fin n → ℝ) (v : Fin n) :
(∑ j,if j=v then 0 else g j)=(∑ j,g j)-g v := by
have hh : (∑ j,g j)=(∑ j,if j=v then 0 else g j)+g v := by
calc
_ = ∑ j, ((if j=v then 0 else g j)+(if j=v then g j else 0)) := by
apply sum_congr rfl; intro j _; split_ifs <;> simp
_ = _ := by rw [sum_add_distrib]; simp
linarith
lemma degree_from_symm (f : Fin n → Fin n → ℝ) (hf : ∀ i j,f i j=f j i) (v : Fin n) :
(completeGraph n).degree (fun e => f e.val.1 e.val.2) v =
∑ j,if j=v then 0 else f v j := by
change (∑ e : Pair n,if e.val.1=v ∨ e.val.2=v then f e.val.1 e.val.2 else 0)=_
rw [sum_pairs (fun i j => if i=v ∨ j=v then f i j else 0)]
have he (i j : Fin n) :
(if i < j then (if i=v ∨ j=v then f i j else 0) else 0) =
(if i=v then (if i < j then f i j else 0) else 0)+
(if j=v then (if i < j then f i j else 0) else 0) := by
by_cases hiv : i=v <;> by_cases hjv : j=v <;> by_cases hij : i < j <;> simp_all
simp_rw [he,sum_add_distrib]
have hleft : (∑ i,∑ j,if i=v then (if i < j then f i j else 0) else 0)=
∑ j,if v < j then f v j else 0 := by
simp [Finset.sum_ite_irrel]
have hright : (∑ i,∑ j,if j=v then (if i < j then f i j else 0) else 0)=
∑ j,if j < v then f v j else 0 := by
simp only [sum_ite_eq',mem_univ,ite_true]
apply sum_congr rfl; intro j _; rw [hf j v]
rw [hleft,hright,←sum_add_distrib]
apply sum_congr rfl; intro j _
rcases lt_trichotomy j v with h|h|h
· simp [h, ne_of_lt h, not_lt_of_ge h.le]
· subst j; simp
· simp [h, ne_of_gt h, not_lt_of_ge h.le]
lemma degree_constant (b : ℝ) (v : Fin n) :
(completeGraph n).degree (fun _ => b) v=((n:ℝ)-1)*b := by
rw [degree_from_symm (fun _ _ => b) (fun _ _ => rfl),sum_skip]
simp; ring
lemma degree_vertex_add (d : Fin n → ℝ) (v : Fin n) :
(completeGraph n).degree (fun e => d e.val.1+d e.val.2) v =
((n:ℝ)-2)*d v+∑ j,d j := by
rw [degree_from_symm (fun i j => d i+d j) (fun _ _ => add_comm _ _),sum_skip]
simp only [sum_add_distrib,sum_const,nsmul_eq_mul,card_univ,Fintype.card_fin]
ring
lemma cut_from_symm (f : Fin n → Fin n → ℝ) (hf : ∀ i j,f i j=f j i)
(S : Finset (Fin n)) :
(completeGraph n).cutMass (fun e => f e.val.1 e.val.2) S =
∑ i∈S,∑ j∈Sᶜ,f i j := by
unfold MatchingEntropy.LooplessGraph.cutMass MatchingEntropy.LooplessGraph.cut
rw [sum_filter]
change (∑ e : Pair n,if (e.val.1∈S ∧ e.val.2∉S) ∨ (e.val.1∉S ∧ e.val.2∈S)
then f e.val.1 e.val.2 else 0)=_
rw [sum_pairs (fun i j => if (i∈S ∧ j∉S) ∨ (i∉S ∧ j∈S) then f i j else 0)]
have he (i j : Fin n) :
(if i < j then (if (i∈S ∧ j∉S) ∨ (i∉S ∧ j∈S) then f i j else 0) else 0) =
(if i < j ∧ i∈S ∧ j∉S then f i j else 0)+
(if i < j ∧ i∉S ∧ j∈S then f i j else 0) := by
by_cases hij : i < j <;> by_cases hi : i∈S <;> by_cases hj : j∈S <;> simp [hij,hi,hj]
simp_rw [he,sum_add_distrib]
have hswap : (∑ i,∑ j,if i < j ∧ i∉S ∧ j∈S then f i j else 0)=
∑ i,∑ j,if j < i ∧ i∈S ∧ j∉S then f i j else 0 := by
rw [sum_comm]
apply sum_congr rfl; intro i _
apply sum_congr rfl; intro j _
rw [hf j i]
by_cases hij : j < i <;> by_cases hi : i∈S <;> by_cases hj : j∈S <;> simp [hij,hi,hj]
rw [hswap,←sum_add_distrib]
simp_rw [←sum_add_distrib]
have hterm (i j : Fin n) :
(if i < j ∧ i∈S ∧ j∉S then f i j else 0)+
(if j < i ∧ i∈S ∧ j∉S then f i j else 0)=
if i∈S ∧ j∉S then f i j else 0 := by
rcases lt_trichotomy i j with h|h|h
· simp [h,not_lt_of_ge h.le]
· subst j; simp
· simp [h,not_lt_of_ge h.le]
simp_rw [hterm]
have hS : S=univ.filter (fun i => i∈S) := by ext i; simp
have hSc : Sᶜ=univ.filter (fun j => j∉S) := by ext j; simp
conv_rhs => rw [hS]; rw [sum_filter]
simp only [ite_and]
apply sum_congr rfl; intro i _
by_cases hi : i∈S
· simp only [hi,ite_true]
rw [←hS,hSc,sum_filter]
· simp only [hi,ite_false,sum_const_zero]
lemma cut_constant (b : ℝ) (S : Finset (Fin n)) :
(completeGraph n).cutMass (fun _ => b) S=(S.card:ℝ)*(Sᶜ.card:ℝ)*b := by
rw [cut_from_symm (fun _ _ => b) (fun _ _ => rfl)]
simp [mul_assoc]
lemma q_le_square : Fintype.card (Pair n) ≤ n^2 := by
calc
_ ≤ Fintype.card (Fin n × Fin n) := Fintype.card_subtype_le _
_ = n^2 := by simp [pow_two]
end
end BinaryMatching
end
end OAI
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