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67aee1d | 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 Mathlib
import OAI.Analysis.HardSphere.Geometry.LocalCheck
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Comparison
variable {I P : Type*} [Fintype I] [DecidableEq I] [MeasurableSpace P]
(avoid : P→P→Prop) (hm : MeasurableSet {p : P×P | avoid p.1 p.2}) (real : I→Prop)
attribute [local instance] Classical.propDecidable
def ownedPairCheck (owner : I→I) (get : I→P×P→P) (hget : ∀i,Measurable (get i))
(i j : I) : LocalCheck (fun _ : I ↦ P×P) where
support := {owner i,owner j}
nonempty := ⟨owner i,by simp⟩
arity := by simpa using Finset.card_insert_le (owner i) ({owner j} : Finset I)
pass x := i≠j→ real i→ real j→avoid (get i (x (owner i))) (get j (x (owner j)))
measurable_pass := by
by_cases hij : i=j
· simp [hij]
by_cases hi : real i
· by_cases hj : real j
· simp only [ne_eq,hij,not_false_eq_true,hi,hj,true_implies]
exact hm.preimage (f:=fun x : I→P×P ↦ (get i (x (owner i)),get j (x (owner j)))) (by fun_prop)
· simp [hj]
· simp [hi]
depends x y h := by rw [h (owner i) (by simp),h (owner j) (by simp)]
def primaryPairCheck (i j : I) : LocalCheck (fun _ : I ↦ P) where
support := {i,j}
nonempty := ⟨i,by simp⟩
arity := by simpa using Finset.card_insert_le i ({j} : Finset I)
pass x := i≠j→ real i→ real j→avoid (x i) (x j)
measurable_pass := by
by_cases hij : i=j
· simp [hij]
by_cases hi : real i
· by_cases hj : real j
· simp only [ne_eq,hij,not_false_eq_true,hi,hj,true_implies]
exact hm.preimage (f:=fun x : I→P ↦ (x i,x j)) (by fun_prop)
· simp [hj]
· simp [hi]
depends x y h := by rw [h i (by simp),h j (by simp)]
def pairedRouteChecks (S : Finset I) (σ : Equiv.Perm I) : Bool×I×I → LocalCheck (fun _ : I ↦ P×P)
| (false,i,j) => ownedPairCheck avoid hm real id (fun _ ↦ Prod.fst) (fun _ ↦ measurable_fst) i j
| (true,i,j) => ownedPairCheck avoid hm real σ (fun k p ↦ if k∈S then p.2 else p.1)
(fun k ↦ by split_ifs <;> fun_prop) i j
omit [Fintype I] in
lemma paired_check_primary (S : Finset I) (σ : Equiv.Perm I) (U : Finset I) (x z : I→P) :
checkWeight (pairedRouteChecks avoid hm real S σ) U (fun i ↦ (x i,z i)) ≤
checkWeight (fun p : I×I ↦ primaryPairCheck avoid hm real p.1 p.2) U x := by
by_cases h : x∈retainedEvent (fun p : I×I ↦ primaryPairCheck avoid hm real p.1 p.2) U
· have he : checkWeight (fun p : I×I ↦ primaryPairCheck avoid hm real p.1 p.2) U x = 1 := by simp [checkWeight,h]
rw [he]
exact checkWeight_le_one _ _ _
· have hn : (fun i ↦ (x i,z i))∉retainedEvent (pairedRouteChecks avoid hm real S σ) U := by
intro hp
apply h
intro c hc
exact hp (false,c.1,c.2) hc
simp only [checkWeight,Set.indicator_of_notMem hn,Set.indicator_of_notMem h,le_refl]
omit [Fintype I] in
lemma perm_mem_of_fix_outside (S : Finset I) (σ : Equiv.Perm I)
(hσ : ∀i,i∉S→σ i=i) (i : I) : σ i∈S ↔ i∈S := by
constructor
· intro h
by_contra hi
rw [hσ i hi] at h
contradiction
· intro hi
by_contra hn
have he : σ (σ i)=σ i := hσ (σ i) hn
have he' : σ i=i := σ.injective he
exact hn (he'.symm ▸ hi)
omit [Fintype I] in
lemma paired_outside_checks (S o : Finset I) (hso : Disjoint S o) (σ : Equiv.Perm I)
(hσ : ∀i,i∉S→σ i=i) (x z : I→P)
(hx : x∈retainedEvent (fun p : I×I ↦ primaryPairCheck avoid hm real p.1 p.2) o) :
(fun i ↦ (x i,z i))∈retainedEvent (pairedRouteChecks avoid hm real S σ) o := by
intro c hc
obtain ⟨b,i,j⟩ := c
cases b
· exact hx (i,j) hc
· have hi : σ i∈o := hc (by simp [pairedRouteChecks,ownedPairCheck])
have hj : σ j∈o := hc (by simp [pairedRouteChecks,ownedPairCheck])
have hin : i∉S := fun his ↦ Finset.disjoint_left.mp hso ((perm_mem_of_fix_outside S σ hσ i).mpr his) hi
have hjn : j∉S := fun hjs ↦ Finset.disjoint_left.mp hso ((perm_mem_of_fix_outside S σ hσ j).mpr hjs) hj
have hio : i∈o := by simpa only [hσ i hin] using hi
have hjo : j∈o := by simpa only [hσ j hjn] using hj
have hre := hx (i,j) (by simpa only [primaryPairCheck,Finset.insert_subset_iff,Finset.singleton_subset_iff] using And.intro hio hjo)
simpa only [pairedRouteChecks,ownedPairCheck,primaryPairCheck,hin,hjn,ite_false,hσ i hin,hσ j hjn] using hre
lemma paired_full_event (S : Finset I) (σ : Equiv.Perm I) (x z : I→P) :
(fun i ↦ (x i,z i))∈retainedEvent (pairedRouteChecks avoid hm real S σ) Finset.univ ↔
(∀i j,i≠j→ real i→ real j→avoid (x i) (x j)) ∧
(∀i j,i≠j→ real i→ real j→avoid (if i∈S then z (σ i) else x (σ i))
(if j∈S then z (σ j) else x (σ j))) := by
constructor
· intro h
exact ⟨fun i j ↦ h (false,i,j) (Finset.subset_univ _),fun i j ↦ h (true,i,j) (Finset.subset_univ _)⟩
· rintro ⟨h₁,h₂⟩ ⟨b,i,j⟩ _
cases b
· exact h₁ i j
· exact h₂ i j
end HardSphere.Comparison
end
end
end
end OAI
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