File size: 6,316 Bytes
e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.RepresentativeIndexSteps
namespace OAI
namespace PartitionPilot.Forcing
universe u
noncomputable def assignmentQueryRelation (I T C D : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ row L A,
z = ZFSet.pair (ZFSet.pair row L) (ZFSet.pair row A) ∧
assignmentOnIndices (assignmentIndices I T L) T A = L) (ZFSet.prod C D)
theorem assignmentQueryRelation_entry (I T C D row L other A : ZFSet.{u}) :
ZFSet.pair (ZFSet.pair row L) (ZFSet.pair other A) ∈ assignmentQueryRelation I T C D ↔
ZFSet.pair row L ∈ C ∧ ZFSet.pair other A ∈ D ∧ row = other ∧
assignmentOnIndices (assignmentIndices I T L) T A = L := by
rw [assignmentQueryRelation, ZFSet.mem_sep, ZFSet.pair_mem_prod]
constructor
· rintro ⟨⟨hq, hp⟩, r, L', A', eq, matchList⟩
obtain ⟨first, second⟩ := ZFSet.pair_inj.1 eq
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 first
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 second
exact ⟨hq, hp, rfl, matchList⟩
· rintro ⟨hq, hp, rfl, matchList⟩
exact ⟨⟨hq, hp⟩, row, L, A, rfl, matchList⟩
noncomputable def realizedAssignmentQueries (I T C D : ZFSet.{u}) : ZFSet.{u} :=
relationImage D C (relationConverse C D (assignmentQueryRelation I T C D))
theorem mem_realizedAssignmentQueries (I T C D q : ZFSet.{u}) :
q ∈ realizedAssignmentQueries I T C D ↔
q ∈ C ∧ ∃ p ∈ D, ZFSet.pair q p ∈ assignmentQueryRelation I T C D := by
rw [realizedAssignmentQueries, mem_relationImage]
constructor
· rintro ⟨hq, p, hp, entry⟩
exact ⟨hq, p, hp, ((pair_mem_relationConverse C D _ q p).1 entry).2.2⟩
· rintro ⟨hq, p, hp, entry⟩
exact ⟨hq, p, hp, (pair_mem_relationConverse C D _ q p).2 ⟨hq, hp, entry⟩⟩
theorem assignmentIndexOverlap_iff (I T L z : ZFSet.{u}) :
(∃ i ∈ I, ∃ w ∈ L,
ZFSet.pair z i ∈ productFirst I (ZFSet.prod T T) ∧
ZFSet.pair w i ∈ productFirst I (ZFSet.prod T T)) ↔
z ∈ stageOneAssignments (assignmentIndices I T L) T := by
constructor
· rintro ⟨i, hi, w, hw, first, second⟩
obtain ⟨j, _, rest, hr, equal⟩ := (mem_productFirst I (ZFSet.prod T T) _).1 first
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal
exact ZFSet.pair_mem_prod.2
⟨(mem_relationImage L I _ _).2 ⟨hi, w, hw, second⟩, hr⟩
· intro member
obtain ⟨i, hi, rest, hr, rfl⟩ := ZFSet.mem_prod.1 member
obtain ⟨hi, w, hw, second⟩ := (mem_relationImage L I _ i).1 hi
exact ⟨i, hi, w, hw, (mem_productFirst I (ZFSet.prod T T) _).2
⟨i, hi, rest, hr, rfl⟩, second⟩
namespace BoundedFormula
open SetFormula
def assignmentQueryOverlapFormula {n} (z I L f : Fin n) : BoundedFormula n :=
exMem I (exMem L.succ (.conj (pairMembershipFormula z.succ.succ 1 f.succ.succ)
(pairMembershipFormula 0 1 f.succ.succ)))
theorem truth_assignmentQueryOverlapFormula {n} (z I L f : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (assignmentQueryOverlapFormula z I L f) env ↔
∃ i ∈ env I, ∃ w ∈ env L,
ZFSet.pair (env z) i ∈ env f ∧ ZFSet.pair w i ∈ env f := by
simp only [assignmentQueryOverlapFormula, truth_exMem, Truth, truth_pairMembershipFormula]
rfl
def assignmentQueryMatchFormula {n} (I L A f : Fin n) : BoundedFormula n :=
.conj (.allMem L (.conj (.mem 0 A.succ)
(assignmentQueryOverlapFormula 0 I.succ L.succ f.succ)))
(.allMem A (.imp (assignmentQueryOverlapFormula 0 I.succ L.succ f.succ) (.mem 0 L.succ)))
theorem truth_assignmentQueryMatchFormula {n} (I L A f : Fin n)
(env : Fin n → ZFSet.{u}) (T : ZFSet.{u})
(projection : env f = productFirst (env I) (ZFSet.prod T T)) :
Truth (assignmentQueryMatchFormula I L A f) env ↔
assignmentOnIndices (assignmentIndices (env I) T (env L)) T (env A) = env L := by
simp only [assignmentQueryMatchFormula, Truth, truth_imp, truth_assignmentQueryOverlapFormula]
change ((∀ z ∈ env L, z ∈ env A ∧ ∃ i ∈ env I, ∃ w ∈ env L,
ZFSet.pair z i ∈ env f ∧ ZFSet.pair w i ∈ env f) ∧
(∀ z ∈ env A, (∃ i ∈ env I, ∃ w ∈ env L,
ZFSet.pair z i ∈ env f ∧ ZFSet.pair w i ∈ env f) → z ∈ env L)) ↔ _
simp only [projection, assignmentIndexOverlap_iff]
constructor
· intro h
apply ZFSet.ext
intro z
rw [mem_assignmentOnIndices]
exact ⟨fun hz => h.2 z hz.1 hz.2, fun hz => h.1 z hz⟩
· intro equal
constructor
· intro z hz
exact (mem_assignmentOnIndices _ _ _ z).1 (equal.symm ▸ hz)
· intro z hz hi
exact equal ▸ (mem_assignmentOnIndices _ _ _ z).2 ⟨hz, hi⟩
def assignmentQueryRelationFormula {n} (z I f : Fin n) : BoundedFormula n :=
exPairFields z (exPairFields 1 (exPairFields 2
(.conj (.equal 3 1) (assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ
2 0 f.succ.succ.succ.succ.succ.succ))))
theorem truth_assignmentQueryRelationFormula {n} (z I f : Fin n)
(env : Fin n → ZFSet.{u}) (T : ZFSet.{u})
(projection : env f = productFirst (env I) (ZFSet.prod T T)) :
Truth (assignmentQueryRelationFormula z I f) env ↔
∃ row L A, env z = ZFSet.pair (ZFSet.pair row L) (ZFSet.pair row A) ∧
assignmentOnIndices (assignmentIndices (env I) T L) T A = L := by
simp only [assignmentQueryRelationFormula, truth_exPairFields]
constructor
· rintro ⟨q, p, hz, row, L, hq, other, A, hp, same, matchList⟩
change q = ZFSet.pair row L at hq
change p = ZFSet.pair other A at hp
change row = other at same
have eq := (truth_assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ 2 0
f.succ.succ.succ.succ.succ.succ
(extend A (extend other (extend L (extend row (extend p (extend q env)))))) T projection).1 matchList
exact ⟨row, L, A, by rw [hz, hq, hp, ← same], eq⟩
· rintro ⟨row, L, A, hz, matchList⟩
exact ⟨ZFSet.pair row L, ZFSet.pair row A, hz, row, L, rfl, row, A, rfl,
rfl, (truth_assignmentQueryMatchFormula I.succ.succ.succ.succ.succ.succ 2 0
f.succ.succ.succ.succ.succ.succ
(extend A (extend row (extend L (extend row
(extend (ZFSet.pair row A) (extend (ZFSet.pair row L) env)))))) T projection).2 matchList⟩
end BoundedFormula
end PartitionPilot.Forcing
end OAI
|