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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.DecodedMeetMaps
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
structure CompressionRangeAccounting (P0 I U R MU Maps Bases Caps : ZFSet.{u}) : Prop where
original_subset : P0 ⊆ U
base_filled : ∀ i ∈ I, ∀ f b,
ZFSet.pair i f ∈ Maps → ZFSet.pair i b ∈ Bases →
∀ x ∈ U, ZFSet.pair x b ∈ R → ∃ y ∈ U, ZFSet.pair y x ∈ f
old_shadow : ∀ i ∈ I, ∀ b k,
ZFSet.pair i b ∈ Bases → ZFSet.pair i k ∈ Caps →
∀ x ∈ P0, ZFSet.pair x k ∈ R → ZFSet.pair x b ∈ R
cap_separation : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → ∀ b c k l,
ZFSet.pair i b ∈ Bases → ZFSet.pair j c ∈ Bases →
ZFSet.pair i k ∈ Caps → ZFSet.pair j l ∈ Caps →
ZFSet.pair k c ∈ R ∨ ZFSet.pair l b ∈ R ∨
∃ z, ZFSet.pair (ZFSet.pair k l) z ∈ MU ∧ ZFSet.pair (ZFSet.pair b c) z ∈ MU
fresh_accounting : ∀ x ∈ U, x ∈ P0 ∨
∃ i ∈ I, ∃ f k b, ∃ y ∈ U,
ZFSet.pair i f ∈ Maps ∧ ZFSet.pair i k ∈ Caps ∧ ZFSet.pair i b ∈ Bases ∧
ZFSet.pair y x ∈ f ∧ ZFSet.pair x k ∈ R ∧ ZFSet.pair x b ∉ R
namespace CompressionRangeAccounting
variable {P0 I U R MU Maps Bases Caps : ZFSet.{u}}
variable (accounting : CompressionRangeAccounting P0 I U R MU Maps Bases Caps)
include accounting
theorem range_completion (meet : MeetOrderCode U R MU) (maps : FunctionGraphOn I Maps)
(embeddings : ∀ i f, ZFSet.pair i f ∈ Maps → MeetEmbeddingCode U U R R MU MU f)
(bases : ZFSet.IsFunc I U Bases) (caps : ZFSet.IsFunc I U Caps) :
∀ i ∈ I, ∀ f k, ZFSet.pair i f ∈ Maps → ZFSet.pair i k ∈ Caps →
∀ x ∈ U, ZFSet.pair x k ∈ R → ∃ y ∈ U, ZFSet.pair y x ∈ f := by
let baseGraph : FunctionGraphOn I Bases := ⟨U, bases⟩
let capGraph : FunctionGraphOn I Caps := ⟨U, caps⟩
let base (i : {i : ZFSet.{u} // i ∈ I}) : {x : ZFSet.{u} // x ∈ U} :=
⟨baseGraph.value i, (ZFSet.pair_mem_prod.1 (bases.1 (baseGraph.value_entry i))).2⟩
let cap (i : {i : ZFSet.{u} // i ∈ I}) : {x : ZFSet.{u} // x ∈ U} :=
⟨capGraph.value i, (ZFSet.pair_mem_prod.1 (caps.1 (capGraph.value_entry i))).2⟩
let embedding (i : {i : ZFSet.{u} // i ∈ I}) :=
embeddings i.val (maps.value i) (maps.value_entry i)
let s (i : {i : ZFSet.{u} // i ∈ I}) := (embedding i).toMeetEmbedding meet meet
have baseFilled : ∀ i x, meet.toMeetOrder.le x (base i) → ∃ y, (s i).toFun y = x := by
intro i x below
obtain ⟨y, hy, entry⟩ := accounting.base_filled i.val i.property
(maps.value i) (base i).val (maps.value_entry i) (baseGraph.value_entry i) x.val x.property below
exact ⟨⟨y, hy⟩, ((embedding i).toMeetEmbedding_eq_iff meet meet ⟨y, hy⟩ x).2 entry⟩
have shadow : ∀ i x, x.val ∈ P0 → meet.toMeetOrder.le x (cap i) →
meet.toMeetOrder.le x (base i) := by
intro i x original below
exact accounting.old_shadow i.val i.property (base i).val (cap i).val
(baseGraph.value_entry i) (capGraph.value_entry i) x.val original below
have separation : ∀ i j, i ≠ j →
meet.toMeetOrder.le (cap i) (base j) ∨ meet.toMeetOrder.le (cap j) (base i) ∨
meet.toMeetOrder.meet (cap i) (cap j) = meet.toMeetOrder.meet (base i) (base j) := by
intro i j different
have rawDifferent : i.val ≠ j.val := fun equal => different (Subtype.ext equal)
rcases accounting.cap_separation i.val i.property j.val j.property rawDifferent
(base i).val (base j).val (cap i).val (cap j).val
(baseGraph.value_entry i) (baseGraph.value_entry j)
(capGraph.value_entry i) (capGraph.value_entry j) with first | second | common
· exact Or.inl first
· exact Or.inr (Or.inl second)
· obtain ⟨z, capsMeet, basesMeet⟩ := common
apply Or.inr ∘ Or.inr
apply Subtype.ext
let operation : FunctionGraphOn (ZFSet.prod U U) MU := ⟨U, meet.operation⟩
exact (operation.unique (meet.toMeetOrder_meet_entry (cap i) (cap j)) capsMeet).trans
(operation.unique (meet.toMeetOrder_meet_entry (base i) (base j)) basesMeet).symm
have accounted : ∀ x : {x : ZFSet.{u} // x ∈ U}, x.val ∈ P0 ∨
∃ j y, (s j).toFun y = x ∧ meet.toMeetOrder.le x (cap j) ∧ ¬ meet.toMeetOrder.le x (base j) := by
intro x
rcases accounting.fresh_accounting x.val x.property with original | fresh
· exact Or.inl original
· obtain ⟨j, hj, f, k, b, y, hy, indexed, capped, based, entry, below, outside⟩ := fresh
let i : {i : ZFSet.{u} // i ∈ I} := ⟨j, hj⟩
have fEq := maps.value_eq_of_entry i indexed
have kEq := capGraph.value_eq_of_entry i capped
have bEq := baseGraph.value_eq_of_entry i based
refine Or.inr ⟨i, ⟨y, hy⟩, ?_, ?_, ?_⟩
· apply ((embedding i).toMeetEmbedding_eq_iff meet meet ⟨y, hy⟩ x).2
exact fEq.symm ▸ entry
· change ZFSet.pair x.val (capGraph.value i) ∈ R
exact kEq.symm ▸ below
· change ZFSet.pair x.val (baseGraph.value i) ∉ R
exact bEq.symm ▸ outside
have complete := Compression.range_completion_from_accounting meet.toMeetOrder
(fun x => x.val ∈ P0) base cap (fun i => (s i).toFun) baseFilled shadow separation accounted
intro i hi f k indexed capped x hx below
let j : {i : ZFSet.{u} // i ∈ I} := ⟨i, hi⟩
have fEq := maps.value_eq_of_entry j indexed
have kEq := capGraph.value_eq_of_entry j capped
obtain ⟨y, equal⟩ := complete j ⟨x, hx⟩ (show ZFSet.pair x (capGraph.value j) ∈ R from kEq.symm ▸ below)
refine ⟨y.val, y.property, ?_⟩
have entry := ((embedding j).toMeetEmbedding_eq_iff meet meet y ⟨x, hx⟩).1 equal
exact fEq ▸ entry
theorem range_iff (meet : MeetOrderCode U R MU) (maps : FunctionGraphOn I Maps)
(embeddings : ∀ i f, ZFSet.pair i f ∈ Maps → MeetEmbeddingCode U U R R MU MU f)
(bases : ZFSet.IsFunc I U Bases) (caps : ZFSet.IsFunc I U Caps)
(bounded : ∀ i ∈ I, ∀ f k, ZFSet.pair i f ∈ Maps → ZFSet.pair i k ∈ Caps →
∀ x y, ZFSet.pair x y ∈ f → ZFSet.pair y k ∈ R)
{i f k x : ZFSet.{u}} (hi : i ∈ I) (indexed : ZFSet.pair i f ∈ Maps)
(capped : ZFSet.pair i k ∈ Caps) :
(∃ y ∈ U, ZFSet.pair y x ∈ f) ↔ x ∈ U ∧ ZFSet.pair x k ∈ R := by
constructor
· rintro ⟨y, _, entry⟩
exact ⟨(ZFSet.pair_mem_prod.1 ((embeddings i f indexed).injection.function.1 entry)).2,
bounded i hi f k indexed capped y x entry⟩
· rintro ⟨hx, below⟩
exact accounting.range_completion meet maps embeddings bases caps i hi f k indexed capped x hx below
end CompressionRangeAccounting
end PartitionPilot.Forcing
end
end OAI
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