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