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import OAI.SetTheory.PartitionPrinciple.Geometry.GroundIndexedClassInclusions
import OAI.SetTheory.PartitionPrinciple.Geometry.GroundPartialCoverPartitions
import OAI.SetTheory.PartitionPrinciple.Models.GroundOrdinalPartitionInjectionAssembly
namespace OAI
open scoped ZFSet
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (N : TransitiveGround.{u})
theorem function_kernel_quotient
(zf : ZFRealization N.sets) {U V f : ZFSet.{u}}
(hU : U ∈ N.sets) (hV : V ∈ N.sets) (hf : f ∈ N.sets)
(function : ZFSet.IsFunc U V f)
(onto : ∀ x ∈ V, ∃ u ∈ U, ZFSet.pair u x ∈ f) :
∃ R ∈ N.sets, EquivalenceRelationCode U R ∧
(∀ u v, ZFSet.pair u v ∈ R ↔ u ∈ U ∧ v ∈ U ∧
∃ x ∈ V, ZFSet.pair u x ∈ f ∧ ZFSet.pair v x ∈ f) ∧
∃ a ∈ N.sets, InjectionCode (equivalenceClasses U R) V a ∧
(∀ x ∈ V, ∃ C ∈ equivalenceClasses U R, ZFSet.pair C x ∈ a) ∧
(∀ u x, ZFSet.pair u x ∈ f →
ZFSet.pair (equivalenceClass U R u) x ∈ a) := by
have pairing : Realize N.sets SetFormula.pairing Fin.elim0 := zf.1.2.2.2.1
have unions : Realize N.sets SetFormula.union Fin.elim0 := zf.1.2.2.2.2.1
have powers : Realize N.sets powerSet Fin.elim0 := zf.1.2.2.2.2.2.1
let inverse := relationConverse U V f
let R := relationCompose U V U f inverse
have inverseGround := N.relationConverse_mem pairing unions powers zf.2.1 hU hV hf
have hR : R ∈ N.sets :=
N.relationCompose_mem pairing unions powers zf.2.1 hU hV hU hf inverseGround
have entries (u v : ZFSet.{u}) : ZFSet.pair u v ∈ R ↔ u ∈ U ∧ v ∈ U ∧
∃ x ∈ V, ZFSet.pair u x ∈ f ∧ ZFSet.pair v x ∈ f := by
simp only [R, inverse, pair_mem_relationCompose, pair_mem_relationConverse]
exact ⟨fun ⟨hu, hv, x, hx, ux, _, _, vx⟩ => ⟨hu, hv, x, hx, ux, vx⟩,
fun ⟨hu, hv, x, hx, ux, vx⟩ => ⟨hu, hv, x, hx, ux, hv, hx, vx⟩⟩
have unique {u x y : ZFSet.{u}} (ux : ZFSet.pair u x ∈ f)
(uy : ZFSet.pair u y ∈ f) : x = y := by
obtain ⟨v, _, unique⟩ := function.2 u (ZFSet.pair_mem_prod.1 (function.1 ux)).1
exact (unique x ux).trans (unique y uy).symm
have equivalence : EquivalenceRelationCode U R := by
refine ⟨fun _ hz => (ZFSet.mem_sep.1 hz).1, ?_, ?_, ?_⟩
· intro u hu
obtain ⟨x, ux, _⟩ := function.2 u hu
exact (entries u u).2 ⟨hu, hu, x, (ZFSet.pair_mem_prod.1 (function.1 ux)).2, ux, ux⟩
· intro u v uv
obtain ⟨hu, hv, x, hx, ux, vx⟩ := (entries u v).1 uv
exact (entries v u).2 ⟨hv, hu, x, hx, vx, ux⟩
· intro u v w uv vw
obtain ⟨hu, _, x, hx, ux, vx⟩ := (entries u v).1 uv
obtain ⟨_, hw, y, _, vy, wy⟩ := (entries v w).1 vw
exact (entries u w).2 ⟨hu, hw, x, hx, ux, (unique vy vx) ▸ wy⟩
let Q := equivalenceClasses U R
let classifier := equivalenceClassifier U R
let backward := relationConverse U Q classifier
let a := relationCompose Q U V backward f
have hQ : Q ∈ N.sets := N.equivalenceClasses_mem powers zf.2.1 hU hR
have hClassifier : classifier ∈ N.sets :=
N.equivalenceClassifier_mem pairing unions powers zf.2.1 hU hR
have hBackward : backward ∈ N.sets :=
N.relationConverse_mem pairing unions powers zf.2.1 hU hQ hClassifier
have ha : a ∈ N.sets := N.relationCompose_mem pairing unions powers zf.2.1 hQ hU hV hBackward hf
have aEntries (C x : ZFSet.{u}) : ZFSet.pair C x ∈ a ↔
C ∈ Q ∧ x ∈ V ∧ ∃ u ∈ U, C = equivalenceClass U R u ∧ ZFSet.pair u x ∈ f := by
change ZFSet.pair C x ∈ relationCompose Q U V backward f ↔ _
rw [pair_mem_relationCompose]
constructor
· rintro ⟨hC, hx, u, hu, cu, ux⟩
have uc := (pair_mem_relationConverse U Q classifier u C).1 cu
have equal := (pair_mem_equivalenceClassifier U R u C).1 uc.2.2
exact ⟨hC, hx, u, hu, equal.2, ux⟩
· rintro ⟨hC, hx, u, hu, equal, ux⟩
exact ⟨hC, hx, u, hu, (pair_mem_relationConverse U Q classifier u C).2
⟨hu, hC, (pair_mem_equivalenceClassifier U R u C).2 ⟨hu, equal⟩⟩, ux⟩
have canonical (u x : ZFSet.{u}) (ux : ZFSet.pair u x ∈ f) :
ZFSet.pair (equivalenceClass U R u) x ∈ a := by
obtain ⟨hu, hx⟩ := ZFSet.pair_mem_prod.1 (function.1 ux)
exact (aEntries _ x).2 ⟨Forcing.equivalenceClass_mem hu, hx, u, hu, rfl, ux⟩
have injection : InjectionCode Q V a := by
refine ⟨⟨fun _ hz => (ZFSet.mem_sep.1 hz).1, ?_⟩, ?_⟩
· intro C hC
obtain ⟨u, hu, rfl⟩ := (mem_equivalenceClasses U R C).1 hC
obtain ⟨x, ux, _⟩ := function.2 u hu
refine ⟨x, canonical u x ux, ?_⟩
intro y cy
obtain ⟨_, _, v, hv, equal, vy⟩ := (aEntries _ y).1 cy
obtain ⟨_, _, z, _, uz, vz⟩ := (entries u v).1 ((equivalence.class_eq_iff hu hv).1 equal)
exact (unique vy vz).trans (unique uz ux)
· intro C D x cx dx
obtain ⟨_, hx, u, hu, rfl, ux⟩ := (aEntries C x).1 cx
obtain ⟨_, _, v, hv, rfl, vx⟩ := (aEntries D x).1 dx
exact (equivalence.class_eq_iff hu hv).2 ((entries u v).2 ⟨hu, hv, x, hx, ux, vx⟩)
refine ⟨R, hR, equivalence, entries, a, ha, injection, ?_, canonical⟩
intro x hx
obtain ⟨u, hu, ux⟩ := onto x hx
exact ⟨equivalenceClass U R u, Forcing.equivalenceClass_mem hu, canonical u x ux⟩
theorem partialCover_quotient_injection
(zf : ZFRealization N.sets) {T B X Y X' Y' h f : ZFSet.{u}}
(quotientInjection : ∀ U ∈ N.sets, ∀ Rf ∈ N.sets, ∀ Rc ∈ N.sets,
RefinementValidityCode T U Rf Rc →
N.groundEmbeddable (equivalenceClasses U Rc) (equivalenceClasses U Rf))
(hT : T ∈ N.sets) (_hX : X ∈ N.sets) (_hY : Y ∈ N.sets)
(hX' : X' ∈ N.sets) (hY' : Y' ∈ N.sets) (hh : h ∈ N.sets) (hf : f ∈ N.sets)
(partialCode : PartialFunctionCode T B h) (function : ZFSet.IsFunc X Y f)
(sourceSubset : X' ⊆ X) (_targetSubset : Y' ⊆ Y)
(maps : ∀ x ∈ X', ∀ y, ZFSet.pair x y ∈ f → y ∈ Y')
(onto : ∀ y ∈ Y', ∃ x ∈ X', ZFSet.pair x y ∈ f)
(covered : ∀ x ∈ X', ∃ u ∈ T, ZFSet.pair u x ∈ h) :
N.groundEmbeddable Y' X' := by
have pairing : Realize N.sets SetFormula.pairing Fin.elim0 := zf.1.2.2.2.1
have unions : Realize N.sets SetFormula.union Fin.elim0 := zf.1.2.2.2.2.1
have powers : Realize N.sets powerSet Fin.elim0 := zf.1.2.2.2.2.2.1
let inverse := relationConverse T X' h
let U := relationImage X' T inverse
have hInverse : inverse ∈ N.sets :=
N.relationConverse_mem pairing unions powers zf.2.1 hT hX' hh
have hU : U ∈ N.sets := N.relationImage_mem zf.2.1 hX' hT hInverse
have uEntries (u : ZFSet.{u}) : u ∈ U ↔ u ∈ T ∧ ∃ x ∈ X', ZFSet.pair u x ∈ h := by
simp only [U, inverse, mem_relationImage, pair_mem_relationConverse]
exact ⟨fun ⟨hu, x, hx, _, _, ux⟩ => ⟨hu, x, hx, ux⟩,
fun ⟨hu, x, hx, ux⟩ => ⟨hu, x, hx, hu, hx, ux⟩⟩
let h' := ZFSet.sep (fun z => z ∈ ZFSet.prod U X') h
have hh' : h' ∈ N.sets := N.injection_restriction_mem zf.2.1 hU hX' hh
have hEntries (u x : ZFSet.{u}) : ZFSet.pair u x ∈ h' ↔
ZFSet.pair u x ∈ h ∧ u ∈ U ∧ x ∈ X' := by
exact ZFSet.mem_sep.trans (and_congr_right fun _ => ZFSet.pair_mem_prod)
have hFunction : ZFSet.IsFunc U X' h' := by
refine ⟨fun _ hz => (ZFSet.mem_sep.1 hz).2, ?_⟩
intro u hu
obtain ⟨_, x, hx, ux⟩ := (uEntries u).1 hu
exact ⟨x, (hEntries u x).2 ⟨ux, hu, hx⟩,
fun y uy => partialCode.unique u y x ((hEntries u y).1 uy).1 ux⟩
have hOnto : ∀ x ∈ X', ∃ u ∈ U, ZFSet.pair u x ∈ h' := by
intro x hx
obtain ⟨u, hu, ux⟩ := covered x hx
have member := (uEntries u).2 ⟨hu, x, hx, ux⟩
exact ⟨u, member, (hEntries u x).2 ⟨ux, member, hx⟩⟩
let f' := ZFSet.sep (fun z => z ∈ ZFSet.prod X' Y') f
have hf' : f' ∈ N.sets := N.injection_restriction_mem zf.2.1 hX' hY' hf
have fEntries (x y : ZFSet.{u}) : ZFSet.pair x y ∈ f' ↔
ZFSet.pair x y ∈ f ∧ x ∈ X' ∧ y ∈ Y' := by
exact ZFSet.mem_sep.trans (and_congr_right fun _ => ZFSet.pair_mem_prod)
have fFunction : ZFSet.IsFunc X' Y' f' := by
refine ⟨fun _ hz => (ZFSet.mem_sep.1 hz).2, ?_⟩
intro x hx
obtain ⟨y, xy, unique⟩ := function.2 x (sourceSubset hx)
exact ⟨y, (fEntries x y).2 ⟨xy, hx, maps x hx y xy⟩,
fun z xz => unique z ((fEntries x z).1 xz).1⟩
let g := relationCompose U X' Y' h' f'
have hg : g ∈ N.sets := N.relationCompose_mem pairing unions powers zf.2.1 hU hX' hY' hh' hf'
have gFunction : ZFSet.IsFunc U Y' g :=
relationCompose_function_of_subset hFunction fFunction (fun _ hx => hx)
have gOnto : ∀ y ∈ Y', ∃ u ∈ U, ZFSet.pair u y ∈ g := by
intro y hy
obtain ⟨x, hx, xy⟩ := onto y hy
obtain ⟨u, hu, ux⟩ := hOnto x hx
exact ⟨u, hu, (pair_mem_relationCompose U X' Y' h' f' u y).2
⟨hu, hy, x, hx, ux, (fEntries x y).2 ⟨xy, hx, hy⟩⟩⟩
obtain ⟨Rf, hRf, fine, fineEntries, a, ha, aInjection, _aOnto, _aCanonical⟩ :=
N.function_kernel_quotient zf hU hX' hh' hFunction hOnto
obtain ⟨Rc, hRc, coarse, coarseEntries, b, hb, bInjection, bOnto, _bCanonical⟩ :=
N.function_kernel_quotient zf hU hY' hg gFunction gOnto
have refines : Rf ⊆ Rc := by
intro z hz
obtain ⟨u, hu, v, hv, rfl⟩ := ZFSet.mem_prod.1 (fine.shape hz)
obtain ⟨_, _, x, hx, ux, vx⟩ := (fineEntries u v).1 hz
obtain ⟨y, xy, _⟩ := fFunction.2 x hx
have hy := (ZFSet.pair_mem_prod.1 (fFunction.1 xy)).2
exact (coarseEntries u v).2 ⟨hu, hv, y, hy,
(pair_mem_relationCompose U X' Y' h' f' u y).2 ⟨hu, hy, x, hx, ux, xy⟩,
(pair_mem_relationCompose U X' Y' h' f' v y).2 ⟨hv, hy, x, hx, vx, xy⟩⟩
obtain ⟨ell, hEll, ellInjection⟩ := quotientInjection U hU Rf hRf Rc hRc
⟨fun u hu => ((uEntries u).1 hu).1, fine, coarse, refines⟩
have fineGround := N.equivalenceClasses_mem powers zf.2.1 hU hRf
have coarseGround := N.equivalenceClasses_mem powers zf.2.1 hU hRc
have imageEq : relationImage (equivalenceClasses U Rc) Y' b = Y' := by
apply ZFSet.ext
intro y
rw [mem_relationImage]
exact ⟨And.left, fun hy => ⟨hy, bOnto y hy⟩⟩
have inverseInjection : InjectionCode Y' (equivalenceClasses U Rc)
(relationConverse (equivalenceClasses U Rc) Y' b) := by
have inverse := bInjection.converse_image
rw [imageEq] at inverse
exact inverse
have inverseGround := N.relationConverse_mem pairing unions powers zf.2.1 coarseGround hY' hb
let first := relationCompose Y' (equivalenceClasses U Rc) (equivalenceClasses U Rf)
(relationConverse (equivalenceClasses U Rc) Y' b) ell
have firstGround : first ∈ N.sets := N.relationCompose_mem pairing unions powers zf.2.1
hY' coarseGround fineGround inverseGround hEll
exact ⟨relationCompose Y' (equivalenceClasses U Rf) X' first a,
N.relationCompose_mem pairing unions powers zf.2.1 hY' fineGround hX' firstGround ha,
(inverseInjection.compose ellInjection).compose aInjection⟩
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula SetRealization
universe u
variable (N : TransitiveGround.{u})
theorem partitionPrinciple_of_rowCover_quotients
(zf : ZFRealization N.sets) {T : ZFSet.{u}} (hT : T ∈ N.sets)
(acwo : ∀ delta ∈ N.sets, ordinalSetProperty delta →
∀ z ∈ N.sets, FunctionGraphOn delta z →
(∀ beta A, ZFSet.pair beta A ∈ z → A.Nonempty) →
∃ c ∈ N.sets, FunctionGraphOn delta c ∧
∀ beta A x, ZFSet.pair beta A ∈ z → ZFSet.pair beta x ∈ c → x ∈ A)
(rowCover : ∀ X ∈ N.sets, ∃ delta ∈ N.sets, ordinalSetProperty delta ∧
∃ h ∈ N.sets, FunctionGraphOn delta h ∧
(∀ i hi, ZFSet.pair i hi ∈ h → PartialFunctionCode T (⋃₀ ⋃₀ hi) hi) ∧
∀ x ∈ X, ∃ i ∈ delta, ∃ hi, ZFSet.pair i hi ∈ h ∧
∃ u ∈ T, ZFSet.pair u x ∈ hi)
(quotientInjection : ∀ U ∈ N.sets, ∀ Rf ∈ N.sets, ∀ Rc ∈ N.sets,
RefinementValidityCode T U Rf Rc →
N.groundEmbeddable (equivalenceClasses U Rc) (equivalenceClasses U Rf)) :
∀ X ∈ N.sets, ∀ Y ∈ N.sets, ∀ f ∈ N.sets, ZFSet.IsFunc X Y f →
(∀ y ∈ Y, ∃ x ∈ X, ZFSet.pair x y ∈ f) → N.groundEmbeddable Y X := by
intro X hX Y hY f hf function onto
obtain ⟨delta, hDelta, ordinal, h, hh, family, partials, cover⟩ := rowCover X hX
obtain ⟨full, _hFull, p, hp, q, hq, fullFunction, pFunction, qFunction,
bounds, coverY, disjointY, disjointX, pieces⟩ :=
N.partialCover_ordinal_partitions zf hDelta ordinal hT hX hY hh hf family function onto cover
apply N.ordinalPartition_injection_assembly zf acwo hDelta ordinal hY hX hp hq
pFunction qFunction bounds coverY disjointY disjointX
intro i hi U V ip iq
obtain ⟨g, ig⟩ := family.total hi
obtain ⟨A, iA⟩ := fullFunction.total hi
obtain ⟨_original, _least, _source, _hRestrict, restrictFunction, restrictOnto, covered⟩ :=
pieces i g A U V ig iA ip iq
have hU : U ∈ N.sets := N.pair_right (N.transitive hp ip)
have hV : V ∈ N.sets := N.pair_right (N.transitive hq iq)
have hg : g ∈ N.sets := N.pair_right (N.transitive hh ig)
apply N.partialCover_quotient_injection zf quotientInjection hT hX hY hV hU hg hf
(partials i g ig) function (bounds i U V ip iq).2 (bounds i U V ip iq).1
· intro x hx y xy
exact (ZFSet.pair_mem_prod.1 (restrictFunction.1
(pair_mem_graphRestriction.2 ⟨xy, hx⟩))).2
· intro y hy
obtain ⟨x, hx, xy⟩ := restrictOnto y hy
exact ⟨x, hx, (pair_mem_graphRestriction.1 xy).1⟩
· exact covered
end PartitionPilot.Forcing.TransitiveGround
end OAI
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