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namespace OAI
namespace PartitionPilot.Forcing
namespace TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem taggedExtension_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
{P N : ZFSet.{u}} (hP : P ∈ M.sets) (hN : N ∈ M.sets) :
taggedExtension P N ∈ M.sets := by
have singleton : ({P} : ZFSet.{u}) ∈ M.sets := by
simpa only [ZFSet.pair_eq_singleton] using M.unorderedPair_mem pairing hP hP
exact M.binary_union_mem pairing unions hP
(M.prod_mem pairing unions powers separation singleton hN)
end TransitiveGround
universe u
namespace BoundedFormula
open SetFormula
def attachmentShadowEntryFormula {n} (z q p μ ell a : Fin n) : BoundedFormula n :=
exMem q (exMem p.succ
(.conj (orderedPairFormula z.succ.succ 1 0)
(exMem q.succ.succ
(.conj (tripleMembershipFormula 2 a.succ.succ.succ 0 μ.succ.succ.succ)
(pairMembershipFormula 0 1 ell.succ.succ.succ)))))
theorem truth_attachmentShadowEntryFormula {n} (z q p μ ell a : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (attachmentShadowEntryFormula z q p μ ell a) env ↔
env z ∈ attachmentShadowGraph (env q) (env p) (env μ) (env ell) (env a) := by
simp only [attachmentShadowEntryFormula, truth_exMem, Truth,
truth_orderedPairFormula, truth_tripleMembershipFormula, truth_pairMembershipFormula,
mem_attachmentShadowGraph]
rfl
end BoundedFormula
namespace TransitiveGround
open SetFormula
variable (M : TransitiveGround.{u})
theorem attachmentIdeal_mem (separation : SeparationSchema M.sets)
{Q rQ a : ZFSet.{u}} (hQ : Q ∈ M.sets) (hrQ : rQ ∈ M.sets)
(ha : a ∈ M.sets) : attachmentIdeal Q rQ a ∈ M.sets := by
let params := extend a (extend rQ Fin.elim0)
have selected := M.bounded_separation_mem separation
(BoundedFormula.pairMembershipFormula 0 1 2) params
(Fin.cases ha (Fin.cases hrQ (fun i => Fin.elim0 i))) hQ
have equal : ZFSet.sep (fun q => BoundedFormula.Truth
(BoundedFormula.pairMembershipFormula 0 1 2) (extend q params)) Q =
attachmentIdeal Q rQ a := by
apply ZFSet.ext
intro q
rw [ZFSet.mem_sep, mem_attachmentIdeal, BoundedFormula.truth_pairMembershipFormula]
rfl
exact equal ▸ selected
theorem freshPart_mem (separation : SeparationSchema M.sets)
{Q rQ a : ZFSet.{u}} (hQ : Q ∈ M.sets) (hrQ : rQ ∈ M.sets)
(ha : a ∈ M.sets) : freshPart Q rQ a ∈ M.sets := by
let params := extend a (extend rQ Fin.elim0)
have selected := M.bounded_separation_mem separation
(.neg (BoundedFormula.pairMembershipFormula 0 1 2)) params
(Fin.cases ha (Fin.cases hrQ (fun i => Fin.elim0 i))) hQ
have equal : ZFSet.sep (fun q => BoundedFormula.Truth
(.neg (BoundedFormula.pairMembershipFormula 0 1 2)) (extend q params)) Q =
freshPart Q rQ a := by
apply ZFSet.ext
intro q
simp only [ZFSet.mem_sep, mem_freshPart, BoundedFormula.Truth,
BoundedFormula.truth_pairMembershipFormula]
rfl
exact equal ▸ selected
theorem attachmentShadowGraph_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
{Q P μQ ell a : ZFSet.{u}} (hQ : Q ∈ M.sets) (hP : P ∈ M.sets)
(hμQ : μQ ∈ M.sets) (hell : ell ∈ M.sets) (ha : a ∈ M.sets) :
attachmentShadowGraph Q P μQ ell a ∈ M.sets := by
let params := extend Q (extend P (extend μQ (extend ell (extend a Fin.elim0))))
have selected := M.bounded_separation_mem separation
(BoundedFormula.attachmentShadowEntryFormula 0 1 2 3 4 5) params
(Fin.cases hQ (Fin.cases hP (Fin.cases hμQ (Fin.cases hell
(Fin.cases ha (fun i => Fin.elim0 i))))))
(M.prod_mem pairing unions powers separation hQ hP)
have equal : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.attachmentShadowEntryFormula 0 1 2 3 4 5) (extend z params))
(ZFSet.prod Q P) = attachmentShadowGraph Q P μQ ell a := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep, BoundedFormula.truth_attachmentShadowEntryFormula]
exact ⟨And.right, fun hz => ⟨(ZFSet.mem_sep.1 hz).1, hz⟩⟩
exact equal ▸ selected
end TransitiveGround
noncomputable def attachmentOldEmbeddingGraph (P : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ x ∈ P, z = ZFSet.pair x x) (ZFSet.prod P P)
noncomputable def attachmentFreshEmbeddingGraph (P N : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ q ∈ N, z = ZFSet.pair q (ZFSet.pair P q))
(ZFSet.prod N (taggedExtension P N))
noncomputable def attachmentTemplateEmbeddingGraph (P Q rQ ell a : ZFSet.{u}) : ZFSet.{u} :=
ell ∪ attachmentFreshEmbeddingGraph P (freshPart Q rQ a)
theorem mem_attachmentOldEmbeddingGraph (P z : ZFSet.{u}) :
z ∈ attachmentOldEmbeddingGraph P ↔ ∃ x ∈ P, z = ZFSet.pair x x := by
constructor
· exact fun h => (ZFSet.mem_sep.1 h).2
· rintro ⟨x, hx, rfl⟩
exact ZFSet.mem_sep.2 ⟨ZFSet.pair_mem_prod.2 ⟨hx, hx⟩, x, hx, rfl⟩
theorem pair_mem_attachmentOldEmbeddingGraph (P x y : ZFSet.{u}) :
ZFSet.pair x y ∈ attachmentOldEmbeddingGraph P ↔ x ∈ P ∧ y = x := by
rw [mem_attachmentOldEmbeddingGraph]
constructor
· rintro ⟨z, hz, eq⟩
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 eq
exact ⟨hz, rfl⟩
· rintro ⟨hx, rfl⟩
exact ⟨_, hx, rfl⟩
theorem attachmentOldEmbeddingGraph_isFunc (P N : ZFSet.{u}) :
ZFSet.IsFunc P (taggedExtension P N) (attachmentOldEmbeddingGraph P) := by
refine ⟨?_, ?_⟩
· intro z hz
obtain ⟨x, hx, rfl⟩ := (mem_attachmentOldEmbeddingGraph P z).1 hz
exact ZFSet.pair_mem_prod.2 ⟨hx, old_mem_taggedExtension hx⟩
· intro x hx
exact ⟨x, (pair_mem_attachmentOldEmbeddingGraph P x x).2 ⟨hx, rfl⟩,
fun y hy => ((pair_mem_attachmentOldEmbeddingGraph P x y).1 hy).2⟩
theorem attachmentOldEmbeddingGraph_injection (P N : ZFSet.{u}) :
InjectionCode P (taggedExtension P N) (attachmentOldEmbeddingGraph P) := by
refine ⟨attachmentOldEmbeddingGraph_isFunc P N, ?_⟩
intro x y z hx hy
exact ((pair_mem_attachmentOldEmbeddingGraph P x z).1 hx).2.symm.trans
((pair_mem_attachmentOldEmbeddingGraph P y z).1 hy).2
theorem mem_attachmentFreshEmbeddingGraph (P N z : ZFSet.{u}) :
z ∈ attachmentFreshEmbeddingGraph P N ↔ ∃ q ∈ N, z = ZFSet.pair q (ZFSet.pair P q) := by
constructor
· exact fun h => (ZFSet.mem_sep.1 h).2
· rintro ⟨q, hq, rfl⟩
exact ZFSet.mem_sep.2
⟨ZFSet.pair_mem_prod.2 ⟨hq, pair_mem_taggedExtension.2 hq⟩, q, hq, rfl⟩
theorem pair_mem_attachmentFreshEmbeddingGraph (P N q y : ZFSet.{u}) :
ZFSet.pair q y ∈ attachmentFreshEmbeddingGraph P N ↔ q ∈ N ∧ y = ZFSet.pair P q := by
rw [mem_attachmentFreshEmbeddingGraph]
constructor
· rintro ⟨r, hr, eq⟩
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 eq
exact ⟨hr, rfl⟩
· rintro ⟨hq, rfl⟩
exact ⟨q, hq, rfl⟩
theorem mem_attachmentTemplateEmbeddingGraph (P Q rQ ell a z : ZFSet.{u}) :
z ∈ attachmentTemplateEmbeddingGraph P Q rQ ell a ↔
z ∈ ell ∨ ∃ q ∈ freshPart Q rQ a, z = ZFSet.pair q (ZFSet.pair P q) := by
rw [attachmentTemplateEmbeddingGraph, ZFSet.mem_union, mem_attachmentFreshEmbeddingGraph]
theorem pair_mem_attachmentTemplateEmbeddingGraph (P Q rQ ell a q y : ZFSet.{u}) :
ZFSet.pair q y ∈ attachmentTemplateEmbeddingGraph P Q rQ ell a ↔
ZFSet.pair q y ∈ ell ∨ (q ∈ freshPart Q rQ a ∧ y = ZFSet.pair P q) := by
rw [attachmentTemplateEmbeddingGraph, ZFSet.mem_union, pair_mem_attachmentFreshEmbeddingGraph]
theorem attachmentTemplateEmbeddingGraph_of_ideal {P Q rQ ell a q y : ZFSet.{u}}
(hq : q ∈ attachmentIdeal Q rQ a) :
ZFSet.pair q y ∈ attachmentTemplateEmbeddingGraph P Q rQ ell a ↔ ZFSet.pair q y ∈ ell := by
rw [pair_mem_attachmentTemplateEmbeddingGraph]
exact ⟨fun h => h.elim id (fun h => False.elim (attachmentIdeal_not_freshPart hq h.1)), Or.inl⟩
theorem attachmentTemplateEmbeddingGraph_of_fresh {P Q rQ ell a q y : ZFSet.{u}}
(function : ZFSet.IsFunc (attachmentIdeal Q rQ a) P ell)
(hq : q ∈ freshPart Q rQ a) :
ZFSet.pair q y ∈ attachmentTemplateEmbeddingGraph P Q rQ ell a ↔ y = ZFSet.pair P q := by
rw [pair_mem_attachmentTemplateEmbeddingGraph]
constructor
· intro h
rcases h with h | h
· exact False.elim (attachmentIdeal_not_freshPart (ZFSet.pair_mem_prod.1 (function.1 h)).1 hq)
· exact h.2
· exact fun eq => Or.inr ⟨hq, eq⟩
theorem attachmentTemplateEmbeddingGraph_isFunc {P Q rQ ell a : ZFSet.{u}}
(function : ZFSet.IsFunc (attachmentIdeal Q rQ a) P ell) :
ZFSet.IsFunc Q (taggedExtension P (freshPart Q rQ a))
(attachmentTemplateEmbeddingGraph P Q rQ ell a) := by
classical
refine ⟨?_, ?_⟩
· intro z hz
rcases (mem_attachmentTemplateEmbeddingGraph P Q rQ ell a z).1 hz with old | ⟨q, hq, rfl⟩
· obtain ⟨q, hq, y, hy, rfl⟩ := ZFSet.mem_prod.1 (function.1 old)
exact ZFSet.pair_mem_prod.2 ⟨((mem_attachmentIdeal Q rQ a q).1 hq).1, old_mem_taggedExtension hy⟩
· exact ZFSet.pair_mem_prod.2
⟨((mem_freshPart Q rQ a q).1 hq).1, pair_mem_taggedExtension.2 hq⟩
· intro q hq
by_cases below : ZFSet.pair q a ∈ rQ
· have ideal := (mem_attachmentIdeal Q rQ a q).2 ⟨hq, below⟩
obtain ⟨y, entry, unique⟩ := function.2 q ideal
exact ⟨y, attachmentTemplateEmbeddingGraph_of_ideal ideal |>.2 entry,
fun z hz => unique z ((attachmentTemplateEmbeddingGraph_of_ideal ideal).1 hz)⟩
· have fresh := (mem_freshPart Q rQ a q).2 ⟨hq, below⟩
exact ⟨ZFSet.pair P q, (attachmentTemplateEmbeddingGraph_of_fresh function fresh).2 rfl,
fun y hy => (attachmentTemplateEmbeddingGraph_of_fresh function fresh).1 hy⟩
theorem attachmentTemplateEmbeddingGraph_injection {P Q rQ ell a : ZFSet.{u}}
(embedding : InjectionCode (attachmentIdeal Q rQ a) P ell) :
InjectionCode Q (taggedExtension P (freshPart Q rQ a))
(attachmentTemplateEmbeddingGraph P Q rQ ell a) := by
refine ⟨attachmentTemplateEmbeddingGraph_isFunc embedding.function, ?_⟩
intro x y z hx hy
rcases (pair_mem_attachmentTemplateEmbeddingGraph P Q rQ ell a x z).1 hx with hx | hx
· rcases (pair_mem_attachmentTemplateEmbeddingGraph P Q rQ ell a y z).1 hy with hy | hy
· exact embedding.injective x y z hx hy
· exact False.elim (pair_tag_not_mem P y (hy.2 ▸ (ZFSet.pair_mem_prod.1 (embedding.function.1 hx)).2))
· rcases (pair_mem_attachmentTemplateEmbeddingGraph P Q rQ ell a y z).1 hy with hy | hy
· exact False.elim (pair_tag_not_mem P x (hx.2 ▸ (ZFSet.pair_mem_prod.1 (embedding.function.1 hy)).2))
· exact (ZFSet.pair_inj.1 (hx.2.symm.trans hy.2)).2
namespace BoundedFormula
open SetFormula
def attachmentOldEmbeddingEntryFormula {n} (z p : Fin n) : BoundedFormula n :=
exMem p (orderedPairFormula z.succ 0 0)
def attachmentFreshEmbeddingEntryFormula {n} (z p d : Fin n) : BoundedFormula n :=
exMem d (exMem z.succ (exMem 0
(.conj (orderedPairFormula 0 p.succ.succ.succ 2)
(orderedPairFormula z.succ.succ.succ 2 0))))
theorem truth_attachmentOldEmbeddingEntryFormula {n} (z p : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (attachmentOldEmbeddingEntryFormula z p) env ↔
∃ x ∈ env p, env z = ZFSet.pair x x := by
simp only [attachmentOldEmbeddingEntryFormula, truth_exMem, truth_orderedPairFormula]
rfl
theorem truth_attachmentFreshEmbeddingEntryFormula {n} (z p d : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (attachmentFreshEmbeddingEntryFormula z p d) env ↔
∃ q ∈ env d, env z = ZFSet.pair q (ZFSet.pair (env p) q) := by
simp only [attachmentFreshEmbeddingEntryFormula, truth_exMem, Truth, truth_orderedPairFormula]
change (∃ q ∈ env d, ∃ a ∈ env z, ∃ b ∈ a,
b = ZFSet.pair (env p) q ∧ env z = ZFSet.pair q b) ↔ _
constructor
· rintro ⟨q, hq, _, _, b, _, rfl, eq⟩
exact ⟨q, hq, eq⟩
· rintro ⟨q, hq, eq⟩
refine ⟨q, hq, {q, ZFSet.pair (env p) q}, ?_, ZFSet.pair (env p) q,
ZFSet.mem_pair.2 (Or.inr rfl), rfl, eq⟩
rw [eq]
exact ZFSet.mem_pair.2 (Or.inr rfl)
end BoundedFormula
namespace TransitiveGround
open SetFormula
variable (M : TransitiveGround.{u})
theorem attachmentOldEmbeddingGraph_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
{P : ZFSet.{u}} (hP : P ∈ M.sets) : attachmentOldEmbeddingGraph P ∈ M.sets := by
let params := extend P Fin.elim0
have selected := M.bounded_separation_mem separation
(BoundedFormula.attachmentOldEmbeddingEntryFormula 0 1) params
(Fin.cases hP (fun i => Fin.elim0 i))
(M.prod_mem pairing unions powers separation hP hP)
have eq : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.attachmentOldEmbeddingEntryFormula 0 1) (extend z params))
(ZFSet.prod P P) = attachmentOldEmbeddingGraph P := by
apply ZFSet.ext
intro z
rw [attachmentOldEmbeddingGraph, ZFSet.mem_sep, ZFSet.mem_sep]
exact and_congr Iff.rfl (BoundedFormula.truth_attachmentOldEmbeddingEntryFormula 0 1 _)
exact eq ▸ selected
theorem attachmentFreshEmbeddingGraph_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
{P N : ZFSet.{u}} (hP : P ∈ M.sets) (hN : N ∈ M.sets) :
attachmentFreshEmbeddingGraph P N ∈ M.sets := by
let params := extend P (extend N Fin.elim0)
have selected := M.bounded_separation_mem separation
(BoundedFormula.attachmentFreshEmbeddingEntryFormula 0 1 2) params
(Fin.cases hP (Fin.cases hN (fun i => Fin.elim0 i)))
(M.prod_mem pairing unions powers separation hN
(M.taggedExtension_mem pairing unions powers separation hP hN))
have eq : ZFSet.sep (fun z => BoundedFormula.Truth
(BoundedFormula.attachmentFreshEmbeddingEntryFormula 0 1 2) (extend z params))
(ZFSet.prod N (taggedExtension P N)) = attachmentFreshEmbeddingGraph P N := by
apply ZFSet.ext
intro z
rw [attachmentFreshEmbeddingGraph, ZFSet.mem_sep, ZFSet.mem_sep]
exact and_congr Iff.rfl (BoundedFormula.truth_attachmentFreshEmbeddingEntryFormula 0 1 2 _)
exact eq ▸ selected
theorem attachmentTemplateEmbeddingGraph_mem
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
{P Q rQ ell a : ZFSet.{u}} (hP : P ∈ M.sets) (hQ : Q ∈ M.sets)
(hrQ : rQ ∈ M.sets) (hell : ell ∈ M.sets) (ha : a ∈ M.sets) :
attachmentTemplateEmbeddingGraph P Q rQ ell a ∈ M.sets :=
M.binary_union_mem pairing unions hell
(M.attachmentFreshEmbeddingGraph_mem pairing unions powers separation hP
(M.freshPart_mem separation hQ hrQ ha))
end TransitiveGround
end PartitionPilot.Forcing
end OAI
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