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import OAI.SetTheory.PartitionPrinciple.Definability.CodedTemplateShadowLaws
namespace OAI
namespace PartitionPilot.Forcing.CodedTemplateAttachment
universe u v w
variable {J : Type v} {P r μ N inside shadow upper : ZFSet.{u}}
variable {old : MeetOrderCode P r μ} {b : Attachment.Templates.{u+1,w,v} old.toMeetOrder J}
variable (d : CodedTemplateAttachment old b N inside shadow upper)
theorem encoded_embedding_order_iff {A r μ graph : ZFSet.{u}}
(source : MeetOrderCode A r μ)
(embedding : MeetEmbedding source.toMeetOrder b.extension)
(function : ZFSet.IsFunc A d.carrier graph)
(entries : ∀ x : {x : ZFSet.{u} // x ∈ A},
ZFSet.pair x.val (d.encode (embedding.toFun x)) ∈ graph)
{x y x' y' : ZFSet.{u}} (hx : ZFSet.pair x x' ∈ graph) (hy : ZFSet.pair y y' ∈ graph) :
ZFSet.pair x' y' ∈ d.order ↔ ZFSet.pair x y ∈ r := by
have hxA := (ZFSet.pair_mem_prod.1 (function.1 hx)).1
have hyA := (ZFSet.pair_mem_prod.1 (function.1 hy)).1
obtain ⟨w, _, uniqueX⟩ := function.2 x hxA
obtain ⟨v, _, uniqueY⟩ := function.2 y hyA
have eqX := (uniqueX x' hx).trans (uniqueX _ (entries ⟨x, hxA⟩)).symm
have eqY := (uniqueY y' hy).trans (uniqueY _ (entries ⟨y, hyA⟩)).symm
rw [eqX, eqY, d.le_encode]
exact embedding.le_iff ⟨x, hxA⟩ ⟨y, hyA⟩
theorem encoded_embedding_code {A r μ graph : ZFSet.{u}}
(source : MeetOrderCode A r μ)
(embedding : MeetEmbedding source.toMeetOrder b.extension)
(function : ZFSet.IsFunc A d.carrier graph)
(entries : ∀ x : {x : ZFSet.{u} // x ∈ A},
ZFSet.pair x.val (d.encode (embedding.toFun x)) ∈ graph) :
MeetEmbeddingCode A d.carrier r d.order μ d.meetGraph graph := by
refine ⟨⟨function, ?_⟩, ?_, ?_⟩
· intro x y z hx hy
have xA := (ZFSet.pair_mem_prod.1 (function.1 hx)).1
have yA := (ZFSet.pair_mem_prod.1 (function.1 hy)).1
have zL := (ZFSet.pair_mem_prod.1 (function.1 hx)).2
have refl := d.partialOrder.refl z zL
exact source.order.antisymm x xA y yA
((d.encoded_embedding_order_iff source embedding function entries hx hy).1 refl)
((d.encoded_embedding_order_iff source embedding function entries hy hx).1 refl)
· intro x y x' y' lower hx hy
exact (d.encoded_embedding_order_iff source embedding function entries hx hy).2 lower
· intro x y z x' y' z' meet hx hy hz
have hxA := (ZFSet.pair_mem_prod.1 (function.1 hx)).1
have hyA := (ZFSet.pair_mem_prod.1 (function.1 hy)).1
have hzA := (ZFSet.pair_mem_prod.1 (function.1 hz)).1
let X : {x : ZFSet.{u} // x ∈ A} := ⟨x, hxA⟩
let Y : {y : ZFSet.{u} // y ∈ A} := ⟨y, hyA⟩
let Z : {z : ZFSet.{u} // z ∈ A} := ⟨z, hzA⟩
have image_eq (t : {t : ZFSet.{u} // t ∈ A}) (v : ZFSet.{u})
(entry : ZFSet.pair t.val v ∈ graph) : v = d.encode (embedding.toFun t) := by
obtain ⟨w, _, unique⟩ := function.2 t.val t.property
exact (unique v entry).trans (unique _ (entries t)).symm
have meet_eq : source.toMeetOrder.meet X Y = Z := by
apply Subtype.ext
obtain ⟨w, _, unique⟩ := source.operation.2 (ZFSet.pair x y)
(ZFSet.pair_mem_prod.2 ⟨hxA, hyA⟩)
exact (unique _ (source.toMeetOrder_meet_entry X Y)).trans (unique z meet).symm
have mapped : embedding.toFun Z =
b.extension.meet (embedding.toFun X) (embedding.toFun Y) := by
rw [← meet_eq]
exact embedding.map_meet X Y
rw [image_eq X x' hx, image_eq Y y' hy, image_eq Z z' hz, mapped]
exact d.encoded_meet_entry (embedding.toFun X) (embedding.toFun Y)
theorem old_embedding_code :
MeetEmbeddingCode P d.carrier r d.order μ d.meetGraph (attachmentOldEmbeddingGraph P) := by
apply d.encoded_embedding_code old b.oldEmbedding
(attachmentOldEmbeddingGraph_isFunc P N)
intro p
exact (pair_mem_attachmentOldEmbeddingGraph P p.val p.val).2 ⟨p.property, rfl⟩
end PartitionPilot.Forcing.CodedTemplateAttachment
namespace PartitionPilot.Forcing.AttachmentBatchInputs
universe u
variable {P rP μP I Qs rs μs As Es Fs Ss : ZFSet.{u}}
variable (d : AttachmentBatchInputs P rP μP I Qs rs μs As Es Fs Ss)
include d
theorem batch_template_graph_entry (i : {i : ZFSet.{u} // i ∈ I})
(q : {q : ZFSet.{u} // q ∈ d.space i}) :
ZFSet.pair q.val (d.coding.encode (d.templates.represent i q)) ∈
batchTemplateGraph I P Qs rs As Es i.val := by
classical
apply (pair_mem_batchTemplateGraph I P Qs rs As Es i.val q.val _).2
apply (d.batch_template_entry_iff i q.val _).2
by_cases below : (d.fiber i).templateOrder.toMeetOrder.le q
⟨d.anchor i, (d.fiber i).anchor_mem⟩
· rw [d.templates.represent_old i q below]
exact Or.inl ((d.fiber i).baseMap_entry ⟨q, below⟩)
· rw [d.templates.represent_new i q below]
exact Or.inr ⟨(mem_freshPart _ _ _ _).2 ⟨q.property, below⟩, rfl⟩
theorem batch_template_graph_entry_iff (i : {i : ZFSet.{u} // i ∈ I})
(q : {q : ZFSet.{u} // q ∈ d.space i}) (y : ZFSet.{u}) :
ZFSet.pair q.val y ∈ batchTemplateGraph I P Qs rs As Es i.val ↔
y = d.coding.encode (d.templates.represent i q) := by
constructor
· intro entry
obtain ⟨v, _, unique⟩ := (d.batch_template_fiber_function i).2 q.val q.property
exact (unique y entry).trans (unique _ (d.batch_template_graph_entry i q)).symm
· intro equal
exact equal.symm ▸ d.batch_template_graph_entry i q
theorem batch_template_embedding_code (i : {i : ZFSet.{u} // i ∈ I}) :
MeetEmbeddingCode (d.space i) d.batchCarrier (d.rel i) d.batchOrder
(d.meet i) d.batchMeetGraph (batchTemplateGraph I P Qs rs As Es i.val) :=
d.coding.encoded_embedding_code (d.fiber i).templateOrder (d.templates.templateEmbedding i)
(d.batch_template_fiber_function i) (d.batch_template_graph_entry i)
theorem batch_template_order_iff (i : {i : ZFSet.{u} // i ∈ I})
{x y x' y' : ZFSet.{u}}
(hx : ZFSet.pair x x' ∈ batchTemplateGraph I P Qs rs As Es i.val)
(hy : ZFSet.pair y y' ∈ batchTemplateGraph I P Qs rs As Es i.val) :
ZFSet.pair x' y' ∈ d.batchOrder ↔ ZFSet.pair x y ∈ d.rel i :=
d.coding.encoded_embedding_order_iff (d.fiber i).templateOrder (d.templates.templateEmbedding i)
(d.batch_template_fiber_function i) (d.batch_template_graph_entry i) hx hy
theorem batch_template_old_agreement (i : {i : ZFSet.{u} // i ∈ I})
{q p : ZFSet.{u}} (entry : ZFSet.pair q p ∈ d.embed i) :
ZFSet.pair q p ∈ batchTemplateGraph I P Qs rs As Es i.val ∧
ZFSet.pair p p ∈ attachmentOldEmbeddingGraph P := by
have shape := ZFSet.pair_mem_prod.1 ((d.fiber i).base_function.1 entry)
exact ⟨(pair_mem_batchTemplateGraph I P Qs rs As Es i.val q p).2
((d.batch_template_entry_iff i q p).2 (Or.inl entry)),
(pair_mem_attachmentOldEmbeddingGraph P p p).2 ⟨shape.2, rfl⟩⟩
end PartitionPilot.Forcing.AttachmentBatchInputs
namespace PartitionPilot.Forcing.AttachmentBatchInputs
universe u
variable {P rP μP I Qs rs μs As Es Fs Ss : ZFSet.{u}}
variable (d : AttachmentBatchInputs P rP μP I Qs rs μs As Es Fs Ss)
noncomputable def batchFreshNode (i : {i : ZFSet.{u} // i ∈ I}) {q : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i)) : d.templates.Fresh :=
⟨i, ⟨⟨q, ((mem_freshPart _ _ _ q).1 hq).1⟩, ((mem_freshPart _ _ _ q).1 hq).2⟩⟩
theorem batch_old_lower_iff (i : {i : ZFSet.{u} // i ∈ I}) {q s p : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i)) (hp : p ∈ P)
(hs : ZFSet.pair (ZFSet.pair i.val q) s ∈ batchAttachmentShadowGraph I P Qs μs As Es) :
ZFSet.pair p (ZFSet.pair P (ZFSet.pair i.val q)) ∈ d.batchOrder ↔ ZFSet.pair p s ∈ rP := by
have eq := d.coding.shadow_value (d.batchFreshNode i hq) s hs
rw [eq]
exact d.coding.old_lower_iff ⟨p, hp⟩ (d.batchFreshNode i hq)
theorem batch_old_upper_iff (i : {i : ZFSet.{u} // i ∈ I}) {q p : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i)) (hp : p ∈ P) :
ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q)) p ∈ d.batchOrder ↔ p ∈ d.filter i :=
d.coding.old_upper_iff (d.batchFreshNode i hq) ⟨p, hp⟩
theorem batch_cross_lower_iff (i j : {i : ZFSet.{u} // i ∈ I}) {q t s : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i))
(ht : t ∈ freshPart (d.space j) (d.rel j) (d.anchor j)) (different : i ≠ j)
(hs : ZFSet.pair (ZFSet.pair j.val t) s ∈ batchAttachmentShadowGraph I P Qs μs As Es) :
ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q)) (ZFSet.pair P (ZFSet.pair j.val t)) ∈
d.batchOrder ↔ s ∈ d.filter i := by
have eq := d.coding.shadow_value (d.batchFreshNode j ht) s hs
rw [eq]
exact d.coding.cross_lower_iff (d.batchFreshNode i hq) (d.batchFreshNode j ht) different
theorem batch_shadow_meet_code (i : {i : ZFSet.{u} // i ∈ I}) {q p s v : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i)) (hp : p ∈ P)
(hs : ZFSet.pair (ZFSet.pair i.val q) s ∈ batchAttachmentShadowGraph I P Qs μs As Es)
(oldMeet : ZFSet.pair (ZFSet.pair s p) v ∈ μP)
(notAbove : ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q)) p ∉ d.batchOrder) :
ZFSet.pair (ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q)) p) v ∈ d.batchMeetGraph := by
let x := d.batchFreshNode i hq
let y : {p : ZFSet.{u} // p ∈ P} := ⟨p, hp⟩
have sameShadow := d.coding.shadow_value x s hs
obtain ⟨m, _, unique⟩ := d.oldOrder.operation.2
(ZFSet.pair (d.templates.shadow x).val p)
(ZFSet.pair_mem_prod.2 ⟨(d.templates.shadow x).property, hp⟩)
have sameMeet : v = (d.oldOrder.toMeetOrder.meet (d.templates.shadow x) y).val :=
(unique v (sameShadow ▸ oldMeet)).trans
(unique _ (d.oldOrder.toMeetOrder_meet_entry (d.templates.shadow x) y)).symm
exact sameMeet.symm ▸ d.coding.encoded_shadow_meet x y notAbove
theorem batch_cross_meet_code (i j : {i : ZFSet.{u} // i ∈ I}) {q t s z v : ZFSet.{u}}
(hq : q ∈ freshPart (d.space i) (d.rel i) (d.anchor i))
(ht : t ∈ freshPart (d.space j) (d.rel j) (d.anchor j)) (different : i ≠ j)
(hs : ZFSet.pair (ZFSet.pair i.val q) s ∈ batchAttachmentShadowGraph I P Qs μs As Es)
(hz : ZFSet.pair (ZFSet.pair j.val t) z ∈ batchAttachmentShadowGraph I P Qs μs As Es)
(oldMeet : ZFSet.pair (ZFSet.pair s z) v ∈ μP)
(notQT : ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q))
(ZFSet.pair P (ZFSet.pair j.val t)) ∉ d.batchOrder)
(notTQ : ZFSet.pair (ZFSet.pair P (ZFSet.pair j.val t))
(ZFSet.pair P (ZFSet.pair i.val q)) ∉ d.batchOrder) :
ZFSet.pair (ZFSet.pair (ZFSet.pair P (ZFSet.pair i.val q))
(ZFSet.pair P (ZFSet.pair j.val t))) v ∈ d.batchMeetGraph := by
let x := d.batchFreshNode i hq
let y := d.batchFreshNode j ht
have sameS := d.coding.shadow_value x s hs
have sameZ := d.coding.shadow_value y z hz
obtain ⟨m, _, unique⟩ := d.oldOrder.operation.2
(ZFSet.pair (d.templates.shadow x).val (d.templates.shadow y).val)
(ZFSet.pair_mem_prod.2 ⟨(d.templates.shadow x).property, (d.templates.shadow y).property⟩)
have sameMeet : v = (d.oldOrder.toMeetOrder.meet (d.templates.shadow x) (d.templates.shadow y)).val :=
(unique v (sameS ▸ sameZ ▸ oldMeet)).trans
(unique _ (d.oldOrder.toMeetOrder_meet_entry (d.templates.shadow x) (d.templates.shadow y))).symm
exact sameMeet.symm ▸ d.coding.encoded_cross_meet x y different notQT notTQ
end PartitionPilot.Forcing.AttachmentBatchInputs
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem batchAttachment_ground_codes
(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 rP μP I Qs rs μs As Es Fs Ss : ZFSet.{u}}
(d : AttachmentBatchInputs P rP μP I Qs rs μs As Es Fs Ss)
(hP : P ∈ M.sets) (hrP : rP ∈ M.sets) (hI : I ∈ M.sets)
(hQs : Qs ∈ M.sets) (hrs : rs ∈ M.sets) (hμs : μs ∈ M.sets)
(hAs : As ∈ M.sets) (hEs : Es ∈ M.sets) (hFs : Fs ∈ M.sets) :
d.batchCarrier ∈ M.sets ∧ d.batchOrder ∈ M.sets ∧ d.batchMeetGraph ∈ M.sets ∧
batchAttachmentShadowGraph I P Qs μs As Es ∈ M.sets ∧
batchAttachmentTemplateGraph I P Qs rs As Es ∈ M.sets ∧
attachmentOldEmbeddingGraph P ∈ M.sets := by
have hN := M.indexedAttachmentFresh_mem pairing unions powers separation hI hQs hrs hAs
have hL := M.taggedExtension_mem pairing unions powers separation hP hN
have inside := M.batchAttachmentInsideGraph_mem pairing unions powers separation hI hQs hrs
have shadow := M.batchAttachmentShadowGraph_mem pairing unions powers separation hI hP hQs hμs hAs hEs
have upper := M.batchAttachmentUpperGraph_mem pairing unions powers separation hI hP hQs hFs
have order := M.codedAttachmentOrder_mem pairing unions powers separation hP hN hrP inside shadow upper
exact ⟨hL, order, M.canonicalMeetGraph_mem pairing unions powers separation hL order,
shadow, M.batchAttachmentTemplateGraph_mem pairing unions powers separation hI hP hQs hrs hAs hEs,
M.attachmentOldEmbeddingGraph_mem pairing unions powers separation hP⟩
theorem exists_batchAttachment
(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 rP μP I Qs rs μs As Es Fs Ss : ZFSet.{u}}
(d : AttachmentBatchInputs P rP μP I Qs rs μs As Es Fs Ss)
(hP : P ∈ M.sets) (hrP : rP ∈ M.sets) (hI : I ∈ M.sets)
(hQs : Qs ∈ M.sets) (hrs : rs ∈ M.sets) (hμs : μs ∈ M.sets)
(hAs : As ∈ M.sets) (hEs : Es ∈ M.sets) (hFs : Fs ∈ M.sets)
(_hSs : Ss ∈ M.sets) :
∃ L ∈ M.sets, ∃ r ∈ M.sets, ∃ μ ∈ M.sets, ∃ shadow ∈ M.sets,
∃ representation ∈ M.sets, ∃ oldMap ∈ M.sets,
L = taggedExtension P (indexedAttachmentFresh I Qs rs As) ∧
r = codedAttachmentOrder P (indexedAttachmentFresh I Qs rs As) rP
(batchAttachmentInsideGraph I Qs rs) shadow (batchAttachmentUpperGraph I P Qs Fs) ∧
μ = canonicalMeetGraph L r ∧
shadow = batchAttachmentShadowGraph I P Qs μs As Es ∧
representation = batchAttachmentTemplateGraph I P Qs rs As Es ∧
oldMap = attachmentOldEmbeddingGraph P ∧
MeetOrderCode L r μ ∧
ZFSet.IsFunc (indexedAttachmentDomain I Qs) P shadow ∧
ZFSet.IsFunc (indexedAttachmentDomain I Qs) L representation ∧
MeetEmbeddingCode P L rP r μP μ oldMap ∧
(∀ p ∈ P, ZFSet.pair p p ∈ oldMap) ∧
(∀ i : {i : ZFSet.{u} // i ∈ I},
batchTemplateGraph I P Qs rs As Es i.val ∈ M.sets ∧
MeetEmbeddingCode (d.space i) L (d.rel i) r (d.meet i) μ
(batchTemplateGraph I P Qs rs As Es i.val)) := by
obtain ⟨hL, hr, hμ, hs, ht, ho⟩ := M.batchAttachment_ground_codes
pairing unions powers separation d hP hrP hI hQs hrs hμs hAs hEs hFs
refine ⟨d.batchCarrier, hL, d.batchOrder, hr, d.batchMeetGraph, hμ,
batchAttachmentShadowGraph I P Qs μs As Es, hs,
batchAttachmentTemplateGraph I P Qs rs As Es, ht,
attachmentOldEmbeddingGraph P, ho, rfl, rfl, rfl, rfl, rfl, rfl,
d.batch_meetOrder, d.batch_shadow_function, d.batch_template_function,
d.coding.old_embedding_code,
(fun p hp => (pair_mem_attachmentOldEmbeddingGraph P p p).2 ⟨hp, rfl⟩), ?_⟩
intro i
exact ⟨M.batchTemplateGraph_mem pairing unions powers separation
hI hP hQs hrs hAs hEs (M.transitive hI i.property), d.batch_template_embedding_code i⟩
end PartitionPilot.Forcing.TransitiveGround
end OAI
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