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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalRewriteSteps
namespace OAI
section
namespace PartitionPilot.Forcing
open SetRealization SetFormula
universe u
variable {J L : Type u} [Group J] (s : Presentation J) (t : Presentation L)
theorem externalWordAtomicCode_iff (xs ys : List (J ⊕ (L × Bool))) :
ExternalWordAtomicCode s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1)
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs)
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ↔
ExternalWordRewrite.Atomic xs ys := by
let a := externalGeneratorAlphabetPresentation s t
have oldOne (g : J) : finiteWordGraph a [.inl g] = wordSingleton (externalActorSymbol (s.encode g)) :=
finiteWordGraph_singleton a _
have oldTwo (g h : J) : finiteWordGraph a [.inl g, .inl h] =
wordDouble (externalActorSymbol (s.encode g)) (externalActorSymbol (s.encode h)) :=
finiteWordGraph_double a _ _
have signedTwo (l : L) (b c : Bool) : finiteWordGraph a [.inr (l, b), .inr (l, c)] =
wordDouble (externalLetterSymbol (t.encode l) b) (externalLetterSymbol (t.encode l) c) :=
finiteWordGraph_double a _ _
constructor
· intro atomic
rcases atomic with ⟨_, left, right⟩ | ⟨g, hg, h, hh, k, hk, product, left, right⟩ |
⟨l, hl, left, right⟩
· have xsEq := finiteWordGraph_injective a (left.trans (oldOne 1).symm)
have ysEq := finiteWordGraph_injective a (right.trans (finiteWordGraph_nil a).symm)
subst xs ys
exact .old_one
· obtain ⟨g, rfl⟩ := (s.mem_carrier g).1 hg
obtain ⟨h, rfl⟩ := (s.mem_carrier h).1 hh
obtain ⟨k, rfl⟩ := (s.mem_carrier k).1 hk
have productEq := ((s.product s).pair_mem_graph s (fun p => p.1 * p.2) (g, h) k).1 product
have xsEq := finiteWordGraph_injective a (left.trans (oldTwo g h).symm)
have ysEq := finiteWordGraph_injective a (right.trans (oldOne k).symm)
subst xs ys k
exact .old_mul g h
· obtain ⟨l, rfl⟩ := (t.mem_carrier l).1 hl
have ysEq := finiteWordGraph_injective a (right.trans (finiteWordGraph_nil a).symm)
subst ys
rcases left with left | left
· have xsEq := finiteWordGraph_injective a (left.trans (signedTwo l false true).symm)
subst xs
exact .cancel l false
· have xsEq := finiteWordGraph_injective a (left.trans (signedTwo l true false).symm)
subst xs
exact .cancel l true
· intro atomic
cases atomic with
| old_one => exact Or.inl ⟨s.encode_mem 1, oldOne 1, finiteWordGraph_nil a⟩
| old_mul g h =>
exact Or.inr (Or.inl ⟨s.encode g, s.encode_mem g, s.encode h, s.encode_mem h,
s.encode (g * h), s.encode_mem (g * h),
((s.product s).pair_mem_graph s (fun p => p.1 * p.2) (g, h) (g * h)).2 rfl,
oldTwo g h, oldOne (g * h)⟩)
| cancel l b =>
refine Or.inr (Or.inr ⟨t.encode l, t.encode_mem l, ?_, finiteWordGraph_nil a⟩)
cases b
· exact Or.inl (signedTwo l false true)
· exact Or.inr (signedTwo l true false)
namespace TransitiveGround
variable (M : TransitiveGround.{u})
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable (hAlphabet : (externalGeneratorAlphabetPresentation s t).carrier ∈ M.sets)
include pairing unions separation infinite hAlphabet
theorem externalWordStepCode_iff (xs ys : List (J ⊕ (L × Bool))) :
ExternalWordStepCode (M.finiteWords (externalGeneratorAlphabet s.carrier t.carrier))
(M.finiteWordAppendGraph (externalGeneratorAlphabet s.carrier t.carrier))
s.carrier t.carrier ((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1)
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs)
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ↔
ExternalWordRewrite.Step xs ys := by
let a := externalGeneratorAlphabetPresentation s t
have word (v : List (J ⊕ (L × Bool))) :=
finiteWordGraph_mem_finiteWords a M pairing unions separation infinite hAlphabet v
have decode (v : ZFSet.{u}) := finiteWords_exact_lists a M pairing unions separation infinite hAlphabet v
have append (v w : List (J ⊕ (L × Bool))) :
ZFSet.pair (ZFSet.pair (finiteWordGraph a v) (finiteWordGraph a w))
(finiteWordGraph a (v ++ w)) ∈ M.finiteWordAppendGraph a.carrier :=
(M.pair_mem_finiteWordAppendGraph _ _ _ _).2
⟨word v, word w, word (v ++ w), finiteWordGraph_append_code a v w⟩
have appendEq (v w z : List (J ⊕ (L × Bool)))
(entry : ZFSet.pair (ZFSet.pair (finiteWordGraph a v) (finiteWordGraph a w))
(finiteWordGraph a z) ∈ M.finiteWordAppendGraph a.carrier) : z = v ++ w :=
finiteWordGraph_injective a ((finiteWordAppendCode_iff a v w _).1
((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 entry).2.2.2)
constructor
· rintro ⟨pre, hp, post, hq, l, hl, r, hr, pl, _, pr, _, atomic, left, right, direction⟩
obtain ⟨pre, rfl⟩ := (decode pre).1 hp
obtain ⟨post, rfl⟩ := (decode post).1 hq
obtain ⟨l, rfl⟩ := (decode l).1 hl
obtain ⟨r, rfl⟩ := (decode r).1 hr
have leftEq := (finiteWordAppendCode_iff a pre l pl).1
((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 left).2.2.2
have rightEq := (finiteWordAppendCode_iff a pre r pr).1
((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 right).2.2.2
subst pl pr
refine ⟨pre, post, l, r, (externalWordAtomicCode_iff s t l r).1 atomic, ?_⟩
rcases direction with ⟨first, second⟩ | ⟨first, second⟩
· exact Or.inl ⟨appendEq _ _ _ first, appendEq _ _ _ second⟩
· exact Or.inr ⟨appendEq _ _ _ first, appendEq _ _ _ second⟩
· rintro ⟨pre, post, l, r, atomic, direction⟩
refine ⟨finiteWordGraph a pre, word pre, finiteWordGraph a post, word post,
finiteWordGraph a l, word l, finiteWordGraph a r, word r,
finiteWordGraph a (pre ++ l), word (pre ++ l),
finiteWordGraph a (pre ++ r), word (pre ++ r),
(externalWordAtomicCode_iff s t l r).2 atomic, append pre l, append pre r, ?_⟩
rcases direction with ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩
· exact Or.inl ⟨append (pre ++ l) post, append (pre ++ r) post⟩
· exact Or.inr ⟨append (pre ++ l) post, append (pre ++ r) post⟩
theorem pair_mem_externalWordSteps (xs ys : List (J ⊕ (L × Bool))) :
ZFSet.pair (finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs)
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ∈
M.externalWordSteps s.carrier t.carrier ((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) ↔
ExternalWordRewrite.Step xs ys := by
rw [externalWordSteps, pair_mem_externalWordStepGraph,
M.externalWordStepCode_iff s t pairing unions separation infinite hAlphabet xs ys]
exact ⟨fun h => h.2.2, fun h => ⟨
finiteWordGraph_mem_finiteWords _ M pairing unions separation infinite hAlphabet xs,
finiteWordGraph_mem_finiteWords _ M pairing unions separation infinite hAlphabet ys, h⟩⟩
end TransitiveGround
end PartitionPilot.Forcing
end
end OAI
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