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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.SetTheory.PartitionPrinciple.Forcing.ConditionQuantifiers
import OAI.SetTheory.PartitionPrinciple.Geometry.AtomicEqualitySteps
namespace OAI
section
namespace PartitionPilot.Forcing.SetFormula
universe u
def atomicApplication {k m : Nat} (E : SetFormula (k + 3))
(params : Fin k → Fin m) (p x y : Fin m) : SetFormula m :=
reindex (extend p (extend x (extend y params))) E
theorem realize_atomicApplication (U : Set ZFSet.{u}) {k m : Nat} (E : SetFormula (k + 3))
(params : Fin k → Fin m) (p x y : Fin m) (env : Fin m → ZFSet.{u}) :
Realize U (atomicApplication E params p x y) env ↔
Realize U E (extend (env p) (extend (env x) (extend (env y) (fun i => env (params i))))) := by
rw [atomicApplication, realize_reindex, map_extend, map_extend, map_extend]
def membershipFormula {k m : Nat} (E : SetFormula (k + 3)) (c o : Fin k)
(params : Fin k → Fin m) (p σ τ : Fin m) : SetFormula m :=
allConditions (params c) <|
imp (pairMembershipFormula 0 p.succ (params o).succ) <|
exConditions (params c).succ <| ex <|
exConditions (params c).succ.succ.succ <|
.conj (pairMembershipFormula 1 0 τ.succ.succ.succ.succ) <|
.conj (pairMembershipFormula 2 3 (params o).succ.succ.succ.succ) <|
.conj (pairMembershipFormula 2 0 (params o).succ.succ.succ.succ) <|
atomicApplication E (fun j => (params j).succ.succ.succ.succ) 2 σ.succ.succ.succ.succ 1
end PartitionPilot.Forcing.SetFormula
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {P : Type u} [Preorder P] (s : Presentation P)
theorem realize_conditionLE (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(carrier : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets)
{m : Nat} (i j o : Fin m) (env : Fin m → ZFSet.{u}) (p q : P)
(hi : env i = s.encode p) (hj : env j = s.encode q) (ho : env o = s.conditionOrder) :
Realize M.sets (pairMembershipFormula i j o) env ↔ p ≤ q := by
have hp : env i ∈ M.sets := hi ▸ M.transitive carrier (s.encode_mem p)
have hq : env j ∈ M.sets := hj ▸ M.transitive carrier (s.encode_mem q)
have horder : env o ∈ M.sets := ho ▸ order
rw [M.realize_pairMembershipFormula pairing i j o env hp hq horder, hi, hj, ho,
s.pair_mem_conditionOrder]
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {P : Type u} [Preorder P] (s : Presentation P)
def RelationMeaning (D : Name.NameDomain s) (R : P → Name s → Name s → Prop) {k : Nat} (E : SetFormula (k + 3))
(constants : Fin k → ZFSet.{u}) : Prop :=
∀ (p : P) (σ τ : D.names),
Realize M.sets E (extend (s.encode p) (extend σ.val.val (extend τ.val.val constants))) ↔
R p σ.val τ.val
theorem realize_relationMembership (D : Name.NameDomain s)
(R : P → Name s → Name s → Prop)
(ground : ∀ τ ∈ D.names, τ.val ∈ M.sets)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(carrier : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets)
{k m : Nat} (E : SetFormula (k + 3)) (c o : Fin k) (params : Fin k → Fin m)
(pi si ti : Fin m) (env : Fin m → ZFSet.{u})
(hc : env (params c) = s.carrier) (ho : env (params o) = s.conditionOrder)
(meaning : M.RelationMeaning s D R E (fun j => env (params j)))
(p : P) (σ τ : D.names) (hp : env pi = s.encode p)
(hσ : env si = σ.val.val) (hτ : env ti = τ.val.val) :
Realize M.sets (membershipFormula E c o params pi si ti) env ↔
Name.RelationMembership R p σ.val τ.val := by
have hτM : env ti ∈ M.sets := hτ ▸ ground τ.val τ.property
rw [membershipFormula, M.realize_allConditions s _ _ env carrier hc]
change (∀ q : P, Realize M.sets _ (extend (s.encode q) env)) ↔
∀ q, q ≤ p → ∃ t, t ≤ q ∧ ∃ ν r,
Name.Entry ν τ.val r ∧ t ≤ r ∧ R t σ.val ν
apply forall_congr'
intro q
rw [realize_imp, M.realize_conditionLE s pairing carrier order 0 pi.succ (params o).succ
(extend (s.encode q) env) q p rfl hp ho]
apply forall_congr'
intro _
rw [M.realize_exConditions s (params c).succ _ (extend (s.encode q) env) carrier hc]
constructor
· rintro ⟨t, ht⟩
obtain ⟨v, hv, rest⟩ := (realize_ex M.sets _ _).1 ht
obtain ⟨r, entry, tq, tr, eq⟩ := (M.realize_exConditions s (params c).succ.succ.succ _
(extend v (extend (s.encode t) (extend (s.encode q) env))) carrier hc).1 rest
let e := extend (s.encode r) (extend v (extend (s.encode t) (extend (s.encode q) env)))
have raw := (M.realize_pairMembershipFormula pairing 1 0 ti.succ.succ.succ.succ e
hv (M.transitive carrier (s.encode_mem r)) hτM).1 entry
change ZFSet.pair v (s.encode r) ∈ env ti at raw
rw [hτ] at raw
let ν : Name s := ⟨v, Name.isName_of_pair_mem τ.val v r raw⟩
have hν : ν ∈ D.names := D.children τ.property raw
have forced := (realize_atomicApplication M.sets E
(fun j => (params j).succ.succ.succ.succ) 2 si.succ.succ.succ.succ 1 e).1 eq
change Realize M.sets E (extend (s.encode t) (extend (env si)
(extend v (fun j => env (params j))))) at forced
rw [hσ] at forced
exact ⟨t, (M.realize_conditionLE s pairing carrier order 2 3
(params o).succ.succ.succ.succ e t q rfl rfl ho).1 tq,
ν, r, raw, (M.realize_conditionLE s pairing carrier order 2 0
(params o).succ.succ.succ.succ e t r rfl rfl ho).1 tr,
(meaning t σ ⟨ν, hν⟩).1 forced⟩
· rintro ⟨t, tq, ν, r, entry, tr, forced⟩
have hν : ν ∈ D.names := D.children τ.property entry
refine ⟨t, ?_⟩
rw [realize_ex]
refine ⟨ν.val, ground ν hν, ?_⟩
apply (M.realize_exConditions s (params c).succ.succ.succ _
(extend ν.val (extend (s.encode t) (extend (s.encode q) env))) carrier hc).2
let e := extend (s.encode r) (extend ν.val (extend (s.encode t) (extend (s.encode q) env)))
refine ⟨r, ?_, ?_, ?_, ?_⟩
· apply (M.realize_pairMembershipFormula pairing 1 0 ti.succ.succ.succ.succ e
(ground ν hν) (M.transitive carrier (s.encode_mem r)) hτM).2
change ZFSet.pair ν.val (s.encode r) ∈ env ti
rw [hτ]
exact entry
· exact (M.realize_conditionLE s pairing carrier order 2 3
(params o).succ.succ.succ.succ e t q rfl rfl ho).2 tq
· exact (M.realize_conditionLE s pairing carrier order 2 0
(params o).succ.succ.succ.succ e t r rfl rfl ho).2 tr
· apply (realize_atomicApplication M.sets E
(fun j => (params j).succ.succ.succ.succ) 2 si.succ.succ.succ.succ 1 e).2
change Realize M.sets E (extend (s.encode t) (extend (env si)
(extend ν.val (fun j => env (params j)))))
rw [hσ]
exact (meaning t σ ⟨ν, hν⟩).2 forced
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.SetFormula
def swapEquality {k : Nat} (E : SetFormula (k + 3)) : SetFormula (k + 3) :=
atomicApplication E (fun j => j.succ.succ.succ) 0 2 1
def matchingDirectionFormula {k m : Nat} (E : SetFormula (k + 3)) (c o : Fin k)
(params : Fin k → Fin m) (p σ τ : Fin m) : SetFormula m :=
.all <| allConditions (params c).succ <|
imp (pairMembershipFormula 1 0 σ.succ.succ) <|
allConditions (params c).succ.succ <|
imp (.conj (pairMembershipFormula 0 p.succ.succ.succ (params o).succ.succ.succ)
(pairMembershipFormula 0 1 (params o).succ.succ.succ)) <|
membershipFormula E c o (fun j => (params j).succ.succ.succ) 0 2 τ.succ.succ.succ
def equalityStepFormula {k m : Nat} (E : SetFormula (k + 3)) (c o : Fin k)
(params : Fin k → Fin m) (p σ τ : Fin m) : SetFormula m :=
.conj (matchingDirectionFormula E c o params p σ τ)
(matchingDirectionFormula (swapEquality E) c o params p τ σ)
end PartitionPilot.Forcing.SetFormula
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {P : Type u} (s : Presentation P)
theorem relationMeaning_swap (D : Name.NameDomain s) (R : P → Name s → Name s → Prop)
{k : Nat} (E : SetFormula (k + 3)) (constants : Fin k → ZFSet.{u})
(meaning : M.RelationMeaning s D R E constants) :
M.RelationMeaning s D (fun p σ τ => R p τ σ) (swapEquality E) constants := by
intro p σ τ
rw [swapEquality, realize_atomicApplication]
exact meaning p τ σ
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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