File size: 4,016 Bytes
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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundDefinableSelections
namespace OAI
namespace PartitionPilot.Forcing.SetFormula
def forcingWitnessFormula {k n m : Nat} (A : ForcingPrimitives k) (φ : SetFormula (n + 2))
(params : Fin k → Fin m) (names : Fin n → Fin m) (z y : Fin m) : SetFormula m :=
ex (exConditions (params A.carrier).succ
(.conj (orderedPairFormula z.succ.succ 1 0)
(.conj (domainApplication A.domain (fun j => (params j).succ.succ) 1)
(.conj (domainApplication A.domain (fun j => (params j).succ.succ) y.succ.succ)
(forcingTranslation A φ (fun j => (params j).succ.succ) 0
(extend 1 (extend y.succ.succ (fun j => (names j).succ.succ))))))))
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_forcingWitnessFormula (D : Name.NameDomain s)
(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 n m : Nat} (A : ForcingPrimitives k) (φ : SetFormula (n + 2))
(params : Fin k → Fin m) (names : Fin n → Fin m) (zi yi : Fin m) (env : Fin m → ZFSet.{u})
(hc : env (params A.carrier) = s.carrier) (ho : env (params A.order) = s.conditionOrder)
(atomic : M.EqualityMeaning s D A.equality (fun j => env (params j)))
(domain : M.DomainMeaning s D A.domain (fun j => env (params j)))
(nameEnv : Assignment D n) (hnames : ∀ i, env (names i) = (nameEnv i).val.val)
(hz : env zi ∈ M.sets) (hy : env yi ∈ M.sets) :
Realize M.sets (forcingWitnessFormula A φ params names zi yi) env ↔
∃ σ : D.names, ∃ p, env zi = ZFSet.pair σ.val.val (s.encode p) ∧
∃ ρ : D.names, ρ.val.val = env yi ∧ Forces D φ (extend σ (extend ρ nameEnv)) p := by
rw [forcingWitnessFormula, realize_ex]
constructor
· rintro ⟨x, hx, body⟩
obtain ⟨p, shape, source, target, forced⟩ := (M.realize_exConditions s (params A.carrier).succ _
(extend x env) carrier hc).1 body
rw [realize_domainApplication] at source target
obtain ⟨σ, eq⟩ := (domain x hx).1 source
obtain ⟨ρ, eqρ⟩ := (domain (env yi) hy).1 target
let e := extend (s.encode p) (extend x env)
have raw := (M.realize_orderedPairFormula pairing zi.succ.succ 1 0 e hz hx
(M.transitive carrier (s.encode_mem p))).1 shape
refine ⟨σ, p, raw.trans (congrArg (fun z => ZFSet.pair z (s.encode p)) eq.symm), ρ, eqρ, ?_⟩
exact (M.realize_forcingTranslation s D ground pairing carrier order A φ
(fun j => (params j).succ.succ) 0 (extend 1 (extend yi.succ.succ (fun j => (names j).succ.succ)))
e hc ho atomic domain p (extend σ (extend ρ nameEnv)) rfl
(Fin.cases eq.symm (Fin.cases eqρ.symm hnames))).1 forced
· rintro ⟨σ, p, shape, ρ, eqρ, forced⟩
refine ⟨σ.val.val, ground σ.val σ.property, ?_⟩
apply (M.realize_exConditions s (params A.carrier).succ _ (extend σ.val.val env) carrier hc).2
let e := extend (s.encode p) (extend σ.val.val env)
refine ⟨p, ?_, ?_, ?_, ?_⟩
· exact (M.realize_orderedPairFormula pairing zi.succ.succ 1 0 e hz (ground σ.val σ.property)
(M.transitive carrier (s.encode_mem p))).2 shape
· rw [realize_domainApplication]
exact (domain σ.val.val (ground σ.val σ.property)).2 ⟨σ, rfl⟩
· rw [realize_domainApplication]
exact (domain (env yi) hy).2 ⟨ρ, eqρ⟩
· exact (M.realize_forcingTranslation s D ground pairing carrier order A φ
(fun j => (params j).succ.succ) 0 (extend 1 (extend yi.succ.succ (fun j => (names j).succ.succ)))
e hc ho atomic domain p (extend σ (extend ρ nameEnv)) rfl
(Fin.cases rfl (Fin.cases eqρ.symm hnames))).2 forced
end PartitionPilot.Forcing.TransitiveGround
end OAI
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