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3277efa | 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 | import OAI.SetTheory.PartitionPrinciple.Forcing.EvaluatedDomains
namespace OAI
section
namespace PartitionPilot.Forcing.Name
open SetRealization
universe u
variable {P : Type u} [Preorder P] {s : Presentation P}
theorem forcesEq_sound (D : NameDomain s) (H : ConditionFilter P)
(generic : H.GenericFor (atomicFamily D)) :
∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, H.mem p → ForcesEq p σ τ →
eval {q | H.mem q} σ = eval {q | H.mem q} τ := by
intro σ
induction σ using inductionOn with
| step σ ih =>
intro hσ τ hτ p hp eq
apply ZFSet.ext
intro value
constructor
· intro hv
obtain ⟨ρ, r, entry, hr, old⟩ := (mem_eval {q | H.mem q} σ value).1 hv
obtain ⟨q, hq, qp, qr⟩ := H.directed hp hr
have hρ := D.children hσ entry
obtain ⟨t, ht, ν, k, target, tk, childEq⟩ := generic_mem_witness D H generic hρ hτ hq
(forcesEq_left eq entry qp qr)
have values := ih ρ r entry hρ ν (D.children hτ target) t ht childEq
exact (mem_eval {q | H.mem q} τ value).2
⟨ν, k, target, H.upward ht tk, values.symm.trans old⟩
· intro hv
obtain ⟨ν, k, entry, hk, old⟩ := (mem_eval {q | H.mem q} τ value).1 hv
obtain ⟨q, hq, qp, qk⟩ := H.directed hp hk
have hν := D.children hτ entry
obtain ⟨t, ht, ρ, r, target, tr, childEq⟩ := generic_mem_witness D H generic hν hσ hq
(forcesEq_right eq entry qp qk)
have values := ih ρ r target (D.children hσ target) ν hν t ht (forcesEq_symm childEq)
exact (mem_eval {q | H.mem q} σ value).2
⟨ρ, r, target, H.upward ht tr, values.trans old⟩
theorem forcesMem_sound (D : NameDomain s) (H : ConditionFilter P)
(generic : H.GenericFor (atomicFamily D)) {σ τ : Name s}
(hσ : σ ∈ D.names) (hτ : τ ∈ D.names) {p : P} (hp : H.mem p)
(h : ForcesMem p σ τ) : eval {q | H.mem q} σ ∈ eval {q | H.mem q} τ := by
obtain ⟨q, hq, ν, r, entry, qr, eq⟩ := generic_mem_witness D H generic hσ hτ hp h
have values := forcesEq_sound D H generic σ hσ ν (D.children hτ entry) q hq eq
exact (mem_eval {q | H.mem q} τ _).2 ⟨ν, r, entry, H.upward hq qr, values.symm⟩
end PartitionPilot.Forcing.Name
end
end OAI
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