| 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 | |