| import OAI.SetTheory.PartitionPrinciple.Geometry.AtomicSoundness |
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| namespace OAI |
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| section |
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| namespace PartitionPilot.Forcing.Name |
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| open SetRealization |
| universe u |
| variable {P : Type u} [Preorder P] {s : Presentation P} |
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| def EqTruth (H : ConditionFilter P) (σ τ : Name s) : Prop := |
| eval {q | H.mem q} σ = eval {q | H.mem q} τ → ∃ p, H.mem p ∧ ForcesEq p σ τ |
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| def MemTruth (H : ConditionFilter P) (σ τ : Name s) : Prop := |
| eval {q | H.mem q} σ ∈ eval {q | H.mem q} τ → ∃ p, H.mem p ∧ ForcesMem p σ τ |
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| theorem memTruth_of_children (H : ConditionFilter P) (σ τ : Name s) |
| (children : ∀ ν r, Entry ν τ r → EqTruth H σ ν) : MemTruth H σ τ := by |
| intro hv |
| obtain ⟨ν, r, entry, hr, values⟩ := (mem_eval {q | H.mem q} τ _).1 hv |
| obtain ⟨p, hp, eq⟩ := children ν r entry values.symm |
| obtain ⟨q, hq, qp, qr⟩ := H.directed hp hr |
| exact ⟨q, hq, forcesMem_of_witness ⟨ν, r, entry, qr, forcesEq_mono eq qp⟩⟩ |
| |
| theorem eqTruth_of_children (D : NameDomain s) (H : ConditionFilter P) |
| (generic : H.GenericFor (atomicFamily D)) (σ τ : Name s) |
| (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) |
| (left : ∀ ρ r, Entry ρ σ r → MemTruth H ρ τ) |
| (right : ∀ ν k, Entry ν τ k → MemTruth H ν σ) : EqTruth H σ τ := by |
| intro values |
| obtain ⟨p, hp, eq | bad | bad⟩ := generic (EqDecisionWitness σ τ) |
| (Or.inr ⟨σ, τ, hσ, hτ, rfl⟩) (eqDecisionWitness_dense σ τ) |
| · exact ⟨p, hp, eq⟩ |
| · obtain ⟨ρ, r, entry, pr, neg⟩ := bad |
| have member : eval {q | H.mem q} ρ ∈ eval {q | H.mem q} τ := by |
| rw [← values] |
| exact (mem_eval {q | H.mem q} σ _).2 ⟨ρ, r, entry, H.upward hp pr, rfl⟩ |
| obtain ⟨q, hq, mem⟩ := left ρ r entry member |
| obtain ⟨t, _, tp, tq⟩ := H.directed hp hq |
| exact False.elim (neg t tp (forcesMem_mono mem tq)) |
| · obtain ⟨ν, k, entry, pk, neg⟩ := bad |
| have member : eval {q | H.mem q} ν ∈ eval {q | H.mem q} σ := by |
| rw [values] |
| exact (mem_eval {q | H.mem q} τ _).2 ⟨ν, k, entry, H.upward hp pk, rfl⟩ |
| obtain ⟨q, hq, mem⟩ := right ν k entry member |
| obtain ⟨t, _, tp, tq⟩ := H.directed hp hq |
| exact False.elim (neg t tp (forcesMem_mono mem tq)) |
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| theorem atomic_truth (D : NameDomain s) (H : ConditionFilter P) |
| (generic : H.GenericFor (atomicFamily D)) : |
| ∀ σ ∈ D.names, ∀ τ ∈ D.names, EqTruth H σ τ ∧ MemTruth H σ τ ∧ MemTruth H τ σ := by |
| intro σ |
| induction σ using inductionOn with |
| | step σ outer => |
| intro hσ τ |
| induction τ using inductionOn with |
| | step τ inner => |
| intro hτ |
| have eq : EqTruth H σ τ := eqTruth_of_children D H generic σ τ hσ hτ |
| (fun ρ r entry => (outer ρ r entry (D.children hσ entry) τ hτ).2.1) |
| (fun ν k entry => (inner ν k entry (D.children hτ entry)).2.2) |
| have mem : MemTruth H σ τ := memTruth_of_children H σ τ |
| (fun ν k entry => (inner ν k entry (D.children hτ entry)).1) |
| have reverse : MemTruth H τ σ := memTruth_of_children H τ σ (by |
| intro ρ r entry values |
| obtain ⟨p, hp, eq⟩ := (outer ρ r entry (D.children hσ entry) τ hτ).1 values.symm |
| exact ⟨p, hp, forcesEq_symm eq⟩) |
| exact ⟨eq, mem, reverse⟩ |
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| theorem eval_eq_iff_forcesEq (D : NameDomain s) (H : ConditionFilter P) |
| (generic : H.GenericFor (atomicFamily D)) {σ τ : Name s} |
| (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) : |
| eval {q | H.mem q} σ = eval {q | H.mem q} τ ↔ ∃ p, H.mem p ∧ ForcesEq p σ τ := by |
| constructor |
| · exact (atomic_truth D H generic σ hσ τ hτ).1 |
| · rintro ⟨p, hp, eq⟩ |
| exact forcesEq_sound D H generic σ hσ τ hτ p hp eq |
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| theorem eval_mem_iff_forcesMem (D : NameDomain s) (H : ConditionFilter P) |
| (generic : H.GenericFor (atomicFamily D)) {σ τ : Name s} |
| (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) : |
| eval {q | H.mem q} σ ∈ eval {q | H.mem q} τ ↔ ∃ p, H.mem p ∧ ForcesMem p σ τ := by |
| constructor |
| · exact (atomic_truth D H generic σ hσ τ hτ).2.1 |
| · rintro ⟨p, hp, mem⟩ |
| exact forcesMem_sound D H generic hσ hτ hp mem |
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| end PartitionPilot.Forcing.Name |
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| end |
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| end OAI |
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