| import OAI.SetTheory.PartitionPrinciple.Forcing.EvaluatedDomains |
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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 EqualityStep (R : P → Name s → Name s → Prop) (p : P) (σ τ : Name s) : Prop := |
| (∀ ρ r, Entry ρ σ r → ∀ q, q ≤ p → q ≤ r → |
| ∃ t, t ≤ q ∧ ∃ ν k, Entry ν τ k ∧ t ≤ k ∧ R t ρ ν) ∧ |
| (∀ ν k, Entry ν τ k → ∀ q, q ≤ p → q ≤ k → |
| ∃ t, t ≤ q ∧ ∃ ρ r, Entry ρ σ r ∧ t ≤ r ∧ R t ρ ν) |
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| def RelationMembership (R : P → Name s → Name s → Prop) (p : P) (σ τ : Name s) : Prop := |
| ∀ q, q ≤ p → ∃ t, t ≤ q ∧ ∃ ν r, Entry ν τ r ∧ t ≤ r ∧ R t σ ν |
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| theorem equalityStep_mono {R S : P → Name s → Name s → Prop} |
| (mono : ∀ p σ τ, R p σ τ → S p σ τ) {p : P} {σ τ : Name s} |
| (h : EqualityStep R p σ τ) : EqualityStep S p σ τ := by |
| constructor |
| · intro ρ r hr q qp qr |
| obtain ⟨t, tq, ν, k, hk, tk, rel⟩ := h.1 ρ r hr q qp qr |
| exact ⟨t, tq, ν, k, hk, tk, mono t ρ ν rel⟩ |
| · intro ν k hk q qp qk |
| obtain ⟨t, tq, ρ, r, hr, tr, rel⟩ := h.2 ν k hk q qp qk |
| exact ⟨t, tq, ρ, r, hr, tr, mono t ρ ν rel⟩ |
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| theorem forcesEq_step (p : P) (σ τ : Name s) : ForcesEq p σ τ ↔ EqualityStep ForcesEq p σ τ := by |
| rw [ForcesEq] |
| simp only [EqualityStep, exists_prop] |
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| theorem equalityStep_dense (R : P → Name s → Name s → Prop) (p : P) (σ τ : Name s) : |
| EqualityStep R p σ τ ↔ |
| (∀ ρ r, Entry ρ σ r → ∀ q, q ≤ p → q ≤ r → RelationMembership R q ρ τ) ∧ |
| (∀ ν k, Entry ν τ k → ∀ q, q ≤ p → q ≤ k → |
| RelationMembership (fun t ρ ν => R t ν ρ) q ν σ) := by |
| constructor |
| · intro h |
| exact ⟨fun ρ r hr q qp qr t tq => h.1 ρ r hr t (tq.trans qp) (tq.trans qr), |
| fun ν k hk q qp qk t tq => h.2 ν k hk t (tq.trans qp) (tq.trans qk)⟩ |
| · rintro ⟨left, right⟩ |
| exact ⟨fun ρ r hr q qp qr => left ρ r hr q qp qr q le_rfl, |
| fun ν k hk q qp qk => right ν k hk q qp qk q le_rfl⟩ |
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| theorem postfixed_equality_sound (D : NameDomain s) (R : P → Name s → Name s → Prop) |
| (post : ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, R p σ τ → EqualityStep R p σ τ) : |
| ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, R p σ τ → ForcesEq p σ τ := by |
| intro σ |
| induction σ using inductionOn with |
| | step σ ih => |
| intro hσ τ hτ p rel |
| have h := post σ hσ τ hτ p rel |
| rw [forcesEq_step] |
| constructor |
| · intro ρ r hr q qp qr |
| obtain ⟨t, tq, ν, k, hk, tk, hrel⟩ := h.1 ρ r hr q qp qr |
| exact ⟨t, tq, ν, k, hk, tk, ih ρ r hr (D.children hσ hr) ν (D.children hτ hk) t hrel⟩ |
| · intro ν k hk q qp qk |
| obtain ⟨t, tq, ρ, r, hr, tr, hrel⟩ := h.2 ν k hk q qp qk |
| exact ⟨t, tq, ρ, r, hr, tr, ih ρ r hr (D.children hσ hr) ν (D.children hτ hk) t hrel⟩ |
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| theorem prefixed_equality_complete (D : NameDomain s) (R : P → Name s → Name s → Prop) |
| (pre : ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, EqualityStep R p σ τ → R p σ τ) : |
| ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, ForcesEq p σ τ → R p σ τ := by |
| intro σ |
| induction σ using inductionOn with |
| | step σ ih => |
| intro hσ τ hτ p forced |
| have h := (forcesEq_step p σ τ).1 forced |
| apply pre σ hσ τ hτ p |
| constructor |
| · intro ρ r hr q qp qr |
| obtain ⟨t, tq, ν, k, hk, tk, hrel⟩ := h.1 ρ r hr q qp qr |
| exact ⟨t, tq, ν, k, hk, tk, ih ρ r hr (D.children hσ hr) ν (D.children hτ hk) t hrel⟩ |
| · intro ν k hk q qp qk |
| obtain ⟨t, tq, ρ, r, hr, tr, hrel⟩ := h.2 ν k hk q qp qk |
| exact ⟨t, tq, ρ, r, hr, tr, ih ρ r hr (D.children hσ hr) ν (D.children hτ hk) t hrel⟩ |
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| theorem fixed_equality_iff (D : NameDomain s) (R : P → Name s → Name s → Prop) |
| (fixed : ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, R p σ τ ↔ EqualityStep R p σ τ) |
| {σ τ : Name s} (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) (p : P) : |
| R p σ τ ↔ ForcesEq p σ τ := |
| ⟨postfixed_equality_sound D R (fun σ hσ τ hτ p => (fixed σ hσ τ hτ p).1) σ hσ τ hτ p, |
| prefixed_equality_complete D R (fun σ hσ τ hτ p => (fixed σ hσ τ hτ p).2) σ hσ τ hτ p⟩ |
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| end PartitionPilot.Forcing.Name |
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| end |
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| end OAI |
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