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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | 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}
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)
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
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⟩
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]
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⟩
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⟩
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⟩
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⟩
end PartitionPilot.Forcing.Name
end
end OAI
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