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namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
namespace Formula
def scope : Formula → ℕ
| .equal i j | .member i j => max (i+1) (j+1)
| .conj φ ψ => max φ.scope ψ.scope
| .neg φ => φ.scope
| .existsMem i φ => max (i+1) (φ.scope-1)
theorem realize_congr (φ : Formula) (d : ZFSet.{u}) (e f : ℕ → ZFSet.{u})
(h : ∀ i, i < φ.scope → e i = f i) : φ.Realize d e ↔ φ.Realize d f := by
induction φ generalizing e f with
| equal i j => simp only [Realize,h i (by simp [scope]),h j (by simp [scope])]
| member i j => simp only [Realize,h i (by simp [scope]),h j (by simp [scope])]
| conj φ ψ ihφ ihψ =>
exact and_congr (ihφ e f (fun i hi => h i (by simp only [scope]; omega)))
(ihψ e f (fun i hi => h i (by simp only [scope]; omega)))
| neg φ ih => exact not_congr (ih e f h)
| existsMem i φ ih =>
simp only [Realize,h i (by simp only [scope]; omega)]
apply exists_congr; intro x
apply and_congr_right; intro _
apply and_congr_right; intro _
apply ih; intro j hj
cases j with
| zero => rfl
| succ j => exact h j (by simp only [scope]; omega)
def width : Formula → ℕ
| .equal i j | .member i j => max (i+1) (j+1)
| .conj φ ψ => max φ.width ψ.width
| .neg φ => φ.width
| .existsMem i φ => max (i+1) φ.width
theorem scope_le_width (φ : Formula) : φ.scope ≤ φ.width := by
induction φ <;> simp only [scope,width] at * <;> omega
end Formula
namespace FiniteTuple
def takeEnv : ℕ → (ℕ → ZFSet.{u}) → List ZFSet.{u}
| 0,_ => []
| n+1,e => e 0 :: takeEnv n (fun i => e (i+1))
@[simp] theorem length_takeEnv (n : ℕ) (e : ℕ → ZFSet.{u}) : (takeEnv n e).length = n := by
induction n generalizing e with
| zero => rfl
| succ n ih => simp only [takeEnv,List.length_cons,ih]
theorem takeEnv_mem (n : ℕ) (e : ℕ → ZFSet.{u}) (h : ZFSet.{u}) (he : ∀ i, e i ∈ h) :
∀ x ∈ takeEnv n e, x ∈ h := by
induction n generalizing e with
| zero => simp [takeEnv]
| succ n ih =>
intro x hx
rcases List.mem_cons.mp hx with rfl|hx
· exact he 0
· exact ih (fun i => e (i+1)) (fun i => he (i+1)) x hx
theorem takeEnv_agree (n : ℕ) (e f : ℕ → ZFSet.{u}) (i : ℕ) (hi : i < n) :
prepend (takeEnv n e) f i = e i := by
induction n generalizing e i with
| zero => omega
| succ n ih => cases i with
| zero => rfl
| succ i => exact ih (fun j => e (j+1)) i (Nat.lt_of_succ_lt_succ hi)
def witnessBody (n i : ℕ) (φ : Formula) : Formula :=
.conj (.member 0 (i+1)) (relativizeBounded (n+1) φ)
theorem eval_witnessBody (n i : ℕ) (φ : Formula) (hn : i < n) (hφ : φ.scope ≤ n+1)
(d x : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(witnessBody n i φ).Eval (cons x (prepend (takeEnv n e) (fun _ => d))) ↔
x ∈ e i ∧ φ.Realize d (cons x e) := by
simp only [witnessBody,Formula.Eval,cons_zero,cons_succ,eval_relativizeBounded]
rw [takeEnv_agree n e _ i hn]
have hd : prepend (takeEnv n e) (fun _ => d) n = d := by
simpa only [length_takeEnv,Nat.add_zero] using prepend_tail (takeEnv n e) (fun _ => d) 0
rw [hd]
apply and_congr_right; intro _
apply φ.realize_congr; intro j hj
cases j with
| zero => rfl
| succ j => exact takeEnv_agree n e _ j (by omega)
def occurrences : Formula → List (ℕ × Formula)
| .equal _ _ | .member _ _ => []
| .conj φ ψ => occurrences φ ++ occurrences ψ
| .neg φ => occurrences φ
| .existsMem i φ => (i,φ) :: occurrences φ
def fragment (n : ℕ) (φ : Formula) : List Formula :=
(occurrences φ).map (fun p => witnessBody n p.1 p.2)
theorem occurrence_width (φ : Formula) (i : ℕ) (ψ : Formula)
(hm : (i,ψ) ∈ occurrences φ) : i < φ.width ∧ ψ.scope ≤ φ.width+1 := by
induction φ with
| equal => simp [occurrences] at hm
| member => simp [occurrences] at hm
| conj φ θ ihφ ihθ =>
rcases List.mem_append.mp hm with hm|hm
· have hh := ihφ hm; simp only [Formula.width]; omega
· have hh := ihθ hm; simp only [Formula.width]; omega
| neg φ ih => exact ih hm
| existsMem j φ ih =>
rcases List.mem_cons.mp hm with hp|hm
· obtain ⟨rfl,rfl⟩ := Prod.mk.inj hp
have hh := Formula.scope_le_width ψ
simp only [Formula.width]; omega
· have hh := ih hm; simp only [Formula.width]; omega
def Witnesses (d h : ZFSet.{u}) (φ : Formula) : Prop :=
∀ i ψ, (i,ψ) ∈ occurrences φ →
∀ e : ℕ → ZFSet.{u}, (∀ j, e j ∈ h) →
(∃ x ∈ d, x ∈ e i ∧ ψ.Realize d (cons x e)) →
∃ x ∈ h, x ∈ e i ∧ ψ.Realize d (cons x e)
theorem realize_of_witnesses (d h : ZFSet.{u}) (hhd : h ⊆ d) (φ : Formula)
(hw : Witnesses d h φ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ h) :
φ.Realize h e ↔ φ.Realize d e := by
induction φ generalizing e with
| equal => rfl
| member => rfl
| conj φ ψ ihφ ihψ =>
have hwφ : Witnesses d h φ := fun i θ hi => hw i θ (List.mem_append_left _ hi)
have hwψ : Witnesses d h ψ := fun i θ hi => hw i θ (List.mem_append_right _ hi)
exact and_congr (ihφ hwφ e he) (ihψ hwψ e he)
| neg φ ih => exact not_congr (ih hw e he)
| existsMem i φ ih =>
have hwφ : Witnesses d h φ := fun j θ hj => hw j θ (List.mem_cons_of_mem _ hj)
have env (x : ZFSet.{u}) (hx : x ∈ h) : ∀ j, cons x e j ∈ h := by
intro j; cases j with
| zero => exact hx
| succ j => exact he j
constructor
· rintro ⟨x,hx,hxi,hφ⟩
exact ⟨x,hhd hx,hxi,(ih hwφ (cons x e) (env x hx)).mp hφ⟩
· intro hφ
obtain ⟨x,hx,hxi,hψ⟩ := hw i φ (List.mem_cons_self) e he hφ
exact ⟨x,hx,hxi,(ih hwφ (cons x e) (env x hx)).mpr hψ⟩
end FiniteTuple
end TuringRigidity.BoundedSetTheory
end OAI
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