Theorem: Arith.Model.DefinedFunction.graph_delta (Arithmetization/Definability/Hierarchy.lean)

State before:
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
m : ℕ
f : (Fin k → M) → M
p : HSemisentence L (k + 1) (𝚺, m)
h : DefinedFunction L (𝚺, m) f p
⊢ HSemiformula.ProperOn M (HSemiformula.graphDelta p)
Tactic:
cases' m with m <;> simp [HSemiformula.graphDelta]
State after:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
⊢ HSemiformula.ProperOn M
    (HSemiformula.mkDelta p
      (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”) ⋯))

State before:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
⊢ HSemiformula.ProperOn M
    (HSemiformula.mkDelta p
      (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”) ⋯))
Tactic:
intro e
State after:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
e : Fin (k + 1) → M
⊢ (Semiformula.Evalbm M e)
      (HSemiformula.mkDelta p
            (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”)
              ⋯)).sigma.val ↔
    (Semiformula.Evalbm M e)
      (HSemiformula.mkDelta p
            (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”)
              ⋯)).pi.val

State before:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
e : Fin (k + 1) → M
⊢ (Semiformula.Evalbm M e)
      (HSemiformula.mkDelta p
            (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”)
              ⋯)).sigma.val ↔
    (Semiformula.Evalbm M e)
      (HSemiformula.mkDelta p
            (HSemiformula.mkPi (“∀' (!((Rew.substs (#0 :> fun x => #x.succ.succ)).hom (HSemiformula.val p)) → #0 = #1)”)
              ⋯)).pi.val
Tactic:
simp [Empty.eq_elim, h.df.iff]
State after:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
e : Fin (k + 1) → M
⊢ (e 0 = f fun x => e x.succ) ↔ (f fun x => e x.succ) = e 0

State before:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
f : (Fin k → M) → M
m : ℕ
p : HSemisentence L (k + 1) (𝚺, m + 1)
h : DefinedFunction L (𝚺, m + 1) f p
e : Fin (k + 1) → M
⊢ (e 0 = f fun x => e x.succ) ↔ (f fun x => e x.succ) = e 0
Tactic:
rw [eq_comm]
State after:
no goals

State before:
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
m : ℕ
f : (Fin k → M) → M
p : HSemisentence L (k + 1) (𝚺, m)
h : DefinedFunction L (𝚺, m) f p
⊢ FirstOrder.Defined (fun v => v 0 = f fun x => v x.succ) (HSemiformula.graphDelta p).val
Tactic:
intro v
State after:
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
m : ℕ
f : (Fin k → M) → M
p : HSemisentence L (k + 1) (𝚺, m)
h : DefinedFunction L (𝚺, m) f p
v : Fin k.succ → M
⊢ (fun v => v 0 = f fun x => v x.succ) v ↔ (Semiformula.Evalbm M v) (HSemiformula.graphDelta p).val

State before:
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ : HierarchySymbol
k : ℕ
P Q : (Fin k → M) → Prop
m : ℕ
f : (Fin k → M) → M
p : HSemisentence L (k + 1) (𝚺, m)
h : DefinedFunction L (𝚺, m) f p
v : Fin k.succ → M
⊢ (fun v => v 0 = f fun x => v x.succ) v ↔ (Semiformula.Evalbm M v) (HSemiformula.graphDelta p).val
Tactic:
simp [h.df.iff]
State after:
no goals