Theorem: Arith.Model.DefinableFunction.term (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
t : Semiterm L M k
⊢ Hierarchy 𝚺 0 (“#0 = !!(Rew.bShift t)”)
Tactic:
simp
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
t : Semiterm L M k
⊢ DefinedWithParam (fun v => v 0 = (fun v => Semiterm.valm M v id t) fun x => v x.succ)
    (HSemiformula.mkSigma (“#0 = !!(Rew.bShift t)”) ⋯)
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
t : Semiterm L M k
v : Fin (k + 1) → M
⊢ (fun v => v 0 = (fun v => Semiterm.valm M v id t) fun x => v x.succ) v ↔
    (Semiformula.Evalm M v id) (HSemiformula.mkSigma (“#0 = !!(Rew.bShift t)”) ⋯).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
t : Semiterm L M k
v : Fin (k + 1) → M
⊢ (fun v => v 0 = (fun v => Semiterm.valm M v id t) fun x => v x.succ) v ↔
    (Semiformula.Evalm M v id) (HSemiformula.mkSigma (“#0 = !!(Rew.bShift t)”) ⋯).val
Tactic:
simp [Semiterm.val_bShift']
State after:
no goals