Theorem: Arith.HSemiformula.ProperOn.ball (Arithmetization/Definability/Hierarchy.lean)

State before:
L : Language
inst✝⁷ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : Structure L M
inst✝ : Structure.ORing L M
k m : ℕ
t : Semiterm L Empty k
p : HSemisentence L (k + 1) (𝚫, m)
hp : ProperOn M p
⊢ ProperOn M (HSemiformula.ball t p)
Tactic:
intro e
State after:
L : Language
inst✝⁷ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : Structure L M
inst✝ : Structure.ORing L M
k m : ℕ
t : Semiterm L Empty k
p : HSemisentence L (k + 1) (𝚫, m)
hp : ProperOn M p
e : Fin k → M
⊢ (Semiformula.Evalbm M e) (HSemiformula.ball t p).sigma.val ↔ (Semiformula.Evalbm M e) (HSemiformula.ball t p).pi.val

State before:
L : Language
inst✝⁷ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : Structure L M
inst✝ : Structure.ORing L M
k m : ℕ
t : Semiterm L Empty k
p : HSemisentence L (k + 1) (𝚫, m)
hp : ProperOn M p
e : Fin k → M
⊢ (Semiformula.Evalbm M e) (HSemiformula.ball t p).sigma.val ↔ (Semiformula.Evalbm M e) (HSemiformula.ball t p).pi.val
Tactic:
simp [HSemiformula.ball, hp.iff]
State after:
no goals