Theorem: Arith.HSemiformula.emb_properWithParam (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
m : ℕ
p : HSemiformula ℒₒᵣ M n (𝚫, m)
⊢ ProperWithParamOn M (emb L p) ↔ ProperWithParamOn M p
Tactic:
rcases p
State after:
case mkDelta
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
m : ℕ
a✝¹ : HSemiformula ℒₒᵣ M n (𝚺, m)
a✝ : HSemiformula ℒₒᵣ M n (𝚷, m)
⊢ ProperWithParamOn M (emb L (a✝¹.mkDelta a✝)) ↔ ProperWithParamOn M (a✝¹.mkDelta a✝)

State before:
case mkDelta
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
m : ℕ
a✝¹ : HSemiformula ℒₒᵣ M n (𝚺, m)
a✝ : HSemiformula ℒₒᵣ M n (𝚷, m)
⊢ ProperWithParamOn M (emb L (a✝¹.mkDelta a✝)) ↔ ProperWithParamOn M (a✝¹.mkDelta a✝)
Tactic:
simp [ProperWithParamOn, emb]
State after:
no goals