Theorem: Arith.HSemiformula.ProperOn.extd (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 : (𝚫, m)-Semisentence n
h : ProperOn M p
⊢ ProperOn M (HSemiformula.extd L 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
m : ℕ
p : (𝚫, m)-Semisentence n
h : ProperOn M p
e : Fin n → M
⊢ (Semiformula.Evalbm M e) (HSemiformula.extd L p).sigma.val ↔ (Semiformula.Evalbm M e) (HSemiformula.extd L 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
m : ℕ
p : (𝚫, m)-Semisentence n
h : ProperOn M p
e : Fin n → M
⊢ (Semiformula.Evalbm M e) (HSemiformula.extd L p).sigma.val ↔ (Semiformula.Evalbm M e) (HSemiformula.extd L p).pi.val
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
e : Fin n → M
a✝¹ : HSemiformula ℒₒᵣ Empty n (𝚺, m)
a✝ : HSemiformula ℒₒᵣ Empty n (𝚷, m)
h : ProperOn M (a✝¹.mkDelta a✝)
⊢ (Semiformula.Evalbm M e) (HSemiformula.extd L (a✝¹.mkDelta a✝)).sigma.val ↔
    (Semiformula.Evalbm M e) (HSemiformula.extd L (a✝¹.mkDelta a✝)).pi.val

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
e : Fin n → M
a✝¹ : HSemiformula ℒₒᵣ Empty n (𝚺, m)
a✝ : HSemiformula ℒₒᵣ Empty n (𝚷, m)
h : ProperOn M (a✝¹.mkDelta a✝)
⊢ (Semiformula.Evalbm M e) (HSemiformula.extd L (a✝¹.mkDelta a✝)).sigma.val ↔
    (Semiformula.Evalbm M e) (HSemiformula.extd L (a✝¹.mkDelta a✝)).pi.val
Tactic:
simpa [HSemiformula.extd] using h.iff e
State after:
no goals