Theorem: Arith.Model.Defined.to_definable (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
p : HSemisentence L k Γ
hP : Defined P p
⊢ DefinedWithParam P (HSemiformula.rew Rew.emb p)
Tactic:
match Γ with
| (𝚺, _) => intro; simp [hP.iff]
| (𝚷, _) => intro; simp [hP.iff]
| (𝚫, _) => exact ⟨
  fun v ↦ by rcases p; simpa [HSemiformula.rew] using hP.proper.rew Rew.emb v,
  by intro; simp [hP.df.iff]⟩
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
snd✝ : ℕ
p : HSemisentence L k (𝚺, snd✝)
hP : Defined P p
⊢ DefinedWithParam P (HSemiformula.rew Rew.emb p)
Tactic:
intro
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
snd✝ : ℕ
p : HSemisentence L k (𝚺, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb 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
snd✝ : ℕ
p : HSemisentence L k (𝚺, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb p).val
Tactic:
simp [hP.iff]
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
snd✝ : ℕ
p : HSemisentence L k (𝚷, snd✝)
hP : Defined P p
⊢ DefinedWithParam P (HSemiformula.rew Rew.emb p)
Tactic:
intro
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
snd✝ : ℕ
p : HSemisentence L k (𝚷, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb 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
snd✝ : ℕ
p : HSemisentence L k (𝚷, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb p).val
Tactic:
simp [hP.iff]
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
snd✝ : ℕ
p : HSemisentence L k (𝚫, snd✝)
hP : Defined P p
⊢ DefinedWithParam P (HSemiformula.rew Rew.emb p)
Tactic:
exact ⟨
fun v ↦ by rcases p; simpa [HSemiformula.rew] using hP.proper.rew Rew.emb v,
by intro; simp [hP.df.iff]⟩
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
snd✝ : ℕ
p : HSemisentence L k (𝚫, snd✝)
hP : Defined P p
v : Fin k → M
⊢ (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb p).sigma.val ↔
    (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb p).pi.val
Tactic:
rcases p
State after:
case mkDelta
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
snd✝ : ℕ
v : Fin k → M
a✝¹ : HSemiformula L Empty k (𝚺, snd✝)
a✝ : HSemiformula L Empty k (𝚷, snd✝)
hP : Defined P (a✝¹.mkDelta a✝)
⊢ (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb (a✝¹.mkDelta a✝)).sigma.val ↔
    (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb (a✝¹.mkDelta a✝)).pi.val

State before:
case mkDelta
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
snd✝ : ℕ
v : Fin k → M
a✝¹ : HSemiformula L Empty k (𝚺, snd✝)
a✝ : HSemiformula L Empty k (𝚷, snd✝)
hP : Defined P (a✝¹.mkDelta a✝)
⊢ (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb (a✝¹.mkDelta a✝)).sigma.val ↔
    (Semiformula.Evalm M v id) (HSemiformula.rew Rew.emb (a✝¹.mkDelta a✝)).pi.val
Tactic:
simpa [HSemiformula.rew] using hP.proper.rew Rew.emb v
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
snd✝ : ℕ
p : HSemisentence L k (𝚫, snd✝)
hP : Defined P p
⊢ FirstOrder.DefinedWithParam P (HSemiformula.rew Rew.emb p).val
Tactic:
intro
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
snd✝ : ℕ
p : HSemisentence L k (𝚫, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb 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
snd✝ : ℕ
p : HSemisentence L k (𝚫, snd✝)
hP : Defined P p
v✝ : Fin k → M
⊢ P v✝ ↔ (Semiformula.Evalm M v✝ id) (HSemiformula.rew Rew.emb p).val
Tactic:
simp [hP.df.iff]
State after:
no goals