Theorem: LO.FirstOrder.Arith.Model.Definable.bex_le₀ (Arithmetization/Definability/Definability.lean)

State before:
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
h : Definable L Γ fun w => P (fun x => w x.succ) (w 0)
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x
Tactic:
rcases hf.bounded with ⟨bf, hbf⟩
State after:
case mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
h : Definable L Γ fun w => P (fun x => w x.succ) (w 0)
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x

State before:
case mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
h : Definable L Γ fun w => P (fun x => w x.succ) (w 0)
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x
Tactic:
rcases hf.definable with ⟨f_graph, hf_graph⟩
State after:
case mk.intro.mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
h : Definable L Γ fun w => P (fun x => w x.succ) (w 0)
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x

State before:
case mk.intro.mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
h : Definable L Γ fun w => P (fun x => w x.succ) (w 0)
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x
Tactic:
rcases h with ⟨p, hp⟩
State after:
case mk.intro.mk.intro.mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
p : HSemiformula L M k.succ Γ
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x

State before:
case mk.intro.mk.intro.mk.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
p : HSemiformula L M k.succ Γ
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
this :
  DefinedWithParam (fun v => ∃ x ≤ Semiterm.valm M v id bf, x = f v ∧ ∃ y ≤ x, P v y)
    (HSemiformula.bex (‘(!!bf + 1)’)
      (f_graph ⋏ HSemiformula.bex (‘(#0 + 1)’) (HSemiformula.rew (Rew.substs (#0 :> fun i => #i.succ.succ)) p)))
⊢ Definable L Γ fun v => ∃ x ≤ f v, P v x
Tactic:
exact .of_iff _ (fun v ↦ ⟨fun h ↦ ⟨f v, hbf v, rfl, h⟩, by rintro ⟨y, hy, rfl, h⟩; exact h⟩) ⟨_, this⟩
State after:
no goals

State before:
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
p : HSemiformula L M k.succ Γ
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
this :
  DefinedWithParam (fun v => ∃ x ≤ Semiterm.valm M v id bf, x = f v ∧ ∃ y ≤ x, P v y)
    (HSemiformula.bex (‘(!!bf + 1)’)
      (f_graph ⋏ HSemiformula.bex (‘(#0 + 1)’) (HSemiformula.rew (Rew.substs (#0 :> fun i => #i.succ.succ)) p)))
v : Fin k → M
⊢ (∃ x ≤ Semiterm.valm M v id bf, x = f v ∧ ∃ y ≤ x, P v y) → ∃ x ≤ f v, P v x
Tactic:
rintro ⟨y, hy, rfl, h⟩
State after:
case intro.intro.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
p : HSemiformula L M k.succ Γ
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
this :
  DefinedWithParam (fun v => ∃ x ≤ Semiterm.valm M v id bf, x = f v ∧ ∃ y ≤ x, P v y)
    (HSemiformula.bex (‘(!!bf + 1)’)
      (f_graph ⋏ HSemiformula.bex (‘(#0 + 1)’) (HSemiformula.rew (Rew.substs (#0 :> fun i => #i.succ.succ)) p)))
v : Fin k → M
hy : f v ≤ Semiterm.valm M v id bf
h : ∃ y ≤ f v, P v y
⊢ ∃ x ≤ f v, P v x

State before:
case intro.intro.intro
k : ℕ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
P✝ Q : (Fin k → M) → Prop
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableBoundedFunction L Γ f
bf : Semiterm L M k
hbf : ∀ (v : Fin k → M), f v ≤ Semiterm.valm M v id bf
f_graph : HSemiformula L M (k + 1) Γ
hf_graph : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) f_graph
p : HSemiformula L M k.succ Γ
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
this :
  DefinedWithParam (fun v => ∃ x ≤ Semiterm.valm M v id bf, x = f v ∧ ∃ y ≤ x, P v y)
    (HSemiformula.bex (‘(!!bf + 1)’)
      (f_graph ⋏ HSemiformula.bex (‘(#0 + 1)’) (HSemiformula.rew (Rew.substs (#0 :> fun i => #i.succ.succ)) p)))
v : Fin k → M
hy : f v ≤ Semiterm.valm M v id bf
h : ∃ y ≤ f v, P v y
⊢ ∃ x ≤ f v, P v x
Tactic:
exact h
State after:
no goals