Theorem: LO.FirstOrder.Rew.q_substs (Logic/FirstOrder/Basic/Syntax/Rew.lean)

State before:
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
⊢ (substs w).q = substs (#0 :> ⇑bShift ∘ w)
Tactic:
ext x
State after:
case hb
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : Fin (n + 1)
⊢ (substs w).q #x = (substs (#0 :> ⇑bShift ∘ w)) #x

case hf
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : ξ
⊢ (substs w).q &x = (substs (#0 :> ⇑bShift ∘ w)) &x

State before:
case hb
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : Fin (n + 1)
⊢ (substs w).q #x = (substs (#0 :> ⇑bShift ∘ w)) #x

case hf
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : ξ
⊢ (substs w).q &x = (substs (#0 :> ⇑bShift ∘ w)) &x
Tactic:
{ cases x using Fin.cases <;> simp }
State after:
case hf
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : ξ
⊢ (substs w).q &x = (substs (#0 :> ⇑bShift ∘ w)) &x

State before:
case hf
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : ξ
⊢ (substs w).q &x = (substs (#0 :> ⇑bShift ∘ w)) &x
Tactic:
{ simp }
State after:
no goals

State before:
case hb
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : Fin (n + 1)
⊢ (substs w).q #x = (substs (#0 :> ⇑bShift ∘ w)) #x
Tactic:
cases x using Fin.cases <;> simp
State after:
no goals

State before:
case hf
L : Language
L' : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ' : Type u_2
ξ₁ : Type u_3
ξ₂ : Type u_4
ξ₃ : Type u_5
n n₁ n₂ n₃ : ℕ
ω✝ ω : Rew L ξ₁ n₁ ξ₂ n₂
n' : ℕ
w : Fin n → Semiterm L ξ n'
x : ξ
⊢ (substs w).q &x = (substs (#0 :> ⇑bShift ∘ w)) &x
Tactic:
simp
State after:
no goals