Theorem: sum_range_succ (PrimeNumberTheoremAnd/Wiener.lean)

State before:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
⊢ ∑ i ∈ Finset.range n, a (i + 1) = ∑ i ∈ Finset.range (n + 1), a i - a 0
Tactic:
have := Finset.sum_range_sub a n
State after:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ i ∈ Finset.range n, (a (i + 1) - a i) = a n - a 0
⊢ ∑ i ∈ Finset.range n, a (i + 1) = ∑ i ∈ Finset.range (n + 1), a i - a 0

State before:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ i ∈ Finset.range n, (a (i + 1) - a i) = a n - a 0
⊢ ∑ i ∈ Finset.range n, a (i + 1) = ∑ i ∈ Finset.range (n + 1), a i - a 0
Tactic:
rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this
State after:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ x ∈ Finset.range n, a (x + 1) = a n - a 0 + ∑ x ∈ Finset.range n, a x
⊢ ∑ i ∈ Finset.range n, a (i + 1) = ∑ i ∈ Finset.range (n + 1), a i - a 0

State before:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ x ∈ Finset.range n, a (x + 1) = a n - a 0 + ∑ x ∈ Finset.range n, a x
⊢ ∑ i ∈ Finset.range n, a (i + 1) = ∑ i ∈ Finset.range (n + 1), a i - a 0
Tactic:
rw [Finset.sum_range_succ, this]
State after:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ x ∈ Finset.range n, a (x + 1) = a n - a 0 + ∑ x ∈ Finset.range n, a x
⊢ a n - a 0 + ∑ x ∈ Finset.range n, a x = ∑ x ∈ Finset.range n, a x + a n - a 0

State before:
n✝ : ℕ
A a✝ b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
a : ℕ → ℝ
n : ℕ
this : ∑ x ∈ Finset.range n, a (x + 1) = a n - a 0 + ∑ x ∈ Finset.range n, a x
⊢ a n - a 0 + ∑ x ∈ Finset.range n, a x = ∑ x ∈ Finset.range n, a x + a n - a 0
Tactic:
ring
State after:
no goals