mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ ((expSeries 𝕜 (tsze R M) (n + 1)) fun x_1 ↦ x).snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)

Tactic:
[

Goal state after:
𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ ((expSeries 𝕜 (tsze R M) (n + 1)) fun x_1 ↦ x).snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ ((expSeries 𝕜 R n) fun x_1 ↦ x.fst) • x.snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)

Tactic:
[

Goal state after:
𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ ((expSeries 𝕜 R n) fun x_1 ↦ x.fst) • x.snd)
    (e • x.snd - ∑ i ∈ Finset.range 1, ((expSeries 𝕜 (tsze R M) i) fun x_1 ↦ x).snd)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
⊢ exp x = inl (exp x.fst) + inr (exp x.fst • x.snd)

Tactic:
[

Goal state after:
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
⊢ exp x = inl (exp x.fst) + inr (exp x.fst • x.snd)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: case neg
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
h : ¬Summable fun n ↦ (expSeries ℚ R n) fun x_1 ↦ x.fst
⊢ ¬Summable fun n ↦ (expSeries ℚ (tsze R M) n) fun x_1 ↦ x

Tactic:
[

Goal state after:
case neg
R : Type u_3
M : Type u_4
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : Module Rᵐᵒᵖ M
inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : IsTopologicalAddGroup M
inst✝⁵ : ContinuousSMul R M
inst✝⁴ : ContinuousSMul Rᵐᵒᵖ M
inst✝³ : Algebra ℚ R
inst✝² : Module ℚ M
inst✝¹ : T2Space R
inst✝ : T2Space M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
h : ¬Summable fun n ↦ (expSeries ℚ R n) fun x_1 ↦ x.fst
⊢ ¬Summable fun n ↦ (expSeries ℚ (tsze R M) n) fun x_1 ↦ x
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖(r₁, m₁) * (r₂, m₂)‖ ≤ ‖(r₁, m₁)‖ * ‖(r₂, m₂)‖

Tactic:
[

Goal state after:
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖(r₁, m₁) * (r₂, m₂)‖ ≤ ‖(r₁, m₁)‖ * ‖(r₂, m₂)‖
Module: Mathlib.Analysis.Normed.Algebra.MatrixExponential

Goal state before:
pp: m : Type u_1
𝔸 : Type u_5
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : m → 𝔸
⊢ exp (diagonal v) = diagonal (exp v)

Tactic:
[

Goal state after:
m : Type u_1
𝔸 : Type u_5
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : m → 𝔸
⊢ exp (diagonal v) = diagonal (exp v)
Module: Mathlib.Analysis.Normed.Algebra.MatrixExponential

Goal state before:
pp: m : Type u_1
n : Type u_2
𝔸 : Type u_5
inst✝⁸ : Fintype m
inst✝⁷ : DecidableEq m
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : m → Matrix n n 𝔸
⊢ exp (blockDiagonal v) = blockDiagonal (exp v)

Tactic:
[

Goal state after:
m : Type u_1
n : Type u_2
𝔸 : Type u_5
inst✝⁸ : Fintype m
inst✝⁷ : DecidableEq m
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq n
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : m → Matrix n n 𝔸
⊢ exp (blockDiagonal v) = blockDiagonal (exp v)
Module: Mathlib.Analysis.Normed.Algebra.MatrixExponential

Goal state before:
pp: m : Type u_1
n' : m → Type u_3
𝔸 : Type u_5
inst✝⁸ : Fintype m
inst✝⁷ : DecidableEq m
inst✝⁶ : (i : m) → Fintype (n' i)
inst✝⁵ : (i : m) → DecidableEq (n' i)
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : (i : m) → Matrix (n' i) (n' i) 𝔸
⊢ exp (blockDiagonal' v) = blockDiagonal' (exp v)

Tactic:
[

Goal state after:
m : Type u_1
n' : m → Type u_3
𝔸 : Type u_5
inst✝⁸ : Fintype m
inst✝⁷ : DecidableEq m
inst✝⁶ : (i : m) → Fintype (n' i)
inst✝⁵ : (i : m) → DecidableEq (n' i)
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : T2Space 𝔸
inst✝ : Algebra ℚ 𝔸
v : (i : m) → Matrix (n' i) (n' i) 𝔸
⊢ exp (blockDiagonal' v) = blockDiagonal' (exp v)
Module: Mathlib.Analysis.Normed.Algebra.MatrixExponential

Goal state before:
pp: m : Type u_1
𝔸 : Type u_5
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : CommRing 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : Algebra ℚ 𝔸
inst✝ : T2Space 𝔸
A : Matrix m m 𝔸
⊢ exp Aᵀ = (exp A)ᵀ

Tactic:
[

Goal state after:
m : Type u_1
𝔸 : Type u_5
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : CommRing 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : IsTopologicalRing 𝔸
inst✝¹ : Algebra ℚ 𝔸
inst✝ : T2Space 𝔸
A : Matrix m m 𝔸
⊢ exp Aᵀ = (exp A)ᵀ
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)! + (-z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
⊢ HasSum (fun n ↦ (z * I) ^ (2 * n) / ↑(2 * n)!) ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)

Tactic:
[

Goal state after:
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)! + (-z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
⊢ HasSum (fun n ↦ (z * I) ^ (2 * n) / ↑(2 * n)!) ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2

Tactic:
[

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: a : ℍ
⊢ ‖star a‖ = ‖a‖

Tactic:
[

Goal state after:
a : ℍ
⊢ ‖star a‖ = ‖a‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)! - (z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
⊢ HasSum (fun n ↦ (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)

Tactic:
[

Goal state after:
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)! - (z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
⊢ HasSum (fun n ↦ (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: x✝¹ x✝ : ℍ
⊢ ‖x✝¹ * x✝‖ = ‖x✝¹‖ * ‖x✝‖

Tactic:
[

Goal state after:
x✝¹ x✝ : ℍ
⊢ ‖x✝¹ * x✝‖ = ‖x✝¹‖ * ‖x✝‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! - (z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) * I / 2 +
      ((-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! - (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) * I / 2

Tactic:
[

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! - (z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) * I / 2 +
      ((-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! - (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) * I / 2
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: x : ℍ
⊢ √(⟪(equivTuple ℝ) x 0, (equivTuple ℝ) x 0⟫ + ⟪(equivTuple ℝ) x 1, (equivTuple ℝ) x 1⟫ +
          ⟪(equivTuple ℝ) x 2, (equivTuple ℝ) x 2⟫ +
        ⟪(equivTuple ℝ) x 3, (equivTuple ℝ) x 3⟫) =
    √(x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)

Tactic:
[

Goal state after:
x : ℍ
⊢ √(⟪(equivTuple ℝ) x 0, (equivTuple ℝ) x 0⟫ + ⟪(equivTuple ℝ) x 1, (equivTuple ℝ) x 1⟫ +
          ⟪(equivTuple ℝ) x 2, (equivTuple ℝ) x 2⟫ +
        ⟪(equivTuple ℝ) x 3, (equivTuple ℝ) x 3⟫) =
    √(x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
⊢ (fun n ↦ (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) = fun n ↦ (z * I) ^ (2 * n) / ↑(2 * n)!

Tactic:
[

Goal state after:
z : ℂ
⊢ (fun n ↦ (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) = fun n ↦ (z * I) ^ (2 * n) / ↑(2 * n)!
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
⊢ (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) = fun n ↦ (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I

Tactic:
[

Goal state after:
z : ℂ
⊢ (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) = fun n ↦ (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖

Tactic:
[

Goal state after:
case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * ‖q‖ ^ (2 * n + 1) / k / ‖q‖
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
hq : q.re = 0
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • q ^ n) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)

Tactic:
[

Goal state after:
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • q ^ n) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto (fun x_1 ↦ φ x x_1) (cobounded (ℝ × ℝ)) (cobounded F)

Tactic:
[

Goal state after:
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto (fun x_1 ↦ φ x x_1) (cobounded (ℝ × ℝ)) (cobounded F)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:

Tactic:
‖y.1‖ * c
        _ ≤ ‖y.1‖ * ‖x - algebraMap ℝ F (y.1⁻¹ * y.2)‖ := by gcongr; exact hbd _
        _ = ‖y.1 • x - algebraMap ℝ F y.2‖ := by
          simp only [← norm_smul, smul_sub, smul_smul, Algebra.algebraMap_eq_smul_one]
          simp_all

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y
h1 : f 1 = 0
⊢ False

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y
h1 : f 1 = 0
⊢ False
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_zero : f 0 = 0
⊢ seminormFromBounded' f 0 = 0

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_zero : f 0 = 0
⊢ seminormFromBounded' f 0 = 0
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_zero : f 0 = 0
⊢ seminormFromBounded' f 0 = 0

Tactic:
seminormFromBounded',

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_zero : f 0 = 0
⊢ ⨆ y, f (0 * y) / f y = 0
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : f (x * 1) ≤ c * f x * f 1
hx : 0 < f x
⊢ 0 ≤ c

Tactic:
rw [mul_one, show c * f x * f 1 = c * f 1 * f x by ring, le_mul_iff_one_le_left hx] at f_mul

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : 1 ≤ c * f 1
hx : 0 < f x
⊢ 0 ≤ c
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h1 : ¬f 1 = 0
| f x / f 1

Tactic:
rw [← mul_one x]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
h1 : ¬f 1 = 0
| f (x * 1) / f 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : f z = 0
hxyz : f (z * (x * y)) = 0
⊢ f (x * y * z) / f z ≤ f (x * z) * seminormFromBounded' f y / f z

Tactic:
[

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : f z = 0
hxyz : f (z * (x * y)) = 0
⊢ f (x * y * z) / f z ≤ f (x * z) * seminormFromBounded' f y / f z
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ seminormFromBounded' f 1 = 1

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ seminormFromBounded' f 1 = 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ seminormFromBounded' f 1 = 1

Tactic:
seminormFromBounded',

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ ⨆ y, f (1 * y) / f y = 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1

Tactic:
[

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1

Tactic:
seminormFromBounded',

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ ⨆ y, f (1 * y) / f y ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfx : f ((x + y) * z) / f z ≤ f (x * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

Tactic:
exact Or.inl <| le_ciSup_of_le (seminormFromBounded_bddAbove_range f_nonneg f_mul x) z hfx

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
⊢ seminormFromBounded' f x = f x

Tactic:
[

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
⊢ seminormFromBounded' f x = f x
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
⊢ seminormFromBounded' f x = f x

Tactic:
seminormFromBounded',

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
⊢ ⨆ y, f (x * y) / f y = f x
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(B.repr ((∑ i, (B.repr x) i • B i) * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y

Tactic:
sum_mul,

Goal state after:
case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(B.repr (∑ i, (B.repr x) i • B i * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y