mathlib-initiative/mathlib-tactics huggingface dataset:
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
m : M
⊢ inl 1 + inr m = 1 + inr m

pp: case hx
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
m : M
⊢ MulOpposite.op (inr m).fst • (inr m).snd = (inr m).fst • (inr m).snd

Tactic:
inl_one

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
m : M
⊢ 1 + inr m = 1 + inr m

case hx
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
m : M
⊢ MulOpposite.op (inr m).fst • (inr m).snd = (inr m).fst • (inr m).snd
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹ : SeminormedRing R
inst✝ : SeminormedAddCommGroup M
x : tsze R M
⊢ ‖WithLp.toLp 1 x‖ = ‖x.fst‖ + ‖x.snd‖

Tactic:
WithLp.prod_norm_eq_add (by norm_num)

Goal state after:
R : Type u_3
M : Type u_4
inst✝¹ : SeminormedRing R
inst✝ : SeminormedAddCommGroup M
x : tsze R M
⊢ (‖(WithLp.toLp 1 x).fst‖ ^ ENNReal.toReal 1 + ‖(WithLp.toLp 1 x).snd‖ ^ ENNReal.toReal 1) ^ (1 / ENNReal.toReal 1) =
    ‖x.fst‖ + ‖x.snd‖
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
⊢ ∃ z, ‖x - (algebraMap ℂ F) z‖ = 0

Tactic:
by
  -- there is a minimizing `z : ℂ`; get it.
  obtain ⟨z, hz⟩ := exists_isMinOn_norm_sub_smul ℂ x
  set M := ‖x - algebraMap ℂ F z‖ with hM
  rcases eq_or_lt_of_le (show 0 ≤ M from norm_nonneg _) with hM₀ | hM₀
    -- minimum is zero: nothing to do
  · exact ⟨z, hM₀.symm⟩
  -- otherwise, use the result from above that `z ↦ ‖x - algebraMap ℂ F z‖` is constant
  -- to derive a contradiction.
  by_contra! H
  have key := norm_sub_eq_norm_sub_of_isMinOn hz H (‖x‖ + M + 1)
  rw [← hM, norm_sub_rev] at key
  replace key := (norm_sub_norm_le ..).trans_eq key
  rw [norm_algebraMap, norm_one, mul_one] at key
  norm_cast at key
  rw [Real.norm_eq_abs, abs_of_nonneg (by positivity)] at key
  linarith only [key]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n : ℕ
⊢ ∃ m < n + 1, p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)

Tactic:
obtain ⟨m, hm_lt, hm⟩ := IsNonarchimedean.add_pow_le hna n x y

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n m : ℕ
hm_lt : m < n + 1
hm : p ((x + y) ^ n) ≤ p (x ^ m) * p (y ^ (n - m))
⊢ ∃ m < n + 1, p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
f : Filter ι
u v : ι → ℝ
inst✝ : f.NeBot
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u * v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f fun x ↦ u x * v x
a : ℝ
a0 : a ≥ 0
au : a < liminf u f
b : ℝ
b0 : b ≥ 0
⊢ b < liminf v f → a * b ≤ liminf (u * v) f

Tactic:
bv

Goal state after:
ι : Type u_1
f : Filter ι
u v : ι → ℝ
inst✝ : f.NeBot
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u * v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f fun x ↦ u x * v x
a : ℝ
a0 : a ≥ 0
au : a < liminf u f
b : ℝ
b0 : b ≥ 0
bv : b < liminf v f
⊢ a * b ≤ liminf (u * v) f
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_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z

Tactic:
← add_div,

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_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ (f (x * z) + f (y * z)) / f z
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
⊢ Tendsto (fun n ↦ (L + ε) ^ (-(↑(n % m1) / ↑n)) * (μ x ^ (n % m1)) ^ (1 * 1 / ↑n)) atTop (𝓝 (1 * 1))

Tactic:
have h_exp : Tendsto (fun n : ℕ => ((n % m1 : ℕ) : ℝ) / (n : ℝ)) atTop (𝓝 0) :=
    tendsto_mod_div_atTop_nhds_zero_nat hm1

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ (L + ε) ^ (-(↑(n % m1) / ↑n)) * (μ x ^ (n % m1)) ^ (1 * 1 / ↑n)) atTop (𝓝 (1 * 1))
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
h : μ (x ^ (↑m1 * (n / ↑m1))) ^ (1 / ↑n) ≤ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n)
hL0' : 0 < L + ε / 2
hm10 : ↑↑m1 ≠ 0
h_lt : 0 < ↑(n / ↑m1) / ↑n
⊢ (μ (x ^ ↑m1) ^ ↑(n / ↑m1)) ^ (1 / ↑n) < (L + ε / 2) ^ (↑(↑m1 * (n / ↑m1) + n % ↑m1) / ↑n + -↑(n % ↑m1) / ↑n)

Tactic:
← add_div,

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
h : μ (x ^ (↑m1 * (n / ↑m1))) ^ (1 / ↑n) ≤ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n)
hL0' : 0 < L + ε / 2
hm10 : ↑↑m1 ≠ 0
h_lt : 0 < ↑(n / ↑m1) / ↑n
⊢ (μ (x ^ ↑m1) ^ ↑(n / ↑m1)) ^ (1 / ↑n) < (L + ε / 2) ^ ((↑(↑m1 * (n / ↑m1) + n % ↑m1) + -↑(n % ↑m1)) / ↑n)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
this : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖
n : ℕ
hn : n = 0
⊢ (if n = 0 then ‖X.coeff n - (C r).coeff n‖ ^ (1 - ↑n)⁻¹ else 0) = if n = 0 then ‖r‖ else 0

Tactic:
rw [if_pos hn, if_pos hn, hn, cast_zero, sub_zero, coeff_X_zero, coeff_C_zero, zero_sub,
        norm_neg, inv_one, rpow_one]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
⊢ μ x ^ (1 / ↑1) ≤ μ x

Tactic:
cast_one,

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
⊢ μ x ^ (1 / 1) ≤ μ x
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
a' : f.SplittingField
ha' : a' ∈ f.rootSet F ∧ a' ≠ a
⊢ 0 < ‖a - a'‖

Tactic:
rw [norm_pos_iff, sub_ne_zero]

Goal state after:
case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
a' : f.SplittingField
ha' : a' ∈ f.rootSet F ∧ a' ≠ a
⊢ a ≠ a'
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ spectralNorm K L x ≤ ⨆ σ, f (σ x)

Tactic:
simp only [spectralNorm]

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
x : L
hf1 : f 1 = 1
p : K[X] := minpoly K x
hp_sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
s : Multiset L
hs : Polynomial.map (algebraMap K L) (minpoly K x) = (Multiset.map (fun x ↦ X - C x) s).prod
h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1
⊢ spectralValue (minpoly K x) ≤ ⨆ σ, f (σ x)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
k : K
y : L
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
g : ↥K⟮y⟯ := AdjoinSimple.gen K y
hgy : k • y = (algebraMap (↥K⟮y⟯) L) (k • g)
h :
  (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) (k • g) =
    k • (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g
⊢ spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
      (k • (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g) =
    ↑‖k‖₊ *
      spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
        ((algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g)

Tactic:
rw [← spectralAlgNorm_of_finiteDimensional_normal_def]

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
k : K
y : L
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
g : ↥K⟮y⟯ := AdjoinSimple.gen K y
hgy : k • y = (algebraMap (↥K⟮y⟯) L) (k • g)
h :
  (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) (k • g) =
    k • (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g
⊢ (spectralAlgNorm_of_finiteDimensional_normal K ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
      (k • (algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g) =
    ↑‖k‖₊ *
      spectralNorm K (↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯)))
        ((algebraMap ↥K⟮y⟯ ↥(normalClosure K (↥K⟮y⟯) (AlgebraicClosure ↥K⟮y⟯))) g)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
m : ℕ
hm : 1 ≤ m
⊢ smoothingFun μ (x ^ m) = smoothingFun μ x ^ m

Tactic:
have hlim : Tendsto (fun n => smoothingSeminormSeq μ x (m * n)) atTop
      (𝓝 (smoothingFun μ x)) :=
    Tendsto.comp (tendsto_smoothingFun_of_map_one_le_one μ hμ1 x) (tendsto_atTop_atTop_of_monotone
      (fun n k hnk ↦ mul_le_mul_right hnk m) (fun n ↦ ⟨n, le_mul_of_one_le_left' hm⟩))

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
m : ℕ
hm : 1 ≤ m
hlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))
⊢ smoothingFun μ (x ^ m) = smoothingFun μ x ^ m
Module: Mathlib.Analysis.Normed.Lp.LpEquiv

Goal state before:
pp: case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ (∑ i, ‖(Equiv.lpPiLp f).ofLp i‖ ^ p.toReal) ^ (1 / p.toReal) = (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
tsum_fintype

Goal state after:
case inr.inr
α : Type u_1
E : α → Type u_2
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
inst✝ : Fintype α
f : ↥(lp E p)
h : 0 < p.toReal
⊢ (∑ i, ‖(Equiv.lpPiLp f).ofLp i‖ ^ p.toReal) ^ (1 / p.toReal) = (∑ b, ‖↑f b‖ ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card

Tactic:
trans (mapAlg K E (minpoly K x)).natDegree

Goal state after:
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).natDegree

K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: case refine_1
ι : Type u_1
R : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
f : ι → R
inst✝ : CompleteSpace R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : ε > 0
r₁ : ℝ
hr₁ : r₁ > 0
s₁ : Finset ι
hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁
s₂ : Finset ι
hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖∏ i ∈ t, (1 + f i) - 1‖ < ε / (2 * r₁)
s : Finset ι := s₁ ∪ s₂
b : Finset ι
hb : s ⊆ b
⊢ ‖(∏ x ∈ s, (1 + f x)) * (∏ x ∈ b \ s, (1 + f x) - 1)‖ < r₁ * (ε / (2 * r₁))

Tactic:
apply (norm_mul_le _ _).trans_lt

Goal state after:
case refine_1
ι : Type u_1
R : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
f : ι → R
inst✝ : CompleteSpace R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : ε > 0
r₁ : ℝ
hr₁ : r₁ > 0
s₁ : Finset ι
hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁
s₂ : Finset ι
hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖∏ i ∈ t, (1 + f i) - 1‖ < ε / (2 * r₁)
s : Finset ι := ⋯
b : Finset ι
hb : s ⊆ b
⊢ ‖∏ x ∈ s, (1 + f x)‖ * ‖∏ x ∈ b \ s, (1 + f x) - 1‖ < r₁ * (ε / (2 * r₁))
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace F
inst✝¹ : T2Space E
inst✝ : T2Space F
u : E →L[𝕜] F
v : F →L[𝕜] E
hvu : (↑v).IsLeftQuasiInverse ↑u
huv : (LinearMap.id.eqLocus (↑u ∘ₗ ↑v)).CoFG
⊢ ∃ E₁ F₁,
    IsClosed[inst✝³] ↑E₁ ∧
      IsClosed[inst✝²] ↑F₁ ∧ E₁.CoFG ∧ F₁.CoFG ∧ ∃ (h : MapsTo ⇑u ↑E₁ ↑F₁), (u.restrict h).IsInvertible

Tactic:
IsLeftQuasiInverse,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace F
inst✝¹ : T2Space E
inst✝ : T2Space F
u : E →L[𝕜] F
v : F →L[𝕜] E
hvu : ↑v ∘ₗ ↑u ≈ LinearMap.id
huv : (LinearMap.id.eqLocus (↑u ∘ₗ ↑v)).CoFG
⊢ ∃ E₁ F₁,
    IsClosed[inst✝³] ↑E₁ ∧
      IsClosed[inst✝²] ↑F₁ ∧ E₁.CoFG ∧ F₁.CoFG ∧ ∃ (h : MapsTo ⇑u ↑E₁ ↑F₁), (u.restrict h).IsInvertible
Module: Mathlib.Analysis.Normed.Ring.InfiniteProd

Goal state before:
pp: case inr
α : Type u_1
R : Type u_2
β : Type u_3
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
inst✝ : CompleteSpace R
g : β → R
bound : β → ℝ
𝓕 : Filter α
f : α → β → R
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝ : 𝓕.NeBot
h_bound_g : ∀ (k : β), ‖g k‖ ≤ bound k
⊢ Tendsto (fun n ↦ ∏' (k : β), (1 + f n k)) 𝓕 (𝓝 (∏' (k : β), (1 + g k)))

Tactic:
have hsum_g : Summable (‖g ·‖) := h_sum.of_nonneg_of_le (fun _ ↦ norm_nonneg _) h_bound_g

Goal state after:
case inr
α : Type u_1
R : Type u_2
β : Type u_3
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
inst✝ : CompleteSpace R
g : β → R
bound : β → ℝ
𝓕 : Filter α
f : α → β → R
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝ : 𝓕.NeBot
h_bound_g : ∀ (k : β), ‖g k‖ ≤ bound k
hsum_g : Summable fun x ↦ ‖g x‖
⊢ Tendsto (fun n ↦ ∏' (k : β), (1 + f n k)) 𝓕 (𝓝 (∏' (k : β), (1 + g k)))
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsUpperSet s
hx : x ∈ closure s
h : ∀ (i : ι), x i < y i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), x i + ε < y i
⊢ y ∈ interior s

Tactic:
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsUpperSet s
hx : x ∈ closure s
h : ∀ (i : ι), x i < y i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), x i + ε < y i
z : ι → ℝ
hz : z ∈ s
hxz : dist x z < ε
⊢ y ∈ interior s
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
⊢ y ∈ interior s

Tactic:
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : dist x z < ε
⊢ y ∈ interior s
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: case neg
δ K x : ℝ
hK : ¬K = 0
⊢ gronwallBound δ K 0 x = δ * rexp (K * x)

Tactic:
· simp only [gronwallBound_of_K_ne_0 hK, zero_div, zero_mul, add_zero]

Goal state after:
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: case neg
δ K ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
hK : 0 ≤ K
x₁ x₂ : ℝ
hx : x₁ ≤ x₂
hK₀ : ¬K = 0
⊢ δ * rexp (K * x₁) + ε / K * (rexp (K * x₁) - 1) ≤ δ * rexp (K * x₂) + ε / K * (rexp (K * x₂) - 1)

Tactic:
have hK_pos : 0 < K := by positivity
    gcongr

Goal state after:
Module: Mathlib.Topology.MetricSpace.Contracting

Goal state before:
pp: α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
s : Set α
hsc : IsComplete s
hsf : MapsTo f s s
hfs : ContractingWith K (MapsTo.restrict f s s hsf)
x : α
hxs : x ∈ s
hx : edist x (f x) ≠ ∞
t : Set α
htc : IsComplete t
htf : MapsTo f t t
hft : ContractingWith K (MapsTo.restrict f t t htf)
y : α
hyt : y ∈ t
hy : edist y (f y) ≠ ∞
hxy : edist x y ≠ ∞
h' : edist (efixedPoint' f hsc hsf hfs x hxs hx) (efixedPoint' f htc htf hft y hyt hy) = ∞
⊢ Metric.edistLtTopSetoid x y

pp: α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
s : Set α
hsc : IsComplete s
hsf : MapsTo f s s
hfs : ContractingWith K (MapsTo.restrict f s s hsf)
x : α
hxs : x ∈ s
hx : edist x (f x) ≠ ∞
t : Set α
htc : IsComplete t
htf : MapsTo f t t
hft : ContractingWith K (MapsTo.restrict f t t htf)
y : α
hyt : y ∈ t
hy : edist y (f y) ≠ ∞
hxy : edist x y ≠ ∞
h' : edist (efixedPoint' f hsc hsf hfs x hxs hx) (efixedPoint' f htc htf hft y hyt hy) = ∞
⊢ Metric.edistLtTopSetoid y (efixedPoint' f htc htf hft y hyt hy)

Tactic:
· exact lt_top_iff_ne_top.2 hxy

Goal state after:
α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
s : Set α
hsc : IsComplete s
hsf : MapsTo f s s
hfs : ContractingWith K (MapsTo.restrict f s s hsf)
x : α
hxs : x ∈ s
hx : edist x (f x) ≠ ∞
t : Set α
htc : IsComplete t
htf : MapsTo f t t
hft : ContractingWith K (MapsTo.restrict f t t htf)
y : α
hyt : y ∈ t
hy : edist y (f y) ≠ ∞
hxy : edist x y ≠ ∞
h' : edist (efixedPoint' f hsc hsf hfs x hxs hx) (efixedPoint' f htc htf hft y hyt hy) = ∞
⊢ Metric.edistLtTopSetoid y (efixedPoint' f htc htf hft y hyt hy)
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup V
inst✝⁸ : InnerProductSpace 𝕜 V
inst✝⁷ : CompleteSpace V
inst✝⁶ : NormedAddCommGroup W
inst✝⁵ : InnerProductSpace 𝕜 W
inst✝⁴ : CompleteSpace W
F : Type u_4
inst✝³ : EquivLike F (V →L[𝕜] V) (W →L[𝕜] W)
inst✝² : NonUnitalAlgEquivClass F 𝕜 (V →L[𝕜] V) (W →L[𝕜] W)
inst✝¹ : StarHomClass F (V →L[𝕜] V) (W →L[𝕜] W)
inst✝ : ContinuousMapClass F (V →L[𝕜] V) (W →L[𝕜] W)
f : F
x y : V →L[𝕜] V
⊢ f x ≤ f y ↔ x ≤ y

Tactic:
by
    obtain ⟨U, hU⟩ := StarAlgEquiv.eq_linearIsometryEquivConjStarAlgEquiv
      (StarAlgEquivClass.toStarAlgEquiv f : _ ≃⋆ₐ[𝕜] _) (map_continuous f)
    have this a : f a = U.conjStarAlgEquiv a := by simpa using! congr($hU a)
    simp_rw [le_def, ← _root_.map_sub, ← isPositive_toLinearMap_iff, this]
    exact LinearMap.isPositive_linearIsometryEquiv_conj_iff U

Goal state after: