mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → Module ℂ (E i)
inst✝¹ : (i : ι) → SMul A (E i)
inst✝ : (i : ι) → CStarModule A (E i)
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
⊢ 0 ≤ ∑ i, ‖x i‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 c
⊢ 1 + x✝ ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : ↑a ≤ ↑b
ha : 0 ≤ ↑a
hb : 0 ≤ ↑b
c : Unitization ℂ A
hc : 0 ≤ c
x : ℝ≥0
x✝ : x ∈ σ ℝ≥0 c
⊢ 1 + x ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
i : ι
y : E i
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ 0 ≤ ‖(equiv A ((j : ι) → E j)).symm (Pi.single i y)‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalCStarAlgebra A
inst✝⁷ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → Module ℂ (E i)
inst✝³ : (i : ι) → SMul A (E i)
inst✝² : (i : ι) → CStarModule A (E i)
inst✝¹ : StarOrderedRing A
inst✝ : DecidableEq ι
i : ι
y : E i
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ 0 ≤ ‖y‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ 0 ≤ ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
x✝ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, (i : ι) → E i) := normedAddCommGroup A
⊢ 0 ≤ ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
x : ℝ≥0
x✝ : x ∈ σₙ ℝ≥0 a
⊢ 0 < (1 + x)⁻¹

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ a = cfcₙ (f ∘ g) a

Tactic:
by
      conv_lhs => rw [← cfcₙ_id ℝ≥0 a]
      refine cfcₙ_congr (Set.InvOn.one_sub_one_add_inv.1.eqOn.symm.mono fun x hx ↦ ?_)
      exact lt_of_le_of_lt (le_nnnorm_of_mem_quasispectrum hx) ha₂

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x✝ : ℝ≥0
a✝ : x✝ ∈ g '' σₙ ℝ≥0 a
⊢ 1 + x✝ ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case neg
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : ¬p a
⊢ LipschitzOnWith 1 (fun f ↦ cfcₙ ((toFun {quasispectrum R a}) f) a)
    {f | ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a) ∧ f 0 = 0}

Tactic:
· simpa [cfcₙ_apply_of_not_predicate a ha] using LipschitzWith.const' 0 |>.lipschitzOnWith

Goal state after:
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case e_f
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
i : m
⊢ ∑ j, ‖diagonal v i j‖₊ = ‖v i‖₊

Tactic:
refine (Finset.sum_eq_single_of_mem _ (Finset.mem_univ i) fun j _hj hij => ?_).trans ?_

Goal state after:
case e_f.refine_1
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
i j : m
_hj : j ∈ Finset.univ
hij : j ≠ i
⊢ ‖diagonal v i j‖₊ = 0

case e_f.refine_2
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
i : m
⊢ ‖diagonal v i i‖₊ = ‖v i‖₊
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊ * Finset.univ.sup fun i ↦ ∑ j, ‖B i j‖₊) ≤
    (Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊) * Finset.univ.sup fun i ↦ ∑ j, ‖B i j‖₊

Tactic:
← Finset.sum_mul,

Goal state after:
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ (Finset.univ.sup fun i ↦ (∑ i_1, ‖A i i_1‖₊) * Finset.univ.sup fun i ↦ ∑ j, ‖B i j‖₊) ≤
    (Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊) * Finset.univ.sup fun i ↦ ∑ j, ‖B i j‖₊
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝ m : A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
ε : ℝ≥0
hε : 0 < ε
⊢ ∃ ia, (0 ≤ ia ∧ ‖ia‖ < 1) ∧ ∀ x ∈ {x | ia ≤ x} ∩ closedBall 0 1, x * m ∈ closedBall m ↑ε

Tactic:
refine ⟨cfcₙ (fun y : ℝ≥0 ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m),
    ⟨cfcₙ_nonneg_of_predicate, norm_cfcₙ_one_sub_one_add_inv_lt_one (ε⁻¹ ^ 2 • m)⟩, ?_⟩

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝ m : A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
ε : ℝ≥0
hε : 0 < ε
⊢ ∀ x ∈ {x | cfcₙ (fun y ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m) ≤ x} ∩ closedBall 0 1, x * m ∈ closedBall m ↑ε
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ ‖replicateRow ι v‖ = ‖toLp 2 v‖

Tactic:
rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]

Goal state after:
m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ (∑ j, ‖replicateRow ι v default j‖ ^ 2) ^ (1 / 2) = (∑ i, ‖(toLp 2 v).ofLp i‖ ^ 2) ^ (1 / 2)
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ ∈ σ ℝ≥0 ↑?a
⊢ x✝ ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False

Tactic:
positivity

Goal state after:
case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ = 0
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case funProp.discharger
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝¹ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
x✝ : ℝ≥0
a✝ : x✝ ∈ Set.univ
⊢ 1 + x✝ ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝² : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
x✝¹ : ℝ≥0
x✝ : x✝¹ ∈ σ ℝ≥0 m
⊢ 1 + ε⁻¹ ^ 2 • x✝¹ ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
n : ℕ
x : A₁ := Classical.choose ⋯
hx₁ : 0 ≤ x
hx₂ : ↑(2 ^ (2 * n)) * ‖x‖ < ‖f x‖
hx : 0 < ‖x‖
⊢ 0 ≤ ‖x‖⁻¹

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
this : star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m = cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m
y : ℝ≥0
hy : y ∈ σ ℝ≥0 m
⊢ 0 < ε ^ 2 + y

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
x_summable : Summable fun n ↦ 2 ^ (-↑n) • x n
n : ℕ
hn : ‖f (∑' (n : ℕ), 2 ^ (-↑n) • x n)‖ < 2 ^ n
this : 2 ^ n * 2 ^ n ≤ ‖f (x n)‖
⊢ 0 < 2 ^ n

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.PositiveLinearMap

Goal state before:
pp: A₁ : Type u_1
A₂ : Type u_2
inst✝⁵ : NonUnitalCStarAlgebra A₁
inst✝⁴ : NonUnitalCStarAlgebra A₂
inst✝³ : PartialOrder A₁
inst✝² : StarOrderedRing A₁
inst✝¹ : PartialOrder A₂
inst✝ : StarOrderedRing A₂
f : A₁ →ₚ[ℂ] A₂
hcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖
x : ℕ → A₁
hx_nonneg : ∀ (x_1 : ℕ), 0 ≤ x x_1
hx_norm : ∀ (x_1 : ℕ), ‖x x_1‖ = 1
hx : ∀ (x_1 : ℕ), 2 ^ (2 * x_1) < ‖f (x x_1)‖
x_summable : Summable fun n ↦ 2 ^ (-↑n) • x n
n : ℕ
hn : ‖f (∑' (n : ℕ), 2 ^ (-↑n) • x n)‖ < 2 ^ n
m : ℕ
⊢ 0 ≤ 2 ^ (-↑m)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case inl
m : Type u_1
n : Type u_2
A : Type u_5
inst✝⁴ : Fintype m
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype n
M : CStarMatrix m n A
C : ℝ≥0
h : ∀ (v : C⋆ᵐᵒᵈ(A, m → A)) (w : C⋆ᵐᵒᵈ(A, n → A)), ‖⟪w, (toCLM M) v⟫_A‖ ≤ ↑C * ‖v‖ * ‖w‖
v : C⋆ᵐᵒᵈ(A, m → A)
h₀ : 0 = ‖(toCLM M) v‖
⊢ 0 ≤ ↑C * ‖v‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
M : CStarMatrix m n A
⊢ 0 ≤ ∑ j, ∑ i, ‖M i j‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ≥0 → ℝ≥0
x✝¹ : ℝ
x✝ : x✝¹ ∈ spectrum ℝ a
⊢ 0 ≤ ↑(f x✝¹.toNNReal)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f (quasispectrum ℂ ↑(ℑ a))
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℑ a) = cfcₙ (fun x ↦ f ↑x.im) a

Tactic:
rw [quasispectrum_imaginaryPart a] at hf

Goal state after:
A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℂ A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : SMulCommClass ℂ A A
inst✝² : StarModule ℂ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMapZero.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ f ↑(ℑ a) = cfcₙ (fun x ↦ f ↑x.im) a
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
⊢ 0 ≤ √‖M * star M‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
f : ℝ≥0 → ℝ≥0
x✝¹ : ℝ
x✝ : x✝¹ ∈ quasispectrum ℝ a
⊢ 0 ≤ ↑(f x✝¹.toNNReal)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case hx
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ 0 ≤ ‖M * star M‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
hmain : ‖M‖ ≤ √‖M * star M‖
⊢ 0 ≤ ‖M * star M‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f (spectrum ℂ ↑(ℑ a))
ha : IsStarNormal a
⊢ cfc f ↑(ℑ a) = cfc (fun x ↦ f ↑x.im) a

Tactic:
rw [spectrum_imaginaryPart a] at hf

Goal state after:
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : ContinuousMap.UniqueHom ℂ A
f : ℂ → ℂ
a : A
hf : ContinuousOn f ((fun x ↦ ↑x.im) '' spectrum ℂ a)
ha : IsStarNormal a
⊢ cfc f ↑(ℑ a) = cfc (fun x ↦ f ↑x.im) a
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by simp [inner_smul_left]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ⟪(toDual 𝕜 E).symm y, x⟫ = y x

Tactic:
← toDual_apply_apply

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ((toDual 𝕜 E) ((toDual 𝕜 E).symm y)) x = y x
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
F : Set (Set E)
hF : F.Nonempty
hAF : ∀ B ∈ F, IsExtreme 𝕜 A B
⊢ IsExtreme 𝕜 A (⋂₀ F)

Tactic:
simpa [sInter_eq_biInter] using isExtreme_biInter hF hAF

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Hom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
hφ : Function.Injective ⇑φ
a✝ : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
hψ : Function.Injective ⇑ψ
a : Unitization ℂ A
⊢ 0 ≤ ‖ψ a‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Hom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
hφ : Function.Injective ⇑φ
a✝ : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
hψ : Function.Injective ⇑ψ
a : Unitization ℂ A
⊢ 0 ≤ ‖a‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
x : f.PreGNS
⊢ 0 ≤ ‖(↑f.toPreGNS ∘ₗ ↑((ContinuousLinearMap.mul ℂ A) a) ∘ₗ ↑f.ofPreGNS) x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
f : A →ₚ[ℂ] ℂ
inst✝ : StarOrderedRing A
a : A
x : f.PreGNS
⊢ 0 ≤ ‖a‖ * ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Extreme

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
he : IsStarProjection e
a : A
ha : 0 ≤ a
ha1 : ‖a‖ ≤ 1
b : A
hb : 0 ≤ b
hb1 : ‖b‖ ≤ 1
x✝ : e ∈ openSegment ℝ a b
t s : ℝ
h0t : 0 < t
h0s : 0 < s
hts : t + s = 1
hlin : t • a + s • b = e
ha01 : ↑a ∈ Icc 0 1
hb01 : ↑b ∈ Icc 0 1
this : t • (↑e * (1 - ↑a) * ↑e) = 0
⊢ 0 ≤ s • b

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: ⊢ 2 ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Composition

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : OpenPartialHomeomorph X Y
e' : OpenPartialHomeomorph Y Z
⊢ (e.symm.restrOpen e'.source ⋯).symm.target = (e'.restrOpen e.target ⋯).source

Tactic:
simp [inter_comm]

Goal state after: