mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : T = 0
⊢ t ^ 2 / (2 * t) > 0

Tactic:
positivity

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
c : ℝ
hc0 : c > 0
hc : ∀ (x : E), c * ‖x‖ ≤ ‖((algebraMap 𝕜 (E →L[𝕜] E)) ((algebraMap ℝ 𝕜) t) - T) x‖
⊢ min (c ^ 2 / (2 * t)) ((t ^ 2 + ‖T‖ ^ 2) / (2 * t)) > 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this✝ : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
hx' : 1 / 2 < x
hx'' : 0 < x
h : 0 ≤ r - ↑n
this : r - ↑n ≤ ↑k
⊢ 0 ≤ |x|

Tactic:
positivity

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
c : ℝ
hc0 : c > 0
x : E
hc : c * ‖x‖ ≤ ‖((algebraMap 𝕜 (E →L[𝕜] E)) ((algebraMap ℝ 𝕜) t) - T) x‖
hx : ¬x = 0
⊢ 0 ≤ c * ‖x‖

Tactic:
positivity

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
c : ℝ
hc0 : c > 0
x : E
hc : c * ‖x‖ ≤ ‖((algebraMap 𝕜 (E →L[𝕜] E)) ((algebraMap ℝ 𝕜) t) - T) x‖
hx : ¬x = 0
⊢ 0 ≤ ‖((algebraMap 𝕜 (E →L[𝕜] E)) ((algebraMap ℝ 𝕜) t) - T) x‖

Tactic:
positivity

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
inst✝ : Nontrivial E
hT' : (algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
⊢ 0 < ‖T‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
⊢ 0 ≤ x ^ (k + l) * f

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this : 0 ≤ C₂
⊢ 0 ≤ 1 + x

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h₁.hx1
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : x ≤ 1
⊢ 0 < (1 + x) / 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x
⊢ 0 < x

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x
⊢ 0 ≤ x ^ (k + l) * f

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x
⊢ 0 ≤ ((1 + x) / 2) ^ (-↑l)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : 1 ≤ x
⊢ 0 < (1 + x) / 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
v : E
⊢ 0 v ≤ ‖v‖ ^ k * ‖f v‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖
C : 𝕜
hC : 1 < ‖C‖
⊢ ‖C‖⁻¹ > 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖
C : 𝕜
hC : 1 < ‖C‖
ε : ℝ
hε : ε > 0
⊢ ε > 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : MeasurableSpace E
μ : Measure E
hμ : μ.HasTemperateGrowth
p : ℝ≥0
hp : ↑p ≠ 0
h_one_add : ∀ (x : E), 0 < 1 + ‖x‖
hp_pos : 0 < ↑p
l : ℕ
hl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ
k : ℕ := ⌈↑l / ↑p⌉₊
hlk : ↑l ≤ ↑k * ↑p
this : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-(↑k * ↑p))‖ₑ ∂μ < ∞
x : E
⊢ 0 ≤ (1 + ‖x‖) ^ (-↑k)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
y_ : ℕ → X := fun n ↦ T (x n)
⊢ ‖μ‖ * c / 2 > 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
y_ : ℕ → X := fun n ↦ T (x n)
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
d : ℝ
hd₀ : d > 0
hd : ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
hyK : 0 ∈ K
hψy : Tendsto (y_ ∘ ψ) atTop (𝓝 0)
hy_eigen' : (T - μ • 1) 0 = 0
this : ∀ᶠ (n : ℕ) in atTop, False
⊢ False

Tactic:
eventually_const

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
y_ : ℕ → X := fun n ↦ T (x n)
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
d : ℝ
hd₀ : d > 0
hd : ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
hyK : 0 ∈ K
hψy : Tendsto (y_ ∘ ψ) atTop (𝓝 0)
hy_eigen' : (T - μ • 1) 0 = 0
this : False
⊢ False
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
c : ℝ
hc₀ : c > 0
hc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) x‖ < ε
φ : ℕ → ℝ
hφ_anti : StrictAnti φ
hφ_pos : ∀ (n : ℕ), 0 < φ n
hφ : Tendsto φ atTop (𝓝 0)
x : ℕ → X
hx_norm_upper : ∀ (n : ℕ), ‖x n‖ ≤ 1
hx_norm_lower : ∀ (n : ℕ), c ≤ ‖x n‖
hx_bound : ∀ (n : ℕ), ‖(T - μ • 1) (x n)‖ < φ n
hx_lim : Tendsto (fun n ↦ (T - μ • 1) (x n)) atTop (𝓝 0)
y_ : ℕ → X := fun n ↦ T (x n)
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 1 ⊆ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hy_lim : Tendsto (fun n ↦ (T - μ • 1) (y_ n)) atTop (𝓝 0)
d : ℝ
hd₀ : d > 0
hd : ∀ᶠ (n : ℕ) in atTop, d ≤ ‖y_ n‖
hyK : 0 ∈ K
hψy : ∀ ε > 0, ∀ᶠ (x : ℕ) in atTop, ‖(y_ ∘ ψ) x‖ < ε
hy_eigen' : (T - μ • 1) 0 = 0
⊢ d > 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ f ∘ ⇑σ ∘ ⇑(sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)

Tactic:
← Equiv.Perm.coe_mul

Goal state after:
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
⊢ f ∘ ⇑(σ * sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K✝ : NNReal
hK✝ : AntilipschitzWith K✝ ⇑S
h₂ : ¬Function.Surjective ⇑S
c : 𝕜
hc : 1 < ‖c‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
hp : Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 (‖c‖ + 1) ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (((fun n ↦ ↑T (f n)) ∘ ψ) n) (((fun n ↦ ↑T (f n)) ∘ ψ) N) < ε
⊢ ‖μ‖ > 0

Tactic:
positivity

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

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
h0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)
f : C(↑(spectrum ℝ A), ℝ)
hf : ↑hA.cfcAux f = 0
h2 : diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0
x : ↑(spectrum ℝ A)
⊢ f x = 0 x

Tactic:
simp only [ContinuousMap.zero_apply]

Goal state after:
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
h0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)
f : C(↑(spectrum ℝ A), ℝ)
hf : ↑hA.cfcAux f = 0
h2 : diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0
x : ↑(spectrum ℝ A)
⊢ f x = 0
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
⊢ (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues))).PosDef ↔
    ∀ (i : n), 0 < hA.eigenvalues i

Tactic:
simp [isUnit_coe.posDef_star_right_conjugate_iff]

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
f g : 𝓓^{n}_{K}(E, F)
⊢ (structureMapLM 𝕜 n 0) f = (structureMapLM 𝕜 n 0) g → f = g

Tactic:
hfg

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n : ℕ∞
K : Compacts E
f g : 𝓓^{n}_{K}(E, F)
hfg : (structureMapLM 𝕜 n 0) f = (structureMapLM 𝕜 n 0) g
⊢ f = g
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA✝ : IsSelfAdjoint A
hA : A.IsHermitian
f : C(↑(spectrum ℝ A), ℝ)
⊢ ‖hA.cfcAux f‖ = ‖f‖

Tactic:
simp only [IsHermitian.cfcAux_apply, Unitary.conjStarAlgAut_apply, ← Unitary.coe_star,
      CStarRing.norm_mul_coe_unitary, CStarRing.norm_coe_unitary_mul, l2_opNorm_diagonal]

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA✝ : IsSelfAdjoint A
hA : A.IsHermitian
f : C(↑(spectrum ℝ A), ℝ)
⊢ ‖RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩‖ = ‖f‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module ℝ A
inst✝¹² : SMulCommClass ℝ A A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁹ : PartialOrder A
inst✝⁸ : StarOrderedRing A
inst✝⁷ : NonnegSpectrumClass ℝ A
inst✝⁶ : IsTopologicalRing A
inst✝⁵ : T2Space A
inst✝⁴ : StarModule ℝ A
R : Type u_3
inst✝³ : Semiring R
inst✝² : SMulWithZero R ℝ≥0
inst✝¹ : SMul R A
inst✝ : IsScalarTower R ℝ≥0 A
r✝ : R
a : A
r : ℝ≥0
⊢ 0 ≤ r

Tactic:
positivity

Goal state after:
Module: Mathlib.Geometry.Manifold.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
dist_f_f : ∀ (x : M), dist (f x) (f x) < ε x
t : M → Set F := fun x ↦ {y | dist y (f x) < ε x ∧ (x ∈ S → y = f x) ∧ (f x = 0 → y = 0)}
this : ∃ g, ∀ (x : M), g x ∈ t x
⊢ ∃ g, (∀ (x : M), dist (g x) (f x) < ε x) ∧ EqOn (⇑g) f S ∧ support ⇑g ⊆ support f

Tactic:
rcases this with ⟨g, hg⟩

Goal state after:
E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
dist_f_f : ∀ (x : M), dist (f x) (f x) < ε x
t : M → Set F := fun x ↦ {y | dist y (f x) < ε x ∧ (x ∈ S → y = f x) ∧ (f x = 0 → y = 0)}
g : C^↑n⟮I, M; 𝓘(ℝ, F), F⟯
hg : ∀ (x : M), g x ∈ t x
⊢ ∃ g, (∀ (x : M), dist (g x) (f x) < ε x) ∧ EqOn (⇑g) f S ∧ support ⇑g ⊆ support f
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
⊢ 0 ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
positivity

Goal state after:
Module: Mathlib.Geometry.Manifold.SmoothApprox

Goal state before:
pp: case pos
E : Type u_1
F : Type u_2
H : Type u_3
M : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f : M → F
ε : M → ℝ
n : ℕ∞
f_cont : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[inst✝⁴, _] ε
ε_pos : ∀ (x : M), 0 < ε x
S U : Set M
hS : IsClosed[inst✝⁴] S
hU : U ∈ 𝓝ˢ S
hfU : ContMDiffOn I 𝓘(ℝ, F) (↑n) f U
dist_f_f : ∀ (x : M), dist (f x) (f x) < ε x
t : M → Set F := fun x ↦ {y | dist y (f x) < ε x ∧ (x ∈ S → y = f x) ∧ (f x = 0 → y = 0)}
t_conv : ∀ (x : M), Convex ℝ (t x)
x : M
hx : x ∉ S
this : ∀ᶠ (y : M) in 𝓝 x, y ∉ S ∧ dist (f x) (f y) < ε y
hx' : f x = 0
⊢ ∀ᶠ (y : M) in 𝓝 x, (y ∉ S ∧ dist (f x) (f y) < ε y) ∧ (f y = 0 → f x = 0)

Tactic:
simpa [hx'] using this

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯ ∈ Lp 𝕜 1 μ

Tactic:
mem_Lp_iff_eLpNorm_lt_top,

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ eLpNorm (↑(AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯)) 1 μ < ∞
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
x✝ : E
⊢ 0 < ε

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
⊢ ∀ (x : E), 0 < ε

Tactic:
intro; positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : ¬f =ᵐ[μ] 0
hs₁ : μ (tsupport f) ≠ ∞
hs₂ : 0 < (μ (tsupport f)).toReal
ε' : ℝ := ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
ε'_def : ε' = ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
⊢ 0 < ε'

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f : 𝓓^{n}_{K}(E, F₁)
hφ : ¬IntegrableOn φ (↑K) μ
⊢ 0 ≤ (∫ (x : E) in ↑K, ‖φ x‖ ∂μ) * ‖B‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K 0) f

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
⊢ 0 ≤ (∫ (x : E) in ↑K, ‖φ x‖ ∂μ) * ‖B‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : ¬f =ᵐ[μ] 0
hs₁ : μ (tsupport f) ≠ ∞
hs₂ : 0 < (μ (tsupport f)).toReal
ε' : ℝ := ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
ε'_def : ε' = ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
hε' : 0 < ε'
hε₂ : ENNReal.ofReal ε' * μ (tsupport f) ^ (1 / p.toReal) ≤ ENNReal.ofReal ε
x✝ : E
⊢ 0 < ε'

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : ¬f =ᵐ[μ] 0
hs₁ : μ (tsupport f) ≠ ∞
hs₂ : 0 < (μ (tsupport f)).toReal
ε' : ℝ := ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
ε'_def : ε' = ε * (μ (tsupport f)).toReal ^ (-(1 / p.toReal))
hε' : 0 < ε'
hε₂ : ENNReal.ofReal ε' * μ (tsupport f) ^ (1 / p.toReal) ≤ ENNReal.ofReal ε
⊢ ∀ (x : E), 0 < ε'

Tactic:
intro; positivity

Goal state after: