mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
⊢ IsClosed[inst✝³] ↑(↑(u.domRestrict A)).range → IsClosed[inst✝³] ↑(↑u).range

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
⊢ IsClosed[inst✝³] ((fun a ↦ u a) '' ↑A) → IsClosed[inst✝³] ↑(↑u).range
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: ι : Type u_2
inst✝ : Fintype ι
x : ι → ℝ
⊢ MonotoneOn (dist x) (Ici x)

Tactic:
simpa using

Goal state after:
ι : Type u_2
inst✝ : Fintype ι
x : ι → ℝ
⊢ MonotoneOn (dist x) (Ici x)
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: ι : Type u_2
inst✝ : Fintype ι
x : ι → ℝ
⊢ AntitoneOn (dist x) (Iic x)

Tactic:
simpa using

Goal state after:
ι : Type u_2
inst✝ : Fintype ι
x : ι → ℝ
⊢ AntitoneOn (dist x) (Iic x)
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: case neg
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : NormedSpace 𝕜 V
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeparatingDual 𝕜 V
inst✝ : SeparatingDual 𝕜 W
f : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W
hV✝ : Subsingleton V
hV : Nontrivial W
⊢ ∃ U, f = U.conjContinuousAlgEquiv

Tactic:
simpa using

Goal state after:
case neg
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : NormedSpace 𝕜 V
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeparatingDual 𝕜 V
inst✝ : SeparatingDual 𝕜 W
f : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W
hV✝ : Subsingleton V
hV : Nontrivial W
⊢ ∃ U, f = U.conjContinuousAlgEquiv
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : NormedSpace 𝕜 V
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeparatingDual 𝕜 V
inst✝ : SeparatingDual 𝕜 W
f : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : StrongDual 𝕜 V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →L[𝕜] W := (apply' W (RingHom.id 𝕜)) z ∘SL (↑f).toContinuousLinearMap ∘SL (smulRightL 𝕜 V V) v
hT : ∀ (x : V), T x = (f (smulRight v x)) z
this : ∀ (A : V →L[𝕜] V) (x : V), T (A x) = (f A) (T x)
d : StrongDual 𝕜 W
hd : d ((f (smulRight v u)) z) = 1
surj : Function.Surjective ⇑T
x y : V
hxy : T x = T y
h_smul : smulRight v x = smulRight v y
⊢ x = y

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : NormedSpace 𝕜 V
inst✝² : NormedSpace 𝕜 W
inst✝¹ : SeparatingDual 𝕜 V
inst✝ : SeparatingDual 𝕜 W
f : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W
hV : Nontrivial V
u : V
hu : u ≠ 0
v : StrongDual 𝕜 V
huv : v u ≠ 0
z : W
hz : ¬(f (smulRight v u)) z = 0 z
T : V →L[𝕜] W := (apply' W (RingHom.id 𝕜)) z ∘SL (↑f).toContinuousLinearMap ∘SL (smulRightL 𝕜 V V) v
hT : ∀ (x : V), T x = (f (smulRight v x)) z
this : ∀ (A : V →L[𝕜] V) (x : V), T (A x) = (f A) (T x)
d : StrongDual 𝕜 W
hd : d ((f (smulRight v u)) z) = 1
surj : Function.Surjective ⇑T
x y : V
hxy : T x = T y
h_smul : smulRight v x = smulRight v y
⊢ x = y
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
e : V ≃L[𝕜] W
α α' : 𝕜
hα : α ≠ 0
hα2 : α' * α' = α⁻¹
he : adjoint ↑e ∘SL ↑e = α • ContinuousLinearMap.id 𝕜 V
he' : ↑e ∘SL adjoint ↑e = α • ContinuousLinearMap.id 𝕜 W
x : V
⊢ ?m.360

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
e : V ≃L[𝕜] W
α α' : 𝕜
hα : α ≠ 0
hα2 : α' * α' = α⁻¹
he : adjoint ↑e ∘SL ↑e = α • ContinuousLinearMap.id 𝕜 V
he' : ↑e ∘SL adjoint ↑e = α • ContinuousLinearMap.id 𝕜 W
x : V
⊢ ?m.360
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
e : V ≃L[𝕜] W
α α' : 𝕜
hα : α ≠ 0
hα2 : α' * α' = α⁻¹
he : adjoint ↑e ∘SL ↑e = α • ContinuousLinearMap.id 𝕜 V
he' : ↑e ∘SL adjoint ↑e = α • ContinuousLinearMap.id 𝕜 W
x : W
⊢ ?m.377

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
e : V ≃L[𝕜] W
α α' : 𝕜
hα : α ≠ 0
hα2 : α' * α' = α⁻¹
he : adjoint ↑e ∘SL ↑e = α • ContinuousLinearMap.id 𝕜 V
he' : ↑e ∘SL adjoint ↑e = α • ContinuousLinearMap.id 𝕜 W
x : W
⊢ ?m.377
Module: Mathlib.Analysis.ODE.Basic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
t₀ ε y : ℝ
hy : y ∈ Metric.ball t₀ ε
⊢ HasDerivAt γ (v y (γ y)) y ↔ HasDerivWithinAt γ (v y (γ y)) (Metric.ball t₀ ε) y

Tactic:
exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: case neg
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝¹ : Subsingleton V
h✝ : Nontrivial W
⊢ ∃ U, f = U.conjStarAlgEquiv

Tactic:
simpa using

Goal state after:
case neg
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝¹ : Subsingleton V
h✝ : Nontrivial W
⊢ ∃ U, f = U.conjStarAlgEquiv
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
⊢ ?m.622

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
⊢ ?m.622
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
⊢ ?m.822

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
⊢ ?m.822
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
hα_re : α = ↑(RCLike.re α)
hα_ne_zero : α ≠ 0
hα_nonneg : 0 ≤ α
hα_pos : 0 < RCLike.re α
x : W
⊢ (↑y ∘SL adjoint ↑y) x = (α • ContinuousLinearMap.id 𝕜 W) x

Tactic:
simpa using

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this : IsUnit (adjoint ↑y ∘SL ↑y)
hα_re : α = ↑(RCLike.re α)
hα_ne_zero : α ≠ 0
hα_nonneg : 0 ≤ α
hα_pos : 0 < RCLike.re α
x : W
⊢ y ((adjoint ↑y) x) = α • x
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
x y : α
h : edist x y ≠ ∞
hy : IsFixedPt f y
⊢ edist x y ≤ edist x (f x) / (1 - ↑K)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x y : α
h : edist x y ≠ ∞
hy : IsFixedPt f y
⊢ edist x y ≤ edist x (f x) / (1 - ↑K)
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
x y : α
hx : IsFixedPt f x
hy : IsFixedPt f y
h : ¬edist x y = ∞
⊢ edist x y ≤ 0

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x y : α
hx : IsFixedPt f x
hy : IsFixedPt f y
h : ¬edist x y = ∞
⊢ edist x y ≤ 0
Module: Mathlib.Analysis.ODE.DiscreteGronwall

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
u b c : ℕ → R
n₀ : ℕ
hu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n
hc : ∀ n ≥ n₀, 0 ≤ c n
n k : ℕ
hk : n₀ ≤ k
ih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c i
hck : 0 ≤ c k
j : ℕ
hj : j ∈ Ico n₀ k
⊢ c k * (b j * ∏ i ∈ Ico (j + 1) k, c i) = b j * ∏ i ∈ Ico (j + 1) (k + 1), c i

Tactic:
prod_Ico_succ_top (by have := mem_Ico.mp hj; omega)

Goal state after:
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
u b c : ℕ → R
n₀ : ℕ
hu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n
hc : ∀ n ≥ n₀, 0 ≤ c n
n k : ℕ
hk : n₀ ≤ k
ih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c i
hck : 0 ≤ c k
j : ℕ
hj : j ∈ Ico n₀ k
⊢ c k * (b j * ∏ i ∈ Ico (j + 1) k, c i) = b j * ((∏ k ∈ Ico (j + 1) k, c k) * c k)
Module: Mathlib.Topology.MetricSpace.Contracting

Goal state before:
pp: α : Type u_1
inst✝ : MetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x y : α
hy : IsFixedPt f y
⊢ dist x y ≤ dist x (f x) / (1 - ↑K)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : MetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x y : α
hy : IsFixedPt f y
⊢ dist x y ≤ dist x (f x) / (1 - ↑K)
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
K a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
ha : f a = 0
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖
x : ℝ
hx : x ∈ Icc a b
⊢ ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + 0

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
K a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
ha : f a = 0
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖
x : ℝ
hx : x ∈ Icc a b
⊢ ∀ (x : ℝ), a ≤ x → x < b → ‖f' x‖ ≤ K * ‖f x‖
Module: Mathlib.Topology.MetricSpace.Contracting

Goal state before:
pp: α : Type u_1
inst✝² : MetricSpace α
K : ℝ≥0
f : α → α
inst✝¹ : Nonempty α
inst✝ : CompleteSpace α
n : ℕ
hf : ContractingWith K f^[n]
x : α := fixedPoint f^[n] hf
hx : f^[n] x = x
this : ¬IsFixedPt f x
⊢ ↑K * dist x (f x) < dist x (f x)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝² : MetricSpace α
K : ℝ≥0
f : α → α
inst✝¹ : Nonempty α
inst✝ : CompleteSpace α
n : ℕ
hf : ContractingWith K f^[n]
x : α := fixedPoint f^[n] hf
hx : f^[n] x = x
this : ¬IsFixedPt f x
⊢ ↑K * dist x (f x) < dist x (f x)
Module: Mathlib.Analysis.ODE.DiscreteGronwall

Goal state before:
pp: case hbc
u b c : ℕ → ℝ
n₀ : ℕ
hun₀ : 0 ≤ u n₀
hu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n
hc : ∀ n ≥ n₀, 0 ≤ c n
hb : ∀ n ≥ n₀, 0 ≤ b n
n : ℕ
hn : n₀ ≤ n
⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ rexp (∑ i ∈ Ico n₀ n, c i)

Tactic:
simpa using

Goal state after:
case hbc
u b c : ℕ → ℝ
n₀ : ℕ
hun₀ : 0 ≤ u n₀
hu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n
hc : ∀ n ≥ n₀, 0 ≤ c n
hb : ∀ n ≥ n₀, 0 ≤ b n
n : ℕ
hn : n₀ ≤ n
⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ ∏ x ∈ Ico n₀ n, rexp (c x)
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case mk.mk
E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L : ℝ≥0
toFun✝¹ : ↑(Icc tmin tmax) → E
lipschitzWith✝¹ : LipschitzWith L toFun✝¹
mem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r
toFun✝ : ↑(Icc tmin tmax) → E
lipschitzWith✝ : LipschitzWith L toFun✝
mem_closedBall₀✝ : toFun✝ t₀ ∈ closedBall x₀ ↑r
h :
  (fun α ↦ { toFun := α.toFun, continuous_toFun := ⋯ })
      { toFun := toFun✝¹, lipschitzWith := lipschitzWith✝¹, mem_closedBall₀ := mem_closedBall₀✝¹ } =
    (fun α ↦ { toFun := α.toFun, continuous_toFun := ⋯ })
      { toFun := toFun✝, lipschitzWith := lipschitzWith✝, mem_closedBall₀ := mem_closedBall₀✝ }
⊢ { toFun := toFun✝¹, lipschitzWith := lipschitzWith✝¹, mem_closedBall₀ := mem_closedBall₀✝¹ } =
    { toFun := toFun✝, lipschitzWith := lipschitzWith✝, mem_closedBall₀ := mem_closedBall₀✝ }

Tactic:
simpa using

Goal state after:
case mk.mk
E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L : ℝ≥0
toFun✝¹ : ↑(Icc tmin tmax) → E
lipschitzWith✝¹ : LipschitzWith L toFun✝¹
mem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r
toFun✝ : ↑(Icc tmin tmax) → E
lipschitzWith✝ : LipschitzWith L toFun✝
mem_closedBall₀✝ : toFun✝ t₀ ∈ closedBall x₀ ↑r
h :
  (fun α ↦ { toFun := α.toFun, continuous_toFun := ⋯ })
      { toFun := toFun✝¹, lipschitzWith := lipschitzWith✝¹, mem_closedBall₀ := mem_closedBall₀✝¹ } =
    (fun α ↦ { toFun := α.toFun, continuous_toFun := ⋯ })
      { toFun := toFun✝, lipschitzWith := lipschitzWith✝, mem_closedBall₀ := mem_closedBall₀✝ }
⊢ toFun✝¹ = toFun✝
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
L : ℝ≥0
α : FunSpace t₀ x₀ 0 L
⊢ α.toFun t₀ = x₀

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
L : ℝ≥0
α : FunSpace t₀ x₀ 0 L
⊢ α.toFun t₀ = x₀
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
hγ : IsIntegralCurveOn γ v s
dt t : ℝ
ht : t ∈ -dt +ᵥ s
⊢ HasDerivWithinAt (γ ∘ fun x ↦ x + dt) ((v ∘ fun x ↦ x + dt) t ((γ ∘ fun x ↦ x + dt) t)) (-dt +ᵥ s) t

Tactic:
rw [comp_apply, hasDerivWithinAt_iff_hasFDerivWithinAt, Function.comp_def,
    hasFDerivWithinAt_comp_add_right, ← hasDerivWithinAt_iff_hasFDerivWithinAt, vadd_neg_vadd]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
hγ : IsIntegralCurveOn γ v s
dt t : ℝ
ht : t ∈ -dt +ᵥ s
⊢ HasDerivWithinAt γ (v (t + dt) ((fun x ↦ γ (x + dt)) t)) s (t + dt)
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
hγ : IsIntegralCurveOn γ v univ
dt : ℝ
⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) univ

Tactic:
simpa using

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
hγ : IsIntegralCurveOn γ v univ
dt : ℝ
⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) univ
Module: Mathlib.Analysis.ODE.ExistUnique

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
α : E → ℝ → E
hα1 : ∀ x ∈ closedBall x₀ ↑r, α x ↑t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWithinAt (α x) (f t (α x t)) (Icc tmin tmax) t
L' : ℝ≥0
hα2 : ∀ t ∈ Icc tmin tmax, LipschitzOnWith L' (fun x ↦ α x t) (closedBall x₀ ↑r)
⊢ ∃ α,
    (∀ x ∈ closedBall x₀ ↑r,
        α (x, ↑t₀) = x ∧
          ∀ t ∈ Icc tmin tmax, HasDerivWithinAt (fun x_1 ↦ α (x, x_1)) (f t (α (x, t))) (Icc tmin tmax) t) ∧
      ContinuousOn α (closedBall x₀ ↑r ×ˢ Icc tmin tmax)

Tactic:
refine ⟨uncurry α, hα1, ?_⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
α : E → ℝ → E
hα1 : ∀ x ∈ closedBall x₀ ↑r, α x ↑t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWithinAt (α x) (f t (α x t)) (Icc tmin tmax) t
L' : ℝ≥0
hα2 : ∀ t ∈ Icc tmin tmax, LipschitzOnWith L' (fun x ↦ α x t) (closedBall x₀ ↑r)
⊢ ContinuousOn (uncurry α) (closedBall x₀ ↑r ×ˢ Icc tmin tmax)
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: p : ℂ[X]
⊢ Integrable (⇑(toAddCircle p)) haarAddCircle

Tactic:
simpa using

Goal state after:
p : ℂ[X]
⊢ Integrable (⇑(toAddCircle p)) haarAddCircle
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hP0 : P.leadingCoeff = 0
⊢ P.degree < Q.degree

Tactic:
leadingCoeff_eq_zero.1 hP0,

Goal state after:
case pos.inl
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)
hP0 : P.leadingCoeff = 0
⊢ degree 0 < Q.degree
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: case inl
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℕ
hb : ↑b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
⊢ ‖if 0 ≤ ↑b then p.coeff (↑b).natAbs else 0‖ ^ 2 = 0

Tactic:
simpa using

Goal state after:
case inl
p : ℂ[X]
this :
  ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
b : ℕ
hb : ↑b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support
⊢ p.coeff b = 0
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)
⊢ Q.comp (-X) ≠ 0

Tactic:
rw [Ne, comp_eq_zero_iff]

Goal state after:
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hQ : Q ≠ 0
h : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)
⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree
hQ : Q ≠ 0
⊢ Q.comp (-X) ≠ 0

Tactic:
rw [Ne, comp_eq_zero_iff]

Goal state after:
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree
hQ : Q ≠ 0
⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : P = 0
⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q

Tactic:
simpa using

Goal state after:
case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
h : P.degree ≤ Q.degree
hp : P = 0
⊢ (fun x ↦ 0) =O[atTop] fun x ↦ eval x Q
Module: Mathlib.RingTheory.PowerSeries.GaussNorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
y : ℕ
hy : v ((coeff y) f) * c ^ y = x✝
⊢ v ((MvPowerSeries.coeff ((Finsupp.uniqueEquiv ()).symm y)) f) * c ^ ((Finsupp.uniqueEquiv ()).symm y) PUnit.unit = x✝

Tactic:
simpa using

Goal state after:
case h
R : Type u_1
inst✝ : Semiring R
v : R → ℝ
c : ℝ
f : R⟦X⟧
h : HasGaussNorm v c f
x✝ : ℝ
y : ℕ
hy : v ((coeff y) f) * c ^ y = x✝
⊢ v ((coeff y) f) * c ^ y = x✝