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

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P * ↑P + 0 + (↑Q * 0 + ↑Q * ↑Pᶜ * (↑Pᶜ * ↑R)) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
mul_zero,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P * ↑P + 0 + (0 + ↑Q * ↑Pᶜ * (↑Pᶜ * ↑R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.Ball.RadialEquiv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
U : Set ℝ
V : Set ↑(sphere 0 r)
hU : IsOpen U
hU₀ : 0 ∉ U
hV : IsOpen[instTopologicalSpaceSubtype] V
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U • Subtype.val '' V)

Tactic:
by
  rw [isOpen_iff_mem_nhds]
  rintro _ ⟨x, hxU, _, ⟨y, hyV, rfl⟩, rfl⟩
  wlog hx₀ : 0 < x generalizing x U
  · replace hx₀ : 0 < -x := by
      rw [not_lt, le_iff_eq_or_lt, ← neg_pos] at hx₀
      exact hx₀.resolve_left <| ne_of_mem_of_not_mem hxU hU₀
    specialize this hU.neg (by simpa) (-x) (by simpa) hx₀
    simp only [neg_smul, nhds_neg, Set.neg_smul, Filter.mem_neg] at this
    simpa using this
  have hr₀ : 0 < r := lt_of_le_of_ne (by simpa using norm_nonneg y.1) hr.symm
  lift x to Ioi (0 : ℝ) using hx₀
  have : V ×ˢ (Ioi (0 : ℝ) ↓∩ U) ∈ 𝓝 (y, x) :=
    prod_mem_nhds (hV.mem_nhds hyV) (hU.preimage_val.mem_nhds hxU)
  replace := image_mem_map (m := Subtype.val ∘ (homeomorphSphereProd E r hr₀).symm) this
  rw [← Filter.map_map, (homeomorphSphereProd _ r hr₀).symm.map_nhds_eq,
    map_nhds_subtype_val, IsOpen.nhdsWithin_eq, homeomorphSphereProd_symm_apply_coe] at this
  · filter_upwards [this]
    rintro _ ⟨⟨a, b⟩, ⟨ha, hb⟩, rfl⟩
    rw [Function.comp_apply, homeomorphSphereProd_symm_apply_coe]
    apply Set.smul_mem_smul
    exacts [hb, mem_image_of_mem _ ha]
  · exact isOpen_compl_singleton
  · simp [x.2.out.ne', ne_zero_of_mem_sphere, hr₀.ne']

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝¹ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this✝ : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
uₐ : ↥A →L[𝕜] ↥(map (↑u) A) := u.restrict ⋯
Φ : (↥S × ↥A) ≃L[𝕜] E := S.prodEquivOfIsTopCompl A S_compl_A
Ψ : (↥(map (↑u) S) × ↥(map (↑u) A)) ≃L[𝕜] F := (map (↑u) S).prodEquivOfIsTopCompl (map (↑u) A) uS_compl_uA
u_eq : ⇑u = ⇑Ψ ∘ ⇑(uₛ.prodMap uₐ) ∘ ⇑Φ.symm
u_restr_eq : ⇑(u.domRestrict A) = ⇑(map (↑u) A).subtypeL ∘ ⇑uₐ
this : IsStrictMap ⇑(uₛ.prodMap uₐ) ↔ IsStrictMap ⇑uₐ
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A)

Tactic:
u_restr_eq,

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝¹ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this✝ : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
uₐ : ↥A →L[𝕜] ↥(map (↑u) A) := u.restrict ⋯
Φ : (↥S × ↥A) ≃L[𝕜] E := S.prodEquivOfIsTopCompl A S_compl_A
Ψ : (↥(map (↑u) S) × ↥(map (↑u) A)) ≃L[𝕜] F := (map (↑u) S).prodEquivOfIsTopCompl (map (↑u) A) uS_compl_uA
u_eq : ⇑u = ⇑Ψ ∘ ⇑(uₛ.prodMap uₐ) ∘ ⇑Φ.symm
u_restr_eq : ⇑(u.domRestrict A) = ⇑(map (↑u) A).subtypeL ∘ ⇑uₐ
this : IsStrictMap ⇑(uₛ.prodMap uₐ) ↔ IsStrictMap ⇑uₐ
⊢ IsStrictMap ⇑u ↔ IsStrictMap (⇑(map (↑u) A).subtypeL ∘ ⇑uₐ)
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
N : Submodule 𝕜 E := A ⊓ (↑u).ker
π : E →L[𝕜] E ⧸ N := N.mkQL
v : E ⧸ N →L[𝕜] F := N.liftQL u ⋯
π_quot : IsOpenQuotientMap ⇑π
u_eq : u = v ∘SL π
B : Submodule 𝕜 (E ⧸ N) := map N.mkQ A
B_cofg : B.CoFG
v_ker : Disjoint (↑v).ker B
comap_B : comap (↑π) B = A
A_mapsTo_B : MapsTo ⇑π ↑A ↑B
B_closed : IsClosed[QuotientModule.Quotient.topologicalSpace N] ↑B
step4_output :
  IsStrictMap ⇑v ∧ IsClosed[inst✝³] ↑(↑v).range ↔
    IsStrictMap ⇑(v.domRestrict B) ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range
range_eq : (↑v).range = (↑u).range
range_restr_eq : (↑(v.domRestrict B)).range = (↑(u.domRestrict A)).range
π' : ↥A →L[𝕜] ↥B := π.restrict A_mapsTo_B
π'_quot : IsOpenQuotientMap ⇑π'
u_restr_eq : u.domRestrict A = v.domRestrict B ∘SL π'
⊢ IsStrictMap ⇑(v.domRestrict B) ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range ↔
    IsStrictMap ⇑(u.domRestrict A) ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range

Tactic:
u_restr_eq,

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
N : Submodule 𝕜 E := A ⊓ (↑u).ker
π : E →L[𝕜] E ⧸ N := N.mkQL
v : E ⧸ N →L[𝕜] F := N.liftQL u ⋯
π_quot : IsOpenQuotientMap ⇑π
u_eq : u = v ∘SL π
B : Submodule 𝕜 (E ⧸ N) := map N.mkQ A
B_cofg : B.CoFG
v_ker : Disjoint (↑v).ker B
comap_B : comap (↑π) B = A
A_mapsTo_B : MapsTo ⇑π ↑A ↑B
B_closed : IsClosed[QuotientModule.Quotient.topologicalSpace N] ↑B
step4_output :
  IsStrictMap ⇑v ∧ IsClosed[inst✝³] ↑(↑v).range ↔
    IsStrictMap ⇑(v.domRestrict B) ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range
range_eq : (↑v).range = (↑u).range
range_restr_eq : (↑(v.domRestrict B)).range = (↑(u.domRestrict A)).range
π' : ↥A →L[𝕜] ↥B := π.restrict A_mapsTo_B
π'_quot : IsOpenQuotientMap ⇑π'
u_restr_eq : u.domRestrict A = v.domRestrict B ∘SL π'
⊢ IsStrictMap ⇑(v.domRestrict B) ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range ↔
    IsStrictMap ⇑(v.domRestrict B ∘SL π') ∧ IsClosed[inst✝³] ↑(↑(v.domRestrict B)).range
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
u v : E →L[𝕜] F
A : Submodule 𝕜 E
inst✝ : A.CoFG
h_eqOn : EqOn ⇑u ⇑v ↑A.topologicalClosure
⊢ IsStrictMap ⇑u ∧ IsClosed[inst✝⁴] ↑(↑u).range ↔ IsStrictMap ⇑v ∧ IsClosed[inst✝⁴] ↑(↑v).range

Tactic:
simp_rw [u.isStrictMap_isClosed_range_iff_restrict _ A.isClosed_topologicalClosure,
    v.isStrictMap_isClosed_range_iff_restrict _ A.isClosed_topologicalClosure,
    LinearMap.coe_range, ContinuousLinearMap.coe_coe, ContinuousLinearMap.coe_domRestrict,
    domRestrict_eq_domRestrict_iff.mpr h_eqOn]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
u : E →L[𝕜] F
A : Submodule 𝕜 F
inst✝ : FiniteDimensional 𝕜 ↥A
A_compl : A.ClosedComplemented
S : Submodule 𝕜 F
A_compl_S : IsTopCompl A S
Φ : (F ⧸ A) ≃L[𝕜] ↥S := A.quotientEquivOfIsTopCompl S A_compl_S
i : ↥S →L[𝕜] F := S.subtypeL
⊢ IsStrictMap ⇑u ∧ IsClosed[inst✝⁴] ↑(↑u).range ↔
    IsStrictMap ⇑(A.mkQL ∘SL u) ∧ IsClosed[QuotientModule.Quotient.topologicalSpace A] ↑(↑(A.mkQL ∘SL u)).range

Tactic:
have i_clemb : IsClosedEmbedding i := S.isClosedEmbedding_subtypeL A_compl_S.symm.isClosed

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
u : E →L[𝕜] F
A : Submodule 𝕜 F
inst✝ : FiniteDimensional 𝕜 ↥A
A_compl : A.ClosedComplemented
S : Submodule 𝕜 F
A_compl_S : IsTopCompl A S
Φ : (F ⧸ A) ≃L[𝕜] ↥S := A.quotientEquivOfIsTopCompl S A_compl_S
i : ↥S →L[𝕜] F := S.subtypeL
i_clemb : IsClosedEmbedding ⇑i
⊢ IsStrictMap ⇑u ∧ IsClosed[inst✝⁴] ↑(↑u).range ↔
    IsStrictMap ⇑(A.mkQL ∘SL u) ∧ IsClosed[QuotientModule.Quotient.topologicalSpace A] ↑(↑(A.mkQL ∘SL u)).range
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 : 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
inj : Function.Injective ⇑T
Tₗ : V ≃ₗ[𝕜] W := LinearEquiv.ofBijective ↑T ⋯
T' : W →L[𝕜] V := (apply' V (RingHom.id 𝕜)) u ∘SL (↑f.symm).toContinuousLinearMap ∘SL (smulRightL 𝕜 W W) d
⊢ ∃ U, ∀ (x : V →L[𝕜] V), f x ∘SL ↑U = ↑U ∘SL x

Tactic:
set TL : V ≃L[𝕜] W := { Tₗ with
    continuous_toFun := T.continuous
    continuous_invFun := by
      change Continuous Tₗ.symm.toLinearMap
      suffices T'.toLinearMap = Tₗ.symm from this ▸ T'.continuous
      simp [LinearMap.ext_iff, ← Tₗ.injective.eq_iff, T', this, hT, hd, Tₗ] }

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 : 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
inj : Function.Injective ⇑T
Tₗ : V ≃ₗ[𝕜] W := LinearEquiv.ofBijective ↑T ⋯
T' : W →L[𝕜] V := (apply' V (RingHom.id 𝕜)) u ∘SL (↑f.symm).toContinuousLinearMap ∘SL (smulRightL 𝕜 W W) d
TL : V ≃L[𝕜] W := { toLinearEquiv := Tₗ, continuous_toFun := ⋯, continuous_invFun := ⋯ }
⊢ ∃ U, ∀ (x : V →L[𝕜] V), f x ∘SL ↑U = ↑U ∘SL x
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:
obtain ⟨U, hU⟩ := StarAlgEquiv.eq_linearIsometryEquivConjStarAlgEquiv
      (StarAlgEquivClass.toStarAlgEquiv f : _ ≃⋆ₐ[𝕜] _) (map_continuous f)

Goal state after:
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
U : V ≃ₗᵢ[𝕜] W
hU : ↑f = U.conjStarAlgEquiv
⊢ f x ≤ f y ↔ x ≤ y
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x y : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
t₁ t₂ : ↑(Icc tmin tmax)
⊢ ‖(∫ (τ : ℝ) in ↑t₀..↑t₁, f τ (α.compProj τ)) - ∫ (τ : ℝ) in ↑t₀..↑t₂, f τ (α.compProj τ)‖ ≤ ↑L * dist t₁ t₂

Tactic:
integral_interval_sub_left (intervalIntegrable_comp_compProj hf _ t₁)
        (intervalIntegrable_comp_compProj hf _ t₂),

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x y : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
t₁ t₂ : ↑(Icc tmin tmax)
⊢ ‖∫ (x : ℝ) in ↑t₂..↑t₁, f x (α.compProj x)‖ ≤ ↑L * dist t₁ t₂
Module: Mathlib.Analysis.ODE.ExistUnique

Goal state before:
pp: case h₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : ℝ → E → E
s : ℝ → Set E
K : ℝ≥0
f g : ℝ → E
a b t₀ : ℝ
hv : ∀ t ∈ Ioo a b, LipschitzOnWith K (v t) (s t)
ht : t₀ ∈ Ioo a b
hf : ContinuousOn f (Icc a b)
hf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t
hfs : ∀ t ∈ Ioo a b, f t ∈ s t
hg : ContinuousOn g (Icc a b)
hg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t
hgs : ∀ t ∈ Ioo a b, g t ∈ s t
heq : f t₀ = g t₀
⊢ EqOn f g (Icc a t₀)

Tactic:
have hss : Ioc a t₀ ⊆ Ioo a b := Ioc_subset_Ioo_right ht.2

Goal state after:
case h₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : ℝ → E → E
s : ℝ → Set E
K : ℝ≥0
f g : ℝ → E
a b t₀ : ℝ
hv : ∀ t ∈ Ioo a b, LipschitzOnWith K (v t) (s t)
ht : t₀ ∈ Ioo a b
hf : ContinuousOn f (Icc a b)
hf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t
hfs : ∀ t ∈ Ioo a b, f t ∈ s t
hg : ContinuousOn g (Icc a b)
hg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t
hgs : ∀ t ∈ Ioo a b, g t ∈ s t
heq : f t₀ = g t₀
hss : Ioc a t₀ ⊆ Ioo a b
⊢ EqOn f g (Icc a t₀)
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 : 𝕜[X]
inst✝ : OrderTopology 𝕜
⊢ (IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ |eval x P|) ↔ P.degree ≤ 0

Tactic:
by
  refine ⟨fun h => ?_, fun h => ⟨|P.coeff 0|, eventually_map.mpr (Eventually.of_forall
    (forall_imp (fun _ => le_of_eq) fun x => congr_arg abs <| _root_.trans (congr_arg (eval x)
    (eq_C_of_degree_le_zero h)) eval_C))⟩⟩
  contrapose! h
  exact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h)

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x y : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
this : Nonempty ↑(Icc tmin tmax)
⊢ ‖toContinuousMap (next hf hx α) - toContinuousMap (next hf hy α)‖ = dist x y

Tactic:
ContinuousMap.norm_eq_iSup_norm

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x y : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
this : Nonempty ↑(Icc tmin tmax)
⊢ ⨆ x_1, ‖(toContinuousMap (next hf hx α) - toContinuousMap (next hf hy α)) x_1‖ = dist x y
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
⊢ degree 0 < Q.degree

Tactic:
degree_zero

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
⊢ ⊥ < Q.degree
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormOneClass A
⊢ X.supNorm = 1

Tactic:
rw [← monomial_one_one_eq_X, supNorm_monomial, norm_one]

Goal state after:
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hy : eval (-y) P = 0
⊢ P.IsRoot (-y)

Tactic:
← IsRoot.def

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hy : P.IsRoot (-y)
⊢ P.IsRoot (-y)
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝ : Ring R
vNeg : ∀ (x : R), v (-x) = v x
f : MvPowerSeries σ R
this : ∀ (t : σ →₀ ℕ), (coeff t) (-f) = -(coeff t) f
⊢ (⨆ t, v ((coeff t) (-f)) * t.prod fun x1 x2 ↦ c x1 ^ x2) = ⨆ t, v ((coeff t) f) * t.prod fun x1 x2 ↦ c x1 ^ x2

Tactic:
simp_rw [this, vNeg]

Goal state after:
Module: Mathlib.Analysis.Real.OfDigits

Goal state before:
pp: case succ
x : ℝ
b : ℕ
inst✝ : NeZero b
hx : x ∈ Set.Ico 0 1
this : b ≠ 0
n : ℕ
ih : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊
⊢ ↑b ^ (n + 1) * (∑ x_1 ∈ Finset.range n, ofDigitsTerm (x.digits b) x_1 + ofDigitsTerm (x.digits b) n) =
    ↑⌊↑b ^ (n + 1) * x⌋₊

Tactic:
mul_add,

Goal state after:
case succ
x : ℝ
b : ℕ
inst✝ : NeZero b
hx : x ∈ Set.Ico 0 1
this : b ≠ 0
n : ℕ
ih : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊
⊢ ↑b ^ (n + 1) * ∑ x_1 ∈ Finset.range n, ofDigitsTerm (x.digits b) x_1 + ↑b ^ (n + 1) * ofDigitsTerm (x.digits b) n =
    ↑⌊↑b ^ (n + 1) * x⌋₊
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: n : ℕ
m : ℤ
hnpos : 0 < n
N : ℤ
D : ℕ
P : D ≠ 0
C : N.natAbs.Coprime D
hxr : ↑{ num := N, den := D, den_nz := P, reduced := C } ^ n = ↑m
hv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y
⊢ False

Tactic:
← cast_pow

Goal state after:
n : ℕ
m : ℤ
hnpos : 0 < n
N : ℤ
D : ℕ
P : D ≠ 0
C : N.natAbs.Coprime D
hxr : ↑({ num := N, den := D, den_nz := P, reduced := C } ^ n) = ↑m
hv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y
⊢ False
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: n : ℕ
m : ℤ
hnpos : 0 < n
N : ℤ
D : ℕ
P : D ≠ 0
C : N.natAbs.Coprime D
hxr : ↑{ num := N, den := D, den_nz := P, reduced := C } ^ n = ↑m
hv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y
⊢ False

Tactic:
rw [← cast_pow] at hxr

Goal state after:
n : ℕ
m : ℤ
hnpos : 0 < n
N : ℤ
D : ℕ
P : D ≠ 0
C : N.natAbs.Coprime D
hxr : ↑({ num := N, den := D, den_nz := P, reduced := C } ^ n) = ↑m
hv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y
⊢ False
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: n : ℕ
p : ℂ[X]
hp : ¬p = 0
hn : n ≤ p.natDegree
S : Multiset (Multiset ℂ) := powersetCard (p.natDegree - n) p.roots
this : ∀ x ∈ S.toFinset, ∏ x_1 ∈ x.toFinset, ‖x_1‖ ^ count x_1 x ≤ ∏ m ∈ p.roots.toFinset, max 1 ‖m‖ ^ count m p.roots
⊢ ∑ x ∈ S.toFinset, ↑(count x S) * (Multiset.map (fun a ↦ max 1 ‖a‖) p.roots).prod =
    ↑(p.natDegree.choose n) * (Multiset.map (fun a ↦ max 1 ‖a‖) p.roots).prod

Tactic:
rw [← Finset.sum_mul]
      congr
      norm_cast
      simp only [mem_powersetCard, mem_toFinset, imp_self, implies_true, sum_count_eq_card,
        card_powersetCard, S, ← Nat.choose_symm hn]
      congr
      exact splits_iff_card_roots.mp <| IsAlgClosed.splits p

Goal state after:
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: case inl
q : ℚ
hq : 0 ≤ q
⊢ Irrational √↑q ↔ ¬IsSquare q ∧ 0 ≤ q

Tactic:
simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]

Goal state after:
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: x : ℝ
h : Irrational x
q : ℚ
hq : q ≠ 0
⊢ Irrational (x / ↑q)

Tactic:
by
  rw [div_eq_mul_inv, ← cast_inv]
  exact h.mul_ratCast (inv_ne_zero hq)

Goal state after:
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: m : ℤ
x : ℝ
⊢ Irrational (x / ↑↑m) ↔ m ≠ 0 ∧ Irrational x

Tactic:
irrational_div_ratCast_iff,

Goal state after:
m : ℤ
x : ℝ
⊢ ↑m ≠ 0 ∧ Irrational x ↔ m ≠ 0 ∧ Irrational x
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: n : ℕ
x : ℝ
⊢ Irrational (x / ↑↑n) ↔ n ≠ 0 ∧ Irrational x

Tactic:
irrational_div_ratCast_iff,

Goal state after:
n : ℕ
x : ℝ
⊢ ↑n ≠ 0 ∧ Irrational x ↔ n ≠ 0 ∧ Irrational x
Module: Mathlib.Analysis.Real.Hyperreal

Goal state before:
pp: x : ℝ*
h0 : x ≠ 0
hi : x⁻¹.Infinitesimal
⊢ x.Infinite

Tactic:
rcases lt_or_gt_of_ne h0 with hn | hp

Goal state after:
case inl
x : ℝ*
h0 : x ≠ 0
hi : x⁻¹.Infinitesimal
hn : x < 0
⊢ x.Infinite

case inr
x : ℝ*
h0 : x ≠ 0
hi : x⁻¹.Infinitesimal
hp : 0 < x
⊢ x.Infinite
Module: Mathlib.Analysis.Real.Hyperreal

Goal state before:
pp: x y : ℝ*
⊢ x.InfinitePos → ¬y.Infinitesimal → y < 0 → (x * y).InfiniteNeg

Tactic:
rw [← infinitePos_neg, ← neg_pos, neg_mul_eq_mul_neg, ← infinitesimal_neg]
  exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos

Goal state after:
Module: Mathlib.Analysis.Real.Pi.Leibniz

Goal state before:
pp: case hf0
⊢ Tendsto (fun i ↦ (2 * ↑i + 1)⁻¹) atTop (𝓝 0)

Tactic:
apply Tendsto.inv_tendsto_atTop

Goal state after:
case hf0
⊢ Tendsto (fun i ↦ 2 * ↑i + 1) atTop atTop