mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
g g' : M' [⋀^ι]→L[R] N
⊢ (g + g').compContinuousLinearMap f = g.compContinuousLinearMap f + g'.compContinuousLinearMap f

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
g g' : M' [⋀^ι]→L[R] N
x✝ : ι → M
⊢ ((g + g').compContinuousLinearMap f) x✝ = (g.compContinuousLinearMap f + g'.compContinuousLinearMap f) x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
c : R
g : M' [⋀^ι]→L[R] N
⊢ (c • g).compContinuousLinearMap f = (RingHom.id R) c • g.compContinuousLinearMap f

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
c : R
g : M' [⋀^ι]→L[R] N
x✝ : ι → M
⊢ ((c • g).compContinuousLinearMap f) x✝ = ((RingHom.id R) c • g.compContinuousLinearMap f) x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : ContinuousMultilinearMap R (fun x ↦ M) N
⊢ (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ 0;
    { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ }) =
    0

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : ContinuousMultilinearMap R (fun x ↦ M) N
x✝ : ι → M
⊢ (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ 0;
      { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ })
      x✝ =
    0 x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f x✝¹ x✝ : ContinuousMultilinearMap R (fun x ↦ M) N
⊢ (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ (x✝¹ + x✝);
    { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ }) =
    (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ x✝¹;
      { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ }) +
      let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ x✝;
      { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ }

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f x✝² x✝¹ : ContinuousMultilinearMap R (fun x ↦ M) N
x✝ : ι → M
⊢ (let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ (x✝² + x✝¹);
      { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ })
      x✝ =
    ((let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ x✝²;
        { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ }) +
        let __ContinuousMultilinearMap := ∑ σ, Equiv.Perm.sign σ • domDomCongr σ x✝¹;
        { toContinuousMultilinearMap := __ContinuousMultilinearMap, map_eq_zero_of_eq' := ⋯ })
      x✝
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
f' : E →L[𝕜] F
x : E
c : 𝕜ˣ
hc : ↑c ≠ 0
⊢ f' ∘SL ↑(ContinuousLinearEquiv.smulLeft c) = c • f'

Tactic:
ext

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
f' : E →L[𝕜] F
x : E
c : 𝕜ˣ
hc : ↑c ≠ 0
x✝ : E
⊢ (f' ∘SL ↑(ContinuousLinearEquiv.smulLeft c)) x✝ = (c • f') x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
f : E [⋀^ι]→L[𝕜] SeparationQuotient F
⊢ (SeparationQuotient.mkCLM 𝕜 F).compContinuousAlternatingMap
      ((SeparationQuotient.outCLM 𝕜 F).compContinuousAlternatingMap f) =
    f

Tactic:
ext

Goal state after:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
f : E [⋀^ι]→L[𝕜] SeparationQuotient F
x✝ : ι → E
⊢ ((SeparationQuotient.mkCLM 𝕜 F).compContinuousAlternatingMap
        ((SeparationQuotient.outCLM 𝕜 F).compContinuousAlternatingMap f))
      x✝ =
    f x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
G : Type u_5
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousConstSMul 𝕜 F
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x ↦ E) F
hf : ∀ (x : G) (v : ι → E) (i j : ι), v i = v j → i ≠ j → (f x) v = 0
x✝¹ x✝ : G
⊢ { toContinuousMultilinearMap := f (x✝¹ + x✝), map_eq_zero_of_eq' := ⋯ } =
    { toContinuousMultilinearMap := f x✝¹, map_eq_zero_of_eq' := ⋯ } +
      { toContinuousMultilinearMap := f x✝, map_eq_zero_of_eq' := ⋯ }

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
G : Type u_5
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousConstSMul 𝕜 F
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x ↦ E) F
hf : ∀ (x : G) (v : ι → E) (i j : ι), v i = v j → i ≠ j → (f x) v = 0
x✝² x✝¹ : G
x✝ : ι → E
⊢ { toContinuousMultilinearMap := f (x✝² + x✝¹), map_eq_zero_of_eq' := ⋯ } x✝ =
    ({ toContinuousMultilinearMap := f x✝², map_eq_zero_of_eq' := ⋯ } +
        { toContinuousMultilinearMap := f x✝¹, map_eq_zero_of_eq' := ⋯ })
      x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
G : Type u_5
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousConstSMul 𝕜 F
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x ↦ E) F
hf : ∀ (x : G) (v : ι → E) (i j : ι), v i = v j → i ≠ j → (f x) v = 0
x✝¹ : 𝕜
x✝ : G
⊢ { toContinuousMultilinearMap := f (x✝¹ • x✝), map_eq_zero_of_eq' := ⋯ } =
    (RingHom.id 𝕜) x✝¹ • { toContinuousMultilinearMap := f x✝, map_eq_zero_of_eq' := ⋯ }

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
G : Type u_5
inst✝³ : AddCommGroup G
inst✝² : Module 𝕜 G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousConstSMul 𝕜 F
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 (fun x ↦ E) F
hf : ∀ (x : G) (v : ι → E) (i j : ι), v i = v j → i ≠ j → (f x) v = 0
x✝² : 𝕜
x✝¹ : G
x✝ : ι → E
⊢ { toContinuousMultilinearMap := f (x✝² • x✝¹), map_eq_zero_of_eq' := ⋯ } x✝ =
    ((RingHom.id 𝕜) x✝² • { toContinuousMultilinearMap := f x✝¹, map_eq_zero_of_eq' := ⋯ }) x✝
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Finite ι
⊢ Continuous compContinuousLinearMapCLM

Tactic:
rcases nonempty_fintype ι
  refine UniformConvergenceCLM.isUniformInducing_postcomp (.id 𝕜)
    (toContinuousMultilinearMapCLM 𝕜 : (E [⋀^ι]→L[𝕜] G) →L[𝕜] _)
    isUniformEmbedding_toContinuousMultilinearMap.isUniformInducing _ |>.isInducing
    |>.continuous_iff |>.mpr ?_
  change Continuous <|
    (toContinuousMultilinearMapCLM 𝕜 : (F [⋀^ι]→L[𝕜] G) →L[𝕜] _).precomp _ ∘
      ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear 𝕜
        (fun _ : ι ↦ E) (fun _ ↦ F) G ∘
      (fun f _ ↦ f)
  fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : ContinuousSMul 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝¹ x✝ : F →L[𝕜] G
⊢ (let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) (x✝¹ + x✝);
    { toLinearMap := __LinearMap, cont := ⋯ }) =
    (let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝¹;
      { toLinearMap := __LinearMap, cont := ⋯ }) +
      let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝;
      { toLinearMap := __LinearMap, cont := ⋯ }

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : ContinuousSMul 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝³ x✝² : F →L[𝕜] G
x✝¹ : E [⋀^ι]→L[𝕜] F
x✝ : ι → E
⊢ ((let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) (x✝³ + x✝²);
        { toLinearMap := __LinearMap, cont := ⋯ })
        x✝¹)
      x✝ =
    (((let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝³;
          { toLinearMap := __LinearMap, cont := ⋯ }) +
          let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝²;
          { toLinearMap := __LinearMap, cont := ⋯ })
        x✝¹)
      x✝
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : ContinuousSMul 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝¹ : 𝕜
x✝ : F →L[𝕜] G
⊢ (let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) (x✝¹ • x✝);
    { toLinearMap := __LinearMap, cont := ⋯ }) =
    (RingHom.id 𝕜) x✝¹ •
      let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝;
      { toLinearMap := __LinearMap, cont := ⋯ }

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : ContinuousSMul 𝕜 E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝³ : 𝕜
x✝² : F →L[𝕜] G
x✝¹ : E [⋀^ι]→L[𝕜] F
x✝ : ι → E
⊢ ((let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) (x✝³ • x✝²);
        { toLinearMap := __LinearMap, cont := ⋯ })
        x✝¹)
      x✝ =
    (((RingHom.id 𝕜) x✝³ •
          let __LinearMap := (compContinuousAlternatingMapₗ 𝕜 E F G) x✝²;
          { toLinearMap := __LinearMap, cont := ⋯ })
        x✝¹)
      x✝
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : IsEmpty ι
⊢ fderivCompContinuousLinearMap = 0

Tactic:
ext

Goal state after:
𝕜 : Type u
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : IsEmpty ι
x✝³ : F [⋀^ι]→L[𝕜] G
x✝² x✝¹ : E →L[𝕜] F
x✝ : ι → E
⊢ ((x✝³.fderivCompContinuousLinearMap x✝²) x✝¹) x✝ = ((0 x✝³ x✝²) x✝¹) x✝
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
f₁ f₂ : F [⋀^ι]→L[𝕜] G
⊢ (f₁ + f₂).fderivCompContinuousLinearMap g = f₁.fderivCompContinuousLinearMap g + f₂.fderivCompContinuousLinearMap g

Tactic:
ext

Goal state after:
𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
f₁ f₂ : F [⋀^ι]→L[𝕜] G
x✝¹ : E →L[𝕜] F
x✝ : ι → E
⊢ (((f₁ + f₂).fderivCompContinuousLinearMap g) x✝¹) x✝ =
    ((f₁.fderivCompContinuousLinearMap g + f₂.fderivCompContinuousLinearMap g) x✝¹) x✝
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
c : 𝕜
f : F [⋀^ι]→L[𝕜] G
⊢ (c • f).fderivCompContinuousLinearMap g = (RingHom.id 𝕜) c • f.fderivCompContinuousLinearMap g

Tactic:
ext

Goal state after:
𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : E →L[𝕜] F
c : 𝕜
f : F [⋀^ι]→L[𝕜] G
x✝¹ : E →L[𝕜] F
x✝ : ι → E
⊢ (((c • f).fderivCompContinuousLinearMap g) x✝¹) x✝ = (((RingHom.id 𝕜) c • f.fderivCompContinuousLinearMap g) x✝¹) x✝
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
⊢ dist x z ≤ dist x y + dist y z

Tactic:
refine induction_on₃ x y z ?_ ?_

Goal state after:
case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
⊢ IsClosed {x | dist x.1 x.2.2 ≤ dist x.1 x.2.1 + dist x.2.1 x.2.2}

case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
x y z : Completion α
⊢ ∀ (a b c : α), dist ↑a ↑c ≤ dist ↑a ↑b + dist ↑b ↑c
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
⊢ s ∈ 𝓤 (Completion α) ↔ ∃ ε > 0, ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s

Tactic:
constructor

Goal state after:
case mp
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
⊢ s ∈ 𝓤 (Completion α) → ∃ ε > 0, ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s

case mpr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
⊢ (∃ ε > 0, ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s) → s ∈ 𝓤 (Completion α)
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t} = {p | ε ≤ dist p.1 p.2} ∪ t

Tactic:
ext

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
x✝ : Completion α × Completion α
⊢ x✝ ∈ {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t} ↔ x✝ ∈ {p | ε ≤ dist p.1 p.2} ∪ t
Module: Mathlib.Topology.MetricSpace.Completion

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α → β
h : Isometry f
x y : Completion α
x✝¹ x✝ : α
⊢ dist (Completion.extension f ↑x✝¹) (Completion.extension f ↑x✝) = dist ↑x✝¹ ↑x✝

Tactic:
simp only [extension_coe h.uniformContinuous, Completion.dist_eq, h.dist_eq]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ HasFTaylorSeriesUpToOn n f p ∅

Tactic:
constructor

Goal state after:
case zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ∀ x ∈ ∅, (p x 0).curry0 = f x

case fderivWithin
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ∀ (m : ℕ), ↑m < n → ∀ x ∈ ∅, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft ∅ x

case cont
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x ↦ p x m) ∅
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
⊢ HasFTaylorSeriesUpToOn N f p s ↔ ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s

Tactic:
constructor

Goal state after:
case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
⊢ HasFTaylorSeriesUpToOn N f p s → ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s

case mpr
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
⊢ (∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s) → HasFTaylorSeriesUpToOn N f p s