ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.uncurryLeft_norm
@[simp]
theorem ContinuousLinearMap.uncurryLeft_norm
    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) :
    ‖f.uncurryLeft‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i => Ei i.succ) G)
Goal: ‖f.uncurryLeft‖ = ‖f‖
-- ContinuousMultilinearMap.uncurryRight_apply
@[simp]
theorem ContinuousMultilinearMap.uncurryRight_apply
    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G))
    (m : ∀ i, Ei i) : f.uncurryRight m = f (init m) (m (last n)) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 (fun i => Ei i.castSucc) (Ei (Fin.last n) →L[𝕜] G)) (m : (i : Fin n.succ) → Ei i)
Goal: f.uncurryRight m = (f (Fin.init m)) (m (Fin.last n))
-- ContinuousMultilinearMap.curryRight_apply
@[simp]
theorem ContinuousMultilinearMap.curryRight_apply (f : ContinuousMultilinearMap 𝕜 Ei G)
    (m : ∀ i : Fin n, Ei <| castSucc i) (x : Ei (last n)) : f.curryRight m x = f (snoc m x) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (m : (i : Fin n) → Ei i.castSucc) (x : Ei (Fin.last n))
Goal: (f.curryRight m) x = f (Fin.snoc m x)
-- ContinuousMultilinearMap.curry_uncurryRight
@[simp]
theorem ContinuousMultilinearMap.curry_uncurryRight
    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G)) :
    f.uncurryRight.curryRight = f 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 (fun i => Ei i.castSucc) (Ei (Fin.last n) →L[𝕜] G))
Goal: f.uncurryRight.curryRight = f
-- ContinuousMultilinearMap.uncurry_curryRight
@[simp]
theorem ContinuousMultilinearMap.uncurry_curryRight (f : ContinuousMultilinearMap 𝕜 Ei G) :
    f.curryRight.uncurryRight = f 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G)
Goal: f.curryRight.uncurryRight = f
-- continuousMultilinearCurryRightEquiv_apply
@[simp]
theorem continuousMultilinearCurryRightEquiv_apply
    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G))
    (v : ∀ i, Ei i) : (continuousMultilinearCurryRightEquiv 𝕜 Ei G) f v = f (init v) (v (last n)) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 (fun i => Ei i.castSucc) (Ei (Fin.last n) →L[𝕜] G)) (v : (i : Fin n.succ) → Ei i)
Goal: ((continuousMultilinearCurryRightEquiv 𝕜 Ei G) f) v = (f (Fin.init v)) (v (Fin.last n))
-- continuousMultilinearCurryRightEquiv_symm_apply
@[simp]
theorem continuousMultilinearCurryRightEquiv_symm_apply (f : ContinuousMultilinearMap 𝕜 Ei G)
    (v : ∀ i : Fin n, Ei <| castSucc i) (x : Ei (last n)) :
    (continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm f v x = f (snoc v x) 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G) (v : (i : Fin n) → Ei i.castSucc) (x : Ei (Fin.last n))
Goal: (((continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm f) v) x = f (Fin.snoc v x)
-- continuousMultilinearCurryRightEquiv_apply'
@[simp]
theorem continuousMultilinearCurryRightEquiv_apply' (f : G[×n]→L[𝕜] G →L[𝕜] G')
    (v : Fin (n + 1) → G) :
    continuousMultilinearCurryRightEquiv' 𝕜 n G G' f v = f (init v) (v (last n)) 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) (G →L[𝕜] G')) (v : Fin (n + 1) → G)
Goal: ((continuousMultilinearCurryRightEquiv' 𝕜 n G G') f) v = (f (Fin.init v)) (v (Fin.last n))
-- continuousMultilinearCurryRightEquiv_symm_apply'
@[simp]
theorem continuousMultilinearCurryRightEquiv_symm_apply' (f : G[×n.succ]→L[𝕜] G')
    (v : Fin n → G) (x : G) :
    (continuousMultilinearCurryRightEquiv' 𝕜 n G G').symm f v x = f (snoc v x) 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') (v : Fin n → G) (x : G)
Goal: (((continuousMultilinearCurryRightEquiv' 𝕜 n G G').symm f) v) x = f (Fin.snoc v x)
-- ContinuousMultilinearMap.curryRight_norm
@[simp]
theorem ContinuousMultilinearMap.curryRight_norm (f : ContinuousMultilinearMap 𝕜 Ei G) :
    ‖f.curryRight‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 Ei G)
Goal: ‖f.curryRight‖ = ‖f‖
-- ContinuousMultilinearMap.uncurryRight_norm
@[simp]
theorem ContinuousMultilinearMap.uncurryRight_norm
    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G)) :
    ‖f.uncurryRight‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {n : ℕ} {Ei : Fin n.succ → Type wEi} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : Fin n.succ) → NormedAddCommGroup (Ei i)] [(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 (fun i => Ei i.castSucc) (Ei (Fin.last n) →L[𝕜] G))
Goal: ‖f.uncurryRight‖ = ‖f‖
-- ContinuousMultilinearMap.curry0_apply
@[simp]
theorem ContinuousMultilinearMap.curry0_apply (x : G') (m : Fin 0 → G) :
    ContinuousMultilinearMap.curry0 𝕜 G x m = x 
Arguments: (𝕜 : Type u) {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (x : G') (m : Fin 0 → G)
Goal: (ContinuousMultilinearMap.curry0 𝕜 G x) m = x
-- ContinuousMultilinearMap.uncurry0_apply
@[simp]
theorem ContinuousMultilinearMap.uncurry0_apply (f : G[×0]→L[𝕜] G') : f.uncurry0 = f 0 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G')
Goal: f.uncurry0 = f 0
-- ContinuousMultilinearMap.apply_zero_curry0
@[simp]
theorem ContinuousMultilinearMap.apply_zero_curry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ContinuousMultilinearMap.curry0 𝕜 G (f x) = f 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') {x : Fin 0 → G}
Goal: ContinuousMultilinearMap.curry0 𝕜 G (f x) = f
-- ContinuousMultilinearMap.uncurry0_curry0
theorem ContinuousMultilinearMap.uncurry0_curry0 (f : G[×0]→L[𝕜] G') :
    ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0 = f 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G')
Goal: ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0 = f
-- ContinuousMultilinearMap.curry0_uncurry0
theorem ContinuousMultilinearMap.curry0_uncurry0 (x : G') :
    (ContinuousMultilinearMap.curry0 𝕜 G x).uncurry0 = x 
Arguments: (𝕜 : Type u) (G : Type wG) {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (x : G')
Goal: (ContinuousMultilinearMap.curry0 𝕜 G x).uncurry0 = x
-- ContinuousMultilinearMap.curry0_norm
@[simp]
theorem ContinuousMultilinearMap.curry0_norm (x : G') :
    ‖ContinuousMultilinearMap.curry0 𝕜 G x‖ = ‖x‖ 
Arguments: (𝕜 : Type u) (G : Type wG) {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (x : G')
Goal: ‖ContinuousMultilinearMap.curry0 𝕜 G x‖ = ‖x‖
-- ContinuousMultilinearMap.fin0_apply_norm
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') {x : Fin 0 → G}
Goal: ‖f x‖ = ‖f‖
-- ContinuousMultilinearMap.uncurry0_norm
theorem ContinuousMultilinearMap.uncurry0_norm (f : G[×0]→L[𝕜] G') : ‖f.uncurry0‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G')
Goal: ‖f.uncurry0‖ = ‖f‖
-- continuousMultilinearCurryFin0_apply
@[simp]
theorem continuousMultilinearCurryFin0_apply (f : G[×0]→L[𝕜] G') :
    continuousMultilinearCurryFin0 𝕜 G G' f = f 0 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G')
Goal: (continuousMultilinearCurryFin0 𝕜 G G') f = f 0
-- continuousMultilinearCurryFin0_symm_apply
@[simp]
theorem continuousMultilinearCurryFin0_symm_apply (x : G') (v : Fin 0 → G) :
    (continuousMultilinearCurryFin0 𝕜 G G').symm x v = x 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (x : G') (v : Fin 0 → G)
Goal: ((continuousMultilinearCurryFin0 𝕜 G G').symm x) v = x
-- continuousMultilinearCurryFin1_apply
@[simp]
theorem continuousMultilinearCurryFin1_apply (f : G[×1]→L[𝕜] G') (x : G) :
    continuousMultilinearCurryFin1 𝕜 G G' f x = f (Fin.snoc 0 x) 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') (x : G)
Goal: ((continuousMultilinearCurryFin1 𝕜 G G') f) x = f (Fin.snoc 0 x)
-- continuousMultilinearCurryFin1_symm_apply
@[simp]
theorem continuousMultilinearCurryFin1_symm_apply (f : G →L[𝕜] G') (v : Fin 1 → G) :
    (continuousMultilinearCurryFin1 𝕜 G G').symm f v = f (v 0) 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →L[𝕜] G') (v : Fin 1 → G)
Goal: ((continuousMultilinearCurryFin1 𝕜 G G').symm f) v = f (v 0)
-- ContinuousMultilinearMap.norm_domDomCongr
@[simp]
theorem norm_domDomCongr (σ : ι ≃ ι') (f : ContinuousMultilinearMap 𝕜 (fun _ : ι => G) G') :
    ‖domDomCongr σ f‖ = ‖f‖ 
Arguments: (𝕜 : Type u) {ι : Type v} {ι' : Type v'} (G : Type wG) (G' : Type wG') [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (σ : ι ≃ ι') (f : ContinuousMultilinearMap 𝕜 (fun x => G) G')
Goal: ‖ContinuousMultilinearMap.domDomCongr σ f‖ = ‖f‖
-- ContinuousMultilinearMap.currySum_apply
@[simp]
theorem currySum_apply (f : ContinuousMultilinearMap 𝕜 (fun _ : ι ⊕ ι' => G) G') (m : ι → G)
    (m' : ι' → G) : f.currySum m m' = f (Sum.elim m m') 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {G : Type wG} {G' : Type wG'} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun x => G) G') (m : ι → G) (m' : ι' → G)
Goal: (f.currySum m) m' = f (Sum.elim m m')
-- ContinuousMultilinearMap.uncurrySum_apply
@[simp]
theorem uncurrySum_apply (f : ContinuousMultilinearMap 𝕜 (fun _ : ι => G)
    (ContinuousMultilinearMap 𝕜 (fun _ : ι' => G) G'))
    (m : ι ⊕ ι' → G) : f.uncurrySum m = f (m ∘ Sum.inl) (m ∘ Sum.inr) 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {G : Type wG} {G' : Type wG'} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 (fun x => G) (ContinuousMultilinearMap 𝕜 (fun x => G) G')) (m : ι ⊕ ι' → G)
Goal: f.uncurrySum m = (f (m ∘ Sum.inl)) (m ∘ Sum.inr)
-- ContinuousMultilinearMap.curryFinFinset_apply
@[simp]
theorem curryFinFinset_apply (hk : s.card = k) (hl : sᶜ.card = l) (f : G[×n]→L[𝕜] G')
    (mk : Fin k → G) (ml : Fin l → G) : curryFinFinset 𝕜 G G' hk hl f mk ml =
      f fun i => Sum.elim mk ml ((finSumEquivOfFinset hk hl).symm i) 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k l : ℕ} {s : Finset (Fin n)} (hk : s.card = k) (hl : sᶜ.card = l) (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') (mk : Fin k → G) (ml : Fin l → G)
Goal: (((ContinuousMultilinearMap.curryFinFinset 𝕜 G G' hk hl) f) mk) ml =
  f fun i => Sum.elim mk ml ((finSumEquivOfFinset hk hl).symm i)
-- ContinuousMultilinearMap.curryFinFinset_symm_apply
@[simp]
theorem curryFinFinset_symm_apply (hk : s.card = k) (hl : sᶜ.card = l)
    (f : G[×k]→L[𝕜] G[×l]→L[𝕜] G') (m : Fin n → G) : (curryFinFinset 𝕜 G G' hk hl).symm f m =
      f (fun i => m <| finSumEquivOfFinset hk hl (Sum.inl i)) fun i =>
        m <| finSumEquivOfFinset hk hl (Sum.inr i) 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k l : ℕ} {s : Finset (Fin n)} (hk : s.card = k) (hl : sᶜ.card = l) (f : ContinuousMultilinearMap 𝕜 (fun i => G) (ContinuousMultilinearMap 𝕜 (fun i => G) G')) (m : Fin n → G)
Goal: ((ContinuousMultilinearMap.curryFinFinset 𝕜 G G' hk hl).symm f) m =
  (f fun i => m ((finSumEquivOfFinset hk hl) (Sum.inl i))) fun i => m ((finSumEquivOfFinset hk hl) (Sum.inr i))
-- ContinuousMultilinearMap.curryFinFinset_symm_apply_piecewise_const
theorem curryFinFinset_symm_apply_piecewise_const (hk : s.card = k) (hl : sᶜ.card = l)
    (f : G[×k]→L[𝕜] G[×l]→L[𝕜] G') (x y : G) :
    (curryFinFinset 𝕜 G G' hk hl).symm f (s.piecewise (fun _ => x) fun _ => y) =
      f (fun _ => x) fun _ => y 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k l : ℕ} {s : Finset (Fin n)} (hk : s.card = k) (hl : sᶜ.card = l) (f : ContinuousMultilinearMap 𝕜 (fun i => G) (ContinuousMultilinearMap 𝕜 (fun i => G) G')) (x y : G)
Goal: ((ContinuousMultilinearMap.curryFinFinset 𝕜 G G' hk hl).symm f) (s.piecewise (fun x_1 => x) fun x => y) =
  (f fun x_1 => x) fun x => y
-- ContinuousMultilinearMap.curryFinFinset_symm_apply_const
@[simp]
theorem curryFinFinset_symm_apply_const (hk : s.card = k) (hl : sᶜ.card = l)
    (f : G[×k]→L[𝕜] G[×l]→L[𝕜] G') (x : G) :
    ((curryFinFinset 𝕜 G G' hk hl).symm f fun _ => x) = f (fun _ => x) fun _ => x 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k l : ℕ} {s : Finset (Fin n)} (hk : s.card = k) (hl : sᶜ.card = l) (f : ContinuousMultilinearMap 𝕜 (fun i => G) (ContinuousMultilinearMap 𝕜 (fun i => G) G')) (x : G)
Goal: (((ContinuousMultilinearMap.curryFinFinset 𝕜 G G' hk hl).symm f) fun x_1 => x) = (f fun x_1 => x) fun x_1 => x
-- ContinuousMultilinearMap.curryFinFinset_apply_const
theorem curryFinFinset_apply_const (hk : s.card = k) (hl : sᶜ.card = l) (f : G[×n]→L[𝕜] G')
    (x y : G) : (curryFinFinset 𝕜 G G' hk hl f (fun _ => x) fun _ => y) =
      f (s.piecewise (fun _ => x) fun _ => y) 
Arguments: {𝕜 : Type u} {n : ℕ} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k l : ℕ} {s : Finset (Fin n)} (hk : s.card = k) (hl : sᶜ.card = l) (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') (x y : G)
Goal: ((((ContinuousMultilinearMap.curryFinFinset 𝕜 G G' hk hl) f) fun x_1 => x) fun x => y) =
  f (s.piecewise (fun x_1 => x) fun x => y)
-- VectorFourier.fourierIntegral_smul_const
theorem fourierIntegral_smul_const (e : AddChar 𝕜 𝕊) (μ : Measure V)
    (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (r : ℂ) :
    fourierIntegral e μ L (r • f) = r • fourierIntegral e μ L f 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (r : ℂ)
Goal: VectorFourier.fourierIntegral e μ L (r • f) = r • VectorFourier.fourierIntegral e μ L f
-- VectorFourier.norm_fourierIntegral_le_integral_norm
theorem norm_fourierIntegral_le_integral_norm (e : AddChar 𝕜 𝕊) (μ : Measure V)
    (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) :
    ‖fourierIntegral e μ L f w‖ ≤ ∫ v : V, ‖f v‖ ∂μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W)
Goal: ‖VectorFourier.fourierIntegral e μ L f w‖ ≤ ∫ (v : V), ‖f v‖ ∂μ
-- VectorFourier.fourierIntegral_comp_add_right
theorem fourierIntegral_comp_add_right [MeasurableAdd V] (e : AddChar 𝕜 𝕊) (μ : Measure V)
    [μ.IsAddRightInvariant] (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V) :
    fourierIntegral e μ L (f ∘ fun v ↦ v + v₀) =
      fun w ↦ e (L v₀ w) • fourierIntegral e μ L f w 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [MeasurableAdd V] (e : AddChar 𝕜 ↥𝕊) (μ : MeasureTheory.Measure V) [μ.IsAddRightInvariant] (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V)
Goal: VectorFourier.fourierIntegral e μ L (f ∘ fun v => v + v₀) = fun w =>
  e ((L v₀) w) • VectorFourier.fourierIntegral e μ L f w
-- VectorFourier.fourierIntegral_convergent_iff
theorem fourierIntegral_convergent_iff (he : Continuous e)
    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (w : W) :
    Integrable (fun v : V ↦ e (-L v w) • f v) μ ↔ Integrable f μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} (he : Continuous ⇑e) (hL : Continuous fun p => (L p.1) p.2) {f : V → E} (w : W)
Goal: MeasureTheory.Integrable (fun v => e (-(L v) w) • f v) μ ↔ MeasureTheory.Integrable f μ
-- VectorFourier.fourierIntegral_add
theorem fourierIntegral_add (he : Continuous e) (hL : Continuous fun p : V × W ↦ L p.1 p.2)
    {f g : V → E} (hf : Integrable f μ) (hg : Integrable g μ) :
    fourierIntegral e μ L f + fourierIntegral e μ L g = fourierIntegral e μ L (f + g) 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} (he : Continuous ⇑e) (hL : Continuous fun p => (L p.1) p.2) {f g : V → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: VectorFourier.fourierIntegral e μ L f + VectorFourier.fourierIntegral e μ L g =
  VectorFourier.fourierIntegral e μ L (f + g)
-- VectorFourier.fourierIntegral_continuous
theorem fourierIntegral_continuous [FirstCountableTopology W] (he : Continuous e)
    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (hf : Integrable f μ) :
    Continuous (fourierIntegral e μ L f) 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} [FirstCountableTopology W] (he : Continuous ⇑e) (hL : Continuous fun p => (L p.1) p.2) {f : V → E} (hf : MeasureTheory.Integrable f μ)
Goal: Continuous (VectorFourier.fourierIntegral e μ L f)
-- VectorFourier.integral_bilin_fourierIntegral_eq_flip
theorem integral_bilin_fourierIntegral_eq_flip
    {f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous e)
    (hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :
    ∫ ξ, M (fourierIntegral e μ L f ξ) (g ξ) ∂ν =
      ∫ x, M (f x) (fourierIntegral e ν L.flip g x) ∂μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} {F : Type u_5} {G : Type u_6} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedAddCommGroup F] [NormedSpace ℂ F] [NormedAddCommGroup G] [NormedSpace ℂ G] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} {ν : MeasureTheory.Measure W} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [SecondCountableTopology V] [CompleteSpace E] [CompleteSpace F] {f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous ⇑e) (hL : Continuous fun p => (L p.1) p.2) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν)
Goal: ∫ (ξ : W), (M (VectorFourier.fourierIntegral e μ L f ξ)) (g ξ) ∂ν =
  ∫ (x : V), (M (f x)) (VectorFourier.fourierIntegral e ν L.flip g x) ∂μ
-- VectorFourier.integral_fourierIntegral_smul_eq_flip
theorem integral_fourierIntegral_smul_eq_flip
    {f : V → ℂ} {g : W → F} (he : Continuous e)
    (hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :
    ∫ ξ, (fourierIntegral e μ L f ξ) • (g ξ) ∂ν =
      ∫ x, (f x) • (fourierIntegral e ν L.flip g x) ∂μ 
Arguments: {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace ℂ F] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W] {e : AddChar 𝕜 ↥𝕊} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} {ν : MeasureTheory.Measure W} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [SecondCountableTopology V] [CompleteSpace F] {f : V → ℂ} {g : W → F} (he : Continuous ⇑e) (hL : Continuous fun p => (L p.1) p.2) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν)
Goal: ∫ (ξ : W), VectorFourier.fourierIntegral e μ L f ξ • g ξ ∂ν =
  ∫ (x : V), f x • VectorFourier.fourierIntegral e ν L.flip g x ∂μ