ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.componentComplMk_mem
theorem componentComplMk_mem (G : SimpleGraph V) {v : V} (vK : v ∉ K) : v ∈ G.componentComplMk vK 
Arguments: {V : Type u} {K : Set V} (G : SimpleGraph V) {v : V} (vK : v ∉ K)
Goal: v ∈ G.componentComplMk vK
-- SimpleGraph.componentComplMk_eq_of_adj
theorem componentComplMk_eq_of_adj (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K)
    (a : G.Adj v w) : G.componentComplMk vK = G.componentComplMk wK 
Arguments: {V : Type u} {K : Set V} (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K) (a : G.Adj v w)
Goal: G.componentComplMk vK = G.componentComplMk wK
-- SimpleGraph.ComponentCompl.ind
@[elab_as_elim] -- Porting note: added
protected theorem ind {β : G.ComponentCompl K → Prop}
    (f : ∀ ⦃v⦄ (hv : v ∉ K), β (G.componentComplMk hv)) : ∀ C : G.ComponentCompl K, β C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {β : G.ComponentCompl K → Prop} (f : ∀ ⦃v : V⦄ (hv : v ∉ K), β (G.componentComplMk hv)) (C : G.ComponentCompl K)
Goal: β C
-- SimpleGraph.ComponentCompl.coe_inj
theorem coe_inj {C D : G.ComponentCompl K} : (C : Set V) = (D : Set V) ↔ C = D 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {C D : G.ComponentCompl K}
Goal: ↑C = ↑D ↔ C = D
-- SimpleGraph.ComponentCompl.nonempty
@[simp]
protected theorem nonempty (C : G.ComponentCompl K) : (C : Set V).Nonempty 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K)
Goal: (↑C).Nonempty
-- SimpleGraph.ComponentCompl.exists_eq_mk
protected theorem exists_eq_mk (C : G.ComponentCompl K) :
    ∃ (v : _) (h : v ∉ K), G.componentComplMk h = C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K)
Goal: ∃ v, ∃ (h : v ∉ K), G.componentComplMk h = C
-- SimpleGraph.ComponentCompl.disjoint_right
protected theorem disjoint_right (C : G.ComponentCompl K) : Disjoint K C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K)
Goal: Disjoint K ↑C
-- SimpleGraph.ComponentCompl.not_mem_of_mem
theorem not_mem_of_mem {C : G.ComponentCompl K} {c : V} (cC : c ∈ C) : c ∉ K 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {C : G.ComponentCompl K} {c : V} (cC : c ∈ C)
Goal: c ∉ K
-- SimpleGraph.ComponentCompl.pairwise_disjoint
protected theorem pairwise_disjoint :
    Pairwise fun C D : G.ComponentCompl K => Disjoint (C : Set V) (D : Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V}
Goal: Pairwise fun C D => Disjoint ↑C ↑D
-- SimpleGraph.ComponentCompl.mem_of_adj
theorem mem_of_adj : ∀ {C : G.ComponentCompl K} (c d : V), c ∈ C → d ∉ K → G.Adj c d → d ∈ C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {C : G.ComponentCompl K} (c d : V)
Goal: c ∈ C → d ∉ K → G.Adj c d → d ∈ C
-- SimpleGraph.ComponentCompl.exists_adj_boundary_pair
theorem exists_adj_boundary_pair (Gc : G.Preconnected) (hK : K.Nonempty) :
    ∀ C : G.ComponentCompl K, ∃ ck : V × V, ck.1 ∈ C ∧ ck.2 ∈ K ∧ G.Adj ck.1 ck.2 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} (Gc : G.Preconnected) (hK : K.Nonempty) (C : G.ComponentCompl K)
Goal: ∃ ck, ck.1 ∈ C ∧ ck.2 ∈ K ∧ G.Adj ck.1 ck.2
-- SimpleGraph.ComponentCompl.subset_hom
theorem subset_hom (C : G.ComponentCompl L) (h : K ⊆ L) : (C : Set V) ⊆ (C.hom h : Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L)
Goal: ↑C ⊆ ↑(SimpleGraph.ComponentCompl.hom h C)
-- SimpleGraph.componentComplMk_mem_hom
theorem _root_.SimpleGraph.componentComplMk_mem_hom
    (G : SimpleGraph V) {v : V} (vK : v ∉ K) (h : L ⊆ K) :
    v ∈ (G.componentComplMk vK).hom h 
Arguments: {V : Type u} {K L : Set V} (G : SimpleGraph V) {v : V} (vK : v ∉ K) (h : L ⊆ K)
Goal: v ∈ SimpleGraph.ComponentCompl.hom h (G.componentComplMk vK)
-- SimpleGraph.ComponentCompl.hom_eq_iff_le
theorem hom_eq_iff_le (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) :
    C.hom h = D ↔ (C : Set V) ⊆ (D : Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K)
Goal: SimpleGraph.ComponentCompl.hom h C = D ↔ ↑C ⊆ ↑D
-- SimpleGraph.ComponentCompl.hom_eq_iff_not_disjoint
theorem hom_eq_iff_not_disjoint (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) :
    C.hom h = D ↔ ¬Disjoint (C : Set V) (D : Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K)
Goal: SimpleGraph.ComponentCompl.hom h C = D ↔ ¬Disjoint ↑C ↑D
-- SimpleGraph.ComponentCompl.hom_refl
theorem hom_refl (C : G.ComponentCompl L) : C.hom (subset_refl L) = C 
Arguments: {V : Type u} {G : SimpleGraph V} {L : Set V} (C : G.ComponentCompl L)
Goal: SimpleGraph.ComponentCompl.hom ⋯ C = C
-- SimpleGraph.ComponentCompl.hom_trans
theorem hom_trans (C : G.ComponentCompl L) (h : K ⊆ L) (h' : M ⊆ K) :
    C.hom (h'.trans h) = (C.hom h).hom h' 
Arguments: {V : Type u} {G : SimpleGraph V} {K L M : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (h' : M ⊆ K)
Goal: SimpleGraph.ComponentCompl.hom ⋯ C = SimpleGraph.ComponentCompl.hom h' (SimpleGraph.ComponentCompl.hom h C)
-- SimpleGraph.ComponentCompl.hom_mk
theorem hom_mk {v : V} (vnL : v ∉ L) (h : K ⊆ L) :
    (G.componentComplMk vnL).hom h = G.componentComplMk (Set.not_mem_subset h vnL) 
Arguments: {V : Type u} {G : SimpleGraph V} {K L : Set V} {v : V} (vnL : v ∉ L) (h : K ⊆ L)
Goal: SimpleGraph.ComponentCompl.hom h (G.componentComplMk vnL) = G.componentComplMk ⋯
-- SimpleGraph.ComponentCompl.hom_infinite
theorem hom_infinite (C : G.ComponentCompl L) (h : K ⊆ L) (Cinf : (C : Set V).Infinite) :
    (C.hom h : Set V).Infinite 
Arguments: {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (Cinf : (↑C).Infinite)
Goal: (↑(SimpleGraph.ComponentCompl.hom h C)).Infinite
-- SimpleGraph.ComponentCompl.infinite_iff_in_all_ranges
theorem infinite_iff_in_all_ranges {K : Finset V} (C : G.ComponentCompl K) :
    C.supp.Infinite ↔ ∀ (L) (h : K ⊆ L), ∃ D : G.ComponentCompl L, D.hom h = C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Finset V} (C : G.ComponentCompl ↑K)
Goal: C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ D, SimpleGraph.ComponentCompl.hom h D = C
-- SimpleGraph.componentComplFunctor_map
@[simps]
def componentComplFunctor : (Finset V)ᵒᵖ ⥤ Type u 
Arguments: {V : Type u} (G : SimpleGraph V)
Goal: ∀ {X Y : (Finset V)ᵒᵖ} (f : X ⟶ Y) (C : G.ComponentCompl ↑(Opposite.unop X)),
  G.componentComplFunctor.map f C = SimpleGraph.ComponentCompl.hom ⋯ C
-- SimpleGraph.componentComplFunctor_obj
@[simps]
def componentComplFunctor : (Finset V)ᵒᵖ ⥤ Type u 
Arguments: {V : Type u} (G : SimpleGraph V) (K : (Finset V)ᵒᵖ)
Goal: G.componentComplFunctor.obj K = G.ComponentCompl ↑(Opposite.unop K)
-- SimpleGraph.end_hom_mk_of_mk
theorem end_hom_mk_of_mk {s} (sec : s ∈ G.end) {K L : (Finset V)ᵒᵖ} (h : L ⟶ K) {v : V}
    (vnL : v ∉ L.unop) (hs : s L = G.componentComplMk vnL) :
    s K = G.componentComplMk (Set.not_mem_subset (le_of_op_hom h : _ ⊆ _) vnL) 
Arguments: {V : Type u} (G : SimpleGraph V) {s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j} (sec : s ∈ G.end) {K L : (Finset V)ᵒᵖ} (h : L ⟶ K) {v : V} (vnL : v ∉ Opposite.unop L) (hs : s L = G.componentComplMk vnL)
Goal: s K = G.componentComplMk ⋯
-- SimpleGraph.infinite_iff_in_eventualRange
theorem infinite_iff_in_eventualRange {K : (Finset V)ᵒᵖ} (C : G.componentComplFunctor.obj K) :
    C.supp.Infinite ↔ C ∈ G.componentComplFunctor.eventualRange K 
Arguments: {V : Type u} (G : SimpleGraph V) {K : (Finset V)ᵒᵖ} (C : G.componentComplFunctor.obj K)
Goal: (SimpleGraph.ComponentCompl.supp C).Infinite ↔ C ∈ G.componentComplFunctor.eventualRange K
-- SimpleGraph.Iso.card_edgeFinset_eq
theorem card_edgeFinset_eq [Fintype G.edgeSet] [Fintype G'.edgeSet] :
    G.edgeFinset.card = G'.edgeFinset.card 
Arguments: {V : Type u_1} {G : SimpleGraph V} {W : Type u_2} {G' : SimpleGraph W} (f : G ≃g G') [Fintype ↑G.edgeSet] [Fintype ↑G'.edgeSet]
Goal: G.edgeFinset.card = G'.edgeFinset.card
-- SimpleGraph.not_adj_replaceVertex_same
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) (s t : V)
Goal: ¬(G.replaceVertex s t).Adj s t
-- SimpleGraph.replaceVertex_self
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) (s : V)
Goal: G.replaceVertex s s = G
-- SimpleGraph.adj_replaceVertex_iff_of_ne_left
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
    (G.replaceVertex s t).Adj s w ↔ G.Adj s w 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) (s : V) {t w : V} (hw : w ≠ t)
Goal: (G.replaceVertex s t).Adj s w ↔ G.Adj s w
-- SimpleGraph.adj_replaceVertex_iff_of_ne_right
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
    (G.replaceVertex s t).Adj t w ↔ G.Adj s w 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) (s : V) {t w : V} (hw : w ≠ t)
Goal: (G.replaceVertex s t).Adj t w ↔ G.Adj s w
-- SimpleGraph.adj_replaceVertex_iff_of_ne
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
    (G.replaceVertex s t).Adj v w ↔ G.Adj v w 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) (s : V) {t v w : V} (hv : v ≠ t) (hw : w ≠ t)
Goal: (G.replaceVertex s t).Adj v w ↔ G.Adj v w
-- SimpleGraph.edgeSet_replaceVertex_of_not_adj
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
    G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} (hn : ¬G.Adj s t)
Goal: (G.replaceVertex s t).edgeSet = G.edgeSet \ G.incidenceSet t ∪ (fun x => s(x, t)) '' G.neighborSet s
-- SimpleGraph.edgeSet_replaceVertex_of_adj
theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
    (G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} (ha : G.Adj s t)
Goal: (G.replaceVertex s t).edgeSet = (G.edgeSet \ G.incidenceSet t ∪ (fun x => s(x, t)) '' G.neighborSet s) \ {s(t, t)}
-- SimpleGraph.edgeFinset_replaceVertex_of_not_adj
theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
    G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] (hn : ¬G.Adj s t)
Goal: (G.replaceVertex s t).edgeFinset =
  G.edgeFinset \ G.incidenceFinset t ∪ Finset.image (fun x => s(x, t)) (G.neighborFinset s)
-- SimpleGraph.edgeFinset_replaceVertex_of_adj
theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
    (G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t))) \ {s(t, t)} 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] (ha : G.Adj s t)
Goal: (G.replaceVertex s t).edgeFinset =
  (G.edgeFinset \ G.incidenceFinset t ∪ Finset.image (fun x => s(x, t)) (G.neighborFinset s)) \ {s(t, t)}
-- SimpleGraph.disjoint_sdiff_neighborFinset_image
lemma disjoint_sdiff_neighborFinset_image :
    Disjoint (G.edgeFinset \ G.incidenceFinset t) ((G.neighborFinset s).image (s(·, t))) 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj]
Goal: Disjoint (G.edgeFinset \ G.incidenceFinset t) (Finset.image (fun x => s(x, t)) (G.neighborFinset s))
-- SimpleGraph.card_edgeFinset_replaceVertex_of_not_adj
theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
    (G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] (hn : ¬G.Adj s t)
Goal: (G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t
-- SimpleGraph.card_edgeFinset_replaceVertex_of_adj
theorem card_edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) :
    (G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t - 1 
Arguments: {V : Type u_1} [DecidableEq V] (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] (ha : G.Adj s t)
Goal: (G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t - 1
-- SimpleGraph.edge_adj
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w 
Arguments: {V : Type u_1} (s t v w : V)
Goal: (SimpleGraph.edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w
-- SimpleGraph.edge_self_eq_bot
lemma edge_self_eq_bot : edge s s = ⊥ 
Arguments: {V : Type u_1} (s : V)
Goal: SimpleGraph.edge s s = ⊥
-- SimpleGraph.sup_edge_self
@[simp]
lemma sup_edge_self : G ⊔ edge s s = G 
Arguments: {V : Type u_1} (G : SimpleGraph V) (s : V)
Goal: G ⊔ SimpleGraph.edge s s = G
-- SimpleGraph.edge_edgeSet_of_ne
lemma edge_edgeSet_of_ne (h : s ≠ t) : (edge s t).edgeSet = {s(s, t)} 
Arguments: {V : Type u_1} {s t : V} (h : s ≠ t)
Goal: (SimpleGraph.edge s t).edgeSet = {s(s, t)}
-- SimpleGraph.sup_edge_of_adj
lemma sup_edge_of_adj (h : G.Adj s t) : G ⊔ edge s t = G 
Arguments: {V : Type u_1} (G : SimpleGraph V) {s t : V} (h : G.Adj s t)
Goal: G ⊔ SimpleGraph.edge s t = G
-- SimpleGraph.edgeFinset_sup_edge
theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
    (G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) 
Arguments: {V : Type u_1} (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] [Fintype ↑(G ⊔ SimpleGraph.edge s t).edgeSet] (hn : ¬G.Adj s t) (h : s ≠ t)
Goal: (G ⊔ SimpleGraph.edge s t).edgeFinset = Finset.cons s(s, t) G.edgeFinset ⋯
-- SimpleGraph.card_edgeFinset_sup_edge
theorem card_edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
    (G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1 
Arguments: {V : Type u_1} (G : SimpleGraph V) {s t : V} [Fintype V] [DecidableRel G.Adj] [Fintype ↑(G ⊔ SimpleGraph.edge s t).edgeSet] (hn : ¬G.Adj s t) (h : s ≠ t)
Goal: (G ⊔ SimpleGraph.edge s t).edgeFinset.card = G.edgeFinset.card + 1
-- iteratedFDerivWithin_zero_fun
@[simp]
theorem iteratedFDerivWithin_zero_fun (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} :
    iteratedFDerivWithin 𝕜 i (fun _ : E ↦ (0 : F)) s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ}
Goal: iteratedFDerivWithin 𝕜 i (fun x => 0) s x = 0
-- iteratedFDeriv_zero_fun
@[simp]
theorem iteratedFDeriv_zero_fun {n : ℕ} : (iteratedFDeriv 𝕜 n fun _ : E ↦ (0 : F)) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ}
Goal: (iteratedFDeriv 𝕜 n fun x => 0) = 0
-- contDiff_zero_fun
theorem contDiff_zero_fun : ContDiff 𝕜 n fun _ : E => (0 : F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞}
Goal: ContDiff 𝕜 n fun x => 0
-- contDiff_const
theorem contDiff_const {c : F} : ContDiff 𝕜 n fun _ : E => c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {c : F}
Goal: ContDiff 𝕜 n fun x => c
-- contDiffOn_const
theorem contDiffOn_const {c : F} {s : Set E} : ContDiffOn 𝕜 n (fun _ : E => c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {c : F} {s : Set E}
Goal: ContDiffOn 𝕜 n (fun x => c) s
-- contDiffAt_const
theorem contDiffAt_const {c : F} : ContDiffAt 𝕜 n (fun _ : E => c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {c : F}
Goal: ContDiffAt 𝕜 n (fun x => c) x
-- contDiffWithinAt_const
theorem contDiffWithinAt_const {c : F} : ContDiffWithinAt 𝕜 n (fun _ : E => c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} {n : ℕ∞} {c : F}
Goal: ContDiffWithinAt 𝕜 n (fun x => c) s x
-- contDiff_of_subsingleton
@[nontriviality]
theorem contDiff_of_subsingleton [Subsingleton F] : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} [Subsingleton F]
Goal: ContDiff 𝕜 n f
-- contDiffAt_of_subsingleton
@[nontriviality]
theorem contDiffAt_of_subsingleton [Subsingleton F] : ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {n : ℕ∞} [Subsingleton F]
Goal: ContDiffAt 𝕜 n f x
-- contDiffWithinAt_of_subsingleton
@[nontriviality]
theorem contDiffWithinAt_of_subsingleton [Subsingleton F] : ContDiffWithinAt 𝕜 n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} [Subsingleton F]
Goal: ContDiffWithinAt 𝕜 n f s x
-- contDiffOn_of_subsingleton
@[nontriviality]
theorem contDiffOn_of_subsingleton [Subsingleton F] : ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} [Subsingleton F]
Goal: ContDiffOn 𝕜 n f s
-- iteratedFDerivWithin_succ_const
theorem iteratedFDerivWithin_succ_const (n : ℕ) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 (n + 1) (fun _ : E ↦ c) s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} (n : ℕ) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 (n + 1) (fun x => c) s x = 0
-- iteratedFDeriv_succ_const
theorem iteratedFDeriv_succ_const (n : ℕ) (c : F) :
    (iteratedFDeriv 𝕜 (n + 1) fun _ : E ↦ c) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : ℕ) (c : F)
Goal: (iteratedFDeriv 𝕜 (n + 1) fun x => c) = 0
-- iteratedFDerivWithin_const_of_ne
theorem iteratedFDerivWithin_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F)
    (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 n (fun _ : E ↦ c) s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} {n : ℕ} (hn : n ≠ 0) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 n (fun x => c) s x = 0
-- iteratedFDeriv_const_of_ne
theorem iteratedFDeriv_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F) :
    (iteratedFDeriv 𝕜 n fun _ : E ↦ c) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (hn : n ≠ 0) (c : F)
Goal: (iteratedFDeriv 𝕜 n fun x => c) = 0
-- IsBoundedLinearMap.contDiff
theorem IsBoundedLinearMap.contDiff (hf : IsBoundedLinearMap 𝕜 f) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} (hf : IsBoundedLinearMap 𝕜 f)
Goal: ContDiff 𝕜 n f
-- ContinuousLinearMap.contDiff
theorem ContinuousLinearMap.contDiff (f : E →L[𝕜] F) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (f : E →L[𝕜] F)
Goal: ContDiff 𝕜 n ⇑f
-- ContinuousLinearEquiv.contDiff
theorem ContinuousLinearEquiv.contDiff (f : E ≃L[𝕜] F) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (f : E ≃L[𝕜] F)
Goal: ContDiff 𝕜 n ⇑f
-- LinearIsometry.contDiff
theorem LinearIsometry.contDiff (f : E →ₗᵢ[𝕜] F) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (f : E →ₗᵢ[𝕜] F)
Goal: ContDiff 𝕜 n ⇑f
-- LinearIsometryEquiv.contDiff
theorem LinearIsometryEquiv.contDiff (f : E ≃ₗᵢ[𝕜] F) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (f : E ≃ₗᵢ[𝕜] F)
Goal: ContDiff 𝕜 n ⇑f
-- contDiff_id
theorem contDiff_id : ContDiff 𝕜 n (id : E → E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞}
Goal: ContDiff 𝕜 n id
-- contDiffWithinAt_id
theorem contDiffWithinAt_id {s x} : ContDiffWithinAt 𝕜 n (id : E → E) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {s : Set E} {x : E}
Goal: ContDiffWithinAt 𝕜 n id s x
-- contDiffAt_id
theorem contDiffAt_id {x} : ContDiffAt 𝕜 n (id : E → E) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {x : E}
Goal: ContDiffAt 𝕜 n id x
-- contDiffOn_id
theorem contDiffOn_id {s} : ContDiffOn 𝕜 n (id : E → E) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {s : Set E}
Goal: ContDiffOn 𝕜 n id s
-- IsBoundedBilinearMap.contDiff
theorem IsBoundedBilinearMap.contDiff (hb : IsBoundedBilinearMap 𝕜 b) : ContDiff 𝕜 n b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {b : E × F → G} {n : ℕ∞} (hb : IsBoundedBilinearMap 𝕜 b)
Goal: ContDiff 𝕜 n b
-- HasFTaylorSeriesUpToOn.continuousLinearMap_comp
theorem HasFTaylorSeriesUpToOn.continuousLinearMap_comp (g : F →L[𝕜] G)
    (hf : HasFTaylorSeriesUpToOn n f p s) :
    HasFTaylorSeriesUpToOn n (g ∘ f) (fun x k => g.compContinuousMultilinearMap (p x k)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (g : F →L[𝕜] G) (hf : HasFTaylorSeriesUpToOn n f p s)
Goal: HasFTaylorSeriesUpToOn n (⇑g ∘ f) (fun x k => g.compContinuousMultilinearMap (p x k)) s
-- ContDiffWithinAt.continuousLinearMap_comp
theorem ContDiffWithinAt.continuousLinearMap_comp (g : F →L[𝕜] G)
    (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (g : F →L[𝕜] G) (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (⇑g ∘ f) s x
-- ContDiffAt.continuousLinearMap_comp
theorem ContDiffAt.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : ContDiffAt 𝕜 n f x) :
    ContDiffAt 𝕜 n (g ∘ f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {x : E} {n : ℕ∞} (g : F →L[𝕜] G) (hf : ContDiffAt 𝕜 n f x)
Goal: ContDiffAt 𝕜 n (⇑g ∘ f) x