ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.pluennecke_petridis_inequality_mul
@[to_additive]
theorem pluennecke_petridis_inequality_mul (C : Finset α)
    (hA : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A B : Finset α} (C : Finset α) (hA : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card)
Goal: (A * B * C).card * A.card ≤ (A * B).card * (A * C).card
-- Finset.pluennecke_petridis_inequality_add
@[to_additive]
theorem pluennecke_petridis_inequality_mul (C : Finset α)
    (hA : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card) :
    (A * B * C).card * A.card ≤ (A * B).card * (A * C).card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A B : Finset α} (C : Finset α) (hA : ∀ A' ⊆ A, (A + B).card * A'.card ≤ (A' + B).card * A.card)
Goal: (A + B + C).card * A.card ≤ (A + B).card * (A + C).card
-- _private.Mathlib.Combinatorics.Additive.PluenneckeRuzsa.0.Finset.add_aux
@[to_additive]
private theorem mul_aux (hA : A.Nonempty) (hAB : A ⊆ B)
    (h : ∀ A' ∈ B.powerset.erase ∅, ((A * C).card : ℚ≥0) / ↑A.card ≤ (A' * C).card / ↑A'.card) :
    ∀ A' ⊆ A, (A * C).card * A'.card ≤ (A' * C).card * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A B C : Finset α} (hA : A.Nonempty) (hAB : A ⊆ B) (h : ∀ A' ∈ B.powerset.erase ∅, ↑(A + C).card / ↑A.card ≤ ↑(A' + C).card / ↑A'.card) (A' : Finset α)
Goal: A' ⊆ A → (A + C).card * A'.card ≤ (A' + C).card * A.card
-- _private.Mathlib.Combinatorics.Additive.PluenneckeRuzsa.0.Finset.mul_aux
@[to_additive]
private theorem mul_aux (hA : A.Nonempty) (hAB : A ⊆ B)
    (h : ∀ A' ∈ B.powerset.erase ∅, ((A * C).card : ℚ≥0) / ↑A.card ≤ (A' * C).card / ↑A'.card) :
    ∀ A' ⊆ A, (A * C).card * A'.card ≤ (A' * C).card * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A B C : Finset α} (hA : A.Nonempty) (hAB : A ⊆ B) (h : ∀ A' ∈ B.powerset.erase ∅, ↑(A * C).card / ↑A.card ≤ ↑(A' * C).card / ↑A'.card) (A' : Finset α)
Goal: A' ⊆ A → (A * C).card * A'.card ≤ (A' * C).card * A.card
-- Finset.ruzsa_triangle_inequality_mul_mul_mul
@[to_additive "**Ruzsa's triangle inequality**. Addition version."]
theorem ruzsa_triangle_inequality_mul_mul_mul (A B C : Finset α) :
    (A * C).card * B.card ≤ (A * B).card * (B * C).card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A * C).card * B.card ≤ (A * B).card * (B * C).card
-- Finset.ruzsa_triangle_inequality_add_add_add
@[to_additive "**Ruzsa's triangle inequality**. Addition version."]
theorem ruzsa_triangle_inequality_mul_mul_mul (A B C : Finset α) :
    (A * C).card * B.card ≤ (A * B).card * (B * C).card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A + C).card * B.card ≤ (A + B).card * (B + C).card
-- Finset.ruzsa_triangle_inequality_add_sub_sub
@[to_additive "**Ruzsa's triangle inequality**. Add-sub-sub version."]
theorem ruzsa_triangle_inequality_mul_div_div (A B C : Finset α) :
    (A * C).card * B.card ≤ (A / B).card * (B / C).card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A + C).card * B.card ≤ (A - B).card * (B - C).card
-- Finset.ruzsa_triangle_inequality_mul_div_div
@[to_additive "**Ruzsa's triangle inequality**. Add-sub-sub version."]
theorem ruzsa_triangle_inequality_mul_div_div (A B C : Finset α) :
    (A * C).card * B.card ≤ (A / B).card * (B / C).card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A * C).card * B.card ≤ (A / B).card * (B / C).card
-- Finset.ruzsa_triangle_inequality_div_mul_div
@[to_additive "**Ruzsa's triangle inequality**. Sub-add-sub version."]
theorem ruzsa_triangle_inequality_div_mul_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A * B).card * (B / C).card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A / C).card * B.card ≤ (A * B).card * (B / C).card
-- Finset.ruzsa_triangle_inequality_sub_add_sub
@[to_additive "**Ruzsa's triangle inequality**. Sub-add-sub version."]
theorem ruzsa_triangle_inequality_div_mul_div (A B C : Finset α) :
    (A / C).card * B.card ≤ (A * B).card * (B / C).card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A - C).card * B.card ≤ (A + B).card * (B - C).card
-- Finset.card_sub_add_le_card_sub_add_card_add
@[to_additive "**Ruzsa's triangle inequality**. Sub-sub-add version."]
theorem card_div_mul_le_card_div_mul_card_mul (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B * C).card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A - C).card * B.card ≤ (A - B).card * (B + C).card
-- Finset.card_div_mul_le_card_div_mul_card_mul
@[to_additive "**Ruzsa's triangle inequality**. Sub-sub-add version."]
theorem card_div_mul_le_card_div_mul_card_mul (A B C : Finset α) :
    (A / C).card * B.card ≤ (A / B).card * (B * C).card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] (A B C : Finset α)
Goal: (A / C).card * B.card ≤ (A / B).card * (B * C).card
-- _private.Mathlib.Combinatorics.Additive.PluenneckeRuzsa.0.Finset.card_mul_pow_le
@[to_additive]
private lemma card_mul_pow_le (hAB : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card)
    (n : ℕ) : (A * B ^ n).card ≤ ((A * B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A B : Finset α} (hAB : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card) (n : ℕ)
Goal: ↑(A * B ^ n).card ≤ (↑(A * B).card / ↑A.card) ^ n * ↑A.card
-- _private.Mathlib.Combinatorics.Additive.PluenneckeRuzsa.0.Finset.card_add_nsmul_le
@[to_additive]
private lemma card_mul_pow_le (hAB : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card)
    (n : ℕ) : (A * B ^ n).card ≤ ((A * B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A B : Finset α} (hAB : ∀ A' ⊆ A, (A + B).card * A'.card ≤ (A' + B).card * A.card) (n : ℕ)
Goal: ↑(A + n • B).card ≤ (↑(A + B).card / ↑A.card) ^ n * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_nsmul_sub_nsmul_add
@[to_additive "The **Plünnecke-Ruzsa inequality**. Addition version. Note that this is genuinely
harder than the subtraction version because we cannot use a double counting argument."]
theorem pluennecke_ruzsa_inequality_pow_div_pow_mul (hA : A.Nonempty) (B : Finset α) (m n : ℕ) :
    ((B ^ m / B ^ n).card) ≤ ((A * B).card / A.card : ℚ≥0) ^ (m + n) * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (m n : ℕ)
Goal: ↑(m • B - n • B).card ≤ (↑(A + B).card / ↑A.card) ^ (m + n) * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_pow_div_pow_mul
@[to_additive "The **Plünnecke-Ruzsa inequality**. Addition version. Note that this is genuinely
harder than the subtraction version because we cannot use a double counting argument."]
theorem pluennecke_ruzsa_inequality_pow_div_pow_mul (hA : A.Nonempty) (B : Finset α) (m n : ℕ) :
    ((B ^ m / B ^ n).card) ≤ ((A * B).card / A.card : ℚ≥0) ^ (m + n) * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (m n : ℕ)
Goal: ↑(B ^ m / B ^ n).card ≤ (↑(A * B).card / ↑A.card) ^ (m + n) * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_nsmul_sub_nsmul_sub
@[to_additive "The **Plünnecke-Ruzsa inequality**. Subtraction version."]
theorem pluennecke_ruzsa_inequality_pow_div_pow_div (hA : A.Nonempty) (B : Finset α) (m n : ℕ) :
    (B ^ m / B ^ n).card ≤ ((A / B).card / A.card : ℚ≥0) ^ (m + n) * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (m n : ℕ)
Goal: ↑(m • B - n • B).card ≤ (↑(A - B).card / ↑A.card) ^ (m + n) * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_pow_div_pow_div
@[to_additive "The **Plünnecke-Ruzsa inequality**. Subtraction version."]
theorem pluennecke_ruzsa_inequality_pow_div_pow_div (hA : A.Nonempty) (B : Finset α) (m n : ℕ) :
    (B ^ m / B ^ n).card ≤ ((A / B).card / A.card : ℚ≥0) ^ (m + n) * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (m n : ℕ)
Goal: ↑(B ^ m / B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ (m + n) * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_pow_mul
@[to_additive "Special case of the **Plünnecke-Ruzsa inequality**. Addition version."]
theorem pluennecke_ruzsa_inequality_pow_mul (hA : A.Nonempty) (B : Finset α) (n : ℕ) :
    (B ^ n).card ≤ ((A * B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (n : ℕ)
Goal: ↑(B ^ n).card ≤ (↑(A * B).card / ↑A.card) ^ n * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_nsmul_add
@[to_additive "Special case of the **Plünnecke-Ruzsa inequality**. Addition version."]
theorem pluennecke_ruzsa_inequality_pow_mul (hA : A.Nonempty) (B : Finset α) (n : ℕ) :
    (B ^ n).card ≤ ((A * B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (n : ℕ)
Goal: ↑(n • B).card ≤ (↑(A + B).card / ↑A.card) ^ n * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_pow_div
@[to_additive "Special case of the **Plünnecke-Ruzsa inequality**. Subtraction version."]
theorem pluennecke_ruzsa_inequality_pow_div (hA : A.Nonempty) (B : Finset α) (n : ℕ) :
    (B ^ n).card ≤ ((A / B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [CommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (n : ℕ)
Goal: ↑(B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ n * ↑A.card
-- Finset.pluennecke_ruzsa_inequality_nsmul_sub
@[to_additive "Special case of the **Plünnecke-Ruzsa inequality**. Subtraction version."]
theorem pluennecke_ruzsa_inequality_pow_div (hA : A.Nonempty) (B : Finset α) (n : ℕ) :
    (B ^ n).card ≤ ((A / B).card / A.card : ℚ≥0) ^ n * A.card 
Arguments: {α : Type u_1} [AddCommGroup α] [DecidableEq α] {A : Finset α} (hA : A.Nonempty) (B : Finset α) (n : ℕ)
Goal: ↑(n • B).card ≤ (↑(A - B).card / ↑A.card) ^ n * ↑A.card
-- SimpleGraph.triangleRemovalBound_pos
lemma triangleRemovalBound_pos (hε : 0 < ε) (hε₁ : ε ≤ 1) : 0 < triangleRemovalBound ε 
Arguments: {ε : ℝ} (hε : 0 < ε) (hε₁ : ε ≤ 1)
Goal: 0 < SimpleGraph.triangleRemovalBound ε
-- SimpleGraph.triangleRemovalBound_nonpos
lemma triangleRemovalBound_nonpos (hε : ε ≤ 0) : triangleRemovalBound ε ≤ 0 
Arguments: {ε : ℝ} (hε : ε ≤ 0)
Goal: SimpleGraph.triangleRemovalBound ε ≤ 0
-- SimpleGraph.triangleRemovalBound_mul_cube_lt
lemma triangleRemovalBound_mul_cube_lt (hε : 0 < ε) :
    triangleRemovalBound ε * ⌈4 / ε⌉₊ ^ 3 < 1 
Arguments: {ε : ℝ} (hε : 0 < ε)
Goal: SimpleGraph.triangleRemovalBound ε * ↑⌈4 / ε⌉₊ ^ 3 < 1
-- _private.Mathlib.Combinatorics.SimpleGraph.Triangle.Removal.0.SimpleGraph.aux
private lemma aux {n k : ℕ} (hk : 0 < k) (hn : k ≤ n) : n < 2 * k * (n / k) 
Arguments: {n k : ℕ} (hk : 0 < k) (hn : k ≤ n)
Goal: n < 2 * k * (n / k)
-- _private.Mathlib.Combinatorics.SimpleGraph.Triangle.Removal.0.SimpleGraph.card_bound
private lemma card_bound (hP₁ : P.IsEquipartition) (hP₃ : P.parts.card ≤ bound (ε / 8) ⌈4/ε⌉₊)
    (hX : s ∈ P.parts) : card α / (2 * bound (ε / 8) ⌈4 / ε⌉₊ : ℝ) ≤ s.card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {s : Finset α} {P : Finpartition Finset.univ} {ε : ℝ} (hP₁ : P.IsEquipartition) (hP₃ : P.parts.card ≤ SzemerediRegularity.bound (ε / 8) ⌈4 / ε⌉₊) (hX : s ∈ P.parts)
Goal: ↑(Fintype.card α) / (2 * ↑(SzemerediRegularity.bound (ε / 8) ⌈4 / ε⌉₊)) ≤ ↑s.card
-- _private.Mathlib.Combinatorics.SimpleGraph.Triangle.Removal.0.SimpleGraph.triangle_removal_aux
private lemma triangle_removal_aux (hε : 0 < ε) (hP₁ : P.IsEquipartition)
    (hP₃ : P.parts.card ≤ bound (ε / 8) ⌈4/ε⌉₊)
    (ht : t ∈ (G.regularityReduced P (ε/8) (ε/4)).cliqueFinset 3) :
    triangleRemovalBound ε * card α ^ 3 ≤ (G.cliqueFinset 3).card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {G : SimpleGraph α} [DecidableRel G.Adj] {t : Finset α} {P : Finpartition Finset.univ} {ε : ℝ} (hε : 0 < ε) (hP₁ : P.IsEquipartition) (hP₃ : P.parts.card ≤ SzemerediRegularity.bound (ε / 8) ⌈4 / ε⌉₊) (ht : t ∈ (SimpleGraph.regularityReduced P G (ε / 8) (ε / 4)).cliqueFinset 3)
Goal: SimpleGraph.triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(G.cliqueFinset 3).card
-- SimpleGraph.regularityReduced_edges_card_aux
lemma regularityReduced_edges_card_aux [Nonempty α] (hε : 0 < ε) (hP : P.IsEquipartition)
    (hPε : P.IsUniform G (ε/8)) (hP' : 4 / ε ≤ P.parts.card) :
    2 * (G.edgeFinset.card - (G.regularityReduced P (ε/8) (ε/4)).edgeFinset.card : ℝ)
      < 2 * ε * (card α ^ 2 : ℕ) 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {G : SimpleGraph α} [DecidableRel G.Adj] {P : Finpartition Finset.univ} {ε : ℝ} [Nonempty α] (hε : 0 < ε) (hP : P.IsEquipartition) (hPε : P.IsUniform G (ε / 8)) (hP' : 4 / ε ≤ ↑P.parts.card)
Goal: 2 * (↑G.edgeFinset.card - ↑(SimpleGraph.regularityReduced P G (ε / 8) (ε / 4)).edgeFinset.card) <
  2 * ε * ↑(Fintype.card α ^ 2)
-- SimpleGraph.FarFromTriangleFree.le_card_cliqueFinset
lemma FarFromTriangleFree.le_card_cliqueFinset (hG : G.FarFromTriangleFree ε) :
    triangleRemovalBound ε * card α ^ 3 ≤ (G.cliqueFinset 3).card 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : ℝ} (hG : G.FarFromTriangleFree ε)
Goal: SimpleGraph.triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(G.cliqueFinset 3).card
-- SimpleGraph.triangle_removal
lemma triangle_removal (hG : (G.cliqueFinset 3).card < triangleRemovalBound ε * card α ^ 3) :
    ∃ G' ≤ G, ∃ _ : DecidableRel G'.Adj,
      (G.edgeFinset.card - G'.edgeFinset.card : ℝ) < ε * (card α^2 : ℕ) ∧ G'.CliqueFree 3 
Arguments: {α : Type u_1} [DecidableEq α] [Fintype α] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : ℝ} (hG : ↑(G.cliqueFinset 3).card < SimpleGraph.triangleRemovalBound ε * ↑(Fintype.card α) ^ 3)
Goal: ∃ G' ≤ G, ∃ x, ↑G.edgeFinset.card - ↑G'.edgeFinset.card < ε * ↑(Fintype.card α ^ 2) ∧ G'.CliqueFree 3
-- contDiff_id'
theorem contDiff_id' : ContDiff K n (fun x : E => x) 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {n : ℕ∞}
Goal: ContDiff K n fun x => x
-- contDiffAt_id'
theorem contDiffAt_id' : ContDiffAt K n (fun x : E => x) x 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {x : E} {n : ℕ∞}
Goal: ContDiffAt K n (fun x => x) x
-- contDiffOn_id'
theorem contDiffOn_id' : ContDiffOn K n (fun x : E => x) s 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {s : Set E} {n : ℕ∞}
Goal: ContDiffOn K n (fun x => x) s
-- ContDiff.comp'
theorem ContDiff.comp' {g : F → G} (hg : ContDiff K n g) (hf : ContDiff K n f) :
    ContDiff K n (fun x => g (f x)) 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace K F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace K G] {f : E → F} {n : ℕ∞} {g : F → G} (hg : ContDiff K n g) (hf : ContDiff K n f)
Goal: ContDiff K n fun x => g (f x)
-- ContDiffAt.comp'
theorem ContDiffAt.comp' {f : E → F} {g : F → G} (hg : ContDiffAt K n g (f x))
    (hf : ContDiffAt K n f x) : ContDiffAt K n (fun x => g (f x)) x 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace K F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace K G] {x : E} {n : ℕ∞} {f : E → F} {g : F → G} (hg : ContDiffAt K n g (f x)) (hf : ContDiffAt K n f x)
Goal: ContDiffAt K n (fun x => g (f x)) x
-- contDiff_pi'
theorem contDiff_pi' (hΦ : ∀ i, ContDiff K n fun x => Φ x i) : ContDiff K n Φ 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {n : ℕ∞} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_8} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace K (F' i)] {Φ : E → (i : ι) → F' i} (hΦ : ∀ (i : ι), ContDiff K n fun x => Φ x i)
Goal: ContDiff K n Φ
-- contDiffOn_pi'
theorem contDiffOn_pi' (hΦ : ∀ i, ContDiffOn K n (fun x => Φ x i) s) : ContDiffOn K n Φ s 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {s : Set E} {n : ℕ∞} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_8} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace K (F' i)] {Φ : E → (i : ι) → F' i} (hΦ : ∀ (i : ι), ContDiffOn K n (fun x => Φ x i) s)
Goal: ContDiffOn K n Φ s
-- contDiffAt_pi'
theorem contDiffAt_pi' (hΦ : ∀ i, ContDiffAt K n (fun x => Φ x i) x) : ContDiffAt K n Φ x 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {x : E} {n : ℕ∞} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_8} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace K (F' i)] {Φ : E → (i : ι) → F' i} (hΦ : ∀ (i : ι), ContDiffAt K n (fun x => Φ x i) x)
Goal: ContDiffAt K n Φ x
-- ContDiffOn.div'
theorem ContDiffOn.div' [CompleteSpace K] {f g : E → K} {n} (hf : ContDiffOn K n f s)
    (hg : ContDiffOn K n g s) (h₀ : ∀ x ∈ s, g x ≠ 0) : ContDiffOn K n (fun x => f x / g x) s 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] {s : Set E} [CompleteSpace K] {f g : E → K} {n : ℕ∞} (hf : ContDiffOn K n f s) (hg : ContDiffOn K n g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: ContDiffOn K n (fun x => f x / g x) s
-- ContDiff.differentiable_iteratedDeriv'
theorem ContDiff.differentiable_iteratedDeriv' {m : ℕ} {f : K → F}
    (hf : ContDiff K (m+1) f) : Differentiable K (iteratedDeriv m f) 
Arguments: {K : Type u_1} [NontriviallyNormedField K] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace K F] {m : ℕ} {f : K → F} (hf : ContDiff K (↑m + 1) f)
Goal: Differentiable K (iteratedDeriv m f)
-- _private.Mathlib.Combinatorics.Additive.ErdosGinzburgZiv.0.totalDegree_f₁_add_totalDegree_f₂
private lemma totalDegree_f₁_add_totalDegree_f₂ {a : ι → ZMod p} :
    (f₁ s a).totalDegree + (f₂ s a).totalDegree < 2 * p - 1 
Arguments: {ι : Type u_1} {p : ℕ} [Fact (Nat.Prime p)] {s : Finset ι} {a : ι → ZMod p}
Goal: (f₁ s a).totalDegree + (f₂ s a).totalDegree < 2 * p - 1
-- _private.Mathlib.Combinatorics.Additive.ErdosGinzburgZiv.0.ZMod.erdos_ginzburg_ziv_prime
private theorem ZMod.erdos_ginzburg_ziv_prime (a : ι → ZMod p) (hs : s.card = 2 * p - 1) :
    ∃ t ⊆ s, t.card = p ∧ ∑ i ∈ t, a i = 0 
Arguments: {ι : Type u_1} {p : ℕ} [Fact (Nat.Prime p)] {s : Finset ι} (a : ι → ZMod p) (hs : s.card = 2 * p - 1)
Goal: ∃ t ⊆ s, t.card = p ∧ ∑ i ∈ t, a i = 0
-- _private.Mathlib.Combinatorics.Additive.ErdosGinzburgZiv.0.Int.erdos_ginzburg_ziv_prime
private theorem Int.erdos_ginzburg_ziv_prime (a : ι → ℤ) (hs : s.card = 2 * p - 1) :
    ∃ t ⊆ s, t.card = p ∧ ↑p ∣ ∑ i ∈ t, a i 
Arguments: {ι : Type u_1} {p : ℕ} [Fact (Nat.Prime p)] {s : Finset ι} (a : ι → ℤ) (hs : s.card = 2 * p - 1)
Goal: ∃ t ⊆ s, t.card = p ∧ ↑p ∣ ∑ i ∈ t, a i
-- Int.erdos_ginzburg_ziv
theorem Int.erdos_ginzburg_ziv (a : ι → ℤ) (hs : 2 * n - 1 ≤ s.card) :
    ∃ t ⊆ s, t.card = n ∧ ↑n ∣ ∑ i ∈ t, a i 
Arguments: {ι : Type u_1} {n : ℕ} {s : Finset ι} (a : ι → ℤ) (hs : 2 * n - 1 ≤ s.card)
Goal: ∃ t ⊆ s, t.card = n ∧ ↑n ∣ ∑ i ∈ t, a i
-- ZMod.erdos_ginzburg_ziv
theorem ZMod.erdos_ginzburg_ziv (a : ι → ZMod n) (hs : 2 * n - 1 ≤ s.card) :
    ∃ t ⊆ s, t.card = n ∧ ∑ i ∈ t, a i = 0 
Arguments: {ι : Type u_1} {n : ℕ} {s : Finset ι} (a : ι → ZMod n) (hs : 2 * n - 1 ≤ s.card)
Goal: ∃ t ⊆ s, t.card = n ∧ ∑ i ∈ t, a i = 0
-- Int.erdos_ginzburg_ziv_multiset
theorem Int.erdos_ginzburg_ziv_multiset (s : Multiset ℤ) (hs : 2 * n - 1 ≤ Multiset.card s) :
    ∃ t ≤ s, Multiset.card t = n ∧ ↑n ∣ t.sum 
Arguments: {n : ℕ} (s : Multiset ℤ) (hs : 2 * n - 1 ≤ Multiset.card s)
Goal: ∃ t ≤ s, Multiset.card t = n ∧ ↑n ∣ t.sum
-- ZMod.erdos_ginzburg_ziv_multiset
theorem ZMod.erdos_ginzburg_ziv_multiset (s : Multiset (ZMod n))
    (hs : 2 * n - 1 ≤ Multiset.card s) : ∃ t ≤ s, Multiset.card t = n ∧ t.sum = 0 
Arguments: {n : ℕ} (s : Multiset (ZMod n)) (hs : 2 * n - 1 ≤ Multiset.card s)
Goal: ∃ t ≤ s, Multiset.card t = n ∧ t.sum = 0
-- _private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.mk_mem_triangleIndices
@[simp]
private lemma mk_mem_triangleIndices : (a, b, c) ∈ triangleIndices A ↔ (a, b) ∈ A ∧ c = a + b 
Arguments: {G : Type u_1} [AddCommGroup G] {A : Finset (G × G)} {a b c : G}
Goal: (a, b, c) ∈ Corners.triangleIndices A ↔ (a, b) ∈ A ∧ c = a + b
-- _private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.card_triangleIndices
@[simp] private lemma card_triangleIndices : (triangleIndices A).card = A.card 
Arguments: {G : Type u_1} [AddCommGroup G] {A : Finset (G × G)}
Goal: (Corners.triangleIndices A).card = A.card
-- _private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.noAccidental
private lemma noAccidental (hs : IsCornerFree (A : Set (G × G))) :
    NoAccidental (triangleIndices A) 
Arguments: {G : Type u_1} [AddCommGroup G] {A : Finset (G × G)} (hs : IsCornerFree ↑A)
Goal: SimpleGraph.TripartiteFromTriangles.NoAccidental (Corners.triangleIndices A)
-- _private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.farFromTriangleFree_graph
private lemma farFromTriangleFree_graph [DecidableEq G] (hε : ε * card G ^ 2 ≤ A.card) :
    (graph <| triangleIndices A).FarFromTriangleFree (ε / 9) 
Arguments: {G : Type u_1} [AddCommGroup G] [Fintype G] {A : Finset (G × G)} {ε : ℝ} [DecidableEq G] (hε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑A.card)
Goal: (SimpleGraph.TripartiteFromTriangles.graph (Corners.triangleIndices A)).FarFromTriangleFree (ε / 9)
-- corners_theorem
theorem corners_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ card G)
    (A : Finset (G × G)) (hAε : ε * card G ^ 2 ≤ A.card) : ¬ IsCornerFree (A : Set (G × G)) 
Arguments: {G : Type u_1} [AddCommGroup G] [Fintype G] (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ Fintype.card G) (A : Finset (G × G)) (hAε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑A.card)
Goal: ¬IsCornerFree ↑A
-- corners_theorem_nat
theorem corners_theorem_nat (hε : 0 < ε) (hn : cornersTheoremBound (ε / 9) ≤ n)
    (A : Finset (ℕ × ℕ)) (hAn : A ⊆ range n ×ˢ range n) (hAε : ε * n ^ 2 ≤ A.card) :
    ¬ IsCornerFree (A : Set (ℕ × ℕ)) 
Arguments: {n : ℕ} {ε : ℝ} (hε : 0 < ε) (hn : cornersTheoremBound (ε / 9) ≤ n) (A : Finset (ℕ × ℕ)) (hAn : A ⊆ Finset.range n ×ˢ Finset.range n) (hAε : ε * ↑n ^ 2 ≤ ↑A.card)
Goal: ¬IsCornerFree ↑A
-- roth_3ap_theorem
theorem roth_3ap_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ card G)
    (A : Finset G) (hAε : ε * card G ≤ A.card) : ¬ ThreeAPFree (A : Set G) 
Arguments: {G : Type u_1} [AddCommGroup G] [Fintype G] (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ Fintype.card G) (A : Finset G) (hAε : ε * ↑(Fintype.card G) ≤ ↑A.card)
Goal: ¬ThreeAPFree ↑A
-- roth_3ap_theorem_nat
theorem roth_3ap_theorem_nat (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound (ε / 3) ≤ n)
    (A : Finset ℕ) (hAn : A ⊆ range n) (hAε : ε * n ≤ A.card) : ¬ ThreeAPFree (A : Set ℕ) 
Arguments: {n : ℕ} (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound (ε / 3) ≤ n) (A : Finset ℕ) (hAn : A ⊆ Finset.range n) (hAε : ε * ↑n ≤ ↑A.card)
Goal: ¬ThreeAPFree ↑A
-- rothNumberNat_isLittleO_id
theorem rothNumberNat_isLittleO_id :
    IsLittleO atTop (fun N ↦ (rothNumberNat N : ℝ)) (fun N ↦ (N : ℝ)) 
Goal: (fun N => ↑(rothNumberNat N)) =o[Filter.atTop] fun N => ↑N
-- Tactic.ReduceModChar.CharP.intCast_eq_mod
lemma CharP.intCast_eq_mod (R : Type _) [Ring R] (p : ℕ) [CharP R p] (k : ℤ) :
    (k : R) = (k % p : ℤ) 
Arguments: (R : Type u_1) [Ring R] (p : ℕ) [CharP R p] (k : ℤ)
Goal: ↑k = ↑(k % ↑p)
-- Tactic.ReduceModChar.CharP.isInt_of_mod
lemma CharP.isInt_of_mod {e' r : ℤ} {α : Type _} [Ring α] {n n' : ℕ} (inst : CharP α n) {e : α}
    (he : IsInt e e') (hn : IsNat n n') (h₂ : IsInt (e' % n') r) : IsInt e r 
Arguments: {e' r : ℤ} {α : Type u_1} [Ring α] {n n' : ℕ} (inst : CharP α n) {e : α} (he : Mathlib.Meta.NormNum.IsInt e e') (hn : Mathlib.Meta.NormNum.IsNat n n') (h₂ : Mathlib.Meta.NormNum.IsInt (e' % ↑n') r)
Goal: Mathlib.Meta.NormNum.IsInt e r