mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.Enumerative.Partition.GenFun

Goal state before:
pp: case refine_3
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : T2Space R
f : ℕ → ℕ → R
d : ℕ
s : Finset ℕ
hs✝ : s ⊇ range d
this :
  ∏ i ∈ s, (1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) =
    ∏ i ∈ Finset.map (addRightEmbedding 1) s, (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1)))
hs : Icc 1 d ⊆ Finset.map (addRightEmbedding 1) s
⊢ ∀ i ∈ univ,
    (Multiset.toFinsupp i.parts).prod f =
      ∏ i_1 ∈ Finset.map (addRightEmbedding 1) s,
        (coeff (i.toFinsuppAntidiag i_1)) (1 + ∑' (j : ℕ), f i_1 (j + 1) • X ^ (i_1 * (j + 1)))

Tactic:
·

Goal state after:
case refine_3
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : T2Space R
f : ℕ → ℕ → R
d : ℕ
s : Finset ℕ
hs✝ : s ⊇ range d
this :
  ∏ i ∈ s, (1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) =
    ∏ i ∈ Finset.map (addRightEmbedding 1) s, (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1)))
hs : Icc 1 d ⊆ Finset.map (addRightEmbedding 1) s
⊢ ∀ i ∈ univ,
    (Multiset.toFinsupp i.parts).prod f =
      ∏ i_1 ∈ Finset.map (addRightEmbedding 1) s,
        (coeff (i.toFinsuppAntidiag i_1)) (1 + ∑' (j : ℕ), f i_1 (j + 1) • X ^ (i_1 * (j + 1)))
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
H : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
κ_add_one_pos : 0 < κ + 1
one_sub_K_pos : 0 < 1 - K
t : ℕ := ⌊(κ + 1) / (1 - K)⌋₊
largeA : ∀ {A : Finset G}, t < #A → κ + 1 < ex A
M : Finset ℝ := {x ∈ map Nat.castEmbedding (Icc (#S) (t * #S)) - K •> map Nat.castEmbedding (Icc 1 t) | κ < x}
A : Finset G
hA : A.Nonempty
hAt : #A ≤ t
⊢ ↑(#(A * S)) ∈ map Nat.castEmbedding (Icc (#S) (t * #S))

pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
H : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
κ_add_one_pos : 0 < κ + 1
one_sub_K_pos : 0 < 1 - K
t : ℕ := ⌊(κ + 1) / (1 - K)⌋₊
largeA : ∀ {A : Finset G}, t < #A → κ + 1 < ex A
M : Finset ℝ := {x ∈ map Nat.castEmbedding (Icc (#S) (t * #S)) - K •> map Nat.castEmbedding (Icc 1 t) | κ < x}
A : Finset G
hA : A.Nonempty
hAt : #A ≤ t
⊢ K * ↑(#A) ∈ K •> map Nat.castEmbedding (Icc 1 t)

Tactic:
·

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
H : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
κ_add_one_pos : 0 < κ + 1
one_sub_K_pos : 0 < 1 - K
t : ℕ := ⌊(κ + 1) / (1 - K)⌋₊
largeA : ∀ {A : Finset G}, t < #A → κ + 1 < ex A
M : Finset ℝ := {x ∈ map Nat.castEmbedding (Icc (#S) (t * #S)) - K •> map Nat.castEmbedding (Icc 1 t) | κ < x}
A : Finset G
hA : A.Nonempty
hAt : #A ≤ t
⊢ ↑(#(A * S)) ∈ map Nat.castEmbedding (Icc (#S) (t * #S))
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
H : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
κ_add_one_pos : 0 < κ + 1
one_sub_K_pos : 0 < 1 - K
t : ℕ := ⌊(κ + 1) / (1 - K)⌋₊
largeA : ∀ {A : Finset G}, t < #A → κ + 1 < ex A
M : Finset ℝ := {x ∈ map Nat.castEmbedding (Icc (#S) (t * #S)) - K •> map Nat.castEmbedding (Icc 1 t) | κ < x}
A : Finset G
hA : A.Nonempty
hAt : #A ≤ t
⊢ K * ↑(#A) ∈ K •> map Nat.castEmbedding (Icc 1 t)

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
H : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
κ_add_one_pos : 0 < κ + 1
one_sub_K_pos : 0 < 1 - K
t : ℕ := ⌊(κ + 1) / (1 - K)⌋₊
largeA : ∀ {A : Finset G}, t < #A → κ + 1 < ex A
M : Finset ℝ := {x ∈ map Nat.castEmbedding (Icc (#S) (t * #S)) - K •> map Nat.castEmbedding (Icc 1 t) | κ < x}
A : Finset G
hA : A.Nonempty
hAt : #A ≤ t
⊢ K * ↑(#A) ∈ K •> map Nat.castEmbedding (Icc 1 t)
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case zero
n : ℕ
x✝ : n + 2 ≠ 0
hk : 0 ∈ Iic (n + 1)
⊢ Even (largeSchroder 0 * (n + 1 - 0).largeSchroder)

pp: case succ
n : ℕ
x✝ : n + 2 ≠ 0
k : ℕ
hk : k + 1 ∈ Iic (n + 1)
⊢ Even ((k + 1).largeSchroder * (n + 1 - (k + 1)).largeSchroder)

Tactic:
·

Goal state after:
case zero
n : ℕ
x✝ : n + 2 ≠ 0
hk : 0 ∈ Iic (n + 1)
⊢ Even (largeSchroder 0 * (n + 1 - 0).largeSchroder)
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case zero
hn : 1 < 0
⊢ (0 + 1).smallSchroder = 3 * smallSchroder 0 + 2 * ∑ i ∈ Ioo 0 (0 - 1), (i + 1).smallSchroder * (0 - i).smallSchroder

pp: case succ.zero
hn : 1 < 0 + 1
⊢ (0 + 1 + 1).smallSchroder =
    3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder

pp: case succ.succ
n : ℕ
hn : 1 < n + 1 + 1
⊢ (n + 1 + 1 + 1).smallSchroder =
    3 * (n + 1 + 1).smallSchroder +
      2 * ∑ i ∈ Ioo 0 (n + 1 + 1 - 1), (i + 1).smallSchroder * (n + 1 + 1 - i).smallSchroder

Tactic:
·

Goal state after:
case zero
hn : 1 < 0
⊢ (0 + 1).smallSchroder = 3 * smallSchroder 0 + 2 * ∑ i ∈ Ioo 0 (0 - 1), (i + 1).smallSchroder * (0 - i).smallSchroder
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case succ.zero
hn : 1 < 0 + 1
⊢ (0 + 1 + 1).smallSchroder =
    3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder

pp: case succ.succ
n : ℕ
hn : 1 < n + 1 + 1
⊢ (n + 1 + 1 + 1).smallSchroder =
    3 * (n + 1 + 1).smallSchroder +
      2 * ∑ i ∈ Ioo 0 (n + 1 + 1 - 1), (i + 1).smallSchroder * (n + 1 + 1 - i).smallSchroder

Tactic:
·

Goal state after:
case succ.zero
hn : 1 < 0 + 1
⊢ (0 + 1 + 1).smallSchroder =
    3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ 2 * (i + 1).smallSchroder * (2 * (n + 1 - i + 1).smallSchroder) =
    2 * (i + 1).smallSchroder * (2 * (n + 2 - i).smallSchroder)

Tactic:
·

Goal state after:
case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ 2 * (i + 1).smallSchroder * (2 * (n + 1 - i + 1).smallSchroder) =
    2 * (i + 1).smallSchroder * (2 * (n + 2 - i).smallSchroder)
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ n + 1 - i ≠ 0

Tactic:
·

Goal state after:
case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ n + 1 - i ≠ 0
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ i ≠ 0

Tactic:
·

Goal state after:
case a
n : ℕ
hn : 1 < n + 1 + 1
i : ℕ
hi : 0 < i ∧ i ≤ n
⊢ i ≠ 0
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: n : ℕ
hn : 1 < n + 1 + 1
⊢ n + 1 ≠ 0

Tactic:
by lia

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Schroder

Goal state before:
pp: n : ℕ
hn : 1 < n + 1 + 1
⊢ 6 * (n + 2).smallSchroder + 4 * ∑ i ∈ Ioo 0 (n + 1), (i + 1).smallSchroder * (n + 2 - i).smallSchroder =
    2 *
      (3 * (n + 1 + 1).smallSchroder +
        2 * ∑ i ∈ Ioo 0 (n + 1 + 1 - 1), (i + 1).smallSchroder * (n + 1 + 1 - i).smallSchroder)

Tactic:
by lia

Goal state after:
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
N : Finset G
hN : IsAtom K S N
n : G
hn : n ∈ N
one_mem_carrier : 1 ∈ n⁻¹ •> N
self_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N
H : Subgroup G := { carrier := n⁻¹ •> ↑N, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
a : G
⊢ a ∈ n⁻¹ •> N ↔ a ∈ ↑H

pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
N : Finset G
hN : IsAtom K S N
n : G
hn : n ∈ N
one_mem_carrier : 1 ∈ n⁻¹ •> N
self_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N
H : Subgroup G := { carrier := n⁻¹ •> ↑N, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
⊢ IsAtom K S (↑H).toFinset

Tactic:
·

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
N : Finset G
hN : IsAtom K S N
n : G
hn : n ∈ N
one_mem_carrier : 1 ∈ n⁻¹ •> N
self_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N
H : Subgroup G := { carrier := n⁻¹ •> ↑N, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
a : G
⊢ a ∈ n⁻¹ •> N ↔ a ∈ ↑H
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
N : Finset G
hN : IsAtom K S N
n : G
hn : n ∈ N
one_mem_carrier : 1 ∈ n⁻¹ •> N
self_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N
H : Subgroup G := { carrier := n⁻¹ •> ↑N, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
⊢ IsAtom K S (↑H).toFinset

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
S : Finset G
hK : K < 1
hS : S.Nonempty
N : Finset G
hN : IsAtom K S N
n : G
hn : n ∈ N
one_mem_carrier : 1 ∈ n⁻¹ •> N
self_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N
H : Subgroup G := { carrier := n⁻¹ •> ↑N, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
⊢ IsAtom K S (↑H).toFinset
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries

Goal state before:
pp: case refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ IsTopologicallyNilpotent X

pp: case refine_2
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Summable fun n ↦ X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1))

pp: case refine_3
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Multipliable fun n ↦ 1 - X ^ (n + k + 1)

pp: case refine_4
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Summable fun k ↦ (-1) ^ k * (X ^ pentagonal (-↑k) - X ^ pentagonal (↑k + 1))

pp: case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ IsTopologicallyNilpotent X
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries

Goal state before:
pp: case refine_2
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Summable fun n ↦ X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1))

pp: case refine_3
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Multipliable fun n ↦ 1 - X ^ (n + k + 1)

pp: case refine_4
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Summable fun k ↦ (-1) ^ k * (X ^ pentagonal (-↑k) - X ^ pentagonal (↑k + 1))

pp: case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Summable fun n ↦ X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1))
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries

Goal state before:
pp: case refine_3
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Multipliable fun n ↦ 1 - X ^ (n + k + 1)

pp: case refine_4
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Summable fun k ↦ (-1) ^ k * (X ^ pentagonal (-↑k) - X ^ pentagonal (↑k + 1))

pp: case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)

Tactic:
·

Goal state after:
case refine_3
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ ∀ (k : ℕ), Multipliable fun n ↦ 1 - X ^ (n + k + 1)
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries

Goal state before:
pp: case refine_4
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Summable fun k ↦ (-1) ^ k * (X ^ pentagonal (-↑k) - X ^ pentagonal (↑k + 1))

pp: case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)

Tactic:
·

Goal state after:
case refine_4
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Summable fun k ↦ (-1) ^ k * (X ^ pentagonal (-↑k) - X ^ pentagonal (↑k + 1))
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries

Goal state before:
pp: case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)

Tactic:
·

Goal state after:
case refine_5
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
a✝ : Nontrivial R
⊢ Tendsto
    (fun k ↦
      (-1) ^ (k + 1) * X ^ ((k + 1) * (3 * k + 4) / 2) *
        ∑' (n : ℕ), X ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - X ^ (k + i + 1)))
    atTop (𝓝 0)
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: n : ℕ
α : Type := Fin (2 * n + 1)
this✝ : Coprime 2 (2 * n + 1)
this : Fact (IsUnit 2)
⊢ n < 2 * n + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case calc_1
n : ℕ
⊢ ↑(n + 3) / 3 - 2 ≤ ↑(2 * (n / 6) + 1)

pp: case calc_2
n : ℕ
⊢ ↑(2 * (n / 6) + 1) * ↑(rothNumberNat (n / 6)) ≤ ↑(ruzsaSzemerediNumberNat (6 * (n / 6) + 3))

Tactic:
·

Goal state after:
case calc_1
n : ℕ
⊢ ↑(n + 3) / 3 - 2 ≤ ↑(2 * (n / 6) + 1)
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case calc_1
n : ℕ
⊢ (↑n + 3) / 3 ≤ 2 * ↑(n / 6) + 1 + 2

Tactic:
← div_add_one (three_ne_zero' ℝ),

Goal state after:
case calc_1
n : ℕ
⊢ ↑n / 3 + 1 ≤ 2 * ↑(n / 6) + 1 + 2
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case calc_1
n : ℕ
⊢ ↑n ≤ 2 * 3 * ↑(n / 6) + (1 + 2 - 1) * 3

Tactic:
add_sub_cancel_left

Goal state after:
case calc_1
n : ℕ
⊢ ↑n ≤ 2 * 3 * ↑(n / 6) + 2 * 3
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case calc_2
n : ℕ
⊢ ↑(2 * (n / 6) + 1) * ↑(rothNumberNat (n / 6)) ≤ ↑(ruzsaSzemerediNumberNat (6 * (n / 6) + 3))

Tactic:
·

Goal state after:
case calc_2
n : ℕ
⊢ ↑(2 * (n / 6) + 1) * ↑(rothNumberNat (n / 6)) ≤ ↑(ruzsaSzemerediNumberNat (6 * (n / 6) + 3))
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case inl
n : ℕ
hn : ↑n / 3 - 2 ≤ 0
⊢ (↑n / 3 - 2) * (↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) ≤ ↑(ruzsaSzemerediNumberNat n)

pp: case inr
n : ℕ
hn : 0 ≤ ↑n / 3 - 2
⊢ (↑n / 3 - 2) * (↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) ≤ ↑(ruzsaSzemerediNumberNat n)

Tactic:
·

Goal state after:
case inl
n : ℕ
hn : ↑n / 3 - 2 ≤ 0
⊢ (↑n / 3 - 2) * (↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) ≤ ↑(ruzsaSzemerediNumberNat n)
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: ⊢ (fun n ↦ ↑n ^ 2 * rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦
    (↑n / 3 - 2) * ↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))

pp: ⊢ (fun n ↦ (↑n / 3 - 2) * ↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) =O[atTop] fun n ↦
    ↑(ruzsaSzemerediNumberNat n)

Tactic:
·

Goal state after:
⊢ (fun n ↦ ↑n ^ 2 * rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦
    (↑n / 3 - 2) * ↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case refine_1
⊢ Nat.cast =O[atTop] fun n ↦ ↑n / 3 - 2

pp: case refine_2
⊢ Nat.cast =O[atTop] fun n ↦ ↑((n - 3) / 6)

pp: case refine_3
⊢ (fun n ↦ rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦ rexp (-4 * √(Real.log ↑((n - 3) / 6)))

Tactic:
·

Goal state after:
case refine_1
⊢ Nat.cast =O[atTop] fun n ↦ ↑n / 3 - 2
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: ⊢ Nat.cast =O[atTop] fun n ↦ ↑n / 3

pp: ⊢ (fun n ↦ ↑n / 3) =O[atTop] fun n ↦ ↑n / 3 - 2

Tactic:
·

Goal state after:
⊢ Nat.cast =O[atTop] fun n ↦ ↑n / 3
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case refine_2
⊢ Nat.cast =O[atTop] fun n ↦ ↑((n - 3) / 6)

pp: case refine_3
⊢ (fun n ↦ rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦ rexp (-4 * √(Real.log ↑((n - 3) / 6)))

Tactic:
·

Goal state after:
case refine_2
⊢ Nat.cast =O[atTop] fun n ↦ ↑((n - 3) / 6)
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case refine_3
⊢ (fun n ↦ rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦ rexp (-4 * √(Real.log ↑((n - 3) / 6)))

Tactic:
·

Goal state after:
case refine_3
⊢ (fun n ↦ rexp (-4 * √(Real.log ↑n))) =O[atTop] fun n ↦ rexp (-4 * √(Real.log ↑((n - 3) / 6)))
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case hx
x : ℕ
hx : 9 ≤ x
⊢ 0 < ↑((x - 3) / 6)

pp: case a
x : ℕ
hx : 9 ≤ x
⊢ (x - 3) / 6 ≤ x

Tactic:
·

Goal state after:
case hx
x : ℕ
hx : 9 ≤ x
⊢ 0 < ↑((x - 3) / 6)
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: case a
x : ℕ
hx : 9 ≤ x
⊢ (x - 3) / 6 ≤ x

Tactic:
·

Goal state after:
case a
x : ℕ
hx : 9 ≤ x
⊢ (x - 3) / 6 ≤ x
Module: Mathlib.Combinatorics.Graph.Subgraph

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
e : β
G H : Graph α β
hHG : H ≤ G
he : e ∈ E(H)
u v : α
huv : H.IsLink e u v
h : G.IsLink e u v
⊢ H.IsLink e u v

pp: case inr
α : Type u_1
β : Type u_2
e : β
G H : Graph α β
hHG : H ≤ G
he : e ∈ E(H)
u v : α
huv : H.IsLink e u v
h : G.IsLink e v u
⊢ H.IsLink e v u

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
e : β
G H : Graph α β
hHG : H ≤ G
he : e ∈ E(H)
u v : α
huv : H.IsLink e u v
h : G.IsLink e u v
⊢ H.IsLink e u v
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

Goal state before:
pp: ⊢ (fun n ↦ (↑n / 3 - 2) * ↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) =O[atTop] fun n ↦
    ↑(ruzsaSzemerediNumberNat n)

Tactic:
·

Goal state after:
⊢ (fun n ↦ (↑n / 3 - 2) * ↑((n - 3) / 6) * rexp (-4 * √(Real.log ↑((n - 3) / 6)))) =O[atTop] fun n ↦
    ↑(ruzsaSzemerediNumberNat n)
Module: Mathlib.Combinatorics.Graph.Subgraph

Goal state before:
pp: α : Type u_1
β : Type u_2
x y : α
e : β
G H : Graph α β
he : G.IsLink e x y
hHG : H ≤c G
tfae_1_to_2 : x ∈ V(H) → y ∈ V(H)
tfae_2_to_3 : y ∈ V(H) → e ∈ E(H)
tfae_3_to_1 : e ∈ E(H) → x ∈ V(H)
⊢ [x ∈ V(H), y ∈ V(H), e ∈ E(H)].TFAE

Tactic:
tfae_finish

Goal state after:
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
h h' : G.IsLink e x y
⊢ y = y

pp: case inr
α : Type u_1
β : Type u_2
y z : α
e : β
G : Graph α β
h : G.IsLink e y y
h' : G.IsLink e y z
⊢ y = z

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
h h' : G.IsLink e x y
⊢ y = y
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
y' : α
h : G.IsLink e x y
h' : G.IsLink e x y'
⊢ x = x ∧ y = y' ∨ x = y' ∧ y = x

pp: case inr
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
x' : α
h : G.IsLink e x y
h' : G.IsLink e x' x
⊢ x = x' ∧ y = x ∨ x = x ∧ y = x'

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
y' : α
h : G.IsLink e x y
h' : G.IsLink e x y'
⊢ x = x ∧ y = y' ∨ x = y' ∧ y = x
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
S : Finset G
ε : ℝ
hε₀ : 0 < ε
hε₁ : ε ≤ 1
hS : S.Nonempty
K : ℝ := ⋯
hK : K < 1
ex : Finset G → ℝ := ⋯
κ : ℝ := ⋯
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
Z : Finset G
hZS : Z ⊆ S
hHZS : ↑H * ↑Z = ↑H * ↑S
hZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z
A : Finset G
hA₁ : #S ≤ #A
hA₂ : ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
calc₁ : ex (↑H).toFinset ≤ (1 - ε / 2) * ↑(#S)
⊢ 0 ≤ 1 - ε / 2

pp: case a
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
S : Finset G
ε : ℝ
hε₀ : 0 < ε
hε₁ : ε ≤ 1
hS : S.Nonempty
K : ℝ := ⋯
hK : K < 1
ex : Finset G → ℝ := ⋯
κ : ℝ := ⋯
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
Z : Finset G
hZS : Z ⊆ S
hHZS : ↑H * ↑Z = ↑H * ↑S
hZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z
A : Finset G
hA₁ : #S ≤ #A
hA₂ : ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
calc₁ : ex (↑H).toFinset ≤ (1 - ε / 2) * ↑(#S)
⊢ S ⊆ (↑H).toFinset * S

Tactic:
·

Goal state after:
case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
S : Finset G
ε : ℝ
hε₀ : 0 < ε
hε₁ : ε ≤ 1
hS : S.Nonempty
K : ℝ := ⋯
hK : K < 1
ex : Finset G → ℝ := ⋯
κ : ℝ := ⋯
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
Z : Finset G
hZS : Z ⊆ S
hHZS : ↑H * ↑Z = ↑H * ↑S
hZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z
A : Finset G
hA₁ : #S ≤ #A
hA₂ : ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
calc₁ : ex (↑H).toFinset ≤ (1 - ε / 2) * ↑(#S)
⊢ 0 ≤ 1 - ε / 2
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
h : G.IsLink e x y
⊢ G.IsLink e x y

pp: case inr
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
h : G.IsLink e x y
⊢ G.IsLink e y x

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
x y : α
e : β
G : Graph α β
h : G.IsLink e x y
⊢ G.IsLink e x y