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

Goal state before:
pp: m : Multiset ℕ
⊢ m.bell * (map (fun j ↦ j !) m).prod * ∏ j ∈ m.toFinset.erase 0, (count j m)! = m.sum !

Tactic:
by
  unfold bell
  rw [← Nat.mul_right_inj (a := ∏ i ∈ m.toFinset, (i * count i m)!) (by positivity)]
  simp only [← mul_assoc, Nat.multinomial_spec]
  rw [mul_assoc, mul_assoc, mul_comm]
  congr
  · rw [mul_comm, mul_assoc, ← Finset.prod_mul_distrib, Finset.prod_multiset_map_count]
    suffices this : _ by
      by_cases hm : 0 ∈ m.toFinset
      · rw [← Finset.prod_erase_mul _ _ hm]
        rw [← Finset.prod_erase_mul _ _ hm]
        simp only [factorial_zero, one_pow, mul_one, zero_mul]
        exact this
      · nth_rewrite 1 [← Finset.erase_eq_of_notMem hm]
        nth_rewrite 3 [← Finset.erase_eq_of_notMem hm]
        exact this
    rw [← Finset.prod_mul_distrib]
    congr! 1 with x hx
    rw [← mul_assoc, bell_mul_eq_lemma]
    simp only [Finset.mem_erase, ne_eq, mem_toFinset] at hx
    simp only [ne_eq, hx.1, not_false_eq_true]
  · rw [Finset.sum_multiset_count]
    simp only [smul_eq_mul, mul_comm]

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
c : Set G
x : G
hc : c ∈ orbit Gᵐᵒᵖ ↑H
hx : ↑H <• x = c
⊢ x ∈ c

Tactic:
by simp only [← hx, mem_rightCoset_iff, mul_inv_cancel, SetLike.mem_coe, one_mem]

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

Goal state before:
pp: case inr.intro
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A : Finset G
hK₁ : 1 < K
hKφ : K < φ
hA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)
hA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)
K_pos hK₀ : 0 < K
hKφ' : 0 < φ - K
hKψ' : 0 < K - ψ
hK₂' : 0 < 2 - K
const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))
A_nonempty : A.Nonempty
S : Finset G := A * A⁻¹
H : Subgroup G := stabilizer G S
S_nonempty : S.Nonempty
this : Finite ↥H
val✝ : Fintype ↥H
H_mul_S : ↑H * ↑S = ↑S
Z : Finset G
hZ : Z ⊆ S ∧ ↑H * ↑Z = ↑H * ↑S ∧ Set.InjOn (fun x ↦ ↑H <• x) ↑Z
H_mul_Z : ↑H * ↑Z = ↑S
H_toFinset_mul_Z : (↑H).toFinset * Z = S
card_H_mul_card_Z : Fintype.card ↥H * #Z = #S
⊢ ∃ H x Z, ↑(#Z) ≤ (2 - K) * K / ((φ - K) * (K - ψ)) ∧ ↑H * ↑Z = ↑A * ↑A⁻¹

Tactic:
refine ⟨H, inferInstance, Z, ?_, mod_cast H_mul_Z⟩

Goal state after:
case inr.intro
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A : Finset G
hK₁ : 1 < K
hKφ : K < φ
hA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)
hA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)
K_pos hK₀ : 0 < K
hKφ' : 0 < φ - K
hKψ' : 0 < K - ψ
hK₂' : 0 < 2 - K
const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))
A_nonempty : A.Nonempty
S : Finset G := A * A⁻¹
H : Subgroup G := stabilizer G S
S_nonempty : S.Nonempty
this : Finite ↥H
val✝ : Fintype ↥H
H_mul_S : ↑H * ↑S = ↑S
Z : Finset G
hZ : Z ⊆ S ∧ ↑H * ↑Z = ↑H * ↑S ∧ Set.InjOn (fun x ↦ ↑H <• x) ↑Z
H_mul_Z : ↑H * ↑Z = ↑S
H_toFinset_mul_Z : (↑H).toFinset * Z = S
card_H_mul_card_Z : Fintype.card ↥H * #Z = #S
⊢ ↑(#Z) ≤ (2 - K) * K / ((φ - K) * (K - ψ))
Module: Mathlib.Combinatorics.Enumerative.IncidenceAlgebra

Goal state before:
pp: case neg
F : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
𝕞 : Type u_4
α : Type u_5
β : Type u_6
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
hab : ¬a = b
h : ¬-∑ x ∈ (Ico a b).attach, muFun 𝕜 a ↑x = 0
⊢ a ≤ b

Tactic:
rw [neg_eq_zero] at h

Goal state after:
case neg
F : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
𝕞 : Type u_4
α : Type u_5
β : Type u_6
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
hab : ¬a = b
h : ¬∑ x ∈ (Ico a b).attach, muFun 𝕜 a ↑x = 0
⊢ a ≤ b
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: case a
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A : Finset G
hK₁ : 1 < K
hKφ : K < φ
hA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)
hA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)
K_pos hK₀ : 0 < K
hKφ' : 0 < φ - K
hKψ' : 0 < K - ψ
hK₂' : 0 < 2 - K
const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))
A_nonempty : A.Nonempty
S : Finset G := ⋯
H : Subgroup G := ⋯
S_nonempty : S.Nonempty
this : Finite ↥H
val✝ : Fintype ↥H
H_mul_S : ↑H * ↑S = ↑S
Z : Finset G
hZ : Z ⊆ S ∧ ↑H * ↑Z = ↑H * ↑S ∧ Set.InjOn (fun x ↦ ↑H <• x) ↑Z
H_mul_Z : ↑H * ↑Z = ↑S
H_toFinset_mul_Z : (↑H).toFinset * Z = S
card_H_mul_card_Z : Fintype.card ↥H * #Z = #S
r : G → ℕ := ⋯
r_inv : ∀ (z : G), r z⁻¹ = r z
l : ℕ := ⋯
hk : l = #({z ∈ S | (K - 1) * ↑(#A) < ↑(r z)})
z : G
hz : z ∈ {z ∈ S | (K - 1) * ↑(#A) < ↑(r z)}
⊢ r z ≤ #A

pp: case h
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A : Finset G
hK₁ : 1 < K
hKφ : K < φ
hA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)
hA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)
K_pos hK₀ : 0 < K
hKφ' : 0 < φ - K
hKψ' : 0 < K - ψ
hK₂' : 0 < 2 - K
const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))
A_nonempty : A.Nonempty
S : Finset G := A * A⁻¹
H : Subgroup G := stabilizer G S
S_nonempty : S.Nonempty
this : Finite ↥H
val✝ : Fintype ↥H
H_mul_S : ↑H * ↑S = ↑S
Z : Finset G
hZ : Z ⊆ S ∧ ↑H * ↑Z = ↑H * ↑S ∧ Set.InjOn (fun x ↦ ↑H <• x) ↑Z
H_mul_Z : ↑H * ↑Z = ↑S
H_toFinset_mul_Z : (↑H).toFinset * Z = S
card_H_mul_card_Z : Fintype.card ↥H * #Z = #S
r : G → ℕ := fun z ↦ A.convolution A⁻¹ z
r_inv : ∀ (z : G), r z⁻¹ = r z
l : ℕ := #({z ∈ S | (K - 1) * ↑(#A) < ↑(r z)})
hk : l = #({z ∈ S | (K - 1) * ↑(#A) < ↑(r z)})
z : G
hz : z ∈ {z ∈ S | ↑(r z) ≤ (K - 1) * ↑(#A)}
⊢ ↑(r z) ≤ (K - 1) * ↑(#A)

Tactic:
· exact convolution_le_card_left

Goal state after:
case h
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
K : ℝ
A : Finset G
hK₁ : 1 < K
hKφ : K < φ
hA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)
hA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)
K_pos hK₀ : 0 < K
hKφ' : 0 < φ - K
hKψ' : 0 < K - ψ
hK₂' : 0 < 2 - K
const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))
A_nonempty : A.Nonempty
S : Finset G := A * A⁻¹
H : Subgroup G := stabilizer G S
S_nonempty : S.Nonempty
this : Finite ↥H
val✝ : Fintype ↥H
H_mul_S : ↑H * ↑S = ↑S
Z : Finset G
hZ : Z ⊆ S ∧ ↑H * ↑Z = ↑H * ↑S ∧ Set.InjOn (fun x ↦ ↑H <• x) ↑Z
H_mul_Z : ↑H * ↑Z = ↑S
H_toFinset_mul_Z : (↑H).toFinset * Z = S
card_H_mul_card_Z : Fintype.card ↥H * #Z = #S
r : G → ℕ := fun z ↦ A.convolution A⁻¹ z
r_inv : ∀ (z : G), r z⁻¹ = r z
l : ℕ := #({z ∈ S | (K - 1) * ↑(#A) < ↑(r z)})
hk : l = #({z ∈ S | (K - 1) * ↑(#A) < ↑(r z)})
z : G
hz : z ∈ {z ∈ S | ↑(r z) ≤ (K - 1) * ↑(#A)}
⊢ ↑(r z) ≤ (K - 1) * ↑(#A)
Module: Mathlib.Combinatorics.Enumerative.IncidenceAlgebra

Goal state before:
pp: case neg
F : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
𝕞 : Type u_4
α : Type u_5
β : Type u_6
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
hab : ¬a = b
h : ¬-∑ x ∈ (Ioc a b).attach, muFun' 𝕜 b ↑x = 0
⊢ a ≤ b

Tactic:
rw [neg_eq_zero] at h

Goal state after:
case neg
F : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
𝕞 : Type u_4
α : Type u_5
β : Type u_6
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
hab : ¬a = b
h : ¬∑ x ∈ (Ioc a b).attach, muFun' 𝕜 b ↑x = 0
⊢ a ≤ b
Module: Mathlib.Combinatorics.Enumerative.IncidenceAlgebra

Goal state before:
pp: case pos
𝕜 : Type u_2
α : Type u_5
inst✝⁴ : AddCommGroup 𝕜
inst✝³ : One 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
hab✝ : ¬a = b
hab : a ≤ b
⊢ ∑ x ∈ Icc a b, (mu' 𝕜) x b = 0

Tactic:
simp [Icc_eq_cons_Ioc hab, mu'_eq_sum_Ioc_of_ne ‹_›]

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

Goal state before:
pp: p q : DyckWord
h✝ : p ≠ 0
h : ¬p = 0
⊢ p.take (p.firstReturn + 1) ⋯ ≠ 0

Tactic:
by rw [← toList_ne_nil, take]; simpa using toList_ne_nil.mpr h

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.Basic

Goal state before:
pp: ⊢ Function.Injective pentagonal

Tactic:
intro x y h
  replace h : (3 * (x + y) - 1) * (x - y) = 0 := by grind [two_mul_natCast_pentagonal]
  cases mul_eq_zero.mp h <;> grind

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.Ring

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
k n : ℕ
x : R
⊢ x ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - x ^ (k + i + 1)) +
      x ^ (3 * k + 5) * (x ^ ((k + 1 + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - x ^ (k + 1 + i + 1))) =
    x ^ ((k + 1) * (n + 1)) * (x ^ (2 * k + (n + 1) + 3) - 1) *
        ((∏ i ∈ Finset.range n, (1 - x ^ (k + i + 2))) * (1 - x ^ (k + n + 2))) -
      x ^ ((k + 1) * n) * (x ^ (2 * k + n + 3) - 1) * ∏ i ∈ Finset.range n, (1 - x ^ (k + i + 2))

Tactic:
rw [Finset.prod_range_succ', Finset.prod_range_succ]

Goal state after:
R : Type u_1
inst✝ : CommRing R
k n : ℕ
x : R
⊢ x ^ ((k + 1) * n) * ((∏ k_1 ∈ Finset.range n, (1 - x ^ (k + (k_1 + 1) + 1))) * (1 - x ^ (k + 0 + 1))) +
      x ^ (3 * k + 5) *
        (x ^ ((k + 1 + 1) * n) * ((∏ x_1 ∈ Finset.range n, (1 - x ^ (k + 1 + x_1 + 1))) * (1 - x ^ (k + 1 + n + 1)))) =
    x ^ ((k + 1) * (n + 1)) * (x ^ (2 * k + (n + 1) + 3) - 1) *
        ((∏ i ∈ Finset.range n, (1 - x ^ (k + i + 2))) * (1 - x ^ (k + n + 2))) -
      x ^ ((k + 1) * n) * (x ^ (2 * k + n + 3) - 1) * ∏ i ∈ Finset.range n, (1 - x ^ (k + i + 2))
Module: Mathlib.Combinatorics.Additive.VerySmallDoubling

Goal state before:
pp: 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
⊢ False

Tactic:
let ex := expansion K S

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

Goal state before:
pp: 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
hA : ∃ A, #S ≤ #A ∧ ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
K : ℝ := 1 - ε / 2
hK : K < 1
⊢ ∃ H x Z, ↑(Fintype.card ↥H) ≤ (2 / ε - 1) * ↑(#S) ∧ ↑(#Z) ≤ 2 / ε - 1 ∧ ↑S ⊆ ↑H * ↑Z

Tactic:
let ex := expansion K S

Goal state after:
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
hA : ∃ A, #S ≤ #A ∧ ↑(#(A * S)) ≤ (2 - ε) * ↑(#S)
K : ℝ := 1 - ε / 2
hK : K < 1
ex : Finset G → ℝ := expansion K S
⊢ ∃ H x Z, ↑(Fintype.card ↥H) ≤ (2 / ε - 1) * ↑(#S) ∧ ↑(#Z) ≤ 2 / ε - 1 ∧ ↑S ⊆ ↑H * ↑Z
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.Ring

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
x : R
hx : IsTopologicallyNilpotent x
hsum : ∀ (k : ℕ), Summable fun n ↦ x ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - x ^ (k + i + 1))
hlhs : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
hrhs : Summable fun k ↦ (-1) ^ k * (x ^ pentagonal (-↑k) - x ^ pentagonal (↑k + 1))
htail :
  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)
h :
  ∀ (n : ℕ),
    ∏' (n : ℕ), (1 - x ^ (n + 1)) -
        (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x =
      ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2))
h1 : ∀ (k : ℕ), pentagonal (↑k + 1) = (k + 1) * (3 * k + 2) / 2
h2 : ∀ (k : ℕ), pentagonal (-↑k) = k * (3 * k + 1) / 2
⊢ Tendsto
    ((fun n ↦
        ∑ x_1 ∈ Finset.range n, (-1) ^ x_1 * (x ^ (x_1 * (3 * x_1 + 1) / 2) - x ^ ((x_1 + 1) * (3 * x_1 + 2) / 2))) ∘
      fun a ↦ a + 1)
    atTop (𝓝 (∏' (n : ℕ), (1 - x ^ (n + 1))))

Tactic:
Function.comp_def,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T2Space R
x : R
hx : IsTopologicallyNilpotent x
hsum : ∀ (k : ℕ), Summable fun n ↦ x ^ ((k + 1) * n) * ∏ i ∈ Finset.range (n + 1), (1 - x ^ (k + i + 1))
hlhs : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
hrhs : Summable fun k ↦ (-1) ^ k * (x ^ pentagonal (-↑k) - x ^ pentagonal (↑k + 1))
htail :
  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)
h :
  ∀ (n : ℕ),
    ∏' (n : ℕ), (1 - x ^ (n + 1)) -
        (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x =
      ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2))
h1 : ∀ (k : ℕ), pentagonal (↑k + 1) = (k + 1) * (3 * k + 2) / 2
h2 : ∀ (k : ℕ), pentagonal (-↑k) = k * (3 * k + 1) / 2
⊢ Tendsto
    (fun x_1 ↦
      ∑ x_2 ∈ Finset.range (x_1 + 1),
        (-1) ^ x_2 * (x ^ (x_2 * (3 * x_2 + 1) / 2) - x ^ ((x_2 + 1) * (3 * x_2 + 2) / 2)))
    atTop (𝓝 (∏' (n : ℕ), (1 - x ^ (n + 1))))
Module: Mathlib.Combinatorics.Graph.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ { vertexSet := V(G₁), IsLink := G₁.IsLink, edgeSet := E(G₁), isLink_symm := ⋯, eq_or_eq_of_isLink_of_isLink := ⋯,
      edge_mem_iff_exists_isLink := ⋯, left_mem_of_isLink := ⋯ } =
    G₂

Tactic:
← G₂.mk_eq_self G₂.edge_mem_iff_exists_isLink

Goal state after:
α : Type u_1
β : Type u_2
G₁ G₂ : Graph α β
hV : V(G₁) = V(G₂)
h : ∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y
⊢ { vertexSet := V(G₁), IsLink := G₁.IsLink, edgeSet := E(G₁), isLink_symm := ⋯, eq_or_eq_of_isLink_of_isLink := ⋯,
      edge_mem_iff_exists_isLink := ⋯, left_mem_of_isLink := ⋯ } =
    { vertexSet := V(G₂), IsLink := G₂.IsLink, edgeSet := E(G₂), isLink_symm := ⋯, eq_or_eq_of_isLink_of_isLink := ⋯,
      edge_mem_iff_exists_isLink := ⋯, left_mem_of_isLink := ⋯ }
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case h
M : Type u_1
inst✝ : Semigroup M
a✝ : Stream' M
m✝ : M
a : Stream' M
m : M
h✝ : FP a.tail m
n : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail
m' : M
⊢ m' ∈ FP (Stream'.drop (n + 1) a) → m * m' ∈ FP a

Tactic:
hm'

Goal state after:
case h
M : Type u_1
inst✝ : Semigroup M
a✝ : Stream' M
m✝ : M
a : Stream' M
m : M
h✝ : FP a.tail m
n : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail
m' : M
hm' : m' ∈ FP (Stream'.drop (n + 1) a)
⊢ m * m' ∈ FP a
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case h
M : Type u_1
inst✝ : Semigroup M
a✝ : Stream' M
m✝ : M
a : Stream' M
m : M
h✝ : FP a.tail m
n : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail
m' : M
⊢ m' ∈ FP (Stream'.drop (n + 1) a) → a.head * m * m' ∈ FP a

Tactic:
hm'

Goal state after:
case h
M : Type u_1
inst✝ : Semigroup M
a✝ : Stream' M
m✝ : M
a : Stream' M
m : M
h✝ : FP a.tail m
n : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail
m' : M
hm' : m' ∈ FP (Stream'.drop (n + 1) a)
⊢ a.head * m * m' ∈ FP a
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case refine_5
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
h : ∃ m ∈ S, m * m = m
⊢ ∃ U, U * U = U ∧ ∀ᶠ (m : M) in ↑U, m ∈ FP a

pp: case refine_1
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ (r : Ultrafilter M), Continuous fun x ↦ x * r

pp: case refine_2
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ S.Nonempty

pp: case refine_3
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ IsCompact S

pp: case refine_4
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ x ∈ S, ∀ y ∈ S, x * y ∈ S

Tactic:
· rcases h with ⟨U, hU, U_idem⟩
    refine ⟨U, U_idem, ?_⟩
    convert! Set.mem_iInter.mp hU 0

Goal state after:
case refine_1
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ (r : Ultrafilter M), Continuous fun x ↦ x * r

case refine_2
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ S.Nonempty

case refine_3
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ IsCompact S

case refine_4
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
⊢ ∀ x ∈ S, ∀ y ∈ S, x * y ∈ S
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
n : ℕ
U : Ultrafilter M
hU : U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop (n + 1) a)}
⊢ ∀ a_1 ∈ FP (Stream'.drop n a).tail, a_1 ∈ FP (Stream'.drop n a)

Tactic:
exact FP.tail _

Goal state after:
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
U : Ultrafilter M
hU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}
V : Ultrafilter M
hV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}
n : ℕ
m : M
hm : m ∈ FP (Stream'.drop n a)
n' : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n' (Stream'.drop n a)), m * m' ∈ FP (Stream'.drop n a)
m' : M
⊢ m' ∈ FP (Stream'.drop (n' + n) a) → m * m' ∈ FP (Stream'.drop n a)

Tactic:
hm'

Goal state after:
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
S : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}
U : Ultrafilter M
hU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}
V : Ultrafilter M
hV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}
n : ℕ
m : M
hm : m ∈ FP (Stream'.drop n a)
n' : ℕ
hn : ∀ m' ∈ FP (Stream'.drop n' (Stream'.drop n a)), m * m' ∈ FP (Stream'.drop n a)
m' : M
hm' : m' ∈ FP (Stream'.drop (n' + n) a)
⊢ m * m' ∈ FP (Stream'.drop n a)
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: M : Type u_1
inst✝ : Semigroup M
a : Stream' M
i j : ℕ
ij : i < j
⊢ a.get i * a.get j ∈ FP (Stream'.drop i a)

Tactic:
rw [← Stream'.head_drop]

Goal state after:
M : Type u_1
inst✝ : Semigroup M
a : Stream' M
i j : ℕ
ij : i < j
⊢ (Stream'.drop i a).head * a.get j ∈ FP (Stream'.drop i a)
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
f : MvPolynomial σ R
hf : m.degree f ≠ 0
⊢ m.toSyn (m.degree (m.subLTerm f)) ≤ m.toSyn (m.degree f) ∧ m.toSyn (m.degree (m.subLTerm f)) ≠ m.toSyn (m.degree f)

Tactic:
refine ⟨degree_sub_LTerm_le f, ?_⟩

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
f : MvPolynomial σ R
hf : m.degree f ≠ 0
⊢ m.toSyn (m.degree (m.subLTerm f)) ≠ m.toSyn (m.degree f)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
I J : Set α
hI : M.Indep I
⊢ (∃ B, (M ／ I).IsBase B ∧ J ⊆ B) ↔ Disjoint J I ∧ ∃ B, M.IsBase B ∧ J ∪ I ⊆ B

Tactic:
hI.contract_isBase_iff,

Goal state after:
α : Type u_1
M : Matroid α
I J : Set α
hI : M.Indep I
⊢ (∃ B, (M.IsBase (B ∪ I) ∧ Disjoint B I) ∧ J ⊆ B) ↔ Disjoint J I ∧ ∃ B, M.IsBase B ∧ J ∪ I ⊆ B
Module: Mathlib.Combinatorics.HalesJewett

Goal state before:
pp: M : Type u_5
κ : Type u_6
inst✝¹ : AddCommMonoid M
S : Finset M
inst✝ : Finite κ
C : M → κ
⊢ ∃ a > 0, ∃ b c, ∀ s ∈ S, C (a • s + b) = c

Tactic:
obtain ⟨ι, _inst, hι⟩ := Line.exists_mono_in_high_dimension S κ
  specialize hι fun v => C <| ∑ i, v i
  obtain ⟨l, c, hl⟩ := hι
  set s : Finset ι := {i | l.idxFun i = none} with hs
  refine ⟨#s, Finset.card_pos.mpr ⟨l.proper.choose, ?_⟩, ∑ i ∈ sᶜ, ((l.idxFun i).map ?_).getD 0,
    c, ?_⟩
  · rw [hs, Finset.mem_filter]
    exact ⟨Finset.mem_univ _, l.proper.choose_spec⟩
  · exact fun m => m
  intro x xs
  rw [← hl ⟨x, xs⟩]
  clear hl; congr
  rw [← Finset.sum_add_sum_compl s]
  congr 1
  · rw [← Finset.sum_const]
    apply Finset.sum_congr rfl
    intro i hi
    rw [hs, Finset.mem_filter] at hi
    rw [l.apply_none _ _ hi.right, Subtype.coe_mk]
  · apply Finset.sum_congr rfl
    intro i hi
    rw [hs, Finset.compl_filter, Finset.mem_filter] at hi
    obtain ⟨y, hy⟩ := Option.ne_none_iff_exists.mp hi.right
    simp [← hy, Option.map_some, Option.getD]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
I J X C : Set α
h : M.IsBasis' I X
hJC : M.IsBasis' J C
h_ind : M.Indep (I \ C ∪ J)
⊢ (M ／ C).IsBasis (I \ C) (X \ C ∩ (M.E \ C))

Tactic:
← sdiff_inter_distrib_right

Goal state after:
α : Type u_1
M : Matroid α
I J X C : Set α
h : M.IsBasis' I X
hJC : M.IsBasis' J C
h_ind : M.Indep (I \ C ∪ J)
⊢ (M ／ C).IsBasis (I \ C) ((X ∩ M.E) \ C)
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : X ⊆ M.loops
⊢ M ／ X = M ＼ X

Tactic:
by
  simp [(empty_isBasis_iff.2 hX).contract_eq_contract_delete]

Goal state after:
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: case of_equiv.Heval
R : Type u_1
inst✝² : CommRing R
σ✝ : Type u_2
inst✝¹ : Finite σ✝
inst✝ : IsDomain R
σ τ : Type u_2
e : σ ≃ τ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial τ R
S : τ → Finset R
Hdeg : ∀ (i : τ), degreeOf i P < #(S i)
Heval : ∀ (x : τ → R), (∀ (i : τ), x i ∈ S i) → (eval x) P = 0
x : σ → R
hx : ∀ (i : σ), x i ∈ S (e i)
⊢ (eval (x ∘ ⇑e.symm)) P = 0

Tactic:
apply Heval

Goal state after:
case of_equiv.Heval
R : Type u_1
inst✝² : CommRing R
σ✝ : Type u_2
inst✝¹ : Finite σ✝
inst✝ : IsDomain R
σ τ : Type u_2
e : σ ≃ τ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial τ R
S : τ → Finset R
Hdeg : ∀ (i : τ), degreeOf i P < #(S i)
Heval : ∀ (x : τ → R), (∀ (i : τ), x i ∈ S i) → (eval x) P = 0
x : σ → R
hx : ∀ (i : σ), x i ∈ S (e i)
⊢ ∀ (i : τ), (x ∘ ⇑e.symm) i ∈ S i
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ (M ／ C).Spanning (X \ C)

Tactic:
contract_spanning_iff',

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : M.Spanning X
C : Set α
hXE : X ⊆ M.E
⊢ M.Spanning (X \ C ∪ C ∩ M.E) ∧ Disjoint (X \ C) C
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: case heval
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := (optionEquivLeft R σ) P
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
y : R
hy : y ∈ S none
⊢ (eval fun x_1 ↦ x_1.elim y x) P = 0

Tactic:
apply Heval

Goal state after:
case heval
R : Type u_1
inst✝³ : CommRing R
σ✝ : Type u_2
inst✝² : Finite σ✝
inst✝¹ : IsDomain R
σ : Type u_2
inst✝ : Fintype σ
h :
  ∀ (P : MvPolynomial σ R) (S : σ → Finset R),
    (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0
P : MvPolynomial (Option σ) R
S : Option σ → Finset R
Hdeg : ∀ (i : Option σ), degreeOf i P < #(S i)
Heval : ∀ (x : Option σ → R), (∀ (i : Option σ), x i ∈ S i) → (eval x) P = 0
Q : Polynomial (MvPolynomial σ R) := ⋯
hQ : Q = (optionEquivLeft R σ) P
x : σ → R
hx : ∀ (i : σ), x i ∈ S (Option.some i)
y : R
hy : y ∈ S none
⊢ ∀ (i : Option σ), i.elim y x ∈ S i
Module: Mathlib.Combinatorics.Quiver.ConnectedComponent

Goal state before:
pp: V : Type u_2
inst✝ : Quiver V
h : IsSStronglyConnected V
⊢ IsStronglyConnected V

Tactic:
intro i j

Goal state after:
V : Type u_2
inst✝ : Quiver V
h : IsSStronglyConnected V
i j : V
⊢ Nonempty (Path i j)