mathlib-initiative/mathlib-tactics huggingface dataset:
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 := 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
z : G
hz : z ∈ S
hrz : (K - 1) * ↑(#A) < ↑(r z)
⊢ z ∈ H

Tactic:
mem_stabilizer_finset'

Goal state after:
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 := 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
z : G
hz : z ∈ S
hrz : (K - 1) * ↑(#A) < ↑(r z)
⊢ ∀ ⦃b : G⦄, b ∈ S → z •> b ∈ S
Module: Mathlib.Combinatorics.Enumerative.DyckWord

Goal state before:
pp: case neg
p : DyckWord
h : ¬p = 0
⊢ p.insidePart.nest + ofTree p.outsidePart.toTree = p

Tactic:
ofTree_toTree p.outsidePart

Goal state after:
case neg
p : DyckWord
h : ¬p = 0
⊢ p.insidePart.nest + p.outsidePart = p
Module: Mathlib.Combinatorics.Enumerative.Partition.Glaisher

Goal state before:
pp: R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : T2Space R
inst✝² : CommSemiring R
inst✝¹ : IsTopologicalSemiring R
p : ℕ → Prop
inst✝ : DecidablePred p
⊢ HasProd (fun i ↦ if p (i + 1) then ∑' (j : ℕ), X ^ ((i + 1) * j) else 1) (PowerSeries.mk fun n ↦ ↑(#(restricted n p)))

Tactic:
convert! hasProd_genFun (fun i c ↦ if p i then (1 : R) else 0) using 1

Goal state after:
case e'_5
R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : T2Space R
inst✝² : CommSemiring R
inst✝¹ : IsTopologicalSemiring R
p : ℕ → Prop
inst✝ : DecidablePred p
⊢ (fun i ↦ if p (i + 1) then ∑' (j : ℕ), X ^ ((i + 1) * j) else 1) = fun i ↦
    1 + ∑' (j : ℕ), (if p (i + 1) then 1 else 0) • X ^ ((i + 1) * (j + 1))

case e'_6
R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : T2Space R
inst✝² : CommSemiring R
inst✝¹ : IsTopologicalSemiring R
p : ℕ → Prop
inst✝ : DecidablePred p
⊢ (PowerSeries.mk fun n ↦ ↑(#(restricted n p))) = genFun fun i c ↦ if p i then 1 else 0
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.Ring

Goal state before:
pp: case succ
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 x_1 ↦ powMulProdOneSubPow k x_1 x
h✝ : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
n : ℕ
ih :
  ∏' (n : ℕ), (1 - x ^ (n + 1)) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x
h : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 2) / 2 = (n + 1) * (3 * n + 4) / 2 + (2 * n + 3)
⊢ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) *
        (1 - x ^ (2 * n + 3) - x ^ (3 * n + 5) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x) =
    ∑ x_1 ∈ Finset.range (n + 1), (-1) ^ x_1 * (x ^ (x_1 * (3 * x_1 + 1) / 2) - x ^ ((x_1 + 1) * (3 * x_1 + 2) / 2)) +
        (-1) ^ (n + 1) * (x ^ ((n + 1) * (3 * (n + 1) + 1) / 2) - x ^ ((n + 1 + 1) * (3 * (n + 1) + 2) / 2)) +
      (-1) ^ (n + 1 + 1) * x ^ ((n + 1 + 1) * (3 * (n + 1) + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x

Tactic:
simp_rw [h]

Goal state after:
case succ
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 x_1 ↦ powMulProdOneSubPow k x_1 x
h✝ : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
n : ℕ
ih :
  ∏' (n : ℕ), (1 - x ^ (n + 1)) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x
h : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 2) / 2 = (n + 1) * (3 * n + 4) / 2 + (2 * n + 3)
⊢ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) *
        (1 - x ^ (2 * n + 3) - x ^ (3 * n + 5) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
        (-1) ^ (n + 1) * (x ^ ((n + 1) * (3 * (n + 1) + 1) / 2) - x ^ ((n + 1) * (3 * n + 4) / 2 + (2 * n + 3))) +
      (-1) ^ (n + 1 + 1) * x ^ ((n + 1 + 1) * (3 * (n + 1) + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x
Module: Mathlib.Combinatorics.Enumerative.Pentagonal.Ring

Goal state before:
pp: case succ
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 x_1 ↦ powMulProdOneSubPow k x_1 x
h✝¹ : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
n : ℕ
ih :
  ∏' (n : ℕ), (1 - x ^ (n + 1)) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x
h✝ : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 2) / 2 = (n + 1) * (3 * n + 4) / 2 + (2 * n + 3)
h : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 4) / 2 = (n + 1) * (3 * n + 4) / 2 + (3 * n + 5)
⊢ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) *
        (1 - x ^ (2 * n + 3) - x ^ (3 * n + 5) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
        (-1) ^ (n + 1) * (x ^ ((n + 1) * (3 * (n + 1) + 1) / 2) - x ^ ((n + 1) * (3 * n + 4) / 2 + (2 * n + 3))) +
      (-1) ^ (n + 1 + 1) * x ^ ((n + 1 + 1) * (3 * (n + 1) + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x

Tactic:
simp_rw [h]

Goal state after:
case succ
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 x_1 ↦ powMulProdOneSubPow k x_1 x
h✝¹ : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)
n : ℕ
ih :
  ∏' (n : ℕ), (1 - x ^ (n + 1)) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) * ∑' (n_1 : ℕ), powMulProdOneSubPow n n_1 x
h✝ : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 2) / 2 = (n + 1) * (3 * n + 4) / 2 + (2 * n + 3)
h : ∀ (n : ℕ), (n + 1 + 1) * (3 * (n + 1) + 4) / 2 = (n + 1) * (3 * n + 4) / 2 + (3 * n + 5)
⊢ ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
      (-1) ^ (n + 1) * x ^ ((n + 1) * (3 * n + 4) / 2) *
        (1 - x ^ (2 * n + 3) - x ^ (3 * n + 5) * ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x) =
    ∑ k ∈ Finset.range (n + 1), (-1) ^ k * (x ^ (k * (3 * k + 1) / 2) - x ^ ((k + 1) * (3 * k + 2) / 2)) +
        (-1) ^ (n + 1) * (x ^ ((n + 1) * (3 * (n + 1) + 1) / 2) - x ^ ((n + 1) * (3 * n + 4) / 2 + (2 * n + 3))) +
      (-1) ^ (n + 1 + 1) * x ^ ((n + 1) * (3 * n + 4) / 2 + (3 * n + 5)) *
        ∑' (n_1 : ℕ), powMulProdOneSubPow (n + 1) n_1 x
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
ex : Finset G → ℝ := expansion K S
κ : ℝ := connectivity K S
⊢ ∃ H x Z, ↑(Fintype.card ↥H) ≤ (2 / ε - 1) * ↑(#S) ∧ ↑(#Z) ≤ 2 / ε - 1 ∧ ↑S ⊆ ↑H * ↑Z

Tactic:
obtain ⟨H, _, hH⟩ := exists_subgroup_isAtom hK hS

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
κ : ℝ := connectivity K S
H : Subgroup G
w✝ : Fintype ↥H
hH : IsAtom K S (↑H).toFinset
⊢ ∃ H x Z, ↑(Fintype.card ↥H) ≤ (2 / ε - 1) * ↑(#S) ∧ ↑(#Z) ≤ 2 / ε - 1 ∧ ↑S ⊆ ↑H * ↑Z
Module: Mathlib.Combinatorics.Extremal.RuzsaSzemeredi

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

Tactic:
refine IsLittleO.right_isBigO_sub ?_

Goal state after:
⊢ (fun n ↦ 2) =o[atTop] fun n ↦ ↑n / 3
Module: Mathlib.Combinatorics.HalesJewett

Goal state before:
pp: case inr
η : Type u_5
α : Type u_6
ι : Type u_7
inst✝ : Nontrivial α
l m : Subspace η α ι
i : ι
hlm : ∀ (x : η → α) (x_1 : ι), ↑l x x_1 = ↑m x x_1
e : η
hl : l.idxFun i = Sum.inr e
x✝ : α ⊕ η
hm : m.idxFun i = x✝
⊢ Sum.inr e = x✝

Tactic:
cases hm : idxFun m i with

Goal state after:
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case h.head'
M : Type u_1
inst✝ : Semigroup M
U : Ultrafilter M
U_idem : U * U = U
exists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty
elem : { s // s ∈ U } → M := fun p ↦ ⋯.some
succ : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * m ∈ ↑p}, ⋯⟩
a : Stream' M
m : M
p : { s // s ∈ U }
⊢ (Stream'.cons (elem p) (Stream'.corec elem succ (succ p))).head ∈ ↑p

Tactic:
Stream'.head_cons

Goal state after:
case h.head'
M : Type u_1
inst✝ : Semigroup M
U : Ultrafilter M
U_idem : U * U = U
exists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty
elem : { s // s ∈ U } → M := fun p ↦ ⋯.some
succ : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * m ∈ ↑p}, ⋯⟩
a : Stream' M
m : M
p : { s // s ∈ U }
⊢ elem p ∈ ↑p
Module: Mathlib.Combinatorics.Hindman

Goal state before:
pp: case H.inl
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih :
  ∀ s_1 ∈ s, ∀ (hs : (s.erase s_1).Nonempty), ∏ i ∈ s.erase s_1, a.get i ∈ FP (Stream'.drop ((s.erase s_1).min' hs) a)
hs : s.Nonempty
h : s.erase (s.min' hs) = ∅
⊢ (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)

Tactic:
rw [h, Finset.prod_empty, mul_one]
    exact FP.head _

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

Goal state before:
pp: α : Type u_1
M : Matroid α
I B : Set α
hI : M.Indep I
⊢ Disjoint B I → B ⊆ M.E → ((M✶ ↾ M.E \ I).IsBase (M.E \ (I ∪ B)) ↔ M✶.IsBase (M.E \ (I ∪ B)))

Tactic:
Spanning.isBase_restrict_iff,

Goal state after:
α : Type u_1
M : Matroid α
I B : Set α
hI : M.Indep I
⊢ Disjoint B I → B ⊆ M.E → (M✶.IsBase (M.E \ (I ∪ B)) ∧ M.E \ (I ∪ B) ⊆ M.E \ I ↔ M✶.IsBase (M.E \ (I ∪ B)))

α : Type u_1
M : Matroid α
I B : Set α
hI : M.Indep I
⊢ M✶.Spanning (M.E \ I)
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case pos
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b✝ : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b✝ i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b✝ i) ≠ 0
hf0 : f = 0
b : ι
⊢ 0 ≤ m.toSyn (m.degree f)

Tactic:
exact bot_le

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

Goal state before:
pp: α : Type u_1
M : Matroid α
e : α
C : Set α
⊢ (M ／ C).IsLoop e ↔ e ∈ M.closure C ∧ e ∉ C

Tactic:
by
  obtain ⟨I, hI⟩ := M.exists_isBasis' C
  rw [hI.contract_eq_contract_delete, delete_isLoop_iff, ← singleton_dep,
    hI.indep.contract_dep_iff, singleton_union, hI.indep.insert_dep_iff, hI.closure_eq_closure]
  by_cases heI : e ∈ I
  · simp [heI, hI.subset heI]
  simp [heI, and_comm]

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

Goal state before:
pp: α : Type u_1
M : Matroid α
C X : Set α
⊢ (M ／ C).closure X = M.closure (X ∪ C) \ C

Tactic:
rw [← sdiff_union_inter (M.closure (X ∪ C) \ C) X, sdiff_sdiff, union_comm C, ← contract_loops_eq,
    union_comm X, ← contract_contract, contract_loops_eq, subset_antisymm_iff, union_subset_iff,
    and_iff_right sdiff_subset, ← sdiff_subset_iff]
  simp only [sdiff_sdiff_right_self, subset_inter_iff, inter_subset_right, and_true]
  refine ⟨fun e ⟨he, he'⟩ ↦ ⟨mem_closure_of_mem' _ (.inr he') (mem_ground_of_mem_closure he).1,
    (closure_subset_ground _ _ he).2⟩, fun e ⟨⟨he, heC⟩, he'⟩ ↦
    mem_closure_of_mem' _ he' ⟨M.closure_subset_ground _ he, heC⟩⟩

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case pos
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ m.toSyn (m.degree (b i * (monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f))) ≤
    m.toSyn (m.degree f)

Tactic:
grw [degree_mul_le, map_add, degree_monomial_le, ← map_add, add_tsub_cancel_of_le hf]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
hc : ¬i = j
⊢ 0 ≤ m.toSyn (m.degree f)

Tactic:
exact bot_le

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Groebner

Goal state before:
pp: case pos
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b✝ : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b✝ i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b✝ i) ≠ 0
hf0 : ¬f = 0
hf : ∀ (i : ι), ¬m.degree (b✝ i) ≤ m.degree f
hf'0 : m.subLTerm f = 0
b : ι
⊢ 0 ≤ m.toSyn (m.degree (m.subLTerm f))

Tactic:
exact bot_le

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: α : Type u_1
M N : Matroid α
h : Disjoint M.E N.E
B : Set α
hB : (M.disjointSum N h).IsBase B
⊢ ∃ BM BN, M.IsBase BM ∧ N.IsBase BN ∧ Disjoint BM BN ∧ B = BM ∪ BN

Tactic:
disjointSum_isBase_iff

Goal state after:
α : Type u_1
M N : Matroid α
h : Disjoint M.E N.E
B : Set α
hB : M.IsBase (B ∩ M.E) ∧ N.IsBase (B ∩ N.E) ∧ B ⊆ M.E ∪ N.E
⊢ ∃ BM BN, M.IsBase BM ∧ N.IsBase BN ∧ Disjoint BM BN ∧ B = BM ∪ BN
Module: Mathlib.Combinatorics.Quiver.Arborescence

Goal state before:
pp: V : Type u
inst✝ : Quiver V
r : V
height : V → ℕ
height_lt : ∀ ⦃a b : V⦄ (a_1 : a ⟶ b), height a < height b
unique_arrow : ∀ ⦃a b c : V⦄ (e : a ⟶ c) (f : b ⟶ c), a = b ∧ e ≍ f
root_or_arrow : ∀ (b : V), b = r ∨ ∃ a, Nonempty (a ⟶ b)
b : V
height_le : ∀ {a b : V} (a_1 : Path a b), height a ≤ height b
p q : Path r b
⊢ p = q

Tactic:
induction p with
      | nil =>
        rcases q with _ | ⟨q, f⟩
        · rfl
        · exact False.elim (lt_irrefl _ (lt_of_le_of_lt (height_le q) (height_lt f)))
      | cons p e ih =>
        rcases q with _ | ⟨q, f⟩
        · exact False.elim (lt_irrefl _ (lt_of_le_of_lt (height_le p) (height_lt e)))
        · rcases unique_arrow e f with ⟨⟨⟩, ⟨⟩⟩
          rw [ih]

Goal state after:
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: case h.Heval
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
S : σ → Finset R
Sne : ∀ (i : σ), (S i).Nonempty
f : MvPolynomial σ R
Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0
this : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
h : σ →₀ MvPolynomial σ R
r : MvPolynomial σ R
hf : f - (linearCombination (MvPolynomial σ R) fun i ↦ Alon.P (S i) i) h = r
hh : ∀ (i : σ), degLex.toSyn (degLex.degree (Alon.P (S i) i * h i)) ≤ degLex.toSyn (degLex.degree f)
hr : ∀ c ∈ r.support, ∀ (i : σ), ¬degLex.degree (Alon.P (S i) i) ≤ c
x : σ → R
hx : ∀ (i : σ), x i ∈ S i
⊢ ∀ x_1 ∈ h.support, (eval x) (h x_1 • Alon.P (S x_1) x_1) = 0

Tactic:
intro i _

Goal state after:
case h.Heval
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
S : σ → Finset R
Sne : ∀ (i : σ), (S i).Nonempty
f : MvPolynomial σ R
Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0
this : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
h : σ →₀ MvPolynomial σ R
r : MvPolynomial σ R
hf : f - (linearCombination (MvPolynomial σ R) fun i ↦ Alon.P (S i) i) h = r
hh : ∀ (i : σ), degLex.toSyn (degLex.degree (Alon.P (S i) i * h i)) ≤ degLex.toSyn (degLex.degree f)
hr : ∀ c ∈ r.support, ∀ (i : σ), ¬degLex.degree (Alon.P (S i) i) ≤ c
x : σ → R
hx : ∀ (i : σ), x i ∈ S i
i : σ
a✝ : i ∈ h.support
⊢ (eval x) (h i • Alon.P (S i) i) = 0
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: case e'_2.e'_6
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
S : σ → Finset R
Sne : ∀ (i : σ), (S i).Nonempty
f : MvPolynomial σ R
Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0
this : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
h : σ →₀ MvPolynomial σ R
r : MvPolynomial σ R
hf : f - (linearCombination (MvPolynomial σ R) fun i ↦ Alon.P (S i) i) h = r
hh : ∀ (i : σ), degLex.toSyn (degLex.degree (Alon.P (S i) i * h i)) ≤ degLex.toSyn (degLex.degree f)
hr : ∀ c ∈ r.support, ∀ (i : σ), ¬degLex.degree (Alon.P (S i) i) ≤ c
x : σ → R
hx : ∀ (i : σ), x i ∈ S i
i : σ
a✝ : i ∈ h.support
⊢ (eval x) (Alon.P (S i) i) = 0

Tactic:
Alon.P,

Goal state after:
case e'_2.e'_6
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
S : σ → Finset R
Sne : ∀ (i : σ), (S i).Nonempty
f : MvPolynomial σ R
Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0
this : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
h : σ →₀ MvPolynomial σ R
r : MvPolynomial σ R
hf : f - (linearCombination (MvPolynomial σ R) fun i ↦ Alon.P (S i) i) h = r
hh : ∀ (i : σ), degLex.toSyn (degLex.degree (Alon.P (S i) i * h i)) ≤ degLex.toSyn (degLex.degree f)
hr : ∀ c ∈ r.support, ∀ (i : σ), ¬degLex.degree (Alon.P (S i) i) ≤ c
x : σ → R
hx : ∀ (i : σ), x i ∈ S i
i : σ
a✝ : i ∈ h.support
⊢ (eval x) (∏ r ∈ S i, (X i - C r)) = 0
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
f : MvPolynomial σ R
t : σ →₀ ℕ
ht : coeff t f ≠ 0
ht' : f.totalDegree = Finsupp.degree t
S : σ → Finset R
htS : ∀ (i : σ), t i < #(S i)
x✝ : LinearOrder σ := ⋯
Heval : ∀ (s : σ → R), (∀ (i : σ), s i ∈ S i) → (eval s) f = 0
h : σ →₀ MvPolynomial σ R
hh : ∀ (i : σ), ((∏ s ∈ S i, (X i - C s)) * h i).totalDegree ≤ f.totalDegree
hf : f = (linearCombination (MvPolynomial σ R) fun i ↦ ∏ r ∈ S i, (X i - C r)) h
⊢ ∀ x ∈ h.support, coeff t (h x • ∏ r ∈ S x, (X x - C r)) = 0

Tactic:
intro i _

Goal state after:
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
f : MvPolynomial σ R
t : σ →₀ ℕ
ht : coeff t f ≠ 0
ht' : f.totalDegree = Finsupp.degree t
S : σ → Finset R
htS : ∀ (i : σ), t i < #(S i)
x✝ : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
Heval : ∀ (s : σ → R), (∀ (i : σ), s i ∈ S i) → (eval s) f = 0
h : σ →₀ MvPolynomial σ R
hh : ∀ (i : σ), ((∏ s ∈ S i, (X i - C s)) * h i).totalDegree ≤ f.totalDegree
hf : f = (linearCombination (MvPolynomial σ R) fun i ↦ ∏ r ∈ S i, (X i - C r)) h
i : σ
a✝ : i ∈ h.support
⊢ coeff t (h i • ∏ r ∈ S i, (X i - C r)) = 0
Module: Mathlib.Combinatorics.Nullstellensatz

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
f : MvPolynomial σ R
t : σ →₀ ℕ
ht : coeff t f ≠ 0
ht' : f.totalDegree = Finsupp.degree t
S : σ → Finset R
htS : ∀ (i : σ), t i < #(S i)
x✝ : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
Heval : ∀ (s : σ → R), (∀ (i : σ), s i ∈ S i) → (eval s) f = 0
h : σ →₀ MvPolynomial σ R
hh : ∀ (i : σ), ((∏ s ∈ S i, (X i - C s)) * h i).totalDegree ≤ f.totalDegree
hf : f = (linearCombination (MvPolynomial σ R) fun i ↦ ∏ r ∈ S i, (X i - C r)) h
i : σ
a✝ : i ∈ h.support
⊢ coeff t (h i • ∏ r ∈ S i, (X i - C r)) = 0

Tactic:
set g := h i * Alon.P (S i) i with hg

Goal state after:
R : Type u_1
inst✝² : CommRing R
σ : Type u_2
inst✝¹ : Finite σ
inst✝ : IsDomain R
f : MvPolynomial σ R
t : σ →₀ ℕ
ht : coeff t f ≠ 0
ht' : f.totalDegree = Finsupp.degree t
S : σ → Finset R
htS : ∀ (i : σ), t i < #(S i)
x✝ : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel
Heval : ∀ (s : σ → R), (∀ (i : σ), s i ∈ S i) → (eval s) f = 0
h : σ →₀ MvPolynomial σ R
hh : ∀ (i : σ), ((∏ s ∈ S i, (X i - C s)) * h i).totalDegree ≤ f.totalDegree
hf : f = (linearCombination (MvPolynomial σ R) fun i ↦ ∏ r ∈ S i, (X i - C r)) h
i : σ
a✝ : i ∈ h.support
g : MvPolynomial σ R := h i * Alon.P (S i) i
hg : g = h i * Alon.P (S i) i
⊢ coeff t (h i • ∏ r ∈ S i, (X i - C r)) = 0
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case neg
α : Type u_1
inst✝² : SemilatticeInf α
s : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
h : a ∉ upperClosure ↑s
⊢ ⊥ ≤ a

Tactic:
exact bot_le

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
u v a : α
hva : Disjoint v a
hua : u ≤ a
⊢ (a ⊔ v) \ v = a

Tactic:
hva.symm.sup_sdiff_cancel_right

Goal state after:
α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
u v a : α
hva : Disjoint v a
hua : u ≤ a
⊢ a = a
Module: Mathlib.Combinatorics.SetFamily.HarrisKleitman

Goal state before:
pp: case empty.inr
α : Type u_1
inst✝ : DecidableEq α
ℬ : Finset (Finset α)
hℬ : IsLowerSet ↑ℬ
hℬs : ℬ = ∅ ∨ ℬ = {∅}
h𝒜 : IsLowerSet ↑{∅}
⊢ #{∅} * #ℬ ≤ 2 ^ #∅ * #({∅} ∩ ℬ)

Tactic:
obtain rfl | rfl := hℬs

Goal state after:
case empty.inr.inl
α : Type u_1
inst✝ : DecidableEq α
h𝒜 : IsLowerSet ↑{∅}
hℬ : IsLowerSet ↑∅
⊢ #{∅} * #∅ ≤ 2 ^ #∅ * #({∅} ∩ ∅)

case empty.inr.inr
α : Type u_1
inst✝ : DecidableEq α
h𝒜 hℬ : IsLowerSet ↑{∅}
⊢ #{∅} * #{∅} ≤ 2 ^ #∅ * #({∅} ∩ {∅})