mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Semigroup

Goal state before:
pp: M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
⊢ ∃ m, m * m = m

Tactic:
obtain ⟨N_closed, ⟨m, hm⟩, N_mul⟩ := hN.prop

Goal state after:
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x ↦ x ∈ S) N
N_closed : IsClosed N
N_mul : ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N
m : M
hm : m ∈ N
⊢ ∃ m, m * m = m
Module: Mathlib.Combinatorics.Hypergraph.Basic

Goal state before:
pp: α : Type u_1
H : Hypergraph α
⊢ ⋃₀ E(H) = V(H) ↔ ∀ x ∈ V(H), ¬H.IsIsolated x

Tactic:
by
  grind [IsIsolated, mem_vertexSet_of_mem_edgeSet]

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

Goal state before:
pp: case refine_1
α : Type u_1
M : Matroid α
B D : Set α
hD : M.Coindep D
h : M.IsBasis B (M.E \ D)
hss : B ⊆ M.E ∧ Disjoint B D
hcl : M.IsBasis B (M.closure (M.E \ D))
⊢ M.IsBase B ∧ Disjoint B D

Tactic:
rw [hD.closure_compl, isBasis_ground_iff] at hcl

Goal state after:
case refine_1
α : Type u_1
M : Matroid α
B D : Set α
hD : M.Coindep D
h : M.IsBasis B (M.E \ D)
hss : B ⊆ M.E ∧ Disjoint B D
hcl : M.IsBase B
⊢ M.IsBase B ∧ Disjoint B D
Module: Mathlib.Combinatorics.Matroid.Minor.Delete

Goal state before:
pp: α : Type u_1
M : Matroid α
⊢ M ＼ M.loops = M.removeLoops

Tactic:
rw [removeLoops, delete_eq_restrict, compl_loops_eq]

Goal state after:
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:
← 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.Combinatorics.Hindman

Goal state before:
pp: case H
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
⊢ a.get (s.min' hs) * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)

Tactic:
← Stream'.head_drop

Goal state after:
case H
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
⊢ (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)
Module: Mathlib.Combinatorics.Matroid.Minor.Order

Goal state before:
pp: α : Type u_1
M : Matroid α
I C D : Set α
hI : (M ／ C ＼ D).Indep I
⊢ M.Indep I

Tactic:
exact hI.of_delete.of_contract

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

Goal state before:
pp: α : Type u_1
M : Matroid α
I X : Set α
hI : M.IsBasis' I X
D : Set α
⊢ Disjoint D (X \ I ∪ I) ∧ M.Dep (D ∪ I) ↔ M.Dep (D ∪ I) ∧ Disjoint X D

Tactic:
sdiff_union_of_subset hI.subset,

Goal state after:
α : Type u_1
M : Matroid α
I X : Set α
hI : M.IsBasis' I X
D : Set α
⊢ Disjoint D X ∧ M.Dep (D ∪ I) ↔ M.Dep (D ∪ I) ∧ Disjoint X D
Module: Mathlib.Combinatorics.Matroid.Sum

Goal state before:
pp: case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h : ∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I

Tactic:
specialize h (t := I ∪ Sigma.mk i '' I₀)

Goal state after:
case refine_1
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
x✝ :
  I ⊆ X ∧
    (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
      X ⊆ univ.sigma fun i ↦ (M i).E
hIX : I ⊆ X
h' : X ⊆ univ.sigma fun i ↦ (M i).E
i : ι
I₀ : Set (α i)
h :
  (∀ (i_1 : ι), (M i_1).Indep (Sigma.mk i_1 ⁻¹' (I ∪ Sigma.mk i '' I₀))) →
    I ∪ Sigma.mk i '' I₀ ⊆ X → I ⊆ I ∪ Sigma.mk i '' I₀ → I = I ∪ Sigma.mk i '' I₀
hI₀ : (M i).Indep I₀
hI₀X : I₀ ⊆ Sigma.mk i ⁻¹' X
hII₀ : Sigma.mk i ⁻¹' I ⊆ I₀
⊢ I₀ ⊆ Sigma.mk i ⁻¹' I
Module: Mathlib.Combinatorics.Matroid.Minor.Contract

Goal state before:
pp: α : Type u_1
M : Matroid α
I J X : Set α
h : M.IsBasis I X
hdj : Disjoint J X
h_ind : M.Indep (I ∪ J)
⊢ (M ／ J).IsBasis I (X \ J)

Tactic:
← (hdj.mono_right h.subset).sdiff_eq_right

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

Goal state before:
pp: α : Type u_1
M : Matroid α
I J X : Set α
h : M.IsBasis I X
hdj : Disjoint J X
h_ind : M.Indep (I ∪ J)
⊢ (M ／ J).IsBasis (I \ J) (X \ J)

Tactic:
exact h.contract_isBasis_of_indep h_ind

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

Goal state before:
pp: α : Type u_1
M : Matroid α
X : Set α
hX : X ⊆ M.coloops
⊢ (M ／ X)✶ = (M ＼ X)✶

Tactic:
dual_delete,

Goal state after:
α : Type u_1
M : Matroid α
X : Set α
hX : X ⊆ M.coloops
⊢ (M ／ X)✶ = M✶ ／ X
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
⊢ False

Tactic:
apply ht

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 := ⋯
Heval : ∀ (s : σ → R), (∀ (i : σ), s i ∈ S i) → (eval s) f = 0
⊢ coeff t f = 0
Module: Mathlib.Combinatorics.Quiver.Schreier

Goal state before:
pp: V : Type u_1
M : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M V
S : Type u_3
ι : S → M
W : Type u_4
inst✝ : MulAction M W
φ : SchreierGraph V ι ⥤q SchreierGraph W ι
φm : φ ⋙q labelling W ι = labelling V ι
v : SchreierGraph V ι
s : S
⊢ φ.obj (ι s • v) = ι s • φ.obj v

Tactic:
by
  -- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'
  let e : v ⟶ ι s • v := ⟨s, rfl⟩
  -- φ.map e is an edge from φ.obj v, and its label is preserved
  have h := (φ.map e).property
  -- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`
  -- We need to show `(φ.map e).val = s`
  have label_eq : (φ.map e).val = s := by
    -- `φm` says `φ ⋙q labelling = labelling`
    -- So `(φ ⋙q labelling).map e = labelling.map e`
    have : (φ ⋙q labelling W ι).map e = (labelling V ι).map e := by
      rw [φm]
    simp only [Prefunctor.comp_map, labelling_map] at this
    exact this
  rw [label_eq] at h
  exact h.symm

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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
s : Finset α
a : α
h : insert a s ∈ 𝓓 a 𝒜
ha : a ∈ s
⊢ s ∈ 𝓓 a 𝒜

Tactic:
insert_eq_of_mem ha

Goal state after:
case pos
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
s : Finset α
a : α
h : s ∈ 𝓓 a 𝒜
ha : a ∈ s
⊢ s ∈ 𝓓 a 𝒜
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: case refine_1
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
h : u ⊆ s ∪ t
⊢ s ∩ u ∪ t ∩ u = u

pp: case refine_2
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
⊢ (∃ a ⊆ s, ∃ b ⊆ t, a ∪ b = u) → u ⊆ s ∪ t

Tactic:
· rwa [← union_inter_distrib_right, inter_eq_right]

Goal state after:
case refine_2
α : Type u_2
inst✝ : DecidableEq α
s t u : Finset α
⊢ (∃ a ⊆ s, ∃ b ⊆ t, a ∪ b = u) → u ⊆ s ∪ t
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_4
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
n : ℕ
h𝒜 : Set.Sized n ↑𝒜
s : Finset α
hs : s ∈ 𝒜
⊢ #sᶜ = Fintype.card α - n

Tactic:
rw [Finset.card_compl, h𝒜 hs]

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

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ u ≤ a

Tactic:
cases ne_of_mem_of_not_mem hb ha hba

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

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
b : α
hb : b ∈ s
h✝ : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ Disjoint v a

Tactic:
cases ne_of_mem_of_not_mem hb ha hba

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #(t \ s) = k

Tactic:
rw [mem_upShadow_iterate_iff_exists_card]
  constructor
  · rintro ⟨u, rfl, hut, htu⟩
    exact ⟨_, htu, sdiff_subset, by rw [sdiff_sdiff_eq_self hut]⟩
  · rintro ⟨s, hs, hst, rfl⟩
    exact ⟨_, rfl, sdiff_subset, by rwa [sdiff_sdiff_eq_self hst]⟩

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ infSum (𝒜 ∪ ℬ) + infSum (𝒜 ⊻ ℬ) = infSum 𝒜 + infSum ℬ

Tactic:
unfold infSum

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∑ s, ↑(#((𝒜 ∪ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
      ∑ s, ↑(#((𝒜 ⊻ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) =
    ∑ s, ↑(#(𝒜.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
      ∑ s, ↑(#(ℬ.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s))
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ (a ⊔ v) \ u ∈ s

Tactic:
cases ne_of_mem_of_not_mem hb ha hba

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ ∂⁺ 𝒜
hts : t ⊆ s
hcardst : #t + k = #s
u : Finset α
hu : u ∈ 𝒜
hut : u ⊆ t
hcardtu : #t = #u + 1
⊢ #u + (k + 1) = #t + k

Tactic:
hcardtu,

Goal state after:
case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ ∂⁺ 𝒜
hts : t ⊆ s
hcardst : #t + k = #s
u : Finset α
hu : u ∈ 𝒜
hut : u ⊆ t
hcardtu : #t = #u + 1
⊢ #u + (k + 1) = #u + 1 + k
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ x ∉ (s ∪ v) \ u

pp: case refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ insert x ((s ∪ v) \ u) ∈ 𝒜

pp: case refine_3
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ ¬∃ a ∉ (s ∪ v) \ u, insert a ((s ∪ v) \ u) ∈ 𝒜'

Tactic:
· exact notMem_sdiff_of_notMem_left (notMem_union.2 ⟨‹x ∉ s›, ‹x ∉ v›⟩)

Goal state after:
case refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ insert x ((s ∪ v) \ u) ∈ 𝒜

case refine_3
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝¹ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝ : v \ u = v
this : x ∉ u
hus : u ⊆ s
⊢ ¬∃ a ∉ (s ∪ v) \ u, insert a ((s ∪ v) \ u) ∈ 𝒜'
Module: Mathlib.Combinatorics.SetFamily.HarrisKleitman

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsLowerSet ↑𝒜ᶜ
hℬ : IsLowerSet ↑ℬ
this : 2 ^ Fintype.card α * #ℬ - #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜ᶜ ∩ ℬ)
⊢ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ

Tactic:
tsub_le_iff_tsub_le,

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsLowerSet ↑𝒜ᶜ
hℬ : IsLowerSet ↑ℬ
this : 2 ^ Fintype.card α * #ℬ - 2 ^ Fintype.card α * #(𝒜ᶜ ∩ ℬ) ≤ #𝒜 * #ℬ
⊢ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝⁴ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝³ : v \ u = v
this✝² : x ∉ u
hus : u ⊆ s
w : α
hwB : w ∉ (s ∪ v) \ u
hw𝒜' : insert w ((s ∪ v) \ u) ∈ 𝒜'
this✝¹ : v ⊆ insert w ((s ∪ v) \ u)
hwu : w ∈ u
z : α
hz : z ∈ v
hxy : IsCompressed (u.erase w) (v.erase z) 𝒜
this✝ : insert w ((s ∪ v) \ u) ∈ 𝒜
this : (insert w ((s ∪ v) \ u) ∪ u.erase w) \ v.erase z ∈ 𝒜
⊢ insert z s = ((s ∪ v) \ u ∪ insert w (u.erase w)) \ v.erase z

Tactic:
insert_erase ‹w ∈ u›,

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝⁴ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝³ : v \ u = v
this✝² : x ∉ u
hus : u ⊆ s
w : α
hwB : w ∉ (s ∪ v) \ u
hw𝒜' : insert w ((s ∪ v) \ u) ∈ 𝒜'
this✝¹ : v ⊆ insert w ((s ∪ v) \ u)
hwu : w ∈ u
z : α
hz : z ∈ v
hxy : IsCompressed (u.erase w) (v.erase z) 𝒜
this✝ : insert w ((s ∪ v) \ u) ∈ 𝒜
this : (insert w ((s ∪ v) \ u) ∪ u.erase w) \ v.erase z ∈ 𝒜
⊢ insert z s = ((s ∪ v) \ u ∪ u) \ v.erase z
Module: Mathlib.Combinatorics.SetFamily.LYM

Goal state before:
pp: case succ
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
k✝ : ℕ
𝒜 : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
k : ℕ
ih :
  k ≤ Fintype.card α →
    ∑ r ∈ range (k + 1), ↑(#(𝒜 # (Fintype.card α - r))) / ↑((Fintype.card α).choose (Fintype.card α - r)) ≤
      ↑(#(falling (Fintype.card α - k) 𝒜)) / ↑((Fintype.card α).choose (Fintype.card α - k))
hk : k + 1 ≤ Fintype.card α
⊢ ↑(#(𝒜 # (Fintype.card α - (k + 1)))) / ↑((Fintype.card α).choose (Fintype.card α - (k + 1))) +
      ∑ x ∈ range (k + 1), ↑(#(𝒜 # (Fintype.card α - x))) / ↑((Fintype.card α).choose (Fintype.card α - x)) ≤
    ↑(#(𝒜 # (Fintype.card α - (k + 1)))) / ↑((Fintype.card α).choose (Fintype.card α - (k + 1))) +
      ↑(#(∂ (falling (Fintype.card α - (k + 1) + 1) 𝒜))) / ↑((Fintype.card α).choose (Fintype.card α - (k + 1)))

Tactic:
← tsub_tsub,

Goal state after:
case succ
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Semifield 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
k✝ : ℕ
𝒜 : Finset (Finset α)
inst✝ : Fintype α
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
k : ℕ
ih :
  k ≤ Fintype.card α →
    ∑ r ∈ range (k + 1), ↑(#(𝒜 # (Fintype.card α - r))) / ↑((Fintype.card α).choose (Fintype.card α - r)) ≤
      ↑(#(falling (Fintype.card α - k) 𝒜)) / ↑((Fintype.card α).choose (Fintype.card α - k))
hk : k + 1 ≤ Fintype.card α
⊢ ↑(#(𝒜 # (Fintype.card α - k - 1))) / ↑((Fintype.card α).choose (Fintype.card α - k - 1)) +
      ∑ x ∈ range (k + 1), ↑(#(𝒜 # (Fintype.card α - x))) / ↑((Fintype.card α).choose (Fintype.card α - x)) ≤
    ↑(#(𝒜 # (Fintype.card α - k - 1))) / ↑((Fintype.card α).choose (Fintype.card α - k - 1)) +
      ↑(#(∂ (falling (Fintype.card α - k - 1 + 1) 𝒜))) / ↑((Fintype.card α).choose (Fintype.card α - k - 1))
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: α : Type u_1
c : Set (Set α)
H : ∀ (a : α), ∃! b, b ∈ c ∧ a ∈ b
x x✝ : α
⊢ (mkClasses c H) x x✝ ↔ (mkClasses c H) x✝ x

Tactic:
rw [Setoid.comm']

Goal state after:
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: case left.a
α : Type u_1
r : Setoid α
f : α → ↑r.classes := fun a ↦ ⟨{x | r x a}, ⋯⟩
f_respects_relation : ∀ (a b : α), r a b → f a = f b
q_b : Quotient r
a : α
h_eq : Quot.lift f f_respects_relation ⟦a⟧ = Quot.lift f f_respects_relation q_b
⊢ ⟦a⟧ = q_b

Tactic:
induction q_b using Quotient.ind with

Goal state after:
case left.a.a
α : Type u_1
r : Setoid α
f : α → ↑r.classes := fun a ↦ ⟨{x | r x a}, ⋯⟩
f_respects_relation : ∀ (a b : α), r a b → f a = f b
a b : α
h_eq : Quot.lift f f_respects_relation ⟦a⟧ = Quot.lift f f_respects_relation ⟦b⟧
⊢ ⟦a⟧ = ⟦b⟧
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: V : Type u
G : SimpleGraph V
inst✝ : Subsingleton V
u v : V
⊢ G.Reachable u v

Tactic:
Subsingleton.allEq u v

Goal state after:
V : Type u
G : SimpleGraph V
inst✝ : Subsingleton V
u v : V
⊢ G.Reachable v v
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: ι : Type u_1
α : Type u_2
s : ι → Set α
hs✝ : IndexedPartition s
β : Type u_3
f : ι → α → β
hs : IndexedPartition s
κ : Type u_4
g : ι → κ
hg : Surjective g
x : α
⊢ x ∈ ⋃ (_ : hs.index x ∈ g ⁻¹' {g (hs.index x)}), s (hs.index x)

Tactic:
simp [hs.mem_index]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex

Goal state before:
pp: V : Type u
G : SimpleGraph V
n : ℕ
⊢ G.chromaticNumber ≤ ↑n ↔ G.Colorable n

Tactic:
refine ⟨fun h ↦ ?_, Colorable.chromaticNumber_le⟩

Goal state after:
V : Type u
G : SimpleGraph V
n : ℕ
h : G.chromaticNumber ≤ ↑n
⊢ G.Colorable n
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex

Goal state before:
pp: V : Type u
G : SimpleGraph V
n : ℕ
h : sInf {n' | G.Colorable n'} ≤ n
m : ℕ
hm : G.Colorable m
this : sInf {n' | G.Colorable n'} ∈ {n' | G.Colorable n'}
⊢ G.Colorable n

Tactic:
Set.mem_ofPred_eq

Goal state after:
V : Type u
G : SimpleGraph V
n : ℕ
h : sInf {n' | G.Colorable n'} ≤ n
m : ℕ
hm : G.Colorable m
this : G.Colorable (sInf {n' | G.Colorable n'})
⊢ G.Colorable n
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
v w : V
hadj : G.Adj v w
⊢ v ∈ t ∧ w ∈ s ∨ v ∈ s ∧ w ∈ t

Tactic:
by
    rw [@and_comm (v ∈ t) (w ∈ s), @and_comm (v ∈ s) (w ∈ t)]
    exact h.mem_of_adj hadj.symm

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hv : v ∈ s
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
⊢ w ∈ t

Tactic:
have nhv : v ∉ t := Set.disjoint_left.mp h.disjoint hv

Goal state after:
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hv : v ∈ s
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
nhv : v ∉ t
⊢ w ∈ t
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
w : V
G : SimpleGraph V
s t : Finset V
inst✝ : Fintype ↑(G.neighborSet w)
h : G.IsBipartiteWith ↑s ↑t
hw : w ∈ t
⊢ G.neighborFinset w ⊆ s

Tactic:
isBipartiteWith_neighborFinset' h hw

Goal state after:
V : Type u_1
w : V
G : SimpleGraph V
s t : Finset V
inst✝ : Fintype ↑(G.neighborSet w)
h : G.IsBipartiteWith ↑s ↑t
hw : w ∈ t
⊢ {v ∈ s | G.Adj v w} ⊆ s
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case inl
V : Type u
G : SimpleGraph V
H K : G.Subgraph
hH : H.Preconnected
hK : K.Preconnected
u : V
hu : u ∈ H.verts
hu' : u ∈ K.verts
v : V
hv : v ∈ H.verts
⊢ (H ⊔ K).coe.Reachable ⟨u, ⋯⟩ ⟨v, ⋯⟩

pp: case inr
V : Type u
G : SimpleGraph V
H K : G.Subgraph
hH : H.Preconnected
hK : K.Preconnected
u : V
hu : u ∈ H.verts
hu' : u ∈ K.verts
v : V
hv : v ∈ K.verts
⊢ (H ⊔ K).coe.Reachable ⟨u, ⋯⟩ ⟨v, ⋯⟩

Tactic:
· exact Reachable.map (Subgraph.inclusion (le_sup_left : H ≤ H ⊔ K)) (hH ⟨u, hu⟩ ⟨v, hv⟩)

Goal state after:
case inr
V : Type u
G : SimpleGraph V
H K : G.Subgraph
hH : H.Preconnected
hK : K.Preconnected
u : V
hu : u ∈ H.verts
hu' : u ∈ K.verts
v : V
hv : v ∈ K.verts
⊢ (H ⊔ K).coe.Reachable ⟨u, ⋯⟩ ⟨v, ⋯⟩
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
v✝ u : V
i : ℕ
p : G.Walk u v✝
hp : p.IsPath
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝¹ : i' < p.length
hl : p.getVert i' = p.getVert i
right✝ : p.getVert (i' + 1) = v
⊢ i' ∈ {i | i ≤ p.length}

Tactic:
Set.mem_ofPred_eq

Goal state after:
V : Type u
G : SimpleGraph V
v✝ u : V
i : ℕ
p : G.Walk u v✝
hp : p.IsPath
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝¹ : i' < p.length
hl : p.getVert i' = p.getVert i
right✝ : p.getVert (i' + 1) = v
⊢ i' ≤ p.length
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case refine_2
V : Type u_1
G : SimpleGraph V
u v : V
h : G.Adj u v
⊢ G.edist u v = 1

Tactic:
exact le_antisymm (edist_le h.toWalk) (Order.one_le_iff_pos.mpr <| edist_pos_of_ne h.ne)

Goal state after: