mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Partition.Equipartition

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
f : ↥s ≃ (t : ↥P.parts) × Fin #↑t
hf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst
g : ↥P.parts ≃ Fin #P.parts
hg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts
z : ↥s → ℕ := fun a ↦ #P.parts * ↑(f a).snd + ↑(g (f a).fst)
gl : ∀ (a : ↥s), ↑(g (f a).fst) < #P.parts
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
a b : ↥s
e : z' a = z' b
⊢ a = b

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
f : ↥s ≃ (t : ↥P.parts) × Fin #↑t
hf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst
g : ↥P.parts ≃ Fin #P.parts
hg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts
z : ↥s → ℕ := fun a ↦ #P.parts * ↑(f a).snd + ↑(g (f a).fst)
gl : ∀ (a : ↥s), ↑(g (f a).fst) < #P.parts
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
a b : ↥s
e : z' a = z' b
⊢ a = b
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
s✝ t u : Finset α
P : Finpartition s✝
a✝ : α
s : Setoid α
x : Finset α
inst✝ : DecidableRel ⇑s
a : α
⊢ a ∈ (image (fun a ↦ {b ∈ x | s a b}) x).sup id ↔ a ∈ x

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
s✝ t u : Finset α
P : Finpartition s✝
a✝ : α
s : Setoid α
x : Finset α
inst✝ : DecidableRel ⇑s
a : α
⊢ a ∈ (image (fun a ↦ {b ∈ x | s a b}) x).sup id ↔ a ∈ x
Module: Mathlib.Order.Partition.Equipartition

Goal state before:
pp: case h
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
f : ↥s ≃ (t : ↥P.parts) × Fin #↑t
hf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst
g : ↥P.parts ≃ Fin #P.parts
hg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts
z : ↥s → ℕ := fun a ↦ #P.parts * ↑(f a).snd + ↑(g (f a).fst)
gl : ∀ (a : ↥s), ↑(g (f a).fst) < #P.parts
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
bij : Function.Bijective z'
a b : ↥s
⊢ P.part ↑a = P.part ↑b ↔ ↑((Equiv.ofBijective z' bij) a) % #P.parts = ↑((Equiv.ofBijective z' bij) b) % #P.parts

Tactic:
[

Goal state after:
case h
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
f : ↥s ≃ (t : ↥P.parts) × Fin #↑t
hf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst
g : ↥P.parts ≃ Fin #P.parts
hg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts
z : ↥s → ℕ := fun a ↦ #P.parts * ↑(f a).snd + ↑(g (f a).fst)
gl : ∀ (a : ↥s), ↑(g (f a).fst) < #P.parts
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
bij : Function.Bijective z'
a b : ↥s
⊢ P.part ↑a = P.part ↑b ↔ ↑((Equiv.ofBijective z' bij) a) % #P.parts = ↑((Equiv.ofBijective z' bij) b) % #P.parts
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
F : Finset (Finset α)
ht : t ∈ F
⊢ ∀ ⦃x : Finset α⦄,
    x ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty} →
      x ∈ image (fun P ↦ {i ∈ s | ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x}) (F.erase t).powerset

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
F : Finset (Finset α)
ht : t ∈ F
⊢ ∀ ⦃x : Finset α⦄,
    x ∈ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty} →
      x ∈ image (fun P ↦ {i ∈ s | ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x}) (F.erase t).powerset
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
x : α × β
⊢ x ∈ interedges r s₂ t₂ → x ∈ interedges r s₁ t₁

Tactic:
[

Goal state after:
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
x : α × β
⊢ x ∈ interedges r s₂ t₂ → x ∈ interedges r s₁ t₁
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝² : (a : α) → DecidablePred (r a)
s : Finset α
t : Finset β
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
x : α
y : β
⊢ (x, y) ∈ interedges r s t ↔
    (x, y) ∈ s.biUnion fun x ↦ map { toFun := fun x_1 ↦ (x, x_1), inj' := ⋯ } ({y ∈ t | r x y})

Tactic:
simp [mem_interedges_iff]

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

Goal state before:
pp: ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝² : (a : α) → DecidablePred (r a)
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s : Finset ι
t : Finset κ
f : ι → Finset α
g : κ → Finset β
⊢ interedges r (s.biUnion f) (t.biUnion g) = (s ×ˢ t).biUnion fun ab ↦ interedges r (f ab.1) (g ab.2)

Tactic:
[

Goal state after:
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝² : (a : α) → DecidablePred (r a)
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
s : Finset ι
t : Finset κ
f : ι → Finset α
g : κ → Finset β
⊢ interedges r (s.biUnion f) (t.biUnion g) = (s ×ˢ t).biUnion fun ab ↦ interedges r (f ab.1) (g ab.2)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise

Goal state before:
pp: case neg
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : ¬(a = 0 ∧ b = 0)
⊢ ∃ Q,
    (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
      (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b

Tactic:
[

Goal state after:
case neg
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : ¬(a = 0 ∧ b = 0)
⊢ ∃ Q,
    (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
      (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s : Finset α
inst✝ : DecidableEq α
P : Finpartition s
t : Finset β
⊢ #(interedges r s t) = ∑ a ∈ P.parts, #(interedges r a t)

Tactic:
[

Goal state after:
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s : Finset α
inst✝ : DecidableEq α
P : Finpartition s
t : Finset β
⊢ #(interedges r s t) = ∑ a ∈ P.parts, #(interedges r a t)
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
t : Finset β
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ #(interedges r s t) = ∑ b ∈ P.parts, #(interedges r s b)

Tactic:
[

Goal state after:
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
t : Finset β
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ #(interedges r s t) = ∑ b ∈ P.parts, #(interedges r s b)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise

Goal state before:
pp: case pos
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : 0 < a ∨ 0 < b
n : ℕ := if 0 < a then m else m + 1
hn : n = if 0 < a then m else m + 1
h : 0 < a
⊢ 0 < m ∧ m ≤ m + 1 ∧ m ≤ a * m + b * (m + 1) ∧ a * m - m + b * (m + 1) = a * m + b * (m + 1) - m

pp: case neg
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : 0 < a ∨ 0 < b
n : ℕ := if 0 < a then m else m + 1
hn : n = if 0 < a then m else m + 1
h : ¬0 < a
⊢ 0 < m + 1 ∧
    m + 1 ≤ m + 1 ∧ m + 1 ≤ a * m + b * (m + 1) ∧ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)

Tactic:
· refine ⟨m_pos, le_succ _, le_add_right (Nat.le_mul_of_pos_left _ ‹0 < a›), ?_⟩
      rw [tsub_add_eq_add_tsub (Nat.le_mul_of_pos_left _ h)]

Goal state after:
case neg
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : 0 < a ∨ 0 < b
n : ℕ := if 0 < a then m else m + 1
hn : n = if 0 < a then m else m + 1
h : ¬0 < a
⊢ 0 < m + 1 ∧
    m + 1 ≤ m + 1 ∧ m + 1 ≤ a * m + b * (m + 1) ∧ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise

Goal state before:
pp: case pos.refine_3
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : 0 < a ∨ 0 < b
n : ℕ := if 0 < a then m else m + 1
hn : n = if 0 < a then m else m + 1
hn₀ : 0 < n
hn₁ : n ≤ m + 1
hn₂ : n ≤ a * m + b * (m + 1)
hn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #s - n
h : ∀ u ∈ P.parts, #u < m + 1
t : Finset α
hts : t ⊆ s
htn : #t = n
ht : t.Nonempty
hcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #(s \ t)
R : Finpartition (s \ t)
hR₁ : ∀ x ∈ R.parts, #x = m ∨ #x = m + 1
left✝ : ∀ x ∈ (P.avoid t).parts, #(x \ {y ∈ R.parts | y ⊆ x}.biUnion id) ≤ m
hR₃ : #({i ∈ R.parts | #i = m + 1}) = if 0 < a then b else b - 1
⊢ #({i ∈ (R.extend ⋯ ⋯ ⋯).parts | #i = m + 1}) = b

Tactic:
[

Goal state after:
case pos.refine_3
α : Type u_1
inst✝ : DecidableEq α
m : ℕ
m_pos : m > 0
s : Finset α
ih :
  ∀ t ⊂ s,
    ∀ {a b : ℕ} {P : Finpartition t},
      a * m + b * (m + 1) = #t →
        ∃ Q,
          (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧
            (∀ x ∈ P.parts, #(x \ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧ #({i ∈ Q.parts | #i = m + 1}) = b
a b : ℕ
P : Finpartition s
hs : a * m + b * (m + 1) = #s
hab : 0 < a ∨ 0 < b
n : ℕ := if 0 < a then m else m + 1
hn : n = if 0 < a then m else m + 1
hn₀ : 0 < n
hn₁ : n ≤ m + 1
hn₂ : n ≤ a * m + b * (m + 1)
hn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #s - n
h : ∀ u ∈ P.parts, #u < m + 1
t : Finset α
hts : t ⊆ s
htn : #t = n
ht : t.Nonempty
hcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #(s \ t)
R : Finpartition (s \ t)
hR₁ : ∀ x ∈ R.parts, #x = m ∨ #x = m + 1
left✝ : ∀ x ∈ (P.avoid t).parts, #(x \ {y ∈ R.parts | y ⊆ x}.biUnion id) ≤ m
hR₃ : #({i ∈ R.parts | #i = m + 1}) = if 0 < a then b else b - 1
⊢ #({i ∈ (R.extend ⋯ ⋯ ⋯).parts | #i = m + 1}) = b
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hA : A.Nonempty
hε : 0 < ε
hP : P.IsEquipartition
hG : P.IsUniform G ε
⊢ #((P.nonUniforms G ε).biUnion fun x ↦ x.1 ×ˢ x.2) ≤ ∑ x ∈ P.nonUniforms G ε, #x.1 * #x.2

Tactic:
[

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hA : A.Nonempty
hε : 0 < ε
hP : P.IsEquipartition
hG : P.IsUniform G ε
⊢ #((P.nonUniforms G ε).biUnion fun x ↦ x.1 ×ˢ x.2) ≤ ∑ x ∈ P.nonUniforms G ε, #x.1 * #x.2
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hA : A.Nonempty
hε : 0 < ε
hP : P.IsEquipartition
hG : P.IsUniform G ε
⊢ ↑(#((P.nonUniforms G ε).biUnion fun x ↦
          match x with
          | (U, V) => U ×ˢ V)) ≤
    ∑ i ∈ P.nonUniforms G ε, ↑(#i.1) * ↑(#i.2)

Tactic:
norm_cast; simp_rw [← card_product]; exact card_biUnion_le

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
hPα' : stepBound #P.parts ≤ Fintype.card α
hPpos : 0 < stepBound #P.parts
⊢ ∑ A ∈ P.parts.attach, #(chunk hP G ε ⋯).parts = stepBound #P.parts

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
hPα' : stepBound #P.parts ≤ Fintype.card α
hPpos : 0 < stepBound #P.parts
⊢ ∑ A ∈ P.parts.attach, #(chunk hP G ε ⋯).parts = stepBound #P.parts
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
⊢ (increment hP G ε).IsEquipartition

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
⊢ (increment hP G ε).IsEquipartition
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
x : V
⊢ (G.deleteIncidenceSet x).edgeSet = G.edgeSet \ G.incidenceSet x

Tactic:
[

Goal state after:
V : Type u_1
G : SimpleGraph V
x : V
⊢ (G.deleteIncidenceSet x).edgeSet = G.edgeSet \ G.incidenceSet x
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
x x✝ : V
⊢ x✝ ∈ (G.deleteIncidenceSet x).support → x✝ ∈ G.support \ {x}

Tactic:
[

Goal state after:
V : Type u_1
G : SimpleGraph V
x x✝ : V
⊢ x✝ ∈ (G.deleteIncidenceSet x).support → x✝ ∈ G.support \ {x}
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
s : Set V
x : V
h : x ∉ s
v₁ v₂ : ↑s
⊢ (induce s (G.deleteIncidenceSet x)).Adj v₁ v₂ ↔ (induce s G).Adj v₁ v₂

Tactic:
[

Goal state after:
V : Type u_1
G : SimpleGraph V
s : Set V
x : V
h : x ∉ s
v₁ v₂ : ↑s
⊢ (induce s (G.deleteIncidenceSet x)).Adj v₁ v₂ ↔ (induce s G).Adj v₁ v₂
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
h_notMem : x ∉ {x}ᶜ
⊢ #(induce {x}ᶜ G).edgeFinset = #(G.deleteIncidenceSet x).edgeFinset

Tactic:
[

Goal state after:
V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
h_notMem : x ∉ {x}ᶜ
⊢ #(induce {x}ᶜ G).edgeFinset = #(G.deleteIncidenceSet x).edgeFinset
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
⊢ #(G.deleteIncidenceSet x).edgeFinset = #G.edgeFinset - G.degree x

Tactic:
[

Goal state after:
V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
⊢ #(G.deleteIncidenceSet x).edgeFinset = #G.edgeFinset - G.degree x
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
hx : {x}.toFinset ⊆ G.support.toFinset
⊢ Fintype.card ↑(G.deleteIncidenceSet x).support ≤ Fintype.card ↑G.support - 1

Tactic:
[

Goal state after:
V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
hx : {x}.toFinset ⊆ G.support.toFinset
⊢ Fintype.card ↑(G.deleteIncidenceSet x).support ≤ Fintype.card ↑G.support - 1
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
hx : {x}.toFinset ⊆ G.support.toFinset
⊢ (G.deleteIncidenceSet x).support.toFinset ⊆ (G.support \ {x}).toFinset

Tactic:
rw [Set.toFinset_subset_toFinset]

Goal state after:
V : Type u_1
inst✝² : Fintype V
inst✝¹ : DecidableEq V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
x : V
hx : {x}.toFinset ⊆ G.support.toFinset
⊢ (G.deleteIncidenceSet x).support ⊆ G.support \ {x}
Module: Mathlib.Combinatorics.SimpleGraph.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
G' : G.Subgraph
⊢ G'.coe.spanningCoe = G'.spanningCoe

Tactic:
ext
  simp only [map_adj, Function.Embedding.subtype_apply, Subtype.exists]
  grind [coe_adj, edge_vert, adj_symm]

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

Goal state before:
pp: case map_rel_iff'
V : Type u_1
W : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
γ : Type u_6
G G₁ G₂ G₃ : SimpleGraph V
H : SimpleGraph W
I : SimpleGraph X
A : SimpleGraph α
B : SimpleGraph β
C : SimpleGraph γ
f : A.Copy B
A' : A.Subgraph
⊢ ∀ {a b : ↑A'.verts},
    (Subgraph.map f.toHom A').coe.Adj ((Equiv.Set.image (⇑f.toHom) A'.verts ⋯) a)
        ((Equiv.Set.image (⇑f.toHom) A'.verts ⋯) b) ↔
      A'.coe.Adj a b

Tactic:
[

Goal state after:
case map_rel_iff'
V : Type u_1
W : Type u_2
X : Type u_3
α : Type u_4
β : Type u_5
γ : Type u_6
G G₁ G₂ G₃ : SimpleGraph V
H : SimpleGraph W
I : SimpleGraph X
A : SimpleGraph α
B : SimpleGraph β
C : SimpleGraph γ
f : A.Copy B
A' : A.Subgraph
⊢ ∀ {a b : ↑A'.verts},
    (Subgraph.map f.toHom A').coe.Adj ((Equiv.Set.image (⇑f.toHom) A'.verts ⋯) a)
        ((Equiv.Set.image (⇑f.toHom) A'.verts ⋯) b) ↔
      A'.coe.Adj a b
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
pp: case pos
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
U : Finset α
hU : U ∈ P.parts
V : Finset α
hV : V ∈ P.parts
hUV : U ≠ V
h₂ : ¬G.IsUniform ε U V
hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U
q :
  G.nonuniformWitness ε U V \ (star hP G ε hU V).biUnion id ⊆
    {B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts | B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion fun B ↦
      B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id
B : Finset α
hB : B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts
h₁ : #U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)
⊢ #(B \ {A ∈ (equitabilise ⋯).parts | A ⊆ B}.biUnion id) ≤ m

pp: case neg
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
U : Finset α
hU : U ∈ P.parts
V : Finset α
hV : V ∈ P.parts
hUV : U ≠ V
h₂ : ¬G.IsUniform ε U V
hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U
q :
  G.nonuniformWitness ε U V \ (star hP G ε hU V).biUnion id ⊆
    {B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts | B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion fun B ↦
      B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id
B : Finset α
hB : B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts
h₁ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)
⊢ #(B \ {A ∈ (equitabilise ⋯).parts | A ⊆ B}.biUnion id) ≤ m

Tactic:
· convert! card_parts_equitabilise_subset_le _ (card_aux₁ h₁) hB

Goal state after:
case neg
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
U : Finset α
hU : U ∈ P.parts
V : Finset α
hV : V ∈ P.parts
hUV : U ≠ V
h₂ : ¬G.IsUniform ε U V
hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U
q :
  G.nonuniformWitness ε U V \ (star hP G ε hU V).biUnion id ⊆
    {B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts | B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion fun B ↦
      B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id
B : Finset α
hB : B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts
h₁ : ¬#U = m * 4 ^ #P.parts + (Fintype.card α / #P.parts - m * 4 ^ #P.parts)
⊢ #(B \ {A ∈ (equitabilise ⋯).parts | A ⊆ B}.biUnion id) ≤ m
Module: Mathlib.Combinatorics.SimpleGraph.Subgraph

Goal state before:
pp: case pos
V : Type u
G : SimpleGraph V
G' : G.Subgraph
v : V
inst✝ : Fintype ↑(G'.neighborSet v)
hG : G'.verts.Subsingleton
hv : v ∈ G'.verts
⊢ G'.degree v = 0

pp: case neg
V : Type u
G : SimpleGraph V
G' : G.Subgraph
v : V
inst✝ : Fintype ↑(G'.neighborSet v)
hG : G'.verts.Subsingleton
hv : v ∉ G'.verts
⊢ G'.degree v = 0

Tactic:
· rw [← G'.coe_degree ⟨v, hv⟩]
    have := (Set.subsingleton_coe _).mpr hG
    exact G'.coe.degree_eq_zero_of_subsingleton ⟨v, hv⟩

Goal state after:
case neg
V : Type u
G : SimpleGraph V
G' : G.Subgraph
v : V
inst✝ : Fintype ↑(G'.neighborSet v)
hG : G'.verts.Subsingleton
hv : v ∉ G'.verts
⊢ G'.degree v = 0
Module: Mathlib.Combinatorics.SimpleGraph.Operations

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
hn : ¬G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : x✝ = t
h✝ : y✝ = t
⊢ False ↔ G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x ↦ s(x, t)) '' G.neighborSet s

pp: case neg
V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
hn : ¬G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : x✝ = t
h✝ : ¬y✝ = t
⊢ G.Adj s y✝ ↔ G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x ↦ s(x, t)) '' G.neighborSet s

pp: case pos
V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
hn : ¬G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : ¬x✝ = t
h✝ : y✝ = t
⊢ G.Adj x✝ s ↔ G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x ↦ s(x, t)) '' G.neighborSet s

pp: case neg
V : Type u_1
G : SimpleGraph V
s t : V
inst✝ : DecidableEq V
hn : ¬G.Adj s t
e : Sym2 V
x✝ y✝ : V
h✝¹ : ¬x✝ = t
h✝ : ¬y✝ = t
⊢ G.Adj x✝ y✝ ↔ G.Adj x✝ y✝ ∧ ¬(G.Adj x✝ y✝ ∧ (t = x✝ ∨ t = y✝)) ∨ s(x✝, y✝) ∈ (fun x ↦ s(x, t)) '' G.neighborSet s

Tactic:
exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by grind]

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

Goal state before:
pp: case right
V : Type u
G : SimpleGraph V
v : V
H : G.Subgraph
h : v ∈ H.verts
⊢ ∀ ⦃v_1 w : V⦄, (G.singletonSubgraph v).Adj v_1 w → H.Adj v_1 w

Tactic:
· exact fun _ _ ↦ False.elim

Goal state after: