mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : EInt
hab : b ≤ a
⊢ IsIso (t.eTruncLTLTToLT a b)

Tactic:
NatTrans.isIso_iff_isIso_app

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : EInt
hab : b ≤ a
⊢ ∀ (X : C), IsIso ((t.eTruncLTLTToLT a b).app X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : EInt
⊢ IsIso (t.eTruncLTGELTSelfToLTGE a b)

Tactic:
NatTrans.isIso_iff_isIso_app

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : EInt
⊢ ∀ (X : C), IsIso ((t.eTruncLTGELTSelfToLTGE a b).app X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b✝ b : EInt
X : C
⊢ IsIso (t.eTruncLTGELTSelfToGELT a b)

Tactic:
NatTrans.isIso_iff_isIso_app

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b✝ b : EInt
X : C
⊢ ∀ (X : C), IsIso ((t.eTruncLTGELTSelfToGELT a b).app X)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
u v : Finset α
⊢ (u, v) ∈ P.nonUniforms G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ ¬G.IsUniform ε u v

Tactic:
rw [nonUniforms, mem_filter, mem_offDiag, and_assoc, and_assoc]

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
⊢ Nonempty α

Tactic:
rw [← Fintype.card_pos_iff]

Goal state after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
⊢ 0 < Fintype.card α
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma

Goal state before:
pp: case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
huniform : P.IsUniform G ε
⊢ #P.parts ≤ stepBound^[i + 1] t

Tactic:
iterate_succ_apply'

Goal state after:
case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
huniform : P.IsUniform G ε
⊢ #P.parts ≤ stepBound (stepBound^[i] t)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma

Goal state before:
pp: case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
huniform : P.IsUniform G ε
⊢ ∃ P,
    P.IsEquipartition ∧
      t ≤ #P.parts ∧ #P.parts ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))

Tactic:
refine ⟨P, hP₁, hP₂, ?_, Or.inl huniform⟩
    rw [iterate_succ_apply']
    exact hP₃.trans (le_stepBound _)

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

Goal state before:
pp: case neg.refine_2.refine_1.inl
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
u : Finset α
hu₁ : u ∈ P.parts
hu₂ : m + 1 ≤ #u
htu : u ⊆ u
htn : #u = n
ht : u.Nonempty
hcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #(s \ u)
R : Finpartition (s \ u)
hR₁ : ∀ x ∈ R.parts, #x = m ∨ #x = m + 1
hR₂ : ∀ x ∈ (P.avoid u).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
⊢ #(u \ (u ∪ {y ∈ R.parts | y ⊆ u}.biUnion id)) ≤ m

Tactic:
sdiff_eq_empty_iff_subset.2 subset_union_left

Goal state after:
case neg.refine_2.refine_1.inl
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
u : Finset α
hu₁ : u ∈ P.parts
hu₂ : m + 1 ≤ #u
htu : u ⊆ u
htn : #u = n
ht : u.Nonempty
hcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #(s \ u)
R : Finpartition (s \ u)
hR₁ : ∀ x ∈ R.parts, #x = m ∨ #x = m + 1
hR₂ : ∀ x ∈ (P.avoid u).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
⊢ #∅ ≤ m
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma

Goal state before:
pp: case neg.refine_2
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
huniform : ¬P.IsUniform G ε
hP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
hεl' : 100 ≤ 4 ^ #P.parts * ε ^ 5
hi : ↑i ≤ 4 / ε ^ 5
hsize : #P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
⊢ stepBound #P.parts ≤ stepBound^[i + 1] t

Tactic:
iterate_succ_apply'

Goal state after:
case neg.refine_2
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
huniform : ¬P.IsUniform G ε
hP₄ : ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
hεl' : 100 ≤ 4 ^ #P.parts * ε ^ 5
hi : ↑i ≤ 4 / ε ^ 5
hsize : #P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊] t
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
⊢ stepBound #P.parts ≤ stepBound (stepBound^[i] t)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      ((∑ x ∈ P.parts.offDiag, if G.IsUniform ε x.1 x.2 then 0 else ε ^ 4 / 3) - #P.parts.offDiag • (ε ^ 5 / 25))

Tactic:
nsmul_eq_mul,

Goal state after:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      ((∑ x ∈ P.parts.offDiag, if G.IsUniform ε x.1 x.2 then 0 else ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25))
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (#({x ∈ P.parts.offDiag | ¬G.IsUniform ε x.1 x.2}) • (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25))

Tactic:
nsmul_eq_mul,

Goal state after:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#({x ∈ P.parts.offDiag | ¬G.IsUniform ε x.1 x.2})) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25))
Module: Mathlib.Combinatorics.SimpleGraph.DeleteEdges

Goal state before:
pp: case refine_1
V : Type u_1
G : SimpleGraph V
𝕜 : Type u_2
inst✝³ : Ring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : Fintype ↑G.edgeSet
p : SimpleGraph V → Prop
r : 𝕜
inst✝ : Fintype (Sym2 V)
h : G.DeleteFar p r
H : SimpleGraph V
x✝ : DecidableRel H.Adj
hHG : H ≤ G
hH : p H
⊢ r ≤ ↑(#G.edgeFinset) - ↑(#H.edgeFinset)

Tactic:
have := h (sdiff_subset (t := H.edgeFinset))
    simp only [deleteEdges_sdiff_eq_of_le hHG, edgeFinset_mono hHG, card_sdiff_of_subset,
      card_le_card, coe_sdiff, coe_edgeFinset, Nat.cast_sub] at this
    exact this hH

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

Goal state before:
pp: case hbc
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ 6 / 7 * ↑(#P.parts) ^ 2 ≤ ↑(#P.parts.offDiag)

Tactic:
rw [← mul_div_right_comm, div_le_iff₀ (by simp), offDiag_card]

Goal state after:
case hbc
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ 6 * ↑(#P.parts) ^ 2 ≤ ↑(#P.parts * #P.parts - #P.parts) * 7
Module: Mathlib.Combinatorics.SimpleGraph.Subgraph

Goal state before:
pp: ι : Sort u_1
V : Type u
G : SimpleGraph V
a b : V
inst✝ : Nonempty ι
f : ι → G.Subgraph
⊢ (⨅ i, f i).Adj a b ↔ ∀ (i : ι), (f i).Adj a b

Tactic:
rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)]

Goal state after:
ι : Sort u_1
V : Type u
G : SimpleGraph V
a b : V
inst✝ : Nonempty ι
f : ι → G.Subgraph
⊢ (∀ G' ∈ Set.range fun i ↦ f i, G'.Adj a b) ↔ ∀ (i : ι), (f i).Adj a b
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

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
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
⊢ ∑ a ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts with a ⊆ G.nonuniformWitness ε U V ∧ a.Nonempty,
      #(a \ {A ∈ (chunk hP G ε hU).parts | A ⊆ a}.biUnion id) ≤
    ∑ B ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts with B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty, m

Tactic:
suffices ∀ B ∈ (atomise U <| P.nonuniformWitnesses G ε U).parts,
        #(B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id) ≤ m by
      gcongr with B hB
      exact this B <| filter_subset _ _ hB

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
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 ∈ (atomise U (P.nonuniformWitnesses G ε U)).parts, #(B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id) ≤ m
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk

Goal state before:
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_aux₂ hP hU h₁) hB

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = #s
hm : m ≠ 0
⊢ (equitabilise h).parts = {u ∈ (equitabilise h).parts | #u = m} ∪ {u ∈ (equitabilise h).parts | #u = m + 1}

Tactic:
← filter_or,

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = #s
hm : m ≠ 0
⊢ (equitabilise h).parts = {a ∈ (equitabilise h).parts | #a = m ∨ #a = m + 1}
Module: Mathlib.Combinatorics.SimpleGraph.Subgraph

Goal state before:
pp: case pos
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
w : V
hvw : G.Adj w w
⊢ (G.subgraphOfAdj hvw).neighborSet w = {w} ∪ {w}

Tactic:
simp [*]

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

Goal state before:
pp: case neg
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
v w : V
hvw : G.Adj v w
h✝ : ¬v = w
⊢ (G.subgraphOfAdj hvw).neighborSet v = {w} ∪ ∅

Tactic:
simp [*]

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

Goal state before:
pp: case pos
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
v w : V
hvw : G.Adj v w
h✝ : ¬w = v
⊢ (G.subgraphOfAdj hvw).neighborSet w = ∅ ∪ {v}

Tactic:
simp [*]

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

Goal state before:
pp: case neg
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u v w : V
hvw : G.Adj v w
h✝¹ : ¬u = v
h✝ : ¬u = w
⊢ (G.subgraphOfAdj hvw).neighborSet u = ∅ ∪ ∅

Tactic:
simp [*]

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
G' : G.Subgraph
s s' : Set V
⊢ G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')

Tactic:
constructor
  · simp
  · simp only [sup_adj, induce_adj, Set.mem_union]
    rintro v w (h | h) <;> simp [h]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Traversal

Goal state before:
pp: case nil.zero
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ nil.getVert 0 ∈ nil.support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Traversal

Goal state before:
pp: case nil.succ
V : Type u
G : SimpleGraph V
u v u✝ : V
n✝ : ℕ
⊢ nil.getVert (n✝ + 1) ∈ nil.support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Traversal

Goal state before:
pp: case cons.zero
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (i : ℕ), p✝.getVert i ∈ p✝.support
⊢ (cons h✝ p✝).getVert 0 ∈ (cons h✝ p✝).support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Traversal

Goal state before:
pp: case cons.succ
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (i : ℕ), p✝.getVert i ∈ p✝.support
n✝ : ℕ
⊢ (cons h✝ p✝).getVert (n✝ + 1) ∈ (cons h✝ p✝).support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
a b u✝ : V
⊢ nil.support.getLast ⋯ = u✝

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
a b u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : p✝.support.getLast ⋯ = w✝
⊢ (cons h✝ p✝).support.getLast ⋯ = w✝

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ u✝ ∈ nil.support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : w✝ ∈ p✝.support
⊢ w✝ ∈ (cons h✝ p✝).support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ u✝ :: List.map (fun x ↦ x.toProd.2) nil.darts = nil.support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : v✝ :: List.map (fun x ↦ x.toProd.2) p✝.darts = p✝.support
⊢ u✝ :: List.map (fun x ↦ x.toProd.2) (cons h✝ p✝).darts = (cons h✝ p✝).support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ List.map (fun x ↦ x.toProd.1) nil.darts ++ [u✝] = nil.support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : List.map (fun x ↦ x.toProd.1) p✝.darts ++ [w✝] = p✝.support
⊢ List.map (fun x ↦ x.toProd.1) (cons h✝ p✝).darts ++ [w✝] = (cons h✝ p✝).support

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ nil.support.length = nil.length + 1

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : p✝.support.length = p✝.length + 1
⊢ (cons h✝ p✝).support.length = (cons h✝ p✝).length + 1

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ nil.darts.length = nil.length

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : p✝.darts.length = p✝.length
⊢ (cons h✝ p✝).darts.length = (cons h✝ p✝).length

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Traversal

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
h₁ : ¬p.Nil
h₂ : 0 < p.darts.length
⊢ p.lastDart h₁ = p.darts[p.darts.length - 1]

Tactic:
by
  simp (disch := grind) [Dart.ext_iff, lastDart_toProd, darts_getElem_eq_getVert,
    p.getVert_of_length_le]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Maps

Goal state before:
pp: case nil
V : Type u
G : SimpleGraph V
u v u✝ : V
⊢ Walk.map Hom.id nil = nil

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Maps

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : Walk.map Hom.id p✝ = p✝
⊢ Walk.map Hom.id (cons h✝ p✝) = cons h✝ p✝

Tactic:
simp [*]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Maps

Goal state before:
pp: case nil
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
f : G →g G'
f' : G' →g G''
u v : V
p : G.Walk u v
u✝ : V
⊢ Walk.map f' (Walk.map f nil) = Walk.map (f'.comp f) nil

Tactic:
simp [*]

Goal state after: