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

Goal state before:
pp: case top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
b : EInt
hab✝ : ⊤ ⟶ b
hab : ⊤ ≤ b
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 b =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊤ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

pp: case bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
b : EInt
hab✝ : ⊥ ⟶ b
hab : ⊥ ≤ b
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 b =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

pp: case coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
b : EInt
a✝ : ℤ
hab✝ : WithBotTop.coe a✝ ⟶ b
hab : WithBotTop.coe a✝ ≤ b
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 b =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
· apply (isZero_zero _).eq_of_src

Goal state after:
case bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
b : EInt
hab✝ : ⊥ ⟶ b
hab : ⊥ ≤ b
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 b =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

case coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
b : EInt
a✝ : ℤ
hab✝ : WithBotTop.coe a✝ ⟶ b
hab : WithBotTop.coe a✝ ≤ b
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 b =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case bot.bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
hab✝ : ⊥ ⟶ ⊥
hab : ⊥ ≤ ⊥
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊥ =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
case bot.bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
hab✝ : ⊥ ⟶ ⊥
hab : True
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊥ =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case bot.coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝ : ℤ
hab✝ : ⊥ ⟶ WithBotTop.coe a✝
hab : ⊥ ≤ WithBotTop.coe a✝
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
case bot.coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝ : ℤ
hab✝ : ⊥ ⟶ WithBotTop.coe a✝
hab : True
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case bot.top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
hab✝ : ⊥ ⟶ ⊤
hab : ⊥ ≤ ⊤
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊤ =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
case bot.top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
hab✝ : ⊥ ⟶ ⊤
hab : True
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊤ =
    WithBotTop.rec 0 t.truncGEδLT 0 ⊥ ≫ (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.bot
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝ : ℤ
hab✝ : WithBotTop.coe a✝ ⟶ ⊥
hab : WithBotTop.coe a✝ ≤ ⊥
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊥ =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝¹ a✝ : ℤ
hab✝ : WithBotTop.coe a✝¹ ⟶ WithBotTop.coe a✝
hab : WithBotTop.coe a✝¹ ≤ WithBotTop.coe a✝
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝¹) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
case coe.coe
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝¹ a✝ : ℤ
hab✝ : WithBotTop.coe a✝¹ ⟶ WithBotTop.coe a✝
hab : a✝¹ ≤ a✝
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝¹) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝ : ℤ
hab✝ : WithBotTop.coe a✝ ⟶ ⊤
hab : WithBotTop.coe a✝ ≤ ⊤
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊤ =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝

Tactic:
simp at hab

Goal state after:
case coe.top
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a✝ : ℤ
hab✝ : WithBotTop.coe a✝ ⟶ ⊤
hab : True
⊢ t.eTruncGE.map hab✝ ≫ WithBotTop.rec 0 t.truncGEδLT 0 ⊤ =
    WithBotTop.rec 0 t.truncGEδLT 0 (WithBotTop.coe a✝) ≫
      (t.eTruncLT ⋙ (Functor.whiskeringRight C C C).obj (shiftFunctor C 1)).map hab✝
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableEq α
a b✝ c : α
P Q R : Finpartition a
hPQ : P ≤ Q
hPR : P ≤ R
b : α
hb : b ∈ P.parts
⊢ ∃ c ∈ (Q ⊓ R).parts, b ≤ c

Tactic:
obtain ⟨c, hc, hbc⟩ := hPQ hb
      obtain ⟨d, hd, hbd⟩ := hPR hb
      have h := _root_.le_inf hbc hbd
      refine
        ⟨c ⊓ d,
          mem_erase_of_ne_of_mem (ne_bot_of_le_ne_bot (P.ne_bot hb) h)
            (mem_image.2 ⟨(c, d), mem_product.2 ⟨hc, hd⟩, rfl⟩),
          h⟩

Goal state after:
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝⁴ : Lattice α
inst✝³ : OrderBot α
inst✝² : IsModularLattice α
inst✝¹ : DecidableEq α
ι : Type u_2
I : Finset ι
s : ι → α
P : (i : ι) → Finpartition (s i)
ha : I.SupIndep s
M : Type u_3
inst✝ : AddCommMonoid M
f : α → M
i : ι
hi : i ∈ ↑I
j : ι
hj : j ∈ ↑I
hij : i ≠ j
⊢ ∀ ⦃a : α⦄, a ∈ (P i).parts → a ∉ (P j).parts

Tactic:
intro p

Goal state after:
α : Type u_1
inst✝⁴ : Lattice α
inst✝³ : OrderBot α
inst✝² : IsModularLattice α
inst✝¹ : DecidableEq α
ι : Type u_2
I : Finset ι
s : ι → α
P : (i : ι) → Finpartition (s i)
ha : I.SupIndep s
M : Type u_3
inst✝ : AddCommMonoid M
f : α → M
i : ι
hi : i ∈ ↑I
j : ι
hj : j ∈ ↑I
hij : i ≠ j
p : α
⊢ p ∈ (P i).parts → p ∉ (P j).parts
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: case inl
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
h : P.parts = ∅
⊢ #s % 0 ≤ 0

Tactic:
parts_eq_empty_iff,

Goal state after:
case inl
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
h : s = ⊥
⊢ #s % 0 ≤ 0
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Bound

Goal state before:
pp: ε : ℝ
hε : 0 < ε
l : ℕ
⊢ log (100 / ε ^ 5) < ↑(initialBound ε l) * log 4

pp: ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 0 < 100 / ε ^ 5

Tactic:
←
    div_lt_iff₀,

Goal state after:
ε : ℝ
hε : 0 < ε
l : ℕ
⊢ log (100 / ε ^ 5) / log 4 < ↑(initialBound ε l)

ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 0 < log 4

ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 0 < 100 / ε ^ 5
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: case refine_2.refine_2
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
hs₂ : s₂.Nonempty
ht₂ : t₂.Nonempty
⊢ 0 ≤ ↑(#t₂) / ↑(#t₁)

pp: case refine_2.refine_3
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
hs₂ : s₂.Nonempty
ht₂ : t₂.Nonempty
⊢ ↑(#t₂) / ↑(#t₁) ≤ 1

Tactic:
· apply div_nonneg <;> exact mod_cast Nat.zero_le _

Goal state after:
case refine_2.refine_3
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
hs₂ : s₂.Nonempty
ht₂ : t₂.Nonempty
⊢ ↑(#t₂) / ↑(#t₁) ≤ 1
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise

Goal state before:
pp: 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
⊢ 0 < n ∧
    n ≤ m + 1 ∧
      n ≤ a * m + b * (m + 1) ∧ (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = #s - n

Tactic:
by
    rw [hn, ← hs]
    split_ifs with h <;> rw [tsub_mul, one_mul]
    · 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)]
    · refine ⟨succ_pos', le_rfl,
        le_add_left (Nat.le_mul_of_pos_left _ <| hab.resolve_left ‹¬0 < a›), ?_⟩
      rw [← add_tsub_assoc_of_le (Nat.le_mul_of_pos_left _ <| hab.resolve_left ‹¬0 < a›)]

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

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ : Finset β
δ : 𝕜
hs : s₂ ⊆ s₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
hs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs₂' : s₂.Nonempty
ht : ∅ ⊆ t₁
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#∅)
⊢ |↑(edgeDensity r s₂ ∅) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2

pp: case inr.inr
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
δ : 𝕜
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
hs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs₂' : s₂.Nonempty
ht₂' : t₂.Nonempty
⊢ |↑(edgeDensity r s₂ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2

Tactic:
· rw [Finset.card_empty, Nat.cast_zero] at ht₂
    simpa [edgeDensity, (nonpos_of_mul_nonpos_right ht₂ hδ₁).antisymm (Nat.cast_nonneg _)] using hδ'

Goal state after:
case inr.inr
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
δ : 𝕜
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
hs₂ : (1 - δ) * ↑(#s₁) ≤ ↑(#s₂)
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs₂' : s₂.Nonempty
ht₂' : t₂.Nonempty
⊢ |↑(edgeDensity r s₂ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2
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
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
⊢ ↑i ≤ 4 / ε ^ 5

Tactic:
have hi : ε ^ 5 / 4 * ↑i ≤ 1 := hP₄.trans (mod_cast P.energy_le_one G)

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
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 : ε ^ 5 / 4 * ↑i ≤ 1
⊢ ↑i ≤ 4 / ε ^ 5
Module: Mathlib.Combinatorics.SimpleGraph.Copy

Goal state before:
pp: 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
g : B.Copy C
f : A.Copy B
⊢ A.Copy C

Tactic:
by
  use g.toHom.comp f.toHom
  rw [Hom.coe_comp]
  exact g.injective.comp f.injective

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

Goal state before:
pp: case refine_4
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H : G.Subgraph
H' : G'.Subgraph
h : H ≤ Subgraph.comap f H'
v w : W
⊢ (Subgraph.map f H).Adj v w → H'.Adj v w

Tactic:
simp only [Relation.Map, map_adj, forall_exists_index, and_imp]
    rintro u u' hu rfl rfl
    exact (h.2 hu).2

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

Goal state before:
pp: V : Type u_1
W : Type u_2
G : SimpleGraph V
H : SimpleGraph W
hH : H ≠ ⊥
⊢ H.Free (G.killCopies H)

Tactic:
rw [killCopies_of_ne_bot hH, deleteEdges, Free, isContained_iff_exists_iso_subgraph]

Goal state after:
V : Type u_1
W : Type u_2
G : SimpleGraph V
H : SimpleGraph W
hH : H ≠ ⊥
⊢ ¬∃ B', Nonempty (H ≃g B'.coe)
Module: Mathlib.Combinatorics.SimpleGraph.Copy

Goal state before:
pp: V : Type u_1
W : Type u_2
G : SimpleGraph V
H : SimpleGraph W
hH : H ≠ ⊥
G' : (G \ fromEdgeSet (⋃ G', ⋃ (hG' : Nonempty (H ≃g G'.coe)), {⋯.some})).Subgraph
hHG' : Nonempty (H ≃g G'.coe)
hG' : (Subgraph.map (Hom.ofLE ⋯) G').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) G').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) G').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hG'.some
e' : Sym2 ↑G'.verts :=
  Sym2.map
    (⇑((Copy.ofLE (G \ fromEdgeSet (⋃ G', ⋃ (hG' : Nonempty (H ≃g G'.coe)), {⋯.some})) G ⋯).isoSubgraphMap G').symm) e
he' : e' ∈ G'.coe.edgeSet
⊢ False

Tactic:
rw [Subgraph.edgeSet_coe] at he'

Goal state after:
V : Type u_1
W : Type u_2
G : SimpleGraph V
H : SimpleGraph W
hH : H ≠ ⊥
G' : (G \ fromEdgeSet (⋃ G', ⋃ (hG' : Nonempty (H ≃g G'.coe)), {⋯.some})).Subgraph
hHG' : Nonempty (H ≃g G'.coe)
hG' : (Subgraph.map (Hom.ofLE ⋯) G').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) G').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) G').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hG'.some
e' : Sym2 ↑G'.verts :=
  Sym2.map
    (⇑((Copy.ofLE (G \ fromEdgeSet (⋃ G', ⋃ (hG' : Nonempty (H ≃g G'.coe)), {⋯.some})) G ⋯).isoSubgraphMap G').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' G'.edgeSet
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Operations

Goal state before:
pp: case inr
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v
⊢ Disjoint G (edge u v) ↔ ¬G.Adj u v

Tactic:
simp [← disjoint_edgeSet, edgeSet_edge_of_ne h]

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
s t : V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
⊢ Fintype ↑(edge s t).edgeSet

Tactic:
rw [edge]; infer_instance

Goal state after:
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
hunif : ¬G.IsUniform ε U V
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hP₁ : 0 < #P.parts
this✝ : 2 ^ #P.parts * ↑m / (↑(#U) * ε) ≤ ε / 10
this : 2 ^ #P.parts = 2 ^ (#P.parts - 1) * 2
⊢ 2 ^ (#P.parts - 1) * ↑m * (↑(#U) * ε) ≤ 2 ^ (#P.parts - 1) * ↑m * (2 * ↑(#(G.nonuniformWitness ε U V)))

Tactic:
gcongr _ * ?_
        exact (G.le_card_nonuniformWitness hunif).trans
          (le_mul_of_one_le_left (cast_nonneg _) one_le_two)

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

Goal state before:
pp: case Adj.mp
V : Type u
G : SimpleGraph V
v w : V
hvw : G.Adj v w
x✝¹ x✝ : V
h : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹
⊢ (⊤.induce {v, w}).Adj x✝¹ x✝

Tactic:
obtain ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ := h

Goal state after:
case Adj.mp.inl
V : Type u
G : SimpleGraph V
v w : V
hvw : G.Adj v w
⊢ (⊤.induce {v, w}).Adj v w

case Adj.mp.inr
V : Type u
G : SimpleGraph V
v w : V
hvw : G.Adj v w
⊢ (⊤.induce {v, w}).Adj w v
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
⊢ u :: p.support.tail = p.support

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
⊢ p.support = u :: p.support.tail

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
u v : V
p : G.Walk u v
⊢ (Walk.map f p).Nil ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
⊢ p.Nil ↔ p.support = [v]

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
⊢ p.darts = [] ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
⊢ p.edges = [] ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
⊢ p.length = 0 ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
⊢ ¬p.Nil ↔ 0 < p.length

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
hinj : Function.Injective ⇑f
u v : V
⊢ Function.Injective (Walk.map f)

Tactic:
intro p

Goal state after:
V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
f : G →g G'
hinj : Function.Injective ⇑f
u v : V
p : G.Walk u v
⊢ ∀ ⦃a₂ : G.Walk u v⦄, Walk.map f p = Walk.map f a₂ → p = a₂
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
v : V
p : G.Walk v v
⊢ p = nil ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
l : List G.Dart
hne : l ≠ []
hchain : List.IsChain G.DartAdj l
⊢ (ofDarts l hne hchain).length = l.length

Tactic:
by
  grind [darts_ofDarts]

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v u' v' w : V
p : G.Walk u v
h : G.Adj v' w
r₁ : G.Walk u' u
r₂ : G.Walk v v'
⊢ ((r₁.append p).append r₂).concat h = (r₁.append p).append (r₂.concat h)

Tactic:
append_concat

Goal state after:
V : Type u_1
G : SimpleGraph V
u v u' v' w : V
p : G.Walk u v
h : G.Adj v' w
r₁ : G.Walk u' u
r₂ : G.Walk v v'
⊢ ((r₁.append p).append r₂).concat h = ((r₁.append p).append r₂).concat h
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v u' v' : V
p : G.Walk u v
⊢ [u', v'] <:+: p.support ∨ [v', u'] <:+: p.support ↔ s(u', v') ∈ p.edges

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩

Goal state after:
case refine_1
V : Type u_1
G : SimpleGraph V
u v u' v' : V
p : G.Walk u v
h : [u', v'] <:+: p.support ∨ [v', u'] <:+: p.support
⊢ s(u', v') ∈ p.edges

case refine_2
V : Type u_1
G : SimpleGraph V
u v u' v' : V
p : G.Walk u v
h : s(u', v') ∈ p.edges
⊢ [u', v'] <:+: p.support ∨ [v', u'] <:+: p.support
Module: Mathlib.Combinatorics.SimpleGraph.Walk.Operations

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v w
⊢ p.reverse.Nil ↔ p.Nil

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
⊢ p.tail.length = p.length - 1

Tactic:
by
  cases p <;> simp

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
n : ℕ
p : G.Walk u v
⊢ p.tail.getVert n = p.getVert (n + 1)

Tactic:
by
  cases p <;> simp

Goal state after: