mathlib-initiative/mathlib-tactics huggingface dataset:
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 : Finpartition a
hb : b ≤ a
px : α
hpx : px ∈ ↑P.parts
right✝¹ : ¬px ⊓ b = ⊥
py : α
hpy : py ∈ ↑P.parts
hxy : px ⊓ b ≠ py ⊓ b
right✝ : ¬py ⊓ b = ⊥
⊢ (Disjoint on id) (px ⊓ b) (py ⊓ b)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a
hb : b ≤ a
px : α
hpx : px ∈ ↑P.parts
right✝¹ : ¬px ⊓ b = ⊥
py : α
hpy : py ∈ ↑P.parts
hxy : px ⊓ b ≠ py ⊓ b
right✝ : ¬py ⊓ b = ⊥
⊢ Disjoint (px ⊓ b) (py ⊓ b)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
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
X : C
n : ℤ
⊢ (t.eTriangleLTGE.obj (WithBotTop.coe n)).obj X ∈ distinguishedTriangles

Tactic:
| coe n =>

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

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
hab : a ≤ b
p : α
hp : p ∈ (P.extendOfLE hab).parts
h : ¬a < b
⊢ p ∈ P.parts

Tactic:
simpa using

Goal state after:
case neg
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
hab : a ≤ b
p : α
hp : p ∈ (P.extendOfLE hab).parts
h : ¬a < b
⊢ p ∈ P.parts
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: 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
n : ℤ
X : C
i : ℤ
h : WithBotTop.coe n ≤ WithBotTop.coe i
⊢ n ≤ ?m.80

Tactic:
simpa using

Goal state after:
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
n : ℤ
X : C
i : ℤ
h : WithBotTop.coe n ≤ WithBotTop.coe i
⊢ n ≤ ?m.80
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
X : C
n : ℤ
inst✝ : t.IsGE X n
j : ℤ
hj : WithBotTop.coe j ≤ WithBotTop.coe n
⊢ j ≤ n

Tactic:
simpa using

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
X : C
n : ℤ
inst✝ : t.IsGE X n
j : ℤ
hj : WithBotTop.coe j ≤ WithBotTop.coe n
⊢ j ≤ n
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 : EInt
X : C
h : a ≤ ⊥
⊢ a = ⊥

Tactic:
simpa using

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 : EInt
X : C
h : a ≤ ⊥
⊢ a = ⊥
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
X : C
b a : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe b
⊢ a ≤ b

Tactic:
simpa using

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
X : C
b a : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe 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
⊢ (image (fun a ↦ {b ∈ x | s a b}) x).SupIndep id

Tactic:
suffices ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c) by
      simp only [supIndep_iff_pairwiseDisjoint, Set.PairwiseDisjoint, Set.Pairwise, coe_image,
        Set.mem_image, mem_coe, ne_eq, onFun, id_eq, disjoint_iff_ne, forall_mem_not_eq,
        forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_filter, not_and, filter_inj',
        not_forall, @not_imp_comm (_ ↔ _), Decidable.not_not]
      intro _ _ _ _ _ _ _ _ ha _ hb
      exact ⟨(s.trans' hb <| s.trans' (s.symm' ha) ·), (s.trans' ha <| s.trans' (s.symm' hb) ·)⟩

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 b c d : α), s a d → s b d → (s a c ↔ s b c)
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
a : α
s : Setoid α
x : Finset α
inst✝ : DecidableRel ⇑s
b : α
⊢ b ∈ (ofSetSetoid s x).part a ↔ a ∈ x ∧ b ∈ x ∧ s a b

Tactic:
suffices (∃ a₁ ∈ x, (b ∈ x ∧ s a₁ b) ∧ a ∈ x ∧ s a₁ a) ↔ a ∈ x ∧ b ∈ x ∧ s a b by
    simpa [mem_part_iff_exists, ofSetSetoid_parts]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
a : α
s : Setoid α
x : Finset α
inst✝ : DecidableRel ⇑s
b : α
⊢ (∃ a₁ ∈ x, (b ∈ x ∧ s a₁ b) ∧ a ∈ x ∧ s a₁ a) ↔ a ∈ x ∧ b ∈ x ∧ s a b
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
X : C
a b : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe b
⊢ a ≤ b

Tactic:
simpa using

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
X : C
a b : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe b
⊢ a ≤ b
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Bound

Goal state before:
pp: case inl
ι : Type u_2
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s t : Finset ι
x : 𝕜
hst : s ⊆ t
f : ι → 𝕜
d : 𝕜
hx : 0 ≤ x
hs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|
ht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2
hscard : 0 = ↑(#s)
⊢ d + ↑(#s) / ↑(#t) * x ^ 2 ≤ (∑ i ∈ t, f i ^ 2) / ↑(#t)

Tactic:
simpa using

Goal state after:
case inl
ι : Type u_2
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s t : Finset ι
x : 𝕜
hst : s ⊆ t
f : ι → 𝕜
d : 𝕜
hx : 0 ≤ x
hs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|
ht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2
hscard : 0 = ↑(#s)
⊢ d ≤ (∑ i ∈ t, f i ^ 2) / ↑(#t)
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
b : EInt
X : C
h : ⊤ ≤ b
⊢ b = ⊤

Tactic:
simpa using

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
b : EInt
X : C
h : ⊤ ≤ b
⊢ b = ⊤
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
a b : ℕ
P : Finpartition s
hs : a * 0 + b * (0 + 1) = #s
⊢ #({i ∈ ⊥.parts | #i = 0 + 1}) = b

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
a b : ℕ
P : Finpartition s
hs : a * 0 + b * (0 + 1) = #s
⊢ #s = b
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
X : C
⊢ (t.eTruncLT.obj b).map ((t.eTruncLTι a).app X) ≫ (t.eTruncLTLTIsoLT a b hab).inv.app X =
    𝟙 ((t.eTruncLT.obj b).obj ((t.eTruncLT.obj a).obj X))

Tactic:
simpa using

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
⊢ (t.eTruncLTLTIsoLT a b hab).inv.app ((t.eTruncLT.obj a).obj X) ≫
      (t.eTruncLT.obj b).map ((t.eTruncLT.obj a).map ((t.eTruncLTι a).app X)) =
    𝟙 ((t.eTruncLT.obj b).obj ((t.eTruncLT.obj a).obj 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
hab : b ≤ a
X : C
⊢ (t.eTruncLT.obj b).map ((t.eTruncLTι a).app X) ≫ (t.eTruncLTLTIsoLT a b hab).inv.app X =
    𝟙 ((t.eTruncLT.obj b).obj ((t.eTruncLT.obj a).obj X))

Tactic:
simpa using (t.eTruncLTLTIsoLT a b hab).hom_inv_id_app X

Goal state after:
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
X : C
⊢ (t.eTruncLTLTIsoLT a b hab).inv.app X ≫ (t.eTruncLT.obj b).map ((t.eTruncLTι a).app X) = 𝟙 ((t.eTruncLT.obj b).obj X)

Tactic:
simpa using

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
⊢ (t.eTruncLTLTIsoLT a b hab).inv.app X ≫ (t.eTruncLT.obj b).map ((t.eTruncLTι a).app X) = 𝟙 ((t.eTruncLT.obj b).obj X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.bot
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
b : ℤ
⊢ IsIso ((t.eTruncLTGELTSelfToLTGE ⊥ (WithBotTop.coe b)).app X)

Tactic:
simpa using

Goal state after:
case coe.bot
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
b : ℤ
⊢ IsIso ((t.truncLT b).map ((t.truncLTι b).app X))
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case top
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 ⊤).app X)

Tactic:
simpa using

Goal state after:
case top
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 (𝟙 ((t.eTruncGE.obj a).obj X))
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case bot
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✝ X : C
⊢ IsIso ((t.eTruncLTGELTSelfToGELT ⊥ b).app X)

Tactic:
simpa using

Goal state after:
case bot
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✝ X : C
⊢ IsIso ((t.eTruncLTι b).app ((t.eTruncLT.obj b).obj X))
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.bot
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✝ X : C
a : ℤ
⊢ IsIso ((t.eTruncLTGELTSelfToGELT (WithBotTop.coe a) ⊥).app X)

Tactic:
simpa using

Goal state after:
case coe.bot
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✝ X : C
a : ℤ
⊢ IsZero ((t.truncGE a).obj (Functor.obj 0 X))
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: case coe.top
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✝ X : C
a : ℤ
⊢ IsIso ((t.eTruncLTGELTSelfToGELT (WithBotTop.coe a) ⊤).app X)

Tactic:
simpa using

Goal state after:
case coe.top
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✝ X : C
a : ℤ
⊢ IsIso (𝟙 ((t.truncGE a).obj 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 : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
hG : G.IsUniform ε s t
hε : ε ≤ 0
⊢ ↑(#s) * ε ≤ ↑(#∅)

Tactic:
simpa using

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
hG : G.IsUniform ε s t
hε : ε ≤ 0
⊢ ↑(#s) * ε ≤ 0
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 : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
hG : G.IsUniform ε s t
hε : ε ≤ 0
⊢ ↑(#t) * ε ≤ ↑(#∅)

Tactic:
simpa using

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
hG : G.IsUniform ε s t
hε : ε ≤ 0
⊢ ↑(#t) * ε ≤ 0
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 : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
hε : 0 < ε
t' : Finset α
ht' : t' ⊆ {b}
ht : ε ≤ ↑(#t')
hs' : ∅ ⊆ {a}
hs : ε ≤ ↑(#∅)
⊢ ε ≤ 0

Tactic:
simpa using

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
hε : 0 < ε
t' : Finset α
ht' : t' ⊆ {b}
ht : ε ≤ ↑(#t')
hs' : ∅ ⊆ {a}
hs : ε ≤ ↑(#∅)
⊢ ε ≤ 0
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 : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
hε : 0 < ε
hs' : {a} ⊆ {a}
hs : ε ≤ ↑(#{a})
ht' : ∅ ⊆ {b}
ht : ε ≤ ↑(#∅)
⊢ ε ≤ 0

Tactic:
simpa using

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
hε : 0 < ε
hs' : {a} ⊆ {a}
hs : ε ≤ ↑(#{a})
ht' : ∅ ⊆ {b}
ht : ε ≤ ↑(#∅)
⊢ ε ≤ 0
Module: Mathlib.Combinatorics.SimpleGraph.Density

Goal state before:
pp: case 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 : Finset α
t₁ t₂ : Finset β
δ : 𝕜
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs : ∅ ⊆ s₁
hs₂ : (1 - δ) * ↑(#s₁) ≤ 0
⊢ |↑(edgeDensity r ∅ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2

Tactic:
simpa using

Goal state after:
case 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 : Finset α
t₁ t₂ : Finset β
δ : 𝕜
ht : t₂ ⊆ t₁
hδ₀ : 0 ≤ δ
hδ₁ : 0 < 1 - δ
ht₂ : (1 - δ) * ↑(#t₁) ≤ ↑(#t₂)
hδ' : 0 ≤ 2 * δ - δ ^ 2
hs : ∅ ⊆ s₁
hs₂ : (1 - δ) * ↑(#s₁) ≤ 0
⊢ δ ^ 2 ≤ 2 * δ
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) ≤ 0
⊢ |↑(edgeDensity r s₂ ∅) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ - δ ^ 2

Tactic:
simpa using

Goal state after:
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) ≤ 0
⊢ δ ^ 2 ≤ 2 * δ
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: case inr.calc_1
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : DecidableEq α
A : Finset α
P : Finpartition A
hP : P.IsEquipartition
h : P.parts.Nonempty
⊢ ↑(#A / #P.parts + 1) ≤ ↑(#A) / ↑(#P.parts) + 1

Tactic:
simpa using

Goal state after:
case inr.calc_1
α : Type u_1
𝕜 : Type u_2
inst✝³ : Field 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
inst✝ : DecidableEq α
A : Finset α
P : Finpartition A
hP : P.IsEquipartition
h : P.parts.Nonempty
⊢ ↑(#A / #P.parts) ≤ ↑(#A) / ↑(#P.parts)
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: case hbc
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.parts * (#P.parts - 1) * (#A / #P.parts + 1) ^ 2 < (#A + #P.parts) ^ 2

Tactic:
exact aux (P.parts_nonempty hA.ne_empty).card_pos

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:
simp_rw [chunk, apply_dite Finpartition.parts, apply_dite card, sum_dite]

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
⊢ ∑ x, #(equitabilise ⋯).parts + ∑ x, #(equitabilise ⋯).parts = stepBound #P.parts
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma

Goal state before:
pp: case refine_1
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
h :
  ∀ (i : ℕ),
    ∃ P,
      P.IsEquipartition ∧ t ≤ #P.parts ∧ #P.parts ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊ + 1] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(P.energy G)
⊢ stepBound^[⌊4 / ε ^ 5⌋₊ + 1] t ≤ bound ε l

pp: case 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
h :
  ∀ (i : ℕ),
    ∃ P,
      P.IsEquipartition ∧ t ≤ #P.parts ∧ #P.parts ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊ + 1] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(P.energy G)
hPenergy : ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(P.energy G)
⊢ 1 < 1

Tactic:
· rw [iterate_succ_apply', stepBound, bound]
      gcongr
      simp

Goal state after:
case 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
h :
  ∀ (i : ℕ),
    ∃ P,
      P.IsEquipartition ∧ t ≤ #P.parts ∧ #P.parts ≤ stepBound^[i] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G))
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[⌊4 / ε ^ 5⌋₊ + 1] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(P.energy G)
hPenergy : ε ^ 5 / 4 * ↑(⌊4 / ε ^ 5⌋₊ + 1) ≤ ↑(P.energy G)
⊢ 1 < 1