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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : OrderBot α
inst✝¹ : DecidableEq α
a b : α
P : Finpartition a
hb : b ≤ a
M : Type u_2
inst✝ : AddCommMonoid M
f : α → M
hf : f ⊥ = 0
⊢ ∑ p ∈ (P.restrict hb).parts, f p = ∑ q ∈ P.parts, f (q ⊓ b)

Tactic:
by

Goal state after:
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
i : EInt
X : C
⊢ (t.eTruncGE.obj i).map ((t.eTruncGEπ i).app X) = (t.eTruncGEπ i).app ((t.eTruncGE.obj i).obj X)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : OrderBot α
inst✝¹ : DecidableEq α
parts : Finset α
hdisjoint : (↑parts).PairwiseDisjoint id
X : Type u_2
inst✝ : AddCommMonoid X
f : α → X
hf : f ⊥ = 0
⊢ ∑ p ∈ (ofPairwiseDisjoint parts hdisjoint).parts, f p = ∑ p ∈ parts, f p

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a✝ b c : α
ι : Type u_2
I : Finset ι
a : ι → α
P : (i : ι) → Finpartition (a i)
ha : I.SupIndep a
⊢ I.SupIndep fun i ↦ (P i).parts.sup id

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a✝ b c : α
ι : Type u_2
I : Finset ι
a : ι → α
P : (i : ι) → Finpartition (a i)
ha : I.SupIndep a
⊢ (I.biUnion fun i ↦ (P i).parts).sup id = I.sup a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a✝ b c : α
ι : Type u_2
I : Finset ι
a : ι → α
P : (i : ι) → Finpartition (a i)
ha : I.SupIndep a
⊢ ⊥ ∉ I.biUnion fun i ↦ (P i).parts

Tactic:
by

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
⊢ ∑ p ∈ (combine P ha).parts, f p = ∑ i ∈ I, ∑ p ∈ (P i).parts, f p

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a b c : α
P✝ : Finpartition a
Q✝ : (i : α) → i ∈ P✝.parts → Finpartition i
P : Finpartition a
Q : (i : α) → i ∈ P.parts → Finpartition i
⊢ P.parts.attach.sup Subtype.val = a

Tactic:
by

Goal state after:
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
i j : EInt
f : i ⟶ j
X : C
⊢ (t.eTruncLT.obj i).map ((t.eTruncLTι j).app X) ≫ (t.eTruncLT.map f).app X =
    (t.eTruncLTι i).app ((t.eTruncLT.obj j).obj X)

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ 0 < 4

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
Q : (i : α) → i ∈ P.parts → Finpartition i
⊢ b ∈ (P.bind Q).parts ↔ ∃ A, ∃ (hA : A ∈ P.parts), b ∈ (Q A hA).parts

Tactic:
by

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

Goal state before:
pp: x✝¹ x✝ : ℕ
h : x✝¹ ≤ x✝
⊢ stepBound x✝¹ ≤ stepBound x✝

Tactic:
by

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

Goal state before:
pp: n : ℕ
⊢ 0 < 4 ^ n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : Semiring α
n : ℕ
⊢ ↑(stepBound n) = ↑n * 4 ^ n

Tactic:
by

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

Goal state before:
pp: ε : ℝ
n : ℕ
h : 100 ≤ 4 ^ n * ε ^ 5
⊢ Odd 5

Tactic:
by

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

Goal state before:
pp: ε : ℝ
n : ℕ
h : 100 ≤ 4 ^ n * ε ^ 5
⊢ 0 ≤ 4 ^ n

Tactic:
by

Goal state after:
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
a : ↥s
⊢ z a < #s

Tactic:
rcases hP.card_parts_eq_average (f a).1.2 with (c | c)

Goal state after:
case inl
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
a : ↥s
c : #↑(f a).fst = #s / #P.parts
⊢ z a < #s

case inr
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
a : ↥s
c : #↑(f a).fst = #s / #P.parts + 1
⊢ z a < #s
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Bound

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
⊢ stepBound #P.parts ≤ #P.parts * 16 ^ #P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a : α
P : Finpartition a
Q : (i : α) → i ∈ P.parts → Finpartition i
⊢ #(P.bind Q).parts = ∑ A ∈ P.parts.attach, #(Q ↑A ⋯).parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
⊢ 0 < m

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
⊢ 0 < ↑m + 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
ε : ℝ
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ 0 < 100

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
ε : ℝ
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ 0 ≤ 4 ^ #P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a
hb : b ≠ ⊥
hab : Disjoint a b
hc : a ⊔ b = c
⊢ (insert b P.parts).SupIndep id

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
ε : ℝ
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ Odd 5

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ 0 ≤ 4 ^ #P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : OrderBot α
inst✝¹ : IsModularLattice α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a
hb : b ≠ ⊥
hab : Disjoint a b
hc : a ⊔ b = c
⊢ (insert b P.parts).sup id = c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b✝ c : α
P : Finpartition a
b : α
⊢ (image (fun x ↦ x \ b) P.parts).sup id = a \ b

Tactic:
by

Goal state after:
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
i : EInt
X : C
⊢ (t.eTriangleLTGE.obj i).obj X ∈ distinguishedTriangles

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a
⊢ c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \ b = c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
hab : a = b
⊢ (P.extendOfLE ⋯).parts = P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ 4 ^ #P.parts ≤ ↑m

Tactic:
by

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
X : C
n : ℤ
inst✝ : t.IsLE X n
⊢ IsZero ((t.eTruncLT.obj ⊥).obj X)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
hab : a < b
⊢ (P.extendOfLE ⋯).parts = insert (b \ a) P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε : ε ≤ 1
⊢ 0 ≤ 100

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε : ε ≤ 1
⊢ 0 < ε ^ 5

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
P : Finpartition univ
ε : ℝ
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε : ε ≤ 1
⊢ 0 ≤ ε

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
⊢ 1 ≤ 4

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
hab : a ≤ b
⊢ P.parts ⊆ (P.extendOfLE hab).parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
h : 1 ≤ 4 ^ #P.parts
⊢ Fintype.card α / #P.parts - Fintype.card α / (#P.parts * 4 ^ #P.parts) * 4 ^ #P.parts + 1 ≤ 4 ^ #P.parts

Tactic:
← Nat.div_div_eq_div_mul

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
h : 1 ≤ 4 ^ #P.parts
⊢ Fintype.card α / #P.parts - Fintype.card α / #P.parts / 4 ^ #P.parts * 4 ^ #P.parts + 1 ≤ 4 ^ #P.parts
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: 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
⊢ p ∈ P.parts ∨ p = b \ a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
ha : a ∈ s
⊢ ∃ t ∈ P.parts, a ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
ha : a ∈ s
⊢ ∃! t, t ∈ P.parts ∧ a ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
ha : a ∈ s
⊢ ∃! t, t ∈ P.parts ∧ a ∈ t

Tactic:
by
  obtain ⟨t, ht, ht'⟩ := P.exists_mem ha
  refine ⟨t, ⟨ht, ht'⟩, ?_⟩
  rintro u ⟨hu, hu'⟩
  exact P.eq_of_mem_parts hu ht hu' ht'

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
X : C
n : ℤ
inst✝ : t.IsLE X n
i : EInt
⊢ t.IsLE ((t.eTruncLT.obj i).obj X) n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
⊢ a + 1 ≤ 4 ^ #P.parts

Tactic:
by

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
X : C
n : ℤ
inst✝ : t.IsGE X n
⊢ IsZero ((t.eTruncGE.obj ⊤).obj X)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hucard : #u = m * 4 ^ #P.parts + a
⊢ (4 ^ #P.parts - a) * m + a * (m + 1) = #u

Tactic:
by

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
X : C
n : ℤ
inst✝ : t.IsGE X n
i : EInt
⊢ t.IsGE ((t.eTruncGE.obj i).obj X) n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t u : Finset α
P : Finpartition s
a : α
parts : Finset (Finset α)
h : ∀ p ∈ parts, p ⊆ s
h' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t
h'' : ∅ ∉ parts
⊢ parts.SupIndep id

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ Fintype.card α / (#P.parts * 4 ^ #P.parts) * 4 ^ #P.parts ≤ Fintype.card α / #P.parts

Tactic:
← Nat.div_div_eq_div_mul

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ Fintype.card α / #P.parts / 4 ^ #P.parts * 4 ^ #P.parts ≤ Fintype.card α / #P.parts
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Bound

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
a : ↥s
c : #↑(f a).fst = #s / #P.parts
⊢ z a < #P.parts * (↑(f a).snd + 1)

Tactic:
by

Goal state after:
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
i : EInt
h : WithBotTop.coe n ≤ i
X : C
⊢ t.IsGE ((t.eTruncGE.obj i).obj X) n

Tactic:
by

Goal state after:
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
h : ⊥ ≤ WithBotTop.coe (n + 1)
⊢ IsZero ((t.eTruncLT.obj ⊥).obj X)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t u : Finset α
P : Finpartition s
a : α
parts : Finset (Finset α)
h : ∀ p ∈ parts, p ⊆ s
h' : ∀ a ∈ s, ∃! t, t ∈ parts ∧ a ∈ t
h'' : ∅ ∉ parts
⊢ parts.sup id = s

Tactic:
by

Goal state after:
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
a : ↥s
c : #↑(f a).fst = #s / #P.parts
⊢ #P.parts * (↑(f a).snd + 1) ≤ #P.parts * (#s / #P.parts)

Tactic:
by

Goal state after:
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 : i ≤ n + 1
⊢ i - 1 ≤ n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
⊢ P.part a ∈ P.parts ↔ a ∈ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
h : a ∉ s
⊢ P.part a = ∅

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
⊢ (P.part a).Nonempty ↔ a ∈ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
⊢ P.part a ⊆ s

Tactic:
by

Goal state after:
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
i : EInt
h : i ≤ WithBotTop.coe (n + 1)
X : C
⊢ t.IsLE ((t.eTruncLT.obj i).obj X) n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
⊢ a ∈ P.part a ↔ a ∈ s

Tactic:
by

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
X : C
n : ℤ
inst✝ : t.IsGE X n
j : ℤ
hj : WithBotTop.coe j ≤ WithBotTop.coe n
⊢ j ≤ n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
P : Finpartition s
a : α
ht : t ∈ P.parts
⊢ P.part a = t ↔ a ∈ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
p : Finset α
hp : p ∈ ↑P.parts
⊢ p ∈ P.part '' ↑s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
⊢ ∃ r ⊆ s, Set.BijOn P.part ↑r ↑P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a b : α
⊢ a ∈ P.part b ↔ ∃ p ∈ P.parts, a ∈ p ∧ b ∈ p

Tactic:
by

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
X : C
n : ℤ
inst✝ : t.IsGE X n
j : EInt
hj : j ≤ WithBotTop.coe n
⊢ IsZero ((t.eTruncLT.obj j).obj X)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t u : Finset α
P : Finpartition s
a : α
x : ↥s
⊢ (fun x ↦ ⟨↑x.snd, ⋯⟩) ((fun x ↦ ⟨⟨P.part ↑x, ⋯⟩, ⟨↑x, ⋯⟩⟩) x) = x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ (4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u

Tactic:
by

Goal state after:
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
a : ↥s
c : #↑(f a).fst = #s / #P.parts + 1
⊢ ↑(f a).snd ≤ #s / #P.parts

Tactic:
by

Goal state after: