mathlib-initiative/mathlib-tactics huggingface dataset:
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 : (t : ↥P.parts) × ↥↑t
⊢ (fun x ↦ ⟨⟨P.part ↑x, ⋯⟩, ⟨↑x, ⋯⟩⟩) ((fun x ↦ ⟨↑x.snd, ⋯⟩) x) = x

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
j : ℤ
hj : n < j
⊢ n ≤ j - 1

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
⊢ ∃ f, ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst

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
⊢ ∑ i ∈ P.parts, #i = #s

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
s : Finset α
⊢ (↑(Finset.map { toFun := singleton, inj' := ⋯ } s)).PairwiseDisjoint id

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
j : ℤ
hj : n < j
this : t.IsLE X (j - 1)
⊢ j - 1 + 1 = j

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
s : Finset α
⊢ (Finset.map { toFun := singleton, inj' := ⋯ } s).sup id = s

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
s : Finset α
⊢ ⊥ ∉ Finset.map { toFun := singleton, inj' := ⋯ } s

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
⊢ 4 ^ #P.parts * (Fintype.card α / (#P.parts * 4 ^ #P.parts)) ≤ #u

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
⊢ 4 ^ #P.parts * (Fintype.card α / #P.parts / 4 ^ #P.parts) ≤ #u
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:
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
s : Finset α
P : Finpartition s
t : Finset α
ht : t ∈ ⊥.parts
⊢ ∃ c ∈ P.parts, t ⊆ c

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.parts ≤ #s

Tactic:
rw [← card_bot s]

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
⊢ #P.parts ≤ #⊥.parts
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
j : EInt
hj : WithBotTop.coe n < j
⊢ IsZero ((t.eTruncGE.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 : Finset α
P : Finpartition s
⊢ #P.parts ≤ #s

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
⊢ 4 ^ #P.parts * ↑m ≤ ↑(#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
inst✝ : IsTriangulated C
a : EInt
X : C
h : a ≤ ⊥
⊢ 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
⊢ #s % #P.parts ≤ #P.parts

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
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
⊢ Fintype.card ↥s = Fintype.card (Fin #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
inst✝ : IsTriangulated C
X : C
b a : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe b
⊢ a ≤ b

Tactic:
by

Goal state after:
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
this : ∀ (a b c d : α), s a d → s b d → (s a c ↔ s b c)
⊢ (image (fun a ↦ {b ∈ x | s a b}) x).SupIndep id

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
s✝ t u : Finset α
P : Finpartition s✝
a✝ : α
s : Setoid α
x : Finset α
inst✝ : DecidableRel ⇑s
a : α
x✝ : a ∈ x
⊢ ∃ i ∈ x, a ∈ x ∧ s i a

Tactic:
by

Goal state after:
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).sup id = x

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
inst✝ : IsTriangulated C
a : EInt
X : C
h : a ≤ ⊤
⊢ IsZero ((t.eTruncGE.obj ⊤).obj ((𝟭 C).obj X))

Tactic:
by

Goal state after:
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
this : ∀ x₁ ∈ x, ∃ x₂ ∈ x, s x₁ x₂
⊢ ⊥ ∉ image (fun a ↦ {b ∈ x | s a b}) x

Tactic:
by

Goal state after:
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

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 1 < 4

Tactic:
by

Goal state after:
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
this : (∃ a₁ ∈ x, (b ∈ x ∧ s a₁ b) ∧ a ∈ x ∧ s a₁ a) ↔ a ∈ x ∧ b ∈ x ∧ s a b
⊢ b ∈ (ofSetSetoid s x).part a ↔ a ∈ x ∧ b ∈ x ∧ s a b

Tactic:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 0 < 100

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 100 < 4 ^ initialBound ε l * ε ^ 5

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
a : α
inst✝¹ : Fintype α
s : Setoid α
inst✝ : DecidableRel ⇑s
b : α
this : b ∈ (ofSetSetoid s univ).part a ↔ a ∈ univ ∧ b ∈ univ ∧ s a b
⊢ b ∈ (ofSetoid s).part a ↔ s a b

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
inst✝ : IsTriangulated C
a : EInt
X : C
h : a ≤ ⊤
⊢ IsZero ((t.eTruncGE.obj ⊤).obj ((t.eTruncGE.obj a).obj X))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
a : α
inst✝¹ : Fintype α
s : Setoid α
inst✝ : DecidableRel ⇑s
b : α
⊢ b ∈ (ofSetoid s).part a ↔ s a b

Tactic:
by

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

Goal state before:
pp: ε : ℝ
l : ℕ
⊢ 0 < 16 ^ stepBound^[⌊4 / ε ^ 5⌋₊] (initialBound ε l)

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
s : Finset α
F : Finset (Finset α)
z : α
Q : Finset (Finset α)
hQ : Q ∈ F.powerset
hz1 : z ∈ id ({i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t})
R : Finset (Finset α)
hR : R ∈ F.powerset
hz2 : z ∈ id ({i ∈ s | ∀ t ∈ F, t ∈ R ↔ i ∈ t})
h : {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t} ≠ {i ∈ s | ∀ t ∈ F, t ∈ R ↔ i ∈ t}
h' : Q = R
⊢ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t} = {i ∈ s | ∀ t ∈ F, t ∈ R ↔ i ∈ t}

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
inst✝ : IsTriangulated C
a b : EInt
h : a ≤ b
X : C
⊢ IsIso ((t.eTruncGE.obj b).map ((t.eTruncGEπ a).app X))

Tactic:
by

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

Goal state before:
pp: 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
hs : x ^ 2 ≤ ((∑ i ∈ s, f i) / ↑(#s)) ^ 2
hs' : ↑(#s) ≠ 0
⊢ ↑(#s) * x ^ 2 ≤ ↑(#s) * ((∑ i ∈ s, f i ^ 2) / ↑(#s))

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
s : Finset α
F : Finset (Finset α)
z : α
Q : Finset (Finset α)
hQ : Q ∈ F.powerset
hz1 : z ∈ s ∧ ∀ t ∈ F, t ∈ Q ↔ z ∈ t
R : Finset (Finset α)
hR : R ∈ F.powerset
hz2 : z ∈ s ∧ ∀ t ∈ F, t ∈ R ↔ z ∈ t
h : {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t} ≠ {i ∈ s | ∀ t ∈ F, t ∈ R ↔ i ∈ t}
⊢ Q = R

Tactic:
mem_powerset

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s✝ t u : Finset α
P : Finpartition s✝
a : α
s : Finset α
F : Finset (Finset α)
z : α
Q : Finset (Finset α)
hQ : Q ⊆ F
hz1 : z ∈ s ∧ ∀ t ∈ F, t ∈ Q ↔ z ∈ t
R : Finset (Finset α)
hR : R ⊆ F
hz2 : z ∈ s ∧ ∀ t ∈ F, t ∈ R ↔ z ∈ t
h : {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t} ≠ {i ∈ s | ∀ t ∈ F, t ∈ R ↔ i ∈ t}
⊢ Q = R
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
⊢ IsZero ((t.eTruncLT.obj ⊥).obj ((t.eTruncLT.obj b).obj X))

Tactic:
by

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

Goal state before:
pp: 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
hs : x ^ 2 ≤ ((∑ i ∈ s, f i) / ↑(#s)) ^ 2
hs' : ↑(#s) ≠ 0
⊢ ∑ i ∈ s, f i ^ 2 ≤ ∑ i ∈ t, f i ^ 2

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
inst✝ : IsTriangulated C
b : EInt
X : C
h : ⊥ ≤ b
⊢ IsZero ((t.eTruncLT.obj ⊥).obj ((𝟭 C).obj X))

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
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
a b : ↥s
this : NeZero #P.parts
e : (f a).snd.1 = (f b).snd.1 ∧ (f a).fst = (f b).fst
⊢ #↑(f a).fst = #↑(f b).fst

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
inst✝ : IsTriangulated C
X : C
a b : ℤ
h : WithBotTop.coe a ≤ WithBotTop.coe b
⊢ a ≤ b

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
s : Finset α
F : Finset (Finset α)
x : Finset α
hx : x ∈ ↑(image (fun Q ↦ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t}) F.powerset)
y : Finset α
hy : y ∈ ↑(image (fun Q ↦ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t}) F.powerset)
h : x ≠ y
z : α
hz1 : z ∈ id x
hz2 : z ∈ id y
⊢ x = y

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
less : ∀ (a : ↥s), z a < #s
z' : ↥s → Fin #s := fun a ↦ ⟨z a, ⋯⟩
⊢ Function.Bijective z'

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
s : Finset α
F : Finset (Finset α)
⊢ (image (fun Q ↦ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t}) F.powerset).sup id = s

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
F : Finset (Finset α)
⊢ t ∈ (atomise s F).parts ↔ t.Nonempty ∧ ∃ Q ⊆ F, {i ∈ s | ∀ u ∈ F, u ∈ Q ↔ i ∈ u} = t

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
inst✝ : IsTriangulated C
b : EInt
X : C
h : ⊤ ≤ b
⊢ b = ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
⊢ (atomise s ∅).parts = {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
inst✝ : IsTriangulated C
a b : EInt
h : a ≤ b
X : C
⊢ IsIso ((t.eTruncLT.obj a).map ((t.eTruncLTι b).app X))

Tactic:
by

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

Goal state before:
pp: 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)
htcard : 0 < ↑(#t)
⊢ |x| ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|

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
inst✝ : IsTriangulated C
a : EInt
X : C
⊢ a ≤ a

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
inst✝ : IsTriangulated C
a : EInt
X : C
⊢ IsIso ((t.eTruncLTι a).app ((t.eTruncLT.obj a).obj X))

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
n : ℤ
inst✝ : t.IsGE 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✝ : DecidableEq α
s t : Finset α
F : Finset (Finset α)
ht : t ∈ F
hts : t ⊆ s
⊢ {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty}.biUnion id = t

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
n : ℤ
inst✝ : t.IsGE X n
i : EInt
⊢ t.IsGE ((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 t : Finset α
F : Finset (Finset α)
ht : t ∈ F
h :
  {u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty} ⊆
    image (fun P ↦ {i ∈ s | ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x}) (F.erase t).powerset
⊢ #({u ∈ (atomise s F).parts | u ⊆ t ∧ u.Nonempty}) ≤ 2 ^ (#F - 1)

Tactic:
by

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

Goal state before:
pp: 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)
htcard : 0 < ↑(#t)
h₁ : x ^ 2 ≤ ((∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)) ^ 2
⊢ x ^ 2 ≤ ((∑ i ∈ s, (f i - (∑ j ∈ t, f j) / ↑(#t))) / ↑(#s)) ^ 2

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
n : ℤ
inst✝ : t.IsLE X n
⊢ IsZero ((t.eTruncGE.obj ⊤).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
n : ℤ
inst✝ : t.IsLE X n
i : EInt
⊢ t.IsLE ((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✝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 : a ≤ b
⊢ IsIso (t.eTruncGEToGEGE a b)

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
inst✝ : IsTriangulated C
a b : EInt
hab : a ≤ b
X : C
⊢ (t.eTruncGE.obj b).map ((t.eTruncGEπ a).app X) ≫ (t.eTruncGEIsoGEGE a b hab).inv.app X =
    𝟙 ((t.eTruncGE.obj b).obj ((𝟭 C).obj X))

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
inst✝ : IsTriangulated C
a b : EInt
hab : a ≤ b
X : C
⊢ (t.eTruncGEIsoGEGE a b hab).inv.app X ≫ (t.eTruncGE.obj b).map ((t.eTruncGEπ a).app X) =
    𝟙 ((t.eTruncGE.obj a ⋙ t.eTruncGE.obj b).obj X)

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, ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ ↑(f a) % #P.parts = ↑(f b) % #P.parts

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
⊢ ∀ a ∈ ↑⊥.parts, #a = 1 ∨ #a = 1 + 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
hs : s ≠ ∅
⊢ (indiscrete hs).IsEquipartition

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
inst✝ : IsTriangulated C
a b : EInt
hab : b ≤ a
⊢ IsIso (t.eTruncLTLTToLT a b)

Tactic:
by

Goal state after: