mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.Additive.CauchyDavenport

Goal state before:
pp: G : Type u_1
inst✝² : DecidableEq G
inst✝¹ : Group G
inst✝ : IsMulTorsionFree G
s t : Finset G
hs : s.Nonempty
ht : t.Nonempty
⊢ #s + #t - 1 ≤ #(s * t)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.CauchyDavenport

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
s t : Finset (ZMod p)
hs : s.Nonempty
ht : t.Nonempty
⊢ min p (#s + #t - 1) ≤ #(s + t)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ↑(#A) * ↑(#(π ε)) = ↑(#A) * ↑(#(V⁻¹ * W))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.CauchyDavenport

Goal state before:
pp: α : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : Mul α
inst✝² : IsCancelMul α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this : s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht}
⊢ #s + #t - 1 ≤ #(s * t)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.CauchyDavenport

Goal state before:
pp: α : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : Mul α
inst✝² : IsCancelMul α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
⊢ #s + #t - 1 ≤ #(s * t)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ↑(#A) * ↑(#(V⁻¹ * W)) ≤ ↑(#(A * V)) * ↑(#(A * W))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x y : G
⊢ Set.MapsTo (fun z ↦ (x⁻¹ * z, z⁻¹ * y)) ↑(x •> A ∩ y •> B) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x⁻¹ * y})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x y : G
⊢ Set.MapsTo (fun ab' ↦ x •> ab'.1) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x⁻¹ * y}) ↑(x •> A ∩ y •> B)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x y : G
⊢ Set.LeftInvOn (fun ab' ↦ x •> ab'.1) (fun z ↦ (x⁻¹ * z, z⁻¹ * y)) ↑(x •> A ∩ y •> B)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x y : G
⊢ Set.RightInvOn (fun ab' ↦ x •> ab'.1) (fun z ↦ (x⁻¹ * z, z⁻¹ * y)) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x⁻¹ * y})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ #(A ∩ x •> B) = A.convolution B⁻¹ x

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ #(x •> A ∩ B) = A.convolution B⁻¹ x⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ #(A ∩ x •> B⁻¹) = A.convolution B x

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ ∀ (i : G × G),
    i ∈ {ab ∈ A ×ˢ B | ab.1 / ab.2 = x} ↔ ((Equiv.refl G).prodCongr (Equiv.inv G)) i ∈ {ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x}

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ Set.MapsTo (fun ab ↦ (ab.1, ab.2⁻¹)) ↑({ab ∈ A ×ˢ B | ab.1 * ab.2⁻¹ = x}) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ↑(#(A * V)) * ↑(#(A * W)) = ↑(#(π ![1, -ε 1, -ε 0])) * ↑(#(π (Fin.cons 1 (tail (tail ε)))))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ Set.MapsTo (fun ab ↦ (ab.1, ab.2⁻¹)) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x}) ↑({ab ∈ A ×ˢ B | ab.1 * ab.2⁻¹ = x})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ Set.LeftInvOn (fun ab ↦ (ab.1, ab.2⁻¹)) (fun ab ↦ (ab.1, ab.2⁻¹)) ↑({ab ∈ A ×ˢ B | ab.1 * ab.2⁻¹ = x})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ Set.RightInvOn (fun ab ↦ (ab.1, ab.2⁻¹)) (fun ab ↦ (ab.1, ab.2⁻¹)) ↑({ab ∈ A ×ˢ B⁻¹ | ab.1 * ab.2 = x})

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ 0 < A.convolution B x ↔ x ∈ A * B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
this : A.convolution B x ≠ 0 ↔ 0 < A.convolution B x
⊢ A.convolution B x ≠ 0 ↔ x ∈ A * B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ A.convolution B x ≠ 0 ↔ x ∈ A * B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ A.convolution B x = 0 ↔ x ∉ A * B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ A.convolution B x ≤ #A

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ A.convolution B x ≤ #B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
x : G
⊢ A.convolution B x⁻¹ = B⁻¹.convolution A⁻¹ x

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
s x : G
⊢ (A <• s).convolution B x = A.convolution (s •> B) x

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
s x : G
⊢ (s •> A).convolution B x = A.convolution B (s⁻¹ * x)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A B : Finset G
s x : G
⊢ A.convolution (B <• s) x = A.convolution B (x * s⁻¹)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : DecidableEq G
inst✝ : Fintype G
B : Finset G
a : G
⊢ univ.convolution B a = #B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Convolution

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : DecidableEq G
inst✝ : Fintype G
A : Finset G
a : G
⊢ A.convolution univ a = #A

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
i : Fin 3
⊢ |![1, -ε 1, -ε 0] i| = 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Defs

Goal state before:
pp: G : Type u_1
inst✝ : AddCommMonoid G
x₁ y₁ x₂ y₂ : G
⊢ ¬IsCorner ∅ x₁ y₁ x₂ y₂

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Defs

Goal state before:
pp: G : Type u_1
inst✝ : AddCommMonoid G
A : Set (G × G)
hA : A.Subsingleton
_x₁ _y₁ _x₂ _y₂ : G
hxyd : IsCorner A _x₁ _y₁ _x₂ _y₂
⊢ _x₁ = _x₂

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ |Fin.cons 1 (tail (tail ε)) 0| = 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Defs

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : AddCommMonoid G
inst✝ : AddCommMonoid H
A : Set (G × G)
s : Set G
t : Set H
f : G → H
x₁ y₁ x₂ y₂ : G
hf : IsAddFreimanHom 2 s t f
hAs : A ⊆ s ×ˢ s
hA : IsCorner A x₁ y₁ x₂ y₂
⊢ IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
d₁ d₂ : G.Dart
⊢ d₁ = d₂ ↔ d₁.toProd = d₂.toProd

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
⊢ ∀ (d₁ d₂ : G.Dart), d₁.edge = d₂.edge ↔ d₁ = d₂ ∨ d₁ = d₂.symm

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
⊢ ∀ {d : G.Dart} {u v : V}, d.edge = s(u, v) ↔ d.toProd = (u, v) ∨ d.toProd = (v, u)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ∀ (i : Fin m), |Fin.cons 1 (tail (tail ε)) i.succ| = 1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
⊢ ∀ {d : G.Dart} {u v : V}, d.edge = s(u, v) ↔ d.toProd.1 = u ∧ d.toProd.2 = v ∨ d.toProd.1 = v ∧ d.toProd.2 = u

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v : V
e₁ e₂ : ↑(G.neighborSet v)
h : G.dartOfNeighborSet v e₁ = G.dartOfNeighborSet v e₂
⊢ ↑e₁ = ↑e₂

Tactic:
by

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Nontrivial V
⊢ Nonempty ⊤.Dart

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ ↑(#(π ![1, -ε 1, -ε 0])) * ↑(#(π (Fin.cons 1 (tail (tail ε))))) ≤ k * ↑(#A) * (k ^ (m - 1) * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Defs

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : AddCommMonoid G
inst✝ : AddCommMonoid H
A : Set (G × G)
s : Set G
t : Set H
f : G → H
x₁ y₁ x₂ y₂ : G
hf : IsAddFreimanIso 2 s t f
hAs : A ⊆ s ×ˢ s
hx₁ : x₁ ∈ s
hy₁ : y₁ ∈ s
hx₂ : x₂ ∈ s
hy₂ : y₂ ∈ s
⊢ IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂) ↔ IsCorner A x₁ y₁ x₂ y₂

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.Corner.Defs

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : AddCommMonoid G
inst✝ : AddCommMonoid H
A : Set (G × G)
s : Set G
t : Set H
f : G → H
hf : IsAddFreimanIso 2 s t f
hAs : A ⊆ s ×ˢ s
⊢ IsCornerFree (Prod.map f f '' A) ↔ IsCornerFree A

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ k * ↑(#A) * (k ^ (m - 1) * ↑(#A)) = ↑(#A) * (k ^ m * ↑(#A))

Tactic:
rw [← pow_sub_one_mul hm₀]

Goal state after:
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ k * ↑(#A) * (k ^ (m - 1) * ↑(#A)) = ↑(#A) * (k ^ (m - 1) * k * ↑(#A))
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
  ∀ (ε : Fin (m + 1) → ℤ),
    (∀ (i : Fin (m + 1)), |ε i| = 1) →
      ↑(#(List.map (fun i ↦ A ^ ε i) (finRange (m + 1))).prod) ≤ k ^ (m + 1 - 2) * ↑(#A)
ε : Fin (m + 1 + 1) → ℤ
hε : ∀ (i : Fin (m + 1 + 1)), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin (m + 1 + 1)), ε i ≠ 0
hA : A.Nonempty
hk : 0 ≤ k
π : {n : ℕ} → (Fin n → ℤ) → Finset G := fun {n} δ ↦ (List.map (fun i ↦ A ^ δ i) (finRange n)).prod
V : Finset G := π ![-ε 1, -ε 0]
W : Finset G := π (tail (tail ε))
⊢ k * ↑(#A) * (k ^ (m - 1) * ↑(#A)) = ↑(#A) * (k ^ m * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m : ℕ
hm : 3 ≤ m
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i ↦ A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
⊢ ↑(#(List.map (fun i ↦ A ^ ε i) (finRange m)).prod) ≤ k ^ (m - 2) * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ 0 ≤ ↑(#(A ^ 3))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ 0 < ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ 0 < ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ ↑(#A) * ↑(#(A⁻¹ * A * A)) = ↑(#A) * ↑(#(A⁻¹ * (A * A)))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ ↑(#A) * ↑(#(A⁻¹ * (A * A))) ≤ ↑(#(A * A)) * ↑(#(A * (A * A)))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ ↑(#(A * A)) * ↑(#(A * (A * A))) = ↑(#(A ^ 2)) * ↑(#(A ^ 3))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ 2 ≤ 3

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ ↑(#(A ^ 2)) * ↑(#(A ^ 3)) ≤ K * ↑(#A) * (K * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ K * ↑(#A) * (K * ↑(#A)) = ↑(#A) * (K ^ 2 * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A⁻¹ * A * A)) ≤ K ^ 2 * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A ^ 3)) ≤ K * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A⁻¹ * A⁻¹ * A)) ≤ K ^ 2 * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A⁻¹ ^ 3)) ≤ K * ↑(#A⁻¹)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A * A⁻¹ * A⁻¹)) ≤ K ^ 2 * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A * A * A⁻¹)) ≤ K ^ 2 * ↑(#A)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ ↑(#A) * ↑(#(A * A⁻¹ * A)) ≤ ↑(#(A * (A * A⁻¹))) * ↑(#(A * A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ ↑(#(A * (A * A⁻¹))) * ↑(#(A * A)) = ↑(#(A * A * A⁻¹)) * ↑(#(A ^ 2))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ 2 ≤ 3

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ ↑(#(A * A * A⁻¹)) * ↑(#(A ^ 2)) ≤ K ^ 2 * ↑(#A) * (K * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
⊢ K ^ 2 * ↑(#A) * (K * ↑(#A)) = ↑(#A) * (K ^ 3 * ↑(#A))

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Additive.SmallTripling

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A * A⁻¹ * A)) ≤ K ^ 3 * ↑(#A)

Tactic:
by

Goal state after: