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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
F₁ : Finset G
hF₁ : ↑(#F₁) ≤ K
hAF₁ : A ^ 2 ⊆ ↑F₁ • A
F₂ : Finset G
hF₂ : ↑(#F₂) ≤ L
hBF₂ : B ^ 2 ⊆ ↑F₂ • B
this : 1 ≤ K
f : G → G → G
hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))
⊢ A ^ m ∩ B ^ n ⊆ ↑F₁ ^ (m - 1) * A ∩ (↑F₂ ^ (n - 1) * B)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
F₁ : Finset G
hF₁ : ↑(#F₁) ≤ K
hAF₁ : A ^ 2 ⊆ ↑F₁ • A
F₂ : Finset G
hF₂ : ↑(#F₂) ≤ L
hBF₂ : B ^ 2 ⊆ ↑F₂ • B
this : 1 ≤ K
f : G → G → G
hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))
⊢ ↑F₁ ^ (m - 1) * A ∩ (↑F₂ ^ (n - 1) * B) = ⋃ a ∈ F₁ ^ (m - 1), ⋃ b ∈ F₂ ^ (n - 1), a • A ∩ b • B

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B C : Finset G
⊢ #(A * C) * #B ≤ #(A * B) * #(B * C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
F₁ : Finset G
hF₁ : ↑(#F₁) ≤ K
hAF₁ : A ^ 2 ⊆ ↑F₁ • A
F₂ : Finset G
hF₂ : ↑(#F₂) ≤ L
hBF₂ : B ^ 2 ⊆ ↑F₂ • B
this : 1 ≤ K
f : G → G → G
hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))
⊢ ⋃ a ∈ F₁ ^ (m - 1), ⋃ b ∈ F₂ ^ (n - 1), a • A ∩ b • B ⊆
    ⋃ a ∈ F₁ ^ (m - 1), ⋃ b ∈ F₂ ^ (n - 1), f a b • (A⁻¹ * A ∩ (B⁻¹ * B))

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B C : Finset G
⊢ #(A * C) * #B ≤ #(A / B) * #(B / C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B C : Finset G
⊢ #(A / C) * #B ≤ #(A * B) * #(B / C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B C : Finset G
⊢ #(A / C) * #B ≤ #(A / B) * #(B * C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B : Finset G
hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A
hA : A.Nonempty
n : ℕ
ih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)
⊢ 0 < ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B : Finset G
hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A
hA : A.Nonempty
n : ℕ
ih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)
⊢ ↑(#(A * B ^ (n + 1))) * ↑(#A) = ↑(#(B ^ n * A * B)) * ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
F₁ : Finset G
hF₁ : ↑(#F₁) ≤ K
hAF₁ : A ^ 2 ⊆ ↑F₁ • A
F₂ : Finset G
hF₂ : ↑(#F₂) ≤ L
hBF₂ : B ^ 2 ⊆ ↑F₂ • B
this : 1 ≤ K
f : G → G → G
hf : ∀ (a b : G), a • A ∩ b • B ⊆ f a b • (A⁻¹ * A ∩ (B⁻¹ * B))
⊢ ⋃ a ∈ F₁ ^ (m - 1), ⋃ b ∈ F₂ ^ (n - 1), f a b • (A⁻¹ * A ∩ (B⁻¹ * B)) =
    ↑(Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) * (A ^ 2 ∩ B ^ 2)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B : Finset G
hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A
hA : A.Nonempty
n : ℕ
ih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)
⊢ ↑(#(A * B)) * ↑(#(B ^ n * A)) ≤ ↑(#(A * B)) * ((↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A))

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B : Finset G
hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A
hA : A.Nonempty
n : ℕ
ih : ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)
⊢ ↑(#(A * B)) * ((↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)) = (↑(#(A * B)) / ↑(#A)) ^ (n + 1) * ↑(#A) * ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (A ^ 2 ∩ B ^ 2)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A B : Finset G
hAB : ∀ A' ⊆ A, #(A * B) * #A' ≤ #(A' * B) * #A
n : ℕ
⊢ ↑(#(A * B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ (A ^ m ∩ B ^ n)⁻¹ = A ^ m ∩ B ^ n

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ 2 ≤ 2 * m

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ 2 ≤ 2 * n

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ A ^ 2 ∩ B ^ 2 ⊆ A ^ m ∩ B ^ n

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ (A ^ m) ^ 2 ∩ (B ^ n) ^ 2 = A ^ (2 * m) ∩ B ^ (2 * n)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ CovBySMul G (K ^ (2 * m - 1) * L ^ (2 * n - 1)) ((A ^ m ∩ B ^ n) ^ 2) (A ^ m ∩ B ^ n)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
this : A * A ⊆ A
x✝ : G
hx : x✝ ∈ A
⊢ x✝⁻¹ ∈ A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ 0 < ↑(#C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(B ^ m * C)) * ↑(#(B ^ n * C)) = ↑(#(C * B ^ m)) * ↑(#(C * B ^ n))

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
⊢ ∃ x, A * A ⊆ x • A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
⊢ x⁻¹ • (A * A) ⊆ A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
⊢ x⁻¹ ∈ A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
hx_inv : x⁻¹ ∈ A
⊢ x * x ∈ A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ ↑(#(C * B ^ m)) * ↑(#(C * B ^ n)) ≤ (↑(#(C * B)) / ↑(#C)) ^ m * ↑(#C) * ((↑(#(C * B)) / ↑(#C)) ^ n * ↑(#C))

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
hx_inv : x⁻¹ ∈ A
hx_sq : x * x ∈ A
⊢ A * A ⊆ x • A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
x : G
hx : A * A ⊆ x • A
hx' : x⁻¹ • (A * A) ⊆ A
hx_inv : x⁻¹ ∈ A
hx_sq : x * x ∈ A
⊢ x • A = x⁻¹ • (x * x) • A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ (↑(#(C * B)) / ↑(#C)) ^ m * ↑(#C) * ((↑(#(C * B)) / ↑(#C)) ^ n * ↑(#C)) =
    (↑(#(C * B)) / ↑(#C)) ^ (m + n) * ↑(#C) * ↑(#C)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
hA : IsApproximateSubgroup 1 A
⊢ ∃ H, ↑H = A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝ : Group G
A : Set G
⊢ (∃ H, ↑H = A) → IsApproximateSubgroup 1 A

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
hA' : A ∈ A.powerset.erase ∅
C : Finset G
hCmin : ∀ x' ∈ A.powerset.erase ∅, ↑(#(C * B)) / ↑(#C) ≤ ↑(#(x' * B)) / ↑(#x')
hC : C.Nonempty
hCA : C ⊆ A
⊢ (↑(#(C * B)) / ↑(#C)) ^ (m + n) * ↑(#C) * ↑(#C) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A) * ↑(#C)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ (mulDysonETransform e x).1 * (mulDysonETransform e x).2 ⊆ x.1 * x.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ #(mulDysonETransform e x).1 + #(mulDysonETransform e x).2 = #x.1 + #x.2

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
m n : ℕ
⊢ ↑(#(B ^ m / B ^ n)) ≤ (↑(#(A / B)) / ↑(#A)) ^ (m + n) * ↑(#A)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ mulDysonETransform e (mulDysonETransform e x) = mulDysonETransform e x

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
n : ℕ
⊢ ↑(#(B ^ n)) ≤ (↑(#(A * B)) / ↑(#A)) ^ n * ↑(#A)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : CommGroup G
A : Finset G
hA : A.Nonempty
B : Finset G
n : ℕ
⊢ ↑(#(B ^ n)) ≤ (↑(#(A / B)) / ↑(#A)) ^ n * ↑(#A)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ e • (mulDysonETransform e x).2 ⊆ (mulDysonETransform e x).1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
x : Finset α × Finset α
⊢ mulETransformLeft 1 x = x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
x : Finset α × Finset α
⊢ mulETransformRight 1 x = x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
⊢ (mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
⊢ (mulETransformRight e x).1 * (mulETransformRight e x).2 ⊆ x.1 * x.2

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: α : Type u_2
inst✝ : Monoid α
n : ℕ∞
⊢ n ≤ minOrder α ↔ ∀ ⦃a : α⦄, a ≠ 1 → IsOfFinOrder a → n ≤ ↑(orderOf a)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: G : Type u_1
inst✝ : Group G
n : ℕ∞
⊢ n ≤ minOrder G ↔ ∀ ⦃s : Subgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsMulTorsionFree G
⊢ minOrder G = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: G : Type u_1
inst✝ : CommGroup G
⊢ minOrder G = ⊤ ↔ IsMulTorsionFree G

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
⊢ #x.1 + #(op e • x.1) = 2 * #x.1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
⊢ #x.2 + #(e⁻¹ • x.2) = 2 * #x.2

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: n : ℕ
hn : n ≠ 0
hn₁ : n ≠ 1
this : Fact (1 < n)
⊢ ↑(n / n.minFac) ≠ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α
⊢ #(mulETransformLeft e x).1 + #(mulETransformLeft e x).2 +
      (#(mulETransformRight e x).1 + #(mulETransformRight e x).2) =
    #x.1 + #x.2 + (#x.1 + #x.2)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ mulETransformLeft e⁻¹ x = (mulETransformRight e x.swap).swap

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : CommGroup α
e : α
x : Finset α × Finset α
⊢ mulETransformRight e⁻¹ x = (mulETransformLeft e x.swap).swap

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
⊢ DevosMulRel x y ↔
    #(x.1 * x.2) < #(y.1 * y.2) ∨
      #(x.1 * x.2) = #(y.1 * y.2) ∧ #y.1 + #y.2 < #x.1 + #x.2 ∨
        #(x.1 * x.2) = #(y.1 * y.2) ∧ #x.1 + #x.2 = #y.1 + #y.2 ∧ #x.1 < #y.1

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
⊢ {x | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn DevosMulRel

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hts : #t < #s
⊢ #((t⁻¹, s⁻¹).1 * (t⁻¹, s⁻¹).2) = #((s, t).1 * (s, t).2) ∧
    #(t⁻¹, s⁻¹).1 + #(t⁻¹, s⁻¹).2 = #(s, t).1 + #(s, t).2 ∧ #(t⁻¹, s⁻¹).1 < #(s, t).1

Tactic:
by

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

Goal state before:
pp: case inl
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hts : #t < #s
⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #t ≤ #(s * t) + 1

pp: case inr
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #t ≤ #(s * t) + 1

Tactic:
· simpa only [← mul_inv_rev, add_comm, card_inv] using
      ih _ _ ht.inv hs.inv
        (devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
          simpa only [← mul_inv_rev, add_comm, card_inv, true_and])

Goal state after:
case inr
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
⊢ minOrder α ≤ ↑(#(s * t)) ∨ #s + #t ≤ #(s * t) + 1
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: n : ℕ
hn : n ≠ 0
hn₁ : n ≠ 1
⊢ minOrder (ZMod n) = ↑n.minFac

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Order.Min

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
⊢ minOrder (ZMod p) = ↑p

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
⊢ op (b⁻¹ * a) • b = a

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
  ∀ (a b : Finset α),
    a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : (s ∩ op g • s).Nonempty
hsg : op g • s = s
⊢ ↑(zpowers g) ⊆ a⁻¹ • ↑s

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
⊢ m ≠ 0

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
i : Fin (m + 1 + 1)
h : ε i = 0
⊢ False

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
⊢ ∀ (i : Fin 3), |1 i| = 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
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
⊢ 0 ≤ ↑(#(List.map (fun i ↦ A ^ 1 i) (finRange 3)).prod)

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
⊢ 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
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))
⊢ 0 < ↑(#A)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
⊢ min (minOrder α) ↑(#s + #t - 1) ≤ ↑(#(s * t))

Tactic:
by

Goal state after: