mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanIso 2 (↑A) (↑B) f
s : Finset α
hsA : ↑s ⊆ ↑A
hcard : #s = mulRothNumber A
hs : ThreeGPFree ↑s
hfs : Set.InjOn f ↑s
this : ThreeGPFree ↑(image f s)
⊢ #(image f s) ≤ mulRothNumber B

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanIso 2 (↑A) (↑B) f
s : Finset α
hsA : ↑s ⊆ ↑A
hcard : #s = mulRothNumber A
hs : ThreeGPFree ↑s
hfs : Set.InjOn f ↑s
this : ThreeGPFree ↑(image f s)
⊢ #(image f s) ≤ mulRothNumber B
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanIso 2 (↑A) (↑B) f
s : Finset α
hsA : ↑s ⊆ ↑A
hcard : #s = mulRothNumber A
hs : ThreeGPFree ↑s
hfs : Set.InjOn f ↑s
this : ThreeGPFree ↑(image f s)
⊢ f '' ↑s ⊆ ↑B

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanIso 2 (↑A) (↑B) f
s : Finset α
hsA : ↑s ⊆ ↑A
hcard : #s = mulRothNumber A
hs : ThreeGPFree ↑s
hfs : Set.InjOn f ↑s
this : ThreeGPFree ↑(image f s)
⊢ f '' ↑s ⊆ ↑B
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs₀ : IsClosed s
hs₁ : StrictConvex 𝕜 s
a : E
ha : a ∈ frontier s
b : E
hb : b ∈ frontier s
c : E
hc : c ∈ frontier s
habc : a + c = b + b
⊢ a + c = b + b

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs₀ : IsClosed s
hs₁ : StrictConvex 𝕜 s
a : E
ha : a ∈ frontier s
b : E
hb : b ∈ frontier s
c : E
hc : c ∈ frontier s
habc : a + c = b + b
⊢ a + c = b + b
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x : E
⊢ ThreeAPFree {x}

Tactic:
]

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x : E
⊢ ThreeAPFree {x}
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : ∃ u_1 ⊆ s, u = map (mulLeftEmbedding a) u_1
hcard : #u = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree ↑u
⊢ mulRothNumber (map (mulLeftEmbedding a) s) ≤ mulRothNumber s

Tactic:
]

Goal state after:
case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : ∃ u_1 ⊆ s, u = map (mulLeftEmbedding a) u_1
hcard : #u = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree ↑u
⊢ mulRothNumber (map (mulLeftEmbedding a) s) ≤ mulRothNumber s
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
x : Fin n → ℕ
hx : x ∈ sphere n d k
⊢ √(∑ i, ‖(toLp 2 (Nat.cast ∘ x)).ofLp i‖ ^ 2) = √↑k

Tactic:
]

Goal state after:
n d k : ℕ
x : Fin n → ℕ
hx : x ∈ sphere n d k
⊢ √(∑ i, ‖(toLp 2 (Nat.cast ∘ x)).ofLp i‖ ^ 2) = √↑k
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #(map (mulLeftEmbedding a) u) = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)
⊢ mulRothNumber (map (mulLeftEmbedding a) s) ≤ mulRothNumber s

Tactic:
]

Goal state after:
case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #(map (mulLeftEmbedding a) u) = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)
⊢ mulRothNumber (map (mulLeftEmbedding a) s) ≤ mulRothNumber s
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
x : Fin n → ℕ
hx : x ∈ ↑(sphere n d k)
⊢ √↑k = √↑k

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #(map (mulLeftEmbedding a) u) = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)
⊢ #u ≤ mulRothNumber s

Tactic:
]

Goal state after:
case refine_1
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #(map (mulLeftEmbedding a) u) = mulRothNumber (map (mulLeftEmbedding a) s)
hu : ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)
⊢ #u ≤ mulRothNumber s
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
a : Fin n.succ → ℕ
⊢ a 0 % d = a 0 % d

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ (map g (map f s)).prod = (map g (map f t)).prod

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ (map g (map f s)).prod = (map g (map f t)).prod
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ s.card = n

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ s.card = n
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #u = mulRothNumber s
hu : ThreeGPFree ↑u
⊢ ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)

Tactic:
]

Goal state after:
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #u = mulRothNumber s
hu : ThreeGPFree ↑u
⊢ ThreeGPFree (⇑(mulLeftEmbedding a) '' ↑u)
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ t.card = n

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ t.card = n
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
x₁ x₂ : Fin n.succ → ℕ
hx₁ : ∀ (i : Fin n.succ), x₁ i < d
hx₂ : ∀ (i : Fin n.succ), x₂ i < d
h : x₁ 0 = x₂ 0 ∧ (map d) (x₁ ∘ Fin.succ) = (map d) (x₂ ∘ Fin.succ)
⊢ x₂ 0 + (map d) (x₂ ∘ Fin.succ) * d = x₂ 0 + (map d) (x₂ ∘ Fin.succ) * d

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
x₁ x₂ : Fin n.succ → ℕ
hx₁ : ∀ (i : Fin n.succ), x₁ i < d
hx₂ : ∀ (i : Fin n.succ), x₂ i < d
h : (map d) x₁ = (map d) x₂
⊢ (map d) x₂ % d = (map d) x₂ % d

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ (map g (map f s)).prod = (map g (map f t)).prod ↔ s.prod = t.prod

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ (map g (map f s)).prod = (map g (map f t)).prod ↔ s.prod = t.prod
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case e'_3
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
u : Finset α
hus : u ⊆ s
hcard : #u = mulRothNumber s
hu : ThreeGPFree ↑u
h : ThreeGPFree ↑(map (mulLeftEmbedding a) u)
⊢ mulRothNumber s = mulRothNumber s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ s.card = n

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ s.card = n
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ t.card = n

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ t.card = n
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
x₁ x₂ : Fin n.succ → ℕ
hx₁ : ∀ (i : Fin n.succ), x₁ i < d
hx₂ : ∀ (i : Fin n.succ), x₂ i < d
h : ∑ x, x₁ x.succ * d ^ ↑x = ∑ x, x₂ x.succ * d ^ ↑x ∨ d = 0
this : x₁ 0 = x₂ 0
⊢ x₁ 0 = x₂ 0 ∧ (map d) (x₁ ∘ Fin.succ) = (map d) (x₂ ∘ Fin.succ)

Tactic:
]

Goal state after:
n d : ℕ
x₁ x₂ : Fin n.succ → ℕ
hx₁ : ∀ (i : Fin n.succ), x₁ i < d
hx₂ : ∀ (i : Fin n.succ), x₂ i < d
h : ∑ x, x₁ x.succ * d ^ ↑x = ∑ x, x₂ x.succ * d ^ ↑x ∨ d = 0
this : x₁ 0 = x₂ 0
⊢ x₁ 0 = x₂ 0 ∧ (map d) (x₁ ∘ Fin.succ) = (map d) (x₂ ∘ Fin.succ)
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
⊢ mulRothNumber (map (mulLeftEmbedding a) s) = mulRothNumber s

Tactic:
mulRothNumber_map_mul_left s a

Goal state after:
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
⊢ mulRothNumber s = mulRothNumber s
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
⊢ mulRothNumber s = mulRothNumber s

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ s.prod = t.prod ↔ s.prod = t.prod

pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ ∀ ⦃x : β⦄, x ∈ map f s → x ∈ B

pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ ∀ ⦃x : β⦄, x ∈ map f t → x ∈ B

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ ∀ ⦃x : β⦄, x ∈ map f s → x ∈ B

α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
⊢ ∀ ⦃x : β⦄, x ∈ map f t → x ∈ B
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: M N : ℕ
⊢ addRothNumber (range M ∪ map (addLeftEmbedding M) (range N)) ≤
    addRothNumber (range M) + addRothNumber (map (addLeftEmbedding M) (range N))

Tactic:
]

Goal state after:
M N : ℕ
⊢ addRothNumber (range M ∪ map (addLeftEmbedding M) (range N)) ≤
    addRothNumber (range M) + addRothNumber (map (addLeftEmbedding M) (range N))
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case inl
a b : ℕ
h : b ≤ a
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: case e'_2
a b : ℕ
h : a ≤ b
⊢ Ico a b = image (⇑(addLeftEmbedding a)) (Ico 0 (b - a))

Tactic:
]

Goal state after:
case e'_2
a b : ℕ
h : a ≤ b
⊢ Ico a b = image (⇑(addLeftEmbedding a)) (Ico 0 (b - a))
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
f : (Fin n → ℕ) →+ EuclideanSpace ℝ (Fin n) := { toFun := fun f ↦ toLp 2 (Nat.cast ∘ f), map_zero' := ⋯, map_add' := ⋯ }
x : Fin n → ℕ
⊢ x ∈ ↑(sphere n d k) → ‖f x‖ = √↑k

Tactic:
]

Goal state after:
n d k : ℕ
f : (Fin n → ℕ) →+ EuclideanSpace ℝ (Fin n) := { toFun := fun f ↦ toLp 2 (Nat.cast ∘ f), map_zero' := ⋯, map_add' := ⋯ }
x : Fin n → ℕ
⊢ x ∈ ↑(sphere n d k) → ‖f x‖ = √↑k
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
⊢ ThreeAPFree (⇑(map (2 * d - 1)) '' ↑(sphere n d k))

Tactic:
]

Goal state after:
n d k : ℕ
⊢ ThreeAPFree (⇑(map (2 * d - 1)) '' ↑(sphere n d k))
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
⊢ ∀ x ∈ ↑(sphere n d k), ∀ y ∈ ↑(sphere n d k), x + y ∈ {x | ∀ (i : Fin n), x i < 2 * d - 1}

Tactic:
]

Goal state after:
n d k : ℕ
⊢ ∀ x ∈ ↑(sphere n d k), ∀ y ∈ ↑(sphere n d k), x + y ∈ {x | ∀ (i : Fin n), x i < 2 * d - 1}
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
a : Fin n → ℕ
ha : a ∈ ↑(sphere n d k)
b : Fin n → ℕ
hb : b ∈ ↑(sphere n d k)
i : Fin n
hai : a i < d
hbi : b i < d
⊢ ((a + b) i + 1).succ ≤ d + d

Tactic:
]

Goal state after:
n d k : ℕ
a : Fin n → ℕ
ha : a ∈ ↑(sphere n d k)
b : Fin n → ℕ
hb : b ∈ ↑(sphere n d k)
i : Fin n
hai : a i < d
hbi : b i < d
⊢ ((a + b) i + 1).succ ≤ d + d
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B₁ B₂ : Set β
f₁ f₂ : α → β
n : ℕ
h₁ : IsMulFreimanHom n A B₁ f₁
h₂ : IsMulFreimanHom n A B₂ f₂
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ (map f₁ t).prod * (map f₂ t).prod = (map f₁ t).prod * (map f₂ t).prod

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
x : Fin n → ℕ
hx : ∀ (i : Fin n), x i < d
⊢ ∑ i, x i ^ 2 ≤ n * (d - 1) ^ 2

Tactic:
]

Goal state after:
n d : ℕ
x : Fin n → ℕ
hx : ∀ (i : Fin n), x i < d
⊢ ∑ i, x i ^ 2 ≤ n * (d - 1) ^ 2
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n✝ : ℕ
inst✝¹ : FunLike F α β
inst✝ : MulHomClass F α β
f : F
hfAB : MapsTo (⇑f) A B
n : ℕ
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n + 1
ht : t.card = n + 1
h : s.prod = t.prod
⊢ (map (⇑f) t).prod = (map (⇑f) t).prod

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
x : Fin n → ℕ
hx : ∀ (i : Fin n), x i < d
this : ∀ (i : Fin n), x i ^ 2 ≤ (d - 1) ^ 2
⊢ n * (d - 1) ^ 2 ≤ n * (d - 1) ^ 2

Tactic:
]

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

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n : ℕ
inst✝¹ : EquivLike F α β
inst✝ : MulEquivClass F α β
f : F
hfAB : BijOn (⇑f) A B
s t : Multiset α
x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
x✝¹ : s.card = n
x✝ : t.card = n
⊢ s.prod = t.prod ↔ s.prod = t.prod

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d : ℕ
⊢ 2 * d * ∑ i ∈ range n, (2 * d + 1) ^ i = 2 * d * ∑ i ∈ range n, (2 * d + 1) ^ i

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : DivisionCommMonoid β
A : Set α
B : Set β
f : α → β
n : ℕ
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ (map f t).prod⁻¹ = (map f t).prod⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: case refine_1
n d : ℕ
x : Fin n → ℕ
hx : x ∈ box n d
⊢ ∑ i, x i ^ 2 ≤ n * (d - 1) ^ 2

Tactic:
]

Goal state after:
case refine_1
n d : ℕ
x : Fin n → ℕ
hx : x ∈ box n d
⊢ ∑ i, x i ^ 2 ≤ n * (d - 1) ^ 2
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoid α
A : Set α
n : ℕ
β : Type u_5
inst✝ : DivisionCommMonoid β
B₁ B₂ : Set β
f₁ f₂ : α → β
h₁ : IsMulFreimanHom n A B₁ f₁
h₂ : IsMulFreimanHom n A B₂ f₂
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ (map f₁ t).prod / (map f₂ t).prod = (map f₁ t).prod / (map f₂ t).prod

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: case refine_2
n d : ℕ
⊢ ↑(d ^ n) ≤ ↑(d ^ n)

pp: case refine_2.ha
n d : ℕ
⊢ ↑(n * (d - 1) ^ 2) + 1 ≠ 0

Tactic:
]

Goal state after:
case refine_2.ha
n d : ℕ
⊢ ↑(n * (d - 1) ^ 2) + 1 ≠ 0
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
⊢ (map val s).sum = (map val t).sum ↔
    (Nat.castRingHom (Fin (n + 1))) (map val s).sum = (Nat.castRingHom (Fin (n + 1))) (map val t).sum

Tactic:
]

Goal state after:
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
⊢ (map val s).sum = (map val t).sum ↔
    (Nat.castRingHom (Fin (n + 1))) (map val s).sum = (Nat.castRingHom (Fin (n + 1))) (map val t).sum
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ (map val u).sum ≤ (map ?f u).card * k

pp: case f
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ Fin (n + 1) → ?m.128

pp: k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ Type ?u.92

Tactic:
]

Goal state after:
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ (map val u).sum ≤ (map ?f u).card * k

case f
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ Fin (n + 1) → ?m.128

k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
u : Multiset (Fin (n + 1))
huk : ∀ x ∈ u, x ≤ ↑k
hu : u.card = m
⊢ Type ?u.92