mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Group.Pointwise.Set.ListOfFn

Goal state before:
pp: case succ
α : Type u_1
inst✝ : Monoid α
n✝ n : ℕ
ih : ∀ {a : α} {s : Fin n → Set α}, a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a
a : α
s : Fin (n + 1) → Set α
⊢ a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a

Tactic:
[

Goal state after:
case succ
α : Type u_1
inst✝ : Monoid α
n✝ n : ℕ
ih : ∀ {a : α} {s : Fin n → Set α}, a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a
a : α
s : Fin (n + 1) → Set α
⊢ a ∈ (List.ofFn s).prod ↔ ∃ f, (List.ofFn fun i ↦ ↑(f i)).prod = a
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
p : γ → Prop
⊢ (∀ z ∈ image₂ f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
p : γ → Prop
⊢ (∀ z ∈ image₂ f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y)
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
p : γ → Prop
⊢ (∃ z ∈ image₂ f s t, p z) ↔ ∃ x ∈ s, ∃ y ∈ t, p (f x y)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
p : γ → Prop
⊢ (∃ z ∈ image₂ f s t, p z) ↔ ∃ x ∈ s, ∃ y ∈ t, p (f x y)
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
u : Finset γ
⊢ image₂ f s t ⊆ u ↔ ∀ a ∈ s, image (fun b ↦ f a b) t ⊆ u

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
u : Finset γ
⊢ image₂ f s t ⊆ u ↔ ∀ a ∈ s, image (fun b ↦ f a b) t ⊆ u
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
u : Finset γ
⊢ image₂ f s t ⊆ u ↔ ∀ b ∈ t, image (fun a ↦ f a b) s ⊆ u

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
u : Finset γ
⊢ image₂ f s t ⊆ u ↔ ∀ b ∈ t, image (fun a ↦ f a b) s ⊆ u
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝¹ : DecidableEq γ
f : α → β → γ
a : α
inst✝ : DecidableEq β
t₁ t₂ : Finset β
hf : Injective (f a)
⊢ image₂ f {a} (t₁ ∩ t₂) = image₂ f {a} t₁ ∩ image₂ f {a} t₂

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝¹ : DecidableEq γ
f : α → β → γ
a : α
inst✝ : DecidableEq β
t₁ t₂ : Finset β
hf : Injective (f a)
⊢ image₂ f {a} (t₁ ∩ t₂) = image₂ f {a} t₁ ∩ image₂ f {a} t₂
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝¹ : DecidableEq γ
f : α → β → γ
b : β
inst✝ : DecidableEq α
s₁ s₂ : Finset α
hf : Injective fun a ↦ f a b
⊢ image₂ f (s₁ ∩ s₂) {b} = image₂ f s₁ {b} ∩ image₂ f s₂ {b}

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝¹ : DecidableEq γ
f : α → β → γ
b : β
inst✝ : DecidableEq α
s₁ s₂ : Finset α
hf : Injective fun a ↦ f a b
⊢ image₂ f (s₁ ∩ s₂) {b} = image₂ f s₁ {b} ∩ image₂ f s₂ {b}
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
⊢ image₂ (fun a b ↦ f b a) t s = image₂ g t s

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
g : β → α → γ
h_comm : ∀ (a : α) (b : β), f a b = g b a
⊢ image₂ (fun a b ↦ f b a) t s = image₂ g t s
Module: Mathlib.Algebra.MonoidAlgebra.MapDomain

Goal state before:
pp: R : Type u_3
S : Type u_4
M : Type u_6
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+ S
⊢ Set.range (map f) = ⇑coeffEquiv ⁻¹' Set.range (mapRange ⇑f ⋯ ∘ ⇑coeffEquiv)

Tactic:
[

Goal state after:
R : Type u_3
S : Type u_4
M : Type u_6
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+ S
⊢ Set.range (map f) = ⇑coeffEquiv ⁻¹' Set.range (mapRange ⇑f ⋯ ∘ ⇑coeffEquiv)
Module: Mathlib.Data.Finset.NAry

Goal state before:
pp: case neg
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s✝ : Finset α
t : Finset β
a : α
s : Finset α
a✝ : a ∉ s
ih : #t ∣ #(image₂ f s t)
hf : ∀ a_1 ∈ insert a s, Injective (f a_1)
hs : ((fun a ↦ image (f a) t) '' ↑(insert a s)).PairwiseDisjoint id
h : ¬Disjoint (image (f a) t) (image₂ f s t)
⊢ #t ∣ #(image (fun b ↦ f a b) t ∪ image₂ f s t)

Tactic:
[

Goal state after:
case neg
α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
s✝ : Finset α
t : Finset β
a : α
s : Finset α
a✝ : a ∉ s
ih : #t ∣ #(image₂ f s t)
hf : ∀ a_1 ∈ insert a s, Injective (f a_1)
hs : ((fun a ↦ image (f a) t) '' ↑(insert a s)).PairwiseDisjoint id
h : ¬Disjoint (image (f a) t) (image₂ f s t)
⊢ #t ∣ #(image (fun b ↦ f a b) t ∪ image₂ f s t)
Module: Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ (f * g).coeff (a0 * b0) = f.coeff a0 * g.coeff b0

Tactic:
[

Goal state after:
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ (f * g).coeff (a0 * b0) = f.coeff a0 * g.coeff b0
Module: Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors

Goal state before:
pp: case refine_1
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ ∀ b ∈ f.coeff.support ×ˢ g.coeff.support,
    b ≠ (a0, b0) → (if b.1 * b.2 = a0 * b0 then f.coeff b.1 * g.coeff b.2 else 0) = 0

Tactic:
[

Goal state after:
case refine_1
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ ∀ b ∈ f.coeff.support ×ˢ g.coeff.support,
    b ≠ (a0, b0) → (if b.1 * b.2 = a0 * b0 then f.coeff b.1 * g.coeff b.2 else 0) = 0
Module: Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors

Goal state before:
pp: case refine_2
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ (a0, b0) ∉ f.coeff.support ×ˢ g.coeff.support → (if True then f.coeff a0 * g.coeff b0 else 0) = 0

Tactic:
[

Goal state after:
case refine_2
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : R[A]
a0 b0 : A
h : UniqueMul f.coeff.support g.coeff.support a0 b0
⊢ (a0, b0) ∉ f.coeff.support ×ˢ g.coeff.support → (if True then f.coeff a0 * g.coeff b0 else 0) = 0
Module: Mathlib.Algebra.MonoidAlgebra.Defs

Goal state before:
pp: R✝ : Type u_1
S : Type u_2
G : Type u_3
M✝ : Type u_4
N : Type u_5
O : Type u_6
ι : Type u_7
inst✝³ : Semiring R✝
x y : R✝[M✝]
r r₁ r₂ : R✝
m m' m₁ m₂ m₁' m₂' : M✝
inst✝² : Mul M✝
R : Type u_8
M : Type u_9
inst✝¹ : Semiring R
inst✝ : Mul M
a b : M
⊢ single (a * b) 1 = single a 1 * single b 1

Tactic:
ext; simp [mul_def, Finsupp.sum_single_index]

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Defs

Goal state before:
pp: R : Type u_1
S : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
ι : Type u_7
inst✝² : Semiring R
x y : R[M]
r r₁ r₂ : R
m m' m₁ m₂ m₁' m₂' : M
inst✝¹ : Monoid M
inst✝ : Monoid N
⊢ ∀ (a : R), IsUnit (singleOneRingHom a) → IsUnit a

Tactic:
[

Goal state after:
R : Type u_1
S : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
ι : Type u_7
inst✝² : Semiring R
x y : R[M]
r r₁ r₂ : R
m m' m₁ m₂ m₁' m₂' : M
inst✝¹ : Monoid M
inst✝ : Monoid N
⊢ ∀ (a : R), IsUnit (singleOneRingHom a) → IsUnit a
Module: Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors

Goal state before:
pp: case H.inr
R : Type u_1
A : Type u_2
inst✝⁴ : Semiring R
inst✝³ : IsCancelAdd R
inst✝² : IsLeftCancelMulZero R
inst✝¹ : Mul A
inst✝ : UniqueProds A
f g₁ g₂ : R[A]
ih :
  ∀ s ∈ g₁.coeff.support ∪ g₂.coeff.support,
    ∀ {g₁_1 g₂_1 : R[A]},
      (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x) g₂_1 →
        g₁_1.coeff.support ∪ g₂_1.coeff.support = (g₁.coeff.support ∪ g₂.coeff.support).erase s → g₁_1 = g₂_1
h✝ : (g₁.coeff.support ∪ g₂.coeff.support).Nonempty
hf : f.coeff.support.Nonempty
af : A
haf : af ∈ f.coeff.support
ag : A
eq : f * (erase ag g₁ + single ag (g₁.coeff ag)) = f * (erase ag g₂ + single ag (g₂.coeff ag))
hag : ag ∈ g₁.coeff.support ∪ g₂.coeff.support
uniq : UniqueMul f.coeff.support (g₁.coeff.support ∪ g₂.coeff.support) af ag
h : f.coeff af * g₂.coeff ag = f.coeff af * g₁.coeff ag
this : g₂.coeff ag = g₁.coeff ag
⊢ erase ag g₁ + single ag (g₁.coeff ag) = erase ag g₂ + single ag (g₂.coeff ag)

Tactic:
this

Goal state after:
case H.inr
R : Type u_1
A : Type u_2
inst✝⁴ : Semiring R
inst✝³ : IsCancelAdd R
inst✝² : IsLeftCancelMulZero R
inst✝¹ : Mul A
inst✝ : UniqueProds A
f g₁ g₂ : R[A]
ih :
  ∀ s ∈ g₁.coeff.support ∪ g₂.coeff.support,
    ∀ {g₁_1 g₂_1 : R[A]},
      (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x) g₂_1 →
        g₁_1.coeff.support ∪ g₂_1.coeff.support = (g₁.coeff.support ∪ g₂.coeff.support).erase s → g₁_1 = g₂_1
h✝ : (g₁.coeff.support ∪ g₂.coeff.support).Nonempty
hf : f.coeff.support.Nonempty
af : A
haf : af ∈ f.coeff.support
ag : A
eq : f * (erase ag g₁ + single ag (g₁.coeff ag)) = f * (erase ag g₂ + single ag (g₁.coeff ag))
hag : ag ∈ g₁.coeff.support ∪ g₂.coeff.support
uniq : UniqueMul f.coeff.support (g₁.coeff.support ∪ g₂.coeff.support) af ag
h : f.coeff af * g₂.coeff ag = f.coeff af * g₁.coeff ag
this : g₂.coeff ag = g₁.coeff ag
⊢ erase ag g₁ + single ag (g₁.coeff ag) = erase ag g₂ + single ag (g₁.coeff ag)
Module: Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors

Goal state before:
pp: case H.inr
R : Type u_1
A : Type u_2
inst✝⁴ : Semiring R
inst✝³ : IsCancelAdd R
inst✝² : IsLeftCancelMulZero R
inst✝¹ : Mul A
inst✝ : UniqueProds A
f g₁ g₂ : R[A]
ih :
  ∀ s ∈ g₁.coeff.support ∪ g₂.coeff.support,
    ∀ {g₁_1 g₂_1 : R[A]},
      (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x) g₂_1 →
        g₁_1.coeff.support ∪ g₂_1.coeff.support = (g₁.coeff.support ∪ g₂.coeff.support).erase s → g₁_1 = g₂_1
h✝ : (g₁.coeff.support ∪ g₂.coeff.support).Nonempty
hf : f.coeff.support.Nonempty
af : A
haf : af ∈ f.coeff.support
ag : A
eq : f * (erase ag g₁ + single ag (g₁.coeff ag)) = f * (erase ag g₂ + single ag (g₁.coeff ag))
hag : ag ∈ g₁.coeff.support ∪ g₂.coeff.support
uniq : UniqueMul f.coeff.support (g₁.coeff.support ∪ g₂.coeff.support) af ag
h : f.coeff af * g₂.coeff ag = f.coeff af * g₁.coeff ag
this : g₂.coeff ag = g₁.coeff ag
⊢ erase ag g₁ + single ag (g₁.coeff ag) = erase ag g₂ + single ag (g₁.coeff ag)

Tactic:
[

Goal state after:
case H.inr
R : Type u_1
A : Type u_2
inst✝⁴ : Semiring R
inst✝³ : IsCancelAdd R
inst✝² : IsLeftCancelMulZero R
inst✝¹ : Mul A
inst✝ : UniqueProds A
f g₁ g₂ : R[A]
ih :
  ∀ s ∈ g₁.coeff.support ∪ g₂.coeff.support,
    ∀ {g₁_1 g₂_1 : R[A]},
      (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x) g₂_1 →
        g₁_1.coeff.support ∪ g₂_1.coeff.support = (g₁.coeff.support ∪ g₂.coeff.support).erase s → g₁_1 = g₂_1
h✝ : (g₁.coeff.support ∪ g₂.coeff.support).Nonempty
hf : f.coeff.support.Nonempty
af : A
haf : af ∈ f.coeff.support
ag : A
eq : f * (erase ag g₁ + single ag (g₁.coeff ag)) = f * (erase ag g₂ + single ag (g₁.coeff ag))
hag : ag ∈ g₁.coeff.support ∪ g₂.coeff.support
uniq : UniqueMul f.coeff.support (g₁.coeff.support ∪ g₂.coeff.support) af ag
h : f.coeff af * g₂.coeff ag = f.coeff af * g₁.coeff ag
this : g₂.coeff ag = g₁.coeff ag
⊢ erase ag g₁ + single ag (g₁.coeff ag) = erase ag g₂ + single ag (g₁.coeff ag)
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
inst✝¹ : Mul G
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
⊢ UniqueMul A B a0 b0 ↔ #({p ∈ A ×ˢ B | p.1 * p.2 = a0 * b0}) ≤ 1

Tactic:
[

Goal state after:
G : Type u_1
inst✝¹ : Mul G
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
⊢ UniqueMul A B a0 b0 ↔ #({p ∈ A ×ˢ B | p.1 * p.2 = a0 * b0}) ≤ 1
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : ∀ ⦃a b c d : G⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d
h : UniqueMul (image (⇑f) A) (image (⇑f) B) (f a0) (f b0)
a b : G
ha : a ∈ A
hb : b ∈ B
ab : a * b = a0 * b0
⊢ f a * f b = f a0 * f b0

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : ∀ ⦃a b c d : G⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d
h : UniqueMul (image (⇑f) A) (image (⇑f) B) (f a0) (f b0)
a b : G
ha : a ∈ A
hb : b ∈ B
ab : a * b = a0 * b0
⊢ f a * f b = f a0 * f b0
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : Function.Injective ⇑f
h : UniqueMul A B a0 b0
x✝¹ x✝ : H
⊢ x✝¹ ∈ image (⇑f) A → x✝ ∈ image (⇑f) B → x✝¹ * x✝ = f a0 * f b0 → x✝¹ = f a0 ∧ x✝ = f b0

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : Function.Injective ⇑f
h : UniqueMul A B a0 b0
x✝¹ x✝ : H
⊢ x✝¹ ∈ image (⇑f) A → x✝ ∈ image (⇑f) B → x✝¹ * x✝ = f a0 * f b0 → x✝¹ = f a0 ∧ x✝ = f b0
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : Function.Injective ⇑f
h : UniqueMul A B a0 b0
a : G
aA : a ∈ A
b : G
bB : b ∈ B
ab : f a * f b = f a0 * f b0
⊢ f a = f a0 ∧ f b = f b0

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝² : Mul G
inst✝¹ : Mul H
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq H
f : G →ₙ* H
hf : Function.Injective ⇑f
h : UniqueMul A B a0 b0
a : G
aA : a ∈ A
b : G
bB : b ∈ B
ab : f a * f b = f a0 * f b0
⊢ f a = f a0 ∧ f b = f b0
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y
⊢ UniqueMul (map f A) (map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0

Tactic:
[

Goal state after:
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y
⊢ UniqueMul (map f A) (map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y
⊢ UniqueMul (map f A) (map f B) (f a0) (f b0) ↔
    UniqueMul (image (⇑{ toFun := ⇑f, map_mul' := mul }) A) (image (⇑{ toFun := ⇑f, map_mul' := mul }) B)
      ({ toFun := ⇑f, map_mul' := mul } a0) ({ toFun := ⇑f, map_mul' := mul } b0)

Tactic:
Finset.map_eq_image

Goal state after:
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y
⊢ UniqueMul (image (⇑f) A) (image (⇑f) B) (f a0) (f b0) ↔
    UniqueMul (image (⇑{ toFun := ⇑f, map_mul' := mul }) A) (image (⇑{ toFun := ⇑f, map_mul' := mul }) B)
      ({ toFun := ⇑f, map_mul' := mul } a0) ({ toFun := ⇑f, map_mul' := mul } b0)
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
inst✝ : Mul G
A B : Finset G
a0 b0 : G
h : UniqueMul A B a0 b0
⊢ UniqueMul (map { toFun := op, inj' := ⋯ } (map { toFun := op, inj' := ⋯ } A))
    (map { toFun := op, inj' := ⋯ } (map { toFun := op, inj' := ⋯ } B)) (op (op a0)) (op (op b0))

Tactic:
[

Goal state after:
G : Type u_1
inst✝ : Mul G
A B : Finset G
a0 b0 : G
h : UniqueMul A B a0 b0
⊢ UniqueMul (map { toFun := op, inj' := ⋯ } (map { toFun := op, inj' := ⋯ } A))
    (map { toFun := op, inj' := ⋯ } (map { toFun := op, inj' := ⋯ } B)) (op (op a0)) (op (op b0))
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: case refl
G : Type u
inst✝¹ : Mul G
inst✝ : UniqueProds G
a b1 b2 : G
he : (fun x ↦ a * x) b1 = (fun x ↦ a * x) b2
this : b1 ∈ {b1, b2}
b : G
hb : b ∈ {b1, b2}
ha : a ∈ {a}
hu : UniqueMul {a} {b1, b2} a b
⊢ b1 = b2

Tactic:
[

Goal state after:
case refl
G : Type u
inst✝¹ : Mul G
inst✝ : UniqueProds G
a b1 b2 : G
he : (fun x ↦ a * x) b1 = (fun x ↦ a * x) b2
this : b1 ∈ {b1, b2}
b : G
hb : b ∈ {b1, b2}
ha : a ∈ {a}
hu : UniqueMul {a} {b1, b2} a b
⊢ b1 = b2
Module: Mathlib.Algebra.Group.Pointwise.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DivisionMonoid α
s t : Finset α
⊢ s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1

Tactic:
[

Goal state after:
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DivisionMonoid α
s t : Finset α
⊢ s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1
Module: Mathlib.Algebra.Group.Pointwise.Finset.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DivisionMonoid α
s : Finset α
⊢ IsUnit ↑s ↔ IsUnit s

Tactic:
[

Goal state after:
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DivisionMonoid α
s : Finset α
⊢ IsUnit ↑s ↔ IsUnit s
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : 1 < #A * #B
⊢ ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2

Tactic:
[

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : 1 < #A * #B
⊢ ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
⊢ 1 < #C ∨ 1 < #D

Tactic:
[

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
⊢ 1 < #C ∨ 1 < #D
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
hcard : 1 < #C ∨ 1 < #D
hC : 1 ∈ C
hD : 1 ∈ D
this : ∃ c ∈ C, ∃ d ∈ D, (c ≠ 1 ∨ d ≠ 1) ∧ UniqueMul C D c d
⊢ ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2

Tactic:
[

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
hcard : 1 < #C ∨ 1 < #D
hC : 1 ∈ C
hD : 1 ∈ D
this : ∃ c ∈ C, ∃ d ∈ D, (c ≠ 1 ∨ d ≠ 1) ∧ UniqueMul C D c d
⊢ ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
Module: Mathlib.Algebra.Group.UniqueProds.Basic

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc✝ : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
hcard : 1 < #C ∨ 1 < #D
hC : 1 ∈ C
hD : 1 ∈ D
a0 : G
ha0 : a0 ∈ A
hc : { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0 ∈ C
b0 : G
hb0 : b0 ∈ B
hd : { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0 ∈ D
hne : { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0 ≠ 1 ∨ { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0 ≠ 1
hu' : UniqueMul C D ({ toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0) ({ toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0)
⊢ (a0, b0) ≠ (a, b)

Tactic:
[

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc✝ : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu : UniqueMul A B a b
C : Finset G := map { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } A
D : Finset G := map { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } B
hcard : 1 < #C ∨ 1 < #D
hC : 1 ∈ C
hD : 1 ∈ D
a0 : G
ha0 : a0 ∈ A
hc : { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0 ∈ C
b0 : G
hb0 : b0 ∈ B
hd : { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0 ∈ D
hne : { toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0 ≠ 1 ∨ { toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0 ≠ 1
hu' : UniqueMul C D ({ toFun := fun x ↦ a⁻¹ * x, inj' := ⋯ } a0) ({ toFun := fun x ↦ x * b⁻¹, inj' := ⋯ } b0)
⊢ (a0, b0) ≠ (a, b)
Module: Mathlib.Algebra.Polynomial.Basic

Goal state before:
pp: R : Type u
a b : R
m✝ n✝ : ℕ
inst✝³ : Semiring R
p q : R[X]
S₁ : Type u_1
S₂ : Type u_2
inst✝² : SMulZeroClass S₁ R
inst✝¹ : SMulZeroClass S₂ R
inst✝ : SMulCommClass S₁ S₂ R
m : S₁
n : S₂
f : R[ℕ]
⊢ m • n • { toFinsupp := f } = n • m • { toFinsupp := f }

Tactic:
[

Goal state after:
R : Type u
a b : R
m✝ n✝ : ℕ
inst✝³ : Semiring R
p q : R[X]
S₁ : Type u_1
S₂ : Type u_2
inst✝² : SMulZeroClass S₁ R
inst✝¹ : SMulZeroClass S₂ R
inst✝ : SMulCommClass S₁ S₂ R
m : S₁
n : S₂
f : R[ℕ]
⊢ m • n • { toFinsupp := f } = n • m • { toFinsupp := f }
Module: Mathlib.Algebra.Polynomial.Basic

Goal state before:
pp: R : Type u
a b : R
m n : ℕ
inst✝⁴ : Semiring R
p q : R[X]
S₁ : Type u_1
S₂ : Type u_2
inst✝³ : SMul S₁ S₂
inst✝² : SMulZeroClass S₁ R
inst✝¹ : SMulZeroClass S₂ R
inst✝ : IsScalarTower S₁ S₂ R
x✝ : S₁
y✝ : S₂
toFinsupp✝ : R[ℕ]
⊢ (x✝ • y✝) • { toFinsupp := toFinsupp✝ } = x✝ • y✝ • { toFinsupp := toFinsupp✝ }

Tactic:
[

Goal state after:
R : Type u
a b : R
m n : ℕ
inst✝⁴ : Semiring R
p q : R[X]
S₁ : Type u_1
S₂ : Type u_2
inst✝³ : SMul S₁ S₂
inst✝² : SMulZeroClass S₁ R
inst✝¹ : SMulZeroClass S₂ R
inst✝ : IsScalarTower S₁ S₂ R
x✝ : S₁
y✝ : S₂
toFinsupp✝ : R[ℕ]
⊢ (x✝ • y✝) • { toFinsupp := toFinsupp✝ } = x✝ • y✝ • { toFinsupp := toFinsupp✝ }
Module: Mathlib.Order.Sublattice

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Lattice α
inst✝ : Lattice β
L : Sublattice α
M : Sublattice β
e : β ≃o α
⊢ map { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } L = M ↔ L = map { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } M

Tactic:
[

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : Lattice α
inst✝ : Lattice β
L : Sublattice α
M : Sublattice β
e : β ≃o α
⊢ map { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } L = M ↔ L = map { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } M
Module: Mathlib.Algebra.Polynomial.Basic

Goal state before:
pp: R : Type u
a b : R
m n : ℕ
inst✝³ : Semiring R
p q : R[X]
α : Type u_1
K : Type u_2
inst✝² : Semiring K
inst✝¹ : DistribSMul α K
inst✝ : IsScalarTower α K K
x✝ : α
toFinsupp✝¹ toFinsupp✝ : K[ℕ]
⊢ (x✝ • { toFinsupp := toFinsupp✝¹ }) • { toFinsupp := toFinsupp✝ } =
    x✝ • { toFinsupp := toFinsupp✝¹ } • { toFinsupp := toFinsupp✝ }

Tactic:
[

Goal state after:
R : Type u
a b : R
m n : ℕ
inst✝³ : Semiring R
p q : R[X]
α : Type u_1
K : Type u_2
inst✝² : Semiring K
inst✝¹ : DistribSMul α K
inst✝ : IsScalarTower α K K
x✝ : α
toFinsupp✝¹ toFinsupp✝ : K[ℕ]
⊢ (x✝ • { toFinsupp := toFinsupp✝¹ }) • { toFinsupp := toFinsupp✝ } =
    x✝ • { toFinsupp := toFinsupp✝¹ } • { toFinsupp := toFinsupp✝ }
Module: Mathlib.Algebra.Polynomial.Basic

Goal state before:
pp: R : Type u
a b : R
m n : ℕ
inst✝³ : Semiring R
p q : R[X]
S : Type u_1
inst✝² : SMulZeroClass S R
inst✝¹ : SMulZeroClass Sᵐᵒᵖ R
inst✝ : IsCentralScalar S R
m✝ : S
toFinsupp✝ : R[ℕ]
⊢ MulOpposite.op m✝ • { toFinsupp := toFinsupp✝ } = m✝ • { toFinsupp := toFinsupp✝ }

Tactic:
[

Goal state after:
R : Type u
a b : R
m n : ℕ
inst✝³ : Semiring R
p q : R[X]
S : Type u_1
inst✝² : SMulZeroClass S R
inst✝¹ : SMulZeroClass Sᵐᵒᵖ R
inst✝ : IsCentralScalar S R
m✝ : S
toFinsupp✝ : R[ℕ]
⊢ MulOpposite.op m✝ • { toFinsupp := toFinsupp✝ } = m✝ • { toFinsupp := toFinsupp✝ }