mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
g : ι → MvPolynomial σ R
b : ι
B : Finset ι
hb : b ∉ B
hb0 : ¬g b = 0
isunit_gb : IsUnit (m.leadingCoeff (g b))
hfd : m.toSyn (m.degree (g b + ∑ b ∈ B, g b)) < m.toSyn (m.degree (g b))
h :
  (∀ b_1 ∈ B, m.degree (g b_1) = m.degree (g b) ∧ IsUnit (m.leadingCoeff (g b_1)) ∨ g b_1 = 0) →
    m.toSyn (m.degree (∑ b ∈ B, g b)) < m.toSyn (m.degree (g b)) →
      ∃ c, ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)
hd : ∀ a ∈ B, m.degree (g a) = m.degree (g b) ∧ IsUnit (m.leadingCoeff (g a)) ∨ g a = 0
⊢ ∑ b' ∈ B, (g b' - (m.leadingCoeff (g b') * ↑isunit_gb.unit⁻¹) • g b) =
    ∑ x ∈ B, ↑isunit_gb.unit⁻¹ • m.sPolynomial (g x) (g b)

Tactic:
·

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
g : ι → MvPolynomial σ R
b : ι
B : Finset ι
hb : b ∉ B
hb0 : ¬g b = 0
isunit_gb : IsUnit (m.leadingCoeff (g b))
hfd : m.toSyn (m.degree (g b + ∑ b ∈ B, g b)) < m.toSyn (m.degree (g b))
h :
  (∀ b_1 ∈ B, m.degree (g b_1) = m.degree (g b) ∧ IsUnit (m.leadingCoeff (g b_1)) ∨ g b_1 = 0) →
    m.toSyn (m.degree (∑ b ∈ B, g b)) < m.toSyn (m.degree (g b)) →
      ∃ c, ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)
hd : ∀ a ∈ B, m.degree (g a) = m.degree (g b) ∧ IsUnit (m.leadingCoeff (g a)) ∨ g a = 0
⊢ ∑ b' ∈ B, (g b' - (m.leadingCoeff (g b') * ↑isunit_gb.unit⁻¹) • g b) =
    ∑ x ∈ B, ↑isunit_gb.unit⁻¹ • m.sPolynomial (g x) (g b)
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case i_surj
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
⊢ Set.SurjOn
    (fun x ↦
      match x with
      | ((i, j), k, l) => (i * k, j * l))
    ↑(m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal) ↑(m * n).divisorsAntidiagonal

pp: case h
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
⊢ ∀ a ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal,
    f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =
      f
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).1 *
        g
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).2

Tactic:
·

Goal state after:
case i_surj
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
⊢ Set.SurjOn
    (fun x ↦
      match x with
      | ((i, j), k, l) => (i * k, j * l))
    ↑(m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal) ↑(m * n).divisorsAntidiagonal
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
b1 b2 : ℕ
h : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0
⊢ ((m = m ∧ m ≠ 0) ∧ n = n ∧ n ≠ 0) ∧ (b1.gcd ((b1, b2).1 * (b1, b2).2), b2.gcd ((b1, b2).1 * (b1, b2).2)) = (b1, b2)

pp: R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
b1 b2 : ℕ
h : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0
⊢ b1 * b2 = n * m

Tactic:
·

Goal state after:
case h
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
b1 b2 : ℕ
h : (b1, b2).1 * (b1, b2).2 = m * n ∧ m * n ≠ 0
⊢ ((m = m ∧ m ≠ 0) ∧ n = n ∧ n ≠ 0) ∧ (b1.gcd ((b1, b2).1 * (b1, b2).2), b2.gcd ((b1, b2).1 * (b1, b2).2)) = (b1, b2)
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case pos
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : σ x = a
hx : ¬σ a = x
⊢ x ∈ ↑s

pp: case pos
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : σ x = σ a
hx : ¬a = x
⊢ x ∈ ↑s

pp: case neg
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : ¬σ x = σ a
hx : ¬σ x = x
⊢ x ∈ ↑s

Tactic:
·

Goal state after:
case pos
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : σ x = a
hx : ¬σ a = x
⊢ x ∈ ↑s
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case pos.inl
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
s : Finset ι
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
x : ι
has : x ∉ s
hamax : ∀ x_1 ∈ s, toLex (g x_1, f x_1) ≤ toLex (g x, f x)
hfg : MonovaryOn f g ↑(insert x s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert x s)
τ : Perm ι := Equiv.trans σ (Equiv.swap x (σ x))
hτ : τ = Equiv.trans σ (Equiv.swap x (σ x))
h₁ : σ x = x
hx : ¬σ x = x
⊢ x ∈ ↑s

pp: case pos.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : σ x = a
hx : ¬σ a = x
hax : x ≠ a
⊢ x ∈ ↑s

Tactic:
·

Goal state after:
case pos.inl
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
s : Finset ι
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
x : ι
has : x ∉ s
hamax : ∀ x_1 ∈ s, toLex (g x_1, f x_1) ≤ toLex (g x, f x)
hfg : MonovaryOn f g ↑(insert x s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert x s)
τ : Perm ι := Equiv.trans σ (Equiv.swap x (σ x))
hτ : τ = Equiv.trans σ (Equiv.swap x (σ x))
h₁ : σ x = x
hx : ¬σ x = x
⊢ x ∈ ↑s
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case pos.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : σ x = a
hx : ¬σ a = x
hax : x ≠ a
⊢ x ∈ ↑s

Tactic:
·

Goal state after:
case pos.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : σ x = a
hx : ¬σ a = x
hax : x ≠ a
⊢ x ∈ ↑s
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
⊢ ∀ a ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal,
    f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =
      f
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).1 *
        g
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).2

Tactic:
·

Goal state after:
case h
R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
m n : ℕ
cop : m.Coprime n
⊢ ∀ a ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal,
    f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) =
      f
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).1 *
        g
          (match a with
            | ((i, j), k, l) => (i * k, j * l)).2
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case pos
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : σ x = σ a
hx : ¬a = x
⊢ x ∈ ↑s

pp: case neg
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : ¬σ x = σ a
hx : ¬σ x = x
⊢ x ∈ ↑s

Tactic:
·

Goal state after:
case pos
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : σ x = σ a
hx : ¬a = x
⊢ x ∈ ↑s
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case neg
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : ¬σ x = σ a
hx : ¬σ x = x
⊢ x ∈ ↑s

Tactic:
·

Goal state after:
case neg
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind : ∀ {σ : Perm ι}, MonovaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
x : ι
h₁ : ¬σ x = a
h₂ : ¬σ x = σ a
hx : ¬σ x = x
⊢ x ∈ ↑s
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case inl
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
n : ℕ
h : 0 ≠ 0 → n ≠ 0 → Coprime 0 n → f (0 * n) = f 0 * f n
hmn : Coprime 0 n
⊢ f (0 * n) = f 0 * f n

pp: case inr
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
m n : ℕ
h : m ≠ 0 → n ≠ 0 → m.Coprime n → f (m * n) = f m * f n
hmn : m.Coprime n
hm : m ≠ 0
⊢ f (m * n) = f m * f n

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
n : ℕ
h : 0 ≠ 0 → n ≠ 0 → Coprime 0 n → f (0 * n) = f 0 * f n
hmn : Coprime 0 n
⊢ f (0 * n) = f 0 * f n
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case insert.inl
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hind : ∑ i ∈ s, f i • g (τ i) ≤ ∑ i ∈ s, f i • g i
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
hτs : {x | τ x ≠ x} ⊆ ↑s
hσa : a = σ a
⊢ f a • g (σ a) + ∑ i ∈ s, f i • g (σ i) ≤ f a • g a + ∑ i ∈ s, f i • g (τ i)

pp: case insert.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hind : ∑ i ∈ s, f i • g (τ i) ≤ ∑ i ∈ s, f i • g i
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
hτs : {x | τ x ≠ x} ⊆ ↑s
hσa : a ≠ σ a
⊢ f a • g (σ a) + ∑ i ∈ s, f i • g (σ i) ≤ f a • g a + ∑ i ∈ s, f i • g (τ i)

Tactic:
·

Goal state after:
case insert.inl
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
σ : Perm ι
hfg : MonovaryOn f g ↑(insert a s)
hσ : {x | σ x ≠ x} ⊆ ↑(insert a s)
τ : Perm ι := Equiv.trans σ (Equiv.swap a (σ a))
hind : ∑ i ∈ s, f i • g (τ i) ≤ ∑ i ∈ s, f i • g i
hτ : τ = Equiv.trans σ (Equiv.swap a (σ a))
hτs : {x | τ x ≠ x} ⊆ ↑s
hσa : a = σ a
⊢ f a • g (σ a) + ∑ i ∈ s, f i • g (σ i) ≤ f a • g a + ∑ i ∈ s, f i • g (τ i)
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
m : ℕ
hm : m ≠ 0
h : m ≠ 0 → 0 ≠ 0 → m.Coprime 0 → f (m * 0) = f m * f 0
hmn : m.Coprime 0
⊢ f (m * 0) = f m * f 0

pp: case inr.inr
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
m n : ℕ
h : m ≠ 0 → n ≠ 0 → m.Coprime n → f (m * n) = f m * f n
hmn : m.Coprime n
hm : m ≠ 0
hn : n ≠ 0
⊢ f (m * n) = f m * f n

Tactic:
·

Goal state after:
case inr.inl
R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
m : ℕ
hm : m ≠ 0
h : m ≠ 0 → 0 ≠ 0 → m.Coprime 0 → f (m * 0) = f m * f 0
hmn : m.Coprime 0
⊢ f (m * 0) = f m * f 0
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case mp
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : f = g
⊢ ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)

pp: case mpr
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)
⊢ f = g

Tactic:
·

Goal state after:
case mp
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : f = g
⊢ ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)
n : ℕ
hn : n = 0
⊢ f n = g n

pp: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)
n : ℕ
hn : ¬n = 0
⊢ f n = g n

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
g : ArithmeticFunction R
hg : g.IsMultiplicative
h : ∀ (p i : ℕ), Nat.Prime p → f (p ^ i) = g (p ^ i)
n : ℕ
hn : n = 0
⊢ f n = g n
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

pp: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : x = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

pp: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : y = 0
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: case inl
ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
f : ι → α
n : ℕ
hf : ∀ i ∈ ∅, 0 ≤ f i
⊢ (∑ i ∈ ∅, f i) ^ (n + 1) / ↑(#∅) ^ n ≤ ∑ i ∈ ∅, f i ^ (n + 1)

pp: case inr
ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
hf : ∀ i ∈ s, 0 ≤ f i
n : ℕ
hs : s.Nonempty
⊢ (∑ i ∈ s, f i) ^ (n + 1) / ↑(#s) ^ n ≤ ∑ i ∈ s, f i ^ (n + 1)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
f : ι → α
n : ℕ
hf : ∀ i ∈ ∅, 0 ≤ f i
⊢ (∑ i ∈ ∅, f i) ^ (n + 1) / ↑(#∅) ^ n ≤ ∑ i ∈ ∅, f i ^ (n + 1)
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case inl
M✝ : Type u_1
inst✝⁴ : Group M✝
inst✝³ : Lattice M✝
M : Type u_2
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
a b : MulArchimedeanOrder M
hab : |val a|ₘ ≤ |val b|ₘ
⊢ a ≤ b ∨ b ≤ a

pp: case inr
M✝ : Type u_1
inst✝⁴ : Group M✝
inst✝³ : Lattice M✝
M : Type u_2
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
a b : MulArchimedeanOrder M
hab : |val b|ₘ ≤ |val a|ₘ
⊢ a ≤ b ∨ b ≤ a

Tactic:
·

Goal state after:
case inl
M✝ : Type u_1
inst✝⁴ : Group M✝
inst✝³ : Lattice M✝
M : Type u_2
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
a b : MulArchimedeanOrder M
hab : |val a|ₘ ≤ |val b|ₘ
⊢ a ≤ b ∨ b ≤ a
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case inr
M✝ : Type u_1
inst✝⁴ : Group M✝
inst✝³ : Lattice M✝
M : Type u_2
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
a b : MulArchimedeanOrder M
hab : |val b|ₘ ≤ |val a|ₘ
⊢ a ≤ b ∨ b ≤ a

Tactic:
·

Goal state after:
case inr
M✝ : Type u_1
inst✝⁴ : Group M✝
inst✝³ : Lattice M✝
M : Type u_2
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
a b : MulArchimedeanOrder M
hab : |val b|ₘ ≤ |val a|ₘ
⊢ a ≤ b ∨ b ≤ a