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
this : -∑ i ∈ B, m.leadingCoeff (g i) = m.leadingCoeff (g b)
⊢ g b + ∑ b ∈ B, g b = ∑ b' ∈ B, (g b' - (m.leadingCoeff (g b') * ↑isunit_gb.unit⁻¹) • g b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.CompleteField

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Field β
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : Archimedean α
a : α
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑(q ^ 2)
hqa : ↑q * ↑q < inducedMap α β a * inducedMap α β a
q' : ℚ
hq' : ↑q < ↑q'
hqa' : ↑q' < a
⊢ ↑q < ↑q'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.CompleteField

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : Field α
inst✝⁵ : LinearOrder α
inst✝⁴ : IsStrictOrderedRing α
inst✝³ : Field β
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : Archimedean α
a : α
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.CompleteField

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
⊢ ∀ (x : α), (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.CompleteField

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
⊢ ∀ (x : α), (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.CompleteField

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
⊢ ∀ {a b : β}, (↑↑(inducedOrderRingHom β γ).toRingHom).toFun a ≤ (↑↑(inducedOrderRingHom β γ).toRingHom).toFun b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Real.Hom

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedAddMonoid R
inst✝³ : Ring S
inst✝² : LinearOrder S
inst✝¹ : IsOrderedAddMonoid S
inst✝ : PosMulMono S
hR : ∀ (r : R), 0 ≤ r → IsSquare r
f : R →+* S
r : R
h : 0 ≤ r
⊢ 0 ≤ f r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Submonoid

Goal state before:
pp: M : Type u_1
S : Type u_2
inst✝⁴ : SetLike S M
inst✝³ : CommMonoid M
inst✝² : PartialOrder M
inst✝¹ : SubmonoidClass S M
inst✝ : MulArchimedean M
H : S
⊢ MulArchimedean ↥H

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
d : m.syn
ι : Type u_3
B : Finset ι
g : ι → MvPolynomial σ R
hd : ∀ b ∈ B, m.toSyn (m.degree (g b)) = d ∧ IsUnit (m.leadingCoeff (g b)) ∨ g b = 0
hfd : m.toSyn (m.degree (∑ b ∈ B, g b)) < d
⊢ ∃ c, ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
f : MvPolynomial σ R
⊢ m.withBotDegree (-f) = m.withBotDegree f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : Field R
f : MvPolynomial σ R
⊢ IsUnit (m.leadingCoeff f) ↔ f ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : Field R
d : m.syn
ι : Type u_3
B : Finset ι
g : ι → MvPolynomial σ R
hd : ∀ b ∈ B, m.toSyn (m.degree (g b)) = d ∨ g b = 0
hfd : m.toSyn (m.degree (∑ b ∈ B, g b)) < d
⊢ ∃ c, ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : Nontrivial R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.degree (X i + C r) = single i 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : Nontrivial R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.degree (X i - C r) = single i 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.Monic (X i + C r)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.Monic (X i - C r)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: 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 | τ x ≠ x} ⊆ ↑s

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
⊢ (f * g).IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
s : Finset ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
⊢ (∑ i ∈ s, f i) • ∑ i ∈ s, g i ≤ #s • ∑ i ∈ s, f i • g i

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : MonoidWithZero R
f : ArithmeticFunction R
⊢ f.IsMultiplicative ↔ f 1 = 1 ∧ ∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → m.Coprime n → f (m * n) = f m * f n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f g : ι → α
hfg : MonovaryOn f g ↑s
⊢ (∑ i ∈ s, f i) * ∑ i ∈ s, g i ≤ ↑(#s) * ∑ i ∈ s, f i * g i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f g : ι → α
hfg : AntivaryOn f g ↑s
⊢ ↑(#s) * ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
hf : ∀ i ∈ s, 0 ≤ f i
⊢ (∑ i ∈ s, f i) ^ (0 + 1) ≤ ↑(#s) ^ 0 * ∑ i ∈ s, f i ^ (0 + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring 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
⊢ (∑ i ∈ s, f i) ^ (n + 1 + 1) = (∑ i ∈ s, f i) ^ (n + 1) * ∑ i ∈ s, f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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 : MulArchimedeanOrder M
⊢ |val a|ₘ ≤ |val a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring 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
⊢ (∑ i ∈ s, f i) ^ (n + 1) * ∑ i ∈ s, f i ≤ (↑(#s) ^ n * ∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring 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
⊢ (↑(#s) ^ n * ∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i = ↑(#s) ^ n * ((∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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 c : MulArchimedeanOrder M
⊢ a ≤ b → b ≤ c → a ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring 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
⊢ ↑(#s) ^ n * ((∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i) ≤ ↑(#s) ^ n * (↑(#s) * ∑ i ∈ s, f i ^ (n + 1) * f i)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

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

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
this : (∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ) → ∃ n, |val b|ₘ ≤ |val a|ₘ ^ n
⊢ (∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ) ↔ (∃ n, |val b|ₘ ≤ |val a|ₘ ^ n) ∧ ¬∃ n, |val a|ₘ ≤ |val b|ₘ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring 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
⊢ ↑(#s) ^ n * (↑(#s) * ∑ i ∈ s, f i ^ (n + 1) * f i) = ↑(#s) ^ (n + 1) * ∑ i ∈ s, f i ^ (n + 1 + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
h✝ : ∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ
h : |val b|ₘ ^ 1 ≤ |val a|ₘ
⊢ |val b|ₘ ≤ |val a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semiring α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
⊢ (∑ i ∈ s, f i) ^ 2 ≤ ↑(#s) * ∑ i ∈ s, f i ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
⊢ a < b ↔ a ≤ b ∧ ¬b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
⊢ ∀ {i : ℕ}, f (i ^ 0) = 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.lcm y ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
⊢ 0 < ↑(#s) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
⊢ |val a|ₘ ≤ |val b|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
⊢ (∑ i ∈ s, f i) ^ (n + 1) / ↑(#s) ^ n ≤ ∑ i ∈ s, f i ^ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
⊢ |val b|ₘ ≤ |val a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: 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
⊢ (Equiv.symm σ) a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: 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
⊢ a ≤ b ∨ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
hs : s.Nonempty
⊢ 0 < ↑(#s) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
hs : s.Nonempty
⊢ 0 < ↑(#s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M✝ : Type u_1
inst✝⁷ : Group M✝
inst✝⁶ : Lattice M✝
M : Type u_2
inst✝⁵ : CommGroup M
inst✝⁴ : LinearOrder M
inst✝³ : IsOrderedMonoid M
a b : M
N : Type u_3
inst✝² : CommGroup N
inst✝¹ : LinearOrder N
inst✝ : IsOrderedMonoid N
f : M →*o N
⊢ Monotone fun a ↦ of (f (val a))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: 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
⊢ ((∑ i ∈ s, f i) / ↑(#s)) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) / ↑(#s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
⊢ mk a = mk b ↔ (∃ m, |b|ₘ ≤ |a|ₘ ^ m) ∧ ∃ n, |a|ₘ ≤ |b|ₘ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.gcd y ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
α : Type u_2
f : M → α
h : ∀ (a b : M), mk a = mk b → f a = f b
a : M
⊢ lift f h (mk a) = f a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
α : Type u_2
f : M → M → α
h : ∀ (a₁ b₁ a₂ b₂ : M), mk a₁ = mk b₁ → mk a₂ = mk b₂ → f a₁ a₂ = f b₁ b₂
a b : M
⊢ lift₂ f h (mk a) (mk b) = f a b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ |a|ₘ ≤ |a⁻¹|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ |a⁻¹|ₘ ≤ |a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
⊢ mk (a / b) = mk (b / a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ |a|ₘ ≤ |(|a|ₘ)|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ |(|a|ₘ)|ₘ ≤ |a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
ha : a ≠ 1
⊢ mk a ≤ mk b ↔ ∃ n, |b|ₘ < |a|ₘ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b a : M
⊢ |1|ₘ ≤ |a|ₘ ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
A : MulArchimedeanClass M
⊢ A ≤ mk 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ mk a = ⊤ → a = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ a = 1 → mk a = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a : M
⊢ ⊤ = mk a ↔ a = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
⊢ ⊤.out = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
a b : M
inst✝ : Nontrivial M
⊢ ∃ x y, x ≠ y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
⊢ AntitoneOn mk (Set.Ici 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
⊢ MonotoneOn mk (Set.Iic 1)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : |a|ₘ ≤ |b|ₘ
⊢ mk b ≤ mk a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁶ : GeneralizedBooleanAlgebra α
inst✝⁵ : LinearOrder ι
inst✝⁴ : LocallyFiniteOrderBot ι
inst✝³ : Add ι
inst✝² : One ι
inst✝¹ : SuccAddOrder ι
inst✝ : NoMaxOrder ι
f : ι → α
i : ι
⊢ disjointed f (i + 1) = f (i + 1) \ (partialSups f) i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
⊢ i ≤ i + 1

Tactic:
← Order.succ_eq_add_one

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
⊢ i ≤ succ i
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
⊢ i ≤ i + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
⊢ i + 1 = i

Tactic:
by

Goal state after: