mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case inl
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |y|ₘ ≤ |z|ₘ
⊢ mk z ≤ mk z

pp: case inl
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |y|ₘ ≤ |z|ₘ
⊢ mk z ≤ mk y

Tactic:
]

Goal state after:
case inl
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |y|ₘ ≤ |z|ₘ
⊢ mk z ≤ mk y
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case inr
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |z|ₘ ≤ |y|ₘ
⊢ mk y ≤ mk y

pp: case inr
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |z|ₘ ≤ |y|ₘ
⊢ mk y ≤ mk z

Tactic:
]

Goal state after:
case inr
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
x y z : M
hy : y ≤ x
hz : x ≤ z
H : |x|ₘ ≤ |max |y|ₘ |z|ₘ|ₘ
h : |z|ₘ ≤ |y|ₘ
⊢ mk y ≤ mk z
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: case neg
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
⊢ (partialSups f) (succ i) = (partialSups f) i ⊔ f (succ i) \ (partialSups f) i

Tactic:
]

Goal state after:
case neg
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
⊢ (partialSups f) (succ i) = (partialSups f) i ⊔ f (succ i) \ (partialSups f) i
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 : mk (a * b) < mk a ∧ mk (a * b) < mk b
⊢ False

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : mk (a * b) < mk a ∧ mk (a * b) < mk b
⊢ False
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
hf : Monotone f
i : ι
⊢ f (succ i) \ f i = f (succ i) \ f i

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
hf : Monotone f
i : ι
⊢ ¬IsMax i

Tactic:
]

Goal state after:
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
hf : Monotone f
i : ι
⊢ ¬IsMax i
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case insert.inr.refine_2
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
σ : Perm ι
hamax : (g (σ a), f (σ a)).1 < (g a, f a).1 ∨ (g (σ a), f (σ a)).1 = (g a, f a).1 ∧ (g (σ a), f (σ a)).2 ≤ (g a, f a).2
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
h1s : (Equiv.symm σ) a ∈ s
⊢ g (σ a) ≤ g a

Tactic:
]

Goal state after:
case insert.inr.refine_2
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
σ : Perm ι
hamax : (g (σ a), f (σ a)).1 < (g a, f a).1 ∨ (g (σ a), f (σ a)).1 = (g a, f a).1 ∧ (g (σ a), f (σ a)).2 ≤ (g a, f a).2
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
h1s : (Equiv.symm σ) a ∈ s
⊢ g (σ a) ≤ g a
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 b ≤ mk a ↔ mk b ≤ mk a

Tactic:
]

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

Goal state before:
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
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ ∏ x_1 ∈ x.primeFactors ∪ y.primeFactors,
      f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
    ∏ x_1 ∈ x.primeFactors ∪ y.primeFactors, f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)

Tactic:
]

Goal state after:
case neg
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_1 ∈ x.primeFactors ∪ y.primeFactors,
      f (x_1 ^ (x.lcm y).factorization x_1) * f (x_1 ^ (x.gcd y).factorization x_1) =
    ∏ x_1 ∈ x.primeFactors ∪ y.primeFactors, f (x_1 ^ x.factorization x_1) * f (x_1 ^ y.factorization x_1)
Module: Mathlib.Algebra.Order.Field.GeomSum

Goal state before:
pp: K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ ∑ i ∈ range n, x ^ i < (1 - x)⁻¹

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ ∑ i ∈ range n, x ^ i < (1 - x)⁻¹
Module: Mathlib.Algebra.Order.Field.GeomSum

Goal state before:
pp: K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 0 < 1 ∧ 0 < x ^ n

pp: K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 1 - x ≠ 0

pp: K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 0 < 1 - x

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 0 < 1 ∧ 0 < x ^ n

K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 1 - x ≠ 0

K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ 0 < 1 - x
Module: Mathlib.Algebra.Order.Field.GeomSum

Goal state before:
pp: K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ x < 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝⁵ : Semifield K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : CanonicallyOrderedAdd K
inst✝¹ : Sub K
inst✝ : OrderedSub K
x : K
h0 : 0 < x
h1 : x < 1
n : ℕ
⊢ x < 1
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 : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |a|ₘ ≤ |a * b|ₘ ^ 2

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |a|ₘ ≤ |a * b|ₘ ^ 2
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 : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |b|ₘ ≤ |a * b|ₘ

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |b|ₘ ≤ |a * b|ₘ
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 : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |b|ₘ * |b|ₘ ≤ |b|ₘ * |b * a|ₘ

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ
⊢ |b|ₘ * |b|ₘ ≤ |b|ₘ * |b * a|ₘ
Module: Mathlib.Algebra.Order.Field.GeomSum

Goal state before:
pp: case inl
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : m ≤ n
⊢ (x ^ m - x ^ n) / (1 - x) ≤ x ^ m / (1 - x)

Tactic:
]

Goal state after:
case inl
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : m ≤ n
⊢ (x ^ m - x ^ n) / (1 - x) ≤ x ^ m / (1 - x)
Module: Mathlib.Algebra.Order.Field.GeomSum

Goal state before:
pp: case inr
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : n < m
⊢ 0 ≤ x ^ m / (1 - x)

pp: case inr
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : n < m
⊢ ¬m < n

Tactic:
]

Goal state after:
case inr
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : n < m
⊢ 0 ≤ x ^ m / (1 - x)

case inr
K : Type u_1
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
x : K
m n : ℕ
hx : 0 ≤ x
h'x : x < 1
hmn : n < m
⊢ ¬m < n
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case cons
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
⊢ mk (a i * ∏ x ∈ s, a x) = mk (a ((Finset.cons i s hi).min' ⋯))

Tactic:
]

Goal state after:
case cons
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
⊢ mk (a i * ∏ x ∈ s, a x) = mk (a ((Finset.cons i s hi).min' ⋯))
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case insert.inr.refine_3
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
h1s : (Equiv.symm σ) a ∈ s
x : ι
hx : ¬σ x = a ∧ x ∈ s
⊢ f x • g (σ x) = f x • g ((Equiv.swap a (σ a)) (σ x))

Tactic:
]

Goal state after:
case insert.inr.refine_3
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
h1s : (Equiv.symm σ) a ∈ s
x : ι
hx : ¬σ x = a ∧ x ∈ s
⊢ f x • g (σ x) = f x • g ((Equiv.swap a (σ a)) (σ x))
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
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi✝ : i ∉ s
hs : s.Nonempty
hi : s.min' hs ∉ s
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi✝)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi✝
h : mk (a i) = mk (a (s.min' hs))
eq : i = s.min' hs
⊢ False

Tactic:
]

Goal state after:
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi✝ : i ∉ s
hs : s.Nonempty
hi : s.min' hs ∉ s
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi✝)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi✝
h : mk (a i) = mk (a (s.min' hs))
eq : i = s.min' hs
⊢ False
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case insert.inr.refine_3
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
h1s : (Equiv.symm σ) a ∈ s
x : ι
hx : ¬σ x = a ∧ x ∈ s
⊢ f x • g (σ x) = f x • g (σ x)

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
h1s : (Equiv.symm σ) a ∈ s
x : ι
hx : ¬σ x = a ∧ x ∈ s
⊢ x ≠ a

Tactic:
]

Goal state after:
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
h1s : (Equiv.symm σ) a ∈ s
x : ι
hx : ¬σ x = a ∧ x ∈ s
⊢ x ≠ a
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.factorization.support ∩ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors

Tactic:
]

Goal state after:
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.factorization.support ∩ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case cons
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
⊢ mk (a i * ∏ x ∈ s, a x) = mk (a ((Finset.cons i s hi).min' ⋯))

Tactic:
]

Goal state after:
case cons
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
⊢ mk (a i * ∏ x ∈ s, a x) = mk (a ((Finset.cons i s hi).min' ⋯))
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommGroupWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hf_lcm : f (x.lcm y) ≠ 0
⊢ f (x.gcd y) = f (x.gcd y)

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommGroupWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hf_gcd : f (x.gcd y) ≠ 0
⊢ f (x.lcm y) = f (x.lcm y)

Tactic:
]

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✝ : PosSMulStrictMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
h : ¬∀ ⦃i : ι⦄, i ∈ ↑s → ∀ ⦃j : ι⦄, j ∈ ↑s → (g ∘ ⇑σ) i < (g ∘ ⇑σ) j → f i ≤ f j
⊢ ¬∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i

Tactic:
]

Goal state after:
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✝ : PosSMulStrictMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
h : ¬∀ ⦃i : ι⦄, i ∈ ↑s → ∀ ⦃j : ι⦄, j ∈ ↑s → (g ∘ ⇑σ) i < (g ∘ ⇑σ) j → f i ≤ f j
⊢ ¬∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case cons.inl
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a i) < mk (∏ i ∈ s, a i)
⊢ mk (a i) = mk (a ((Finset.cons i s hi).min' ⋯))

Tactic:
]

Goal state after:
case cons.inl
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a i) < mk (∏ i ∈ s, a i)
⊢ mk (a i) = mk (a ((Finset.cons i s hi).min' ⋯))
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
ι : Type u_2
f : ι → ArithmeticFunction R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, (f i).IsMultiplicative) → (∏ i ∈ s, f i).IsMultiplicative
hf : ∀ i ∈ cons a s ha, (f i).IsMultiplicative
⊢ (f a * ∏ x ∈ s, f x).IsMultiplicative

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
ι : Type u_2
f : ι → ArithmeticFunction R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, (f i).IsMultiplicative) → (∏ i ∈ s, f i).IsMultiplicative
hf : ∀ i ∈ cons a s ha, (f i).IsMultiplicative
⊢ (f a * ∏ x ∈ s, f x).IsMultiplicative
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
f : ArithmeticFunction R
hf : f.IsMultiplicative
k : ℕ
hk : (f ^ k).IsMultiplicative
⊢ (f ^ k * f).IsMultiplicative

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
f : ArithmeticFunction R
hf : f.IsMultiplicative
k : ℕ
hk : (f ^ k).IsMultiplicative
⊢ (f ^ k * f).IsMultiplicative
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case 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✝ : PosSMulStrictMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
x : ι
hx : x ∈ ↑s
y : ι
hy : y ∈ ↑s
hgxy : (g ∘ ⇑σ) x < (g ∘ ⇑σ) y
hfxy : f y < f x
τ : Perm ι := Equiv.trans (Equiv.swap x y) σ
hτs : {x | τ x ≠ x} ⊆ ↑s
hxy : x ≠ y
z : ι
hz : z ≠ y ∧ z ≠ x ∧ z ∈ s
⊢ f z • g (σ z) = f z • g (σ z)

Tactic:
]

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

Goal state before:
pp: case inr
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a i) < mk (a (s.min' hs))
y : ι
hy : y ∈ Finset.cons i s hi
hmem : y ∈ s
⊢ (mk ∘ a) i < (mk ∘ a) (s.min' hs)

Tactic:
]

Goal state after:
case inr
M : Type u_1
inst✝³ : CommGroup M
inst✝² : LinearOrder M
inst✝¹ : IsOrderedMonoid M
ι : Type u_2
inst✝ : LinearOrder ι
s✝ : Finset ι
a : ι → M
i : ι
s : Finset ι
hi : i ∉ s
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons i s hi)
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
hminmem : s.min' hs ∈ Finset.cons i s hi
hne : mk (a i) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a i) < mk (a (s.min' hs))
y : ι
hy : y ∈ Finset.cons i s hi
hmem : y ∈ s
⊢ (mk ∘ a) i < (mk ∘ a) (s.min' hs)