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

Goal state before:
pp: case refine_1
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
⊢ ∑ x ∈ s, f x • g ((Equiv.symm σ) x) = ∑ i ∈ s, f i • g (σ⁻¹ i)

Tactic:
]

Goal state after:
case refine_1
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
⊢ ∑ x ∈ s, f x • g ((Equiv.symm σ) x) = ∑ i ∈ s, f i • g (σ⁻¹ i)
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case e'_7
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s
⊢ g = g

Tactic:
]

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

Goal state before:
pp: case cons.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 (∏ i ∈ s, a i) < mk (a i)
⊢ mk (a (s.min' hs)) = mk (a ((Finset.cons i s hi).min' ⋯))

Tactic:
]

Goal state after:
case cons.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 (∏ i ∈ s, a i) < mk (a i)
⊢ mk (a (s.min' hs)) = mk (a ((Finset.cons i s hi).min' ⋯))
Module: Mathlib.Algebra.Order.Rearrangement

Goal state before:
pp: case e'_8
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn f (g ∘ ⇑σ⁻¹) ↑s
⊢ ↑s = ↑s

Tactic:
]

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

Goal state before:
pp: case e'_6
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn (f ∘ ⇑σ) g ↑s
⊢ f = f

Tactic:
]

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

Goal state before:
pp: case e'_8
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn (f ∘ ⇑σ) g ↑s
⊢ ⇑(Equiv.symm σ) '' ↑s = ↑s

Tactic:
]

Goal state after:
case e'_8
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
hσinv : {x | σ⁻¹ x ≠ x} ⊆ ↑s
h : MonovaryOn (f ∘ ⇑σ) g ↑s
⊢ ⇑(Equiv.symm σ) '' ↑s = ↑s
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
s : Finset ι
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
y : ι
hi : y ∉ s
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons y s hi)
hminmem : s.min' hs ∈ Finset.cons y s hi
hne : mk (a y) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a (s.min' hs)) < mk (a y)
hy : y ∈ Finset.cons y s hi
⊢ (mk ∘ a) (s.min' hs) < (mk ∘ a) y

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
s : Finset ι
hs : s.Nonempty
ih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
ih : mk (∏ i ∈ s, a i) = mk (a (s.min' hs))
y : ι
hi : y ∉ s
hmono : StrictMonoOn (mk ∘ a) ↑(Finset.cons y s hi)
hminmem : s.min' hs ∈ Finset.cons y s hi
hne : mk (a y) ≠ mk (∏ i ∈ s, a i)
hlt : mk (a (s.min' hs)) < mk (a y)
hy : y ∈ Finset.cons y s hi
⊢ (mk ∘ a) (s.min' hs) < (mk ∘ a) y
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 < mk b
hpos : 1 ≤ a
h : a⁻¹ < b ∧ b < a
⊢ b < a

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 < mk b
hpos : 1 ≤ a
h : a⁻¹ < b ∧ b < a
⊢ b < 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
h✝ : mk a < mk b
hneg : a ≤ 1
h : a < b ∧ b < a⁻¹
⊢ a < b

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 < mk b
hneg : a ≤ 1
h : a < b ∧ b < a⁻¹
⊢ a < b
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case neg
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → mk a = mk b
x y : M
hy : 1 < y
hx : 1 < x
hxy : mk x = mk y
left✝ : ∃ m, |y|ₘ ≤ |x|ₘ ^ m
m : ℕ
hm : x ≤ y ^ m
⊢ ∃ n, x ≤ y ^ n

Tactic:
]

Goal state after:
case neg
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
h : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → mk a = mk b
x y : M
hy : 1 < y
hx : 1 < x
hxy : mk x = mk y
left✝ : ∃ m, |y|ₘ ≤ |x|ₘ ^ m
m : ℕ
hm : x ≤ y ^ m
⊢ ∃ n, x ≤ y ^ n
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝⁵ : CommGroup M
inst✝⁴ : LinearOrder M
inst✝³ : IsOrderedMonoid M
a b : M
N : Type u_2
inst✝² : CommGroup N
inst✝¹ : LinearOrder N
inst✝ : IsOrderedMonoid N
f : M →*o N
h : mk a = mk b
⊢ (orderHom f) (mk b) = (orderHom f) (mk b)

Tactic:
]

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

Goal state before:
pp: M : Type u_1
inst✝⁵ : CommGroup M
inst✝⁴ : LinearOrder M
inst✝³ : IsOrderedMonoid M
a b : M
N : Type u_2
inst✝² : CommGroup N
inst✝¹ : LinearOrder N
inst✝ : IsOrderedMonoid N
f : M →*o N
h : mk a ≤ mk b
⊢ (orderHom f) (mk a) ≤ (orderHom f) (mk b)

Tactic:
]

Goal state after:
M : Type u_1
inst✝⁵ : CommGroup M
inst✝⁴ : LinearOrder M
inst✝³ : IsOrderedMonoid M
a b : M
N : Type u_2
inst✝² : CommGroup N
inst✝¹ : LinearOrder N
inst✝ : IsOrderedMonoid N
f : M →*o N
h : mk a ≤ mk b
⊢ (orderHom f) (mk a) ≤ (orderHom f) (mk b)
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝⁵ : CommGroup M
inst✝⁴ : LinearOrder M
inst✝³ : IsOrderedMonoid M
N : Type u_2
inst✝² : CommGroup N
inst✝¹ : LinearOrder N
inst✝ : IsOrderedMonoid N
f : M →*o N
⊢ mk 1 = mk 1

Tactic:
]

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
s : UpperSet (MulArchimedeanClass M)
a b : M
ha : mk a ∈ ↑s
hb : mk b ∈ ↑s
⊢ mk (a * b) ∈ ↑s

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
s : UpperSet (MulArchimedeanClass M)
a b : M
ha : mk a ∈ ↑s
hb : mk b ∈ ↑s
⊢ mk (a * b) ∈ ↑s
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
s t : UpperSet (MulArchimedeanClass M)
hst : s < t
⊢ ↑(subsemigroup t) ⊂ ↑(subsemigroup s)

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s t : UpperSet (MulArchimedeanClass M)
hst : s < t
⊢ ↑(subsemigroup t) ⊂ ↑(subsemigroup s)
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
s : UpperSet (MulArchimedeanClass M)
hs : ¬s = ⊤
⊢ mk 1 ∈ ↑s

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
s : UpperSet (MulArchimedeanClass M)
hs : ¬s = ⊤
⊢ mk 1 ∈ ↑s
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
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subgroup t) ⊂ ↑(subgroup s)

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subgroup t) ⊂ ↑(subgroup s)
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
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subgroup t) ⊂ ↑(subsemigroup s)

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subgroup t) ⊂ ↑(subsemigroup s)
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
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subsemigroup t) ⊂ ↑(subsemigroup s)

Tactic:
]

Goal state after:
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s : UpperSet (MulArchimedeanClass M)
hs : s ∈ Set.Iio ⊤
t : UpperSet (MulArchimedeanClass M)
ht : t ∈ Set.Iio ⊤
hst : s < t
⊢ ↑(subsemigroup t) ⊂ ↑(subsemigroup s)
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
t : UpperSet (MulArchimedeanClass M)
hst : ⊤ ≤ t
⊢ subgroup ⊤ ≤ subgroup ⊤

Tactic:
]

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

Goal state before:
pp: case inr.inr
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s t : UpperSet (MulArchimedeanClass M)
hst : s ≤ t
hs : s ≠ ⊤
ht : t ≠ ⊤
⊢ s ≤ t

Tactic:
]

Goal state after:
case inr.inr
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
s t : UpperSet (MulArchimedeanClass M)
hst : s ≤ t
hs : s ≠ ⊤
ht : t ≠ ⊤
⊢ s ≤ t
Module: Mathlib.Algebra.Order.Floor.Div

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
a : ℕ
ha : a ≤ 0
b : ℕ
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: case inr
K : Type u_2
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a : K
n : ℕ
hn : n > 0
⊢ ⌊a / ↑n⌋₊ = ⌊a / ↑n * ↑n⌋₊ / n

Tactic:
]

Goal state after:
case inr
K : Type u_2
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a : K
n : ℕ
hn : n > 0
⊢ ⌊a / ↑n⌋₊ = ⌊a / ↑n * ↑n⌋₊ / n
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: case inr
K : Type u_2
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a : K
n : ℕ
hn : n > 0
⊢ ⌊a / ↑n⌋₊ = ⌊a / ↑n⌋₊

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
m n : ℕ
⊢ m = m

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb₀ : 0 < b
hb : b < 1
hba : ↑⌈b / (1 - b)⌉₊ ≤ a
⊢ ⌈b / (1 - b)⌉₊ ≤ ⌊a⌋₊

Tactic:
]

Goal state after:
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb₀ : 0 < b
hb : b < 1
hba : ↑⌈b / (1 - b)⌉₊ ≤ a
⊢ ⌈b / (1 - b)⌉₊ ≤ ⌊a⌋₊
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hab : a ≤ (b - 1)⁻¹
⊢ ↑⌈(b - 1)⁻¹⌉₊ / b < a

pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hab : a ≤ (b - 1)⁻¹
⊢ 0 < b

Tactic:
]

Goal state after:
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hab : a ≤ (b - 1)⁻¹
⊢ ↑⌈(b - 1)⁻¹⌉₊ / b < a

K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hab : a ≤ (b - 1)⁻¹
⊢ 0 < b
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: case neg
M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a✝ b✝ : M
α : Type u_2
f : { a // a ≠ 1 } → α
h : ∀ (a b : { a // a ≠ 1 }), mk ↑a ⋯ = mk ↑b ⋯ → f a = f b
x✝ : FiniteMulArchimedeanClass M
A : MulArchimedeanClass M
hA : A ≠ ⊤
a b : M
h' : MulArchimedeanClass.mk a = MulArchimedeanClass.mk b
ha : ¬a = 1
hb : ¬b = 1
⊢ ↑(f ⟨b, hb⟩) = ↑(f ⟨b, hb⟩)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: case inr
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ ↑⌈a⌉₊ < b * a

Tactic:
]

Goal state after:
case inr
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ ↑⌈a⌉₊ < b * a
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ a + 1 = a + 1

pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ b - 1 ≠ 0

Tactic:
]

Goal state after:
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ b - 1 ≠ 0
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ b * a = b * a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: case inl
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊

pp: case inl.hb
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ b ≠ 0

Tactic:
]

Goal state after:
case inl
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊

case inl.hb
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ b ≠ 0
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: case coe
n : ℕ
x✝ : ℝ≥0
hn₀ : ¬n = 0
⊢ n ≤ ⌈x✝⌉₊ ↔ n ≤ ⌈x✝⌉₊

pp: case coe
n : ℕ
x✝ : ℝ≥0
hn₀ : ¬n = 0
⊢ 1 ≤ n

Tactic:
]

Goal state after:
case coe
n : ℕ
x✝ : ℝ≥0
hn₀ : ¬n = 0
⊢ 1 ≤ n
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a : K
ha : 1 ≤ a
⊢ 2⁻¹ * a < ↑⌊a⌋₊

Tactic:
]

Goal state after:
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a : K
ha : 1 ≤ a
⊢ 2⁻¹ * a < ↑⌊a⌋₊
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌊↑(r + ↑n)⌋ₑ = ↑(⌊r⌋₊ + n)

Tactic:
]

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌊↑(r + ↑n)⌋ₑ = ↑(⌊r⌋₊ + n)
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: R : Type u_3
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
r : R
n d : ℕ
h : IsNNRat r n d
res : ℕ
hres : n / d = res
⊢ n / d = n / d

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌈↑(r + ↑n)⌉ₑ = ↑(⌈r⌉₊ + n)

Tactic:
]

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌈↑(r + ↑n)⌉ₑ = ↑(⌈r⌉₊ + n)
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: R : Type u_3
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
n d : ℕ
hd : Invertible ↑d
⊢ ↑(Int.negOfNat n) * ⅟↑d < 1

Tactic:
]

Goal state after:
R : Type u_3
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
n d : ℕ
hd : Invertible ↑d
⊢ ↑(Int.negOfNat n) * ⅟↑d < 1
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌊↑(r - ↑n)⌋ₑ = ↑(⌊r⌋₊ - n)

Tactic:
]

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌊↑(r - ↑n)⌋ₑ = ↑(⌊r⌋₊ - n)
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌈↑(r - ↑n)⌉ₑ = ↑(⌈r⌉₊ - n)

Tactic:
]

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌈↑(r - ↑n)⌉ₑ = ↑(⌈r⌉₊ - n)
Module: Mathlib.Algebra.Order.Group.Cone

Goal state before:
pp: S : Type u_1
G : Type u_2
inst✝² : CommGroup G
inst✝¹ : SetLike S G
C : S
inst✝ : GroupConeClass S G
a b c : G
nab : b / a ∈ C
nbc : c / b ∈ C
⊢ c / a ∈ C

Tactic:
simpa using mul_mem nbc nab

Goal state after:
Module: Mathlib.Algebra.Order.Group.DenselyOrdered

Goal state before:
pp: M : Type u_2
inst✝⁴ : LinearOrder M
inst✝³ : DenselyOrdered M
x : M
inst✝² : CommMonoid M
inst✝¹ : ExistsMulOfLE M
inst✝ : IsOrderedCancelMonoid M
hx : 1 < x
n : ℕ
y : M
hy : 1 < y
hyx : y ^ (n + 1) < x
z : M
hz : 1 < z
hzy : z ^ 2 ≤ y
⊢ (z ^ 2) ^ (n + 1) = (z ^ 2) ^ (n + 1)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Cyclic

Goal state before:
pp: case inr.inl
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH : IsCyclic ↥H
ha : ⊥ = H
⊢ ∃ a < 1, Subgroup.zpowers a = H

Tactic:
]

Goal state after:
case inr.inl
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH : IsCyclic ↥H
ha : ⊥ = H
⊢ ∃ a < 1, Subgroup.zpowers a = H
Module: Mathlib.Algebra.Order.Group.Cyclic

Goal state before:
pp: case right
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH : IsCyclic ↥H
a : G
ha : Subgroup.zpowers a = H
ha1 : 1 < a
⊢ H = H

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Int.Sum

Goal state before:
pp: s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ ∑ x ∈ s ∩ r, x + ∑ x ∈ s \ r, x ≤ ∑ x ∈ s ∩ r, x + #(s \ r) • (c - ↑(#s))

Tactic:
]

Goal state after:
s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ ∑ x ∈ s ∩ r, x + ∑ x ∈ s \ r, x ≤ ∑ x ∈ s ∩ r, x + #(s \ r) • (c - ↑(#s))
Module: Mathlib.Algebra.Order.Group.Cyclic

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH✝ : IsCyclic ↥H
g : G
hg : g < 1
hH : Subgroup.zpowers g = Subgroup.zpowers (Subgroup.genLTOne H)
hg' : ¬IsOfFinOrder g
⊢ g = Subgroup.genLTOne H

Tactic:
]

Goal state after:
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH✝ : IsCyclic ↥H
g : G
hg : g < 1
hH : Subgroup.zpowers g = Subgroup.zpowers (Subgroup.genLTOne H)
hg' : ¬IsOfFinOrder g
⊢ g = Subgroup.genLTOne H
Module: Mathlib.Algebra.Order.Group.Int.Sum

Goal state before:
pp: s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ ∑ x ∈ r ∩ s, x + #(r \ s) • (c - ↑(#s)) = ∑ x ∈ r ∩ s, x + #(r \ s) • (c - ↑(#s))

pp: s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ #s = #r

Tactic:
]

Goal state after:
s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ #s = #r
Module: Mathlib.Algebra.Order.Group.Cyclic

Goal state before:
pp: case inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH✝ : IsCyclic ↥H
g : G
hg : 1 < Subgroup.genLTOne H
hH : Subgroup.zpowers g = Subgroup.zpowers (Subgroup.genLTOne H)
hg' : ¬IsOfFinOrder g
h : g⁻¹ = Subgroup.genLTOne H
⊢ g = Subgroup.genLTOne H

Tactic:
]

Goal state after:
case inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
H : Subgroup G
inst✝ : Nontrivial ↥H
hH✝ : IsCyclic ↥H
g : G
hg : 1 < Subgroup.genLTOne H
hH : Subgroup.zpowers g = Subgroup.zpowers (Subgroup.genLTOne H)
hg' : ¬IsOfFinOrder g
h : g⁻¹ = Subgroup.genLTOne H
⊢ g = Subgroup.genLTOne H
Module: Mathlib.Algebra.Order.Group.Int.Sum

Goal state before:
pp: s : Finset ℤ
c : ℤ
hs : ∀ x ∈ s, x ≤ c
r : Finset ℤ := Ioc (c - ↑(#s)) c
⊢ #s = #s

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Group.Cyclic

Goal state before:
pp: case inl
G : Type u_1
inst✝² : CommGroup G
inst✝¹ : LinearOrder G
inst✝ : IsOrderedMonoid G
g1 g2 : G
hg1 : g1 < 1
hg2 : g2 < 1
h : Subgroup.zpowers g1 = ⊥
h' : Subgroup.zpowers g2 = ⊥
⊢ g1 = g2

Tactic:
]

Goal state after:
case inl
G : Type u_1
inst✝² : CommGroup G
inst✝¹ : LinearOrder G
inst✝ : IsOrderedMonoid G
g1 g2 : G
hg1 : g1 < 1
hg2 : g2 < 1
h : Subgroup.zpowers g1 = ⊥
h' : Subgroup.zpowers g2 = ⊥
⊢ g1 = g2