mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Hom.Set

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : Preorder β
f g : α ≃o β
⊢ univ = univ

Tactic:
]

Goal state after:
Module: Mathlib.Order.Hom.Set

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder β
inst✝¹ : WellFoundedLT β
inst✝ : Preorder α
f g : α ≃o β
⊢ symm ?m.34 = g.symm.symm

pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder β
inst✝¹ : WellFoundedLT β
inst✝ : Preorder α
f g : α ≃o β
⊢ β ≃o α

Tactic:
]

Goal state after:
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : Preorder α
inst✝ : IsDirectedOrder α
s t : Set α
a : α
ha : a ∈ upperBounds s
b : α
hb : b ∈ upperBounds t
c : α
hca : a ≤ c
hcb : b ≤ c
⊢ (upperBounds s ∩ upperBounds t).Nonempty

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Preorder α
inst✝ : IsDirectedOrder α
s t : Set α
a : α
ha : a ∈ upperBounds s
b : α
hb : b ∈ upperBounds t
c : α
hca : a ≤ c
hcb : b ≤ c
⊢ (upperBounds s ∩ upperBounds t).Nonempty
Module: Mathlib.Order.Hom.Set

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder α
inst✝¹ : WellFoundedGT α
inst✝ : Preorder β
f g : α ≃o β
⊢ OrderIso.dual ?m.34 = g.dual.dual

pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder α
inst✝¹ : WellFoundedGT α
inst✝ : Preorder β
f g : α ≃o β
⊢ αᵒᵈ ≃o βᵒᵈ

Tactic:
]

Goal state after:
Module: Mathlib.Order.Hom.Set

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder β
inst✝¹ : WellFoundedGT β
inst✝ : Preorder α
f g : α ≃o β
⊢ OrderIso.dual ?m.34 = g.dual.dual

pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : LinearOrder β
inst✝¹ : WellFoundedGT β
inst✝ : Preorder α
f g : α ≃o β
⊢ αᵒᵈ ≃o βᵒᵈ

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.Basic

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PartialOrder α
a b : α
s : Set α
ho : Ioo a b ⊆ s
hc : s ⊆ Icc a b
ha : ¬a ∈ s
hb : ¬b ∈ s
⊢ s ⊆ (Icc a b \ {b}) \ {a}

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : PartialOrder α
a b : α
s : Set α
ho : Ioo a b ⊆ s
hc : s ⊆ Icc a b
ha : ¬a ∈ s
hb : ¬b ∈ s
⊢ s ⊆ (Icc a b \ {b}) \ {a}
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: γ : Type u_3
inst✝ : SemilatticeSup γ
s : Set γ
x₀ : γ
⊢ (∃ x, ∀ y ∈ s, y ≤ x) ↔ ∃ x, ∀ y ∈ s, y ≤ x

pp: γ : Type u_3
inst✝ : SemilatticeSup γ
s : Set γ
x₀ : γ
⊢ Monotone fun x ↦ ∀ y ∈ s, y ≤ x

Tactic:
]

Goal state after:
γ : Type u_3
inst✝ : SemilatticeSup γ
s : Set γ
x₀ : γ
⊢ Monotone fun x ↦ ∀ y ∈ s, y ≤ x
Module: Mathlib.Order.Interval.Set.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a b c : α
hab : a ≤ b
hbc : b ≤ c
⊢ Ici a ∩ Iic c ∩ Icc b b = {b}

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
a b c : α
hab : a ≤ b
hbc : b ≤ c
⊢ Ici a ∩ Iic c ∩ Icc b b = {b}
Module: Mathlib.Order.Interval.Set.Basic

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a b c d : α
h : a ≤ b
heq : ¬a ≤ b
contra : ¬c ≤ d
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
a b c d : α
h : a ≤ b
heq : ¬a ≤ b
contra : ¬c ≤ d
⊢ False
Module: Mathlib.Order.Interval.Set.Basic

Goal state before:
pp: α : Type u_1
inst✝ : SemilatticeInf α
a b c : α
⊢ Ioi a ∩ Iic (b ⊓ c) = Ioi a ∩ Iic (b ⊓ c)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: case neg
γ : Type u_3
inst✝ : LinearOrder γ
i : γ
h_exists_lt : ∀ j ∈ upperBounds (Iio i), i ≤ j
⊢ ∀ x ∈ upperBounds (Iio i), i ≤ x

Tactic:
]

Goal state after:
case neg
γ : Type u_3
inst✝ : LinearOrder γ
i : γ
h_exists_lt : ∀ j ∈ upperBounds (Iio i), i ≤ j
⊢ ∀ x ∈ upperBounds (Iio i), i ≤ x
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: γ : Type u_3
inst✝¹ : PartialOrder γ
inst✝ : OrderTop γ
⊢ {⊤} = {⊤}

Tactic:
]

Goal state after:
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : Preorder α
s : Set α
a : α
inst✝ : NoBotOrder α
hs : IsLUB s a
h : s = ∅
x✝ : α
⊢ x✝ ∈ univ

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Preorder α
s : Set α
a : α
inst✝ : NoBotOrder α
hs : IsLUB s a
h : s = ∅
x✝ : α
⊢ x✝ ∈ univ
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: γ : Type u_3
inst✝ : SemilatticeSup γ
a b : γ
s : Set γ
hs : IsLUB s b
⊢ IsLUB ({a} ∪ s) (a ⊔ b)

Tactic:
]

Goal state after:
γ : Type u_3
inst✝ : SemilatticeSup γ
a b : γ
s : Set γ
hs : IsLUB s b
⊢ IsLUB ({a} ∪ s) (a ⊔ b)
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: γ : Type u_3
inst✝ : LinearOrder γ
a b : γ
s : Set γ
hs : IsGreatest s b
⊢ IsGreatest ({a} ∪ s) (max a b)

Tactic:
]

Goal state after:
γ : Type u_3
inst✝ : LinearOrder γ
a b : γ
s : Set γ
hs : IsGreatest s b
⊢ IsGreatest ({a} ∪ s) (max a b)
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
s : Set α
⊢ Ici a ∩ upperBounds s = Ici a ∩ upperBounds s

Tactic:
]

Goal state after:
Module: Mathlib.Order.Bounds.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
t : Set β
hs : Nonempty ↑s
ht : Nonempty ↑t
x✝ : α × β
⊢ x✝ ∈ upperBounds (s ×ˢ t) ↔ x✝ ∈ upperBounds s ×ˢ upperBounds t

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
t : Set β
hs : Nonempty ↑s
ht : Nonempty ↑t
x✝ : α × β
⊢ x✝ ∈ upperBounds (s ×ˢ t) ↔ x✝ ∈ upperBounds s ×ˢ upperBounds t
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℕ
inst✝³ : Semiring A
inst✝² : PartialOrder A
inst✝¹ : IsOrderedRing A
inst✝ : Nontrivial A
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < ↑n

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℕ
inst✝³ : Semiring A
inst✝² : PartialOrder A
inst✝¹ : IsOrderedRing A
inst✝ : Nontrivial A
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < ↑n
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℕ
inst✝³ : AddMonoidWithOne A
inst✝² : PartialOrder A
inst✝¹ : AddLeftMono A
inst✝ : ZeroLEOneClass A
h'' : NeZero 1
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < ↑n

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℕ
inst✝³ : AddMonoidWithOne A
inst✝² : PartialOrder A
inst✝¹ : AddLeftMono A
inst✝ : ZeroLEOneClass A
h'' : NeZero 1
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < ↑n
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℕ
inst✝² : Semiring A
inst✝¹ : PartialOrder A
inst✝ : IsOrderedRing A
h : NormNum.IsNat e n
⊢ 0 ≤ ↑n

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℕ
inst✝² : Semiring A
inst✝¹ : PartialOrder A
inst✝ : IsOrderedRing A
h : NormNum.IsNat e n
⊢ 0 ≤ ↑n
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℕ
inst✝³ : AddMonoidWithOne A
inst✝² : PartialOrder A
inst✝¹ : AddLeftMono A
inst✝ : ZeroLEOneClass A
h : NormNum.IsNat e n
⊢ 0 ≤ ↑n

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℕ
inst✝³ : AddMonoidWithOne A
inst✝² : PartialOrder A
inst✝¹ : AddLeftMono A
inst✝ : ZeroLEOneClass A
h : NormNum.IsNat e n
⊢ 0 ≤ ↑n
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℕ
inst✝² : Ring A
inst✝¹ : PartialOrder A
inst✝ : IsStrictOrderedRing A
h : NormNum.IsInt e (Int.negOfNat n)
w : Nat.ble 1 n = true
⊢ -↑n ≠ 0

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℕ
inst✝² : Ring A
inst✝¹ : PartialOrder A
inst✝ : IsStrictOrderedRing A
h : NormNum.IsInt e (Int.negOfNat n)
w : Nat.ble 1 n = true
⊢ -↑n ≠ 0
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n d : ℕ
inst✝² : Semiring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (0 < n) = true
pos_invOf_d : 0 < ⅟↑d
pos_n : 0 < ↑n
⊢ 0 < ↑n * ⅟↑d

Tactic:
]

Goal state after:
A : Type u_1
e : A
n d : ℕ
inst✝² : Semiring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (0 < n) = true
pos_invOf_d : 0 < ⅟↑d
pos_n : 0 < ↑n
⊢ 0 < ↑n * ⅟↑d
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (0 < n) = true
pos_invOf_d : 0 < ⅟↑d
pos_n : 0 < ↑n
⊢ 0 < ↑n * ⅟↑d

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (0 < n) = true
pos_invOf_d : 0 < ⅟↑d
pos_n : 0 < ↑n
⊢ 0 < ↑n * ⅟↑d
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n d : ℕ
inst✝¹ : Semiring A
inst✝ : LinearOrder A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n = 0) = true
⊢ 0 ≤ ↑0 * ⅟↑d

Tactic:
]

Goal state after:
A : Type u_1
e : A
n d : ℕ
inst✝¹ : Semiring A
inst✝ : LinearOrder A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n = 0) = true
⊢ 0 ≤ ↑0 * ⅟↑d
Module: Mathlib.Tactic.Abel

Goal state before:
pp: α : Type u_1
inst✝ : SubtractionMonoid α
a b c : α
h : a + -b = c
⊢ c = c

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝¹ : Ring A
inst✝ : LinearOrder A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n = 0) = true
⊢ 0 ≤ ↑0 * ⅟↑d

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝¹ : Ring A
inst✝ : LinearOrder A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n = 0) = true
⊢ 0 ≤ ↑0 * ⅟↑d
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n < 0) = true
pos_invOf_d : 0 < ⅟↑d
neg_n : ↑n < 0
⊢ e ≠ 0

Tactic:
have neg := mul_neg_of_neg_of_pos neg_n pos_invOf_d

Goal state after:
A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n < 0) = true
pos_invOf_d : 0 < ⅟↑d
neg_n : ↑n < 0
neg : ↑n * ⅟↑d < 0
⊢ e ≠ 0
Module: Mathlib.Tactic.Abel

Goal state before:
pp: α : Type u_1
inst✝ : AddCommMonoid α
l r tl tr t : α
prl : l = tl
prr : r = tr
prt : tl + tr = t
⊢ t = t

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.Positivity.Core

Goal state before:
pp: A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n < 0) = true
pos_invOf_d : 0 < ⅟↑d
neg_n : ↑n < 0
neg : ↑n * ⅟↑d < 0
⊢ ↑n * ⅟↑d ≠ 0

Tactic:
]

Goal state after:
A : Type u_1
e : A
n : ℤ
d : ℕ
inst✝² : Ring A
inst✝¹ : LinearOrder A
inst✝ : IsStrictOrderedRing A
inv : Invertible ↑d
eq : e = ↑n * ⅟↑d
h : decide (n < 0) = true
pos_invOf_d : 0 < ⅟↑d
neg_n : ↑n < 0
neg : ↑n * ⅟↑d < 0
⊢ ↑n * ⅟↑d ≠ 0
Module: Mathlib.Tactic.Abel

Goal state before:
pp: α : Type u_1
inst✝ : AddCommGroup α
l r tl tr t : α
prl : l = tl
prr : r = tr
prt : tl + tr = t
⊢ t = t

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type ?u.3
inst✝ : Semiring α
k : ℕ
b c : α
hc1 : c * ↑k * b = 1
hc2 : ↑k * b * c = 1
⊢ Invertible b

Tactic:
]

Goal state after:
α : Type ?u.3
inst✝ : Semiring α
k : ℕ
b c : α
hc1 : c * ↑k * b = 1
hc2 : ↑k * b * c = 1
⊢ Invertible b
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type ?u.3
inst✝ : Semiring α
k : ℕ
b c : α
hc1 : c * ↑k * b = 1
hc2 : b * (c * ↑k) = 1
⊢ Invertible b

Tactic:
]

Goal state after:
α : Type ?u.3
inst✝ : Semiring α
k : ℕ
b c : α
hc1 : c * ↑k * b = 1
hc2 : b * (c * ↑k) = 1
⊢ Invertible b
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: case eq
α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * db + nb * da = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ⅟↑da + ↑nb * ⅟↑db = ↑k * ↑nc * ⅟↑db * ⅟↑da
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑db * ⅟↑da = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc

Tactic:
]

Goal state after:
case eq
α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * db + nb * da = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ⅟↑da + ↑nb * ⅟↑db = ↑k * ↑nc * ⅟↑db * ⅟↑da
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑db * ⅟↑da = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
hb : 0 < b
⊢ b ≤ a ↔ b ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
hb : 0 < b
⊢ a ≤ b ↔ a ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
hb : 0 < b
⊢ b < a ↔ b < a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
hb : 0 < b
⊢ a < b ↔ a < b

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
ha : 0 < a
h : a < b
⊢ a < b

Tactic:
]

Goal state after:
α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b : α
ha : 0 < a
h : a < b
⊢ a < b
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
h1 : 0 < a
h2 : a < 1
⊢ a < 1

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
h1 : 0 < a
h2 : a < 1
⊢ a < 1
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
h1 : 0 < a
h2 : a ≤ 1
⊢ a ≤ 1

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
h1 : 0 < a
h2 : a ≤ 1
⊢ a ≤ 1
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
⊢ 0 ≤ a ↔ 0 ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a : α
inst✝ : IsStrictOrderedRing α
⊢ 0 < a ↔ 0 < a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : IsStrictOrderedRing α
a : α
⊢ a = a

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.List.Defs

Goal state before:
pp: case cons
M : Type u_2
inst✝¹ : Mul M
inst✝ : One M
l✝ : List M
p : M → Prop
hom : ∀ (a b : M), p a → p b → p (a * b)
unit : p 1
a : M
l : List M
ih : (∀ (x : M), x ∈ l → p x) → p l.prod
base : ∀ (x : M), x ∈ a :: l → p x
⊢ p (a * l.prod)

Tactic:
]

Goal state after:
case cons
M : Type u_2
inst✝¹ : Mul M
inst✝ : One M
l✝ : List M
p : M → Prop
hom : ∀ (a b : M), p a → p b → p (a * b)
unit : p 1
a : M
l : List M
ih : (∀ (x : M), x ∈ l → p x) → p l.prod
base : ∀ (x : M), x ∈ a :: l → p x
⊢ p (a * l.prod)
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a b c d e : α
inst✝ : IsStrictOrderedRing α
h : a / b ≤ c / d
he : 0 ≤ e
⊢ a / b * (1 / e) ≤ c / d * (1 / e)

Tactic:
]

Goal state after:
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
a b c d e : α
inst✝ : IsStrictOrderedRing α
h : a / b ≤ c / d
he : 0 ≤ e
⊢ a / b * (1 / e) ≤ c / d * (1 / e)
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b c d : α
h : a * b / c ≤ d
hc : 0 < c
⊢ b * a ≤ d * c

Tactic:
]

Goal state after:
α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b c d : α
h : a * b / c ≤ d
hc : 0 < c
⊢ b * a ≤ d * c
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: case eq
α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * ↑db + nb * ↑da = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ⅟↑da + ↑nb * ⅟↑db = ↑k * ↑nc * ⅟↑db * ⅟↑da
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑db * ⅟↑da = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc

Tactic:
]

Goal state after:
case eq
α : Type u_1
inst✝ : Ring α
na nb nc : ℤ
da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * ↑db + nb * ↑da = ↑k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ⅟↑da + ↑nb * ⅟↑db = ↑k * ↑nc * ⅟↑db * ⅟↑da
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑db * ⅟↑da = ↑nc * ↑k * ↑dc * ⅟↑db * ⅟↑da * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b c d : α
h : a * b / c ≤ d
hc : 0 < c
⊢ a * b / c ≤ d

Tactic:
]

Goal state after:
α : Type u_2
inst✝² : Semifield α
inst✝¹ : PartialOrder α
inst✝ : PosMulReflectLT α
a b c d : α
h : a * b / c ≤ d
hc : 0 < c
⊢ a * b / c ≤ d
Module: Mathlib.Algebra.BigOperators.Group.List.Defs

Goal state before:
pp: case neg
M : Type u_2
inst✝ : MulOneClass M
l✝ : List M
p : M → Prop
hom : ∀ (a b : M), p a → p b → p (a * b)
a : M
l : List M
ih : l ≠ [] → (∀ (x : M), x ∈ l → p x) → p l.prod
hl : a :: l ≠ []
base : ∀ (x : M), x ∈ a :: l → p x
hl_empty : ¬l = []
⊢ p (a * l.prod)

Tactic:
]

Goal state after:
case neg
M : Type u_2
inst✝ : MulOneClass M
l✝ : List M
p : M → Prop
hom : ∀ (a b : M), p a → p b → p (a * b)
a : M
l : List M
ih : l ≠ [] → (∀ (x : M), x ∈ l → p x) → p l.prod
hl : a :: l ≠ []
base : ∀ (x : M), x ∈ a :: l → p x
hl_empty : ¬l = []
⊢ p (a * l.prod)
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
num✝ : ℤ
denom✝ : ℕ
h : Invertible ↑denom✝
⊢ -↑num✝ * ⅟↑denom✝ = ↑num✝.neg * ⅟↑denom✝

Tactic:
← Int.cast_neg

Goal state after:
α : Type u_1
inst✝ : Ring α
num✝ : ℤ
denom✝ : ℕ
h : Invertible ↑denom✝
⊢ ↑(-num✝) * ⅟↑denom✝ = ↑num✝.neg * ⅟↑denom✝
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
num✝ : ℤ
denom✝ : ℕ
h : Invertible ↑denom✝
⊢ ↑(-num✝) * ⅟↑denom✝ = ↑num✝.neg * ⅟↑denom✝

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Ring α
num✝ : ℤ
denom✝ : ℕ
h : Invertible ↑denom✝
⊢ ↑(-num✝) * ⅟↑denom✝ = ↑num✝.neg * ⅟↑denom✝
Module: Mathlib.Algebra.BigOperators.Group.List.Defs

Goal state before:
pp: case zero
M : Type u_2
inst✝ : Monoid M
a : M
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.List.Defs

Goal state before:
pp: case succ
M : Type u_2
inst✝ : Monoid M
a : M
n : ℕ
ih : (replicate n a).prod = a ^ n
⊢ a * a ^ n = a * a ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
f : α → α → α
a b : α
na nb nc : ℤ
da db dc k : ℕ
hf : f = HSub.hSub
ra : IsRat a na da
rb : IsRat b nb db
h₁ : (na.mul ↑db).sub (nb.mul ↑da) = (↑k).mul nc
h₂ : da.mul db = k.mul dc
⊢ IsRat (a + -b) nc dc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Ring α
f : α → α → α
a b : α
na nb nc : ℤ
da db dc k : ℕ
hf : f = HSub.hSub
ra : IsRat a na da
rb : IsRat b nb db
h₁ : (na.mul ↑db).sub (nb.mul ↑da) = (↑k).mul nc
h₂ : da.mul db = k.mul dc
⊢ IsRat (a + -b) nc dc
Module: Mathlib.Algebra.BigOperators.Group.List.Defs

Goal state before:
pp: M : Type u_2
inst✝ : Monoid M
l : List M
m : M
h : ∀ (x : M), x ∈ l → x = m
⊢ l.prod = l.prod

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Ring α
f : α → α → α
a b : α
na nb nc : ℤ
da db dc k : ℕ
hf : f = HSub.hSub
ra : IsRat a na da
rb : IsRat b nb db
h₁ : (na.mul ↑db).sub (nb.mul ↑da) = (↑k).mul nc
h₂ : da.mul db = k.mul dc
⊢ (na.mul ↑db).add (-(nb * ↑da)) = (↑k).mul nc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Ring α
f : α → α → α
a b : α
na nb nc : ℤ
da db dc k : ℕ
hf : f = HSub.hSub
ra : IsRat a na da
rb : IsRat b nb db
h₁ : (na.mul ↑db).sub (nb.mul ↑da) = (↑k).mul nc
h₂ : da.mul db = k.mul dc
⊢ (na.mul ↑db).add (-(nb * ↑da)) = (↑k).mul nc
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ↑da * ↑db * ⅟↑db * ⅟↑da * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑k * ↑nc * ⅟↑da * ⅟↑db = ↑nc * ⅟↑dc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ↑da * ↑db * ⅟↑db * ⅟↑da * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑k * ↑nc * ⅟↑da * ⅟↑db = ↑nc * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: α : Type u_4
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b : α
hb✝ : 0 ≤ b
h : ∀ (ε : α), 1 < ε → a ≤ b * ε
hb : 0 < b
x : α
hbx : b < x
⊢ x = x

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: case inr
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
s : Set α
hs : IsGLB s b
ha : 0 ≤ 0
⊢ IsGLB {0} 0

Tactic:
]

Goal state after:
case inr
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
s : Set α
hs : IsGLB s b
ha : 0 ≤ 0
⊢ IsGLB {0} 0
Module: Mathlib.Tactic.NormNum.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑da * ⅟↑db = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : Semiring α
na nb nc da db dc k : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h₁✝ : na * nb = k * nc
h₂✝ : da * db = k * dc
this✝ : Invertible ↑(da * db)
this : Invertible ↑dc
H : ∀ (a : α), a * ⅟↑da * ⅟↑db = a * ⅟↑db * ⅟↑da
h₁ : ↑na * ↑nb = ↑k * ↑nc
h₂ : ↑nc * ⅟↑dc = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
⊢ ↑nc * ↑k * ⅟↑da * ⅟↑db = ↑nc * ↑k * ↑dc * ⅟↑da * ⅟↑db * ⅟↑dc
Module: Mathlib.Algebra.Order.Field.Basic

Goal state before:
pp: case inr.inl
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
ha : 0 ≤ 0
hs : IsBot b
hb : 0 ≤ b
⊢ IsBot 0

Tactic:
]

Goal state after:
case inr.inl
α : Type u_2
inst✝³ : Semifield α
inst✝² : PartialOrder α
inst✝¹ : PosMulReflectLT α
b : α
inst✝ : IsStrictOrderedRing α
ha : 0 ≤ 0
hs : IsBot b
hb : 0 ≤ b
⊢ IsBot 0