mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ a * a⁻¹ ≤ a * b ↔ 1 ≤ a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ b * a ≤ b * b⁻¹ ↔ b * a ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ 1 ≤ b⁻¹ * a ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ b ≤ a * 1 ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a : α
⊢ 1 < a⁻¹ ↔ a < 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a : α
⊢ a⁻¹ < 1 ↔ 1 < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ b < a⁻¹ * c ↔ a * b < c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ b⁻¹ * a < c ↔ a < b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b : α
⊢ a * a⁻¹ < a * b ↔ 1 < a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b : α
⊢ b * a < b * b⁻¹ ↔ b * a < 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b : α
⊢ 1 < b⁻¹ * a ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b : α
⊢ b < a * 1 ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a : α
⊢ a⁻¹ ≤ 1 ↔ 1 ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a : α
⊢ 1 ≤ a⁻¹ ↔ a ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a⁻¹ * a ≤ b * a ↔ 1 ≤ b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a * b ≤ b⁻¹ * b ↔ a * b ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b c : α
⊢ a * b⁻¹ * b ≤ c * b ↔ a ≤ c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b c : α
⊢ c * b ≤ a * b⁻¹ * b ↔ c * b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a ≤ 1 * b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ 1 ≤ a * b⁻¹ ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ b ≤ 1 * a ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a : α
⊢ a⁻¹ < 1 ↔ 1 < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a : α
⊢ 1 < a⁻¹ ↔ a < 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ a⁻¹ * a < b * a ↔ 1 < b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ a * b < b⁻¹ * b ↔ a * b < 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b c : α
⊢ a * b⁻¹ < c ↔ a < c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b c : α
⊢ c * b < a * b⁻¹ * b ↔ c * b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ a * b⁻¹ < 1 ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ 1 < a * b⁻¹ ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ b < 1 * a ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ a / b ≤ a ↔ 1 ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b : α
⊢ a ≤ a / b ↔ b ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LE α
inst✝¹ : MulLeftMono α
a b : α
inst✝ : MulRightMono α
⊢ a⁻¹ ≤ b⁻¹ ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LE α
inst✝¹ : MulLeftMono α
a b c d : α
inst✝ : MulRightMono α
⊢ a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b : α
⊢ a / b < a ↔ 1 < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE

Goal state before:
pp: α : Type u
inst✝³ : MulOneClass α
inst✝² : Preorder α
inst✝¹ : ExistsMulOfLE α
a b : α
inst✝ : MulLeftReflectLE α
h : a ≤ b
⊢ ∃ c, 1 ≤ c ∧ a * c = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE

Goal state before:
pp: α : Type u
inst✝³ : MulOneClass α
inst✝² : Preorder α
inst✝¹ : ExistsMulOfLE α
a b : α
inst✝ : MulLeftReflectLT α
h : a < b
⊢ ∃ c, 1 < c ∧ a * c = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
a b : α
inst✝ : MulRightStrictMono α
⊢ a⁻¹ < b⁻¹ ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
a b : α
inst✝ : MulRightStrictMono α
⊢ a⁻¹ < b ↔ b⁻¹ < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE

Goal state before:
pp: α : Type u
inst✝⁴ : MulOneClass α
inst✝³ : Preorder α
inst✝² : ExistsMulOfLE α
a b : α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
⊢ (∃ c, 1 ≤ c ∧ a * c = b) → a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
a b : α
inst✝ : MulRightStrictMono α
⊢ a < b⁻¹ ↔ b < a⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE

Goal state before:
pp: α : Type u
inst✝⁴ : MulOneClass α
inst✝³ : Preorder α
inst✝² : ExistsMulOfLE α
a b : α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulLeftReflectLT α
⊢ (∃ c, 1 < c ∧ a * c = b) → a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
a b c d : α
inst✝ : MulRightStrictMono α
⊢ a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c : α
⊢ c⁻¹ * a ≤ b ↔ a ≤ b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE

Goal state before:
pp: α : Type u
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : Monoid α
inst✝¹ : ExistsMulOfLE α
inst✝ : MulLeftReflectLT α
a b : α
h : ∀ (ε : α), 1 < ε → a ≤ b * ε
x : α
hxb : b < x
⊢ a ≤ x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c : α
⊢ a * b⁻¹ ≤ c ↔ a ≤ b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c d : α
⊢ a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ c⁻¹ * a < b ↔ a < b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ a * b⁻¹ < c ↔ a < b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c d : α
⊢ a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b c : α
⊢ a / c ≤ b / c ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ 1 ≤ a / b ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a / b ≤ 1 ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b c : α
⊢ a ≤ c / b ↔ a * b ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b c : α
⊢ a / c ≤ b ↔ a ≤ b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LE α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
b c a : α
⊢ a / b ≤ a / c ↔ c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c d : α
⊢ a / b ≤ c / d ↔ a * d ≤ c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : Preorder α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
a b c d : α
h₁ : a ≤ b
h₂ : c ≤ d
⊢ a * c ≤ b * d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c : α
⊢ b ≤ c / a ↔ a * b ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : Preorder α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddLeftMono α
⊢ 0 ≤ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c : α
⊢ a / b ≤ c ↔ a ≤ b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : Preorder α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddLeftMono α
⊢ 0 ≤ 3

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : Preorder α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddLeftMono α
⊢ 0 ≤ 4

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LE α
inst✝ : MulLeftMono α
a b c : α
⊢ a⁻¹ ≤ b / c ↔ c ≤ a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : LE α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddLeftMono α
⊢ 1 + 0 ≤ 1 + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : MulLeftMono α
this : MulRightMono α
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₂ * b₂ ≤ a₁ * b₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : MulLeftMono α
this : MulRightMono α
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₂ * b₂ ≤ a₁ * b₁

Tactic:
by rintro ⟨rfl, rfl⟩; rfl

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝³ : AddMonoidWithOne α
inst✝² : LE α
inst✝¹ : ZeroLEOneClass α
inst✝ : AddRightMono α
⊢ 0 + 1 ≤ 1 + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddMonoidWithOne α
inst✝³ : PartialOrder α
inst✝² : ZeroLEOneClass α
inst✝¹ : NeZero 1
inst✝ : AddLeftMono α
⊢ 0 < 3

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddMonoidWithOne α
inst✝³ : PartialOrder α
inst✝² : ZeroLEOneClass α
inst✝¹ : NeZero 1
inst✝ : AddLeftMono α
⊢ 0 < 4

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
⊢ a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : Preorder α
inst✝ : MulLeftMono α
a b c d : α
hab : a ≤ b
hcd : c ≤ d
⊢ a / d ≤ b / c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.NatCast

Goal state before:
pp: α : Type u_1
inst✝⁴ : AddMonoidWithOne α
inst✝³ : PartialOrder α
inst✝² : ZeroLEOneClass α
inst✝¹ : NeZero 1
inst✝ : AddLeftStrictMono α
⊢ 1 < 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b c : α
⊢ a / c < b / c ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ 1 < a / b ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b c d : α
hac : a ≤ c
hbd : b ≤ d
⊢ a * b = c * d ↔ a = c ∧ b = d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b : α
⊢ a / b < 1 ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : Mul α
inst✝¹ : PartialOrder α
inst✝ : MulRightStrictMono α
a b c : α
h : b ≤ c ∨ c ≤ b
⊢ c * a = b * a ↔ c = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b c : α
⊢ a < c / b ↔ a * b < c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : Mul α
inst✝¹ : PartialOrder α
inst✝ : MulLeftStrictMono α
a b c : α
h : b ≤ c ∨ c ≤ b
⊢ a * c = a * b ↔ c = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LT α
inst✝ : MulRightStrictMono α
a b c : α
⊢ a / c < b ↔ a < b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
b c a : α
⊢ a / b < a / c ↔ c < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LT α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b c : α
⊢ a⁻¹ < b / c ↔ c < a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c d : α
⊢ a / b < c / d ↔ a * d < c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: case inr.inr.inr.inr
α : Type u_1
inst✝³ : Mul α
inst✝² : LinearOrder α
a b c d : α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
h : a * b = c * d
hca : c < a
hdb : d < b
⊢ a = c ∧ b = d ∨ a < c ∨ b < d

Tactic:
exact False.elim <| ne_of_lt (mul_lt_mul_of_lt_of_lt hca hdb) h.symm

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ b < c / a ↔ a * b < c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : LinearOrder α
a b c d : α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
h : a * b = c * d
⊢ a = c ∧ b = d ∨ a < c ∨ b < d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LT α
inst✝ : MulLeftStrictMono α
a b c : α
⊢ a / b < c ↔ a < b * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : Preorder α
inst✝ : MulLeftStrictMono α
a b c d : α
hab : a < b
hcd : c < d
⊢ a / d < b / c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : LinearOrder α
a b c d : α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
h : a * b < c * d
⊢ min a b < max c d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : CommGroup α
inst✝¹ : LinearOrder α
inst✝ : MulLeftMono α
a b c d : α
⊢ a / d < b / c → a < b ∨ c < d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LinearOrder α
inst✝ : MulRightMono α
a b : α
⊢ cmp (a / b) 1 = cmp a b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : LinearOrder α
a b c d : α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightMono α
h : a * b ≤ c * d
⊢ min a b ≤ max c d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝² : Group α
inst✝¹ : LinearOrder α
inst✝ : MulLeftMono α
a b : α
h : ∀ (ε : α), 1 < ε → a < b * ε
h₁ : b < a
⊢ a < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : LinearOrder α
a b c d : α
inst✝¹ : MulLeftMono α
inst✝ : MulRightStrictMono α
h : a * b ≤ c * d
⊢ min a b ≤ max c d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LE α
inst✝ : MulLeftReflectLE α
a b : α
h : a ≤ a * b
⊢ a * 1 ≤ a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LE α
inst✝ : MulLeftReflectLE α
a b : α
h : a * b ≤ a
⊢ a * b ≤ a * 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LE α
inst✝ : MulRightReflectLE α
a b : α
h : b ≤ a * b
⊢ 1 * b ≤ a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LE α
inst✝ : MulRightReflectLE α
a b : α
h : a * b ≤ b
⊢ a * b ≤ 1 * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LE α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
a b : α
⊢ a ≤ a * b ↔ a * 1 ≤ a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Group.Unbundled.Basic

Goal state before:
pp: α : Type u
inst✝³ : Group α
inst✝² : LinearOrder α
inst✝¹ : MulLeftMono α
a b : α
inst✝ : MulRightMono α
⊢ a / b ≤ a⁻¹ * b ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LE α
inst✝¹ : MulRightMono α
inst✝ : MulRightReflectLE α
a b : α
⊢ a ≤ b * a ↔ 1 * a ≤ b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LE α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
a b : α
⊢ a * b ≤ a ↔ a * b ≤ a * 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LE α
inst✝¹ : MulRightMono α
inst✝ : MulRightReflectLE α
a b : α
⊢ a * b ≤ b ↔ a * b ≤ 1 * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LT α
inst✝ : MulLeftReflectLT α
a b : α
h : a < a * b
⊢ a * 1 < a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Unbundled.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MulOneClass α
inst✝¹ : LT α
inst✝ : MulLeftReflectLT α
a b : α
h : a * b < a
⊢ a * b < a * 1

Tactic:
by

Goal state after: