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Basic.lean
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Basic.lean
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/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
/-!
# Ordered groups
This file develops the basics of unbundled ordered groups.
## Implementation details
Unfortunately, the number of `'` appended to lemmas in this file
may differ between the multiplicative and the additive version of a lemma.
The reason is that we did not want to change existing names in the library.
-/
assert_not_exists OrderedCommMonoid
open Function
universe u
variable {α : Type u}
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
/-- Uses `left` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a]
simp
/-- Uses `left` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by
rw [← mul_le_mul_iff_left a]
simp
@[to_additive (attr := simp)]
theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by
rw [← mul_le_mul_iff_left a]
simp
@[to_additive (attr := simp)]
theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c := by
rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]
@[to_additive neg_le_iff_add_nonneg']
theorem inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b :=
(mul_le_mul_iff_left a).symm.trans <| by rw [mul_inv_self]
@[to_additive]
theorem le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 :=
(mul_le_mul_iff_left b).symm.trans <| by rw [mul_inv_self]
@[to_additive]
theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a := by
rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
@[to_additive]
theorem inv_mul_le_one_iff : a⁻¹ * b ≤ 1 ↔ b ≤ a :=
-- Porting note: why is the `_root_` needed?
_root_.trans inv_mul_le_iff_le_mul <| by rw [mul_one]
end TypeclassesLeftLE
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
/-- Uses `left` co(ntra)variant. -/
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
/-- Uses `left` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
@[to_additive (attr := simp)]
theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c := by
rw [← mul_lt_mul_iff_left a]
simp
@[to_additive (attr := simp)]
theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
@[to_additive]
theorem inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < a * b :=
(mul_lt_mul_iff_left a).symm.trans <| by rw [mul_inv_self]
@[to_additive]
theorem lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 :=
(mul_lt_mul_iff_left b).symm.trans <| by rw [mul_inv_self]
@[to_additive]
theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a := by
rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
@[to_additive]
theorem inv_mul_lt_one_iff : a⁻¹ * b < 1 ↔ b < a :=
_root_.trans inv_mul_lt_iff_lt_mul <| by rw [mul_one]
end TypeclassesLeftLT
section TypeclassesRightLE
variable [LE α] [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
/-- Uses `right` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_right a]
simp
/-- Uses `right` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by
rw [← mul_le_mul_iff_right a]
simp
@[to_additive neg_le_iff_add_nonneg]
theorem inv_le_iff_one_le_mul : a⁻¹ ≤ b ↔ 1 ≤ b * a :=
(mul_le_mul_iff_right a).symm.trans <| by rw [inv_mul_self]
@[to_additive]
theorem le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans <| by rw [inv_mul_self]
@[to_additive (attr := simp)]
theorem mul_inv_le_iff_le_mul : a * b⁻¹ ≤ c ↔ a ≤ c * b :=
(mul_le_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel_right]
@[to_additive (attr := simp)]
theorem le_mul_inv_iff_mul_le : c ≤ a * b⁻¹ ↔ c * b ≤ a :=
(mul_le_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel_right]
-- Porting note (#10618): `simp` can prove this
@[to_additive]
theorem mul_inv_le_one_iff_le : a * b⁻¹ ≤ 1 ↔ a ≤ b :=
mul_inv_le_iff_le_mul.trans <| by rw [one_mul]
@[to_additive]
theorem le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ b ≤ a := by
rw [← mul_le_mul_iff_right b, one_mul, inv_mul_cancel_right]
@[to_additive]
theorem mul_inv_le_one_iff : b * a⁻¹ ≤ 1 ↔ b ≤ a :=
_root_.trans mul_inv_le_iff_le_mul <| by rw [one_mul]
end TypeclassesRightLE
section TypeclassesRightLT
variable [LT α] [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
/-- Uses `right` co(ntra)variant. -/
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
/-- Uses `right` co(ntra)variant. -/
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
@[to_additive]
theorem inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a :=
(mul_lt_mul_iff_right a).symm.trans <| by rw [inv_mul_self]
@[to_additive]
theorem lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 :=
(mul_lt_mul_iff_right b).symm.trans <| by rw [inv_mul_self]
@[to_additive (attr := simp)]
theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b := by
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right]
@[to_additive (attr := simp)]
theorem lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * b < a :=
(mul_lt_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel_right]
-- Porting note (#10618): `simp` can prove this
@[to_additive]
theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b := by
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
@[to_additive]
theorem lt_mul_inv_iff_lt : 1 < a * b⁻¹ ↔ b < a := by
rw [← mul_lt_mul_iff_right b, one_mul, inv_mul_cancel_right]
@[to_additive]
theorem mul_inv_lt_one_iff : b * a⁻¹ < 1 ↔ b < a :=
_root_.trans mul_inv_lt_iff_lt_mul <| by rw [one_mul]
end TypeclassesRightLT
section TypeclassesLeftRightLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by
rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b]
simp
alias ⟨le_of_neg_le_neg, _⟩ := neg_le_neg_iff
@[to_additive]
theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b := by
rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
@[to_additive (attr := simp)]
theorem div_le_self_iff (a : α) {b : α} : a / b ≤ a ↔ 1 ≤ b := by
simp [div_eq_mul_inv]
@[to_additive (attr := simp)]
theorem le_div_self_iff (a : α) {b : α} : a ≤ a / b ↔ b ≤ 1 := by
simp [div_eq_mul_inv]
alias ⟨_, sub_le_self⟩ := sub_le_self_iff
end TypeclassesLeftRightLE
section TypeclassesLeftRightLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
@[to_additive neg_lt]
theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by rw [← inv_lt_inv_iff, inv_inv]
@[to_additive lt_neg]
theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by rw [← inv_lt_inv_iff, inv_inv]
alias ⟨lt_inv_of_lt_inv, _⟩ := lt_inv'
attribute [to_additive] lt_inv_of_lt_inv
alias ⟨inv_lt_of_inv_lt', _⟩ := inv_lt'
attribute [to_additive neg_lt_of_neg_lt] inv_lt_of_inv_lt'
@[to_additive]
theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b := by
rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
@[to_additive (attr := simp)]
theorem div_lt_self_iff (a : α) {b : α} : a / b < a ↔ 1 < b := by
simp [div_eq_mul_inv]
alias ⟨_, sub_lt_self⟩ := sub_lt_self_iff
end TypeclassesLeftRightLT
section Preorder
variable [Preorder α]
section LeftLE
variable [CovariantClass α α (· * ·) (· ≤ ·)] {a : α}
@[to_additive]
theorem Left.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (Left.inv_le_one_iff.mpr h) h
alias neg_le_self := Left.neg_le_self
@[to_additive]
theorem Left.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (Left.one_le_inv_iff.mpr h)
end LeftLE
section LeftLT
variable [CovariantClass α α (· * ·) (· < ·)] {a : α}
@[to_additive]
theorem Left.inv_lt_self (h : 1 < a) : a⁻¹ < a :=
(Left.inv_lt_one_iff.mpr h).trans h
alias neg_lt_self := Left.neg_lt_self
@[to_additive]
theorem Left.self_lt_inv (h : a < 1) : a < a⁻¹ :=
lt_trans h (Left.one_lt_inv_iff.mpr h)
end LeftLT
section RightLE
variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a : α}
@[to_additive]
theorem Right.inv_le_self (h : 1 ≤ a) : a⁻¹ ≤ a :=
le_trans (Right.inv_le_one_iff.mpr h) h
@[to_additive]
theorem Right.self_le_inv (h : a ≤ 1) : a ≤ a⁻¹ :=
le_trans h (Right.one_le_inv_iff.mpr h)
end RightLE
section RightLT
variable [CovariantClass α α (swap (· * ·)) (· < ·)] {a : α}
@[to_additive]
theorem Right.inv_lt_self (h : 1 < a) : a⁻¹ < a :=
(Right.inv_lt_one_iff.mpr h).trans h
@[to_additive]
theorem Right.self_lt_inv (h : a < 1) : a < a⁻¹ :=
lt_trans h (Right.one_lt_inv_iff.mpr h)
end RightLT
end Preorder
end Group
section CommGroup
variable [CommGroup α]
section LE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive]
theorem inv_mul_le_iff_le_mul' : c⁻¹ * a ≤ b ↔ a ≤ b * c := by rw [inv_mul_le_iff_le_mul, mul_comm]
-- Porting note: `simp` simplifies LHS to `a ≤ c * b`
@[to_additive]
theorem mul_inv_le_iff_le_mul' : a * b⁻¹ ≤ c ↔ a ≤ b * c := by
rw [← inv_mul_le_iff_le_mul, mul_comm]
@[to_additive add_neg_le_add_neg_iff]
theorem mul_inv_le_mul_inv_iff' : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b := by
rw [mul_comm c, mul_inv_le_inv_mul_iff, mul_comm]
end LE
section LT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c d : α}
@[to_additive]
theorem inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c := by rw [inv_mul_lt_iff_lt_mul, mul_comm]
-- Porting note: `simp` simplifies LHS to `a < c * b`
@[to_additive]
theorem mul_inv_lt_iff_le_mul' : a * b⁻¹ < c ↔ a < b * c := by
rw [← inv_mul_lt_iff_lt_mul, mul_comm]
@[to_additive add_neg_lt_add_neg_iff]
theorem mul_inv_lt_mul_inv_iff' : a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b := by
rw [mul_comm c, mul_inv_lt_inv_mul_iff, mul_comm]
end LT
end CommGroup
alias ⟨one_le_of_inv_le_one, _⟩ := Left.inv_le_one_iff
attribute [to_additive] one_le_of_inv_le_one
alias ⟨le_one_of_one_le_inv, _⟩ := Left.one_le_inv_iff
attribute [to_additive nonpos_of_neg_nonneg] le_one_of_one_le_inv
alias ⟨lt_of_inv_lt_inv, _⟩ := inv_lt_inv_iff
attribute [to_additive] lt_of_inv_lt_inv
alias ⟨one_lt_of_inv_lt_one, _⟩ := Left.inv_lt_one_iff
attribute [to_additive] one_lt_of_inv_lt_one
alias inv_lt_one_iff_one_lt := Left.inv_lt_one_iff
attribute [to_additive] inv_lt_one_iff_one_lt
alias inv_lt_one' := Left.inv_lt_one_iff
attribute [to_additive neg_lt_zero] inv_lt_one'
alias ⟨inv_of_one_lt_inv, _⟩ := Left.one_lt_inv_iff
attribute [to_additive neg_of_neg_pos] inv_of_one_lt_inv
alias ⟨_, one_lt_inv_of_inv⟩ := Left.one_lt_inv_iff
attribute [to_additive neg_pos_of_neg] one_lt_inv_of_inv
alias ⟨mul_le_of_le_inv_mul, _⟩ := le_inv_mul_iff_mul_le
attribute [to_additive] mul_le_of_le_inv_mul
alias ⟨_, le_inv_mul_of_mul_le⟩ := le_inv_mul_iff_mul_le
attribute [to_additive] le_inv_mul_of_mul_le
alias ⟨_, inv_mul_le_of_le_mul⟩ := inv_mul_le_iff_le_mul
-- Porting note: was `inv_mul_le_iff_le_mul`
attribute [to_additive] inv_mul_le_of_le_mul
alias ⟨mul_lt_of_lt_inv_mul, _⟩ := lt_inv_mul_iff_mul_lt
attribute [to_additive] mul_lt_of_lt_inv_mul
alias ⟨_, lt_inv_mul_of_mul_lt⟩ := lt_inv_mul_iff_mul_lt
attribute [to_additive] lt_inv_mul_of_mul_lt
alias ⟨lt_mul_of_inv_mul_lt, inv_mul_lt_of_lt_mul⟩ := inv_mul_lt_iff_lt_mul
attribute [to_additive] lt_mul_of_inv_mul_lt
attribute [to_additive] inv_mul_lt_of_lt_mul
alias lt_mul_of_inv_mul_lt_left := lt_mul_of_inv_mul_lt
attribute [to_additive] lt_mul_of_inv_mul_lt_left
alias inv_le_one' := Left.inv_le_one_iff
attribute [to_additive neg_nonpos] inv_le_one'
alias one_le_inv' := Left.one_le_inv_iff
attribute [to_additive neg_nonneg] one_le_inv'
alias one_lt_inv' := Left.one_lt_inv_iff
attribute [to_additive neg_pos] one_lt_inv'
-- Most of the lemmas that are primed in this section appear in ordered_field.
-- I (DT) did not try to minimise the assumptions.
section Group
variable [Group α] [LE α]
section Right
variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
@[to_additive]
theorem div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b := by
simpa only [div_eq_mul_inv] using mul_le_mul_iff_right _
@[to_additive (attr := gcongr) sub_le_sub_right]
theorem div_le_div_right' (h : a ≤ b) (c : α) : a / c ≤ b / c :=
(div_le_div_iff_right c).2 h
@[to_additive (attr := simp) sub_nonneg]
theorem one_le_div' : 1 ≤ a / b ↔ b ≤ a := by
rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨le_of_sub_nonneg, sub_nonneg_of_le⟩ := sub_nonneg
@[to_additive sub_nonpos]
theorem div_le_one' : a / b ≤ 1 ↔ a ≤ b := by
rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨le_of_sub_nonpos, sub_nonpos_of_le⟩ := sub_nonpos
@[to_additive]
theorem le_div_iff_mul_le : a ≤ c / b ↔ a * b ≤ c := by
rw [← mul_le_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨add_le_of_le_sub_right, le_sub_right_of_add_le⟩ := le_sub_iff_add_le
@[to_additive]
theorem div_le_iff_le_mul : a / c ≤ b ↔ a ≤ b * c := by
rw [← mul_le_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]
-- Note: we intentionally don't have `@[simp]` for the additive version,
-- since the LHS simplifies with `tsub_le_iff_right`
attribute [simp] div_le_iff_le_mul
-- TODO: Should we get rid of `sub_le_iff_le_add` in favor of
-- (a renamed version of) `tsub_le_iff_right`?
-- see Note [lower instance priority]
instance (priority := 100) AddGroup.toOrderedSub {α : Type*} [AddGroup α] [LE α]
[CovariantClass α α (swap (· + ·)) (· ≤ ·)] : OrderedSub α :=
⟨fun _ _ _ => sub_le_iff_le_add⟩
end Right
section Left
variable [CovariantClass α α (· * ·) (· ≤ ·)]
variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
@[to_additive]
theorem div_le_div_iff_left (a : α) : a / b ≤ a / c ↔ c ≤ b := by
rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_le_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_le_inv_iff]
@[to_additive (attr := gcongr) sub_le_sub_left]
theorem div_le_div_left' (h : a ≤ b) (c : α) : c / b ≤ c / a :=
(div_le_div_iff_left c).2 h
end Left
end Group
section CommGroup
variable [CommGroup α]
section LE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive sub_le_sub_iff]
theorem div_le_div_iff' : a / b ≤ c / d ↔ a * d ≤ c * b := by
simpa only [div_eq_mul_inv] using mul_inv_le_mul_inv_iff'
@[to_additive]
theorem le_div_iff_mul_le' : b ≤ c / a ↔ a * b ≤ c := by rw [le_div_iff_mul_le, mul_comm]
alias ⟨add_le_of_le_sub_left, le_sub_left_of_add_le⟩ := le_sub_iff_add_le'
@[to_additive]
theorem div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c := by rw [div_le_iff_le_mul, mul_comm]
alias ⟨le_add_of_sub_left_le, sub_left_le_of_le_add⟩ := sub_le_iff_le_add'
@[to_additive (attr := simp)]
theorem inv_le_div_iff_le_mul : b⁻¹ ≤ a / c ↔ c ≤ a * b :=
le_div_iff_mul_le.trans inv_mul_le_iff_le_mul'
@[to_additive]
theorem inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b := by rw [inv_le_div_iff_le_mul, mul_comm]
@[to_additive]
theorem div_le_comm : a / b ≤ c ↔ a / c ≤ b :=
div_le_iff_le_mul'.trans div_le_iff_le_mul.symm
@[to_additive]
theorem le_div_comm : a ≤ b / c ↔ c ≤ b / a :=
le_div_iff_mul_le'.trans le_div_iff_mul_le.symm
end LE
section Preorder
variable [Preorder α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := gcongr) sub_le_sub]
theorem div_le_div'' (hab : a ≤ b) (hcd : c ≤ d) : a / d ≤ b / c := by
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_le_inv_mul_iff, mul_comm]
exact mul_le_mul' hab hcd
end Preorder
end CommGroup
-- Most of the lemmas that are primed in this section appear in ordered_field.
-- I (DT) did not try to minimise the assumptions.
section Group
variable [Group α] [LT α]
section Right
variable [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
@[to_additive (attr := simp)]
theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b := by
simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _
@[to_additive (attr := gcongr) sub_lt_sub_right]
theorem div_lt_div_right' (h : a < b) (c : α) : a / c < b / c :=
(div_lt_div_iff_right c).2 h
@[to_additive (attr := simp) sub_pos]
theorem one_lt_div' : 1 < a / b ↔ b < a := by
rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨lt_of_sub_pos, sub_pos_of_lt⟩ := sub_pos
@[to_additive (attr := simp) sub_neg "For `a - -b = a + b`, see `sub_neg_eq_add`."]
theorem div_lt_one' : a / b < 1 ↔ a < b := by
rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨lt_of_sub_neg, sub_neg_of_lt⟩ := sub_neg
alias sub_lt_zero := sub_neg
@[to_additive]
theorem lt_div_iff_mul_lt : a < c / b ↔ a * b < c := by
rw [← mul_lt_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨add_lt_of_lt_sub_right, lt_sub_right_of_add_lt⟩ := lt_sub_iff_add_lt
@[to_additive]
theorem div_lt_iff_lt_mul : a / c < b ↔ a < b * c := by
rw [← mul_lt_mul_iff_right c, div_eq_mul_inv, inv_mul_cancel_right]
alias ⟨lt_add_of_sub_right_lt, sub_right_lt_of_lt_add⟩ := sub_lt_iff_lt_add
end Right
section Left
variable [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c : α}
@[to_additive (attr := simp)]
theorem div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b := by
rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_lt_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_lt_inv_iff]
@[to_additive (attr := simp)]
theorem inv_lt_div_iff_lt_mul : a⁻¹ < b / c ↔ c < a * b := by
rw [div_eq_mul_inv, lt_mul_inv_iff_mul_lt, inv_mul_lt_iff_lt_mul]
@[to_additive (attr := gcongr) sub_lt_sub_left]
theorem div_lt_div_left' (h : a < b) (c : α) : c / b < c / a :=
(div_lt_div_iff_left c).2 h
end Left
end Group
section CommGroup
variable [CommGroup α]
section LT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c d : α}
@[to_additive sub_lt_sub_iff]
theorem div_lt_div_iff' : a / b < c / d ↔ a * d < c * b := by
simpa only [div_eq_mul_inv] using mul_inv_lt_mul_inv_iff'
@[to_additive]
theorem lt_div_iff_mul_lt' : b < c / a ↔ a * b < c := by rw [lt_div_iff_mul_lt, mul_comm]
alias ⟨add_lt_of_lt_sub_left, lt_sub_left_of_add_lt⟩ := lt_sub_iff_add_lt'
@[to_additive]
theorem div_lt_iff_lt_mul' : a / b < c ↔ a < b * c := by rw [div_lt_iff_lt_mul, mul_comm]
alias ⟨lt_add_of_sub_left_lt, sub_left_lt_of_lt_add⟩ := sub_lt_iff_lt_add'
@[to_additive]
theorem inv_lt_div_iff_lt_mul' : b⁻¹ < a / c ↔ c < a * b :=
lt_div_iff_mul_lt.trans inv_mul_lt_iff_lt_mul'
@[to_additive]
theorem div_lt_comm : a / b < c ↔ a / c < b :=
div_lt_iff_lt_mul'.trans div_lt_iff_lt_mul.symm
@[to_additive]
theorem lt_div_comm : a < b / c ↔ c < b / a :=
lt_div_iff_mul_lt'.trans lt_div_iff_mul_lt.symm
end LT
section Preorder
variable [Preorder α] [CovariantClass α α (· * ·) (· < ·)] {a b c d : α}
@[to_additive (attr := gcongr) sub_lt_sub]
theorem div_lt_div'' (hab : a < b) (hcd : c < d) : a / d < b / c := by
rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_lt_inv_mul_iff, mul_comm]
exact mul_lt_mul_of_lt_of_lt hab hcd
end Preorder
section LinearOrder
variable [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive] lemma lt_or_lt_of_div_lt_div : a / d < b / c → a < b ∨ c < d := by
contrapose!; exact fun h ↦ div_le_div'' h.1 h.2
end LinearOrder
end CommGroup
section LinearOrder
variable [Group α] [LinearOrder α]
@[to_additive (attr := simp) cmp_sub_zero]
theorem cmp_div_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] (a b : α) :
cmp (a / b) 1 = cmp a b := by rw [← cmp_mul_right' _ _ b, one_mul, div_mul_cancel]
variable [CovariantClass α α (· * ·) (· ≤ ·)]
section VariableNames
variable {a b c : α}
@[to_additive]
theorem le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b :=
le_of_not_lt fun h₁ => lt_irrefl a (by simpa using h _ (lt_inv_mul_iff_lt.mpr h₁))
@[to_additive]
theorem le_iff_forall_one_lt_lt_mul : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε :=
⟨fun h _ => lt_mul_of_le_of_one_lt h, le_of_forall_one_lt_lt_mul⟩
/- I (DT) introduced this lemma to prove (the additive version `sub_le_sub_flip` of)
`div_le_div_flip` below. Now I wonder what is the point of either of these lemmas... -/
@[to_additive]
theorem div_le_inv_mul_iff [CovariantClass α α (swap (· * ·)) (· ≤ ·)] :
a / b ≤ a⁻¹ * b ↔ a ≤ b := by
rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff]
exact
⟨fun h => not_lt.mp fun k => not_lt.mpr h (mul_lt_mul_of_lt_of_lt k k), fun h =>
mul_le_mul' h h⟩
-- What is the point of this lemma? See comment about `div_le_inv_mul_iff` above.
-- Note: we intentionally don't have `@[simp]` for the additive version,
-- since the LHS simplifies with `tsub_le_iff_right`
@[to_additive]
theorem div_le_div_flip {α : Type*} [CommGroup α] [LinearOrder α]
[CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} : a / b ≤ b / a ↔ a ≤ b := by
rw [div_eq_mul_inv b, mul_comm]
exact div_le_inv_mul_iff
end VariableNames
end LinearOrder
section
variable {β : Type*} [Group α] [Preorder α] [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (swap (· * ·)) (· ≤ ·)] [Preorder β] {f : β → α} {s : Set β}
@[to_additive]
theorem Monotone.inv (hf : Monotone f) : Antitone fun x => (f x)⁻¹ := fun _ _ hxy =>
inv_le_inv_iff.2 (hf hxy)
@[to_additive]
theorem Antitone.inv (hf : Antitone f) : Monotone fun x => (f x)⁻¹ := fun _ _ hxy =>
inv_le_inv_iff.2 (hf hxy)
@[to_additive]
theorem MonotoneOn.inv (hf : MonotoneOn f s) : AntitoneOn (fun x => (f x)⁻¹) s :=
fun _ hx _ hy hxy => inv_le_inv_iff.2 (hf hx hy hxy)
@[to_additive]
theorem AntitoneOn.inv (hf : AntitoneOn f s) : MonotoneOn (fun x => (f x)⁻¹) s :=
fun _ hx _ hy hxy => inv_le_inv_iff.2 (hf hx hy hxy)
end
section
variable {β : Type*} [Group α] [Preorder α] [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)] [Preorder β] {f : β → α} {s : Set β}
@[to_additive]
theorem StrictMono.inv (hf : StrictMono f) : StrictAnti fun x => (f x)⁻¹ := fun _ _ hxy =>
inv_lt_inv_iff.2 (hf hxy)
@[to_additive]
theorem StrictAnti.inv (hf : StrictAnti f) : StrictMono fun x => (f x)⁻¹ := fun _ _ hxy =>
inv_lt_inv_iff.2 (hf hxy)
@[to_additive]
theorem StrictMonoOn.inv (hf : StrictMonoOn f s) : StrictAntiOn (fun x => (f x)⁻¹) s :=
fun _ hx _ hy hxy => inv_lt_inv_iff.2 (hf hx hy hxy)
@[to_additive]
theorem StrictAntiOn.inv (hf : StrictAntiOn f s) : StrictMonoOn (fun x => (f x)⁻¹) s :=
fun _ hx _ hy hxy => inv_lt_inv_iff.2 (hf hx hy hxy)
end