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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, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.MinMax
import Mathlib.Tactic.Contrapose
#align_import algebra.order.monoid.lemmas from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# Ordered monoids
This file develops the basics of ordered monoids.
## 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.
## Remark
Almost no monoid is actually present in this file: most assumptions have been generalized to
`Mul` or `MulOneClass`.
-/
-- TODO: If possible, uniformize lemma names, taking special care of `'`,
-- after the `ordered`-refactor is done.
open Function
variable {α β : Type*}
section Mul
variable [Mul α]
section LE
variable [LE α]
/- The prime on this lemma is present only on the multiplicative version. The unprimed version
is taken by the analogous lemma for semiring, with an extra non-negativity assumption. -/
@[to_additive (attr := gcongr) add_le_add_left]
theorem mul_le_mul_left' [CovariantClass α α (· * ·) (· ≤ ·)] {b c : α} (bc : b ≤ c) (a : α) :
a * b ≤ a * c :=
CovariantClass.elim _ bc
#align mul_le_mul_left' mul_le_mul_left'
#align add_le_add_left add_le_add_left
@[to_additive le_of_add_le_add_left]
theorem le_of_mul_le_mul_left' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
(bc : a * b ≤ a * c) :
b ≤ c :=
ContravariantClass.elim _ bc
#align le_of_mul_le_mul_left' le_of_mul_le_mul_left'
#align le_of_add_le_add_left le_of_add_le_add_left
/- The prime on this lemma is present only on the multiplicative version. The unprimed version
is taken by the analogous lemma for semiring, with an extra non-negativity assumption. -/
@[to_additive (attr := gcongr) add_le_add_right]
theorem mul_le_mul_right' [i : CovariantClass α α (swap (· * ·)) (· ≤ ·)] {b c : α} (bc : b ≤ c)
(a : α) :
b * a ≤ c * a :=
i.elim a bc
#align mul_le_mul_right' mul_le_mul_right'
#align add_le_add_right add_le_add_right
@[to_additive le_of_add_le_add_right]
theorem le_of_mul_le_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
(bc : b * a ≤ c * a) :
b ≤ c :=
i.elim a bc
#align le_of_mul_le_mul_right' le_of_mul_le_mul_right'
#align le_of_add_le_add_right le_of_add_le_add_right
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_left [CovariantClass α α (· * ·) (· ≤ ·)]
[ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b c : α} :
a * b ≤ a * c ↔ b ≤ c :=
rel_iff_cov α α (· * ·) (· ≤ ·) a
#align mul_le_mul_iff_left mul_le_mul_iff_left
#align add_le_add_iff_left add_le_add_iff_left
@[to_additive (attr := simp)]
theorem mul_le_mul_iff_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
[ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b c : α} :
b * a ≤ c * a ↔ b ≤ c :=
rel_iff_cov α α (swap (· * ·)) (· ≤ ·) a
#align mul_le_mul_iff_right mul_le_mul_iff_right
#align add_le_add_iff_right add_le_add_iff_right
end LE
section LT
variable [LT α]
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_left [CovariantClass α α (· * ·) (· < ·)]
[ContravariantClass α α (· * ·) (· < ·)] (a : α) {b c : α} :
a * b < a * c ↔ b < c :=
rel_iff_cov α α (· * ·) (· < ·) a
#align mul_lt_mul_iff_left mul_lt_mul_iff_left
#align add_lt_add_iff_left add_lt_add_iff_left
@[to_additive (attr := simp)]
theorem mul_lt_mul_iff_right [CovariantClass α α (swap (· * ·)) (· < ·)]
[ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b c : α} :
b * a < c * a ↔ b < c :=
rel_iff_cov α α (swap (· * ·)) (· < ·) a
#align mul_lt_mul_iff_right mul_lt_mul_iff_right
#align add_lt_add_iff_right add_lt_add_iff_right
@[to_additive (attr := gcongr) add_lt_add_left]
theorem mul_lt_mul_left' [CovariantClass α α (· * ·) (· < ·)] {b c : α} (bc : b < c) (a : α) :
a * b < a * c :=
CovariantClass.elim _ bc
#align mul_lt_mul_left' mul_lt_mul_left'
#align add_lt_add_left add_lt_add_left
@[to_additive lt_of_add_lt_add_left]
theorem lt_of_mul_lt_mul_left' [ContravariantClass α α (· * ·) (· < ·)] {a b c : α}
(bc : a * b < a * c) :
b < c :=
ContravariantClass.elim _ bc
#align lt_of_mul_lt_mul_left' lt_of_mul_lt_mul_left'
#align lt_of_add_lt_add_left lt_of_add_lt_add_left
@[to_additive (attr := gcongr) add_lt_add_right]
theorem mul_lt_mul_right' [i : CovariantClass α α (swap (· * ·)) (· < ·)] {b c : α} (bc : b < c)
(a : α) :
b * a < c * a :=
i.elim a bc
#align mul_lt_mul_right' mul_lt_mul_right'
#align add_lt_add_right add_lt_add_right
@[to_additive lt_of_add_lt_add_right]
theorem lt_of_mul_lt_mul_right' [i : ContravariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
(bc : b * a < c * a) :
b < c :=
i.elim a bc
#align lt_of_mul_lt_mul_right' lt_of_mul_lt_mul_right'
#align lt_of_add_lt_add_right lt_of_add_lt_add_right
end LT
section Preorder
variable [Preorder α]
@[to_additive (attr := gcongr)]
theorem mul_lt_mul_of_lt_of_lt [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d :=
calc
a * c < a * d := mul_lt_mul_left' h₂ a
_ < b * d := mul_lt_mul_right' h₁ d
#align mul_lt_mul_of_lt_of_lt mul_lt_mul_of_lt_of_lt
#align add_lt_add_of_lt_of_lt add_lt_add_of_lt_of_lt
alias add_lt_add := add_lt_add_of_lt_of_lt
#align add_lt_add add_lt_add
@[to_additive]
theorem mul_lt_mul_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a ≤ b) (h₂ : c < d) :
a * c < b * d :=
(mul_le_mul_right' h₁ _).trans_lt (mul_lt_mul_left' h₂ b)
#align mul_lt_mul_of_le_of_lt mul_lt_mul_of_le_of_lt
#align add_lt_add_of_le_of_lt add_lt_add_of_le_of_lt
@[to_additive]
theorem mul_lt_mul_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (h₁ : a < b) (h₂ : c ≤ d) :
a * c < b * d :=
(mul_le_mul_left' h₂ _).trans_lt (mul_lt_mul_right' h₁ d)
#align mul_lt_mul_of_lt_of_le mul_lt_mul_of_lt_of_le
#align add_lt_add_of_lt_of_le add_lt_add_of_lt_of_le
/-- Only assumes left strict covariance. -/
@[to_additive "Only assumes left strict covariance"]
theorem Left.mul_lt_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α} (h₁ : a < b) (h₂ : c < d) :
a * c < b * d :=
mul_lt_mul_of_le_of_lt h₁.le h₂
#align left.mul_lt_mul Left.mul_lt_mul
#align left.add_lt_add Left.add_lt_add
/-- Only assumes right strict covariance. -/
@[to_additive "Only assumes right strict covariance"]
theorem Right.mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
(h₁ : a < b) (h₂ : c < d) :
a * c < b * d :=
mul_lt_mul_of_lt_of_le h₁ h₂.le
#align right.mul_lt_mul Right.mul_lt_mul
#align right.add_lt_add Right.add_lt_add
@[to_additive (attr := gcongr) add_le_add]
theorem mul_le_mul' [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) :
a * c ≤ b * d :=
(mul_le_mul_left' h₂ _).trans (mul_le_mul_right' h₁ d)
#align mul_le_mul' mul_le_mul'
#align add_le_add add_le_add
@[to_additive]
theorem mul_le_mul_three [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e)
(h₃ : c ≤ f) :
a * b * c ≤ d * e * f :=
mul_le_mul' (mul_le_mul' h₁ h₂) h₃
#align mul_le_mul_three mul_le_mul_three
#align add_le_add_three add_le_add_three
@[to_additive]
theorem mul_lt_of_mul_lt_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b < c)
(hle : d ≤ b) :
a * d < c :=
(mul_le_mul_left' hle a).trans_lt h
#align mul_lt_of_mul_lt_left mul_lt_of_mul_lt_left
#align add_lt_of_add_lt_left add_lt_of_add_lt_left
@[to_additive]
theorem mul_le_of_mul_le_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a * b ≤ c)
(hle : d ≤ b) :
a * d ≤ c :=
@act_rel_of_rel_of_act_rel _ _ _ (· ≤ ·) _ _ a _ _ _ hle h
#align mul_le_of_mul_le_left mul_le_of_mul_le_left
#align add_le_of_add_le_left add_le_of_add_le_left
@[to_additive]
theorem mul_lt_of_mul_lt_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
(h : a * b < c) (hle : d ≤ a) :
d * b < c :=
(mul_le_mul_right' hle b).trans_lt h
#align mul_lt_of_mul_lt_right mul_lt_of_mul_lt_right
#align add_lt_of_add_lt_right add_lt_of_add_lt_right
@[to_additive]
theorem mul_le_of_mul_le_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
(h : a * b ≤ c) (hle : d ≤ a) :
d * b ≤ c :=
(mul_le_mul_right' hle b).trans h
#align mul_le_of_mul_le_right mul_le_of_mul_le_right
#align add_le_of_add_le_right add_le_of_add_le_right
@[to_additive]
theorem lt_mul_of_lt_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a < b * c)
(hle : c ≤ d) :
a < b * d :=
h.trans_le (mul_le_mul_left' hle b)
#align lt_mul_of_lt_mul_left lt_mul_of_lt_mul_left
#align lt_add_of_lt_add_left lt_add_of_lt_add_left
@[to_additive]
theorem le_mul_of_le_mul_left [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α} (h : a ≤ b * c)
(hle : c ≤ d) :
a ≤ b * d :=
@rel_act_of_rel_of_rel_act _ _ _ (· ≤ ·) _ _ b _ _ _ hle h
#align le_mul_of_le_mul_left le_mul_of_le_mul_left
#align le_add_of_le_add_left le_add_of_le_add_left
@[to_additive]
theorem lt_mul_of_lt_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
(h : a < b * c) (hle : b ≤ d) :
a < d * c :=
h.trans_le (mul_le_mul_right' hle c)
#align lt_mul_of_lt_mul_right lt_mul_of_lt_mul_right
#align lt_add_of_lt_add_right lt_add_of_lt_add_right
@[to_additive]
theorem le_mul_of_le_mul_right [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
(h : a ≤ b * c) (hle : b ≤ d) :
a ≤ d * c :=
h.trans (mul_le_mul_right' hle c)
#align le_mul_of_le_mul_right le_mul_of_le_mul_right
#align le_add_of_le_add_right le_add_of_le_add_right
end Preorder
section PartialOrder
variable [PartialOrder α]
@[to_additive]
theorem mul_left_cancel'' [ContravariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (h : a * b = a * c) :
b = c :=
(le_of_mul_le_mul_left' h.le).antisymm (le_of_mul_le_mul_left' h.ge)
#align mul_left_cancel'' mul_left_cancel''
#align add_left_cancel'' add_left_cancel''
@[to_additive]
theorem mul_right_cancel'' [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
(h : a * b = c * b) :
a = c :=
(le_of_mul_le_mul_right' h.le).antisymm (le_of_mul_le_mul_right' h.ge)
#align mul_right_cancel'' mul_right_cancel''
#align add_right_cancel'' add_right_cancel''
@[to_additive] lemma mul_le_mul_iff_of_ge [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂ := by
haveI := covariantClass_le_of_lt α α (· * ·)
haveI := covariantClass_le_of_lt α α (swap (· * ·))
refine ⟨fun h ↦ ?_, by rintro ⟨rfl, rfl⟩; rfl⟩
simp only [eq_iff_le_not_lt, ha, hb, true_and]
refine ⟨fun ha ↦ h.not_lt ?_, fun hb ↦ h.not_lt ?_⟩
exacts [mul_lt_mul_of_lt_of_le ha hb, mul_lt_mul_of_le_of_lt ha hb]
#align add_le_add_iff_of_ge add_le_add_iff_of_geₓ
#align mul_le_mul_iff_of_ge mul_le_mul_iff_of_geₓ
@[to_additive] theorem mul_eq_mul_iff_eq_and_eq [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) :
a * b = c * d ↔ a = c ∧ b = d := by
haveI := covariantClass_le_of_lt α α (· * ·)
haveI := covariantClass_le_of_lt α α (swap (· * ·))
rw [le_antisymm_iff, eq_true (mul_le_mul' hac hbd), true_and, mul_le_mul_iff_of_ge hac hbd]
#align mul_eq_mul_iff_eq_and_eq mul_eq_mul_iff_eq_and_eqₓ
#align add_eq_add_iff_eq_and_eq add_eq_add_iff_eq_and_eqₓ
end PartialOrder
section LinearOrder
variable [LinearOrder α] {a b c d : α}
@[to_additive] lemma min_lt_max_of_mul_lt_mul
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
(h : a * b < c * d) : min a b < max c d := by
simp_rw [min_lt_iff, lt_max_iff]; contrapose! h; exact mul_le_mul' h.1.1 h.2.2
#align min_lt_max_of_mul_lt_mul min_lt_max_of_mul_lt_mulₓ
#align min_lt_max_of_add_lt_add min_lt_max_of_add_lt_addₓ
@[to_additive] lemma Left.min_le_max_of_mul_le_mul
[CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
(h : a * b ≤ c * d) : min a b ≤ max c d := by
simp_rw [min_le_iff, le_max_iff]; contrapose! h; exact mul_lt_mul_of_le_of_lt h.1.1.le h.2.2
@[to_additive] lemma Right.min_le_max_of_mul_le_mul
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
(h : a * b ≤ c * d) : min a b ≤ max c d := by
simp_rw [min_le_iff, le_max_iff]; contrapose! h; exact mul_lt_mul_of_lt_of_le h.1.1 h.2.2.le
@[to_additive] lemma min_le_max_of_mul_le_mul
[CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
(h : a * b ≤ c * d) : min a b ≤ max c d :=
let _ := covariantClass_le_of_lt α α (swap (· * ·))
Left.min_le_max_of_mul_le_mul h
#align min_le_max_of_add_le_add min_le_max_of_add_le_add
#align min_le_max_of_mul_le_mul min_le_max_of_mul_le_mul
end LinearOrder
section LinearOrder
variable [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
@[to_additive max_add_add_le_max_add_max]
theorem max_mul_mul_le_max_mul_max' : max (a * b) (c * d) ≤ max a c * max b d :=
max_le (mul_le_mul' (le_max_left _ _) <| le_max_left _ _) <|
mul_le_mul' (le_max_right _ _) <| le_max_right _ _
#align max_mul_mul_le_max_mul_max' max_mul_mul_le_max_mul_max'
#align max_add_add_le_max_add_max max_add_add_le_max_add_max
--TODO: Also missing `min_mul_min_le_min_mul_mul`
@[to_additive min_add_min_le_min_add_add]
theorem min_mul_min_le_min_mul_mul' : min a c * min b d ≤ min (a * b) (c * d) :=
le_min (mul_le_mul' (min_le_left _ _) <| min_le_left _ _) <|
mul_le_mul' (min_le_right _ _) <| min_le_right _ _
#align min_mul_min_le_min_mul_mul' min_mul_min_le_min_mul_mul'
#align min_add_min_le_min_add_add min_add_min_le_min_add_add
end LinearOrder
end Mul
-- using one
section MulOneClass
variable [MulOneClass α]
section LE
variable [LE α]
@[to_additive le_add_of_nonneg_right]
theorem le_mul_of_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
a ≤ a * b :=
calc
a = a * 1 := (mul_one a).symm
_ ≤ a * b := mul_le_mul_left' h a
#align le_mul_of_one_le_right' le_mul_of_one_le_right'
#align le_add_of_nonneg_right le_add_of_nonneg_right
@[to_additive add_le_of_nonpos_right]
theorem mul_le_of_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
a * b ≤ a :=
calc
a * b ≤ a * 1 := mul_le_mul_left' h a
_ = a := mul_one a
#align mul_le_of_le_one_right' mul_le_of_le_one_right'
#align add_le_of_nonpos_right add_le_of_nonpos_right
@[to_additive le_add_of_nonneg_left]
theorem le_mul_of_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : 1 ≤ b) :
a ≤ b * a :=
calc
a = 1 * a := (one_mul a).symm
_ ≤ b * a := mul_le_mul_right' h a
#align le_mul_of_one_le_left' le_mul_of_one_le_left'
#align le_add_of_nonneg_left le_add_of_nonneg_left
@[to_additive add_le_of_nonpos_left]
theorem mul_le_of_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (h : b ≤ 1) :
b * a ≤ a :=
calc
b * a ≤ 1 * a := mul_le_mul_right' h a
_ = a := one_mul a
#align mul_le_of_le_one_left' mul_le_of_le_one_left'
#align add_le_of_nonpos_left add_le_of_nonpos_left
@[to_additive]
theorem one_le_of_le_mul_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a ≤ a * b) :
1 ≤ b :=
le_of_mul_le_mul_left' <| by simpa only [mul_one]
#align one_le_of_le_mul_right one_le_of_le_mul_right
#align nonneg_of_le_add_right nonneg_of_le_add_right
@[to_additive]
theorem le_one_of_mul_le_right [ContravariantClass α α (· * ·) (· ≤ ·)] {a b : α} (h : a * b ≤ a) :
b ≤ 1 :=
le_of_mul_le_mul_left' <| by simpa only [mul_one]
#align le_one_of_mul_le_right le_one_of_mul_le_right
#align nonpos_of_add_le_right nonpos_of_add_le_right
@[to_additive]
theorem one_le_of_le_mul_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
(h : b ≤ a * b) :
1 ≤ a :=
le_of_mul_le_mul_right' <| by simpa only [one_mul]
#align one_le_of_le_mul_left one_le_of_le_mul_left
#align nonneg_of_le_add_left nonneg_of_le_add_left
@[to_additive]
theorem le_one_of_mul_le_left [ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
(h : a * b ≤ b) :
a ≤ 1 :=
le_of_mul_le_mul_right' <| by simpa only [one_mul]
#align le_one_of_mul_le_left le_one_of_mul_le_left
#align nonpos_of_add_le_left nonpos_of_add_le_left
@[to_additive (attr := simp) le_add_iff_nonneg_right]
theorem le_mul_iff_one_le_right' [CovariantClass α α (· * ·) (· ≤ ·)]
[ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
a ≤ a * b ↔ 1 ≤ b :=
Iff.trans (by rw [mul_one]) (mul_le_mul_iff_left a)
#align le_mul_iff_one_le_right' le_mul_iff_one_le_right'
#align le_add_iff_nonneg_right le_add_iff_nonneg_right
@[to_additive (attr := simp) le_add_iff_nonneg_left]
theorem le_mul_iff_one_le_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
[ContravariantClass α α (swap (· * ·)) (· ≤ ·)] (a : α) {b : α} :
a ≤ b * a ↔ 1 ≤ b :=
Iff.trans (by rw [one_mul]) (mul_le_mul_iff_right a)
#align le_mul_iff_one_le_left' le_mul_iff_one_le_left'
#align le_add_iff_nonneg_left le_add_iff_nonneg_left
@[to_additive (attr := simp) add_le_iff_nonpos_right]
theorem mul_le_iff_le_one_right' [CovariantClass α α (· * ·) (· ≤ ·)]
[ContravariantClass α α (· * ·) (· ≤ ·)] (a : α) {b : α} :
a * b ≤ a ↔ b ≤ 1 :=
Iff.trans (by rw [mul_one]) (mul_le_mul_iff_left a)
#align mul_le_iff_le_one_right' mul_le_iff_le_one_right'
#align add_le_iff_nonpos_right add_le_iff_nonpos_right
@[to_additive (attr := simp) add_le_iff_nonpos_left]
theorem mul_le_iff_le_one_left' [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
[ContravariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} :
a * b ≤ b ↔ a ≤ 1 :=
Iff.trans (by rw [one_mul]) (mul_le_mul_iff_right b)
#align mul_le_iff_le_one_left' mul_le_iff_le_one_left'
#align add_le_iff_nonpos_left add_le_iff_nonpos_left
end LE
section LT
variable [LT α]
@[to_additive lt_add_of_pos_right]
theorem lt_mul_of_one_lt_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : 1 < b) :
a < a * b :=
calc
a = a * 1 := (mul_one a).symm
_ < a * b := mul_lt_mul_left' h a
#align lt_mul_of_one_lt_right' lt_mul_of_one_lt_right'
#align lt_add_of_pos_right lt_add_of_pos_right
@[to_additive add_lt_of_neg_right]
theorem mul_lt_of_lt_one_right' [CovariantClass α α (· * ·) (· < ·)] (a : α) {b : α} (h : b < 1) :
a * b < a :=
calc
a * b < a * 1 := mul_lt_mul_left' h a
_ = a := mul_one a
#align mul_lt_of_lt_one_right' mul_lt_of_lt_one_right'
#align add_lt_of_neg_right add_lt_of_neg_right
@[to_additive lt_add_of_pos_left]
theorem lt_mul_of_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
(h : 1 < b) :
a < b * a :=
calc
a = 1 * a := (one_mul a).symm
_ < b * a := mul_lt_mul_right' h a
#align lt_mul_of_one_lt_left' lt_mul_of_one_lt_left'
#align lt_add_of_pos_left lt_add_of_pos_left
@[to_additive add_lt_of_neg_left]
theorem mul_lt_of_lt_one_left' [CovariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α}
(h : b < 1) :
b * a < a :=
calc
b * a < 1 * a := mul_lt_mul_right' h a
_ = a := one_mul a
#align mul_lt_of_lt_one_left' mul_lt_of_lt_one_left'
#align add_lt_of_neg_left add_lt_of_neg_left
@[to_additive]
theorem one_lt_of_lt_mul_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a < a * b) :
1 < b :=
lt_of_mul_lt_mul_left' <| by simpa only [mul_one]
#align one_lt_of_lt_mul_right one_lt_of_lt_mul_right
#align pos_of_lt_add_right pos_of_lt_add_right
@[to_additive]
theorem lt_one_of_mul_lt_right [ContravariantClass α α (· * ·) (· < ·)] {a b : α} (h : a * b < a) :
b < 1 :=
lt_of_mul_lt_mul_left' <| by simpa only [mul_one]
#align lt_one_of_mul_lt_right lt_one_of_mul_lt_right
#align neg_of_add_lt_right neg_of_add_lt_right
@[to_additive]
theorem one_lt_of_lt_mul_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
(h : b < a * b) :
1 < a :=
lt_of_mul_lt_mul_right' <| by simpa only [one_mul]
#align one_lt_of_lt_mul_left one_lt_of_lt_mul_left
#align pos_of_lt_add_left pos_of_lt_add_left
@[to_additive]
theorem lt_one_of_mul_lt_left [ContravariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
(h : a * b < b) :
a < 1 :=
lt_of_mul_lt_mul_right' <| by simpa only [one_mul]
#align lt_one_of_mul_lt_left lt_one_of_mul_lt_left
#align neg_of_add_lt_left neg_of_add_lt_left
@[to_additive (attr := simp) lt_add_iff_pos_right]
theorem lt_mul_iff_one_lt_right' [CovariantClass α α (· * ·) (· < ·)]
[ContravariantClass α α (· * ·) (· < ·)] (a : α) {b : α} :
a < a * b ↔ 1 < b :=
Iff.trans (by rw [mul_one]) (mul_lt_mul_iff_left a)
#align lt_mul_iff_one_lt_right' lt_mul_iff_one_lt_right'
#align lt_add_iff_pos_right lt_add_iff_pos_right
@[to_additive (attr := simp) lt_add_iff_pos_left]
theorem lt_mul_iff_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)]
[ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α} : a < b * a ↔ 1 < b :=
Iff.trans (by rw [one_mul]) (mul_lt_mul_iff_right a)
#align lt_mul_iff_one_lt_left' lt_mul_iff_one_lt_left'
#align lt_add_iff_pos_left lt_add_iff_pos_left
@[to_additive (attr := simp) add_lt_iff_neg_left]
theorem mul_lt_iff_lt_one_left' [CovariantClass α α (· * ·) (· < ·)]
[ContravariantClass α α (· * ·) (· < ·)] {a b : α} :
a * b < a ↔ b < 1 :=
Iff.trans (by rw [mul_one]) (mul_lt_mul_iff_left a)
#align mul_lt_iff_lt_one_left' mul_lt_iff_lt_one_left'
#align add_lt_iff_neg_left add_lt_iff_neg_left
@[to_additive (attr := simp) add_lt_iff_neg_right]
theorem mul_lt_iff_lt_one_right' [CovariantClass α α (swap (· * ·)) (· < ·)]
[ContravariantClass α α (swap (· * ·)) (· < ·)] {a : α} (b : α) : a * b < b ↔ a < 1 :=
Iff.trans (by rw [one_mul]) (mul_lt_mul_iff_right b)
#align mul_lt_iff_lt_one_right' mul_lt_iff_lt_one_right'
#align add_lt_iff_neg_right add_lt_iff_neg_right
end LT
section Preorder
variable [Preorder α]
/-! Lemmas of the form `b ≤ c → a ≤ 1 → b * a ≤ c`,
which assume left covariance. -/
@[to_additive]
theorem mul_le_of_le_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
(ha : a ≤ 1) :
b * a ≤ c :=
calc
b * a ≤ b * 1 := mul_le_mul_left' ha b
_ = b := mul_one b
_ ≤ c := hbc
#align mul_le_of_le_of_le_one mul_le_of_le_of_le_one
#align add_le_of_le_of_nonpos add_le_of_le_of_nonpos
@[to_additive]
theorem mul_lt_of_le_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
(ha : a < 1) :
b * a < c :=
calc
b * a < b * 1 := mul_lt_mul_left' ha b
_ = b := mul_one b
_ ≤ c := hbc
#align mul_lt_of_le_of_lt_one mul_lt_of_le_of_lt_one
#align add_lt_of_le_of_neg add_lt_of_le_of_neg
@[to_additive]
theorem mul_lt_of_lt_of_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
(ha : a ≤ 1) :
b * a < c :=
calc
b * a ≤ b * 1 := mul_le_mul_left' ha b
_ = b := mul_one b
_ < c := hbc
#align mul_lt_of_lt_of_le_one mul_lt_of_lt_of_le_one
#align add_lt_of_lt_of_nonpos add_lt_of_lt_of_nonpos
@[to_additive]
theorem mul_lt_of_lt_of_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
(ha : a < 1) :
b * a < c :=
calc
b * a < b * 1 := mul_lt_mul_left' ha b
_ = b := mul_one b
_ < c := hbc
#align mul_lt_of_lt_of_lt_one mul_lt_of_lt_of_lt_one
#align add_lt_of_lt_of_neg add_lt_of_lt_of_neg
@[to_additive]
theorem mul_lt_of_lt_of_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
(ha : a < 1) :
b * a < c :=
mul_lt_of_lt_of_le_one hbc ha.le
#align mul_lt_of_lt_of_lt_one' mul_lt_of_lt_of_lt_one'
#align add_lt_of_lt_of_neg' add_lt_of_lt_of_neg'
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.mul_le_one`. -/
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonpos`."]
theorem Left.mul_le_one [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a ≤ 1) (hb : b ≤ 1) :
a * b ≤ 1 :=
mul_le_of_le_of_le_one ha hb
#align left.mul_le_one Left.mul_le_one
#align left.add_nonpos Left.add_nonpos
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.mul_lt_one_of_le_of_lt`. -/
@[to_additive Left.add_neg_of_nonpos_of_neg
"Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg_of_nonpos_of_neg`."]
theorem Left.mul_lt_one_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a ≤ 1)
(hb : b < 1) :
a * b < 1 :=
mul_lt_of_le_of_lt_one ha hb
#align left.mul_lt_one_of_le_of_lt Left.mul_lt_one_of_le_of_lt
#align left.add_neg_of_nonpos_of_neg Left.add_neg_of_nonpos_of_neg
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.mul_lt_one_of_lt_of_le`. -/
@[to_additive Left.add_neg_of_neg_of_nonpos
"Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg_of_neg_of_nonpos`."]
theorem Left.mul_lt_one_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1)
(hb : b ≤ 1) :
a * b < 1 :=
mul_lt_of_lt_of_le_one ha hb
#align left.mul_lt_one_of_lt_of_le Left.mul_lt_one_of_lt_of_le
#align left.add_neg_of_neg_of_nonpos Left.add_neg_of_neg_of_nonpos
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.mul_lt_one`. -/
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg`."]
theorem Left.mul_lt_one [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
a * b < 1 :=
mul_lt_of_lt_of_lt_one ha hb
#align left.mul_lt_one Left.mul_lt_one
#align left.add_neg Left.add_neg
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.mul_lt_one'`. -/
@[to_additive "Assumes left covariance.
The lemma assuming right covariance is `Right.add_neg'`."]
theorem Left.mul_lt_one' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : a < 1) (hb : b < 1) :
a * b < 1 :=
mul_lt_of_lt_of_lt_one' ha hb
#align left.mul_lt_one' Left.mul_lt_one'
#align left.add_neg' Left.add_neg'
/-! Lemmas of the form `b ≤ c → 1 ≤ a → b ≤ c * a`,
which assume left covariance. -/
@[to_additive]
theorem le_mul_of_le_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b ≤ c)
(ha : 1 ≤ a) :
b ≤ c * a :=
calc
b ≤ c := hbc
_ = c * 1 := (mul_one c).symm
_ ≤ c * a := mul_le_mul_left' ha c
#align le_mul_of_le_of_one_le le_mul_of_le_of_one_le
#align le_add_of_le_of_nonneg le_add_of_le_of_nonneg
@[to_additive]
theorem lt_mul_of_le_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b ≤ c)
(ha : 1 < a) :
b < c * a :=
calc
b ≤ c := hbc
_ = c * 1 := (mul_one c).symm
_ < c * a := mul_lt_mul_left' ha c
#align lt_mul_of_le_of_one_lt lt_mul_of_le_of_one_lt
#align lt_add_of_le_of_pos lt_add_of_le_of_pos
@[to_additive]
theorem lt_mul_of_lt_of_one_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
(ha : 1 ≤ a) :
b < c * a :=
calc
b < c := hbc
_ = c * 1 := (mul_one c).symm
_ ≤ c * a := mul_le_mul_left' ha c
#align lt_mul_of_lt_of_one_le lt_mul_of_lt_of_one_le
#align lt_add_of_lt_of_nonneg lt_add_of_lt_of_nonneg
@[to_additive]
theorem lt_mul_of_lt_of_one_lt [CovariantClass α α (· * ·) (· < ·)] {a b c : α} (hbc : b < c)
(ha : 1 < a) :
b < c * a :=
calc
b < c := hbc
_ = c * 1 := (mul_one c).symm
_ < c * a := mul_lt_mul_left' ha c
#align lt_mul_of_lt_of_one_lt lt_mul_of_lt_of_one_lt
#align lt_add_of_lt_of_pos lt_add_of_lt_of_pos
@[to_additive]
theorem lt_mul_of_lt_of_one_lt' [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (hbc : b < c)
(ha : 1 < a) :
b < c * a :=
lt_mul_of_lt_of_one_le hbc ha.le
#align lt_mul_of_lt_of_one_lt' lt_mul_of_lt_of_one_lt'
#align lt_add_of_lt_of_pos' lt_add_of_lt_of_pos'
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.one_le_mul`. -/
@[to_additive Left.add_nonneg "Assumes left covariance.
The lemma assuming right covariance is `Right.add_nonneg`."]
theorem Left.one_le_mul [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
1 ≤ a * b :=
le_mul_of_le_of_one_le ha hb
#align left.one_le_mul Left.one_le_mul
#align left.add_nonneg Left.add_nonneg
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.one_lt_mul_of_le_of_lt`. -/
@[to_additive Left.add_pos_of_nonneg_of_pos
"Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos_of_nonneg_of_pos`."]
theorem Left.one_lt_mul_of_le_of_lt [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : 1 ≤ a)
(hb : 1 < b) :
1 < a * b :=
lt_mul_of_le_of_one_lt ha hb
#align left.one_lt_mul_of_le_of_lt Left.one_lt_mul_of_le_of_lt
#align left.add_pos_of_nonneg_of_pos Left.add_pos_of_nonneg_of_pos
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.one_lt_mul_of_lt_of_le`. -/
@[to_additive Left.add_pos_of_pos_of_nonneg
"Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos_of_pos_of_nonneg`."]
theorem Left.one_lt_mul_of_lt_of_le [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : 1 < a)
(hb : 1 ≤ b) :
1 < a * b :=
lt_mul_of_lt_of_one_le ha hb
#align left.one_lt_mul_of_lt_of_le Left.one_lt_mul_of_lt_of_le
#align left.add_pos_of_pos_of_nonneg Left.add_pos_of_pos_of_nonneg
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.one_lt_mul`. -/
@[to_additive Left.add_pos "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos`."]
theorem Left.one_lt_mul [CovariantClass α α (· * ·) (· < ·)] {a b : α} (ha : 1 < a) (hb : 1 < b) :
1 < a * b :=
lt_mul_of_lt_of_one_lt ha hb
#align left.one_lt_mul Left.one_lt_mul
#align left.add_pos Left.add_pos
/-- Assumes left covariance.
The lemma assuming right covariance is `Right.one_lt_mul'`. -/
@[to_additive Left.add_pos' "Assumes left covariance.
The lemma assuming right covariance is `Right.add_pos'`."]
theorem Left.one_lt_mul' [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : 1 < a) (hb : 1 < b) :
1 < a * b :=
lt_mul_of_lt_of_one_lt' ha hb
#align left.one_lt_mul' Left.one_lt_mul'
#align left.add_pos' Left.add_pos'
/-! Lemmas of the form `a ≤ 1 → b ≤ c → a * b ≤ c`,
which assume right covariance. -/
@[to_additive]
theorem mul_le_of_le_one_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
(hbc : b ≤ c) :
a * b ≤ c :=
calc
a * b ≤ 1 * b := mul_le_mul_right' ha b
_ = b := one_mul b
_ ≤ c := hbc
#align mul_le_of_le_one_of_le mul_le_of_le_one_of_le
#align add_le_of_nonpos_of_le add_le_of_nonpos_of_le
@[to_additive]
theorem mul_lt_of_lt_one_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
(hbc : b ≤ c) :
a * b < c :=
calc
a * b < 1 * b := mul_lt_mul_right' ha b
_ = b := one_mul b
_ ≤ c := hbc
#align mul_lt_of_lt_one_of_le mul_lt_of_lt_one_of_le
#align add_lt_of_neg_of_le add_lt_of_neg_of_le
@[to_additive]
theorem mul_lt_of_le_one_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a ≤ 1)
(hb : b < c) :
a * b < c :=
calc
a * b ≤ 1 * b := mul_le_mul_right' ha b
_ = b := one_mul b
_ < c := hb
#align mul_lt_of_le_one_of_lt mul_lt_of_le_one_of_lt
#align add_lt_of_nonpos_of_lt add_lt_of_nonpos_of_lt
@[to_additive]
theorem mul_lt_of_lt_one_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : a < 1)
(hb : b < c) :
a * b < c :=
calc
a * b < 1 * b := mul_lt_mul_right' ha b
_ = b := one_mul b
_ < c := hb
#align mul_lt_of_lt_one_of_lt mul_lt_of_lt_one_of_lt
#align add_lt_of_neg_of_lt add_lt_of_neg_of_lt
@[to_additive]
theorem mul_lt_of_lt_one_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : a < 1)
(hbc : b < c) :
a * b < c :=
mul_lt_of_le_one_of_lt ha.le hbc
#align mul_lt_of_lt_one_of_lt' mul_lt_of_lt_one_of_lt'
#align add_lt_of_neg_of_lt' add_lt_of_neg_of_lt'
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_le_one`. -/
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonpos`."]
theorem Right.mul_le_one [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a ≤ 1)
(hb : b ≤ 1) :
a * b ≤ 1 :=
mul_le_of_le_one_of_le ha hb
#align right.mul_le_one Right.mul_le_one
#align right.add_nonpos Right.add_nonpos
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one_of_lt_of_le`. -/
@[to_additive Right.add_neg_of_neg_of_nonpos
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg_of_neg_of_nonpos`."]
theorem Right.mul_lt_one_of_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α}
(ha : a < 1) (hb : b ≤ 1) :
a * b < 1 :=
mul_lt_of_lt_one_of_le ha hb
#align right.mul_lt_one_of_lt_of_le Right.mul_lt_one_of_lt_of_le
#align right.add_neg_of_neg_of_nonpos Right.add_neg_of_neg_of_nonpos
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one_of_le_of_lt`. -/
@[to_additive Right.add_neg_of_nonpos_of_neg
"Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg_of_nonpos_of_neg`."]
theorem Right.mul_lt_one_of_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α}
(ha : a ≤ 1) (hb : b < 1) :
a * b < 1 :=
mul_lt_of_le_one_of_lt ha hb
#align right.mul_lt_one_of_le_of_lt Right.mul_lt_one_of_le_of_lt
#align right.add_neg_of_nonpos_of_neg Right.add_neg_of_nonpos_of_neg
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one`. -/
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg`."]
theorem Right.mul_lt_one [CovariantClass α α (swap (· * ·)) (· < ·)] {a b : α} (ha : a < 1)
(hb : b < 1) :
a * b < 1 :=
mul_lt_of_lt_one_of_lt ha hb
#align right.mul_lt_one Right.mul_lt_one
#align right.add_neg Right.add_neg
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.mul_lt_one'`. -/
@[to_additive "Assumes right covariance.
The lemma assuming left covariance is `Left.add_neg'`."]
theorem Right.mul_lt_one' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : a < 1)
(hb : b < 1) :
a * b < 1 :=
mul_lt_of_lt_one_of_lt' ha hb
#align right.mul_lt_one' Right.mul_lt_one'
#align right.add_neg' Right.add_neg'
/-! Lemmas of the form `1 ≤ a → b ≤ c → b ≤ a * c`,
which assume right covariance. -/
@[to_additive]
theorem le_mul_of_one_le_of_le [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
(hbc : b ≤ c) :
b ≤ a * c :=
calc
b ≤ c := hbc
_ = 1 * c := (one_mul c).symm
_ ≤ a * c := mul_le_mul_right' ha c
#align le_mul_of_one_le_of_le le_mul_of_one_le_of_le
#align le_add_of_nonneg_of_le le_add_of_nonneg_of_le
@[to_additive]
theorem lt_mul_of_one_lt_of_le [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
(hbc : b ≤ c) :
b < a * c :=
calc
b ≤ c := hbc
_ = 1 * c := (one_mul c).symm
_ < a * c := mul_lt_mul_right' ha c
#align lt_mul_of_one_lt_of_le lt_mul_of_one_lt_of_le
#align lt_add_of_pos_of_le lt_add_of_pos_of_le
@[to_additive]
theorem lt_mul_of_one_le_of_lt [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 ≤ a)
(hbc : b < c) :
b < a * c :=
calc
b < c := hbc
_ = 1 * c := (one_mul c).symm
_ ≤ a * c := mul_le_mul_right' ha c
#align lt_mul_of_one_le_of_lt lt_mul_of_one_le_of_lt
#align lt_add_of_nonneg_of_lt lt_add_of_nonneg_of_lt
@[to_additive]
theorem lt_mul_of_one_lt_of_lt [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α} (ha : 1 < a)
(hbc : b < c) :
b < a * c :=
calc
b < c := hbc
_ = 1 * c := (one_mul c).symm
_ < a * c := mul_lt_mul_right' ha c
#align lt_mul_of_one_lt_of_lt lt_mul_of_one_lt_of_lt
#align lt_add_of_pos_of_lt lt_add_of_pos_of_lt
@[to_additive]
theorem lt_mul_of_one_lt_of_lt' [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α} (ha : 1 < a)
(hbc : b < c) :
b < a * c :=
lt_mul_of_one_le_of_lt ha.le hbc
#align lt_mul_of_one_lt_of_lt' lt_mul_of_one_lt_of_lt'
#align lt_add_of_pos_of_lt' lt_add_of_pos_of_lt'
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.one_le_mul`. -/
@[to_additive Right.add_nonneg "Assumes right covariance.
The lemma assuming left covariance is `Left.add_nonneg`."]
theorem Right.one_le_mul [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b : α} (ha : 1 ≤ a)
(hb : 1 ≤ b) :
1 ≤ a * b :=
le_mul_of_one_le_of_le ha hb
#align right.one_le_mul Right.one_le_mul
#align right.add_nonneg Right.add_nonneg
/-- Assumes right covariance.
The lemma assuming left covariance is `Left.one_lt_mul_of_lt_of_le`. -/