statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
contravariant_class_swap_add_lt [contravariant_class M M (swap (+)) (<)] :
contravariant_class {x : M // 0 < x} {x : M // 0 < x} (swap (+)) (<) | ⟨λ x y z h, subtype.coe_lt_coe.1 $ lt_of_add_lt_add_right h⟩ | instance | positive.contravariant_class_swap_add_lt | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_add_le [contravariant_class M M (+) (≤)] :
contravariant_class {x : M // 0 < x} {x : M // 0 < x} (+) (≤) | ⟨λ x y z h, subtype.coe_le_coe.1 $ le_of_add_le_add_left h⟩ | instance | positive.contravariant_class_add_le | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_swap_add_le [contravariant_class M M (swap (+)) (≤)] :
contravariant_class {x : M // 0 < x} {x : M // 0 < x} (swap (+)) (≤) | ⟨λ x y z h, subtype.coe_le_coe.1 $ le_of_add_le_add_right h⟩ | instance | positive.contravariant_class_swap_add_le | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_le [add_monoid M] [partial_order M] [covariant_class M M (+) (<)] :
covariant_class {x : M // 0 < x} {x : M // 0 < x} (+) (≤) | ⟨λ x, strict_mono.monotone $ λ _ _ h, add_lt_add_left h _⟩ | instance | positive.covariant_class_add_le | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"add_monoid",
"covariant_class",
"strict_mono.monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : {x : R // 0 < x}) : ↑(x * y) = (x * y : R) | rfl | lemma | positive.coe_mul | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (x : {x : R // 0 < x}) (n : ℕ) : ↑(x ^ n) = (x ^ n : R) | rfl | lemma | positive.coe_pow | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [nontrivial R] : ((1 : {x : R // 0 < x}) : R) = 1 | rfl | lemma | positive.coe_one | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_one : |(1 : α)| = 1 | abs_of_pos zero_lt_one | lemma | abs_one | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_of_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_two : |(2 : α)| = 2 | abs_of_pos zero_lt_two | lemma | abs_two | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_of_pos",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_mul (a b : α) : |a * b| = |a| * |b| | begin
rw [abs_eq (mul_nonneg (abs_nonneg a) (abs_nonneg b))],
cases le_total a 0 with ha ha; cases le_total b 0 with hb hb;
simp only [abs_of_nonpos, abs_of_nonneg, true_or, or_true, eq_self_iff_true,
neg_mul, mul_neg, neg_neg, *]
end | lemma | abs_mul | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_eq",
"abs_nonneg",
"abs_of_nonneg",
"abs_of_nonpos",
"mul_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_hom : α →*₀ α | ⟨abs, abs_zero, abs_one, abs_mul⟩ | def | abs_hom | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_one",
"abs_zero"
] | `abs` as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_mul_abs_self (a : α) : |a| * |a| = a * a | abs_by_cases (λ x, x * x = a * a) rfl (neg_mul_neg a a) | lemma | abs_mul_abs_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_by_cases",
"neg_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_mul_self (a : α) : |a * a| = a * a | by rw [abs_mul, abs_mul_abs_self] | lemma | abs_mul_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_mul",
"abs_mul_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_self : |a| = a ↔ 0 ≤ a | by simp [abs_eq_max_neg] | lemma | abs_eq_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_eq_max_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_neg_self : |a| = -a ↔ a ≤ 0 | by simp [abs_eq_max_neg] | lemma | abs_eq_neg_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_eq_max_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cases (a : α) : (|a| = a ∧ 0 ≤ a) ∨ (|a| = -a ∧ a < 0) | begin
by_cases 0 ≤ a,
{ left,
exact ⟨abs_eq_self.mpr h, h⟩ },
{ right,
push_neg at h,
exact ⟨abs_eq_neg_self.mpr (le_of_lt h), h⟩ }
end | lemma | abs_cases | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [] | For an element `a` of a linear ordered ring, either `abs a = a` and `0 ≤ a`,
or `abs a = -a` and `a < 0`.
Use cases on this lemma to automate linarith in inequalities | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
max_zero_add_max_neg_zero_eq_abs_self (a : α) :
max a 0 + max (-a) 0 = |a| | begin
symmetry,
rcases le_total 0 a with ha|ha;
simp [ha],
end | lemma | max_zero_add_max_neg_zero_eq_abs_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_iff_mul_self_eq : |a| = |b| ↔ a * a = b * b | begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact (mul_self_inj (abs_nonneg a) (abs_nonneg b)).symm,
end | lemma | abs_eq_iff_mul_self_eq | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_mul_abs_self",
"abs_nonneg",
"mul_self_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt_iff_mul_self_lt : |a| < |b| ↔ a * a < b * b | begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_lt_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end | lemma | abs_lt_iff_mul_self_lt | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_mul_abs_self",
"abs_nonneg",
"mul_self_lt_mul_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_iff_mul_self_le : |a| ≤ |b| ↔ a * a ≤ b * b | begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_le_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end | lemma | abs_le_iff_mul_self_le | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_mul_abs_self",
"abs_nonneg",
"mul_self_le_mul_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_one_iff_mul_self_le_one : |a| ≤ 1 ↔ a * a ≤ 1 | by simpa only [abs_one, one_mul] using @abs_le_iff_mul_self_le α _ a 1 | lemma | abs_le_one_iff_mul_self_le_one | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_le_iff_mul_self_le",
"abs_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_sq (a b : α) : |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b | begin
rw abs_mul_abs_self,
simp only [mul_add, add_comm, add_left_comm, mul_comm, sub_eq_add_neg,
mul_one, mul_neg, neg_add_rev, neg_neg],
end | lemma | abs_sub_sq | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_mul_abs_self",
"mul_comm",
"mul_neg",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b | by { cases abs_choice a with h h; simp only [h, neg_dvd] } | lemma | abs_dvd | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_choice",
"neg_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_dvd_self (a : α) : |a| ∣ a | (abs_dvd a a).mpr (dvd_refl a) | lemma | abs_dvd_self | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_dvd",
"dvd_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b | by { cases abs_choice b with h h; simp only [h, dvd_neg] } | lemma | dvd_abs | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_choice",
"dvd_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_dvd_abs (a : α) : a ∣ |a| | (dvd_abs a a).mpr (dvd_refl a) | lemma | self_dvd_abs | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"dvd_abs",
"dvd_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_dvd_abs (a b : α) : |a| ∣ |b| ↔ a ∣ b | (abs_dvd _ _).trans (dvd_abs _ _) | lemma | abs_dvd_abs | algebra.order.ring | src/algebra/order/ring/abs.lean | [
"algebra.order.ring.defs",
"algebra.ring.divisibility",
"algebra.order.group.abs"
] | [
"abs_dvd",
"dvd_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_comm_semiring (α : Type*) extends
canonically_ordered_add_monoid α, comm_semiring α | (eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : α}, a * b = 0 → a = 0 ∨ b = 0) | class | canonically_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"canonically_ordered_add_monoid",
"comm_semiring"
] | A canonically ordered commutative semiring is an ordered, commutative semiring in which `a ≤ b`
iff there exists `c` with `b = a + c`. This is satisfied by the natural numbers, for example, but
not the integers or other ordered groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_add_mul_le_mul_add_mul (hab : a ≤ b) (hcd : c ≤ d) : a * d + b * c ≤ a * c + b * d | begin
obtain ⟨b, rfl⟩ := exists_add_of_le hab,
obtain ⟨d, rfl⟩ := exists_add_of_le hcd,
rw [mul_add, add_right_comm, mul_add, ←add_assoc],
exact add_le_add_left (mul_le_mul_of_nonneg_right hab $ (le_add_iff_nonneg_right _).1 hcd) _,
end | lemma | mul_add_mul_le_mul_add_mul | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_le_mul_of_nonneg_right"
] | Binary **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_add_mul_le_mul_add_mul' (hba : b ≤ a) (hdc : d ≤ c) : a • d + b • c ≤ a • c + b • d | by { rw [add_comm (a • d), add_comm (a • c)], exact mul_add_mul_le_mul_add_mul hba hdc } | lemma | mul_add_mul_le_mul_add_mul' | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_add_mul_le_mul_add_mul"
] | Binary **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_add_mul_lt_mul_add_mul (hab : a < b) (hcd : c < d) : a * d + b * c < a * c + b * d | begin
obtain ⟨b, rfl⟩ := exists_add_of_le hab.le,
obtain ⟨d, rfl⟩ := exists_add_of_le hcd.le,
rw [mul_add, add_right_comm, mul_add, ←add_assoc],
exact add_lt_add_left (mul_lt_mul_of_pos_right hab $ (lt_add_iff_pos_right _).1 hcd) _,
end | lemma | mul_add_mul_lt_mul_add_mul | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_lt_mul_of_pos_right"
] | Binary strict **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_add_mul_lt_mul_add_mul' (hba : b < a) (hdc : d < c) : a • d + b • c < a • c + b • d | by { rw [add_comm (a • d), add_comm (a • c)], exact mul_add_mul_lt_mul_add_mul hba hdc } | lemma | mul_add_mul_lt_mul_add_mul' | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_add_mul_lt_mul_add_mul"
] | Binary **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_no_zero_divisors : no_zero_divisors α | ⟨λ a b h, canonically_ordered_comm_semiring.eq_zero_or_eq_zero_of_mul_eq_zero h⟩ | instance | canonically_ordered_comm_semiring.to_no_zero_divisors | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_covariant_mul_le : covariant_class α α (*) (≤) | begin
refine ⟨λ a b c h, _⟩,
rcases exists_add_of_le h with ⟨c, rfl⟩,
rw mul_add,
apply self_le_add_right
end | instance | canonically_ordered_comm_semiring.to_covariant_mul_le | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_comm_monoid : ordered_comm_monoid α | { mul_le_mul_left := λ _ _, mul_le_mul_left',
.. ‹canonically_ordered_comm_semiring α› } | instance | canonically_ordered_comm_semiring.to_ordered_comm_monoid | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_le_mul_left",
"mul_le_mul_left'",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ordered_comm_semiring : ordered_comm_semiring α | { zero_le_one := zero_le _,
mul_le_mul_of_nonneg_left := λ a b c h _, mul_le_mul_left' h _,
mul_le_mul_of_nonneg_right := λ a b c h _, mul_le_mul_right' h _,
..‹canonically_ordered_comm_semiring α› } | instance | canonically_ordered_comm_semiring.to_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_le_mul_left'",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_le_mul_right'",
"ordered_comm_semiring",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos : 0 < a * b ↔ (0 < a) ∧ (0 < b) | by simp only [pos_iff_ne_zero, ne.def, mul_eq_zero, not_or_distrib] | lemma | canonically_ordered_comm_semiring.mul_pos | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_eq_zero",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tsub (h : add_le_cancellable (a * c)) :
a * (b - c) = a * b - a * c | begin
cases total_of (≤) b c with hbc hcb,
{ rw [tsub_eq_zero_iff_le.2 hbc, mul_zero, tsub_eq_zero_iff_le.2 (mul_le_mul_left' hbc a)] },
{ apply h.eq_tsub_of_add_eq, rw [← mul_add, tsub_add_cancel_of_le hcb] }
end | lemma | add_le_cancellable.mul_tsub | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_le_mul_left'",
"mul_tsub",
"mul_zero",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_mul (h : add_le_cancellable (b * c)) : (a - b) * c = a * c - b * c | by { simp only [mul_comm _ c] at *, exact h.mul_tsub } | lemma | add_le_cancellable.tsub_mul | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [
"mul_comm",
"tsub_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_tsub (a b c : α) : a * (b - c) = a * b - a * c | contravariant.add_le_cancellable.mul_tsub | lemma | mul_tsub | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_mul (a b c : α) : (a - b) * c = a * c - b * c | contravariant.add_le_cancellable.tsub_mul | lemma | tsub_mul | algebra.order.ring | src/algebra/order/ring/canonical.lean | [
"algebra.order.ring.defs",
"algebra.order.sub.canonical",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring.to_char_zero [strict_ordered_semiring α] : char_zero α | ⟨strict_mono.injective $ strict_mono_nat_of_lt_succ $ λ n,
by { rw [nat.cast_succ], apply lt_add_one }⟩ | instance | strict_ordered_semiring.to_char_zero | algebra.order.ring | src/algebra/order/ring/char_zero.lean | [
"algebra.char_zero.defs",
"algebra.order.ring.defs"
] | [
"char_zero",
"lt_add_one",
"nat.cast_succ",
"strict_mono_nat_of_lt_succ",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive_cone (α : Type*) [ring α] extends add_comm_group.positive_cone α | (one_nonneg : nonneg 1)
(mul_pos : ∀ (a b), pos a → pos b → pos (a * b)) | structure | ring.positive_cone | algebra.order.ring | src/algebra/order/ring/cone.lean | [
"algebra.order.ring.defs"
] | [
"add_comm_group.positive_cone",
"ring"
] | A positive cone in a ring consists of a positive cone in underlying `add_comm_group`,
which contains `1` and such that the positive elements are closed under multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
total_positive_cone (α : Type*) [ring α]
extends positive_cone α, add_comm_group.total_positive_cone α | structure | ring.total_positive_cone | algebra.order.ring | src/algebra/order/ring/cone.lean | [
"algebra.order.ring.defs"
] | [
"add_comm_group.total_positive_cone",
"ring"
] | A total positive cone in a nontrivial ring induces a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive_cone.one_pos (C : positive_cone α) : C.pos 1 | (C.pos_iff _).2 ⟨C.one_nonneg, λ h, one_ne_zero $ C.nonneg_antisymm C.one_nonneg h⟩ | lemma | ring.positive_cone.one_pos | algebra.order.ring | src/algebra/order/ring/cone.lean | [
"algebra.order.ring.defs"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_ring.mk_of_positive_cone (C : positive_cone α) : strict_ordered_ring α | { exists_pair_ne := ⟨0, 1, λ h, by simpa [←h, C.pos_iff] using C.one_pos⟩,
zero_le_one := by { change C.nonneg (1 - 0), convert C.one_nonneg, simp, },
mul_pos := λ x y xp yp, begin
change C.pos (x*y - 0),
convert C.mul_pos x y (by { convert xp, simp, }) (by { convert yp, simp, }),
simp,
end,
..‹ring... | def | strict_ordered_ring.mk_of_positive_cone | algebra.order.ring | src/algebra/order/ring/cone.lean | [
"algebra.order.ring.defs"
] | [
"exists_pair_ne",
"ordered_add_comm_group.mk_of_positive_cone",
"strict_ordered_ring",
"zero_le_one"
] | Construct a `strict_ordered_ring` by designating a positive cone in an existing `ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_ring.mk_of_positive_cone (C : total_positive_cone α) : linear_ordered_ring α | { ..strict_ordered_ring.mk_of_positive_cone C.to_positive_cone,
..linear_ordered_add_comm_group.mk_of_positive_cone C.to_total_positive_cone, } | def | linear_ordered_ring.mk_of_positive_cone | algebra.order.ring | src/algebra/order/ring/cone.lean | [
"algebra.order.ring.defs"
] | [
"linear_ordered_add_comm_group.mk_of_positive_cone",
"linear_ordered_ring",
"strict_ordered_ring.mk_of_positive_cone"
] | Construct a `linear_ordered_ring` by
designating a positive cone in an existing `ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_one_le_two_mul [has_le α] [semiring α] [covariant_class α α (+) (≤)]
{a : α} (a1 : 1 ≤ a) :
a + 1 ≤ 2 * a | calc a + 1 ≤ a + a : add_le_add_left a1 a
... = 2 * a : (two_mul _).symm | lemma | add_one_le_two_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"covariant_class",
"semiring",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring (α : Type u) extends semiring α, ordered_add_comm_monoid α | (zero_le_one : (0 : α) ≤ 1)
(mul_le_mul_of_nonneg_left : ∀ a b c : α, a ≤ b → 0 ≤ c → c * a ≤ c * b)
(mul_le_mul_of_nonneg_right : ∀ a b c : α, a ≤ b → 0 ≤ c → a * c ≤ b * c) | class | ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"ordered_add_comm_monoid",
"semiring",
"zero_le_one"
] | An `ordered_semiring` is a semiring with a partial order such that addition is monotone and
multiplication by a nonnegative number is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_semiring (α : Type u) extends ordered_semiring α, comm_semiring α | class | ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"comm_semiring",
"ordered_semiring"
] | An `ordered_comm_semiring` is a commutative semiring with a partial order such that addition is
monotone and multiplication by a nonnegative number is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_ring (α : Type u) extends ring α, ordered_add_comm_group α | (zero_le_one : 0 ≤ (1 : α))
(mul_nonneg : ∀ a b : α, 0 ≤ a → 0 ≤ b → 0 ≤ a * b) | class | ordered_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"ordered_add_comm_group",
"ring",
"zero_le_one"
] | An `ordered_ring` is a ring with a partial order such that addition is monotone and
multiplication by a nonnegative number is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_ring (α : Type u) extends ordered_ring α, comm_ring α | class | ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"comm_ring",
"ordered_ring"
] | An `ordered_comm_ring` is a commutative ring with a partial order such that addition is monotone
and multiplication by a nonnegative number is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring (α : Type u)
extends semiring α, ordered_cancel_add_comm_monoid α, nontrivial α | (zero_le_one : (0 : α) ≤ 1)
(mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b)
(mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c) | class | strict_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"nontrivial",
"ordered_cancel_add_comm_monoid",
"semiring",
"zero_le_one"
] | A `strict_ordered_semiring` is a nontrivial semiring with a partial order such that addition is
strictly monotone and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_semiring (α : Type u) extends strict_ordered_semiring α, comm_semiring α | class | strict_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"comm_semiring",
"strict_ordered_semiring"
] | A `strict_ordered_comm_semiring` is a commutative semiring with a partial order such that
addition is strictly monotone and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_ring (α : Type u) extends ring α, ordered_add_comm_group α, nontrivial α | (zero_le_one : 0 ≤ (1 : α))
(mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b) | class | strict_ordered_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"nontrivial",
"ordered_add_comm_group",
"ring",
"zero_le_one"
] | A `strict_ordered_ring` is a ring with a partial order such that addition is strictly monotone
and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_ring (α : Type*) extends strict_ordered_ring α, comm_ring α | class | strict_ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"comm_ring",
"strict_ordered_ring"
] | A `strict_ordered_comm_ring` is a commutative ring with a partial order such that addition is
strictly monotone and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semiring (α : Type u)
extends strict_ordered_semiring α, linear_ordered_add_comm_monoid α | class | linear_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"linear_ordered_add_comm_monoid",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
linear_ordered_comm_semiring (α : Type*)
extends strict_ordered_comm_semiring α, linear_ordered_semiring α | class | linear_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"linear_ordered_semiring",
"strict_ordered_comm_semiring"
] | A `linear_ordered_comm_semiring` is a nontrivial commutative semiring with a linear order such
that addition is monotone and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_ring (α : Type u) extends strict_ordered_ring α, linear_order α | class | linear_ordered_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_ordered_ring"
] | A `linear_ordered_ring` is a ring with a linear order such that addition is monotone and
multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_ring (α : Type u) extends linear_ordered_ring α, comm_monoid α | class | linear_ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"comm_monoid",
"linear_ordered_ring"
] | A `linear_ordered_comm_ring` is a commutative ring with a linear order such that addition is
monotone and multiplication by a positive number is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring.zero_le_one_class : zero_le_one_class α | { ..‹ordered_semiring α› } | instance | ordered_semiring.zero_le_one_class | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring.to_pos_mul_mono : pos_mul_mono α | ⟨λ x a b h, ordered_semiring.mul_le_mul_of_nonneg_left _ _ _ h x.2⟩ | instance | ordered_semiring.to_pos_mul_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring.to_mul_pos_mono : mul_pos_mono α | ⟨λ x a b h, ordered_semiring.mul_le_mul_of_nonneg_right _ _ _ h x.2⟩ | instance | ordered_semiring.to_mul_pos_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_mono : monotone (bit1 : α → α) | λ a b h, add_le_add_right (bit0_mono h) _ | lemma | bit1_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"bit0_mono",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_nonneg (H : 0 ≤ a) : ∀ (n : ℕ), 0 ≤ a ^ n | | 0 := by { rw pow_zero, exact zero_le_one}
| (n+1) := by { rw pow_succ, exact mul_nonneg H (pow_nonneg _) } | lemma | pow_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"pow_succ",
"pow_zero",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_mul_two_add (a2 : 2 ≤ a) (b0 : 0 ≤ b) : a + (2 + b) ≤ a * (2 + b) | calc a + (2 + b) ≤ a + (a + a * b) :
add_le_add_left (add_le_add a2 $ le_mul_of_one_le_left b0 $ one_le_two.trans a2) a
... ≤ a * (2 + b) : by rw [mul_add, mul_two, add_assoc] | lemma | add_le_mul_two_add | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"le_mul_of_one_le_left",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_mul_of_one_le_of_one_le (ha : 1 ≤ a) (hb : 1 ≤ b) : (1 : α) ≤ a * b | left.one_le_mul_of_le_of_le ha hb $ zero_le_one.trans ha | lemma | one_le_mul_of_one_le_of_one_le | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"left.one_le_mul_of_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_mul_left_of_nonneg (ha : 0 ≤ a) : monotone (λ x, a * x) | λ b c h, mul_le_mul_of_nonneg_left h ha | lemma | monotone_mul_left_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_mul_right_of_nonneg (ha : 0 ≤ a) : monotone (λ x, x * a) | λ b c h, mul_le_mul_of_nonneg_right h ha | lemma | monotone_mul_right_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul_const (hf : monotone f) (ha : 0 ≤ a) : monotone (λ x, f x * a) | (monotone_mul_right_of_nonneg ha).comp hf | lemma | monotone.mul_const | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"monotone_mul_right_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.const_mul (hf : monotone f) (ha : 0 ≤ a) : monotone (λ x, a * f x) | (monotone_mul_left_of_nonneg ha).comp hf | lemma | monotone.const_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"monotone_mul_left_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.mul_const (hf : antitone f) (ha : 0 ≤ a) : antitone (λ x, f x * a) | (monotone_mul_right_of_nonneg ha).comp_antitone hf | lemma | antitone.mul_const | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"monotone_mul_right_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.const_mul (hf : antitone f) (ha : 0 ≤ a) : antitone (λ x, a * f x) | (monotone_mul_left_of_nonneg ha).comp_antitone hf | lemma | antitone.const_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"monotone_mul_left_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul (hf : monotone f) (hg : monotone g) (hf₀ : ∀ x, 0 ≤ f x) (hg₀ : ∀ x, 0 ≤ g x) :
monotone (f * g) | λ b c h, mul_le_mul (hf h) (hg h) (hg₀ _) (hf₀ _) | lemma | monotone.mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_pos [nontrivial α] (h : 0 ≤ a) : 0 < bit1 a | zero_lt_one.trans_le $ bit1_zero.symm.trans_le $ bit1_mono h | lemma | bit1_pos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"bit1_mono",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_pos' (h : 0 < a) : 0 < bit1 a | by { nontriviality, exact bit1_pos h.le } | lemma | bit1_pos' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"bit1_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_one (ha : a ≤ 1) (hb' : 0 ≤ b) (hb : b ≤ 1) : a * b ≤ 1 | one_mul (1 : α) ▸ mul_le_mul ha hb hb' zero_le_one | lemma | mul_le_one | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul",
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_mul_of_le_of_lt (ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b | hb.trans_le $ le_mul_of_one_le_left (zero_le_one.trans hb.le) ha | lemma | one_lt_mul_of_le_of_lt | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"le_mul_of_one_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_mul_of_lt_of_le (ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b | ha.trans_le $ le_mul_of_one_le_right (zero_le_one.trans ha.le) hb | lemma | one_lt_mul_of_lt_of_le | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"le_mul_of_one_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_one_of_nonneg_of_lt_one_left (ha₀ : 0 ≤ a) (ha : a < 1) (hb : b ≤ 1) : a * b < 1 | (mul_le_of_le_one_right ha₀ hb).trans_lt ha | lemma | mul_lt_one_of_nonneg_of_lt_one_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_of_le_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_one_of_nonneg_of_lt_one_right (ha : a ≤ 1) (hb₀ : 0 ≤ b) (hb : b < 1) : a * b < 1 | (mul_le_of_le_one_left hb₀ ha).trans_lt hb | lemma | mul_lt_one_of_nonneg_of_lt_one_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_of_le_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_ring.to_ordered_semiring : ordered_semiring α | { mul_le_mul_of_nonneg_left := λ a b c h hc,
by simpa only [mul_sub, sub_nonneg] using ordered_ring.mul_nonneg _ _ hc (sub_nonneg.2 h),
mul_le_mul_of_nonneg_right := λ a b c h hc,
by simpa only [sub_mul, sub_nonneg] using ordered_ring.mul_nonneg _ _ (sub_nonneg.2 h) hc,
..‹ordered_ring α›, ..ring.to_semirin... | instance | ordered_ring.to_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"ordered_semiring",
"ring.to_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_left (h : b ≤ a) (hc : c ≤ 0) : c * a ≤ c * b | by simpa only [neg_mul, neg_le_neg_iff] using mul_le_mul_of_nonneg_left h (neg_nonneg.2 hc) | lemma | mul_le_mul_of_nonpos_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c | by simpa only [mul_neg, neg_le_neg_iff] using mul_le_mul_of_nonneg_right h (neg_nonneg.2 hc) | lemma | mul_le_mul_of_nonpos_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_right",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_nonneg_of_nonpos_of_nonpos (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a * b | by simpa only [zero_mul] using mul_le_mul_of_nonpos_right ha hb | lemma | mul_nonneg_of_nonpos_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_right",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonneg_of_nonpos (hca : c ≤ a) (hbd : b ≤ d) (hc : 0 ≤ c) (hb : b ≤ 0) :
a * b ≤ c * d | (mul_le_mul_of_nonpos_right hca hb).trans $ mul_le_mul_of_nonneg_left hbd hc | lemma | mul_le_mul_of_nonneg_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonneg_of_nonpos' (hca : c ≤ a) (hbd : b ≤ d) (ha : 0 ≤ a) (hd : d ≤ 0) :
a * b ≤ c * d | (mul_le_mul_of_nonneg_left hbd ha).trans $ mul_le_mul_of_nonpos_right hca hd | lemma | mul_le_mul_of_nonneg_of_nonpos' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_of_nonneg (hac : a ≤ c) (hdb : d ≤ b) (hc : c ≤ 0) (hb : 0 ≤ b) :
a * b ≤ c * d | (mul_le_mul_of_nonneg_right hac hb).trans $ mul_le_mul_of_nonpos_left hdb hc | lemma | mul_le_mul_of_nonpos_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_right",
"mul_le_mul_of_nonpos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_of_nonneg' (hca : c ≤ a) (hbd : b ≤ d) (ha : 0 ≤ a) (hd : d ≤ 0) :
a * b ≤ c * d | (mul_le_mul_of_nonneg_left hbd ha).trans $ mul_le_mul_of_nonpos_right hca hd | lemma | mul_le_mul_of_nonpos_of_nonneg' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_of_nonpos (hca : c ≤ a) (hdb : d ≤ b) (hc : c ≤ 0) (hb : b ≤ 0) :
a * b ≤ c * d | (mul_le_mul_of_nonpos_right hca hb).trans $ mul_le_mul_of_nonpos_left hdb hc | lemma | mul_le_mul_of_nonpos_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_left",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonpos_of_nonpos' (hca : c ≤ a) (hdb : d ≤ b) (ha : a ≤ 0) (hd : d ≤ 0) :
a * b ≤ c * d | (mul_le_mul_of_nonpos_left hdb ha).trans $ mul_le_mul_of_nonpos_right hca hd | lemma | mul_le_mul_of_nonpos_of_nonpos' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_left",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_one_left (hb : b ≤ 0) (h : a ≤ 1) : b ≤ a * b | by simpa only [one_mul] using mul_le_mul_of_nonpos_right h hb | lemma | le_mul_of_le_one_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_right",
"one_mul"
] | Variant of `mul_le_of_le_one_left` for `b` non-positive instead of non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_one_le_left (hb : b ≤ 0) (h : 1 ≤ a) : a * b ≤ b | by simpa only [one_mul] using mul_le_mul_of_nonpos_right h hb | lemma | mul_le_of_one_le_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_right",
"one_mul"
] | Variant of `le_mul_of_one_le_left` for `b` non-positive instead of non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_of_le_one_right (ha : a ≤ 0) (h : b ≤ 1) : a ≤ a * b | by simpa only [mul_one] using mul_le_mul_of_nonpos_left h ha | lemma | le_mul_of_le_one_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_left",
"mul_one"
] | Variant of `mul_le_of_le_one_right` for `a` non-positive instead of non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_one_le_right (ha : a ≤ 0) (h : 1 ≤ b) : a * b ≤ a | by simpa only [mul_one] using mul_le_mul_of_nonpos_left h ha | lemma | mul_le_of_one_le_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonpos_left",
"mul_one"
] | Variant of `le_mul_of_one_le_right` for `a` non-positive instead of non-negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_mul_left {a : α} (ha : a ≤ 0) : antitone ((*) a) | λ b c b_le_c, mul_le_mul_of_nonpos_left b_le_c ha | lemma | antitone_mul_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"mul_le_mul_of_nonpos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_mul_right {a : α} (ha : a ≤ 0) : antitone (λ x, x * a) | λ b c b_le_c, mul_le_mul_of_nonpos_right b_le_c ha | lemma | antitone_mul_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"mul_le_mul_of_nonpos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.const_mul_of_nonpos (hf : monotone f) (ha : a ≤ 0) : antitone (λ x, a * f x) | (antitone_mul_left ha).comp_monotone hf | lemma | monotone.const_mul_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"antitone_mul_left",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul_const_of_nonpos (hf : monotone f) (ha : a ≤ 0) : antitone (λ x, f x * a) | (antitone_mul_right ha).comp_monotone hf | lemma | monotone.mul_const_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"antitone_mul_right",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.const_mul_of_nonpos (hf : antitone f) (ha : a ≤ 0) : monotone (λ x, a * f x) | (antitone_mul_left ha).comp hf | lemma | antitone.const_mul_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"antitone_mul_left",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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