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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|---|---|---|---|---|---|---|---|---|---|---|
left.one_lt_mul_of_le_of_lt [covariant_class α α (*) (<)]
{a b : α} (ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b | lt_mul_of_le_of_one_lt ha hb | lemma | left.one_lt_mul_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_le_of_one_lt"
] | Assumes left covariance.
The lemma assuming right covariance is `right.one_lt_mul_of_le_of_lt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.one_lt_mul_of_lt_of_le [covariant_class α α (*) (≤)]
{a b : α} (ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b | lt_mul_of_lt_of_one_le ha hb | lemma | left.one_lt_mul_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_lt_of_one_le"
] | Assumes left covariance.
The lemma assuming right covariance is `right.one_lt_mul_of_lt_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.one_lt_mul [covariant_class α α (*) (<)]
{a b : α} (ha : 1 < a) (hb : 1 < b) : 1 < a * b | lt_mul_of_lt_of_one_lt ha hb | lemma | left.one_lt_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_lt_of_one_lt"
] | Assumes left covariance.
The lemma assuming right covariance is `right.one_lt_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.one_lt_mul' [covariant_class α α (*) (≤)]
{a b : α} (ha : 1 < a) (hb : 1 < b) : 1 < a * b | lt_mul_of_lt_of_one_lt' ha hb | lemma | left.one_lt_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_lt_of_one_lt'"
] | Assumes left covariance.
The lemma assuming right covariance is `right.one_lt_mul'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_le_one_of_le [covariant_class 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 | lemma | mul_le_of_le_one_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_of_le [covariant_class 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 | lemma | mul_lt_of_lt_one_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_one_of_lt [covariant_class 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 | lemma | mul_lt_of_le_one_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_of_lt [covariant_class 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 | lemma | mul_lt_of_lt_one_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_of_lt' [covariant_class 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 | lemma | mul_lt_of_lt_one_of_lt' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_le_one_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_le_one [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 | mul_le_of_le_one_of_le ha hb | lemma | right.mul_le_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_one_of_le"
] | Assumes right covariance.
The lemma assuming left covariance is `left.mul_le_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_one_of_lt_of_le [covariant_class α α (swap (*)) (<)]
{a b : α} (ha : a < 1) (hb : b ≤ 1) : a * b < 1 | mul_lt_of_lt_one_of_le ha hb | lemma | right.mul_lt_one_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_one_of_le"
] | Assumes right covariance.
The lemma assuming left covariance is `left.mul_lt_one_of_lt_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_one_of_le_of_lt [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : a ≤ 1) (hb : b < 1) : a * b < 1 | mul_lt_of_le_one_of_lt ha hb | lemma | right.mul_lt_one_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_le_one_of_lt"
] | Assumes right covariance.
The lemma assuming left covariance is `left.mul_lt_one_of_le_of_lt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_one [covariant_class α α (swap (*)) (<)]
{a b : α} (ha : a < 1) (hb : b < 1) : a * b < 1 | mul_lt_of_lt_one_of_lt ha hb | lemma | right.mul_lt_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_one_of_lt"
] | Assumes right covariance.
The lemma assuming left covariance is `left.mul_lt_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_one' [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : a < 1) (hb : b < 1) : a * b < 1 | mul_lt_of_lt_one_of_lt' ha hb | lemma | right.mul_lt_one' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_one_of_lt'"
] | Assumes right covariance.
The lemma assuming left covariance is `left.mul_lt_one'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_of_one_le_of_le [covariant_class 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 | lemma | le_mul_of_one_le_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_of_le [covariant_class 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 | lemma | lt_mul_of_one_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_le_of_lt [covariant_class 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 | lemma | lt_mul_of_one_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_of_lt [covariant_class 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 | lemma | lt_mul_of_one_lt_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_of_lt' [covariant_class 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 | lemma | lt_mul_of_one_lt_of_lt' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_one_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.one_le_mul [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b | le_mul_of_one_le_of_le ha hb | lemma | right.one_le_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_one_le_of_le"
] | Assumes right covariance.
The lemma assuming left covariance is `left.one_le_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul_of_lt_of_le [covariant_class α α (swap (*)) (<)]
{a b : α} (ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b | lt_mul_of_one_lt_of_le ha hb | lemma | right.one_lt_mul_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_one_lt_of_le"
] | Assumes right covariance.
The lemma assuming left covariance is `left.one_lt_mul_of_lt_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul_of_le_of_lt [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b | lt_mul_of_one_le_of_lt ha hb | lemma | right.one_lt_mul_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_one_le_of_lt"
] | Assumes right covariance.
The lemma assuming left covariance is `left.one_lt_mul_of_le_of_lt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul [covariant_class α α (swap (*)) (<)]
{a b : α} (ha : 1 < a) (hb : 1 < b) : 1 < a * b | lt_mul_of_one_lt_of_lt ha hb | lemma | right.one_lt_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_one_lt_of_lt"
] | Assumes right covariance.
The lemma assuming left covariance is `left.one_lt_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul' [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : 1 < a) (hb : 1 < b) : 1 < a * b | lt_mul_of_one_lt_of_lt' ha hb | lemma | right.one_lt_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_one_lt_of_lt'"
] | Assumes right covariance.
The lemma assuming left covariance is `left.one_lt_mul'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_of_mul_lt_of_one_le_left [covariant_class α α (*) (≤)]
{a b c : α} (h : a * b < c) (hle : 1 ≤ b) : a < c | (le_mul_of_one_le_right' hle).trans_lt h | lemma | lt_of_mul_lt_of_one_le_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_one_le_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_of_one_le_left [covariant_class α α (*) (≤)]
{a b c : α} (h : a * b ≤ c) (hle : 1 ≤ b) : a ≤ c | (le_mul_of_one_le_right' hle).trans h | lemma | le_of_mul_le_of_one_le_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_one_le_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lt_mul_of_le_one_left [covariant_class α α (*) (≤)]
{a b c : α} (h : a < b * c) (hle : c ≤ 1) : a < b | h.trans_le (mul_le_of_le_one_right' hle) | lemma | lt_of_lt_mul_of_le_one_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_one_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_le_mul_of_le_one_left [covariant_class α α (*) (≤)]
{a b c : α} (h : a ≤ b * c) (hle : c ≤ 1) : a ≤ b | h.trans (mul_le_of_le_one_right' hle) | lemma | le_of_le_mul_of_le_one_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_one_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_of_one_le_right [covariant_class α α (swap (*)) (≤)]
{a b c : α} (h : a * b < c) (hle : 1 ≤ a) : b < c | (le_mul_of_one_le_left' hle).trans_lt h | lemma | lt_of_mul_lt_of_one_le_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_of_one_le_right [covariant_class α α (swap (*)) (≤)]
{a b c : α} (h : a * b ≤ c) (hle : 1 ≤ a) : b ≤ c | (le_mul_of_one_le_left' hle).trans h | lemma | le_of_mul_le_of_one_le_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lt_mul_of_le_one_right [covariant_class α α (swap (*)) (≤)]
{a b c : α} (h : a < b * c) (hle : b ≤ 1) : a < c | h.trans_le (mul_le_of_le_one_left' hle) | lemma | lt_of_lt_mul_of_le_one_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_one_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_le_mul_of_le_one_right [covariant_class α α (swap (*)) (≤)]
{a b c : α} (h : a ≤ b * c) (hle : b ≤ 1) : a ≤ c | h.trans (mul_le_of_le_one_left' hle) | lemma | le_of_le_mul_of_le_one_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_one_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
{a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) : a * b = 1 ↔ a = 1 ∧ b = 1 | iff.intro
(assume hab : a * b = 1,
have a ≤ 1, from hab ▸ le_mul_of_le_of_one_le le_rfl hb,
have a = 1, from le_antisymm this ha,
have b ≤ 1, from hab ▸ le_mul_of_one_le_of_le ha le_rfl,
have b = 1, from le_antisymm this hb,
and.intro ‹a = 1› ‹b = 1›)
(assume ⟨ha', hb'⟩, by rw [ha', hb', mul_one]) | lemma | mul_eq_one_iff' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_le_of_one_le",
"le_mul_of_one_le_of_le",
"le_rfl",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_of_ge [covariant_class α α (*) (≤)]
[covariant_class α α (swap (*)) (≤)] [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (<)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂ | begin
refine ⟨λ h, _, by { rintro ⟨rfl, rfl⟩, refl }⟩,
simp only [eq_iff_le_not_lt, ha, hb, true_and],
refine ⟨λ ha, h.not_lt _, λ hb, h.not_lt _⟩,
{ exact mul_lt_mul_of_lt_of_le ha hb },
{ exact mul_lt_mul_of_le_of_lt ha hb }
end | lemma | mul_le_mul_iff_of_ge | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"eq_iff_le_not_lt",
"mul_lt_mul_of_le_of_lt",
"mul_lt_mul_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_one_le_mul_left (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : a = 1 | ha.eq_of_not_lt $ λ h, hab.not_lt $ mul_lt_one_of_lt_of_le h hb | lemma | eq_one_of_one_le_mul_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_mul_le_one_left (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : a = 1 | ha.eq_of_not_gt $ λ h, hab.not_lt $ one_lt_mul_of_lt_of_le' h hb | lemma | eq_one_of_mul_le_one_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_one_le_mul_right (ha : a ≤ 1) (hb : b ≤ 1) (hab : 1 ≤ a * b) : b = 1 | hb.eq_of_not_lt $ λ h, hab.not_lt $ right.mul_lt_one_of_le_of_lt ha h | lemma | eq_one_of_one_le_mul_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"right.mul_lt_one_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_mul_le_one_right (ha : 1 ≤ a) (hb : 1 ≤ b) (hab : a * b ≤ 1) : b = 1 | hb.eq_of_not_gt $ λ h, hab.not_lt $ right.one_lt_mul_of_le_of_lt ha h | lemma | eq_one_of_mul_le_one_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"right.one_lt_mul_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_square_le [covariant_class α α (*) (<)]
(a : α) : ∃ (b : α), b * b ≤ a | begin
by_cases h : a < 1,
{ use a,
have : a*a < a*1,
exact mul_lt_mul_left' h a,
rw mul_one at this,
exact le_of_lt this },
{ use 1,
push_neg at h,
rwa mul_one }
end | lemma | exists_square_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant.to_left_cancel_semigroup
[contravariant_class α α (*) (≤)] :
left_cancel_semigroup α | { mul_left_cancel := λ a b c, mul_left_cancel''
..‹semigroup α› } | def | contravariant.to_left_cancel_semigroup | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"left_cancel_semigroup",
"mul_left_cancel",
"mul_left_cancel''"
] | A semigroup with a partial order and satisfying `left_cancel_semigroup`
(i.e. `a * c < b * c → a < b`) is a `left_cancel semigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant.to_right_cancel_semigroup
[contravariant_class α α (swap (*)) (≤)] :
right_cancel_semigroup α | { mul_right_cancel := λ a b c, mul_right_cancel''
..‹semigroup α› } | def | contravariant.to_right_cancel_semigroup | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"mul_right_cancel",
"mul_right_cancel''",
"right_cancel_semigroup"
] | A semigroup with a partial order and satisfying `right_cancel_semigroup`
(i.e. `a * c < b * c → a < b`) is a `right_cancel semigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_eq_mul_iff_eq_and_eq
[covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (≤)]
[contravariant_class α α (*) (≤)] [contravariant_class α α (swap (*)) (≤)]
{a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) : a * b = c * d ↔ a = c ∧ b = d | begin
refine ⟨λ h, _, λ h, congr_arg2 (*) h.1 h.2⟩,
rcases hac.eq_or_lt with rfl | hac,
{ exact ⟨rfl, mul_left_cancel'' h⟩ },
rcases eq_or_lt_of_le hbd with rfl | hbd,
{ exact ⟨mul_right_cancel'' h, rfl⟩ },
exact ((left.mul_lt_mul hac hbd).ne h).elim,
end | lemma | left.mul_eq_mul_iff_eq_and_eq | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"congr_arg2",
"contravariant_class",
"covariant_class",
"eq_or_lt_of_le",
"left.mul_lt_mul",
"mul_left_cancel''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_eq_mul_iff_eq_and_eq
[covariant_class α α (*) (≤)] [contravariant_class α α (*) (≤)]
[covariant_class α α (swap (*)) (<)] [contravariant_class α α (swap (*)) (≤)]
{a b c d : α} (hac : a ≤ c) (hbd : b ≤ d) : a * b = c * d ↔ a = c ∧ b = d | begin
refine ⟨λ h, _, λ h, congr_arg2 (*) h.1 h.2⟩,
rcases hac.eq_or_lt with rfl | hac,
{ exact ⟨rfl, mul_left_cancel'' h⟩ },
rcases eq_or_lt_of_le hbd with rfl | hbd,
{ exact ⟨mul_right_cancel'' h, rfl⟩ },
exact ((right.mul_lt_mul hac hbd).ne h).elim,
end | lemma | right.mul_eq_mul_iff_eq_and_eq | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"congr_arg2",
"contravariant_class",
"covariant_class",
"eq_or_lt_of_le",
"mul_left_cancel''",
"right.mul_lt_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.const_mul' [covariant_class α α (*) (≤)] (hf : monotone f) (a : α) :
monotone (λ x, a * f x) | λ x y h, mul_le_mul_left' (hf h) a | lemma | monotone.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.const_mul' [covariant_class α α (*) (≤)] (hf : monotone_on f s) (a : α) :
monotone_on (λ x, a * f x) s | λ x hx y hy h, mul_le_mul_left' (hf hx hy h) a | lemma | monotone_on.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone_on",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.const_mul' [covariant_class α α (*) (≤)] (hf : antitone f) (a : α) :
antitone (λ x, a * f x) | λ x y h, mul_le_mul_left' (hf h) a | lemma | antitone.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone",
"covariant_class",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.const_mul' [covariant_class α α (*) (≤)] (hf : antitone_on f s) (a : α) :
antitone_on (λ x, a * f x) s | λ x hx y hy h, mul_le_mul_left' (hf hx hy h) a | lemma | antitone_on.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone_on",
"covariant_class",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul_const' [covariant_class α α (swap (*)) (≤)]
(hf : monotone f) (a : α) : monotone (λ x, f x * a) | λ x y h, mul_le_mul_right' (hf h) a | lemma | monotone.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.mul_const' [covariant_class α α (swap (*)) (≤)]
(hf : monotone_on f s) (a : α) : monotone_on (λ x, f x * a) s | λ x hx y hy h, mul_le_mul_right' (hf hx hy h) a | lemma | monotone_on.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone_on",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.mul_const' [covariant_class α α (swap (*)) (≤)]
(hf : antitone f) (a : α) : antitone (λ x, f x * a) | λ x y h, mul_le_mul_right' (hf h) a | lemma | antitone.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone",
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.mul_const' [covariant_class α α (swap (*)) (≤)]
(hf : antitone_on f s) (a : α) : antitone_on (λ x, f x * a) s | λ x hx y hy h, mul_le_mul_right' (hf hx hy h) a | lemma | antitone_on.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone_on",
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
(hf : monotone f) (hg : monotone g) : monotone (λ x, f x * g x) | λ x y h, mul_le_mul' (hf h) (hg h) | lemma | monotone.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone",
"mul_le_mul'"
] | The product of two monotone functions is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.mul' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
(hf : monotone_on f s) (hg : monotone_on g s) : monotone_on (λ x, f x * g x) s | λ x hx y hy h, mul_le_mul' (hf hx hy h) (hg hx hy h) | lemma | monotone_on.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone_on",
"mul_le_mul'"
] | The product of two monotone functions is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.mul' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
(hf : antitone f) (hg : antitone g) : antitone (λ x, f x * g x) | λ x y h, mul_le_mul' (hf h) (hg h) | lemma | antitone.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone",
"covariant_class",
"mul_le_mul'"
] | The product of two antitone functions is antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.mul' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
(hf : antitone_on f s) (hg : antitone_on g s) : antitone_on (λ x, f x * g x) s | λ x hx y hy h, mul_le_mul' (hf hx hy h) (hg hx hy h) | lemma | antitone_on.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone_on",
"covariant_class",
"mul_le_mul'"
] | The product of two antitone functions is antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono.const_mul' (hf : strict_mono f) (c : α) :
strict_mono (λ x, c * f x) | λ a b ab, mul_lt_mul_left' (hf ab) c | lemma | strict_mono.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_left'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.const_mul' (hf : strict_mono_on f s) (c : α) :
strict_mono_on (λ x, c * f x) s | λ a ha b hb ab, mul_lt_mul_left' (hf ha hb ab) c | lemma | strict_mono_on.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_left'",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.const_mul' (hf : strict_anti f) (c : α) :
strict_anti (λ x, c * f x) | λ a b ab, mul_lt_mul_left' (hf ab) c | lemma | strict_anti.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_left'",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.const_mul' (hf : strict_anti_on f s) (c : α) :
strict_anti_on (λ x, c * f x) s | λ a ha b hb ab, mul_lt_mul_left' (hf ha hb ab) c | lemma | strict_anti_on.const_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_left'",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul_const' (hf : strict_mono f) (c : α) :
strict_mono (λ x, f x * c) | λ a b ab, mul_lt_mul_right' (hf ab) c | lemma | strict_mono.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_right'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.mul_const' (hf : strict_mono_on f s) (c : α) :
strict_mono_on (λ x, f x * c) s | λ a ha b hb ab, mul_lt_mul_right' (hf ha hb ab) c | lemma | strict_mono_on.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_right'",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.mul_const' (hf : strict_anti f) (c : α) :
strict_anti (λ x, f x * c) | λ a b ab, mul_lt_mul_right' (hf ab) c | lemma | strict_anti.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_right'",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.mul_const' (hf : strict_anti_on f s) (c : α) :
strict_anti_on (λ x, f x * c) s | λ a ha b hb ab, mul_lt_mul_right' (hf ha hb ab) c | lemma | strict_anti_on.mul_const' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_lt_mul_right'",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
(hf : strict_mono f) (hg : strict_mono g) :
strict_mono (λ x, f x * g x) | λ a b ab, mul_lt_mul_of_lt_of_lt (hf ab) (hg ab) | lemma | strict_mono.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_lt",
"strict_mono"
] | The product of two strictly monotone functions is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.mul' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
(hf : strict_mono_on f s) (hg : strict_mono_on g s) :
strict_mono_on (λ x, f x * g x) s | λ a ha b hb ab, mul_lt_mul_of_lt_of_lt (hf ha hb ab) (hg ha hb ab) | lemma | strict_mono_on.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_lt",
"strict_mono_on"
] | The product of two strictly monotone functions is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti.mul' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
(hf : strict_anti f) (hg : strict_anti g) :
strict_anti (λ x, f x * g x) | λ a b ab, mul_lt_mul_of_lt_of_lt (hf ab) (hg ab) | lemma | strict_anti.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_lt",
"strict_anti"
] | The product of two strictly antitone functions is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on.mul' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
(hf : strict_anti_on f s) (hg : strict_anti_on g s) :
strict_anti_on (λ x, f x * g x) s | λ a ha b hb ab, mul_lt_mul_of_lt_of_lt (hf ha hb ab) (hg ha hb ab) | lemma | strict_anti_on.mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_lt",
"strict_anti_on"
] | The product of two strictly antitone functions is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.mul_strict_mono' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (≤)]
{f g : β → α} (hf : monotone f) (hg : strict_mono g) :
strict_mono (λ x, f x * g x) | λ x y h, mul_lt_mul_of_le_of_lt (hf h.le) (hg h) | lemma | monotone.mul_strict_mono' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone",
"mul_lt_mul_of_le_of_lt",
"strict_mono"
] | The product of a monotone function and a strictly monotone function is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.mul_strict_mono' [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (≤)] {f g : β → α}
(hf : monotone_on f s) (hg : strict_mono_on g s) :
strict_mono_on (λ x, f x * g x) s | λ x hx y hy h, mul_lt_mul_of_le_of_lt (hf hx hy h.le) (hg hx hy h) | lemma | monotone_on.mul_strict_mono' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone_on",
"mul_lt_mul_of_le_of_lt",
"strict_mono_on"
] | The product of a monotone function and a strictly monotone function is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.mul_strict_anti' [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (≤)]
{f g : β → α} (hf : antitone f) (hg : strict_anti g) :
strict_anti (λ x, f x * g x) | λ x y h, mul_lt_mul_of_le_of_lt (hf h.le) (hg h) | lemma | antitone.mul_strict_anti' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone",
"covariant_class",
"mul_lt_mul_of_le_of_lt",
"strict_anti"
] | The product of a antitone function and a strictly antitone function is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.mul_strict_anti' [covariant_class α α (*) (<)]
[covariant_class α α (swap (*)) (≤)] {f g : β → α}
(hf : antitone_on f s) (hg : strict_anti_on g s) :
strict_anti_on (λ x, f x * g x) s | λ x hx y hy h, mul_lt_mul_of_le_of_lt (hf hx hy h.le) (hg hx hy h) | lemma | antitone_on.mul_strict_anti' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone_on",
"covariant_class",
"mul_lt_mul_of_le_of_lt",
"strict_anti_on"
] | The product of a antitone function and a strictly antitone function is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono.mul_monotone' (hf : strict_mono f) (hg : monotone g) :
strict_mono (λ x, f x * g x) | λ x y h, mul_lt_mul_of_lt_of_le (hf h) (hg h.le) | lemma | strict_mono.mul_monotone' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"monotone",
"mul_lt_mul_of_lt_of_le",
"strict_mono"
] | The product of a strictly monotone function and a monotone function is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.mul_monotone' (hf : strict_mono_on f s) (hg : monotone_on g s) :
strict_mono_on (λ x, f x * g x) s | λ x hx y hy h, mul_lt_mul_of_lt_of_le (hf hx hy h) (hg hx hy h.le) | lemma | strict_mono_on.mul_monotone' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"monotone_on",
"mul_lt_mul_of_lt_of_le",
"strict_mono_on"
] | The product of a strictly monotone function and a monotone function is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti.mul_antitone' (hf : strict_anti f) (hg : antitone g) :
strict_anti (λ x, f x * g x) | λ x y h, mul_lt_mul_of_lt_of_le (hf h) (hg h.le) | lemma | strict_anti.mul_antitone' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone",
"mul_lt_mul_of_lt_of_le",
"strict_anti"
] | The product of a strictly antitone function and a antitone function is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on.mul_antitone' (hf : strict_anti_on f s) (hg : antitone_on g s) :
strict_anti_on (λ x, f x * g x) s | λ x hx y hy h, mul_lt_mul_of_lt_of_le (hf hx hy h) (hg hx hy h.le) | lemma | strict_anti_on.mul_antitone' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"antitone_on",
"mul_lt_mul_of_lt_of_le",
"strict_anti_on"
] | The product of a strictly antitone function and a antitone function is strictly antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cmp_mul_left' {α : Type*} [has_mul α] [linear_order α] [covariant_class α α (*) (<)]
(a b c : α) : cmp (a * b) (a * c) = cmp b c | (strict_mono_id.const_mul' a).cmp_map_eq b c | lemma | cmp_mul_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_mul_right' {α : Type*} [has_mul α] [linear_order α] [covariant_class α α (swap (*)) (<)]
(a b c : α) : cmp (a * c) (b * c) = cmp a b | (strict_mono_id.mul_const' c).cmp_map_eq a b | lemma | cmp_mul_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_cancellable [has_mul α] [has_le α] (a : α) : Prop | ∀ ⦃b c⦄, a * b ≤ a * c → b ≤ c | def | mul_le_cancellable | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [] | An element `a : α` is `mul_le_cancellable` if `x ↦ a * x` is order-reflecting.
We will make a separate version of many lemmas that require `[contravariant_class α α (*) (≤)]` with
`mul_le_cancellable` assumptions instead. These lemmas can then be instantiated to specific types,
like `ennreal`, where we can replace the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant.mul_le_cancellable [has_mul α] [has_le α] [contravariant_class α α (*) (≤)]
{a : α} : mul_le_cancellable a | λ b c, le_of_mul_le_mul_left' | lemma | contravariant.mul_le_cancellable | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_left'",
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_cancellable_one [monoid α] [has_le α] : mul_le_cancellable (1 : α) | λ a b, by simpa only [one_mul] using id | lemma | mul_le_cancellable_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"monoid",
"mul_le_cancellable",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective [has_mul α] [partial_order α] {a : α} (ha : mul_le_cancellable a) :
injective ((*) a) | λ b c h, le_antisymm (ha h.le) (ha h.ge) | lemma | mul_le_cancellable.injective | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj [has_mul α] [partial_order α] {a b c : α} (ha : mul_le_cancellable a) :
a * b = a * c ↔ b = c | ha.injective.eq_iff | lemma | mul_le_cancellable.inj | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_left [comm_semigroup α] [partial_order α] {a : α}
(ha : mul_le_cancellable a) : injective (* a) | λ b c h, ha.injective $ by rwa [mul_comm a, mul_comm a] | lemma | mul_le_cancellable.injective_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"comm_semigroup",
"mul_comm",
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_left [comm_semigroup α] [partial_order α] {a b c : α}
(hc : mul_le_cancellable c) : a * c = b * c ↔ a = b | hc.injective_left.eq_iff | lemma | mul_le_cancellable.inj_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"comm_semigroup",
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_left [has_mul α] [covariant_class α α (*) (≤)]
{a b c : α} (ha : mul_le_cancellable a) : a * b ≤ a * c ↔ b ≤ c | ⟨λ h, ha h, λ h, mul_le_mul_left' h a⟩ | lemma | mul_le_cancellable.mul_le_mul_iff_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_cancellable",
"mul_le_mul_iff_left",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_right [comm_semigroup α] [covariant_class α α (*) (≤)]
{a b c : α} (ha : mul_le_cancellable a) : b * a ≤ c * a ↔ b ≤ c | by rw [mul_comm b, mul_comm c, ha.mul_le_mul_iff_left] | lemma | mul_le_cancellable.mul_le_mul_iff_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"comm_semigroup",
"covariant_class",
"mul_comm",
"mul_le_cancellable",
"mul_le_mul_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_right [mul_one_class α] [covariant_class α α (*) (≤)]
{a b : α} (ha : mul_le_cancellable a) : a ≤ a * b ↔ 1 ≤ b | iff.trans (by rw [mul_one]) ha.mul_le_mul_iff_left | lemma | mul_le_cancellable.le_mul_iff_one_le_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_iff_one_le_right",
"mul_le_cancellable",
"mul_one",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_right [mul_one_class α] [covariant_class α α (*) (≤)]
{a b : α} (ha : mul_le_cancellable a) : a * b ≤ a ↔ b ≤ 1 | iff.trans (by rw [mul_one]) ha.mul_le_mul_iff_left | lemma | mul_le_cancellable.mul_le_iff_le_one_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_cancellable",
"mul_le_iff_le_one_right",
"mul_one",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_left [comm_monoid α] [covariant_class α α (*) (≤)]
{a b : α} (ha : mul_le_cancellable a) : a ≤ b * a ↔ 1 ≤ b | by rw [mul_comm, ha.le_mul_iff_one_le_right] | lemma | mul_le_cancellable.le_mul_iff_one_le_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"comm_monoid",
"covariant_class",
"le_mul_iff_one_le_left",
"mul_comm",
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_left [comm_monoid α] [covariant_class α α (*) (≤)]
{a b : α} (ha : mul_le_cancellable a) : b * a ≤ a ↔ b ≤ 1 | by rw [mul_comm, ha.mul_le_iff_le_one_right] | lemma | mul_le_cancellable.mul_le_iff_le_one_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"comm_monoid",
"covariant_class",
"mul_comm",
"mul_le_cancellable",
"mul_le_iff_le_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_mono [covariant_class α α (+) (≤)] [covariant_class α α (swap (+)) (≤)] :
monotone (bit0 : α → α) | λ a b h, add_le_add h h | lemma | bit0_mono | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_strict_mono [covariant_class α α (+) (<)] [covariant_class α α (swap (+)) (<)] :
strict_mono (bit0 : α → α) | λ a b h, add_lt_add h h | lemma | bit0_strict_mono | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fn_min_mul_fn_max [linear_order α] [comm_semigroup β] (f : α → β) (n m : α) :
f (min n m) * f (max n m) = f n * f m | by { cases le_total n m with h h; simp [h, mul_comm] } | lemma | fn_min_mul_fn_max | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"comm_semigroup",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_max [linear_order α] [comm_semigroup α] (n m : α) :
min n m * max n m = n * m | fn_min_mul_fn_max id n m | lemma | min_mul_max | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"comm_semigroup",
"fn_min_mul_fn_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_mul_left (a b c : α) : min (a * b) (a * c) = a * min b c | (monotone_id.const_mul' a).map_min.symm | lemma | min_mul_mul_left | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_mul_mul_left (a b c : α) : max (a * b) (a * c) = a * max b c | (monotone_id.const_mul' a).map_max.symm | lemma | max_mul_mul_left | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_mul_right (a b c : α) : min (a * c) (b * c) = min a b * c | (monotone_id.mul_const' c).map_min.symm | lemma | min_mul_mul_right | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_mul_mul_right (a b c : α) : max (a * c) (b * c) = max a b * c | (monotone_id.mul_const' c).map_max.symm | lemma | max_mul_mul_right | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_or_lt_of_mul_lt_mul [covariant_class α α (*) (≤)]
[covariant_class α α (swap (*)) (≤)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ | by { contrapose!, exact λ h, mul_le_mul' h.1 h.2 } | lemma | lt_or_lt_of_mul_lt_mul | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_or_lt_of_mul_le_mul [covariant_class α α (*) (≤)]
[covariant_class α α (swap (*)) (<)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ | by { contrapose!, exact λ h, mul_lt_mul_of_lt_of_le h.1 h.2 } | lemma | le_or_lt_of_mul_le_mul | algebra.order.monoid | src/algebra/order/monoid/min_max.lean | [
"order.min_max",
"algebra.order.monoid.lemmas"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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