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
value |
|---|---|---|---|---|---|---|---|---|---|---|
mul_nonneg_of_three (a b c : α) :
0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a | by iterate 3 { rw mul_nonneg_iff };
have := le_total 0 a; have := le_total 0 b; have := le_total 0 c; itauto | lemma | mul_nonneg_of_three | 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... | [
"itauto",
"mul_nonneg_iff"
] | Out of three elements of a `linear_ordered_ring`, two must have the same sign. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_nonpos_iff : a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b | by rw [← neg_nonneg, neg_mul_eq_mul_neg, mul_nonneg_iff, neg_nonneg, neg_nonpos] | lemma | mul_nonpos_iff | 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_nonneg_iff",
"neg_mul_eq_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_nonneg (a : α) : 0 ≤ a * a | (le_total 0 a).elim (λ h, mul_nonneg h h) (λ h, mul_nonneg_of_nonpos_of_nonpos h h) | lemma | mul_self_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_nonneg_of_nonpos_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_le_self_iff : -a ≤ a ↔ 0 ≤ a | by simp [neg_le_iff_add_nonneg, ← two_mul, mul_nonneg_iff, zero_le_one, (zero_lt_two' α).not_le] | lemma | neg_le_self_iff | 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_nonneg_iff",
"two_mul",
"zero_le_one",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_lt_self_iff : -a < a ↔ 0 < a | by simp [neg_lt_iff_pos_add, ← two_mul, mul_pos_iff, zero_lt_one, (zero_lt_two' α).not_lt] | lemma | neg_lt_self_iff | 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_iff",
"two_mul",
"zero_lt_one",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_neg_self_iff : a ≤ -a ↔ a ≤ 0 | calc a ≤ -a ↔ -(-a) ≤ -a : by rw neg_neg
... ↔ 0 ≤ -a : neg_le_self_iff
... ↔ a ≤ 0 : neg_nonneg | lemma | le_neg_self_iff | 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... | [
"neg_le_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_neg_self_iff : a < -a ↔ a < 0 | calc a < -a ↔ -(-a) < -a : by rw neg_neg
... ↔ 0 < -a : neg_lt_self_iff
... ↔ a < 0 : neg_pos | lemma | lt_neg_self_iff | 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... | [
"neg_lt_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_lt_zero : -1 < (0:α) | neg_lt_zero.2 zero_lt_one | lemma | neg_one_lt_zero | 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_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left_of_neg {a b c : α} (h : c < 0) : c * a ≤ c * b ↔ b ≤ a | (strict_anti_mul_left h).le_iff_le | lemma | mul_le_mul_left_of_neg | 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_anti_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_right_of_neg {a b c : α} (h : c < 0) : a * c ≤ b * c ↔ b ≤ a | (strict_anti_mul_right h).le_iff_le | lemma | mul_le_mul_right_of_neg | 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_anti_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_left_of_neg {a b c : α} (h : c < 0) : c * a < c * b ↔ b < a | (strict_anti_mul_left h).lt_iff_lt | lemma | mul_lt_mul_left_of_neg | 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_anti_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_right_of_neg {a b c : α} (h : c < 0) : a * c < b * c ↔ b < a | (strict_anti_mul_right h).lt_iff_lt | lemma | mul_lt_mul_right_of_neg | 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_anti_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_of_nonpos_left (h : c * a < c * b) (hc : c ≤ 0) : b < a | lt_of_mul_lt_mul_left (by rwa [neg_mul, neg_mul, neg_lt_neg_iff]) $ neg_nonneg.2 hc | lemma | lt_of_mul_lt_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... | [
"lt_of_mul_lt_mul_left",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_of_nonpos_right (h : a * c < b * c) (hc : c ≤ 0) : b < a | lt_of_mul_lt_mul_right (by rwa [mul_neg, mul_neg, neg_lt_neg_iff]) $ neg_nonneg.2 hc | lemma | lt_of_mul_lt_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... | [
"lt_of_mul_lt_mul_right",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_mul_neg_left {a : α} (ha : a < 0) (b c : α) : cmp (a * b) (a * c) = cmp c b | (strict_anti_mul_left ha).cmp_map_eq b c | lemma | cmp_mul_neg_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... | [
"strict_anti_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_mul_neg_right {a : α} (ha : a < 0) (b c : α) : cmp (b * a) (c * a) = cmp c b | (strict_anti_mul_right ha).cmp_map_eq b c | lemma | cmp_mul_neg_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... | [
"strict_anti_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_lt (a : α) : a - 1 < a | sub_lt_iff_lt_add.2 (lt_add_one a) | lemma | sub_one_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... | [
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_pos {a : α} : 0 < a * a ↔ a ≠ 0 | begin
split,
{ rintro h rfl, rw mul_zero at h, exact h.false },
{ intro h,
cases h.lt_or_lt with h h,
exacts [mul_pos_of_neg_of_neg h h, mul_pos h h] }
end | lemma | mul_self_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... | [
"mul_pos_of_neg_of_neg",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_le_mul_self_of_le_of_neg_le {x y : α} (h₁ : x ≤ y) (h₂ : -x ≤ y) : x * x ≤ y * y | (le_total 0 x).elim (λ h, mul_le_mul h₁ h₁ h (h.trans h₁))
(λ h, le_of_eq_of_le (neg_mul_neg x x).symm
(mul_le_mul h₂ h₂ (neg_nonneg.mpr h) ((neg_nonneg.mpr h).trans h₂))) | lemma | mul_self_le_mul_self_of_le_of_neg_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_of_eq_of_le",
"mul_le_mul",
"neg_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_mul_nonpos_left {a b : α} (h : a * b ≤ 0) (hb : b < 0) : 0 ≤ a | le_of_not_gt (λ ha, absurd h (mul_pos_of_neg_of_neg ha hb).not_le) | lemma | nonneg_of_mul_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_pos_of_neg_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_mul_nonpos_right {a b : α} (h : a * b ≤ 0) (ha : a < 0) : 0 ≤ b | le_of_not_gt (λ hb, absurd h (mul_pos_of_neg_of_neg ha hb).not_le) | lemma | nonneg_of_mul_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_pos_of_neg_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mul_neg_left {a b : α} (h : a * b < 0) (hb : b ≤ 0) : 0 < a | lt_of_not_ge (λ ha, absurd h (mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt) | lemma | pos_of_mul_neg_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_nonneg_of_nonpos_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mul_neg_right {a b : α} (h : a * b < 0) (ha : a ≤ 0) : 0 < b | lt_of_not_ge (λ hb, absurd h (mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt) | lemma | pos_of_mul_neg_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_nonneg_of_nonpos_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_iff_pos_of_mul_neg (hab : a * b < 0) : a < 0 ↔ 0 < b | ⟨pos_of_mul_neg_right hab ∘ le_of_lt, neg_of_mul_neg_left hab ∘ le_of_lt⟩ | lemma | neg_iff_pos_of_mul_neg | 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... | [
"neg_of_mul_neg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_iff_neg_of_mul_neg (hab : a * b < 0) : 0 < a ↔ b < 0 | ⟨neg_of_mul_neg_right hab ∘ le_of_lt, pos_of_mul_neg_left hab ∘ le_of_lt⟩ | lemma | pos_iff_neg_of_mul_neg | 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_of_mul_neg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_add_mul_self_eq_zero {x y : α} : x * x + y * y = 0 ↔ x = 0 ∧ y = 0 | by rw [add_eq_zero_iff', mul_self_eq_zero, mul_self_eq_zero]; apply mul_self_nonneg | lemma | mul_self_add_mul_self_eq_zero | 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_self_eq_zero",
"mul_self_nonneg"
] | The sum of two squares is zero iff both elements are zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_mul_self_add_mul_self_eq_zero (h : a * a + b * b = 0) : a = 0 | (mul_self_add_mul_self_eq_zero.mp h).left | lemma | eq_zero_of_mul_self_add_mul_self_eq_zero | 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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_ring.to_strict_ordered_comm_ring [d : linear_ordered_comm_ring α] :
strict_ordered_comm_ring α | { ..d } | instance | linear_ordered_comm_ring.to_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... | [
"linear_ordered_comm_ring",
"strict_ordered_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_ring.to_linear_ordered_comm_semiring [d : linear_ordered_comm_ring α] :
linear_ordered_comm_semiring α | { .. d, ..linear_ordered_ring.to_linear_ordered_semiring } | instance | linear_ordered_comm_ring.to_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_comm_ring",
"linear_ordered_comm_semiring",
"linear_ordered_ring.to_linear_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_mul_mul_le_max_mul_max (b c : α) (ha : 0 ≤ a) (hd: 0 ≤ d) :
max (a * b) (d * c) ≤ max a c * max d b | have ba : b * a ≤ max d b * max c a, from
mul_le_mul (le_max_right d b) (le_max_right c a) ha (le_trans hd (le_max_left d b)),
have cd : c * d ≤ max a c * max b d, from
mul_le_mul (le_max_right a c) (le_max_right b d) hd (le_trans ha (le_max_left a c)),
max_le
(by simpa [mul_comm, max_comm] using ba)
(by simpa ... | lemma | max_mul_mul_le_max_mul_max | 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_comm",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring [ordered_semiring α] [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] (f : β → α) (hf : injective f) (zero : f 0 = 0)
(one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) ... | { zero_le_one := show f 0 ≤ f 1, by simp only [zero, one, zero_le_one],
mul_le_mul_of_nonneg_left := λ a b c h hc, show f (c * a) ≤ f (c * b),
by { rw [mul, mul], refine mul_le_mul_of_nonneg_left h _, rwa ←zero },
mul_le_mul_of_nonneg_right := λ a b c h hc, show f (a * c) ≤ f (b * c),
by { rw [mul, mul], re... | def | function.injective.ordered_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"ordered_semiring",
"zero_le_one"
] | Pullback an `ordered_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_semiring [ordered_comm_semiring α] [has_zero β] [has_one β] [has_add β]
[has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] (f : β → α) (hf : injective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (nsmul : ∀ x (n : ℕ), f (n • x) =... | { ..hf.comm_semiring f zero one add mul nsmul npow nat_cast,
..hf.ordered_semiring f zero one add mul nsmul npow nat_cast } | def | function.injective.ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"ordered_comm_semiring"
] | Pullback an `ordered_comm_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_ring [ordered_ring α] [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β]
[has_int_cast β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * ... | { mul_nonneg := λ a b ha hb, show f 0 ≤ f (a * b),
by { rw [zero, mul], apply mul_nonneg; rwa ← zero },
..hf.ordered_semiring f zero one add mul nsmul npow nat_cast,
..hf.ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.ordered_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"ordered_ring"
] | Pullback an `ordered_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_ring [ordered_comm_ring α] [has_zero β] [has_one β] [has_add β]
[has_mul β] [has_neg β] [has_sub β] [has_pow β ℕ] [has_smul ℕ β] [has_smul ℤ β] [has_nat_cast β]
[has_int_cast β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y... | { ..hf.ordered_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast,
..hf.comm_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"ordered_comm_ring"
] | Pullback an `ordered_comm_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_semiring [strict_ordered_semiring α] [has_zero β] [has_one β]
[has_add β] [has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] (f : β → α)
(hf : injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (nsmul : ∀ x (n : ℕ), f (n • ... | { mul_lt_mul_of_pos_left := λ a b c h hc, show f (c * a) < f (c * b),
by simpa only [mul, zero] using mul_lt_mul_of_pos_left ‹f a < f b› (by rwa ←zero),
mul_lt_mul_of_pos_right := λ a b c h hc, show f (a * c) < f (b * c),
by simpa only [mul, zero] using mul_lt_mul_of_pos_right ‹f a < f b› (by rwa ←zero),
..... | def | function.injective.strict_ordered_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"pullback_nonzero",
"strict_ordered_semiring"
] | Pullback a `strict_ordered_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_semiring [strict_ordered_comm_semiring α] [has_zero β] [has_one β]
[has_add β] [has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] (f : β → α)
(hf : injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (nsmul : ∀ x (n : ℕ... | { ..hf.comm_semiring f zero one add mul nsmul npow nat_cast,
..hf.strict_ordered_semiring f zero one add mul nsmul npow nat_cast } | def | function.injective.strict_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"strict_ordered_comm_semiring"
] | Pullback a `strict_ordered_comm_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_ring [strict_ordered_ring α] [has_zero β] [has_one β] [has_add β]
[has_mul β] [has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β]
[has_int_cast β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x... | { mul_pos := λ a b a0 b0, show f 0 < f (a * b), by { rw [zero, mul], apply mul_pos; rwa ← zero },
..hf.strict_ordered_semiring f zero one add mul nsmul npow nat_cast,
..hf.ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.strict_ordered_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"strict_ordered_ring"
] | Pullback a `strict_ordered_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_ring [strict_ordered_comm_ring α] [has_zero β]
[has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_pow β ℕ] [has_smul ℕ β]
[has_smul ℤ β] [has_nat_cast β] [has_int_cast β] (f : β → α) (hf : injective f) (zero : f 0 = 0)
(one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀... | { ..hf.strict_ordered_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast,
..hf.comm_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.strict_ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"strict_ordered_comm_ring"
] | Pullback a `strict_ordered_comm_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_semiring [linear_ordered_semiring α] [has_zero β] [has_one β]
[has_add β] [has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β] [has_sup β] [has_inf β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(ns... | { .. linear_order.lift f hf hsup hinf,
.. hf.strict_ordered_semiring f zero one add mul nsmul npow nat_cast } | def | function.injective.linear_ordered_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_inf",
"has_nat_cast",
"has_smul",
"has_sup",
"linear_order.lift",
"linear_ordered_semiring"
] | Pullback a `linear_ordered_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_comm_semiring [linear_ordered_comm_semiring α]
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_pow β ℕ] [has_smul ℕ β] [has_nat_cast β]
[has_sup β] [has_inf β] (f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * ... | { ..hf.linear_ordered_semiring f zero one add mul nsmul npow nat_cast hsup hinf,
..hf.strict_ordered_comm_semiring f zero one add mul nsmul npow nat_cast } | def | function.injective.linear_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_inf",
"has_nat_cast",
"has_smul",
"has_sup",
"linear_ordered_comm_semiring"
] | Pullback a `linear_ordered_semiring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_ring [linear_ordered_ring α] [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β]
[has_int_cast β] [has_sup β] [has_inf β] (f : β → α) (hf : injective f) (zero : f 0 = 0)
(one : f 1 = 1) (add : ∀ x y, f (x + y) = f x +... | { .. linear_order.lift f hf hsup hinf,
.. hf.strict_ordered_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.linear_ordered_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_inf",
"has_int_cast",
"has_nat_cast",
"has_smul",
"has_sup",
"linear_order.lift",
"linear_ordered_ring"
] | Pullback a `linear_ordered_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_comm_ring [linear_ordered_comm_ring α] [has_zero β]
[has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_pow β ℕ] [has_smul ℕ β]
[has_smul ℤ β] [has_nat_cast β] [has_int_cast β] [has_sup β] [has_inf β] (f : β → α)
(hf : injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x +... | { .. linear_order.lift f hf hsup hinf,
.. hf.strict_ordered_comm_ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast } | def | function.injective.linear_ordered_comm_ring | algebra.order.ring | src/algebra/order/ring/inj_surj.lean | [
"algebra.order.ring.defs",
"algebra.order.monoid.cancel.basic",
"algebra.ring.inj_surj"
] | [
"has_inf",
"has_int_cast",
"has_nat_cast",
"has_smul",
"has_sup",
"linear_order.lift",
"linear_ordered_comm_ring"
] | Pullback a `linear_ordered_comm_ring` under an injective map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_mul_mono : Prop | covariant_class α≥0 α (λ x y, x * y) (≤) | abbreviation | pos_mul_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class"
] | `pos_mul_mono α` is an abbreviation for `covariant_class α≥0 α (λ x y, x * y) (≤)`,
expressing that multiplication by nonnegative elements on the left is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pos_mono : Prop | covariant_class α≥0 α (λ x y, y * x) (≤) | abbreviation | mul_pos_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class"
] | `mul_pos_mono α` is an abbreviation for `covariant_class α≥0 α (λ x y, y * x) (≤)`,
expressing that multiplication by nonnegative elements on the right is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_mul_strict_mono : Prop | covariant_class α>0 α (λ x y, x * y) (<) | abbreviation | pos_mul_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class"
] | `pos_mul_strict_mono α` is an abbreviation for `covariant_class α>0 α (λ x y, x * y) (<)`,
expressing that multiplication by positive elements on the left is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pos_strict_mono : Prop | covariant_class α>0 α (λ x y, y * x) (<) | abbreviation | mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class"
] | `mul_pos_strict_mono α` is an abbreviation for `covariant_class α>0 α (λ x y, y * x) (<)`,
expressing that multiplication by positive elements on the right is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_mul_reflect_lt : Prop | contravariant_class α≥0 α (λ x y, x * y) (<) | abbreviation | pos_mul_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class"
] | `pos_mul_reflect_lt α` is an abbreviation for `contravariant_class α≥0 α (λ x y, x * y) (<)`,
expressing that multiplication by nonnegative elements on the left is strictly reverse monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pos_reflect_lt : Prop | contravariant_class α≥0 α (λ x y, y * x) (<) | abbreviation | mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class"
] | `mul_pos_reflect_lt α` is an abbreviation for `contravariant_class α≥0 α (λ x y, y * x) (<)`,
expressing that multiplication by nonnegative elements on the right is strictly reverse monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_mul_mono_rev : Prop | contravariant_class α>0 α (λ x y, x * y) (≤) | abbreviation | pos_mul_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class"
] | `pos_mul_mono_rev α` is an abbreviation for `contravariant_class α>0 α (λ x y, x * y) (≤)`,
expressing that multiplication by positive elements on the left is reverse monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_pos_mono_rev : Prop | contravariant_class α>0 α (λ x y, y * x) (≤) | abbreviation | mul_pos_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class"
] | `mul_pos_mono_rev α` is an abbreviation for `contravariant_class α>0 α (λ x y, y * x) (≤)`,
expressing that multiplication by positive elements on the right is reverse monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pos_mul_mono.to_covariant_class_pos_mul_le [pos_mul_mono α] :
covariant_class α>0 α (λ x y, x * y) (≤) | ⟨λ a b c bc, @covariant_class.elim α≥0 α (λ x y, x * y) (≤) _ ⟨_, a.2.le⟩ _ _ bc⟩ | instance | pos_mul_mono.to_covariant_class_pos_mul_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono.to_covariant_class_pos_mul_le [mul_pos_mono α] :
covariant_class α>0 α (λ x y, y * x) (≤) | ⟨λ a b c bc, @covariant_class.elim α≥0 α (λ x y, y * x) (≤) _ ⟨_, a.2.le⟩ _ _ bc⟩ | instance | mul_pos_mono.to_covariant_class_pos_mul_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_reflect_lt.to_contravariant_class_pos_mul_lt [pos_mul_reflect_lt α] :
contravariant_class α>0 α (λ x y, x * y) (<) | ⟨λ a b c bc, @contravariant_class.elim α≥0 α (λ x y, x * y) (<) _ ⟨_, a.2.le⟩ _ _ bc⟩ | instance | pos_mul_reflect_lt.to_contravariant_class_pos_mul_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_reflect_lt.to_contravariant_class_pos_mul_lt [mul_pos_reflect_lt α] :
contravariant_class α>0 α (λ x y, y * x) (<) | ⟨λ a b c bc, @contravariant_class.elim α≥0 α (λ x y, y * x) (<) _ ⟨_, a.2.le⟩ _ _ bc⟩ | instance | mul_pos_reflect_lt.to_contravariant_class_pos_mul_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class",
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonneg_left [pos_mul_mono α] (h : b ≤ c) (a0 : 0 ≤ a) : a * b ≤ a * c | @covariant_class.elim α≥0 α (λ x y, x * y) (≤) _ ⟨a, a0⟩ _ _ h | lemma | mul_le_mul_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_nonneg_right [mul_pos_mono α] (h : b ≤ c) (a0 : 0 ≤ a) : b * a ≤ c * a | @covariant_class.elim α≥0 α (λ x y, y * x) (≤) _ ⟨a, a0⟩ _ _ h | lemma | mul_le_mul_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_pos_left [pos_mul_strict_mono α] (bc : b < c) (a0 : 0 < a) : a * b < a * c | @covariant_class.elim α>0 α (λ x y, x * y) (<) _ ⟨a, a0⟩ _ _ bc | lemma | mul_lt_mul_of_pos_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_pos_right [mul_pos_strict_mono α] (bc : b < c) (a0 : 0 < a) : b * a < c * a | @covariant_class.elim α>0 α (λ x y, y * x) (<) _ ⟨a, a0⟩ _ _ bc | lemma | mul_lt_mul_of_pos_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_left [pos_mul_reflect_lt α] (h : a * b < a * c) (a0 : 0 ≤ a) : b < c | @contravariant_class.elim α≥0 α (λ x y, x * y) (<) _ ⟨a, a0⟩ _ _ h | lemma | lt_of_mul_lt_mul_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_right [mul_pos_reflect_lt α] (h : b * a < c * a) (a0 : 0 ≤ a) : b < c | @contravariant_class.elim α≥0 α (λ x y, y * x) (<) _ ⟨a, a0⟩ _ _ h | lemma | lt_of_mul_lt_mul_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_left [pos_mul_mono_rev α] (bc : a * b ≤ a * c) (a0 : 0 < a) : b ≤ c | @contravariant_class.elim α>0 α (λ x y, x * y) (≤) _ ⟨a, a0⟩ _ _ bc | lemma | le_of_mul_le_mul_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_right [mul_pos_mono_rev α] (bc : b * a ≤ c * a) (a0 : 0 < a) : b ≤ c | @contravariant_class.elim α>0 α (λ x y, y * x) (≤) _ ⟨a, a0⟩ _ _ bc | lemma | le_of_mul_le_mul_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_left [pos_mul_strict_mono α] [pos_mul_reflect_lt α]
(a0 : 0 < a) :
a * b < a * c ↔ b < c | @rel_iff_cov α>0 α (λ x y, x * y) (<) _ _ ⟨a, a0⟩ _ _ | lemma | mul_lt_mul_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_reflect_lt",
"pos_mul_strict_mono",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_right [mul_pos_strict_mono α] [mul_pos_reflect_lt α]
(a0 : 0 < a) :
b * a < c * a ↔ b < c | @rel_iff_cov α>0 α (λ x y, y * x) (<) _ _ ⟨a, a0⟩ _ _ | lemma | mul_lt_mul_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_reflect_lt",
"mul_pos_strict_mono",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left [pos_mul_mono α] [pos_mul_mono_rev α]
(a0 : 0 < a) :
a * b ≤ a * c ↔ b ≤ c | @rel_iff_cov α>0 α (λ x y, x * y) (≤) _ _ ⟨a, a0⟩ _ _ | lemma | mul_le_mul_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono",
"pos_mul_mono_rev",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_right [mul_pos_mono α] [mul_pos_mono_rev α]
(a0 : 0 < a) :
b * a ≤ c * a ↔ b ≤ c | @rel_iff_cov α>0 α (λ x y, y * x) (≤) _ _ ⟨a, a0⟩ _ _ | lemma | mul_le_mul_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"mul_pos_mono_rev",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_pos_of_nonneg [pos_mul_strict_mono α] [mul_pos_mono α]
(h₁ : a ≤ b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 ≤ d) : a * c < b * d | (mul_lt_mul_of_pos_left h₂ a0).trans_le (mul_le_mul_of_nonneg_right h₁ d0) | lemma | mul_lt_mul_of_pos_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_lt_mul_of_pos_left",
"mul_pos_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_le_of_le' [pos_mul_strict_mono α] [mul_pos_mono α]
(h₁ : a ≤ b) (h₂ : c < d) (b0 : 0 < b) (c0 : 0 ≤ c) : a * c < b * d | (mul_le_mul_of_nonneg_right h₁ c0).trans_lt (mul_lt_mul_of_pos_left h₂ b0) | lemma | mul_lt_mul_of_le_of_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_lt_mul_of_pos_left",
"mul_pos_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_nonneg_of_pos [pos_mul_mono α] [mul_pos_strict_mono α]
(h₁ : a < b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 < d) : a * c < b * d | (mul_le_mul_of_nonneg_left h₂ a0).trans_lt (mul_lt_mul_of_pos_right h₁ d0) | lemma | mul_lt_mul_of_nonneg_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_le_of_lt' [pos_mul_mono α] [mul_pos_strict_mono α]
(h₁ : a < b) (h₂ : c ≤ d) (b0 : 0 ≤ b) (c0 : 0 < c) : a * c < b * d | (mul_lt_mul_of_pos_right h₁ c0).trans_le (mul_le_mul_of_nonneg_left h₂ b0) | lemma | mul_lt_mul_of_le_of_lt' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_pos_of_pos [pos_mul_strict_mono α] [mul_pos_strict_mono α]
(h₁ : a < b) (h₂ : c < d) (a0 : 0 < a) (d0 : 0 < d) : a * c < b * d | (mul_lt_mul_of_pos_left h₂ a0).trans (mul_lt_mul_of_pos_right h₁ d0) | lemma | mul_lt_mul_of_pos_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_lt_of_lt' [pos_mul_strict_mono α] [mul_pos_strict_mono α]
(h₁ : a < b) (h₂ : c < d) (b0 : 0 < b) (c0 : 0 < c) : a * c < b * d | (mul_lt_mul_of_pos_right h₁ c0).trans (mul_lt_mul_of_pos_left h₂ b0) | lemma | mul_lt_mul_of_lt_of_lt' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_mul_lt_of_nonneg_left [pos_mul_mono α] (h : a * b < c) (hdb : d ≤ b) (ha : 0 ≤ a) :
a * d < c | (mul_le_mul_of_nonneg_left hdb ha).trans_lt h | lemma | mul_lt_of_mul_lt_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_mul_of_nonneg_left [pos_mul_mono α] (h : a < b * c) (hcd : c ≤ d) (hb : 0 ≤ b) :
a < b * d | h.trans_le $ mul_le_mul_of_nonneg_left hcd hb | lemma | lt_mul_of_lt_mul_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_mul_lt_of_nonneg_right [mul_pos_mono α] (h : a * b < c) (hda : d ≤ a) (hb : 0 ≤ b) :
d * b < c | (mul_le_mul_of_nonneg_right hda hb).trans_lt h | lemma | mul_lt_of_mul_lt_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_mul_of_nonneg_right [mul_pos_mono α] (h : a < b * c) (hbd : b ≤ d) (hc : 0 ≤ c) :
a < d * c | h.trans_le $ mul_le_mul_of_nonneg_right hbd hc | lemma | lt_mul_of_lt_mul_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_strict_mono.to_pos_mul_mono_rev [pos_mul_strict_mono α] : pos_mul_mono_rev α | ⟨λ x a b h, le_of_not_lt $ λ h', h.not_lt $ mul_lt_mul_of_pos_left h' x.prop⟩ | instance | pos_mul_strict_mono.to_pos_mul_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"pos_mul_mono_rev",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_strict_mono.to_mul_pos_mono_rev [mul_pos_strict_mono α] : mul_pos_mono_rev α | ⟨λ x a b h, le_of_not_lt $ λ h', h.not_lt $ mul_lt_mul_of_pos_right h' x.prop⟩ | instance | mul_pos_strict_mono.to_mul_pos_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_right",
"mul_pos_mono_rev",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_rev.to_pos_mul_strict_mono [pos_mul_mono_rev α] : pos_mul_strict_mono α | ⟨λ x a b h, lt_of_not_le $ λ h', h.not_le $ le_of_mul_le_mul_of_pos_left h' x.prop⟩ | lemma | pos_mul_mono_rev.to_pos_mul_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_not_le",
"pos_mul_mono_rev",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_rev.to_mul_pos_strict_mono [mul_pos_mono_rev α] : mul_pos_strict_mono α | ⟨λ x a b h, lt_of_not_le $ λ h', h.not_le $ le_of_mul_le_mul_of_pos_right h' x.prop⟩ | lemma | mul_pos_mono_rev.to_mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_not_le",
"mul_pos_mono_rev",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_strict_mono_iff_pos_mul_mono_rev : pos_mul_strict_mono α ↔ pos_mul_mono_rev α | ⟨@pos_mul_strict_mono.to_pos_mul_mono_rev _ _ _ _, @pos_mul_mono_rev.to_pos_mul_strict_mono _ _ _ _⟩ | lemma | pos_mul_strict_mono_iff_pos_mul_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono_rev",
"pos_mul_mono_rev.to_pos_mul_strict_mono",
"pos_mul_strict_mono",
"pos_mul_strict_mono.to_pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_strict_mono_iff_mul_pos_mono_rev : mul_pos_strict_mono α ↔ mul_pos_mono_rev α | ⟨@mul_pos_strict_mono.to_mul_pos_mono_rev _ _ _ _, @mul_pos_mono_rev.to_mul_pos_strict_mono _ _ _ _⟩ | lemma | mul_pos_strict_mono_iff_mul_pos_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono_rev",
"mul_pos_mono_rev.to_mul_pos_strict_mono",
"mul_pos_strict_mono",
"mul_pos_strict_mono.to_mul_pos_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_reflect_lt.to_pos_mul_mono [pos_mul_reflect_lt α] : pos_mul_mono α | ⟨λ x a b h, le_of_not_lt $ λ h', h.not_lt $ lt_of_mul_lt_mul_left h' x.prop⟩ | lemma | pos_mul_reflect_lt.to_pos_mul_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_left",
"pos_mul_mono",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_reflect_lt.to_mul_pos_mono [mul_pos_reflect_lt α] : mul_pos_mono α | ⟨λ x a b h, le_of_not_lt $ λ h', h.not_lt $ lt_of_mul_lt_mul_right h' x.prop⟩ | lemma | mul_pos_reflect_lt.to_mul_pos_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_right",
"mul_pos_mono",
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono.to_pos_mul_reflect_lt [pos_mul_mono α] : pos_mul_reflect_lt α | ⟨λ x a b h, lt_of_not_le $ λ h', h.not_le $ mul_le_mul_of_nonneg_left h' x.prop⟩ | lemma | pos_mul_mono.to_pos_mul_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_not_le",
"mul_le_mul_of_nonneg_left",
"pos_mul_mono",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono.to_mul_pos_reflect_lt [mul_pos_mono α] : mul_pos_reflect_lt α | ⟨λ x a b h, lt_of_not_le $ λ h', h.not_le $ mul_le_mul_of_nonneg_right h' x.prop⟩ | lemma | mul_pos_mono.to_mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_not_le",
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_iff_pos_mul_reflect_lt : pos_mul_mono α ↔ pos_mul_reflect_lt α | ⟨@pos_mul_mono.to_pos_mul_reflect_lt _ _ _ _, @pos_mul_reflect_lt.to_pos_mul_mono _ _ _ _⟩ | lemma | pos_mul_mono_iff_pos_mul_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono",
"pos_mul_mono.to_pos_mul_reflect_lt",
"pos_mul_reflect_lt",
"pos_mul_reflect_lt.to_pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_iff_mul_pos_reflect_lt : mul_pos_mono α ↔ mul_pos_reflect_lt α | ⟨@mul_pos_mono.to_mul_pos_reflect_lt _ _ _ _, @mul_pos_reflect_lt.to_mul_pos_mono _ _ _ _⟩ | lemma | mul_pos_mono_iff_mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"mul_pos_mono.to_mul_pos_reflect_lt",
"mul_pos_reflect_lt",
"mul_pos_reflect_lt.to_mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.mul_pos [pos_mul_strict_mono α] (ha : 0 < a) (hb : 0 < b) : 0 < a * b | by simpa only [mul_zero] using mul_lt_mul_of_pos_left hb ha | lemma | left.mul_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_zero",
"pos_mul_strict_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_neg_of_pos_of_neg [pos_mul_strict_mono α] (ha : 0 < a) (hb : b < 0) : a * b < 0 | by simpa only [mul_zero] using mul_lt_mul_of_pos_left hb ha | lemma | mul_neg_of_pos_of_neg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_zero",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_mul_left [pos_mul_strict_mono α] [pos_mul_reflect_lt α] (h : 0 < c) :
0 < c * b ↔ 0 < b | by { convert mul_lt_mul_left h, simp } | lemma | zero_lt_mul_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_left",
"pos_mul_reflect_lt",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_pos [mul_pos_strict_mono α] (ha : 0 < a) (hb : 0 < b) : 0 < a * b | by simpa only [zero_mul] using mul_lt_mul_of_pos_right ha hb | lemma | right.mul_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"zero_mul"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_neg_of_neg_of_pos [mul_pos_strict_mono α] (ha : a < 0) (hb : 0 < b) : a * b < 0 | by simpa only [zero_mul] using mul_lt_mul_of_pos_right ha hb | lemma | mul_neg_of_neg_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_mul_right [mul_pos_strict_mono α] [mul_pos_reflect_lt α] (h : 0 < c) :
0 < b * c ↔ 0 < b | by { convert mul_lt_mul_right h, simp } | lemma | zero_lt_mul_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_right",
"mul_pos_reflect_lt",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.mul_nonneg [pos_mul_mono α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b | by simpa only [mul_zero] using mul_le_mul_of_nonneg_left hb ha | lemma | left.mul_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_zero",
"pos_mul_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_nonpos_of_nonneg_of_nonpos [pos_mul_mono α] (ha : 0 ≤ a) (hb : b ≤ 0) : a * b ≤ 0 | by simpa only [mul_zero] using mul_le_mul_of_nonneg_left hb ha | lemma | mul_nonpos_of_nonneg_of_nonpos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_zero",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_nonneg [mul_pos_mono α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b | by simpa only [zero_mul] using mul_le_mul_of_nonneg_right ha hb | lemma | right.mul_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"zero_mul"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_nonpos_of_nonpos_of_nonneg [mul_pos_mono α] (ha : a ≤ 0) (hb : 0 ≤ b) : a * b ≤ 0 | by simpa only [zero_mul] using mul_le_mul_of_nonneg_right ha hb | lemma | mul_nonpos_of_nonpos_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mul_pos_right [pos_mul_reflect_lt α] (h : 0 < a * b) (ha : 0 ≤ a) : 0 < b | lt_of_mul_lt_mul_left ((mul_zero a).symm ▸ h : a * 0 < a * b) ha | lemma | pos_of_mul_pos_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_left",
"mul_zero",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_mul_pos_left [mul_pos_reflect_lt α] (h : 0 < a * b) (hb : 0 ≤ b) : 0 < a | lt_of_mul_lt_mul_right ((zero_mul b).symm ▸ h : 0 * b < a * b) hb | lemma | pos_of_mul_pos_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_right",
"mul_pos_reflect_lt",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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