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 |
|---|---|---|---|---|---|---|---|---|---|---|
antitone.mul_const_of_nonpos (hf : antitone f) (ha : a ≤ 0) : monotone (λ x, f x * a) | (antitone_mul_right ha).comp hf | lemma | antitone.mul_const_of_nonpos | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"antitone_mul_right",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.mul_monotone (hf : antitone f) (hg : monotone g) (hf₀ : ∀ x, f x ≤ 0)
(hg₀ : ∀ x, 0 ≤ g x) :
antitone (f * g) | λ b c h, mul_le_mul_of_nonpos_of_nonneg (hf h) (hg h) (hf₀ _) (hg₀ _) | lemma | antitone.mul_monotone | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"monotone",
"mul_le_mul_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul_antitone (hf : monotone f) (hg : antitone g) (hf₀ : ∀ x, 0 ≤ f x)
(hg₀ : ∀ x, g x ≤ 0) :
antitone (f * g) | λ b c h, mul_le_mul_of_nonneg_of_nonpos (hf h) (hg h) (hf₀ _) (hg₀ _) | lemma | monotone.mul_antitone | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"monotone",
"mul_le_mul_of_nonneg_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.mul (hf : antitone f) (hg : antitone g) (hf₀ : ∀ x, f x ≤ 0) (hg₀ : ∀ x, g x ≤ 0) :
monotone (f * g) | λ b c h, mul_le_mul_of_nonpos_of_nonpos (hf h) (hg h) (hf₀ _) (hg₀ _) | lemma | antitone.mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"antitone",
"monotone",
"mul_le_mul_of_nonpos_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_exists_nonneg_add (a b : α) : a ≤ b ↔ ∃ c ≥ 0, b = a + c | ⟨λ h, ⟨b - a, sub_nonneg.mpr h, by simp⟩,
λ ⟨c, hc, h⟩, by { rw [h, le_add_iff_nonneg_right], exact hc }⟩ | lemma | le_iff_exists_nonneg_add | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_ring.to_ordered_comm_semiring : ordered_comm_semiring α | { ..ordered_ring.to_ordered_semiring, ..‹ordered_comm_ring α› } | instance | ordered_comm_ring.to_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... | [
"ordered_comm_semiring",
"ordered_ring.to_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring.to_pos_mul_strict_mono : pos_mul_strict_mono α | ⟨λ x a b h, strict_ordered_semiring.mul_lt_mul_of_pos_left _ _ _ h x.prop⟩ | instance | strict_ordered_semiring.to_pos_mul_strict_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring.to_mul_pos_strict_mono : mul_pos_strict_mono α | ⟨λ x a b h, strict_ordered_semiring.mul_lt_mul_of_pos_right _ _ _ h x.prop⟩ | instance | strict_ordered_semiring.to_mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring.to_ordered_semiring' [@decidable_rel α (≤)] : ordered_semiring α | { mul_le_mul_of_nonneg_left := λ a b c hab hc, begin
obtain rfl | hab := decidable.eq_or_lt_of_le hab,
{ refl },
obtain rfl | hc := decidable.eq_or_lt_of_le hc,
{ simp },
{ exact (mul_lt_mul_of_pos_left hab hc).le }
end,
mul_le_mul_of_nonneg_right := λ a b c hab hc, begin
obtain rfl | hab :=... | def | strict_ordered_semiring.to_ordered_semiring' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"ordered_semiring"
] | A choice-free version of `strict_ordered_semiring.to_ordered_semiring` to avoid using choice in
basic `nat` lemmas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_semiring.to_ordered_semiring : ordered_semiring α | { mul_le_mul_of_nonneg_left := λ _ _ _, begin
letI := @strict_ordered_semiring.to_ordered_semiring' α _ (classical.dec_rel _),
exact mul_le_mul_of_nonneg_left,
end,
mul_le_mul_of_nonneg_right := λ _ _ _, begin
letI := @strict_ordered_semiring.to_ordered_semiring' α _ (classical.dec_rel _),
exact mul... | instance | strict_ordered_semiring.to_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"classical.dec_rel",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"ordered_semiring",
"strict_ordered_semiring.to_ordered_semiring'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul (hac : a < c) (hbd : b ≤ d) (hb : 0 < b) (hc : 0 ≤ c) : a * b < c * d | (mul_lt_mul_of_pos_right hac hb).trans_le $ mul_le_mul_of_nonneg_left hbd hc | lemma | mul_lt_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_left",
"mul_lt_mul_of_pos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul' (hac : a ≤ c) (hbd : b < d) (hb : 0 ≤ b) (hc : 0 < c) : a * b < c * d | (mul_le_mul_of_nonneg_right hac hb).trans_lt $ mul_lt_mul_of_pos_left hbd hc | lemma | mul_lt_mul' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_of_nonneg_right",
"mul_lt_mul_of_pos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_pos (H : 0 < a) : ∀ (n : ℕ), 0 < a ^ n | | 0 := by { nontriviality, rw pow_zero, exact zero_lt_one }
| (n+1) := by { rw pow_succ, exact mul_pos H (pow_pos _) } | theorem | pow_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... | [
"pow_succ",
"pow_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_lt_mul_self (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b | mul_lt_mul' h2.le h2 h1 $ h1.trans_lt h2 | lemma | mul_self_lt_mul_self | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_mul_self : strict_mono_on (λ x : α, x * x) {x | 0 ≤ x} | λ x hx y hy hxy, mul_self_lt_mul_self hx hxy | lemma | strict_mono_on_mul_self | 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_lt_mul_self",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable.mul_lt_mul'' [@decidable_rel α (≤)]
(h1 : a < c) (h2 : b < d) (h3 : 0 ≤ a) (h4 : 0 ≤ b) : a * b < c * d | h4.lt_or_eq_dec.elim
(λ b0, mul_lt_mul h1 h2.le b0 $ h3.trans h1.le)
(λ b0, by rw [← b0, mul_zero]; exact
mul_pos (h3.trans_lt h1) (h4.trans_lt h2)) | lemma | decidable.mul_lt_mul'' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul'' : a < c → b < d → 0 ≤ a → 0 ≤ b → a * b < c * d | by classical; exact decidable.mul_lt_mul'' | lemma | mul_lt_mul'' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"decidable.mul_lt_mul''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_left (hn : 0 < a) (hm : 1 < b) : a < b * a | by { convert mul_lt_mul_of_pos_right hm hn, rw one_mul } | lemma | lt_mul_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_pos_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_right (hn : 0 < a) (hm : 1 < b) : a < a * b | by { convert mul_lt_mul_of_pos_left hm hn, rw mul_one } | lemma | lt_mul_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_pos_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_self (hn : 1 < a) : a < a * a | lt_mul_left (hn.trans_le' zero_le_one) hn | lemma | lt_mul_self | 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_mul_left",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_mul_left_of_pos (ha : 0 < a) : strict_mono (λ x, a * x) | assume b c b_lt_c, mul_lt_mul_of_pos_left b_lt_c ha | lemma | strict_mono_mul_left_of_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_lt_mul_of_pos_left",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_mul_right_of_pos (ha : 0 < a) : strict_mono (λ x, x * a) | assume b c b_lt_c, mul_lt_mul_of_pos_right b_lt_c ha | lemma | strict_mono_mul_right_of_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_lt_mul_of_pos_right",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul_const (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, (f x) * a) | (strict_mono_mul_right_of_pos ha).comp hf | lemma | strict_mono.mul_const | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_mono",
"strict_mono_mul_right_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.const_mul (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, a * (f x)) | (strict_mono_mul_left_of_pos ha).comp hf | lemma | strict_mono.const_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_mono",
"strict_mono_mul_left_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.mul_const (hf : strict_anti f) (ha : 0 < a) : strict_anti (λ x, f x * a) | (strict_mono_mul_right_of_pos ha).comp_strict_anti hf | lemma | strict_anti.mul_const | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_anti",
"strict_mono_mul_right_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.const_mul (hf : strict_anti f) (ha : 0 < a) : strict_anti (λ x, a * f x) | (strict_mono_mul_left_of_pos ha).comp_strict_anti hf | lemma | strict_anti.const_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_anti",
"strict_mono_mul_left_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul_monotone (hf : strict_mono f) (hg : monotone g) (hf₀ : ∀ x, 0 ≤ f x)
(hg₀ : ∀ x, 0 < g x) :
strict_mono (f * g) | λ b c h, mul_lt_mul (hf h) (hg h.le) (hg₀ _) (hf₀ _) | lemma | strict_mono.mul_monotone | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"mul_lt_mul",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.mul_strict_mono (hf : monotone f) (hg : strict_mono g) (hf₀ : ∀ x, 0 < f x)
(hg₀ : ∀ x, 0 ≤ g x) :
strict_mono (f * g) | λ b c h, mul_lt_mul' (hf h.le) (hg h) (hg₀ _) (hf₀ _) | lemma | monotone.mul_strict_mono | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone",
"mul_lt_mul'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul (hf : strict_mono f) (hg : strict_mono g) (hf₀ : ∀ x, 0 ≤ f x)
(hg₀ : ∀ x, 0 ≤ g x) :
strict_mono (f * g) | λ b c h, mul_lt_mul'' (hf h) (hg h) (hf₀ _) (hg₀ _) | lemma | strict_mono.mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul''",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_two_mul_self (ha : 0 < a) : a < 2 * a | lt_mul_of_one_lt_left ha one_lt_two | lemma | lt_two_mul_self | 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_mul_of_one_lt_left",
"one_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring.to_no_max_order : no_max_order α | ⟨λ a, ⟨a + 1, lt_add_of_pos_right _ one_pos⟩⟩ | instance | strict_ordered_semiring.to_no_max_order | 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... | [
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_comm_semiring.to_ordered_comm_semiring' [@decidable_rel α (≤)] :
ordered_comm_semiring α | { ..‹strict_ordered_comm_semiring α›, ..strict_ordered_semiring.to_ordered_semiring' } | def | strict_ordered_comm_semiring.to_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... | [
"ordered_comm_semiring",
"strict_ordered_semiring.to_ordered_semiring'"
] | A choice-free version of `strict_ordered_comm_semiring.to_ordered_comm_semiring` to avoid using
choice in basic `nat` lemmas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_semiring.to_ordered_comm_semiring : ordered_comm_semiring α | { ..‹strict_ordered_comm_semiring α›, ..strict_ordered_semiring.to_ordered_semiring } | instance | strict_ordered_comm_semiring.to_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... | [
"ordered_comm_semiring",
"strict_ordered_semiring.to_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_ring.to_strict_ordered_semiring : strict_ordered_semiring α | { le_of_add_le_add_left := @le_of_add_le_add_left α _ _ _,
mul_lt_mul_of_pos_left := λ a b c h hc,
by simpa only [mul_sub, sub_pos] using strict_ordered_ring.mul_pos _ _ hc (sub_pos.2 h),
mul_lt_mul_of_pos_right := λ a b c h hc,
by simpa only [sub_mul, sub_pos] using strict_ordered_ring.mul_pos _ _ (sub_pos... | instance | strict_ordered_ring.to_strict_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_pos_left",
"mul_lt_mul_of_pos_right",
"ring.to_semiring",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_ring.to_ordered_ring' [@decidable_rel α (≤)] : ordered_ring α | { mul_nonneg := λ a b ha hb, begin
obtain ha | ha := decidable.eq_or_lt_of_le ha,
{ rw [←ha, zero_mul] },
obtain hb | hb := decidable.eq_or_lt_of_le hb,
{ rw [←hb, mul_zero] },
{ exact (strict_ordered_ring.mul_pos _ _ ha hb).le }
end,
..‹strict_ordered_ring α›, ..ring.to_semiring } | def | strict_ordered_ring.to_ordered_ring' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_zero",
"ordered_ring",
"ring.to_semiring",
"zero_mul"
] | A choice-free version of `strict_ordered_ring.to_ordered_ring` to avoid using choice in basic
`int` lemmas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_ring.to_ordered_ring : ordered_ring α | { mul_nonneg := λ a b, begin
letI := @strict_ordered_ring.to_ordered_ring' α _ (classical.dec_rel _),
exact mul_nonneg,
end,
..‹strict_ordered_ring α› } | instance | strict_ordered_ring.to_ordered_ring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"classical.dec_rel",
"ordered_ring",
"strict_ordered_ring.to_ordered_ring'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_neg_left (h : b < a) (hc : c < 0) : c * a < c * b | by simpa only [neg_mul, neg_lt_neg_iff] using mul_lt_mul_of_pos_left h (neg_pos_of_neg hc) | lemma | mul_lt_mul_of_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_lt_mul_of_pos_left",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_neg_right (h : b < a) (hc : c < 0) : a * c < b * c | by simpa only [mul_neg, neg_lt_neg_iff] using mul_lt_mul_of_pos_right h (neg_pos_of_neg hc) | lemma | mul_lt_mul_of_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_lt_mul_of_pos_right",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_of_neg_of_neg {a b : α} (ha : a < 0) (hb : b < 0) : 0 < a * b | by simpa only [zero_mul] using mul_lt_mul_of_neg_right ha hb | lemma | mul_pos_of_neg_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... | [
"mul_lt_mul_of_neg_right",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_one_left (hb : b < 0) (h : a < 1) : b < a * b | by simpa only [one_mul] using mul_lt_mul_of_neg_right h hb | lemma | lt_mul_of_lt_one_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_neg_right",
"one_mul"
] | Variant of `mul_lt_of_lt_one_left` for `b` negative instead of positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_lt_of_one_lt_left (hb : b < 0) (h : 1 < a) : a * b < b | by simpa only [one_mul] using mul_lt_mul_of_neg_right h hb | lemma | mul_lt_of_one_lt_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_lt_mul_of_neg_right",
"one_mul"
] | Variant of `lt_mul_of_one_lt_left` for `b` negative instead of positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_mul_of_lt_one_right (ha : a < 0) (h : b < 1) : a < a * b | by simpa only [mul_one] using mul_lt_mul_of_neg_left h ha | lemma | lt_mul_of_lt_one_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_neg_left",
"mul_one"
] | Variant of `mul_lt_of_lt_one_right` for `a` negative instead of positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_lt_of_one_lt_right (ha : a < 0) (h : 1 < b) : a * b < a | by simpa only [mul_one] using mul_lt_mul_of_neg_left h ha | lemma | mul_lt_of_one_lt_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_lt_mul_of_neg_left",
"mul_one"
] | Variant of `lt_mul_of_lt_one_right` for `a` negative instead of positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_mul_left {a : α} (ha : a < 0) : strict_anti ((*) a) | λ b c b_lt_c, mul_lt_mul_of_neg_left b_lt_c ha | lemma | strict_anti_mul_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_neg_left",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_mul_right {a : α} (ha : a < 0) : strict_anti (λ x, x * a) | λ b c b_lt_c, mul_lt_mul_of_neg_right b_lt_c ha | lemma | strict_anti_mul_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_of_neg_right",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.const_mul_of_neg (hf : strict_mono f) (ha : a < 0) : strict_anti (λ x, a * f x) | (strict_anti_mul_left ha).comp_strict_mono hf | lemma | strict_mono.const_mul_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",
"strict_anti_mul_left",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.mul_const_of_neg (hf : strict_mono f) (ha : a < 0) : strict_anti (λ x, f x * a) | (strict_anti_mul_right ha).comp_strict_mono hf | lemma | strict_mono.mul_const_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",
"strict_anti_mul_right",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.const_mul_of_neg (hf : strict_anti f) (ha : a < 0) : strict_mono (λ x, a * f x) | (strict_anti_mul_left ha).comp hf | lemma | strict_anti.const_mul_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",
"strict_anti_mul_left",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.mul_const_of_neg (hf : strict_anti f) (ha : a < 0) : strict_mono (λ x, f x * a) | (strict_anti_mul_right ha).comp hf | lemma | strict_anti.mul_const_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",
"strict_anti_mul_right",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_comm_ring.to_ordered_comm_ring' [@decidable_rel α (≤)] : ordered_comm_ring α | { ..‹strict_ordered_comm_ring α›, ..strict_ordered_ring.to_ordered_ring' } | def | strict_ordered_comm_ring.to_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... | [
"ordered_comm_ring",
"strict_ordered_ring.to_ordered_ring'"
] | A choice-free version of `strict_ordered_comm_ring.to_ordered_comm_semiring'` to avoid using
choice in basic `int` lemmas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_ordered_comm_ring.to_strict_ordered_comm_semiring :
strict_ordered_comm_semiring α | { ..‹strict_ordered_comm_ring α›, ..strict_ordered_ring.to_strict_ordered_semiring } | instance | strict_ordered_comm_ring.to_strict_ordered_comm_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"strict_ordered_comm_semiring",
"strict_ordered_ring.to_strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_comm_ring.to_ordered_comm_ring : ordered_comm_ring α | { ..‹strict_ordered_comm_ring α›, ..strict_ordered_ring.to_ordered_ring } | instance | strict_ordered_comm_ring.to_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... | [
"ordered_comm_ring",
"strict_ordered_ring.to_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semiring.to_pos_mul_reflect_lt : pos_mul_reflect_lt α | ⟨λ a b c, (monotone_mul_left_of_nonneg a.2).reflect_lt⟩ | instance | linear_ordered_semiring.to_pos_mul_reflect_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... | [
"monotone_mul_left_of_nonneg",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semiring.to_mul_pos_reflect_lt : mul_pos_reflect_lt α | ⟨λ a b c, (monotone_mul_right_of_nonneg a.2).reflect_lt⟩ | instance | linear_ordered_semiring.to_mul_pos_reflect_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... | [
"monotone_mul_right_of_nonneg",
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg (hab : 0 ≤ a * b) :
(0 ≤ a ∧ 0 ≤ b) ∨ (a ≤ 0 ∧ b ≤ 0) | begin
refine decidable.or_iff_not_and_not.2 _,
simp only [not_and, not_le], intros ab nab, apply not_lt_of_le hab _,
rcases lt_trichotomy 0 a with (ha|rfl|ha),
exacts [mul_neg_of_pos_of_neg ha (ab ha.le), ((ab le_rfl).asymm (nab le_rfl)).elim,
mul_neg_of_neg_of_pos ha (nab ha.le)]
end | lemma | nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg | 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_rfl",
"mul_neg_of_neg_of_pos",
"mul_neg_of_pos_of_neg",
"not_and",
"not_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_mul_nonneg_left (h : 0 ≤ a * b) (hb : 0 < b) : 0 ≤ a | le_of_not_gt $ λ ha, (mul_neg_of_neg_of_pos ha hb).not_le h | lemma | nonneg_of_mul_nonneg_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_neg_of_neg_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_mul_nonneg_right (h : 0 ≤ a * b) (ha : 0 < a) : 0 ≤ b | le_of_not_gt $ λ hb, (mul_neg_of_pos_of_neg ha hb).not_le h | lemma | nonneg_of_mul_nonneg_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_neg_of_pos_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_mul_neg_left (h : a * b < 0) (hb : 0 ≤ b) : a < 0 | lt_of_not_ge $ λ ha, (mul_nonneg ha hb).not_lt h | lemma | neg_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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_mul_neg_right (h : a * b < 0) (ha : 0 ≤ a) : b < 0 | lt_of_not_ge $ λ hb, (mul_nonneg ha hb).not_lt h | lemma | neg_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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos_of_mul_nonpos_left (h : a * b ≤ 0) (hb : 0 < b) : a ≤ 0 | le_of_not_gt (assume ha : a > 0, (mul_pos ha hb).not_le h) | lemma | nonpos_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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos_of_mul_nonpos_right (h : a * b ≤ 0) (ha : 0 < a) : b ≤ 0 | le_of_not_gt (assume hb : b > 0, (mul_pos ha hb).not_le h) | lemma | nonpos_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... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_mul_left (h : 0 < c) : 0 ≤ c * b ↔ 0 ≤ b | by { convert mul_le_mul_left h, simp } | lemma | zero_le_mul_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_mul_right (h : 0 < c) : 0 ≤ b * c ↔ 0 ≤ b | by { convert mul_le_mul_right h, simp } | lemma | zero_le_mul_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_mul_of_left_le_right (a2 : 2 ≤ a) (ab : a ≤ b) : a + b ≤ a * b | have 0 < b, from
calc 0 < 2 : zero_lt_two
... ≤ a : a2
... ≤ b : ab,
calc a + b ≤ b + b : add_le_add_right ab b
... = 2 * b : (two_mul b).symm
... ≤ a * b : (mul_le_mul_right this).mpr a2 | lemma | add_le_mul_of_left_le_right | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_right",
"two_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_mul_of_right_le_left (b2 : 2 ≤ b) (ba : b ≤ a) : a + b ≤ a * b | have 0 < a, from
calc 0 < 2 : zero_lt_two
... ≤ b : b2
... ≤ a : ba,
calc a + b ≤ a + a : add_le_add_left ba a
... = a * 2 : (mul_two a).symm
... ≤ a * b : (mul_le_mul_left this).mpr b2 | lemma | add_le_mul_of_right_le_left | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_left",
"mul_two",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_mul (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ a * b | if hab : a ≤ b then add_le_mul_of_left_le_right a2 hab
else add_le_mul_of_right_le_left b2 (le_of_not_le hab) | lemma | add_le_mul | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"add_le_mul_of_left_le_right",
"add_le_mul_of_right_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_mul' (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ b * a | (le_of_eq (add_comm _ _)).trans (add_le_mul b2 a2) | lemma | add_le_mul' | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"add_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_le_bit0 : bit0 a ≤ bit0 b ↔ a ≤ b | by rw [bit0, bit0, ← two_mul, ← two_mul, mul_le_mul_left (zero_lt_two : 0 < (2:α))] | lemma | bit0_le_bit0 | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_le_mul_left",
"two_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_lt_bit0 : bit0 a < bit0 b ↔ a < b | by rw [bit0, bit0, ← two_mul, ← two_mul, mul_lt_mul_left (zero_lt_two : 0 < (2:α))] | lemma | bit0_lt_bit0 | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"mul_lt_mul_left",
"two_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_le_bit1 : bit1 a ≤ bit1 b ↔ a ≤ b | (add_le_add_iff_right 1).trans bit0_le_bit0 | lemma | bit1_le_bit1 | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"bit0_le_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_lt_bit1 : bit1 a < bit1 b ↔ a < b | (add_lt_add_iff_right 1).trans bit0_lt_bit0 | lemma | bit1_lt_bit1 | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"bit0_lt_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_bit1 : (1 : α) ≤ bit1 a ↔ 0 ≤ a | by rw [bit1, le_add_iff_nonneg_left, bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))] | lemma | one_le_bit1 | 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... | [
"two_mul",
"zero_le_mul_left",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_bit1 : (1 : α) < bit1 a ↔ 0 < a | by rw [bit1, lt_add_iff_pos_left, bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))] | lemma | one_lt_bit1 | 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... | [
"two_mul",
"zero_lt_mul_left",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_bit0 : (0 : α) ≤ bit0 a ↔ 0 ≤ a | by rw [bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))] | lemma | zero_le_bit0 | 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... | [
"two_mul",
"zero_le_mul_left",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_bit0 : (0 : α) < bit0 a ↔ 0 < a | by rw [bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))] | lemma | zero_lt_bit0 | 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... | [
"two_mul",
"zero_lt_mul_left",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_nonneg_iff_right_nonneg_of_pos (ha : 0 < a) : 0 ≤ a * b ↔ 0 ≤ b | ⟨λ h, nonneg_of_mul_nonneg_right h ha, mul_nonneg ha.le⟩ | theorem | mul_nonneg_iff_right_nonneg_of_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... | [
"nonneg_of_mul_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_nonneg_iff_left_nonneg_of_pos (hb : 0 < b) : 0 ≤ a * b ↔ 0 ≤ a | ⟨λ h, nonneg_of_mul_nonneg_left h hb, λ h, mul_nonneg h hb.le⟩ | theorem | mul_nonneg_iff_left_nonneg_of_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... | [
"nonneg_of_mul_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos_of_mul_nonneg_left (h : 0 ≤ a * b) (hb : b < 0) : a ≤ 0 | le_of_not_gt (λ ha, absurd h (mul_neg_of_pos_of_neg ha hb).not_le) | lemma | nonpos_of_mul_nonneg_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_neg_of_pos_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonpos_of_mul_nonneg_right (h : 0 ≤ a * b) (ha : a < 0) : b ≤ 0 | le_of_not_gt (λ hb, absurd h (mul_neg_of_neg_of_pos ha hb).not_le) | lemma | nonpos_of_mul_nonneg_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_neg_of_neg_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.inv_pos {u : αˣ} : (0 : α) < ↑u⁻¹ ↔ (0 : α) < u | have ∀ {u : αˣ}, (0 : α) < u → (0 : α) < ↑u⁻¹ := λ u h,
(zero_lt_mul_left h).mp $ u.mul_inv.symm ▸ zero_lt_one,
⟨this, this⟩ | lemma | units.inv_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... | [
"zero_lt_mul_left",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.inv_neg {u : αˣ} : ↑u⁻¹ < (0 : α) ↔ ↑u < (0 : α) | have ∀ {u : αˣ}, ↑u < (0 : α) → ↑u⁻¹ < (0 : α) := λ u h,
neg_of_mul_pos_right (by exact (u.mul_inv.symm ▸ zero_lt_one)) h.le,
⟨this, this⟩ | lemma | units.inv_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_pos_right",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_mul_pos_left (ha : 0 < a) (b c : α) : cmp (a * b) (a * c) = cmp b c | (strict_mono_mul_left_of_pos ha).cmp_map_eq b c | lemma | cmp_mul_pos_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_mono_mul_left_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cmp_mul_pos_right (ha : 0 < a) (b c : α) : cmp (b * a) (c * a) = cmp b c | (strict_mono_mul_right_of_pos ha).cmp_map_eq b c | lemma | cmp_mul_pos_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_mono_mul_right_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_max_of_nonneg (b c : α) (ha : 0 ≤ a) : a * max b c = max (a * b) (a * c) | (monotone_mul_left_of_nonneg ha).map_max | lemma | mul_max_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone_mul_left_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_min_of_nonneg (b c : α) (ha : 0 ≤ a) : a * min b c = min (a * b) (a * c) | (monotone_mul_left_of_nonneg ha).map_min | lemma | mul_min_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone_mul_left_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : max a b * c = max (a * c) (b * c) | (monotone_mul_right_of_nonneg hc).map_max | lemma | max_mul_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone_mul_right_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : min a b * c = min (a * c) (b * c) | (monotone_mul_right_of_nonneg hc).map_min | lemma | min_mul_of_nonneg | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"monotone_mul_right_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_of_one_le {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c) : a ≤ b | le_of_mul_le_mul_right (h.trans $ le_mul_of_one_le_right hb hc) $ zero_lt_one.trans_le hc | lemma | le_of_mul_le_of_one_le | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"le_mul_of_one_le_right",
"le_of_mul_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_le_nonneg_of_sq_le_sq {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b) : a ≤ b | le_of_not_gt $ λ hab, (mul_self_lt_mul_self hb hab).not_le h | lemma | nonneg_le_nonneg_of_sq_le_sq | 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_lt_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_le_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a ≤ b ↔ a * a ≤ b * b | ⟨mul_self_le_mul_self h1, nonneg_le_nonneg_of_sq_le_sq h2⟩ | lemma | mul_self_le_mul_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... | [
"nonneg_le_nonneg_of_sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_lt_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a < b ↔ a * a < b * b | ((@strict_mono_on_mul_self α _).lt_iff_lt h1 h2).symm | lemma | mul_self_lt_mul_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... | [
"strict_mono_on_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_inj {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a * a = b * b ↔ a = b | (@strict_mono_on_mul_self α _).eq_iff_eq h1 h2 | lemma | mul_self_inj | 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_mono_on_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_semiring.to_linear_ordered_cancel_add_comm_monoid
[linear_ordered_comm_semiring α] : linear_ordered_cancel_add_comm_monoid α | { ..‹linear_ordered_comm_semiring α› } | instance | linear_ordered_comm_semiring.to_linear_ordered_cancel_add_comm_monoid | 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_cancel_add_comm_monoid",
"linear_ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_ring.to_linear_ordered_semiring : linear_ordered_semiring α | { ..‹linear_ordered_ring α›, ..strict_ordered_ring.to_strict_ordered_semiring } | instance | linear_ordered_ring.to_linear_ordered_semiring | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"linear_ordered_semiring",
"strict_ordered_ring.to_strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_ring.to_linear_ordered_add_comm_group : linear_ordered_add_comm_group α | { ..‹linear_ordered_ring α› } | instance | linear_ordered_ring.to_linear_ordered_add_comm_group | algebra.order.ring | src/algebra/order/ring/defs.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.cancel.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.monoid.nat_cast",
"algebra.order.monoid.with_zero.defs",
"algebra.order.ring.lemmas",
"algebra.ring.defs",
"order.min_max",
"tactic.nontriviality",
"data.pi.algebra",
"algebra.gr... | [
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_ring.no_zero_divisors : no_zero_divisors α | { eq_zero_or_eq_zero_of_mul_eq_zero :=
begin
intros a b hab,
refine decidable.or_iff_not_and_not.2 (λ h, _), revert hab,
cases lt_or_gt_of_ne h.1 with ha ha; cases lt_or_gt_of_ne h.2 with hb hb,
exacts [(mul_pos_of_neg_of_neg ha hb).ne.symm, (mul_neg_of_neg_of_pos ha hb).ne,
(mul_neg... | instance | linear_ordered_ring.no_zero_divisors | 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_neg_of_neg_of_pos",
"mul_neg_of_pos_of_neg",
"mul_pos_of_neg_of_neg",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_ring.is_domain : is_domain α | { mul_left_cancel_of_ne_zero := λ a b c ha h,
begin
rw [← sub_eq_zero, ← mul_sub] at h,
exact sub_eq_zero.1 ((eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left ha)
end,
mul_right_cancel_of_ne_zero := λ a b c hb h,
begin
rw [← sub_eq_zero, ← sub_mul] at h,
exact sub_eq_zero.1 ((eq_zero_or_eq_zero... | instance | linear_ordered_ring.is_domain | 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... | [
"is_domain",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_iff : 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 | ⟨pos_and_pos_or_neg_and_neg_of_mul_pos,
λ h, h.elim (and_imp.2 mul_pos) (and_imp.2 mul_pos_of_neg_of_neg)⟩ | lemma | mul_pos_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_of_neg_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_iff : a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b | by rw [← neg_pos, neg_mul_eq_mul_neg, mul_pos_iff, neg_pos, neg_lt_zero] | lemma | mul_neg_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",
"neg_mul_eq_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_nonneg_iff : 0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | ⟨nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg,
λ h, h.elim (and_imp.2 mul_nonneg) (and_imp.2 mul_nonneg_of_nonpos_of_nonpos)⟩ | lemma | mul_nonneg_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_of_nonpos_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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