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 |
|---|---|---|---|---|---|---|---|---|---|---|
le_kstar : a ≤ a∗ | le_trans (le_mul_of_one_le_left' one_le_kstar) kstar_mul_le_kstar | lemma | le_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_mul_le_kstar",
"le_mul_of_one_le_left'",
"one_le_kstar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_mono : monotone (has_kstar.kstar : α → α) | λ a b h, kstar_le_of_mul_le_left one_le_kstar $ kstar_mul_le (h.trans le_kstar) $
mul_kstar_le_kstar | lemma | kstar_mono | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_le_of_mul_le_left",
"kstar_mul_le",
"le_kstar",
"monotone",
"mul_kstar_le_kstar",
"one_le_kstar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_eq_one : a∗ = 1 ↔ a ≤ 1 | ⟨le_kstar.trans_eq, λ h, one_le_kstar.antisymm' $ kstar_le_of_mul_le_left le_rfl $ by rwa one_mul⟩ | lemma | kstar_eq_one | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_le_of_mul_le_left",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_zero : (0 : α)∗ = 1 | kstar_eq_one.2 zero_le_one | lemma | kstar_zero | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_one : (1 : α)∗ = 1 | kstar_eq_one.2 le_rfl | lemma | kstar_one | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_mul_kstar (a : α) : a∗ * a∗ = a∗ | (mul_kstar_le le_rfl $ kstar_mul_le_kstar).antisymm $ le_mul_of_one_le_left' one_le_kstar | lemma | kstar_mul_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_mul_le_kstar",
"le_mul_of_one_le_left'",
"le_rfl",
"mul_kstar_le",
"one_le_kstar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_eq_self : a∗ = a ↔ a * a = a ∧ 1 ≤ a | ⟨λ h, ⟨by rw [←h, kstar_mul_kstar], one_le_kstar.trans_eq h⟩, λ h,
(kstar_le_of_mul_le_left h.2 h.1.le).antisymm le_kstar⟩ | lemma | kstar_eq_self | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"kstar_le_of_mul_le_left",
"kstar_mul_kstar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_idem (a : α) : a∗∗ = a∗ | kstar_eq_self.2 ⟨kstar_mul_kstar _, one_le_kstar⟩ | lemma | kstar_idem | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_kstar : ∀ {n : ℕ}, a ^ n ≤ a∗ | | 0 := (pow_zero _).trans_le one_le_kstar
| (n + 1) := by {rw pow_succ, exact (mul_le_mul_left' pow_le_kstar _).trans mul_kstar_le_kstar } | lemma | pow_le_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"mul_kstar_le_kstar",
"mul_le_mul_left'",
"one_le_kstar",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_def (a : α × β) : a∗ = (a.1∗, a.2∗) | rfl | lemma | prod.kstar_def | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_kstar (a : α × β) : a∗.1 = a.1∗ | rfl | lemma | prod.fst_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_kstar (a : α × β) : a∗.2 = a.2∗ | rfl | lemma | prod.snd_kstar | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_def (a : Π i, π i) : a∗ = λ i, (a i)∗ | rfl | lemma | pi.kstar_def | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kstar_apply (a : Π i, π i) (i : ι) : a∗ i = (a i)∗ | rfl | lemma | pi.kstar_apply | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
idem_semiring [idem_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_bot 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... | { add_eq_sup := λ a b, hf $ by erw [sup, add, add_eq_sup],
bot := ⊥,
bot_le := λ a, bot.trans_le $ @bot_le _ _ _ $ f a,
..hf.semiring f zero one add mul nsmul npow nat_cast, ..hf.semilattice_sup _ sup, ..‹has_bot β› } | def | function.injective.idem_semiring | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"add_eq_sup",
"bot_le",
"has_bot",
"has_nat_cast",
"has_smul",
"has_sup",
"idem_semiring"
] | Pullback an `idem_semiring` instance along an injective function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
idem_comm_semiring [idem_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_bot 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 ... | { ..hf.comm_semiring f zero one add mul nsmul npow nat_cast,
..hf.idem_semiring f zero one add mul nsmul npow nat_cast sup bot } | def | function.injective.idem_comm_semiring | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"has_bot",
"has_nat_cast",
"has_smul",
"has_sup",
"idem_comm_semiring"
] | Pullback an `idem_comm_semiring` instance along an injective function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kleene_algebra [kleene_algebra 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_bot β] [has_kstar 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 ... | { one_le_kstar := λ a, one.trans_le $ by { erw kstar, exact one_le_kstar },
mul_kstar_le_kstar := λ a, by { change f _ ≤ _, erw [mul, kstar], exact mul_kstar_le_kstar },
kstar_mul_le_kstar := λ a, by { change f _ ≤ _, erw [mul, kstar], exact kstar_mul_le_kstar },
mul_kstar_le_self := λ a b (h : f _ ≤ _),
by {... | def | function.injective.kleene_algebra | algebra.order | src/algebra/order/kleene.lean | [
"algebra.order.ring.canonical",
"algebra.ring.pi",
"algebra.ring.prod",
"order.hom.complete_lattice"
] | [
"has_bot",
"has_kstar",
"has_nat_cast",
"has_smul",
"has_sup",
"kleene_algebra",
"kstar_mul_le_kstar",
"kstar_mul_le_self",
"mul_kstar_le_kstar",
"mul_kstar_le_self",
"one_le_kstar"
] | Pullback an `idem_comm_semiring` instance along an injective function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_sup [covariant_class α α (*) (≤)] (a b c : α) : c * (a ⊔ b) = (c * a) ⊔ (c * b) | (order_iso.mul_left _).map_sup _ _ | lemma | mul_sup | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul [covariant_class α α (*) (≤)] (a b c : α) : (a ⊔ b) * c = (a * c) ⊔ (b * c) | (order_iso.mul_right _).map_sup _ _ | lemma | sup_mul | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inf [covariant_class α α (*) (≤)] (a b c : α) : c * (a ⊓ b) = (c * a) ⊓ (c * b) | (order_iso.mul_left _).map_inf _ _ | lemma | mul_inf | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_mul [covariant_class α α (*) (≤)] (a b c : α) : (a ⊓ b) * c = (a * c) ⊓ (b * c) | (order_iso.mul_right _).map_inf _ _ | lemma | inf_mul | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_sup_eq_inv_inf_inv [covariant_class α α (*) (≤)] (a b : α) : (a ⊔ b)⁻¹ = a⁻¹ ⊓ b⁻¹ | (order_iso.inv α).map_sup _ _ | lemma | inv_sup_eq_inv_inf_inv | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inf_eq_sup_inv [covariant_class α α (*) (≤)] (a b : α) : (a ⊓ b)⁻¹ = a⁻¹ ⊔ b⁻¹ | (order_iso.inv α).map_inf _ _ | lemma | inv_inf_eq_sup_inv | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_mul_sup [covariant_class α α (*) (≤)] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b | calc (a ⊓ b) * (a ⊔ b) = (a ⊓ b) * ((a * b) * (b⁻¹ ⊔ a⁻¹)) :
by rw [mul_sup b⁻¹ a⁻¹ (a * b), mul_inv_cancel_right, mul_inv_cancel_comm]
... = (a ⊓ b) * ((a * b) * (a ⊓ b)⁻¹) : by rw [inv_inf_eq_sup_inv, sup_comm]
... = a * b : by rw [mul_comm, inv_mul_cancel_right] | lemma | inf_mul_sup | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"inv_inf_eq_sup_inv",
"inv_mul_cancel_right",
"mul_comm",
"mul_inv_cancel_comm",
"mul_inv_cancel_right",
"mul_sup",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one_lattice_has_pos_part : has_pos_part (α) | ⟨λ a, a ⊔ 1⟩ | instance | lattice_ordered_comm_group.has_one_lattice_has_pos_part | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"has_pos_part"
] | Let `α` be a lattice ordered commutative group with identity `1`. For an element `a` of type `α`,
the element `a ⊔ 1` is said to be the *positive component* of `a`, denoted `a⁺`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
m_pos_part_def (a : α) : a⁺ = a ⊔ 1 | rfl | lemma | lattice_ordered_comm_group.m_pos_part_def | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one_lattice_has_neg_part : has_neg_part (α) | ⟨λ a, a⁻¹ ⊔ 1⟩ | instance | lattice_ordered_comm_group.has_one_lattice_has_neg_part | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"has_neg_part"
] | Let `α` be a lattice ordered commutative group with identity `1`. For an element `a` of type `α`,
the element `(-a) ⊔ 1` is said to be the *negative component* of `a`, denoted `a⁻`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
m_neg_part_def (a : α) : a⁻ = a⁻¹ ⊔ 1 | rfl | lemma | lattice_ordered_comm_group.m_neg_part_def | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_one : (1 : α)⁺ = 1 | sup_idem | lemma | lattice_ordered_comm_group.pos_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one : (1 : α)⁻ = 1 | by rw [m_neg_part_def, inv_one, sup_idem] | lemma | lattice_ordered_comm_group.neg_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"inv_one",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_inv_inf_one [covariant_class α α (*) (≤)] (a : α) : a⁻ = (a ⊓ 1)⁻¹ | by rw [m_neg_part_def, ← inv_inj, inv_sup_eq_inv_inf_inv, inv_inv, inv_inv, inv_one] | lemma | lattice_ordered_comm_group.neg_eq_inv_inf_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"inv_inj",
"inv_inv",
"inv_one",
"inv_sup_eq_inv_inf_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mabs (a : α) : a ≤ |a| | le_sup_left | lemma | lattice_ordered_comm_group.le_mabs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_abs (a : α) : a⁻¹ ≤ |a| | le_sup_right | lemma | lattice_ordered_comm_group.inv_le_abs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pos (a : α) : 1 ≤ a⁺ | le_sup_right | lemma | lattice_ordered_comm_group.one_le_pos | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_neg (a : α) : 1 ≤ a⁻ | le_sup_right | lemma | lattice_ordered_comm_group.one_le_neg | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_le_one_iff {a : α} : a⁺ ≤ 1 ↔ a ≤ 1 | by rw [m_pos_part_def, sup_le_iff, and_iff_left le_rfl] | lemma | lattice_ordered_comm_group.pos_le_one_iff | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_rfl",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_le_one_iff {a : α} : a⁻ ≤ 1 ↔ a⁻¹ ≤ 1 | by rw [m_neg_part_def, sup_le_iff, and_iff_left le_rfl] | lemma | lattice_ordered_comm_group.neg_le_one_iff | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_rfl",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_eq_one_iff {a : α} : a⁺ = 1 ↔ a ≤ 1 | sup_eq_right | lemma | lattice_ordered_comm_group.pos_eq_one_iff | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"sup_eq_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_one_iff' {a : α} : a⁻ = 1 ↔ a⁻¹ ≤ 1 | sup_eq_right | lemma | lattice_ordered_comm_group.neg_eq_one_iff' | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"sup_eq_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_one_iff [covariant_class α α has_mul.mul has_le.le] {a : α} : a⁻ = 1 ↔ 1 ≤ a | by rw [le_antisymm_iff, neg_le_one_iff, inv_le_one', and_iff_left (one_le_neg _)] | lemma | lattice_ordered_comm_group.neg_eq_one_iff | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
m_le_pos (a : α) : a ≤ a⁺ | le_sup_left | lemma | lattice_ordered_comm_group.m_le_pos | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_neg (a : α) : a⁻¹ ≤ a⁻ | le_sup_left | lemma | lattice_ordered_comm_group.inv_le_neg | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_pos_inv (a : α) : a⁻ = (a⁻¹)⁺ | rfl | lemma | lattice_ordered_comm_group.neg_eq_pos_inv | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_eq_neg_inv (a : α) : a⁺ = (a⁻¹)⁻ | by rw [neg_eq_pos_inv, inv_inv] | lemma | lattice_ordered_comm_group.pos_eq_neg_inv | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_div_neg [covariant_class α α (*) (≤)] (a : α) : a⁺ / a⁻ = a | begin
symmetry,
rw div_eq_mul_inv,
apply eq_mul_inv_of_mul_eq,
rw [m_neg_part_def, mul_sup, mul_one, mul_right_inv, sup_comm, m_pos_part_def],
end | lemma | lattice_ordered_comm_group.pos_div_neg | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"eq_mul_inv_of_mul_eq",
"mul_one",
"mul_right_inv",
"mul_sup",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_inf_neg_eq_one [covariant_class α α (*) (≤)] (a : α) : a⁺ ⊓ a⁻ = 1 | by rw [←mul_right_inj (a⁻)⁻¹, mul_inf, mul_one, mul_left_inv, mul_comm,
← div_eq_mul_inv, pos_div_neg, neg_eq_inv_inf_one, inv_inv] | lemma | lattice_ordered_comm_group.pos_inf_neg_eq_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inv_inv",
"mul_comm",
"mul_inf",
"mul_left_inv",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_mul_pos_div [covariant_class α α (*) (≤)] (a b : α) : a ⊔ b = b * (a / b)⁺ | calc a ⊔ b = (b * (a / b)) ⊔ (b * 1) : by rw [mul_one b, div_eq_mul_inv, mul_comm a,
mul_inv_cancel_left]
... = b * ((a / b) ⊔ 1) : by rw ← mul_sup (a / b) 1 b | lemma | lattice_ordered_comm_group.sup_eq_mul_pos_div | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"mul_comm",
"mul_inv_cancel_left",
"mul_one",
"mul_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq_div_pos_div [covariant_class α α (*) (≤)] (a b : α) : a ⊓ b = a / (a / b)⁺ | calc a ⊓ b = (a * 1) ⊓ (a * (b / a)) : by { rw [mul_one a, div_eq_mul_inv, mul_comm b,
mul_inv_cancel_left], }
... = a * (1 ⊓ (b / a)) : by rw ← mul_inf 1 (b / a) a
... = a * ((b / a) ⊓ 1) : by rw inf_comm
... = a * ((a / b)⁻¹ ⊓ 1) : by { rw div_eq_mul_inv, nth_rewrite 0 ← inv_inv b,
rw [← mul_inv, mul_co... | lemma | lattice_ordered_comm_group.inf_eq_div_pos_div | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inf_comm",
"inv_inv",
"inv_one",
"inv_sup_eq_inv_inf_inv",
"mul_comm",
"mul_inf",
"mul_inv",
"mul_inv_cancel_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
m_le_iff_pos_le_neg_ge [covariant_class α α (*) (≤)] (a b : α) : a ≤ b ↔ a⁺ ≤ b⁺ ∧ b⁻ ≤ a⁻ | begin
split; intro h,
{ split,
{ exact sup_le (h.trans (m_le_pos b)) (one_le_pos b), },
{ rw ← inv_le_inv_iff at h,
exact sup_le (h.trans (inv_le_neg a)) (one_le_neg a), } },
{ rw [← pos_div_neg a, ← pos_div_neg b],
exact div_le_div'' h.1 h.2, }
end | lemma | lattice_ordered_comm_group.m_le_iff_pos_le_neg_ge | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_le_div''",
"inv_le_inv_iff",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
m_neg_abs [covariant_class α α (*) (≤)] (a : α) : |a|⁻ = 1 | begin
refine le_antisymm _ _,
{ rw ← pos_inf_neg_eq_one a,
apply le_inf,
{ rw pos_eq_neg_inv,
exact ((m_le_iff_pos_le_neg_ge _ _).mp (inv_le_abs a)).right, },
{ exact and.right (iff.elim_left (m_le_iff_pos_le_neg_ge _ _) (le_mabs a)), } },
{ exact one_le_neg _, }
end | lemma | lattice_ordered_comm_group.m_neg_abs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
m_pos_abs [covariant_class α α (*) (≤)] (a : α) : |a|⁺ = |a| | begin
nth_rewrite 1 ← pos_div_neg (|a|),
rw div_eq_mul_inv,
symmetry,
rw [mul_right_eq_self, inv_eq_one],
exact m_neg_abs a,
end | lemma | lattice_ordered_comm_group.m_pos_abs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inv_eq_one",
"mul_right_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_abs [covariant_class α α (*) (≤)] (a : α) : 1 ≤ |a| | by { rw ← m_pos_abs, exact one_le_pos _, } | lemma | lattice_ordered_comm_group.one_le_abs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_neg [covariant_class α α (*) (≤)] (a : α) : |a| = a⁺ * a⁻ | begin
rw [m_pos_part_def, sup_mul, one_mul, m_neg_part_def, mul_sup, mul_one, mul_inv_self, sup_assoc,
←@sup_assoc _ _ a, sup_eq_right.2 le_sup_right],
exact (sup_eq_left.2 $ one_le_abs a).symm,
end | lemma | lattice_ordered_comm_group.pos_mul_neg | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"le_sup_right",
"mul_inv_self",
"mul_one",
"mul_sup",
"one_mul",
"sup_assoc",
"sup_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_div_inf_eq_abs_div [covariant_class α α (*) (≤)] (a b : α) :
(a ⊔ b) / (a ⊓ b) = |b / a| | by
rw [sup_eq_mul_pos_div, inf_comm, inf_eq_div_pos_div, div_eq_mul_inv, div_eq_mul_inv b ((b / a)⁺),
mul_inv_rev, inv_inv, mul_comm, ← mul_assoc, inv_mul_cancel_right, pos_eq_neg_inv (a / b),
div_eq_mul_inv a b, mul_inv_rev, ← div_eq_mul_inv, inv_inv, ← pos_mul_neg] | lemma | lattice_ordered_comm_group.sup_div_inf_eq_abs_div | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inf_comm",
"inv_inv",
"inv_mul_cancel_right",
"mul_assoc",
"mul_comm",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sq_eq_mul_mul_abs_div [covariant_class α α (*) (≤)] (a b : α) :
(a ⊔ b)^2 = a * b * |b / a| | by rw [← inf_mul_sup a b, ← sup_div_inf_eq_abs_div, div_eq_mul_inv, ← mul_assoc, mul_comm,
mul_assoc, ← pow_two, inv_mul_cancel_left] | lemma | lattice_ordered_comm_group.sup_sq_eq_mul_mul_abs_div | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inf_mul_sup",
"inv_mul_cancel_left",
"mul_assoc",
"mul_comm",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sq_eq_mul_div_abs_div [covariant_class α α (*) (≤)] (a b : α) :
(a ⊓ b)^2 = a * b / |b / a| | by rw [← inf_mul_sup a b, ← sup_div_inf_eq_abs_div, div_eq_mul_inv, div_eq_mul_inv,
mul_inv_rev, inv_inv, mul_assoc, mul_inv_cancel_comm_assoc, ← pow_two] | lemma | lattice_ordered_comm_group.inf_sq_eq_mul_div_abs_div | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"div_eq_mul_inv",
"inf_mul_sup",
"inv_inv",
"mul_assoc",
"mul_inv_cancel_comm_assoc",
"mul_inv_rev",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lattice_ordered_comm_group_to_distrib_lattice (α : Type u)
[s: lattice α] [comm_group α] [covariant_class α α (*) (≤)] : distrib_lattice α | { le_sup_inf :=
begin
intros,
rw [← mul_le_mul_iff_left (x ⊓ (y ⊓ z)), inf_mul_sup x (y ⊓ z),
← inv_mul_le_iff_le_mul, le_inf_iff],
split,
{ rw [inv_mul_le_iff_le_mul, ← inf_mul_sup x y],
apply mul_le_mul',
{ apply inf_le_inf_left, apply inf_le_left, },
{ apply inf_le_left, } }... | def | lattice_ordered_comm_group.lattice_ordered_comm_group_to_distrib_lattice | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"comm_group",
"covariant_class",
"distrib_lattice",
"inf_le_inf_left",
"inf_le_left",
"inf_le_right",
"inf_mul_sup",
"inv_mul_le_iff_le_mul",
"lattice",
"le_inf_iff",
"le_sup_inf",
"mul_le_mul'",
"mul_le_mul_iff_left"
] | Every lattice ordered commutative group is a distributive lattice | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_div_sup_mul_abs_div_inf [covariant_class α α (*) (≤)] (a b c : α) : | |(a ⊔ c) / (b ⊔ c)| * |(a ⊓ c) / (b ⊓ c)| = |a / b| :=
begin
letI : distrib_lattice α := lattice_ordered_comm_group_to_distrib_lattice α,
calc |(a ⊔ c) / (b ⊔ c)| * |(a ⊓ c) / (b ⊓ c)| =
((b ⊔ c ⊔ (a ⊔ c)) / ((b ⊔ c) ⊓ (a ⊔ c))) * |(a ⊓ c) / (b ⊓ c)| : by rw sup_div_inf_eq_abs_div
... = (b ⊔ c ⊔ (a ⊔ c)) / ((... | theorem | lattice_ordered_comm_group.abs_div_sup_mul_abs_div_inf | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"distrib_lattice",
"div_eq_mul_inv",
"div_mul_div_comm",
"inf_assoc",
"inf_comm",
"inf_mul_sup",
"inf_right_idem",
"inf_sup_right",
"mul_assoc",
"mul_comm",
"mul_inv_cancel_left",
"mul_inv_rev",
"sup_assoc",
"sup_comm",
"sup_inf_right",
"sup_right_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_one_le (a : α) (h : 1 ≤ a) : a⁺ = a | by { rw m_pos_part_def, exact sup_of_le_left h, } | lemma | lattice_ordered_comm_group.pos_of_one_le | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_eq_self_of_one_lt_pos {α} [linear_order α] [comm_group α]
{x : α} (hx : 1 < x⁺) : x⁺ = x | begin
rw [m_pos_part_def, right_lt_sup, not_le] at hx,
rw [m_pos_part_def, sup_eq_left],
exact hx.le
end | lemma | lattice_ordered_comm_group.pos_eq_self_of_one_lt_pos | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"comm_group",
"right_lt_sup",
"sup_eq_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_le_one (a : α) (h : a ≤ 1) : a⁺ = 1 | pos_eq_one_iff.mpr h | lemma | lattice_ordered_comm_group.pos_of_le_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_one_le_inv (a : α) (h : 1 ≤ a⁻¹) : a⁻ = a⁻¹ | by { rw neg_eq_pos_inv, exact pos_of_one_le _ h, } | lemma | lattice_ordered_comm_group.neg_of_one_le_inv | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_inv_le_one (a : α) (h : a⁻¹ ≤ 1) : a⁻ = 1 | neg_eq_one_iff'.mpr h | lemma | lattice_ordered_comm_group.neg_of_inv_le_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_le_one [covariant_class α α (*) (≤)] (a : α) (h : a ≤ 1) : a⁻ = a⁻¹ | sup_eq_left.2 $ one_le_inv'.2 h | lemma | lattice_ordered_comm_group.neg_of_le_one | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_one_le [covariant_class α α (*) (≤)] (a : α) (h : 1 ≤ a) : a⁻ = 1 | neg_eq_one_iff.mpr h | lemma | lattice_ordered_comm_group.neg_of_one_le | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mabs_of_one_le [covariant_class α α (*) (≤)] (a : α) (h : 1 ≤ a) : |a| = a | sup_eq_left.2 $ left.inv_le_self h | lemma | lattice_ordered_comm_group.mabs_of_one_le | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"left.inv_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mabs_mabs [covariant_class α α (*) (≤)] (a : α) : | |a| | = |a| | mabs_of_one_le _ (one_le_abs _) | lemma | lattice_ordered_comm_group.mabs_mabs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class"
] | The unary operation of taking the absolute value is idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mabs_sup_div_sup_le_mabs [covariant_class α α (*) (≤)] (a b c : α) : | |(a ⊔ c) / (b ⊔ c)| ≤ |a / b| :=
begin
apply le_of_mul_le_of_one_le_left,
{ rw abs_div_sup_mul_abs_div_inf, },
{ exact one_le_abs _, },
end | lemma | lattice_ordered_comm_group.mabs_sup_div_sup_le_mabs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"le_of_mul_le_of_one_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mabs_inf_div_inf_le_mabs [covariant_class α α (*) (≤)] (a b c : α) : | |(a ⊓ c) / (b ⊓ c)| ≤ |a / b| :=
begin
apply le_of_mul_le_of_one_le_right,
{ rw abs_div_sup_mul_abs_div_inf, },
{ exact one_le_abs _, },
end | lemma | lattice_ordered_comm_group.mabs_inf_div_inf_le_mabs | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"le_of_mul_le_of_one_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
m_Birkhoff_inequalities [covariant_class α α (*) (≤)] (a b c : α) : | |(a ⊔ c) / (b ⊔ c)| ⊔ |(a ⊓ c) / (b ⊓ c)| ≤ |a / b| :=
sup_le (mabs_sup_div_sup_le_mabs a b c) (mabs_inf_div_inf_le_mabs a b c) | theorem | lattice_ordered_comm_group.m_Birkhoff_inequalities | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mabs_mul_le [covariant_class α α (*) (≤)] (a b : α) : |a * b| ≤ |a| * |b| | begin
apply sup_le,
{ exact mul_le_mul' (le_mabs a) (le_mabs b), },
{ rw mul_inv,
exact mul_le_mul' (inv_le_abs _) (inv_le_abs _), }
end | lemma | lattice_ordered_comm_group.mabs_mul_le | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"covariant_class",
"mul_inv",
"mul_le_mul'",
"sup_le"
] | The absolute value satisfies the triangle inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_inv_comm (a b : α) : |a/b| = |b/a| | begin
unfold has_abs.abs,
rw [inv_div a b, ← inv_inv (a / b), inv_div, sup_comm],
end | lemma | lattice_ordered_comm_group.abs_inv_comm | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"inv_div",
"inv_inv",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_abs_div_abs_le [covariant_class α α (*) (≤)] (a b : α) : | |a| / |b| | ≤ |a / b| | begin
rw [abs_eq_sup_inv, sup_le_iff],
split,
{ apply div_le_iff_le_mul.2,
convert mabs_mul_le (a/b) b,
rw div_mul_cancel',
exact covariant_swap_mul_le_of_covariant_mul_le α, },
{ rw [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_le_iff_le_mul, abs_inv_comm],
convert mabs_mul_le (b/a) a,
{ ... | lemma | lattice_ordered_comm_group.abs_abs_div_abs_le | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [
"abs_eq_sup_inv",
"covariant_class",
"covariant_swap_mul_le_of_covariant_mul_le",
"div_eq_mul_inv",
"div_mul_cancel'",
"inv_inv",
"mul_inv_le_iff_le_mul",
"mul_inv_rev",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_solid (s : set β) : Prop | ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, |y| ≤ |x| → y ∈ s | def | lattice_ordered_add_comm_group.is_solid | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | A subset `s ⊆ β`, with `β` an `add_comm_group` with a `lattice` structure, is solid if for
all `x ∈ s` and all `y ∈ β` such that `|y| ≤ |x|`, then `y ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
solid_closure (s : set β) : set β | {y | ∃ x ∈ s, |y| ≤ |x|} | def | lattice_ordered_add_comm_group.solid_closure | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | The solid closure of a subset `s` is the smallest superset of `s` that is solid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_solid_solid_closure (s : set β) : is_solid (solid_closure s) | λ x ⟨y, hy, hxy⟩ z hzx, ⟨y, hy, hzx.trans hxy⟩ | lemma | lattice_ordered_add_comm_group.is_solid_solid_closure | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
solid_closure_min (s t : set β) (h1 : s ⊆ t) (h2 : is_solid t) : solid_closure s ⊆ t | λ _ ⟨_, hy, hxy⟩, h2 (h1 hy) hxy | lemma | lattice_ordered_add_comm_group.solid_closure_min | algebra.order | src/algebra/order/lattice_group.lean | [
"algebra.group_power.basic",
"algebra.order.group.abs",
"tactic.nth_rewrite",
"order.closure"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_neg_iff_of_pos (hc : 0 < c) :
c • a < 0 ↔ a < 0 | begin
rw [←neg_neg a, smul_neg, neg_neg_iff_pos, neg_neg_iff_pos],
exact smul_pos_iff_of_pos hc,
end | lemma | smul_neg_iff_of_pos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_neg",
"smul_pos_iff_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_lt_smul_of_neg (h : a < b) (hc : c < 0) :
c • b < c • a | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff],
exact smul_lt_smul_of_pos h (neg_pos_of_neg hc),
end | lemma | smul_lt_smul_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul_of_nonpos (h : a ≤ b) (hc : c ≤ 0) :
c • b ≤ c • a | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff],
exact smul_le_smul_of_nonneg h (neg_nonneg_of_nonpos hc),
end | lemma | smul_le_smul_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_smul_eq_smul_of_neg_of_le (hab : c • a = c • b) (hc : c < 0) (h : a ≤ b) :
a = b | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_inj] at hab,
exact eq_of_smul_eq_smul_of_pos_of_le hab (neg_pos_of_neg hc) h,
end | lemma | eq_of_smul_eq_smul_of_neg_of_le | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"eq_of_smul_eq_smul_of_pos_of_le",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_smul_lt_smul_of_nonpos (h : c • a < c • b) (hc : c ≤ 0) :
b < a | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff] at h,
exact lt_of_smul_lt_smul_of_nonneg h (neg_nonneg_of_nonpos hc),
end | lemma | lt_of_smul_lt_smul_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"lt_of_smul_lt_smul_of_nonneg",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_lt_smul_iff_of_neg (hc : c < 0) :
c • a < c • b ↔ b < a | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff],
exact smul_lt_smul_iff_of_pos (neg_pos_of_neg hc),
end | lemma | smul_lt_smul_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_lt_smul_iff_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_neg_iff_of_neg (hc : c < 0) :
c • a < 0 ↔ 0 < a | begin
rw [←neg_neg c, neg_smul, neg_neg_iff_pos],
exact smul_pos_iff_of_pos (neg_pos_of_neg hc),
end | lemma | smul_neg_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_pos_iff_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pos_iff_of_neg (hc : c < 0) :
0 < c • a ↔ a < 0 | begin
rw [←neg_neg c, neg_smul, neg_pos],
exact smul_neg_iff_of_pos (neg_pos_of_neg hc),
end | lemma | smul_pos_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_neg_iff_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonpos_of_nonpos_of_nonneg (hc : c ≤ 0) (ha : 0 ≤ a) : c • a ≤ 0 | calc
c • a ≤ c • 0 : smul_le_smul_of_nonpos ha hc
... = 0 : smul_zero c | lemma | smul_nonpos_of_nonpos_of_nonneg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_le_smul_of_nonpos",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg_of_nonpos_of_nonpos (hc : c ≤ 0) (ha : a ≤ 0) : 0 ≤ c • a | @smul_nonpos_of_nonpos_of_nonneg k Mᵒᵈ _ _ _ _ _ _ hc ha | lemma | smul_nonneg_of_nonpos_of_nonpos | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_nonpos_of_nonpos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_smul_left (hc : c ≤ 0) : antitone (has_smul.smul c : M → M) | λ a b h, smul_le_smul_of_nonpos h hc | lemma | antitone_smul_left | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone",
"smul_le_smul_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_smul_left (hc : c < 0) : strict_anti (has_smul.smul c : M → M) | λ a b h, smul_lt_smul_of_neg h hc | lemma | strict_anti_smul_left | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_lt_smul_of_neg",
"strict_anti"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_add_smul_le_smul_add_smul [contravariant_class M M (+) (≤)] {a b : k} {c d : M}
(hab : a ≤ b) (hcd : c ≤ d) :
a • d + b • c ≤ a • c + b • d | begin
obtain ⟨b, rfl⟩ := exists_add_of_le hab,
obtain ⟨d, rfl⟩ := exists_add_of_le hcd,
rw [smul_add, add_right_comm, smul_add, ←add_assoc, add_smul _ _ d],
rw le_add_iff_nonneg_right at hab hcd,
exact add_le_add_left (le_add_of_nonneg_right $ smul_nonneg hab hcd) _,
end | lemma | smul_add_smul_le_smul_add_smul | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"add_smul",
"contravariant_class",
"smul_add",
"smul_nonneg"
] | Binary **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_smul_le_smul_add_smul' [contravariant_class M M (+) (≤)] {a b : k} {c d : M}
(hba : b ≤ a) (hdc : d ≤ c) : a • d + b • c ≤ a • c + b • d | by { rw [add_comm (a • d), add_comm (a • c)], exact smul_add_smul_le_smul_add_smul hba hdc } | lemma | smul_add_smul_le_smul_add_smul' | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"contravariant_class",
"smul_add_smul_le_smul_add_smul"
] | Binary **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_smul_lt_smul_add_smul [covariant_class M M (+) (<)] [contravariant_class M M (+) (<)]
{a b : k} {c d : M} (hab : a < b) (hcd : c < d) : a • d + b • c < a • c + b • d | begin
obtain ⟨b, rfl⟩ := exists_add_of_le hab.le,
obtain ⟨d, rfl⟩ := exists_add_of_le hcd.le,
rw [smul_add, add_right_comm, smul_add, ←add_assoc, add_smul _ _ d],
rw lt_add_iff_pos_right at hab hcd,
exact add_lt_add_left (lt_add_of_pos_right _ $ smul_pos hab hcd) _,
end | lemma | smul_add_smul_lt_smul_add_smul | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"add_smul",
"contravariant_class",
"covariant_class",
"smul_add"
] | Binary strict **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_smul_lt_smul_add_smul' [covariant_class M M (+) (<)]
[contravariant_class M M (+) (<)] {a b : k} {c d : M} (hba : b < a) (hdc : d < c) :
a • d + b • c < a • c + b • d | by { rw [add_comm (a • d), add_comm (a • c)], exact smul_add_smul_lt_smul_add_smul hba hdc } | lemma | smul_add_smul_lt_smul_add_smul' | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"contravariant_class",
"covariant_class",
"smul_add_smul_lt_smul_add_smul"
] | Binary strict **rearrangement inequality**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_le_smul_iff_of_neg (hc : c < 0) : c • a ≤ c • b ↔ b ≤ a | begin
rw [←neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff],
exact smul_le_smul_iff_of_pos (neg_pos_of_neg hc),
end | lemma | smul_le_smul_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"neg_smul",
"smul_le_smul_iff_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_smul_le_iff_of_neg (h : c < 0) : c⁻¹ • a ≤ b ↔ c • b ≤ a | by { rw [←smul_le_smul_iff_of_neg h, smul_inv_smul₀ h.ne], apply_instance } | lemma | inv_smul_le_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_smul_lt_iff_of_neg (h : c < 0) : c⁻¹ • a < b ↔ c • b < a | by { rw [←smul_lt_smul_iff_of_neg h, smul_inv_smul₀ h.ne], apply_instance } | lemma | inv_smul_lt_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inv_le_iff_of_neg (h : c < 0) : a ≤ c⁻¹ • b ↔ b ≤ c • a | by { rw [←smul_le_smul_iff_of_neg h, smul_inv_smul₀ h.ne], apply_instance } | lemma | smul_inv_le_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inv_lt_iff_of_neg (h : c < 0) : a < c⁻¹ • b ↔ b < c • a | by { rw [←smul_lt_smul_iff_of_neg h, smul_inv_smul₀ h.ne], apply_instance } | lemma | smul_inv_lt_iff_of_neg | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.smul_left_dual {c : k} (hc : c < 0) : M ≃o Mᵒᵈ | { to_fun := λ b, order_dual.to_dual (c • b),
inv_fun := λ b, c⁻¹ • (order_dual.of_dual b),
left_inv := inv_smul_smul₀ hc.ne,
right_inv := smul_inv_smul₀ hc.ne,
map_rel_iff' := λ b₁ b₂, smul_le_smul_iff_of_neg hc } | def | order_iso.smul_left_dual | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"inv_fun",
"inv_smul_smul₀",
"order_dual.of_dual",
"order_dual.to_dual",
"smul_inv_smul₀",
"smul_le_smul_iff_of_neg"
] | Left scalar multiplication as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_lower_bounds_subset_upper_bounds_smul (hc : c ≤ 0) :
c • lower_bounds s ⊆ upper_bounds (c • s) | (antitone_smul_left hc).image_lower_bounds_subset_upper_bounds_image | lemma | smul_lower_bounds_subset_upper_bounds_smul | algebra.order | src/algebra/order/module.lean | [
"algebra.order.smul"
] | [
"antitone_smul_left",
"lower_bounds",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.