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 |
|---|---|---|---|---|---|---|---|---|---|---|
contravariant_class_mul_lt {α : Type u} [has_mul α] [partial_order α]
[contravariant_class α α (*) (<)] :
contravariant_class (with_zero α) (with_zero α) (*) (<) | begin
refine ⟨λ a b c h, _⟩,
have := ((zero_le _).trans_lt h).ne',
lift a to α using left_ne_zero_of_mul this,
lift c to α using right_ne_zero_of_mul this,
induction b using with_zero.rec_zero_coe,
exacts [zero_lt_coe _, coe_lt_coe.mpr (lt_of_mul_lt_mul_left' $ coe_lt_coe.mp h)]
end | instance | with_zero.contravariant_class_mul_lt | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/basic.lean | [
"algebra.order.monoid.with_zero.defs",
"algebra.group_with_zero.basic"
] | [
"contravariant_class",
"left_ne_zero_of_mul",
"lift",
"lt_of_mul_lt_mul_left'",
"right_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_monoid_with_zero (α : Type*)
extends linear_ordered_comm_monoid α, comm_monoid_with_zero α | (zero_le_one : (0 : α) ≤ 1) | class | linear_ordered_comm_monoid_with_zero | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"comm_monoid_with_zero",
"linear_ordered_comm_monoid",
"zero_le_one"
] | A linearly ordered commutative monoid with a zero element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_comm_monoid_with_zero.to_zero_le_one_class
[linear_ordered_comm_monoid_with_zero α] : zero_le_one_class α | { ..‹linear_ordered_comm_monoid_with_zero α› } | instance | linear_ordered_comm_monoid_with_zero.to_zero_le_one_class | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"linear_ordered_comm_monoid_with_zero",
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_add_monoid.to_zero_le_one_class [canonically_ordered_add_monoid α]
[has_one α] : zero_le_one_class α | ⟨zero_le 1⟩ | instance | canonically_ordered_add_monoid.to_zero_le_one_class | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"canonically_ordered_add_monoid",
"zero_le_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le [preorder α] (a : with_zero α) : 0 ≤ a | bot_le | lemma | with_zero.zero_le | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lt_coe [preorder α] (a : α) : (0 : with_zero α) < a | with_bot.bot_lt_coe a | lemma | with_zero.zero_lt_coe | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"with_bot.bot_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_bot [preorder α] : (0 : with_zero α) = ⊥ | rfl | lemma | with_zero.zero_eq_bot | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_coe [preorder α] {a b : α} : (a : with_zero α) < b ↔ a < b | with_bot.coe_lt_coe | lemma | with_zero.coe_lt_coe | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"with_bot.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe [preorder α] {a b : α} : (a : with_zero α) ≤ b ↔ a ≤ b | with_bot.coe_le_coe | lemma | with_zero.coe_le_coe | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"with_bot.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_mul_le {α : Type u} [has_mul α] [preorder α]
[covariant_class α α (*) (≤)] :
covariant_class (with_zero α) (with_zero α) (*) (≤) | begin
refine ⟨λ a b c hbc, _⟩,
induction a using with_zero.rec_zero_coe, { exact zero_le _ },
induction b using with_zero.rec_zero_coe, { exact zero_le _ },
rcases with_bot.coe_le_iff.1 hbc with ⟨c, rfl, hbc'⟩,
rw [← coe_mul, ← coe_mul, coe_le_coe],
exact mul_le_mul_left' hbc' a
end | instance | with_zero.covariant_class_mul_le | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"covariant_class",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_max_iff [linear_order α] {a b c : α} :
(a : with_zero α) ≤ max b c ↔ a ≤ max b c | by simp only [with_zero.coe_le_coe, le_max_iff] | lemma | with_zero.le_max_iff | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"le_max_iff",
"with_zero.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_iff [linear_order α] {a b c : α} :
min (a : with_zero α) b ≤ c ↔ min a b ≤ c | by simp only [with_zero.coe_le_coe, min_le_iff] | lemma | with_zero.min_le_iff | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"min_le_iff",
"with_zero.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_le [add_zero_class α] [preorder α]
[covariant_class α α (+) (≤)] (h : ∀ a : α, 0 ≤ a) :
covariant_class (with_zero α) (with_zero α) (+) (≤) | begin
refine ⟨λ a b c hbc, _⟩,
induction a using with_zero.rec_zero_coe,
{ rwa [zero_add, zero_add] },
induction b using with_zero.rec_zero_coe,
{ rw [add_zero],
induction c using with_zero.rec_zero_coe,
{ rw [add_zero], exact le_rfl },
{ rw [← coe_add, coe_le_coe],
exact le_add_of_nonneg_ri... | lemma | with_zero.covariant_class_add_le | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"add_zero_class",
"covariant_class",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_add_comm_monoid [ordered_add_comm_monoid α]
(zero_le : ∀ a : α, 0 ≤ a) : ordered_add_comm_monoid (with_zero α) | { add_le_add_left := @add_le_add_left _ _ _ (with_zero.covariant_class_add_le zero_le),
..with_zero.partial_order,
..with_zero.add_comm_monoid, .. } | def | with_zero.ordered_add_comm_monoid | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"ordered_add_comm_monoid",
"with_zero.covariant_class_add_le"
] | If `0` is the least element in `α`, then `with_zero α` is an `ordered_add_comm_monoid`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_zero.has_exists_add_of_le {α} [has_add α] [preorder α] [has_exists_add_of_le α] :
has_exists_add_of_le (with_zero α) | ⟨λ a b, begin
apply with_zero.cases_on a,
{ exact λ _, ⟨b, (zero_add b).symm⟩ },
apply with_zero.cases_on b,
{ exact λ b' h, (with_bot.not_coe_le_bot _ h).elim },
rintro a' b' h,
obtain ⟨c, rfl⟩ := exists_add_of_le (with_zero.coe_le_coe.1 h),
exact ⟨c, rfl⟩,
end⟩ | instance | with_zero.has_exists_add_of_le | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"has_exists_add_of_le",
"with_bot.not_coe_le_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_zero.canonically_ordered_add_monoid {α : Type u} [canonically_ordered_add_monoid α] :
canonically_ordered_add_monoid (with_zero α) | { le_self_add := λ a b, begin
apply with_zero.cases_on a,
{ exact bot_le },
apply with_zero.cases_on b,
{ exact λ b', le_rfl },
{ exact λ a' b', with_zero.coe_le_coe.2 le_self_add }
end,
.. with_zero.order_bot,
.. with_zero.ordered_add_comm_monoid zero_le, ..with_zero.has_exists_add_of_le } | instance | with_zero.canonically_ordered_add_monoid | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"bot_le",
"canonically_ordered_add_monoid",
"le_rfl",
"with_zero.has_exists_add_of_le",
"with_zero.ordered_add_comm_monoid"
] | Adding a new zero to a canonically ordered additive monoid produces another one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_zero.canonically_linear_ordered_add_monoid
(α : Type*) [canonically_linear_ordered_add_monoid α] :
canonically_linear_ordered_add_monoid (with_zero α) | { .. with_zero.canonically_ordered_add_monoid,
.. with_zero.linear_order } | instance | with_zero.canonically_linear_ordered_add_monoid | algebra.order.monoid.with_zero | src/algebra/order/monoid/with_zero/defs.lean | [
"algebra.group.with_one.defs",
"algebra.order.monoid.canonical.defs",
"algebra.order.zero_le_one"
] | [
"canonically_linear_ordered_add_monoid",
"with_zero.canonically_ordered_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_inv : has_inv {x : α // 0 ≤ x} | ⟨λ x, ⟨x⁻¹, inv_nonneg.2 x.2⟩⟩ | instance | nonneg.has_inv | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (a : {x : α // 0 ≤ x}) : ((a⁻¹ : {x : α // 0 ≤ x}) : α) = a⁻¹ | rfl | lemma | nonneg.coe_inv | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mk (hx : 0 ≤ x) : (⟨x, hx⟩ : {x : α // 0 ≤ x})⁻¹ = ⟨x⁻¹, inv_nonneg.2 hx⟩ | rfl | lemma | nonneg.inv_mk | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_div : has_div {x : α // 0 ≤ x} | ⟨λ x y, ⟨x / y, div_nonneg x.2 y.2⟩⟩ | instance | nonneg.has_div | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"div_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (a b : {x : α // 0 ≤ x}) :
((a / b : {x : α // 0 ≤ x}) : α) = a / b | rfl | lemma | nonneg.coe_div | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_div_mk (hx : 0 ≤ x) (hy : 0 ≤ y) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) / ⟨y, hy⟩ = ⟨x / y, div_nonneg hx hy⟩ | rfl | lemma | nonneg.mk_div_mk | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"div_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zpow : has_pow {x : α // 0 ≤ x} ℤ | ⟨λ a n, ⟨a ^ n, zpow_nonneg a.2 _⟩⟩ | instance | nonneg.has_zpow | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"zpow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (a : {x : α // 0 ≤ x}) (n : ℤ) :
((a ^ n : {x : α // 0 ≤ x}) : α) = a ^ n | rfl | lemma | nonneg.coe_zpow | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_zpow (hx : 0 ≤ x) (n : ℤ) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) ^ n = ⟨x ^ n, zpow_nonneg hx n⟩ | rfl | lemma | nonneg.mk_zpow | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"zpow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semifield : linear_ordered_semifield {x : α // 0 ≤ x} | subtype.coe_injective.linear_ordered_semifield _ nonneg.coe_zero nonneg.coe_one nonneg.coe_add
nonneg.coe_mul nonneg.coe_inv nonneg.coe_div (λ _ _, rfl) nonneg.coe_pow nonneg.coe_zpow
nonneg.coe_nat_cast (λ _ _, rfl) (λ _ _, rfl) | instance | nonneg.linear_ordered_semifield | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"linear_ordered_semifield",
"nonneg.coe_add",
"nonneg.coe_div",
"nonneg.coe_inv",
"nonneg.coe_mul",
"nonneg.coe_nat_cast",
"nonneg.coe_one",
"nonneg.coe_pow",
"nonneg.coe_zero",
"nonneg.coe_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_semifield [linear_ordered_field α] :
canonically_linear_ordered_semifield {x : α // 0 ≤ x} | { ..nonneg.linear_ordered_semifield, ..nonneg.canonically_ordered_comm_semiring } | instance | nonneg.canonically_linear_ordered_semifield | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"canonically_linear_ordered_semifield",
"linear_ordered_field",
"nonneg.canonically_ordered_comm_semiring",
"nonneg.linear_ordered_semifield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_group_with_zero [linear_ordered_field α] :
linear_ordered_comm_group_with_zero {x : α // 0 ≤ x} | infer_instance | instance | nonneg.linear_ordered_comm_group_with_zero | algebra.order.nonneg | src/algebra/order/nonneg/field.lean | [
"algebra.order.field.basic",
"algebra.order.field.canonical.defs",
"algebra.order.field.inj_surj",
"algebra.order.nonneg.ring"
] | [
"linear_ordered_comm_group_with_zero",
"linear_ordered_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean [ordered_add_comm_monoid α] [archimedean α] : archimedean {x : α // 0 ≤ x} | ⟨λ x y hy,
let ⟨n, hr⟩ := archimedean.arch (x : α) (hy : (0 : α) < y) in
⟨n, show (x : α) ≤ (n • y : {x : α // 0 ≤ x}), by simp [*, -nsmul_eq_mul, nsmul_coe]⟩⟩ | instance | nonneg.archimedean | algebra.order.nonneg | src/algebra/order/nonneg/floor.lean | [
"algebra.order.nonneg.ring",
"algebra.order.archimedean"
] | [
"archimedean",
"nsmul_eq_mul",
"ordered_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_semiring [ordered_semiring α] [floor_semiring α] : floor_semiring {r : α // 0 ≤ r} | { floor := λ a, ⌊(a : α)⌋₊,
ceil := λ a, ⌈(a : α)⌉₊,
floor_of_neg := λ a ha, floor_semiring.floor_of_neg ha,
gc_floor := λ a n ha, begin
refine (floor_semiring.gc_floor (show 0 ≤ (a : α), from ha)).trans _,
rw [←subtype.coe_le_coe, nonneg.coe_nat_cast]
end,
gc_ceil := λ a n, begin
refine (floor_se... | instance | nonneg.floor_semiring | algebra.order.nonneg | src/algebra/order/nonneg/floor.lean | [
"algebra.order.nonneg.ring",
"algebra.order.archimedean"
] | [
"floor_semiring",
"nonneg.coe_nat_cast",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_floor_coe [ordered_semiring α] [floor_semiring α] (a : {r : α // 0 ≤ r}) :
⌊(a : α)⌋₊ = ⌊a⌋₊ | rfl | lemma | nonneg.nat_floor_coe | algebra.order.nonneg | src/algebra/order/nonneg/floor.lean | [
"algebra.order.nonneg.ring",
"algebra.order.archimedean"
] | [
"floor_semiring",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_ceil_coe [ordered_semiring α] [floor_semiring α] (a : {r : α // 0 ≤ r}) :
⌈(a : α)⌉₊ = ⌈a⌉₊ | rfl | lemma | nonneg.nat_ceil_coe | algebra.order.nonneg | src/algebra/order/nonneg/floor.lean | [
"algebra.order.nonneg.ring",
"algebra.order.archimedean"
] | [
"floor_semiring",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_bot [preorder α] {a : α} : order_bot {x : α // a ≤ x} | { ..set.Ici.order_bot } | instance | nonneg.order_bot | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"order_bot"
] | This instance uses data fields from `subtype.partial_order` to help type-class inference.
The `set.Ici` data fields are definitionally equal, but that requires unfolding semireducible
definitions, so type-class inference won't see this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_eq [preorder α] {a : α} : (⊥ : {x : α // a ≤ x}) = ⟨a, le_rfl⟩ | rfl | lemma | nonneg.bot_eq | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_max_order [partial_order α] [no_max_order α] {a : α} : no_max_order {x : α // a ≤ x} | set.Ici.no_max_order | instance | nonneg.no_max_order | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"no_max_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semilattice_sup [semilattice_sup α] {a : α} : semilattice_sup {x : α // a ≤ x} | set.Ici.semilattice_sup | instance | nonneg.semilattice_sup | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semilattice_inf [semilattice_inf α] {a : α} : semilattice_inf {x : α // a ≤ x} | set.Ici.semilattice_inf | instance | nonneg.semilattice_inf | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_lattice [distrib_lattice α] {a : α} : distrib_lattice {x : α // a ≤ x} | set.Ici.distrib_lattice | instance | nonneg.distrib_lattice | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"distrib_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
densely_ordered [preorder α] [densely_ordered α] {a : α} :
densely_ordered {x : α // a ≤ x} | show densely_ordered (Ici a), from set.densely_ordered | instance | nonneg.densely_ordered | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"densely_ordered"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conditionally_complete_linear_order
[conditionally_complete_linear_order α] {a : α} :
conditionally_complete_linear_order {x : α // a ≤ x} | { .. @ord_connected_subset_conditionally_complete_linear_order α (set.Ici a) _ ⟨⟨a, le_rfl⟩⟩ _ } | def | nonneg.conditionally_complete_linear_order | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"conditionally_complete_linear_order",
"ord_connected_subset_conditionally_complete_linear_order",
"set.Ici"
] | If `Sup ∅ ≤ a` then `{x : α // a ≤ x}` is a `conditionally_complete_linear_order`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conditionally_complete_linear_order_bot
[conditionally_complete_linear_order α] {a : α} (h : Sup ∅ ≤ a) :
conditionally_complete_linear_order_bot {x : α // a ≤ x} | { cSup_empty := (function.funext_iff.1
(@subset_Sup_def α (set.Ici a) _ ⟨⟨a, le_rfl⟩⟩) ∅).trans $ subtype.eq $
by { rw bot_eq, cases h.lt_or_eq with h2 h2, { simp [h2.not_le] }, simp [h2] },
..nonneg.order_bot,
..nonneg.conditionally_complete_linear_order } | def | nonneg.conditionally_complete_linear_order_bot | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"cSup_empty",
"conditionally_complete_linear_order",
"conditionally_complete_linear_order_bot",
"nonneg.conditionally_complete_linear_order",
"nonneg.order_bot",
"set.Ici",
"subset_Sup_def"
] | If `Sup ∅ ≤ a` then `{x : α // a ≤ x}` is a `conditionally_complete_linear_order_bot`.
This instance uses data fields from `subtype.linear_order` to help type-class inference.
The `set.Ici` data fields are definitionally equal, but that requires unfolding semireducible
definitions, so type-class inference won't see th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited [preorder α] {a : α} : inhabited {x : α // a ≤ x} | ⟨⟨a, le_rfl⟩⟩ | instance | nonneg.inhabited | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero [has_zero α] [preorder α] : has_zero {x : α // 0 ≤ x} | ⟨⟨0, le_rfl⟩⟩ | instance | nonneg.has_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero [has_zero α] [preorder α] : ((0 : {x : α // 0 ≤ x}) : α) = 0 | rfl | lemma | nonneg.coe_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero [has_zero α] [preorder α] {x : α} (hx : 0 ≤ x) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) = 0 ↔ x = 0 | subtype.ext_iff | lemma | nonneg.mk_eq_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_add [add_zero_class α] [preorder α] [covariant_class α α (+) (≤)] :
has_add {x : α // 0 ≤ x} | ⟨λ x y, ⟨x + y, add_nonneg x.2 y.2⟩⟩ | instance | nonneg.has_add | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_zero_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_mk [add_zero_class α] [preorder α] [covariant_class α α (+) (≤)] {x y : α}
(hx : 0 ≤ x) (hy : 0 ≤ y) : (⟨x, hx⟩ : {x : α // 0 ≤ x}) + ⟨y, hy⟩ = ⟨x + y, add_nonneg hx hy⟩ | rfl | lemma | nonneg.mk_add_mk | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_zero_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add [add_zero_class α] [preorder α] [covariant_class α α (+) (≤)]
(a b : {x : α // 0 ≤ x}) : ((a + b : {x : α // 0 ≤ x}) : α) = a + b | rfl | lemma | nonneg.coe_add | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_zero_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul [add_monoid α] [preorder α] [covariant_class α α (+) (≤)] :
has_smul ℕ {x : α // 0 ≤ x} | ⟨λ n x, ⟨n • x, nsmul_nonneg x.prop n⟩⟩ | instance | nonneg.has_nsmul | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_monoid",
"covariant_class",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_mk [add_monoid α] [preorder α] [covariant_class α α (+) (≤)] (n : ℕ)
{x : α} (hx : 0 ≤ x) : (n • ⟨x, hx⟩ : {x : α // 0 ≤ x}) = ⟨n • x, nsmul_nonneg hx n⟩ | rfl | lemma | nonneg.nsmul_mk | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_monoid",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul [add_monoid α] [preorder α] [covariant_class α α (+) (≤)]
(n : ℕ) (a : {x : α // 0 ≤ x}) : ((n • a : {x : α // 0 ≤ x}) : α) = n • a | rfl | lemma | nonneg.coe_nsmul | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_monoid",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_add_comm_monoid [ordered_add_comm_monoid α] :
ordered_add_comm_monoid {x : α // 0 ≤ x} | subtype.coe_injective.ordered_add_comm_monoid _ rfl (λ x y, rfl) (λ _ _, rfl) | instance | nonneg.ordered_add_comm_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_monoid [linear_ordered_add_comm_monoid α] :
linear_ordered_add_comm_monoid {x : α // 0 ≤ x} | subtype.coe_injective.linear_ordered_add_comm_monoid _ rfl (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | nonneg.linear_ordered_add_comm_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"linear_ordered_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_add_comm_monoid [ordered_cancel_add_comm_monoid α] :
ordered_cancel_add_comm_monoid {x : α // 0 ≤ x} | subtype.coe_injective.ordered_cancel_add_comm_monoid _ rfl (λ x y, rfl) (λ _ _, rfl) | instance | nonneg.ordered_cancel_add_comm_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_cancel_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_cancel_add_comm_monoid [linear_ordered_cancel_add_comm_monoid α] :
linear_ordered_cancel_add_comm_monoid {x : α // 0 ≤ x} | subtype.coe_injective.linear_ordered_cancel_add_comm_monoid _ rfl (λ x y, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) | instance | nonneg.linear_ordered_cancel_add_comm_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"linear_ordered_cancel_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom [ordered_add_comm_monoid α] : {x : α // 0 ≤ x} →+ α | ⟨coe, nonneg.coe_zero, nonneg.coe_add⟩ | def | nonneg.coe_add_monoid_hom | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"nonneg.coe_zero",
"ordered_add_comm_monoid"
] | Coercion `{x : α // 0 ≤ x} → α` as a `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nsmul_coe [ordered_add_comm_monoid α] (n : ℕ) (r : {x : α // 0 ≤ x}) :
↑(n • r) = n • (r : α) | nonneg.coe_add_monoid_hom.map_nsmul _ _ | lemma | nonneg.nsmul_coe | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one [ordered_semiring α] : has_one {x : α // 0 ≤ x} | { one := ⟨1, zero_le_one⟩ } | instance | nonneg.has_one | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [ordered_semiring α] : ((1 : {x : α // 0 ≤ x}) : α) = 1 | rfl | lemma | nonneg.coe_one | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_one [ordered_semiring α] {x : α} (hx : 0 ≤ x) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) = 1 ↔ x = 1 | subtype.ext_iff | lemma | nonneg.mk_eq_one | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul [ordered_semiring α] : has_mul {x : α // 0 ≤ x} | { mul := λ x y, ⟨x * y, mul_nonneg x.2 y.2⟩ } | instance | nonneg.has_mul | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [ordered_semiring α] (a b : {x : α // 0 ≤ x}) :
((a * b : {x : α // 0 ≤ x}) : α) = a * b | rfl | lemma | nonneg.coe_mul | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk [ordered_semiring α] {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) * ⟨y, hy⟩ = ⟨x * y, mul_nonneg hx hy⟩ | rfl | lemma | nonneg.mk_mul_mk | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_with_one [ordered_semiring α] : add_monoid_with_one {x : α // 0 ≤ x} | { nat_cast := λ n, ⟨n, nat.cast_nonneg n⟩,
nat_cast_zero := by simp [nat.cast],
nat_cast_succ := λ _, by simp [nat.cast]; refl,
.. nonneg.has_one, .. nonneg.ordered_add_comm_monoid } | instance | nonneg.add_monoid_with_one | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"add_monoid_with_one",
"nat.cast",
"nat.cast_nonneg",
"nonneg.has_one",
"nonneg.ordered_add_comm_monoid",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast [ordered_semiring α] (n : ℕ) : ((↑n : {x : α // 0 ≤ x}) : α) = n | rfl | lemma | nonneg.coe_nat_cast | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_nat_cast [ordered_semiring α] (n : ℕ) :
(⟨n, n.cast_nonneg⟩ : {x : α // 0 ≤ x}) = n | rfl | lemma | nonneg.mk_nat_cast | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pow [ordered_semiring α] : has_pow {x : α // 0 ≤ x} ℕ | { pow := λ x n, ⟨x ^ n, pow_nonneg x.2 n⟩ } | instance | nonneg.has_pow | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow [ordered_semiring α] (a : {x : α // 0 ≤ x}) (n : ℕ) :
(↑(a ^ n) : α) = a ^ n | rfl | lemma | nonneg.coe_pow | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pow [ordered_semiring α] {x : α} (hx : 0 ≤ x) (n : ℕ) :
(⟨x, hx⟩ : {x : α // 0 ≤ x}) ^ n = ⟨x ^ n, pow_nonneg hx n⟩ | rfl | lemma | nonneg.mk_pow | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_semiring [ordered_semiring α] : ordered_semiring {x : α // 0 ≤ x} | subtype.coe_injective.ordered_semiring _
rfl rfl (λ x y, rfl) (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | nonneg.ordered_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_semiring [strict_ordered_semiring α] :
strict_ordered_semiring {x : α // 0 ≤ x} | subtype.coe_injective.strict_ordered_semiring _
rfl rfl (λ x y, rfl) (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | nonneg.strict_ordered_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_semiring [ordered_comm_semiring α] : ordered_comm_semiring {x : α // 0 ≤ x} | subtype.coe_injective.ordered_comm_semiring _
rfl rfl (λ x y, rfl) (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | nonneg.ordered_comm_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_ordered_comm_semiring [strict_ordered_comm_semiring α] :
strict_ordered_comm_semiring {x : α // 0 ≤ x} | subtype.coe_injective.strict_ordered_comm_semiring _
rfl rfl (λ x y, rfl) (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) | instance | nonneg.strict_ordered_comm_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"strict_ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero [ordered_semiring α] : monoid_with_zero {x : α // 0 ≤ x} | by apply_instance | instance | nonneg.monoid_with_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"monoid_with_zero",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid_with_zero [ordered_comm_semiring α] : comm_monoid_with_zero {x : α // 0 ≤ x} | by apply_instance | instance | nonneg.comm_monoid_with_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"comm_monoid_with_zero",
"ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring [ordered_semiring α] : semiring {x : α // 0 ≤ x} | infer_instance | instance | nonneg.semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"ordered_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring [ordered_comm_semiring α] : comm_semiring {x : α // 0 ≤ x} | infer_instance | instance | nonneg.comm_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"comm_semiring",
"ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial [linear_ordered_semiring α] : nontrivial {x : α // 0 ≤ x} | ⟨ ⟨0, 1, λ h, zero_ne_one (congr_arg subtype.val h)⟩ ⟩ | instance | nonneg.nontrivial | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"linear_ordered_semiring",
"nontrivial",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semiring [linear_ordered_semiring α] :
linear_ordered_semiring {x : α // 0 ≤ x} | subtype.coe_injective.linear_ordered_semiring _
rfl rfl (λ x y, rfl) (λ x y, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)(λ _ _, rfl) (λ _ _, rfl) | instance | nonneg.linear_ordered_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"linear_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_monoid_with_zero [linear_ordered_comm_ring α] :
linear_ordered_comm_monoid_with_zero {x : α // 0 ≤ x} | { mul_le_mul_left := λ a b h c, mul_le_mul_of_nonneg_left h c.2,
..nonneg.linear_ordered_semiring,
..nonneg.ordered_comm_semiring } | instance | nonneg.linear_ordered_comm_monoid_with_zero | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"linear_ordered_comm_monoid_with_zero",
"linear_ordered_comm_ring",
"mul_le_mul_left",
"mul_le_mul_of_nonneg_left",
"nonneg.linear_ordered_semiring",
"nonneg.ordered_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom [ordered_semiring α] : {x : α // 0 ≤ x} →+* α | ⟨coe, nonneg.coe_one, nonneg.coe_mul, nonneg.coe_zero, nonneg.coe_add⟩ | def | nonneg.coe_ring_hom | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"nonneg.coe_mul",
"nonneg.coe_one",
"nonneg.coe_zero",
"ordered_semiring"
] | Coercion `{x : α // 0 ≤ x} → α` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
canonically_ordered_add_monoid [ordered_ring α] :
canonically_ordered_add_monoid {x : α // 0 ≤ x} | { le_self_add := λ a b, le_add_of_nonneg_right b.2,
exists_add_of_le := λ a b h,
⟨⟨b - a, sub_nonneg_of_le h⟩, subtype.ext (add_sub_cancel'_right _ _).symm⟩,
..nonneg.ordered_add_comm_monoid,
..nonneg.order_bot } | instance | nonneg.canonically_ordered_add_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"canonically_ordered_add_monoid",
"nonneg.order_bot",
"nonneg.ordered_add_comm_monoid",
"ordered_ring",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_comm_semiring [ordered_comm_ring α] [no_zero_divisors α] :
canonically_ordered_comm_semiring {x : α // 0 ≤ x} | { eq_zero_or_eq_zero_of_mul_eq_zero := by { rintro ⟨a, ha⟩ ⟨b, hb⟩, simp },
..nonneg.canonically_ordered_add_monoid,
..nonneg.ordered_comm_semiring } | instance | nonneg.canonically_ordered_comm_semiring | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"canonically_ordered_comm_semiring",
"no_zero_divisors",
"nonneg.canonically_ordered_add_monoid",
"nonneg.ordered_comm_semiring",
"ordered_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_add_monoid [linear_ordered_ring α] :
canonically_linear_ordered_add_monoid {x : α // 0 ≤ x} | { ..subtype.linear_order _, ..nonneg.canonically_ordered_add_monoid } | instance | nonneg.canonically_linear_ordered_add_monoid | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"canonically_linear_ordered_add_monoid",
"linear_ordered_ring",
"nonneg.canonically_ordered_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonneg (a : α) : {x : α // 0 ≤ x} | ⟨max a 0, le_max_right _ _⟩ | def | nonneg.to_nonneg | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | The function `a ↦ max a 0` of type `α → {x : α // 0 ≤ x}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_nonneg {a : α} : (to_nonneg a : α) = max a 0 | rfl | lemma | nonneg.coe_to_nonneg | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonneg_of_nonneg {a : α} (h : 0 ≤ a) : to_nonneg a = ⟨a, h⟩ | by simp [to_nonneg, h] | lemma | nonneg.to_nonneg_of_nonneg | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonneg_coe {a : {x : α // 0 ≤ x}} : to_nonneg (a : α) = a | by { cases a with a ha, exact to_nonneg_of_nonneg ha } | lemma | nonneg.to_nonneg_coe | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonneg_le {a : α} {b : {x : α // 0 ≤ x}} : to_nonneg a ≤ b ↔ a ≤ b | by { cases b with b hb, simp [to_nonneg, hb] } | lemma | nonneg.to_nonneg_le | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonneg_lt {a : {x : α // 0 ≤ x}} {b : α} : a < to_nonneg b ↔ ↑a < b | by { cases a with a ha, simp [to_nonneg, ha.not_lt] } | lemma | nonneg.to_nonneg_lt | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sub [has_sub α] : has_sub {x : α // 0 ≤ x} | ⟨λ x y, to_nonneg (x - y)⟩ | instance | nonneg.has_sub | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sub_mk [has_sub α] {x y : α}
(hx : 0 ≤ x) (hy : 0 ≤ y) : (⟨x, hx⟩ : {x : α // 0 ≤ x}) - ⟨y, hy⟩ = to_nonneg (x - y) | rfl | lemma | nonneg.mk_sub_mk | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ordered_sub [linear_ordered_ring α] : has_ordered_sub {x : α // 0 ≤ x} | ⟨by { rintro ⟨a, ha⟩ ⟨b, hb⟩ ⟨c, hc⟩, simp only [sub_le_iff_le_add, subtype.mk_le_mk, mk_sub_mk,
mk_add_mk, to_nonneg_le, subtype.coe_mk]}⟩ | instance | nonneg.has_ordered_sub | algebra.order.nonneg | src/algebra/order/nonneg/ring.lean | [
"data.nat.cast.basic",
"algebra.order.ring.defs",
"algebra.order.ring.inj_surj",
"algebra.group_power.order",
"order.complete_lattice_intervals",
"order.lattice_intervals"
] | [
"has_ordered_sub",
"linear_ordered_ring",
"subtype.coe_mk",
"subtype.mk_le_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (x : {x : K // 0 < x}) : ↑x⁻¹ = (x⁻¹ : K) | rfl | lemma | positive.coe_inv | algebra.order.positive | src/algebra/order/positive/field.lean | [
"algebra.order.field.basic",
"algebra.order.positive.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (x : {x : K // 0 < x}) (n : ℤ) : ↑(x ^ n) = (x ^ n : K) | rfl | lemma | positive.coe_zpow | algebra.order.positive | src/algebra/order/positive/field.lean | [
"algebra.order.field.basic",
"algebra.order.positive.ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : {x : M // 0 < x}) : ↑(x + y) = (x + y : M) | rfl | lemma | positive.coe_add | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_add_lt : covariant_class {x : M // 0 < x} {x : M // 0 < x} (+) (<) | ⟨λ x y z hyz, subtype.coe_lt_coe.1 $ add_lt_add_left hyz _⟩ | instance | positive.covariant_class_add_lt | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_lt [covariant_class M M (swap (+)) (<)] :
covariant_class {x : M // 0 < x} {x : M // 0 < x} (swap (+)) (<) | ⟨λ x y z hyz, subtype.coe_lt_coe.1 $ add_lt_add_right hyz _⟩ | instance | positive.covariant_class_swap_add_lt | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_add_lt [contravariant_class M M (+) (<)] :
contravariant_class {x : M // 0 < x} {x : M // 0 < x} (+) (<) | ⟨λ x y z h, subtype.coe_lt_coe.1 $ lt_of_add_lt_add_left h⟩ | instance | positive.contravariant_class_add_lt | algebra.order.positive | src/algebra/order/positive/ring.lean | [
"algebra.order.ring.defs",
"algebra.ring.inj_surj"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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