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 |
|---|---|---|---|---|---|---|---|---|---|---|
_root_.add_monoid_hom.with_top_map
{M N : Type*} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
with_top M →+ with_top N | { to_fun := with_top.map f,
.. f.to_zero_hom.with_top_map, .. f.to_add_hom.with_top_map } | def | add_monoid_hom.with_top_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_zero_class",
"with_top",
"with_top.map"
] | A version of `with_top.map` for `add_monoid_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one [has_one α] : ((1 : α) : with_bot α) = 1 | rfl | lemma | with_bot.coe_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_one [has_one α] {a : α} : (a : with_bot α) = 1 ↔ a = 1 | with_top.coe_eq_one | lemma | with_bot.coe_eq_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot",
"with_top.coe_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unbot_one [has_one α] : (1 : with_bot α).unbot coe_ne_bot = 1 | rfl | lemma | with_bot.unbot_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unbot_one' [has_one α] (d : α) : (1 : with_bot α).unbot' d = 1 | rfl | lemma | with_bot.unbot_one' | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_coe [has_one α] [has_le α] {a : α} : 1 ≤ (a : with_bot α) ↔ 1 ≤ a | coe_le_coe | lemma | with_bot.one_le_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_one [has_one α] [has_le α] {a : α} : (a : with_bot α) ≤ 1 ↔ a ≤ 1 | coe_le_coe | lemma | with_bot.coe_le_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_coe [has_one α] [has_lt α] {a : α} : 1 < (a : with_bot α) ↔ 1 < a | coe_lt_coe | lemma | with_bot.one_lt_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_one [has_one α] [has_lt α] {a : α} : (a : with_bot α) < 1 ↔ a < 1 | coe_lt_coe | lemma | with_bot.coe_lt_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one {β} [has_one α] (f : α → β) :
(1 : with_bot α).map f = (f 1 : with_bot β) | rfl | lemma | with_bot.map_one | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"map_one",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat [add_monoid_with_one α] (n : ℕ) : ((n : α) : with_bot α) = n | rfl | lemma | with_bot.coe_nat | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_ne_bot [add_monoid_with_one α] (n : ℕ) : (n : with_bot α) ≠ ⊥ | coe_ne_bot | lemma | with_bot.nat_ne_bot | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_ne_nat [add_monoid_with_one α] (n : ℕ) : (⊥ : with_bot α) ≠ n | bot_ne_coe | lemma | with_bot.bot_ne_nat | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_monoid_with_one",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (a b : α) : ((a + b : α) : with_bot α) = a + b | rfl | lemma | with_bot.coe_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit0 : ((bit0 x : α) : with_bot α) = bit0 x | rfl | lemma | with_bot.coe_bit0 | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit1 [has_one α] {a : α} : ((bit1 a : α) : with_bot α) = bit1 a | rfl | lemma | with_bot.coe_bit1 | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_add (a : with_bot α) : ⊥ + a = ⊥ | rfl | lemma | with_bot.bot_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_bot (a : with_bot α) : a + ⊥ = ⊥ | by cases a; refl | lemma | with_bot.add_bot | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_bot : a + b = ⊥ ↔ a = ⊥ ∨ b = ⊥ | with_top.add_eq_top | lemma | with_bot.add_eq_bot | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top.add_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_ne_bot : a + b ≠ ⊥ ↔ a ≠ ⊥ ∧ b ≠ ⊥ | with_top.add_ne_top | lemma | with_bot.add_ne_bot | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top.add_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_lt_add [has_lt α] {a b : with_bot α} : ⊥ < a + b ↔ ⊥ < a ∧ ⊥ < b | @with_top.add_lt_top αᵒᵈ _ _ _ _ | lemma | with_bot.bot_lt_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_bot",
"with_top.add_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_coe : a + b = x ↔ ∃ (a' b' : α), ↑a' = a ∧ ↑b' = b ∧ a' + b' = x | with_top.add_eq_coe | lemma | with_bot.add_eq_coe | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top.add_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_coe_eq_bot_iff : a + y = ⊥ ↔ a = ⊥ | with_top.add_coe_eq_top_iff | lemma | with_bot.add_coe_eq_bot_iff | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top.add_coe_eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_eq_bot_iff : ↑x + b = ⊥ ↔ b = ⊥ | with_top.coe_add_eq_top_iff | lemma | with_bot.coe_add_eq_bot_iff | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"with_top.coe_add_eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add {F} [has_add β] [add_hom_class F α β] (f : F) (a b : with_bot α) :
(a + b).map f = a.map f + b.map f | with_top.map_add f a b | lemma | with_bot.map_add | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_hom_class",
"with_bot",
"with_top.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.one_hom.with_bot_map {M N : Type*} [has_one M] [has_one N] (f : one_hom M N) :
one_hom (with_bot M) (with_bot N) | { to_fun := with_bot.map f,
map_one' := by rw [with_bot.map_one, map_one, coe_one] } | def | one_hom.with_bot_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"map_one",
"one_hom",
"with_bot",
"with_bot.map",
"with_bot.map_one"
] | A version of `with_bot.map` for `one_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.add_hom.with_bot_map
{M N : Type*} [has_add M] [has_add N] (f : add_hom M N) :
add_hom (with_bot M) (with_bot N) | { to_fun := with_bot.map f,
map_add' := with_bot.map_add f } | def | add_hom.with_bot_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_hom",
"with_bot",
"with_bot.map",
"with_bot.map_add"
] | A version of `with_bot.map` for `add_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.add_monoid_hom.with_bot_map
{M N : Type*} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
with_bot M →+ with_bot N | { to_fun := with_bot.map f,
.. f.to_zero_hom.with_bot_map, .. f.to_add_hom.with_bot_map } | def | add_monoid_hom.with_bot_map | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"add_zero_class",
"with_bot",
"with_bot.map"
] | A version of `with_bot.map` for `add_monoid_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_class_add_le [covariant_class α α (+) (≤)] :
covariant_class (with_bot α) (with_bot α) (+) (≤) | @order_dual.covariant_class_add_le (with_top αᵒᵈ) _ _ _ | instance | with_bot.covariant_class_add_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_bot",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_class_swap_add_le [covariant_class α α (swap (+)) (≤)] :
covariant_class (with_bot α) (with_bot α) (swap (+)) (≤) | @order_dual.covariant_class_swap_add_le (with_top αᵒᵈ) _ _ _ | instance | with_bot.covariant_class_swap_add_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_bot",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_add_lt [contravariant_class α α (+) (<)] :
contravariant_class (with_bot α) (with_bot α) (+) (<) | @order_dual.contravariant_class_add_lt (with_top αᵒᵈ) _ _ _ | instance | with_bot.contravariant_class_add_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_bot",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_class_swap_add_lt [contravariant_class α α (swap (+)) (<)] :
contravariant_class (with_bot α) (with_bot α) (swap (+)) (<) | @order_dual.contravariant_class_swap_add_lt (with_top αᵒᵈ) _ _ _ | instance | with_bot.contravariant_class_swap_add_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_bot",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_add_le_add_left [contravariant_class α α (+) (≤)] (ha : a ≠ ⊥)
(h : a + b ≤ a + c) : b ≤ c | @with_top.le_of_add_le_add_left αᵒᵈ _ _ _ _ _ _ ha h | lemma | with_bot.le_of_add_le_add_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_top.le_of_add_le_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_add_le_add_right [contravariant_class α α (swap (+)) (≤)] (ha : a ≠ ⊥)
(h : b + a ≤ c + a) : b ≤ c | @with_top.le_of_add_le_add_right αᵒᵈ _ _ _ _ _ _ ha h | lemma | with_bot.le_of_add_le_add_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"with_top.le_of_add_le_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_left [covariant_class α α (+) (<)] (ha : a ≠ ⊥) (h : b < c) :
a + b < a + c | @with_top.add_lt_add_left αᵒᵈ _ _ _ _ _ _ ha h | lemma | with_bot.add_lt_add_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_right [covariant_class α α (swap (+)) (<)] (ha : a ≠ ⊥) (h : b < c) :
b + a < c + a | @with_top.add_lt_add_right αᵒᵈ _ _ _ _ _ _ ha h | lemma | with_bot.add_lt_add_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_iff_left [covariant_class α α (+) (≤)] [contravariant_class α α (+) (≤)]
(ha : a ≠ ⊥) : a + b ≤ a + c ↔ b ≤ c | ⟨with_bot.le_of_add_le_add_left ha, λ h, add_le_add_left h a⟩ | lemma | with_bot.add_le_add_iff_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_add_iff_right [covariant_class α α (swap (+)) (≤)]
[contravariant_class α α (swap (+)) (≤)] (ha : a ≠ ⊥) : b + a ≤ c + a ↔ b ≤ c | ⟨with_bot.le_of_add_le_add_right ha, λ h, add_le_add_right h a⟩ | lemma | with_bot.add_le_add_iff_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_iff_left [covariant_class α α (+) (<)] [contravariant_class α α (+) (<)]
(ha : a ≠ ⊥) : a + b < a + c ↔ b < c | ⟨lt_of_add_lt_add_left, with_bot.add_lt_add_left ha⟩ | lemma | with_bot.add_lt_add_iff_left | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class",
"with_bot.add_lt_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_iff_right [covariant_class α α (swap (+)) (<)]
[contravariant_class α α (swap (+)) (<)] (ha : a ≠ ⊥) : b + a < c + a ↔ b < c | ⟨lt_of_add_lt_add_right, with_bot.add_lt_add_right ha⟩ | lemma | with_bot.add_lt_add_iff_right | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"contravariant_class",
"covariant_class",
"with_bot.add_lt_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_of_le_of_lt [covariant_class α α (+) (<)]
[covariant_class α α (swap (+)) (≤)] (hb : b ≠ ⊥) (hab : a ≤ b) (hcd : c < d) : a + c < b + d | @with_top.add_lt_add_of_le_of_lt αᵒᵈ _ _ _ _ _ _ _ _ hb hab hcd | lemma | with_bot.add_lt_add_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_lt_add_of_lt_of_le [covariant_class α α (+) (≤)]
[covariant_class α α (swap (+)) (<)] (hd : d ≠ ⊥) (hab : a < b) (hcd : c ≤ d) : a + c < b + d | @with_top.add_lt_add_of_lt_of_le αᵒᵈ _ _ _ _ _ _ _ _ hd hab hcd | lemma | with_bot.add_lt_add_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/with_top.lean | [
"algebra.hom.group",
"algebra.order.monoid.order_dual",
"algebra.order.monoid.with_zero.basic",
"data.nat.cast.defs"
] | [
"covariant_class",
"with_top.add_lt_add_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.ordered_cancel_comm_monoid {β : Type*}
[has_one β] [has_mul β] [has_pow β ℕ]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
ordered_cancel_comm_monoid β | { le_of_mul_le_mul_left := λ a b c (bc : f (a * b) ≤ f (a * c)),
(mul_le_mul_iff_left (f a)).mp (by rwa [← mul, ← mul]),
..hf.ordered_comm_monoid f one mul npow } | def | function.injective.ordered_cancel_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/basic.lean | [
"algebra.order.monoid.basic",
"algebra.order.monoid.cancel.defs"
] | [
"le_of_mul_le_mul_left",
"mul_le_mul_iff_left",
"ordered_cancel_comm_monoid"
] | Pullback an `ordered_cancel_comm_monoid` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.linear_ordered_cancel_comm_monoid {β : Type*}
[has_one β] [has_mul β] [has_pow β ℕ] [has_sup β] [has_inf β]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(hsup : ∀ x y, f (x ⊔ y) = max (f x) (f y)) (hinf ... | { ..hf.linear_ordered_comm_monoid f one mul npow hsup hinf,
..hf.ordered_cancel_comm_monoid f one mul npow } | def | function.injective.linear_ordered_cancel_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/basic.lean | [
"algebra.order.monoid.basic",
"algebra.order.monoid.cancel.defs"
] | [
"has_inf",
"has_sup",
"linear_ordered_cancel_comm_monoid"
] | Pullback a `linear_ordered_cancel_comm_monoid` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_cancel_add_comm_monoid (α : Type u) extends add_comm_monoid α, partial_order α | (add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b)
(le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c) | class | ordered_cancel_add_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"add_comm_monoid"
] | An ordered cancellative additive commutative monoid
is an additive commutative monoid with a partial order,
in which addition is cancellative and monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_cancel_comm_monoid (α : Type u) extends comm_monoid α, partial_order α | (mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b)
(le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c) | class | ordered_cancel_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"comm_monoid",
"le_of_mul_le_mul_left",
"mul_le_mul_left"
] | An ordered cancellative commutative monoid
is a commutative monoid with a partial order,
in which multiplication is cancellative and monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_cancel_comm_monoid.to_contravariant_class_le_left :
contravariant_class α α (*) (≤) | ⟨ordered_cancel_comm_monoid.le_of_mul_le_mul_left⟩ | instance | ordered_cancel_comm_monoid.to_contravariant_class_le_left | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left : ∀ a b c : α, a * b < a * c → b < c | λ a b c h, lt_of_le_not_le
(ordered_cancel_comm_monoid.le_of_mul_le_mul_left a b c h.le) $
mt (λ h, ordered_cancel_comm_monoid.mul_le_mul_left _ _ h _) (not_le_of_gt h) | lemma | ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.to_contravariant_class_left
(M : Type*) [ordered_cancel_comm_monoid M] :
contravariant_class M M (*) (<) | { elim := λ a b c, ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left _ _ _ } | instance | ordered_cancel_comm_monoid.to_contravariant_class_left | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"contravariant_class",
"ordered_cancel_comm_monoid",
"ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.to_contravariant_class_right
(M : Type*) [ordered_cancel_comm_monoid M] :
contravariant_class M M (swap (*)) (<) | contravariant_swap_mul_lt_of_contravariant_mul_lt M | instance | ordered_cancel_comm_monoid.to_contravariant_class_right | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"contravariant_class",
"contravariant_swap_mul_lt_of_contravariant_mul_lt",
"ordered_cancel_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.to_ordered_comm_monoid : ordered_comm_monoid α | { ..‹ordered_cancel_comm_monoid α› } | instance | ordered_cancel_comm_monoid.to_ordered_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_cancel_comm_monoid.to_cancel_comm_monoid : cancel_comm_monoid α | { mul_left_cancel := λ a b c h,
(le_of_mul_le_mul_left' h.le).antisymm $ le_of_mul_le_mul_left' h.ge,
..‹ordered_cancel_comm_monoid α› } | instance | ordered_cancel_comm_monoid.to_cancel_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"cancel_comm_monoid",
"le_of_mul_le_mul_left'",
"mul_left_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_cancel_add_comm_monoid (α : Type u)
extends ordered_cancel_add_comm_monoid α, linear_ordered_add_comm_monoid α | class | linear_ordered_cancel_add_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"linear_ordered_add_comm_monoid",
"ordered_cancel_add_comm_monoid"
] | A linearly ordered cancellative additive commutative monoid
is an additive commutative monoid with a decidable linear order
in which addition is cancellative and monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_cancel_comm_monoid (α : Type u)
extends ordered_cancel_comm_monoid α, linear_ordered_comm_monoid α | class | linear_ordered_cancel_comm_monoid | algebra.order.monoid.cancel | src/algebra/order/monoid/cancel/defs.lean | [
"algebra.order.monoid.defs"
] | [
"linear_ordered_comm_monoid",
"ordered_cancel_comm_monoid"
] | A linearly ordered cancellative commutative monoid
is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_exists_mul_of_le (α : Type u) [has_mul α] [has_le α] : Prop | (exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a * c) | class | has_exists_mul_of_le | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | An `ordered_comm_monoid` with one-sided 'division' in the sense that
if `a ≤ b`, there is some `c` for which `a * c = b`. This is a weaker version
of the condition on canonical orderings defined by `canonically_ordered_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_exists_add_of_le (α : Type u) [has_add α] [has_le α] : Prop | (exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a + c) | class | has_exists_add_of_le | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | An `ordered_add_comm_monoid` with one-sided 'subtraction' in the sense that
if `a ≤ b`, then there is some `c` for which `a + c = b`. This is a weaker version
of the condition on canonical orderings defined by `canonically_ordered_add_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.has_exists_mul_of_le (α : Type u) [group α] [has_le α] : has_exists_mul_of_le α | ⟨λ a b hab, ⟨a⁻¹ * b, (mul_inv_cancel_left _ _).symm⟩⟩ | instance | group.has_exists_mul_of_le | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"group",
"has_exists_mul_of_le",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b | by { obtain ⟨c, rfl⟩ := exists_mul_of_le h.le, exact ⟨c, one_lt_of_lt_mul_right h, rfl⟩ } | lemma | exists_one_lt_mul_of_lt' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_lt_of_lt_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b | le_of_forall_le_of_dense $ λ x hxb, by { obtain ⟨ε, rfl⟩ := exists_mul_of_le hxb.le,
exact h _ ((lt_mul_iff_one_lt_right' b).1 hxb) } | lemma | le_of_forall_one_lt_le_mul | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_of_forall_le_of_dense",
"lt_mul_iff_one_lt_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_one_lt_lt_mul' (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b | le_of_forall_one_lt_le_mul $ λ ε hε, (h _ hε).le | lemma | le_of_forall_one_lt_lt_mul' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_of_forall_one_lt_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε | ⟨λ h ε, lt_mul_of_le_of_one_lt h, le_of_forall_one_lt_lt_mul'⟩ | lemma | le_iff_forall_one_lt_lt_mul' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"lt_mul_of_le_of_one_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_add_monoid (α : Type*) extends ordered_add_comm_monoid α, has_bot α | (bot_le : ∀ x : α, ⊥ ≤ x)
(exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ c, b = a + c)
(le_self_add : ∀ a b : α, a ≤ a + b) | class | canonically_ordered_add_monoid | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"bot_le",
"has_bot",
"ordered_add_comm_monoid"
] | A canonically ordered additive monoid is an ordered commutative additive monoid
in which the ordering coincides with the subtractibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a + c`.
This is satisfied by the natural numbers, for example, but not
the integers or other nontrivial `orde... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
canonically_ordered_add_monoid.to_order_bot (α : Type u)
[h : canonically_ordered_add_monoid α] : order_bot α | { ..h } | instance | canonically_ordered_add_monoid.to_order_bot | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_monoid (α : Type*) extends ordered_comm_monoid α, has_bot α | (bot_le : ∀ x : α, ⊥ ≤ x)
(exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ c, b = a * c)
(le_self_mul : ∀ a b : α, a ≤ a * b) | class | canonically_ordered_monoid | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"bot_le",
"has_bot",
"le_self_mul",
"ordered_comm_monoid"
] | A canonically ordered monoid is an ordered commutative monoid
in which the ordering coincides with the divisibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a * c`.
Examples seem rare; it seems more likely that the `order_dual`
of a naturally-occurring lattice satisfies this than the la... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
canonically_ordered_monoid.to_order_bot (α : Type u)
[h : canonically_ordered_monoid α] : order_bot α | { ..h } | instance | canonically_ordered_monoid.to_order_bot | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_monoid",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_ordered_monoid.has_exists_mul_of_le (α : Type u)
[h : canonically_ordered_monoid α] : has_exists_mul_of_le α | { ..h } | instance | canonically_ordered_monoid.has_exists_mul_of_le | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_monoid",
"has_exists_mul_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_mul : a ≤ a * c | canonically_ordered_monoid.le_self_mul _ _ | lemma | le_self_mul | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_self : a ≤ b * a | by { rw mul_comm, exact le_self_mul } | lemma | le_mul_self | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_self_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_mul_right (a b : α) : a ≤ a * b | le_self_mul | lemma | self_le_mul_right | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_self_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_mul_left (a b : α) : a ≤ b * a | le_mul_self | lemma | self_le_mul_left | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_left : a * b ≤ c → a ≤ c | le_self_mul.trans | lemma | le_of_mul_le_left | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_right : a * b ≤ c → b ≤ c | le_mul_self.trans | lemma | le_of_mul_le_right | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_left : a ≤ b → a ≤ b * c | le_self_mul.trans' | lemma | le_mul_of_le_left | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_right : a ≤ c → a ≤ b * c | le_mul_self.trans' | lemma | le_mul_of_le_right | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_exists_mul : a ≤ b ↔ ∃ c, b = a * c | ⟨exists_mul_of_le, by { rintro ⟨c, rfl⟩, exact le_self_mul }⟩ | lemma | le_iff_exists_mul | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_self_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_exists_mul' : a ≤ b ↔ ∃ c, b = c * a | by simpa only [mul_comm _ a] using le_iff_exists_mul | lemma | le_iff_exists_mul' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_iff_exists_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le (a : α) : 1 ≤ a | le_iff_exists_mul.mpr ⟨a, (one_mul _).symm⟩ | lemma | one_le | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_one : (⊥ : α) = 1 | le_antisymm bot_le (one_le ⊥) | lemma | bot_eq_one | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"bot_le",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff : a * b = 1 ↔ a = 1 ∧ b = 1 | mul_eq_one_iff' (one_le _) (one_le _) | lemma | mul_eq_one_iff | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"mul_eq_one_iff'",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff_eq_one : a ≤ 1 ↔ a = 1 | (one_le a).le_iff_eq | lemma | le_one_iff_eq_one | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_iff_ne_one : 1 < a ↔ a ≠ 1 | (one_le a).lt_iff_ne.trans ne_comm | lemma | one_lt_iff_ne_one | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"ne_comm",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_or_one_lt : a = 1 ∨ 1 < a | (one_le a).eq_or_lt.imp_left eq.symm | lemma | eq_one_or_one_lt | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_mul_iff : 1 < a * b ↔ 1 < a ∨ 1 < b | by simp only [one_lt_iff_ne_one, ne.def, mul_eq_one_iff, not_and_distrib] | lemma | one_lt_mul_iff | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"mul_eq_one_iff",
"not_and_distrib",
"one_lt_iff_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_one_lt_mul_of_lt (h : a < b) : ∃ c (hc : 1 < c), a * c = b | begin
obtain ⟨c, hc⟩ := le_iff_exists_mul.1 h.le,
refine ⟨c, one_lt_iff_ne_one.2 _, hc.symm⟩,
rintro rfl,
simpa [hc, lt_irrefl] using h
end | lemma | exists_one_lt_mul_of_lt | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_left (h : a ≤ c) : a ≤ b * c | calc a = 1 * a : by simp
... ≤ b * c : mul_le_mul' (one_le _) h | lemma | le_mul_left | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"mul_le_mul'",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_right (h : a ≤ b) : a ≤ b * c | calc a = a * 1 : by simp
... ≤ b * c : mul_le_mul' h (one_le _) | lemma | le_mul_right | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"mul_le_mul'",
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_iff_exists_mul [covariant_class α α (*) (<)] : a < b ↔ ∃ c > 1, b = a * c | begin
simp_rw [lt_iff_le_and_ne, and_comm, le_iff_exists_mul, ← exists_and_distrib_left, exists_prop],
apply exists_congr, intro c,
rw [and.congr_left_iff, gt_iff_lt], rintro rfl,
split,
{ rw [one_lt_iff_ne_one], apply mt, rintro rfl, rw [mul_one] },
{ rw [← (self_le_mul_right a c).lt_iff_ne], apply lt_mul_... | lemma | lt_iff_exists_mul | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"and.congr_left_iff",
"covariant_class",
"exists_and_distrib_left",
"exists_prop",
"gt_iff_lt",
"le_iff_exists_mul",
"lt_iff_le_and_ne",
"lt_mul_of_one_lt_right'",
"mul_one",
"one_lt_iff_ne_one",
"self_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_gt {M : Type*} [canonically_ordered_add_monoid M] {n m : M} (h : n < m) : 0 < m | lt_of_le_of_lt (zero_le _) h | lemma | pos_of_gt | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos {M} (a : M) [canonically_ordered_add_monoid M] [ne_zero a] : 0 < a | (zero_le a).lt_of_ne $ ne_zero.out.symm | lemma | ne_zero.pos | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_gt {M} [canonically_ordered_add_monoid M] {x y : M} (h : x < y) : ne_zero y | of_pos $ pos_of_gt h | lemma | ne_zero.of_gt | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid",
"ne_zero",
"pos_of_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_gt' {M} [canonically_ordered_add_monoid M] [has_one M] {y : M}
[fact (1 < y)] : ne_zero y | of_gt $ fact.out $ 1 < y | instance | ne_zero.of_gt' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid",
"fact",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0 {M} [canonically_ordered_add_monoid M] {x : M} [ne_zero x] : ne_zero (bit0 x) | of_pos $ bit0_pos $ ne_zero.pos x | instance | ne_zero.bit0 | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"bit0_pos",
"canonically_ordered_add_monoid",
"ne_zero",
"ne_zero.pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_add_monoid (α : Type*)
extends canonically_ordered_add_monoid α, linear_order α | class | canonically_linear_ordered_add_monoid | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_add_monoid"
] | A canonically linear-ordered additive monoid is a canonically ordered additive monoid
whose ordering is a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_monoid (α : Type*)
extends canonically_ordered_monoid α, linear_order α | class | canonically_linear_ordered_monoid | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"canonically_ordered_monoid"
] | A canonically linear-ordered monoid is a canonically ordered monoid
whose ordering is a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonically_linear_ordered_monoid.semilattice_sup : semilattice_sup α | { ..linear_order.to_lattice } | instance | canonically_linear_ordered_monoid.semilattice_sup | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"linear_order.to_lattice",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_distrib (a b c : α) : min a (b * c) = min a (min a b * min a c) | begin
cases le_total a b with hb hb,
{ simp [hb, le_mul_right] },
{ cases le_total a c with hc hc,
{ simp [hc, le_mul_left] },
{ simp [hb, hc] } }
end | lemma | min_mul_distrib | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"le_mul_left",
"le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_distrib' (a b c : α) : min (a * b) c = min (min a c * min b c) c | by simpa [min_comm _ c] using min_mul_distrib c a b | lemma | min_mul_distrib' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"min_mul_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_min (a : α) : min 1 a = 1 | min_eq_left (one_le a) | lemma | one_min | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_one (a : α) : min a 1 = 1 | min_eq_right (one_le a) | lemma | min_one | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_one' : (⊥ : α) = 1 | bot_eq_one | lemma | bot_eq_one' | algebra.order.monoid.canonical | src/algebra/order/monoid/canonical/defs.lean | [
"order.bounded_order",
"order.min_max",
"algebra.ne_zero",
"algebra.order.monoid.defs"
] | [
"bot_eq_one"
] | In a linearly ordered monoid, we are happy for `bot_eq_one` to be a `@[simp]` lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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