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
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|---|---|---|---|---|---|---|---|---|---|---|
strict_mono.inv (hf : strict_mono f) : strict_anti (λ x, (f x)⁻¹) | λ x y hxy, inv_lt_inv_iff.2 (hf hxy) | lemma | strict_mono.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti.inv (hf : strict_anti f) : strict_mono (λ x, (f x)⁻¹) | λ x y hxy, inv_lt_inv_iff.2 (hf hxy) | lemma | strict_anti.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.inv (hf : strict_mono_on f s) :
strict_anti_on (λ x, (f x)⁻¹) s | λ x hx y hy hxy, inv_lt_inv_iff.2 (hf hx hy hxy) | lemma | strict_mono_on.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on.inv (hf : strict_anti_on f s) :
strict_mono_on (λ x, (f x)⁻¹) s | λ x hx y hy hxy, inv_lt_inv_iff.2 (hf hx hy hxy) | lemma | strict_anti_on.inv | algebra.order.group | src/algebra/order/group/defs.lean | [
"order.hom.basic",
"algebra.order.sub.defs",
"algebra.order.monoid.cancel.defs"
] | [
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_lt_one_mul_le (h : ∀ ε < 1, a * ε ≤ b) : a ≤ b | @le_of_forall_one_lt_le_mul αᵒᵈ _ _ _ _ _ _ _ _ h | lemma | le_of_forall_lt_one_mul_le | algebra.order.group | src/algebra/order/group/densely_ordered.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.group.defs",
"algebra.order.monoid.order_dual"
] | [
"le_of_forall_one_lt_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_one_lt_div_le (h : ∀ ε : α, 1 < ε → a / ε ≤ b) : a ≤ b | le_of_forall_lt_one_mul_le $ λ ε ε1,
by simpa only [div_eq_mul_inv, inv_inv] using h ε⁻¹ (left.one_lt_inv_iff.2 ε1) | lemma | le_of_forall_one_lt_div_le | algebra.order.group | src/algebra/order/group/densely_ordered.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.group.defs",
"algebra.order.monoid.order_dual"
] | [
"div_eq_mul_inv",
"inv_inv",
"le_of_forall_lt_one_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_forall_one_lt_le_mul : a ≤ b ↔ ∀ ε, 1 < ε → a ≤ b * ε | ⟨λ h ε ε_pos, le_mul_of_le_of_one_le h ε_pos.le, le_of_forall_one_lt_le_mul⟩ | lemma | le_iff_forall_one_lt_le_mul | algebra.order.group | src/algebra/order/group/densely_ordered.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.group.defs",
"algebra.order.monoid.order_dual"
] | [
"le_mul_of_le_of_one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_forall_lt_one_mul_le : a ≤ b ↔ ∀ ε < 1, a * ε ≤ b | @le_iff_forall_one_lt_le_mul αᵒᵈ _ _ _ _ _ _ | lemma | le_iff_forall_lt_one_mul_le | algebra.order.group | src/algebra/order/group/densely_ordered.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.group.defs",
"algebra.order.monoid.order_dual"
] | [
"le_iff_forall_one_lt_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.ordered_comm_group [ordered_comm_group α] {β : Type*}
[has_one β] [has_mul β] [has_inv β] [has_div β] [has_pow β ℕ] [has_pow β ℤ]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y)
(inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y)
... | { ..partial_order.lift f hf,
..hf.ordered_comm_monoid f one mul npow,
..hf.comm_group f one mul inv div npow zpow } | def | function.injective.ordered_comm_group | algebra.order.group | src/algebra/order/group/inj_surj.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.basic",
"algebra.order.group.instances"
] | [
"ordered_comm_group",
"partial_order.lift"
] | Pullback an `ordered_comm_group` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.linear_ordered_comm_group [linear_ordered_comm_group α] {β : Type*}
[has_one β] [has_mul β] [has_inv β] [has_div β] [has_pow β ℕ] [has_pow β ℤ]
[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)
(inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
... | { ..linear_order.lift f hf hsup hinf,
..hf.ordered_comm_group f one mul inv div npow zpow } | def | function.injective.linear_ordered_comm_group | algebra.order.group | src/algebra/order/group/inj_surj.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.basic",
"algebra.order.group.instances"
] | [
"has_inf",
"has_sup",
"linear_order.lift",
"linear_ordered_comm_group"
] | Pullback a `linear_ordered_comm_group` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
max_one_div_max_inv_one_eq_self (a : α) :
max a 1 / max a⁻¹ 1 = a | by { rcases le_total a 1 with h|h; simp [h] } | lemma | max_one_div_max_inv_one_eq_self | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_inv_inv' (a b : α) : min (a⁻¹) (b⁻¹) = (max a b)⁻¹ | eq.symm $ @monotone.map_max α αᵒᵈ _ _ has_inv.inv a b $ λ a b, inv_le_inv_iff.mpr | lemma | min_inv_inv' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"monotone.map_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_inv_inv' (a b : α) : max (a⁻¹) (b⁻¹) = (min a b)⁻¹ | eq.symm $ @monotone.map_min α αᵒᵈ _ _ has_inv.inv a b $ λ a b, inv_le_inv_iff.mpr | lemma | max_inv_inv' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"monotone.map_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c | by simpa only [div_eq_mul_inv] using min_mul_mul_right a b (c⁻¹) | lemma | min_div_div_right' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"div_eq_mul_inv",
"min_mul_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c | by simpa only [div_eq_mul_inv] using max_mul_mul_right a b (c⁻¹) | lemma | max_div_div_right' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"div_eq_mul_inv",
"max_mul_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c | by simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv'] | lemma | min_div_div_left' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"div_eq_mul_inv",
"min_inv_inv'",
"min_mul_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c | by simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv'] | lemma | max_div_div_left' | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"div_eq_mul_inv",
"max_inv_inv'",
"max_mul_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_sub_max_le_max (a b c d : α) : max a b - max c d ≤ max (a - c) (b - d) | begin
simp only [sub_le_iff_le_add, max_le_iff], split,
calc a = a - c + c : (sub_add_cancel a c).symm
... ≤ max (a - c) (b - d) + max c d : add_le_add (le_max_left _ _) (le_max_left _ _),
calc b = b - d + d : (sub_add_cancel b d).symm
... ≤ max (a - c) (b - d) + max c d : add_le_add (le_max_right _ _) (le_ma... | lemma | max_sub_max_le_max | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"max_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_max_sub_max_le_max (a b c d : α) : |max a b - max c d| ≤ max (|a - c|) (|b - d|) | begin
refine abs_sub_le_iff.2 ⟨_, _⟩,
{ exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) },
{ rw [abs_sub_comm a c, abs_sub_comm b d],
exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) }
end | lemma | abs_max_sub_max_le_max | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"abs_sub_comm",
"le_abs_self",
"max_le_max",
"max_sub_max_le_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_min_sub_min_le_max (a b c d : α) : |min a b - min c d| ≤ max (|a - c|) (|b - d|) | by simpa only [max_neg_neg, neg_sub_neg, abs_sub_comm]
using abs_max_sub_max_le_max (-a) (-b) (-c) (-d) | lemma | abs_min_sub_min_le_max | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"abs_max_sub_max_le_max",
"abs_sub_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_max_sub_max_le_abs (a b c : α) : |max a c - max b c| ≤ |a - b| | by simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg _)]
using abs_max_sub_max_le_max a c b c | lemma | abs_max_sub_max_le_abs | algebra.order.group | src/algebra/order/group/min_max.lean | [
"algebra.order.group.abs",
"algebra.order.monoid.min_max"
] | [
"abs_max_sub_max_le_max",
"abs_nonneg",
"abs_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.inv : α ≃o αᵒᵈ | { to_equiv := (equiv.inv α).trans order_dual.to_dual,
map_rel_iff' := λ a b, @inv_le_inv_iff α _ _ _ _ _ _ } | def | order_iso.inv | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"equiv.inv",
"inv_le_inv_iff",
"order_dual.to_dual"
] | `x ↦ x⁻¹` as an order-reversing equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a | (order_iso.inv α).symm_apply_le | lemma | inv_le' | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ | (order_iso.inv α).le_symm_apply | lemma | le_inv' | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.div_left (a : α) : α ≃o αᵒᵈ | { to_equiv := (equiv.div_left a).trans order_dual.to_dual,
map_rel_iff' := λ x y, @div_le_div_iff_left α _ _ _ _ _ _ _ } | def | order_iso.div_left | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"div_le_div_iff_left",
"equiv.div_left",
"order_dual.to_dual"
] | `x ↦ a / x` as an order-reversing equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_right (a : α) : α ≃o α | { map_rel_iff' := λ _ _, mul_le_mul_iff_right a, to_equiv := equiv.mul_right a } | def | order_iso.mul_right | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"equiv.mul_right",
"mul_le_mul_iff_right"
] | `equiv.mul_right` as an `order_iso`. See also `order_embedding.mul_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_right_symm (a : α) :
(order_iso.mul_right a).symm = order_iso.mul_right a⁻¹ | by { ext x, refl } | lemma | order_iso.mul_right_symm | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"order_iso.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.div_right (a : α) : α ≃o α | { to_equiv := equiv.div_right a,
map_rel_iff' := λ x y, div_le_div_iff_right a } | def | order_iso.div_right | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"div_le_div_iff_right",
"equiv.div_right"
] | `x ↦ x / a` as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_left (a : α) : α ≃o α | { map_rel_iff' := λ _ _, mul_le_mul_iff_left a, to_equiv := equiv.mul_left a } | def | order_iso.mul_left | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"equiv.mul_left",
"mul_le_mul_iff_left"
] | `equiv.mul_left` as an `order_iso`. See also `order_embedding.mul_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso.mul_left_symm (a : α) :
(order_iso.mul_left a).symm = order_iso.mul_left a⁻¹ | by { ext x, refl } | lemma | order_iso.mul_left_symm | algebra.order.group | src/algebra/order/group/order_iso.lean | [
"algebra.order.group.defs",
"algebra.hom.equiv.units.basic"
] | [
"order_iso.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.ordered_comm_group [ordered_comm_monoid α] : ordered_comm_group αˣ | { mul_le_mul_left := λ a b h c, (mul_le_mul_left' (h : (a : α) ≤ b) _ : (c : α) * a ≤ c * b),
.. units.partial_order,
.. units.comm_group } | instance | units.ordered_comm_group | algebra.order.group | src/algebra/order/group/units.lean | [
"algebra.order.group.defs",
"algebra.order.monoid.defs",
"algebra.order.monoid.units"
] | [
"mul_le_mul_left",
"mul_le_mul_left'",
"ordered_comm_group",
"ordered_comm_monoid"
] | The units of an ordered commutative monoid form an ordered commutative group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top.linear_ordered_add_comm_group_with_top :
linear_ordered_add_comm_group_with_top (with_top α) | { neg := option.map (λ a : α, -a),
neg_top := @option.map_none _ _ (λ a : α, -a),
add_neg_cancel := begin
rintro (a | a) ha,
{ exact (ha rfl).elim },
{ exact with_top.coe_add.symm.trans (with_top.coe_eq_coe.2 (add_neg_self a)) }
end,
.. with_top.linear_ordered_add_comm_monoid_with_... | instance | with_top.linear_ordered_add_comm_group_with_top | algebra.order.group | src/algebra/order/group/with_top.lean | [
"algebra.order.group.instances",
"algebra.order.monoid.with_top"
] | [
"linear_ordered_add_comm_group_with_top",
"option.map_none",
"option.nontrivial",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top.coe_neg (a : α) : ((-a : α) : with_top α) = -a | rfl | lemma | with_top.coe_neg | algebra.order.group | src/algebra/order/group/with_top.lean | [
"algebra.order.group.instances",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_hom_class (F : Type*) (α β : out_param $ Type*) [has_zero β] [has_le β]
extends fun_like F α (λ _, β) | (map_nonneg (f : F) : ∀ a, 0 ≤ f a) | class | nonneg_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"fun_like",
"map_nonneg"
] | `nonneg_hom_class F α β` states that `F` is a type of nonnegative morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subadditive_hom_class (F : Type*) (α β : out_param $ Type*) [has_add α] [has_add β] [has_le β]
extends fun_like F α (λ _, β) | (map_add_le_add (f : F) : ∀ a b, f (a + b) ≤ f a + f b) | class | subadditive_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"fun_like"
] | `subadditive_hom_class F α β` states that `F` is a type of subadditive morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submultiplicative_hom_class (F : Type*) (α β : out_param $ Type*) [has_mul α] [has_mul β]
[has_le β] extends fun_like F α (λ _, β) | (map_mul_le_mul (f : F) : ∀ a b, f (a * b) ≤ f a * f b) | class | submultiplicative_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"fun_like"
] | `submultiplicative_hom_class F α β` states that `F` is a type of submultiplicative morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_add_hom_class (F : Type*) (α β : out_param $ Type*) [has_mul α] [has_add β] [has_le β]
extends fun_like F α (λ _, β) | (map_mul_le_add (f : F) : ∀ a b, f (a * b) ≤ f a + f b) | class | mul_le_add_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"fun_like"
] | `mul_le_add_hom_class F α β` states that `F` is a type of subadditive morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarchimedean_hom_class (F : Type*) (α β : out_param $ Type*) [has_add α] [linear_order β]
extends fun_like F α (λ _, β) | (map_add_le_max (f : F) : ∀ a b, f (a + b) ≤ max (f a) (f b)) | class | nonarchimedean_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"fun_like"
] | `nonarchimedean_hom_class F α β` states that `F` is a type of non-archimedean morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_map_mul_map_div [group α] [comm_semigroup β] [has_le β]
[submultiplicative_hom_class F α β] (f : F) (a b : α) : f a ≤ f b * f (a / b) | by simpa only [mul_comm, div_mul_cancel'] using map_mul_le_mul f (a / b) b | lemma | le_map_mul_map_div | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"comm_semigroup",
"div_mul_cancel'",
"group",
"mul_comm",
"submultiplicative_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_add_map_div [group α] [add_comm_semigroup β] [has_le β]
[mul_le_add_hom_class F α β] (f : F) (a b : α) : f a ≤ f b + f (a / b) | by simpa only [add_comm, div_mul_cancel'] using map_mul_le_add f (a / b) b | lemma | le_map_add_map_div | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_comm_semigroup",
"div_mul_cancel'",
"group",
"mul_le_add_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_div_mul_map_div [group α] [comm_semigroup β] [has_le β]
[submultiplicative_hom_class F α β] (f : F) (a b c: α) : f (a / c) ≤ f (a / b) * f (b / c) | by simpa only [div_mul_div_cancel'] using map_mul_le_mul f (a / b) (b / c) | lemma | le_map_div_mul_map_div | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"comm_semigroup",
"div_mul_div_cancel'",
"group",
"submultiplicative_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_div_add_map_div [group α] [add_comm_semigroup β] [has_le β]
[mul_le_add_hom_class F α β] (f : F) (a b c: α) : f (a / c) ≤ f (a / b) + f (b / c) | by simpa only [div_mul_div_cancel'] using map_mul_le_add f (a / b) (b / c) | lemma | le_map_div_add_map_div | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_comm_semigroup",
"div_mul_div_cancel'",
"group",
"mul_le_add_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_seminorm_class (F : Type*) (α β : out_param $ Type*) [add_group α]
[ordered_add_comm_monoid β] extends subadditive_hom_class F α β | (map_zero (f : F) : f 0 = 0)
(map_neg_eq_map (f : F) (a : α) : f (-a) = f a) | class | add_group_seminorm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group",
"ordered_add_comm_monoid",
"subadditive_hom_class"
] | `add_group_seminorm_class F α` states that `F` is a type of `β`-valued seminorms on the additive
group `α`.
You should extend this class when you extend `add_group_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_seminorm_class (F : Type*) (α β : out_param $ Type*) [group α]
[ordered_add_comm_monoid β] extends mul_le_add_hom_class F α β | (map_one_eq_zero (f : F) : f 1 = 0)
(map_inv_eq_map (f : F) (a : α) : f a⁻¹ = f a) | class | group_seminorm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"group",
"mul_le_add_hom_class",
"ordered_add_comm_monoid"
] | `group_seminorm_class F α` states that `F` is a type of `β`-valued seminorms on the group `α`.
You should extend this class when you extend `group_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_norm_class (F : Type*) (α β : out_param $ Type*) [add_group α]
[ordered_add_comm_monoid β] extends add_group_seminorm_class F α β | (eq_zero_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 0) | class | add_group_norm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group",
"add_group_seminorm_class",
"ordered_add_comm_monoid"
] | `add_group_norm_class F α` states that `F` is a type of `β`-valued norms on the additive group
`α`.
You should extend this class when you extend `add_group_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_norm_class (F : Type*) (α β : out_param $ Type*) [group α] [ordered_add_comm_monoid β]
extends group_seminorm_class F α β | (eq_one_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 1) | class | group_norm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"group",
"group_seminorm_class",
"ordered_add_comm_monoid"
] | `group_norm_class F α` states that `F` is a type of `β`-valued norms on the group `α`.
You should extend this class when you extend `group_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_seminorm_class.to_zero_hom_class [add_group α] [ordered_add_comm_monoid β]
[add_group_seminorm_class F α β] :
zero_hom_class F α β | { ..‹add_group_seminorm_class F α β› } | instance | add_group_seminorm_class.to_zero_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group",
"add_group_seminorm_class",
"ordered_add_comm_monoid",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div_le_add : f (x / y) ≤ f x + f y | by { rw [div_eq_mul_inv, ←map_inv_eq_map f y], exact map_mul_le_add _ _ _ } | lemma | map_div_le_add | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div_rev : f (x / y) = f (y / x) | by rw [←inv_div, map_inv_eq_map] | lemma | map_div_rev | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_add_map_div' : f x ≤ f y + f (y / x) | by simpa only [add_comm, map_div_rev, div_mul_cancel'] using map_mul_le_add f (x / y) y | lemma | le_map_add_map_div' | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"div_mul_cancel'",
"map_div_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_map_le_div [group α] [linear_ordered_add_comm_group β]
[group_seminorm_class F α β] (f : F) (x y : α) : |f x - f y| ≤ f (x / y) | begin
rw [abs_sub_le_iff, sub_le_iff_le_add', sub_le_iff_le_add'],
exact ⟨le_map_add_map_div _ _ _, le_map_add_map_div' _ _ _⟩
end | lemma | abs_sub_map_le_div | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"abs_sub_le_iff",
"group",
"group_seminorm_class",
"le_map_add_map_div'",
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_seminorm_class.to_nonneg_hom_class [group α] [linear_ordered_add_comm_monoid β]
[group_seminorm_class F α β] :
nonneg_hom_class F α β | { map_nonneg := λ f a, (nsmul_nonneg_iff two_ne_zero).1 $
by { rw [two_nsmul, ←map_one_eq_zero f, ←div_self' a], exact map_div_le_add _ _ _ },
..‹group_seminorm_class F α β› } | instance | group_seminorm_class.to_nonneg_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"group",
"group_seminorm_class",
"linear_ordered_add_comm_monoid",
"map_div_le_add",
"map_nonneg",
"nonneg_hom_class",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff_eq_one : f x = 0 ↔ x = 1 | ⟨eq_one_of_map_eq_zero _, by { rintro rfl, exact map_one_eq_zero _ }⟩ | lemma | map_eq_zero_iff_eq_one | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_zero_iff_ne_one : f x ≠ 0 ↔ x ≠ 1 | (map_eq_zero_iff_eq_one _).not | lemma | map_ne_zero_iff_ne_one | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"map_eq_zero_iff_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pos_of_ne_one [group α] [linear_ordered_add_comm_monoid β]
[group_norm_class F α β] (f : F) {x : α} (hx : x ≠ 1) : 0 < f x | (map_nonneg _ _).lt_of_ne $ ((map_ne_zero_iff_ne_one _).2 hx).symm | lemma | map_pos_of_ne_one | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"group",
"group_norm_class",
"linear_ordered_add_comm_monoid",
"map_ne_zero_iff_ne_one",
"map_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_seminorm_class (F : Type*) (α β : out_param $ Type*) [non_unital_non_assoc_ring α]
[ordered_semiring β] extends add_group_seminorm_class F α β, submultiplicative_hom_class F α β | class | ring_seminorm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group_seminorm_class",
"non_unital_non_assoc_ring",
"ordered_semiring",
"submultiplicative_hom_class"
] | `ring_seminorm_class F α` states that `F` is a type of `β`-valued seminorms on the ring `α`.
You should extend this class when you extend `ring_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_norm_class (F : Type*) (α β : out_param $ Type*) [non_unital_non_assoc_ring α]
[ordered_semiring β] extends ring_seminorm_class F α β, add_group_norm_class F α β | class | ring_norm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group_norm_class",
"non_unital_non_assoc_ring",
"ordered_semiring",
"ring_seminorm_class"
] | `ring_norm_class F α` states that `F` is a type of `β`-valued norms on the ring `α`.
You should extend this class when you extend `ring_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_seminorm_class (F : Type*) (α β : out_param $ Type*) [non_assoc_ring α]
[ordered_semiring β] extends add_group_seminorm_class F α β, monoid_with_zero_hom_class F α β | class | mul_ring_seminorm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group_seminorm_class",
"monoid_with_zero_hom_class",
"non_assoc_ring",
"ordered_semiring"
] | `mul_ring_seminorm_class F α` states that `F` is a type of `β`-valued multiplicative seminorms
on the ring `α`.
You should extend this class when you extend `mul_ring_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_norm_class (F : Type*) (α β : out_param $ Type*) [non_assoc_ring α]
[ordered_semiring β] extends mul_ring_seminorm_class F α β, add_group_norm_class F α β | class | mul_ring_norm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"add_group_norm_class",
"mul_ring_seminorm_class",
"non_assoc_ring",
"ordered_semiring"
] | `mul_ring_norm_class F α` states that `F` is a type of `β`-valued multiplicative norms on the
ring `α`.
You should extend this class when you extend `mul_ring_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_seminorm_class.to_nonneg_hom_class [non_unital_non_assoc_ring α]
[linear_ordered_semiring β] [ring_seminorm_class F α β] : nonneg_hom_class F α β | add_group_seminorm_class.to_nonneg_hom_class | instance | ring_seminorm_class.to_nonneg_hom_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"linear_ordered_semiring",
"non_unital_non_assoc_ring",
"nonneg_hom_class",
"ring_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_seminorm_class.to_ring_seminorm_class [non_assoc_ring α] [ordered_semiring β]
[mul_ring_seminorm_class F α β] : ring_seminorm_class F α β | { map_mul_le_mul := λ f a b, (map_mul _ _ _).le,
..‹mul_ring_seminorm_class F α β› } | instance | mul_ring_seminorm_class.to_ring_seminorm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"map_mul",
"mul_ring_seminorm_class",
"non_assoc_ring",
"ordered_semiring",
"ring_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ring_norm_class.to_ring_norm_class [non_assoc_ring α] [ordered_semiring β]
[mul_ring_norm_class F α β] : ring_norm_class F α β | { ..‹mul_ring_norm_class F α β›, ..mul_ring_seminorm_class.to_ring_seminorm_class } | instance | mul_ring_norm_class.to_ring_norm_class | algebra.order.hom | src/algebra/order/hom/basic.lean | [
"algebra.group_power.order"
] | [
"mul_ring_norm_class",
"mul_ring_seminorm_class.to_ring_seminorm_class",
"non_assoc_ring",
"ordered_semiring",
"ring_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_add_monoid_hom (α β : Type*) [preorder α] [preorder β] [add_zero_class α]
[add_zero_class β] extends α →+ β | (monotone' : monotone to_fun) | structure | order_add_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"add_zero_class",
"monotone"
] | `α →+o β` is the type of monotone functions `α → β` that preserve the `ordered_add_comm_monoid`
structure.
`order_add_monoid_hom` is also used for ordered group homomorphisms.
When possible, instead of parametrizing results over `(f : α →+o β)`,
you should parametrize over `(F : Type*) [order_add_monoid_hom_class F α... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_add_monoid_hom_class (F : Type*) (α β : out_param $ Type*) [preorder α] [preorder β]
[add_zero_class α] [add_zero_class β]
extends add_monoid_hom_class F α β | (monotone (f : F) : monotone f) | class | order_add_monoid_hom_class | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"add_monoid_hom_class",
"add_zero_class",
"monotone"
] | `order_add_monoid_hom_class F α β` states that `F` is a type of ordered monoid homomorphisms.
You should also extend this typeclass when you extend `order_add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monoid_hom (α β : Type*) [preorder α] [preorder β] [mul_one_class α]
[mul_one_class β]
extends α →* β | (monotone' : monotone to_fun) | structure | order_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monotone",
"mul_one_class"
] | `α →*o β` is the type of functions `α → β` that preserve the `ordered_comm_monoid` structure.
`order_monoid_hom` is also used for ordered group homomorphisms.
When possible, instead of parametrizing results over `(f : α →*o β)`,
you should parametrize over `(F : Type*) [order_monoid_hom_class F α β] (f : F)`.
When y... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monoid_hom_class (F : Type*) (α β : out_param $ Type*)
[preorder α] [preorder β] [mul_one_class α] [mul_one_class β]
extends monoid_hom_class F α β | (monotone (f : F) : monotone f) | class | order_monoid_hom_class | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monoid_hom_class",
"monotone",
"mul_one_class"
] | `order_monoid_hom_class F α β` states that `F` is a type of ordered monoid homomorphisms.
You should also extend this typeclass when you extend `order_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monoid_hom_class.to_order_hom_class [order_monoid_hom_class F α β] :
order_hom_class F α β | { map_rel := order_monoid_hom_class.monotone,
.. ‹order_monoid_hom_class F α β› } | instance | order_monoid_hom_class.to_order_hom_class | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_hom_class",
"order_monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_monoid_with_zero_hom (α β : Type*) [preorder α] [preorder β]
[mul_zero_one_class α] [mul_zero_one_class β]
extends α →*₀ β | (monotone' : monotone to_fun) | structure | order_monoid_with_zero_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monotone",
"mul_zero_one_class"
] | `order_monoid_with_zero_hom α β` is the type of functions `α → β` that preserve
the `monoid_with_zero` structure.
`order_monoid_with_zero_hom` is also used for group homomorphisms.
When possible, instead of parametrizing results over `(f : α →+ β)`,
you should parametrize over `(F : Type*) [order_monoid_with_zero_hom... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monoid_with_zero_hom_class (F : Type*) (α β : out_param $ Type*)
[preorder α] [preorder β] [mul_zero_one_class α] [mul_zero_one_class β]
extends monoid_with_zero_hom_class F α β | (monotone (f : F) : monotone f) | class | order_monoid_with_zero_hom_class | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monoid_with_zero_hom_class",
"monotone",
"mul_zero_one_class"
] | `order_monoid_with_zero_hom_class F α β` states that `F` is a type of
ordered monoid with zero homomorphisms.
You should also extend this typeclass when you extend `order_monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monoid_with_zero_hom_class.to_order_monoid_hom_class
[order_monoid_with_zero_hom_class F α β] : order_monoid_hom_class F α β | { .. ‹order_monoid_with_zero_hom_class F α β› } | instance | order_monoid_with_zero_hom_class.to_order_monoid_hom_class | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_hom_class",
"order_monoid_with_zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nonneg (ha : 0 ≤ a) : 0 ≤ f a | by { rw ←map_zero f, exact order_hom_class.mono _ ha } | lemma | map_nonneg | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_hom_class.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nonpos (ha : a ≤ 0) : f a ≤ 0 | by { rw ←map_zero f, exact order_hom_class.mono _ ha } | lemma | map_nonpos | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_hom_class.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_iff_map_nonneg : monotone (f : α → β) ↔ ∀ a, 0 ≤ a → 0 ≤ f a | ⟨λ h a, by { rw ←map_zero f, apply h }, λ h a b hl,
by { rw [←sub_add_cancel b a, map_add f], exact le_add_of_nonneg_left (h _ $ sub_nonneg.2 hl) }⟩ | lemma | monotone_iff_map_nonneg | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_iff_map_nonpos : antitone (f : α → β) ↔ ∀ a, 0 ≤ a → f a ≤ 0 | monotone_to_dual_comp_iff.symm.trans $ monotone_iff_map_nonneg _ | lemma | antitone_iff_map_nonpos | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"antitone",
"monotone_iff_map_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_iff_map_nonpos : monotone (f : α → β) ↔ ∀ a ≤ 0, f a ≤ 0 | antitone_comp_of_dual_iff.symm.trans $ antitone_iff_map_nonpos _ | lemma | monotone_iff_map_nonpos | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"antitone_iff_map_nonpos",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_iff_map_nonneg : antitone (f : α → β) ↔ ∀ a ≤ 0, 0 ≤ f a | monotone_comp_of_dual_iff.symm.trans $ monotone_iff_map_nonneg _ | lemma | antitone_iff_map_nonneg | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"antitone",
"monotone_iff_map_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_iff_map_pos : strict_mono (f : α → β) ↔ ∀ a, 0 < a → 0 < f a | ⟨λ h a, by { rw ←map_zero f, apply h }, λ h a b hl,
by { rw [←sub_add_cancel b a, map_add f], exact lt_add_of_pos_left _ (h _ $ sub_pos.2 hl) }⟩ | lemma | strict_mono_iff_map_pos | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_iff_map_neg : strict_anti (f : α → β) ↔ ∀ a, 0 < a → f a < 0 | strict_mono_to_dual_comp_iff.symm.trans $ strict_mono_iff_map_pos _ | lemma | strict_anti_iff_map_neg | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"strict_anti",
"strict_mono_iff_map_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_iff_map_neg : strict_mono (f : α → β) ↔ ∀ a < 0, f a < 0 | strict_anti_comp_of_dual_iff.symm.trans $ strict_anti_iff_map_neg _ | lemma | strict_mono_iff_map_neg | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"strict_anti_iff_map_neg",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_iff_map_pos : strict_anti (f : α → β) ↔ ∀ a < 0, 0 < f a | strict_mono_comp_of_dual_iff.symm.trans $ strict_mono_iff_map_pos _ | lemma | strict_anti_iff_map_pos | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"strict_anti",
"strict_mono_iff_map_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | order_monoid_hom.ext | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : α →*o β) : f.to_fun = (f : α → β) | rfl | lemma | order_monoid_hom.to_fun_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α →* β) (h) : (order_monoid_hom.mk f h : α → β) = f | rfl | lemma | order_monoid_hom.coe_mk | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : α →*o β) (h) : order_monoid_hom.mk (f : α →* β) h = f | by { ext, refl } | lemma | order_monoid_hom.mk_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_hom (f : α →*o β) : α →o β | { ..f } | def | order_monoid_hom.to_order_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Reinterpret an ordered monoid homomorphism as an order homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_monoid_hom (f : α →*o β) : ((f : α →* β) : α → β) = f | rfl | lemma | order_monoid_hom.coe_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_hom (f : α →*o β) : ((f : α →o β) : α → β) = f | rfl | lemma | order_monoid_hom.coe_order_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom_injective : injective (to_monoid_hom : _ → α →* β) | λ f g h, ext $ by convert fun_like.ext_iff.1 h | lemma | order_monoid_hom.to_monoid_hom_injective | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_hom_injective : injective (to_order_hom : _ → α →o β) | λ f g h, ext $ by convert fun_like.ext_iff.1 h | lemma | order_monoid_hom.to_order_hom_injective | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →*o β) (f' : α → β) (h : f' = f) : α →*o β | { to_fun := f',
monotone' := h.symm.subst f.monotone',
..f.to_monoid_hom.copy f' h } | def | order_monoid_hom.copy | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Copy of an `order_monoid_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →*o β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | order_monoid_hom.coe_copy | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →*o β) (f' : α → β) (h : f' = f) :
f.copy f' h = f | fun_like.ext' h | lemma | order_monoid_hom.copy_eq | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →*o α | { ..monoid_hom.id α, ..order_hom.id } | def | order_monoid_hom.id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monoid_hom.id",
"order_hom.id"
] | The identity map as an ordered monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(order_monoid_hom.id α) = id | rfl | lemma | order_monoid_hom.coe_id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : β →*o γ) (g : α →*o β) : α →*o γ | { ..f.to_monoid_hom.comp (g : α →* β), ..f.to_order_hom.comp (g : α →o β) } | def | order_monoid_hom.comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Composition of `order_monoid_hom`s as an `order_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : β →*o γ) (g : α →*o β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | order_monoid_hom.coe_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : β →*o γ) (g : α →*o β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | order_monoid_hom.comp_apply | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_monoid_hom (f : β →*o γ) (g : α →*o β) :
(f.comp g : α →* γ) = (f : β →* γ).comp g | rfl | lemma | order_monoid_hom.coe_comp_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_order_hom (f : β →*o γ) (g : α →*o β) :
(f.comp g : α →o γ) = (f : β →o γ).comp g | rfl | lemma | order_monoid_hom.coe_comp_order_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : γ →*o δ) (g : β →*o γ) (h : α →*o β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | order_monoid_hom.comp_assoc | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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