fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
unique_units : unique ((associates α)ˣ) :=
{ default := 1, uniq := associates.units_eq_one } | unique_units : unique ((associates α)ˣ) | { default := 1, uniq := associates.units_eq_one } | instance | associates.unique_units | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.units_eq_one",
"unique"
] | null | 698 | 699 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_unit_eq_one (u : (associates α)ˣ): (u : associates α) = 1 :=
by simp | coe_unit_eq_one (u : (associates α)ˣ): (u : associates α) = 1 | by simp | theorem | associates.coe_unit_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates"
] | null | 701 | 702 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff_eq_one (a : associates α) : is_unit a ↔ a = 1 :=
iff.intro
(assume ⟨u, h⟩, h ▸ coe_unit_eq_one _)
(assume h, h.symm ▸ is_unit_one) | is_unit_iff_eq_one (a : associates α) : is_unit a ↔ a = 1 | iff.intro
(assume ⟨u, h⟩, h ▸ coe_unit_eq_one _)
(assume h, h.symm ▸ is_unit_one) | theorem | associates.is_unit_iff_eq_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"is_unit",
"is_unit_one"
] | null | 704 | 707 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff_eq_bot {a : associates α} : is_unit a ↔ a = ⊥ :=
by rw [associates.is_unit_iff_eq_one, bot_eq_one] | is_unit_iff_eq_bot {a : associates α} : is_unit a ↔ a = ⊥ | by rw [associates.is_unit_iff_eq_one, bot_eq_one] | lemma | associates.is_unit_iff_eq_bot | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.is_unit_iff_eq_one",
"bot_eq_one",
"is_unit"
] | null | 709 | 710 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_mk {a : α} : is_unit (associates.mk a) ↔ is_unit a :=
calc is_unit (associates.mk a) ↔ a ~ᵤ 1 :
by rw [is_unit_iff_eq_one, one_eq_mk_one, mk_eq_mk_iff_associated]
... ↔ is_unit a : associated_one_iff_is_unit | is_unit_mk {a : α} : is_unit (associates.mk a) ↔ is_unit a | calc is_unit (associates.mk a) ↔ a ~ᵤ 1 :
by rw [is_unit_iff_eq_one, one_eq_mk_one, mk_eq_mk_iff_associated]
... ↔ is_unit a : associated_one_iff_is_unit | theorem | associates.is_unit_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_one_iff_is_unit",
"associates.mk",
"is_unit"
] | null | 712 | 715 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mono {a b c d : associates α} (h₁ : a ≤ b) (h₂ : c ≤ d) :
a * c ≤ b * d :=
let ⟨x, hx⟩ := h₁, ⟨y, hy⟩ := h₂ in
⟨x * y, by simp [hx, hy, mul_comm, mul_assoc, mul_left_comm]⟩ | mul_mono {a b c d : associates α} (h₁ : a ≤ b) (h₂ : c ≤ d) :
a * c ≤ b * d | let ⟨x, hx⟩ := h₁, ⟨y, hy⟩ := h₂ in
⟨x * y, by simp [hx, hy, mul_comm, mul_assoc, mul_left_comm]⟩ | theorem | associates.mul_mono | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | null | 719 | 722 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le {a : associates α} : 1 ≤ a :=
dvd.intro _ (one_mul a) | one_le {a : associates α} : 1 ≤ a | dvd.intro _ (one_mul a) | theorem | associates.one_le | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd.intro",
"one_le",
"one_mul"
] | null | 724 | 725 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_right {a b : associates α} : a ≤ a * b := ⟨b, rfl⟩ | le_mul_right {a b : associates α} : a ≤ a * b | ⟨b, rfl⟩ | theorem | associates.le_mul_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_mul_right"
] | null | 727 | 727 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_left {a b : associates α} : a ≤ b * a :=
by rw [mul_comm]; exact le_mul_right | le_mul_left {a b : associates α} : a ≤ b * a | by rw [mul_comm]; exact le_mul_right | theorem | associates.le_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_mul_left",
"le_mul_right",
"mul_comm"
] | null | 729 | 730 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: order_bot (associates α) :=
{ bot := 1,
bot_le := assume a, one_le } | : order_bot (associates α) | { bot := 1,
bot_le := assume a, one_le } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"bot_le",
"one_le",
"order_bot"
] | null | 732 | 734 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_mk_le_mk {a b : α} : associates.mk a ≤ associates.mk b → a ∣ b
| ⟨c', hc'⟩ := (quotient.induction_on c' $ assume c hc,
let ⟨d, hd⟩ := (quotient.exact hc).symm in
⟨(↑d) * c,
calc b = (a * c) * ↑d : hd.symm
... = a * (↑d * c) : by ac_refl⟩) hc' | dvd_of_mk_le_mk {a b : α} : associates.mk a ≤ associates.mk b → a ∣ b
| ⟨c', hc'⟩ | (quotient.induction_on c' $ assume c hc,
let ⟨d, hd⟩ := (quotient.exact hc).symm in
⟨(↑d) * c,
calc b = (a * c) * ↑d : hd.symm
... = a * (↑d * c) : by ac_refl⟩) hc' | theorem | associates.dvd_of_mk_le_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 738 | 743 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_le_mk_of_dvd {a b : α} : a ∣ b → associates.mk a ≤ associates.mk b :=
assume ⟨c, hc⟩, ⟨associates.mk c, by simp [hc]; refl⟩ | mk_le_mk_of_dvd {a b : α} : a ∣ b → associates.mk a ≤ associates.mk b | assume ⟨c, hc⟩, ⟨associates.mk c, by simp [hc]; refl⟩ | theorem | associates.mk_le_mk_of_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 745 | 746 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_le_mk_iff_dvd_iff {a b : α} : associates.mk a ≤ associates.mk b ↔ a ∣ b :=
iff.intro dvd_of_mk_le_mk mk_le_mk_of_dvd | mk_le_mk_iff_dvd_iff {a b : α} : associates.mk a ≤ associates.mk b ↔ a ∣ b | iff.intro dvd_of_mk_le_mk mk_le_mk_of_dvd | theorem | associates.mk_le_mk_iff_dvd_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 748 | 749 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_dvd_mk {a b : α} : associates.mk a ∣ associates.mk b ↔ a ∣ b :=
iff.intro dvd_of_mk_le_mk mk_le_mk_of_dvd | mk_dvd_mk {a b : α} : associates.mk a ∣ associates.mk b ↔ a ∣ b | iff.intro dvd_of_mk_le_mk mk_le_mk_of_dvd | theorem | associates.mk_dvd_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 751 | 752 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[has_zero α] [monoid α] : has_zero (associates α) := ⟨⟦ 0 ⟧⟩ | [has_zero α] [monoid α] : has_zero (associates α) | ⟨⟦ 0 ⟧⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"monoid"
] | null | 756 | 756 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_zero α] [monoid α] : has_top (associates α) := ⟨0⟩ | [has_zero α] [monoid α] : has_top (associates α) | ⟨0⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"has_top",
"monoid"
] | null | 757 | 757 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero {a : α} : associates.mk a = 0 ↔ a = 0 :=
⟨assume h, (associated_zero_iff_eq_zero a).1 $ quotient.exact h, assume h, h.symm ▸ rfl⟩ | mk_eq_zero {a : α} : associates.mk a = 0 ↔ a = 0 | ⟨assume h, (associated_zero_iff_eq_zero a).1 $ quotient.exact h, assume h, h.symm ▸ rfl⟩ | theorem | associates.mk_eq_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_zero_iff_eq_zero",
"associates.mk"
] | null | 762 | 763 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_ne_zero {a : α} : associates.mk a ≠ 0 ↔ a ≠ 0 :=
not_congr mk_eq_zero | mk_ne_zero {a : α} : associates.mk a ≠ 0 ↔ a ≠ 0 | not_congr mk_eq_zero | theorem | associates.mk_ne_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | null | 765 | 766 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[nontrivial α] : nontrivial (associates α) :=
⟨⟨0, 1,
assume h,
have (0 : α) ~ᵤ 1, from quotient.exact h,
have (0 : α) = 1, from ((associated_zero_iff_eq_zero 1).1 this.symm).symm,
zero_ne_one this⟩⟩ | [nontrivial α] : nontrivial (associates α) | ⟨⟨0, 1,
assume h,
have (0 : α) ~ᵤ 1, from quotient.exact h,
have (0 : α) = 1, from ((associated_zero_iff_eq_zero 1).1 this.symm).symm,
zero_ne_one this⟩⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_zero_iff_eq_zero",
"associates",
"nontrivial",
"zero_ne_one"
] | null | 768 | 773 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_non_zero_rep {a : associates α} : a ≠ 0 → ∃ a0 : α, a0 ≠ 0 ∧ associates.mk a0 = a :=
quotient.induction_on a (λ b nz, ⟨b, mt (congr_arg quotient.mk) nz, rfl⟩) | exists_non_zero_rep {a : associates α} : a ≠ 0 → ∃ a0 : α, a0 ≠ 0 ∧ associates.mk a0 = a | quotient.induction_on a (λ b nz, ⟨b, mt (congr_arg quotient.mk) nz, rfl⟩) | lemma | associates.exists_non_zero_rep | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk"
] | null | 775 | 776 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: comm_monoid_with_zero (associates α) :=
{ zero_mul := by { rintro ⟨a⟩, show associates.mk (0 * a) = associates.mk 0, rw [zero_mul] },
mul_zero := by { rintro ⟨a⟩, show associates.mk (a * 0) = associates.mk 0, rw [mul_zero] },
.. associates.comm_monoid, .. associates.has_zero } | : comm_monoid_with_zero (associates α) | { zero_mul := by { rintro ⟨a⟩, show associates.mk (0 * a) = associates.mk 0, rw [zero_mul] },
mul_zero := by { rintro ⟨a⟩, show associates.mk (a * 0) = associates.mk 0, rw [mul_zero] },
.. associates.comm_monoid, .. associates.has_zero } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk",
"comm_monoid_with_zero",
"mul_zero",
"zero_mul"
] | null | 784 | 787 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: order_top (associates α) :=
{ top := 0,
le_top := assume a, ⟨0, (mul_zero a).symm⟩ } | : order_top (associates α) | { top := 0,
le_top := assume a, ⟨0, (mul_zero a).symm⟩ } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_top",
"mul_zero",
"order_top"
] | null | 789 | 791 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: bounded_order (associates α) :=
{ .. associates.order_top,
.. associates.order_bot } | : bounded_order (associates α) | { .. associates.order_top,
.. associates.order_bot } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"bounded_order"
] | null | 793 | 795 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[decidable_rel ((∣) : α → α → Prop)] :
decidable_rel ((∣) : associates α → associates α → Prop) :=
λ a b, quotient.rec_on_subsingleton₂ a b (λ a b, decidable_of_iff' _ mk_dvd_mk) | [decidable_rel ((∣) : α → α → Prop)] :
decidable_rel ((∣) : associates α → associates α → Prop) | λ a b, quotient.rec_on_subsingleton₂ a b (λ a b, decidable_of_iff' _ mk_dvd_mk) | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"decidable_of_iff'"
] | null | 797 | 799 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.le_or_le {p : associates α} (hp : prime p) {a b : associates α} (h : p ≤ a * b) :
p ≤ a ∨ p ≤ b :=
hp.2.2 a b h | prime.le_or_le {p : associates α} (hp : prime p) {a b : associates α} (h : p ≤ a * b) :
p ≤ a ∨ p ≤ b | hp.2.2 a b h | lemma | associates.prime.le_or_le | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"prime"
] | null | 801 | 803 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_mk (p : α) : prime (associates.mk p) ↔ _root_.prime p :=
begin
rw [prime, _root_.prime, forall_associated],
transitivity,
{ apply and_congr, refl,
apply and_congr, refl,
apply forall_congr, assume a,
exact forall_associated },
apply and_congr mk_ne_zero,
apply and_congr,
{ rw [is_unit_mk],... | prime_mk (p : α) : prime (associates.mk p) ↔ _root_.prime p | begin
rw [prime, _root_.prime, forall_associated],
transitivity,
{ apply and_congr, refl,
apply and_congr, refl,
apply forall_congr, assume a,
exact forall_associated },
apply and_congr mk_ne_zero,
apply and_congr,
{ rw [is_unit_mk], },
refine forall₂_congr (λ a b, _),
rw [mk_mul_mk, mk_dvd_... | lemma | associates.prime_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"forall₂_congr",
"prime"
] | null | 805 | 818 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_mk (a : α) : irreducible (associates.mk a) ↔ irreducible a :=
begin
simp only [irreducible_iff, is_unit_mk],
apply and_congr iff.rfl,
split,
{ rintro h x y rfl,
simpa [is_unit_mk] using h (associates.mk x) (associates.mk y) rfl },
{ intros h x y,
refine quotient.induction_on₂ x y (assume x... | irreducible_mk (a : α) : irreducible (associates.mk a) ↔ irreducible a | begin
simp only [irreducible_iff, is_unit_mk],
apply and_congr iff.rfl,
split,
{ rintro h x y rfl,
simpa [is_unit_mk] using h (associates.mk x) (associates.mk y) rfl },
{ intros h x y,
refine quotient.induction_on₂ x y (assume x y a_eq, _),
rcases quotient.exact a_eq.symm with ⟨u, a_eq⟩,
rw mu... | theorem | associates.irreducible_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"irreducible",
"irreducible_iff",
"is_unit",
"mul_assoc"
] | null | 820 | 833 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_dvd_not_unit_mk_iff {a b : α} :
dvd_not_unit (associates.mk a) (associates.mk b) ↔
dvd_not_unit a b :=
begin
rw [dvd_not_unit, dvd_not_unit, mk_ne_zero],
apply and_congr_right, intro ane0,
split,
{ contrapose!, rw forall_associated,
intros h x hx hbax,
rw [mk_mul_mk, mk_eq_mk_iff_associated] at h... | mk_dvd_not_unit_mk_iff {a b : α} :
dvd_not_unit (associates.mk a) (associates.mk b) ↔
dvd_not_unit a b | begin
rw [dvd_not_unit, dvd_not_unit, mk_ne_zero],
apply and_congr_right, intro ane0,
split,
{ contrapose!, rw forall_associated,
intros h x hx hbax,
rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax,
cases hbax with u hu,
apply h (x * ↑u⁻¹),
{ rw is_unit_mk at hx,
rw associated.is_unit_... | theorem | associates.mk_dvd_not_unit_mk_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.is_unit_iff",
"associates.mk",
"dvd_not_unit",
"mul_assoc"
] | null | 835 | 856 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit_of_lt {a b : associates α} (hlt : a < b) :
dvd_not_unit a b :=
begin
split, { rintro rfl, apply not_lt_of_le _ hlt, apply dvd_zero },
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩,
refine ⟨x, _, rfl⟩,
contrapose! ndvd,
rcases ndvd with ⟨u, rfl⟩,
simp,
end | dvd_not_unit_of_lt {a b : associates α} (hlt : a < b) :
dvd_not_unit a b | begin
split, { rintro rfl, apply not_lt_of_le _ hlt, apply dvd_zero },
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩,
refine ⟨x, _, rfl⟩,
contrapose! ndvd,
rcases ndvd with ⟨u, rfl⟩,
simp,
end | theorem | associates.dvd_not_unit_of_lt | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd_not_unit",
"dvd_zero",
"not_lt_of_le"
] | null | 858 | 867 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_iff_prime_iff :
(∀ a : α, irreducible a ↔ prime a) ↔ (∀ a : (associates α), irreducible a ↔ prime a) :=
by simp_rw [forall_associated, irreducible_mk, prime_mk] | irreducible_iff_prime_iff :
(∀ a : α, irreducible a ↔ prime a) ↔ (∀ a : (associates α), irreducible a ↔ prime a) | by simp_rw [forall_associated, irreducible_mk, prime_mk] | theorem | associates.irreducible_iff_prime_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"irreducible",
"prime"
] | null | 869 | 871 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: partial_order (associates α) :=
{ le_antisymm := λ a' b', quotient.induction_on₂ a' b' (λ a b hab hba,
quot.sound $ associated_of_dvd_dvd (dvd_of_mk_le_mk hab) (dvd_of_mk_le_mk hba))
.. associates.preorder } | : partial_order (associates α) | { le_antisymm := λ a' b', quotient.induction_on₂ a' b' (λ a b hab hba,
quot.sound $ associated_of_dvd_dvd (dvd_of_mk_le_mk hab) (dvd_of_mk_le_mk hba))
.. associates.preorder } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_of_dvd_dvd",
"associates"
] | null | 878 | 881 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: ordered_comm_monoid (associates α) :=
{ mul_le_mul_left := λ a b ⟨d, hd⟩ c, hd.symm ▸ mul_assoc c a d ▸ le_mul_right,
..associates.comm_monoid,
..associates.partial_order} | : ordered_comm_monoid (associates α) | { mul_le_mul_left := λ a b ⟨d, hd⟩ c, hd.symm ▸ mul_assoc c a d ▸ le_mul_right,
..associates.comm_monoid,
..associates.partial_order} | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_mul_right",
"mul_assoc",
"mul_le_mul_left",
"ordered_comm_monoid"
] | null | 883 | 886 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: no_zero_divisors (associates α) :=
⟨λ x y,
(quotient.induction_on₂ x y $ assume a b h,
have a * b = 0, from (associated_zero_iff_eq_zero _).1 (quotient.exact h),
have a = 0 ∨ b = 0, from mul_eq_zero.1 this,
this.imp (assume h, h.symm ▸ rfl) (assume h, h.symm ▸ rfl))⟩ | : no_zero_divisors (associates α) | ⟨λ x y,
(quotient.induction_on₂ x y $ assume a b h,
have a * b = 0, from (associated_zero_iff_eq_zero _).1 (quotient.exact h),
have a = 0 ∨ b = 0, from mul_eq_zero.1 this,
this.imp (assume h, h.symm ▸ rfl) (assume h, h.symm ▸ rfl))⟩ | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_zero_iff_eq_zero",
"associates",
"no_zero_divisors"
] | null | 888 | 893 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: cancel_comm_monoid_with_zero (associates α) :=
{ mul_left_cancel_of_ne_zero :=
begin
rintros ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h,
rcases quotient.exact' h with ⟨u, hu⟩,
rw [mul_assoc] at hu,
exact quotient.sound' ⟨u, mul_left_cancel₀ (mk_ne_zero.1 ha) hu⟩
end,
.. (infer_instance : comm_monoid_with_zero... | : cancel_comm_monoid_with_zero (associates α) | { mul_left_cancel_of_ne_zero :=
begin
rintros ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h,
rcases quotient.exact' h with ⟨u, hu⟩,
rw [mul_assoc] at hu,
exact quotient.sound' ⟨u, mul_left_cancel₀ (mk_ne_zero.1 ha) hu⟩
end,
.. (infer_instance : comm_monoid_with_zero (associates α)) } | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"cancel_comm_monoid_with_zero",
"comm_monoid_with_zero",
"mul_assoc",
"mul_left_cancel₀",
"quotient.exact'",
"quotient.sound'"
] | null | 895 | 903 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_left (a b c : associates α) (ha : a ≠ 0) :
a * b ≤ a * c → b ≤ c
| ⟨d, hd⟩ := ⟨d, mul_left_cancel₀ ha $ by rwa ← mul_assoc⟩ | le_of_mul_le_mul_left (a b c : associates α) (ha : a ≠ 0) :
a * b ≤ a * c → b ≤ c
| ⟨d, hd⟩ | ⟨d, mul_left_cancel₀ ha $ by rwa ← mul_assoc⟩ | lemma | associates.le_of_mul_le_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_of_mul_le_mul_left",
"mul_left_cancel₀"
] | null | 905 | 907 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_or_eq_of_le_of_prime :
∀(p m : associates α), prime p → m ≤ p → (m = 1 ∨ m = p)
| _ m ⟨hp0, hp1, h⟩ ⟨d, rfl⟩ :=
match h m d dvd_rfl with
| or.inl h := classical.by_cases (assume : m = 0, by simp [this]) $
assume : m ≠ 0,
have m * d ≤ m * 1, by simpa using h,
have d ≤ 1, from associates.le_of_mul_le_mul_left... | one_or_eq_of_le_of_prime :
∀(p m : associates α), prime p → m ≤ p → (m = 1 ∨ m = p)
| _ m ⟨hp0, hp1, h⟩ ⟨d, rfl⟩ | match h m d dvd_rfl with
| or.inl h := classical.by_cases (assume : m = 0, by simp [this]) $
assume : m ≠ 0,
have m * d ≤ m * 1, by simpa using h,
have d ≤ 1, from associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this,
have d = 1, from bot_unique this,
by simp [this]
| or.inr h := classical.by_cases (assume : d... | lemma | associates.one_or_eq_of_le_of_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.le_of_mul_le_mul_left",
"bot_unique",
"dvd_rfl",
"mul_comm",
"prime"
] | null | 909 | 923 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: canonically_ordered_monoid (associates α) :=
{ exists_mul_of_le := λ a b, id,
le_self_mul := λ a b, ⟨b, rfl⟩,
..associates.cancel_comm_monoid_with_zero,
..associates.bounded_order,
..associates.ordered_comm_monoid} | : canonically_ordered_monoid (associates α) | { exists_mul_of_le := λ a b, id,
le_self_mul := λ a b, ⟨b, rfl⟩,
..associates.cancel_comm_monoid_with_zero,
..associates.bounded_order,
..associates.ordered_comm_monoid} | instance | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"canonically_ordered_monoid",
"le_self_mul"
] | null | 925 | 930 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit_iff_lt {a b : associates α} :
dvd_not_unit a b ↔ a < b :=
dvd_and_not_dvd_iff.symm | dvd_not_unit_iff_lt {a b : associates α} :
dvd_not_unit a b ↔ a < b | dvd_and_not_dvd_iff.symm | theorem | associates.dvd_not_unit_iff_lt | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd_not_unit"
] | null | 932 | 934 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_one_iff {p : associates α} : p ≤ 1 ↔ p = 1 :=
by rw [← associates.bot_eq_one, le_bot_iff] | le_one_iff {p : associates α} : p ≤ 1 ↔ p = 1 | by rw [← associates.bot_eq_one, le_bot_iff] | lemma | associates.le_one_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.bot_eq_one",
"le_bot_iff"
] | null | 936 | 937 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit.is_unit_of_irreducible_right [comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) (hq : irreducible q) : is_unit p :=
begin
obtain ⟨hp', x, hx, hx'⟩ := h,
exact or.resolve_right ((irreducible_iff.1 hq).right p x hx') hx
end | dvd_not_unit.is_unit_of_irreducible_right [comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) (hq : irreducible q) : is_unit p | begin
obtain ⟨hp', x, hx, hx'⟩ := h,
exact or.resolve_right ((irreducible_iff.1 hq).right p x hx') hx
end | lemma | dvd_not_unit.is_unit_of_irreducible_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"irreducible",
"is_unit"
] | null | 945 | 950 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_irreducible_of_not_unit_dvd_not_unit [comm_monoid_with_zero α] {p q : α}
(hp : ¬is_unit p) (h : dvd_not_unit p q) : ¬ irreducible q :=
mt h.is_unit_of_irreducible_right hp | not_irreducible_of_not_unit_dvd_not_unit [comm_monoid_with_zero α] {p q : α}
(hp : ¬is_unit p) (h : dvd_not_unit p q) : ¬ irreducible q | mt h.is_unit_of_irreducible_right hp | lemma | not_irreducible_of_not_unit_dvd_not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"irreducible",
"is_unit"
] | null | 952 | 954 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit.not_unit [comm_monoid_with_zero α] {p q : α}
(hp : dvd_not_unit p q) : ¬ is_unit q :=
begin
obtain ⟨-, x, hx, rfl⟩ := hp,
exact λ hc, hx (is_unit_iff_dvd_one.mpr (dvd_of_mul_left_dvd (is_unit_iff_dvd_one.mp hc))),
end | dvd_not_unit.not_unit [comm_monoid_with_zero α] {p q : α}
(hp : dvd_not_unit p q) : ¬ is_unit q | begin
obtain ⟨-, x, hx, rfl⟩ := hp,
exact λ hc, hx (is_unit_iff_dvd_one.mpr (dvd_of_mul_left_dvd (is_unit_iff_dvd_one.mp hc))),
end | lemma | dvd_not_unit.not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"dvd_of_mul_left_dvd",
"is_unit"
] | null | 956 | 961 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit_of_dvd_not_unit_associated [comm_monoid_with_zero α]
[nontrivial α] {p q r : α} (h : dvd_not_unit p q) (h' : associated q r) : dvd_not_unit p r :=
begin
obtain ⟨u, rfl⟩ := associated.symm h',
obtain ⟨hp, x, hx⟩ := h,
refine ⟨hp, x * ↑(u⁻¹), dvd_not_unit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _... | dvd_not_unit_of_dvd_not_unit_associated [comm_monoid_with_zero α]
[nontrivial α] {p q r : α} (h : dvd_not_unit p q) (h' : associated q r) : dvd_not_unit p r | begin
obtain ⟨u, rfl⟩ := associated.symm h',
obtain ⟨hp, x, hx⟩ := h,
refine ⟨hp, x * ↑(u⁻¹), dvd_not_unit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _ _⟩, _⟩,
rw [← mul_assoc, ← hx.right, mul_assoc, units.mul_inv, mul_one]
end | lemma | dvd_not_unit_of_dvd_not_unit_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated.symm",
"comm_monoid_with_zero",
"dvd_not_unit",
"dvd_not_unit.not_unit",
"mul_assoc",
"mul_comm",
"mul_one",
"ne_zero",
"nontrivial",
"units.mul_inv"
] | null | 963 | 970 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_of_associated_mul [cancel_comm_monoid_with_zero α]
{p b : α} (h : associated (p * b) p) (hp : p ≠ 0) : is_unit b :=
begin
cases h with a ha,
refine is_unit_of_mul_eq_one b a ((mul_right_inj' hp).mp _),
rwa [← mul_assoc, mul_one],
end | is_unit_of_associated_mul [cancel_comm_monoid_with_zero α]
{p b : α} (h : associated (p * b) p) (hp : p ≠ 0) : is_unit b | begin
cases h with a ha,
refine is_unit_of_mul_eq_one b a ((mul_right_inj' hp).mp _),
rwa [← mul_assoc, mul_one],
end | lemma | is_unit_of_associated_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"is_unit",
"is_unit_of_mul_eq_one",
"mul_assoc",
"mul_one",
"mul_right_inj'"
] | null | 976 | 982 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit.not_associated [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : ¬ associated p q :=
begin
rintro ⟨a, rfl⟩,
obtain ⟨hp, x, hx, hx'⟩ := h,
rcases (mul_right_inj' hp).mp hx' with rfl,
exact hx a.is_unit,
end | dvd_not_unit.not_associated [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : ¬ associated p q | begin
rintro ⟨a, rfl⟩,
obtain ⟨hp, x, hx, hx'⟩ := h,
rcases (mul_right_inj' hp).mp hx' with rfl,
exact hx a.is_unit,
end | lemma | dvd_not_unit.not_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"dvd_not_unit",
"mul_right_inj'"
] | null | 984 | 991 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit.ne [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : p ≠ q :=
begin
by_contra hcontra,
obtain ⟨hp, x, hx', hx''⟩ := h,
conv_lhs at hx'' {rw [← hcontra, ← mul_one p]},
rw (mul_left_cancel₀ hp hx'').symm at hx',
exact hx' is_unit_one,
end | dvd_not_unit.ne [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : p ≠ q | begin
by_contra hcontra,
obtain ⟨hp, x, hx', hx''⟩ := h,
conv_lhs at hx'' {rw [← hcontra, ← mul_one p]},
rw (mul_left_cancel₀ hp hx'').symm at hx',
exact hx' is_unit_one,
end | lemma | dvd_not_unit.ne | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"by_contra",
"cancel_comm_monoid_with_zero",
"dvd_not_unit",
"is_unit_one",
"mul_left_cancel₀",
"mul_one"
] | null | 993 | 1,001 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_injective_of_not_unit [cancel_comm_monoid_with_zero α] {q : α}
(hq : ¬ is_unit q) (hq' : q ≠ 0): function.injective (λ (n : ℕ), q^n) :=
begin
refine injective_of_lt_imp_ne (λ n m h, dvd_not_unit.ne ⟨pow_ne_zero n hq', q^(m - n), _, _⟩),
{ exact not_is_unit_of_not_is_unit_dvd hq (dvd_pow (dvd_refl _) (nat.sub_... | pow_injective_of_not_unit [cancel_comm_monoid_with_zero α] {q : α}
(hq : ¬ is_unit q) (hq' : q ≠ 0): function.injective (λ (n : ℕ), q^n) | begin
refine injective_of_lt_imp_ne (λ n m h, dvd_not_unit.ne ⟨pow_ne_zero n hq', q^(m - n), _, _⟩),
{ exact not_is_unit_of_not_is_unit_dvd hq (dvd_pow (dvd_refl _) (nat.sub_pos_of_lt h).ne') },
{ exact (pow_mul_pow_sub q h.le).symm }
end | lemma | pow_injective_of_not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_not_unit.ne",
"dvd_pow",
"dvd_refl",
"injective_of_lt_imp_ne",
"is_unit",
"not_is_unit_of_not_is_unit_dvd",
"pow_mul_pow_sub"
] | null | 1,003 | 1,009 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_prime_pow [cancel_comm_monoid_with_zero α] {p q : α} (hp : prime p) (n : ℕ) :
q ∣ p^n ↔ ∃ i ≤ n, associated q (p ^ i) :=
begin
induction n with n ih generalizing q,
{ simp [← is_unit_iff_dvd_one, associated_one_iff_is_unit] },
refine ⟨λ h, _, λ ⟨i, hi, hq⟩, hq.dvd.trans (pow_dvd_pow p hi)⟩,
rw pow_succ at... | dvd_prime_pow [cancel_comm_monoid_with_zero α] {p q : α} (hp : prime p) (n : ℕ) :
q ∣ p^n ↔ ∃ i ≤ n, associated q (p ^ i) | begin
induction n with n ih generalizing q,
{ simp [← is_unit_iff_dvd_one, associated_one_iff_is_unit] },
refine ⟨λ h, _, λ ⟨i, hi, hq⟩, hq.dvd.trans (pow_dvd_pow p hi)⟩,
rw pow_succ at h,
rcases hp.left_dvd_or_dvd_right_of_dvd_mul h with (⟨q, rfl⟩ | hno),
{ rw [mul_dvd_mul_iff_left hp.ne_zero, ih] at h,
... | lemma | dvd_prime_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_one_iff_is_unit",
"cancel_comm_monoid_with_zero",
"ih",
"is_unit_iff_dvd_one",
"mul_dvd_mul_iff_left",
"pow_dvd_pow",
"pow_succ",
"prime"
] | null | 1,011 | 1,025 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above_inv : bdd_above s⁻¹ ↔ bdd_below s := (order_iso.inv G).bdd_above_preimage | bdd_above_inv : bdd_above s⁻¹ ↔ bdd_below s | (order_iso.inv G).bdd_above_preimage | lemma | bdd_above_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below",
"order_iso.inv"
] | null | 30 | 31 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_below_inv : bdd_below s⁻¹ ↔ bdd_above s := (order_iso.inv G).bdd_below_preimage | bdd_below_inv : bdd_below s⁻¹ ↔ bdd_above s | (order_iso.inv G).bdd_below_preimage | lemma | bdd_below_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below",
"order_iso.inv"
] | null | 33 | 34 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above.inv (h : bdd_above s) : bdd_below s⁻¹ := bdd_below_inv.2 h | bdd_above.inv (h : bdd_above s) : bdd_below s⁻¹ | bdd_below_inv.2 h | lemma | bdd_above.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below"
] | null | 36 | 37 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_below.inv (h : bdd_below s) : bdd_above s⁻¹ := bdd_above_inv.2 h | bdd_below.inv (h : bdd_below s) : bdd_above s⁻¹ | bdd_above_inv.2 h | lemma | bdd_below.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below"
] | null | 39 | 40 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lub_inv : is_lub s⁻¹ a ↔ is_glb s a⁻¹ := (order_iso.inv G).is_lub_preimage | is_lub_inv : is_lub s⁻¹ a ↔ is_glb s a⁻¹ | (order_iso.inv G).is_lub_preimage | lemma | is_lub_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | null | 42 | 43 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lub_inv' : is_lub s⁻¹ a⁻¹ ↔ is_glb s a := (order_iso.inv G).is_lub_preimage' | is_lub_inv' : is_lub s⁻¹ a⁻¹ ↔ is_glb s a | (order_iso.inv G).is_lub_preimage' | lemma | is_lub_inv' | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | null | 45 | 46 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_glb.inv (h : is_glb s a) : is_lub s⁻¹ a⁻¹ := is_lub_inv'.2 h | is_glb.inv (h : is_glb s a) : is_lub s⁻¹ a⁻¹ | is_lub_inv'.2 h | lemma | is_glb.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub"
] | null | 48 | 49 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_glb_inv : is_glb s⁻¹ a ↔ is_lub s a⁻¹ := (order_iso.inv G).is_glb_preimage | is_glb_inv : is_glb s⁻¹ a ↔ is_lub s a⁻¹ | (order_iso.inv G).is_glb_preimage | lemma | is_glb_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | null | 51 | 52 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_glb_inv' : is_glb s⁻¹ a⁻¹ ↔ is_lub s a := (order_iso.inv G).is_glb_preimage' | is_glb_inv' : is_glb s⁻¹ a⁻¹ ↔ is_lub s a | (order_iso.inv G).is_glb_preimage' | lemma | is_glb_inv' | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | null | 54 | 55 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lub.inv (h : is_lub s a) : is_glb s⁻¹ a⁻¹ := is_glb_inv'.2 h | is_lub.inv (h : is_lub s a) : is_glb s⁻¹ a⁻¹ | is_glb_inv'.2 h | lemma | is_lub.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub"
] | null | 57 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mem_upper_bounds_mul {s t : set M} {a b : M} (ha : a ∈ upper_bounds s)
(hb : b ∈ upper_bounds t) :
a * b ∈ upper_bounds (s * t) :=
forall_image2_iff.2 $ λ x hx y hy, mul_le_mul' (ha hx) (hb hy) | mul_mem_upper_bounds_mul {s t : set M} {a b : M} (ha : a ∈ upper_bounds s)
(hb : b ∈ upper_bounds t) :
a * b ∈ upper_bounds (s * t) | forall_image2_iff.2 $ λ x hx y hy, mul_le_mul' (ha hx) (hb hy) | lemma | mul_mem_upper_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"mul_le_mul'",
"upper_bounds"
] | null | 67 | 70 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_upper_bounds_mul (s t : set M) :
upper_bounds s * upper_bounds t ⊆ upper_bounds (s * t) :=
image2_subset_iff.2 $ λ x hx y hy, mul_mem_upper_bounds_mul hx hy | subset_upper_bounds_mul (s t : set M) :
upper_bounds s * upper_bounds t ⊆ upper_bounds (s * t) | image2_subset_iff.2 $ λ x hx y hy, mul_mem_upper_bounds_mul hx hy | lemma | subset_upper_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"mul_mem_upper_bounds_mul",
"upper_bounds"
] | null | 72 | 74 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mem_lower_bounds_mul {s t : set M} {a b : M} (ha : a ∈ lower_bounds s)
(hb : b ∈ lower_bounds t) : a * b ∈ lower_bounds (s * t) :=
@mul_mem_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ _ _ ha hb | mul_mem_lower_bounds_mul {s t : set M} {a b : M} (ha : a ∈ lower_bounds s)
(hb : b ∈ lower_bounds t) : a * b ∈ lower_bounds (s * t) | @mul_mem_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ _ _ ha hb | lemma | mul_mem_lower_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"lower_bounds",
"mul_mem_upper_bounds_mul"
] | null | 76 | 78 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_lower_bounds_mul (s t : set M) :
lower_bounds s * lower_bounds t ⊆ lower_bounds (s * t) :=
@subset_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ | subset_lower_bounds_mul (s t : set M) :
lower_bounds s * lower_bounds t ⊆ lower_bounds (s * t) | @subset_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ | lemma | subset_lower_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"lower_bounds",
"subset_upper_bounds_mul"
] | null | 80 | 82 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above.mul {s t : set M} (hs : bdd_above s) (ht : bdd_above t) :
bdd_above (s * t) :=
(hs.mul ht).mono (subset_upper_bounds_mul s t) | bdd_above.mul {s t : set M} (hs : bdd_above s) (ht : bdd_above t) :
bdd_above (s * t) | (hs.mul ht).mono (subset_upper_bounds_mul s t) | lemma | bdd_above.mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"subset_upper_bounds_mul"
] | null | 84 | 86 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_below.mul {s t : set M} (hs : bdd_below s) (ht : bdd_below t) :
bdd_below (s * t) :=
(hs.mul ht).mono (subset_lower_bounds_mul s t) | bdd_below.mul {s t : set M} (hs : bdd_below s) (ht : bdd_below t) :
bdd_below (s * t) | (hs.mul ht).mono (subset_lower_bounds_mul s t) | lemma | bdd_below.mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_below",
"subset_lower_bounds_mul"
] | null | 88 | 90 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
csupr_mul (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) * a = ⨆ i, f i * a :=
(order_iso.mul_right a).map_csupr hf | csupr_mul (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) * a = ⨆ i, f i * a | (order_iso.mul_right a).map_csupr hf | lemma | csupr_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"order_iso.mul_right",
"set.range"
] | null | 101 | 103 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
csupr_div (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) / a = ⨆ i, f i / a :=
by simp only [div_eq_mul_inv, csupr_mul hf] | csupr_div (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) / a = ⨆ i, f i / a | by simp only [div_eq_mul_inv, csupr_mul hf] | lemma | csupr_div | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"csupr_mul",
"div_eq_mul_inv",
"set.range"
] | null | 105 | 107 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_csupr (hf : bdd_above (set.range f)) (a : G) :
a * (⨆ i, f i) = ⨆ i, a * f i :=
(order_iso.mul_left a).map_csupr hf | mul_csupr (hf : bdd_above (set.range f)) (a : G) :
a * (⨆ i, f i) = ⨆ i, a * f i | (order_iso.mul_left a).map_csupr hf | lemma | mul_csupr | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"order_iso.mul_left",
"set.range"
] | null | 116 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant : Prop := ∀ (m) {n₁ n₂}, r n₁ n₂ → r (μ m n₁) (μ m n₂) | covariant : Prop | ∀ (m) {n₁ n₂}, r n₁ n₂ → r (μ m n₁) (μ m n₂) | def | covariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | `covariant` is useful to formulate succintly statements about the interactions between an
action of a Type on another one and a relation on the acted-upon Type.
See the `covariant_class` doc-string for its meaning. | 77 | 77 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant : Prop := ∀ (m) {n₁ n₂}, r (μ m n₁) (μ m n₂) → r n₁ n₂ | contravariant : Prop | ∀ (m) {n₁ n₂}, r (μ m n₁) (μ m n₂) → r n₁ n₂ | def | contravariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | `contravariant` is useful to formulate succintly statements about the interactions between an
action of a Type on another one and a relation on the acted-upon Type.
See the `contravariant_class` doc-string for its meaning. | 83 | 83 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_class : Prop :=
(elim : covariant M N μ r) | covariant_class : Prop | (elim : covariant M N μ r) | class | covariant_class | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | Given an action `μ` of a Type `M` on a Type `N` and a relation `r` on `N`, informally, the
`covariant_class` says that "the action `μ` preserves the relation `r`."
More precisely, the `covariant_class` is a class taking two Types `M N`, together with an "action"
`μ : M → N → N` and a relation `r : N → N → Prop`. Its ... | 96 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_class : Prop :=
(elim : contravariant M N μ r) | contravariant_class : Prop | (elim : contravariant M N μ r) | class | contravariant_class | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | Given an action `μ` of a Type `M` on a Type `N` and a relation `r` on `N`, informally, the
`contravariant_class` says that "if the result of the action `μ` on a pair satisfies the
relation `r`, then the initial pair satisfied the relation `r`."
More precisely, the `contravariant_class` is a class taking two Types `M N... | 111 | 112 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iff_cov [covariant_class M N μ r] [contravariant_class M N μ r] (m : M) {a b : N} :
r (μ m a) (μ m b) ↔ r a b :=
⟨contravariant_class.elim _, covariant_class.elim _⟩ | rel_iff_cov [covariant_class M N μ r] [contravariant_class M N μ r] (m : M) {a b : N} :
r (μ m a) (μ m b) ↔ r a b | ⟨contravariant_class.elim _, covariant_class.elim _⟩ | lemma | rel_iff_cov | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class"
] | null | 114 | 116 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant.flip (h : covariant M N μ r) : covariant M N μ (flip r) :=
λ a b c hbc, h a hbc | covariant.flip (h : covariant M N μ r) : covariant M N μ (flip r) | λ a b c hbc, h a hbc | lemma | covariant.flip | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | null | 122 | 123 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant.flip (h : contravariant M N μ r) : contravariant M N μ (flip r) :=
λ a b c hbc, h a hbc | contravariant.flip (h : contravariant M N μ r) : contravariant M N μ (flip r) | λ a b c hbc, h a hbc | lemma | contravariant.flip | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | null | 125 | 126 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
act_rel_act_of_rel (m : M) {a b : N} (ab : r a b) :
r (μ m a) (μ m b) :=
covariant_class.elim _ ab | act_rel_act_of_rel (m : M) {a b : N} (ab : r a b) :
r (μ m a) (μ m b) | covariant_class.elim _ ab | lemma | act_rel_act_of_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | null | 133 | 135 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.covariant_iff_contravariant [group N] :
covariant N N (*) r ↔ contravariant N N (*) r :=
begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c],
exact h a⁻¹ bc },
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c] at bc,
exact h a⁻¹ bc... | group.covariant_iff_contravariant [group N] :
covariant N N (*) r ↔ contravariant N N (*) r | begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c],
exact h a⁻¹ bc },
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c] at bc,
exact h a⁻¹ bc }
end | lemma | group.covariant_iff_contravariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant",
"group",
"inv_mul_cancel_left"
] | null | 137 | 146 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.covconv [group N] [covariant_class N N (*) r] :
contravariant_class N N (*) r :=
⟨group.covariant_iff_contravariant.mp covariant_class.elim⟩ | group.covconv [group N] [covariant_class N N (*) r] :
contravariant_class N N (*) r | ⟨group.covariant_iff_contravariant.mp covariant_class.elim⟩ | instance | group.covconv | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"group"
] | null | 148 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.covariant_swap_iff_contravariant_swap [group N] :
covariant N N (swap (*)) r ↔ contravariant N N (swap (*)) r :=
begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a],
exact h a⁻¹ bc },
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c ... | group.covariant_swap_iff_contravariant_swap [group N] :
covariant N N (swap (*)) r ↔ contravariant N N (swap (*)) r | begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a],
exact h a⁻¹ bc },
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a] at bc,
exact h a⁻¹ bc }
end | lemma | group.covariant_swap_iff_contravariant_swap | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant",
"group",
"mul_inv_cancel_right"
] | null | 153 | 162 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.covconv_swap [group N] [covariant_class N N (swap (*)) r] :
contravariant_class N N (swap (*)) r :=
⟨group.covariant_swap_iff_contravariant_swap.mp covariant_class.elim⟩ | group.covconv_swap [group N] [covariant_class N N (swap (*)) r] :
contravariant_class N N (swap (*)) r | ⟨group.covariant_swap_iff_contravariant_swap.mp covariant_class.elim⟩ | instance | group.covconv_swap | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"group"
] | null | 164 | 167 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (μ m b) c) :
r (μ m a) c :=
trans (act_rel_act_of_rel m ab) rl | act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (μ m b) c) :
r (μ m a) c | trans (act_rel_act_of_rel m ab) rl | lemma | act_rel_of_rel_of_act_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | null | 173 | 175 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_act_of_rel_of_rel_act (ab : r a b) (rr : r c (μ m a)) :
r c (μ m b) :=
trans rr (act_rel_act_of_rel _ ab) | rel_act_of_rel_of_rel_act (ab : r a b) (rr : r c (μ m a)) :
r c (μ m b) | trans rr (act_rel_act_of_rel _ ab) | lemma | rel_act_of_rel_of_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | null | 177 | 179 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
act_rel_act_of_rel_of_rel (ab : r a b) (cd : r c d) :
r (mu a c) (mu b d) :=
trans (act_rel_act_of_rel c ab : _) (act_rel_act_of_rel b cd) | act_rel_act_of_rel_of_rel (ab : r a b) (cd : r c d) :
r (mu a c) (mu b d) | trans (act_rel_act_of_rel c ab : _) (act_rel_act_of_rel b cd) | lemma | act_rel_act_of_rel_of_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | null | 190 | 192 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_of_act_rel_act (m : M) {a b : N} (ab : r (μ m a) (μ m b)) :
r a b :=
contravariant_class.elim _ ab | rel_of_act_rel_act (m : M) {a b : N} (ab : r (μ m a) (μ m b)) :
r a b | contravariant_class.elim _ ab | lemma | rel_of_act_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | null | 199 | 201 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
act_rel_of_act_rel_of_rel_act_rel (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
r (μ m a) c :=
trans ab (rel_of_act_rel_act m rl) | act_rel_of_act_rel_of_rel_act_rel (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
r (μ m a) c | trans ab (rel_of_act_rel_act m rl) | lemma | act_rel_of_act_rel_of_rel_act_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"rel_of_act_rel_act"
] | null | 207 | 209 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_act_of_act_rel_act_of_rel_act (ab : r (μ m a) (μ m b)) (rr : r b (μ m c)) :
r a (μ m c) :=
trans (rel_of_act_rel_act m ab) rr | rel_act_of_act_rel_act_of_rel_act (ab : r (μ m a) (μ m b)) (rr : r b (μ m c)) :
r a (μ m c) | trans (rel_of_act_rel_act m ab) rr | lemma | rel_act_of_act_rel_act_of_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"rel_of_act_rel_act"
] | null | 211 | 213 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant.monotone_of_const [covariant_class M N μ (≤)] (m : M) : monotone (μ m) :=
λ a b ha, covariant_class.elim m ha | covariant.monotone_of_const [covariant_class M N μ (≤)] (m : M) : monotone (μ m) | λ a b ha, covariant_class.elim m ha | lemma | covariant.monotone_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant_class",
"monotone"
] | The partial application of a constant to a covariant operator is monotone. | 225 | 226 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.covariant_of_const [covariant_class M N μ (≤)] (hf : monotone f) (m : M) :
monotone (λ n, f (μ m n)) :=
hf.comp $ covariant.monotone_of_const m | monotone.covariant_of_const [covariant_class M N μ (≤)] (hf : monotone f) (m : M) :
monotone (λ n, f (μ m n)) | hf.comp $ covariant.monotone_of_const m | lemma | monotone.covariant_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant.monotone_of_const",
"covariant_class",
"monotone"
] | A monotone function remains monotone when composed with the partial application
of a covariant operator. E.g., `∀ (m : ℕ), monotone f → monotone (λ n, f (m + n))`. | 230 | 232 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : monotone f) (m : N) :
monotone (λ n, f (μ n m)) :=
hf.comp $ covariant.monotone_of_const m | monotone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : monotone f) (m : N) :
monotone (λ n, f (μ n m)) | hf.comp $ covariant.monotone_of_const m | lemma | monotone.covariant_of_const' | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant.monotone_of_const",
"covariant_class",
"monotone"
] | Same as `monotone.covariant_of_const`, but with the constant on the other side of
the operator. E.g., `∀ (m : ℕ), monotone f → monotone (λ n, f (n + m))`. | 236 | 239 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.covariant_of_const [covariant_class M N μ (≤)] (hf : antitone f) (m : M) :
antitone (λ n, f (μ m n)) :=
hf.comp_monotone $ covariant.monotone_of_const m | antitone.covariant_of_const [covariant_class M N μ (≤)] (hf : antitone f) (m : M) :
antitone (λ n, f (μ m n)) | hf.comp_monotone $ covariant.monotone_of_const m | lemma | antitone.covariant_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"antitone",
"covariant.monotone_of_const",
"covariant_class"
] | Dual of `monotone.covariant_of_const` | 242 | 244 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : antitone f) (m : N) :
antitone (λ n, f (μ n m)) :=
hf.comp_monotone $ covariant.monotone_of_const m | antitone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : antitone f) (m : N) :
antitone (λ n, f (μ n m)) | hf.comp_monotone $ covariant.monotone_of_const m | lemma | antitone.covariant_of_const' | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"antitone",
"covariant.monotone_of_const",
"covariant_class"
] | Dual of `monotone.covariant_of_const'` | 247 | 250 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_le_of_covariant_lt [partial_order N] :
covariant M N μ (<) → covariant M N μ (≤) :=
begin
refine λ h a b c bc, _,
rcases le_iff_eq_or_lt.mp bc with rfl | bc,
{ exact rfl.le },
{ exact (h _ bc).le }
end | covariant_le_of_covariant_lt [partial_order N] :
covariant M N μ (<) → covariant M N μ (≤) | begin
refine λ h a b c bc, _,
rcases le_iff_eq_or_lt.mp bc with rfl | bc,
{ exact rfl.le },
{ exact (h _ bc).le }
end | lemma | covariant_le_of_covariant_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | null | 254 | 261 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_lt_of_contravariant_le [partial_order N] :
contravariant M N μ (≤) → contravariant M N μ (<) :=
begin
refine λ h a b c bc, lt_iff_le_and_ne.mpr ⟨h a bc.le, _⟩,
rintro rfl,
exact lt_irrefl _ bc,
end | contravariant_lt_of_contravariant_le [partial_order N] :
contravariant M N μ (≤) → contravariant M N μ (<) | begin
refine λ h a b c bc, lt_iff_le_and_ne.mpr ⟨h a bc.le, _⟩,
rintro rfl,
exact lt_irrefl _ bc,
end | lemma | contravariant_lt_of_contravariant_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | null | 263 | 269 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_le_iff_contravariant_lt [linear_order N] :
covariant M N μ (≤) ↔ contravariant M N μ (<) :=
⟨ λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k)),
λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k))⟩ | covariant_le_iff_contravariant_lt [linear_order N] :
covariant M N μ (≤) ↔ contravariant M N μ (<) | ⟨ λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k)),
λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k))⟩ | lemma | covariant_le_iff_contravariant_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant"
] | null | 271 | 274 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_lt_iff_contravariant_le [linear_order N] :
covariant M N μ (<) ↔ contravariant M N μ (≤) :=
⟨ λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k)),
λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k))⟩ | covariant_lt_iff_contravariant_le [linear_order N] :
covariant M N μ (<) ↔ contravariant M N μ (≤) | ⟨ λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k)),
λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k))⟩ | lemma | covariant_lt_iff_contravariant_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant"
] | null | 276 | 279 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_flip_mul_iff [comm_semigroup N] :
covariant N N (flip (*)) (r) ↔ covariant N N (*) (r) :=
by rw is_symm_op.flip_eq | covariant_flip_mul_iff [comm_semigroup N] :
covariant N N (flip (*)) (r) ↔ covariant N N (*) (r) | by rw is_symm_op.flip_eq | lemma | covariant_flip_mul_iff | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"covariant",
"is_symm_op.flip_eq"
] | null | 281 | 284 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_flip_mul_iff [comm_semigroup N] :
contravariant N N (flip (*)) (r) ↔ contravariant N N (*) (r) :=
by rw is_symm_op.flip_eq | contravariant_flip_mul_iff [comm_semigroup N] :
contravariant N N (flip (*)) (r) ↔ contravariant N N (*) (r) | by rw is_symm_op.flip_eq | lemma | contravariant_flip_mul_iff | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"contravariant",
"is_symm_op.flip_eq"
] | null | 286 | 289 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_mul_lt_of_covariant_mul_le [has_mul N] [linear_order N]
[covariant_class N N (*) (≤)] : contravariant_class N N (*) (<) :=
{ elim := (covariant_le_iff_contravariant_lt N N (*)).mp covariant_class.elim } | contravariant_mul_lt_of_covariant_mul_le [has_mul N] [linear_order N]
[covariant_class N N (*) (≤)] : contravariant_class N N (*) (<) | { elim := (covariant_le_iff_contravariant_lt N N (*)).mp covariant_class.elim } | instance | contravariant_mul_lt_of_covariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"covariant_le_iff_contravariant_lt"
] | null | 291 | 294 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_mul_lt_of_contravariant_mul_le [has_mul N] [linear_order N]
[contravariant_class N N (*) (≤)] : covariant_class N N (*) (<) :=
{ elim := (covariant_lt_iff_contravariant_le N N (*)).mpr contravariant_class.elim } | covariant_mul_lt_of_contravariant_mul_le [has_mul N] [linear_order N]
[contravariant_class N N (*) (≤)] : covariant_class N N (*) (<) | { elim := (covariant_lt_iff_contravariant_le N N (*)).mpr contravariant_class.elim } | instance | covariant_mul_lt_of_contravariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"covariant_lt_iff_contravariant_le"
] | null | 296 | 299 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_swap_mul_le_of_covariant_mul_le [comm_semigroup N] [has_le N]
[covariant_class N N (*) (≤)] : covariant_class N N (swap (*)) (≤) :=
{ elim := (covariant_flip_mul_iff N (≤)).mpr covariant_class.elim } | covariant_swap_mul_le_of_covariant_mul_le [comm_semigroup N] [has_le N]
[covariant_class N N (*) (≤)] : covariant_class N N (swap (*)) (≤) | { elim := (covariant_flip_mul_iff N (≤)).mpr covariant_class.elim } | instance | covariant_swap_mul_le_of_covariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"covariant_class",
"covariant_flip_mul_iff"
] | null | 301 | 304 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_swap_mul_le_of_contravariant_mul_le [comm_semigroup N] [has_le N]
[contravariant_class N N (*) (≤)] : contravariant_class N N (swap (*)) (≤) :=
{ elim := (contravariant_flip_mul_iff N (≤)).mpr contravariant_class.elim } | contravariant_swap_mul_le_of_contravariant_mul_le [comm_semigroup N] [has_le N]
[contravariant_class N N (*) (≤)] : contravariant_class N N (swap (*)) (≤) | { elim := (contravariant_flip_mul_iff N (≤)).mpr contravariant_class.elim } | instance | contravariant_swap_mul_le_of_contravariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"contravariant_class",
"contravariant_flip_mul_iff"
] | null | 306 | 309 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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