statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
eq_on_sclosure {f g : F} {s : set R} (h : set.eq_on (f : R → S) (g : R → S) s) :
set.eq_on f g (closure s) | show closure s ≤ eq_slocus f g, from closure_le.2 h | lemma | non_unital_ring_hom.eq_on_sclosure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"set.eq_on"
] | If two non-unital ring homomorphisms are equal on a set, then they are equal on its
non-unital subsemiring closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_eq_on_stop {f g : F} (h : set.eq_on (f : R → S) (g : R → S)
(⊤ : non_unital_subsemiring R)) : f = g | fun_like.ext _ _ (λ x, h trivial) | lemma | non_unital_ring_hom.eq_of_eq_on_stop | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"fun_like.ext",
"non_unital_subsemiring",
"set.eq_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq_on_sdense {s : set R} (hs : closure s = ⊤) {f g : F}
(h : s.eq_on (f : R → S) (g : R → S)) :
f = g | eq_of_eq_on_stop $ hs ▸ eq_on_sclosure h | lemma | non_unital_ring_hom.eq_of_eq_on_sdense | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sclosure_preimage_le (f : F) (s : set S) :
closure ((f : R → S) ⁻¹' s) ≤ (closure s).comap f | closure_le.2 $ λ x hx, set_like.mem_coe.2 $ mem_comap.2 $ subset_closure hx | lemma | non_unital_ring_hom.sclosure_preimage_le | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sclosure (f : F) (s : set R) :
(closure s).map f = closure ((f : R → S) '' s) | le_antisymm
(map_le_iff_le_comap.2 $ le_trans (closure_mono $ set.subset_preimage_image _ _)
(sclosure_preimage_le _ _))
(closure_le.2 $ set.image_subset _ subset_closure) | lemma | non_unital_ring_hom.map_sclosure | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"closure",
"closure_mono",
"set.image_subset",
"set.subset_preimage_image",
"subset_closure"
] | The image under a ring homomorphism of the subsemiring generated by a set equals
the subsemiring generated by the image of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion {S T : non_unital_subsemiring R} (h : S ≤ T) : S →ₙ+* T | cod_restrict (subtype S) _ (λ x, h x.2) | def | non_unital_subsemiring.inclusion | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring"
] | The non-unital ring homomorphism associated to an inclusion of
non-unital subsemirings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
srange_subtype (s : non_unital_subsemiring R) : (subtype s).srange = s | set_like.coe_injective $ (coe_srange _).trans subtype.range_coe | lemma | non_unital_subsemiring.srange_subtype | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"set_like.coe_injective",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_fst : (fst R S).srange = ⊤ | non_unital_ring_hom.srange_top_of_surjective (fst R S) prod.fst_surjective | lemma | non_unital_subsemiring.range_fst | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_ring_hom.srange_top_of_surjective",
"prod.fst_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_snd : (snd R S).srange = ⊤ | non_unital_ring_hom.srange_top_of_surjective (snd R S) $ prod.snd_surjective | lemma | non_unital_subsemiring.range_snd | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_ring_hom.srange_top_of_surjective",
"prod.snd_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_subsemiring_congr (h : s = t) : s ≃+* t | { map_mul' := λ _ _, rfl, map_add' := λ _ _, rfl, ..equiv.set_congr $ congr_arg _ h } | def | ring_equiv.non_unital_subsemiring_congr | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"equiv.set_congr"
] | Makes the identity isomorphism from a proof two non-unital subsemirings of a multiplicative
monoid are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sof_left_inverse' {g : S → R} {f : F} (h : function.left_inverse g f) :
R ≃+* srange f | { to_fun := srange_restrict f,
inv_fun := λ x, g (subtype (srange f) x),
left_inv := h,
right_inv := λ x, subtype.ext $
let ⟨x', hx'⟩ := non_unital_ring_hom.mem_srange.mp x.prop in
show f (g x) = x, by rw [←hx', h x'],
..(srange_restrict f) } | def | ring_equiv.sof_left_inverse' | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"inv_fun",
"subtype.ext"
] | Restrict a non-unital ring homomorphism with a left inverse to a ring isomorphism to its
`non_unital_ring_hom.srange`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sof_left_inverse'_apply
{g : S → R} {f : F} (h : function.left_inverse g f) (x : R) :
↑(sof_left_inverse' h x) = f x | rfl | lemma | ring_equiv.sof_left_inverse'_apply | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sof_left_inverse'_symm_apply
{g : S → R} {f : F} (h : function.left_inverse g f) (x : srange f) :
(sof_left_inverse' h).symm x = g x | rfl | lemma | ring_equiv.sof_left_inverse'_symm_apply | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_subsemiring_map (e : R ≃+* S) (s : non_unital_subsemiring R) :
s ≃+* non_unital_subsemiring.map e.to_non_unital_ring_hom s | { ..e.to_add_equiv.add_submonoid_map s.to_add_submonoid,
..e.to_mul_equiv.subsemigroup_map s.to_subsemigroup } | def | ring_equiv.non_unital_subsemiring_map | ring_theory.non_unital_subsemiring | src/ring_theory/non_unital_subsemiring/basic.lean | [
"algebra.ring.equiv",
"algebra.ring.prod",
"data.set.finite",
"group_theory.submonoid.membership",
"group_theory.subsemigroup.membership",
"group_theory.subsemigroup.centralizer"
] | [
"non_unital_subsemiring",
"non_unital_subsemiring.map"
] | Given an equivalence `e : R ≃+* S` of non-unital semirings and a non-unital subsemiring
`s` of `R`, `non_unital_subsemiring_map e s` is the induced equivalence between `s` and
`s.map e` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_eqv : setoid (R × S) | { r := λ rs rs', ∃ (u : S) (v : R), rs'.1 * u = rs.1 * v
∧ (rs'.2 : R) * u = rs.2 * v,
iseqv :=
begin
refine ⟨_, _, _⟩,
{ rintro ⟨r,s⟩, use 1, use 1, simp [submonoid.one_mem] },
{ rintros ⟨r, s⟩ ⟨r', s'⟩ ⟨u, v, hru, hsu⟩,
rcases ore_condition (s : R) s' with ⟨r₂, s₂, h₁... | def | ore_localization.ore_eqv | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"submonoid.coe_mul",
"submonoid.one_mem"
] | The setoid on `R × S` used for the Ore localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_localization (R : Type*) [monoid R] (S : submonoid R) [ore_set S] | quotient (ore_localization.ore_eqv R S) | def | ore_localization | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"monoid",
"ore_localization.ore_eqv",
"submonoid"
] | The ore localization of a monoid and a submonoid fulfilling the ore condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div (r : R) (s : S) : R[S⁻¹] | quotient.mk (r, s) | def | ore_localization.ore_div | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | The division in the ore localization `R[S⁻¹]`, as a fraction of an element of `R` and `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ind {β : R [S ⁻¹] → Prop} (c : ∀ (r : R) (s : S), β (r /ₒ s)) : ∀ q, β q | by { apply quotient.ind, rintro ⟨r, s⟩, exact c r s } | lemma | ore_localization.ind | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ore_div_eq_iff {r₁ r₂ : R} {s₁ s₂ : S} :
r₁ /ₒ s₁ = r₂ /ₒ s₂ ↔ (∃ (u : S) (v : R), r₂ * u = r₁ * v ∧ (s₂ : R) * u = s₁ * v) | quotient.eq' | lemma | ore_localization.ore_div_eq_iff | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
expand (r : R) (s : S) (t : R) (hst : (s : R) * t ∈ S) :
r /ₒ s = (r * t) /ₒ (⟨s * t, hst⟩) | by { apply quotient.sound, refine ⟨s, t * s, _, _⟩; dsimp; rw [mul_assoc]; refl } | lemma | ore_localization.expand | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc"
] | A fraction `r /ₒ s` is equal to its expansion by an arbitrary factor `t` if `s * t ∈ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
expand' (r : R) (s s' : S) : r /ₒ s = (r * s') /ₒ (s * s') | ore_localization.expand r s s' (by norm_cast; apply set_like.coe_mem) | lemma | ore_localization.expand' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.expand",
"set_like.coe_mem"
] | A fraction is equal to its expansion by an factor from s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_num_factor_eq {r r' r₁ r₂ : R} {s t : S}
(h : r * t = r' * t) : (r₁ * r * r₂) /ₒ s = (r₁ * r' * r₂) /ₒ s | begin
rcases ore_condition r₂ t with ⟨r₂',t', hr₂⟩,
calc (r₁ * r * r₂) /ₒ s = (r₁ * r * r₂ * t') /ₒ (s * t') : ore_localization.expand _ _ t' _
... = ((r₁ * r) * (r₂ * t')) /ₒ (s * t') : by simp [←mul_assoc]
... = ((r₁ * r) * (t * r₂')) /ₒ (s * t') : by rw hr₂
... ... | lemma | ore_localization.eq_of_num_factor_eq | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.expand"
] | Fractions which differ by a factor of the numerator can be proven equal if
those factors expand to equal elements of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_expand {C : Sort*} (P : R → S → C)
(hP : ∀ (r t : R) (s : S) (ht : ((s : R) * t) ∈ S), P r s = P (r * t) ⟨s * t, ht⟩) :
R[S⁻¹] → C | quotient.lift (λ (p : R × S), P p.1 p.2) $ λ p q pq,
begin
cases p with r₁ s₁, cases q with r₂ s₂, rcases pq with ⟨u, v, hr₂, hs₂⟩,
dsimp at *,
have s₁vS : (s₁ : R) * v ∈ S,
{ rw [←hs₂, ←S.coe_mul], exact set_like.coe_mem (s₂ * u) },
replace hs₂ : s₂ * u = ⟨(s₁ : R) * v, s₁vS⟩, { ext, simp [hs₂] },
rw [hP r... | def | ore_localization.lift_expand | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"set_like.coe_mem"
] | A function or predicate over `R` and `S` can be lifted to `R[S⁻¹]` if it is invariant
under expansion on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_expand_of {C : Sort*} {P : R → S → C}
{hP : ∀ (r t : R) (s : S) (ht : ((s : R) * t) ∈ S), P r s = P (r * t) ⟨s * t, ht⟩}
(r : R) (s : S) :
lift_expand P hP (r /ₒ s) = P r s | rfl | lemma | ore_localization.lift_expand_of | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift₂_expand {C : Sort*} (P : R → S → R → S → C)
(hP : ∀ (r₁ t₁ : R) (s₁ : S) (ht₁ : (s₁ : R) * t₁ ∈ S)
(r₂ t₂ : R) (s₂ : S) (ht₂ : (s₂ : R) * t₂ ∈ S),
P r₁ s₁ r₂ s₂ = P (r₁ * t₁) ⟨s₁ * t₁, ht₁⟩ (r₂ * t₂) ⟨s₂ * t₂, ht₂⟩) : R[S⁻¹] → R[S⁻¹] → C | lift_expand
(λ r₁ s₁, lift_expand (P r₁ s₁) $ λ r₂ t₂ s₂ ht₂, by simp [hP r₁ 1 s₁ (by simp) r₂ t₂ s₂ ht₂]) $
λ r₁ t₁ s₁ ht₁,
begin
ext x, induction x using ore_localization.ind with r₂ s₂,
rw [lift_expand_of, lift_expand_of, hP r₁ t₁ s₁ ht₁ r₂ 1 s₂ (by simp)], simp,
end | def | ore_localization.lift₂_expand | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind"
] | A version of `lift_expand` used to simultaneously lift functions with two arguments
in ``R[S⁻¹]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift₂_expand_of {C : Sort*} {P : R → S → R → S → C}
{hP : ∀ (r₁ t₁ : R) (s₁ : S) (ht₁ : (s₁ : R) * t₁ ∈ S)
(r₂ t₂ : R) (s₂ : S) (ht₂ : (s₂ : R) * t₂ ∈ S),
P r₁ s₁ r₂ s₂ = P (r₁ * t₁) ⟨s₁ * t₁, ht₁⟩ (r₂ * t₂) ⟨s₂ * t₂, ht₂⟩}
(r₁ : R) (s₁ : S) (r₂ : R) (s₂ : S) :
lift₂_expand P hP (r₁ /ₒ s₁) (r₂ /ₒ s₂... | rfl | lemma | ore_localization.lift₂_expand_of | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul' (r₁ : R) (s₁ : S) (r₂ : R) (s₂ : S) : R[S⁻¹] | (r₁ * ore_num r₂ s₁) /ₒ (s₂ * ore_denom r₂ s₁) | def | ore_localization.mul' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul'_char (r₁ r₂ : R) (s₁ s₂ : S) (u : S) (v : R) (huv : r₂ * (u : R) = s₁ * v) :
mul' r₁ s₁ r₂ s₂ = (r₁ * v) /ₒ (s₂ * u) | begin
simp only [mul'],
have h₀ := ore_eq r₂ s₁, set v₀ := ore_num r₂ s₁, set u₀ := ore_denom r₂ s₁,
rcases ore_condition (u₀ : R) u with ⟨r₃, s₃, h₃⟩,
have :=
calc (s₁ : R) * (v * r₃) = r₂ * u * r₃ : by assoc_rw ←huv; symmetry; apply mul_assoc
... = r₂ * u₀ * s₃ : by assoc_rw ←h₃; refl... | lemma | ore_localization.mul'_char | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul : R[S⁻¹] → R[S⁻¹] → R[S⁻¹] | lift₂_expand mul' $ λ r₂ p s₂ hp r₁ r s₁ hr,
begin
have h₁ := ore_eq r₁ s₂, set r₁' := ore_num r₁ s₂, set s₂' := ore_denom r₁ s₂,
rcases ore_condition (↑s₂ * r₁') ⟨s₂ * p, hp⟩ with ⟨p', s_star, h₂⟩, dsimp at h₂,
rcases ore_condition r (s₂' * s_star) with ⟨p_flat, s_flat, h₃⟩, simp only [S.coe_mul] at h₃,
have :... | def | ore_localization.mul | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"ore_localization.eq_of_num_factor_eq",
"ore_localization.expand",
"set_like.coe_mem",
"set_like.coe_mk",
"submonoid.coe_mul"
] | The multiplication on the Ore localization of monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div_mul_ore_div {r₁ r₂ : R} {s₁ s₂ : S} :
(r₁ /ₒ s₁) * (r₂ /ₒ s₂) = (r₁ * ore_num r₂ s₁) /ₒ (s₂ * ore_denom r₂ s₁) | rfl | lemma | ore_localization.ore_div_mul_ore_div | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ore_div_mul_char (r₁ r₂ : R) (s₁ s₂ : S) (r' : R) (s' : S)
(huv : r₂ * (s' : R) = s₁ * r') : (r₁ /ₒ s₁) * (r₂ /ₒ s₂) = (r₁ * r') /ₒ (s₂ * s') | mul'_char r₁ r₂ s₁ s₂ s' r' huv | lemma | ore_localization.ore_div_mul_char | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | A characterization lemma for the multiplication on the Ore localization, allowing for a choice
of Ore numerator and Ore denominator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div_mul_char' (r₁ r₂ : R) (s₁ s₂ : S) :
Σ' r' : R, Σ' s' : S, r₂ * (s' : R) = s₁ * r'
∧ (r₁ /ₒ s₁) * (r₂ /ₒ s₂) = (r₁ * r') /ₒ (s₂ * s') | ⟨ore_num r₂ s₁, ore_denom r₂ s₁, ore_eq r₂ s₁, ore_div_mul_ore_div⟩ | def | ore_localization.ore_div_mul_char' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | Another characterization lemma for the multiplication on the Ore localizaion delivering
Ore witnesses and conditions bundled in a sigma type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_def : (1 : R[S⁻¹]) = 1 /ₒ 1 | rfl | lemma | ore_localization.one_def | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_one' {r : R} (hr : r ∈ S) : r /ₒ ⟨r, hr⟩ = 1 | by { rw [ore_localization.one_def, ore_div_eq_iff], exact ⟨⟨r, hr⟩, 1, by simp, by simp⟩ } | lemma | ore_localization.div_eq_one' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.one_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_one {s : S} : (s : R) /ₒ s = 1 | by { cases s; apply ore_localization.div_eq_one' } | lemma | ore_localization.div_eq_one | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"div_eq_one",
"ore_localization.div_eq_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul (x : R[S⁻¹]) : 1 * x = x | begin
induction x using ore_localization.ind with r s,
simp [ore_localization.one_def, ore_div_mul_char (1 : R) r (1 : S) s r 1 (by simp)]
end | lemma | ore_localization.one_mul | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"one_mul",
"ore_localization.ind",
"ore_localization.one_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one (x : R[S⁻¹]) : x * 1 = x | begin
induction x using ore_localization.ind with r s,
simp [ore_localization.one_def, ore_div_mul_char r 1 s 1 1 s (by simp)]
end | lemma | ore_localization.mul_one | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_one",
"ore_localization.ind",
"ore_localization.one_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_assoc (x y z : R[S⁻¹]) : x * y * z = x * (y * z) | begin
induction x using ore_localization.ind with r₁ s₁,
induction y using ore_localization.ind with r₂ s₂,
induction z using ore_localization.ind with r₃ s₃,
rcases ore_div_mul_char' r₁ r₂ s₁ s₂ with ⟨ra, sa, ha, ha'⟩, rw ha', clear ha',
rcases ore_div_mul_char' r₂ r₃ s₂ s₃ with ⟨rb, sb, hb, hb'⟩, rw hb', cl... | lemma | ore_localization.mul_assoc | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"ore_localization.ind",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv (s s' : S) : ((s : R) /ₒ s') * (s' /ₒ s) = 1 | by simp [ore_div_mul_char (s :R) s' s' s 1 1 (by simp)] | lemma | ore_localization.mul_inv | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div {r : R} {s t : S} : (r /ₒ s) * (1 /ₒ t) = r /ₒ (t * s) | by simp [ore_div_mul_char r 1 s t 1 s (by simp)] | lemma | ore_localization.mul_one_div | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel {r : R} {s t : S} : (r /ₒ s) * (s /ₒ t) = r /ₒ t | by simp [ore_div_mul_char r s s t 1 1 (by simp)] | lemma | ore_localization.mul_cancel | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_cancel' {r₁ r₂ : R} {s t : S} : (r₁ /ₒ s) * ((s * r₂) /ₒ t) = (r₁ * r₂) /ₒ t | by simp [ore_div_mul_char r₁ (s * r₂) s t r₂ 1 (by simp)] | lemma | ore_localization.mul_cancel' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_one_mul {p r : R} {s : S} :
(r /ₒ 1) * (p /ₒ s) = (r * p) /ₒ s | --TODO use coercion r ↦ r /ₒ 1
by simp [ore_div_mul_char r p 1 s p 1 (by simp)] | lemma | ore_localization.div_one_mul | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numerator_unit (s : S) : units R[S⁻¹] | { val := (s : R) /ₒ 1,
inv := (1 : R) /ₒ s,
val_inv := ore_localization.mul_inv s 1,
inv_val := ore_localization.mul_inv 1 s } | def | ore_localization.numerator_unit | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.mul_inv",
"units"
] | The fraction `s /ₒ 1` as a unit in `R[S⁻¹]`, where `s : S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numerator_hom : R →* R[S⁻¹] | { to_fun := λ r, r /ₒ 1,
map_one' := rfl,
map_mul' := λ r₁ r₂, div_one_mul.symm } | def | ore_localization.numerator_hom | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | The multiplicative homomorphism from `R` to `R[S⁻¹]`, mapping `r : R` to the
fraction `r /ₒ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numerator_hom_apply {r : R} : numerator_hom r = r /ₒ (1 : S) | rfl | lemma | ore_localization.numerator_hom_apply | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numerator_is_unit (s : S) : is_unit ((numerator_hom (s : R)) : R[S⁻¹]) | ⟨numerator_unit s, rfl⟩ | lemma | ore_localization.numerator_is_unit | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_mul_hom : R[S⁻¹] →* T | { to_fun := λ x, x.lift_expand (λ r s, (f r) * ((fS s)⁻¹ : units T)) $ λ r t s ht,
begin
have : ((fS ⟨s * t, ht⟩) : T) = fS s * f t,
{ simp only [←hf, set_like.coe_mk, monoid_hom.map_mul] },
conv_rhs { rw [monoid_hom.map_mul, ←mul_one (f r), ←units.coe_one, ←mul_left_inv (fS s)],
rw [units.coe_mul, ... | def | ore_localization.universal_mul_hom | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"monoid_hom.map_mul",
"mul_assoc",
"mul_inv",
"mul_one",
"one_mul",
"ore_localization.ind",
"ore_localization.one_def",
"set_like.coe_mk",
"units",
"units.coe_mul",
"units.mul_inv"
] | The universal lift from a morphism `R →* T`, which maps elements of `S` to units of `T`,
to a morphism `R[S⁻¹] →* T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
universal_mul_hom_apply {r : R} {s : S} :
universal_mul_hom f fS hf (r /ₒ s) = (f r) * ((fS s)⁻¹ : units T) | rfl | lemma | ore_localization.universal_mul_hom_apply | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_mul_hom_commutes {r : R} : universal_mul_hom f fS hf (numerator_hom r) = f r | by simp [numerator_hom_apply, universal_mul_hom_apply] | lemma | ore_localization.universal_mul_hom_commutes | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_mul_hom_unique (φ : R[S⁻¹] →* T) (huniv : ∀ (r : R), φ (numerator_hom r) = f r) :
φ = universal_mul_hom f fS hf | begin
ext, induction x using ore_localization.ind with r s,
rw [universal_mul_hom_apply, ←huniv r, numerator_hom_apply, ←mul_one (φ (r /ₒ s)),
←units.coe_one, ←mul_right_inv (fS s), units.coe_mul, ←mul_assoc,
←hf, ←huniv, ←φ.map_mul, numerator_hom_apply, ore_localization.mul_cancel],
end | lemma | ore_localization.universal_mul_hom_unique | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind",
"ore_localization.mul_cancel",
"units.coe_mul"
] | The universal morphism `universal_mul_hom` is unique. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div_mul_ore_div_comm {r₁ r₂ : R} {s₁ s₂ : S} :
(r₁ /ₒ s₁) * (r₂ /ₒ s₂) = (r₁ * r₂) /ₒ (s₁ * s₂) | by rw [ore_div_mul_char r₁ r₂ s₁ s₂ r₂ s₁ (by simp [mul_comm]), mul_comm s₂] | lemma | ore_localization.ore_div_mul_ore_div_comm | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_map : S.localization_map R[S⁻¹] | { to_fun := numerator_hom,
map_one' := rfl,
map_mul' := λ r₁ r₂, by simp,
map_units' := numerator_is_unit,
surj' := λ z,
begin
induction z using ore_localization.ind with r s,
use (r, s), dsimp,
rw [numerator_hom_apply, numerator_hom_apply], simp
end,
eq_iff_exists' := λ r₁ r₂,
begin
dsi... | def | ore_localization.localization_map | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_comm",
"one_mul",
"ore_localization.ind"
] | The morphism `numerator_hom` is a monoid localization map in the case of commutative `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_monoid_localization : localization S ≃* R[S⁻¹] | localization.mul_equiv_of_quotient (ore_localization.localization_map R S) | def | ore_localization.equiv_monoid_localization | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"localization",
"localization.mul_equiv_of_quotient",
"ore_localization.localization_map"
] | If `R` is commutative, Ore localization and monoid localization are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add'' (r₁ : R) (s₁ : S) (r₂ : R) (s₂ : S) : R[S⁻¹] | (r₁ * ore_denom (s₁ : R) s₂ + r₂ * ore_num s₁ s₂) /ₒ (s₁ * ore_denom s₁ s₂) | def | ore_localization.add'' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add''_char
(r₁ : R) (s₁ : S) (r₂ : R) (s₂ : S)
(rb : R) (sb : S) (hb : (s₁ : R) * sb = (s₂ : R) * rb) :
add'' r₁ s₁ r₂ s₂ = (r₁ * sb + r₂ * rb) /ₒ (s₁ * sb) | begin
simp only [add''],
have ha := ore_eq (s₁ : R) s₂,
set! ra := ore_num (s₁ : R) s₂ with h, rw ←h at *, clear h, -- r tilde
set! sa := ore_denom (s₁ : R) s₂ with h, rw ←h at *, clear h, -- s tilde
rcases ore_condition (sa : R) sb with ⟨rc, sc, hc⟩, -- s*, r*
have : (s₂ : R) * (rb * rc) = s₂ * (ra * sc),
... | lemma | ore_localization.add''_char | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add' (r₂ : R) (s₂ : S) : R[S⁻¹] → R[S⁻¹] | --plus tilde
quotient.lift (λ (r₁s₁ : R × S), add'' r₁s₁.1 r₁s₁.2 r₂ s₂) $
begin
rintros ⟨r₁', s₁'⟩ ⟨r₁, s₁⟩ ⟨sb, rb, hb, hb'⟩, -- s*, r*
rcases ore_condition (s₁' : R) s₂ with ⟨rc, sc, hc⟩, --s~~, r~~
rcases ore_condition rb sc with ⟨rd, sd, hd⟩, -- s#, r#
dsimp at *, rw add''_char _ _ _ _ rc sc hc,
have : ↑... | def | ore_localization.add' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_one",
"submonoid.coe_mul",
"submonoid.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add'_comm (r₁ r₂ : R) (s₁ s₂ : S) : add' r₁ s₁ (r₂ /ₒ s₂) = add' r₂ s₂ (r₁ /ₒ s₁) | begin
simp only [add', ore_div, add'', quotient.lift_mk, quotient.eq],
have hb := ore_eq ↑s₂ s₁, set rb := ore_num ↑s₂ s₁ with h, -- r~~
rw ←h, clear h, set sb := ore_denom ↑s₂ s₁ with h, rw ←h, clear h, -- s~~
have ha := ore_eq ↑s₁ s₂, set ra := ore_num ↑s₁ s₂ with h, -- r~
rw ←h, clear h, set sa := ore_... | lemma | ore_localization.add'_comm | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_assoc",
"quotient.eq",
"quotient.lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add : R[S⁻¹] → R[S⁻¹] → R[S⁻¹] | λ x,
quotient.lift (λ rs : R × S, add' rs.1 rs.2 x)
begin
rintros ⟨r₁, s₁⟩ ⟨r₂, s₂⟩ hyz,
induction x using ore_localization.ind with r₃ s₃,
dsimp, rw [add'_comm, add'_comm r₂],
simp [(/ₒ), quotient.sound hyz],
end | def | ore_localization.add | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind"
] | The addition on the Ore localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div_add_ore_div {r r' : R} {s s' : S} :
r /ₒ s + r' /ₒ s' = (r * ore_denom (s : R) s' + r' * ore_num s s') /ₒ (s * ore_denom s s') | rfl | lemma | ore_localization.ore_div_add_ore_div | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ore_div_add_char {r r' : R} (s s' : S) (rb : R) (sb : S)
(h : (s : R) * sb = s' * rb) :
r /ₒ s + r' /ₒ s' = (r * sb + r' * rb) /ₒ (s * sb) | add''_char r s r' s' rb sb h | lemma | ore_localization.ore_div_add_char | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | A characterization of the addition on the Ore localizaion, allowing for arbitrary Ore
numerator and Ore denominator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_div_add_char' (r r' : R) (s s' : S) :
Σ' r'' : R, Σ' s'' : S, (s : R) * s'' = s' * r'' ∧
r /ₒ s + r' /ₒs' = (r * s'' + r' * r'') /ₒ (s * s'') | ⟨ore_num s s', ore_denom s s', ore_eq s s', ore_div_add_ore_div⟩ | def | ore_localization.ore_div_add_char' | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | Another characterization of the addition on the Ore localization, bundling up all witnesses
and conditions into a sigma type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_ore_div {r r' : R} {s : S} : (r /ₒ s) + (r' /ₒ s) = (r + r') /ₒ s | by simp [ore_div_add_char s s 1 1 (by simp)] | lemma | ore_localization.add_ore_div | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_assoc (x y z : R[S⁻¹]) :
(x + y) + z = x + (y + z) | begin
induction x using ore_localization.ind with r₁ s₁,
induction y using ore_localization.ind with r₂ s₂,
induction z using ore_localization.ind with r₃ s₃,
rcases ore_div_add_char' r₁ r₂ s₁ s₂ with ⟨ra, sa, ha, ha'⟩, rw ha', clear ha',
rcases ore_div_add_char' r₂ r₃ s₂ s₃ with ⟨rb, sb, hb, hb'⟩, rw hb', cl... | lemma | ore_localization.add_assoc | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.expand",
"ore_localization.expand'",
"ore_localization.ind",
"subtype.coe_eq_of_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : R[S⁻¹] | 0 /ₒ 1 | def | ore_localization.zero | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_def : (0 : R[S⁻¹]) = 0 /ₒ 1 | rfl | lemma | ore_localization.zero_def | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_div_eq_zero (s : S) : 0 /ₒ s = 0 | by { rw [ore_localization.zero_def, ore_div_eq_iff], exact ⟨s, 1, by simp⟩ } | lemma | ore_localization.zero_div_eq_zero | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.zero_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_add (x : R[S⁻¹]) : 0 + x = x | begin
induction x using ore_localization.ind,
rw [←zero_div_eq_zero, add_ore_div], simp
end | lemma | ore_localization.zero_add | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm (x y : R[S⁻¹]) : x + y = y + x | begin
induction x using ore_localization.ind,
induction y using ore_localization.ind,
change add' _ _ (_ /ₒ _) = _, apply add'_comm
end | lemma | ore_localization.add_comm | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mul (x : R[S⁻¹]) : 0 * x = 0 | begin
induction x using ore_localization.ind with r s,
rw [ore_localization.zero_def, ore_div_mul_char 0 r 1 s r 1 (by simp)], simp
end | lemma | ore_localization.zero_mul | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind",
"ore_localization.zero_def",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero (x : R[S⁻¹]) : x * 0 = 0 | begin
induction x using ore_localization.ind with r s,
rw [ore_localization.zero_def, ore_div_mul_char r 0 s 1 0 1 (by simp)], simp
end | lemma | ore_localization.mul_zero | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_zero",
"ore_localization.ind",
"ore_localization.zero_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_distrib (x y z : R[S⁻¹]) : x * (y + z) = x * y + x * z | begin
induction x using ore_localization.ind with r₁ s₁,
induction y using ore_localization.ind with r₂ s₂,
induction z using ore_localization.ind with r₃ s₃,
rcases ore_div_add_char' r₂ r₃ s₂ s₃ with ⟨ra, sa, ha, q⟩, rw q, clear q,
rw ore_localization.expand' r₂ s₂ sa,
rcases ore_div_mul_char' r₁ (r₂ * sa)... | lemma | ore_localization.left_distrib | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"left_distrib",
"mul_assoc",
"mul_one",
"ore_localization.expand",
"ore_localization.expand'",
"ore_localization.ind",
"ore_localization.mul_cancel'",
"set_like.coe_mem",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_distrib (x y z : R[S⁻¹]) : (x + y) * z = x * z + y * z | begin
induction x using ore_localization.ind with r₁ s₁,
induction y using ore_localization.ind with r₂ s₂,
induction z using ore_localization.ind with r₃ s₃,
rcases ore_div_add_char' r₁ r₂ s₁ s₂ with ⟨ra, sa, ha, ha'⟩, rw ha', clear ha', norm_cast at ha,
rw ore_localization.expand' r₁ s₁ sa,
rw ore_localiz... | lemma | ore_localization.right_distrib | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.expand",
"ore_localization.expand'",
"ore_localization.ind",
"right_distrib",
"set_like.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_hom : R[S⁻¹] →+* T | { map_zero' :=
begin
rw [monoid_hom.to_fun_eq_coe, ore_localization.zero_def, universal_mul_hom_apply],
simp
end,
map_add' := λ x y,
begin
induction x using ore_localization.ind with r₁ s₁,
induction y using ore_localization.ind with r₂ s₂,
rcases ore_div_add_char' r₁ r₂ s₁ s₂ with ⟨r₃, s₃, ... | def | ore_localization.universal_hom | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"monoid_hom.map_mul",
"monoid_hom.to_fun_eq_coe",
"mul_assoc",
"mul_inv_rev",
"mul_one",
"mul_right_inv",
"one_mul",
"ore_localization.ind",
"ore_localization.zero_def",
"ring_hom.coe_monoid_hom",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.to_monoid_hom_eq_coe",
"set_like.coe_mem",
... | The universal lift from a ring homomorphism `f : R →+* T`, which maps elements in `S` to
units of `T`, to a ring homomorphism `R[S⁻¹] →+* T`. This extends the construction on
monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
universal_hom_apply {r : R} {s : S} :
universal_hom f fS hf (r /ₒ s) = (f r) * ((fS s)⁻¹ : units T) | rfl | lemma | ore_localization.universal_hom_apply | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_hom_commutes {r : R} : universal_hom f fS hf (numerator_hom r) = f r | by simp [numerator_hom_apply, universal_hom_apply] | lemma | ore_localization.universal_hom_commutes | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_hom_unique (φ : R[S⁻¹] →+* T) (huniv : ∀ (r : R), φ (numerator_hom r) = f r) :
φ = universal_hom f fS hf | ring_hom.coe_monoid_hom_injective $
universal_mul_hom_unique (ring_hom.to_monoid_hom f) fS hf ↑φ huniv | lemma | ore_localization.universal_hom_unique | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ring_hom.coe_monoid_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg : R[S⁻¹] → R[S⁻¹] | lift_expand (λ (r : R) (s : S), (- r) /ₒ s) $
λ r t s ht, by rw [neg_mul_eq_neg_mul, ←ore_localization.expand] | def | ore_localization.neg | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"neg_mul_eq_neg_mul"
] | Negation on the Ore localization is defined via negation on the numerator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_def (r : R) (s : S) : - (r /ₒ s) = (- r) /ₒ s | rfl | lemma | ore_localization.neg_def | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_neg (x : R[S⁻¹]) : (- x) + x = 0 | by induction x using ore_localization.ind with r s; simp | lemma | ore_localization.add_left_neg | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"ore_localization.ind"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numerator_hom_inj (hS : S ≤ R⁰) : function.injective (numerator_hom : R → R[S⁻¹]) | λ r₁ r₂ h,
begin
rw [numerator_hom_apply, numerator_hom_apply, ore_div_eq_iff] at h,
rcases h with ⟨u, v, h₁, h₂⟩,
simp only [S.coe_one, one_mul] at h₂,
rwa [←h₂, mul_cancel_right_mem_non_zero_divisor (hS (set_like.coe_mem u)), eq_comm] at h₁,
end | lemma | ore_localization.numerator_hom_inj | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_cancel_right_mem_non_zero_divisor",
"one_mul",
"set_like.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_of_non_zero_divisors [nontrivial R] (hS : S ≤ R⁰) : nontrivial R[S⁻¹] | ⟨⟨0, 1, λ h,
begin
rw [ore_localization.one_def, ore_localization.zero_def] at h,
apply non_zero_divisors.coe_ne_zero 1 (numerator_hom_inj hS h).symm
end⟩⟩ | lemma | ore_localization.nontrivial_of_non_zero_divisors | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"non_zero_divisors.coe_ne_zero",
"nontrivial",
"ore_localization.one_def",
"ore_localization.zero_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv : R[R⁰⁻¹] → R[R⁰⁻¹] | lift_expand (λ r s, if hr: r = (0 : R) then (0 : R[R⁰⁻¹])
else (s /ₒ ⟨r, λ _, eq_zero_of_ne_zero_of_mul_right_eq_zero hr⟩))
begin
intros r t s hst,
by_cases hr : r = 0,
{ simp [hr] },
{ by_cases ht : t = 0,
{ exfalso, apply non_zero_divisors.coe_ne_zero ⟨_, hst⟩, simp [ht, mul_zero] },
{ simp only [hr... | def | ore_localization.inv | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"eq_zero_of_ne_zero_of_mul_right_eq_zero",
"mul_eq_zero",
"mul_zero",
"non_zero_divisors.coe_ne_zero",
"ore_localization.expand",
"set_like.coe_mk"
] | The inversion of Ore fractions for a ring without zero divisors, satisying `0⁻¹ = 0` and
`(r /ₒ r')⁻¹ = r' /ₒ r` for `r ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_def {r : R} {s : R⁰} :
(r /ₒ s)⁻¹ = if hr: r = (0 : R) then (0 : R[R⁰⁻¹])
else (s /ₒ ⟨r, λ _, eq_zero_of_ne_zero_of_mul_right_eq_zero hr⟩) | rfl | lemma | ore_localization.inv_def | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"eq_zero_of_ne_zero_of_mul_right_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel (x : R[R⁰⁻¹]) (h : x ≠ 0) : x * x⁻¹ = 1 | begin
induction x using ore_localization.ind with r s,
rw [ore_localization.inv_def, ore_localization.one_def],
by_cases hr : r = 0,
{ exfalso, apply h, simp [hr] },
{ simp [hr], apply ore_localization.div_eq_one' }
end | lemma | ore_localization.mul_inv_cancel | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"mul_inv_cancel",
"ore_localization.div_eq_one'",
"ore_localization.ind",
"ore_localization.inv_def",
"ore_localization.one_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zero : (0 : R[R⁰⁻¹])⁻¹ = 0 | by { rw [ore_localization.zero_def, ore_localization.inv_def], simp } | lemma | ore_localization.inv_zero | ring_theory.ore_localization | src/ring_theory/ore_localization/basic.lean | [
"group_theory.monoid_localization",
"ring_theory.non_zero_divisors",
"ring_theory.ore_localization.ore_set",
"tactic.noncomm_ring"
] | [
"inv_zero",
"ore_localization.inv_def",
"ore_localization.zero_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ore_set {R : Type*} [monoid R] (S : submonoid R) | (ore_left_cancel : ∀ (r₁ r₂ : R) (s : S), ↑s * r₁ = s * r₂ → ∃ s' : S, r₁ * s' = r₂ * s')
(ore_num : R → S → R)
(ore_denom : R → S → S)
(ore_eq : ∀ (r : R) (s : S), r * ore_denom r s = s * ore_num r s) | class | ore_localization.ore_set | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [
"monoid",
"submonoid"
] | A submonoid `S` of a monoid `R` is (right) Ore if common factors on the left can be turned
into common factors on the right, and if each pair of `r : R` and `s : S` admits an Ore numerator
`v : R` and an Ore denominator `u : S` such that `r * u = s * v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_left_cancel (r₁ r₂ : R) (s : S) (h : ↑s * r₁ = s * r₂) : ∃ s' : S, r₁ * s' = r₂ * s' | ore_set.ore_left_cancel r₁ r₂ s h | lemma | ore_localization.ore_left_cancel | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [] | Common factors on the left can be turned into common factors on the right, a weak form of
cancellability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_num (r : R) (s : S) : R | ore_set.ore_num r s | def | ore_localization.ore_num | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [] | The Ore numerator of a fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_denom (r : R) (s : S) : S | ore_set.ore_denom r s | def | ore_localization.ore_denom | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [] | The Ore denominator of a fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_eq (r : R) (s : S) : r * (ore_denom r s) = s * (ore_num r s) | ore_set.ore_eq r s | lemma | ore_localization.ore_eq | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [] | The Ore condition of a fraction, expressed in terms of `ore_num` and `ore_denom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_condition (r : R) (s : S) : Σ' r' : R, Σ' s' : S, r * s' = s * r' | ⟨ore_num r s, ore_denom r s, ore_eq r s⟩ | def | ore_localization.ore_condition | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [] | The Ore condition bundled in a sigma type. This is useful in situations where we want to obtain
both witnesses and the condition for a given fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_set_bot : ore_set (⊥ : submonoid R) | { ore_left_cancel := λ _ _ s h,
⟨s, begin
rcases s with ⟨s, hs⟩,
rw submonoid.mem_bot at hs,
subst hs,
rw [set_like.coe_mk, one_mul, one_mul] at h,
subst h
end⟩,
ore_num := λ r _, r,
ore_denom := λ _ s, s,
ore_eq := λ _ s, by { rcases s with ⟨s, hs⟩, r... | instance | ore_localization.ore_set_bot | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [
"one_mul",
"set_like.coe_mk",
"submonoid",
"submonoid.mem_bot"
] | The trivial submonoid is an Ore set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_set_comm {R} [comm_monoid R] (S : submonoid R) : ore_set S | { ore_left_cancel := λ m n s h, ⟨s, by rw [mul_comm n s, mul_comm m s, h]⟩,
ore_num := λ r _, r,
ore_denom := λ _ s, s,
ore_eq := λ r s, by rw mul_comm } | instance | ore_localization.ore_set_comm | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [
"comm_monoid",
"mul_comm",
"submonoid"
] | Every submonoid of a commutative monoid is an Ore set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_set_of_cancel_monoid_with_zero
{R : Type*} [cancel_monoid_with_zero R] {S : submonoid R}
(ore_num : R → S → R) (ore_denom : R → S → S)
(ore_eq : ∀ (r : R) (s : S), r * (ore_denom r s) = s * (ore_num r s)) :
ore_set S | { ore_left_cancel := λ r₁ r₂ s h, ⟨s, mul_eq_mul_right_iff.mpr (mul_eq_mul_left_iff.mp h)⟩,
ore_num := ore_num,
ore_denom := ore_denom,
ore_eq := ore_eq } | def | ore_localization.ore_set_of_cancel_monoid_with_zero | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [
"cancel_monoid_with_zero",
"submonoid"
] | Cancellability in monoids with zeros can act as a replacement for the `ore_left_cancel`
condition of an ore set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ore_set_of_no_zero_divisors
{R : Type*} [ring R] [no_zero_divisors R] {S : submonoid R}
(ore_num : R → S → R) (ore_denom : R → S → S)
(ore_eq : ∀ (r : R) (s : S), r * (ore_denom r s) = s * (ore_num r s)) :
ore_set S | begin
letI : cancel_monoid_with_zero R := no_zero_divisors.to_cancel_monoid_with_zero,
exact ore_set_of_cancel_monoid_with_zero ore_num ore_denom ore_eq
end | def | ore_localization.ore_set_of_no_zero_divisors | ring_theory.ore_localization | src/ring_theory/ore_localization/ore_set.lean | [
"algebra.ring.regular",
"group_theory.submonoid.basic"
] | [
"cancel_monoid_with_zero",
"no_zero_divisors",
"no_zero_divisors.to_cancel_monoid_with_zero",
"ring",
"submonoid"
] | In rings without zero divisors, the first (cancellability) condition is always fulfilled,
it suffices to give a proof for the Ore condition itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_le (n : with_bot ℕ) : submodule R R[X] | ⨅ k : ℕ, ⨅ h : ↑k > n, (lcoeff R k).ker | def | polynomial.degree_le | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"submodule",
"with_bot"
] | The `R`-submodule of `R[X]` consisting of polynomials of degree ≤ `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_lt (n : ℕ) : submodule R R[X] | ⨅ k : ℕ, ⨅ h : k ≥ n, (lcoeff R k).ker | def | polynomial.degree_lt | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | The `R`-submodule of `R[X]` consisting of polynomials of degree < `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_degree_le {n : with_bot ℕ} {f : R[X]} :
f ∈ degree_le R n ↔ degree f ≤ n | by simp only [degree_le, submodule.mem_infi, degree_le_iff_coeff_zero, linear_map.mem_ker]; refl | theorem | polynomial.mem_degree_le | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"linear_map.mem_ker",
"submodule.mem_infi",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_le_mono {m n : with_bot ℕ} (H : m ≤ n) :
degree_le R m ≤ degree_le R n | λ f hf, mem_degree_le.2 (le_trans (mem_degree_le.1 hf) H) | theorem | polynomial.degree_le_mono | ring_theory.polynomial | src/ring_theory/polynomial/basic.lean | [
"algebra.char_p.basic",
"algebra.geom_sum",
"data.mv_polynomial.comm_ring",
"data.mv_polynomial.equiv",
"ring_theory.polynomial.content",
"ring_theory.unique_factorization_domain"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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