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
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unop_map_list_prod [semiring R] [semiring S] (f : R ≃+* Sᵐᵒᵖ) (l : list R) :
mul_opposite.unop (f l.prod) = (l.map (mul_opposite.unop ∘ f)).reverse.prod | unop_map_list_prod f l | lemma | ring_equiv.unop_map_list_prod | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"mul_opposite.unop",
"semiring",
"unop_map_list_prod"
] | An isomorphism into the opposite ring acts on the product by acting on the reversed elements | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_multiset_prod [comm_semiring R] [comm_semiring S] (f : R ≃+* S)
(s : multiset R) : f s.prod = (s.map f).prod | map_multiset_prod f s | lemma | ring_equiv.map_multiset_prod | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"comm_semiring",
"map_multiset_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_multiset_sum [non_assoc_semiring R] [non_assoc_semiring S]
(f : R ≃+* S) (s : multiset R) : f s.sum = (s.map f).sum | map_multiset_sum f s | lemma | ring_equiv.map_multiset_sum | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"multiset",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod [comm_semiring R] [comm_semiring S] (g : R ≃+* S) (f : α → R)
(s : finset α) : g (∏ x in s, f x) = ∏ x in s, g (f x) | map_prod g f s | lemma | ring_equiv.map_prod | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"comm_semiring",
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum [non_assoc_semiring R] [non_assoc_semiring S]
(g : R ≃+* S) (f : α → R) (s : finset α) : g (∑ x in s, f x) = ∑ x in s, g (f x) | map_sum g f s | lemma | ring_equiv.map_sum | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"finset",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : multiset α → α | foldr (*) (λ x y z, by simp [mul_left_comm]) 1 | def | multiset.prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_left_comm",
"multiset"
] | Product of a multiset given a commutative monoid structure on `α`.
`prod {a, b, c} = a * b * c` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_eq_foldr (s : multiset α) : prod s = foldr (*) (λ x y z, by simp [mul_left_comm]) 1 s | rfl | lemma | multiset.prod_eq_foldr | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_left_comm",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_foldl (s : multiset α) : prod s = foldl (*) (λ x y z, by simp [mul_right_comm]) 1 s | (foldr_swap _ _ _ _).trans (by simp [mul_comm]) | lemma | multiset.prod_eq_foldl | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_comm",
"mul_right_comm",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod (l : list α) : prod ↑l = l.prod | prod_eq_foldl _ | lemma | multiset.coe_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_to_list (s : multiset α) : s.to_list.prod = s.prod | begin
conv_rhs { rw ←coe_to_list s },
rw coe_prod,
end | lemma | multiset.prod_to_list | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero : @prod α _ 0 = 1 | rfl | lemma | multiset.prod_zero | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_cons (a : α) (s) : prod (a ::ₘ s) = a * prod s | foldr_cons _ _ _ _ _ | lemma | multiset.prod_cons | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_erase [decidable_eq α] (h : a ∈ s) : a * (s.erase a).prod = s.prod | by rw [← s.coe_to_list, coe_erase, coe_prod, coe_prod, list.prod_erase (mem_to_list.2 h)] | lemma | multiset.prod_erase | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_erase"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_erase [decidable_eq ι] {a : ι} (h : a ∈ m) :
f a * ((m.erase a).map f).prod = (m.map f).prod | by rw [← m.coe_to_list, coe_erase, coe_map, coe_map, coe_prod, coe_prod,
list.prod_map_erase f (mem_to_list.2 h)] | lemma | multiset.prod_map_erase | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_map_erase"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_singleton (a : α) : prod {a} = a | by simp only [mul_one, prod_cons, ←cons_zero, eq_self_iff_true, prod_zero] | lemma | multiset.prod_singleton | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pair (a b : α) : ({a, b} : multiset α).prod = a * b | by rw [insert_eq_cons, prod_cons, prod_singleton] | lemma | multiset.prod_pair | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_add (s t : multiset α) : prod (s + t) = prod s * prod t | quotient.induction_on₂ s t $ λ l₁ l₂, by simp | lemma | multiset.prod_add | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_nsmul (m : multiset α) : ∀ (n : ℕ), (n • m).prod = m.prod ^ n | | 0 := by { rw [zero_nsmul, pow_zero], refl }
| (n + 1) :=
by rw [add_nsmul, one_nsmul, pow_add, pow_one, prod_add, prod_nsmul n] | lemma | multiset.prod_nsmul | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset",
"pow_add",
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_replicate (n : ℕ) (a : α) : (replicate n a).prod = a ^ n | by simp [replicate, list.prod_replicate] | lemma | multiset.prod_replicate | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_replicate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_eq_pow_single [decidable_eq ι] (i : ι) (hf : ∀ i' ≠ i, i' ∈ m → f i' = 1) :
(m.map f).prod = f i ^ m.count i | begin
induction m using quotient.induction_on with l,
simp [list.prod_map_eq_pow_single i f hf],
end | lemma | multiset.prod_map_eq_pow_single | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_map_eq_pow_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_pow_single [decidable_eq α] (a : α) (h : ∀ a' ≠ a, a' ∈ s → a' = 1) :
s.prod = a ^ (s.count a) | begin
induction s using quotient.induction_on with l,
simp [list.prod_eq_pow_single a h],
end | lemma | multiset.prod_eq_pow_single | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_eq_pow_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_count [decidable_eq α] (a : α) : a ^ s.count a = (s.filter (eq a)).prod | by rw [filter_eq, prod_replicate] | lemma | multiset.pow_count | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_hom [comm_monoid β] (s : multiset α) {F : Type*} [monoid_hom_class F α β] (f : F) :
(s.map f).prod = f s.prod | quotient.induction_on s $ λ l, by simp only [l.prod_hom f, quot_mk_to_coe, coe_map, coe_prod] | lemma | multiset.prod_hom | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"monoid_hom_class",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_hom' [comm_monoid β] (s : multiset ι) {F : Type*} [monoid_hom_class F α β] (f : F)
(g : ι → α) : (s.map $ λ i, f $ g i).prod = f (s.map g).prod | by { convert (s.map g).prod_hom f, exact (map_map _ _ _).symm } | lemma | multiset.prod_hom' | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"monoid_hom_class",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_hom₂ [comm_monoid β] [comm_monoid γ] (s : multiset ι) (f : α → β → γ)
(hf : ∀ a b c d, f (a * b) (c * d) = f a c * f b d) (hf' : f 1 1 = 1) (f₁ : ι → α) (f₂ : ι → β) :
(s.map $ λ i, f (f₁ i) (f₂ i)).prod = f (s.map f₁).prod (s.map f₂).prod | quotient.induction_on s $ λ l,
by simp only [l.prod_hom₂ f hf hf', quot_mk_to_coe, coe_map, coe_prod] | lemma | multiset.prod_hom₂ | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_hom_rel [comm_monoid β] (s : multiset ι) {r : α → β → Prop} {f : ι → α} {g : ι → β}
(h₁ : r 1 1) (h₂ : ∀ ⦃a b c⦄, r b c → r (f a * b) (g a * c)) :
r (s.map f).prod (s.map g).prod | quotient.induction_on s $ λ l,
by simp only [l.prod_hom_rel h₁ h₂, quot_mk_to_coe, coe_map, coe_prod] | lemma | multiset.prod_hom_rel | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_one : prod (m.map (λ i, (1 : α))) = 1 | by rw [map_const, prod_replicate, one_pow] | lemma | multiset.prod_map_one | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_mul : (m.map $ λ i, f i * g i).prod = (m.map f).prod * (m.map g).prod | m.prod_hom₂ (*) mul_mul_mul_comm (mul_one _) _ _ | lemma | multiset.prod_map_mul | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_mul_mul_comm",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_neg [has_distrib_neg α] (s : multiset α) :
(s.map has_neg.neg).prod = (-1) ^ s.card * s.prod | by { refine quotient.ind _ s, simp } | lemma | multiset.prod_map_neg | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"has_distrib_neg",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_pow {n : ℕ} : (m.map $ λ i, f i ^ n).prod = (m.map f).prod ^ n | m.prod_hom' (pow_monoid_hom n : α →* α) f | lemma | multiset.prod_map_pow | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"pow_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_prod_map (m : multiset β) (n : multiset γ) {f : β → γ → α} :
prod (m.map $ λ a, prod $ n.map $ λ b, f a b) = prod (n.map $ λ b, prod $ m.map $ λ a, f a b) | multiset.induction_on m (by simp) (λ a m ih, by simp [ih]) | lemma | multiset.prod_map_prod_map | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"ih",
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_induction (p : α → Prop) (s : multiset α) (p_mul : ∀ a b, p a → p b → p (a * b))
(p_one : p 1) (p_s : ∀ a ∈ s, p a) :
p s.prod | begin
rw prod_eq_foldr,
exact foldr_induction (*) (λ x y z, by simp [mul_left_comm]) 1 p s p_mul p_one p_s,
end | lemma | multiset.prod_induction | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_left_comm",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_induction_nonempty (p : α → Prop) (p_mul : ∀ a b, p a → p b → p (a * b))
(hs : s ≠ ∅) (p_s : ∀ a ∈ s, p a) :
p s.prod | begin
revert s,
refine multiset.induction _ _,
{ intro h,
exfalso,
simpa using h },
intros a s hs hsa hpsa,
rw prod_cons,
by_cases hs_empty : s = ∅,
{ simp [hs_empty, hpsa a] },
have hps : ∀ x, x ∈ s → p x, from λ x hxs, hpsa x (mem_cons_of_mem hxs),
exact p_mul a s.prod (hpsa a (mem_cons_self... | lemma | multiset.prod_induction_nonempty | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mem_cons_of_mem",
"multiset.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_dvd_prod_of_le (h : s ≤ t) : s.prod ∣ t.prod | by { obtain ⟨z, rfl⟩ := exists_add_of_le h, simp only [prod_add, dvd_mul_right] } | lemma | multiset.prod_dvd_prod_of_le | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"dvd_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_dvd_prod_of_dvd [comm_monoid β] {S : multiset α} (g1 g2 : α → β)
(h : ∀ a ∈ S, g1 a ∣ g2 a) :
(multiset.map g1 S).prod ∣ (multiset.map g2 S).prod | begin
apply multiset.induction_on' S, { simp },
intros a T haS _ IH,
simp [mul_dvd_mul (h a haS) IH]
end | lemma | multiset.prod_dvd_prod_of_dvd | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"mul_dvd_mul",
"multiset",
"multiset.induction_on'",
"multiset.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_monoid_hom : multiset α →+ α | { to_fun := sum,
map_zero' := sum_zero,
map_add' := sum_add } | def | multiset.sum_add_monoid_hom | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | `multiset.sum`, the sum of the elements of a multiset, promoted to a morphism of
`add_comm_monoid`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sum_add_monoid_hom : (sum_add_monoid_hom : multiset α → α) = sum | rfl | lemma | multiset.coe_sum_add_monoid_hom | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_zero {s : multiset α} (h : (0 : α) ∈ s) : s.prod = 0 | begin
rcases multiset.exists_cons_of_mem h with ⟨s', hs'⟩,
simp [hs', multiset.prod_cons]
end | lemma | multiset.prod_eq_zero | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset",
"multiset.exists_cons_of_mem",
"multiset.prod_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_zero_iff : s.prod = 0 ↔ (0 : α) ∈ s | quotient.induction_on s $ λ l, by { rw [quot_mk_to_coe, coe_prod], exact list.prod_eq_zero_iff } | lemma | multiset.prod_eq_zero_iff | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ne_zero (h : (0 : α) ∉ s) : s.prod ≠ 0 | mt prod_eq_zero_iff.1 h | lemma | multiset.prod_ne_zero | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_inv' (m : multiset α) : (m.map has_inv.inv).prod = m.prod⁻¹ | m.prod_hom (inv_monoid_hom : α →* α) | lemma | multiset.prod_map_inv' | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"inv_monoid_hom",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_inv : (m.map $ λ i, (f i)⁻¹).prod = (m.map f).prod ⁻¹ | by { convert (m.map f).prod_map_inv', rw map_map } | lemma | multiset.prod_map_inv | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_div : (m.map $ λ i, f i / g i).prod = (m.map f).prod / (m.map g).prod | m.prod_hom₂ (/) mul_div_mul_comm (div_one _) _ _ | lemma | multiset.prod_map_div | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"div_one",
"mul_div_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_zpow {n : ℤ} : (m.map $ λ i, f i ^ n).prod = (m.map f).prod ^ n | by { convert (m.map f).prod_hom (zpow_group_hom _ : α →* α), rw map_map, refl } | lemma | multiset.prod_map_zpow | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"zpow_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_map_mul_left : sum (s.map (λ i, a * f i)) = a * sum (s.map f) | multiset.induction_on s (by simp) (λ i s ih, by simp [ih, mul_add]) | lemma | multiset.sum_map_mul_left | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"ih",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_map_mul_right : sum (s.map (λ i, f i * a)) = sum (s.map f) * a | multiset.induction_on s (by simp) (λ a s ih, by simp [ih, add_mul]) | lemma | multiset.sum_map_mul_right | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"ih",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_sum {a : α} {s : multiset α} : (∀ x ∈ s, a ∣ x) → a ∣ s.sum | multiset.induction_on s (λ _, dvd_zero _)
(λ x s ih h, by { rw sum_cons, exact dvd_add
(h _ (mem_cons_self _ _)) (ih $ λ y hy, h _ $ mem_cons.2 $ or.inr hy) }) | lemma | multiset.dvd_sum | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"dvd_add",
"dvd_zero",
"ih",
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_prod_of_one_le : (∀ x ∈ s, (1 : α) ≤ x) → 1 ≤ s.prod | quotient.induction_on s $ λ l hl, by simpa using list.one_le_prod_of_one_le hl | lemma | multiset.one_le_prod_of_one_le | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.one_le_prod_of_one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_le_prod : (∀ x ∈ s, (1 : α) ≤ x) → ∀ x ∈ s, x ≤ s.prod | quotient.induction_on s $ λ l hl x hx, by simpa using list.single_le_prod hl x hx | lemma | multiset.single_le_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.single_le_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_pow_card (s : multiset α) (n : α) (h : ∀ x ∈ s, x ≤ n) : s.prod ≤ n ^ s.card | begin
induction s using quotient.induction_on,
simpa using list.prod_le_pow_card _ _ h,
end | lemma | multiset.prod_le_pow_card | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.prod_le_pow_card",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
all_one_of_le_one_le_of_prod_eq_one :
(∀ x ∈ s, (1 : α) ≤ x) → s.prod = 1 → ∀ x ∈ s, x = (1 : α) | begin
apply quotient.induction_on s,
simp only [quot_mk_to_coe, coe_prod, mem_coe],
exact λ l, list.all_one_of_le_one_le_of_prod_eq_one,
end | lemma | multiset.all_one_of_le_one_le_of_prod_eq_one | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"list.all_one_of_le_one_le_of_prod_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_of_rel_le (h : s.rel (≤) t) : s.prod ≤ t.prod | begin
induction h with _ _ _ _ rh _ rt,
{ refl },
{ rw [prod_cons, prod_cons],
exact mul_le_mul' rh rt }
end | lemma | multiset.prod_le_prod_of_rel_le | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_le_prod_map {s : multiset ι} (f : ι → α) (g : ι → α) (h : ∀ i, i ∈ s → f i ≤ g i) :
(s.map f).prod ≤ (s.map g).prod | prod_le_prod_of_rel_le $ rel_map.2 $ rel_refl_of_refl_on h | lemma | multiset.prod_map_le_prod_map | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_le_prod (f : α → α) (h : ∀ x, x ∈ s → f x ≤ x) : (s.map f).prod ≤ s.prod | prod_le_prod_of_rel_le $ rel_map_left.2 $ rel_refl_of_refl_on h | lemma | multiset.prod_map_le_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_map (f : α → α) (h : ∀ x, x ∈ s → x ≤ f x) : s.prod ≤ (s.map f).prod | @prod_map_le_prod αᵒᵈ _ _ f h | lemma | multiset.prod_le_prod_map | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_card_le_prod (h : ∀ x ∈ s, a ≤ x) : a ^ s.card ≤ s.prod | by { rw [←multiset.prod_replicate, ←multiset.map_const], exact prod_map_le_prod _ h } | lemma | multiset.pow_card_le_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_nonneg [ordered_comm_semiring α] {m : multiset α} (h : ∀ a ∈ m, (0 : α) ≤ a) :
0 ≤ m.prod | begin
revert h,
refine m.induction_on _ _,
{ rintro -, rw prod_zero, exact zero_le_one },
intros a s hs ih,
rw prod_cons,
exact mul_nonneg (ih _ $ mem_cons_self _ _) (hs $ λ a ha, ih _ $ mem_cons_of_mem ha),
end | lemma | multiset.prod_nonneg | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"ih",
"mem_cons_of_mem",
"multiset",
"ordered_comm_semiring",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one [comm_monoid α] {m : multiset α} (h : ∀ x ∈ m, x = (1 : α)) : m.prod = 1 | begin
induction m using quotient.induction_on with l,
simp [list.prod_eq_one h],
end | lemma | multiset.prod_eq_one | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"list.prod_eq_one",
"multiset"
] | Slightly more general version of `multiset.prod_eq_one_iff` for a non-ordered `monoid` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_prod_of_mem [canonically_ordered_monoid α] {m : multiset α} {a : α} (h : a ∈ m) :
a ≤ m.prod | begin
obtain ⟨m', rfl⟩ := exists_cons_of_mem h,
rw [prod_cons],
exact _root_.le_mul_right (le_refl a),
end | lemma | multiset.le_prod_of_mem | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"canonically_ordered_monoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_prod_of_submultiplicative_on_pred [comm_monoid α] [ordered_comm_monoid β]
(f : α → β) (p : α → Prop) (h_one : f 1 = 1) (hp_one : p 1)
(h_mul : ∀ a b, p a → p b → f (a * b) ≤ f a * f b)
(hp_mul : ∀ a b, p a → p b → p (a * b)) (s : multiset α) (hps : ∀ a, a ∈ s → p a) :
f s.prod ≤ (s.map f).prod | begin
revert s,
refine multiset.induction _ _,
{ simp [le_of_eq h_one] },
intros a s hs hpsa,
have hps : ∀ x, x ∈ s → p x, from λ x hx, hpsa x (mem_cons_of_mem hx),
have hp_prod : p s.prod, from prod_induction p s hp_mul hp_one hps,
rw [prod_cons, map_cons, prod_cons],
exact (h_mul a s.prod (hpsa a (mem... | lemma | multiset.le_prod_of_submultiplicative_on_pred | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"map_cons",
"mem_cons_of_mem",
"mul_le_mul_left'",
"multiset",
"multiset.induction",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_prod_of_submultiplicative [comm_monoid α] [ordered_comm_monoid β]
(f : α → β) (h_one : f 1 = 1) (h_mul : ∀ a b, f (a * b) ≤ f a * f b) (s : multiset α) :
f s.prod ≤ (s.map f).prod | le_prod_of_submultiplicative_on_pred f (λ i, true) h_one trivial (λ x y _ _ , h_mul x y) (by simp)
s (by simp) | lemma | multiset.le_prod_of_submultiplicative | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"multiset",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_prod_nonempty_of_submultiplicative_on_pred [comm_monoid α] [ordered_comm_monoid β]
(f : α → β) (p : α → Prop) (h_mul : ∀ a b, p a → p b → f (a * b) ≤ f a * f b)
(hp_mul : ∀ a b, p a → p b → p (a * b)) (s : multiset α) (hs_nonempty : s ≠ ∅)
(hs : ∀ a, a ∈ s → p a) :
f s.prod ≤ (s.map f).prod | begin
revert s,
refine multiset.induction _ _,
{ intro h,
exfalso,
exact h rfl },
rintros a s hs hsa_nonempty hsa_prop,
rw [prod_cons, map_cons, prod_cons],
by_cases hs_empty : s = ∅,
{ simp [hs_empty] },
have hsa_restrict : (∀ x, x ∈ s → p x), from λ x hx, hsa_prop x (mem_cons_of_mem hx),
hav... | lemma | multiset.le_prod_nonempty_of_submultiplicative_on_pred | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"map_cons",
"mem_cons_of_mem",
"mul_le_mul_left'",
"multiset",
"multiset.induction",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_prod_nonempty_of_submultiplicative [comm_monoid α] [ordered_comm_monoid β]
(f : α → β) (h_mul : ∀ a b, f (a * b) ≤ f a * f b) (s : multiset α) (hs_nonempty : s ≠ ∅) :
f s.prod ≤ (s.map f).prod | le_prod_nonempty_of_submultiplicative_on_pred f (λ i, true) (by simp [h_mul]) (by simp) s
hs_nonempty (by simp) | lemma | multiset.le_prod_nonempty_of_submultiplicative | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"multiset",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_map_singleton (s : multiset α) : (s.map (λ a, ({a} : multiset α))).sum = s | multiset.induction_on s (by simp) (by simp) | lemma | multiset.sum_map_singleton | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset",
"multiset.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sum_le_sum_abs [linear_ordered_add_comm_group α] {s : multiset α} :
abs s.sum ≤ (s.map abs).sum | le_sum_of_subadditive _ abs_zero abs_add s | lemma | multiset.abs_sum_le_sum_abs | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"abs_add",
"abs_zero",
"linear_ordered_add_comm_group",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_nat_mod (s : multiset ℕ) (n : ℕ) : s.sum % n = (s.map (% n)).sum % n | by induction s using multiset.induction; simp [nat.add_mod, *] | lemma | multiset.sum_nat_mod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset",
"multiset.induction",
"nat.add_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_nat_mod (s : multiset ℕ) (n : ℕ) : s.prod % n = (s.map (% n)).prod % n | by induction s using multiset.induction; simp [nat.mul_mod, *] | lemma | multiset.prod_nat_mod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"multiset",
"multiset.induction",
"nat.mul_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_int_mod (s : multiset ℤ) (n : ℤ) : s.sum % n = (s.map (% n)).sum % n | by induction s using multiset.induction; simp [int.add_mod, *] | lemma | multiset.sum_int_mod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"int.add_mod",
"multiset",
"multiset.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_int_mod (s : multiset ℤ) (n : ℤ) : s.prod % n = (s.map (% n)).prod % n | by induction s using multiset.induction; simp [int.mul_mod, *] | lemma | multiset.prod_int_mod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"int.mul_mod",
"multiset",
"multiset.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_multiset_prod [comm_monoid α] [comm_monoid β] {F : Type*} [monoid_hom_class F α β]
(f : F) (s : multiset α) : f s.prod = (s.map f).prod | (s.prod_hom f).symm | lemma | map_multiset_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"monoid_hom_class",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_multiset_prod [comm_monoid α] [comm_monoid β] (f : α →* β)
(s : multiset α) : f s.prod = (s.map f).prod | (s.prod_hom f).symm | lemma | monoid_hom.map_multiset_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/basic.lean | [
"data.list.big_operators.basic",
"data.multiset.basic"
] | [
"comm_monoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_prod [comm_monoid α] {s : multiset α} {a : α} : a ∈ s → a ∣ s.prod | quotient.induction_on s (λ l a h, by simpa using list.dvd_prod h) a | lemma | multiset.dvd_prod | algebra.big_operators.multiset | src/algebra/big_operators/multiset/lemmas.lean | [
"data.list.big_operators.lemmas",
"algebra.big_operators.multiset.basic"
] | [
"comm_monoid",
"list.dvd_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one_iff [canonically_ordered_monoid α] {m : multiset α} :
m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) | quotient.induction_on m $ λ l, by simpa using list.prod_eq_one_iff l | lemma | multiset.prod_eq_one_iff | algebra.big_operators.multiset | src/algebra/big_operators/multiset/lemmas.lean | [
"data.list.big_operators.lemmas",
"algebra.big_operators.multiset.basic"
] | [
"canonically_ordered_monoid",
"list.prod_eq_one_iff",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_sum_right (s : multiset α) (a : α) (h : ∀ b ∈ s, commute a b) :
commute a s.sum | begin
induction s using quotient.induction_on,
rw [quot_mk_to_coe, coe_sum],
exact commute.list_sum_right _ _ h,
end | lemma | commute.multiset_sum_right | algebra.big_operators.multiset | src/algebra/big_operators/multiset/lemmas.lean | [
"data.list.big_operators.lemmas",
"algebra.big_operators.multiset.basic"
] | [
"commute",
"commute.list_sum_right",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_sum_left (s : multiset α) (b : α) (h : ∀ a ∈ s, commute a b) :
commute s.sum b | (commute.multiset_sum_right _ _ $ λ a ha, (h _ ha).symm).symm | lemma | commute.multiset_sum_left | algebra.big_operators.multiset | src/algebra/big_operators/multiset/lemmas.lean | [
"data.list.big_operators.lemmas",
"algebra.big_operators.multiset.basic"
] | [
"commute",
"commute.multiset_sum_right",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
BoolRing | bundled boolean_ring | def | BoolRing | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"boolean_ring"
] | The category of Boolean rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (α : Type*) [boolean_ring α] : BoolRing | bundled.of α | def | BoolRing.of | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"BoolRing",
"boolean_ring"
] | Construct a bundled `BoolRing` from a `boolean_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (α : Type*) [boolean_ring α] : ↥(of α) = α | rfl | lemma | BoolRing.coe_of | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"boolean_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_CommRing : has_forget₂ BoolRing CommRing | bundled_hom.forget₂ _ _ | instance | BoolRing.has_forget_to_CommRing | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"BoolRing",
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β | { hom := e,
inv := e.symm,
hom_inv_id' := by { ext, exact e.symm_apply_apply _ },
inv_hom_id' := by { ext, exact e.apply_symm_apply _ } } | def | BoolRing.iso.mk | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [] | Constructs an isomorphism of Boolean rings from a ring isomorphism between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
BoolRing.has_forget_to_BoolAlg : has_forget₂ BoolRing BoolAlg | { forget₂ := { obj := λ X, BoolAlg.of (as_boolalg X), map := λ X Y, ring_hom.as_boolalg } } | instance | BoolRing.has_forget_to_BoolAlg | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"BoolAlg",
"BoolAlg.of",
"BoolRing",
"as_boolalg",
"ring_hom.as_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
BoolAlg.has_forget_to_BoolRing : has_forget₂ BoolAlg BoolRing | { forget₂ := { obj := λ X, BoolRing.of (as_boolring X),
map := λ X Y, bounded_lattice_hom.as_boolring } } | instance | BoolAlg.has_forget_to_BoolRing | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"BoolAlg",
"BoolRing",
"BoolRing.of",
"as_boolring",
"bounded_lattice_hom.as_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
BoolRing_equiv_BoolAlg : BoolRing ≌ BoolAlg | equivalence.mk (forget₂ BoolRing BoolAlg) (forget₂ BoolAlg BoolRing)
(nat_iso.of_components (λ X, BoolRing.iso.mk $ (ring_equiv.as_boolring_as_boolalg X).symm) $
λ X Y f, rfl)
(nat_iso.of_components (λ X, BoolAlg.iso.mk $ order_iso.as_boolalg_as_boolring X) $
λ X Y f, rfl) | def | BoolRing_equiv_BoolAlg | algebra.category | src/algebra/category/BoolRing.lean | [
"algebra.category.Ring.basic",
"algebra.ring.boolean_ring",
"order.category.BoolAlg"
] | [
"BoolAlg",
"BoolAlg.iso.mk",
"BoolRing",
"BoolRing.iso.mk",
"order_iso.as_boolalg_as_boolring",
"ring_equiv.as_boolring_as_boolalg"
] | The equivalence between Boolean rings and Boolean algebras. This is actually an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
GroupWithZero | bundled group_with_zero | def | GroupWithZero | algebra.category | src/algebra/category/GroupWithZero.lean | [
"category_theory.category.Bipointed",
"algebra.category.Mon.basic"
] | [
"group_with_zero"
] | The category of groups with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (α : Type*) [group_with_zero α] : GroupWithZero | bundled.of α | def | GroupWithZero.of | algebra.category | src/algebra/category/GroupWithZero.lean | [
"category_theory.category.Bipointed",
"algebra.category.Mon.basic"
] | [
"GroupWithZero",
"group_with_zero"
] | Construct a bundled `GroupWithZero` from a `group_with_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_Bipointed : has_forget₂ GroupWithZero Bipointed | { forget₂ := { obj := λ X, ⟨X, 0, 1⟩, map := λ X Y f, ⟨f, f.map_zero', f.map_one'⟩ } } | instance | GroupWithZero.has_forget_to_Bipointed | algebra.category | src/algebra/category/GroupWithZero.lean | [
"category_theory.category.Bipointed",
"algebra.category.Mon.basic"
] | [
"Bipointed",
"GroupWithZero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Mon : has_forget₂ GroupWithZero Mon | { forget₂ := { obj := λ X, ⟨X⟩, map := λ X Y, monoid_with_zero_hom.to_monoid_hom } } | instance | GroupWithZero.has_forget_to_Mon | algebra.category | src/algebra/category/GroupWithZero.lean | [
"category_theory.category.Bipointed",
"algebra.category.Mon.basic"
] | [
"GroupWithZero",
"Mon"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.mk {α β : GroupWithZero.{u}} (e : α ≃* β) : α ≅ β | { hom := e,
inv := e.symm,
hom_inv_id' := by { ext, exact e.symm_apply_apply _ },
inv_hom_id' := by { ext, exact e.apply_symm_apply _ } } | def | GroupWithZero.iso.mk | algebra.category | src/algebra/category/GroupWithZero.lean | [
"category_theory.category.Bipointed",
"algebra.category.Mon.basic"
] | [] | Constructs an isomorphism of groups with zero from a group isomorphism between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Algebra | (carrier : Type v)
[is_ring : ring carrier]
[is_algebra : algebra R carrier] | structure | Algebra | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra",
"ring"
] | The category of R-algebras and their morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_Ring : has_forget₂ (Algebra.{v} R) Ring.{v} | { forget₂ :=
{ obj := λ A, Ring.of A,
map := λ A₁ A₂ f, alg_hom.to_ring_hom f, } } | instance | Algebra.has_forget_to_Ring | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Ring.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Module : has_forget₂ (Algebra.{v} R) (Module.{v} R) | { forget₂ :=
{ obj := λ M, Module.of R M,
map := λ M₁ M₂ f, alg_hom.to_linear_map f, } } | instance | Algebra.has_forget_to_Module | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Module.of",
"alg_hom.to_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (X : Type v) [ring X] [algebra R X] : Algebra.{v} R | ⟨X⟩ | def | Algebra.of | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra",
"ring"
] | The object in the category of R-algebras associated to a type equipped with the appropriate
typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {R : Type u} [comm_ring R] {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y]
(f : X →ₐ[R] Y) : of R X ⟶ of R Y | f | def | Algebra.of_hom | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra",
"comm_ring",
"ring"
] | Typecheck a `alg_hom` as a morphism in `Algebra R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {R : Type u} [comm_ring R]
{X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y] (f : X →ₐ[R] Y) (x : X) :
of_hom f x = f x | rfl | lemma | Algebra.of_hom_apply | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra",
"comm_ring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (X : Type u) [ring X] [algebra R X] : (of R X : Type u) = X | rfl | lemma | Algebra.coe_of | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_self_iso (M : Algebra.{v} R) : Algebra.of R M ≅ M | { hom := 𝟙 M, inv := 𝟙 M } | def | Algebra.of_self_iso | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"Algebra.of"
] | Forgetting to the underlying type and then building the bundled object returns the original
algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (m : M) : (𝟙 M : M → M) m = m | rfl | lemma | Algebra.id_apply | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp (f : M ⟶ N) (g : N ⟶ U) :
((f ≫ g) : M → U) = g ∘ f | rfl | lemma | Algebra.coe_comp | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free : Type u ⥤ Algebra.{u} R | { obj := λ S,
{ carrier := free_algebra R S,
is_ring := algebra.semiring_to_ring R },
map := λ S T f, free_algebra.lift _ $ (free_algebra.ι _) ∘ f,
-- obviously can fill the next two goals, but it is slow
map_id' := by { intros X, ext1, simp only [free_algebra.ι_comp_lift], refl },
map_comp' := by { intro... | def | Algebra.free | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"algebra.semiring_to_ring",
"category_theory.coe_comp",
"free",
"free_algebra",
"free_algebra.lift",
"free_algebra.lift_ι_apply",
"free_algebra.ι",
"free_algebra.ι_comp_lift"
] | The "free algebra" functor, sending a type `S` to the free algebra on `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj : free.{u} R ⊣ forget (Algebra.{u} R) | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X A, (free_algebra.lift _).symm,
-- Relying on `obviously` to fill out these proofs is very slow :(
hom_equiv_naturality_left_symm' := by { intros, ext,
simp only [free_map, equiv.symm_symm, free_algebra.lift_ι_apply, category_theory.coe_comp,
function.comp_app,... | def | Algebra.adj | algebra.category.Algebra | src/algebra/category/Algebra/basic.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.free_algebra",
"algebra.category.Ring.basic",
"algebra.category.Module.basic"
] | [
"adj",
"category_theory.coe_comp",
"equiv.symm_symm",
"free_algebra.lift",
"free_algebra.lift_symm_apply",
"free_algebra.lift_ι_apply"
] | The free/forget adjunction for `R`-algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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