statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
geom_sum_mul [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (x - 1) = x ^ n - 1 | begin
have := (commute.one_right x).geom_sum₂_mul n,
rw [one_pow, geom_sum₂_with_one] at this,
exact this
end | theorem | geom_sum_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.one_right",
"geom_sum₂_mul",
"geom_sum₂_with_one",
"one_pow",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_geom_sum [ring α] (x : α) (n : ℕ) :
(x - 1) * (∑ i in range n, x ^ i) = x ^ n - 1 | op_injective $ by simpa using geom_sum_mul (op x) n | lemma | mul_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_mul_neg [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n | begin
have := congr_arg has_neg.neg (geom_sum_mul x n),
rw [neg_sub, ← mul_neg, neg_sub] at this,
exact this
end | theorem | geom_sum_mul_neg | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_mul",
"mul_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_geom_sum [ring α] (x : α) (n : ℕ) :
(1 - x) * (∑ i in range n, x ^ i) = 1 - x ^ n | op_injective $ by simpa using geom_sum_mul_neg (op x) n | lemma | mul_neg_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_mul_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_comm {α : Type u} [semiring α] {x y : α} (n : ℕ)
(h : commute x y) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) | begin
cases n, { simp },
simp only [nat.succ_eq_add_one, nat.add_sub_cancel],
rw ← finset.sum_flip,
refine finset.sum_congr rfl (λ i hi, _),
simpa [nat.sub_sub_self (nat.succ_le_succ_iff.mp (finset.mem_range.mp hi))] using h.pow_pow _ _
end | lemma | commute.geom_sum₂_comm | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_comm {α : Type u} [comm_semiring α] (x y : α) (n : ℕ) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) | (commute.all x y).geom_sum₂_comm n | lemma | geom_sum₂_comm | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_semiring",
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂ [division_ring α] {x y : α} (h' : commute x y) (h : x ≠ y)
(n : ℕ) : (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) | have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h'.geom_sum₂_mul, mul_div_cancel _ this] | theorem | commute.geom_sum₂ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"division_ring",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom₂_sum [field α] {x y : α} (h : x ≠ y) (n : ℕ) :
(∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) | (commute.all x y).geom_sum₂ h n | theorem | geom₂_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.all",
"field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_eq [division_ring α] {x : α} (h : x ≠ 1) (n : ℕ) :
(∑ i in range n, x ^ i) = (x ^ n - 1) / (x - 1) | have x - 1 ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← geom_sum_mul, mul_div_cancel _ this] | theorem | geom_sum_eq | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"division_ring",
"geom_sum_mul",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.mul_geom_sum₂_Ico [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) | begin
rw [sum_Ico_eq_sub _ hmn],
have : ∑ k in range m, x ^ k * y ^ (n - 1 - k)
= ∑ k in range m, x ^ k * (y ^ (n - m) * y ^ (m - 1 - k)),
{ refine sum_congr rfl (λ j j_in, _),
rw ← pow_add,
congr,
rw [mem_range, nat.lt_iff_add_one_le, add_comm] at j_in,
have h' : n - m + (m - (1 + j)) = n - (... | theorem | commute.mul_geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"add_tsub_cancel_of_le",
"commute",
"finset.Ico",
"mul_assoc",
"nat.lt_iff_add_one_le",
"pow_add",
"pow_mul_comm",
"ring",
"tsub_add_eq_tsub_tsub",
"tsub_add_tsub_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_succ_eq {α : Type u} [ring α] {x y : α}
(h : commute x y) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) | begin
simp_rw [mul_sum, sum_range_succ_comm, tsub_self, pow_zero, mul_one, add_right_inj, ←mul_assoc,
(h.symm.pow_right _).eq, mul_assoc, ←pow_succ],
refine sum_congr rfl (λ i hi, _),
suffices : n - 1 - i + 1 = n - i, { rw this },
cases n,
{ exact absurd (list.mem_range.mp hi) i.not_lt_zero },
{ rw [tsu... | theorem | commute.geom_sum₂_succ_eq | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"mul_assoc",
"mul_one",
"nat.le_pred_of_lt",
"pow_zero",
"ring",
"tsub_add_cancel_of_le",
"tsub_add_eq_add_tsub",
"tsub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_succ_eq {α : Type u} [comm_ring α] (x y : α) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) | (commute.all x y).geom_sum₂_succ_eq | theorem | geom_sum₂_succ_eq | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_ring",
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_geom_sum₂_Ico [comm_ring α] (x y : α) {m n : ℕ} (hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) | (commute.all x y).mul_geom_sum₂_Ico hmn | theorem | mul_geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_ring",
"commute.all",
"finset.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_Ico_mul [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ (n - m) * x ^ m | begin
apply op_injective,
simp only [op_sub, op_mul, op_pow, op_sum],
have : ∑ k in Ico m n, op y ^ (n - 1 - k) * op x ^ k
= ∑ k in Ico m n, op x ^ k * op y ^ (n - 1 - k),
{ refine sum_congr rfl (λ k k_in, _),
apply commute.pow_pow (commute.op h.symm) },
rw this,
exact (commute.op h).mul_geom_sum₂_I... | theorem | commute.geom_sum₂_Ico_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"commute.op",
"commute.pow_pow",
"finset.Ico",
"mul_geom_sum₂_Ico",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_Ico_mul [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (x - 1) = x^n - x^m | by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul, geom_sum_mul, sub_sub_sub_cancel_right] | theorem | geom_sum_Ico_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"finset.Ico",
"geom_sum_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_Ico_mul_neg [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (1 - x) = x^m - x^n | by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left] | theorem | geom_sum_Ico_mul_neg | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"finset.Ico",
"geom_sum_mul_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_Ico [division_ring α] {x y : α} (h : commute x y) (hxy : x ≠ y)
{m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) | have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h.geom_sum₂_Ico_mul hmn, mul_div_cancel _ this] | theorem | commute.geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"division_ring",
"finset.Ico",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_Ico [field α] {x y : α} (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) | (commute.all x y).geom_sum₂_Ico hxy hmn | theorem | geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.all",
"field",
"finset.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_Ico [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1) | by simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same,
sub_sub_sub_cancel_right] | theorem | geom_sum_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_sub_div_same",
"division_ring",
"finset.Ico",
"geom_sum_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_Ico' [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) | by { simp only [geom_sum_Ico hx hmn], convert neg_div_neg_eq (x^m - x^n) (1-x); abel } | theorem | geom_sum_Ico' | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"division_ring",
"finset.Ico",
"geom_sum_Ico",
"neg_div_neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_Ico_le_of_lt_one [linear_ordered_field α]
{x : α} (hx : 0 ≤ x) (h'x : x < 1) {m n : ℕ} :
∑ i in Ico m n, x ^ i ≤ x ^ m / (1 - x) | begin
rcases le_or_lt m n with hmn | hmn,
{ rw geom_sum_Ico' h'x.ne hmn,
apply div_le_div (pow_nonneg hx _) _ (sub_pos.2 h'x) le_rfl,
simpa using pow_nonneg hx _ },
{ rw [Ico_eq_empty, sum_empty],
{ apply div_nonneg (pow_nonneg hx _),
simpa using h'x.le },
{ simpa using hmn.le } },
end | lemma | geom_sum_Ico_le_of_lt_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_le_div",
"div_nonneg",
"geom_sum_Ico'",
"le_rfl",
"linear_ordered_field",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_inv [division_ring α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
(∑ i in range n, x⁻¹ ^ i) = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) | have h₁ : x⁻¹ ≠ 1, by rwa [inv_eq_one_div, ne.def, div_eq_iff_mul_eq hx0, one_mul],
have h₂ : x⁻¹ - 1 ≠ 0, from mt sub_eq_zero.1 h₁,
have h₃ : x - 1 ≠ 0, from mt sub_eq_zero.1 hx1,
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
nat.rec_on n (by simp)
(λ n h, by rw [pow_succ, mul_inv_rev, ←mul_assoc, h, mul_assoc, mul_i... | lemma | geom_sum_inv | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_eq_iff_mul_eq",
"division_ring",
"geom_sum_eq",
"inv_eq_one_div",
"inv_mul_cancel",
"mul_assoc",
"mul_inv_cancel",
"mul_inv_rev",
"mul_right_inj'",
"one_mul",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.map_geom_sum [semiring α] [semiring β] (x : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i) = ∑ i in range n, (f x) ^ i | by simp [f.map_sum] | theorem | ring_hom.map_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.map_geom_sum₂ [semiring α] [semiring β] (x y : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) =
∑ i in range n, (f x) ^ i * ((f y) ^ (n - 1 - i)) | by simp [f.map_sum] | theorem | ring_hom.map_geom_sum₂ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.pred_mul_geom_sum_le (a b n : ℕ) :
(b - 1) * ∑ i in range n.succ, a/b^i ≤ a * b - a/b^n | calc
(b - 1) * (∑ i in range n.succ, a/b^i)
= ∑ i in range n, a/b^(i + 1) * b + a * b
- (∑ i in range n, a/b^i + a/b^n)
: by rw [tsub_mul, mul_comm, sum_mul, one_mul, sum_range_succ',
sum_range_succ, pow_zero, nat.div_one]
... ≤ ∑ i in range n, a/b^i + a * b - (∑ i in range n, a/b^i + ... | lemma | nat.pred_mul_geom_sum_le | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"add_tsub_add_eq_tsub_left",
"mul_comm",
"one_mul",
"pow_succ'",
"pow_zero",
"tsub_le_tsub_right",
"tsub_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.geom_sum_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in range n, a/b^i ≤ a * b/(b - 1) | begin
refine (nat.le_div_iff_mul_le $ tsub_pos_of_lt hb).2 _,
cases n,
{ rw [sum_range_zero, zero_mul],
exact nat.zero_le _ },
rw mul_comm,
exact (nat.pred_mul_geom_sum_le a b n).trans tsub_le_self,
end | lemma | nat.geom_sum_le | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"mul_comm",
"nat.pred_mul_geom_sum_le",
"tsub_le_self",
"tsub_pos_of_lt",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.geom_sum_Ico_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in Ico 1 n, a/b^i ≤ a/(b - 1) | begin
cases n,
{ rw [Ico_eq_empty_of_le (zero_le_one' ℕ), sum_empty],
exact nat.zero_le _ },
rw ←add_le_add_iff_left a,
calc
a + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i
= a/b^0 + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i : by rw [pow_zero, nat.div_one]
... = ∑ i in range n.succ, a/b^i : begin
rw [... | lemma | nat.geom_sum_Ico_le | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"add_tsub_cancel_of_le",
"mul_one",
"nat.geom_sum_le",
"pow_zero",
"tsub_pos_of_lt",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_pos [strict_ordered_semiring α] (hx : 0 ≤ x) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i | sum_pos' (λ k hk, pow_nonneg hx _) ⟨0, mem_range.2 hn.bot_lt, by simp⟩ | lemma | geom_sum_pos | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"pow_nonneg",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_pos_and_lt_one [strict_ordered_ring α] (hx : x < 0) (hx' : 0 < x + 1) (hn : 1 < n) :
0 < ∑ i in range n, x ^ i ∧ ∑ i in range n, x ^ i < 1 | begin
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ rw geom_sum_two,
exact ⟨hx', (add_lt_iff_neg_right _).2 hx⟩ },
clear hn n,
intros n hn ihn,
rw [geom_sum_succ, add_lt_iff_neg_right, ← neg_lt_iff_pos_add', neg_mul_eq_neg_mul],
exact ⟨mul_lt_one_of_nonneg_of_lt_one_left (neg_nonneg.2 hx.le)
... | lemma | geom_sum_pos_and_lt_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ",
"geom_sum_two",
"mul_neg_of_neg_of_pos",
"nat.le_induction",
"neg_mul_eq_neg_mul",
"strict_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_alternating_of_le_neg_one [strict_ordered_ring α] (hx : x + 1 ≤ 0) (n : ℕ) :
if even n then ∑ i in range n, x ^ i ≤ 0 else 1 ≤ ∑ i in range n, x ^ i | begin
have hx0 : x ≤ 0 := (le_add_of_nonneg_right zero_le_one).trans hx,
induction n with n ih,
{ simp only [even_zero, geom_sum_zero, le_refl] },
simp only [nat.even_add_one, geom_sum_succ],
split_ifs at ih,
{ rw [if_neg (not_not_intro h), le_add_iff_nonneg_left],
exact mul_nonneg_of_nonpos_of_nonpos h... | lemma | geom_sum_alternating_of_le_neg_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ",
"geom_sum_zero",
"ih",
"mul_le_mul_of_nonpos_left",
"mul_nonneg_of_nonpos_of_nonpos",
"mul_one",
"nat.even_add_one",
"strict_ordered_ring",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_alternating_of_lt_neg_one [strict_ordered_ring α] (hx : x + 1 < 0) (hn : 1 < n) :
if even n then ∑ i in range n, x ^ i < 0 else 1 < ∑ i in range n, x ^ i | begin
have hx0 : x < 0, from ((le_add_iff_nonneg_right _).2 zero_le_one).trans_lt hx,
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ simp only [geom_sum_two, hx, true_or, even_bit0, if_true_left_eq_or] },
clear hn n,
intros n hn ihn,
simp only [nat.even_add_one, geom_sum_succ],
by_cases hn' : even... | lemma | geom_sum_alternating_of_lt_neg_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"even_bit0",
"geom_sum_succ",
"geom_sum_two",
"mul_lt_mul_of_neg_left",
"mul_one",
"mul_pos_of_neg_of_neg",
"nat.even_add_one",
"nat.le_induction",
"strict_ordered_ring",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_pos' [linear_ordered_ring α] (hx : 0 < x + 1) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i | begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
obtain hx' | hx' := lt_or_le x 0,
{ exact (geom_sum_pos_and_lt_one hx' hx n.one_lt_succ_succ).1 },
{ exact geom_sum_pos hx' (by simp only [nat.succ_ne_zero, ne.def, not_false_iff]) }
end | lemma | geom_sum_pos' | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_pos",
"geom_sum_pos_and_lt_one",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.geom_sum_pos [linear_ordered_ring α] (h : odd n) :
0 < ∑ i in range n, x ^ i | begin
rcases n with (_ | _ | k),
{ exact ((show ¬ odd 0, from dec_trivial) h).elim },
{ simp only [geom_sum_one, zero_lt_one] },
rw nat.odd_iff_not_even at h,
rcases lt_trichotomy (x + 1) 0 with hx | hx | hx,
{ have := geom_sum_alternating_of_lt_neg_one hx k.one_lt_succ_succ,
simp only [h, if_false] at ... | lemma | odd.geom_sum_pos | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_alternating_of_lt_neg_one",
"geom_sum_one",
"geom_sum_pos'",
"linear_ordered_ring",
"nat.odd_iff_not_even",
"neg_one_geom_sum",
"odd",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_pos_iff [linear_ordered_ring α] (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i ↔ odd n ∨ 0 < x + 1 | begin
refine ⟨λ h, _, _⟩,
{ rw [or_iff_not_imp_left, ←not_le, ←nat.even_iff_not_odd],
refine λ hn hx, h.not_le _,
simpa [if_pos hn] using geom_sum_alternating_of_le_neg_one hx n },
{ rintro (hn | hx'),
{ exact hn.geom_sum_pos },
{ exact geom_sum_pos' hx' hn } }
end | lemma | geom_sum_pos_iff | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_alternating_of_le_neg_one",
"geom_sum_pos'",
"linear_ordered_ring",
"odd",
"or_iff_not_imp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_ne_zero [linear_ordered_ring α] (hx : x ≠ -1) (hn : n ≠ 0) :
∑ i in range n, x ^ i ≠ 0 | begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
rw [ne.def, eq_neg_iff_add_eq_zero, ←ne.def] at hx,
obtain h | h := hx.lt_or_lt,
{ have := geom_sum_alternating_of_lt_neg_one h n.one_lt_succ_succ,
split_ifs at this,
{ exact this.ne },
{ exact (zero_lt_one.trans this).ne' } },
{ exact (... | lemma | geom_sum_ne_zero | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_alternating_of_lt_neg_one",
"geom_sum_pos'",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_eq_zero_iff_neg_one [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i = 0 ↔ x = -1 ∧ even n | begin
refine ⟨λ h, _, λ ⟨h, hn⟩, by simp only [h, hn, neg_one_geom_sum, if_true]⟩,
contrapose! h,
obtain rfl | hx := eq_or_ne x (-1),
{ simp only [h rfl, neg_one_geom_sum, if_false, ne.def, not_false_iff, one_ne_zero] },
{ exact geom_sum_ne_zero hx hn }
end | lemma | geom_sum_eq_zero_iff_neg_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"eq_or_ne",
"geom_sum_ne_zero",
"linear_ordered_ring",
"neg_one_geom_sum",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_neg_iff [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i < 0 ↔ even n ∧ x + 1 < 0 | by rw [← not_iff_not, not_lt, le_iff_lt_or_eq, eq_comm,
or_congr (geom_sum_pos_iff hn) (geom_sum_eq_zero_iff_neg_one hn), nat.odd_iff_not_even,
← add_eq_zero_iff_eq_neg, not_and, not_lt, le_iff_lt_or_eq, eq_comm,
← imp_iff_not_or, or_comm, and_comm, decidable.and_or_imp, or_comm] | lemma | geom_sum_neg_iff | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"decidable.and_or_imp",
"geom_sum_eq_zero_iff_neg_one",
"geom_sum_pos_iff",
"imp_iff_not_or",
"linear_ordered_ring",
"nat.odd_iff_not_even",
"not_and",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_monoid (A : ι → Type*) | sigma A | def | graded_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | A type alias of sigma types for graded monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {A : ι → Type*} : Π i, A i → graded_monoid A | sigma.mk | def | graded_monoid.mk | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid"
] | Construct an element of a graded monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_one [has_zero ι] | (one : A 0) | class | graded_monoid.ghas_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | A graded version of `has_one`, which must be of grade 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_one.to_has_one [has_zero ι] [ghas_one A] : has_one (graded_monoid A) | ⟨⟨_, ghas_one.one⟩⟩ | instance | graded_monoid.ghas_one.to_has_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid"
] | `ghas_one` implies `has_one (graded_monoid A)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_mul [has_add ι] | (mul {i j} : A i → A j → A (i + j)) | class | graded_monoid.ghas_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | A graded version of `has_mul`. Multiplication combines grades additively, like
`add_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_mul.to_has_mul [has_add ι] [ghas_mul A] :
has_mul (graded_monoid A) | ⟨λ (x y : graded_monoid A), ⟨_, ghas_mul.mul x.snd y.snd⟩⟩ | instance | graded_monoid.ghas_mul.to_has_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid"
] | `ghas_mul` implies `has_mul (graded_monoid A)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_mul_mk [has_add ι] [ghas_mul A] {i j} (a : A i) (b : A j) :
mk i a * mk j b = mk (i + j) (ghas_mul.mul a b) | rfl | lemma | graded_monoid.mk_mul_mk | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gnpow_rec : Π (n : ℕ) {i}, A i → A (n • i) | | 0 i a := cast (congr_arg A (zero_nsmul i).symm) ghas_one.one
| (n + 1) i a := cast (congr_arg A (succ_nsmul i n).symm) (ghas_mul.mul a $ gnpow_rec _ a) | def | graded_monoid.gmonoid.gnpow_rec | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | A default implementation of power on a graded monoid, like `npow_rec`.
`gmonoid.gnpow` should be used instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gnpow_rec_zero (a : graded_monoid A) : graded_monoid.mk _ (gnpow_rec 0 a.snd) = 1 | sigma.ext (zero_nsmul _) (heq_of_cast_eq _ rfl).symm | lemma | graded_monoid.gmonoid.gnpow_rec_zero | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid",
"graded_monoid.mk",
"heq_of_cast_eq",
"sigma.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_gnpow_rec_zero_tac : tactic unit | `[apply graded_monoid.gmonoid.gnpow_rec_zero] | def | graded_monoid.gmonoid.apply_gnpow_rec_zero_tac | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.gmonoid.gnpow_rec_zero"
] | Tactic used to autofill `graded_monoid.gmonoid.gnpow_zero'` when the default
`graded_monoid.gmonoid.gnpow_rec` is used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gnpow_rec_succ (n : ℕ) (a : graded_monoid A) :
(graded_monoid.mk _ $ gnpow_rec n.succ a.snd) = a * ⟨_, gnpow_rec n a.snd⟩ | sigma.ext (succ_nsmul _ _) (heq_of_cast_eq _ rfl).symm | lemma | graded_monoid.gmonoid.gnpow_rec_succ | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid",
"graded_monoid.mk",
"heq_of_cast_eq",
"sigma.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_gnpow_rec_succ_tac : tactic unit | `[apply graded_monoid.gmonoid.gnpow_rec_succ] | def | graded_monoid.gmonoid.apply_gnpow_rec_succ_tac | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.gmonoid.gnpow_rec_succ"
] | Tactic used to autofill `graded_monoid.gmonoid.gnpow_succ'` when the default
`graded_monoid.gmonoid.gnpow_rec` is used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmonoid [add_monoid ι] extends ghas_mul A, ghas_one A | (one_mul (a : graded_monoid A) : 1 * a = a)
(mul_one (a : graded_monoid A) : a * 1 = a)
(mul_assoc (a b c : graded_monoid A) : a * b * c = a * (b * c))
(gnpow : Π (n : ℕ) {i}, A i → A (n • i) := gmonoid.gnpow_rec)
(gnpow_zero' : Π (a : graded_monoid A), graded_monoid.mk _ (gnpow 0 a.snd) = 1
. gmonoid.apply_gnpow_rec... | class | graded_monoid.gmonoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"graded_monoid",
"graded_monoid.mk",
"mul_assoc",
"mul_one",
"one_mul"
] | A graded version of `monoid`.
Like `monoid.npow`, this has an optional `gmonoid.gnpow` field to allow definitional control of
natural powers of a graded monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmonoid.to_monoid [add_monoid ι] [gmonoid A] :
monoid (graded_monoid A) | { one := (1), mul := (*),
npow := λ n a, graded_monoid.mk _ (gmonoid.gnpow n a.snd),
npow_zero' := λ a, gmonoid.gnpow_zero' a,
npow_succ' := λ n a, gmonoid.gnpow_succ' n a,
one_mul := gmonoid.one_mul, mul_one := gmonoid.mul_one, mul_assoc := gmonoid.mul_assoc } | instance | graded_monoid.gmonoid.to_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"graded_monoid",
"graded_monoid.mk",
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | `gmonoid` implies a `monoid (graded_monoid A)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_pow [add_monoid ι] [gmonoid A] {i} (a : A i) (n : ℕ) :
mk i a ^ n = mk (n • i) (gmonoid.gnpow _ a) | begin
induction n with n,
{ rw [pow_zero],
exact (gmonoid.gnpow_zero' ⟨_, a⟩).symm, },
{ rw [pow_succ, n_ih, mk_mul_mk],
exact (gmonoid.gnpow_succ' n ⟨_, a⟩).symm, },
end | lemma | graded_monoid.mk_pow | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcomm_monoid [add_comm_monoid ι] extends gmonoid A | (mul_comm (a : graded_monoid A) (b : graded_monoid A) : a * b = b * a) | class | graded_monoid.gcomm_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_comm_monoid",
"graded_monoid",
"mul_comm"
] | A graded version of `comm_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcomm_monoid.to_comm_monoid [add_comm_monoid ι] [gcomm_monoid A] :
comm_monoid (graded_monoid A) | { mul_comm := gcomm_monoid.mul_comm, ..gmonoid.to_monoid A } | instance | graded_monoid.gcomm_monoid.to_comm_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_comm_monoid",
"comm_monoid",
"graded_monoid",
"mul_comm"
] | `gcomm_monoid` implies a `comm_monoid (graded_monoid A)`, although this is only used as an
instance locally to define notation in `gmonoid` and similar typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.has_one : has_one (A 0) | ⟨ghas_one.one⟩ | instance | graded_monoid.grade_zero.has_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | `1 : A 0` is the value provided in `ghas_one.one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.has_smul (i : ι) : has_smul (A 0) (A i) | { smul := λ x y, (zero_add i).rec (ghas_mul.mul x y) } | instance | graded_monoid.grade_zero.has_smul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"has_smul"
] | `(•) : A 0 → A i → A i` is the value provided in `graded_monoid.ghas_mul.mul`, composed with
an `eq.rec` to turn `A (0 + i)` into `A i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.has_mul : has_mul (A 0) | { mul := (•) } | instance | graded_monoid.grade_zero.has_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | `(*) : A 0 → A 0 → A 0` is the value provided in `graded_monoid.ghas_mul.mul`, composed with
an `eq.rec` to turn `A (0 + 0)` into `A 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_zero_smul {i} (a : A 0) (b : A i) : mk _ (a • b) = mk _ a * mk _ b | sigma.ext (zero_add _).symm $ eq_rec_heq _ _ | lemma | graded_monoid.mk_zero_smul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"sigma.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.smul_eq_mul (a b : A 0) : a • b = a * b | rfl | lemma | graded_monoid.grade_zero.smul_eq_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_zero_pow (a : A 0) (n : ℕ) : mk _ (a ^ n) = mk _ a ^ n | sigma.ext (nsmul_zero n).symm $ eq_rec_heq _ _ | lemma | graded_monoid.mk_zero_pow | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"sigma.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.monoid : monoid (A 0) | function.injective.monoid (mk 0) sigma_mk_injective rfl mk_zero_smul mk_zero_pow | instance | graded_monoid.grade_zero.monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"function.injective.monoid",
"monoid",
"sigma_mk_injective"
] | The `monoid` structure derived from `gmonoid A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.comm_monoid : comm_monoid (A 0) | function.injective.comm_monoid (mk 0) sigma_mk_injective rfl mk_zero_smul mk_zero_pow | instance | graded_monoid.grade_zero.comm_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"comm_monoid",
"function.injective.comm_monoid",
"sigma_mk_injective"
] | The `comm_monoid` structure derived from `gcomm_monoid A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_zero_monoid_hom : A 0 →* (graded_monoid A) | { to_fun := mk 0, map_one' := rfl, map_mul' := mk_zero_smul } | def | graded_monoid.mk_zero_monoid_hom | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid"
] | `graded_monoid.mk 0` is a `monoid_hom`, using the `graded_monoid.grade_zero.monoid` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.mul_action {i} : mul_action (A 0) (A i) | begin
letI := mul_action.comp_hom (graded_monoid A) (mk_zero_monoid_hom A),
exact function.injective.mul_action (mk i) sigma_mk_injective mk_zero_smul,
end | instance | graded_monoid.grade_zero.mul_action | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"function.injective.mul_action",
"graded_monoid",
"mul_action",
"mul_action.comp_hom",
"sigma_mk_injective"
] | Each grade `A i` derives a `A 0`-action structure from `gmonoid A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list.dprod_index (l : list α) (fι : α → ι) : ι | l.foldr (λ i b, fι i + b) 0 | def | list.dprod_index | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | The index used by `list.dprod`. Propositionally this is equal to `(l.map fι).sum`, but
definitionally it needs to have a different form to avoid introducing `eq.rec`s in `list.dprod`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list.dprod_index_nil (fι : α → ι) : ([] : list α).dprod_index fι = 0 | rfl | lemma | list.dprod_index_nil | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.dprod_index_cons (a : α) (l : list α) (fι : α → ι) :
(a :: l).dprod_index fι = fι a + l.dprod_index fι | rfl | lemma | list.dprod_index_cons | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.dprod_index_eq_map_sum (l : list α) (fι : α → ι) :
l.dprod_index fι = (l.map fι).sum | begin
dunfold list.dprod_index,
induction l,
{ simp, },
{ simp [l_ih], },
end | lemma | list.dprod_index_eq_map_sum | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"list.dprod_index"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.dprod (l : list α) (fι : α → ι) (fA : Π a, A (fι a)) :
A (l.dprod_index fι) | l.foldr_rec_on _ _ graded_monoid.ghas_one.one (λ i x a ha, graded_monoid.ghas_mul.mul (fA a) x) | def | list.dprod | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | A dependent product for graded monoids represented by the indexed family of types `A i`.
This is a dependent version of `(l.map fA).prod`.
For a list `l : list α`, this computes the product of `fA a` over `a`, where each `fA` is of type
`A (fι a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list.dprod_nil (fι : α → ι) (fA : Π a, A (fι a)) :
(list.nil : list α).dprod fι fA = graded_monoid.ghas_one.one | rfl | lemma | list.dprod_nil | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.dprod_cons (fι : α → ι) (fA : Π a, A (fι a)) (a : α) (l : list α) :
(a :: l).dprod fι fA = (graded_monoid.ghas_mul.mul (fA a) (l.dprod fι fA) : _) | rfl | lemma | list.dprod_cons | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_monoid.mk_list_dprod (l : list α) (fι : α → ι) (fA : Π a, A (fι a)) :
graded_monoid.mk _ (l.dprod fι fA) = (l.map (λ a, graded_monoid.mk (fι a) (fA a))).prod | begin
induction l,
{ simp, refl },
{ simp [←l_ih, graded_monoid.mk_mul_mk, list.prod_cons],
refl, },
end | lemma | graded_monoid.mk_list_dprod | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.mk",
"graded_monoid.mk_mul_mk",
"list.prod_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_monoid.list_prod_map_eq_dprod (l : list α) (f : α → graded_monoid A) :
(l.map f).prod = graded_monoid.mk _ (l.dprod (λ i, (f i).1) (λ i, (f i).2)) | begin
rw [graded_monoid.mk_list_dprod, graded_monoid.mk],
simp_rw sigma.eta,
end | lemma | graded_monoid.list_prod_map_eq_dprod | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid",
"graded_monoid.mk",
"graded_monoid.mk_list_dprod",
"sigma.eta"
] | A variant of `graded_monoid.mk_list_dprod` for rewriting in the other direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_monoid.list_prod_of_fn_eq_dprod {n : ℕ} (f : fin n → graded_monoid A) :
(list.of_fn f).prod =
graded_monoid.mk _ ((list.fin_range n).dprod (λ i, (f i).1) (λ i, (f i).2)) | by rw [list.of_fn_eq_map, graded_monoid.list_prod_map_eq_dprod] | lemma | graded_monoid.list_prod_of_fn_eq_dprod | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid",
"graded_monoid.list_prod_map_eq_dprod",
"graded_monoid.mk",
"list.fin_range",
"list.of_fn",
"list.of_fn_eq_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one.ghas_one [has_zero ι] [has_one R] : graded_monoid.ghas_one (λ i : ι, R) | { one := 1 } | instance | has_one.ghas_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.ghas_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul.ghas_mul [has_add ι] [has_mul R] : graded_monoid.ghas_mul (λ i : ι, R) | { mul := λ i j, (*) } | instance | has_mul.ghas_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.ghas_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid.gmonoid [add_monoid ι] [monoid R] : graded_monoid.gmonoid (λ i : ι, R) | { one_mul := λ a, sigma.ext (zero_add _) (heq_of_eq (one_mul _)),
mul_one := λ a, sigma.ext (add_zero _) (heq_of_eq (mul_one _)),
mul_assoc := λ a b c, sigma.ext (add_assoc _ _ _) (heq_of_eq (mul_assoc _ _ _)),
gnpow := λ n i a, a ^ n,
gnpow_zero' := λ a, sigma.ext (zero_nsmul _) (heq_of_eq (monoid.npow_zero' _... | instance | monoid.gmonoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"graded_monoid.gmonoid",
"has_mul.ghas_mul",
"has_one.ghas_one",
"monoid",
"mul_assoc",
"mul_one",
"one_mul",
"sigma.ext"
] | If all grades are the same type and themselves form a monoid, then there is a trivial grading
structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_monoid.gcomm_monoid [add_comm_monoid ι] [comm_monoid R] :
graded_monoid.gcomm_monoid (λ i : ι, R) | { mul_comm := λ a b, sigma.ext (add_comm _ _) (heq_of_eq (mul_comm _ _)),
..monoid.gmonoid ι } | instance | comm_monoid.gcomm_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_comm_monoid",
"comm_monoid",
"graded_monoid.gcomm_monoid",
"monoid.gmonoid",
"mul_comm",
"sigma.ext"
] | If all grades are the same type and themselves form a commutative monoid, then there is a
trivial grading structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list.dprod_monoid {α} [add_monoid ι] [monoid R] (l : list α) (fι : α → ι)
(fA : α → R) :
(l.dprod fι fA : (λ i : ι, R) _) = ((l.map fA).prod : _) | begin
induction l,
{ rw [list.dprod_nil, list.map_nil, list.prod_nil], refl },
{ rw [list.dprod_cons, list.map_cons, list.prod_cons, l_ih], refl },
end | lemma | list.dprod_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"list.dprod_cons",
"list.dprod_nil",
"list.map_nil",
"list.prod_cons",
"list.prod_nil",
"monoid"
] | When all the indexed types are the same, the dependent product is just the regular product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.has_graded_one {S : Type*} [set_like S R] [has_one R] [has_zero ι]
(A : ι → S) : Prop | (one_mem : (1 : R) ∈ A 0) | class | set_like.has_graded_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like"
] | A version of `graded_monoid.ghas_one` for internally graded objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.one_mem_graded {S : Type*} [set_like S R] [has_one R] [has_zero ι] (A : ι → S)
[set_like.has_graded_one A] : (1 : R) ∈ A 0 | set_like.has_graded_one.one_mem | lemma | set_like.one_mem_graded | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like",
"set_like.has_graded_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.ghas_one {S : Type*} [set_like S R] [has_one R] [has_zero ι] (A : ι → S)
[set_like.has_graded_one A] : graded_monoid.ghas_one (λ i, A i) | { one := ⟨1, set_like.one_mem_graded _⟩ } | instance | set_like.ghas_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.ghas_one",
"set_like",
"set_like.has_graded_one",
"set_like.one_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.coe_ghas_one {S : Type*} [set_like S R] [has_one R] [has_zero ι] (A : ι → S)
[set_like.has_graded_one A] : ↑(@graded_monoid.ghas_one.one _ (λ i, A i) _ _) = (1 : R) | rfl | lemma | set_like.coe_ghas_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like",
"set_like.has_graded_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.has_graded_mul {S : Type*} [set_like S R] [has_mul R] [has_add ι]
(A : ι → S) : Prop | (mul_mem : ∀ ⦃i j⦄ {gi gj}, gi ∈ A i → gj ∈ A j → gi * gj ∈ A (i + j)) | class | set_like.has_graded_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like"
] | A version of `graded_monoid.ghas_one` for internally graded objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.mul_mem_graded {S : Type*} [set_like S R] [has_mul R] [has_add ι] {A : ι → S}
[set_like.has_graded_mul A] ⦃i j⦄ {gi gj} (hi : gi ∈ A i) (hj : gj ∈ A j) :
gi * gj ∈ A (i + j) | set_like.has_graded_mul.mul_mem hi hj | lemma | set_like.mul_mem_graded | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like",
"set_like.has_graded_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.ghas_mul {S : Type*} [set_like S R] [has_mul R] [has_add ι] (A : ι → S)
[set_like.has_graded_mul A] :
graded_monoid.ghas_mul (λ i, A i) | { mul := λ i j a b, ⟨(a * b : R), set_like.mul_mem_graded a.prop b.prop⟩ } | instance | set_like.ghas_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"graded_monoid.ghas_mul",
"set_like",
"set_like.has_graded_mul",
"set_like.mul_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.coe_ghas_mul {S : Type*} [set_like S R] [has_mul R] [has_add ι] (A : ι → S)
[set_like.has_graded_mul A] {i j : ι} (x : A i) (y : A j) :
↑(@graded_monoid.ghas_mul.mul _ (λ i, A i) _ _ _ _ x y) = (x * y : R) | rfl | lemma | set_like.coe_ghas_mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like",
"set_like.has_graded_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.graded_monoid {S : Type*} [set_like S R] [monoid R] [add_monoid ι]
(A : ι → S) extends set_like.has_graded_one A, set_like.has_graded_mul A : Prop | class | set_like.graded_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"monoid",
"set_like",
"set_like.has_graded_mul",
"set_like.has_graded_one"
] | A version of `graded_monoid.gmonoid` for internally graded objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_graded (n : ℕ) {r : R} {i : ι} (h : r ∈ A i) : r ^ n ∈ A (n • i) | begin
induction n,
{ rw [pow_zero, zero_nsmul], exact one_mem_graded _ },
{ rw [pow_succ', succ_nsmul'], exact mul_mem_graded n_ih h },
end | lemma | set_like.pow_mem_graded | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod_map_mem_graded {ι'} (l : list ι') (i : ι' → ι) (r : ι' → R)
(h : ∀ j ∈ l, r j ∈ A (i j)) :
(l.map r).prod ∈ A (l.map i).sum | begin
induction l,
{ rw [list.map_nil, list.map_nil, list.prod_nil, list.sum_nil],
exact one_mem_graded _ },
{ rw [list.map_cons, list.map_cons, list.prod_cons, list.sum_cons],
exact mul_mem_graded
(h _ $ list.mem_cons_self _ _) (l_ih $ λ j hj, h _ $ list.mem_cons_of_mem _ hj) },
end | lemma | set_like.list_prod_map_mem_graded | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"list.map_nil",
"list.prod_cons",
"list.prod_nil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod_of_fn_mem_graded {n} (i : fin n → ι) (r : fin n → R) (h : ∀ j, r j ∈ A (i j)) :
(list.of_fn r).prod ∈ A (list.of_fn i).sum | begin
rw [list.of_fn_eq_map, list.of_fn_eq_map],
exact list_prod_map_mem_graded _ _ _ (λ _ _, h _),
end | lemma | set_like.list_prod_of_fn_mem_graded | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"list.of_fn",
"list.of_fn_eq_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.gmonoid {S : Type*} [set_like S R] [monoid R] [add_monoid ι] (A : ι → S)
[set_like.graded_monoid A] :
graded_monoid.gmonoid (λ i, A i) | { one_mul := λ ⟨i, a, h⟩, sigma.subtype_ext (zero_add _) (one_mul _),
mul_one := λ ⟨i, a, h⟩, sigma.subtype_ext (add_zero _) (mul_one _),
mul_assoc := λ ⟨i, a, ha⟩ ⟨j, b, hb⟩ ⟨k, c, hc⟩,
sigma.subtype_ext (add_assoc _ _ _) (mul_assoc _ _ _),
gnpow := λ n i a, ⟨a ^ n, set_like.pow_mem_graded n a.prop⟩,
gnpow... | instance | set_like.gmonoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"graded_monoid.gmonoid",
"monoid",
"mul_assoc",
"mul_one",
"one_mul",
"pow_succ",
"pow_zero",
"set_like",
"set_like.ghas_mul",
"set_like.ghas_one",
"set_like.graded_monoid",
"set_like.pow_mem_graded",
"sigma.subtype_ext"
] | Build a `gmonoid` instance for a collection of subobjects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.coe_gnpow {S : Type*} [set_like S R] [monoid R] [add_monoid ι] (A : ι → S)
[set_like.graded_monoid A] {i : ι} (x : A i) (n : ℕ) :
↑(@graded_monoid.gmonoid.gnpow _ (λ i, A i) _ _ n _ x) = (x ^ n : R) | rfl | lemma | set_like.coe_gnpow | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"monoid",
"set_like",
"set_like.graded_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.gcomm_monoid {S : Type*} [set_like S R] [comm_monoid R] [add_comm_monoid ι]
(A : ι → S) [set_like.graded_monoid A] :
graded_monoid.gcomm_monoid (λ i, A i) | { mul_comm := λ ⟨i, a, ha⟩ ⟨j, b, hb⟩, sigma.subtype_ext (add_comm _ _) (mul_comm _ _),
..set_like.gmonoid A} | instance | set_like.gcomm_monoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_comm_monoid",
"comm_monoid",
"graded_monoid.gcomm_monoid",
"mul_comm",
"set_like",
"set_like.gmonoid",
"set_like.graded_monoid",
"sigma.subtype_ext"
] | Build a `gcomm_monoid` instance for a collection of subobjects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.coe_list_dprod (A : ι → S) [set_like.graded_monoid A]
(fι : α → ι) (fA : Π a, A (fι a)) (l : list α) :
↑(l.dprod fι fA : (λ i, ↥(A i)) _) = (list.prod (l.map (λ a, fA a)) : R) | begin
induction l,
{ rw [list.dprod_nil, coe_ghas_one, list.map_nil, list.prod_nil] },
{ rw [list.dprod_cons, coe_ghas_mul, list.map_cons, list.prod_cons, l_ih], },
end | lemma | set_like.coe_list_dprod | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"list.dprod_cons",
"list.dprod_nil",
"list.map_nil",
"list.prod",
"list.prod_cons",
"list.prod_nil",
"set_like.graded_monoid"
] | Coercing a dependent product of subtypes is the same as taking the regular product of the
coercions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.list_dprod_eq (A : ι → S) [set_like.graded_monoid A]
(fι : α → ι) (fA : Π a, A (fι a)) (l : list α) :
(l.dprod fι fA : (λ i, ↥(A i)) _) =
⟨list.prod (l.map (λ a, fA a)), (l.dprod_index_eq_map_sum fι).symm ▸
list_prod_map_mem_graded l _ _ (λ i hi, (fA i).prop)⟩ | subtype.ext $ set_like.coe_list_dprod _ _ _ _ | lemma | set_like.list_dprod_eq | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like.coe_list_dprod",
"set_like.graded_monoid",
"subtype.ext"
] | A version of `list.coe_dprod_set_like` with `subtype.mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.is_homogeneous (A : ι → S) (a : R) : Prop | ∃ i, a ∈ A i | def | set_like.is_homogeneous | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [] | An element `a : R` is said to be homogeneous if there is some `i : ι` such that `a ∈ A i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.is_homogeneous_coe {A : ι → S} {i} (x : A i) :
set_like.is_homogeneous A (x : R) | ⟨i, x.prop⟩ | lemma | set_like.is_homogeneous_coe | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.is_homogeneous_one [has_zero ι] [has_one R]
(A : ι → S) [set_like.has_graded_one A] : set_like.is_homogeneous A (1 : R) | ⟨0, set_like.one_mem_graded _⟩ | lemma | set_like.is_homogeneous_one | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like.has_graded_one",
"set_like.is_homogeneous",
"set_like.one_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.is_homogeneous.mul [has_add ι] [has_mul R] {A : ι → S}
[set_like.has_graded_mul A] {a b : R} :
set_like.is_homogeneous A a → set_like.is_homogeneous A b → set_like.is_homogeneous A (a * b) | | ⟨i, hi⟩ ⟨j, hj⟩ := ⟨i + j, set_like.mul_mem_graded hi hj⟩ | lemma | set_like.is_homogeneous.mul | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"set_like.has_graded_mul",
"set_like.is_homogeneous",
"set_like.mul_mem_graded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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