fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
graded_monoid (A : ι → Type*) := sigma A | 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. | 99 | 99 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{A : ι → Type*} [inhabited ι] [inhabited (A default)]: inhabited (graded_monoid A) :=
sigma.inhabited | {A : ι → Type*} [inhabited ι] [inhabited (A default)]: inhabited (graded_monoid A) | sigma.inhabited | instance | 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"
] | null | 103 | 104 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {A : ι → Type*} : Π i, A i → graded_monoid A := sigma.mk | 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. | 107 | 107 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_one [has_zero ι] :=
(one : A 0) | 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. | 115 | 116 | true | 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⟩⟩ | 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)` | 119 | 120 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_mul [has_add ι] :=
(mul {i j} : A i → A j → A (i + j)) | 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`. | 124 | 125 | true | 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⟩⟩ | 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)`. | 128 | 130 | true | 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 | 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"
] | [] | null | 132 | 134 | true | 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) | 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. | 142 | 144 | true | 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 | 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"
] | null | 146 | 147 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_gnpow_rec_zero_tac : tactic unit := `[apply graded_monoid.gmonoid.gnpow_rec_zero] | 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. | 151 | 151 | true | 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 | 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"
] | null | 153 | 155 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_gnpow_rec_succ_tac : tactic unit := `[apply graded_monoid.gmonoid.gnpow_rec_succ] | 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. | 159 | 159 | true | 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_m... | 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. | 167 | 176 | true | 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_as... | 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)`. | 179 | 185 | true | 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 | 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"
] | null | 187 | 195 | true | 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) | 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`. | 198 | 199 | true | 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 } | 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. | 203 | 205 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.has_one : has_one (A 0) :=
⟨ghas_one.one⟩ | 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`. | 224 | 226 | true | 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) } | 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`. | 236 | 237 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.has_mul : has_mul (A 0) :=
{ mul := (•) } | 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`. | 242 | 243 | true | 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 _ _ | 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"
] | null | 247 | 248 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.smul_eq_mul (a b : A 0) : a • b = a * b := rfl | 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"
] | [] | null | 250 | 250 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: has_pow (A 0) ℕ :=
{ pow := λ x n, (nsmul_zero n).rec (gmonoid.gnpow n x : A (n • 0)) } | : has_pow (A 0) ℕ | { pow := λ x n, (nsmul_zero n).rec (gmonoid.gnpow n x : A (n • 0)) } | instance | 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"
] | [] | null | 258 | 259 | true | 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 _ _ | 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"
] | null | 263 | 264 | true | 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 | 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`. | 269 | 270 | true | 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 | 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`. | 278 | 279 | true | 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 } | 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. | 288 | 289 | true | 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 | 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`. | 292 | 296 | true | 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 | 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`. | 312 | 313 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
list.dprod_index_nil (fι : α → ι) : ([] : list α).dprod_index fι = 0 := rfl | 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"
] | [] | null | 315 | 315 | true | 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 | 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"
] | [] | null | 316 | 317 | true | 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 | 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"
] | null | 319 | 326 | true | 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) | 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)`. | 333 | 335 | true | 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 | 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"
] | [] | null | 337 | 338 | true | 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 | 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"
] | [] | null | 342 | 343 | true | 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 | 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"
] | null | 345 | 352 | true | 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 | 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. | 355 | 360 | true | 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] | 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"
] | null | 362 | 365 | true | 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 } | 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"
] | null | 374 | 376 | true | 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, (*) } | 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"
] | null | 378 | 380 | true | 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 ^ ... | 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. | 384 | 393 | true | 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 } | 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. | 397 | 400 | true | 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 | 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. | 403 | 410 | true | 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) | 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. | 421 | 423 | true | 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 | 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"
] | null | 425 | 426 | true | 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 _⟩ } | 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"
] | null | 428 | 430 | true | 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 | 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"
] | null | 432 | 433 | true | 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)) | 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. | 436 | 438 | true | 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 | 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"
] | null | 440 | 443 | true | 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⟩ } | 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"
] | null | 445 | 448 | true | 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 | 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"
] | null | 450 | 452 | true | 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 | 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. | 455 | 456 | false | 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 | 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"
] | null | 462 | 467 | true | 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... | 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"
] | null | 469 | 479 | true | 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 | 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"
] | null | 481 | 486 | true | 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⟩ ⟨... | 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. | 491 | 502 | true | 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 | 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"
] | null | 504 | 506 | true | 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} | 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. | 509 | 513 | true | 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... | 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. | 521 | 528 | true | 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_d... | 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`. | 533 | 538 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.is_homogeneous (A : ι → S) (a : R) : Prop := ∃ i, a ∈ A i | 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`. | 549 | 549 | true | 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⟩ | 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"
] | null | 551 | 553 | true | 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 _⟩ | 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"
] | null | 555 | 557 | true | 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⟩ | 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"
] | null | 559 | 562 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.homogeneous_submonoid [add_monoid ι] [monoid R]
(A : ι → S) [set_like.graded_monoid A] : submonoid R :=
{ carrier := { a | set_like.is_homogeneous A a },
one_mem' := set_like.is_homogeneous_one A,
mul_mem' := λ a b, set_like.is_homogeneous.mul } | set_like.homogeneous_submonoid [add_monoid ι] [monoid R]
(A : ι → S) [set_like.graded_monoid A] : submonoid R | { carrier := { a | set_like.is_homogeneous A a },
one_mem' := set_like.is_homogeneous_one A,
mul_mem' := λ a b, set_like.is_homogeneous.mul } | def | set_like.homogeneous_submonoid | 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.graded_monoid",
"set_like.is_homogeneous",
"set_like.is_homogeneous.mul",
"set_like.is_homogeneous_one",
"submonoid"
] | When `A` is a `set_like.graded_monoid A`, then the homogeneous elements forms a submonoid. | 565 | 569 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_smul [has_add ι] :=
(smul {i j} : A i → M j → M (i + j)) | ghas_smul [has_add ι] | (smul {i j} : A i → M j → M (i + j)) | class | graded_monoid.ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | A graded version of `has_smul`. Scalar multiplication combines grades additively, i.e.
if `a ∈ A i` and `m ∈ M j`, then `a • b` must be in `M (i + j)` | 63 | 64 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_mul.to_ghas_smul [has_add ι] [ghas_mul A] : ghas_smul A A :=
{ smul := λ _ _, ghas_mul.mul } | ghas_mul.to_ghas_smul [has_add ι] [ghas_mul A] : ghas_smul A A | { smul := λ _ _, ghas_mul.mul } | instance | graded_monoid.ghas_mul.to_ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | A graded version of `has_mul.to_has_smul` | 67 | 68 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_smul.to_has_smul [has_add ι] [ghas_smul A M] :
has_smul (graded_monoid A) (graded_monoid M) :=
⟨λ (x : graded_monoid A) (y : graded_monoid M), ⟨_, ghas_smul.smul x.snd y.snd⟩⟩ | ghas_smul.to_has_smul [has_add ι] [ghas_smul A M] :
has_smul (graded_monoid A) (graded_monoid M) | ⟨λ (x : graded_monoid A) (y : graded_monoid M), ⟨_, ghas_smul.smul x.snd y.snd⟩⟩ | instance | graded_monoid.ghas_smul.to_has_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"graded_monoid",
"has_smul"
] | null | 70 | 72 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_smul_mk [has_add ι] [ghas_smul A M] {i j} (a : A i) (b : M j) :
mk i a • mk j b = mk (i + j) (ghas_smul.smul a b) :=
rfl | mk_smul_mk [has_add ι] [ghas_smul A M] {i j} (a : A i) (b : M j) :
mk i a • mk j b = mk (i + j) (ghas_smul.smul a b) | rfl | lemma | graded_monoid.mk_smul_mk | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | null | 74 | 76 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmul_action [add_monoid ι] [gmonoid A] extends ghas_smul A M :=
(one_smul (b : graded_monoid M) : (1 : graded_monoid A) • b = b)
(mul_smul (a a' : graded_monoid A) (b : graded_monoid M) : (a * a') • b = a • a' • b) | gmul_action [add_monoid ι] [gmonoid A] extends ghas_smul A M | (one_smul (b : graded_monoid M) : (1 : graded_monoid A) • b = b)
(mul_smul (a a' : graded_monoid A) (b : graded_monoid M) : (a * a') • b = a • a' • b) | class | graded_monoid.gmul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"graded_monoid",
"one_smul"
] | A graded version of `mul_action`. | 79 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmonoid.to_gmul_action [add_monoid ι] [gmonoid A] :
gmul_action A A :=
{ one_smul := gmonoid.one_mul,
mul_smul := gmonoid.mul_assoc,
..ghas_mul.to_ghas_smul _ } | gmonoid.to_gmul_action [add_monoid ι] [gmonoid A] :
gmul_action A A | { one_smul := gmonoid.one_mul,
mul_smul := gmonoid.mul_assoc,
..ghas_mul.to_ghas_smul _ } | instance | graded_monoid.gmonoid.to_gmul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"one_smul"
] | The graded version of `monoid.to_mul_action`. | 84 | 88 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmul_action.to_mul_action [add_monoid ι] [gmonoid A] [gmul_action A M] :
mul_action (graded_monoid A) (graded_monoid M) :=
{ one_smul := gmul_action.one_smul,
mul_smul := gmul_action.mul_smul } | gmul_action.to_mul_action [add_monoid ι] [gmonoid A] [gmul_action A M] :
mul_action (graded_monoid A) (graded_monoid M) | { one_smul := gmul_action.one_smul,
mul_smul := gmul_action.mul_smul } | instance | graded_monoid.gmul_action.to_mul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"graded_monoid",
"mul_action",
"one_smul"
] | null | 90 | 93 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.has_graded_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) : Prop :=
(smul_mem : ∀ ⦃i j : ι⦄ {ai bj}, ai ∈ A i → bj ∈ B j → ai • bj ∈ B (i + j)) | set_like.has_graded_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) : Prop | (smul_mem : ∀ ⦃i j : ι⦄ {ai bj}, ai ∈ A i → bj ∈ B j → ai • bj ∈ B (i + j)) | class | set_like.has_graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like"
] | A version of `graded_monoid.ghas_smul` for internally graded objects. | 106 | 108 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B] :
graded_monoid.ghas_smul (λ i, A i) (λ i, B i) :=
{ smul := λ i j a b, ⟨(a : R) • b, set_like.has_graded_smul.smul_mem a.2 b.2⟩ } | set_like.ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B] :
graded_monoid.ghas_smul (λ i, A i) (λ i, B i) | { smul := λ i j a b, ⟨(a : R) • b, set_like.has_graded_smul.smul_mem a.2 b.2⟩ } | instance | set_like.ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"graded_monoid.ghas_smul",
"has_smul",
"set_like",
"set_like.has_graded_smul"
] | null | 110 | 113 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.coe_ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B]
{i j : ι} (x : A i) (y : B j) :
(@graded_monoid.ghas_smul.smul ι (λ i, A i) (λ i, B i) _ _ i j x y : M) = ((x : R) • y) :=
rfl | set_like.coe_ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B]
{i j : ι} (x : A i) (y : B j) :
(@graded_monoid.ghas_smul.smul ι (λ i, A i) (λ i, B i) _ _ i j x y : M) = ((x : R) • y) | rfl | lemma | set_like.coe_ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like",
"set_like.has_graded_smul"
] | null | 115 | 119 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.has_graded_mul.to_has_graded_smul [add_monoid ι] [monoid R]
{S : Type*} [set_like S R] (A : ι → S) [set_like.graded_monoid A] :
set_like.has_graded_smul A A :=
{ smul_mem := λ i j ai bj hi hj, set_like.graded_monoid.mul_mem hi hj, } | set_like.has_graded_mul.to_has_graded_smul [add_monoid ι] [monoid R]
{S : Type*} [set_like S R] (A : ι → S) [set_like.graded_monoid A] :
set_like.has_graded_smul A A | { smul_mem := λ i j ai bj hi hj, set_like.graded_monoid.mul_mem hi hj, } | instance | set_like.has_graded_mul.to_has_graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"monoid",
"set_like",
"set_like.graded_monoid",
"set_like.has_graded_smul"
] | Internally graded version of `has_mul.to_has_smul`. | 122 | 125 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.is_homogeneous.graded_smul [has_add ι] [has_smul R M] {A : ι → S} {B : ι → N}
[set_like.has_graded_smul A B] {a : R} {b : M} :
set_like.is_homogeneous A a → set_like.is_homogeneous B b → set_like.is_homogeneous B (a • b)
| ⟨i, hi⟩ ⟨j, hj⟩ := ⟨i + j, set_like.has_graded_smul.smul_mem hi hj⟩ | set_like.is_homogeneous.graded_smul [has_add ι] [has_smul R M] {A : ι → S} {B : ι → N}
[set_like.has_graded_smul A B] {a : R} {b : M} :
set_like.is_homogeneous A a → set_like.is_homogeneous B b → set_like.is_homogeneous B (a • b)
| ⟨i, hi⟩ ⟨j, hj⟩ | ⟨i + j, set_like.has_graded_smul.smul_mem hi hj⟩ | lemma | set_like.is_homogeneous.graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like.has_graded_smul",
"set_like.is_homogeneous"
] | null | 133 | 136 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator {M} [has_zero M] (s : set α) (f : α → M) : α → M
| x := by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 0 | indicator {M} [has_zero M] (s : set α) (f : α → M) : α → M
| x | by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 0 | def | set.indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | `indicator s f a` is `f a` if `a ∈ s`, `0` otherwise. | 45 | 46 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator (s : set α) (f : α → M) : α → M
| x := by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 1 | mul_indicator (s : set α) (f : α → M) : α → M
| x | by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 1 | def | set.mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | `mul_indicator s f a` is `f a` if `a ∈ s`, `1` otherwise. | 49 | 51 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piecewise_eq_mul_indicator [decidable_pred (∈ s)] :
s.piecewise f 1 = s.mul_indicator f :=
funext $ λ x, @if_congr _ _ _ _ (id _) _ _ _ _ iff.rfl rfl rfl | piecewise_eq_mul_indicator [decidable_pred (∈ s)] :
s.piecewise f 1 = s.mul_indicator f | funext $ λ x, @if_congr _ _ _ _ (id _) _ _ _ _ iff.rfl rfl rfl | lemma | set.piecewise_eq_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 53 | 55 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply (s : set α) (f : α → M) (a : α) [decidable (a ∈ s)] :
mul_indicator s f a = if a ∈ s then f a else 1 := by convert rfl | mul_indicator_apply (s : set α) (f : α → M) (a : α) [decidable (a ∈ s)] :
mul_indicator s f a = if a ∈ s then f a else 1 | by convert rfl | lemma | set.mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 57 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_of_mem (h : a ∈ s) (f : α → M) :
mul_indicator s f a = f a :=
by { letI := classical.dec (a ∈ s), exact if_pos h } | mul_indicator_of_mem (h : a ∈ s) (f : α → M) :
mul_indicator s f a = f a | by { letI := classical.dec (a ∈ s), exact if_pos h } | lemma | set.mul_indicator_of_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | null | 60 | 62 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_of_not_mem (h : a ∉ s) (f : α → M) :
mul_indicator s f a = 1 :=
by { letI := classical.dec (a ∈ s), exact if_neg h } | mul_indicator_of_not_mem (h : a ∉ s) (f : α → M) :
mul_indicator s f a = 1 | by { letI := classical.dec (a ∈ s), exact if_neg h } | lemma | set.mul_indicator_of_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | null | 64 | 66 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_eq_one_or_self (s : set α) (f : α → M) (a : α) :
mul_indicator s f a = 1 ∨ mul_indicator s f a = f a :=
begin
by_cases h : a ∈ s,
{ exact or.inr (mul_indicator_of_mem h f) },
{ exact or.inl (mul_indicator_of_not_mem h f) }
end | mul_indicator_eq_one_or_self (s : set α) (f : α → M) (a : α) :
mul_indicator s f a = 1 ∨ mul_indicator s f a = f a | begin
by_cases h : a ∈ s,
{ exact or.inr (mul_indicator_of_mem h f) },
{ exact or.inl (mul_indicator_of_not_mem h f) }
end | lemma | set.mul_indicator_eq_one_or_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 68 | 74 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_eq_self :
s.mul_indicator f a = f a ↔ (a ∉ s → f a = 1) :=
by letI := classical.dec (a ∈ s); exact ite_eq_left_iff.trans (by rw [@eq_comm _ (f a)]) | mul_indicator_apply_eq_self :
s.mul_indicator f a = f a ↔ (a ∉ s → f a = 1) | by letI := classical.dec (a ∈ s); exact ite_eq_left_iff.trans (by rw [@eq_comm _ (f a)]) | lemma | set.mul_indicator_apply_eq_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | null | 76 | 78 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_eq_self : s.mul_indicator f = f ↔ mul_support f ⊆ s :=
by simp only [funext_iff, subset_def, mem_mul_support, mul_indicator_apply_eq_self, not_imp_comm] | mul_indicator_eq_self : s.mul_indicator f = f ↔ mul_support f ⊆ s | by simp only [funext_iff, subset_def, mem_mul_support, mul_indicator_apply_eq_self, not_imp_comm] | lemma | set.mul_indicator_eq_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"not_imp_comm"
] | null | 80 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_eq_self_of_superset (h1 : s.mul_indicator f = f) (h2 : s ⊆ t) :
t.mul_indicator f = f :=
by { rw mul_indicator_eq_self at h1 ⊢, exact subset.trans h1 h2 } | mul_indicator_eq_self_of_superset (h1 : s.mul_indicator f = f) (h2 : s ⊆ t) :
t.mul_indicator f = f | by { rw mul_indicator_eq_self at h1 ⊢, exact subset.trans h1 h2 } | lemma | set.mul_indicator_eq_self_of_superset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 83 | 85 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_eq_one :
mul_indicator s f a = 1 ↔ (a ∈ s → f a = 1) :=
by letI := classical.dec (a ∈ s); exact ite_eq_right_iff | mul_indicator_apply_eq_one :
mul_indicator s f a = 1 ↔ (a ∈ s → f a = 1) | by letI := classical.dec (a ∈ s); exact ite_eq_right_iff | lemma | set.mul_indicator_apply_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec",
"ite_eq_right_iff"
] | null | 87 | 89 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_eq_one :
mul_indicator s f = (λ x, 1) ↔ disjoint (mul_support f) s :=
by simp only [funext_iff, mul_indicator_apply_eq_one, set.disjoint_left, mem_mul_support,
not_imp_not] | mul_indicator_eq_one :
mul_indicator s f = (λ x, 1) ↔ disjoint (mul_support f) s | by simp only [funext_iff, mul_indicator_apply_eq_one, set.disjoint_left, mem_mul_support,
not_imp_not] | lemma | set.mul_indicator_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"not_imp_not",
"set.disjoint_left"
] | null | 91 | 94 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_eq_one' :
mul_indicator s f = 1 ↔ disjoint (mul_support f) s :=
mul_indicator_eq_one | mul_indicator_eq_one' :
mul_indicator s f = 1 ↔ disjoint (mul_support f) s | mul_indicator_eq_one | lemma | set.mul_indicator_eq_one' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint"
] | null | 96 | 98 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_ne_one {a : α} :
s.mul_indicator f a ≠ 1 ↔ a ∈ s ∩ mul_support f :=
by simp only [ne.def, mul_indicator_apply_eq_one, not_imp, mem_inter_iff, mem_mul_support] | mul_indicator_apply_ne_one {a : α} :
s.mul_indicator f a ≠ 1 ↔ a ∈ s ∩ mul_support f | by simp only [ne.def, mul_indicator_apply_eq_one, not_imp, mem_inter_iff, mem_mul_support] | lemma | set.mul_indicator_apply_ne_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"not_imp"
] | null | 100 | 102 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_support_mul_indicator :
function.mul_support (s.mul_indicator f) = s ∩ function.mul_support f :=
ext $ λ x, by simp [function.mem_mul_support, mul_indicator_apply_eq_one] | mul_support_mul_indicator :
function.mul_support (s.mul_indicator f) = s ∩ function.mul_support f | ext $ λ x, by simp [function.mem_mul_support, mul_indicator_apply_eq_one] | lemma | set.mul_support_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"function.mem_mul_support",
"function.mul_support"
] | null | 104 | 106 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_mul_indicator_ne_one (h : mul_indicator s f a ≠ 1) : a ∈ s :=
not_imp_comm.1 (λ hn, mul_indicator_of_not_mem hn f) h | mem_of_mul_indicator_ne_one (h : mul_indicator s f a ≠ 1) : a ∈ s | not_imp_comm.1 (λ hn, mul_indicator_of_not_mem hn f) h | lemma | set.mem_of_mul_indicator_ne_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | If a multiplicative indicator function is not equal to `1` at a point, then that point is in the
set. | 110 | 113 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_mul_indicator : eq_on (mul_indicator s f) f s :=
λ x hx, mul_indicator_of_mem hx f | eq_on_mul_indicator : eq_on (mul_indicator s f) f s | λ x hx, mul_indicator_of_mem hx f | lemma | set.eq_on_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 115 | 116 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_support_mul_indicator_subset : mul_support (s.mul_indicator f) ⊆ s :=
λ x hx, hx.imp_symm (λ h, mul_indicator_of_not_mem h f) | mul_support_mul_indicator_subset : mul_support (s.mul_indicator f) ⊆ s | λ x hx, hx.imp_symm (λ h, mul_indicator_of_not_mem h f) | lemma | set.mul_support_mul_indicator_subset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 118 | 119 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul_support : mul_indicator (mul_support f) f = f :=
mul_indicator_eq_self.2 subset.rfl | mul_indicator_mul_support : mul_indicator (mul_support f) f = f | mul_indicator_eq_self.2 subset.rfl | lemma | set.mul_indicator_mul_support | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 121 | 122 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_range_comp {ι : Sort*} (f : ι → α) (g : α → M) :
mul_indicator (range f) g ∘ f = g ∘ f :=
by letI := classical.dec_pred (∈ range f); exact piecewise_range_comp _ _ _ | mul_indicator_range_comp {ι : Sort*} (f : ι → α) (g : α → M) :
mul_indicator (range f) g ∘ f = g ∘ f | by letI := classical.dec_pred (∈ range f); exact piecewise_range_comp _ _ _ | lemma | set.mul_indicator_range_comp | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | null | 124 | 126 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_congr (h : eq_on f g s) :
mul_indicator s f = mul_indicator s g :=
funext $ λx, by { simp only [mul_indicator], split_ifs, { exact h h_1 }, refl } | mul_indicator_congr (h : eq_on f g s) :
mul_indicator s f = mul_indicator s g | funext $ λx, by { simp only [mul_indicator], split_ifs, { exact h h_1 }, refl } | lemma | set.mul_indicator_congr | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 128 | 130 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_univ (f : α → M) : mul_indicator (univ : set α) f = f :=
mul_indicator_eq_self.2 $ subset_univ _ | mul_indicator_univ (f : α → M) : mul_indicator (univ : set α) f = f | mul_indicator_eq_self.2 $ subset_univ _ | lemma | set.mul_indicator_univ | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 132 | 133 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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