statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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coe_stream_nth_rat_eq :
((int_fract_pair.stream q n).map (mapFr coe) : option $ int_fract_pair K)
= int_fract_pair.stream v n | begin
induction n with n IH,
case nat.zero : { simp [int_fract_pair.stream, (coe_of_rat_eq v_eq_q)] },
case nat.succ :
{ rw v_eq_q at IH,
cases stream_q_nth_eq : (int_fract_pair.stream q n) with ifp_n,
case option.none : { simp [int_fract_pair.stream, IH.symm, v_eq_q, stream_q_nth_eq] },
case option... | lemma | generalized_continued_fraction.int_fract_pair.coe_stream_nth_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_stream_rat_eq :
((int_fract_pair.stream q).map (option.map (mapFr coe)) : stream $ option $ int_fract_pair K) =
int_fract_pair.stream v | by { funext n, exact (int_fract_pair.coe_stream_nth_rat_eq v_eq_q n) } | lemma | generalized_continued_fraction.int_fract_pair.coe_stream_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [
"stream"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_h_rat_eq : (↑((of q).h : ℚ) : K) = (of v).h | begin
unfold of int_fract_pair.seq1,
rw ←(int_fract_pair.coe_of_rat_eq v_eq_q),
simp
end | lemma | generalized_continued_fraction.coe_of_h_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_s_nth_rat_eq :
(((of q).s.nth n).map (pair.map coe) : option $ pair K) = (of v).s.nth n | begin
simp only [of, int_fract_pair.seq1, seq.map_nth, seq.nth_tail],
simp only [seq.nth],
rw [←(int_fract_pair.coe_stream_rat_eq v_eq_q)],
rcases succ_nth_stream_eq : (int_fract_pair.stream q (n + 1)) with _ | ⟨_, _⟩;
simp [stream.map, stream.nth, succ_nth_stream_eq]
end | lemma | generalized_continued_fraction.coe_of_s_nth_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [
"stream.map",
"stream.nth"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_s_rat_eq : (((of q).s).map (pair.map coe) : seq $ pair K) = (of v).s | by { ext n, rw ←(coe_of_s_nth_rat_eq v_eq_q), refl } | lemma | generalized_continued_fraction.coe_of_s_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_rat_eq :
(⟨(of q).h, (of q).s.map (pair.map coe)⟩ : generalized_continued_fraction K) = of v | begin
cases gcf_v_eq : (of v) with h s, subst v,
obtain rfl : ↑⌊↑q⌋ = h, by { injection gcf_v_eq },
simp [coe_of_h_rat_eq rfl, coe_of_s_rat_eq rfl, gcf_v_eq]
end | lemma | generalized_continued_fraction.coe_of_rat_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [
"generalized_continued_fraction"
] | Given `(v : K), (q : ℚ), and v = q`, we have that `gcf.of q = gcf.of v` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_terminates_iff_of_rat_terminates {v : K} {q : ℚ} (v_eq_q : v = (q : K)) :
(of v).terminates ↔ (of q).terminates | begin
split;
intro h;
cases h with n h;
use n;
simp only [seq.terminated_at, (coe_of_s_nth_rat_eq v_eq_q n).symm] at h ⊢;
cases ((of q).s.nth n);
trivial
end | lemma | generalized_continued_fraction.of_terminates_iff_of_rat_terminates | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_inv_fr_num_lt_num_of_pos (q_pos : 0 < q) :
(int_fract_pair.of q⁻¹).fr.num < q.num | rat.fract_inv_num_lt_num_of_pos q_pos | lemma | generalized_continued_fraction.int_fract_pair.of_inv_fr_num_lt_num_of_pos | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [
"rat.fract_inv_num_lt_num_of_pos"
] | Shows that for any `q : ℚ` with `0 < q < 1`, the numerator of the fractional part of
`int_fract_pair.of q⁻¹` is smaller than the numerator of `q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_succ_nth_fr_num_lt_nth_fr_num_rat {ifp_n ifp_succ_n : int_fract_pair ℚ}
(stream_nth_eq : int_fract_pair.stream q n = some ifp_n)
(stream_succ_nth_eq : int_fract_pair.stream q (n + 1) = some ifp_succ_n) :
ifp_succ_n.fr.num < ifp_n.fr.num | begin
obtain ⟨ifp_n', stream_nth_eq', ifp_n_fract_ne_zero, int_fract_pair.of_eq_ifp_succ_n⟩ :
∃ ifp_n', int_fract_pair.stream q n = some ifp_n' ∧ ifp_n'.fr ≠ 0
∧ int_fract_pair.of ifp_n'.fr⁻¹ = ifp_succ_n, from
succ_nth_stream_eq_some_iff.elim_left stream_succ_nth_eq,
have : ifp_n = ifp_n', by injecti... | lemma | generalized_continued_fraction.int_fract_pair.stream_succ_nth_fr_num_lt_nth_fr_num_rat | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | Shows that the sequence of numerators of the fractional parts of the stream is strictly
antitone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_nth_fr_num_le_fr_num_sub_n_rat : ∀ {ifp_n : int_fract_pair ℚ},
int_fract_pair.stream q n = some ifp_n → ifp_n.fr.num ≤ (int_fract_pair.of q).fr.num - n | begin
induction n with n IH,
case nat.zero
{ assume ifp_zero stream_zero_eq,
have : int_fract_pair.of q = ifp_zero, by injection stream_zero_eq,
simp [le_refl, this.symm] },
case nat.succ
{ assume ifp_succ_n stream_succ_nth_eq,
suffices : ifp_succ_n.fr.num + 1 ≤ (int_fract_pair.of q).fr.num - n, b... | lemma | generalized_continued_fraction.int_fract_pair.stream_nth_fr_num_le_fr_num_sub_n_rat | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nth_stream_eq_none_of_rat (q : ℚ) : ∃ (n : ℕ), int_fract_pair.stream q n = none | begin
let fract_q_num := (int.fract q).num, let n := fract_q_num.nat_abs + 1,
cases stream_nth_eq : (int_fract_pair.stream q n) with ifp,
{ use n, exact stream_nth_eq },
{ -- arrive at a contradiction since the numerator decreased num + 1 times but every fractional
-- value is nonnegative.
have ifp_fr_n... | lemma | generalized_continued_fraction.int_fract_pair.exists_nth_stream_eq_none_of_rat | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [
"int.fract",
"int.fract_nonneg",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminates_of_rat (q : ℚ) : (of q).terminates | exists.elim (int_fract_pair.exists_nth_stream_eq_none_of_rat q)
( assume n stream_nth_eq_none,
exists.intro n
( have int_fract_pair.stream q (n + 1) = none, from
int_fract_pair.stream_is_seq q stream_nth_eq_none,
(of_terminated_at_n_iff_succ_nth_int_fract_pair_stream_eq_none.elim_right this) ) ) | theorem | generalized_continued_fraction.terminates_of_rat | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | The continued fraction of a rational number terminates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminates_iff_rat (v : K) : (of v).terminates ↔ ∃ (q : ℚ), v = (q : K) | iff.intro
( assume terminates_v : (of v).terminates,
show ∃ (q : ℚ), v = (q : K), from exists_rat_eq_of_terminates terminates_v )
( assume exists_q_eq_v : ∃ (q : ℚ), v = (↑q : K),
exists.elim exists_q_eq_v
( assume q,
assume v_eq_q : v = ↑q,
have (of q).terminates, from terminates_of_rat q,
(of_termin... | theorem | generalized_continued_fraction.terminates_iff_rat | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/terminates_iff_rat.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.computation.correctness_terminating",
"data.rat.floor"
] | [] | The continued fraction `generalized_continued_fraction.of v` terminates if and only if `v ∈ ℚ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_zero (v : K) : int_fract_pair.stream v 0 = some (int_fract_pair.of v) | rfl | lemma | generalized_continued_fraction.int_fract_pair.stream_zero | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stream_eq_none_of_fr_eq_zero {ifp_n : int_fract_pair K}
(stream_nth_eq : int_fract_pair.stream v n = some ifp_n) (nth_fr_eq_zero : ifp_n.fr = 0) :
int_fract_pair.stream v (n + 1) = none | begin
cases ifp_n with _ fr,
change fr = 0 at nth_fr_eq_zero,
simp [int_fract_pair.stream, stream_nth_eq, nth_fr_eq_zero]
end | lemma | generalized_continued_fraction.int_fract_pair.stream_eq_none_of_fr_eq_zero | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_stream_eq_none_iff : int_fract_pair.stream v (n + 1) = none
↔ (int_fract_pair.stream v n = none ∨ ∃ ifp, int_fract_pair.stream v n = some ifp ∧ ifp.fr = 0) | begin
rw [int_fract_pair.stream],
cases int_fract_pair.stream v n; simp [imp_false]
end | lemma | generalized_continued_fraction.int_fract_pair.succ_nth_stream_eq_none_iff | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"imp_false"
] | Gives a recurrence to compute the `n + 1`th value of the sequence of integer and fractional
parts of a value in case of termination. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_nth_stream_eq_some_iff {ifp_succ_n : int_fract_pair K} :
int_fract_pair.stream v (n + 1) = some ifp_succ_n
↔ ∃ (ifp_n : int_fract_pair K), int_fract_pair.stream v n = some ifp_n
∧ ifp_n.fr ≠ 0
∧ int_fract_pair.of ifp_n.fr⁻¹ = ifp_succ_n | by simp [int_fract_pair.stream, ite_eq_iff] | lemma | generalized_continued_fraction.int_fract_pair.succ_nth_stream_eq_some_iff | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"ite_eq_iff"
] | Gives a recurrence to compute the `n + 1`th value of the sequence of integer and fractional
parts of a value in case of non-termination. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_succ_of_some {p : int_fract_pair K}
(h : int_fract_pair.stream v n = some p) (h' : p.fr ≠ 0) :
int_fract_pair.stream v (n + 1) = some (int_fract_pair.of (p.fr)⁻¹) | succ_nth_stream_eq_some_iff.mpr ⟨p, h, h', rfl⟩ | lemma | generalized_continued_fraction.int_fract_pair.stream_succ_of_some | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | An easier to use version of one direction of
`generalized_continued_fraction.int_fract_pair.succ_nth_stream_eq_some_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_succ_of_int (a : ℤ) (n : ℕ) : int_fract_pair.stream (a : K) (n + 1) = none | begin
induction n with n ih,
{ refine int_fract_pair.stream_eq_none_of_fr_eq_zero (int_fract_pair.stream_zero (a : K)) _,
simp only [int_fract_pair.of, int.fract_int_cast], },
{ exact int_fract_pair.succ_nth_stream_eq_none_iff.mpr (or.inl ih), }
end | lemma | generalized_continued_fraction.int_fract_pair.stream_succ_of_int | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"ih",
"int.fract_int_cast"
] | The stream of `int_fract_pair`s of an integer stops after the first term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_succ_nth_stream_of_fr_zero {ifp_succ_n : int_fract_pair K}
(stream_succ_nth_eq : int_fract_pair.stream v (n + 1) = some ifp_succ_n)
(succ_nth_fr_eq_zero : ifp_succ_n.fr = 0) :
∃ ifp_n : int_fract_pair K, int_fract_pair.stream v n = some ifp_n ∧ ifp_n.fr⁻¹ = ⌊ifp_n.fr⁻¹⌋ | begin
-- get the witness from `succ_nth_stream_eq_some_iff` and prove that it has the additional
-- properties
rcases (succ_nth_stream_eq_some_iff.mp stream_succ_nth_eq) with
⟨ifp_n, seq_nth_eq, nth_fr_ne_zero, rfl⟩,
refine ⟨ifp_n, seq_nth_eq, _⟩,
simpa only [int_fract_pair.of, int.fract, sub_eq_zero] usi... | lemma | generalized_continued_fraction.int_fract_pair.exists_succ_nth_stream_of_fr_zero | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"int.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stream_succ (h : int.fract v ≠ 0) (n : ℕ) :
int_fract_pair.stream v (n + 1) = int_fract_pair.stream (int.fract v)⁻¹ n | begin
induction n with n ih,
{ have H : (int_fract_pair.of v).fr = int.fract v := rfl,
rw [stream_zero, stream_succ_of_some (stream_zero v) (ne_of_eq_of_ne H h), H], },
{ cases eq_or_ne (int_fract_pair.stream (int.fract v)⁻¹ n) none with hnone hsome,
{ rw hnone at ih,
rw [succ_nth_stream_eq_none_iff... | lemma | generalized_continued_fraction.int_fract_pair.stream_succ | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"eq_or_ne",
"ih",
"int.fract"
] | A recurrence relation that expresses the `(n+1)`th term of the stream of `int_fract_pair`s
of `v` for non-integer `v` in terms of the `n`th term of the stream associated to
the inverse of the fractional part of `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_fract_pair.seq1_fst_eq_of : (int_fract_pair.seq1 v).fst = int_fract_pair.of v | rfl | lemma | generalized_continued_fraction.int_fract_pair.seq1_fst_eq_of | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | The head term of the sequence with head of `v` is just the integer part of `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_h_eq_int_fract_pair_seq1_fst_b : (of v).h = (int_fract_pair.seq1 v).fst.b | by { cases aux_seq_eq : (int_fract_pair.seq1 v), simp [of, aux_seq_eq] } | lemma | generalized_continued_fraction.of_h_eq_int_fract_pair_seq1_fst_b | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_h_eq_floor : (of v).h = ⌊v⌋ | by simp [of_h_eq_int_fract_pair_seq1_fst_b, int_fract_pair.of] | lemma | generalized_continued_fraction.of_h_eq_floor | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | The head term of the gcf of `v` is `⌊v⌋`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_fract_pair.nth_seq1_eq_succ_nth_stream :
(int_fract_pair.seq1 v).snd.nth n = (int_fract_pair.stream v) (n + 1) | rfl | lemma | generalized_continued_fraction.int_fract_pair.nth_seq1_eq_succ_nth_stream | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_terminated_at_iff_int_fract_pair_seq1_terminated_at :
(of v).terminated_at n ↔ (int_fract_pair.seq1 v).snd.terminated_at n | option.map_eq_none | lemma | generalized_continued_fraction.of_terminated_at_iff_int_fract_pair_seq1_terminated_at | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"option.map_eq_none"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_terminated_at_n_iff_succ_nth_int_fract_pair_stream_eq_none :
(of v).terminated_at n ↔ int_fract_pair.stream v (n + 1) = none | by rw [of_terminated_at_iff_int_fract_pair_seq1_terminated_at, stream.seq.terminated_at,
int_fract_pair.nth_seq1_eq_succ_nth_stream] | lemma | generalized_continued_fraction.of_terminated_at_n_iff_succ_nth_int_fract_pair_stream_eq_none | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"stream.seq.terminated_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_fract_pair.exists_succ_nth_stream_of_gcf_of_nth_eq_some {gp_n : pair K}
(s_nth_eq : (of v).s.nth n = some gp_n) :
∃ (ifp : int_fract_pair K), int_fract_pair.stream v (n + 1) = some ifp ∧ (ifp.b : K) = gp_n.b | begin
obtain ⟨ifp, stream_succ_nth_eq, gp_n_eq⟩ :
∃ ifp, int_fract_pair.stream v (n + 1) = some ifp ∧ pair.mk 1 (ifp.b : K) = gp_n, by
{ unfold of int_fract_pair.seq1 at s_nth_eq,
rwa [seq.map_tail, seq.nth_tail, seq.map_nth, option.map_eq_some'] at s_nth_eq },
cases gp_n_eq,
injection gp_n_eq with ... | lemma | generalized_continued_fraction.int_fract_pair.exists_succ_nth_stream_of_gcf_of_nth_eq_some | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"option.map_eq_some'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nth_of_eq_some_of_succ_nth_int_fract_pair_stream {ifp_succ_n : int_fract_pair K}
(stream_succ_nth_eq : int_fract_pair.stream v (n + 1) = some ifp_succ_n) :
(of v).s.nth n = some ⟨1, ifp_succ_n.b⟩ | begin
unfold of int_fract_pair.seq1,
rw [seq.map_tail, seq.nth_tail, seq.map_nth],
simp [seq.nth, stream_succ_nth_eq]
end | lemma | generalized_continued_fraction.nth_of_eq_some_of_succ_nth_int_fract_pair_stream | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | Shows how the entries of the sequence of the computed continued fraction can be obtained by the
integer parts of the stream of integer and fractional parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_of_eq_some_of_nth_int_fract_pair_stream_fr_ne_zero {ifp_n : int_fract_pair K}
(stream_nth_eq : int_fract_pair.stream v n = some ifp_n) (nth_fr_ne_zero : ifp_n.fr ≠ 0) :
(of v).s.nth n = some ⟨1, (int_fract_pair.of ifp_n.fr⁻¹).b⟩ | have int_fract_pair.stream v (n + 1) = some (int_fract_pair.of ifp_n.fr⁻¹), by
{ cases ifp_n, simp [int_fract_pair.stream, stream_nth_eq, nth_fr_ne_zero] },
nth_of_eq_some_of_succ_nth_int_fract_pair_stream this | lemma | generalized_continued_fraction.nth_of_eq_some_of_nth_int_fract_pair_stream_fr_ne_zero | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | Shows how the entries of the sequence of the computed continued fraction can be obtained by the
fractional parts of the stream of integer and fractional parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_s_head_aux (v : K) :
(of v).s.nth 0 = (int_fract_pair.stream v 1).bind (some ∘ λ p, {a := 1, b := p.b}) | begin
rw [of, int_fract_pair.seq1, of._match_1],
simp only [seq.map_tail, seq.map, seq.tail, seq.head, seq.nth, stream.map],
rw [← stream.nth_succ, stream.nth, option.map],
end | lemma | generalized_continued_fraction.of_s_head_aux | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"stream.map",
"stream.nth",
"stream.nth_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_s_head (h : fract v ≠ 0) : (of v).s.head = some ⟨1, ⌊(fract v)⁻¹⌋⟩ | begin
change (of v).s.nth 0 = _,
rw [of_s_head_aux, stream_succ_of_some (stream_zero v) h, option.bind],
refl,
end | lemma | generalized_continued_fraction.of_s_head | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | This gives the first pair of coefficients of the continued fraction of a non-integer `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_s_of_int (a : ℤ) : (of (a : K)).s = seq.nil | begin
have h : ∀ n, (of (a : K)).s.nth n = none,
{ intro n,
induction n with n ih,
{ rw [of_s_head_aux, stream_succ_of_int, option.bind], },
{ exact (of (a : K)).s.prop ih, } },
exact seq.ext (λ n, (h n).trans (seq.nth_nil n).symm),
end | lemma | generalized_continued_fraction.of_s_of_int | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"ih"
] | If `a` is an integer, then the coefficient sequence of its continued fraction is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_s_succ (n : ℕ) : (of v).s.nth (n + 1) = (of (fract v)⁻¹).s.nth n | begin
cases eq_or_ne (fract v) 0 with h h,
{ obtain ⟨a, rfl⟩ : ∃ a : ℤ, v = a := ⟨⌊v⌋, eq_of_sub_eq_zero h⟩,
rw [fract_int_cast, inv_zero, of_s_of_int, ← cast_zero, of_s_of_int, seq.nth_nil,
seq.nth_nil], },
cases eq_or_ne ((of (fract v)⁻¹).s.nth n) none with h₁ h₁,
{ rwa [h₁, ← terminated_at_iff_s... | lemma | generalized_continued_fraction.of_s_succ | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"eq_or_ne",
"inv_zero"
] | Recurrence for the `generalized_continued_fraction.of` an element `v` of `K` in terms of
that of the inverse of the fractional part of `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_s_tail : (of v).s.tail = (of (fract v)⁻¹).s | seq.ext $ λ n, seq.nth_tail (of v).s n ▸ of_s_succ v n | lemma | generalized_continued_fraction.of_s_tail | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [] | This expresses the tail of the coefficient sequence of the `generalized_continued_fraction.of`
an element `v` of `K` as the coefficient sequence of that of the inverse of the
fractional part of `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents'_of_int (a : ℤ) : (of (a : K)).convergents' n = a | begin
induction n with n ih,
{ simp only [zeroth_convergent'_eq_h, of_h_eq_floor, floor_int_cast], },
{ rw [convergents', of_h_eq_floor, floor_int_cast, add_right_eq_self],
exact convergents'_aux_succ_none ((of_s_of_int K a).symm ▸ seq.nth_nil 0) _, }
end | lemma | generalized_continued_fraction.convergents'_of_int | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"ih"
] | If `a` is an integer, then the `convergents'` of its continued fraction expansion
are all equal to `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents'_succ :
(of v).convergents' (n + 1) = ⌊v⌋ + 1 / (of (fract v)⁻¹).convergents' n | begin
cases eq_or_ne (fract v) 0 with h h,
{ obtain ⟨a, rfl⟩ : ∃ a : ℤ, v = a := ⟨⌊v⌋, eq_of_sub_eq_zero h⟩,
rw [convergents'_of_int, fract_int_cast, inv_zero, ← cast_zero,
convergents'_of_int, cast_zero, div_zero, add_zero, floor_int_cast], },
{ rw [convergents', of_h_eq_floor, add_right_inj, converg... | lemma | generalized_continued_fraction.convergents'_succ | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/translations.lean | [
"algebra.continued_fractions.computation.basic",
"algebra.continued_fractions.translations"
] | [
"div_zero",
"eq_or_ne",
"inv_zero"
] | The recurrence relation for the `convergents'` of the continued fraction expansion
of an element `v` of `K` in terms of the convergents of the inverse of its fractional part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
galgebra | (to_fun : R →+ A 0)
(map_one : to_fun 1 = graded_monoid.ghas_one.one)
(map_mul : ∀ r s,
graded_monoid.mk _ (to_fun (r * s)) = ⟨_, graded_monoid.ghas_mul.mul (to_fun r) (to_fun s)⟩)
(commutes : ∀ r x, graded_monoid.mk _ (to_fun r) * x = x * ⟨_, to_fun r⟩)
(smul_def : ∀ r (x : graded_monoid A), graded_monoid.mk x.1 (r ... | class | direct_sum.galgebra | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"graded_monoid",
"graded_monoid.mk",
"map_mul",
"map_one"
] | A graded version of `algebra`. An instance of `direct_sum.galgebra R A` endows `(⨁ i, A i)`
with an `R`-algebra structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_apply (r : R) :
algebra_map R (⨁ i, A i) r = direct_sum.of A 0 (galgebra.to_fun r) | rfl | lemma | direct_sum.algebra_map_apply | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"algebra_map",
"algebra_map_apply",
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_to_add_monoid_hom :
↑(algebra_map R (⨁ i, A i)) = (direct_sum.of A 0).comp (galgebra.to_fun : R →+ A 0) | rfl | lemma | direct_sum.algebra_map_to_add_monoid_hom | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"algebra_map",
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_algebra
(f : Π i, A i →ₗ[R] B) (hone : f _ (graded_monoid.ghas_one.one) = 1)
(hmul : ∀ {i j} (ai : A i) (aj : A j), f _ (graded_monoid.ghas_mul.mul ai aj) = f _ ai * f _ aj)
(hcommutes : ∀ r, (f 0) (galgebra.to_fun r) = (algebra_map R B) r) :
(⨁ i, A i) →ₐ[R] B | { to_fun := to_semiring (λ i, (f i).to_add_monoid_hom) hone @hmul,
commutes' := λ r, (direct_sum.to_semiring_of _ _ _ _ _).trans (hcommutes r),
.. to_semiring (λ i, (f i).to_add_monoid_hom) hone @hmul} | def | direct_sum.to_algebra | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"algebra_map",
"direct_sum.to_semiring_of"
] | A family of `linear_map`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`
describes an `alg_hom` on `⨁ i, A i`. This is a stronger version of `direct_sum.to_semiring`.
Of particular interest is the case when `A i` are bundled subojects, `f` is the family of
coercions such as `submodule.subtype (A i)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_ext' ⦃f g : (⨁ i, A i) →ₐ[R] B⦄
(h : ∀ i, f.to_linear_map.comp (lof _ _ A i) = g.to_linear_map.comp (lof _ _ A i)) : f = g | alg_hom.to_linear_map_injective $ direct_sum.linear_map_ext _ h | lemma | direct_sum.alg_hom_ext' | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"alg_hom.to_linear_map_injective",
"direct_sum.linear_map_ext"
] | Two `alg_hom`s out of a direct sum are equal if they agree on the generators.
See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_ext ⦃f g : (⨁ i, A i) →ₐ[R] B⦄ (h : ∀ i x, f (of A i x) = g (of A i x)) : f = g | alg_hom_ext' R A $ λ i, linear_map.ext $ h i | lemma | direct_sum.alg_hom_ext | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.direct_sum_galgebra {R A : Type*} [decidable_eq ι]
[add_monoid ι] [comm_semiring R] [semiring A] [algebra R A] :
direct_sum.galgebra R (λ i : ι, A) | { to_fun := (algebra_map R A).to_add_monoid_hom,
map_one := (algebra_map R A).map_one,
map_mul := λ a b, sigma.ext (zero_add _).symm (heq_of_eq $ (algebra_map R A).map_mul a b),
commutes := λ r ⟨ai, a⟩, sigma.ext ((zero_add _).trans (add_zero _).symm)
(heq_of_eq $ algebra.commu... | instance | algebra.direct_sum_galgebra | algebra.direct_sum | src/algebra/direct_sum/algebra.lean | [
"algebra.algebra.basic",
"algebra.direct_sum.module",
"algebra.direct_sum.ring"
] | [
"add_monoid",
"algebra",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"comm_semiring",
"direct_sum.galgebra",
"map_mul",
"map_one",
"semiring",
"sigma.ext"
] | A direct sum of copies of a `algebra` inherits the algebra structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum [Π i, add_comm_monoid (β i)] : Type* | Π₀ i, β i | def | direct_sum | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid"
] | `direct_sum β` is the direct sum of a family of additive commutative monoids `β i`.
Note: `open_locale direct_sum` will enable the notation `⨁ i, β i` for `direct_sum β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_apply (g₁ g₂ : ⨁ i, β i) (i : ι) : (g₁ - g₂) i = g₁ i - g₂ i | rfl | lemma | direct_sum.sub_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (i : ι) : (0 : ⨁ i, β i) i = 0 | rfl | lemma | direct_sum.zero_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (g₁ g₂ : ⨁ i, β i) (i : ι) : (g₁ + g₂) i = g₁ i + g₂ i | rfl | lemma | direct_sum.add_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (s : finset ι) : (Π i : (↑s : set ι), β i.1) →+ ⨁ i, β i | { to_fun := dfinsupp.mk s,
map_add' := λ _ _, dfinsupp.mk_add,
map_zero' := dfinsupp.mk_zero, } | def | direct_sum.mk | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.mk",
"dfinsupp.mk_add",
"dfinsupp.mk_zero",
"finset"
] | `mk β s x` is the element of `⨁ i, β i` that is zero outside `s`
and has coefficient `x i` for `i` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (i : ι) : β i →+ ⨁ i, β i | dfinsupp.single_add_hom β i | def | direct_sum.of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.single_add_hom"
] | `of i` is the natural inclusion map from `β i` to `⨁ i, β i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_eq_same (i : ι) (x : β i) : (of _ i x) i = x | dfinsupp.single_eq_same | lemma | direct_sum.of_eq_same | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.single_eq_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq_of_ne (i j : ι) (x : β i) (h : i ≠ j) : (of _ i x) j = 0 | dfinsupp.single_eq_of_ne h | lemma | direct_sum.of_eq_of_ne | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.single_eq_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_zero [Π (i : ι) (x : β i), decidable (x ≠ 0)] :
(0 : ⨁ i, β i).support = ∅ | dfinsupp.support_zero | lemma | direct_sum.support_zero | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.support_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_of [Π (i : ι) (x : β i), decidable (x ≠ 0)]
(i : ι) (x : β i) (h : x ≠ 0) :
(of _ i x).support = {i} | dfinsupp.support_single_ne_zero h | lemma | direct_sum.support_of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.support_single_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_of_subset [Π (i : ι) (x : β i), decidable (x ≠ 0)] {i : ι} {b : β i} :
(of _ i b).support ⊆ {i} | dfinsupp.support_single_subset | lemma | direct_sum.support_of_subset | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.support_single_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_support_of [Π (i : ι) (x : β i), decidable (x ≠ 0)] (x : ⨁ i, β i) :
∑ i in x.support, of β i (x i) = x | dfinsupp.sum_single | lemma | direct_sum.sum_support_of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sum_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_injective (s : finset ι) : function.injective (mk β s) | dfinsupp.mk_injective s | theorem | direct_sum.mk_injective | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.mk_injective",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective (i : ι) : function.injective (of β i) | dfinsupp.single_injective | theorem | direct_sum.of_injective | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.single_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {C : (⨁ i, β i) → Prop}
(x : ⨁ i, β i) (H_zero : C 0)
(H_basic : ∀ (i : ι) (x : β i), C (of β i x))
(H_plus : ∀ x y, C x → C y → C (x + y)) : C x | begin
apply dfinsupp.induction x H_zero,
intros i b f h1 h2 ih,
solve_by_elim
end | theorem | direct_sum.induction_on | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.induction",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_hom_ext {γ : Type*} [add_monoid γ] ⦃f g : (⨁ i, β i) →+ γ⦄
(H : ∀ (i : ι) (y : β i), f (of _ i y) = g (of _ i y)) : f = g | dfinsupp.add_hom_ext H | lemma | direct_sum.add_hom_ext | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_monoid",
"dfinsupp.add_hom_ext"
] | If two additive homomorphisms from `⨁ i, β i` are equal on each `of β i y`,
then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_hom_ext' {γ : Type*} [add_monoid γ] ⦃f g : (⨁ i, β i) →+ γ⦄
(H : ∀ (i : ι), f.comp (of _ i) = g.comp (of _ i)) : f = g | add_hom_ext $ λ i, add_monoid_hom.congr_fun $ H i | lemma | direct_sum.add_hom_ext' | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_monoid"
] | If two additive homomorphisms from `⨁ i, β i` are equal on each `of β i y`,
then they are equal.
See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_monoid : (⨁ i, β i) →+ γ | (dfinsupp.lift_add_hom φ) | def | direct_sum.to_add_monoid | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.lift_add_hom"
] | `to_add_monoid φ` is the natural homomorphism from `⨁ i, β i` to `γ`
induced by a family `φ` of homomorphisms `β i → γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_monoid_of (i) (x : β i) : to_add_monoid φ (of β i x) = φ i x | dfinsupp.lift_add_hom_apply_single φ i x | lemma | direct_sum.to_add_monoid_of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.lift_add_hom_apply_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid.unique (f : ⨁ i, β i) :
ψ f = to_add_monoid (λ i, ψ.comp (of β i)) f | by {congr, ext, simp [to_add_monoid, of]} | theorem | direct_sum.to_add_monoid.unique | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_add_monoid : (⨁ i, γ →+ β i) →+ (γ →+ ⨁ i, β i) | to_add_monoid $ λ i, add_monoid_hom.comp_hom (of β i) | def | direct_sum.from_add_monoid | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | `from_add_monoid φ` is the natural homomorphism from `γ` to `⨁ i, β i`
induced by a family `φ` of homomorphisms `γ → β i`.
Note that this is not an isomorphism. Not every homomorphism `γ →+ ⨁ i, β i` arises in this way. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_add_monoid_of (i : ι) (f : γ →+ β i) :
from_add_monoid (of _ i f) = (of _ i).comp f | by { rw [from_add_monoid, to_add_monoid_of], refl } | lemma | direct_sum.from_add_monoid_of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_add_monoid_of_apply (i : ι) (f : γ →+ β i) (x : γ) :
from_add_monoid (of _ i f) x = of _ i (f x) | by rw [from_add_monoid_of, add_monoid_hom.coe_comp] | lemma | direct_sum.from_add_monoid_of_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_to_set (S T : set ι) (H : S ⊆ T) :
(⨁ (i : S), β i) →+ (⨁ (i : T), β i) | to_add_monoid $ λ i, of (λ (i : subtype T), β i) ⟨↑i, H i.prop⟩ | def | direct_sum.set_to_set | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique [∀ i, subsingleton (β i)] : unique (⨁ i, β i) | dfinsupp.unique | instance | direct_sum.unique | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_of_is_empty [is_empty ι] : unique (⨁ i, β i) | dfinsupp.unique_of_is_empty | instance | direct_sum.unique_of_is_empty | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.unique_of_is_empty",
"is_empty",
"unique"
] | A direct sum over an empty type is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (M : Type v) (ι : Type* := punit) [add_comm_monoid M] [unique ι] :
(⨁ (_ : ι), M) ≃+ M | { to_fun := direct_sum.to_add_monoid (λ _, add_monoid_hom.id M),
inv_fun := of (λ _, M) default,
left_inv := λ x, direct_sum.induction_on x
(by rw [add_monoid_hom.map_zero, add_monoid_hom.map_zero])
(λ p x, by rw [unique.default_eq p, to_add_monoid_of]; refl)
(λ x y ihx ihy, by rw [add_monoid_hom.map_ad... | def | direct_sum.id | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"direct_sum.induction_on",
"direct_sum.to_add_monoid",
"inv_fun",
"unique",
"unique.default_eq"
] | The natural equivalence between `⨁ _ : ι, M` and `M` when `unique ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_congr_left (h : ι ≃ κ) : (⨁ i, β i) ≃+ ⨁ k, β (h.symm k) | { map_add' := dfinsupp.comap_domain'_add _ _,
..dfinsupp.equiv_congr_left h } | def | direct_sum.equiv_congr_left | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.comap_domain'_add",
"dfinsupp.equiv_congr_left"
] | Reindexing terms of a direct sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_congr_left_apply (h : ι ≃ κ) (f : ⨁ i, β i) (k : κ) :
equiv_congr_left h f k = f (h.symm k) | dfinsupp.comap_domain'_apply _ _ _ _ | lemma | direct_sum.equiv_congr_left_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.comap_domain'_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv_prod_direct_sum : (⨁ i, α i) ≃+ α none × ⨁ i, α (some i) | { map_add' := dfinsupp.equiv_prod_dfinsupp_add, ..dfinsupp.equiv_prod_dfinsupp } | def | direct_sum.add_equiv_prod_direct_sum | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.equiv_prod_dfinsupp",
"dfinsupp.equiv_prod_dfinsupp_add"
] | Isomorphism obtained by separating the term of index `none` of a direct sum over `option ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_curry : (⨁ (i : Σ i, _), δ i.1 i.2) →+ ⨁ i j, δ i j | { to_fun := @dfinsupp.sigma_curry _ _ δ _,
map_zero' := dfinsupp.sigma_curry_zero,
map_add' := λ f g, @dfinsupp.sigma_curry_add _ _ δ _ f g } | def | direct_sum.sigma_curry | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sigma_curry",
"dfinsupp.sigma_curry_add",
"dfinsupp.sigma_curry_zero"
] | The natural map between `⨁ (i : Σ i, α i), δ i.1 i.2` and `⨁ i (j : α i), δ i j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_curry_apply (f : ⨁ (i : Σ i, _), δ i.1 i.2) (i : ι) (j : α i) :
sigma_curry f i j = f ⟨i, j⟩ | @dfinsupp.sigma_curry_apply _ _ δ _ f i j | lemma | direct_sum.sigma_curry_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sigma_curry_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_uncurry [Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)] :
(⨁ i j, δ i j) →+ ⨁ (i : Σ i, _), δ i.1 i.2 | { to_fun := dfinsupp.sigma_uncurry,
map_zero' := dfinsupp.sigma_uncurry_zero,
map_add' := dfinsupp.sigma_uncurry_add } | def | direct_sum.sigma_uncurry | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sigma_uncurry",
"dfinsupp.sigma_uncurry_add",
"dfinsupp.sigma_uncurry_zero"
] | The natural map between `⨁ i (j : α i), δ i j` and `Π₀ (i : Σ i, α i), δ i.1 i.2`, inverse of
`curry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_uncurry_apply [Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)]
(f : ⨁ i j, δ i j) (i : ι) (j : α i) :
sigma_uncurry f ⟨i, j⟩ = f i j | dfinsupp.sigma_uncurry_apply f i j | lemma | direct_sum.sigma_uncurry_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sigma_uncurry_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_curry_equiv
[Π i, decidable_eq (α i)] [Π i j, decidable_eq (δ i j)] :
(⨁ (i : Σ i, _), δ i.1 i.2) ≃+ ⨁ i j, δ i j | { ..sigma_curry, ..dfinsupp.sigma_curry_equiv } | def | direct_sum.sigma_curry_equiv | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"dfinsupp.sigma_curry_equiv"
] | The natural map between `⨁ (i : Σ i, α i), δ i.1 i.2` and `⨁ i (j : α i), δ i j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add_monoid_hom {M S : Type*} [decidable_eq ι] [add_comm_monoid M]
[set_like S M] [add_submonoid_class S M] (A : ι → S) : (⨁ i, A i) →+ M | to_add_monoid (λ i, add_submonoid_class.subtype (A i)) | def | direct_sum.coe_add_monoid_hom | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"add_submonoid_class",
"set_like"
] | The canonical embedding from `⨁ i, A i` to `M` where `A` is a collection of `add_submonoid M`
indexed by `ι`.
When `S = submodule _ M`, this is available as a `linear_map`, `direct_sum.coe_linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add_monoid_hom_of {M S : Type*} [decidable_eq ι] [add_comm_monoid M]
[set_like S M] [add_submonoid_class S M] (A : ι → S) (i : ι) (x : A i) :
direct_sum.coe_add_monoid_hom A (of (λ i, A i) i x) = x | to_add_monoid_of _ _ _ | lemma | direct_sum.coe_add_monoid_hom_of | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"add_submonoid_class",
"direct_sum.coe_add_monoid_hom",
"set_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of_apply {M S : Type*} [decidable_eq ι] [add_comm_monoid M]
[set_like S M] [add_submonoid_class S M] {A : ι → S} (i j : ι) (x : A i) :
(of _ i x j : M) = if i = j then x else 0 | begin
obtain rfl | h := decidable.eq_or_ne i j,
{ rw [direct_sum.of_eq_same, if_pos rfl], },
{ rw [direct_sum.of_eq_of_ne _ _ _ _ h, if_neg h, zero_mem_class.coe_zero], },
end | lemma | direct_sum.coe_of_apply | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"add_submonoid_class",
"decidable.eq_or_ne",
"direct_sum.of_eq_of_ne",
"direct_sum.of_eq_same",
"set_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_internal {M S : Type*} [decidable_eq ι] [add_comm_monoid M]
[set_like S M] [add_submonoid_class S M] (A : ι → S) : Prop | function.bijective (direct_sum.coe_add_monoid_hom A) | def | direct_sum.is_internal | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"add_submonoid_class",
"direct_sum.coe_add_monoid_hom",
"set_like"
] | The `direct_sum` formed by a collection of additive submonoids (or subgroups, or submodules) of
`M` is said to be internal if the canonical map `(⨁ i, A i) →+ M` is bijective.
For the alternate statement in terms of independence and spanning, see
`direct_sum.subgroup_is_internal_iff_independent_and_supr_eq_top` and
`d... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_internal.add_submonoid_supr_eq_top {M : Type*} [decidable_eq ι] [add_comm_monoid M]
(A : ι → add_submonoid M)
(h : is_internal A) : supr A = ⊤ | begin
rw [add_submonoid.supr_eq_mrange_dfinsupp_sum_add_hom, add_monoid_hom.mrange_top_iff_surjective],
exact function.bijective.surjective h,
end | lemma | direct_sum.is_internal.add_submonoid_supr_eq_top | algebra.direct_sum | src/algebra/direct_sum/basic.lean | [
"data.dfinsupp.basic",
"group_theory.submonoid.operations"
] | [
"add_comm_monoid",
"add_submonoid",
"add_submonoid.supr_eq_mrange_dfinsupp_sum_add_hom",
"function.bijective.surjective",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposition | (decompose' : M → ⨁ i, ℳ i)
(left_inv : function.left_inverse (direct_sum.coe_add_monoid_hom ℳ) decompose' )
(right_inv : function.right_inverse (direct_sum.coe_add_monoid_hom ℳ) decompose') | class | direct_sum.decomposition | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.coe_add_monoid_hom"
] | A decomposition is an equivalence between an additive monoid `M` and a direct sum of additive
submonoids `ℳ i` of that `M`, such that the "recomposition" is canonical. This definition also
works for additive groups and modules.
This is a version of `direct_sum.is_internal` which comes with a constructive inverse to th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomposition.is_internal : direct_sum.is_internal ℳ | ⟨decomposition.right_inv.injective, decomposition.left_inv.surjective⟩ | lemma | direct_sum.decomposition.is_internal | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.is_internal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose : M ≃ ⨁ i, ℳ i | { to_fun := decomposition.decompose',
inv_fun := direct_sum.coe_add_monoid_hom ℳ,
left_inv := decomposition.left_inv,
right_inv := decomposition.right_inv } | def | direct_sum.decompose | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.coe_add_monoid_hom",
"inv_fun"
] | If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
to a direct sum of components. This is the canonical spelling of the `decompose'` field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomposition.induction_on {p : M → Prop}
(h_zero : p 0) (h_homogeneous : ∀ {i} (m : ℳ i), p (m : M))
(h_add : ∀ (m m' : M), p m → p m' → p (m + m')) : ∀ m, p m | begin
let ℳ' : ι → add_submonoid M :=
λ i, (⟨ℳ i, λ _ _, add_mem_class.add_mem, zero_mem_class.zero_mem _⟩ : add_submonoid M),
haveI t : direct_sum.decomposition ℳ' :=
{ decompose' := direct_sum.decompose ℳ,
left_inv := λ _, (decompose ℳ).left_inv _,
right_inv := λ _, (decompose ℳ).right_inv _, },
h... | lemma | direct_sum.decomposition.induction_on | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"add_submonoid",
"direct_sum.decompose",
"direct_sum.decomposition",
"direct_sum.is_internal.add_submonoid_supr_eq_top",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposition.decompose'_eq : decomposition.decompose' = decompose ℳ | rfl | lemma | direct_sum.decomposition.decompose'_eq | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_of {i : ι} (x : ℳ i) :
(decompose ℳ).symm (direct_sum.of _ i x) = x | direct_sum.coe_add_monoid_hom_of ℳ _ _ | lemma | direct_sum.decompose_symm_of | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.coe_add_monoid_hom_of",
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_coe {i : ι} (x : ℳ i) :
decompose ℳ (x : M) = direct_sum.of _ i x | by rw [←decompose_symm_of, equiv.apply_symm_apply] | lemma | direct_sum.decompose_coe | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.of",
"equiv.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_of_mem {x : M} {i : ι} (hx : x ∈ ℳ i) :
decompose ℳ x = direct_sum.of (λ i, ℳ i) i ⟨x, hx⟩ | decompose_coe _ ⟨x, hx⟩ | lemma | direct_sum.decompose_of_mem | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_of_mem_same {x : M} {i : ι} (hx : x ∈ ℳ i) :
(decompose ℳ x i : M) = x | by rw [decompose_of_mem _ hx, direct_sum.of_eq_same, subtype.coe_mk] | lemma | direct_sum.decompose_of_mem_same | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.of_eq_same",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_of_mem_ne {x : M} {i j : ι} (hx : x ∈ ℳ i) (hij : i ≠ j):
(decompose ℳ x j : M) = 0 | by rw [decompose_of_mem _ hx, direct_sum.of_eq_of_ne _ _ _ _ hij,
zero_mem_class.coe_zero] | lemma | direct_sum.decompose_of_mem_ne | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.of_eq_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_add_equiv : M ≃+ ⨁ i, ℳ i | add_equiv.symm
{ map_add' := map_add (direct_sum.coe_add_monoid_hom ℳ),
..(decompose ℳ).symm } | def | direct_sum.decompose_add_equiv | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"direct_sum.coe_add_monoid_hom"
] | If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
an additive monoid to a direct sum of components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decompose_zero : decompose ℳ (0 : M) = 0 | map_zero (decompose_add_equiv ℳ) | lemma | direct_sum.decompose_zero | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_zero : (decompose ℳ).symm 0 = (0 : M) | map_zero (decompose_add_equiv ℳ).symm | lemma | direct_sum.decompose_symm_zero | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_add (x y : M) : decompose ℳ (x + y) = decompose ℳ x + decompose ℳ y | map_add (decompose_add_equiv ℳ) x y | lemma | direct_sum.decompose_add | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_add (x y : ⨁ i, ℳ i) :
(decompose ℳ).symm (x + y) = (decompose ℳ).symm x + (decompose ℳ).symm y | map_add (decompose_add_equiv ℳ).symm x y | lemma | direct_sum.decompose_symm_add | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_sum {ι'} (s : finset ι') (f : ι' → M) :
decompose ℳ (∑ i in s, f i) = ∑ i in s, decompose ℳ (f i) | map_sum (decompose_add_equiv ℳ) f s | lemma | direct_sum.decompose_sum | algebra.direct_sum | src/algebra/direct_sum/decomposition.lean | [
"algebra.direct_sum.module",
"algebra.module.submodule.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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