statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y | by rw [←pow_two, ←pow_two, ←pow_two, add_sq] | lemma | char_two.add_mul_self | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"add_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_sum_sq (l : list R) : l.sum ^ 2 = (l.map (^ 2)).sum | list_sum_pow_char _ _ | lemma | char_two.list_sum_sq | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"list_sum_pow_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_sum_mul_self (l : list R) : l.sum * l.sum = (list.map (λ x, x * x) l).sum | by simp_rw [←pow_two, list_sum_sq] | lemma | char_two.list_sum_mul_self | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_sum_sq (l : multiset R) : l.sum ^ 2 = (l.map (^ 2)).sum | multiset_sum_pow_char _ _ | lemma | char_two.multiset_sum_sq | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"multiset",
"multiset_sum_pow_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_sum_mul_self (l : multiset R) : l.sum * l.sum = (multiset.map (λ x, x * x) l).sum | by simp_rw [←pow_two, multiset_sum_sq] | lemma | char_two.multiset_sum_mul_self | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"multiset",
"multiset.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_sq (s : finset ι) (f : ι → R) :
(∑ i in s, f i) ^ 2 = ∑ i in s, f i ^ 2 | sum_pow_char _ _ _ | lemma | char_two.sum_sq | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"finset",
"sum_pow_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mul_self (s : finset ι) (f : ι → R) :
(∑ i in s, f i) * (∑ i in s, f i) = ∑ i in s, f i * f i | by simp_rw [←pow_two, sum_sq] | lemma | char_two.sum_mul_self | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_eq_one_iff [nontrivial R]: (-1 : R) = 1 ↔ ring_char R = 2 | begin
refine ⟨λ h, _, λ h, @@char_two.neg_eq _ (ring_char.of_eq h) 1⟩,
rw [eq_comm, ←sub_eq_zero, sub_neg_eq_add, ← nat.cast_one, ← nat.cast_add] at h,
exact ((nat.dvd_prime nat.prime_two).mp (ring_char.dvd h)).resolve_left char_p.ring_char_ne_one
end | lemma | neg_one_eq_one_iff | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"char_p.ring_char_ne_one",
"char_two.neg_eq",
"nat.cast_add",
"nat.cast_one",
"nat.dvd_prime",
"nat.prime_two",
"nontrivial",
"ring_char",
"ring_char.dvd",
"ring_char.of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_of_neg_one [nontrivial R] :
order_of (-1 : R) = if ring_char R = 2 then 1 else 2 | begin
split_ifs,
{ rw [neg_one_eq_one_iff.2 h, order_of_one] },
apply order_of_eq_prime,
{ simp },
simpa [neg_one_eq_one_iff] using h
end | lemma | order_of_neg_one | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"neg_one_eq_one_iff",
"nontrivial",
"order_of",
"order_of_eq_prime",
"order_of_one",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_zero (R : Type*) [add_monoid_with_one R] : Prop | (cast_injective : function.injective (coe : ℕ → R)) | class | char_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"add_monoid_with_one"
] | Typeclass for monoids with characteristic zero.
(This is usually stated on fields but it makes sense for any additive monoid with 1.)
*Warning*: for a semiring `R`, `char_zero R` and `char_p R 0` need not coincide.
* `char_zero R` requires an injection `ℕ ↪ R`;
* `char_p R 0` asks that only `0 : ℕ` maps to `0 : R` u... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero_of_inj_zero {R : Type*} [add_group_with_one R]
(H : ∀ n:ℕ, (n:R) = 0 → n = 0) : char_zero R | ⟨λ m n h, begin
induction m with m ih generalizing n, { rw H n, rw [← h, nat.cast_zero] },
cases n with n, { apply H, rw [h, nat.cast_zero], },
simp_rw [nat.cast_succ, add_right_cancel_iff] at h, rwa ih,
end⟩ | theorem | char_zero_of_inj_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"add_group_with_one",
"char_zero",
"ih",
"nat.cast_succ",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_injective : function.injective (coe : ℕ → R) | char_zero.cast_injective | theorem | nat.cast_injective | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_inj {m n : ℕ} : (m : R) = n ↔ m = n | cast_injective.eq_iff | theorem | nat.cast_inj | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_eq_zero {n : ℕ} : (n : R) = 0 ↔ n = 0 | by rw [← cast_zero, cast_inj] | theorem | nat.cast_eq_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_ne_zero {n : ℕ} : (n : R) ≠ 0 ↔ n ≠ 0 | not_congr cast_eq_zero | theorem | nat.cast_ne_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_add_one_ne_zero (n : ℕ) : (n + 1 : R) ≠ 0 | by exact_mod_cast n.succ_ne_zero | lemma | nat.cast_add_one_ne_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_eq_one {n : ℕ} : (n : R) = 1 ↔ n = 1 | by rw [←cast_one, cast_inj] | theorem | nat.cast_eq_one | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_ne_one {n : ℕ} : (n : R) ≠ 1 ↔ n ≠ 1 | cast_eq_one.not | theorem | nat.cast_ne_one | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_zero {M} {n : ℕ}
[ne_zero n] [add_monoid_with_one M] [char_zero M] : ne_zero (n : M) | ⟨nat.cast_ne_zero.mpr out⟩ | instance | ne_zero.char_zero | algebra.char_zero | src/algebra/char_zero/defs.lean | [
"data.int.cast.defs"
] | [
"add_monoid_with_one",
"char_zero",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_zero.infinite : infinite M | infinite.of_injective coe nat.cast_injective | instance | char_zero.infinite | algebra.char_zero | src/algebra/char_zero/infinite.lean | [
"algebra.char_zero.defs",
"data.fintype.card"
] | [
"infinite",
"infinite.of_injective",
"nat.cast_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_embedding : ℕ ↪ R | ⟨coe, cast_injective⟩ | def | nat.cast_embedding | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [] | `nat.cast` as an embedding into monoids of characteristic `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cast_pow_eq_one {R : Type*} [semiring R] [char_zero R] (q : ℕ) (n : ℕ) (hn : n ≠ 0) :
(q : R) ^ n = 1 ↔ q = 1 | by { rw [←cast_pow, cast_eq_one], exact pow_eq_one_iff hn } | lemma | nat.cast_pow_eq_one | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"char_zero",
"pow_eq_one_iff",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_div_char_zero {k : Type*} [division_semiring k] [char_zero k] {m n : ℕ}
(n_dvd : n ∣ m) : ((m / n : ℕ) : k) = m / n | begin
rcases eq_or_ne n 0 with rfl | hn,
{ simp },
{ exact cast_div n_dvd (cast_ne_zero.2 hn), },
end | theorem | nat.cast_div_char_zero | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"char_zero",
"division_semiring",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_zero.ne_zero.two : ne_zero (2 : M) | ⟨have ((2:ℕ):M) ≠ 0, from nat.cast_ne_zero.2 dec_trivial, by rwa [nat.cast_two] at this⟩ | instance | char_zero.ne_zero.two | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"nat.cast_two",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self_eq_zero {a : R} : a + a = 0 ↔ a = 0 | by simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or] | lemma | add_self_eq_zero | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"mul_eq_zero",
"two_mul",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_eq_zero {a : R} : bit0 a = 0 ↔ a = 0 | add_self_eq_zero | lemma | bit0_eq_zero | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"add_self_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_bit0 {a : R} : 0 = bit0 a ↔ a = 0 | by { rw [eq_comm], exact bit0_eq_zero } | lemma | zero_eq_bit0 | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"bit0_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_ne_zero : bit0 a ≠ 0 ↔ a ≠ 0 | bit0_eq_zero.not | lemma | bit0_ne_zero | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_bit0 : 0 ≠ bit0 a ↔ a ≠ 0 | zero_eq_bit0.not | lemma | zero_ne_bit0 | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_self_iff {a : R} : -a = a ↔ a = 0 | neg_eq_iff_add_eq_zero.trans add_self_eq_zero | lemma | neg_eq_self_iff | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"add_self_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_neg_self_iff {a : R} : a = -a ↔ a = 0 | eq_neg_iff_add_eq_zero.trans add_self_eq_zero | lemma | eq_neg_self_iff | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"add_self_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_mul_inj {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) : n = 0 ∨ a = b | begin
rw [←sub_eq_zero, ←mul_sub, mul_eq_zero, sub_eq_zero] at h,
exact_mod_cast h,
end | lemma | nat_mul_inj | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"mul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_mul_inj' {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) (w : n ≠ 0) : a = b | by simpa [w] using nat_mul_inj h | lemma | nat_mul_inj' | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"nat_mul_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_injective : function.injective (bit0 : R → R) | λ a b h, begin
dsimp [bit0] at h,
simp only [(two_mul a).symm, (two_mul b).symm] at h,
refine nat_mul_inj' _ two_ne_zero,
exact_mod_cast h,
end | lemma | bit0_injective | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"nat_mul_inj'",
"two_mul",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_injective : function.injective (bit1 : R → R) | λ a b h, begin
simp only [bit1, add_left_inj] at h,
exact bit0_injective h,
end | lemma | bit1_injective | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"bit0_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_eq_bit0 {a b : R} : bit0 a = bit0 b ↔ a = b | bit0_injective.eq_iff | lemma | bit0_eq_bit0 | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_eq_bit1 {a b : R} : bit1 a = bit1 b ↔ a = b | bit1_injective.eq_iff | lemma | bit1_eq_bit1 | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_eq_one {a : R} : bit1 a = 1 ↔ a = 0 | by rw [show (1 : R) = bit1 0, by simp, bit1_eq_bit1] | lemma | bit1_eq_one | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"bit1_eq_bit1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_bit1 {a : R} : 1 = bit1 a ↔ a = 0 | by { rw [eq_comm], exact bit1_eq_one } | lemma | one_eq_bit1 | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"bit1_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_add_self (a : R) : (a + a) / 2 = a | by rw [← mul_two, mul_div_cancel a two_ne_zero] | lemma | half_add_self | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"mul_div_cancel",
"mul_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_halves' (a : R) : a / 2 + a / 2 = a | by rw [← add_div, half_add_self] | lemma | add_halves' | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"add_div",
"half_add_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_half (a : R) : a - a / 2 = a / 2 | by rw [sub_eq_iff_eq_add, add_halves'] | lemma | sub_half | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"add_halves'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_sub (a : R) : a / 2 - a = - (a / 2) | by rw [← neg_sub, sub_half] | lemma | half_sub | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"sub_half"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.char_zero (ϕ : R →+* S) [hS : char_zero S] : char_zero R | ⟨λ a b h, char_zero.cast_injective (by rw [←map_nat_cast ϕ, ←map_nat_cast ϕ, h])⟩ | lemma | ring_hom.char_zero | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"char_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.char_zero_iff {ϕ : R →+* S} (hϕ : function.injective ϕ) :
char_zero R ↔ char_zero S | ⟨λ hR, ⟨by introsI a b h; rwa [← @nat.cast_inj R, ← hϕ.eq_iff, map_nat_cast ϕ, map_nat_cast ϕ]⟩,
λ hS, by exactI ϕ.char_zero⟩ | lemma | ring_hom.char_zero_iff | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"char_zero",
"map_nat_cast",
"nat.cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.injective_nat (f : ℕ →+* R) [char_zero R] :
function.injective f | subsingleton.elim (nat.cast_ring_hom _) f ▸ nat.cast_injective | lemma | ring_hom.injective_nat | algebra.char_zero | src/algebra/char_zero/lemmas.lean | [
"data.nat.cast.field",
"algebra.group_power.lemmas"
] | [
"char_zero",
"nat.cast_injective",
"nat.cast_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz : z ≠ 0) :
z • r ∈ add_subgroup.zmultiples p ↔
∃ k : fin z.nat_abs, r - (k : ℕ) • (p / z : R) ∈ add_subgroup.zmultiples p | begin
rw [add_subgroup.mem_zmultiples_iff],
simp_rw [add_subgroup.mem_zmultiples_iff, div_eq_mul_inv, ←smul_mul_assoc, eq_sub_iff_add_eq],
have hz' : (z : R) ≠ 0 := int.cast_ne_zero.mpr hz,
conv_rhs { simp only [←(mul_right_injective₀ hz').eq_iff] { single_pass := tt}, },
simp_rw [←zsmul_eq_mul, smul_add, ←mu... | lemma | add_subgroup.zsmul_mem_zmultiples_iff_exists_sub_div | algebra.char_zero | src/algebra/char_zero/quotient.lean | [
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples",
"div_eq_mul_inv",
"fin.coe_mk",
"int.abs_eq_nat_abs",
"int.div_add_mod",
"int.mod_lt",
"int.mod_nonneg",
"int.to_nat_of_nonneg",
"mul_inv_cancel",
"mul_one",
"mul_right_injective₀",
"smul_add",
"smul_smul",
"to_nat",
"zsmul_eq_mul"
] | `z • r` is a multiple of `p` iff `r` is `pk/z` above a multiple of `p`, where `0 ≤ k < |z|`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :
n • r ∈ add_subgroup.zmultiples p ↔
∃ k : fin n, r - (k : ℕ) • (p / n : R) ∈ add_subgroup.zmultiples p | begin
simp_rw [←coe_nat_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (int.coe_nat_ne_zero.mpr hn),
int.cast_coe_nat],
refl,
end | lemma | add_subgroup.nsmul_mem_zmultiples_iff_exists_sub_div | algebra.char_zero | src/algebra/char_zero/quotient.lean | [
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples",
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_zsmul_eq_zsmul_iff {ψ θ : R ⧸ add_subgroup.zmultiples p} {z : ℤ} (hz : z ≠ 0) :
z • ψ = z • θ ↔ (∃ k : fin z.nat_abs, ψ = θ + (k : ℕ) • (p / z : R)) | begin
induction ψ using quotient.induction_on',
induction θ using quotient.induction_on',
have : (quotient.mk' : R → R ⧸ add_subgroup.zmultiples p) = coe := rfl,
simp only [this],
simp_rw [←coe_zsmul, ←coe_nsmul, ←coe_add, quotient_add_group.eq_iff_sub_mem, ←smul_sub,
←sub_sub, add_subgroup.zsmul_mem_zmul... | lemma | quotient_add_group.zmultiples_zsmul_eq_zsmul_iff | algebra.char_zero | src/algebra/char_zero/quotient.lean | [
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples",
"add_subgroup.zsmul_mem_zmultiples_iff_exists_sub_div",
"quotient.induction_on'",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_nsmul_eq_nsmul_iff {ψ θ : R ⧸ add_subgroup.zmultiples p} {n : ℕ} (hz : n ≠ 0) :
n • ψ = n • θ ↔ (∃ k : fin n, ψ = θ + (k : ℕ) • (p / n : R)) | begin
simp_rw [←coe_nat_zsmul ψ, ←coe_nat_zsmul θ,
zmultiples_zsmul_eq_zsmul_iff (int.coe_nat_ne_zero.mpr hz), int.cast_coe_nat],
refl,
end | lemma | quotient_add_group.zmultiples_nsmul_eq_nsmul_iff | algebra.char_zero | src/algebra/char_zero/quotient.lean | [
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples",
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_continued_fraction.pair | (a : α) (b : α) | structure | generalized_continued_fraction.pair | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | We collect a partial numerator `aᵢ` and partial denominator `bᵢ` in a pair `⟨aᵢ,bᵢ⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {β : Type*} (f : α → β) (gp : pair α) : pair β | ⟨f gp.a, f gp.b⟩ | def | generalized_continued_fraction.pair.map | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Maps a function `f` on both components of a given pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_generalized_continued_fraction_pair : has_coe (pair α) (pair β) | ⟨map coe⟩ | instance | generalized_continued_fraction.pair.has_coe_to_generalized_continued_fraction_pair | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Coerce a pair by elementwise coercion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_generalized_continued_fraction_pair {a b : α} :
(↑(pair.mk a b) : pair β) = pair.mk (a : β) (b : β) | rfl | lemma | generalized_continued_fraction.pair.coe_to_generalized_continued_fraction_pair | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_continued_fraction | (h : α) (s : seq $ pair α) | structure | generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | A *generalised continued fraction* (gcf) is a potentially infinite expression of the form
$$
h + \dfrac{a_0}
{b_0 + \dfrac{a_1}
{b_1 + \dfrac{a_2}
{b_2 + \dfrac{a_3}
{b_3 + \dots}}}}
$$
where ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_integer (a : α) : generalized_continued_fraction α | ⟨a, seq.nil⟩ | def | generalized_continued_fraction.of_integer | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Constructs a generalized continued fraction without fractional part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_numerators (g : generalized_continued_fraction α) : seq α | g.s.map pair.a | def | generalized_continued_fraction.partial_numerators | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Returns the sequence of partial numerators `aᵢ` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_denominators (g : generalized_continued_fraction α) : seq α | g.s.map pair.b | def | generalized_continued_fraction.partial_denominators | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Returns the sequence of partial denominators `bᵢ` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminated_at (g : generalized_continued_fraction α) (n : ℕ) : Prop | g.s.terminated_at n | def | generalized_continued_fraction.terminated_at | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | A gcf terminated at position `n` if its sequence terminates at position `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminated_at_decidable (g : generalized_continued_fraction α) (n : ℕ) :
decidable (g.terminated_at n) | by { unfold terminated_at, apply_instance } | instance | generalized_continued_fraction.terminated_at_decidable | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | It is decidable whether a gcf terminated at a given position. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminates (g : generalized_continued_fraction α) : Prop | g.s.terminates | def | generalized_continued_fraction.terminates | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | A gcf terminates if its sequence terminates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_generalized_continued_fraction :
has_coe (generalized_continued_fraction α) (generalized_continued_fraction β) | ⟨λ g, ⟨(g.h : β), (g.s.map coe : seq $ pair β)⟩⟩ | instance | generalized_continued_fraction.has_coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Coerce a gcf by elementwise coercion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_generalized_continued_fraction {g : generalized_continued_fraction α} :
(↑(g : generalized_continued_fraction α) : generalized_continued_fraction β) =
⟨(g.h : β), (g.s.map coe : seq $ pair β)⟩ | rfl | lemma | generalized_continued_fraction.coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_continued_fraction.is_simple_continued_fraction
(g : generalized_continued_fraction α) [has_one α] : Prop | ∀ (n : ℕ) (aₙ : α), g.partial_numerators.nth n = some aₙ → aₙ = 1 | def | generalized_continued_fraction.is_simple_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | A generalized continued fraction is a *simple continued fraction* if all partial numerators are
equal to one.
$$
h + \dfrac{1}
{b_0 + \dfrac{1}
{b_1 + \dfrac{1}
{b_2 + \dfrac{1}
{b_3 + \dots}}... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_continued_fraction [has_one α] | {g : generalized_continued_fraction α // g.is_simple_continued_fraction} | def | simple_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | A *simple continued fraction* (scf) is a generalized continued fraction (gcf) whose partial
numerators are equal to one.
$$
h + \dfrac{1}
{b_0 + \dfrac{1}
{b_1 + \dfrac{1}
{b_2 + \dfrac{1}
{b_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_integer (a : α) : simple_continued_fraction α | ⟨generalized_continued_fraction.of_integer a, λ n aₙ h, by cases h⟩ | def | simple_continued_fraction.of_integer | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"simple_continued_fraction"
] | Constructs a simple continued fraction without fractional part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_generalized_continued_fraction :
has_coe (simple_continued_fraction α) (generalized_continued_fraction α) | by {unfold simple_continued_fraction, apply_instance} | instance | simple_continued_fraction.has_coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"simple_continued_fraction"
] | Lift a scf to a gcf using the inclusion map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_generalized_continued_fraction {s : simple_continued_fraction α} :
(↑s : generalized_continued_fraction α) = s.val | rfl | lemma | simple_continued_fraction.coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"simple_continued_fraction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple_continued_fraction.is_continued_fraction [has_one α] [has_zero α] [has_lt α]
(s : simple_continued_fraction α) : Prop | ∀ (n : ℕ) (bₙ : α),
(↑s : generalized_continued_fraction α).partial_denominators.nth n = some bₙ → 0 < bₙ | def | simple_continued_fraction.is_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"simple_continued_fraction"
] | A simple continued fraction is a *(regular) continued fraction* ((r)cf) if all partial denominators
`bᵢ` are positive, i.e. `0 < bᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continued_fraction [has_one α] [has_zero α] [has_lt α] | {s : simple_continued_fraction α // s.is_continued_fraction} | def | continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"simple_continued_fraction"
] | A *(regular) continued fraction* ((r)cf) is a simple continued fraction (scf) whose partial
denominators are all positive. It is the subtype of scfs that satisfy
`simple_continued_fraction.is_continued_fraction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_integer (a : α) : continued_fraction α | ⟨simple_continued_fraction.of_integer a, λ n bₙ h, by cases h⟩ | def | continued_fraction.of_integer | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"continued_fraction"
] | Constructs a continued fraction without fractional part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_simple_continued_fraction :
has_coe (continued_fraction α) (simple_continued_fraction α) | by {unfold continued_fraction, apply_instance} | instance | continued_fraction.has_coe_to_simple_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"continued_fraction",
"simple_continued_fraction"
] | Lift a cf to a scf using the inclusion map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_simple_continued_fraction {c : continued_fraction α} :
(↑c : simple_continued_fraction α) = c.val | rfl | lemma | continued_fraction.coe_to_simple_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"continued_fraction",
"simple_continued_fraction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_generalized_continued_fraction :
has_coe (continued_fraction α) (generalized_continued_fraction α) | ⟨λ c, ↑(↑c : simple_continued_fraction α)⟩ | instance | continued_fraction.has_coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"continued_fraction",
"generalized_continued_fraction",
"simple_continued_fraction"
] | Lift a cf to a scf using the inclusion map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_generalized_continued_fraction {c : continued_fraction α} :
(↑c : generalized_continued_fraction α) = c.val | rfl | lemma | continued_fraction.coe_to_generalized_continued_fraction | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"continued_fraction",
"generalized_continued_fraction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
next_numerator (a b ppredA predA : K) : K | b * predA + a * ppredA | def | generalized_continued_fraction.next_numerator | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Returns the next numerator `Aₙ = bₙ₋₁ * Aₙ₋₁ + aₙ₋₁ * Aₙ₋₂`, where `predA` is `Aₙ₋₁`,
`ppredA` is `Aₙ₋₂`, `a` is `aₙ₋₁`, and `b` is `bₙ₋₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
next_denominator (aₙ bₙ ppredB predB : K) : K | bₙ * predB + aₙ * ppredB | def | generalized_continued_fraction.next_denominator | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Returns the next denominator `Bₙ = bₙ₋₁ * Bₙ₋₁ + aₙ₋₁ * Bₙ₋₂``, where `predB` is `Bₙ₋₁` and
`ppredB` is `Bₙ₋₂`, `a` is `aₙ₋₁`, and `b` is `bₙ₋₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
next_continuants (a b : K) (ppred pred : pair K) : pair K | ⟨next_numerator a b ppred.a pred.a, next_denominator a b ppred.b pred.b⟩ | def | generalized_continued_fraction.next_continuants | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Returns the next continuants `⟨Aₙ, Bₙ⟩` using `next_numerator` and `next_denominator`, where `pred`
is `⟨Aₙ₋₁, Bₙ₋₁⟩`, `ppred` is `⟨Aₙ₋₂, Bₙ₋₂⟩`, `a` is `aₙ₋₁`, and `b` is `bₙ₋₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuants_aux (g : generalized_continued_fraction K) : stream (pair K) | | 0 := ⟨1, 0⟩
| 1 := ⟨g.h, 1⟩
| (n + 2) :=
match g.s.nth n with
| none := continuants_aux (n + 1)
| some gp := next_continuants gp.a gp.b (continuants_aux n) (continuants_aux $ n + 1)
end | def | generalized_continued_fraction.continuants_aux | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"stream"
] | Returns the continuants `⟨Aₙ₋₁, Bₙ₋₁⟩` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuants (g : generalized_continued_fraction K) : stream (pair K) | g.continuants_aux.tail | def | generalized_continued_fraction.continuants | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"stream"
] | Returns the continuants `⟨Aₙ, Bₙ⟩` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numerators (g : generalized_continued_fraction K) : stream K | g.continuants.map pair.a | def | generalized_continued_fraction.numerators | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"stream"
] | Returns the numerators `Aₙ` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denominators (g : generalized_continued_fraction K) : stream K | g.continuants.map pair.b | def | generalized_continued_fraction.denominators | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"stream"
] | Returns the denominators `Bₙ` of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents (g : generalized_continued_fraction K) : stream K | λ (n : ℕ), (g.numerators n) / (g.denominators n) | def | generalized_continued_fraction.convergents | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction",
"stream"
] | Returns the convergents `Aₙ / Bₙ` of `g`, where `Aₙ, Bₙ` are the nth continuants of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents'_aux : seq (pair K) → ℕ → K | | s 0 := 0
| s (n + 1) := match s.head with
| none := 0
| some gp := gp.a / (gp.b + convergents'_aux s.tail n)
end | def | generalized_continued_fraction.convergents'_aux | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [] | Returns the approximation of the fraction described by the given sequence up to a given position n.
For example, `convergents'_aux [(1, 2), (3, 4), (5, 6)] 2 = 1 / (2 + 3 / 4)` and
`convergents'_aux [(1, 2), (3, 4), (5, 6)] 0 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents' (g : generalized_continued_fraction K) (n : ℕ) : K | g.h + convergents'_aux g.s n | def | generalized_continued_fraction.convergents' | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Returns the convergents of `g` by evaluating the fraction described by `g` up to a given
position `n`. For example, `convergents' [9; (1, 2), (3, 4), (5, 6)] 2 = 9 + 1 / (2 + 3 / 4)` and
`convergents' [9; (1, 2), (3, 4), (5, 6)] 0 = 9` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {g g' : generalized_continued_fraction α} :
g = g' ↔ g.h = g'.h ∧ g.s = g'.s | by { cases g, cases g', simp } | lemma | generalized_continued_fraction.ext_iff | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | Two gcfs `g` and `g'` are equal if and only if their components are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {g g' : generalized_continued_fraction α} (hyp : g.h = g'.h ∧ g.s = g'.s) :
g = g' | generalized_continued_fraction.ext_iff.elim_right hyp | lemma | generalized_continued_fraction.ext | algebra.continued_fractions | src/algebra/continued_fractions/basic.lean | [
"data.seq.seq",
"algebra.field.defs"
] | [
"generalized_continued_fraction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuants_aux_recurrence
{gp ppred pred : pair K} (nth_s_eq : g.s.nth n = some gp)
(nth_conts_aux_eq : g.continuants_aux n = ppred)
(succ_nth_conts_aux_eq : g.continuants_aux (n + 1) = pred) :
g.continuants_aux (n + 2) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ | by simp [*, continuants_aux, next_continuants, next_denominator, next_numerator] | lemma | generalized_continued_fraction.continuants_aux_recurrence | algebra.continued_fractions | src/algebra/continued_fractions/continuants_recurrence.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuants_recurrence_aux
{gp ppred pred : pair K} (nth_s_eq : g.s.nth n = some gp)
(nth_conts_aux_eq : g.continuants_aux n = ppred)
(succ_nth_conts_aux_eq : g.continuants_aux (n + 1) = pred) :
g.continuants (n + 1) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ | by simp [nth_cont_eq_succ_nth_cont_aux,
(continuants_aux_recurrence nth_s_eq nth_conts_aux_eq succ_nth_conts_aux_eq)] | lemma | generalized_continued_fraction.continuants_recurrence_aux | algebra.continued_fractions | src/algebra/continued_fractions/continuants_recurrence.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuants_recurrence
{gp ppred pred : pair K}
(succ_nth_s_eq : g.s.nth (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred)
(succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ | begin
rw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq,
exact (continuants_recurrence_aux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq)
end | theorem | generalized_continued_fraction.continuants_recurrence | algebra.continued_fractions | src/algebra/continued_fractions/continuants_recurrence.lean | [
"algebra.continued_fractions.translations"
] | [] | Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂` and `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numerators_recurrence {gp : pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.nth (n + 1) = some gp)
(nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA | begin
obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.a = ppredA,
from exists_conts_a_of_num nth_num_eq,
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.a = predA, from
exists_conts_a_of_num succ_nth_num_eq,
rw [num_eq... | lemma | generalized_continued_fraction.numerators_recurrence | algebra.continued_fractions | src/algebra/continued_fractions/continuants_recurrence.lean | [
"algebra.continued_fractions.translations"
] | [] | Shows that `Aₙ = bₙ * Aₙ₋₁ + aₙ * Aₙ₋₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denominators_recurrence {gp : pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.nth (n + 1) = some gp)
(nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppredB | begin
obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.b = ppredB,
from exists_conts_b_of_denom nth_denom_eq,
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.b = predB, from
exists_conts_b_of_denom succ_nth_denom_eq,
rw... | lemma | generalized_continued_fraction.denominators_recurrence | algebra.continued_fractions | src/algebra/continued_fractions/continuants_recurrence.lean | [
"algebra.continued_fractions.translations"
] | [] | Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_seq (s : seq $ pair K) (n : ℕ) : seq (pair K) | match prod.mk (s.nth n) (s.nth (n + 1)) with
| ⟨some gp_n, some gp_succ_n⟩ := seq.nats.zip_with
-- return the squashed value at position `n`; otherwise, do nothing.
(λ n' gp, if n' = n then ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ else gp) s
| _ := s
end | def | generalized_continued_fraction.squash_seq | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [] | Given a sequence of gcf.pairs `s = [(a₀, bₒ), (a₁, b₁), ...]`, `squash_seq s n`
combines `⟨aₙ, bₙ⟩` and `⟨aₙ₊₁, bₙ₊₁⟩` at position `n` to `⟨aₙ, bₙ + aₙ₊₁ / bₙ₊₁⟩`. For example,
`squash_seq s 0 = [(a₀, bₒ + a₁ / b₁), (a₁, b₁),...]`.
If `s.terminated_at (n + 1)`, then `squash_seq s n = s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_seq_eq_self_of_terminated (terminated_at_succ_n : s.terminated_at (n + 1)) :
squash_seq s n = s | begin
change s.nth (n + 1) = none at terminated_at_succ_n,
cases s_nth_eq : (s.nth n);
simp only [*, squash_seq]
end | lemma | generalized_continued_fraction.squash_seq_eq_self_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [] | If the sequence already terminated at position `n + 1`, nothing gets squashed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_seq_nth_of_not_terminated {gp_n gp_succ_n : pair K}
(s_nth_eq : s.nth n = some gp_n) (s_succ_nth_eq : s.nth (n + 1) = some gp_succ_n) :
(squash_seq s n).nth n = some ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ | by simp [*, squash_seq] | lemma | generalized_continued_fraction.squash_seq_nth_of_not_terminated | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [] | If the sequence has not terminated before position `n + 1`, the value at `n + 1` gets
squashed into position `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_seq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squash_seq s n).nth m = s.nth m | begin
cases s_succ_nth_eq : s.nth (n + 1),
case option.none { rw (squash_seq_eq_self_of_terminated s_succ_nth_eq) },
case option.some
{ obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.nth n = some gp_n, from
s.ge_stable n.le_succ s_succ_nth_eq,
obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.nth m = some gp_m, from
s.g... | lemma | generalized_continued_fraction.squash_seq_nth_of_lt | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [] | The values before the squashed position stay the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_seq_succ_n_tail_eq_squash_seq_tail_n :
(squash_seq s (n + 1)).tail = squash_seq s.tail n | begin
cases s_succ_succ_nth_eq : s.nth (n + 2) with gp_succ_succ_n,
case option.none
{ have : squash_seq s (n + 1) = s, from squash_seq_eq_self_of_terminated s_succ_succ_nth_eq,
cases s_succ_nth_eq : (s.nth (n + 1));
simp only [squash_seq, seq.nth_tail, s_succ_nth_eq, s_succ_succ_nth_eq] },
case option.... | lemma | generalized_continued_fraction.squash_seq_succ_n_tail_eq_squash_seq_tail_n | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [
"option.map₂_none_right"
] | Squashing at position `n + 1` and taking the tail is the same as squashing the tail of the
sequence at position `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squash_seq :
convergents'_aux s (n + 2) = convergents'_aux (squash_seq s n) (n + 1) | begin
cases s_succ_nth_eq : (s.nth $ n + 1) with gp_succ_n,
case option.none
{ rw [(squash_seq_eq_self_of_terminated s_succ_nth_eq),
(convergents'_aux_stable_step_of_terminated s_succ_nth_eq)] },
case option.some
{ induction n with m IH generalizing s gp_succ_n,
case nat.zero
{ obtain ⟨gp_head... | lemma | generalized_continued_fraction.succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squash_seq | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [
"zero_le_one"
] | The auxiliary function `convergents'_aux` returns the same value for a sequence and the
corresponding squashed sequence at the squashed position. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_gcf (g : generalized_continued_fraction K) : ℕ → generalized_continued_fraction K | | 0 := match g.s.nth 0 with
| none := g
| some gp := ⟨g.h + gp.a / gp.b, g.s⟩
end
| (n + 1) := ⟨g.h, squash_seq g.s n⟩ | def | generalized_continued_fraction.squash_gcf | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [
"generalized_continued_fraction"
] | Given a gcf `g = [h; (a₀, bₒ), (a₁, b₁), ...]`, we have
- `squash_nth.gcf g 0 = [h + a₀ / b₀); (a₀, bₒ), ...]`,
- `squash_nth.gcf g (n + 1) = ⟨g.h, squash_seq g.s n⟩` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squash_gcf_eq_self_of_terminated (terminated_at_n : terminated_at g n) :
squash_gcf g n = g | begin
cases n,
case nat.zero
{ change g.s.nth 0 = none at terminated_at_n,
simp only [convergents', squash_gcf, convergents'_aux, terminated_at_n] },
case nat.succ
{ cases g, simp [(squash_seq_eq_self_of_terminated terminated_at_n), squash_gcf] }
end | lemma | generalized_continued_fraction.squash_gcf_eq_self_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/convergents_equiv.lean | [
"algebra.continued_fractions.continuants_recurrence",
"algebra.continued_fractions.terminated_stable",
"tactic.field_simp",
"tactic.ring"
] | [] | If the gcf already terminated at position `n`, nothing gets squashed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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