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
value |
|---|---|---|---|---|---|---|---|---|---|---|
squash_gcf_nth_of_lt {m : ℕ} (m_lt_n : m < n) :
(squash_gcf g (n + 1)).s.nth m = g.s.nth m | by simp only [squash_gcf, (squash_seq_nth_of_lt m_lt_n)] | lemma | generalized_continued_fraction.squash_gcf_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 |
succ_nth_convergent'_eq_squash_gcf_nth_convergent' :
g.convergents' (n + 1) = (squash_gcf g n).convergents' n | begin
cases n,
case nat.zero
{ cases g_s_head_eq : (g.s.nth 0);
simp [g_s_head_eq, squash_gcf, convergents', convergents'_aux, seq.head] },
case nat.succ
{ simp only [succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squash_seq,
convergents', squash_gcf] }
end | lemma | generalized_continued_fraction.succ_nth_convergent'_eq_squash_gcf_nth_convergent' | 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"
] | [] | `convergents'` returns the same value for a gcf and the corresponding squashed gcf at the
squashed position. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuants_aux_eq_continuants_aux_squash_gcf_of_le {m : ℕ} :
m ≤ n → continuants_aux g m = (squash_gcf g n).continuants_aux m | nat.strong_induction_on m
(begin
clear m,
assume m IH m_le_n,
cases m with m',
{ refl },
{ cases n with n',
{ exact (m'.not_succ_le_zero m_le_n).elim }, -- 1 ≰ 0
{ cases m' with m'',
{ refl },
{ -- get some inequalities to instantiate the IH for m'' and m'' + 1
have m'_lt_n : m'' +... | lemma | generalized_continued_fraction.continuants_aux_eq_continuants_aux_squash_gcf_of_le | 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"
] | [
"lt_add_one",
"zero_lt_two"
] | The auxiliary continuants before the squashed position stay the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_nth_convergent_eq_squash_gcf_nth_convergent [field K]
(nth_part_denom_ne_zero : ∀ {b : K}, g.partial_denominators.nth n = some b → b ≠ 0) :
g.convergents (n + 1) = (squash_gcf g n).convergents n | begin
cases decidable.em (g.terminated_at n) with terminated_at_n not_terminated_at_n,
{ have : squash_gcf g n = g, from squash_gcf_eq_self_of_terminated terminated_at_n,
simp only [this, (convergents_stable_of_terminated n.le_succ terminated_at_n)] },
{ obtain ⟨⟨a, b⟩, s_nth_eq⟩ : ∃ gp_n, g.s.nth n = some gp... | lemma | generalized_continued_fraction.succ_nth_convergent_eq_squash_gcf_nth_convergent | 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"
] | [
"field",
"mul_div_cancel",
"mul_div_cancel_left",
"ring"
] | The convergents coincide in the expected way at the squashed position if the partial denominator
at the squashed position is not zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents_eq_convergents' [linear_ordered_field K]
(s_pos : ∀ {gp : pair K} {m : ℕ}, m < n → g.s.nth m = some gp → 0 < gp.a ∧ 0 < gp.b) :
g.convergents n = g.convergents' n | begin
induction n with n IH generalizing g,
case nat.zero { simp },
case nat.succ
{ let g' := squash_gcf g n, -- first replace the rhs with the squashed computation
suffices : g.convergents (n + 1) = g'.convergents' n, by
rwa [succ_nth_convergent'_eq_squash_gcf_nth_convergent'],
cases decidable.em... | theorem | generalized_continued_fraction.convergents_eq_convergents' | 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"
] | [
"div_pos",
"linear_ordered_field",
"lt_add_one"
] | Shows that the recurrence relation (`convergents`) and direct evaluation (`convergents'`) of the
gcf coincide at position `n` if the sequence of fractions contains strictly positive values only.
Requiring positivity of all values is just one possible condition to obtain this result.
For example, the dual - sequences wi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convergents_eq_convergents' [linear_ordered_field K] {c : continued_fraction K} :
(↑c : generalized_continued_fraction K).convergents =
(↑c : generalized_continued_fraction K).convergents' | begin
ext n,
apply convergents_eq_convergents',
assume gp m m_lt_n s_nth_eq,
exact ⟨zero_lt_one.trans_le ((c : simple_continued_fraction K).property m gp.a
(part_num_eq_s_a s_nth_eq)).symm.le,
c.property m gp.b $ part_denom_eq_s_b s_nth_eq⟩
end | theorem | continued_fraction.convergents_eq_convergents' | 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"
] | [
"continued_fraction",
"generalized_continued_fraction",
"linear_ordered_field",
"simple_continued_fraction"
] | Shows that the recurrence relation (`convergents`) and direct evaluation (`convergents'`) of a
(regular) continued fraction coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminated_stable (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.terminated_at m | g.s.terminated_stable n_le_m terminated_at_n | lemma | generalized_continued_fraction.terminated_stable | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | If a gcf terminated at position `n`, it also terminated at `m ≥ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuants_aux_stable_step_of_terminated (terminated_at_n : g.terminated_at n) :
g.continuants_aux (n + 2) = g.continuants_aux (n + 1) | by { rw [terminated_at_iff_s_none] at terminated_at_n,
simp only [terminated_at_n, continuants_aux] } | lemma | generalized_continued_fraction.continuants_aux_stable_step_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuants_aux_stable_of_terminated (n_lt_m : n < m)
(terminated_at_n : g.terminated_at n) :
g.continuants_aux m = g.continuants_aux (n + 1) | begin
refine nat.le_induction rfl (λ k hnk hk, _) _ n_lt_m,
rcases nat.exists_eq_add_of_lt hnk with ⟨k, rfl⟩,
refine (continuants_aux_stable_step_of_terminated _).trans hk,
exact terminated_stable (nat.le_add_right _ _) terminated_at_n
end | lemma | generalized_continued_fraction.continuants_aux_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [
"nat.exists_eq_add_of_lt",
"nat.le_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents'_aux_stable_step_of_terminated {s : seq $ pair K}
(terminated_at_n : s.terminated_at n) :
convergents'_aux s (n + 1) = convergents'_aux s n | begin
change s.nth n = none at terminated_at_n,
induction n with n IH generalizing s,
case nat.zero
{ simp only [convergents'_aux, terminated_at_n, seq.head] },
case nat.succ
{ cases s_head_eq : s.head with gp_head,
case option.none { simp only [convergents'_aux, s_head_eq] },
case option.some
{... | lemma | generalized_continued_fraction.convergents'_aux_stable_step_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents'_aux_stable_of_terminated
{s : seq $ pair K} (n_le_m : n ≤ m)
(terminated_at_n : s.terminated_at n) :
convergents'_aux s m = convergents'_aux s n | begin
induction n_le_m with m n_le_m IH,
{ refl },
{ refine (convergents'_aux_stable_step_of_terminated _).trans IH,
exact s.terminated_stable n_le_m terminated_at_n }
end | lemma | generalized_continued_fraction.convergents'_aux_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuants_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.continuants m = g.continuants n | by simp only [nth_cont_eq_succ_nth_cont_aux,
(continuants_aux_stable_of_terminated (nat.pred_le_iff.elim_left n_le_m) terminated_at_n)] | lemma | generalized_continued_fraction.continuants_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numerators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.numerators m = g.numerators n | by simp only [num_eq_conts_a, (continuants_stable_of_terminated n_le_m terminated_at_n)] | lemma | generalized_continued_fraction.numerators_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denominators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.denominators m = g.denominators n | by simp only [denom_eq_conts_b, (continuants_stable_of_terminated n_le_m terminated_at_n)] | lemma | generalized_continued_fraction.denominators_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.convergents m = g.convergents n | by simp only [convergents, (denominators_stable_of_terminated n_le_m terminated_at_n),
(numerators_stable_of_terminated n_le_m terminated_at_n)] | lemma | generalized_continued_fraction.convergents_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents'_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.terminated_at n) :
g.convergents' m = g.convergents' n | by simp only [convergents', (convergents'_aux_stable_of_terminated n_le_m terminated_at_n)] | lemma | generalized_continued_fraction.convergents'_stable_of_terminated | algebra.continued_fractions | src/algebra/continued_fractions/terminated_stable.lean | [
"algebra.continued_fractions.translations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminated_at_iff_s_terminated_at : g.terminated_at n ↔ g.s.terminated_at n | by refl | lemma | generalized_continued_fraction.terminated_at_iff_s_terminated_at | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminated_at_iff_s_none : g.terminated_at n ↔ g.s.nth n = none | by refl | lemma | generalized_continued_fraction.terminated_at_iff_s_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
part_num_none_iff_s_none : g.partial_numerators.nth n = none ↔ g.s.nth n = none | by cases s_nth_eq : (g.s.nth n); simp [partial_numerators, s_nth_eq] | lemma | generalized_continued_fraction.part_num_none_iff_s_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminated_at_iff_part_num_none : g.terminated_at n ↔ g.partial_numerators.nth n = none | by rw [terminated_at_iff_s_none, part_num_none_iff_s_none] | lemma | generalized_continued_fraction.terminated_at_iff_part_num_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
part_denom_none_iff_s_none : g.partial_denominators.nth n = none ↔ g.s.nth n = none | by cases s_nth_eq : (g.s.nth n); simp [partial_denominators, s_nth_eq] | lemma | generalized_continued_fraction.part_denom_none_iff_s_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
terminated_at_iff_part_denom_none : g.terminated_at n ↔ g.partial_denominators.nth n = none | by rw [terminated_at_iff_s_none, part_denom_none_iff_s_none] | lemma | generalized_continued_fraction.terminated_at_iff_part_denom_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
part_num_eq_s_a {gp : pair α} (s_nth_eq : g.s.nth n = some gp) :
g.partial_numerators.nth n = some gp.a | by simp [partial_numerators, s_nth_eq] | lemma | generalized_continued_fraction.part_num_eq_s_a | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
part_denom_eq_s_b {gp : pair α} (s_nth_eq : g.s.nth n = some gp) :
g.partial_denominators.nth n = some gp.b | by simp [partial_denominators, s_nth_eq] | lemma | generalized_continued_fraction.part_denom_eq_s_b | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_s_a_of_part_num {a : α} (nth_part_num_eq : g.partial_numerators.nth n = some a) :
∃ gp, g.s.nth n = some gp ∧ gp.a = a | by simpa [partial_numerators, seq.map_nth] using nth_part_num_eq | lemma | generalized_continued_fraction.exists_s_a_of_part_num | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_s_b_of_part_denom {b : α} (nth_part_denom_eq : g.partial_denominators.nth n = some b) :
∃ gp, g.s.nth n = some gp ∧ gp.b = b | by simpa [partial_denominators, seq.map_nth] using nth_part_denom_eq | lemma | generalized_continued_fraction.exists_s_b_of_part_denom | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nth_cont_eq_succ_nth_cont_aux : g.continuants n = g.continuants_aux (n + 1) | rfl | lemma | generalized_continued_fraction.nth_cont_eq_succ_nth_cont_aux | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_eq_conts_a : g.numerators n = (g.continuants n).a | rfl | lemma | generalized_continued_fraction.num_eq_conts_a | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_eq_conts_b : g.denominators n = (g.continuants n).b | rfl | lemma | generalized_continued_fraction.denom_eq_conts_b | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergent_eq_num_div_denom : g.convergents n = g.numerators n / g.denominators n | rfl | lemma | generalized_continued_fraction.convergent_eq_num_div_denom | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergent_eq_conts_a_div_conts_b :
g.convergents n = (g.continuants n).a / (g.continuants n).b | rfl | lemma | generalized_continued_fraction.convergent_eq_conts_a_div_conts_b | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_conts_a_of_num {A : K} (nth_num_eq : g.numerators n = A) :
∃ conts, g.continuants n = conts ∧ conts.a = A | by simpa | lemma | generalized_continued_fraction.exists_conts_a_of_num | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_conts_b_of_denom {B : K} (nth_denom_eq : g.denominators n = B) :
∃ conts, g.continuants n = conts ∧ conts.b = B | by simpa | lemma | generalized_continued_fraction.exists_conts_b_of_denom | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_continuant_aux_eq_one_zero : g.continuants_aux 0 = ⟨1, 0⟩ | rfl | lemma | generalized_continued_fraction.zeroth_continuant_aux_eq_one_zero | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_continuant_aux_eq_h_one : g.continuants_aux 1 = ⟨g.h, 1⟩ | rfl | lemma | generalized_continued_fraction.first_continuant_aux_eq_h_one | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_continuant_eq_h_one : g.continuants 0 = ⟨g.h, 1⟩ | rfl | lemma | generalized_continued_fraction.zeroth_continuant_eq_h_one | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_numerator_eq_h : g.numerators 0 = g.h | rfl | lemma | generalized_continued_fraction.zeroth_numerator_eq_h | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_denominator_eq_one : g.denominators 0 = 1 | rfl | lemma | generalized_continued_fraction.zeroth_denominator_eq_one | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_convergent_eq_h : g.convergents 0 = g.h | by simp [convergent_eq_num_div_denom, num_eq_conts_a, denom_eq_conts_b, div_one] | lemma | generalized_continued_fraction.zeroth_convergent_eq_h | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [
"div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
second_continuant_aux_eq {gp : pair K} (zeroth_s_eq : g.s.nth 0 = some gp) :
g.continuants_aux 2 = ⟨gp.b * g.h + gp.a, gp.b⟩ | by simp [zeroth_s_eq, continuants_aux, next_continuants, next_denominator, next_numerator] | lemma | generalized_continued_fraction.second_continuant_aux_eq | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_continuant_eq {gp : pair K} (zeroth_s_eq : g.s.nth 0 = some gp) :
g.continuants 1 = ⟨gp.b * g.h + gp.a, gp.b⟩ | by simp [nth_cont_eq_succ_nth_cont_aux, (second_continuant_aux_eq zeroth_s_eq)] | lemma | generalized_continued_fraction.first_continuant_eq | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_numerator_eq {gp : pair K} (zeroth_s_eq : g.s.nth 0 = some gp) :
g.numerators 1 = gp.b * g.h + gp.a | by simp[num_eq_conts_a, (first_continuant_eq zeroth_s_eq)] | lemma | generalized_continued_fraction.first_numerator_eq | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_denominator_eq {gp : pair K} (zeroth_s_eq : g.s.nth 0 = some gp) :
g.denominators 1 = gp.b | by simp[denom_eq_conts_b, (first_continuant_eq zeroth_s_eq)] | lemma | generalized_continued_fraction.first_denominator_eq | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_convergent'_aux_eq_zero {s : seq $ pair K} : convergents'_aux s 0 = 0 | rfl | lemma | generalized_continued_fraction.zeroth_convergent'_aux_eq_zero | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zeroth_convergent'_eq_h : g.convergents' 0 = g.h | by simp [convergents'] | lemma | generalized_continued_fraction.zeroth_convergent'_eq_h | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents'_aux_succ_none {s : seq (pair K)} (h : s.head = none) (n : ℕ) :
convergents'_aux s (n + 1) = 0 | by rw [convergents'_aux, h, convergents'_aux._match_1] | lemma | generalized_continued_fraction.convergents'_aux_succ_none | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents'_aux_succ_some {s : seq (pair K)} {p : pair K} (h : s.head = some p) (n : ℕ) :
convergents'_aux s (n + 1) = p.a / (p.b + convergents'_aux s.tail n) | by rw [convergents'_aux, h, convergents'_aux._match_1] | lemma | generalized_continued_fraction.convergents'_aux_succ_some | algebra.continued_fractions | src/algebra/continued_fractions/translations.lean | [
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nth_stream_fr_nonneg_lt_one {ifp_n : int_fract_pair K}
(nth_stream_eq : int_fract_pair.stream v n = some ifp_n) :
0 ≤ ifp_n.fr ∧ ifp_n.fr < 1 | begin
cases n,
case nat.zero
{ have : int_fract_pair.of v = ifp_n, by injection nth_stream_eq,
rw [←this, int_fract_pair.of],
exact ⟨fract_nonneg _, fract_lt_one _⟩ },
case nat.succ
{ rcases (succ_nth_stream_eq_some_iff.elim_left nth_stream_eq) with ⟨_, _, _, ifp_of_eq_ifp_n⟩,
rw [←ifp_of_eq_ifp_n... | lemma | generalized_continued_fraction.int_fract_pair.nth_stream_fr_nonneg_lt_one | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the fractional parts of the stream are in `[0,1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_stream_fr_nonneg {ifp_n : int_fract_pair K}
(nth_stream_eq : int_fract_pair.stream v n = some ifp_n) :
0 ≤ ifp_n.fr | (nth_stream_fr_nonneg_lt_one nth_stream_eq).left | lemma | generalized_continued_fraction.int_fract_pair.nth_stream_fr_nonneg | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the fractional parts of the stream are nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_stream_fr_lt_one {ifp_n : int_fract_pair K}
(nth_stream_eq : int_fract_pair.stream v n = some ifp_n) :
ifp_n.fr < 1 | (nth_stream_fr_nonneg_lt_one nth_stream_eq).right | lemma | generalized_continued_fraction.int_fract_pair.nth_stream_fr_lt_one | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the fractional parts of the stream are smaller than one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le_succ_nth_stream_b {ifp_succ_n : int_fract_pair K}
(succ_nth_stream_eq : int_fract_pair.stream v (n + 1) = some ifp_succ_n) :
1 ≤ ifp_succ_n.b | begin
obtain ⟨ifp_n, nth_stream_eq, stream_nth_fr_ne_zero, ⟨-⟩⟩ :
∃ ifp_n, int_fract_pair.stream v 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 succ_nth_stream_eq,
suffices : 1 ≤ ifp_n.fr⁻¹, { rw_mod_cast [le_floor], assumption }... | lemma | generalized_continued_fraction.int_fract_pair.one_le_succ_nth_stream_b | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"one_le_inv"
] | Shows that the integer parts of the stream are at least one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_nth_stream_b_le_nth_stream_fr_inv {ifp_n ifp_succ_n : int_fract_pair K}
(nth_stream_eq : int_fract_pair.stream v n = some ifp_n)
(succ_nth_stream_eq : int_fract_pair.stream v (n + 1) = some ifp_succ_n) :
(ifp_succ_n.b : K) ≤ ifp_n.fr⁻¹ | begin
suffices : (⌊ifp_n.fr⁻¹⌋ : K) ≤ ifp_n.fr⁻¹,
{ cases ifp_n with _ ifp_n_fr,
have : ifp_n_fr ≠ 0,
{ intro h, simpa [h, int_fract_pair.stream, nth_stream_eq] using succ_nth_stream_eq },
have : int_fract_pair.of ifp_n_fr⁻¹ = ifp_succ_n,
{ simpa [this, int_fract_pair.stream, nth_stream_eq, option.c... | lemma | generalized_continued_fraction.int_fract_pair.succ_nth_stream_b_le_nth_stream_fr_inv | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"option.coe_def"
] | Shows that the `n + 1`th integer part `bₙ₊₁` of the stream is smaller or equal than the inverse of
the `n`th fractional part `frₙ` of the stream.
This result is straight-forward as `bₙ₊₁` is defined as the floor of `1 / frₙ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_one_le_nth_part_denom {b : K}
(nth_part_denom_eq : (of v).partial_denominators.nth n = some b) :
1 ≤ b | begin
obtain ⟨gp_n, nth_s_eq, ⟨-⟩⟩ : ∃ gp_n, (of v).s.nth n = some gp_n ∧ gp_n.b = b, from
exists_s_b_of_part_denom nth_part_denom_eq,
obtain ⟨ifp_n, succ_nth_stream_eq, ifp_n_b_eq_gp_n_b⟩ :
∃ ifp, int_fract_pair.stream v (n + 1) = some ifp ∧ (ifp.b : K) = gp_n.b, from
int_fract_pair.exists_succ_nth_... | lemma | generalized_continued_fraction.of_one_le_nth_part_denom | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the integer parts of the continued fraction are at least one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_part_num_eq_one_and_exists_int_part_denom_eq {gp : generalized_continued_fraction.pair K}
(nth_s_eq : (of v).s.nth n = some gp) :
gp.a = 1 ∧ ∃ (z : ℤ), gp.b = (z : K) | begin
obtain ⟨ifp, stream_succ_nth_eq, -⟩ :
∃ ifp, int_fract_pair.stream v (n + 1) = some ifp ∧ _,
from int_fract_pair.exists_succ_nth_stream_of_gcf_of_nth_eq_some nth_s_eq,
have : gp = ⟨1, ifp.b⟩, by
{ have : (of v).s.nth n = some ⟨1, ifp.b⟩, from
nth_of_eq_some_of_succ_nth_int_fract_pair_stream ... | lemma | generalized_continued_fraction.of_part_num_eq_one_and_exists_int_part_denom_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"generalized_continued_fraction.pair"
] | Shows that the partial numerators `aᵢ` of the continued fraction are equal to one and the partial
denominators `bᵢ` correspond to integers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_part_num_eq_one {a : K} (nth_part_num_eq : (of v).partial_numerators.nth n = some a) :
a = 1 | begin
obtain ⟨gp, nth_s_eq, gp_a_eq_a_n⟩ : ∃ gp, (of v).s.nth n = some gp ∧ gp.a = a, from
exists_s_a_of_part_num nth_part_num_eq,
have : gp.a = 1, from (of_part_num_eq_one_and_exists_int_part_denom_eq nth_s_eq).left,
rwa gp_a_eq_a_n at this
end | lemma | generalized_continued_fraction.of_part_num_eq_one | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the partial numerators `aᵢ` are equal to one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_int_eq_of_part_denom {b : K}
(nth_part_denom_eq : (of v).partial_denominators.nth n = some b) :
∃ (z : ℤ), b = (z : K) | begin
obtain ⟨gp, nth_s_eq, gp_b_eq_b_n⟩ : ∃ gp, (of v).s.nth n = some gp ∧ gp.b = b, from
exists_s_b_of_part_denom nth_part_denom_eq,
have : ∃ (z : ℤ), gp.b = (z : K), from
(of_part_num_eq_one_and_exists_int_part_denom_eq nth_s_eq).right,
rwa gp_b_eq_b_n at this
end | lemma | generalized_continued_fraction.exists_int_eq_of_part_denom | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the partial denominators `bᵢ` correspond to an integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fib_le_of_continuants_aux_b : (n ≤ 1 ∨ ¬(of v).terminated_at (n - 2)) →
(fib n : K) ≤ ((of v).continuants_aux n).b | nat.strong_induction_on n
begin
clear n,
assume n IH hyp,
rcases n with _|_|n,
{ simp [fib_add_two, continuants_aux] }, -- case n = 0
{ simp [fib_add_two, continuants_aux] }, -- case n = 1
{ let g := of v, -- case 2 ≤ n
have : ¬(n + 2 ≤ 1), by linarith,
have not_terminated_at_n : ¬g.terminated_at n... | lemma | generalized_continued_fraction.fib_le_of_continuants_aux_b | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"one_mul",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nth_fib_le_of_nth_denom (hyp: n = 0 ∨ ¬(of v).terminated_at (n - 1)) :
(fib (n + 1) : K) ≤ (of v).denominators n | begin
rw [denom_eq_conts_b, nth_cont_eq_succ_nth_cont_aux],
have : (n + 1) ≤ 1 ∨ ¬(of v).terminated_at (n - 1), by
{ cases n,
case nat.zero : { exact (or.inl $ le_refl 1) },
case nat.succ : { exact or.inr (or.resolve_left hyp n.succ_ne_zero) } },
exact (fib_le_of_continuants_aux_b this)
end | lemma | generalized_continued_fraction.succ_nth_fib_le_of_nth_denom | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that the `n`th denominator is greater than or equal to the `n + 1`th fibonacci number,
that is `nat.fib (n + 1) ≤ Bₙ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_le_of_continuants_aux_b : 0 ≤ ((of v).continuants_aux n).b | begin
let g := of v,
induction n with n IH,
case nat.zero: { refl },
case nat.succ:
{ cases (decidable.em $ g.terminated_at (n - 1)) with terminated not_terminated,
{ cases n, -- terminating case
{ simp [zero_le_one] },
{ have : g.continuants_aux (n + 2) = g.continuants_aux (n + 1), from
... | lemma | generalized_continued_fraction.zero_le_of_continuants_aux_b | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_of_denom : 0 ≤ (of v).denominators n | by { rw [denom_eq_conts_b, nth_cont_eq_succ_nth_cont_aux], exact zero_le_of_continuants_aux_b } | lemma | generalized_continued_fraction.zero_le_of_denom | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that all denominators are nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_succ_succ_nth_continuants_aux_b {b : K}
(nth_part_denom_eq : (of v).partial_denominators.nth n = some b) :
b * ((of v).continuants_aux $ n + 1).b ≤ ((of v).continuants_aux $ n + 2).b | begin
obtain ⟨gp_n, nth_s_eq, rfl⟩ : ∃ gp_n, (of v).s.nth n = some gp_n ∧ gp_n.b = b,
from exists_s_b_of_part_denom nth_part_denom_eq,
simp [of_part_num_eq_one (part_num_eq_s_a nth_s_eq), zero_le_of_continuants_aux_b,
generalized_continued_fraction.continuants_aux_recurrence nth_s_eq rfl rfl]
end | lemma | generalized_continued_fraction.le_of_succ_succ_nth_continuants_aux_b | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"generalized_continued_fraction.continuants_aux_recurrence"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_succ_nth_denom {b : K}
(nth_part_denom_eq : (of v).partial_denominators.nth n = some b) :
b * (of v).denominators n ≤ (of v).denominators (n + 1) | begin
rw [denom_eq_conts_b, nth_cont_eq_succ_nth_cont_aux],
exact (le_of_succ_succ_nth_continuants_aux_b nth_part_denom_eq)
end | theorem | generalized_continued_fraction.le_of_succ_nth_denom | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | Shows that `bₙ * Bₙ ≤ Bₙ₊₁`, where `bₙ` is the `n`th partial denominator and `Bₙ₊₁` and `Bₙ` are
the `n + 1`th and `n`th denominator of the continued fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_denom_mono : (of v).denominators n ≤ (of v).denominators (n + 1) | begin
let g := of v,
cases (decidable.em $ g.partial_denominators.terminated_at n) with terminated not_terminated,
{ have : g.partial_denominators.nth n = none, by rwa stream.seq.terminated_at at terminated,
have : g.terminated_at n, from
terminated_at_iff_part_denom_none.elim_right (by rwa stream.seq.t... | theorem | generalized_continued_fraction.of_denom_mono | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"mul_le_mul_of_nonneg_right",
"stream.seq.terminated_at"
] | Shows that the sequence of denominators is monotone, that is `Bₙ ≤ Bₙ₊₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
determinant_aux (hyp: n = 0 ∨ ¬(of v).terminated_at (n - 1)) :
((of v).continuants_aux n).a * ((of v).continuants_aux (n + 1)).b
- ((of v).continuants_aux n).b * ((of v).continuants_aux (n + 1)).a
= (-1)^n | begin
induction n with n IH,
case nat.zero { simp [continuants_aux] },
case nat.succ
{ -- set up some shorthand notation
let g := of v,
let conts := continuants_aux g (n + 2),
set pred_conts := continuants_aux g (n + 1) with pred_conts_eq,
set ppred_conts := continuants_aux g n with ppred_conts_... | lemma | generalized_continued_fraction.determinant_aux | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"pow_succ",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
determinant (not_terminated_at_n : ¬(of v).terminated_at n) :
(of v).numerators n * (of v).denominators (n + 1)
- (of v).denominators n * (of v).numerators (n + 1)
= (-1)^(n + 1) | (determinant_aux $ or.inr $ not_terminated_at_n) | lemma | generalized_continued_fraction.determinant | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [] | The determinant formula `Aₙ * Bₙ₊₁ - Bₙ * Aₙ₊₁ = (-1)^(n + 1)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_convergents_eq {ifp : int_fract_pair K}
(stream_nth_eq : int_fract_pair.stream v n = some ifp) :
let g | of v in
let B := (g.continuants_aux (n + 1)).b in
let pB := (g.continuants_aux n).b in
v - g.convergents n = if ifp.fr = 0 then 0 else (-1)^n / (B * (ifp.fr⁻¹ * B + pB)) :=
begin
-- set up some shorthand notation
let g := of v,
let conts := g.continuants_aux (n + 1),
let pred_conts := g.continuants_aux n,... | lemma | generalized_continued_fraction.sub_convergents_eq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"div_sub_div",
"generalized_continued_fraction.comp_exact_value",
"inv_pos",
"ring",
"zero_lt_one"
] | This lemma follows from the finite correctness proof, the determinant equality, and
by simplifying the difference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_sub_convergents_le (not_terminated_at_n : ¬(of v).terminated_at n) : | |v - (of v).convergents n|
≤ 1 / (((of v).denominators n) * ((of v).denominators $ n + 1)) :=
begin
-- shorthand notation
let g := of v,
let nextConts := g.continuants_aux (n + 2),
set conts := continuants_aux g (n + 1) with conts_eq,
set pred_conts := continuants_aux g n with pred_conts_eq,
-- change the... | theorem | generalized_continued_fraction.abs_sub_convergents_le | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"abs_div",
"abs_neg_one_pow",
"abs_of_pos",
"div_le_div_of_le_left",
"mul_le_mul_left",
"zero_le_one"
] | Shows that `|v - Aₙ / Bₙ| ≤ 1 / (Bₙ * Bₙ₊₁)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_sub_convergents_le' {b : K}
(nth_part_denom_eq : (of v).partial_denominators.nth n = some b) : | |v - (of v).convergents n| ≤ 1 / (b * ((of v).denominators n) * ((of v).denominators n)) :=
begin
have not_terminated_at_n : ¬(of v).terminated_at n,
by simp [terminated_at_iff_part_denom_none, nth_part_denom_eq],
refine (abs_sub_convergents_le not_terminated_at_n).trans _,
-- One can show that `0 < (generali... | lemma | generalized_continued_fraction.abs_sub_convergents_le' | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximations.lean | [
"algebra.continued_fractions.computation.correctness_terminating",
"data.nat.fib",
"tactic.solve_by_elim"
] | [
"div_zero",
"generalized_continued_fraction.of",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"mul_zero",
"one_div_le_one_div_of_le",
"zero_mul"
] | Shows that `|v - Aₙ / Bₙ| ≤ 1 / (bₙ * Bₙ * Bₙ)`. This bound is worse than the one shown in
`gcf.abs_sub_convergents_le`, but sometimes it is easier to apply and sufficient for one's use case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generalized_continued_fraction.of_is_simple_continued_fraction :
(of v).is_simple_continued_fraction | (λ _ _ nth_part_num_eq, of_part_num_eq_one nth_part_num_eq) | lemma | generalized_continued_fraction.of_is_simple_continued_fraction | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple_continued_fraction.of : simple_continued_fraction K | ⟨of v, generalized_continued_fraction.of_is_simple_continued_fraction v⟩ | def | simple_continued_fraction.of | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"generalized_continued_fraction.of_is_simple_continued_fraction",
"simple_continued_fraction"
] | Creates the simple continued fraction of a value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_continued_fraction.of_is_continued_fraction :
(simple_continued_fraction.of v).is_continued_fraction | (λ _ denom nth_part_denom_eq, lt_of_lt_of_le zero_lt_one
(of_one_le_nth_part_denom nth_part_denom_eq)) | lemma | simple_continued_fraction.of_is_continued_fraction | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"simple_continued_fraction.of",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continued_fraction.of : continued_fraction K | ⟨simple_continued_fraction.of v, simple_continued_fraction.of_is_continued_fraction v⟩ | def | continued_fraction.of | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"continued_fraction",
"simple_continued_fraction.of_is_continued_fraction"
] | Creates the continued fraction of a value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_convergents_eq_convergents' :
(of v).convergents = (of v).convergents' | @continued_fraction.convergents_eq_convergents' _ _ (continued_fraction.of v) | lemma | generalized_continued_fraction.of_convergents_eq_convergents' | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"continued_fraction.convergents_eq_convergents'",
"continued_fraction.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergents_succ (n : ℕ) :
(of v).convergents (n + 1) = ⌊v⌋ + 1 / (of (int.fract v)⁻¹).convergents n | by rw [of_convergents_eq_convergents', convergents'_succ, of_convergents_eq_convergents'] | lemma | generalized_continued_fraction.convergents_succ | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"int.fract"
] | 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 |
of_convergence_epsilon :
∀ (ε > (0 : K)), ∃ (N : ℕ), ∀ (n ≥ N), |v - (of v).convergents n| < ε | begin
assume ε ε_pos,
-- use the archimedean property to obtian a suitable N
rcases (exists_nat_gt (1 / ε) : ∃ (N' : ℕ), 1 / ε < N') with ⟨N', one_div_ε_lt_N'⟩,
let N := max N' 5, -- set minimum to 5 to have N ≤ fib N work
existsi N,
assume n n_ge_N,
let g := of v,
cases decidable.em (g.terminated_at n)... | theorem | generalized_continued_fraction.of_convergence_epsilon | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"div_lt_iff",
"div_lt_iff'",
"exists_nat_gt",
"le_mul_self",
"mul_le_mul",
"mul_le_mul_left",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_convergence [order_topology K] :
filter.tendsto ((of v).convergents) filter.at_top $ nhds v | by simpa [linear_ordered_add_comm_group.tendsto_nhds, abs_sub_comm] using (of_convergence_epsilon v) | theorem | generalized_continued_fraction.of_convergence | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/approximation_corollaries.lean | [
"algebra.continued_fractions.computation.approximations",
"algebra.continued_fractions.convergents_equiv",
"algebra.order.archimedean",
"algebra.algebra.basic",
"topology.order.basic"
] | [
"abs_sub_comm",
"filter.at_top",
"filter.tendsto",
"linear_ordered_add_comm_group.tendsto_nhds",
"nhds",
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_fract_pair | (b : ℤ) (fr : K) | structure | generalized_continued_fraction.int_fract_pair | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | We collect an integer part `b = ⌊v⌋` and fractional part `fr = v - ⌊v⌋` of a value `v` in a pair
`⟨b, fr⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited [inhabited K] : inhabited (int_fract_pair K) | ⟨⟨0, default⟩⟩ | instance | generalized_continued_fraction.int_fract_pair.inhabited | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mapFr {β : Type*} (f : K → β) (gp : int_fract_pair K) : int_fract_pair β | ⟨gp.b, f gp.fr⟩ | def | generalized_continued_fraction.int_fract_pair.mapFr | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | Maps a function `f` on the fractional components of a given pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_int_fract_pair : has_coe (int_fract_pair K) (int_fract_pair β) | ⟨mapFr coe⟩ | instance | generalized_continued_fraction.int_fract_pair.has_coe_to_int_fract_pair | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | Coerce a pair by coercing the fractional component. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_int_fract_pair {b : ℤ} {fr : K} :
(↑(int_fract_pair.mk b fr) : int_fract_pair β) = int_fract_pair.mk b (↑fr : β) | rfl | lemma | generalized_continued_fraction.int_fract_pair.coe_to_int_fract_pair | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (v : K) : int_fract_pair K | ⟨⌊v⌋, int.fract v⟩ | def | generalized_continued_fraction.int_fract_pair.of | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [
"int.fract"
] | Creates the integer and fractional part of a value `v`, i.e. `⟨⌊v⌋, v - ⌊v⌋⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream (v : K) : stream $ option (int_fract_pair K) | | 0 := some (int_fract_pair.of v)
| (n + 1) := (stream n).bind $ λ ap_n,
if ap_n.fr = 0 then none else some (int_fract_pair.of ap_n.fr⁻¹) | def | generalized_continued_fraction.int_fract_pair.stream | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [
"stream"
] | Creates the stream of integer and fractional parts of a value `v` needed to obtain the continued
fraction representation of `v` in `generalized_continued_fraction.of`. More precisely, given a value
`v : K`, it recursively computes a stream of option `ℤ × K` pairs as follows:
- `stream v 0 = some ⟨⌊v⌋, v - ⌊v⌋⟩`
- `stre... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stream_is_seq (v : K) : (int_fract_pair.stream v).is_seq | by { assume _ hyp, simp [int_fract_pair.stream, hyp] } | lemma | generalized_continued_fraction.int_fract_pair.stream_is_seq | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [] | Shows that `int_fract_pair.stream` has the sequence property, that is once we return `none` at
position `n`, we also return `none` at `n + 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq1 (v : K) : stream.seq1 $ int_fract_pair K | ⟨ int_fract_pair.of v,--the head
stream.seq.tail -- take the tail of `int_fract_pair.stream` since the first element is already in
-- the head
-- create a sequence from `int_fract_pair.stream`
⟨ int_fract_pair.stream v, -- the underlying stream
@stream_is_seq _ _ _ v ⟩ ⟩ | def | generalized_continued_fraction.int_fract_pair.seq1 | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [
"stream.seq.tail",
"stream.seq1"
] | Uses `int_fract_pair.stream` to create a sequence with head (i.e. `seq1`) of integer and fractional
parts of a value `v`. The first value of `int_fract_pair.stream` is never `none`, so we can safely
extract it and put the tail of the stream in the sequence part.
This is just an intermediate representation and users sh... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of [linear_ordered_field K] [floor_ring K] (v : K) :
generalized_continued_fraction K | let ⟨h, s⟩ := int_fract_pair.seq1 v in -- get the sequence of integer and fractional parts.
⟨ h.b, -- the head is just the first integer part
s.map (λ p, ⟨1, p.b⟩) ⟩ | def | generalized_continued_fraction.of | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/basic.lean | [
"algebra.order.floor",
"algebra.continued_fractions.basic"
] | [
"floor_ring",
"generalized_continued_fraction",
"linear_ordered_field"
] | Returns the `generalized_continued_fraction` of a value. In fact, the returned gcf is also
a `continued_fraction` that terminates if and only if `v` is rational (those proofs will be
added in a future commit).
The continued fraction representation of `v` is given by `[⌊v⌋; b₀, b₁, b₂,...]`, where
`[b₀; b₁, b₂,...]` re... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_exact_value
(pconts conts : pair K) (fr : K) : K | -- if the fractional part is zero, we exactly approximated the value by the last continuants
if fr = 0 then conts.a / conts.b
-- otherwise, we have to include the fractional part in a final continuants step.
else let exact_conts := next_continuants 1 fr⁻¹ pconts conts in
exact_conts.a / exact_conts.b | def | generalized_continued_fraction.comp_exact_value | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [] | Given two continuants `pconts` and `conts` and a value `fr`, this function returns
- `conts.a / conts.b` if `fr = 0`
- `exact_conts.a / exact_conts.b` where `exact_conts = next_continuants 1 fr⁻¹ pconts conts`
otherwise.
This function can be used to compute the exact value approxmated by a continued fraction
`genera... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_exact_value_correctness_of_stream_eq_some_aux_comp {a : K} (b c : K)
(fract_a_ne_zero : int.fract a ≠ 0) :
((⌊a⌋ : K) * b + c) / (int.fract a) + b = (b * a + c) / int.fract a | by { field_simp [fract_a_ne_zero], rw int.fract, ring } | lemma | generalized_continued_fraction.comp_exact_value_correctness_of_stream_eq_some_aux_comp | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [
"int.fract",
"ring"
] | Just a computational lemma we need for the next main proof. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_exact_value_correctness_of_stream_eq_some :
∀ {ifp_n : int_fract_pair K}, int_fract_pair.stream v n = some ifp_n →
v = comp_exact_value ((of v).continuants_aux n) ((of v).continuants_aux $ n + 1) ifp_n.fr | begin
let g := of v,
induction n with n IH,
{ assume ifp_zero stream_zero_eq, -- nat.zero
have : int_fract_pair.of v = ifp_zero, by
{ have : int_fract_pair.stream v 0 = some (int_fract_pair.of v), from rfl,
simpa only [this] using stream_zero_eq },
cases this,
cases decidable.em (int.fract v... | lemma | generalized_continued_fraction.comp_exact_value_correctness_of_stream_eq_some | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [
"int.fract",
"int.fract_add_floor",
"inv_eq_one_div"
] | Shows the correctness of `comp_exact_value` in case the continued fraction
`generalized_continued_fraction.of v` did not terminate at position `n`. That is, we obtain the
value `v` if we pass the two successive (auxiliary) continuants at positions `n` and `n + 1` as well
as the fractional part at `int_fract_pair.stream... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_correctness_of_nth_stream_eq_none
(nth_stream_eq_none : int_fract_pair.stream v n = none) :
v = (of v).convergents (n - 1) | begin
induction n with n IH,
case nat.zero { contradiction }, -- int_fract_pair.stream v 0 ≠ none
case nat.succ
{ rename nth_stream_eq_none succ_nth_stream_eq_none,
let g := of v,
change v = g.convergents n,
have : int_fract_pair.stream v n = none
∨ ∃ ifp, int_fract_pair.stream v n = some ifp ... | lemma | generalized_continued_fraction.of_correctness_of_nth_stream_eq_none | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [] | The convergent of `generalized_continued_fraction.of v` at step `n - 1` is exactly `v` if the
`int_fract_pair.stream` of the corresponding continued fraction terminated at step `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_correctness_of_terminated_at (terminated_at_n : (of v).terminated_at n) :
v = (of v).convergents n | have int_fract_pair.stream v (n + 1) = none, from
of_terminated_at_n_iff_succ_nth_int_fract_pair_stream_eq_none.elim_left terminated_at_n,
of_correctness_of_nth_stream_eq_none this | theorem | generalized_continued_fraction.of_correctness_of_terminated_at | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [] | If `generalized_continued_fraction.of v` terminated at step `n`, then the `n`th convergent is
exactly `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_correctness_of_terminates (terminates : (of v).terminates) :
∃ (n : ℕ), v = (of v).convergents n | exists.elim terminates
( assume n terminated_at_n,
exists.intro n (of_correctness_of_terminated_at terminated_at_n) ) | lemma | generalized_continued_fraction.of_correctness_of_terminates | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [] | If `generalized_continued_fraction.of v` terminates, then there is `n : ℕ` such that the `n`th
convergent is exactly `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_correctness_at_top_of_terminates (terminates : (of v).terminates) :
∀ᶠ n in at_top, v = (of v).convergents n | begin
rw eventually_at_top,
obtain ⟨n, terminated_at_n⟩ : ∃ n, (of v).terminated_at n,
from terminates,
use n,
assume m m_geq_n,
rw (convergents_stable_of_terminated m_geq_n terminated_at_n),
exact of_correctness_of_terminated_at terminated_at_n
end | lemma | generalized_continued_fraction.of_correctness_at_top_of_terminates | algebra.continued_fractions.computation | src/algebra/continued_fractions/computation/correctness_terminating.lean | [
"algebra.continued_fractions.computation.translations",
"algebra.continued_fractions.terminated_stable",
"algebra.continued_fractions.continuants_recurrence",
"order.filter.at_top_bot",
"tactic.field_simp"
] | [] | If `generalized_continued_fraction.of v` terminates, then its convergents will eventually always
be `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_gcf_pair_rat_eq_of_nth_conts_aux :
∃ (conts : pair ℚ),
(of v).continuants_aux n = (conts.map coe : pair K) | nat.strong_induction_on n
begin
clear n,
let g := of v,
assume n IH,
rcases n with _|_|n,
-- n = 0
{ suffices : ∃ (gp : pair ℚ), pair.mk (1 : K) 0 = gp.map coe, by simpa [continuants_aux],
use (pair.mk 1 0),
simp },
-- n = 1
{ suffices : ∃ (conts : pair ℚ), pair.mk g.h 1 = conts.map coe, by
... | lemma | generalized_continued_fraction.exists_gcf_pair_rat_eq_of_nth_conts_aux | 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"
] | [
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_gcf_pair_rat_eq_nth_conts :
∃ (conts : pair ℚ), (of v).continuants n = (conts.map coe : pair K) | by { rw [nth_cont_eq_succ_nth_cont_aux], exact (exists_gcf_pair_rat_eq_of_nth_conts_aux v $ n + 1) } | lemma | generalized_continued_fraction.exists_gcf_pair_rat_eq_nth_conts | 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_rat_eq_nth_numerator : ∃ (q : ℚ), (of v).numerators n = (q : K) | begin
rcases (exists_gcf_pair_rat_eq_nth_conts v n) with ⟨⟨a, _⟩, nth_cont_eq⟩,
use a,
simp [num_eq_conts_a, nth_cont_eq],
end | lemma | generalized_continued_fraction.exists_rat_eq_nth_numerator | 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_rat_eq_nth_denominator : ∃ (q : ℚ), (of v).denominators n = (q : K) | begin
rcases (exists_gcf_pair_rat_eq_nth_conts v n) with ⟨⟨_, b⟩, nth_cont_eq⟩,
use b,
simp [denom_eq_conts_b, nth_cont_eq]
end | lemma | generalized_continued_fraction.exists_rat_eq_nth_denominator | 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_rat_eq_nth_convergent : ∃ (q : ℚ), (of v).convergents n = (q : K) | begin
rcases (exists_rat_eq_nth_numerator v n) with ⟨Aₙ, nth_num_eq⟩,
rcases (exists_rat_eq_nth_denominator v n) with ⟨Bₙ, nth_denom_eq⟩,
use (Aₙ / Bₙ),
simp [nth_num_eq, nth_denom_eq, convergent_eq_num_div_denom]
end | lemma | generalized_continued_fraction.exists_rat_eq_nth_convergent | 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"
] | [] | Every finite convergent corresponds to a rational number. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_rat_eq_of_terminates
(terminates : (of v).terminates) :
∃ (q : ℚ), v = ↑q | begin
obtain ⟨n, v_eq_conv⟩ : ∃ n, v = (of v).convergents n, from
of_correctness_of_terminates terminates,
obtain ⟨q, conv_eq_q⟩ :
∃ (q : ℚ), (of v).convergents n = (↑q : K), from exists_rat_eq_nth_convergent v n,
have : v = (↑q : K), from eq.trans v_eq_conv conv_eq_q,
use [q, this]
end | theorem | generalized_continued_fraction.exists_rat_eq_of_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"
] | [] | Every terminating continued fraction corresponds to a rational number. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_rat_eq :
((int_fract_pair.of q).mapFr coe : int_fract_pair K) = int_fract_pair.of v | by simp [int_fract_pair.of, v_eq_q] | lemma | generalized_continued_fraction.int_fract_pair.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.