statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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f_add_nat_eq (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (n : ℕ) :
f (x + n) = f x + ∑ (m : ℕ) in finset.range n, log (x + m) | begin
induction n with n hn,
{ simp },
{ have : x + n.succ = (x + n) + 1,
{ push_cast, ring },
rw [this, hf_feq, hn],
rw [finset.range_succ, finset.sum_insert (finset.not_mem_range_self)],
abel,
linarith [(nat.cast_nonneg n : 0 ≤ (n:ℝ))] },
end | lemma | real.bohr_mollerup.f_add_nat_eq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"finset.not_mem_range_self",
"finset.range",
"finset.range_succ",
"nat.cast_nonneg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
f_add_nat_le
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hn : n ≠ 0) (hx : 0 < x) (hx' : x ≤ 1) :
f (n + x) ≤ f n + x * log n | begin
have hn': 0 < (n:ℝ) := nat.cast_pos.mpr (nat.pos_of_ne_zero hn),
have : f n + x * log n = (1 - x) * f n + x * f (n + 1),
{ rw [hf_feq hn'], ring, },
rw [this, (by ring : (n:ℝ) + x = (1 - x) * n + x * (n + 1))],
simpa only [smul_eq_mul] using hf_conv.2 hn' (by linarith : 0 < (n + 1 : ℝ))
(by linarith... | lemma | real.bohr_mollerup.f_add_nat_le | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"ring",
"smul_eq_mul"
] | Linear upper bound for `f (x + n)` on unit interval | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f_add_nat_ge
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hn : 2 ≤ n) (hx : 0 < x) :
f n + x * log (n - 1) ≤ f (n + x) | begin
have npos : 0 < (n:ℝ) - 1,
{ rw [←nat.cast_one, sub_pos, nat.cast_lt], linarith, },
have c := (convex_on_iff_slope_mono_adjacent.mp $ hf_conv).2
npos (by linarith : 0 < (n:ℝ) + x) (by linarith : (n:ℝ) - 1 < (n:ℝ)) (by linarith),
rw [add_sub_cancel', sub_sub_cancel, div_one] at c,
have : f (↑n - 1) =... | lemma | real.bohr_mollerup.f_add_nat_ge | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"div_one",
"le_div_iff",
"mul_comm",
"nat.cast_lt",
"ring"
] | Linear lower bound for `f (x + n)` on unit interval | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_gamma_seq_add_one (x : ℝ) (n : ℕ) :
log_gamma_seq (x + 1) n = log_gamma_seq x (n + 1) + log x - (x + 1) * (log (n + 1) - log n) | begin
dsimp only [nat.factorial_succ, log_gamma_seq],
conv_rhs { rw [finset.sum_range_succ', nat.cast_zero, add_zero], },
rw [nat.cast_mul, log_mul], rotate,
{ rw nat.cast_ne_zero, exact nat.succ_ne_zero n },
{ rw nat.cast_ne_zero, exact nat.factorial_ne_zero n, },
have : ∑ (m : ℕ) in finset.range (n + 1),... | lemma | real.bohr_mollerup.log_gamma_seq_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"finset.range",
"nat.cast_add_one",
"nat.cast_mul",
"nat.cast_ne_zero",
"nat.cast_zero",
"nat.factorial_ne_zero",
"nat.factorial_succ",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_log_gamma_seq
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hx : 0 < x) (hx' : x ≤ 1) (n : ℕ) :
f x ≤ f 1 + x * log (n + 1) - x * log n + log_gamma_seq x n | begin
rw [log_gamma_seq, ←add_sub_assoc, le_sub_iff_add_le, ←f_add_nat_eq @hf_feq hx, add_comm x],
refine (f_add_nat_le hf_conv @hf_feq (nat.add_one_ne_zero n) hx hx').trans (le_of_eq _),
rw [f_nat_eq @hf_feq (by linarith : n + 1 ≠ 0), nat.add_sub_cancel, nat.cast_add_one],
ring,
end | lemma | real.bohr_mollerup.le_log_gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"nat.cast_add_one",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ge_log_gamma_seq
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hx : 0 < x) (hn : n ≠ 0) :
f 1 + log_gamma_seq x n ≤ f x | begin
dsimp [log_gamma_seq],
rw [←add_sub_assoc, sub_le_iff_le_add, ←f_add_nat_eq @hf_feq hx, add_comm x _],
refine le_trans (le_of_eq _) (f_add_nat_ge hf_conv @hf_feq _ hx),
{ rw [f_nat_eq @hf_feq, nat.add_sub_cancel, nat.cast_add_one, add_sub_cancel],
{ ring },
{ exact nat.succ_ne_zero _} },
{ apply... | lemma | real.bohr_mollerup.ge_log_gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"nat.cast_add_one",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_gamma_seq_of_le_one
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hx : 0 < x) (hx' : x ≤ 1) :
tendsto (log_gamma_seq x) at_top (𝓝 $ f x - f 1) | begin
refine tendsto_of_tendsto_of_tendsto_of_le_of_le' _ tendsto_const_nhds _ _,
show ∀ᶠ (n : ℕ) in at_top, log_gamma_seq x n ≤ f x - f 1,
{ refine eventually.mp (eventually_ne_at_top 0) (eventually_of_forall (λ n hn, _)),
exact le_sub_iff_add_le'.mpr (ge_log_gamma_seq hf_conv @hf_feq hx hn) },
show ∀ᶠ (n ... | lemma | real.bohr_mollerup.tendsto_log_gamma_seq_of_le_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"ring",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_gamma_seq
(hf_conv : convex_on ℝ (Ioi 0) f) (hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = f y + log y)
(hx : 0 < x) :
tendsto (log_gamma_seq x) at_top (𝓝 $ f x - f 1) | begin
suffices : ∀ (m : ℕ), ↑m < x → x ≤ m + 1 →
tendsto (log_gamma_seq x) at_top (𝓝 $ f x - f 1),
{ refine this (⌈x - 1⌉₊) _ _,
{ rcases lt_or_le x 1,
{ rwa [nat.ceil_eq_zero.mpr (by linarith : x - 1 ≤ 0), nat.cast_zero] },
{ convert nat.ceil_lt_add_one (by linarith : 0 ≤ x - 1),
abel ... | lemma | real.bohr_mollerup.tendsto_log_gamma_seq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"convex_on",
"nat.cast_nonneg",
"nat.cast_succ",
"nat.cast_zero",
"nat.ceil_lt_add_one",
"nat.le_ceil",
"ring",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_Gamma {x : ℝ} (hx : 0 < x) :
tendsto (log_gamma_seq x) at_top (𝓝 $ log (Gamma x)) | begin
have : log (Gamma x) = (log ∘ Gamma) x - (log ∘ Gamma) 1,
{ simp_rw [function.comp_app, Gamma_one, log_one, sub_zero] },
rw this,
refine bohr_mollerup.tendsto_log_gamma_seq convex_on_log_Gamma (λ y hy, _) hx,
rw [function.comp_app, Gamma_add_one hy.ne', log_mul hy.ne' (Gamma_pos_of_pos hy).ne', add_comm... | lemma | real.bohr_mollerup.tendsto_log_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_Gamma_of_log_convex {f : ℝ → ℝ}
(hf_conv : convex_on ℝ (Ioi 0) (log ∘ f))
(hf_feq : ∀ {y:ℝ}, 0 < y → f (y + 1) = y * f y)
(hf_pos : ∀ {y:ℝ}, 0 < y → 0 < f y)
(hf_one : f 1 = 1) :
eq_on f Gamma (Ioi (0:ℝ)) | begin
suffices : eq_on (log ∘ f) (log ∘ Gamma) (Ioi (0:ℝ)),
from λ x hx, log_inj_on_pos (hf_pos hx) (Gamma_pos_of_pos hx) (this hx),
intros x hx,
have e1 := bohr_mollerup.tendsto_log_gamma_seq hf_conv _ hx,
{ rw [function.comp_app log f 1, hf_one, log_one, sub_zero] at e1,
exact tendsto_nhds_unique e1 (... | lemma | real.eq_Gamma_of_log_convex | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"convex_on",
"ring",
"tendsto_nhds_unique"
] | The **Bohr-Mollerup theorem**: the Gamma function is the *unique* log-convex, positive-valued
function on the positive reals which satisfies `f 1 = 1` and `f (x + 1) = x * f x` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Gamma_two : Gamma 2 = 1 | by simpa using Gamma_nat_eq_factorial 1 | lemma | real.Gamma_two | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_three_div_two_lt_one : Gamma (3 / 2) < 1 | begin
-- This can also be proved using the closed-form evaluation of `Gamma (1 / 2)` in
-- `analysis.special_functions.gaussian`, but we give a self-contained proof using log-convexity
-- to avoid unnecessary imports.
have A : (0:ℝ) < 3/2, by norm_num,
have := bohr_mollerup.f_add_nat_le convex_on_log_Gamma (λ... | lemma | real.Gamma_three_div_two_lt_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"div_neg_of_neg_of_pos",
"div_sub'",
"mul_comm",
"mul_one",
"nat.cast_two",
"one_half_pos",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_strict_mono_on_Ici : strict_mono_on Gamma (Ici 2) | begin
convert convex_on_Gamma.strict_mono_of_lt (by norm_num : (0:ℝ) < 3/2)
(by norm_num : (3/2 : ℝ) < 2) (Gamma_two.symm ▸ Gamma_three_div_two_lt_one),
symmetry,
rw inter_eq_right_iff_subset,
exact λ x hx, two_pos.trans_le hx,
end | lemma | real.Gamma_strict_mono_on_Ici | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
doubling_Gamma (s : ℝ) : ℝ | Gamma (s / 2) * Gamma (s / 2 + 1 / 2) * 2 ^ (s - 1) / sqrt π | def | real.doubling_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma"
] | Auxiliary definition for the doubling formula (we'll show this is equal to `Gamma s`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
doubling_Gamma_add_one (s : ℝ) (hs : s ≠ 0) :
doubling_Gamma (s + 1) = s * doubling_Gamma s | begin
rw [doubling_Gamma, doubling_Gamma, (by abel : s + 1 - 1 = s - 1 + 1), add_div, add_assoc,
add_halves (1 : ℝ), Gamma_add_one (div_ne_zero hs two_ne_zero), rpow_add two_pos, rpow_one],
ring,
end | lemma | real.doubling_Gamma_add_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"add_div",
"add_halves",
"div_ne_zero",
"ring",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
doubling_Gamma_one : doubling_Gamma 1 = 1 | by simp_rw [doubling_Gamma, Gamma_one_half_eq, add_halves (1 : ℝ), sub_self, Gamma_one, mul_one,
rpow_zero, mul_one, div_self (sqrt_ne_zero'.mpr pi_pos)] | lemma | real.doubling_Gamma_one | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"add_halves",
"div_self",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_doubling_Gamma_eq :
eq_on (log ∘ doubling_Gamma) (λ s, log (Gamma (s / 2)) + log (Gamma (s / 2 + 1 / 2))
+ s * log 2 - log (2 * sqrt π)) (Ioi 0) | begin
intros s hs,
have h1 : sqrt π ≠ 0, from sqrt_ne_zero'.mpr pi_pos,
have h2 : Gamma (s / 2) ≠ 0, from (Gamma_pos_of_pos $ div_pos hs two_pos).ne',
have h3 : Gamma (s / 2 + 1 / 2) ≠ 0,
from (Gamma_pos_of_pos $ add_pos (div_pos hs two_pos) one_half_pos).ne',
have h4 : (2 : ℝ) ^ (s - 1) ≠ 0, from (rpow_p... | lemma | real.log_doubling_Gamma_eq | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"div_pos",
"mul_ne_zero",
"one_half_pos",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
doubling_Gamma_log_convex_Ioi : convex_on ℝ (Ioi (0:ℝ)) (log ∘ doubling_Gamma) | begin
refine (((convex_on.add _ _).add _).add_const _).congr log_doubling_Gamma_eq.symm,
{ convert convex_on_log_Gamma.comp_affine_map
(distrib_mul_action.to_linear_map ℝ ℝ (1 / 2 : ℝ)).to_affine_map,
{ simpa only [zero_div] using (preimage_const_mul_Ioi (0 : ℝ) one_half_pos).symm, },
{ ext1 x,
... | lemma | real.doubling_Gamma_log_convex_Ioi | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"affine_map.const",
"convex_Ioi",
"convex_on",
"convex_on.add",
"convex_on.subset",
"convex_on_id",
"distrib_mul_action.to_linear_map",
"div_self",
"mul_comm",
"mul_one_div",
"neg_div",
"one_half_pos",
"one_lt_two",
"smul_eq_mul",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
doubling_Gamma_eq_Gamma {s : ℝ} (hs : 0 < s) : doubling_Gamma s = Gamma s | begin
refine eq_Gamma_of_log_convex doubling_Gamma_log_convex_Ioi
(λ y hy, doubling_Gamma_add_one y hy.ne') (λ y hy, _) doubling_Gamma_one hs,
apply_rules [mul_pos, Gamma_pos_of_pos, add_pos, inv_pos_of_pos,
rpow_pos_of_pos, two_pos, one_pos, sqrt_pos_of_pos pi_pos]
end | lemma | real.doubling_Gamma_eq_Gamma | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Gamma_mul_Gamma_add_half_of_pos {s : ℝ} (hs : 0 < s) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * 2 ^ (1 - 2 * s) * sqrt π | begin
rw [←(doubling_Gamma_eq_Gamma (mul_pos two_pos hs)),
doubling_Gamma, mul_div_cancel_left _ (two_ne_zero' ℝ),
(by abel : 1 - 2 * s = -(2 * s - 1)), rpow_neg zero_le_two],
field_simp [(sqrt_pos_of_pos pi_pos).ne', (rpow_pos_of_pos two_pos (2 * s - 1)).ne'],
ring,
end | lemma | real.Gamma_mul_Gamma_add_half_of_pos | analysis.special_functions.gamma | src/analysis/special_functions/gamma/bohr_mollerup.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.gaussian"
] | [
"Gamma",
"mul_div_cancel_left",
"ring",
"two_ne_zero'",
"zero_le_two"
] | Legendre's doubling formula for the Gamma function, for positive real arguments. Note that
we shall later prove this for all `s` as `real.Gamma_mul_Gamma_add_half` (superseding this result)
but this result is needed as an intermediate step. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
logb (b x : ℝ) : ℝ | log x / log b | def | real.logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_div_log : log x / log b = logb b x | rfl | lemma | real.log_div_log | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_zero : logb b 0 = 0 | by simp [logb] | lemma | real.logb_zero | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_one : logb b 1 = 0 | by simp [logb] | lemma | real.logb_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_abs (x : ℝ) : logb b (|x|) = logb b x | by rw [logb, logb, log_abs] | lemma | real.logb_abs | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_neg_eq_logb (x : ℝ) : logb b (-x) = logb b x | by rw [← logb_abs x, ← logb_abs (-x), abs_neg] | lemma | real.logb_neg_eq_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_mul (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x * y) = logb b x + logb b y | by simp_rw [logb, log_mul hx hy, add_div] | lemma | real.logb_mul | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"add_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_div (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x / y) = logb b x - logb b y | by simp_rw [logb, log_div hx hy, sub_div] | lemma | real.logb_div | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_inv (x : ℝ) : logb b (x⁻¹) = -logb b x | by simp [logb, neg_div] | lemma | real.logb_inv | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"neg_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_logb (a b : ℝ) : (logb a b)⁻¹ = logb b a | by simp_rw [logb, inv_div] | lemma | real.inv_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_logb_mul_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
(logb (a * b) c)⁻¹ = (logb a c)⁻¹ + (logb b c)⁻¹ | by simp_rw inv_logb; exact logb_mul h₁ h₂ | theorem | real.inv_logb_mul_base | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_logb_div_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
(logb (a / b) c)⁻¹ = (logb a c)⁻¹ - (logb b c)⁻¹ | by simp_rw inv_logb; exact logb_div h₁ h₂ | theorem | real.inv_logb_div_base | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_mul_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
logb (a * b) c = ((logb a c)⁻¹ + (logb b c)⁻¹)⁻¹ | by rw [←inv_logb_mul_base h₁ h₂ c, inv_inv] | theorem | real.logb_mul_base | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_div_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
logb (a / b) c = ((logb a c)⁻¹ - (logb b c)⁻¹)⁻¹ | by rw [←inv_logb_div_base h₁ h₂ c, inv_inv] | theorem | real.logb_div_base | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_logb {a b c : ℝ} (h₁ : b ≠ 0) (h₂ : b ≠ 1) (h₃ : b ≠ -1) :
logb a b * logb b c = logb a c | begin
unfold logb,
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr ⟨h₁, h₂, h₃⟩)],
end | theorem | real.mul_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_mul_div_cancel",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_logb {a b c : ℝ} (h₁ : c ≠ 0) (h₂ : c ≠ 1) (h₃ : c ≠ -1) :
logb a c / logb b c = logb a b | begin
unfold logb,
-- TODO: div_div_div_cancel_left is missing for `group_with_zero`,
rw [div_div_div_eq, mul_comm, mul_div_mul_right _ _ (log_ne_zero.mpr ⟨h₁, h₂, h₃⟩)],
end | theorem | real.div_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_div_div_eq",
"mul_comm",
"mul_div_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_b_ne_zero : log b ≠ 0 | begin
have b_ne_zero : b ≠ 0, linarith,
have b_ne_minus_one : b ≠ -1, linarith,
simp [b_ne_one, b_ne_zero, b_ne_minus_one],
end | lemma | real.log_b_ne_zero | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_rpow :
logb b (b ^ x) = x | begin
rw [logb, div_eq_iff, log_rpow b_pos],
exact log_b_ne_zero b_pos b_ne_one,
end | lemma | real.logb_rpow | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_logb_eq_abs (hx : x ≠ 0) : b ^ (logb b x) = |x| | begin
apply log_inj_on_pos,
simp only [set.mem_Ioi],
apply rpow_pos_of_pos b_pos,
simp only [abs_pos, mem_Ioi, ne.def, hx, not_false_iff],
rw [log_rpow b_pos, logb, log_abs],
field_simp [log_b_ne_zero b_pos b_ne_one],
end | lemma | real.rpow_logb_eq_abs | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_pos",
"set.mem_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_logb (hx : 0 < x) : b ^ (logb b x) = x | by { rw rpow_logb_eq_abs b_pos b_ne_one (hx.ne'), exact abs_of_pos hx, } | lemma | real.rpow_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_logb_of_neg (hx : x < 0) : b ^ (logb b x) = -x | by { rw rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx), exact abs_of_neg hx } | lemma | real.rpow_logb_of_neg | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_logb : surj_on (logb b) (Ioi 0) univ | λ x _, ⟨rpow b x, rpow_pos_of_pos b_pos x, logb_rpow b_pos b_ne_one⟩ | lemma | real.surj_on_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_surjective : surjective (logb b) | λ x, ⟨b ^ x, logb_rpow b_pos b_ne_one⟩ | lemma | real.logb_surjective | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_logb : range (logb b) = univ | (logb_surjective b_pos b_ne_one).range_eq | lemma | real.range_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_logb' : surj_on (logb b) (Iio 0) univ | begin
intros x x_in_univ,
use -b ^ x,
split,
{ simp only [right.neg_neg_iff, set.mem_Iio], apply rpow_pos_of_pos b_pos, },
{ rw [logb_neg_eq_logb, logb_rpow b_pos b_ne_one], },
end | lemma | real.surj_on_logb' | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"set.mem_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
b_pos : 0 < b | by linarith | lemma | real.b_pos | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
b_ne_one : b ≠ 1 | by linarith | lemma | real.b_ne_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_le_logb (h : 0 < x) (h₁ : 0 < y) :
logb b x ≤ logb b y ↔ x ≤ y | by { rw [logb, logb, div_le_div_right (log_pos hb), log_le_log h h₁], } | lemma | real.logb_le_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_le_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y | by { rw [logb, logb, div_lt_div_right (log_pos hb)], exact log_lt_log hx hxy, } | lemma | real.logb_lt_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_lt_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ x < y | by { rw [logb, logb, div_lt_div_right (log_pos hb)], exact log_lt_log_iff hx hy, } | lemma | real.logb_lt_logb_iff | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_lt_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_le_iff_le_rpow (hx : 0 < x) : logb b x ≤ y ↔ x ≤ b ^ y | by rw [←rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one hb) hx] | lemma | real.logb_le_iff_le_rpow | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_iff_lt_rpow (hx : 0 < x) : logb b x < y ↔ x < b ^ y | by rw [←rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one hb) hx] | lemma | real.logb_lt_iff_lt_rpow | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_logb_iff_rpow_le (hy : 0 < y) : x ≤ logb b y ↔ b ^ x ≤ y | by rw [←rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one hb) hy] | lemma | real.le_logb_iff_rpow_le | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_logb_iff_rpow_lt (hy : 0 < y) : x < logb b y ↔ b ^ x < y | by rw [←rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one hb) hy] | lemma | real.lt_logb_iff_rpow_lt | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x | by { rw ← @logb_one b, rw logb_lt_logb_iff hb zero_lt_one hx, } | lemma | real.logb_pos_iff | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_pos (hx : 1 < x) : 0 < logb b x | by { rw logb_pos_iff hb (lt_trans zero_lt_one hx), exact hx, } | lemma | real.logb_pos | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 | by { rw ← logb_one, exact logb_lt_logb_iff hb h zero_lt_one, } | lemma | real.logb_neg_iff | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_neg (h0 : 0 < x) (h1 : x < 1) : logb b x < 0 | (logb_neg_iff hb h0).2 h1 | lemma | real.logb_neg | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonneg_iff (hx : 0 < x) : 0 ≤ logb b x ↔ 1 ≤ x | by rw [← not_lt, logb_neg_iff hb hx, not_lt] | lemma | real.logb_nonneg_iff | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonneg (hx : 1 ≤ x) : 0 ≤ logb b x | (logb_nonneg_iff hb (zero_lt_one.trans_le hx)).2 hx | lemma | real.logb_nonneg | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonpos_iff (hx : 0 < x) : logb b x ≤ 0 ↔ x ≤ 1 | by rw [← not_lt, logb_pos_iff hb hx, not_lt] | lemma | real.logb_nonpos_iff | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonpos_iff' (hx : 0 ≤ x) : logb b x ≤ 0 ↔ x ≤ 1 | begin
rcases hx.eq_or_lt with (rfl|hx),
{ simp [le_refl, zero_le_one] },
exact logb_nonpos_iff hb hx,
end | lemma | real.logb_nonpos_iff' | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonpos (hx : 0 ≤ x) (h'x : x ≤ 1) : logb b x ≤ 0 | (logb_nonpos_iff' hb hx).2 h'x | lemma | real.logb_nonpos | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_logb : strict_mono_on (logb b) (set.Ioi 0) | λ x hx y hy hxy, logb_lt_logb hb hx hxy | lemma | real.strict_mono_on_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"set.Ioi",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_logb : strict_anti_on (logb b) (set.Iio 0) | begin
rintros x (hx : x < 0) y (hy : y < 0) hxy,
rw [← logb_abs y, ← logb_abs x],
refine logb_lt_logb hb (abs_pos.2 hy.ne) _,
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff],
end | lemma | real.strict_anti_on_logb | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_of_neg",
"set.Iio",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_inj_on_pos : set.inj_on (logb b) (set.Ioi 0) | (strict_mono_on_logb hb).inj_on | lemma | real.logb_inj_on_pos | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"set.Ioi",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_pos_of_logb_eq_zero (h₁ : 0 < x) (h₂ : logb b x = 0) :
x = 1 | logb_inj_on_pos hb (set.mem_Ioi.2 h₁) (set.mem_Ioi.2 zero_lt_one)
(h₂.trans real.logb_one.symm) | lemma | real.eq_one_of_pos_of_logb_eq_zero | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_ne_zero_of_pos_of_ne_one (hx_pos : 0 < x) (hx : x ≠ 1) :
logb b x ≠ 0 | mt (eq_one_of_pos_of_logb_eq_zero hb hx_pos) hx | lemma | real.logb_ne_zero_of_pos_of_ne_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_logb_at_top : tendsto (logb b) at_top at_top | tendsto.at_top_div_const (log_pos hb) tendsto_log_at_top | lemma | real.tendsto_logb_at_top | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) :
logb b x ≤ logb b y ↔ y ≤ x | by { rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log h₁ h], } | lemma | real.logb_le_logb_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_le_div_right_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x | by { rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)], exact log_lt_log hx hxy, } | lemma | real.logb_lt_logb_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_lt_div_right_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :
logb b x < logb b y ↔ y < x | by { rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)], exact log_lt_log_iff hy hx } | lemma | real.logb_lt_logb_iff_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_lt_div_right_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_le_iff_le_rpow_of_base_lt_one (hx : 0 < x) : logb b x ≤ y ↔ b ^ y ≤ x | by rw [←rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx] | lemma | real.logb_le_iff_le_rpow_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_lt_iff_lt_rpow_of_base_lt_one (hx : 0 < x) : logb b x < y ↔ b ^ y < x | by rw [←rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx] | lemma | real.logb_lt_iff_lt_rpow_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_logb_iff_rpow_le_of_base_lt_one (hy : 0 < y) : x ≤ logb b y ↔ y ≤ b ^ x | by rw [←rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy] | lemma | real.le_logb_iff_rpow_le_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_logb_iff_rpow_lt_of_base_lt_one (hy : 0 < y) : x < logb b y ↔ y < b ^ x | by rw [←rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy] | lemma | real.lt_logb_iff_rpow_lt_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_pos_iff_of_base_lt_one (hx : 0 < x) : 0 < logb b x ↔ x < 1 | by rw [← @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one zero_lt_one hx] | lemma | real.logb_pos_iff_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_pos_of_base_lt_one (hx : 0 < x) (hx' : x < 1) : 0 < logb b x | by { rw logb_pos_iff_of_base_lt_one b_pos b_lt_one hx, exact hx', } | lemma | real.logb_pos_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_neg_iff_of_base_lt_one (h : 0 < x) : logb b x < 0 ↔ 1 < x | by rw [← @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one h zero_lt_one] | lemma | real.logb_neg_iff_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_neg_of_base_lt_one (h1 : 1 < x) : logb b x < 0 | (logb_neg_iff_of_base_lt_one b_pos b_lt_one (lt_trans zero_lt_one h1)).2 h1 | lemma | real.logb_neg_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonneg_iff_of_base_lt_one (hx : 0 < x) : 0 ≤ logb b x ↔ x ≤ 1 | by rw [← not_lt, logb_neg_iff_of_base_lt_one b_pos b_lt_one hx, not_lt] | lemma | real.logb_nonneg_iff_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonneg_of_base_lt_one (hx : 0 < x) (hx' : x ≤ 1) : 0 ≤ logb b x | by {rw [logb_nonneg_iff_of_base_lt_one b_pos b_lt_one hx], exact hx' } | lemma | real.logb_nonneg_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_nonpos_iff_of_base_lt_one (hx : 0 < x) : logb b x ≤ 0 ↔ 1 ≤ x | by rw [← not_lt, logb_pos_iff_of_base_lt_one b_pos b_lt_one hx, not_lt] | lemma | real.logb_nonpos_iff_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_logb_of_base_lt_one : strict_anti_on (logb b) (set.Ioi 0) | λ x hx y hy hxy, logb_lt_logb_of_base_lt_one b_pos b_lt_one hx hxy | lemma | real.strict_anti_on_logb_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"set.Ioi",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_logb_of_base_lt_one : strict_mono_on (logb b) (set.Iio 0) | begin
rintros x (hx : x < 0) y (hy : y < 0) hxy,
rw [← logb_abs y, ← logb_abs x],
refine logb_lt_logb_of_base_lt_one b_pos b_lt_one (abs_pos.2 hy.ne) _,
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff],
end | lemma | real.strict_mono_on_logb_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"abs_of_neg",
"set.Iio",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_inj_on_pos_of_base_lt_one : set.inj_on (logb b) (set.Ioi 0) | (strict_anti_on_logb_of_base_lt_one b_pos b_lt_one).inj_on | lemma | real.logb_inj_on_pos_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"set.Ioi",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_pos_of_logb_eq_zero_of_base_lt_one (h₁ : 0 < x) (h₂ : logb b x = 0) :
x = 1 | logb_inj_on_pos_of_base_lt_one b_pos b_lt_one (set.mem_Ioi.2 h₁) (set.mem_Ioi.2 zero_lt_one)
(h₂.trans real.logb_one.symm) | lemma | real.eq_one_of_pos_of_logb_eq_zero_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_ne_zero_of_pos_of_ne_one_of_base_lt_one (hx_pos : 0 < x) (hx : x ≠ 1) :
logb b x ≠ 0 | mt (eq_one_of_pos_of_logb_eq_zero_of_base_lt_one b_pos b_lt_one hx_pos) hx | lemma | real.logb_ne_zero_of_pos_of_ne_one_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_logb_at_top_of_base_lt_one : tendsto (logb b) at_top at_bot | begin
rw tendsto_at_top_at_bot,
intro e,
use 1 ⊔ b ^ e,
intro a,
simp only [and_imp, sup_le_iff],
intro ha,
rw logb_le_iff_le_rpow_of_base_lt_one b_pos b_lt_one,
tauto,
exact lt_of_lt_of_le zero_lt_one ha,
end | lemma | real.tendsto_logb_at_top_of_base_lt_one | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"and_imp",
"sup_le_iff",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) : ⌊logb b r⌋ = int.log b r | begin
obtain rfl | hr := hr.eq_or_lt,
{ rw [logb_zero, int.log_zero_right, int.floor_zero] },
have hb1' : 1 < (b : ℝ) := nat.one_lt_cast.mpr hb,
apply le_antisymm,
{ rw [←int.zpow_le_iff_le_log hb hr, ←rpow_int_cast b],
refine le_of_le_of_eq _ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr),
exact rp... | lemma | real.floor_logb_nat_cast | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"int.floor_le",
"int.floor_zero",
"int.le_floor",
"int.log",
"int.log_zero_right",
"int.zpow_log_le_self",
"le_of_le_of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_logb_nat_cast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) : ⌈logb b r⌉ = int.clog b r | begin
obtain rfl | hr := hr.eq_or_lt,
{ rw [logb_zero, int.clog_zero_right, int.ceil_zero] },
have hb1' : 1 < (b : ℝ) := nat.one_lt_cast.mpr hb,
apply le_antisymm,
{ rw [int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpow_int_cast],
refine int.self_le_zpow_clog hb r },
{ rw [←int.le_zpow_iff_clog_le hb hr, ←... | lemma | real.ceil_logb_nat_cast | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"int.ceil_le",
"int.ceil_zero",
"int.clog",
"int.clog_zero_right",
"int.le_ceil",
"int.self_le_zpow_clog"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_eq_zero :
logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 | begin
simp_rw [logb, div_eq_zero_iff, log_eq_zero],
tauto,
end | lemma | real.logb_eq_zero | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"div_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
logb_prod {α : Type*} (s : finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0):
logb b (∏ i in s, f i) = ∑ i in s, logb b (f i) | begin
classical,
induction s using finset.induction_on with a s ha ih,
{ simp },
simp only [finset.mem_insert, forall_eq_or_imp] at hf,
simp [ha, ih hf.2, logb_mul hf.1 (finset.prod_ne_zero_iff.2 hf.2)],
end | lemma | real.logb_prod | analysis.special_functions.log | src/analysis/special_functions/log/base.lean | [
"analysis.special_functions.pow.real",
"data.int.log"
] | [
"finset",
"finset.induction_on",
"finset.mem_insert",
"forall_eq_or_imp",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log (x : ℝ) : ℝ | if hx : x = 0 then 0 else exp_order_iso.symm ⟨|x|, abs_pos.2 hx⟩ | def | real.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [] | The real logarithm function, equal to the inverse of the exponential for `x > 0`,
to `log |x|` for `x < 0`, and to `0` for `0`. We use this unconventional extension to
`(-∞, 0]` as it gives the formula `log (x * y) = log x + log y` for all nonzero `x` and `y`, and
the derivative of `log` is `1/x` away from `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_of_ne_zero (hx : x ≠ 0) : log x = exp_order_iso.symm ⟨|x|, abs_pos.2 hx⟩ | dif_neg hx | lemma | real.log_of_ne_zero | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_of_pos (hx : 0 < x) : log x = exp_order_iso.symm ⟨x, hx⟩ | by { rw [log_of_ne_zero hx.ne'], congr, exact abs_of_pos hx } | lemma | real.log_of_pos | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_log_eq_abs (hx : x ≠ 0) : exp (log x) = |x| | by rw [log_of_ne_zero hx, ← coe_exp_order_iso_apply, order_iso.apply_symm_apply, subtype.coe_mk] | lemma | real.exp_log_eq_abs | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp",
"order_iso.apply_symm_apply",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_log (hx : 0 < x) : exp (log x) = x | by { rw exp_log_eq_abs hx.ne', exact abs_of_pos hx } | lemma | real.exp_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_of_pos",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_log_of_neg (hx : x < 0) : exp (log x) = -x | by { rw exp_log_eq_abs (ne_of_lt hx), exact abs_of_neg hx } | lemma | real.exp_log_of_neg | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_of_neg",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_exp_log (x : ℝ) : x ≤ exp (log x) | begin
by_cases h_zero : x = 0,
{ rw [h_zero, log, dif_pos rfl, exp_zero], exact zero_le_one, },
{ rw exp_log_eq_abs h_zero, exact le_abs_self _, },
end | lemma | real.le_exp_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp",
"exp_zero",
"le_abs_self",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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