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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|---|---|---|---|---|---|---|---|---|---|---|
log_exp (x : ℝ) : log (exp x) = x | exp_injective $ exp_log (exp_pos x) | lemma | real.log_exp | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_log : surj_on log (Ioi 0) univ | λ x _, ⟨exp x, exp_pos x, log_exp x⟩ | lemma | real.surj_on_log | 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_surjective : surjective log | λ x, ⟨exp x, log_exp x⟩ | lemma | real.log_surjective | 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 | |
range_log : range log = univ | log_surjective.range_eq | lemma | real.range_log | 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_zero : log 0 = 0 | dif_pos rfl | lemma | real.log_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_one : log 1 = 0 | exp_injective $ by rw [exp_log zero_lt_one, exp_zero] | lemma | real.log_one | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_abs (x : ℝ) : log (|x|) = log x | begin
by_cases h : x = 0,
{ simp [h] },
{ rw [← exp_eq_exp, exp_log_eq_abs h, exp_log_eq_abs (abs_pos.2 h).ne', abs_abs] }
end | lemma | real.log_abs | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_abs",
"exp_eq_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_neg_eq_log (x : ℝ) : log (-x) = log x | by rw [← log_abs x, ← log_abs (-x), abs_neg] | lemma | real.log_neg_eq_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_log {x : ℝ} (hx : 0 < x) : sinh (log x) = (x - x⁻¹) / 2 | by rw [sinh_eq, exp_neg, exp_log hx] | lemma | real.sinh_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_log {x : ℝ} (hx : 0 < x) : cosh (log x) = (x + x⁻¹) / 2 | by rw [cosh_eq, exp_neg, exp_log hx] | lemma | real.cosh_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_log' : surj_on log (Iio 0) univ | λ x _, ⟨-exp x, neg_lt_zero.2 $ exp_pos x, by rw [log_neg_eq_log, log_exp]⟩ | lemma | real.surj_on_log' | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_mul (hx : x ≠ 0) (hy : y ≠ 0) : log (x * y) = log x + log y | exp_injective $
by rw [exp_log_eq_abs (mul_ne_zero hx hy), exp_add, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_mul] | lemma | real.log_mul | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_mul",
"exp_add",
"mul_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_div (hx : x ≠ 0) (hy : y ≠ 0) : log (x / y) = log x - log y | exp_injective $
by rw [exp_log_eq_abs (div_ne_zero hx hy), exp_sub, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_div] | lemma | real.log_div | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_div",
"div_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_inv (x : ℝ) : log (x⁻¹) = -log x | begin
by_cases hx : x = 0, { simp [hx] },
rw [← exp_eq_exp, exp_log_eq_abs (inv_ne_zero hx), exp_neg, exp_log_eq_abs hx, abs_inv]
end | lemma | real.log_inv | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_inv",
"exp_eq_exp",
"exp_neg",
"inv_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_le_log (h : 0 < x) (h₁ : 0 < y) : log x ≤ log y ↔ x ≤ y | by rw [← exp_le_exp, exp_log h, exp_log h₁] | lemma | real.log_le_log | 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_lt_log (hx : 0 < x) : x < y → log x < log y | by { intro h, rwa [← exp_lt_exp, exp_log hx, exp_log (lt_trans hx h)] } | lemma | real.log_lt_log | 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_lt_log_iff (hx : 0 < x) (hy : 0 < y) : log x < log y ↔ x < y | by { rw [← exp_lt_exp, exp_log hx, exp_log hy] } | lemma | real.log_lt_log_iff | 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_le_iff_le_exp (hx : 0 < x) : log x ≤ y ↔ x ≤ exp y | by rw [←exp_le_exp, exp_log hx] | lemma | real.log_le_iff_le_exp | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_lt_iff_lt_exp (hx : 0 < x) : log x < y ↔ x < exp y | by rw [←exp_lt_exp, exp_log hx] | lemma | real.log_lt_iff_lt_exp | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_log_iff_exp_le (hy : 0 < y) : x ≤ log y ↔ exp x ≤ y | by rw [←exp_le_exp, exp_log hy] | lemma | real.le_log_iff_exp_le | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_log_iff_exp_lt (hy : 0 < y) : x < log y ↔ exp x < y | by rw [←exp_lt_exp, exp_log hy] | lemma | real.lt_log_iff_exp_lt | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_pos_iff (hx : 0 < x) : 0 < log x ↔ 1 < x | by { rw ← log_one, exact log_lt_log_iff zero_lt_one hx } | lemma | real.log_pos_iff | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_pos (hx : 1 < x) : 0 < log x | (log_pos_iff (lt_trans zero_lt_one hx)).2 hx | lemma | real.log_pos | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_neg_iff (h : 0 < x) : log x < 0 ↔ x < 1 | by { rw ← log_one, exact log_lt_log_iff h zero_lt_one } | lemma | real.log_neg_iff | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_neg (h0 : 0 < x) (h1 : x < 1) : log x < 0 | (log_neg_iff h0).2 h1 | lemma | real.log_neg | 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_nonneg_iff (hx : 0 < x) : 0 ≤ log x ↔ 1 ≤ x | by rw [← not_lt, log_neg_iff hx, not_lt] | lemma | real.log_nonneg_iff | 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_nonneg (hx : 1 ≤ x) : 0 ≤ log x | (log_nonneg_iff (zero_lt_one.trans_le hx)).2 hx | lemma | real.log_nonneg | 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_nonpos_iff (hx : 0 < x) : log x ≤ 0 ↔ x ≤ 1 | by rw [← not_lt, log_pos_iff hx, not_lt] | lemma | real.log_nonpos_iff | 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_nonpos_iff' (hx : 0 ≤ x) : log x ≤ 0 ↔ x ≤ 1 | begin
rcases hx.eq_or_lt with (rfl|hx),
{ simp [le_refl, zero_le_one] },
exact log_nonpos_iff hx
end | lemma | real.log_nonpos_iff' | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_nonpos (hx : 0 ≤ x) (h'x : x ≤ 1) : log x ≤ 0 | (log_nonpos_iff' hx).2 h'x | lemma | real.log_nonpos | 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 | |
strict_mono_on_log : strict_mono_on log (set.Ioi 0) | λ x hx y hy hxy, log_lt_log hx hxy | lemma | real.strict_mono_on_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"set.Ioi",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_log : strict_anti_on log (set.Iio 0) | begin
rintros x (hx : x < 0) y (hy : y < 0) hxy,
rw [← log_abs y, ← log_abs x],
refine log_lt_log (abs_pos.2 hy.ne) _,
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]
end | lemma | real.strict_anti_on_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_of_neg",
"set.Iio",
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_inj_on_pos : set.inj_on log (set.Ioi 0) | strict_mono_on_log.inj_on | lemma | real.log_inj_on_pos | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"set.Ioi",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_lt_sub_one_of_pos (hx1 : 0 < x) (hx2 : x ≠ 1) : log x < x - 1 | begin
have h : log x ≠ 0,
{ rw [← log_one, log_inj_on_pos.ne_iff hx1 zero_lt_one],
exact hx2 },
linarith [add_one_lt_exp_of_nonzero h, exp_log hx1],
end | lemma | real.log_lt_sub_one_of_pos | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_pos_of_log_eq_zero {x : ℝ} (h₁ : 0 < x) (h₂ : log x = 0) : x = 1 | log_inj_on_pos (set.mem_Ioi.2 h₁) (set.mem_Ioi.2 zero_lt_one) (h₂.trans real.log_one.symm) | lemma | real.eq_one_of_pos_of_log_eq_zero | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_ne_zero_of_pos_of_ne_one {x : ℝ} (hx_pos : 0 < x) (hx : x ≠ 1) : log x ≠ 0 | mt (eq_one_of_pos_of_log_eq_zero hx_pos) hx | lemma | real.log_ne_zero_of_pos_of_ne_one | 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_eq_zero {x : ℝ} : log x = 0 ↔ x = 0 ∨ x = 1 ∨ x = -1 | begin
split,
{ intros h,
rcases lt_trichotomy x 0 with x_lt_zero | rfl | x_gt_zero,
{ refine or.inr (or.inr (neg_eq_iff_eq_neg.mp _)),
rw [←log_neg_eq_log x] at h,
exact eq_one_of_pos_of_log_eq_zero (neg_pos.mpr x_lt_zero) h, },
{ exact or.inl rfl },
{ exact or.inr (or.inl (eq_one_of_pos... | lemma | real.log_eq_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_ne_zero {x : ℝ} : log x ≠ 0 ↔ x ≠ 0 ∧ x ≠ 1 ∧ x ≠ -1 | by simpa only [not_or_distrib] using log_eq_zero.not | lemma | real.log_ne_zero | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_pow (x : ℝ) (n : ℕ) : log (x ^ n) = n * log x | begin
induction n with n ih,
{ simp },
rcases eq_or_ne x 0 with rfl | hx,
{ simp },
rw [pow_succ', log_mul (pow_ne_zero _ hx) hx, ih, nat.cast_succ, add_mul, one_mul],
end | lemma | real.log_pow | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"eq_or_ne",
"ih",
"nat.cast_succ",
"one_mul",
"pow_ne_zero",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_zpow (x : ℝ) (n : ℤ) : log (x ^ n) = n * log x | begin
induction n,
{ rw [int.of_nat_eq_coe, zpow_coe_nat, log_pow, int.cast_coe_nat] },
rw [zpow_neg_succ_of_nat, log_inv, log_pow, int.cast_neg_succ_of_nat, nat.cast_add_one,
neg_mul_eq_neg_mul],
end | lemma | real.log_zpow | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"int.cast_coe_nat",
"int.cast_neg_succ_of_nat",
"nat.cast_add_one",
"neg_mul_eq_neg_mul",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_sqrt {x : ℝ} (hx : 0 ≤ x) : log (sqrt x) = log x / 2 | by { rw [eq_div_iff, mul_comm, ← nat.cast_two, ← log_pow, sq_sqrt hx], exact two_ne_zero } | lemma | real.log_sqrt | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"eq_div_iff",
"mul_comm",
"nat.cast_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_le_sub_one_of_pos {x : ℝ} (hx : 0 < x) : log x ≤ x - 1 | begin
rw le_sub_iff_add_le,
convert add_one_le_exp (log x),
rw exp_log hx,
end | lemma | real.log_le_sub_one_of_pos | 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 | |
abs_log_mul_self_lt (x: ℝ) (h1 : 0 < x) (h2 : x ≤ 1) : |log x * x| < 1 | begin
have : 0 < 1/x := by simpa only [one_div, inv_pos] using h1,
replace := log_le_sub_one_of_pos this,
replace : log (1 / x) < 1/x := by linarith,
rw [log_div one_ne_zero h1.ne', log_one, zero_sub, lt_div_iff h1] at this,
have aux : 0 ≤ -log x * x,
{ refine mul_nonneg _ h1.le, rw ←log_inv, apply log_nonn... | lemma | real.abs_log_mul_self_lt | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"abs_neg",
"abs_of_nonneg",
"aux",
"inv_one",
"inv_pos",
"le_inv",
"lt_div_iff",
"neg_mul",
"one_div",
"one_ne_zero",
"zero_lt_one"
] | Bound for `|log x * x|` in the interval `(0, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_log_at_top : tendsto log at_top at_top | tendsto_comp_exp_at_top.1 $ by simpa only [log_exp] using tendsto_id | lemma | real.tendsto_log_at_top | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [] | The real logarithm function tends to `+∞` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_log_nhds_within_zero : tendsto log (𝓝[≠] 0) at_bot | begin
rw [← (show _ = log, from funext log_abs)],
refine tendsto.comp _ tendsto_abs_nhds_within_zero,
simpa [← tendsto_comp_exp_at_bot] using tendsto_id
end | lemma | real.tendsto_log_nhds_within_zero | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"tendsto_abs_nhds_within_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_log : continuous_on log {0}ᶜ | begin
rw [continuous_on_iff_continuous_restrict, restrict],
conv in (log _) { rw [log_of_ne_zero (show (x : ℝ) ≠ 0, from x.2)] },
exact exp_order_iso.symm.continuous.comp (continuous_subtype_coe.norm.subtype_mk _)
end | lemma | real.continuous_on_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_on",
"continuous_on_iff_continuous_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_log : continuous (λ x : {x : ℝ // x ≠ 0}, log x) | continuous_on_iff_continuous_restrict.1 $ continuous_on_log.mono $ λ x hx, hx | lemma | real.continuous_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_log' : continuous (λ x : {x : ℝ // 0 < x}, log x) | continuous_on_iff_continuous_restrict.1 $ continuous_on_log.mono $ λ x hx, ne_of_gt hx | lemma | real.continuous_log' | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_log (hx : x ≠ 0) : continuous_at log x | (continuous_on_log x hx).continuous_at $ is_open.mem_nhds is_open_compl_singleton hx | lemma | real.continuous_at_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_at",
"is_open.mem_nhds",
"is_open_compl_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_log_iff : continuous_at log x ↔ x ≠ 0 | begin
refine ⟨_, continuous_at_log⟩,
rintros h rfl,
exact not_tendsto_nhds_of_tendsto_at_bot tendsto_log_nhds_within_zero _
(h.tendsto.mono_left inf_le_left)
end | lemma | real.continuous_at_log_iff | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_at",
"inf_le_left",
"not_tendsto_nhds_of_tendsto_at_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_prod {α : Type*} (s : finset α) (f : α → ℝ) (hf : ∀ x ∈ s, f x ≠ 0):
log (∏ i in s, f i) = ∑ i in s, log (f i) | begin
induction s using finset.cons_induction_on with a s ha ih,
{ simp },
{ rw [finset.forall_mem_cons] at hf,
simp [ih hf.2, log_mul hf.1 (finset.prod_ne_zero_iff.2 hf.2)] }
end | lemma | real.log_prod | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"finset",
"finset.cons_induction_on",
"finset.forall_mem_cons",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_nat_eq_sum_factorization (n : ℕ) : log n = n.factorization.sum (λ p t, t * log p) | begin
rcases eq_or_ne n 0 with rfl | hn,
{ simp },
nth_rewrite 0 [←nat.factorization_prod_pow_eq_self hn],
rw [finsupp.prod, nat.cast_prod, log_prod _ _ (λ p hp, _), finsupp.sum],
{ simp_rw [nat.cast_pow, log_pow] },
{ norm_cast,
exact pow_ne_zero _ (nat.prime_of_mem_factorization hp).ne_zero },
end | lemma | real.log_nat_eq_sum_factorization | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"eq_or_ne",
"finsupp.prod",
"nat.cast_pow",
"nat.cast_prod",
"nat.prime_of_mem_factorization",
"ne_zero",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_log_div_mul_add_at_top (a b : ℝ) (n : ℕ) (ha : a ≠ 0) :
tendsto (λ x, log x ^ n / (a * x + b)) at_top (𝓝 0) | ((tendsto_div_pow_mul_exp_add_at_top a b n ha.symm).comp tendsto_log_at_top).congr'
(by filter_upwards [eventually_gt_at_top (0 : ℝ)] with x hx using by simp [exp_log hx]) | lemma | real.tendsto_pow_log_div_mul_add_at_top | 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 | |
is_o_pow_log_id_at_top {n : ℕ} : (λ x, log x ^ n) =o[at_top] id | begin
rw asymptotics.is_o_iff_tendsto',
{ simpa using tendsto_pow_log_div_mul_add_at_top 1 0 n one_ne_zero },
filter_upwards [eventually_ne_at_top (0 : ℝ)] with x h₁ h₂ using (h₁ h₂).elim,
end | lemma | real.is_o_pow_log_id_at_top | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"asymptotics.is_o_iff_tendsto'",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_log_id_at_top : log =o[at_top] id | is_o_pow_log_id_at_top.congr_left (λ x, pow_one _) | lemma | real.is_o_log_id_at_top | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.log {f : α → ℝ} {l : filter α} {x : ℝ} (h : tendsto f l (𝓝 x)) (hx : x ≠ 0) :
tendsto (λ x, log (f x)) l (𝓝 (log x)) | (continuous_at_log hx).tendsto.comp h | lemma | filter.tendsto.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.log (hf : continuous f) (h₀ : ∀ x, f x ≠ 0) : continuous (λ x, log (f x)) | continuous_on_log.comp_continuous hf h₀ | lemma | continuous.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.log (hf : continuous_at f a) (h₀ : f a ≠ 0) :
continuous_at (λ x, log (f x)) a | hf.log h₀ | lemma | continuous_at.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.log (hf : continuous_within_at f s a) (h₀ : f a ≠ 0) :
continuous_within_at (λ x, log (f x)) s a | hf.log h₀ | lemma | continuous_within_at.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.log (hf : continuous_on f s) (h₀ : ∀ x ∈ s, f x ≠ 0) :
continuous_on (λ x, log (f x)) s | λ x hx, (hf x hx).log (h₀ x hx) | lemma | continuous_on.log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_comp_add_sub_log (y : ℝ) :
tendsto (λ x:ℝ, log (x + y) - log x) at_top (𝓝 0) | begin
refine tendsto.congr' (_ : ∀ᶠ (x : ℝ) in at_top, log (1 + y / x) = _) _,
{ refine eventually.mp ((eventually_ne_at_top 0).and (eventually_gt_at_top (-y)))
(eventually_of_forall (λ x hx, _)),
rw ← log_div _ hx.1,
{ congr' 1,
field_simp [hx.1] },
{ linarith [hx.2] } },
{ suffices : tend... | lemma | real.tendsto_log_comp_add_sub_log | 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 | |
tendsto_log_nat_add_one_sub_log : tendsto (λ (k : ℕ), log (k + 1) - log k) at_top (𝓝 0) | (tendsto_log_comp_add_sub_log 1).comp tendsto_coe_nat_at_top_at_top | lemma | real.tendsto_log_nat_add_one_sub_log | analysis.special_functions.log | src/analysis/special_functions/log/basic.lean | [
"analysis.special_functions.exp",
"data.nat.factorization.basic"
] | [
"tendsto_coe_nat_at_top_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_log_of_pos (hx : 0 < x) : has_strict_deriv_at log x⁻¹ x | have has_strict_deriv_at log (exp $ log x)⁻¹ x,
from (has_strict_deriv_at_exp $ log x).of_local_left_inverse (continuous_at_log hx.ne')
(ne_of_gt $ exp_pos _) $ eventually.mono (lt_mem_nhds hx) @exp_log,
by rwa [exp_log hx] at this | lemma | real.has_strict_deriv_at_log_of_pos | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"exp",
"has_strict_deriv_at",
"has_strict_deriv_at_exp",
"lt_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_log (hx : x ≠ 0) : has_strict_deriv_at log x⁻¹ x | begin
cases hx.lt_or_lt with hx hx,
{ convert (has_strict_deriv_at_log_of_pos (neg_pos.mpr hx)).comp x (has_strict_deriv_at_neg x),
{ ext y, exact (log_neg_eq_log y).symm },
{ field_simp [hx.ne] } },
{ exact has_strict_deriv_at_log_of_pos hx }
end | lemma | real.has_strict_deriv_at_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_log (hx : x ≠ 0) : has_deriv_at log x⁻¹ x | (has_strict_deriv_at_log hx).has_deriv_at | lemma | real.has_deriv_at_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_log (hx : x ≠ 0) : differentiable_at ℝ log x | (has_deriv_at_log hx).differentiable_at | lemma | real.differentiable_at_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_log : differentiable_on ℝ log {0}ᶜ | λ x hx, (differentiable_at_log hx).differentiable_within_at | lemma | real.differentiable_on_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_on",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_log_iff : differentiable_at ℝ log x ↔ x ≠ 0 | ⟨λ h, continuous_at_log_iff.1 h.continuous_at, differentiable_at_log⟩ | lemma | real.differentiable_at_log_iff | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_log (x : ℝ) : deriv log x = x⁻¹ | if hx : x = 0 then
by rw [deriv_zero_of_not_differentiable_at (mt differentiable_at_log_iff.1 (not_not.2 hx)), hx,
inv_zero]
else (has_deriv_at_log hx).deriv | lemma | real.deriv_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"inv_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_log' : deriv log = has_inv.inv | funext deriv_log | lemma | real.deriv_log' | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_log {n : ℕ∞} : cont_diff_on ℝ n log {0}ᶜ | begin
suffices : cont_diff_on ℝ ⊤ log {0}ᶜ, from this.of_le le_top,
refine (cont_diff_on_top_iff_deriv_of_open is_open_compl_singleton).2 _,
simp [differentiable_on_log, cont_diff_on_inv]
end | lemma | real.cont_diff_on_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_on",
"cont_diff_on_inv",
"cont_diff_on_top_iff_deriv_of_open",
"is_open_compl_singleton",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_log {n : ℕ∞} : cont_diff_at ℝ n log x ↔ x ≠ 0 | ⟨λ h, continuous_at_log_iff.1 h.continuous_at,
λ hx, (cont_diff_on_log x hx).cont_diff_at $
is_open.mem_nhds is_open_compl_singleton hx⟩ | lemma | real.cont_diff_at_log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_at",
"is_open.mem_nhds",
"is_open_compl_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.log (hf : has_deriv_within_at f f' s x) (hx : f x ≠ 0) :
has_deriv_within_at (λ y, log (f y)) (f' / (f x)) s x | begin
rw div_eq_inv_mul,
exact (has_deriv_at_log hx).comp_has_deriv_within_at x hf
end | lemma | has_deriv_within_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.log (hf : has_deriv_at f f' x) (hx : f x ≠ 0) :
has_deriv_at (λ y, log (f y)) (f' / f x) x | begin
rw ← has_deriv_within_at_univ at *,
exact hf.log hx
end | lemma | has_deriv_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.log (hf : has_strict_deriv_at f f' x) (hx : f x ≠ 0) :
has_strict_deriv_at (λ y, log (f y)) (f' / f x) x | begin
rw div_eq_inv_mul,
exact (has_strict_deriv_at_log hx).comp x hf
end | lemma | has_strict_deriv_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"div_eq_inv_mul",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within.log (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, log (f x)) s x = (deriv_within f s x) / (f x) | (hf.has_deriv_within_at.log hx).deriv_within hxs | lemma | deriv_within.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.log (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
deriv (λx, log (f x)) x = (deriv f x) / (f x) | (hf.has_deriv_at.log hx).deriv | lemma | deriv.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.log (hf : has_fderiv_within_at f f' s x) (hx : f x ≠ 0) :
has_fderiv_within_at (λ x, log (f x)) ((f x)⁻¹ • f') s x | (has_deriv_at_log hx).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.log (hf : has_fderiv_at f f' x) (hx : f x ≠ 0) :
has_fderiv_at (λ x, log (f x)) ((f x)⁻¹ • f') x | (has_deriv_at_log hx).comp_has_fderiv_at x hf | lemma | has_fderiv_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.log (hf : has_strict_fderiv_at f f' x) (hx : f x ≠ 0) :
has_strict_fderiv_at (λ x, log (f x)) ((f x)⁻¹ • f') x | (has_strict_deriv_at_log hx).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.log (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0) :
differentiable_within_at ℝ (λx, log (f x)) s x | (hf.has_fderiv_within_at.log hx).differentiable_within_at | lemma | differentiable_within_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.log (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
differentiable_at ℝ (λx, log (f x)) x | (hf.has_fderiv_at.log hx).differentiable_at | lemma | differentiable_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.log {n} (hf : cont_diff_at ℝ n f x) (hx : f x ≠ 0) :
cont_diff_at ℝ n (λ x, log (f x)) x | (cont_diff_at_log.2 hx).comp x hf | lemma | cont_diff_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.log {n} (hf : cont_diff_within_at ℝ n f s x) (hx : f x ≠ 0) :
cont_diff_within_at ℝ n (λ x, log (f x)) s x | (cont_diff_at_log.2 hx).comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.log {n} (hf : cont_diff_on ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on ℝ n (λ x, log (f x)) s | λ x hx, (hf x hx).log (hs x hx) | lemma | cont_diff_on.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.log {n} (hf : cont_diff ℝ n f) (h : ∀ x, f x ≠ 0) :
cont_diff ℝ n (λ x, log (f x)) | cont_diff_iff_cont_diff_at.2 $ λ x, hf.cont_diff_at.log (h x) | lemma | cont_diff.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.log (hf : differentiable_on ℝ f s) (hx : ∀ x ∈ s, f x ≠ 0) :
differentiable_on ℝ (λx, log (f x)) s | λx h, (hf x h).log (hx x h) | lemma | differentiable_on.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.log (hf : differentiable ℝ f) (hx : ∀ x, f x ≠ 0) :
differentiable ℝ (λx, log (f x)) | λx, (hf x).log (hx x) | lemma | differentiable.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within.log (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, log (f x)) s x = (f x)⁻¹ • fderiv_within ℝ f s x | (hf.has_fderiv_within_at.log hx).fderiv_within hxs | lemma | fderiv_within.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv.log (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
fderiv ℝ (λx, log (f x)) x = (f x)⁻¹ • fderiv ℝ f x | (hf.has_fderiv_at.log hx).fderiv | lemma | fderiv.log | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_mul_log_one_plus_div_at_top (t : ℝ) :
tendsto (λ x, x * log (1 + t / x)) at_top (𝓝 t) | begin
have h₁ : tendsto (λ h, h⁻¹ * log (1 + t * h)) (𝓝[≠] 0) (𝓝 t),
{ simpa [has_deriv_at_iff_tendsto_slope, slope_fun_def] using
(((has_deriv_at_id (0 : ℝ)).const_mul t).const_add 1).log (by simp) },
have h₂ : tendsto (λ x : ℝ, x⁻¹) at_top (𝓝[≠] 0) :=
tendsto_inv_at_top_zero'.mono_right (nhds_withi... | lemma | real.tendsto_mul_log_one_plus_div_at_top | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"has_deriv_at_id",
"has_deriv_at_iff_tendsto_slope",
"inv_inv",
"nhds_within_mono",
"slope_fun_def"
] | The function `x * log (1 + t / x)` tends to `t` at `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) : | |((∑ i in range n, x^(i+1)/(i+1)) + log (1-x))| ≤ (|x|)^(n+1) / (1 - |x|) :=
begin
/- For the proof, we show that the derivative of the function to be estimated is small,
and then apply the mean value inequality. -/
let F : ℝ → ℝ := λ x, ∑ i in range n, x^(i+1)/(i+1) + log (1-x),
-- First step: compute the deri... | lemma | real.abs_log_sub_add_sum_range_le | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"abs_div",
"abs_neg",
"abs_nonneg",
"convex.norm_image_sub_le_of_norm_deriv_le",
"convex_Icc",
"deriv",
"differentiable_at",
"div_le_div",
"div_mul_eq_mul_div",
"geom_sum_eq",
"le_abs_self",
"mul_div_cancel_left",
"nat.cast_add_one_pos",
"neg_le_abs_self",
"pow_abs",
"pow_le_pow_of_le_... | A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
where the main point of the bound is that it tends to `0`. The goal is to deduce the series
expansion of the logarithm, in `has_sum_pow_div_log_of_abs_lt_1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_pow_div_log_of_abs_lt_1 {x : ℝ} (h : |x| < 1) :
has_sum (λ (n : ℕ), x ^ (n + 1) / (n + 1)) (-log (1 - x)) | begin
rw summable.has_sum_iff_tendsto_nat,
show tendsto (λ (n : ℕ), ∑ (i : ℕ) in range n, x ^ (i + 1) / (i + 1)) at_top (𝓝 (-log (1 - x))),
{ rw [tendsto_iff_norm_tendsto_zero],
simp only [norm_eq_abs, sub_neg_eq_add],
refine squeeze_zero (λ n, abs_nonneg _) (abs_log_sub_add_sum_range_le h) _,
suffic... | theorem | real.has_sum_pow_div_log_of_abs_lt_1 | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"abs_div",
"abs_nonneg",
"abs_of_nonneg",
"div_le_div_of_le_left",
"has_sum",
"mul_le_of_le_one_right",
"nat.cast_add_one_pos",
"pow_abs",
"pow_nonneg",
"pow_succ",
"pow_succ'",
"squeeze_zero",
"summable",
"summable.has_sum_iff_tendsto_nat",
"summable_geometric_of_lt_1",
"summable_of_n... | Power series expansion of the logarithm around `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_log_sub_log_of_abs_lt_1 {x : ℝ} (h : |x| < 1) :
has_sum (λ k : ℕ, (2 : ℝ) * (1 / (2 * k + 1)) * x ^ (2 * k + 1)) (log (1 + x) - log(1 - x)) | begin
let term := λ n : ℕ, (-1) * ((-x) ^ (n + 1) / ((n : ℝ) + 1)) + x ^ (n + 1) / (n + 1),
have h_term_eq_goal : term ∘ (*) 2 = λ k : ℕ, 2 * (1 / (2 * k + 1)) * x ^ (2 * k + 1),
{ ext n,
dsimp [term],
rw [odd.neg_pow (⟨n, rfl⟩ : odd (2 * n + 1)) x],
push_cast,
ring_nf, },
rw [← h_term_eq_goal, ... | lemma | real.has_sum_log_sub_log_of_abs_lt_1 | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"abs_neg",
"even.neg_pow",
"has_sum",
"mul_right_injective₀",
"nat.even_add_one",
"neg_one_mul",
"odd",
"odd.neg_pow",
"range_two_mul",
"two_ne_zero'"
] | Power series expansion of `log(1 + x) - log(1 - x)` for `|x| < 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_log_one_add_inv {a : ℝ} (h : 0 < a) :
has_sum (λ k : ℕ, (2 : ℝ) * (1 / (2 * k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1))
(log (1 + a⁻¹)) | begin
have h₁ : |1 / (2 * a + 1)| < 1,
{ rw [abs_of_pos, div_lt_one],
{ linarith, },
{ linarith, },
{ exact div_pos one_pos (by linarith), }, },
convert has_sum_log_sub_log_of_abs_lt_1 h₁,
have h₂ : (2 : ℝ) * a + 1 ≠ 0 := by linarith,
have h₃ := h.ne',
rw ← log_div,
{ congr,
field_simp,
... | theorem | real.has_sum_log_one_add_inv | analysis.special_functions.log | src/analysis/special_functions/log/deriv.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv",
"analysis.special_functions.log.basic",
"analysis.special_functions.exp_deriv"
] | [
"abs_of_pos",
"div_lt_one",
"div_pos",
"has_sum"
] | Expansion of `log (1 + a⁻¹)` as a series in powers of `1 / (2 * a + 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_mul_self_monotone_on : monotone_on (λ x : ℝ, log x * x) {x | 1 ≤ x} | begin
-- TODO: can be strengthened to exp (-1) ≤ x
simp only [monotone_on, mem_set_of_eq],
intros x hex y hey hxy,
have x_pos : 0 < x := lt_of_lt_of_le zero_lt_one hex,
have y_pos : 0 < y := lt_of_lt_of_le zero_lt_one hey,
refine mul_le_mul ((log_le_log x_pos y_pos).mpr hxy) hxy (le_of_lt x_pos) _,
rwa [l... | lemma | real.log_mul_self_monotone_on | analysis.special_functions.log | src/analysis/special_functions/log/monotone.lean | [
"analysis.special_functions.pow.real"
] | [
"monotone_on",
"mul_le_mul",
"real.exp_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_div_self_antitone_on : antitone_on (λ x : ℝ, log x / x) {x | exp 1 ≤ x} | begin
simp only [antitone_on, mem_set_of_eq],
intros x hex y hey hxy,
have x_pos : 0 < x := (exp_pos 1).trans_le hex,
have y_pos : 0 < y := (exp_pos 1).trans_le hey,
have hlogx : 1 ≤ log x := by rwa le_log_iff_exp_le x_pos,
have hyx : 0 ≤ y / x - 1 := by rwa [le_sub_iff_add_le, le_div_iff x_pos, zero_add, o... | lemma | real.log_div_self_antitone_on | analysis.special_functions.log | src/analysis/special_functions/log/monotone.lean | [
"analysis.special_functions.pow.real"
] | [
"antitone_on",
"div_le_iff",
"div_pos",
"exp",
"le_div_iff",
"le_mul_of_one_le_left",
"one_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_div_self_rpow_antitone_on {a : ℝ} (ha : 0 < a) :
antitone_on (λ x : ℝ, log x / x ^ a) {x | exp (1 / a) ≤ x} | begin
simp only [antitone_on, mem_set_of_eq],
intros x hex y hey hxy,
have x_pos : 0 < x := lt_of_lt_of_le (exp_pos (1 / a)) hex,
have y_pos : 0 < y := by linarith,
have x_nonneg : 0 ≤ x := le_trans (le_of_lt (exp_pos (1 / a))) hex,
have y_nonneg : 0 ≤ y := by linarith,
nth_rewrite 0 ←rpow_one y,
nth_re... | lemma | real.log_div_self_rpow_antitone_on | analysis.special_functions.log | src/analysis/special_functions/log/monotone.lean | [
"analysis.special_functions.pow.real"
] | [
"antitone_on",
"div_eq_mul_one_div",
"exp",
"mul_div_assoc",
"mul_le_mul_left",
"real.exp_eq_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_div_sqrt_antitone_on :
antitone_on (λ x : ℝ, log x / sqrt x) {x | exp 2 ≤ x} | begin
simp_rw sqrt_eq_rpow,
convert @log_div_self_rpow_antitone_on (1 / 2) (by norm_num),
norm_num,
end | lemma | real.log_div_sqrt_antitone_on | analysis.special_functions.log | src/analysis/special_functions/log/monotone.lean | [
"analysis.special_functions.pow.real"
] | [
"antitone_on",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_rpow_at_top {y : ℝ} (hy : 0 < y) : tendsto (λ x : ℝ, x ^ y) at_top at_top | begin
rw tendsto_at_top_at_top,
intro b,
use (max b 0) ^ (1/y),
intros x hx,
exact le_of_max_le_left
(by { convert rpow_le_rpow (rpow_nonneg_of_nonneg (le_max_right b 0) (1/y)) hx (le_of_lt hy),
rw [← rpow_mul (le_max_right b 0), (eq_div_iff (ne_of_gt hy)).mp rfl, rpow_one] }),
end | lemma | tendsto_rpow_at_top | analysis.special_functions.pow | src/analysis/special_functions/pow/asymptotics.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"eq_div_iff",
"le_of_max_le_left"
] | The function `x ^ y` tends to `+∞` at `+∞` for any positive real `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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