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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rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z | nnreal.eq $ real.rpow_add' x.2 h | lemma | nnreal.rpow_add' | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_add'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z | nnreal.eq $ real.rpow_mul x.2 y z | lemma | nnreal.rpow_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ -y = (x ^ y)⁻¹ | nnreal.eq $ real.rpow_neg x.2 _ | lemma | nnreal.rpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x ⁻¹ | by simp [rpow_neg] | lemma | nnreal.rpow_neg_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z | nnreal.eq $ real.rpow_sub (pos_iff_ne_zero.2 hx) y z | lemma | nnreal.rpow_sub | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) :
x ^ (y - z) = x ^ y / x ^ z | nnreal.eq $ real.rpow_sub' x.2 h | lemma | nnreal.rpow_sub' | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_sub'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x | by field_simp [← rpow_mul] | lemma | nnreal.rpow_inv_rpow_self | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x | by field_simp [← rpow_mul] | lemma | nnreal.rpow_self_rpow_inv | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_rpow (x : ℝ≥0) (y : ℝ) : (x⁻¹) ^ y = (x ^ y)⁻¹ | nnreal.eq $ real.inv_rpow x.2 y | lemma | nnreal.inv_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.inv_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_rpow (x y : ℝ≥0) (z : ℝ) : (x / y) ^ z = x ^ z / y ^ z | nnreal.eq $ real.div_rpow x.2 y.2 z | lemma | nnreal.div_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.div_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1/(2:ℝ)) | begin
refine nnreal.eq _,
push_cast,
exact real.sqrt_eq_rpow x.1,
end | lemma | nnreal.sqrt_eq_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.sqrt_eq_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nat_cast (x : ℝ≥0) (n : ℕ) : x ^ (n : ℝ) = x ^ n | nnreal.eq $ by simpa only [coe_rpow, coe_pow] using real.rpow_nat_cast x n | lemma | nnreal.rpow_nat_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_two (x : ℝ≥0) : x ^ (2 : ℝ) = x ^ 2 | by { rw ← rpow_nat_cast, simp only [nat.cast_bit0, nat.cast_one] } | lemma | nnreal.rpow_two | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nat.cast_bit0",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow {x y : ℝ≥0} {z : ℝ} : (x*y)^z = x^z * y^z | nnreal.eq $ real.mul_rpow x.2 y.2 | lemma | nnreal.mul_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.eq",
"real.mul_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow {x y : ℝ≥0} {z: ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x^z ≤ y^z | real.rpow_le_rpow x.2 h₁ h₂ | lemma | nnreal.rpow_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow {x y : ℝ≥0} {z: ℝ} (h₁ : x < y) (h₂ : 0 < z) : x^z < y^z | real.rpow_lt_rpow x.2 h₁ h₂ | lemma | nnreal.rpow_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z < y ^ z ↔ x < y | real.rpow_lt_rpow_iff x.2 y.2 hz | lemma | nnreal.rpow_lt_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_rpow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y | real.rpow_le_rpow_iff x.2 y.2 hz | lemma | nnreal.rpow_le_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_rpow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_one_div_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ≤ y ^ (1 / z) ↔ x ^ z ≤ y | by rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne'] | lemma | nnreal.le_rpow_one_div_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one_div_le_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ (1 / z) ≤ y ↔ x ≤ y ^ z | by rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne'] | lemma | nnreal.rpow_one_div_le_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_lt {x : ℝ≥0} {y z : ℝ} (hx : 1 < x) (hyz : y < z) : x^y < x^z | real.rpow_lt_rpow_of_exponent_lt hx hyz | lemma | nnreal.rpow_lt_rpow_of_exponent_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_rpow_of_exponent_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_le {x : ℝ≥0} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) : x^y ≤ x^z | real.rpow_le_rpow_of_exponent_le hx hyz | lemma | nnreal.rpow_le_rpow_of_exponent_le | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_rpow_of_exponent_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_gt {x : ℝ≥0} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) :
x^y < x^z | real.rpow_lt_rpow_of_exponent_gt hx0 hx1 hyz | lemma | nnreal.rpow_lt_rpow_of_exponent_gt | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_rpow_of_exponent_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_ge {x : ℝ≥0} {y z : ℝ} (hx0 : 0 < x) (hx1 : x ≤ 1) (hyz : z ≤ y) :
x^y ≤ x^z | real.rpow_le_rpow_of_exponent_ge hx0 hx1 hyz | lemma | nnreal.rpow_le_rpow_of_exponent_ge | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_rpow_of_exponent_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_pos {p : ℝ} {x : ℝ≥0} (hx_pos : 0 < x) : 0 < x^p | begin
have rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x^p,
{ intros p hp_pos,
rw ←zero_rpow hp_pos.ne',
exact rpow_lt_rpow hx_pos hp_pos },
rcases lt_trichotomy 0 p with hp_pos|rfl|hp_neg,
{ exact rpow_pos_of_nonneg hp_pos },
{ simp only [zero_lt_one, rpow_zero] },
{ rw [←neg_neg p, rpow_neg, inv_p... | lemma | nnreal.rpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"inv_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one {x : ℝ≥0} {z : ℝ} (hx1 : x < 1) (hz : 0 < z) : x^z < 1 | real.rpow_lt_one (coe_nonneg x) hx1 hz | lemma | nnreal.rpow_lt_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_one {x : ℝ≥0} {z : ℝ} (hx2 : x ≤ 1) (hz : 0 ≤ z) : x^z ≤ 1 | real.rpow_le_one x.2 hx2 hz | lemma | nnreal.rpow_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x^z < 1 | real.rpow_lt_one_of_one_lt_of_neg hx hz | lemma | nnreal.rpow_lt_one_of_one_lt_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_lt_one_of_one_lt_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_one_of_one_le_of_nonpos {x : ℝ≥0} {z : ℝ} (hx : 1 ≤ x) (hz : z ≤ 0) : x^z ≤ 1 | real.rpow_le_one_of_one_le_of_nonpos hx hz | lemma | nnreal.rpow_le_one_of_one_le_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.rpow_le_one_of_one_le_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow {x : ℝ≥0} {z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x^z | real.one_lt_rpow hx hz | lemma | nnreal.one_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.one_lt_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow {x : ℝ≥0} {z : ℝ} (h : 1 ≤ x) (h₁ : 0 ≤ z) : 1 ≤ x^z | real.one_le_rpow h h₁ | lemma | nnreal.one_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.one_le_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1)
(hz : z < 0) : 1 < x^z | real.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz | lemma | nnreal.one_lt_rpow_of_pos_of_lt_one_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.one_lt_rpow_of_pos_of_lt_one_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow_of_pos_of_le_one_of_nonpos {x : ℝ≥0} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1)
(hz : z ≤ 0) : 1 ≤ x^z | real.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 hz | lemma | nnreal.one_le_rpow_of_pos_of_le_one_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.one_le_rpow_of_pos_of_le_one_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_self_of_le_one {x : ℝ≥0} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x | begin
rcases eq_bot_or_bot_lt x with rfl | (h : 0 < x),
{ have : z ≠ 0 := by linarith,
simp [this] },
nth_rewrite 1 ←nnreal.rpow_one x,
exact nnreal.rpow_le_rpow_of_exponent_ge h hx h_one_le,
end | lemma | nnreal.rpow_le_self_of_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"eq_bot_or_bot_lt",
"nnreal.rpow_le_rpow_of_exponent_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_injective {x : ℝ} (hx : x ≠ 0) : function.injective (λ y : ℝ≥0, y^x) | λ y z hyz, by simpa only [rpow_inv_rpow_self hx] using congr_arg (λ y, y ^ (1 / x)) hyz | lemma | nnreal.rpow_left_injective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x ^ z = y ^ z ↔ x = y | (rpow_left_injective hz).eq_iff | lemma | nnreal.rpow_eq_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_surjective {x : ℝ} (hx : x ≠ 0) : function.surjective (λ y : ℝ≥0, y^x) | λ y, ⟨y ^ x⁻¹, by simp_rw [←rpow_mul, _root_.inv_mul_cancel hx, rpow_one]⟩ | lemma | nnreal.rpow_left_surjective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_bijective {x : ℝ} (hx : x ≠ 0) : function.bijective (λ y : ℝ≥0, y^x) | ⟨rpow_left_injective hx, rpow_left_surjective hx⟩ | lemma | nnreal.rpow_left_bijective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_rpow_one_div_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x = y ^ (1 / z) ↔ x ^ z = y | by rw [← rpow_eq_rpow_iff hz, rpow_self_rpow_inv hz] | lemma | nnreal.eq_rpow_one_div_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one_div_eq_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x ^ (1 / z) = y ↔ x = y ^ z | by rw [← rpow_eq_rpow_iff hz, rpow_self_rpow_inv hz] | lemma | nnreal.rpow_one_div_eq_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_nat_rpow_nat_inv (x : ℝ≥0) {n : ℕ} (hn : n ≠ 0) :
(x ^ n) ^ (n⁻¹ : ℝ) = x | by { rw [← nnreal.coe_eq, coe_rpow, nnreal.coe_pow], exact real.pow_nat_rpow_nat_inv x.2 hn } | lemma | nnreal.pow_nat_rpow_nat_inv | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.coe_eq",
"nnreal.coe_pow",
"real.pow_nat_rpow_nat_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nat_inv_pow_nat (x : ℝ≥0) {n : ℕ} (hn : n ≠ 0) :
(x ^ (n⁻¹ : ℝ)) ^ n = x | by { rw [← nnreal.coe_eq, nnreal.coe_pow, coe_rpow], exact real.rpow_nat_inv_pow_nat x.2 hn } | lemma | nnreal.rpow_nat_inv_pow_nat | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.coe_eq",
"nnreal.coe_pow",
"real.rpow_nat_inv_pow_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.real.to_nnreal_rpow_of_nonneg {x y : ℝ} (hx : 0 ≤ x) :
real.to_nnreal (x ^ y) = (real.to_nnreal x) ^ y | begin
nth_rewrite 0 ← real.coe_to_nnreal x hx,
rw [←nnreal.coe_rpow, real.to_nnreal_coe],
end | lemma | real.to_nnreal_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"real.coe_to_nnreal",
"real.to_nnreal",
"real.to_nnreal_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow : ℝ≥0∞ → ℝ → ℝ≥0∞ | | (some x) y := if x = 0 ∧ y < 0 then ⊤ else (x ^ y : ℝ≥0)
| none y := if 0 < y then ⊤ else if y = 0 then 1 else 0 | def | ennreal.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | The real power function `x^y` on extended nonnegative reals, defined for `x : ℝ≥0∞` and
`y : ℝ` as the restriction of the real power function if `0 < x < ⊤`, and with the natural values
for `0` and `⊤` (i.e., `0 ^ x = 0` for `x > 0`, `1` for `x = 0` and `⊤` for `x < 0`, and
`⊤ ^ x = 1 / 0 ^ x`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_eq_pow (x : ℝ≥0∞) (y : ℝ) : rpow x y = x ^ y | rfl | lemma | ennreal.rpow_eq_pow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_zero {x : ℝ≥0∞} : x ^ (0 : ℝ) = 1 | by cases x; { dsimp only [(^), rpow], simp [lt_irrefl] } | lemma | ennreal.rpow_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_rpow_def (y : ℝ) : (⊤ : ℝ≥0∞) ^ y = if 0 < y then ⊤ else if y = 0 then 1 else 0 | rfl | lemma | ennreal.top_rpow_def | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_rpow_of_pos {y : ℝ} (h : 0 < y) : (⊤ : ℝ≥0∞) ^ y = ⊤ | by simp [top_rpow_def, h] | lemma | ennreal.top_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_rpow_of_neg {y : ℝ} (h : y < 0) : (⊤ : ℝ≥0∞) ^ y = 0 | by simp [top_rpow_def, asymm h, ne_of_lt h] | lemma | ennreal.top_rpow_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_of_pos {y : ℝ} (h : 0 < y) : (0 : ℝ≥0∞) ^ y = 0 | begin
rw [← ennreal.coe_zero, ← ennreal.some_eq_coe],
dsimp only [(^), rpow],
simp [h, asymm h, ne_of_gt h],
end | lemma | ennreal.zero_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.coe_zero",
"ennreal.some_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_of_neg {y : ℝ} (h : y < 0) : (0 : ℝ≥0∞) ^ y = ⊤ | begin
rw [← ennreal.coe_zero, ← ennreal.some_eq_coe],
dsimp only [(^), rpow],
simp [h, ne_of_gt h],
end | lemma | ennreal.zero_rpow_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.coe_zero",
"ennreal.some_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_def (y : ℝ) : (0 : ℝ≥0∞) ^ y = if 0 < y then 0 else if y = 0 then 1 else ⊤ | begin
rcases lt_trichotomy 0 y with H|rfl|H,
{ simp [H, ne_of_gt, zero_rpow_of_pos, lt_irrefl] },
{ simp [lt_irrefl] },
{ simp [H, asymm H, ne_of_lt, zero_rpow_of_neg] }
end | lemma | ennreal.zero_rpow_def | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_mul_self (y : ℝ) : (0 : ℝ≥0∞) ^ y * 0 ^ y = 0 ^ y | by { rw zero_rpow_def, split_ifs, exacts [zero_mul _, one_mul _, top_mul_top] } | lemma | ennreal.zero_rpow_mul_self | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rpow_of_ne_zero {x : ℝ≥0} (h : x ≠ 0) (y : ℝ) :
(x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) | begin
rw [← ennreal.some_eq_coe],
dsimp only [(^), rpow],
simp [h]
end | lemma | ennreal.coe_rpow_of_ne_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.some_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rpow_of_nonneg (x : ℝ≥0) {y : ℝ} (h : 0 ≤ y) :
(x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) | begin
by_cases hx : x = 0,
{ rcases le_iff_eq_or_lt.1 h with H|H,
{ simp [hx, H.symm] },
{ simp [hx, zero_rpow_of_pos H, nnreal.zero_rpow (ne_of_gt H)] } },
{ exact coe_rpow_of_ne_zero hx _ }
end | lemma | ennreal.coe_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.zero_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rpow_def (x : ℝ≥0) (y : ℝ) :
(x : ℝ≥0∞) ^ y = if x = 0 ∧ y < 0 then ⊤ else (x ^ y : ℝ≥0) | rfl | lemma | ennreal.coe_rpow_def | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one (x : ℝ≥0∞) : x ^ (1 : ℝ) = x | begin
cases x,
{ exact dif_pos zero_lt_one },
{ change ite _ _ _ = _,
simp only [nnreal.rpow_one, some_eq_coe, ite_eq_right_iff, top_ne_coe, and_imp],
exact λ _, zero_le_one.not_lt }
end | lemma | ennreal.rpow_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"and_imp",
"ite_eq_right_iff",
"nnreal.rpow_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_rpow (x : ℝ) : (1 : ℝ≥0∞) ^ x = 1 | by { rw [← coe_one, coe_rpow_of_ne_zero one_ne_zero], simp } | lemma | ennreal.one_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_zero_iff {x : ℝ≥0∞} {y : ℝ} :
x ^ y = 0 ↔ (x = 0 ∧ 0 < y) ∨ (x = ⊤ ∧ y < 0) | begin
cases x,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [H, top_rpow_of_neg, top_rpow_of_pos, le_of_lt] },
{ by_cases h : x = 0,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] },
{ simp [coe_rpow_of_ne_zero h, h] } }
end | lemma | ennreal.rpow_eq_zero_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_top_iff {x : ℝ≥0∞} {y : ℝ} :
x ^ y = ⊤ ↔ (x = 0 ∧ y < 0) ∨ (x = ⊤ ∧ 0 < y) | begin
cases x,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [H, top_rpow_of_neg, top_rpow_of_pos, le_of_lt] },
{ by_cases h : x = 0,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] },
{ simp [coe_rpow_of_ne_zero h, h] } }
end | lemma | ennreal.rpow_eq_top_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_top_iff_of_pos {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y) : x ^ y = ⊤ ↔ x = ⊤ | by simp [rpow_eq_top_iff, hy, asymm hy] | lemma | ennreal.rpow_eq_top_iff_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_top_of_nonneg (x : ℝ≥0∞) {y : ℝ} (hy0 : 0 ≤ y) : x ^ y = ⊤ → x = ⊤ | begin
rw ennreal.rpow_eq_top_iff,
intro h,
cases h,
{ exfalso, rw lt_iff_not_ge at h, exact h.right hy0, },
{ exact h.left, },
end | lemma | ennreal.rpow_eq_top_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_ne_top_of_nonneg {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤) : x ^ y ≠ ⊤ | mt (ennreal.rpow_eq_top_of_nonneg x hy0) h | lemma | ennreal.rpow_ne_top_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_eq_top_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_top_of_nonneg {x : ℝ≥0∞} {y : ℝ} (hy0 : 0 ≤ y) (h : x ≠ ⊤) : x ^ y < ⊤ | lt_top_iff_ne_top.mpr (ennreal.rpow_ne_top_of_nonneg hy0 h) | lemma | ennreal.rpow_lt_top_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_ne_top_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y + z) = x ^ y * x ^ z | begin
cases x, { exact (h'x rfl).elim },
have : x ≠ 0 := λ h, by simpa [h] using hx,
simp [coe_rpow_of_ne_zero this, nnreal.rpow_add this]
end | lemma | ennreal.rpow_add | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg (x : ℝ≥0∞) (y : ℝ) : x ^ -y = (x ^ y)⁻¹ | begin
cases x,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [top_rpow_of_pos, top_rpow_of_neg, H, neg_pos.mpr] },
{ by_cases h : x = 0,
{ rcases lt_trichotomy y 0 with H|H|H;
simp [h, zero_rpow_of_pos, zero_rpow_of_neg, H, neg_pos.mpr] },
{ have A : x ^ y ≠ 0, by simp [h],
simp [coe_rpow_of... | lemma | ennreal.rpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y - z) = x ^ y / x ^ z | by rw [sub_eq_add_neg, rpow_add _ _ hx h'x, rpow_neg, div_eq_mul_inv] | lemma | ennreal.rpow_sub | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg_one (x : ℝ≥0∞) : x ^ (-1 : ℝ) = x ⁻¹ | by simp [rpow_neg] | lemma | ennreal.rpow_neg_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_mul (x : ℝ≥0∞) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z | begin
cases x,
{ rcases lt_trichotomy y 0 with Hy|Hy|Hy;
rcases lt_trichotomy z 0 with Hz|Hz|Hz;
simp [Hy, Hz, zero_rpow_of_neg, zero_rpow_of_pos, top_rpow_of_neg, top_rpow_of_pos,
mul_pos_of_neg_of_neg, mul_neg_of_neg_of_pos, mul_neg_of_pos_of_neg] },
{ by_cases h : x = 0,
{ rcases lt_trich... | lemma | ennreal.rpow_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"mul_neg_of_neg_of_pos",
"mul_neg_of_pos_of_neg",
"mul_pos_of_neg_of_neg",
"nnreal.rpow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nat_cast (x : ℝ≥0∞) (n : ℕ) : x ^ (n : ℝ) = x ^ n | begin
cases x,
{ cases n;
simp [top_rpow_of_pos (nat.cast_add_one_pos _), top_pow (nat.succ_pos _)] },
{ simp [coe_rpow_of_nonneg _ (nat.cast_nonneg n)] }
end | lemma | ennreal.rpow_nat_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nat.cast_add_one_pos",
"nat.cast_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_two (x : ℝ≥0∞) : x ^ (2 : ℝ) = x ^ 2 | by { rw ← rpow_nat_cast, simp only [nat.cast_bit0, nat.cast_one] } | lemma | ennreal.rpow_two | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nat.cast_bit0",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow_eq_ite (x y : ℝ≥0∞) (z : ℝ) :
(x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z | begin
rcases eq_or_ne z 0 with rfl|hz, { simp },
replace hz := hz.lt_or_lt,
wlog hxy : x ≤ y,
{ convert this y x z hz (le_of_not_le hxy) using 2; simp only [mul_comm, and_comm, or_comm], },
rcases eq_or_ne x 0 with rfl|hx0,
{ induction y using with_top.rec_top_coe; cases hz with hz hz; simp [*, hz.not_lt] }... | lemma | ennreal.mul_rpow_eq_ite | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"bot_unique",
"eq_or_ne",
"mul_comm",
"mul_ne_zero",
"nnreal.mul_rpow",
"top_unique",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow_of_ne_top {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) (z : ℝ) :
(x * y) ^ z = x^z * y^z | by simp [*, mul_rpow_eq_ite] | lemma | ennreal.mul_rpow_of_ne_top | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_rpow (x y : ℝ≥0) (z : ℝ) :
((x : ℝ≥0∞) * y) ^ z = x^z * y^z | mul_rpow_of_ne_top coe_ne_top coe_ne_top z | lemma | ennreal.coe_mul_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow_of_ne_zero {x y : ℝ≥0∞} (hx : x ≠ 0) (hy : y ≠ 0) (z : ℝ) :
(x * y) ^ z = x ^ z * y ^ z | by simp [*, mul_rpow_eq_ite] | lemma | ennreal.mul_rpow_of_ne_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow_of_nonneg (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z) :
(x * y) ^ z = x ^ z * y ^ z | by simp [hz.not_lt, mul_rpow_eq_ite] | lemma | ennreal.mul_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_rpow (x : ℝ≥0∞) (y : ℝ) : (x⁻¹) ^ y = (x ^ y)⁻¹ | begin
rcases eq_or_ne y 0 with rfl|hy, { simp only [rpow_zero, inv_one] },
replace hy := hy.lt_or_lt,
rcases eq_or_ne x 0 with rfl|h0, { cases hy; simp * },
rcases eq_or_ne x ⊤ with rfl|h_top, { cases hy; simp * },
apply ennreal.eq_inv_of_mul_eq_one_left,
rw [← mul_rpow_of_ne_zero (ennreal.inv_ne_zero.2 h_t... | lemma | ennreal.inv_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.eq_inv_of_mul_eq_one_left",
"ennreal.inv_mul_cancel",
"eq_or_ne",
"inv_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_rpow_of_nonneg (x y : ℝ≥0∞) {z : ℝ} (hz : 0 ≤ z) :
(x / y) ^ z = x ^ z / y ^ z | by rw [div_eq_mul_inv, mul_rpow_of_nonneg _ _ hz, inv_rpow, div_eq_mul_inv] | lemma | ennreal.div_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_rpow_of_pos {z : ℝ} (h : 0 < z) : strict_mono (λ x : ℝ≥0∞, x ^ z) | begin
intros x y hxy,
lift x to ℝ≥0 using ne_top_of_lt hxy,
rcases eq_or_ne y ∞ with rfl|hy,
{ simp only [top_rpow_of_pos h, coe_rpow_of_nonneg _ h.le, coe_lt_top] },
{ lift y to ℝ≥0 using hy,
simp only [coe_rpow_of_nonneg _ h.le, nnreal.rpow_lt_rpow (coe_lt_coe.1 hxy) h, coe_lt_coe] }
end | lemma | ennreal.strict_mono_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"eq_or_ne",
"lift",
"ne_top_of_lt",
"nnreal.rpow_lt_rpow",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_rpow_of_nonneg {z : ℝ} (h : 0 ≤ z) : monotone (λ x : ℝ≥0∞, x ^ z) | h.eq_or_lt.elim (λ h0, h0 ▸ by simp only [rpow_zero, monotone_const])
(λ h0, (strict_mono_rpow_of_pos h0).monotone) | lemma | ennreal.monotone_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"monotone",
"monotone_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_rpow (y : ℝ) (hy : 0 < y) : ℝ≥0∞ ≃o ℝ≥0∞ | (strict_mono_rpow_of_pos hy).order_iso_of_right_inverse (λ x, x ^ y) (λ x, x ^ (1 / y))
(λ x, by { dsimp, rw [←rpow_mul, one_div_mul_cancel hy.ne.symm, rpow_one] }) | def | ennreal.order_iso_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"one_div_mul_cancel"
] | Bundles `λ x : ℝ≥0∞, x ^ y` into an order isomorphism when `y : ℝ` is positive,
where the inverse is `λ x : ℝ≥0∞, x ^ (1 / y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_rpow_symm_apply (y : ℝ) (hy : 0 < y) :
(order_iso_rpow y hy).symm = order_iso_rpow (1 / y) (one_div_pos.2 hy) | by { simp only [order_iso_rpow, one_div_one_div], refl } | lemma | ennreal.order_iso_rpow_symm_apply | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"one_div_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow {x y : ℝ≥0∞} {z : ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x^z ≤ y^z | monotone_rpow_of_nonneg h₂ h₁ | lemma | ennreal.rpow_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow {x y : ℝ≥0∞} {z : ℝ} (h₁ : x < y) (h₂ : 0 < z) : x^z < y^z | strict_mono_rpow_of_pos h₂ h₁ | lemma | ennreal.rpow_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y | (strict_mono_rpow_of_pos hz).le_iff_le | lemma | ennreal.rpow_le_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ z < y ^ z ↔ x < y | (strict_mono_rpow_of_pos hz).lt_iff_lt | lemma | ennreal.rpow_lt_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_one_div_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ≤ y ^ (1 / z) ↔ x ^ z ≤ y | begin
nth_rewrite 0 ←rpow_one x,
nth_rewrite 0 ←@_root_.mul_inv_cancel _ _ z hz.ne',
rw [rpow_mul, ←one_div, @rpow_le_rpow_iff _ _ (1/z) (by simp [hz])],
end | lemma | ennreal.le_rpow_one_div_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_rpow_one_div_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x < y ^ (1 / z) ↔ x ^ z < y | begin
nth_rewrite 0 ←rpow_one x,
nth_rewrite 0 ←@_root_.mul_inv_cancel _ _ z (ne_of_lt hz).symm,
rw [rpow_mul, ←one_div, @rpow_lt_rpow_iff _ _ (1/z) (by simp [hz])],
end | lemma | ennreal.lt_rpow_one_div_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one_div_le_iff {x y : ℝ≥0∞} {z : ℝ} (hz : 0 < z) : x ^ (1 / z) ≤ y ↔ x ≤ y ^ z | begin
nth_rewrite 0 ← ennreal.rpow_one y,
nth_rewrite 1 ← @_root_.mul_inv_cancel _ _ z hz.ne.symm,
rw [ennreal.rpow_mul, ← one_div, ennreal.rpow_le_rpow_iff (one_div_pos.2 hz)],
end | lemma | ennreal.rpow_one_div_le_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_le_rpow_iff",
"ennreal.rpow_mul",
"ennreal.rpow_one",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_lt {x : ℝ≥0∞} {y z : ℝ} (hx : 1 < x) (hx' : x ≠ ⊤) (hyz : y < z) :
x^y < x^z | begin
lift x to ℝ≥0 using hx',
rw [one_lt_coe_iff] at hx,
simp [coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
nnreal.rpow_lt_rpow_of_exponent_lt hx hyz]
end | lemma | ennreal.rpow_lt_rpow_of_exponent_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"lift",
"nnreal.rpow_lt_rpow_of_exponent_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_le {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) : x^y ≤ x^z | begin
cases x,
{ rcases lt_trichotomy y 0 with Hy|Hy|Hy;
rcases lt_trichotomy z 0 with Hz|Hz|Hz;
simp [Hy, Hz, top_rpow_of_neg, top_rpow_of_pos, le_refl];
linarith },
{ simp only [one_le_coe_iff, some_eq_coe] at hx,
simp [coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
nn... | lemma | ennreal.rpow_le_rpow_of_exponent_le | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_le_rpow_of_exponent_le",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_gt {x : ℝ≥0∞} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) :
x^y < x^z | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx1 le_top),
simp only [coe_lt_one_iff, coe_pos] at hx0 hx1,
simp [coe_rpow_of_ne_zero (ne_of_gt hx0), nnreal.rpow_lt_rpow_of_exponent_gt hx0 hx1 hyz]
end | lemma | ennreal.rpow_lt_rpow_of_exponent_gt | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"le_top",
"lift",
"nnreal.rpow_lt_rpow_of_exponent_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_ge {x : ℝ≥0∞} {y z : ℝ} (hx1 : x ≤ 1) (hyz : z ≤ y) :
x^y ≤ x^z | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx1 coe_lt_top),
by_cases h : x = 0,
{ rcases lt_trichotomy y 0 with Hy|Hy|Hy;
rcases lt_trichotomy z 0 with Hz|Hz|Hz;
simp [Hy, Hz, h, zero_rpow_of_neg, zero_rpow_of_pos, le_refl];
linarith },
{ rw [coe_le_one_iff] at hx1,
simp [coe_rpow_of_ne_... | lemma | ennreal.rpow_le_rpow_of_exponent_ge | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"lift",
"nnreal.rpow_le_rpow_of_exponent_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_self_of_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x | begin
nth_rewrite 1 ←ennreal.rpow_one x,
exact ennreal.rpow_le_rpow_of_exponent_ge hx h_one_le,
end | lemma | ennreal.rpow_le_self_of_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_le_rpow_of_exponent_ge"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_self_of_one_le {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (h_one_le : 1 ≤ z) : x ≤ x ^ z | begin
nth_rewrite 0 ←ennreal.rpow_one x,
exact ennreal.rpow_le_rpow_of_exponent_le hx h_one_le,
end | lemma | ennreal.le_rpow_self_of_one_le | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_le_rpow_of_exponent_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_pos_of_nonneg {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hp_nonneg : 0 ≤ p) : 0 < x^p | begin
by_cases hp_zero : p = 0,
{ simp [hp_zero, zero_lt_one], },
{ rw ←ne.def at hp_zero,
have hp_pos := lt_of_le_of_ne hp_nonneg hp_zero.symm,
rw ←zero_rpow_of_pos hp_pos, exact rpow_lt_rpow hx_pos hp_pos, },
end | lemma | ennreal.rpow_pos_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_pos {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hx_ne_top : x ≠ ⊤) : 0 < x^p | begin
cases lt_or_le 0 p with hp_pos hp_nonpos,
{ exact rpow_pos_of_nonneg hx_pos (le_of_lt hp_pos), },
{ rw [←neg_neg p, rpow_neg, ennreal.inv_pos],
exact rpow_ne_top_of_nonneg (right.nonneg_neg_iff.mpr hp_nonpos) hx_ne_top, },
end | lemma | ennreal.rpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.inv_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one {x : ℝ≥0∞} {z : ℝ} (hx : x < 1) (hz : 0 < z) : x^z < 1 | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx le_top),
simp only [coe_lt_one_iff] at hx,
simp [coe_rpow_of_nonneg _ (le_of_lt hz), nnreal.rpow_lt_one hx hz],
end | lemma | ennreal.rpow_lt_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"le_top",
"lift",
"nnreal.rpow_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (hz : 0 ≤ z) : x^z ≤ 1 | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx coe_lt_top),
simp only [coe_le_one_iff] at hx,
simp [coe_rpow_of_nonneg _ hz, nnreal.rpow_le_one hx hz],
end | lemma | ennreal.rpow_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"lift",
"nnreal.rpow_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x^z < 1 | begin
cases x,
{ simp [top_rpow_of_neg hz, zero_lt_one] },
{ simp only [some_eq_coe, one_lt_coe_iff] at hx,
simp [coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
nnreal.rpow_lt_one_of_one_lt_of_neg hx hz] },
end | lemma | ennreal.rpow_lt_one_of_one_lt_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_lt_one_of_one_lt_of_neg",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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