statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
cast_int_mul_left (h : commute a b) (m : ℤ) : commute ((m : R) * a) b | h.cast_int_mul_left m | lemma | commute.cast_int_mul_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_cast_int_mul (h : commute a b) (m n : ℤ) : commute (m * a : R) (n * b : R) | h.cast_int_mul_cast_int_mul m n | lemma | commute.cast_int_mul_cast_int_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_left : commute (m : R) a | int.cast_commute _ _ | lemma | commute.cast_int_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"int.cast_commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_right : commute a m | int.commute_cast _ _ | lemma | commute.cast_int_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"int.commute_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_cast_int_mul : commute a (n * a : R) | (commute.refl a).cast_int_mul_right n | theorem | commute.self_cast_int_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_self : commute ((n : R) * a) a | (commute.refl a).cast_int_mul_left n | theorem | commute.cast_int_mul_self | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_cast_int_mul_cast_int_mul : commute (m * a : R) (n * a : R) | (commute.refl a).cast_int_mul_cast_int_mul m n | theorem | commute.self_cast_int_mul_cast_int_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.to_add_pow (a : multiplicative ℕ) (b : ℕ) : to_add (a ^ b) = to_add a * b | begin
induction b with b ih,
{ erw [pow_zero, to_add_one, mul_zero] },
{ simp [*, pow_succ, add_comm, nat.mul_succ] }
end | lemma | nat.to_add_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"mul_zero",
"multiplicative",
"pow_succ",
"pow_zero",
"to_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.of_add_mul (a b : ℕ) : of_add (a * b) = of_add a ^ b | (nat.to_add_pow _ _).symm | lemma | nat.of_add_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"nat.to_add_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.to_add_pow (a : multiplicative ℤ) (b : ℕ) : to_add (a ^ b) = to_add a * b | by induction b; simp [*, mul_add, pow_succ, add_comm] | lemma | int.to_add_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"multiplicative",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.to_add_zpow (a : multiplicative ℤ) (b : ℤ) : to_add (a ^ b) = to_add a * b | int.induction_on b (by simp)
(by simp [zpow_add, mul_add] {contextual := tt})
(by simp [zpow_add, mul_add, sub_eq_add_neg, -int.add_neg_one] {contextual := tt}) | lemma | int.to_add_zpow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.add_neg_one",
"int.induction_on",
"multiplicative",
"zpow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.of_add_mul (a b : ℤ) : of_add (a * b) = of_add a ^ b | (int.to_add_zpow _ _).symm | lemma | int.of_add_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.to_add_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_pow (u : Mˣ) (x : M) (n : ℕ) : (↑u * x * ↑(u⁻¹))^n = u * x^n * ↑(u⁻¹) | (divp_eq_iff_mul_eq.2 ((u.mk_semiconj_by x).pow_right n).eq.symm).symm | lemma | units.conj_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"conj_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_pow' (u : Mˣ) (x : M) (n : ℕ) : (↑(u⁻¹) * x * u)^n = ↑(u⁻¹) * x^n * u | (u⁻¹).conj_pow x n | lemma | units.conj_pow' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"conj_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_pow [monoid M] (x : M) (n : ℕ) : op (x ^ n) = (op x) ^ n | rfl | lemma | mul_opposite.op_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monoid"
] | Moving to the opposite monoid commutes with taking powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_pow [monoid M] (x : Mᵐᵒᵖ) (n : ℕ) : unop (x ^ n) = (unop x) ^ n | rfl | lemma | mul_opposite.unop_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_zpow [div_inv_monoid M] (x : M) (z : ℤ) : op (x ^ z) = (op x) ^ z | rfl | lemma | mul_opposite.op_zpow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"div_inv_monoid"
] | Moving to the opposite group or group_with_zero commutes with taking powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_zpow [div_inv_monoid M] (x : Mᵐᵒᵖ) (z : ℤ) : unop (x ^ z) = (unop x) ^ z | rfl | lemma | mul_opposite.unop_zpow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_of_le_left' [covariant_class M M (swap (*)) (≤)] {a b : M} (hab : a ≤ b) :
∀ i : ℕ, a ^ i ≤ b ^ i | | 0 := by simp
| (k+1) := by { rw [pow_succ, pow_succ],
exact mul_le_mul' hab (pow_le_pow_of_le_left' k) } | lemma | pow_le_pow_of_le_left' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"mul_le_mul'",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pow_of_one_le' {a : M} (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n | | 0 := by simp
| (k + 1) := by { rw pow_succ, exact one_le_mul H (one_le_pow_of_one_le' k) } | theorem | one_le_pow_of_one_le' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_one' {a : M} (H : a ≤ 1) (n : ℕ) : a ^ n ≤ 1 | @one_le_pow_of_one_le' Mᵒᵈ _ _ _ _ H n | lemma | pow_le_one' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_of_one_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m | let ⟨k, hk⟩ := nat.le.dest h in
calc a ^ n ≤ a ^ n * a ^ k : le_mul_of_one_le_right' (one_le_pow_of_one_le' ha _)
... = a ^ m : by rw [← hk, pow_add] | theorem | pow_le_pow' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"le_mul_of_one_le_right'",
"one_le_pow_of_one_le'",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_of_le_one' {a : M} {n m : ℕ} (ha : a ≤ 1) (h : n ≤ m) : a ^ m ≤ a ^ n | @pow_le_pow' Mᵒᵈ _ _ _ _ _ _ ha h | theorem | pow_le_pow_of_le_one' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_le_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k | begin
rcases nat.exists_eq_succ_of_ne_zero hk with ⟨l, rfl⟩,
clear hk,
induction l with l IH,
{ simpa using ha },
{ rw pow_succ,
exact one_lt_mul'' ha IH }
end | theorem | one_lt_pow' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_one' {a : M} (ha : a < 1) {k : ℕ} (hk : k ≠ 0) : a ^ k < 1 | @one_lt_pow' Mᵒᵈ _ _ _ _ ha k hk | lemma | pow_lt_one' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_lt_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow' [covariant_class M M (*) (<)] {a : M} {n m : ℕ} (ha : 1 < a) (h : n < m) :
a ^ n < a ^ m | begin
rcases nat.le.dest h with ⟨k, rfl⟩, clear h,
rw [pow_add, pow_succ', mul_assoc, ← pow_succ],
exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)
end | theorem | pow_lt_pow' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"lt_mul_of_one_lt_right'",
"mul_assoc",
"one_lt_pow'",
"pow_add",
"pow_succ",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_strict_mono_left [covariant_class M M (*) (<)] {a : M} (ha : 1 < a) :
strict_mono ((^) a : ℕ → M) | λ m n, pow_lt_pow' ha | lemma | pow_strict_mono_left | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"pow_lt_pow'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x^n | | 0 := (pow_zero x).ge
| (n + 1) := by { rw pow_succ, exact left.one_le_mul hx left.one_le_pow_of_le } | lemma | left.one_le_pow_of_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"left.one_le_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x^n ≤ 1 | | 0 := (pow_zero _).le
| (n + 1) := by { rw pow_succ, exact left.mul_le_one hx left.pow_le_one_of_le } | lemma | left.pow_le_one_of_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"left.mul_le_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x^n | | 0 := (pow_zero _).ge
| (n + 1) := by { rw pow_succ, exact right.one_le_mul hx right.one_le_pow_of_le } | lemma | right.one_le_pow_of_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_succ",
"pow_zero",
"right.one_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x^n ≤ 1 | | 0 := (pow_zero _).le
| (n + 1) := by { rw pow_succ, exact right.mul_le_one hx right.pow_le_one_of_le } | lemma | right.pow_le_one_of_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_succ",
"pow_zero",
"right.mul_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono.pow_right' (hf : strict_mono f) : ∀ {n : ℕ}, n ≠ 0 → strict_mono (λ a, f a ^ n) | | 0 hn := (hn rfl).elim
| 1 hn := by simpa
| (nat.succ $ nat.succ n) hn :=
by { simp_rw pow_succ _ (n + 1), exact hf.mul' (strict_mono.pow_right' n.succ_ne_zero) } | lemma | strict_mono.pow_right' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_succ",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_strict_mono_right' {n : ℕ} (hn : n ≠ 0) : strict_mono (λ a : M, a ^ n) | strict_mono_id.pow_right' hn | lemma | pow_strict_mono_right' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"strict_mono"
] | See also `pow_strict_mono_right` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.pow_right {f : β → M} (hf : monotone f) : ∀ n : ℕ, monotone (λ a, f a ^ n) | | 0 := by simpa using monotone_const
| (n + 1) := by { simp_rw pow_succ, exact hf.mul' (monotone.pow_right _) } | lemma | monotone.pow_right | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"monotone",
"monotone_const",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mono_right (n : ℕ) : monotone (λ a : M, a ^ n) | monotone_id.pow_right _ | lemma | pow_mono_right | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.pow_lt_one_of_lt [covariant_class M M (*) (<)] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 1) :
x^n < 1 | nat.le_induction ((pow_one _).trans_lt h) (λ n _ ih, by { rw pow_succ, exact mul_lt_one h ih }) _
(nat.succ_le_iff.2 hn) | lemma | left.pow_lt_one_of_lt | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"ih",
"nat.le_induction",
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.pow_lt_one_of_lt [covariant_class M M (swap (*)) (<)] {n : ℕ} {x : M}
(hn : 0 < n) (h : x < 1) :
x^n < 1 | nat.le_induction ((pow_one _).trans_lt h)
(λ n _ ih, by { rw pow_succ, exact right.mul_lt_one h ih }) _ (nat.succ_le_iff.2 hn) | lemma | right.pow_lt_one_of_lt | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"ih",
"nat.le_induction",
"pow_one",
"pow_succ",
"right.mul_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 ≤ x ^ n ↔ 1 ≤ x | ⟨le_imp_le_of_lt_imp_lt $ λ h, pow_lt_one' h hn, λ h, one_le_pow_of_one_le' h n⟩ | lemma | one_le_pow_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_of_one_le'",
"pow_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n ≤ 1 ↔ x ≤ 1 | @one_le_pow_iff Mᵒᵈ _ _ _ _ _ hn | lemma | pow_le_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_pow_iff {x : M} {n : ℕ} (hn : n ≠ 0) : 1 < x ^ n ↔ 1 < x | lt_iff_lt_of_le_iff_le (pow_le_one_iff hn) | lemma | one_lt_pow_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lt_iff_lt_of_le_iff_le",
"pow_le_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n < 1 ↔ x < 1 | lt_iff_lt_of_le_iff_le (one_le_pow_iff hn) | lemma | pow_lt_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lt_iff_lt_of_le_iff_le",
"one_le_pow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n = 1 ↔ x = 1 | by simp only [le_antisymm_iff, pow_le_one_iff hn, one_le_pow_iff hn] | lemma | pow_eq_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_iff",
"pow_le_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_iff' (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n | (pow_strict_mono_left ha).le_iff_le | lemma | pow_le_pow_iff' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_iff' (ha : 1 < a) : a ^ m < a ^ n ↔ m < n | (pow_strict_mono_left ha).lt_iff_lt | lemma | pow_lt_pow_iff' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_pow_lt_pow' {a b : M} (n : ℕ) : a ^ n < b ^ n → a < b | (pow_mono_right _).reflect_lt | lemma | lt_of_pow_lt_pow' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_lt_max_of_mul_lt_mul {a b c d : M} (h : a * b < c * d) : min a b < max c d | lt_of_pow_lt_pow' 2 $ by { simp_rw pow_two, exact (mul_le_mul' inf_le_left
inf_le_right).trans_lt (h.trans_le $ mul_le_mul' le_sup_left le_sup_right) } | lemma | min_lt_max_of_mul_lt_mul | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"inf_le_left",
"inf_le_right",
"le_sup_left",
"le_sup_right",
"lt_of_pow_lt_pow'",
"mul_le_mul'",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_lt_of_mul_lt_sq {a b c : M} (h : a * b < c ^ 2) : min a b < c | by simpa using min_lt_max_of_mul_lt_mul (h.trans_eq $ pow_two _) | lemma | min_lt_of_mul_lt_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"min_lt_max_of_mul_lt_mul",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_max_of_sq_lt_mul {a b c : M} (h : a ^ 2 < b * c) : a < max b c | by simpa using min_lt_max_of_mul_lt_mul ((pow_two _).symm.trans_lt h) | lemma | lt_max_of_sq_lt_mul | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"min_lt_max_of_mul_lt_mul",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_pow_le_pow' {a b : M} {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ b ^ n → a ≤ b | (pow_strict_mono_right' hn).le_iff_le.1 | lemma | le_of_pow_le_pow' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_of_mul_le_sq {a b c : M} (h : a * b ≤ c ^ 2) : min a b ≤ c | by simpa using min_le_max_of_mul_le_mul (h.trans_eq $ pow_two _) | lemma | min_le_of_mul_le_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"min_le_max_of_mul_le_mul",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_max_of_sq_le_mul {a b c : M} (h : a ^ 2 ≤ b * c) : a ≤ max b c | by simpa using min_le_max_of_mul_le_mul ((pow_two _).symm.trans_le h) | lemma | le_max_of_sq_le_mul | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"min_le_max_of_mul_le_mul",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.pow_lt_one_iff [covariant_class M M (*) (<)] {n : ℕ} {x : M} (hn : 0 < n) :
x^n < 1 ↔ x < 1 | by { haveI := has_mul.to_covariant_class_left M, exact pow_lt_one_iff hn.ne' } | lemma | left.pow_lt_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"has_mul.to_covariant_class_left",
"pow_lt_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.pow_lt_one_iff [covariant_class M M (swap (*)) (<)] {n : ℕ} {x : M} (hn : 0 < n) :
x^n < 1 ↔ x < 1 | ⟨λ H, not_le.mp $ λ k, H.not_le $ by { haveI := has_mul.to_covariant_class_right M,
exact right.one_le_pow_of_le k }, right.pow_lt_one_of_lt hn⟩ | lemma | right.pow_lt_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"covariant_class",
"has_mul.to_covariant_class_right",
"right.one_le_pow_of_le",
"right.pow_lt_one_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_zpow {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) :
1 ≤ x ^ n | begin
lift n to ℕ using hn,
rw zpow_coe_nat,
apply one_le_pow_of_one_le' H,
end | theorem | one_le_zpow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lift",
"one_le_pow_of_one_le'",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_pos {a : R} (H : 0 < a) (n : ℕ) : 0 < a ^ n | pos_iff_ne_zero.2 $ pow_ne_zero _ H.ne' | theorem | canonically_ordered_comm_semiring.pow_pos | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_ne_zero",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_pow_le_one : ∀ n : ℕ, (0 : R) ^ n ≤ 1 | | 0 := (pow_zero _).le
| (n + 1) := by { rw [zero_pow n.succ_pos], exact zero_le_one } | lemma | zero_pow_le_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_zero",
"zero_le_one",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y ^ n ≤ (x + y) ^ n | begin
rcases nat.exists_eq_succ_of_ne_zero hn with ⟨k, rfl⟩,
induction k with k ih, { simp only [pow_one] },
let n := k.succ,
have h1 := add_nonneg (mul_nonneg hx (pow_nonneg hy n)) (mul_nonneg hy (pow_nonneg hx n)),
have h2 := add_nonneg hx hy,
calc x^n.succ + y^n.succ
≤ x*x^n + y*y^n + (x*y^n + y*x^n)... | theorem | pow_add_pow_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"ih",
"mul_le_mul_of_nonneg_left",
"pow_nonneg",
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_one : ∀ (n : ℕ) (h₀ : 0 ≤ a) (h₁ : a ≤ 1), a ^ n ≤ 1 | | 0 h₀ h₁ := (pow_zero a).le
| (n + 1) h₀ h₁ := (pow_succ' a n).le.trans (mul_le_one (pow_le_one n h₀ h₁) h₀ h₁) | lemma | pow_le_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_le_one",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_one (h₀ : 0 ≤ a) (h₁ : a < 1) : ∀ {n : ℕ} (hn : n ≠ 0), a ^ n < 1 | | 0 h := (h rfl).elim
| (n + 1) h :=
by { rw pow_succ, exact mul_lt_one_of_nonneg_of_lt_one_left h₀ h₁ (pow_le_one _ h₀ h₁.le) } | lemma | pow_lt_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_lt_one_of_nonneg_of_lt_one_left",
"pow_le_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pow_of_one_le (H : 1 ≤ a) : ∀ (n : ℕ), 1 ≤ a ^ n | | 0 := by rw [pow_zero]
| (n+1) := by { rw pow_succ, simpa only [mul_one] using mul_le_mul H (one_le_pow_of_one_le n)
zero_le_one (le_trans zero_le_one H) } | theorem | one_le_pow_of_one_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_le_mul",
"mul_one",
"pow_succ",
"pow_zero",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mono (h : 1 ≤ a) : monotone (λ n : ℕ, a ^ n) | monotone_nat_of_le_succ $ λ n,
by { rw pow_succ, exact le_mul_of_one_le_left (pow_nonneg (zero_le_one.trans h) _) h } | lemma | pow_mono | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"le_mul_of_one_le_left",
"monotone",
"monotone_nat_of_le_succ",
"pow_nonneg",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m | pow_mono ha h | theorem | pow_le_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_pow (ha : 1 ≤ a) (h : m ≠ 0) : a ≤ a ^ m | (pow_one a).symm.trans_le (pow_le_pow ha $ pos_iff_ne_zero.mpr h) | theorem | le_self_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_le_pow",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_of_le_left {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) : ∀ i : ℕ, a^i ≤ b^i | | 0 := by simp
| (k+1) := by { rw [pow_succ, pow_succ],
exact mul_le_mul hab (pow_le_pow_of_le_left _) (pow_nonneg ha _) (le_trans ha hab) } | lemma | pow_le_pow_of_le_left | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_le_mul",
"pow_nonneg",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_pow (ha : 1 < a) : ∀ {n : ℕ} (hn : n ≠ 0), 1 < a ^ n | | 0 h := (h rfl).elim
| (n + 1) h :=
by { rw pow_succ, exact one_lt_mul_of_lt_of_le ha (one_le_pow_of_one_le ha.le _) } | lemma | one_lt_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_of_one_le",
"one_lt_mul_of_lt_of_le",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_of_lt_left (h : x < y) (hx : 0 ≤ x) : ∀ {n : ℕ}, 0 < n → x ^ n < y ^ n | | 0 hn := hn.false.elim
| (n + 1) _ := by simpa only [pow_succ'] using
mul_lt_mul_of_le_of_le' (pow_le_pow_of_le_left hx h.le _) h (pow_pos (hx.trans_lt h) _) hx | lemma | pow_lt_pow_of_lt_left | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_lt_mul_of_le_of_le'",
"pow_le_pow_of_le_left",
"pow_pos",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_pow (hn : 0 < n) : strict_mono_on (λ x : R, x ^ n) (set.Ici 0) | λ x hx y hy h, pow_lt_pow_of_lt_left h hx hn | lemma | strict_mono_on_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_lt_pow_of_lt_left",
"set.Ici",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_strict_mono_right (h : 1 < a) : strict_mono (λ n : ℕ, a ^ n) | have 0 < a := zero_le_one.trans_lt h,
strict_mono_nat_of_lt_succ $ λ n, by simpa only [one_mul, pow_succ]
using mul_lt_mul h (le_refl (a ^ n)) (pow_pos this _) this.le | lemma | pow_strict_mono_right | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_lt_mul",
"one_mul",
"pow_pos",
"pow_succ",
"strict_mono",
"strict_mono_nat_of_lt_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow (h : 1 < a) (h2 : n < m) : a ^ n < a ^ m | pow_strict_mono_right h h2 | lemma | pow_lt_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_iff (h : 1 < a) : a ^ n < a ^ m ↔ n < m | (pow_strict_mono_right h).lt_iff_lt | lemma | pow_lt_pow_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_iff (h : 1 < a) : a ^ n ≤ a ^ m ↔ n ≤ m | (pow_strict_mono_right h).le_iff_le | lemma | pow_le_pow_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_pow (h₀ : 0 < a) (h₁ : a < 1) : strict_anti (λ n : ℕ, a ^ n) | strict_anti_nat_of_succ_lt $ λ n,
by simpa only [pow_succ, one_mul] using mul_lt_mul h₁ le_rfl (pow_pos h₀ n) zero_le_one | lemma | strict_anti_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"le_rfl",
"mul_lt_mul",
"one_mul",
"pow_pos",
"pow_succ",
"strict_anti",
"strict_anti_nat_of_succ_lt",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_iff_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) : a ^ m < a ^ n ↔ n < m | (strict_anti_pow h₀ h₁).lt_iff_lt | lemma | pow_lt_pow_iff_of_lt_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"strict_anti_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_of_lt_one (h : 0 < a) (ha : a < 1) {i j : ℕ} (hij : i < j) : a ^ j < a ^ i | (pow_lt_pow_iff_of_lt_one h ha).2 hij | lemma | pow_lt_pow_of_lt_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_lt_pow_iff_of_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_self_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n) : a ^ n < a | calc a ^ n < a ^ 1 : pow_lt_pow_of_lt_one h₀ h₁ hn
... = a : pow_one _ | lemma | pow_lt_self_of_lt_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_lt_pow_of_lt_one",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_pos_of_pos (ha : 0 < a) : 0 < a ^ 2 | by { rw sq, exact mul_pos ha ha } | lemma | sq_pos_of_pos | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0_pos_of_neg (ha : a < 0) (n : ℕ) : 0 < a ^ bit0 n | begin
rw pow_bit0',
exact pow_pos (mul_pos_of_neg_of_neg ha ha) _,
end | lemma | pow_bit0_pos_of_neg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_pos_of_neg_of_neg",
"pow_bit0'",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1_neg (ha : a < 0) (n : ℕ) : a ^ bit1 n < 0 | begin
rw [bit1, pow_succ],
exact mul_neg_of_neg_of_pos ha (pow_bit0_pos_of_neg ha n),
end | lemma | pow_bit1_neg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_neg_of_neg_of_pos",
"pow_bit0_pos_of_neg",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_pos_of_neg (ha : a < 0) : 0 < a ^ 2 | pow_bit0_pos_of_neg ha _ | lemma | sq_pos_of_neg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_bit0_pos_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_one_iff_of_nonneg {a : R} (ha : 0 ≤ a) {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ 1 ↔ a ≤ 1 | begin
refine ⟨_, pow_le_one n ha⟩,
rw [←not_lt, ←not_lt],
exact mt (λ h, one_lt_pow h hn),
end | lemma | pow_le_one_iff_of_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_lt_pow",
"pow_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pow_iff_of_nonneg {a : R} (ha : 0 ≤ a) {n : ℕ} (hn : n ≠ 0) : 1 ≤ a ^ n ↔ 1 ≤ a | begin
refine ⟨_, λ h, one_le_pow_of_one_le h n⟩,
rw [←not_lt, ←not_lt],
exact mt (λ h, pow_lt_one ha h hn),
end | lemma | one_le_pow_iff_of_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_of_one_le",
"pow_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_pow_iff_of_nonneg {a : R} (ha : 0 ≤ a) {n : ℕ} (hn : n ≠ 0) : 1 < a ^ n ↔ 1 < a | lt_iff_lt_of_le_iff_le (pow_le_one_iff_of_nonneg ha hn) | lemma | one_lt_pow_iff_of_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lt_iff_lt_of_le_iff_le",
"pow_le_one_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_one_iff_of_nonneg {a : R} (ha : 0 ≤ a) {n : ℕ} (hn : n ≠ 0) : a ^ n < 1 ↔ a < 1 | lt_iff_lt_of_le_iff_le (one_le_pow_iff_of_nonneg ha hn) | lemma | pow_lt_one_iff_of_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lt_iff_lt_of_le_iff_le",
"one_le_pow_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_le_one_iff {a : R} (ha : 0 ≤ a) : a^2 ≤ 1 ↔ a ≤ 1 | pow_le_one_iff_of_nonneg ha (nat.succ_ne_zero _) | lemma | sq_le_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_le_one_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_lt_one_iff {a : R} (ha : 0 ≤ a) : a^2 < 1 ↔ a < 1 | pow_lt_one_iff_of_nonneg ha (nat.succ_ne_zero _) | lemma | sq_lt_one_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_lt_one_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_sq_iff {a : R} (ha : 0 ≤ a) : 1 ≤ a^2 ↔ 1 ≤ a | one_le_pow_iff_of_nonneg ha (nat.succ_ne_zero _) | lemma | one_le_sq_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_le_pow_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_sq_iff {a : R} (ha : 0 ≤ a) : 1 < a^2 ↔ 1 < a | one_lt_pow_iff_of_nonneg ha (nat.succ_ne_zero _) | lemma | one_lt_sq_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_lt_pow_iff_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_left_inj {x y : R} {n : ℕ} (Hxpos : 0 ≤ x) (Hypos : 0 ≤ y) (Hnpos : 0 < n) :
x ^ n = y ^ n ↔ x = y | (@strict_mono_on_pow R _ _ Hnpos).eq_iff_eq Hxpos Hypos | theorem | pow_left_inj | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"strict_mono_on_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_pow_lt_pow {a b : R} (n : ℕ) (hb : 0 ≤ b) (h : a ^ n < b ^ n) : a < b | lt_of_not_ge $ λ hn, not_lt_of_ge (pow_le_pow_of_le_left hb hn _) h | lemma | lt_of_pow_lt_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_le_pow_of_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_pow_le_pow {a b : R} (n : ℕ) (hb : 0 ≤ b) (hn : 0 < n) (h : a ^ n ≤ b ^ n) : a ≤ b | le_of_not_lt $ λ h1, not_le_of_lt (pow_lt_pow_of_lt_left h1 hb hn) h | lemma | le_of_pow_le_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"not_le_of_lt",
"pow_lt_pow_of_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_sq {a b : R} (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ 2 = b ^ 2 ↔ a = b | pow_left_inj ha hb dec_trivial | lemma | sq_eq_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_self_lt_mul_self (hb : 0 ≤ b) : a * a < b * b → a < b | by { simp_rw ←sq, exact lt_of_pow_lt_pow _ hb } | lemma | lt_of_mul_self_lt_mul_self | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"lt_of_pow_lt_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_abs (a : R) (n : ℕ) : |a| ^ n = |a ^ n| | ((abs_hom.to_monoid_hom : R →* R).map_pow a n).symm | lemma | pow_abs | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_neg_one_pow (n : ℕ) : |(-1 : R) ^ n| = 1 | by rw [←pow_abs, abs_neg, abs_one, one_pow] | lemma | abs_neg_one_pow | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_neg",
"abs_one",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_pow_eq_one (a : R) {n : ℕ} (h : 0 < n) : | |a ^ n| = 1 ↔ |a| = 1 :=
by { convert pow_left_inj (abs_nonneg a) zero_le_one h,
exacts [(pow_abs _ _).symm, (one_pow _).symm] } | lemma | abs_pow_eq_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_nonneg",
"one_pow",
"pow_abs",
"pow_left_inj",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0_nonneg (a : R) (n : ℕ) : 0 ≤ a ^ bit0 n | by { rw pow_bit0, exact mul_self_nonneg _ } | theorem | pow_bit0_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_self_nonneg",
"pow_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_nonneg (a : R) : 0 ≤ a ^ 2 | pow_bit0_nonneg a 1 | theorem | sq_nonneg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_bit0_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0_pos {a : R} (h : a ≠ 0) (n : ℕ) : 0 < a ^ bit0 n | (pow_bit0_nonneg a n).lt_of_ne (pow_ne_zero _ h).symm | theorem | pow_bit0_pos | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_bit0_nonneg",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_pos_of_ne_zero (a : R) (h : a ≠ 0) : 0 < a ^ 2 | pow_bit0_pos h 1 | theorem | sq_pos_of_ne_zero | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_bit0_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0_pos_iff (a : R) {n : ℕ} (hn : n ≠ 0) : 0 < a ^ bit0 n ↔ a ≠ 0 | begin
refine ⟨λ h, _, λ h, pow_bit0_pos h n⟩,
rintro rfl,
rw zero_pow (nat.zero_lt_bit0 hn) at h,
exact lt_irrefl _ h,
end | theorem | pow_bit0_pos_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_bit0_pos",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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