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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|---|---|---|---|---|---|---|---|---|---|---|
unop_neg [has_neg α] (x : αᵐᵒᵖ) : unop (-x) = -unop x | rfl | lemma | mul_opposite.unop_neg | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_mul [has_mul α] (x y : α) : op (x * y) = op y * op x | rfl | lemma | mul_opposite.op_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_mul [has_mul α] (x y : αᵐᵒᵖ) : unop (x * y) = unop y * unop x | rfl | lemma | mul_opposite.unop_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_inv [has_inv α] (x : α) : op (x⁻¹) = (op x)⁻¹ | rfl | lemma | mul_opposite.op_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_inv [has_inv α] (x : αᵐᵒᵖ) : unop (x⁻¹) = (unop x)⁻¹ | rfl | lemma | mul_opposite.unop_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_sub [has_sub α] (x y : α) : op (x - y) = op x - op y | rfl | lemma | mul_opposite.op_sub | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_sub [has_sub α] (x y : αᵐᵒᵖ) : unop (x - y) = unop x - unop y | rfl | lemma | mul_opposite.unop_sub | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_smul {R : Type*} [has_smul R α] (c : R) (a : α) :
op (c • a) = c • op a | rfl | lemma | mul_opposite.op_smul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_smul {R : Type*} [has_smul R α] (c : R) (a : αᵐᵒᵖ) :
unop (c • a) = c • unop a | rfl | lemma | mul_opposite.unop_smul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_eq_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop = (0 : α) ↔ a = (0 : αᵐᵒᵖ) | unop_injective.eq_iff' rfl | lemma | mul_opposite.unop_eq_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_eq_zero_iff [has_zero α] (a : α) : op a = (0 : αᵐᵒᵖ) ↔ a = (0 : α) | op_injective.eq_iff' rfl | lemma | mul_opposite.op_eq_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_ne_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop ≠ (0 : α) ↔ a ≠ (0 : αᵐᵒᵖ) | not_congr $ unop_eq_zero_iff a | lemma | mul_opposite.unop_ne_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ne_zero_iff [has_zero α] (a : α) : op a ≠ (0 : αᵐᵒᵖ) ↔ a ≠ (0 : α) | not_congr $ op_eq_zero_iff a | lemma | mul_opposite.op_ne_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_eq_one_iff [has_one α] (a : αᵐᵒᵖ) : a.unop = 1 ↔ a = 1 | unop_injective.eq_iff' rfl | lemma | mul_opposite.unop_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_eq_one_iff [has_one α] (a : α) : op a = 1 ↔ a = 1 | op_injective.eq_iff' rfl | lemma | mul_opposite.op_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_one [has_one α] : unop 1 = (1 : α) | rfl | lemma | add_opposite.unop_one | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_eq_one_iff [has_one α] {a : α} : op a = 1 ↔ a = 1 | op_injective.eq_iff' op_one | lemma | add_opposite.op_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_eq_one_iff [has_one α] {a : αᵃᵒᵖ} : unop a = 1 ↔ a = 1 | unop_injective.eq_iff' unop_one | lemma | add_opposite.unop_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_mul [has_mul α] (a b : α) : op (a * b) = op a * op b | rfl | lemma | add_opposite.op_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_mul [has_mul α] (a b : αᵃᵒᵖ) : unop (a * b) = unop a * unop b | rfl | lemma | add_opposite.unop_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_inv [has_inv α] (a : α) : op a⁻¹ = (op a)⁻¹ | rfl | lemma | add_opposite.op_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_inv [has_inv α] (a : αᵃᵒᵖ) : unop a⁻¹ = (unop a)⁻¹ | rfl | lemma | add_opposite.unop_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_div [has_div α] (a b : α) : op (a / b) = op a / op b | rfl | lemma | add_opposite.op_div | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_div [has_div α] (a b : αᵃᵒᵖ) : unop (a / b) = unop a / unop b | rfl | lemma | add_opposite.unop_div | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square (a : α) : Prop | ∃ r, a = r * r | def | is_square | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | An element `a` of a type `α` with multiplication satisfies `is_square a` if `a = r * r`,
for some `r : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_square_mul_self (m : α) : is_square (m * m) | ⟨m, rfl⟩ | lemma | is_square_mul_self | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_op_iff (a : α) : is_square (op a) ↔ is_square a | ⟨λ ⟨c, hc⟩, ⟨unop c, by rw [← unop_mul, ← hc, unop_op]⟩, λ ⟨c, hc⟩, by simp [hc]⟩ | lemma | is_square_op_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_one [mul_one_class α] : is_square (1 : α) | ⟨1, (mul_one _).symm⟩ | lemma | is_square_one | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"mul_one",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square.map [mul_one_class α] [mul_one_class β] [monoid_hom_class F α β] {m : α} (f : F) :
is_square m → is_square (f m) | by { rintro ⟨m, rfl⟩, exact ⟨f m, by simp⟩ } | lemma | is_square.map | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"monoid_hom_class",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_iff_exists_sq (m : α) : is_square m ↔ ∃ c, m = c ^ 2 | by simp [is_square, pow_two] | lemma | is_square_iff_exists_sq | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square.pow (n : ℕ) : is_square a → is_square (a ^ n) | by { rintro ⟨a, rfl⟩, exact ⟨a ^ n, (commute.refl _).mul_pow _⟩ } | lemma | is_square.pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"commute.refl",
"is_square",
"mul_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.is_square_pow : even n → ∀ a : α, is_square (a ^ n) | by { rintro ⟨n, rfl⟩ a, exact ⟨a ^ n, pow_add _ _ _⟩ } | lemma | even.is_square_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_sq (a : α) : is_square (a ^ 2) | ⟨a, pow_two _⟩ | lemma | is_square_sq | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.neg_pow : even n → ∀ a : α, (-a) ^ n = a ^ n | by { rintro ⟨c, rfl⟩ a, simp_rw [←two_mul, pow_mul, neg_sq] } | lemma | even.neg_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"neg_sq",
"pow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.neg_one_pow (h : even n) : (-1 : α) ^ n = 1 | by rw [h.neg_pow, one_pow] | lemma | even.neg_one_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square.mul [comm_semigroup α] {a b : α} :
is_square a → is_square b → is_square (a * b) | by { rintro ⟨a, rfl⟩ ⟨b, rfl⟩, exact ⟨a * b, mul_mul_mul_comm _ _ _ _⟩ } | lemma | is_square.mul | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"comm_semigroup",
"is_square",
"mul_mul_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_zero [mul_zero_class α] : is_square (0 : α) | ⟨0, (mul_zero _).symm⟩ | lemma | is_square_zero | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"is_square",
"mul_zero",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_inv : is_square a⁻¹ ↔ is_square a | begin
refine ⟨λ h, _, λ h, _⟩,
{ rw [← is_square_op_iff, ← inv_inv a],
exact h.map (mul_equiv.inv' α) },
{ exact ((is_square_op_iff a).mpr h).map (mul_equiv.inv' α).symm }
end | lemma | is_square_inv | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"inv_inv",
"is_square",
"is_square_op_iff",
"mul_equiv.inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square.zpow (n : ℤ) : is_square a → is_square (a ^ n) | by { rintro ⟨a, rfl⟩, exact ⟨a ^ n, (commute.refl _).mul_zpow _⟩ } | lemma | is_square.zpow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"commute.refl",
"is_square",
"mul_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.neg_zpow : even n → ∀ a : α, (-a) ^ n = a ^ n | by { rintro ⟨c, rfl⟩ a, exact zpow_bit0_neg _ _ } | lemma | even.neg_zpow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"zpow_bit0_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.neg_one_zpow (h : even n) : (-1 : α) ^ n = 1 | by rw [h.neg_zpow, one_zpow] | lemma | even.neg_one_zpow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"one_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_abs [subtraction_monoid α] [linear_order α] {a : α} : even (|a|) ↔ even a | by cases abs_choice a; simp only [h, even_neg] | lemma | even_abs | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"abs_choice",
"subtraction_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square.div [division_comm_monoid α] {a b : α} (ha : is_square a) (hb : is_square b) :
is_square (a / b) | by { rw div_eq_mul_inv, exact ha.mul hb.inv } | lemma | is_square.div | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"div_eq_mul_inv",
"division_comm_monoid",
"is_square"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.is_square_zpow [group α] {n : ℤ} : even n → ∀ a : α, is_square (a ^ n) | by { rintro ⟨n, rfl⟩ a, exact ⟨a ^ n, zpow_add _ _ _⟩ } | lemma | even.is_square_zpow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"group",
"is_square",
"zpow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.tsub [canonically_linear_ordered_add_monoid α] [has_sub α] [has_ordered_sub α]
[contravariant_class α α (+) (≤)] {m n : α} (hm : even m) (hn : even n) : even (m - n) | begin
obtain ⟨a, rfl⟩ := hm,
obtain ⟨b, rfl⟩ := hn,
refine ⟨a - b, _⟩,
obtain h | h := le_total a b,
{ rw [tsub_eq_zero_of_le h, tsub_eq_zero_of_le (add_le_add h h), add_zero] },
{ exact (tsub_add_tsub_comm h h).symm }
end | lemma | even.tsub | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"canonically_linear_ordered_add_monoid",
"contravariant_class",
"has_ordered_sub",
"tsub_add_tsub_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_iff_exists_bit0 [has_add α] {a : α} : even a ↔ ∃ b, a = bit0 b | iff.rfl | lemma | even_iff_exists_bit0 | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_iff_exists_two_mul (m : α) : even m ↔ ∃ c, m = 2 * c | by simp [even_iff_exists_two_nsmul] | lemma | even_iff_exists_two_mul | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_iff_two_dvd {a : α} : even a ↔ 2 ∣ a | by simp [even, has_dvd.dvd, two_mul] | lemma | even_iff_two_dvd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.trans_dvd (hm : even m) (hn : m ∣ n) : even n | even_iff_two_dvd.2 $ hm.two_dvd.trans hn | theorem | even.trans_dvd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_dvd.dvd.even (hn : m ∣ n) (hm : even m) : even n | hm.trans_dvd hn | theorem | has_dvd.dvd.even | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_two_mul (α : Type*) [semiring α] :
set.range (λ x : α, 2 * x) = {a | even a} | by { ext x, simp [eq_comm, two_mul, even] } | lemma | range_two_mul | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"semiring",
"set.range",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_bit0 (a : α) : even (bit0 a) | ⟨a, rfl⟩ | lemma | even_bit0 | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_two : even (2 : α) | ⟨1, rfl⟩ | lemma | even_two | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.mul_left (hm : even m) (n) : even (n * m) | hm.map (add_monoid_hom.mul_left n) | lemma | even.mul_left | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"add_monoid_hom.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.mul_right (hm : even m) (n) : even (m * n) | hm.map (add_monoid_hom.mul_right n) | lemma | even.mul_right | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"add_monoid_hom.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_two_mul (m : α) : even (2 * m) | ⟨m, two_mul _⟩ | lemma | even_two_mul | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.pow_of_ne_zero (hm : even m) : ∀ {a : ℕ}, a ≠ 0 → even (m ^ a) | | 0 a0 := (a0 rfl).elim
| (a + 1) _ := by { rw pow_succ, exact hm.mul_right _ } | lemma | even.pow_of_ne_zero | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd (a : α) : Prop | ∃ k, a = 2*k + 1 | def | odd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [] | An element `a` of a semiring is odd if there exists `k` such `a = 2*k + 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
odd_iff_exists_bit1 {a : α} : odd a ↔ ∃ b, a = bit1 b | exists_congr $ λ b, by { rw two_mul, refl } | lemma | odd_iff_exists_bit1 | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_bit1 (a : α) : odd (bit1 a) | odd_iff_exists_bit1.2 ⟨a, rfl⟩ | lemma | odd_bit1 | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_two_mul_add_one (α : Type*) [semiring α] :
set.range (λ x : α, 2 * x + 1) = {a | odd a} | by { ext x, simp [odd, eq_comm] } | lemma | range_two_mul_add_one | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"semiring",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.add_odd : even m → odd n → odd (m + n) | by { rintro ⟨m, rfl⟩ ⟨n, rfl⟩, exact ⟨m + n, by rw [mul_add, ← two_mul, add_assoc]⟩ } | lemma | even.add_odd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.add_even (hm : odd m) (hn : even n) : odd (m + n) | by { rw add_comm, exact hn.add_odd hm } | lemma | odd.add_even | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.add_odd : odd m → odd n → even (m + n) | begin
rintro ⟨m, rfl⟩ ⟨n, rfl⟩,
refine ⟨n + m + 1, _⟩,
rw [two_mul, two_mul],
ac_refl
end | lemma | odd.add_odd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_one : odd (1 : α) | ⟨0, (zero_add _).symm.trans (congr_arg (+ (1 : α)) (mul_zero _).symm)⟩ | lemma | odd_one | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"mul_zero",
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_two_mul_add_one (m : α) : odd (2 * m + 1) | ⟨m, rfl⟩ | lemma | odd_two_mul_add_one | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.map [ring_hom_class F α β] (f : F) : odd m → odd (f m) | by { rintro ⟨m, rfl⟩, exact ⟨f m, by simp [two_mul]⟩ } | lemma | odd.map | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"ring_hom_class",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.mul : odd m → odd n → odd (m * n) | begin
rintro ⟨m, rfl⟩ ⟨n, rfl⟩,
refine ⟨2 * m * n + n + m, _⟩,
rw [mul_add, add_mul, mul_one, ← add_assoc, one_mul, mul_assoc, ← mul_add, ← mul_add, ← mul_assoc,
← nat.cast_two, ← nat.cast_comm],
end | lemma | odd.mul | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"mul_assoc",
"mul_one",
"nat.cast_comm",
"nat.cast_two",
"odd",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow (hm : odd m) : ∀ {a : ℕ}, odd (m ^ a) | | 0 := by { rw pow_zero, exact odd_one }
| (a + 1) := by { rw pow_succ, exact hm.mul odd.pow } | lemma | odd.pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"odd_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.neg_pow : odd n → ∀ a : α, (-a) ^ n = - a ^ n | by { rintro ⟨c, rfl⟩ a, simp_rw [pow_add, pow_mul, neg_sq, pow_one, mul_neg] } | lemma | odd.neg_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"mul_neg",
"neg_sq",
"odd",
"pow_add",
"pow_mul",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.neg_one_pow (h : odd n) : (-1 : α) ^ n = -1 | by rw [h.neg_pow, one_pow] | lemma | odd.neg_one_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pos [nontrivial α] {n : α} (hn : odd n) : 0 < n | begin
obtain ⟨k, rfl⟩ := hn,
rw [pos_iff_ne_zero, ne.def, add_eq_zero_iff, not_and'],
exact λ h, (one_ne_zero h).elim
end | lemma | odd.pos | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"nontrivial",
"not_and'",
"odd",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even_neg_two : even (- 2 : α) | by simp only [even_neg, even_two] | lemma | even_neg_two | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"even_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.neg (hp : odd a) : odd (-a) | begin
obtain ⟨k, hk⟩ := hp,
use -(k + 1),
rw [mul_neg, mul_add, neg_add, add_assoc, two_mul (1 : α), neg_add,
neg_add_cancel_right, ←neg_add, hk],
end | lemma | odd.neg | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"mul_neg",
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_neg : odd (-a) ↔ odd a | ⟨λ h, neg_neg a ▸ h.neg, odd.neg⟩ | lemma | odd_neg | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_neg_one : odd (- 1 : α) | by simp | lemma | odd_neg_one | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.sub_even (ha : odd a) (hb : even b) : odd (a - b) | by { rw sub_eq_add_neg, exact ha.add_even hb.neg } | lemma | odd.sub_even | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.sub_odd (ha : even a) (hb : odd b) : odd (a - b) | by { rw sub_eq_add_neg, exact ha.add_odd hb.neg } | lemma | even.sub_odd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.sub_odd (ha : odd a) (hb : odd b) : even (a - b) | by { rw sub_eq_add_neg, exact ha.add_odd hb.neg } | lemma | odd.sub_odd | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd_abs [linear_order α] : odd (abs a) ↔ odd a | by cases abs_choice a with h h; simp only [h, odd_neg] | lemma | odd_abs | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"abs_choice",
"odd",
"odd_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.pow_nonneg (hn : even n) (a : R) : 0 ≤ a ^ n | by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_nonneg a k | lemma | even.pow_nonneg | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"pow_bit0_nonneg",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.pow_pos (hn : even n) (ha : a ≠ 0) : 0 < a ^ n | by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_pos ha k | lemma | even.pow_pos | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"pow_bit0_pos",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_nonpos (hn : odd n) (ha : a ≤ 0) : a ^ n ≤ 0 | by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_nonpos_iff.mpr ha | lemma | odd.pow_nonpos | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_neg (hn : odd n) (ha : a < 0) : a ^ n < 0 | by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_neg_iff.mpr ha | lemma | odd.pow_neg | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_nonneg_iff (hn : odd n) : 0 ≤ a ^ n ↔ 0 ≤ a | ⟨λ h, le_of_not_lt (λ ha, h.not_lt $ hn.pow_neg ha), λ ha, pow_nonneg ha n⟩ | lemma | odd.pow_nonneg_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_nonpos_iff (hn : odd n) : a ^ n ≤ 0 ↔ a ≤ 0 | ⟨λ h, le_of_not_lt (λ ha, h.not_lt $ pow_pos ha _), hn.pow_nonpos⟩ | lemma | odd.pow_nonpos_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_pos_iff (hn : odd n) : 0 < a ^ n ↔ 0 < a | ⟨λ h, lt_of_not_le (λ ha, h.not_le $ hn.pow_nonpos ha), λ ha, pow_pos ha n⟩ | lemma | odd.pow_pos_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"lt_of_not_le",
"odd",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.pow_neg_iff (hn : odd n) : a ^ n < 0 ↔ a < 0 | ⟨λ h, lt_of_not_le (λ ha, h.not_le $ pow_nonneg ha _), hn.pow_neg⟩ | lemma | odd.pow_neg_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"lt_of_not_le",
"odd",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.pow_pos_iff (hn : even n) (h₀ : 0 < n) : 0 < a ^ n ↔ a ≠ 0 | ⟨λ h ha, by { rw [ha, zero_pow h₀] at h, exact lt_irrefl 0 h }, hn.pow_pos⟩ | lemma | even.pow_pos_iff | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
even.pow_abs {p : ℕ} (hp : even p) (a : R) : |a| ^ p = a ^ p | begin
rw [←abs_pow, abs_eq_self],
exact hp.pow_nonneg _
end | lemma | even.pow_abs | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"abs_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0_abs (a : R) (p : ℕ) : |a| ^ bit0 p = a ^ bit0 p | (even_bit0 _).pow_abs _ | lemma | pow_bit0_abs | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"even_bit0",
"pow_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.strict_mono_pow (hn : odd n) : strict_mono (λ a : R, a ^ n) | by cases hn with k hk; simpa only [hk, two_mul] using strict_mono_pow_bit1 _ | lemma | odd.strict_mono_pow | algebra | src/algebra/parity.lean | [
"algebra.group_power.lemmas"
] | [
"odd",
"strict_mono",
"strict_mono_pow_bit1",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup_pempty : semigroup pempty.{u+1} | { mul := λ x y, by cases x,
mul_assoc := λ x y z, by cases x } | instance | semigroup_pempty | algebra | src/algebra/pempty_instances.lean | [
"algebra.group.defs",
"tactic.to_additive"
] | [
"mul_assoc",
"semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic [has_add α] (f : α → β) (c : α) : Prop | ∀ x : α, f (x + c) = f x | def | function.periodic | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [] | A function `f` is said to be `periodic` with period `c` if for all `x`, `f (x + c) = f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.funext [has_add α] (h : periodic f c) :
(λ x, f (x + c)) = f | funext h | lemma | function.periodic.funext | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.comp [has_add α] (h : periodic f c) (g : β → γ) :
periodic (g ∘ f) c | by simp * at * | lemma | function.periodic.comp | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.comp_add_hom [has_add α] [has_add γ]
(h : periodic f c) (g : add_hom γ α) (g_inv : α → γ) (hg : right_inverse g_inv g) :
periodic (f ∘ g) (g_inv c) | λ x, by simp only [hg c, h (g x), add_hom.map_add, comp_app] | lemma | function.periodic.comp_add_hom | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [
"add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.mul [has_add α] [has_mul β] (hf : periodic f c) (hg : periodic g c) :
periodic (f * g) c | by simp * at * | lemma | function.periodic.mul | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.div [has_add α] [has_div β] (hf : periodic f c) (hg : periodic g c) :
periodic (f / g) c | by simp * at * | lemma | function.periodic.div | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.list.periodic_prod [has_add α] [monoid β]
(l : list (α → β)) (hl : ∀ f ∈ l, periodic f c) :
periodic l.prod c | begin
induction l with g l ih hl,
{ simp, },
{ rw [list.forall_mem_cons] at hl,
simpa only [list.prod_cons] using hl.1.mul (ih hl.2) }
end | lemma | list.periodic_prod | algebra | src/algebra/periodic.lean | [
"algebra.big_operators.basic",
"algebra.field.opposite",
"algebra.module.basic",
"algebra.order.archimedean",
"data.int.parity",
"group_theory.coset",
"group_theory.subgroup.zpowers",
"group_theory.submonoid.membership"
] | [
"ih",
"list.forall_mem_cons",
"list.prod_cons",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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