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
value |
|---|---|---|---|---|---|---|---|---|---|---|
_root_.multiset.periodic_prod [has_add α] [comm_monoid β]
(s : multiset (α → β)) (hs : ∀ f ∈ s, periodic f c) :
periodic s.prod c | s.prod_to_list ▸ s.to_list.periodic_prod $ λ f hf, hs f $ multiset.mem_to_list.mp hf | lemma | multiset.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"
] | [
"comm_monoid",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.periodic_prod [has_add α] [comm_monoid β]
{ι : Type*} {f : ι → α → β} (s : finset ι) (hs : ∀ i ∈ s, periodic (f i) c) :
periodic (∏ i in s, f i) c | s.prod_to_list f ▸ (s.to_list.map f).periodic_prod (by simpa [-periodic]) | lemma | finset.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"
] | [
"comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.smul [has_add α] [has_smul γ β] (h : periodic f c) (a : γ) :
periodic (a • f) c | by simp * at * | lemma | function.periodic.smul | 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"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_smul [add_monoid α] [group γ] [distrib_mul_action γ α]
(h : periodic f c) (a : γ) :
periodic (λ x, f (a • x)) (a⁻¹ • c) | λ x, by simpa only [smul_add, smul_inv_smul] using h (a • x) | lemma | function.periodic.const_smul | 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_monoid",
"distrib_mul_action",
"group",
"smul_add",
"smul_inv_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_smul₀ [add_comm_monoid α] [division_semiring γ] [module γ α]
(h : periodic f c) (a : γ) :
periodic (λ x, f (a • x)) (a⁻¹ • c) | begin
intro x,
by_cases ha : a = 0, { simp only [ha, zero_smul] },
simpa only [smul_add, smul_inv_smul₀ ha] using h (a • x),
end | lemma | function.periodic.const_smul₀ | 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_comm_monoid",
"division_semiring",
"module",
"smul_add",
"smul_inv_smul₀",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_mul [division_semiring α] (h : periodic f c) (a : α) :
periodic (λ x, f (a * x)) (a⁻¹ * c) | h.const_smul₀ a | lemma | function.periodic.const_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"
] | [
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_inv_smul [add_monoid α] [group γ] [distrib_mul_action γ α]
(h : periodic f c) (a : γ) :
periodic (λ x, f (a⁻¹ • x)) (a • c) | by simpa only [inv_inv] using h.const_smul a⁻¹ | lemma | function.periodic.const_inv_smul | 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_monoid",
"distrib_mul_action",
"group",
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_inv_smul₀ [add_comm_monoid α] [division_semiring γ] [module γ α]
(h : periodic f c) (a : γ) :
periodic (λ x, f (a⁻¹ • x)) (a • c) | by simpa only [inv_inv] using h.const_smul₀ a⁻¹ | lemma | function.periodic.const_inv_smul₀ | 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_comm_monoid",
"division_semiring",
"inv_inv",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_inv_mul [division_semiring α] (h : periodic f c) (a : α) :
periodic (λ x, f (a⁻¹ * x)) (a * c) | h.const_inv_smul₀ a | lemma | function.periodic.const_inv_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"
] | [
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.mul_const [division_semiring α] (h : periodic f c) (a : α) :
periodic (λ x, f (x * a)) (c * a⁻¹) | h.const_smul₀ $ mul_opposite.op a | lemma | function.periodic.mul_const | 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"
] | [
"division_semiring",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.mul_const' [division_semiring α]
(h : periodic f c) (a : α) :
periodic (λ x, f (x * a)) (c / a) | by simpa only [div_eq_mul_inv] using h.mul_const a | lemma | function.periodic.mul_const' | 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"
] | [
"div_eq_mul_inv",
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.mul_const_inv [division_semiring α] (h : periodic f c) (a : α) :
periodic (λ x, f (x * a⁻¹)) (c * a) | h.const_inv_smul₀ $ mul_opposite.op a | lemma | function.periodic.mul_const_inv | 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"
] | [
"division_semiring",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.div_const [division_semiring α] (h : periodic f c) (a : α) :
periodic (λ x, f (x / a)) (c * a) | by simpa only [div_eq_mul_inv] using h.mul_const_inv a | lemma | function.periodic.div_const | 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"
] | [
"div_eq_mul_inv",
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.add_period [add_semigroup α] (h1 : periodic f c₁) (h2 : periodic f c₂) :
periodic f (c₁ + c₂) | by simp [*, ← add_assoc] at * | lemma | function.periodic.add_period | 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_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_eq [add_group α] (h : periodic f c) (x : α) :
f (x - c) = f x | by simpa only [sub_add_cancel] using (h (x - c)).symm | lemma | function.periodic.sub_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_eq' [add_comm_group α] (h : periodic f c) :
f (c - x) = f (-x) | by simpa only [sub_eq_neg_add] using h (-x) | lemma | function.periodic.sub_eq' | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.neg [add_group α] (h : periodic f c) :
periodic f (-c) | by simpa only [sub_eq_add_neg, periodic] using h.sub_eq | lemma | function.periodic.neg | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_period [add_group α] (h1 : periodic f c₁) (h2 : periodic f c₂) :
periodic f (c₁ - c₂) | by simpa only [sub_eq_add_neg] using h1.add_period h2.neg | lemma | function.periodic.sub_period | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_add [add_semigroup α] (h : periodic f c) (a : α) :
periodic (λ x, f (a + x)) c | λ x, by simpa [add_assoc] using h (a + x) | lemma | function.periodic.const_add | 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_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.add_const [add_comm_semigroup α] (h : periodic f c) (a : α) :
periodic (λ x, f (x + a)) c | by simpa only [add_comm] using h.const_add a | lemma | function.periodic.add_const | 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_comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.const_sub [add_comm_group α] (h : periodic f c) (a : α) :
periodic (λ x, f (a - x)) c | λ x, by simp only [← sub_sub, h.sub_eq] | lemma | function.periodic.const_sub | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_const [add_comm_group α] (h : periodic f c) (a : α) :
periodic (λ x, f (x - a)) c | by simpa only [sub_eq_add_neg] using h.add_const (-a) | lemma | function.periodic.sub_const | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nsmul [add_monoid α] (h : periodic f c) (n : ℕ) :
periodic f (n • c) | by induction n; simp [nat.succ_eq_add_one, add_nsmul, ← add_assoc, zero_nsmul, *] at * | lemma | function.periodic.nsmul | 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_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nat_mul [semiring α] (h : periodic f c) (n : ℕ) :
periodic f (n * c) | by simpa only [nsmul_eq_mul] using h.nsmul n | lemma | function.periodic.nat_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"
] | [
"nsmul_eq_mul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.neg_nsmul [add_group α] (h : periodic f c) (n : ℕ) :
periodic f (-(n • c)) | (h.nsmul n).neg | lemma | function.periodic.neg_nsmul | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.neg_nat_mul [ring α] (h : periodic f c) (n : ℕ) :
periodic f (-(n * c)) | (h.nat_mul n).neg | lemma | function.periodic.neg_nat_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"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_nsmul_eq [add_group α] (h : periodic f c) (n : ℕ) :
f (x - n • c) = f x | by simpa only [sub_eq_add_neg] using h.neg_nsmul n x | lemma | function.periodic.sub_nsmul_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_nat_mul_eq [ring α] (h : periodic f c) (n : ℕ) :
f (x - n * c) = f x | by simpa only [nsmul_eq_mul] using h.sub_nsmul_eq n | lemma | function.periodic.sub_nat_mul_eq | 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"
] | [
"nsmul_eq_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nsmul_sub_eq [add_comm_group α] (h : periodic f c) (n : ℕ) :
f (n • c - x) = f (-x) | (h.nsmul n).sub_eq' | lemma | function.periodic.nsmul_sub_eq | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nat_mul_sub_eq [ring α] (h : periodic f c) (n : ℕ) :
f (n * c - x) = f (-x) | by simpa only [sub_eq_neg_add] using h.nat_mul n (-x) | lemma | function.periodic.nat_mul_sub_eq | 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"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.zsmul [add_group α] (h : periodic f c) (n : ℤ) :
periodic f (n • c) | begin
cases n,
{ simpa only [int.of_nat_eq_coe, coe_nat_zsmul] using h.nsmul n },
{ simpa only [zsmul_neg_succ_of_nat] using (h.nsmul n.succ).neg },
end | lemma | function.periodic.zsmul | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.int_mul [ring α] (h : periodic f c) (n : ℤ) :
periodic f (n * c) | by simpa only [zsmul_eq_mul] using h.zsmul n | lemma | function.periodic.int_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"
] | [
"ring",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_zsmul_eq [add_group α] (h : periodic f c) (n : ℤ) :
f (x - n • c) = f x | (h.zsmul n).sub_eq x | lemma | function.periodic.sub_zsmul_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_int_mul_eq [ring α] (h : periodic f c) (n : ℤ) :
f (x - n * c) = f x | (h.int_mul n).sub_eq x | lemma | function.periodic.sub_int_mul_eq | 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"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.zsmul_sub_eq [add_comm_group α] (h : periodic f c) (n : ℤ) :
f (n • c - x) = f (-x) | (h.zsmul _).sub_eq' | lemma | function.periodic.zsmul_sub_eq | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.int_mul_sub_eq [ring α] (h : periodic f c) (n : ℤ) :
f (n * c - x) = f (-x) | (h.int_mul _).sub_eq' | lemma | function.periodic.int_mul_sub_eq | 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"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.eq [add_zero_class α] (h : periodic f c) :
f c = f 0 | by simpa only [zero_add] using h 0 | lemma | function.periodic.eq | 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_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.neg_eq [add_group α] (h : periodic f c) :
f (-c) = f 0 | h.neg.eq | lemma | function.periodic.neg_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nsmul_eq [add_monoid α] (h : periodic f c) (n : ℕ) :
f (n • c) = f 0 | (h.nsmul n).eq | lemma | function.periodic.nsmul_eq | 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_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.nat_mul_eq [semiring α] (h : periodic f c) (n : ℕ) :
f (n * c) = f 0 | (h.nat_mul n).eq | lemma | function.periodic.nat_mul_eq | 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"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.zsmul_eq [add_group α] (h : periodic f c) (n : ℤ) :
f (n • c) = f 0 | (h.zsmul n).eq | lemma | function.periodic.zsmul_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.int_mul_eq [ring α] (h : periodic f c) (n : ℤ) :
f (n * c) = f 0 | (h.int_mul n).eq | lemma | function.periodic.int_mul_eq | 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"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.exists_mem_Ico₀ [linear_ordered_add_comm_group α] [archimedean α]
(h : periodic f c) (hc : 0 < c) (x) :
∃ y ∈ set.Ico 0 c, f x = f y | let ⟨n, H, _⟩ := exists_unique_zsmul_near_of_pos' hc x in
⟨x - n • c, H, (h.sub_zsmul_eq n).symm⟩ | lemma | function.periodic.exists_mem_Ico₀ | 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"
] | [
"archimedean",
"exists_unique_zsmul_near_of_pos'",
"linear_ordered_add_comm_group",
"set.Ico"
] | If a function `f` is `periodic` with positive period `c`, then for all `x` there exists some
`y ∈ Ico 0 c` such that `f x = f y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.exists_mem_Ico [linear_ordered_add_comm_group α] [archimedean α]
(h : periodic f c) (hc : 0 < c) (x a) :
∃ y ∈ set.Ico a (a + c), f x = f y | let ⟨n, H, _⟩ := exists_unique_add_zsmul_mem_Ico hc x a in
⟨x + n • c, H, (h.zsmul n x).symm⟩ | lemma | function.periodic.exists_mem_Ico | 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"
] | [
"archimedean",
"exists_unique_add_zsmul_mem_Ico",
"linear_ordered_add_comm_group",
"set.Ico"
] | If a function `f` is `periodic` with positive period `c`, then for all `x` there exists some
`y ∈ Ico a (a + c)` such that `f x = f y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.exists_mem_Ioc [linear_ordered_add_comm_group α] [archimedean α]
(h : periodic f c) (hc : 0 < c) (x a) :
∃ y ∈ set.Ioc a (a + c), f x = f y | let ⟨n, H, _⟩ := exists_unique_add_zsmul_mem_Ioc hc x a in
⟨x + n • c, H, (h.zsmul n x).symm⟩ | lemma | function.periodic.exists_mem_Ioc | 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"
] | [
"archimedean",
"exists_unique_add_zsmul_mem_Ioc",
"linear_ordered_add_comm_group",
"set.Ioc"
] | If a function `f` is `periodic` with positive period `c`, then for all `x` there exists some
`y ∈ Ioc a (a + c)` such that `f x = f y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.image_Ioc [linear_ordered_add_comm_group α] [archimedean α]
(h : periodic f c) (hc : 0 < c) (a : α) :
f '' set.Ioc a (a + c) = set.range f | (set.image_subset_range _ _).antisymm $ set.range_subset_iff.2 $ λ x,
let ⟨y, hy, hyx⟩ := h.exists_mem_Ioc hc x a in ⟨y, hy, hyx.symm⟩ | lemma | function.periodic.image_Ioc | 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"
] | [
"archimedean",
"linear_ordered_add_comm_group",
"set.Ioc",
"set.image_subset_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic_with_period_zero [add_zero_class α]
(f : α → β) :
periodic f 0 | λ x, by rw add_zero | lemma | function.periodic_with_period_zero | 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_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.map_vadd_zmultiples [add_comm_group α] (hf : periodic f c)
(a : add_subgroup.zmultiples c) (x : α) :
f (a +ᵥ x) = f x | by { rcases a with ⟨_, m, rfl⟩, simp [add_subgroup.vadd_def, add_comm _ x, hf.zsmul m x] } | lemma | function.periodic.map_vadd_zmultiples | 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_comm_group",
"add_subgroup.zmultiples"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.map_vadd_multiples [add_comm_monoid α] (hf : periodic f c)
(a : add_submonoid.multiples c) (x : α) :
f (a +ᵥ x) = f x | by { rcases a with ⟨_, m, rfl⟩, simp [add_submonoid.vadd_def, add_comm _ x, hf.nsmul m x] } | lemma | function.periodic.map_vadd_multiples | 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_comm_monoid",
"add_submonoid.multiples"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.lift [add_group α] (h : periodic f c) (x : α ⧸ add_subgroup.zmultiples c) : β | quotient.lift_on' x f $
λ a b h', (begin
rw quotient_add_group.left_rel_apply at h',
obtain ⟨k, hk⟩ := h',
exact (h.zsmul k _).symm.trans (congr_arg f (add_eq_of_eq_neg_add hk)),
end) | def | function.periodic.lift | 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_group",
"add_subgroup.zmultiples",
"quotient.lift_on'"
] | Lift a periodic function to a function from the quotient group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.lift_coe [add_group α] (h : periodic f c) (a : α) :
h.lift (a : α ⧸ add_subgroup.zmultiples c) = f a | rfl | lemma | function.periodic.lift_coe | 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_group",
"add_subgroup.zmultiples"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic [has_add α] [has_neg β] (f : α → β) (c : α) : Prop | ∀ x : α, f (x + c) = -f x | def | function.antiperiodic | 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 `antiperiodic` with antiperiod `c` if for all `x`,
`f (x + c) = -f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antiperiodic.funext [has_add α] [has_neg β] (h : antiperiodic f c) :
(λ x, f (x + c)) = -f | funext h | lemma | function.antiperiodic.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 | |
antiperiodic.funext' [has_add α] [has_involutive_neg β] (h : antiperiodic f c) :
(λ x, -f (x + c)) = f | neg_eq_iff_eq_neg.mpr h.funext | lemma | function.antiperiodic.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"
] | [
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.periodic [semiring α] [has_involutive_neg β] (h : antiperiodic f c) :
periodic f (2 * c) | by simp [two_mul, ← add_assoc, h _] | lemma | function.antiperiodic.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"
] | [
"has_involutive_neg",
"semiring",
"two_mul"
] | If a function is `antiperiodic` with antiperiod `c`, then it is also `periodic` with period
`2 * c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antiperiodic.eq [add_zero_class α] [has_neg β] (h : antiperiodic f c) :
f c = -f 0 | by simpa only [zero_add] using h 0 | lemma | function.antiperiodic.eq | 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_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.nat_even_mul_periodic [semiring α] [has_involutive_neg β]
(h : antiperiodic f c) (n : ℕ) :
periodic f (n * (2 * c)) | h.periodic.nat_mul n | lemma | function.antiperiodic.nat_even_mul_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"
] | [
"has_involutive_neg",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.nat_odd_mul_antiperiodic [semiring α] [has_involutive_neg β]
(h : antiperiodic f c) (n : ℕ) :
antiperiodic f (n * (2 * c) + c) | λ x, by rw [← add_assoc, h, h.periodic.nat_mul] | lemma | function.antiperiodic.nat_odd_mul_antiperiodic | 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"
] | [
"has_involutive_neg",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.int_even_mul_periodic [ring α] [has_involutive_neg β]
(h : antiperiodic f c) (n : ℤ) :
periodic f (n * (2 * c)) | h.periodic.int_mul n | lemma | function.antiperiodic.int_even_mul_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"
] | [
"has_involutive_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.int_odd_mul_antiperiodic [ring α] [has_involutive_neg β]
(h : antiperiodic f c) (n : ℤ) :
antiperiodic f (n * (2 * c) + c) | λ x, by rw [← add_assoc, h, h.periodic.int_mul] | lemma | function.antiperiodic.int_odd_mul_antiperiodic | 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"
] | [
"has_involutive_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.sub_eq [add_group α] [has_involutive_neg β]
(h : antiperiodic f c) (x : α) :
f (x - c) = -f x | by rw [← neg_eq_iff_eq_neg, ← h (x - c), sub_add_cancel] | lemma | function.antiperiodic.sub_eq | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.sub_eq' [add_comm_group α] [has_neg β] (h : antiperiodic f c) :
f (c - x) = -f (-x) | by simpa only [sub_eq_neg_add] using h (-x) | lemma | function.antiperiodic.sub_eq' | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.neg [add_group α] [has_involutive_neg β]
(h : antiperiodic f c) :
antiperiodic f (-c) | by simpa only [sub_eq_add_neg, antiperiodic] using h.sub_eq | lemma | function.antiperiodic.neg | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.neg_eq [add_group α] [has_involutive_neg β]
(h : antiperiodic f c) :
f (-c) = -f 0 | by simpa only [zero_add] using h.neg 0 | lemma | function.antiperiodic.neg_eq | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.nat_mul_eq_of_eq_zero [ring α] [neg_zero_class β]
(h : antiperiodic f c) (hi : f 0 = 0) : ∀ n : ℕ, f (n * c) = 0 | | 0 := by rwa [nat.cast_zero, zero_mul]
| (n + 1) := by simp [add_mul, antiperiodic.nat_mul_eq_of_eq_zero n, h _] | lemma | function.antiperiodic.nat_mul_eq_of_eq_zero | 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"
] | [
"nat.cast_zero",
"neg_zero_class",
"ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.int_mul_eq_of_eq_zero [ring α] [subtraction_monoid β]
(h : antiperiodic f c) (hi : f 0 = 0) : ∀ n : ℤ, f (n * c) = 0 | | (n : ℕ) := by rwa [int.cast_coe_nat, h.nat_mul_eq_of_eq_zero]
| -[1+n] := by rw [int.cast_neg_succ_of_nat, neg_mul, ← mul_neg, h.neg.nat_mul_eq_of_eq_zero hi] | lemma | function.antiperiodic.int_mul_eq_of_eq_zero | 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"
] | [
"int.cast_coe_nat",
"int.cast_neg_succ_of_nat",
"mul_neg",
"neg_mul",
"ring",
"subtraction_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_add [add_semigroup α] [has_neg β] (h : antiperiodic f c) (a : α) :
antiperiodic (λ x, f (a + x)) c | λ x, by simpa [add_assoc] using h (a + x) | lemma | function.antiperiodic.const_add | 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_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.add_const [add_comm_semigroup α] [has_neg β] (h : antiperiodic f c) (a : α) :
antiperiodic (λ x, f (x + a)) c | λ x, by simpa only [add_right_comm] using h (x + a) | lemma | function.antiperiodic.add_const | 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_comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_sub [add_comm_group α] [has_involutive_neg β] (h : antiperiodic f c)
(a : α) : antiperiodic (λ x, f (a - x)) c | λ x, by simp only [← sub_sub, h.sub_eq] | lemma | function.antiperiodic.const_sub | 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_comm_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.sub_const [add_comm_group α] [has_neg β] (h : antiperiodic f c) (a : α) :
antiperiodic (λ x, f (x - a)) c | by simpa only [sub_eq_add_neg] using h.add_const (-a) | lemma | function.antiperiodic.sub_const | 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_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.smul [has_add α] [monoid γ] [add_group β] [distrib_mul_action γ β]
(h : antiperiodic f c) (a : γ) :
antiperiodic (a • f) c | by simp * at * | lemma | function.antiperiodic.smul | 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_group",
"distrib_mul_action",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_smul [add_monoid α] [has_neg β] [group γ] [distrib_mul_action γ α]
(h : antiperiodic f c) (a : γ) :
antiperiodic (λ x, f (a • x)) (a⁻¹ • c) | λ x, by simpa only [smul_add, smul_inv_smul] using h (a • x) | lemma | function.antiperiodic.const_smul | 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_monoid",
"distrib_mul_action",
"group",
"smul_add",
"smul_inv_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_smul₀ [add_comm_monoid α] [has_neg β] [division_semiring γ] [module γ α]
(h : antiperiodic f c) {a : γ} (ha : a ≠ 0) :
antiperiodic (λ x, f (a • x)) (a⁻¹ • c) | λ x, by simpa only [smul_add, smul_inv_smul₀ ha] using h (a • x) | lemma | function.antiperiodic.const_smul₀ | 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_comm_monoid",
"division_semiring",
"module",
"smul_add",
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_mul [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (a * x)) (a⁻¹ * c) | h.const_smul₀ ha | lemma | function.antiperiodic.const_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"
] | [
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_inv_smul [add_monoid α] [has_neg β] [group γ] [distrib_mul_action γ α]
(h : antiperiodic f c) (a : γ) :
antiperiodic (λ x, f (a⁻¹ • x)) (a • c) | by simpa only [inv_inv] using h.const_smul a⁻¹ | lemma | function.antiperiodic.const_inv_smul | 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_monoid",
"distrib_mul_action",
"group",
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_inv_smul₀
[add_comm_monoid α] [has_neg β] [division_semiring γ] [module γ α]
(h : antiperiodic f c) {a : γ} (ha : a ≠ 0) :
antiperiodic (λ x, f (a⁻¹ • x)) (a • c) | by simpa only [inv_inv] using h.const_smul₀ (inv_ne_zero ha) | lemma | function.antiperiodic.const_inv_smul₀ | 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_comm_monoid",
"division_semiring",
"inv_inv",
"inv_ne_zero",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.const_inv_mul [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (a⁻¹ * x)) (a * c) | h.const_inv_smul₀ ha | lemma | function.antiperiodic.const_inv_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"
] | [
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.mul_const [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (x * a)) (c * a⁻¹) | h.const_smul₀ $ (mul_opposite.op_ne_zero_iff a).mpr ha | lemma | function.antiperiodic.mul_const | 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"
] | [
"division_semiring",
"mul_opposite.op_ne_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.mul_const' [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (x * a)) (c / a) | by simpa only [div_eq_mul_inv] using h.mul_const ha | lemma | function.antiperiodic.mul_const' | 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"
] | [
"div_eq_mul_inv",
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.mul_const_inv [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (x * a⁻¹)) (c * a) | h.const_inv_smul₀ $ (mul_opposite.op_ne_zero_iff a).mpr ha | lemma | function.antiperiodic.mul_const_inv | 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"
] | [
"division_semiring",
"mul_opposite.op_ne_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.div_inv [division_semiring α] [has_neg β]
(h : antiperiodic f c) {a : α} (ha : a ≠ 0) :
antiperiodic (λ x, f (x / a)) (c * a) | by simpa only [div_eq_mul_inv] using h.mul_const_inv ha | lemma | function.antiperiodic.div_inv | 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"
] | [
"div_eq_mul_inv",
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.add [add_group α] [has_involutive_neg β]
(h1 : antiperiodic f c₁) (h2 : antiperiodic f c₂) :
periodic f (c₁ + c₂) | by simp [*, ← add_assoc] at * | lemma | function.antiperiodic.add | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.sub [add_group α] [has_involutive_neg β]
(h1 : antiperiodic f c₁) (h2 : antiperiodic f c₂) :
periodic f (c₁ - c₂) | by simpa only [sub_eq_add_neg] using h1.add h2.neg | lemma | function.antiperiodic.sub | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.add_antiperiod [add_group α] [has_neg β]
(h1 : periodic f c₁) (h2 : antiperiodic f c₂) :
antiperiodic f (c₁ + c₂) | by simp [*, ← add_assoc] at * | lemma | function.periodic.add_antiperiod | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_antiperiod [add_group α] [has_involutive_neg β]
(h1 : periodic f c₁) (h2 : antiperiodic f c₂) :
antiperiodic f (c₁ - c₂) | by simpa only [sub_eq_add_neg] using h1.add_antiperiod h2.neg | lemma | function.periodic.sub_antiperiod | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.add_antiperiod_eq [add_group α] [has_neg β]
(h1 : periodic f c₁) (h2 : antiperiodic f c₂) :
f (c₁ + c₂) = -f 0 | (h1.add_antiperiod h2).eq | lemma | function.periodic.add_antiperiod_eq | 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_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic.sub_antiperiod_eq [add_group α] [has_involutive_neg β]
(h1 : periodic f c₁) (h2 : antiperiodic f c₂) :
f (c₁ - c₂) = -f 0 | (h1.sub_antiperiod h2).eq | lemma | function.periodic.sub_antiperiod_eq | 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_group",
"has_involutive_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.mul [has_add α] [has_mul β] [has_distrib_neg β]
(hf : antiperiodic f c) (hg : antiperiodic g c) :
periodic (f * g) c | by simp * at * | lemma | function.antiperiodic.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"
] | [
"has_distrib_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antiperiodic.div [has_add α] [division_monoid β] [has_distrib_neg β]
(hf : antiperiodic f c) (hg : antiperiodic g c) :
periodic (f / g) c | by simp [*, neg_div_neg_eq] at * | lemma | function.antiperiodic.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"
] | [
"division_monoid",
"has_distrib_neg",
"neg_div_neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.fract_periodic (α) [linear_ordered_ring α] [floor_ring α] :
function.periodic int.fract (1 : α) | by exact_mod_cast λ a, int.fract_add_int a 1 | lemma | int.fract_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"
] | [
"floor_ring",
"function.periodic",
"int.fract",
"int.fract_add_int",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq : (1 : punit) = star | rfl | lemma | punit.one_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq : x * y = star | rfl | lemma | punit.mul_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq : x / y = star | rfl | lemma | punit.div_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq : x⁻¹ = star | rfl | lemma | punit.inv_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq : gcd x y = star | rfl | lemma | punit.gcd_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq : lcm x y = star | rfl | lemma | punit.lcm_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_unit_eq : norm_unit x = 1 | rfl | lemma | punit.norm_unit_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq (r : R) : r • y = star | rfl | lemma | punit.smul_eq | algebra | src/algebra/punit_instances.lean | [
"algebra.module.basic",
"algebra.gcd_monoid.basic",
"algebra.group_ring_action.basic",
"group_theory.group_action.defs",
"order.complete_boolean_algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrim [ring R] (a b c : R) : R | b^2 - 4 * a * c | def | discrim | algebra | src/algebra/quadratic_discriminant.lean | [
"algebra.char_p.invertible",
"order.filter.at_top_bot",
"tactic.linarith",
"tactic.field_simp",
"tactic.linear_combination"
] | [
"ring"
] | Discriminant of a quadratic | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrim_neg [ring R] (a b c : R) : discrim (-a) (-b) (-c) = discrim a b c | by simp [discrim] | lemma | discrim_neg | algebra | src/algebra/quadratic_discriminant.lean | [
"algebra.char_p.invertible",
"order.filter.at_top_bot",
"tactic.linarith",
"tactic.field_simp",
"tactic.linear_combination"
] | [
"discrim",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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