fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 | 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"
] | null | 137 | 140 | true | 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 | 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"
] | null | 142 | 144 | true | 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 | 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"
] | null | 146 | 148 | true | 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 * | 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"
] | null | 150 | 152 | true | 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 | 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"
] | null | 154 | 156 | true | 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) | 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"
] | null | 158 | 160 | true | 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 | 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"
] | null | 162 | 164 | true | 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 | 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"
] | null | 166 | 168 | true | 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) | 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"
] | null | 170 | 172 | true | 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 | 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"
] | null | 174 | 176 | true | 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] | 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"
] | null | 178 | 180 | true | 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) | 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"
] | null | 182 | 184 | true | 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 * | 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"
] | null | 186 | 188 | true | 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 | 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"
] | null | 190 | 192 | true | 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 | 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"
] | null | 194 | 196 | true | 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 | 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"
] | null | 198 | 200 | true | 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 | 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"
] | null | 202 | 204 | true | 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 | 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"
] | null | 206 | 208 | true | 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' | 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"
] | null | 210 | 212 | true | 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) | 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"
] | null | 214 | 216 | true | 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 | 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"
] | null | 218 | 224 | true | 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 | 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"
] | null | 226 | 228 | true | 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 | 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"
] | null | 230 | 232 | true | 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 | 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"
] | null | 234 | 236 | true | 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' | 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"
] | null | 238 | 240 | true | 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' | 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"
] | null | 242 | 244 | true | 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 | 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"
] | null | 246 | 248 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic.neg_eq [add_group α] (h : periodic f c) :
f (-c) = f 0 :=
h.neg.eq | 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"
] | null | 250 | 252 | true | 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 | 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"
] | null | 254 | 256 | true | 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 | 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"
] | null | 258 | 260 | true | 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 | 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"
] | null | 262 | 264 | true | 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 | 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"
] | null | 266 | 268 | true | 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⟩ | 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`. | 272 | 276 | true | 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⟩ | 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`. | 280 | 284 | true | 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⟩ | 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`. | 288 | 292 | true | 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⟩ | 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"
] | null | 294 | 298 | true | 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 | 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"
] | null | 300 | 303 | true | 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] } | 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"
] | null | 305 | 308 | true | 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] } | 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"
] | null | 310 | 313 | true | 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) | 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. | 316 | 322 | true | 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 | 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"
] | null | 324 | 326 | true | 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 | 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`. | 332 | 333 | true | 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 | 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"
] | [] | null | 335 | 337 | true | 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 | 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"
] | null | 339 | 341 | true | 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 _] | 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`. | 345 | 347 | true | 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 | 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"
] | null | 349 | 351 | true | 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 | 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"
] | null | 353 | 356 | true | 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] | 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"
] | null | 358 | 361 | true | 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 | 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"
] | null | 363 | 366 | true | 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] | 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"
] | null | 368 | 371 | true | 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] | 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"
] | null | 373 | 376 | true | 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) | 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"
] | null | 378 | 380 | true | 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 | 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"
] | null | 382 | 385 | true | 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 | 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"
] | null | 387 | 390 | true | 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 _] | 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"
] | null | 392 | 395 | true | 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] | 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"
] | null | 397 | 400 | true | 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) | 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"
] | null | 402 | 404 | true | 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) | 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"
] | null | 406 | 408 | true | 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] | 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"
] | null | 410 | 412 | true | 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) | 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"
] | null | 414 | 416 | true | 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 * | 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"
] | null | 418 | 421 | true | 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) | 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"
] | null | 423 | 426 | true | 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) | 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"
] | null | 428 | 431 | true | 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 | 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"
] | null | 433 | 436 | true | 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 | 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"
] | null | 438 | 441 | true | 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) | 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"
] | null | 443 | 447 | true | 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 | 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"
] | null | 449 | 452 | true | 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 | 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"
] | null | 454 | 457 | true | 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 | 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"
] | null | 459 | 462 | true | 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 | 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"
] | null | 464 | 467 | true | 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 | 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"
] | null | 469 | 472 | true | 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 * | 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"
] | null | 474 | 477 | true | 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 | 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"
] | null | 479 | 482 | true | 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 * | 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"
] | null | 484 | 487 | true | 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 | 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"
] | null | 489 | 492 | true | 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 | 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"
] | null | 494 | 497 | true | 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 | 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"
] | null | 499 | 502 | true | 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 * | 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"
] | null | 504 | 507 | true | 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 * | 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"
] | null | 509 | 512 | true | 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 | 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"
] | null | 516 | 518 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: comm_group punit :=
by refine_struct
{ mul := λ _ _, star,
one := star,
inv := λ _, star,
div := λ _ _, star,
npow := λ _ _, star,
zpow := λ _ _, star,
.. };
intros; exact subsingleton.elim _ _ | : comm_group punit | by refine_struct
{ mul := λ _ _, star,
one := star,
inv := λ _, star,
div := λ _ _, star,
npow := λ _ _, star,
zpow := λ _ _, star,
.. };
intros; exact subsingleton.elim _ _ | instance | 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"
] | [
"comm_group"
] | null | 28 | 38 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq : (1 : punit) = star := rfl | 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"
] | [] | null | 40 | 40 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq : x * y = star := rfl | 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"
] | [] | null | 41 | 41 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_eq : x / y = star := rfl | 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"
] | [] | null | 43 | 43 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_eq : x⁻¹ = star := rfl | 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"
] | [] | null | 45 | 45 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: comm_ring punit :=
by refine
{ nat_cast := λ _, punit.star,
.. punit.comm_group,
.. punit.add_comm_group,
.. };
intros; exact subsingleton.elim _ _ | : comm_ring punit | by refine
{ nat_cast := λ _, punit.star,
.. punit.comm_group,
.. punit.add_comm_group,
.. };
intros; exact subsingleton.elim _ _ | instance | 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"
] | [
"comm_ring"
] | null | 47 | 53 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: cancel_comm_monoid_with_zero punit :=
by refine
{ .. punit.comm_ring,
.. };
intros; exact subsingleton.elim _ _ | : cancel_comm_monoid_with_zero punit | by refine
{ .. punit.comm_ring,
.. };
intros; exact subsingleton.elim _ _ | instance | 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"
] | [
"cancel_comm_monoid_with_zero"
] | null | 55 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: normalized_gcd_monoid punit :=
by refine
{ gcd := λ _ _, star,
lcm := λ _ _, star,
norm_unit := λ x, 1,
gcd_dvd_left := λ _ _, ⟨star, subsingleton.elim _ _⟩,
gcd_dvd_right := λ _ _, ⟨star, subsingleton.elim _ _⟩,
dvd_gcd := λ _ _ _ _ _, ⟨star, subsingleton.elim _ _⟩,
gcd_mul_lcm := λ _ _, ⟨1, subsingleton... | : normalized_gcd_monoid punit | by refine
{ gcd := λ _ _, star,
lcm := λ _ _, star,
norm_unit := λ x, 1,
gcd_dvd_left := λ _ _, ⟨star, subsingleton.elim _ _⟩,
gcd_dvd_right := λ _ _, ⟨star, subsingleton.elim _ _⟩,
dvd_gcd := λ _ _ _ _ _, ⟨star, subsingleton.elim _ _⟩,
gcd_mul_lcm := λ _ _, ⟨1, subsingleton.elim _ _⟩,
.. };
intros; exact... | instance | 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"
] | [
"gcd_mul_lcm",
"normalized_gcd_monoid"
] | null | 61 | 71 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq : gcd x y = star := rfl | 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"
] | [] | null | 73 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcm_eq : lcm x y = star := rfl | 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"
] | [] | null | 74 | 74 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_unit_eq : norm_unit x = 1 := rfl | 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"
] | [] | null | 75 | 75 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: canonically_ordered_add_monoid punit :=
by refine
{ exists_add_of_le := λ _ _ _, ⟨star, subsingleton.elim _ _⟩,
.. punit.comm_ring, .. punit.complete_boolean_algebra, .. };
intros; trivial | : canonically_ordered_add_monoid punit | by refine
{ exists_add_of_le := λ _ _ _, ⟨star, subsingleton.elim _ _⟩,
.. punit.comm_ring, .. punit.complete_boolean_algebra, .. };
intros; trivial | instance | 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"
] | [
"canonically_ordered_add_monoid"
] | null | 77 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: linear_ordered_cancel_add_comm_monoid punit :=
{ le_of_add_le_add_left := λ _ _ _ _, trivial,
.. punit.canonically_ordered_add_monoid, ..punit.linear_order } | : linear_ordered_cancel_add_comm_monoid punit | { le_of_add_le_add_left := λ _ _ _ _, trivial,
.. punit.canonically_ordered_add_monoid, ..punit.linear_order } | instance | 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"
] | [
"linear_ordered_cancel_add_comm_monoid"
] | null | 83 | 85 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: linear_ordered_add_comm_monoid_with_top punit :=
{ top_add' := λ _, rfl,
..punit.complete_boolean_algebra,
..punit.linear_ordered_cancel_add_comm_monoid } | : linear_ordered_add_comm_monoid_with_top punit | { top_add' := λ _, rfl,
..punit.complete_boolean_algebra,
..punit.linear_ordered_cancel_add_comm_monoid } | instance | 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"
] | [
"linear_ordered_add_comm_monoid_with_top"
] | null | 87 | 90 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: has_smul R punit := ⟨λ _ _, star⟩ | : has_smul R punit | ⟨λ _ _, star⟩ | instance | 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"
] | [
"has_smul"
] | null | 92 | 92 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq (r : R) : r • y = star := rfl | 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"
] | [] | null | 94 | 94 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: is_central_scalar R punit := ⟨λ _ _, rfl⟩ | : is_central_scalar R punit | ⟨λ _ _, rfl⟩ | instance | 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"
] | [
"is_central_scalar"
] | null | 96 | 96 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: smul_comm_class R S punit := ⟨λ _ _ _, rfl⟩ | : smul_comm_class R S punit | ⟨λ _ _ _, rfl⟩ | instance | 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"
] | [
"smul_comm_class"
] | null | 97 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_smul R S] : is_scalar_tower R S punit := ⟨λ _ _ _, rfl⟩ | [has_smul R S] : is_scalar_tower R S punit | ⟨λ _ _ _, rfl⟩ | instance | 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"
] | [
"has_smul",
"is_scalar_tower"
] | null | 98 | 98 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_zero R] : smul_with_zero R punit :=
by refine { ..punit.has_smul, .. };
intros; exact subsingleton.elim _ _ | [has_zero R] : smul_with_zero R punit | by refine { ..punit.has_smul, .. };
intros; exact subsingleton.elim _ _ | instance | 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"
] | [
"smul_with_zero"
] | null | 100 | 102 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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