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 |
|---|---|---|---|---|---|---|---|---|---|---|
to_lex_rat_cast [has_rat_cast α] (n : ℚ) : to_lex (n : α) = n | rfl | lemma | to_lex_rat_cast | algebra.field | src/algebra/field/basic.lean | [
"algebra.field.defs",
"algebra.group_with_zero.units.lemmas",
"algebra.hom.ring",
"algebra.ring.commute"
] | [
"has_rat_cast",
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_rat_cast [has_rat_cast α] (n : ℚ) : (of_lex n : α) = n | rfl | lemma | of_lex_rat_cast | algebra.field | src/algebra/field/basic.lean | [
"algebra.field.defs",
"algebra.group_with_zero.units.lemmas",
"algebra.hom.ring",
"algebra.ring.commute"
] | [
"has_rat_cast",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat.cast_rec [has_lift_t ℕ K] [has_lift_t ℤ K] [has_mul K] [has_inv K] : ℚ → K | | ⟨a, b, _, _⟩ := ↑a * (↑b)⁻¹ | def | rat.cast_rec | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [] | The default definition of the coercion `(↑(a : ℚ) : K)` for a division ring `K`
is defined as `(a / b : K) = (a : K) * (b : K)⁻¹`.
Use `coe` instead of `rat.cast_rec` for better definitional behaviour. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_rat_cast (K : Type u) | (rat_cast : ℚ → K) | class | has_rat_cast | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [] | Type class for the canonical homomorphism `ℚ → K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
qsmul_rec (coe : ℚ → K) [has_mul K] (a : ℚ) (x : K) : K | coe a * x | def | qsmul_rec | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [] | The default definition of the scalar multiplication `(a : ℚ) • (x : K)` for a division ring `K`
is given by `a • x = (↑ a) * x`.
Use `(a : ℚ) • (x : K)` instead of `qsmul_rec` for better definitional behaviour. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_semiring (α : Type*) extends semiring α, group_with_zero α | class | division_semiring | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"group_with_zero",
"semiring"
] | A `division_semiring` is a `semiring` with multiplicative inverses for nonzero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_ring (K : Type u) extends ring K, div_inv_monoid K, nontrivial K, has_rat_cast K | (mul_inv_cancel : ∀ {a : K}, a ≠ 0 → a * a⁻¹ = 1)
(inv_zero : (0 : K)⁻¹ = 0)
(rat_cast := rat.cast_rec)
(rat_cast_mk : ∀ (a : ℤ) (b : ℕ) h1 h2, rat_cast ⟨a, b, h1, h2⟩ = a * b⁻¹ . try_refl_tac)
(qsmul : ℚ → K → K := qsmul_rec rat_cast)
(qsmul_eq_mul' : ∀ (a : ℚ) (x : K), qsmul a x = rat_cast a * x . try_refl_tac) | class | division_ring | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"div_inv_monoid",
"has_rat_cast",
"inv_zero",
"mul_inv_cancel",
"nontrivial",
"qsmul_rec",
"rat.cast_rec",
"ring",
"try_refl_tac"
] | A `division_ring` is a `ring` with multiplicative inverses for nonzero elements.
An instance of `division_ring K` includes maps `rat_cast : ℚ → K` and `qsmul : ℚ → K → K`.
If the division ring has positive characteristic p, we define `rat_cast (1 / p) = 1 / 0 = 0`
for consistency with our division by zero convention.
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_ring.to_division_semiring [division_ring α] : division_semiring α | { ..‹division_ring α›, ..(infer_instance : semiring α) } | instance | division_ring.to_division_semiring | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"division_ring",
"division_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semifield (α : Type*) extends comm_semiring α, division_semiring α, comm_group_with_zero α | class | semifield | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"comm_group_with_zero",
"comm_semiring",
"division_semiring"
] | A `semifield` is a `comm_semiring` with multiplicative inverses for nonzero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
field (K : Type u) extends comm_ring K, division_ring K | class | field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"comm_ring",
"division_ring"
] | A `field` is a `comm_ring` with multiplicative inverses for nonzero elements.
An instance of `field K` includes maps `of_rat : ℚ → K` and `qsmul : ℚ → K → K`.
If the field has positive characteristic p, we define `of_rat (1 / p) = 1 / 0 = 0`
for consistency with our division by zero convention.
The fields `of_rat` and... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_coe {K : Type*} [has_rat_cast K] : has_coe_t ℚ K | ⟨has_rat_cast.rat_cast⟩ | instance | rat.cast_coe | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"has_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_mk' (a b h1 h2) : ((⟨a, b, h1, h2⟩ : ℚ) : K) = a * b⁻¹ | division_ring.rat_cast_mk _ _ _ _ | theorem | rat.cast_mk' | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_def : ∀ (r : ℚ), (r : K) = r.num / r.denom | | ⟨a, b, h1, h2⟩ := (cast_mk' _ _ _ _).trans (div_eq_mul_inv _ _).symm | theorem | rat.cast_def | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_division_ring : has_smul ℚ K | ⟨division_ring.qsmul⟩ | instance | rat.smul_division_ring | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def (a : ℚ) (x : K) : a • x = ↑a * x | division_ring.qsmul_eq_mul' a x | lemma | rat.smul_def | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_one_eq_coe (A : Type*) [division_ring A] (m : ℚ) :
m • (1 : A) = ↑m | by rw [rat.smul_def, mul_one] | lemma | rat.smul_one_eq_coe | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"division_ring",
"mul_one",
"rat.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
field.to_semifield : semifield K | { .. ‹field K›, .. (infer_instance : semiring K) } | instance | field.to_semifield | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"semifield",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field (R : Type u) [semiring R] : Prop | (exists_pair_ne : ∃ (x y : R), x ≠ y)
(mul_comm : ∀ (x y : R), x * y = y * x)
(mul_inv_cancel : ∀ {a : R}, a ≠ 0 → ∃ b, a * b = 1) | structure | is_field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"exists_pair_ne",
"mul_comm",
"mul_inv_cancel",
"semiring"
] | A predicate to express that a (semi)ring is a (semi)field.
This is mainly useful because such a predicate does not contain data,
and can therefore be easily transported along ring isomorphisms.
Additionaly, this is useful when trying to prove that
a particular ring structure extends to a (semi)field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semifield.to_is_field (R : Type u) [semifield R] : is_field R | { mul_inv_cancel := λ a ha, ⟨a⁻¹, mul_inv_cancel ha⟩,
..‹semifield R› } | lemma | semifield.to_is_field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"is_field",
"mul_inv_cancel",
"semifield"
] | Transferring from `semifield` to `is_field`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field.to_is_field (R : Type u) [field R] : is_field R | semifield.to_is_field _ | lemma | field.to_is_field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"field",
"is_field",
"semifield.to_is_field"
] | Transferring from `field` to `is_field`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_field.nontrivial {R : Type u} [semiring R] (h : is_field R) : nontrivial R | ⟨h.exists_pair_ne⟩ | lemma | is_field.nontrivial | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"is_field",
"nontrivial",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_field_of_subsingleton (R : Type u) [semiring R] [subsingleton R] :
¬is_field R | λ h, let ⟨x, y, h⟩ := h.exists_pair_ne in h (subsingleton.elim _ _) | lemma | not_is_field_of_subsingleton | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"is_field",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field.to_semifield {R : Type u} [semiring R] (h : is_field R) : semifield R | { inv := λ a, if ha : a = 0 then 0 else classical.some (is_field.mul_inv_cancel h ha),
inv_zero := dif_pos rfl,
mul_inv_cancel := λ a ha,
begin
convert classical.some_spec (is_field.mul_inv_cancel h ha),
exact dif_neg ha
end,
.. ‹semiring R›, ..h } | def | is_field.to_semifield | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"inv_zero",
"is_field",
"mul_inv_cancel",
"semifield",
"semiring"
] | Transferring from `is_field` to `semifield`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_field.to_field {R : Type u} [ring R] (h : is_field R) : field R | { .. ‹ring R›, ..is_field.to_semifield h } | def | is_field.to_field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"field",
"is_field",
"is_field.to_semifield",
"ring"
] | Transferring from `is_field` to `field`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniq_inv_of_is_field (R : Type u) [ring R] (hf : is_field R) :
∀ (x : R), x ≠ 0 → ∃! (y : R), x * y = 1 | begin
intros x hx,
apply exists_unique_of_exists_of_unique,
{ exact hf.mul_inv_cancel hx },
{ intros y z hxy hxz,
calc y = y * (x * z) : by rw [hxz, mul_one]
... = (x * y) * z : by rw [← mul_assoc, hf.mul_comm y x]
... = z : by rw [hxy, one_mul] }
end | lemma | uniq_inv_of_is_field | algebra.field | src/algebra/field/defs.lean | [
"data.rat.init",
"algebra.ring.defs"
] | [
"is_field",
"mul_assoc",
"mul_one",
"one_mul",
"ring"
] | For each field, and for each nonzero element of said field, there is a unique inverse.
Since `is_field` doesn't remember the data of an `inv` function and as such,
a lemma that there is a unique inverse could be useful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_rat_cast [has_rat_cast α] (q : ℚ) : op (q : α) = q | rfl | lemma | mul_opposite.op_rat_cast | algebra.field | src/algebra/field/opposite.lean | [
"algebra.field.defs",
"algebra.ring.opposite",
"data.int.cast.lemmas"
] | [
"has_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_rat_cast [has_rat_cast α] (q : ℚ) : unop (q : αᵐᵒᵖ) = q | rfl | lemma | mul_opposite.unop_rat_cast | algebra.field | src/algebra/field/opposite.lean | [
"algebra.field.defs",
"algebra.ring.opposite",
"data.int.cast.lemmas"
] | [
"has_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1_neg (a : α) (n : ℤ) : (-a) ^ bit1 n = - a ^ bit1 n | by rw [zpow_bit1', zpow_bit1', neg_mul_neg, neg_mul_eq_mul_neg] | lemma | zpow_bit1_neg | algebra.field | src/algebra/field/power.lean | [
"algebra.field.defs",
"algebra.group_with_zero.power",
"algebra.parity"
] | [
"neg_mul_eq_mul_neg",
"neg_mul_neg",
"zpow_bit1'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.neg_zpow (h : odd n) (a : α) : (-a) ^ n = - a ^ n | by { obtain ⟨k, rfl⟩ := h.exists_bit1, exact zpow_bit1_neg _ _ } | lemma | odd.neg_zpow | algebra.field | src/algebra/field/power.lean | [
"algebra.field.defs",
"algebra.group_with_zero.power",
"algebra.parity"
] | [
"odd",
"zpow_bit1_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.neg_one_zpow (h : odd n) : (-1 : α) ^ n = -1 | by rw [h.neg_zpow, one_zpow] | lemma | odd.neg_one_zpow | algebra.field | src/algebra/field/power.lean | [
"algebra.field.defs",
"algebra.group_with_zero.power",
"algebra.parity"
] | [
"odd",
"one_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
up_rat_cast [has_rat_cast α] (q : ℚ) : up (q : α) = q | rfl | lemma | ulift.up_rat_cast | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"has_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
down_rat_cast [has_rat_cast α] (q : ℚ) : down (q : ulift α) = q | rfl | lemma | ulift.down_rat_cast | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"has_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_semiring [division_semiring α] : division_semiring (ulift α) | by refine down_injective.division_semiring down _ _ _ _ _ _ _ _ _ _; intros; refl | instance | ulift.division_semiring | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semifield [semifield α] : semifield (ulift α) | { ..ulift.division_semiring, ..ulift.comm_group_with_zero } | instance | ulift.semifield | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"semifield",
"ulift.comm_group_with_zero",
"ulift.division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_ring [division_ring α] : division_ring (ulift α) | { ..ulift.division_semiring, ..ulift.add_group } | instance | ulift.division_ring | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"division_ring",
"ulift.division_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
field [field α] : field (ulift α) | { ..ulift.semifield, ..ulift.division_ring } | instance | ulift.field | algebra.field | src/algebra/field/ulift.lean | [
"algebra.field.basic",
"algebra.ring.ulift"
] | [
"field",
"ulift.division_ring",
"ulift.semifield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_monoid (α) | list α | def | free_monoid | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | Free monoid over a given alphabet. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list : free_monoid α ≃ list α | equiv.refl _ | def | free_monoid.to_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"equiv.refl",
"free_monoid"
] | The identity equivalence between `free_monoid α` and `list α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_list : list α ≃ free_monoid α | equiv.refl _ | def | free_monoid.of_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"equiv.refl",
"free_monoid"
] | The identity equivalence between `list α` and `free_monoid α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list_symm : (@to_list α).symm = of_list | rfl | lemma | free_monoid.to_list_symm | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_symm : (@of_list α).symm = to_list | rfl | lemma | free_monoid.of_list_symm | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_of_list (l : list α) : to_list (of_list l) = l | rfl | lemma | free_monoid.to_list_of_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"to_list_of_list"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_to_list (xs : free_monoid α) : of_list (to_list xs) = xs | rfl | lemma | free_monoid.of_list_to_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_comp_of_list : @to_list α ∘ of_list = id | rfl | lemma | free_monoid.to_list_comp_of_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_comp_to_list : @of_list α ∘ to_list = id | rfl | lemma | free_monoid.of_list_comp_to_list | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_one : (1 : free_monoid α).to_list = [] | rfl | lemma | free_monoid.to_list_one | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_nil : of_list ([] : list α) = 1 | rfl | lemma | free_monoid.of_list_nil | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"of_list_nil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_mul (xs ys : free_monoid α) : (xs * ys).to_list = xs.to_list ++ ys.to_list | rfl | lemma | free_monoid.to_list_mul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_append (xs ys : list α) :
of_list (xs ++ ys) = of_list xs * of_list ys | rfl | lemma | free_monoid.of_list_append | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"of_list_append"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_prod (xs : list (free_monoid α)) : to_list xs.prod = (xs.map to_list).join | by induction xs; simp [*, list.join] | lemma | free_monoid.to_list_prod | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_join (xs : list (list α)) : of_list xs.join = (xs.map of_list).prod | to_list.injective $ by simp | lemma | free_monoid.of_list_join | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (x : α) : free_monoid α | of_list [x] | def | free_monoid.of | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | Embeds an element of `α` into `free_monoid α` as a singleton list. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_list_of (x : α) : to_list (of x) = [x] | rfl | lemma | free_monoid.to_list_of | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_singleton (x : α) : of_list [x] = of x | rfl | lemma | free_monoid.of_list_singleton | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_cons (x : α) (xs : list α) :
of_list (x :: xs) = of x * of_list xs | rfl | lemma | free_monoid.of_list_cons | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"of_list_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_of_mul (x : α) (xs : free_monoid α) :
to_list (of x * xs) = x :: xs.to_list | rfl | lemma | free_monoid.to_list_of_mul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective : function.injective (@of α) | list.singleton_injective | lemma | free_monoid.of_injective | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"list.singleton_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec_on {C : free_monoid α → Sort*} (xs : free_monoid α) (h0 : C 1)
(ih : Π x xs, C xs → C (of x * xs)) : C xs | list.rec_on xs h0 ih | def | free_monoid.rec_on | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | Recursor for `free_monoid` using `1` and `free_monoid.of x * xs` instead of `[]` and
`x :: xs`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rec_on_one {C : free_monoid α → Sort*} (h0 : C 1)
(ih : Π x xs, C xs → C (of x * xs)) :
@rec_on α C 1 h0 ih = h0 | rfl | lemma | free_monoid.rec_on_one | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec_on_of_mul {C : free_monoid α → Sort*} (x : α) (xs : free_monoid α)
(h0 : C 1) (ih : Π x xs, C xs → C (of x * xs)) :
@rec_on α C (of x * xs) h0 ih = ih x xs (rec_on xs h0 ih) | rfl | lemma | free_monoid.rec_on_of_mul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cases_on {C : free_monoid α → Sort*} (xs : free_monoid α) (h0 : C 1)
(ih : Π x xs, C (of x * xs)) : C xs | list.cases_on xs h0 ih | def | free_monoid.cases_on | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | A version of `list.cases_on` for `free_monoid` using `1` and `free_monoid.of x * xs` instead of
`[]` and `x :: xs`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cases_on_one {C : free_monoid α → Sort*} (h0 : C 1)
(ih : Π x xs, C (of x * xs)) :
@cases_on α C 1 h0 ih = h0 | rfl | lemma | free_monoid.cases_on_one | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cases_on_of_mul {C : free_monoid α → Sort*} (x : α) (xs : free_monoid α)
(h0 : C 1) (ih : Π x xs, C (of x * xs)) :
@cases_on α C (of x * xs) h0 ih = ih x xs | rfl | lemma | free_monoid.cases_on_of_mul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_eq ⦃f g : free_monoid α →* M⦄ (h : ∀ x, f (of x) = g (of x)) :
f = g | monoid_hom.ext $ λ l, rec_on l (f.map_one.trans g.map_one.symm) $
λ x xs hxs, by simp only [h, hxs, monoid_hom.map_mul] | lemma | free_monoid.hom_eq | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"monoid_hom.ext",
"monoid_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_aux {M} [monoid M] (l : list M) : M | l.rec_on 1 (λ x xs (_ : M), list.foldl (*) x xs) | def | free_monoid.prod_aux | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"monoid"
] | A variant of `list.prod` that has `[x].prod = x` true definitionally.
The purpose is to make `free_monoid.lift_eval_of` true by `rfl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_aux_eq : ∀ l : list M, free_monoid.prod_aux l = l.prod | | [] := rfl
| (x :: xs) := congr_arg (λ x, list.foldl (*) x xs) (one_mul _).symm | lemma | free_monoid.prod_aux_eq | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid.prod_aux",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : (α → M) ≃ (free_monoid α →* M) | { to_fun := λ f, ⟨λ l, free_monoid.prod_aux (l.to_list.map f), rfl,
λ l₁ l₂, by simp only [prod_aux_eq, to_list_mul, list.map_append, list.prod_append]⟩,
inv_fun := λ f x, f (of x),
left_inv := λ f, rfl,
right_inv := λ f, hom_eq $ λ x, rfl } | def | free_monoid.lift | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"free_monoid.prod_aux",
"inv_fun",
"lift",
"list.prod_append"
] | Equivalence between maps `α → M` and monoid homomorphisms `free_monoid α →* M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_symm_apply (f : free_monoid α →* M) : lift.symm f = f ∘ of | rfl | lemma | free_monoid.lift_symm_apply | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_apply (f : α → M) (l : free_monoid α) : lift f l = (l.to_list.map f).prod | prod_aux_eq _ | lemma | free_monoid.lift_apply | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of (f : α → M) : lift f ∘ of = f | rfl | lemma | free_monoid.lift_comp_of | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_eval_of (f : α → M) (x : α) : lift f (of x) = f x | rfl | lemma | free_monoid.lift_eval_of | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_restrict (f : free_monoid α →* M) : lift (f ∘ of) = f | lift.apply_symm_apply f | lemma | free_monoid.lift_restrict | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_lift (g : M →* N) (f : α → M) : g.comp (lift f) = lift (g ∘ f) | by { ext, simp } | lemma | free_monoid.comp_lift | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_map_lift (g : M →* N) (f : α → M) (x : free_monoid α) : g (lift f x) = lift (g ∘ f) x | monoid_hom.ext_iff.1 (comp_lift g f) x | lemma | free_monoid.hom_map_lift | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_action (f : α → β → β) : mul_action (free_monoid α) β | { smul := λ l b, l.to_list.foldr f b,
one_smul := λ x, rfl,
mul_smul := λ xs ys b, list.foldr_append _ _ _ _ } | def | free_monoid.mk_mul_action | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"list.foldr_append",
"mul_action",
"one_smul"
] | Define a multiplicative action of `free_monoid α` on `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : α → β → β) (l : free_monoid α) (b : β) :
(by haveI := mk_mul_action f; exact l • b = l.to_list.foldr f b) | rfl | lemma | free_monoid.smul_def | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_smul (f : α → β → β) (l : list α) (b : β) :
(by haveI := mk_mul_action f; exact (of_list l) • b = l.foldr f b) | rfl | lemma | free_monoid.of_list_smul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_smul (f : α → β → β) (x : α) (y : β) :
(by haveI := mk_mul_action f; exact of x • y) = f x y | rfl | lemma | free_monoid.of_smul | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) : free_monoid α →* free_monoid β | { to_fun := λ l, of_list $ l.to_list.map f,
map_one' := rfl,
map_mul' := λ l₁ l₂, list.map_append _ _ _ } | def | free_monoid.map | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | The unique monoid homomorphism `free_monoid α →* free_monoid β` that sends
each `of x` to `of (f x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_of (f : α → β) (x : α) : map f (of x) = of (f x) | rfl | lemma | free_monoid.map_of | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_list_map (f : α → β) (xs : free_monoid α) :
(map f xs).to_list = xs.to_list.map f | rfl | lemma | free_monoid.to_list_map | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_map (f : α → β) (xs : list α) :
of_list (xs.map f) = map f (of_list xs) | rfl | lemma | free_monoid.of_list_map | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_comp_eq_map (f : α → β) :
lift (λ x, of (f x)) = map f | hom_eq $ λ x, rfl | lemma | free_monoid.lift_of_comp_eq_map | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (g : β → γ) (f : α → β) : map (g ∘ f) = (map g).comp (map f) | hom_eq $ λ x, rfl | lemma | free_monoid.map_comp | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (@id α) = monoid_hom.id (free_monoid α) | hom_eq $ λ x, rfl | lemma | free_monoid.map_id | algebra.free_monoid | src/algebra/free_monoid/basic.lean | [
"data.list.big_operators.basic"
] | [
"free_monoid",
"map_id",
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countp (p : α → Prop) [decidable_pred p] : free_add_monoid α →+ ℕ | ⟨list.countp p, list.countp_nil p, list.countp_append _⟩ | def | free_add_monoid.countp | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"list.countp_append",
"list.countp_nil"
] | `list.countp` as a bundled additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countp_of (x : α) : countp p (of x) = if p x then 1 else 0 | rfl | lemma | free_add_monoid.countp_of | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countp_apply (l : free_add_monoid α) : countp p l = list.countp p l | rfl | lemma | free_add_monoid.countp_apply | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"list.countp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
count [decidable_eq α] (x : α) : free_add_monoid α →+ ℕ | countp (eq x) | def | free_add_monoid.count | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [] | `list.count` as a bundled additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
count_of [decidable_eq α] (x y : α) : count x (of y) = pi.single x 1 y | by simp only [count, countp_of, pi.single_apply, eq_comm] | lemma | free_add_monoid.count_of | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
count_apply [decidable_eq α] (x : α) (l : free_add_monoid α) :
count x l = list.count x l | rfl | lemma | free_add_monoid.count_apply | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"list.count"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countp (p : α → Prop) [decidable_pred p] : free_monoid α →* multiplicative ℕ | (free_add_monoid.countp p).to_multiplicative | def | free_monoid.countp | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"free_add_monoid.countp",
"free_monoid",
"multiplicative"
] | `list.countp` as a bundled multiplicative monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countp_of' (x : α) :
countp p (of x) = if p x then multiplicative.of_add 1 else multiplicative.of_add 0 | rfl | lemma | free_monoid.countp_of' | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countp_of (x : α) : countp p (of x) = if p x then multiplicative.of_add 1 else 1 | by rw [countp_of', of_add_zero] | lemma | free_monoid.countp_of | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"multiplicative.of_add",
"of_add_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countp_apply (l : free_add_monoid α) :
countp p l = multiplicative.of_add (list.countp p l) | rfl | lemma | free_monoid.countp_apply | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"list.countp",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
count [decidable_eq α] (x : α) : free_monoid α →* multiplicative ℕ | countp (eq x) | def | free_monoid.count | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"free_monoid",
"multiplicative"
] | `list.count` as a bundled additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
count_apply [decidable_eq α] (x : α) (l : free_add_monoid α) :
count x l = multiplicative.of_add (list.count x l) | rfl | lemma | free_monoid.count_apply | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"list.count",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
count_of [decidable_eq α] (x y : α) :
count x (of y) = @pi.mul_single α (λ _, multiplicative ℕ) _ _ x (multiplicative.of_add 1) y | by simp only [count, countp_of, pi.mul_single_apply, eq_comm] | lemma | free_monoid.count_of | algebra.free_monoid | src/algebra/free_monoid/count.lean | [
"algebra.free_monoid.basic",
"data.list.count"
] | [
"multiplicative",
"multiplicative.of_add",
"pi.mul_single",
"pi.mul_single_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalization_monoid (α : Type*)
[cancel_comm_monoid_with_zero α] | (norm_unit : α → αˣ)
(norm_unit_zero : norm_unit 0 = 1)
(norm_unit_mul : ∀{a b}, a ≠ 0 → b ≠ 0 → norm_unit (a * b) = norm_unit a * norm_unit b)
(norm_unit_coe_units : ∀(u : αˣ), norm_unit u = u⁻¹) | class | normalization_monoid | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [
"cancel_comm_monoid_with_zero"
] | Normalization monoid: multiplying with `norm_unit` gives a normal form for associated
elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_unit_one : norm_unit (1:α) = 1 | norm_unit_coe_units 1 | theorem | norm_unit_one | algebra.gcd_monoid | src/algebra/gcd_monoid/basic.lean | [
"algebra.associated",
"algebra.group_power.lemmas",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.