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 |
|---|---|---|---|---|---|---|---|---|---|---|
tail_mul (x y : free_semigroup α) : (x * y).2 = x.2 ++ (y.1 :: y.2) | rfl | lemma | free_semigroup.tail_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk (x y : α) (L1 L2 : list α) :
mk x L1 * mk y L2 = mk x (L1 ++ y :: L2) | rfl | lemma | free_semigroup.mk_mul_mk | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (x : α) : free_semigroup α | ⟨x, []⟩ | def | free_semigroup.of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | The embedding `α → free_semigroup α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length (x : free_semigroup α) : ℕ | x.tail.length + 1 | def | free_semigroup.length | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | Length of an element of free semigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_mul (x y : free_semigroup α) :
(x * y).length = x.length + y.length | by simp [length, ← add_assoc, add_right_comm] | lemma | free_semigroup.length_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_of (x : α) : (of x).length = 1 | rfl | lemma | free_semigroup.length_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec_on_mul {C : free_semigroup α → Sort l} (x)
(ih1 : ∀ x, C (of x)) (ih2 : ∀ x y, C (of x) → C y → C (of x * y)) :
C x | free_semigroup.rec_on x $ λ f s, list.rec_on s ih1 (λ hd tl ih f, ih2 f ⟨hd, tl⟩ (ih1 f) (ih hd)) f | def | free_semigroup.rec_on_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"ih"
] | Recursor for free semigroup using `of` and `*`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext {β : Type v} [has_mul β] {f g : free_semigroup α →ₙ* β} (h : f ∘ of = g ∘ of) :
f = g | fun_like.ext _ _ $ λ x, free_semigroup.rec_on_mul x (congr_fun h) $
λ x y hx hy, by simp only [map_mul, *] | lemma | free_semigroup.hom_ext | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"free_semigroup.rec_on_mul",
"fun_like.ext",
"hom_ext",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : (α → β) ≃ (free_semigroup α →ₙ* β) | { to_fun := λ f,
{ to_fun := λ x, x.2.foldl (λ a b, a * f b) (f x.1),
map_mul' := λ x y, by simp only [head_mul, tail_mul, ← list.foldl_map f, list.foldl_append,
list.foldl_cons, list.foldl_assoc] },
inv_fun := λ f, f ∘ of,
left_inv := λ f, rfl,
right_inv := λ f, hom_ext rfl } | def | free_semigroup.lift | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"hom_ext",
"inv_fun",
"lift",
"list.foldl_append",
"list.foldl_assoc",
"list.foldl_cons",
"list.foldl_map"
] | Lifts a function `α → β` to a semigroup homomorphism `free_semigroup α → β` given
a semigroup `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp_of' (f : free_semigroup α →ₙ* β) : lift (f ∘ of) = f | hom_ext rfl | lemma | free_semigroup.lift_comp_of' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"hom_ext",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_mul (x y) : lift f (of x * y) = f x * lift f y | by rw [map_mul, lift_of] | lemma | free_semigroup.lift_of_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"lift",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : free_semigroup α →ₙ* free_semigroup β | lift $ of ∘ f | def | free_semigroup.map | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"lift"
] | The unique semigroup homomorphism that sends `of x` to `of (f x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
length_map (x) : (map f x).length = x.length | free_semigroup.rec_on_mul x (λ x, rfl) $ λ x y hx hy, by simp only [map_mul, length_mul, *] | lemma | free_semigroup.length_map | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup.rec_on_mul",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec_on_pure {C : free_semigroup α → Sort l} (x)
(ih1 : ∀ x, C (pure x)) (ih2 : ∀ x y, C (pure x) → C y → C (pure x * y)) :
C x | free_semigroup.rec_on_mul x ih1 ih2 | def | free_semigroup.rec_on_pure | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"free_semigroup.rec_on_mul"
] | Recursor that uses `pure` instead of `of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pure (f : α → β) (x) : (f <$> pure x : free_semigroup β) = pure (f x) | rfl | lemma | free_semigroup.map_pure | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul' (f : α → β) (x y : free_semigroup α) :
(f <$> (x * y)) = (f <$> x * f <$> y) | map_mul (map f) _ _ | lemma | free_semigroup.map_mul' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_bind (f : α → free_semigroup β) (x) :
(pure x >>= f) = f x | rfl | lemma | free_semigroup.pure_bind | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bind (f : α → free_semigroup β) (x y : free_semigroup α) :
(x * y >>= f) = ((x >>= f) * (y >>= f)) | map_mul (lift f) _ _ | lemma | free_semigroup.mul_bind | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"lift",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_seq {f : α → β} {x : free_semigroup α} :
pure f <*> x = f <$> x | rfl | lemma | free_semigroup.pure_seq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_seq {f g : free_semigroup (α → β)} {x : free_semigroup α} :
(f * g) <*> x = (f <*> x) * (g <*> x) | mul_bind _ _ _ | lemma | free_semigroup.mul_seq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse {m : Type u → Type u} [applicative m]
{α β : Type u} (F : α → m β) (x : free_semigroup α) : m (free_semigroup β) | rec_on_pure x (λ x, pure <$> F x) (λ x y ihx ihy, (*) <$> ihx <*> ihy) | def | free_semigroup.traverse | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | `free_semigroup` is traversable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
traverse_pure (x) :traverse F (pure x : free_semigroup α) = pure <$> F x | rfl | lemma | free_semigroup.traverse_pure | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_pure' : traverse F ∘ pure = λ x, (pure <$> F x : m (free_semigroup β)) | rfl | lemma | free_semigroup.traverse_pure' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_mul (x y : free_semigroup α) :
traverse F (x * y) = (*) <$> traverse F x <*> traverse F y | let ⟨x, L1⟩ := x, ⟨y, L2⟩ := y in
list.rec_on L1 (λ x, rfl) (λ hd tl ih x,
show (*) <$> pure <$> F x <*> traverse F ((mk hd tl) * (mk y L2)) =
(*) <$> ((*) <$> pure <$> F x <*> traverse F (mk hd tl)) <*> traverse F (mk y L2),
by rw ih; simp only [(∘), (mul_assoc _ _ _).symm] with functor_norm) x | lemma | free_semigroup.traverse_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup",
"ih",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_mul' :
function.comp (traverse F) ∘ @has_mul.mul (free_semigroup α) _ =
λ x y, (*) <$> traverse F x <*> traverse F y | funext $ λ x, funext $ λ y, traverse_mul F x y | lemma | free_semigroup.traverse_mul' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_eq (x) : free_semigroup.traverse F x = traverse F x | rfl | lemma | free_semigroup.traverse_eq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup.traverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_map_seq (x y : free_semigroup α) :
((*) <$> x <*> y : id (free_semigroup α)) = (x * y : free_semigroup α) | rfl | lemma | free_semigroup.mul_map_seq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_free_semigroup : free_magma α →ₙ* free_semigroup α | free_magma.lift free_semigroup.of | def | free_magma.to_free_semigroup | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"free_magma.lift",
"free_semigroup",
"free_semigroup.of"
] | The canonical multiplicative morphism from `free_magma α` to `free_semigroup α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_free_semigroup_of (x : α) :
to_free_semigroup (of x) = free_semigroup.of x | rfl | lemma | free_magma.to_free_semigroup_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_free_semigroup_comp_of :
@to_free_semigroup α ∘ of = free_semigroup.of | rfl | lemma | free_magma.to_free_semigroup_comp_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_free_semigroup_comp_map (f : α → β) :
to_free_semigroup.comp (map f) = (free_semigroup.map f).comp to_free_semigroup | by { ext1, refl } | lemma | free_magma.to_free_semigroup_comp_map | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_semigroup.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_free_semigroup_map (f : α → β) (x : free_magma α) :
(map f x).to_free_semigroup = free_semigroup.map f x.to_free_semigroup | fun_like.congr_fun (to_free_semigroup_comp_map f) x | lemma | free_magma.to_free_semigroup_map | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"free_semigroup.map",
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
length_to_free_semigroup (x : free_magma α) :
x.to_free_semigroup.length = x.length | free_magma.rec_on_mul x (λ x, rfl) $ λ x y hx hy,
by rw [map_mul, free_semigroup.length_mul, length, hx, hy] | lemma | free_magma.length_to_free_semigroup | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"free_magma.rec_on_mul",
"free_semigroup.length_mul",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_magma_assoc_quotient_equiv (α : Type u) :
magma.assoc_quotient (free_magma α) ≃* free_semigroup α | (magma.assoc_quotient.lift free_magma.to_free_semigroup).to_mul_equiv
(free_semigroup.lift (magma.assoc_quotient.of ∘ free_magma.of))
(by { ext, refl }) (by { ext1, refl }) | def | free_magma_assoc_quotient_equiv | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"free_magma.to_free_semigroup",
"free_semigroup",
"free_semigroup.lift",
"magma.assoc_quotient",
"magma.assoc_quotient.lift",
"magma.assoc_quotient.of"
] | Isomorphism between `magma.assoc_quotient (free_magma α)` and `free_semigroup α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre
| of : X → pre
| of_scalar : R → pre
| add : pre → pre → pre
| mul : pre → pre → pre | inductive | free_algebra.pre | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | This inductive type is used to express representatives of the free algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_generator : has_coe X (pre R X) | ⟨of⟩ | def | free_algebra.pre.has_coe_generator | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | Coercion from `X` to `pre R X`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_semiring : has_coe R (pre R X) | ⟨of_scalar⟩ | def | free_algebra.pre.has_coe_semiring | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | Coercion from `R` to `pre R X`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_mul : has_mul (pre R X) | ⟨mul⟩ | def | free_algebra.pre.has_mul | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | Multiplication in `pre R X` defined as `pre.mul`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_add : has_add (pre R X) | ⟨add⟩ | def | free_algebra.pre.has_add | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | Addition in `pre R X` defined as `pre.add`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero : has_zero (pre R X) | ⟨of_scalar 0⟩ | def | free_algebra.pre.has_zero | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | Zero in `pre R X` defined as the image of `0` from `R`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_one : has_one (pre R X) | ⟨of_scalar 1⟩ | def | free_algebra.pre.has_one | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [] | One in `pre R X` defined as the image of `1` from `R`. Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_smul : has_smul R (pre R X) | ⟨λ r m, mul (of_scalar r) m⟩ | def | free_algebra.pre.has_smul | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"has_smul"
] | Scalar multiplication defined as multiplication by the image of elements from `R`.
Note: Used for notation only. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_fun {A : Type*} [semiring A] [algebra R A] (f : X → A) : pre R X → A | λ t, pre.rec_on t f (algebra_map _ _) (λ _ _, (+)) (λ _ _, (*)) | def | free_algebra.lift_fun | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra",
"algebra_map",
"semiring"
] | Given a function from `X` to an `R`-algebra `A`, `lift_fun` provides a lift of `f` to a function
from `pre R X` to `A`. This is mainly used in the construction of `free_algebra.lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel : (pre R X) → (pre R X) → Prop
-- force `of_scalar` to be a central semiring morphism
| add_scalar {r s : R} : rel ↑(r + s) (↑r + ↑s)
| mul_scalar {r s : R} : rel ↑(r * s) (↑r * ↑s)
| central_scalar {r : R} {a : pre R X} : rel (r * a) (a * r)
-- commutative additive semigroup
| add_assoc {a b c : pre R X} : rel (a ... | inductive | free_algebra.rel | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"left_distrib",
"mul_assoc",
"mul_one",
"mul_zero",
"one_mul",
"rel",
"right_distrib",
"zero_mul"
] | An inductively defined relation on `pre R X` used to force the initial algebra structure on
the associated quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_algebra | quot (free_algebra.rel R X) | def | free_algebra | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra.rel"
] | The free algebra for the type `X` over the commutative semiring `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι : X → free_algebra R X | λ m, quot.mk _ m | def | free_algebra.ι | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra"
] | The canonical function `X → free_algebra R X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mk_eq_ι (m : X) : quot.mk (free_algebra.rel R X) m = ι R m | by rw [ι] | lemma | free_algebra.quot_mk_eq_ι | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra.rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux (f : X → A) : (free_algebra R X →ₐ[R] A) | { to_fun := λ a, quot.lift_on a (lift_fun _ _ f) $ λ a b h,
begin
induction h,
{ exact (algebra_map R A).map_add h_r h_s, },
{ exact (algebra_map R A).map_mul h_r h_s },
{ apply algebra.commutes },
{ change _ + _ + _ = _ + (_ + _),
rw add_assoc },
{ change _ + _ = _ + _,
rw add_com... | def | free_algebra.lift_aux | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra.commutes",
"algebra_map",
"free_algebra",
"left_distrib",
"map_mul",
"mul_assoc",
"right_distrib"
] | Internal definition used to define `lift` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (X → A) ≃ (free_algebra R X →ₐ[R] A) | { to_fun := lift_aux R,
inv_fun := λ F, F ∘ (ι R),
left_inv := λ f, by {ext, rw [ι], refl},
right_inv := λ F, by
{ ext x,
rcases x,
induction x,
case pre.of :
{ change ((F : free_algebra R X → A) ∘ (ι R)) _ = _,
rw [ι],
refl },
case pre.of_scalar :
{ change algebra_map _ _ x ... | def | free_algebra.lift | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"alg_hom.commutes",
"alg_hom.map_add",
"alg_hom.map_mul",
"algebra_map",
"free_algebra",
"inv_fun",
"lift"
] | Given a function `f : X → A` where `A` is an `R`-algebra, `lift R f` is the unique lift
of `f` to a morphism of `R`-algebras `free_algebra R X → A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_aux_eq (f : X → A) : lift_aux R f = lift R f | by { rw [lift], refl } | lemma | free_algebra.lift_aux_eq | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_symm_apply (F : free_algebra R X →ₐ[R] A) : (lift R).symm F = F ∘ (ι R) | by { rw [lift], refl } | lemma | free_algebra.lift_symm_apply | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_comp_lift (f : X → A) :
(lift R f : free_algebra R X → A) ∘ (ι R) = f | by { ext, rw [ι, lift], refl } | theorem | free_algebra.ι_comp_lift | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ι_apply (f : X → A) (x) :
lift R f (ι R x) = f x | by { rw [ι, lift], refl } | theorem | free_algebra.lift_ι_apply | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (f : X → A) (g : free_algebra R X →ₐ[R] A) :
(g : free_algebra R X → A) ∘ (ι R) = f ↔ g = lift R f | by { rw [← (lift R).symm_apply_eq, lift], refl } | theorem | free_algebra.lift_unique | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift",
"lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_ι (g : free_algebra R X →ₐ[R] A) :
lift R ((g : free_algebra R X → A) ∘ (ι R)) = g | by { rw ←lift_symm_apply, exact (lift R).apply_symm_apply g } | theorem | free_algebra.lift_comp_ι | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {f g : free_algebra R X →ₐ[R] A}
(w : ((f : free_algebra R X → A) ∘ (ι R)) = ((g : free_algebra R X → A) ∘ (ι R))) : f = g | begin
rw [←lift_symm_apply, ←lift_symm_apply] at w,
exact (lift R).symm.injective w,
end | theorem | free_algebra.hom_ext | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"hom_ext",
"lift"
] | See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_monoid_algebra_free_monoid : free_algebra R X ≃ₐ[R] monoid_algebra R (free_monoid X) | alg_equiv.of_alg_hom
(lift R (λ x, (monoid_algebra.of R (free_monoid X)) (free_monoid.of x)))
((monoid_algebra.lift R (free_monoid X) (free_algebra R X)) (free_monoid.lift (ι R)))
begin
apply monoid_algebra.alg_hom_ext, intro x,
apply free_monoid.rec_on x,
{ simp, refl, },
{ intros x y ih, simp at ih, simp ... | def | free_algebra.equiv_monoid_algebra_free_monoid | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"alg_equiv.of_alg_hom",
"free_algebra",
"free_monoid",
"free_monoid.lift",
"free_monoid.of",
"free_monoid.rec_on",
"ih",
"lift",
"monoid_algebra",
"monoid_algebra.alg_hom_ext",
"monoid_algebra.lift",
"monoid_algebra.of"
] | The free algebra on `X` is "just" the monoid algebra on the free monoid on `X`.
This would be useful when constructing linear maps out of a free algebra,
for example. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_inv : free_algebra R X →ₐ[R] R | lift R (0 : X → R) | def | free_algebra.algebra_map_inv | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift"
] | The left-inverse of `algebra_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_left_inverse :
function.left_inverse algebra_map_inv (algebra_map R $ free_algebra R X) | λ x, by simp [algebra_map_inv] | lemma | free_algebra.algebra_map_left_inverse | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra_map",
"free_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_inj (x y : R) :
algebra_map R (free_algebra R X) x = algebra_map R (free_algebra R X) y ↔ x = y | algebra_map_left_inverse.injective.eq_iff | lemma | free_algebra.algebra_map_inj | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra_map",
"free_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_zero_iff (x : R) : algebra_map R (free_algebra R X) x = 0 ↔ x = 0 | map_eq_zero_iff (algebra_map _ _) algebra_map_left_inverse.injective | lemma | free_algebra.algebra_map_eq_zero_iff | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra_map",
"free_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_one_iff (x : R) : algebra_map R (free_algebra R X) x = 1 ↔ x = 1 | map_eq_one_iff (algebra_map _ _) algebra_map_left_inverse.injective | lemma | free_algebra.algebra_map_eq_one_iff | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra_map",
"free_algebra",
"map_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_injective [nontrivial R] : function.injective (ι R : X → free_algebra R X) | λ x y hoxy, classical.by_contradiction $ by classical; exact assume hxy : x ≠ y,
let f : free_algebra R X →ₐ[R] R :=
lift R (λ z, if x = z then (1 : R) else 0) in
have hfx1 : f (ι R x) = 1, from (lift_ι_apply _ _).trans $ if_pos rfl,
have hfy1 : f (ι R y) = 1, from hoxy ▸ hfx1,
have hfy0 : f (ι R y) = 0, fr... | lemma | free_algebra.ι_injective | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"lift",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_inj [nontrivial R] (x y : X) : ι R x = ι R y ↔ x = y | ι_injective.eq_iff | lemma | free_algebra.ι_inj | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_ne_algebra_map [nontrivial R] (x : X) (r : R) : ι R x ≠ algebra_map R _ r | λ h,
let f0 : free_algebra R X →ₐ[R] R := lift R 0 in
let f1 : free_algebra R X →ₐ[R] R := lift R 1 in
have hf0 : f0 (ι R x) = 0, from lift_ι_apply _ _,
have hf1 : f1 (ι R x) = 1, from lift_ι_apply _ _,
begin
rw [h, f0.commutes, algebra.id.map_eq_self] at hf0,
rw [h, f1.commutes, algebra.id.map_eq_sel... | lemma | free_algebra.ι_ne_algebra_map | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra.id.map_eq_self",
"algebra_map",
"free_algebra",
"lift",
"nontrivial",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_ne_zero [nontrivial R] (x : X) : ι R x ≠ 0 | ι_ne_algebra_map x 0 | lemma | free_algebra.ι_ne_zero | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_ne_one [nontrivial R] (x : X) : ι R x ≠ 1 | ι_ne_algebra_map x 1 | lemma | free_algebra.ι_ne_one | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction {C : free_algebra R X → Prop}
(h_grade0 : ∀ r, C (algebra_map R (free_algebra R X) r))
(h_grade1 : ∀ x, C (ι R x))
(h_mul : ∀ a b, C a → C b → C (a * b))
(h_add : ∀ a b, C a → C b → C (a + b))
(a : free_algebra R X) :
C a | begin
-- the arguments are enough to construct a subalgebra, and a mapping into it from X
let s : subalgebra R (free_algebra R X) :=
{ carrier := C,
mul_mem' := h_mul,
add_mem' := h_add,
algebra_map_mem' := h_grade0, },
let of : X → s := subtype.coind (ι R) h_grade1,
-- the mapping through the sub... | lemma | free_algebra.induction | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"alg_hom.ext_iff",
"alg_hom.id",
"algebra_map",
"free_algebra",
"lift",
"subalgebra",
"subtype.coind",
"subtype.prop"
] | An induction principle for the free algebra.
If `C` holds for the `algebra_map` of `r : R` into `free_algebra R X`, the `ι` of `x : X`, and is
preserved under addition and muliplication, then it holds for all of `free_algebra R X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_non_unital_non_assoc_algebra | monoid_algebra R (free_magma X) | abbreviation | free_non_unital_non_assoc_algebra | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_magma",
"monoid_algebra"
] | The free non-unital, non-associative algebra on the type `X` with coefficients in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : X → free_non_unital_non_assoc_algebra R X | (monoid_algebra.of_magma R _) ∘ free_magma.of | def | free_non_unital_non_assoc_algebra.of | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"monoid_algebra.of_magma"
] | The embedding of `X` into the free algebra with coefficients in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (X → A) ≃ (free_non_unital_non_assoc_algebra R X →ₙₐ[R] A) | free_magma.lift.trans (monoid_algebra.lift_magma R) | def | free_non_unital_non_assoc_algebra.lift | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"lift",
"monoid_algebra.lift_magma"
] | The functor `X ↦ free_non_unital_non_assoc_algebra R X` from the category of types to the
category of non-unital, non-associative algebras over `R` is adjoint to the forgetful functor in the
other direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_symm_apply (F : free_non_unital_non_assoc_algebra R X →ₙₐ[R] A) :
(lift R).symm F = F ∘ (of R) | rfl | lemma | free_non_unital_non_assoc_algebra.lift_symm_apply | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_lift (f : X → A) : (lift R f) ∘ (of R) = f | (lift R).left_inv f | lemma | free_non_unital_non_assoc_algebra.of_comp_lift | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique
(f : X → A) (F : free_non_unital_non_assoc_algebra R X →ₙₐ[R] A) :
F ∘ (of R) = f ↔ F = lift R f | (lift R).symm_apply_eq | lemma | free_non_unital_non_assoc_algebra.lift_unique | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"lift",
"lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_apply (f : X → A) (x) : lift R f (of R x) = f x | congr_fun (of_comp_lift _ f) x | lemma | free_non_unital_non_assoc_algebra.lift_of_apply | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of (F : free_non_unital_non_assoc_algebra R X →ₙₐ[R] A) :
lift R (F ∘ (of R)) = F | (lift R).apply_symm_apply F | lemma | free_non_unital_non_assoc_algebra.lift_comp_of | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {F₁ F₂ : free_non_unital_non_assoc_algebra R X →ₙₐ[R] A}
(h : ∀ x, F₁ (of R x) = F₂ (of R x)) : F₁ = F₂ | (lift R).symm.injective $ funext h | lemma | free_non_unital_non_assoc_algebra.hom_ext | algebra | src/algebra/free_non_unital_non_assoc_algebra.lean | [
"algebra.free",
"algebra.monoid_algebra.basic"
] | [
"free_non_unital_non_assoc_algebra",
"hom_ext",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_succ {x : α} {n : ℕ} :
∑ i in range (n + 1), x ^ i = x * ∑ i in range n, x ^ i + 1 | by simp only [mul_sum, ←pow_succ, sum_range_succ', pow_zero] | lemma | geom_sum_succ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_succ' {x : α} {n : ℕ} :
∑ i in range (n + 1), x ^ i = x ^ n + ∑ i in range n, x ^ i | (sum_range_succ _ _).trans (add_comm _ _) | lemma | geom_sum_succ' | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_zero (x : α) :
∑ i in range 0, x ^ i = 0 | rfl | theorem | geom_sum_zero | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_one (x : α) :
∑ i in range 1, x ^ i = 1 | by simp [geom_sum_succ'] | theorem | geom_sum_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_two {x : α} : ∑ i in range 2, x ^ i = x + 1 | by simp [geom_sum_succ'] | lemma | geom_sum_two | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_geom_sum : ∀ {n}, ∑ i in range n, (0 : α) ^ i = if n = 0 then 0 else 1 | | 0 := by simp
| 1 := by simp
| (n+2) := by { rw geom_sum_succ', simp [zero_geom_sum] } | lemma | zero_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_geom_sum (n : ℕ) : ∑ i in range n, (1 : α) ^ i = n | by simp | lemma | one_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_geom_sum (x : α) (n : ℕ) :
op (∑ i in range n, x ^ i) = ∑ i in range n, (op x) ^ i | by simp | lemma | op_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_geom_sum₂ (x y : α) (n : ℕ) :
op (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) =
∑ i in range n, (op y) ^ i * ((op x) ^ (n - 1 - i)) | begin
simp only [op_sum, op_mul, op_pow],
rw ← sum_range_reflect,
refine sum_congr rfl (λ j j_in, _),
rw [mem_range, nat.lt_iff_add_one_le] at j_in,
congr,
apply tsub_tsub_cancel_of_le,
exact le_tsub_of_add_le_right j_in
end | lemma | op_geom_sum₂ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"le_tsub_of_add_le_right",
"nat.lt_iff_add_one_le",
"tsub_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_with_one (x : α) (n : ℕ) :
∑ i in range n, x ^ i * (1 ^ (n - 1 - i)) = ∑ i in range n, x ^ i | sum_congr rfl (λ i _, by { rw [one_pow, mul_one] }) | theorem | geom_sum₂_with_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"mul_one",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_mul_add {x y : α} (h : commute x y) (n : ℕ) :
(∑ i in range n, (x + y) ^ i * (y ^ (n - 1 - i))) * x + y ^ n = (x + y) ^ n | begin
let f := λ (m i : ℕ), (x + y) ^ i * y ^ (m - 1 - i),
change (∑ i in range n, (f n) i) * x + y ^ n = (x + y) ^ n,
induction n with n ih,
{ rw [range_zero, sum_empty, zero_mul, zero_add, pow_zero, pow_zero] },
{ have f_last : f (n + 1) n = (x + y) ^ n :=
by { dsimp [f],
rw [← tsub_add_eq_ts... | theorem | commute.geom_sum₂_mul_add | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"add_tsub_cancel_of_le",
"add_tsub_cancel_right",
"commute",
"commute.refl",
"ih",
"mul_assoc",
"mul_one",
"pow_succ",
"pow_zero",
"tsub_add_eq_tsub_tsub",
"tsub_self",
"zero_mul"
] | $x^n-y^n = (x-y) \sum x^ky^{n-1-k}$ reformulated without `-` signs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_one_geom_sum [ring α] {n : ℕ} :
∑ i in range n, (-1 : α) ^ i = if even n then 0 else 1 | begin
induction n with k hk,
{ simp },
{ simp only [geom_sum_succ', nat.even_add_one, hk],
split_ifs,
{ rw [h.neg_one_pow, add_zero] },
{ rw [(nat.odd_iff_not_even.2 h).neg_one_pow, neg_add_self] } }
end | lemma | neg_one_geom_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"geom_sum_succ'",
"nat.even_add_one",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_self {α : Type*} [comm_ring α] (x : α) (n : ℕ) :
∑ i in range n, x ^ i * (x ^ (n - 1 - i)) = n * x ^ (n-1) | calc ∑ i in finset.range n, x ^ i * x ^ (n - 1 - i)
= ∑ i in finset.range n, x ^ (i + (n - 1 - i)) : by simp_rw [← pow_add]
... = ∑ i in finset.range n, x ^ (n - 1) : finset.sum_congr rfl
(λ i hi, congr_arg _ $ add_tsub_cancel_of_le $ nat.le_pred_of_lt $ finset.mem_range.1 hi)
... = (finset.range n).card • (x ^ ... | theorem | geom_sum₂_self | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"add_tsub_cancel_of_le",
"comm_ring",
"finset.card_range",
"finset.range",
"nat.le_pred_of_lt",
"nsmul_eq_mul",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_mul_add [comm_semiring α] (x y : α) (n : ℕ) :
(∑ i in range n, (x + y) ^ i * (y ^ (n - 1 - i))) * x + y ^ n = (x + y) ^ n | (commute.all x y).geom_sum₂_mul_add n | theorem | geom_sum₂_mul_add | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_semiring",
"commute.all"
] | $x^n-y^n = (x-y) \sum x^ky^{n-1-k}$ reformulated without `-` signs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_mul_add [semiring α] (x : α) (n : ℕ) :
(∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n | begin
have := (commute.one_right x).geom_sum₂_mul_add n,
rw [one_pow, geom_sum₂_with_one] at this,
exact this
end | theorem | geom_sum_mul_add | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.one_right",
"geom_sum₂_mul_add",
"geom_sum₂_with_one",
"one_pow",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.geom_sum₂_mul [ring α] {x y : α} (h : commute x y) (n : ℕ) :
(∑ i in range n, x ^ i * (y ^ (n - 1 - i))) * (x - y) = x ^ n - y ^ n | begin
have := (h.sub_left (commute.refl y)).geom_sum₂_mul_add n,
rw [sub_add_cancel] at this,
rw [← this, add_sub_cancel]
end | theorem | commute.geom_sum₂_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"commute.refl",
"geom_sum₂_mul_add",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.mul_neg_geom_sum₂ [ring α] {x y : α} (h : commute x y) (n : ℕ) :
(y - x) * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = y ^ n - x ^ n | begin
apply op_injective,
simp only [op_mul, op_sub, op_geom_sum₂, op_pow],
exact (commute.op h.symm).geom_sum₂_mul n
end | lemma | commute.mul_neg_geom_sum₂ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"commute.op",
"geom_sum₂_mul",
"op_geom_sum₂",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.mul_geom_sum₂ [ring α] {x y : α} (h : commute x y) (n : ℕ) :
(x - y) * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = x ^ n - y ^ n | by rw [← neg_sub (y ^ n), ← h.mul_neg_geom_sum₂, ← neg_mul, neg_sub] | lemma | commute.mul_geom_sum₂ | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"neg_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum₂_mul [comm_ring α] (x y : α) (n : ℕ) :
(∑ i in range n, x ^ i * (y ^ (n - 1 - i))) * (x - y) = x ^ n - y ^ n | (commute.all x y).geom_sum₂_mul n | theorem | geom_sum₂_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_ring",
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_dvd_pow_sub_pow [comm_ring α] (x y : α) (n : ℕ) : x - y ∣ x ^ n - y ^ n | dvd.intro_left _ (geom_sum₂_mul x y n) | theorem | sub_dvd_pow_sub_pow | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_ring",
"dvd.intro_left",
"geom_sum₂_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_sub_dvd_pow_sub_pow (x y n : ℕ) : x - y ∣ x ^ n - y ^ n | begin
cases le_or_lt y x with h,
{ have : y ^ n ≤ x ^ n := nat.pow_le_pow_of_le_left h _,
exact_mod_cast sub_dvd_pow_sub_pow (x : ℤ) ↑y n },
{ have : x ^ n ≤ y ^ n := nat.pow_le_pow_of_le_left h.le _,
exact (nat.sub_eq_zero_of_le this).symm ▸ dvd_zero (x - y) }
end | theorem | nat_sub_dvd_pow_sub_pow | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"dvd_zero",
"nat.pow_le_pow_of_le_left",
"sub_dvd_pow_sub_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.add_dvd_pow_add_pow [comm_ring α] (x y : α) {n : ℕ} (h : odd n) :
x + y ∣ x ^ n + y ^ n | begin
have h₁ := geom_sum₂_mul x (-y) n,
rw [odd.neg_pow h y, sub_neg_eq_add, sub_neg_eq_add] at h₁,
exact dvd.intro_left _ h₁,
end | theorem | odd.add_dvd_pow_add_pow | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"comm_ring",
"dvd.intro_left",
"geom_sum₂_mul",
"odd",
"odd.neg_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
odd.nat_add_dvd_pow_add_pow (x y : ℕ) {n : ℕ} (h : odd n) : x + y ∣ x ^ n + y ^ n | by exact_mod_cast odd.add_dvd_pow_add_pow (x : ℤ) ↑y h | theorem | odd.nat_add_dvd_pow_add_pow | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"odd",
"odd.add_dvd_pow_add_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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