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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 ... | 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. | 105 | 128 | false | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_algebra := quot (free_algebra.rel R X) | 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`. | 135 | 135 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: semiring (free_algebra R X) :=
{ add := quot.map₂ (+) (λ _ _ _, rel.add_compat_right) (λ _ _ _, rel.add_compat_left),
add_assoc := by { rintros ⟨⟩ ⟨⟩ ⟨⟩, exact quot.sound rel.add_assoc },
zero := quot.mk _ 0,
zero_add := by { rintro ⟨⟩, exact quot.sound rel.zero_add },
add_zero := begin
rintros ⟨⟩,
ch... | : semiring (free_algebra R X) | { add := quot.map₂ (+) (λ _ _ _, rel.add_compat_right) (λ _ _ _, rel.add_compat_left),
add_assoc := by { rintros ⟨⟩ ⟨⟩ ⟨⟩, exact quot.sound rel.add_assoc },
zero := quot.mk _ 0,
zero_add := by { rintro ⟨⟩, exact quot.sound rel.zero_add },
add_zero := begin
rintros ⟨⟩,
change quot.mk _ _ = _,
rw [quo... | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"left_distrib",
"mul_assoc",
"mul_one",
"mul_zero",
"one_mul",
"quot.map₂",
"right_distrib",
"semiring",
"zero_mul"
] | null | 143 | 162 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: inhabited (free_algebra R X) := ⟨0⟩ | : inhabited (free_algebra R X) | ⟨0⟩ | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra"
] | null | 164 | 164 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: has_smul R (free_algebra R X) :=
{ smul := λ r, quot.map ((*) ↑r) (λ a b, rel.mul_compat_right) } | : has_smul R (free_algebra R X) | { smul := λ r, quot.map ((*) ↑r) (λ a b, rel.mul_compat_right) } | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"has_smul",
"quot.map"
] | null | 166 | 167 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: algebra R (free_algebra R X) :=
{ to_fun := λ r, quot.mk _ r,
map_one' := rfl,
map_mul' := λ _ _, quot.sound rel.mul_scalar,
map_zero' := rfl,
map_add' := λ _ _, quot.sound rel.add_scalar,
commutes' := λ _, by { rintros ⟨⟩, exact quot.sound rel.central_scalar },
smul_def' := λ _ _, rfl } | : algebra R (free_algebra R X) | { to_fun := λ r, quot.mk _ r,
map_one' := rfl,
map_mul' := λ _ _, quot.sound rel.mul_scalar,
map_zero' := rfl,
map_add' := λ _ _, quot.sound rel.add_scalar,
commutes' := λ _, by { rintros ⟨⟩, exact quot.sound rel.central_scalar },
smul_def' := λ _ _, rfl } | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra",
"free_algebra"
] | null | 169 | 176 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{S : Type*} [comm_ring S] : ring (free_algebra S X) := algebra.semiring_to_ring S | {S : Type*} [comm_ring S] : ring (free_algebra S X) | algebra.semiring_to_ring S | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"algebra.semiring_to_ring",
"comm_ring",
"free_algebra",
"ring"
] | null | 178 | 178 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι : X → free_algebra R X := λ m, quot.mk _ m | 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`. | 185 | 185 | true | 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] | 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"
] | null | 187 | 188 | true | 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_... | 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` | 193 | 231 | true | 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 ... | 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`. | 237 | 257 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_aux_eq (f : X → A) : lift_aux R f = lift R f :=
by { rw [lift], refl } | 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"
] | null | 259 | 260 | true | 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 } | 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"
] | null | 262 | 264 | true | 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 } | 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"
] | null | 268 | 271 | true | 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 } | 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"
] | null | 273 | 276 | true | 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 } | 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"
] | null | 278 | 281 | true | 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 } | 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"
] | null | 292 | 295 | true | 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 | 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]. | 298 | 304 | true | 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, ... | 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. | 312 | 323 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[nontrivial R] : nontrivial (free_algebra R X) :=
equiv_monoid_algebra_free_monoid.surjective.nontrivial | [nontrivial R] : nontrivial (free_algebra R X) | equiv_monoid_algebra_free_monoid.surjective.nontrivial | instance | algebra | src/algebra/free_algebra.lean | [
"algebra.algebra.subalgebra.basic",
"algebra.monoid_algebra.basic"
] | [
"free_algebra",
"nontrivial"
] | null | 325 | 326 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_inv : free_algebra R X →ₐ[R] R :=
lift R (0 : X → R) | 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`. | 331 | 332 | true | 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] | 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"
] | null | 334 | 336 | true | 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 | 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"
] | null | 338 | 340 | true | 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 | 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"
] | null | 342 | 343 | true | 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 | 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"
] | null | 345 | 346 | true | 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... | 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"
] | null | 349 | 356 | true | 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 | 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"
] | null | 358 | 359 | true | 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... | 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"
] | null | 361 | 371 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_ne_zero [nontrivial R] (x : X) : ι R x ≠ 0 :=
ι_ne_algebra_map x 0 | 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"
] | null | 373 | 374 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_ne_one [nontrivial R] (x : X) : ι R x ≠ 1 :=
ι_ne_algebra_map x 1 | 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"
] | null | 376 | 377 | true | 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 mapp... | 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`. | 392 | 416 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_non_unital_non_assoc_algebra := monoid_algebra R (free_magma X) | 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`. | 48 | 48 | true | 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 | 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`. | 55 | 56 | true | 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) | 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. | 64 | 65 | true | 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 | 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"
] | null | 67 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_comp_lift (f : X → A) : (lift R f) ∘ (of R) = f :=
(lift R).left_inv f | 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"
] | null | 71 | 72 | true | 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 | 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"
] | null | 74 | 77 | true | 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 | 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"
] | null | 79 | 80 | true | 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 | 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"
] | null | 82 | 84 | true | 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 | 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"
] | null | 86 | 88 | true | 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] | 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"
] | null | 43 | 45 | true | 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 _ _) | 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"
] | [] | null | 47 | 49 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_zero (x : α) :
∑ i in range 0, x ^ i = 0 := rfl | 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"
] | [] | null | 51 | 52 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_one (x : α) :
∑ i in range 1, x ^ i = 1 :=
by simp [geom_sum_succ'] | 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'"
] | null | 54 | 56 | true | 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'] | 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'"
] | null | 58 | 59 | true | 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] } | 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'"
] | null | 61 | 64 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_geom_sum (n : ℕ) : ∑ i in range n, (1 : α) ^ i = n :=
by simp | 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"
] | [] | null | 66 | 67 | true | 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 | 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"
] | [] | null | 69 | 71 | true | 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 tsu... | 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"
] | null | 73 | 84 | true | 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] }) | 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"
] | null | 86 | 88 | true | 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_emp... | 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. | 91 | 117 | true | 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 | 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"
] | null | 121 | 130 | true | 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 ... | 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"
] | null | 132 | 139 | true | 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 | 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. | 142 | 144 | true | 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 | 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"
] | null | 146 | 152 | true | 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 | 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"
] | null | 154 | 160 | true | 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 | 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"
] | null | 162 | 168 | true | 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] | 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"
] | null | 170 | 172 | true | 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 | 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"
] | null | 174 | 176 | true | 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) | 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"
] | null | 178 | 179 | true | 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 ▸ ... | 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"
] | null | 181 | 188 | true | 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 | 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"
] | null | 190 | 196 | true | 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 | 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"
] | null | 198 | 199 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_mul [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (x - 1) = x ^ n - 1 :=
begin
have := (commute.one_right x).geom_sum₂_mul n,
rw [one_pow, geom_sum₂_with_one] at this,
exact this
end | geom_sum_mul [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (x - 1) = x ^ n - 1 | begin
have := (commute.one_right x).geom_sum₂_mul n,
rw [one_pow, geom_sum₂_with_one] at this,
exact this
end | theorem | geom_sum_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.one_right",
"geom_sum₂_mul",
"geom_sum₂_with_one",
"one_pow",
"ring"
] | null | 201 | 207 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_geom_sum [ring α] (x : α) (n : ℕ) :
(x - 1) * (∑ i in range n, x ^ i) = x ^ n - 1 :=
op_injective $ by simpa using geom_sum_mul (op x) n | mul_geom_sum [ring α] (x : α) (n : ℕ) :
(x - 1) * (∑ i in range n, x ^ i) = x ^ n - 1 | op_injective $ by simpa using geom_sum_mul (op x) n | lemma | mul_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_mul",
"ring"
] | null | 209 | 211 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_mul_neg [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n :=
begin
have := congr_arg has_neg.neg (geom_sum_mul x n),
rw [neg_sub, ← mul_neg, neg_sub] at this,
exact this
end | geom_sum_mul_neg [ring α] (x : α) (n : ℕ) :
(∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n | begin
have := congr_arg has_neg.neg (geom_sum_mul x n),
rw [neg_sub, ← mul_neg, neg_sub] at this,
exact this
end | theorem | geom_sum_mul_neg | 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_mul",
"mul_neg",
"ring"
] | null | 213 | 219 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_neg_geom_sum [ring α] (x : α) (n : ℕ) :
(1 - x) * (∑ i in range n, x ^ i) = 1 - x ^ n :=
op_injective $ by simpa using geom_sum_mul_neg (op x) n | mul_neg_geom_sum [ring α] (x : α) (n : ℕ) :
(1 - x) * (∑ i in range n, x ^ i) = 1 - x ^ n | op_injective $ by simpa using geom_sum_mul_neg (op x) n | lemma | mul_neg_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_mul_neg",
"ring"
] | null | 221 | 223 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.geom_sum₂_comm {α : Type u} [semiring α] {x y : α} (n : ℕ)
(h : commute x y) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) :=
begin
cases n, { simp },
simp only [nat.succ_eq_add_one, nat.add_sub_cancel],
rw ← finset.sum_flip,
refine finset.sum_congr rfl (λ i hi,... | commute.geom_sum₂_comm {α : Type u} [semiring α] {x y : α} (n : ℕ)
(h : commute x y) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) | begin
cases n, { simp },
simp only [nat.succ_eq_add_one, nat.add_sub_cancel],
rw ← finset.sum_flip,
refine finset.sum_congr rfl (λ i hi, _),
simpa [nat.sub_sub_self (nat.succ_le_succ_iff.mp (finset.mem_range.mp hi))] using h.pow_pow _ _
end | lemma | commute.geom_sum₂_comm | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"semiring"
] | null | 225 | 234 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum₂_comm {α : Type u} [comm_semiring α] (x y : α) (n : ℕ) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) :=
(commute.all x y).geom_sum₂_comm n | geom_sum₂_comm {α : Type u} [comm_semiring α] (x y : α) (n : ℕ) :
∑ i in range n, x ^ i * y ^ (n - 1 - i) = ∑ i in range n, y ^ i * x ^ (n - 1 - i) | (commute.all x y).geom_sum₂_comm n | lemma | geom_sum₂_comm | 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"
] | null | 236 | 238 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.geom_sum₂ [division_ring α] {x y : α} (h' : commute x y) (h : x ≠ y)
(n : ℕ) : (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) :=
have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h'.geom_sum₂_mul, mul_div_cancel _ this] | commute.geom_sum₂ [division_ring α] {x y : α} (h' : commute x y) (h : x ≠ y)
(n : ℕ) : (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) | have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h'.geom_sum₂_mul, mul_div_cancel _ this] | theorem | commute.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",
"division_ring",
"mul_div_cancel"
] | null | 240 | 243 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom₂_sum [field α] {x y : α} (h : x ≠ y) (n : ℕ) :
(∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) :=
(commute.all x y).geom_sum₂ h n | geom₂_sum [field α] {x y : α} (h : x ≠ y) (n : ℕ) :
(∑ i in range n, x ^ i * (y ^ (n - 1 - i))) = (x ^ n - y ^ n) / (x - y) | (commute.all x y).geom_sum₂ h n | theorem | geom₂_sum | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.all",
"field"
] | null | 245 | 247 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_eq [division_ring α] {x : α} (h : x ≠ 1) (n : ℕ) :
(∑ i in range n, x ^ i) = (x ^ n - 1) / (x - 1) :=
have x - 1 ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← geom_sum_mul, mul_div_cancel _ this] | geom_sum_eq [division_ring α] {x : α} (h : x ≠ 1) (n : ℕ) :
(∑ i in range n, x ^ i) = (x ^ n - 1) / (x - 1) | have x - 1 ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← geom_sum_mul, mul_div_cancel _ this] | theorem | geom_sum_eq | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"division_ring",
"geom_sum_mul",
"mul_div_cancel"
] | null | 249 | 252 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.mul_geom_sum₂_Ico [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) :=
begin
rw [sum_Ico_eq_sub _ hmn],
have : ∑ k in range m, x ^ k * y ^ (n - 1 - k)
= ∑ k in range m, x ^ k * (y ^ (n - m) * y ^ (m... | commute.mul_geom_sum₂_Ico [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) | begin
rw [sum_Ico_eq_sub _ hmn],
have : ∑ k in range m, x ^ k * y ^ (n - 1 - k)
= ∑ k in range m, x ^ k * (y ^ (n - m) * y ^ (m - 1 - k)),
{ refine sum_congr rfl (λ j j_in, _),
rw ← pow_add,
congr,
rw [mem_range, nat.lt_iff_add_one_le, add_comm] at j_in,
have h' : n - m + (m - (1 + j)) = n - (... | theorem | commute.mul_geom_sum₂_Ico | 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",
"commute",
"finset.Ico",
"mul_assoc",
"nat.lt_iff_add_one_le",
"pow_add",
"pow_mul_comm",
"ring",
"tsub_add_eq_tsub_tsub",
"tsub_add_tsub_cancel"
] | null | 254 | 273 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.geom_sum₂_succ_eq {α : Type u} [ring α] {x y : α}
(h : commute x y) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) :=
begin
simp_rw [mul_sum, sum_range_succ_comm, tsub_self, pow_zero, mul_one, add_right_inj, ←mul_assoc,
(h.symm.pow_r... | commute.geom_sum₂_succ_eq {α : Type u} [ring α] {x y : α}
(h : commute x y) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) | begin
simp_rw [mul_sum, sum_range_succ_comm, tsub_self, pow_zero, mul_one, add_right_inj, ←mul_assoc,
(h.symm.pow_right _).eq, mul_assoc, ←pow_succ],
refine sum_congr rfl (λ i hi, _),
suffices : n - 1 - i + 1 = n - i, { rw this },
cases n,
{ exact absurd (list.mem_range.mp hi) i.not_lt_zero },
{ rw [tsu... | theorem | commute.geom_sum₂_succ_eq | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"mul_assoc",
"mul_one",
"nat.le_pred_of_lt",
"pow_zero",
"ring",
"tsub_add_cancel_of_le",
"tsub_add_eq_add_tsub",
"tsub_self"
] | null | 275 | 288 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum₂_succ_eq {α : Type u} [comm_ring α] (x y : α) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) :=
(commute.all x y).geom_sum₂_succ_eq | geom_sum₂_succ_eq {α : Type u} [comm_ring α] (x y : α) {n : ℕ} :
∑ i in range (n + 1), x ^ i * (y ^ (n - i)) =
x ^ n + y * (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) | (commute.all x y).geom_sum₂_succ_eq | theorem | geom_sum₂_succ_eq | 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"
] | null | 290 | 293 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_geom_sum₂_Ico [comm_ring α] (x y : α) {m n : ℕ} (hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) :=
(commute.all x y).mul_geom_sum₂_Ico hmn | mul_geom_sum₂_Ico [comm_ring α] (x y : α) {m n : ℕ} (hmn : m ≤ n) :
(x - y) * (∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) = x ^ n - x ^ m * y ^ (n - m) | (commute.all x y).mul_geom_sum₂_Ico hmn | theorem | mul_geom_sum₂_Ico | 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",
"finset.Ico"
] | null | 295 | 297 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.geom_sum₂_Ico_mul [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ (n - m) * x ^ m :=
begin
apply op_injective,
simp only [op_sub, op_mul, op_pow, op_sum],
have : ∑ k in Ico m n, op y ^ (n - 1 - k) * op x ^ k
=... | commute.geom_sum₂_Ico_mul [ring α] {x y : α} (h : commute x y) {m n : ℕ}
(hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ (n - m) * x ^ m | begin
apply op_injective,
simp only [op_sub, op_mul, op_pow, op_sum],
have : ∑ k in Ico m n, op y ^ (n - 1 - k) * op x ^ k
= ∑ k in Ico m n, op x ^ k * op y ^ (n - 1 - k),
{ refine sum_congr rfl (λ k k_in, _),
apply commute.pow_pow (commute.op h.symm) },
rw this,
exact (commute.op h).mul_geom_sum₂_I... | theorem | commute.geom_sum₂_Ico_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.op",
"commute.pow_pow",
"finset.Ico",
"mul_geom_sum₂_Ico",
"ring"
] | null | 299 | 311 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_Ico_mul [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (x - 1) = x^n - x^m :=
by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul, geom_sum_mul, sub_sub_sub_cancel_right] | geom_sum_Ico_mul [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (x - 1) = x^n - x^m | by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul, geom_sum_mul, sub_sub_sub_cancel_right] | theorem | geom_sum_Ico_mul | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"finset.Ico",
"geom_sum_mul",
"ring"
] | null | 313 | 316 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_Ico_mul_neg [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (1 - x) = x^m - x^n :=
by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left] | geom_sum_Ico_mul_neg [ring α] (x : α) {m n : ℕ} (hmn : m ≤ n) :
(∑ i in finset.Ico m n, x ^ i) * (1 - x) = x^m - x^n | by rw [sum_Ico_eq_sub _ hmn, sub_mul,
geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left] | theorem | geom_sum_Ico_mul_neg | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"finset.Ico",
"geom_sum_mul_neg",
"ring"
] | null | 318 | 321 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commute.geom_sum₂_Ico [division_ring α] {x y : α} (h : commute x y) (hxy : x ≠ y)
{m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) :=
have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h.geom_sum₂_Ico_mul hmn, mul_div_can... | commute.geom_sum₂_Ico [division_ring α] {x y : α} (h : commute x y) (hxy : x ≠ y)
{m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) | have x - y ≠ 0, by simp [*, -sub_eq_add_neg, sub_eq_iff_eq_add] at *,
by rw [← h.geom_sum₂_Ico_mul hmn, mul_div_cancel _ this] | theorem | commute.geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute",
"division_ring",
"finset.Ico",
"mul_div_cancel"
] | null | 323 | 327 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum₂_Ico [field α] {x y : α} (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) :=
(commute.all x y).geom_sum₂_Ico hxy hmn | geom_sum₂_Ico [field α] {x y : α} (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m ) / (x - y) | (commute.all x y).geom_sum₂_Ico hxy hmn | theorem | geom_sum₂_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"commute.all",
"field",
"finset.Ico"
] | null | 329 | 331 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_Ico [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1) :=
by simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same,
sub_sub_sub_cancel_right] | geom_sum_Ico [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1) | by simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same,
sub_sub_sub_cancel_right] | theorem | geom_sum_Ico | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_sub_div_same",
"division_ring",
"finset.Ico",
"geom_sum_eq"
] | null | 333 | 336 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_Ico' [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) :=
by { simp only [geom_sum_Ico hx hmn], convert neg_div_neg_eq (x^m - x^n) (1-x); abel } | geom_sum_Ico' [division_ring α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i in finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) | by { simp only [geom_sum_Ico hx hmn], convert neg_div_neg_eq (x^m - x^n) (1-x); abel } | theorem | geom_sum_Ico' | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"division_ring",
"finset.Ico",
"geom_sum_Ico",
"neg_div_neg_eq"
] | null | 338 | 340 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_Ico_le_of_lt_one [linear_ordered_field α]
{x : α} (hx : 0 ≤ x) (h'x : x < 1) {m n : ℕ} :
∑ i in Ico m n, x ^ i ≤ x ^ m / (1 - x) :=
begin
rcases le_or_lt m n with hmn | hmn,
{ rw geom_sum_Ico' h'x.ne hmn,
apply div_le_div (pow_nonneg hx _) _ (sub_pos.2 h'x) le_rfl,
simpa using pow_nonneg hx _ }... | geom_sum_Ico_le_of_lt_one [linear_ordered_field α]
{x : α} (hx : 0 ≤ x) (h'x : x < 1) {m n : ℕ} :
∑ i in Ico m n, x ^ i ≤ x ^ m / (1 - x) | begin
rcases le_or_lt m n with hmn | hmn,
{ rw geom_sum_Ico' h'x.ne hmn,
apply div_le_div (pow_nonneg hx _) _ (sub_pos.2 h'x) le_rfl,
simpa using pow_nonneg hx _ },
{ rw [Ico_eq_empty, sum_empty],
{ apply div_nonneg (pow_nonneg hx _),
simpa using h'x.le },
{ simpa using hmn.le } },
end | lemma | geom_sum_Ico_le_of_lt_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_le_div",
"div_nonneg",
"geom_sum_Ico'",
"le_rfl",
"linear_ordered_field",
"pow_nonneg"
] | null | 342 | 354 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_inv [division_ring α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
(∑ i in range n, x⁻¹ ^ i) = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) :=
have h₁ : x⁻¹ ≠ 1, by rwa [inv_eq_one_div, ne.def, div_eq_iff_mul_eq hx0, one_mul],
have h₂ : x⁻¹ - 1 ≠ 0, from mt sub_eq_zero.1 h₁,
have h₃ : x - 1 ≠ 0, from mt sub_eq_zero.1 hx1,... | geom_sum_inv [division_ring α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
(∑ i in range n, x⁻¹ ^ i) = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) | have h₁ : x⁻¹ ≠ 1, by rwa [inv_eq_one_div, ne.def, div_eq_iff_mul_eq hx0, one_mul],
have h₂ : x⁻¹ - 1 ≠ 0, from mt sub_eq_zero.1 h₁,
have h₃ : x - 1 ≠ 0, from mt sub_eq_zero.1 hx1,
have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=
nat.rec_on n (by simp)
(λ n h, by rw [pow_succ, mul_inv_rev, ←mul_assoc, h, mul_assoc, mul_i... | lemma | geom_sum_inv | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"div_eq_iff_mul_eq",
"division_ring",
"geom_sum_eq",
"inv_eq_one_div",
"inv_mul_cancel",
"mul_assoc",
"mul_inv_cancel",
"mul_inv_rev",
"mul_right_inj'",
"one_mul",
"pow_succ"
] | null | 356 | 370 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_geom_sum [semiring α] [semiring β] (x : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i) = ∑ i in range n, (f x) ^ i :=
by simp [f.map_sum] | ring_hom.map_geom_sum [semiring α] [semiring β] (x : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i) = ∑ i in range n, (f x) ^ i | by simp [f.map_sum] | theorem | ring_hom.map_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"
] | [
"semiring"
] | null | 374 | 376 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.map_geom_sum₂ [semiring α] [semiring β] (x y : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) =
∑ i in range n, (f x) ^ i * ((f y) ^ (n - 1 - i)) :=
by simp [f.map_sum] | ring_hom.map_geom_sum₂ [semiring α] [semiring β] (x y : α) (n : ℕ) (f : α →+* β) :
f (∑ i in range n, x ^ i * (y ^ (n - 1 - i))) =
∑ i in range n, (f x) ^ i * ((f y) ^ (n - 1 - i)) | by simp [f.map_sum] | theorem | ring_hom.map_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"
] | [
"semiring"
] | null | 378 | 381 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.pred_mul_geom_sum_le (a b n : ℕ) :
(b - 1) * ∑ i in range n.succ, a/b^i ≤ a * b - a/b^n :=
calc
(b - 1) * (∑ i in range n.succ, a/b^i)
= ∑ i in range n, a/b^(i + 1) * b + a * b
- (∑ i in range n, a/b^i + a/b^n)
: by rw [tsub_mul, mul_comm, sum_mul, one_mul, sum_range_succ',
sum_ran... | nat.pred_mul_geom_sum_le (a b n : ℕ) :
(b - 1) * ∑ i in range n.succ, a/b^i ≤ a * b - a/b^n | calc
(b - 1) * (∑ i in range n.succ, a/b^i)
= ∑ i in range n, a/b^(i + 1) * b + a * b
- (∑ i in range n, a/b^i + a/b^n)
: by rw [tsub_mul, mul_comm, sum_mul, one_mul, sum_range_succ',
sum_range_succ, pow_zero, nat.div_one]
... ≤ ∑ i in range n, a/b^i + a * b - (∑ i in range n, a/b^i + ... | lemma | nat.pred_mul_geom_sum_le | 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_add_eq_tsub_left",
"mul_comm",
"one_mul",
"pow_succ'",
"pow_zero",
"tsub_le_tsub_right",
"tsub_mul"
] | null | 385 | 399 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.geom_sum_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in range n, a/b^i ≤ a * b/(b - 1) :=
begin
refine (nat.le_div_iff_mul_le $ tsub_pos_of_lt hb).2 _,
cases n,
{ rw [sum_range_zero, zero_mul],
exact nat.zero_le _ },
rw mul_comm,
exact (nat.pred_mul_geom_sum_le a b n).trans tsub_le_self,
end | nat.geom_sum_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in range n, a/b^i ≤ a * b/(b - 1) | begin
refine (nat.le_div_iff_mul_le $ tsub_pos_of_lt hb).2 _,
cases n,
{ rw [sum_range_zero, zero_mul],
exact nat.zero_le _ },
rw mul_comm,
exact (nat.pred_mul_geom_sum_le a b n).trans tsub_le_self,
end | lemma | nat.geom_sum_le | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"mul_comm",
"nat.pred_mul_geom_sum_le",
"tsub_le_self",
"tsub_pos_of_lt",
"zero_mul"
] | null | 401 | 410 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.geom_sum_Ico_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in Ico 1 n, a/b^i ≤ a/(b - 1) :=
begin
cases n,
{ rw [Ico_eq_empty_of_le (zero_le_one' ℕ), sum_empty],
exact nat.zero_le _ },
rw ←add_le_add_iff_left a,
calc
a + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i
= a/b^0 + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i... | nat.geom_sum_Ico_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i in Ico 1 n, a/b^i ≤ a/(b - 1) | begin
cases n,
{ rw [Ico_eq_empty_of_le (zero_le_one' ℕ), sum_empty],
exact nat.zero_le _ },
rw ←add_le_add_iff_left a,
calc
a + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i
= a/b^0 + ∑ (i : ℕ) in Ico 1 n.succ, a/b^i : by rw [pow_zero, nat.div_one]
... = ∑ i in range n.succ, a/b^i : begin
rw [... | lemma | nat.geom_sum_Ico_le | 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",
"mul_one",
"nat.geom_sum_le",
"pow_zero",
"tsub_pos_of_lt",
"zero_le_one'"
] | null | 412 | 431 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_pos [strict_ordered_semiring α] (hx : 0 ≤ x) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i :=
sum_pos' (λ k hk, pow_nonneg hx _) ⟨0, mem_range.2 hn.bot_lt, by simp⟩ | geom_sum_pos [strict_ordered_semiring α] (hx : 0 ≤ x) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i | sum_pos' (λ k hk, pow_nonneg hx _) ⟨0, mem_range.2 hn.bot_lt, by simp⟩ | lemma | geom_sum_pos | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"pow_nonneg",
"strict_ordered_semiring"
] | null | 437 | 439 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_pos_and_lt_one [strict_ordered_ring α] (hx : x < 0) (hx' : 0 < x + 1) (hn : 1 < n) :
0 < ∑ i in range n, x ^ i ∧ ∑ i in range n, x ^ i < 1 :=
begin
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ rw geom_sum_two,
exact ⟨hx', (add_lt_iff_neg_right _).2 hx⟩ },
clear hn n,
intros n hn ihn,
... | geom_sum_pos_and_lt_one [strict_ordered_ring α] (hx : x < 0) (hx' : 0 < x + 1) (hn : 1 < n) :
0 < ∑ i in range n, x ^ i ∧ ∑ i in range n, x ^ i < 1 | begin
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ rw geom_sum_two,
exact ⟨hx', (add_lt_iff_neg_right _).2 hx⟩ },
clear hn n,
intros n hn ihn,
rw [geom_sum_succ, add_lt_iff_neg_right, ← neg_lt_iff_pos_add', neg_mul_eq_neg_mul],
exact ⟨mul_lt_one_of_nonneg_of_lt_one_left (neg_nonneg.2 hx.le)
... | lemma | geom_sum_pos_and_lt_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",
"geom_sum_two",
"mul_neg_of_neg_of_pos",
"nat.le_induction",
"neg_mul_eq_neg_mul",
"strict_ordered_ring"
] | null | 441 | 452 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_alternating_of_le_neg_one [strict_ordered_ring α] (hx : x + 1 ≤ 0) (n : ℕ) :
if even n then ∑ i in range n, x ^ i ≤ 0 else 1 ≤ ∑ i in range n, x ^ i :=
begin
have hx0 : x ≤ 0 := (le_add_of_nonneg_right zero_le_one).trans hx,
induction n with n ih,
{ simp only [even_zero, geom_sum_zero, le_refl] },
si... | geom_sum_alternating_of_le_neg_one [strict_ordered_ring α] (hx : x + 1 ≤ 0) (n : ℕ) :
if even n then ∑ i in range n, x ^ i ≤ 0 else 1 ≤ ∑ i in range n, x ^ i | begin
have hx0 : x ≤ 0 := (le_add_of_nonneg_right zero_le_one).trans hx,
induction n with n ih,
{ simp only [even_zero, geom_sum_zero, le_refl] },
simp only [nat.even_add_one, geom_sum_succ],
split_ifs at ih,
{ rw [if_neg (not_not_intro h), le_add_iff_nonneg_left],
exact mul_nonneg_of_nonpos_of_nonpos h... | lemma | geom_sum_alternating_of_le_neg_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",
"geom_sum_zero",
"ih",
"mul_le_mul_of_nonpos_left",
"mul_nonneg_of_nonpos_of_nonpos",
"mul_one",
"nat.even_add_one",
"strict_ordered_ring",
"zero_le_one"
] | null | 454 | 467 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_alternating_of_lt_neg_one [strict_ordered_ring α] (hx : x + 1 < 0) (hn : 1 < n) :
if even n then ∑ i in range n, x ^ i < 0 else 1 < ∑ i in range n, x ^ i :=
begin
have hx0 : x < 0, from ((le_add_iff_nonneg_right _).2 zero_le_one).trans_lt hx,
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ simp... | geom_sum_alternating_of_lt_neg_one [strict_ordered_ring α] (hx : x + 1 < 0) (hn : 1 < n) :
if even n then ∑ i in range n, x ^ i < 0 else 1 < ∑ i in range n, x ^ i | begin
have hx0 : x < 0, from ((le_add_iff_nonneg_right _).2 zero_le_one).trans_lt hx,
refine nat.le_induction _ _ n (show 2 ≤ n, from hn),
{ simp only [geom_sum_two, hx, true_or, even_bit0, if_true_left_eq_or] },
clear hn n,
intros n hn ihn,
simp only [nat.even_add_one, geom_sum_succ],
by_cases hn' : even... | lemma | geom_sum_alternating_of_lt_neg_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"even_bit0",
"geom_sum_succ",
"geom_sum_two",
"mul_lt_mul_of_neg_left",
"mul_one",
"mul_pos_of_neg_of_neg",
"nat.even_add_one",
"nat.le_induction",
"strict_ordered_ring",
"zero_le_one"
] | null | 469 | 485 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_pos' [linear_ordered_ring α] (hx : 0 < x + 1) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i :=
begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
obtain hx' | hx' := lt_or_le x 0,
{ exact (geom_sum_pos_and_lt_one hx' hx n.one_lt_succ_succ).1 },
{ exact geom_sum_pos hx' (by simp only [nat.succ_ne... | geom_sum_pos' [linear_ordered_ring α] (hx : 0 < x + 1) (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i | begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
obtain hx' | hx' := lt_or_le x 0,
{ exact (geom_sum_pos_and_lt_one hx' hx n.one_lt_succ_succ).1 },
{ exact geom_sum_pos hx' (by simp only [nat.succ_ne_zero, ne.def, not_false_iff]) }
end | lemma | geom_sum_pos' | 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_pos",
"geom_sum_pos_and_lt_one",
"linear_ordered_ring"
] | null | 487 | 496 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
odd.geom_sum_pos [linear_ordered_ring α] (h : odd n) :
0 < ∑ i in range n, x ^ i :=
begin
rcases n with (_ | _ | k),
{ exact ((show ¬ odd 0, from dec_trivial) h).elim },
{ simp only [geom_sum_one, zero_lt_one] },
rw nat.odd_iff_not_even at h,
rcases lt_trichotomy (x + 1) 0 with hx | hx | hx,
{ have := geo... | odd.geom_sum_pos [linear_ordered_ring α] (h : odd n) :
0 < ∑ i in range n, x ^ i | begin
rcases n with (_ | _ | k),
{ exact ((show ¬ odd 0, from dec_trivial) h).elim },
{ simp only [geom_sum_one, zero_lt_one] },
rw nat.odd_iff_not_even at h,
rcases lt_trichotomy (x + 1) 0 with hx | hx | hx,
{ have := geom_sum_alternating_of_lt_neg_one hx k.one_lt_succ_succ,
simp only [h, if_false] at ... | lemma | odd.geom_sum_pos | 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_alternating_of_lt_neg_one",
"geom_sum_one",
"geom_sum_pos'",
"linear_ordered_ring",
"nat.odd_iff_not_even",
"neg_one_geom_sum",
"odd",
"zero_lt_one"
] | null | 498 | 511 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_pos_iff [linear_ordered_ring α] (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i ↔ odd n ∨ 0 < x + 1 :=
begin
refine ⟨λ h, _, _⟩,
{ rw [or_iff_not_imp_left, ←not_le, ←nat.even_iff_not_odd],
refine λ hn hx, h.not_le _,
simpa [if_pos hn] using geom_sum_alternating_of_le_neg_one hx n },
{ rintro (hn | hx'... | geom_sum_pos_iff [linear_ordered_ring α] (hn : n ≠ 0) :
0 < ∑ i in range n, x ^ i ↔ odd n ∨ 0 < x + 1 | begin
refine ⟨λ h, _, _⟩,
{ rw [or_iff_not_imp_left, ←not_le, ←nat.even_iff_not_odd],
refine λ hn hx, h.not_le _,
simpa [if_pos hn] using geom_sum_alternating_of_le_neg_one hx n },
{ rintro (hn | hx'),
{ exact hn.geom_sum_pos },
{ exact geom_sum_pos' hx' hn } }
end | lemma | geom_sum_pos_iff | 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_alternating_of_le_neg_one",
"geom_sum_pos'",
"linear_ordered_ring",
"odd",
"or_iff_not_imp_left"
] | null | 513 | 523 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_ne_zero [linear_ordered_ring α] (hx : x ≠ -1) (hn : n ≠ 0) :
∑ i in range n, x ^ i ≠ 0 :=
begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
rw [ne.def, eq_neg_iff_add_eq_zero, ←ne.def] at hx,
obtain h | h := hx.lt_or_lt,
{ have := geom_sum_alternating_of_lt_neg_one h n.one_lt_succ_succ,
... | geom_sum_ne_zero [linear_ordered_ring α] (hx : x ≠ -1) (hn : n ≠ 0) :
∑ i in range n, x ^ i ≠ 0 | begin
obtain _ | _ | n := n,
{ cases hn rfl },
{ simp },
rw [ne.def, eq_neg_iff_add_eq_zero, ←ne.def] at hx,
obtain h | h := hx.lt_or_lt,
{ have := geom_sum_alternating_of_lt_neg_one h n.one_lt_succ_succ,
split_ifs at this,
{ exact this.ne },
{ exact (zero_lt_one.trans this).ne' } },
{ exact (... | lemma | geom_sum_ne_zero | 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_alternating_of_lt_neg_one",
"geom_sum_pos'",
"linear_ordered_ring"
] | null | 525 | 538 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_eq_zero_iff_neg_one [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i = 0 ↔ x = -1 ∧ even n :=
begin
refine ⟨λ h, _, λ ⟨h, hn⟩, by simp only [h, hn, neg_one_geom_sum, if_true]⟩,
contrapose! h,
obtain rfl | hx := eq_or_ne x (-1),
{ simp only [h rfl, neg_one_geom_sum, if_false, ne.def, not_f... | geom_sum_eq_zero_iff_neg_one [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i = 0 ↔ x = -1 ∧ even n | begin
refine ⟨λ h, _, λ ⟨h, hn⟩, by simp only [h, hn, neg_one_geom_sum, if_true]⟩,
contrapose! h,
obtain rfl | hx := eq_or_ne x (-1),
{ simp only [h rfl, neg_one_geom_sum, if_false, ne.def, not_false_iff, one_ne_zero] },
{ exact geom_sum_ne_zero hx hn }
end | lemma | geom_sum_eq_zero_iff_neg_one | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"eq_or_ne",
"geom_sum_ne_zero",
"linear_ordered_ring",
"neg_one_geom_sum",
"one_ne_zero"
] | null | 540 | 548 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sum_neg_iff [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i < 0 ↔ even n ∧ x + 1 < 0 :=
by rw [← not_iff_not, not_lt, le_iff_lt_or_eq, eq_comm,
or_congr (geom_sum_pos_iff hn) (geom_sum_eq_zero_iff_neg_one hn), nat.odd_iff_not_even,
← add_eq_zero_iff_eq_neg, not_and, not_lt, le_iff_lt_o... | geom_sum_neg_iff [linear_ordered_ring α] (hn : n ≠ 0) :
∑ i in range n, x ^ i < 0 ↔ even n ∧ x + 1 < 0 | by rw [← not_iff_not, not_lt, le_iff_lt_or_eq, eq_comm,
or_congr (geom_sum_pos_iff hn) (geom_sum_eq_zero_iff_neg_one hn), nat.odd_iff_not_even,
← add_eq_zero_iff_eq_neg, not_and, not_lt, le_iff_lt_or_eq, eq_comm,
← imp_iff_not_or, or_comm, and_comm, decidable.and_or_imp, or_comm] | lemma | geom_sum_neg_iff | algebra | src/algebra/geom_sum.lean | [
"algebra.big_operators.order",
"algebra.big_operators.ring",
"algebra.big_operators.intervals",
"tactic.abel",
"data.nat.parity"
] | [
"decidable.and_or_imp",
"geom_sum_eq_zero_iff_neg_one",
"geom_sum_pos_iff",
"imp_iff_not_or",
"linear_ordered_ring",
"nat.odd_iff_not_even",
"not_and",
"not_iff_not"
] | null | 550 | 555 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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