mathlib3 documentation

data.list.big_operators.basic

Sums and products from lists #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file provides basic results about list.prod, list.sum, which calculate the product and sum of elements of a list and list.alternating_prod, list.alternating_sum, their alternating counterparts. These are defined in data.list.defs.

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@[simp]
theorem list.sum_nil {M : Type u_3} [add_monoid M] :
list.nil.sum = 0
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@[simp]
theorem list.prod_nil {M : Type u_3} [monoid M] :
list.nil.prod = 1
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theorem list.sum_singleton {M : Type u_3} [add_monoid M] {a : M} :
[a].sum = a
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theorem list.prod_singleton {M : Type u_3} [monoid M] {a : M} :
[a].prod = a
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@[simp]
theorem list.sum_cons {M : Type u_3} [add_monoid M] {l : list M} {a : M} :
(a :: l).sum = a + l.sum
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@[simp]
theorem list.prod_cons {M : Type u_3} [monoid M] {l : list M} {a : M} :
(a :: l).prod = a * l.prod
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@[simp]
theorem list.prod_append {M : Type u_3} [monoid M] {l₁ l₂ : list M} :
(l₁ ++ l₂).prod = l₁.prod * l₂.prod
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@[simp]
theorem list.sum_append {M : Type u_3} [add_monoid M] {l₁ l₂ : list M} :
(l₁ ++ l₂).sum = l₁.sum + l₂.sum
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theorem list.sum_concat {M : Type u_3} [add_monoid M] {l : list M} {a : M} :
(l.concat a).sum = l.sum + a
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theorem list.prod_concat {M : Type u_3} [monoid M] {l : list M} {a : M} :
(l.concat a).prod = l.prod * a
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@[simp]
theorem list.sum_join {M : Type u_3} [add_monoid M] {l : list (list M)} :
l.join.sum = (list.map list.sum l).sum
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@[simp]
theorem list.prod_join {M : Type u_3} [monoid M] {l : list (list M)} :
l.join.prod = (list.map list.prod l).prod
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theorem list.sum_eq_foldr {M : Type u_3} [add_monoid M] {l : list M} :
l.sum = list.foldr has_add.add 0 l
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theorem list.prod_eq_foldr {M : Type u_3} [monoid M] {l : list M} :
l.prod = list.foldr has_mul.mul 1 l
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@[simp]
theorem list.prod_replicate {M : Type u_3} [monoid M] (n : ℕ) (a : M) :
(list.replicate n a).prod = a ^ n
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@[simp]
theorem list.sum_replicate {M : Type u_3} [add_monoid M] (n : ℕ) (a : M) :
(list.replicate n a).sum = n • a
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theorem list.prod_eq_pow_card {M : Type u_3} [monoid M] (l : list M) (m : M) (h : ∀ (x : M), x ∈ l → x = m) :
l.prod = m ^ l.length
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theorem list.sum_eq_card_nsmul {M : Type u_3} [add_monoid M] (l : list M) (m : M) (h : ∀ (x : M), x ∈ l → x = m) :
l.sum = l.length • m
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theorem list.prod_hom_rel {ι : Type u_1} {M : Type u_3} {N : Type u_4} [monoid M] [monoid N] (l : list ι) {r : M → N → Prop} {f : ι → M} {g : ι → N} (h₁ : r 1 1) (h₂ : ∀ ⦃i : ι⦄ ⦃a : M⦄ ⦃b : N⦄, r a b → r (f i * a) (g i * b)) :
r (list.map f l).prod (list.map g l).prod
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theorem list.sum_hom_rel {ι : Type u_1} {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] (l : list ι) {r : M → N → Prop} {f : ι → M} {g : ι → N} (h₁ : r 0 0) (h₂ : ∀ ⦃i : ι⦄ ⦃a : M⦄ ⦃b : N⦄, r a b → r (f i + a) (g i + b)) :
r (list.map f l).sum (list.map g l).sum
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theorem list.sum_hom {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] (l : list M) {F : Type u_1} [add_monoid_hom_class F M N] (f : F) :
(list.map ⇑f l).sum = ⇑f l.sum
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theorem list.prod_hom {M : Type u_3} {N : Type u_4} [monoid M] [monoid N] (l : list M) {F : Type u_1} [monoid_hom_class F M N] (f : F) :
(list.map ⇑f l).prod = ⇑f l.prod
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theorem list.prod_hom₂ {ι : Type u_1} {M : Type u_3} {N : Type u_4} {P : Type u_5} [monoid M] [monoid N] [monoid P] (l : list ι) (f : M → N → P) (hf : ∀ (a b : M) (c d : N), f (a * b) (c * d) = f a c * f b d) (hf' : f 1 1 = 1) (f₁ : ι → M) (f₂ : ι → N) :
(list.map (λ (i : ι), f (f₁ i) (f₂ i)) l).prod = f (list.map f₁ l).prod (list.map f₂ l).prod
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theorem list.sum_hom₂ {ι : Type u_1} {M : Type u_3} {N : Type u_4} {P : Type u_5} [add_monoid M] [add_monoid N] [add_monoid P] (l : list ι) (f : M → N → P) (hf : ∀ (a b : M) (c d : N), f (a + b) (c + d) = f a c + f b d) (hf' : f 0 0 = 0) (f₁ : ι → M) (f₂ : ι → N) :
(list.map (λ (i : ι), f (f₁ i) (f₂ i)) l).sum = f (list.map f₁ l).sum (list.map f₂ l).sum
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@[simp]
theorem list.prod_map_mul {ι : Type u_1} {α : Type u_2} [comm_monoid α] {l : list ι} {f g : ι → α} :
(list.map (λ (i : ι), f i * g i) l).prod = (list.map f l).prod * (list.map g l).prod
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@[simp]
theorem list.sum_map_add {ι : Type u_1} {α : Type u_2} [add_comm_monoid α] {l : list ι} {f g : ι → α} :
(list.map (λ (i : ι), f i + g i) l).sum = (list.map f l).sum + (list.map g l).sum
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@[simp]
theorem list.prod_map_neg {α : Type u_1} [comm_monoid α] [has_distrib_neg α] (l : list α) :
(list.map has_neg.neg l).prod = (-1) ^ l.length * l.prod
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theorem list.prod_map_hom {ι : Type u_1} {M : Type u_3} {N : Type u_4} [monoid M] [monoid N] (L : list ι) (f : ι → M) {G : Type u_2} [monoid_hom_class G M N] (g : G) :
(list.map (⇑g ∘ f) L).prod = ⇑g (list.map f L).prod
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theorem list.sum_map_hom {ι : Type u_1} {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] (L : list ι) (f : ι → M) {G : Type u_2} [add_monoid_hom_class G M N] (g : G) :
(list.map (⇑g ∘ f) L).sum = ⇑g (list.map f L).sum
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theorem list.sum_is_add_unit {M : Type u_3} [add_monoid M] {L : list M} (u : ∀ (m : M), m ∈ L → is_add_unit m) :
is_add_unit L.sum
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theorem list.prod_is_unit {M : Type u_3} [monoid M] {L : list M} (u : ∀ (m : M), m ∈ L → is_unit m) :
is_unit L.prod
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theorem list.prod_is_unit_iff {α : Type u_1} [comm_monoid α] {L : list α} :
is_unit L.prod ↔ ∀ (m : α), m ∈ L → is_unit m
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theorem list.sum_is_add_unit_iff {α : Type u_1} [add_comm_monoid α] {L : list α} :
is_add_unit L.sum ↔ ∀ (m : α), m ∈ L → is_add_unit m
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@[simp]
theorem list.sum_take_add_sum_drop {M : Type u_3} [add_monoid M] (L : list M) (i : ℕ) :
(list.take i L).sum + (list.drop i L).sum = L.sum
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@[simp]
theorem list.prod_take_mul_prod_drop {M : Type u_3} [monoid M] (L : list M) (i : ℕ) :
(list.take i L).prod * (list.drop i L).prod = L.prod
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@[simp]
theorem list.sum_take_succ {M : Type u_3} [add_monoid M] (L : list M) (i : ℕ) (p : i < L.length) :
(list.take (i + 1) L).sum = (list.take i L).sum + L.nth_le i p
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@[simp]
theorem list.prod_take_succ {M : Type u_3} [monoid M] (L : list M) (i : ℕ) (p : i < L.length) :
(list.take (i + 1) L).prod = (list.take i L).prod * L.nth_le i p
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theorem list.length_pos_of_prod_ne_one {M : Type u_3} [monoid M] (L : list M) (h : L.prod ≠ 1) :
0 < L.length

A list with product not one must have positive length.

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theorem list.length_pos_of_sum_ne_zero {M : Type u_3} [add_monoid M] (L : list M) (h : L.sum ≠ 0) :
0 < L.length

A list with sum not zero must have positive length.

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theorem list.length_pos_of_one_lt_prod {M : Type u_3} [monoid M] [preorder M] (L : list M) (h : 1 < L.prod) :
0 < L.length

A list with product greater than one must have positive length.

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theorem list.length_pos_of_sum_pos {M : Type u_3} [add_monoid M] [preorder M] (L : list M) (h : 0 < L.sum) :
0 < L.length

A list with positive sum must have positive length.

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theorem list.length_pos_of_prod_lt_one {M : Type u_3} [monoid M] [preorder M] (L : list M) (h : L.prod < 1) :
0 < L.length

A list with product less than one must have positive length.

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theorem list.length_pos_of_sum_neg {M : Type u_3} [add_monoid M] [preorder M] (L : list M) (h : L.sum < 0) :
0 < L.length

A list with negative sum must have positive length.

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theorem list.prod_update_nth {M : Type u_3} [monoid M] (L : list M) (n : ℕ) (a : M) :
(L.update_nth n a).prod = (list.take n L).prod * ite (n < L.length) a 1 * (list.drop (n + 1) L).prod
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theorem list.sum_update_nth {M : Type u_3} [add_monoid M] (L : list M) (n : ℕ) (a : M) :
(L.update_nth n a).sum = (list.take n L).sum + ite (n < L.length) a 0 + (list.drop (n + 1) L).sum
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theorem list.nth_zero_add_tail_sum {M : Type u_3} [add_monoid M] (l : list M) :
(l.nth 0).get_or_else 0 + l.tail.sum = l.sum

We'd like to state this as L.head + L.tail.sum = L.sum, but because L.head relies on an inhabited instance to return a garbage value on the empty list, this is not possible. Instead, we write the statement in terms of (L.nth 0).get_or_else 0.

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theorem list.nth_zero_mul_tail_prod {M : Type u_3} [monoid M] (l : list M) :
(l.nth 0).get_or_else 1 * l.tail.prod = l.prod

We'd like to state this as L.head * L.tail.prod = L.prod, but because L.head relies on an inhabited instance to return a garbage value on the empty list, this is not possible. Instead, we write the statement in terms of (L.nth 0).get_or_else 1.

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theorem list.head_add_tail_sum_of_ne_nil {M : Type u_3} [add_monoid M] [inhabited M] (l : list M) (h : l ≠ list.nil) :
l.head + l.tail.sum = l.sum

Same as nth_zero_add_tail_sum, but avoiding the list.head garbage complication by requiring the list to be nonempty.

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theorem list.head_mul_tail_prod_of_ne_nil {M : Type u_3} [monoid M] [inhabited M] (l : list M) (h : l ≠ list.nil) :
l.head * l.tail.prod = l.prod

Same as nth_zero_mul_tail_prod, but avoiding the list.head garbage complication by requiring the list to be nonempty.

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theorem add_commute.list_sum_right {M : Type u_3} [add_monoid M] (l : list M) (y : M) (h : ∀ (x : M), x ∈ l → add_commute y x) :
add_commute y l.sum
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theorem commute.list_prod_right {M : Type u_3} [monoid M] (l : list M) (y : M) (h : ∀ (x : M), x ∈ l → commute y x) :
commute y l.prod
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theorem add_commute.list_sum_left {M : Type u_3} [add_monoid M] (l : list M) (y : M) (h : ∀ (x : M), x ∈ l → add_commute x y) :
add_commute l.sum y
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theorem commute.list_prod_left {M : Type u_3} [monoid M] (l : list M) (y : M) (h : ∀ (x : M), x ∈ l → commute x y) :
commute l.prod y
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theorem list.forall₂.sum_le_sum {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] {l₁ l₂ : list M} (h : list.forall₂ has_le.le l₁ l₂) :
l₁.sum ≤ l₂.sum
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theorem list.forall₂.prod_le_prod' {M : Type u_3} [monoid M] [preorder M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] {l₁ l₂ : list M} (h : list.forall₂ has_le.le l₁ l₂) :
l₁.prod ≤ l₂.prod
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theorem list.sublist.sum_le_sum {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] {l₁ l₂ : list M} (h : l₁ <+ l₂) (h₁ : ∀ (a : M), a ∈ l₂ → 0 ≤ a) :
l₁.sum ≤ l₂.sum

If l₁ is a sublist of l₂ and all elements of l₂ are nonnegative, then l₁.sum ≤ l₂.sum. One can prove a stronger version assuming ∀ a ∈ l₂.diff l₁, 0 ≤ a instead of ∀ a ∈ l₂, 0 ≤ a but this lemma is not yet in mathlib.

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theorem list.sublist.prod_le_prod' {M : Type u_3} [monoid M] [preorder M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] {l₁ l₂ : list M} (h : l₁ <+ l₂) (h₁ : ∀ (a : M), a ∈ l₂ → 1 ≤ a) :
l₁.prod ≤ l₂.prod

If l₁ is a sublist of l₂ and all elements of l₂ are greater than or equal to one, then l₁.prod ≤ l₂.prod. One can prove a stronger version assuming ∀ a ∈ l₂.diff l₁, 1 ≤ a instead of ∀ a ∈ l₂, 1 ≤ a but this lemma is not yet in mathlib.

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theorem list.sublist_forall₂.sum_le_sum {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] {l₁ l₂ : list M} (h : list.sublist_forall₂ has_le.le l₁ l₂) (h₁ : ∀ (a : M), a ∈ l₂ → 0 ≤ a) :
l₁.sum ≤ l₂.sum
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theorem list.sublist_forall₂.prod_le_prod' {M : Type u_3} [monoid M] [preorder M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] {l₁ l₂ : list M} (h : list.sublist_forall₂ has_le.le l₁ l₂) (h₁ : ∀ (a : M), a ∈ l₂ → 1 ≤ a) :
l₁.prod ≤ l₂.prod
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theorem list.sum_le_sum {ι : Type u_1} {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] {l : list ι} {f g : ι → M} (h : ∀ (i : ι), i ∈ l → f i ≤ g i) :
(list.map f l).sum ≤ (list.map g l).sum
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theorem list.prod_le_prod' {ι : Type u_1} {M : Type u_3} [monoid M] [preorder M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] {l : list ι} {f g : ι → M} (h : ∀ (i : ι), i ∈ l → f i ≤ g i) :
(list.map f l).prod ≤ (list.map g l).prod
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theorem list.prod_lt_prod' {ι : Type u_1} {M : Type u_3} [monoid M] [preorder M] [covariant_class M M has_mul.mul has_lt.lt] [covariant_class M M has_mul.mul has_le.le] [covariant_class M M (function.swap has_mul.mul) has_lt.lt] [covariant_class M M (function.swap has_mul.mul) has_le.le] {l : list ι} (f g : ι → M) (h₁ : ∀ (i : ι), i ∈ l → f i ≤ g i) (h₂ : ∃ (i : ι) (H : i ∈ l), f i < g i) :
(list.map f l).prod < (list.map g l).prod
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theorem list.sum_lt_sum {ι : Type u_1} {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M has_add.add has_lt.lt] [covariant_class M M has_add.add has_le.le] [covariant_class M M (function.swap has_add.add) has_lt.lt] [covariant_class M M (function.swap has_add.add) has_le.le] {l : list ι} (f g : ι → M) (h₁ : ∀ (i : ι), i ∈ l → f i ≤ g i) (h₂ : ∃ (i : ι) (H : i ∈ l), f i < g i) :
(list.map f l).sum < (list.map g l).sum
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theorem list.sum_lt_sum_of_ne_nil {ι : Type u_1} {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M has_add.add has_lt.lt] [covariant_class M M has_add.add has_le.le] [covariant_class M M (function.swap has_add.add) has_lt.lt] [covariant_class M M (function.swap has_add.add) has_le.le] {l : list ι} (hl : l ≠ list.nil) (f g : ι → M) (hlt : ∀ (i : ι), i ∈ l → f i < g i) :
(list.map f l).sum < (list.map g l).sum
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theorem list.prod_lt_prod_of_ne_nil {ι : Type u_1} {M : Type u_3} [monoid M] [preorder M] [covariant_class M M has_mul.mul has_lt.lt] [covariant_class M M has_mul.mul has_le.le] [covariant_class M M (function.swap has_mul.mul) has_lt.lt] [covariant_class M M (function.swap has_mul.mul) has_le.le] {l : list ι} (hl : l ≠ list.nil) (f g : ι → M) (hlt : ∀ (i : ι), i ∈ l → f i < g i) :
(list.map f l).prod < (list.map g l).prod
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theorem list.prod_le_pow_card {M : Type u_3} [monoid M] [preorder M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] (l : list M) (n : M) (h : ∀ (x : M), x ∈ l → x ≤ n) :
l.prod ≤ n ^ l.length
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theorem list.sum_le_card_nsmul {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] (l : list M) (n : M) (h : ∀ (x : M), x ∈ l → x ≤ n) :
l.sum ≤ l.length • n
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theorem list.exists_lt_of_prod_lt' {ι : Type u_1} {M : Type u_3} [monoid M] [linear_order M] [covariant_class M M (function.swap has_mul.mul) has_le.le] [covariant_class M M has_mul.mul has_le.le] {l : list ι} (f g : ι → M) (h : (list.map f l).prod < (list.map g l).prod) :
∃ (i : ι) (H : i ∈ l), f i < g i
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theorem list.exists_lt_of_sum_lt {ι : Type u_1} {M : Type u_3} [add_monoid M] [linear_order M] [covariant_class M M (function.swap has_add.add) has_le.le] [covariant_class M M has_add.add has_le.le] {l : list ι} (f g : ι → M) (h : (list.map f l).sum < (list.map g l).sum) :
∃ (i : ι) (H : i ∈ l), f i < g i
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theorem list.exists_le_of_sum_le {ι : Type u_1} {M : Type u_3} [add_monoid M] [linear_order M] [covariant_class M M has_add.add has_lt.lt] [covariant_class M M has_add.add has_le.le] [covariant_class M M (function.swap has_add.add) has_lt.lt] [covariant_class M M (function.swap has_add.add) has_le.le] {l : list ι} (hl : l ≠ list.nil) (f g : ι → M) (h : (list.map f l).sum ≤ (list.map g l).sum) :
∃ (x : ι) (H : x ∈ l), f x ≤ g x
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theorem list.exists_le_of_prod_le' {ι : Type u_1} {M : Type u_3} [monoid M] [linear_order M] [covariant_class M M has_mul.mul has_lt.lt] [covariant_class M M has_mul.mul has_le.le] [covariant_class M M (function.swap has_mul.mul) has_lt.lt] [covariant_class M M (function.swap has_mul.mul) has_le.le] {l : list ι} (hl : l ≠ list.nil) (f g : ι → M) (h : (list.map f l).prod ≤ (list.map g l).prod) :
∃ (x : ι) (H : x ∈ l), f x ≤ g x
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theorem list.sum_nonneg {M : Type u_3} [add_monoid M] [preorder M] [covariant_class M M has_add.add has_le.le] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 0 ≤ x) :
0 ≤ l.sum
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theorem list.one_le_prod_of_one_le {M : Type u_3} [monoid M] [preorder M] [covariant_class M M has_mul.mul has_le.le] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 1 ≤ x) :
1 ≤ l.prod
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theorem list.prod_eq_zero {M₀ : Type u_6} [monoid_with_zero M₀] {L : list M₀} (h : 0 ∈ L) :
L.prod = 0

If zero is an element of a list L, then list.prod L = 0. If the domain is a nontrivial monoid with zero with no divisors, then this implication becomes an iff, see list.prod_eq_zero_iff.

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@[simp]
theorem list.prod_eq_zero_iff {M₀ : Type u_6} [monoid_with_zero M₀] [nontrivial M₀] [no_zero_divisors M₀] {L : list M₀} :
L.prod = 0 ↔ 0 ∈ L

Product of elements of a list L equals zero if and only if 0 ∈ L. See also list.prod_eq_zero for an implication that needs weaker typeclass assumptions.

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theorem list.prod_ne_zero {M₀ : Type u_6} [monoid_with_zero M₀] [nontrivial M₀] [no_zero_divisors M₀] {L : list M₀} (hL : 0 ∉ L) :
L.prod ≠ 0
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theorem list.prod_inv_reverse {G : Type u_7} [group G] (L : list G) :
(L.prod)⁻¹ = (list.map (λ (x : G), x⁻¹) L).reverse.prod

This is the list.prod version of mul_inv_rev

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theorem list.sum_neg_reverse {G : Type u_7} [add_group G] (L : list G) :
-L.sum = (list.map (λ (x : G), -x) L).reverse.sum

This is the list.sum version of add_neg_rev

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theorem list.prod_reverse_noncomm {G : Type u_7} [group G] (L : list G) :
L.reverse.prod = ((list.map (λ (x : G), x⁻¹) L).prod)⁻¹

A non-commutative variant of list.prod_reverse

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theorem list.sum_reverse_noncomm {G : Type u_7} [add_group G] (L : list G) :
L.reverse.sum = -(list.map (λ (x : G), -x) L).sum

A non-commutative variant of list.sum_reverse

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@[simp]
theorem list.sum_drop_succ {G : Type u_7} [add_group G] (L : list G) (i : ℕ) (p : i < L.length) :
(list.drop (i + 1) L).sum = -L.nth_le i p + (list.drop i L).sum

Counterpart to list.sum_take_succ when we have an negation operation

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@[simp]
theorem list.prod_drop_succ {G : Type u_7} [group G] (L : list G) (i : ℕ) (p : i < L.length) :
(list.drop (i + 1) L).prod = (L.nth_le i p)⁻¹ * (list.drop i L).prod

Counterpart to list.prod_take_succ when we have an inverse operation

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theorem list.sum_neg {G : Type u_7} [add_comm_group G] (L : list G) :
-L.sum = (list.map (λ (x : G), -x) L).sum

This is the list.sum version of add_neg

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theorem list.prod_inv {G : Type u_7} [comm_group G] (L : list G) :
(L.prod)⁻¹ = (list.map (λ (x : G), x⁻¹) L).prod

This is the list.prod version of mul_inv

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theorem list.prod_update_nth' {G : Type u_7} [comm_group G] (L : list G) (n : ℕ) (a : G) :
(L.update_nth n a).prod = L.prod * dite (n < L.length) (λ (hn : n < L.length), (L.nth_le n hn)⁻¹ * a) (λ (hn : ¬n < L.length), 1)

Alternative version of list.prod_update_nth when the list is over a group

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theorem list.sum_update_nth' {G : Type u_7} [add_comm_group G] (L : list G) (n : ℕ) (a : G) :
(L.update_nth n a).sum = L.sum + dite (n < L.length) (λ (hn : n < L.length), -L.nth_le n hn + a) (λ (hn : ¬n < L.length), 0)

Alternative version of list.sum_update_nth when the list is over a group

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theorem list.eq_of_sum_take_eq {M : Type u_3} [add_left_cancel_monoid M] {L L' : list M} (h : L.length = L'.length) (h' : ∀ (i : ℕ), i ≤ L.length → (list.take i L).sum = (list.take i L').sum) :
L = L'
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theorem list.eq_of_prod_take_eq {M : Type u_3} [left_cancel_monoid M] {L L' : list M} (h : L.length = L'.length) (h' : ∀ (i : ℕ), i ≤ L.length → (list.take i L).prod = (list.take i L').prod) :
L = L'
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theorem list.monotone_sum_take {M : Type u_3} [canonically_ordered_add_monoid M] (L : list M) :
monotone (λ (i : ℕ), (list.take i L).sum)
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theorem list.monotone_prod_take {M : Type u_3} [canonically_ordered_monoid M] (L : list M) :
monotone (λ (i : ℕ), (list.take i L).prod)
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theorem list.one_lt_prod_of_one_lt {M : Type u_3} [ordered_comm_monoid M] (l : list M) (hl : ∀ (x : M), x ∈ l → 1 < x) (hl₂ : l ≠ list.nil) :
1 < l.prod
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theorem list.sum_pos {M : Type u_3} [ordered_add_comm_monoid M] (l : list M) (hl : ∀ (x : M), x ∈ l → 0 < x) (hl₂ : l ≠ list.nil) :
0 < l.sum
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theorem list.single_le_sum {M : Type u_3} [ordered_add_comm_monoid M] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 0 ≤ x) (x : M) (H : x ∈ l) :
x ≤ l.sum
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theorem list.single_le_prod {M : Type u_3} [ordered_comm_monoid M] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 1 ≤ x) (x : M) (H : x ∈ l) :
x ≤ l.prod
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theorem list.all_zero_of_le_zero_le_of_sum_eq_zero {M : Type u_3} [ordered_add_comm_monoid M] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 0 ≤ x) (hl₂ : l.sum = 0) {x : M} (hx : x ∈ l) :
x = 0
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theorem list.all_one_of_le_one_le_of_prod_eq_one {M : Type u_3} [ordered_comm_monoid M] {l : list M} (hl₁ : ∀ (x : M), x ∈ l → 1 ≤ x) (hl₂ : l.prod = 1) {x : M} (hx : x ∈ l) :
x = 1
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theorem list.sum_eq_zero {M : Type u_3} [add_monoid M] {l : list M} (hl : ∀ (x : M), x ∈ l → x = 0) :
l.sum = 0

Slightly more general version of list.sum_eq_zero_iff for a non-ordered add_monoid

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theorem list.prod_eq_one {M : Type u_3} [monoid M] {l : list M} (hl : ∀ (x : M), x ∈ l → x = 1) :
l.prod = 1

Slightly more general version of list.prod_eq_one_iff for a non-ordered monoid

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theorem list.exists_mem_ne_one_of_prod_ne_one {M : Type u_3} [monoid M] {l : list M} (h : l.prod ≠ 1) :
∃ (x : M) (H : x ∈ l), x ≠ 1
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theorem list.exists_mem_ne_zero_of_sum_ne_zero {M : Type u_3} [add_monoid M] {l : list M} (h : l.sum ≠ 0) :
∃ (x : M) (H : x ∈ l), x ≠ 0
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theorem list.sum_le_foldr_max {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] [linear_order N] (f : M → N) (h0 : f 0 ≤ 0) (hadd : ∀ (x y : M), f (x + y) ≤ linear_order.max (f x) (f y)) (l : list M) :
f l.sum ≤ list.foldr linear_order.max 0 (list.map f l)
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@[simp]
theorem list.sum_erase {M : Type u_3} [decidable_eq M] [add_comm_monoid M] {a : M} {l : list M} :
a ∈ l → a + (l.erase a).sum = l.sum
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@[simp]
theorem list.prod_erase {M : Type u_3} [decidable_eq M] [comm_monoid M] {a : M} {l : list M} :
a ∈ l → a * (l.erase a).prod = l.prod
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@[simp]
theorem list.prod_map_erase {ι : Type u_1} {M : Type u_3} [decidable_eq ι] [comm_monoid M] (f : ι → M) {a : ι} {l : list ι} :
a ∈ l → f a * (list.map f (l.erase a)).prod = (list.map f l).prod
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@[simp]
theorem list.sum_map_erase {ι : Type u_1} {M : Type u_3} [decidable_eq ι] [add_comm_monoid M] (f : ι → M) {a : ι} {l : list ι} :
a ∈ l → f a + (list.map f (l.erase a)).sum = (list.map f l).sum
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theorem list.sum_const_nat (m n : ℕ) :
(list.replicate m n).sum = m * n
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theorem list.prod_pos {R : Type u_8} [strict_ordered_semiring R] (l : list R) (h : ∀ (a : R), a ∈ l → 0 < a) :
0 < l.prod

The product of a list of positive natural numbers is positive, and likewise for any nontrivial ordered semiring.

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@[simp]
theorem canonically_ordered_comm_semiring.list_prod_pos {α : Type u_1} [canonically_ordered_comm_semiring α] [nontrivial α] {l : list α} :
0 < l.prod ↔ ∀ (x : α), x ∈ l → 0 < x

A variant of list.prod_pos for canonically_ordered_comm_semiring.

Several lemmas about sum/head/tail for list ℕ. These are hard to generalize well, as they rely on the fact that default ℕ = 0. If desired, we could add a class stating that default = 0.

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theorem list.head_add_tail_sum (L : list ℕ) :
L.head + L.tail.sum = L.sum

This relies on default ℕ = 0.

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theorem list.head_le_sum (L : list ℕ) :
L.head ≤ L.sum

This relies on default ℕ = 0.

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theorem list.tail_sum (L : list ℕ) :
L.tail.sum = L.sum - L.head

This relies on default ℕ = 0.

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@[simp]
theorem list.alternating_sum_nil {α : Type u_2} [has_zero α] [has_add α] [has_neg α] :
list.nil.alternating_sum = 0
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@[simp]
theorem list.alternating_prod_nil {α : Type u_2} [has_one α] [has_mul α] [has_inv α] :
list.nil.alternating_prod = 1
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@[simp]
theorem list.alternating_prod_singleton {α : Type u_2} [has_one α] [has_mul α] [has_inv α] (a : α) :
[a].alternating_prod = a
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@[simp]
theorem list.alternating_sum_singleton {α : Type u_2} [has_zero α] [has_add α] [has_neg α] (a : α) :
[a].alternating_sum = a
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theorem list.alternating_prod_cons_cons' {α : Type u_2} [has_one α] [has_mul α] [has_inv α] (a b : α) (l : list α) :
(a :: b :: l).alternating_prod = a * b⁻¹ * l.alternating_prod
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theorem list.alternating_sum_cons_cons' {α : Type u_2} [has_zero α] [has_add α] [has_neg α] (a b : α) (l : list α) :
(a :: b :: l).alternating_sum = a + -b + l.alternating_sum
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theorem list.alternating_prod_cons_cons {α : Type u_2} [div_inv_monoid α] (a b : α) (l : list α) :
(a :: b :: l).alternating_prod = a / b * l.alternating_prod
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theorem list.alternating_sum_cons_cons {α : Type u_2} [sub_neg_monoid α] (a b : α) (l : list α) :
(a :: b :: l).alternating_sum = a - b + l.alternating_sum
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theorem list.alternating_prod_cons' {α : Type u_2} [comm_group α] (a : α) (l : list α) :
(a :: l).alternating_prod = a * (l.alternating_prod)⁻¹
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theorem list.alternating_sum_cons' {α : Type u_2} [add_comm_group α] (a : α) (l : list α) :
(a :: l).alternating_sum = a + -l.alternating_sum
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@[simp]
theorem list.alternating_prod_cons {α : Type u_2} [comm_group α] (a : α) (l : list α) :
(a :: l).alternating_prod = a / l.alternating_prod
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@[simp]
theorem list.alternating_sum_cons {α : Type u_2} [add_comm_group α] (a : α) (l : list α) :
(a :: l).alternating_sum = a - l.alternating_sum
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theorem list.sum_nat_mod (l : list ℕ) (n : ℕ) :
l.sum % n = (list.map (λ (_x : ℕ), _x % n) l).sum % n
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theorem list.prod_nat_mod (l : list ℕ) (n : ℕ) :
l.prod % n = (list.map (λ (_x : ℕ), _x % n) l).prod % n
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theorem list.sum_int_mod (l : list ℤ) (n : ℤ) :
l.sum % n = (list.map (λ (_x : ℤ), _x % n) l).sum % n
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theorem list.prod_int_mod (l : list ℤ) (n : ℤ) :
l.prod % n = (list.map (λ (_x : ℤ), _x % n) l).prod % n
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theorem map_list_sum {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] {F : Type u_1} [add_monoid_hom_class F M N] (f : F) (l : list M) :
⇑f l.sum = (list.map ⇑f l).sum
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theorem map_list_prod {M : Type u_3} {N : Type u_4} [monoid M] [monoid N] {F : Type u_1} [monoid_hom_class F M N] (f : F) (l : list M) :
⇑f l.prod = (list.map ⇑f l).prod
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@[protected]
theorem monoid_hom.map_list_prod {M : Type u_3} {N : Type u_4} [monoid M] [monoid N] (f : M →* N) (l : list M) :
⇑f l.prod = (list.map ⇑f l).prod

Deprecated, use _root_.map_list_prod instead.

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@[protected]
theorem add_monoid_hom.map_list_sum {M : Type u_3} {N : Type u_4} [add_monoid M] [add_monoid N] (f : M →+ N) (l : list M) :
⇑f l.sum = (list.map ⇑f l).sum

Deprecated, use _root_.map_list_sum instead.