statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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le_prod_of_submultiplicative (f : M → N) (h_one : f 1 = 1)
(h_mul : ∀ x y, f (x * y) ≤ f x * f y) (s : finset ι) (g : ι → M) :
f (∏ i in s, g i) ≤ ∏ i in s, f (g i) | begin
refine le_trans (multiset.le_prod_of_submultiplicative f h_one h_mul _) _,
rw multiset.map_map,
refl,
end | lemma | finset.le_prod_of_submultiplicative | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"multiset.le_prod_of_submultiplicative",
"multiset.map_map"
] | If `f : M → N` is a submultiplicative function, `f (x * y) ≤ f x * f y`, `f 1 = 1`, and `g i`,
`i ∈ s`, is a finite family of elements of `M`, then `f (∏ i in s, g i) ≤ ∏ i in s, f (g i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_le_prod' (h : ∀ i ∈ s, f i ≤ g i) : ∏ i in s, f i ≤ ∏ i in s, g i | multiset.prod_map_le_prod_map f g h | lemma | finset.prod_le_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"multiset.prod_map_le_prod_map"
] | In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`∏ i in s, f i ≤ ∏ i in s, g i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le_prod' (h : ∀i ∈ s, 1 ≤ f i) : 1 ≤ (∏ i in s, f i) | le_trans (by rw prod_const_one) (prod_le_prod' h) | lemma | finset.one_le_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_prod'' (h : ∀ (i : ι), 1 ≤ f i) : 1 ≤ ∏ (i : ι) in s, f i | finset.one_le_prod' (λ i hi, h i) | lemma | finset.one_le_prod'' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.one_le_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_one' (h : ∀i ∈ s, f i ≤ 1) : (∏ i in s, f i) ≤ 1 | (prod_le_prod' h).trans_eq (by rw prod_const_one) | lemma | finset.prod_le_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
∏ i in s, f i ≤ ∏ i in t, f i | by classical;
calc (∏ i in s, f i) ≤ (∏ i in t \ s, f i) * (∏ i in s, f i) :
le_mul_of_one_le_left' $ one_le_prod' $ by simpa only [mem_sdiff, and_imp]
... = ∏ i in t \ s ∪ s, f i : (prod_union sdiff_disjoint).symm
... = ∏ i in t, f i : by rw [sdiff_union_of_subset h] | lemma | finset.prod_le_prod_of_subset_of_one_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"and_imp",
"le_mul_of_one_le_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_set_of_one_le' (hf : ∀ x, 1 ≤ f x) : monotone (λ s, ∏ x in s, f x) | λ s t hst, prod_le_prod_of_subset_of_one_le' hst $ λ x _ _, hf x | lemma | finset.prod_mono_set_of_one_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_univ_prod_of_one_le' [fintype ι] {s : finset ι} (w : ∀ x, 1 ≤ f x) :
∏ x in s, f x ≤ ∏ x, f x | prod_le_prod_of_subset_of_one_le' (subset_univ s) (λ a _ _, w a) | lemma | finset.prod_le_univ_prod_of_one_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one_iff_of_one_le' : (∀ i ∈ s, 1 ≤ f i) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1) | begin
classical,
apply finset.induction_on s,
exact λ _, ⟨λ _ _, false.elim, λ _, rfl⟩,
assume a s ha ih H,
have : ∀ i ∈ s, 1 ≤ f i, from λ _, H _ ∘ mem_insert_of_mem,
rw [prod_insert ha, mul_eq_one_iff' (H _ $ mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
end | lemma | finset.prod_eq_one_iff_of_one_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.induction_on",
"ih",
"mul_eq_one_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one_iff_of_le_one' : (∀ i ∈ s, f i ≤ 1) → (∏ i in s, f i = 1 ↔ ∀ i ∈ s, f i = 1) | @prod_eq_one_iff_of_one_le' _ Nᵒᵈ _ _ _ | lemma | finset.prod_eq_one_iff_of_le_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_le_prod' (hf : ∀ i ∈ s, 1 ≤ f i) {a} (h : a ∈ s) : f a ≤ (∏ x in s, f x) | calc f a = ∏ i in {a}, f i : prod_singleton.symm
... ≤ ∏ i in s, f i :
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) $ λ i hi _, hf i hi | lemma | finset.single_le_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_pow_card (s : finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
s.prod f ≤ n ^ s.card | begin
refine (multiset.prod_le_pow_card (s.val.map f) n _).trans _,
{ simpa using h },
{ simp }
end | lemma | finset.prod_le_pow_card | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"multiset.prod_le_pow_card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_card_le_prod (s : finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, n ≤ f x) :
n ^ s.card ≤ s.prod f | @finset.prod_le_pow_card _ Nᵒᵈ _ _ _ _ h | lemma | finset.pow_card_le_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"finset.prod_le_pow_card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_bUnion_le_card_mul [decidable_eq β] (s : finset ι) (f : ι → finset β) (n : ℕ)
(h : ∀ a ∈ s, (f a).card ≤ n) :
(s.bUnion f).card ≤ s.card * n | card_bUnion_le.trans $ sum_le_card_nsmul _ _ _ h | lemma | finset.card_bUnion_le_card_mul | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : finset ι'}
{g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, (1 : N) ≤ ∏ x in s.filter (λ x, g x = y), f x) :
∏ y in t, ∏ x in s.filter (λ x, g x = y), f x ≤ ∏ x in s, f x | calc (∏ y in t, ∏ x in s.filter (λ x, g x = y), f x) ≤
(∏ y in t ∪ s.image g, ∏ x in s.filter (λ x, g x = y), f x) :
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) $ λ y hyts, h y
... = ∏ x in s, f x :
prod_fiberwise_of_maps_to (λ x hx, mem_union.2 $ or.inr $ mem_image_of_mem _ hx) _ | lemma | finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : finset ι'}
{g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, (∏ x in s.filter (λ x, g x = y), f x) ≤ 1) :
(∏ x in s, f x) ≤ ∏ y in t, ∏ x in s.filter (λ x, g x = y), f x | @prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nᵒᵈ _ _ _ _ _ _ _ h | lemma | finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sum_le_sum_abs {G : Type*} [linear_ordered_add_comm_group G] (f : ι → G) (s : finset ι) : | |∑ i in s, f i| ≤ ∑ i in s, |f i| :=
le_sum_of_subadditive _ abs_zero abs_add s f | lemma | finset.abs_sum_le_sum_abs | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"abs_add",
"abs_zero",
"finset",
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sum_of_nonneg {G : Type*} [linear_ordered_add_comm_group G] {f : ι → G} {s : finset ι}
(hf : ∀ i ∈ s, 0 ≤ f i) : | |∑ (i : ι) in s, f i| = ∑ (i : ι) in s, f i :=
by rw abs_of_nonneg (finset.sum_nonneg hf) | lemma | finset.abs_sum_of_nonneg | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"abs_of_nonneg",
"finset",
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sum_of_nonneg' {G : Type*} [linear_ordered_add_comm_group G] {f : ι → G} {s : finset ι}
(hf : ∀ i, 0 ≤ f i) : | |∑ (i : ι) in s, f i| = ∑ (i : ι) in s, f i :=
by rw abs_of_nonneg (finset.sum_nonneg' hf) | lemma | finset.abs_sum_of_nonneg' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"abs_of_nonneg",
"finset",
"linear_ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_prod {R : Type*} [linear_ordered_comm_ring R] {f : ι → R} {s : finset ι} : | |∏ x in s, f x| = ∏ x in s, |f x| :=
(abs_hom.to_monoid_hom : R →* R).map_prod _ _ | lemma | finset.abs_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"linear_ordered_comm_ring",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_mul_card_image_of_maps_to {f : α → β} {s : finset α} {t : finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, (s.filter (λ x, f x = a)).card ≤ n) :
s.card ≤ n * t.card | calc s.card = (∑ a in t, (s.filter (λ x, f x = a)).card) : card_eq_sum_card_fiberwise Hf
... ≤ (∑ _ in t, n) : sum_le_sum hn
... = _ : by simp [mul_comm] | theorem | finset.card_le_mul_card_image_of_maps_to | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_mul_card_image {f : α → β} (s : finset α)
(n : ℕ) (hn : ∀ a ∈ s.image f, (s.filter (λ x, f x = a)).card ≤ n) :
s.card ≤ n * (s.image f).card | card_le_mul_card_image_of_maps_to (λ x, mem_image_of_mem _) n hn | theorem | finset.card_le_mul_card_image | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_card_image_le_card_of_maps_to {f : α → β} {s : finset α} {t : finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter (λ x, f x = a)).card) :
n * t.card ≤ s.card | calc n * t.card = (∑ _ in t, n) : by simp [mul_comm]
... ≤ (∑ a in t, (s.filter (λ x, f x = a)).card) : sum_le_sum hn
... = s.card : by rw ← card_eq_sum_card_fiberwise Hf | theorem | finset.mul_card_image_le_card_of_maps_to | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_card_image_le_card {f : α → β} (s : finset α)
(n : ℕ) (hn : ∀ a ∈ s.image f, n ≤ (s.filter (λ x, f x = a)).card) :
n * (s.image f).card ≤ s.card | mul_card_image_le_card_of_maps_to (λ x, mem_image_of_mem _) n hn | theorem | finset.mul_card_image_le_card | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_card_inter_le (h : ∀ a ∈ s, (B.filter $ (∈) a).card ≤ n) :
∑ t in B, (s ∩ t).card ≤ s.card * n | begin
refine le_trans _ (s.sum_le_card_nsmul _ _ h),
simp_rw [←filter_mem_eq_inter, card_eq_sum_ones, sum_filter],
exact sum_comm.le,
end | lemma | finset.sum_card_inter_le | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | If every element belongs to at most `n` finsets, then the sum of their sizes is at most `n`
times how many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_card_le [fintype α] (h : ∀ a, (B.filter $ (∈) a).card ≤ n) :
∑ s in B, s.card ≤ fintype.card α * n | calc ∑ s in B, s.card = ∑ s in B, (univ ∩ s).card : by simp_rw univ_inter
... ≤ fintype.card α * n : sum_card_inter_le (λ a _, h a) | lemma | finset.sum_card_le | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"fintype",
"fintype.card"
] | If every element belongs to at most `n` finsets, then the sum of their sizes is at most `n`
times how many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter $ (∈) a).card) :
s.card * n ≤ ∑ t in B, (s ∩ t).card | begin
apply (s.card_nsmul_le_sum _ _ h).trans,
simp_rw [←filter_mem_eq_inter, card_eq_sum_ones, sum_filter],
exact sum_comm.le,
end | lemma | finset.le_sum_card_inter | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | If every element belongs to at least `n` finsets, then the sum of their sizes is at least `n`
times how many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sum_card [fintype α] (h : ∀ a, n ≤ (B.filter $ (∈) a).card) :
fintype.card α * n ≤ ∑ s in B, s.card | calc fintype.card α * n ≤ ∑ s in B, (univ ∩ s).card : le_sum_card_inter (λ a _, h a)
... = ∑ s in B, s.card : by simp_rw univ_inter | lemma | finset.le_sum_card | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"fintype",
"fintype.card"
] | If every element belongs to at least `n` finsets, then the sum of their sizes is at least `n`
times how many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_card_inter (h : ∀ a ∈ s, (B.filter $ (∈) a).card = n) :
∑ t in B, (s ∩ t).card = s.card * n | (sum_card_inter_le $ λ a ha, (h a ha).le).antisymm (le_sum_card_inter $ λ a ha, (h a ha).ge) | lemma | finset.sum_card_inter | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | If every element belongs to exactly `n` finsets, then the sum of their sizes is `n` times how
many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_card [fintype α] (h : ∀ a, (B.filter $ (∈) a).card = n) :
∑ s in B, s.card = fintype.card α * n | by simp_rw [fintype.card, ←sum_card_inter (λ a _, h a), univ_inter] | lemma | finset.sum_card | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"fintype",
"fintype.card"
] | If every element belongs to exactly `n` finsets, then the sum of their sizes is `n` times how
many they are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_le_card_bUnion {s : finset ι} {f : ι → finset α} (hs : (s : set ι).pairwise_disjoint f)
(hf : ∀ i ∈ s, (f i).nonempty) :
s.card ≤ (s.bUnion f).card | by { rw [card_bUnion hs, card_eq_sum_ones], exact sum_le_sum (λ i hi, (hf i hi).card_pos) } | lemma | finset.card_le_card_bUnion | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_card_bUnion_add_card_fiber {s : finset ι} {f : ι → finset α}
(hs : (s : set ι).pairwise_disjoint f) :
s.card ≤ (s.bUnion f).card + (s.filter $ λ i, f i = ∅).card | begin
rw [←finset.filter_card_add_filter_neg_card_eq_card (λ i, f i = ∅), add_comm],
exact add_le_add_right ((card_le_card_bUnion (hs.subset $ filter_subset _ _) $ λ i hi,
nonempty_of_ne_empty $ (mem_filter.1 hi).2).trans $ card_le_of_subset $
bUnion_subset_bUnion_of_subset_left _ $ filter_subset _ _) _,
en... | lemma | finset.card_le_card_bUnion_add_card_fiber | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_card_bUnion_add_one {s : finset ι} {f : ι → finset α} (hf : injective f)
(hs : (s : set ι).pairwise_disjoint f) :
s.card ≤ (s.bUnion f).card + 1 | (card_le_card_bUnion_add_card_fiber hs).trans $ add_le_add_left (card_le_one.2 $ λ i hi j hj, hf $
(mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _ | lemma | finset.card_le_card_bUnion_add_one | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_one_iff' : ∏ x in s, f x = 1 ↔ ∀ x ∈ s, f x = 1 | prod_eq_one_iff_of_one_le' $ λ x hx, one_le (f x) | lemma | finset.prod_eq_one_iff' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_of_subset' (h : s ⊆ t) : ∏ x in s, f x ≤ ∏ x in t, f x | prod_le_prod_of_subset_of_one_le' h $ assume x h₁ h₂, one_le _ | lemma | finset.prod_le_prod_of_subset' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_set' (f : ι → M) : monotone (λ s, ∏ x in s, f x) | λ s₁ s₂ hs, prod_le_prod_of_subset' hs | lemma | finset.prod_mono_set' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_of_ne_one' (h : ∀ x ∈ s, f x ≠ 1 → x ∈ t) :
∏ x in s, f x ≤ ∏ x in t, f x | by classical;
calc ∏ x in s, f x = (∏ x in s.filter (λ x, f x = 1), f x) * ∏ x in s.filter (λ x, f x ≠ 1), f x :
by rw [← prod_union, filter_union_filter_neg_eq];
exact disjoint_filter.2 (assume _ _ h n_h, n_h h)
... ≤ (∏ x in t, f x) : mul_le_of_le_one_of_le
(prod_le_one' $ by simp only [mem_filter,... | lemma | finset.prod_le_prod_of_ne_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"and_imp",
"mul_le_of_le_one_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_prod' (Hle : ∀ i ∈ s, f i ≤ g i) (Hlt : ∃ i ∈ s, f i < g i) :
∏ i in s, f i < ∏ i in s, g i | begin
classical,
rcases Hlt with ⟨i, hi, hlt⟩,
rw [← insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)],
exact mul_lt_mul_of_lt_of_le hlt (prod_le_prod' $ λ j hj, Hle j $ mem_of_mem_erase hj)
end | theorem | finset.prod_lt_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"mul_lt_mul_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_prod_of_nonempty' (hs : s.nonempty) (Hlt : ∀ i ∈ s, f i < g i) :
∏ i in s, f i < ∏ i in s, g i | begin
apply prod_lt_prod',
{ intros i hi, apply le_of_lt (Hlt i hi) },
cases hs with i hi,
exact ⟨i, hi, Hlt i hi⟩,
end | lemma | finset.prod_lt_prod_of_nonempty' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_prod_of_subset' (h : s ⊆ t) {i : ι} (ht : i ∈ t) (hs : i ∉ s) (hlt : 1 < f i)
(hle : ∀ j ∈ t, j ∉ s → 1 ≤ f j) :
∏ j in s, f j < ∏ j in t, f j | by classical;
calc ∏ j in s, f j < ∏ j in insert i s, f j :
begin
rw prod_insert hs,
exact lt_mul_of_one_lt_left' (∏ j in s, f j) hlt,
end
... ≤ ∏ j in t, f j :
begin
apply prod_le_prod_of_subset_of_one_le',
{ simp [finset.insert_subset, h, ht] },
{ assume x hx h'x,
simp only [mem_insert, not_or_distrib] ... | lemma | finset.prod_lt_prod_of_subset' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.insert_subset",
"lt_mul_of_one_lt_left'",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_lt_prod' {i j : ι} (hij : j ≠ i) (hi : i ∈ s) (hj : j ∈ s) (hlt : 1 < f j)
(hle : ∀ k ∈ s, k ≠ i → 1 ≤ f k) :
f i < ∏ k in s, f k | calc f i = ∏ k in {i}, f k : prod_singleton.symm
... < ∏ k in s, f k :
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt $
λ k hks hki, hle k hks (mt mem_singleton.2 hki) | lemma | finset.single_lt_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_prod (h : ∀i ∈ s, 1 < f i) (hs : s.nonempty) :
1 < (∏ i in s, f i) | lt_of_le_of_lt (by rw prod_const_one) $ prod_lt_prod_of_nonempty' hs h | lemma | finset.one_lt_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_one (h : ∀i ∈ s, f i < 1) (hs : s.nonempty) :
(∏ i in s, f i) < 1 | (prod_lt_prod_of_nonempty' hs h).trans_le (by rw prod_const_one) | lemma | finset.prod_lt_one | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_prod' (h : ∀ i ∈ s, 1 ≤ f i) (hs : ∃ i ∈ s, 1 < f i) :
1 < (∏ i in s, f i) | prod_const_one.symm.trans_lt $ prod_lt_prod' h hs | lemma | finset.one_lt_prod' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_one' (h : ∀ i ∈ s, f i ≤ 1) (hs : ∃ i ∈ s, f i < 1) :
∏ i in s, f i < 1 | prod_const_one.le.trans_lt' $ prod_lt_prod' h hs | lemma | finset.prod_lt_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
∏ i in s, f i = ∏ i in s, g i ↔ ∀ i ∈ s, f i = g i | begin
classical,
revert h,
refine finset.induction_on s (λ _, ⟨λ _ _, false.elim, λ _, rfl⟩) (λ a s ha ih H, _),
specialize ih (λ i, H i ∘ finset.mem_insert_of_mem),
rw [finset.prod_insert ha, finset.prod_insert ha, finset.forall_mem_insert, ←ih],
exact mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a)) (... | lemma | finset.prod_eq_prod_iff_of_le | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.forall_mem_insert",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.prod_insert",
"finset.prod_le_prod'",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lt_of_prod_lt' (Hlt : ∏ i in s, f i < ∏ i in s, g i) :
∃ i ∈ s, f i < g i | begin
contrapose! Hlt with Hle,
exact prod_le_prod' Hle
end | theorem | finset.exists_lt_of_prod_lt' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le_of_prod_le' (hs : s.nonempty) (Hle : ∏ i in s, f i ≤ ∏ i in s, g i) :
∃ i ∈ s, f i ≤ g i | begin
contrapose! Hle with Hlt,
exact prod_lt_prod_of_nonempty' hs Hlt
end | theorem | finset.exists_le_of_prod_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_one_lt_of_prod_one_of_exists_ne_one' (f : ι → M)
(h₁ : ∏ i in s, f i = 1) (h₂ : ∃ i ∈ s, f i ≠ 1) :
∃ i ∈ s, 1 < f i | begin
contrapose! h₁,
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂,
apply ne_of_lt,
calc ∏ j in s, f j < ∏ j in s, 1 : prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
... = 1 : prod_const_one
end | lemma | finset.exists_one_lt_of_prod_one_of_exists_ne_one' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_nonneg (h0 : ∀ i ∈ s, 0 ≤ f i) : 0 ≤ ∏ i in s, f i | prod_induction f (λ i, 0 ≤ i) (λ _ _ ha hb, mul_nonneg ha hb) zero_le_one h0 | lemma | finset.prod_nonneg | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ g i) :
∏ i in s, f i ≤ ∏ i in s, g i | begin
induction s using finset.induction with a s has ih h,
{ simp },
{ simp only [prod_insert has], apply mul_le_mul,
{ exact h1 a (mem_insert_self a s) },
{ apply ih (λ x H, h0 _ _) (λ x H, h1 _ _); exact (mem_insert_of_mem H) },
{ apply prod_nonneg (λ x H, h0 x (mem_insert_of_mem H)) },
{ apply... | lemma | finset.prod_le_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.induction",
"ih",
"mul_le_mul"
] | If all `f i`, `i ∈ s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_le_one (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ 1) :
∏ i in s, f i ≤ 1 | begin
convert ← prod_le_prod h0 h1,
exact finset.prod_const_one
end | lemma | finset.prod_le_one | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.prod_const_one"
] | If each `f i`, `i ∈ s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_add_prod_le {i : ι} {f g h : ι → R}
(hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j)
(hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) (hg : ∀ i ∈ s, 0 ≤ g i) (hh : ∀ i ∈ s, 0 ≤ h i) :
∏ i in s, g i + ∏ i in s, h i ≤ ∏ i in s, f i | begin
simp_rw [prod_eq_mul_prod_diff_singleton hi],
refine le_trans _ (mul_le_mul_of_nonneg_right h2i _),
{ rw [right_distrib],
apply add_le_add; apply mul_le_mul_of_nonneg_left; try { apply_assumption; assumption };
apply prod_le_prod; simp * { contextual := tt } },
{ apply prod_nonneg, simp only [an... | lemma | finset.prod_add_prod_le | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"and_imp",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"right_distrib"
] | If `g, h ≤ f` and `g i + h i ≤ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_pos (h0 : ∀ i ∈ s, 0 < f i) : 0 < ∏ i in s, f i | prod_induction f (λ x, 0 < x) (λ _ _ ha hb, mul_pos ha hb) zero_lt_one h0 | lemma | finset.prod_pos | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.canonically_ordered_comm_semiring.multiset_prod_pos [nontrivial R] {m : multiset R} :
0 < m.prod ↔ (∀ x ∈ m, (0 : R) < x) | begin
induction m using quotient.induction_on,
rw [multiset.quot_mk_to_coe, multiset.coe_prod],
exact canonically_ordered_comm_semiring.list_prod_pos,
end | lemma | canonically_ordered_comm_semiring.multiset_prod_pos | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"canonically_ordered_comm_semiring.list_prod_pos",
"multiset",
"multiset.coe_prod",
"multiset.quot_mk_to_coe",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.canonically_ordered_comm_semiring.prod_pos [nontrivial R] :
0 < ∏ i in s, f i ↔ (∀ i ∈ s, (0 : R) < f i) | canonically_ordered_comm_semiring.multiset_prod_pos.trans $ by simp | lemma | canonically_ordered_comm_semiring.prod_pos | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"nontrivial"
] | Note that the name is to match `canonically_ordered_comm_semiring.mul_pos`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_add_prod_le' (hi : i ∈ s) (h2i : g i + h i ≤ f i)
(hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) :
∏ i in s, g i + ∏ i in s, h i ≤ ∏ i in s, f i | begin
classical, simp_rw [prod_eq_mul_prod_diff_singleton hi],
refine le_trans _ (mul_le_mul_right' h2i _),
rw [right_distrib],
apply add_le_add; apply mul_le_mul_left'; apply prod_le_prod';
simp only [and_imp, mem_sdiff, mem_singleton]; intros; apply_assumption; assumption
end | lemma | finset.prod_add_prod_le' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"and_imp",
"mul_le_mul_left'",
"mul_le_mul_right'",
"right_distrib"
] | If `g, h ≤ f` and `g i + h i ≤ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `canonically_ordered_comm_semiring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mono' [ordered_comm_monoid M] : monotone (λ f : ι → M, ∏ i, f i) | λ f g hfg, finset.prod_le_prod' $ λ x _, hfg x | lemma | fintype.prod_mono' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.prod_le_prod'",
"monotone",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_strict_mono' [ordered_cancel_comm_monoid M] : strict_mono (λ f : ι → M, ∏ x, f x) | λ f g hfg, let ⟨hle, i, hlt⟩ := pi.lt_def.mp hfg in
finset.prod_lt_prod' (λ i _, hle i) ⟨i, finset.mem_univ i, hlt⟩ | lemma | fintype.prod_strict_mono' | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset.mem_univ",
"finset.prod_lt_prod'",
"ordered_cancel_comm_monoid",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lt_top [comm_monoid_with_zero R] [no_zero_divisors R] [nontrivial R] [decidable_eq R]
[has_lt R] {s : finset ι} {f : ι → with_top R} (h : ∀ i ∈ s, f i ≠ ⊤) :
∏ i in s, f i < ⊤ | prod_induction f (λ a, a < ⊤) (λ a b h₁ h₂, mul_lt_top' h₁ h₂) (coe_lt_top 1) $
λ a ha, with_top.lt_top_iff_ne_top.2 (h a ha) | lemma | with_top.prod_lt_top | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"comm_monoid_with_zero",
"finset",
"no_zero_divisors",
"nontrivial",
"with_top"
] | A product of finite numbers is still finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_eq_top_iff [add_comm_monoid M] {s : finset ι} {f : ι → with_top M} :
∑ i in s, f i = ⊤ ↔ ∃ i ∈ s, f i = ⊤ | by induction s using finset.cons_induction; simp [*, or_and_distrib_right, exists_or_distrib] | lemma | with_top.sum_eq_top_iff | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"add_comm_monoid",
"exists_or_distrib",
"finset",
"finset.cons_induction",
"or_and_distrib_right",
"with_top"
] | A sum of numbers is infinite iff one of them is infinite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lt_top_iff [add_comm_monoid M] [has_lt M] {s : finset ι} {f : ι → with_top M} :
∑ i in s, f i < ⊤ ↔ ∀ i ∈ s, f i < ⊤ | by simp only [with_top.lt_top_iff_ne_top, ne.def, sum_eq_top_iff, not_exists] | lemma | with_top.sum_lt_top_iff | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"add_comm_monoid",
"finset",
"not_exists",
"with_top",
"with_top.lt_top_iff_ne_top"
] | A sum of finite numbers is still finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_lt_top [add_comm_monoid M] [has_lt M] {s : finset ι} {f : ι → with_top M}
(h : ∀ i ∈ s, f i ≠ ⊤) : (∑ i in s, f i) < ⊤ | sum_lt_top_iff.2 $ λ i hi, with_top.lt_top_iff_ne_top.2 (h i hi) | lemma | with_top.sum_lt_top | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"add_comm_monoid",
"finset",
"with_top"
] | A sum of finite numbers is still finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absolute_value.sum_le [semiring R] [ordered_semiring S]
(abv : absolute_value R S) (s : finset ι) (f : ι → R) :
abv (∑ i in s, f i) ≤ ∑ i in s, abv (f i) | finset.le_sum_of_subadditive abv (map_zero _) abv.add_le _ _ | lemma | absolute_value.sum_le | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"absolute_value",
"finset",
"ordered_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_absolute_value.abv_sum [semiring R] [ordered_semiring S] (abv : R → S)
[is_absolute_value abv] (f : ι → R) (s : finset ι) :
abv (∑ i in s, f i) ≤ ∑ i in s, abv (f i) | (is_absolute_value.to_absolute_value abv).sum_le _ _ | lemma | is_absolute_value.abv_sum | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"finset",
"is_absolute_value",
"is_absolute_value.to_absolute_value",
"ordered_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absolute_value.map_prod [comm_semiring R] [nontrivial R] [linear_ordered_comm_ring S]
(abv : absolute_value R S) (f : ι → R) (s : finset ι) :
abv (∏ i in s, f i) = ∏ i in s, abv (f i) | abv.to_monoid_hom.map_prod f s | lemma | absolute_value.map_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"absolute_value",
"comm_semiring",
"finset",
"linear_ordered_comm_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_absolute_value.map_prod [comm_semiring R] [nontrivial R] [linear_ordered_comm_ring S]
(abv : R → S) [is_absolute_value abv] (f : ι → R) (s : finset ι) :
abv (∏ i in s, f i) = ∏ i in s, abv (f i) | (is_absolute_value.to_absolute_value abv).map_prod _ _ | lemma | is_absolute_value.map_prod | algebra.big_operators | src/algebra/big_operators/order.lean | [
"algebra.order.absolute_value",
"algebra.order.ring.with_top",
"algebra.big_operators.basic",
"data.fintype.card"
] | [
"comm_semiring",
"finset",
"is_absolute_value",
"is_absolute_value.to_absolute_value",
"linear_ordered_comm_ring",
"map_prod",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod_apply {α : Type*} {β : α → Type*} [Πa, monoid (β a)] (a : α) (l : list (Πa, β a)) :
l.prod a = (l.map (λf:Πa, β a, f a)).prod | (eval_monoid_hom β a).map_list_prod _ | lemma | pi.list_prod_apply | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"map_list_prod",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_apply {α : Type*} {β : α → Type*} [∀a, comm_monoid (β a)] (a : α)
(s : multiset (Πa, β a)) : s.prod a = (s.map (λf:Πa, β a, f a)).prod | (eval_monoid_hom β a).map_multiset_prod _ | lemma | pi.multiset_prod_apply | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"map_multiset_prod",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀a, comm_monoid (β a)] (a : α)
(s : finset γ) (g : γ → Πa, β a) : (∏ c in s, g c) a = ∏ c in s, g c a | (pi.eval_monoid_hom β a).map_prod _ _ | lemma | finset.prod_apply | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"finset",
"map_prod",
"pi.eval_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀a, comm_monoid (β a)]
(s : finset γ) (g : γ → Πa, β a) : (∏ c in s, g c) = (λ a, ∏ c in s, g c a) | funext (λ a, finset.prod_apply _ _ _) | lemma | finset.prod_fn | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"finset",
"finset.prod_apply"
] | An 'unapplied' analogue of `finset.prod_apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [fintype γ]
[∀a, comm_monoid (β a)] (a : α) (g : γ → Πa, β a) : (∏ c, g c) a = ∏ c, g c a | finset.prod_apply a finset.univ g | lemma | fintype.prod_apply | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"finset.prod_apply",
"finset.univ",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_prod {α β γ : Type*} [comm_monoid α] [comm_monoid β] (s : finset γ)
(f : γ → α) (g : γ → β) : (∏ x in s, f x, ∏ x in s, g x) = ∏ x in s, (f x, g x) | by haveI := classical.dec_eq γ; exact
finset.induction_on s rfl (by simp [prod.ext_iff] {contextual := tt}) | lemma | prod_mk_prod | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"classical.dec_eq",
"comm_monoid",
"finset",
"finset.induction_on",
"prod.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.univ_prod_mul_single [fintype I] (f : Π i, Z i) :
∏ i, pi.mul_single i (f i) = f | by { ext a, simp } | lemma | finset.univ_prod_mul_single | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"fintype",
"pi.mul_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.functions_ext [finite I] (G : Type*) [comm_monoid G]
(g h : (Π i, Z i) →* G) (H : ∀ i x, g (pi.mul_single i x) = h (pi.mul_single i x)) : g = h | begin
casesI nonempty_fintype I,
ext k,
rw [← finset.univ_prod_mul_single k, g.map_prod, h.map_prod],
simp only [H]
end | lemma | monoid_hom.functions_ext | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"finite",
"finset.univ_prod_mul_single",
"nonempty_fintype",
"pi.mul_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.functions_ext' [finite I] (M : Type*) [comm_monoid M]
(g h : (Π i, Z i) →* M)
(H : ∀ i, g.comp (monoid_hom.single Z i) = h.comp (monoid_hom.single Z i)) :
g = h | g.functions_ext M h $ λ i, monoid_hom.congr_fun (H i) | lemma | monoid_hom.functions_ext' | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"comm_monoid",
"finite",
"monoid_hom.congr_fun",
"monoid_hom.single"
] | This is used as the ext lemma instead of `monoid_hom.functions_ext` for reasons explained in
note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.functions_ext [finite I] (G : Type*) [non_assoc_semiring G]
(g h : (Π i, f i) →+* G) (H : ∀ (i : I) (x : f i), g (single i x) = h (single i x)) : g = h | ring_hom.coe_add_monoid_hom_injective $
@add_monoid_hom.functions_ext I _ f _ _ G _ (g : (Π i, f i) →+ G) h H | lemma | ring_hom.functions_ext | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"finite",
"non_assoc_semiring",
"ring_hom.coe_add_monoid_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod : (∏ c in s, f c).1 = ∏ c in s, (f c).1 | (monoid_hom.fst α β).map_prod f s | lemma | prod.fst_prod | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"map_prod",
"monoid_hom.fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_prod : (∏ c in s, f c).2 = ∏ c in s, (f c).2 | (monoid_hom.snd α β).map_prod f s | lemma | prod.snd_prod | algebra.big_operators | src/algebra/big_operators/pi.lean | [
"data.fintype.card",
"algebra.group.prod",
"algebra.big_operators.basic",
"algebra.ring.pi"
] | [
"map_prod",
"monoid_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pow_eq_pow_sum {x : β} {f : α → ℕ} :
∀ {s : finset α}, (∏ i in s, x ^ (f i)) = x ^ (∑ x in s, f x) | begin
apply finset.induction,
{ simp },
{ assume a s has H,
rw [finset.prod_insert has, finset.sum_insert has, pow_add, H] }
end | lemma | finset.prod_pow_eq_pow_sum | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset",
"finset.induction",
"finset.prod_insert",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mul : (∑ x in s, f x) * b = ∑ x in s, f x * b | add_monoid_hom.map_sum (add_monoid_hom.mul_right b) _ s | lemma | finset.sum_mul | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"add_monoid_hom.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sum : b * (∑ x in s, f x) = ∑ x in s, b * f x | add_monoid_hom.map_sum (add_monoid_hom.mul_left b) _ s | lemma | finset.mul_sum | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"add_monoid_hom.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mul_sum {ι₁ : Type*} {ι₂ : Type*} (s₁ : finset ι₁) (s₂ : finset ι₂)
(f₁ : ι₁ → β) (f₂ : ι₂ → β) :
(∑ x₁ in s₁, f₁ x₁) * (∑ x₂ in s₂, f₂ x₂) = ∑ p in s₁ ×ˢ s₂, f₁ p.1 * f₂ p.2 | by { rw [sum_product, sum_mul, sum_congr rfl], intros, rw mul_sum } | lemma | finset.sum_mul_sum | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mul_boole [decidable_eq α] (s : finset α) (f : α → β) (a : α) :
(∑ x in s, (f x * ite (a = x) 1 0)) = ite (a ∈ s) (f a) 0 | by simp | lemma | finset.sum_mul_boole | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_boole_mul [decidable_eq α] (s : finset α) (f : α → β) (a : α) :
(∑ x in s, (ite (a = x) 1 0) * f x) = ite (a ∈ s) (f a) 0 | by simp | lemma | finset.sum_boole_mul | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_div [division_semiring β] {s : finset α} {f : α → β} {b : β} :
(∑ x in s, f x) / b = ∑ x in s, f x / b | by simp only [div_eq_mul_inv, sum_mul] | lemma | finset.sum_div | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"div_eq_mul_inv",
"division_semiring",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sum {δ : α → Type*} [decidable_eq α] [∀a, decidable_eq (δ a)]
{s : finset α} {t : Πa, finset (δ a)} {f : Πa, δ a → β} :
(∏ a in s, ∑ b in (t a), f a b) =
∑ p in (s.pi t), ∏ x in s.attach, f x.1 (p x.1 x.2) | begin
induction s using finset.induction with a s ha ih,
{ rw [pi_empty, sum_singleton], refl },
{ have h₁ : ∀x ∈ t a, ∀y ∈ t a, ∀h : x ≠ y,
disjoint (image (pi.cons s a x) (pi s t)) (image (pi.cons s a y) (pi s t)),
{ assume x hx y hy h,
simp only [disjoint_iff_ne, mem_image],
rintros _ ⟨... | lemma | finset.prod_sum | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"disjoint",
"finset",
"finset.induction",
"ih"
] | The product over a sum can be written as a sum over the product of sets, `finset.pi`.
`finset.prod_univ_sum` is an alternative statement when the product is over `univ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_add (f g : α → β) (s : finset α) :
∏ a in s, (f a + g a) = ∑ t in s.powerset, ((∏ a in t, f a) * (∏ a in (s \ t), g a)) | calc ∏ a in s, (f a + g a)
= ∏ a in s, ∑ p in ({true, false} : finset Prop), if p then f a else g a : by simp
... = ∑ p in (s.pi (λ _, {true, false}) : finset (Π a ∈ s, Prop)),
∏ a in s.attach, if p a.1 a.2 then f a.1 else g a.1 : prod_sum
... = ∑ t in s.powerset, (∏ a in t, f a) * (∏ a in (s \ t), g a) : b... | lemma | finset.prod_add | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"congr_arg2",
"eq_iff_iff",
"exists_prop",
"finset",
"function.funext_iff",
"subtype.coe_mk"
] | The product of `f a + g a` over all of `s` is the sum
over the powerset of `s` of the product of `f` over a subset `t` times
the product of `g` over the complement of `t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_add_ordered {ι R : Type*} [comm_semiring R] [linear_order ι] (s : finset ι)
(f g : ι → R) :
(∏ i in s, (f i + g i)) = (∏ i in s, f i) +
∑ i in s, g i * (∏ j in s.filter (< i), (f j + g j)) * ∏ j in s.filter (λ j, i < j), f j | begin
refine finset.induction_on_max s (by simp) _,
clear s, intros a s ha ihs,
have ha' : a ∉ s, from λ ha', (ha a ha').false,
rw [prod_insert ha', prod_insert ha', sum_insert ha', filter_insert, if_neg (lt_irrefl a),
filter_true_of_mem ha, ihs, add_mul, mul_add, mul_add, add_assoc],
congr' 1, rw add_com... | lemma | finset.prod_add_ordered | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_semiring",
"finset",
"finset.induction_on_max",
"mul_left_comm",
"mul_one"
] | `∏ i, (f i + g i) = (∏ i, f i) + ∑ i, g i * (∏ j < i, f j + g j) * (∏ j > i, f j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_sub_ordered {ι R : Type*} [comm_ring R] [linear_order ι] (s : finset ι) (f g : ι → R) :
(∏ i in s, (f i - g i)) = (∏ i in s, f i) -
∑ i in s, g i * (∏ j in s.filter (< i), (f j - g j)) * ∏ j in s.filter (λ j, i < j), f j | begin
simp only [sub_eq_add_neg],
convert prod_add_ordered s f (λ i, -g i),
simp,
end | lemma | finset.prod_sub_ordered | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_ring",
"finset"
] | `∏ i, (f i - g i) = (∏ i, f i) - ∑ i, g i * (∏ j < i, f j - g j) * (∏ j > i, f j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_one_sub_ordered {ι R : Type*} [comm_ring R] [linear_order ι] (s : finset ι) (f : ι → R) :
(∏ i in s, (1 - f i)) = 1 - ∑ i in s, f i * ∏ j in s.filter (< i), (1 - f j) | by { rw prod_sub_ordered, simp } | lemma | finset.prod_one_sub_ordered | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_ring",
"finset"
] | `∏ i, (1 - f i) = 1 - ∑ i, f i * (∏ j < i, 1 - f j)`. This formula is useful in construction of
a partition of unity from a collection of “bump” functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_pow_mul_eq_add_pow
{α R : Type*} [comm_semiring R] (a b : R) (s : finset α) :
(∑ t in s.powerset, a ^ t.card * b ^ (s.card - t.card)) = (a + b) ^ s.card | begin
rw [← prod_const, prod_add],
refine finset.sum_congr rfl (λ t ht, _),
rw [prod_const, prod_const, ← card_sdiff (mem_powerset.1 ht)]
end | lemma | finset.sum_pow_mul_eq_add_pow | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_semiring",
"finset"
] | Summing `a^s.card * b^(n-s.card)` over all finite subsets `s` of a `finset`
gives `(a + b)^s.card`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_sum {b : β} {s : finset α} {f : α → β}
(h : ∀ x ∈ s, b ∣ f x) : b ∣ ∑ x in s, f x | multiset.dvd_sum (λ y hy, by rcases multiset.mem_map.1 hy with ⟨x, hx, rfl⟩; exact h x hx) | theorem | finset.dvd_sum | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset",
"multiset.dvd_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_nat_cast (s : finset α) (f : α → ℕ) :
↑(∏ x in s, f x : ℕ) = (∏ x in s, (f x : β)) | (nat.cast_ring_hom β).map_prod f s | lemma | finset.prod_nat_cast | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"finset",
"map_prod",
"nat.cast_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_range_cast_nat_sub (n k : ℕ) :
∏ i in range k, (n - i : R) = (∏ i in range k, (n - i) : ℕ) | begin
rw prod_nat_cast,
cases le_or_lt k n with hkn hnk,
{ exact prod_congr rfl (λ i hi, (nat.cast_sub $ (mem_range.1 hi).le.trans hkn).symm) },
{ rw ← mem_range at hnk,
rw [prod_eq_zero hnk, prod_eq_zero hnk]; simp }
end | lemma | finset.prod_range_cast_nat_sub | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"nat.cast_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_powerset_insert [decidable_eq α] [comm_monoid β] {s : finset α} {x : α} (h : x ∉ s)
(f : finset α → β) :
(∏ a in (insert x s).powerset, f a) =
(∏ a in s.powerset, f a) * (∏ t in s.powerset, f (insert x t)) | begin
rw [powerset_insert, finset.prod_union, finset.prod_image],
{ assume t₁ h₁ t₂ h₂ heq,
rw [← finset.erase_insert (not_mem_of_mem_powerset_of_not_mem h₁ h),
← finset.erase_insert (not_mem_of_mem_powerset_of_not_mem h₂ h), heq] },
{ rw finset.disjoint_iff_ne,
assume t₁ h₁ t₂ h₂,
rcases fins... | lemma | finset.prod_powerset_insert | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_monoid",
"finset",
"finset.disjoint_iff_ne",
"finset.erase_insert",
"finset.prod_image",
"finset.prod_union"
] | A product over all subsets of `s ∪ {x}` is obtained by multiplying the product over all subsets
of `s`, and over all subsets of `s` to which one adds `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_powerset [comm_monoid β] (s : finset α) (f : finset α → β) :
∏ t in powerset s, f t = ∏ j in range (card s + 1), ∏ t in powerset_len j s, f t | by rw [powerset_card_disj_Union, prod_disj_Union] | lemma | finset.prod_powerset | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"comm_monoid",
"finset"
] | A product over `powerset s` is equal to the double product over sets of subsets of `s` with
`card s = k`, for `k = 1, ..., card s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_range_succ_mul_sum_range_succ [non_unital_non_assoc_semiring β] (n k : ℕ) (f g : ℕ → β) :
(∑ i in range (n+1), f i) * (∑ i in range (k+1), g i) =
(∑ i in range n, f i) * (∑ i in range k, g i) +
f n * (∑ i in range k, g i) +
(∑ i in range n, f i) * g k +
f n * g k | by simp only [add_mul, mul_add, add_assoc, sum_range_succ] | lemma | finset.sum_range_succ_mul_sum_range_succ | algebra.big_operators | src/algebra/big_operators/ring.lean | [
"algebra.big_operators.basic",
"algebra.field.defs",
"data.finset.pi",
"data.finset.powerset"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_list_prod [semiring R] [semiring S] (f : R ≃+* S) (l : list R) :
f l.prod = (l.map f).prod | map_list_prod f l | lemma | ring_equiv.map_list_prod | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"map_list_prod",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_list_sum [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S)
(l : list R) : f l.sum = (l.map f).sum | map_list_sum f l | lemma | ring_equiv.map_list_sum | algebra.big_operators | src/algebra/big_operators/ring_equiv.lean | [
"algebra.big_operators.basic",
"algebra.ring.equiv"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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