statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
of_add_sum (s : finset ι) (f : ι → α) :
of_add (∑ i in s, f i) = ∏ i in s, of_add (f i) | rfl | lemma | of_add_sum | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_prod (s : finset ι) (f : ι → multiplicative α) :
to_add (∏ i in s, f i) = ∑ i in s, to_add (f i) | rfl | lemma | to_add_prod | algebra.big_operators | src/algebra/big_operators/basic.lean | [
"algebra.big_operators.multiset.lemmas",
"algebra.group.pi",
"algebra.group_power.lemmas",
"algebra.hom.equiv.basic",
"algebra.ring.opposite",
"data.finset.sum",
"data.fintype.basic",
"data.finset.sigma",
"data.multiset.powerset",
"data.set.pairwise.basic"
] | [
"finset",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_range [comm_monoid β] {n : ℕ} (f : ℕ → β) :
∏ i in finset.range n, f i = ∏ i : fin n, f i | prod_bij'
(λ k w, ⟨k, mem_range.mp w⟩)
(λ a ha, mem_univ _)
(λ a ha, congr_arg _ (fin.coe_mk _).symm)
(λ a m, a)
(λ a m, mem_range.mpr a.prop)
(λ a ha, fin.coe_mk _)
(λ a ha, fin.eta _ _) | theorem | finset.prod_range | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"fin.coe_mk",
"fin.eta",
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_def [comm_monoid β] {n : ℕ} (f : fin n → β) :
∏ i, f i = ((list.fin_range n).map f).prod | by simp [univ_def] | theorem | fin.prod_univ_def | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"list.fin_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_of_fn [comm_monoid β] {n : ℕ} (f : fin n → β) :
(list.of_fn f).prod = ∏ i, f i | by rw [list.of_fn_eq_map, prod_univ_def] | theorem | fin.prod_of_fn | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"list.of_fn",
"list.of_fn_eq_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_zero [comm_monoid β] (f : fin 0 → β) : ∏ i, f i = 1 | rfl | theorem | fin.prod_univ_zero | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | A product of a function `f : fin 0 → β` is `1` because `fin 0` is empty | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_univ_succ_above [comm_monoid β] {n : ℕ} (f : fin (n + 1) → β) (x : fin (n + 1)) :
∏ i, f i = f x * ∏ i : fin n, f (x.succ_above i) | by rw [univ_succ_above, prod_cons, finset.prod_map, rel_embedding.coe_fn_to_embedding] | theorem | fin.prod_univ_succ_above | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"finset.prod_map",
"rel_embedding.coe_fn_to_embedding"
] | A product of a function `f : fin (n + 1) → β` over all `fin (n + 1)`
is the product of `f x`, for some `x : fin (n + 1)` times the remaining product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_univ_succ [comm_monoid β] {n : ℕ} (f : fin (n + 1) → β) :
∏ i, f i = f 0 * ∏ i : fin n, f i.succ | prod_univ_succ_above f 0 | theorem | fin.prod_univ_succ | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | A product of a function `f : fin (n + 1) → β` over all `fin (n + 1)`
is the product of `f 0` plus the remaining product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_univ_cast_succ [comm_monoid β] {n : ℕ} (f : fin (n + 1) → β) :
∏ i, f i = (∏ i : fin n, f i.cast_succ) * f (last n) | by simpa [mul_comm] using prod_univ_succ_above f (last n) | theorem | fin.prod_univ_cast_succ | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"mul_comm"
] | A product of a function `f : fin (n + 1) → β` over all `fin (n + 1)`
is the product of `f (fin.last n)` plus the remaining product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_cons [comm_monoid β] {n : ℕ} (x : β) (f : fin n → β) :
∏ i : fin n.succ, (cons x f : fin n.succ → β) i = x * ∏ i : fin n, f i | by simp_rw [prod_univ_succ, cons_zero, cons_succ] | lemma | fin.prod_cons | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_one [comm_monoid β] (f : fin 1 → β) :
∏ i, f i = f 0 | by simp | theorem | fin.prod_univ_one | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_two [comm_monoid β] (f : fin 2 → β) :
∏ i, f i = f 0 * f 1 | by simp [prod_univ_succ] | theorem | fin.prod_univ_two | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_three [comm_monoid β] (f : fin 3 → β) :
∏ i, f i = f 0 * f 1 * f 2 | by { rw [prod_univ_cast_succ, prod_univ_two], refl } | theorem | fin.prod_univ_three | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_four [comm_monoid β] (f : fin 4 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 | by { rw [prod_univ_cast_succ, prod_univ_three], refl } | theorem | fin.prod_univ_four | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_five [comm_monoid β] (f : fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 | by { rw [prod_univ_cast_succ, prod_univ_four], refl } | theorem | fin.prod_univ_five | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_six [comm_monoid β] (f : fin 6 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 | by { rw [prod_univ_cast_succ, prod_univ_five], refl } | theorem | fin.prod_univ_six | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_seven [comm_monoid β] (f : fin 7 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 | by { rw [prod_univ_cast_succ, prod_univ_six], refl } | theorem | fin.prod_univ_seven | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_eight [comm_monoid β] (f : fin 8 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 | by { rw [prod_univ_cast_succ, prod_univ_seven], refl } | theorem | fin.prod_univ_eight | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [comm_semiring R] (a b : R) :
∑ s : finset (fin n), a ^ s.card * b ^ (n - s.card) = (a + b) ^ n | by simpa using fintype.sum_pow_mul_eq_add_pow (fin n) a b | lemma | fin.sum_pow_mul_eq_add_pow | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_semiring",
"finset",
"fintype.sum_pow_mul_eq_add_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_const [comm_monoid α] (n : ℕ) (x : α) : ∏ i : fin n, x = x ^ n | by simp | lemma | fin.prod_const | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_const [add_comm_monoid α] (n : ℕ) (x : α) : ∑ i : fin n, x = n • x | by simp | lemma | fin.sum_const | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_Ioi_zero {M : Type*} [comm_monoid M] {n : ℕ} {v : fin n.succ → M} :
∏ i in Ioi 0, v i = ∏ j : fin n, v j.succ | by rw [Ioi_zero_eq_map, finset.prod_map, rel_embedding.coe_fn_to_embedding, coe_succ_embedding] | lemma | fin.prod_Ioi_zero | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"finset.prod_map",
"rel_embedding.coe_fn_to_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_Ioi_succ {M : Type*} [comm_monoid M] {n : ℕ} (i : fin n) (v : fin n.succ → M) :
∏ j in Ioi i.succ, v j = ∏ j in Ioi i, v j.succ | by rw [Ioi_succ, finset.prod_map, rel_embedding.coe_fn_to_embedding, coe_succ_embedding] | lemma | fin.prod_Ioi_succ | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"finset.prod_map",
"rel_embedding.coe_fn_to_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_congr' {M : Type*} [comm_monoid M] {a b : ℕ} (f : fin b → M) (h : a = b) :
∏ (i : fin a), f (cast h i) = ∏ (i : fin b), f i | by { subst h, congr, ext, congr, ext, rw coe_cast, } | lemma | fin.prod_congr' | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_univ_add {M : Type*} [comm_monoid M] {a b : ℕ} (f : fin (a+b) → M) :
∏ (i : fin (a+b)), f i =
(∏ (i : fin a), f (cast_add b i)) * ∏ (i : fin b), f (nat_add a i) | begin
rw fintype.prod_equiv fin_sum_fin_equiv.symm f (λ i, f (fin_sum_fin_equiv.to_fun i)), swap,
{ intro x,
simp only [equiv.to_fun_as_coe, equiv.apply_symm_apply], },
apply fintype.prod_sum_type,
end | lemma | fin.prod_univ_add | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"equiv.apply_symm_apply",
"equiv.to_fun_as_coe",
"fintype.prod_equiv",
"fintype.prod_sum_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_trunc {M : Type*} [comm_monoid M] {a b : ℕ} (f : fin (a+b) → M)
(hf : ∀ (j : fin b), f (nat_add a j) = 1) :
∏ (i : fin (a+b)), f i =
∏ (i : fin a), f (cast_le (nat.le.intro rfl) i) | by simpa only [prod_univ_add, fintype.prod_eq_one _ hf, mul_one] | lemma | fin.prod_trunc | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_monoid",
"fintype.prod_eq_one",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_prod (f : fin n → α) (i : fin (n + 1)) : α | ((list.of_fn f).take i).prod | def | fin.partial_prod | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"list.of_fn"
] | For `f = (a₁, ..., aₙ)` in `αⁿ`, `partial_prod f` is `(1, a₁, a₁a₂, ..., a₁...aₙ)` in `αⁿ⁺¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_prod_zero (f : fin n → α) :
partial_prod f 0 = 1 | by simp [partial_prod] | lemma | fin.partial_prod_zero | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_prod_succ (f : fin n → α) (j : fin n) :
partial_prod f j.succ = partial_prod f j.cast_succ * (f j) | by simp [partial_prod, list.take_succ, list.of_fn_nth_val, dif_pos j.is_lt, ←option.coe_def] | lemma | fin.partial_prod_succ | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"list.of_fn_nth_val",
"list.take_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_prod_succ' (f : fin (n + 1) → α) (j : fin (n + 1)) :
partial_prod f j.succ = f 0 * partial_prod (fin.tail f) j | by simpa [partial_prod] | lemma | fin.partial_prod_succ' | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.tail"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_prod_left_inv {G : Type*} [group G] (f : fin (n + 1) → G) :
f 0 • partial_prod (λ i : fin n, (f i)⁻¹ * f i.succ) = f | funext $ λ x, fin.induction_on x (by simp) (λ x hx,
begin
simp only [coe_eq_cast_succ, pi.smul_apply, smul_eq_mul] at hx ⊢,
rw [partial_prod_succ, ←mul_assoc, hx, mul_inv_cancel_left],
end) | lemma | fin.partial_prod_left_inv | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.induction_on",
"group",
"mul_inv_cancel_left",
"pi.smul_apply",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_prod_right_inv {G : Type*} [group G]
(f : fin n → G) (i : fin n) :
(partial_prod f i.cast_succ)⁻¹ * partial_prod f i.succ = f i | begin
cases i with i hn,
induction i with i hi generalizing hn,
{ simp [-fin.succ_mk, partial_prod_succ] },
{ specialize hi (lt_trans (nat.lt_succ_self i) hn),
simp only [fin.coe_eq_cast_succ, fin.succ_mk, fin.cast_succ_mk] at hi ⊢,
rw [←fin.succ_mk _ _ (lt_trans (nat.lt_succ_self _) hn), ←fin.succ_mk],... | lemma | fin.partial_prod_right_inv | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.cast_succ_mk",
"fin.coe_eq_cast_succ",
"fin.succ_mk",
"group",
"inv_mul_cancel_left",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_partial_prod_mul_eq_contract_nth {G : Type*} [group G]
(g : fin (n + 1) → G) (j : fin (n + 1)) (k : fin n) :
(partial_prod g (j.succ.succ_above k.cast_succ))⁻¹ * partial_prod g (j.succ_above k).succ
= j.contract_nth has_mul.mul g k | begin
rcases lt_trichotomy (k : ℕ) j with (h|h|h),
{ rwa [succ_above_below, succ_above_below, partial_prod_right_inv, contract_nth_apply_of_lt],
{ assumption },
{ rw [cast_succ_lt_iff_succ_le, succ_le_succ_iff, le_iff_coe_le_coe],
exact le_of_lt h }},
{ rwa [succ_above_below, succ_above_above, parti... | lemma | fin.inv_partial_prod_mul_eq_contract_nth | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"group",
"inv_mul_cancel_left"
] | Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.
Then if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.
If `k = j`, it says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ₊₁ = gₖgₖ₊₁`.
If `k > j`, it says `(g₀g₁...gₖ)⁻¹ * g₀g₁...gₖ₊₁ = gₖ₊₁.`
Useful for defining group cohomology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_function_fin_equiv {m n : ℕ} : (fin n → fin m) ≃ fin (m ^ n) | equiv.of_right_inverse_of_card_le
(le_of_eq $ by simp_rw [fintype.card_fun, fintype.card_fin])
(λ f, ⟨∑ i, f i * m ^ (i : ℕ), begin
induction n with n ih generalizing f,
{ simp },
cases m,
{ exact is_empty_elim (f $ fin.last _) },
simp_rw [fin.sum_univ_cast_succ, fin.coe_cast_succ, fin.coe_last]... | def | fin_function_fin_equiv | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"equiv.of_right_inverse_of_card_le",
"fin.cast",
"fin.coe_cast_succ",
"fin.coe_last",
"fin.coe_mk",
"fin.coe_succ",
"fin.coe_zero",
"fin.ext_iff",
"fin.forall_iff",
"fin.is_le",
"fin.last",
"fintype.card_fin",
"fintype.card_fun",
"ih",
"is_empty_elim",
"mul_le_mul_right'",
"mul_left_... | Equivalence between `fin n → fin m` and `fin (m ^ n)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_function_fin_equiv_apply {m n : ℕ} (f : fin n → fin m):
(fin_function_fin_equiv f : ℕ) = ∑ (i : fin n), ↑(f i) * m ^ (i : ℕ) | rfl | lemma | fin_function_fin_equiv_apply | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin_function_fin_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_function_fin_equiv_single {m n : ℕ} [ne_zero m] (i : fin n) (j : fin m) :
(fin_function_fin_equiv (pi.single i j) : ℕ) = j * m ^ (i : ℕ) | begin
rw [fin_function_fin_equiv_apply, fintype.sum_eq_single i, pi.single_eq_same],
rintro x hx,
rw [pi.single_eq_of_ne hx, fin.coe_zero, zero_mul],
end | lemma | fin_function_fin_equiv_single | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.coe_zero",
"fin_function_fin_equiv",
"fin_function_fin_equiv_apply",
"ne_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_pi_fin_equiv {m : ℕ} {n : fin m → ℕ} :
(Π i : fin m, fin (n i)) ≃ fin (∏ i : fin m, n i) | equiv.of_right_inverse_of_card_le
(le_of_eq $ by simp_rw [fintype.card_pi, fintype.card_fin])
(λ f, ⟨∑ i, f i * ∏ j, n (fin.cast_le i.is_lt.le j), begin
induction m with m ih generalizing f,
{ simp },
rw [fin.prod_univ_cast_succ, fin.sum_univ_cast_succ],
suffices : ∀ (n : fin m → ℕ) (nn : ℕ) (f : Π ... | def | fin_pi_fin_equiv | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"equiv.of_right_inverse_of_card_le",
"fin.cast",
"fin.cast_le",
"fin.coe_mk",
"fin.coe_succ",
"fin.coe_zero",
"fin.cons",
"fin.cons_induction",
"fin.cons_succ",
"fin.cons_zero",
"fin.ext_iff",
"fin.forall_iff",
"fin.init",
"fin.init_snoc",
"fin.is_le",
"fin.is_lt",
"fin.last",
"fin... | Equivalence between `Π i : fin m, fin (n i)` and `fin (∏ i : fin m, n i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_pi_fin_equiv_apply {m : ℕ} {n : fin m → ℕ} (f : Π i : fin m, fin (n i)):
(fin_pi_fin_equiv f : ℕ) = ∑ i, f i * ∏ j, n (fin.cast_le i.is_lt.le j) | rfl | lemma | fin_pi_fin_equiv_apply | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.cast_le",
"fin_pi_fin_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_pi_fin_equiv_single {m : ℕ} {n : fin m → ℕ} [Π i, ne_zero (n i)]
(i : fin m) (j : fin (n i)) :
(fin_pi_fin_equiv (pi.single i j : Π i : fin m, fin (n i)) : ℕ)
= j * ∏ j, n (fin.cast_le i.is_lt.le j) | begin
rw [fin_pi_fin_equiv_apply, fintype.sum_eq_single i, pi.single_eq_same],
rintro x hx,
rw [pi.single_eq_of_ne hx, fin.coe_zero, zero_mul],
end | lemma | fin_pi_fin_equiv_single | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"fin.cast_le",
"fin.coe_zero",
"fin_pi_fin_equiv",
"fin_pi_fin_equiv_apply",
"ne_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_take_of_fn {n : ℕ} (f : fin n → α) (i : ℕ) :
((of_fn f).take i).prod = ∏ j in finset.univ.filter (λ (j : fin n), j.val < i), f j | begin
have A : ∀ (j : fin n), ¬ ((j : ℕ) < 0) := λ j, not_lt_bot,
induction i with i IH, { simp [A] },
by_cases h : i < n,
{ have : i < length (of_fn f), by rwa [length_of_fn f],
rw prod_take_succ _ _ this,
have A : ((finset.univ : finset (fin n)).filter (λ j, j.val < i + 1))
= ((finset.univ : fin... | lemma | list.prod_take_of_fn | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"filter",
"finset",
"finset.filter",
"finset.prod_union",
"finset.univ",
"nat.lt_succ_iff_lt_or_eq",
"not_lt_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_of_fn {n : ℕ} {f : fin n → α} :
(of_fn f).prod = ∏ i, f i | begin
convert prod_take_of_fn f n,
{ rw [take_all_of_le (le_of_eq (length_of_fn f))] },
{ have : ∀ (j : fin n), (j : ℕ) < n := λ j, j.is_lt,
simp [this] }
end | lemma | list.prod_of_fn | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alternating_sum_eq_finset_sum {G : Type*} [add_comm_group G] :
∀ (L : list G), alternating_sum L = ∑ i : fin L.length, (-1 : ℤ) ^ (i : ℕ) • L.nth_le i i.is_lt | | [] := by { rw [alternating_sum, finset.sum_eq_zero], rintro ⟨i, ⟨⟩⟩ }
| (g :: []) := by simp
| (g :: h :: L) :=
calc g + -h + L.alternating_sum
= g + -h + ∑ i : fin L.length, (-1 : ℤ) ^ (i : ℕ) • L.nth_le i i.2 :
congr_arg _ (alternating_sum_eq_finset_sum _)
... = ∑ i : fin (L.length + 2), (-1 : ℤ) ^ (i : ℕ... | lemma | list.alternating_sum_eq_finset_sum | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"add_comm_group",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alternating_prod_eq_finset_prod {G : Type*} [comm_group G] :
∀ (L : list G), alternating_prod L = ∏ i : fin L.length, (L.nth_le i i.2) ^ ((-1 : ℤ) ^ (i : ℕ)) | | [] := by { rw [alternating_prod, finset.prod_eq_one], rintro ⟨i, ⟨⟩⟩ }
| (g :: []) :=
begin
show g = ∏ i : fin 1, [g].nth_le i i.2 ^ (-1 : ℤ) ^ (i : ℕ),
rw [fin.prod_univ_succ], simp,
end
| (g :: h :: L) :=
calc g * h⁻¹ * L.alternating_prod
= g * h⁻¹ * ∏ i : fin L.length, L.nth_le i i.2 ^ (-1 : ℤ) ^ (i : ℕ) :... | lemma | list.alternating_prod_eq_finset_prod | algebra.big_operators | src/algebra/big_operators/fin.lean | [
"data.fintype.big_operators",
"data.fintype.fin",
"data.list.fin_range",
"logic.equiv.fin"
] | [
"comm_group",
"fin.prod_univ_succ",
"finset.prod_eq_one",
"mul_assoc",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsum {M α} [add_comm_monoid M] (f : α → M) : M | if h : (support (f ∘ plift.down)).finite then ∑ i in h.to_finset, f i.down else 0 | def | finsum | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"add_comm_monoid",
"finite"
] | Sum of `f x` as `x` ranges over the elements of the support of `f`, if it's finite. Zero
otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod (f : α → M) : M | if h : (mul_support (f ∘ plift.down)).finite then ∏ i in h.to_finset, f i.down else 1 | def | finprod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite"
] | Product of `f x` as `x` ranges over the elements of the multiplicative support of `f`, if it's
finite. One otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_eq_prod_plift_of_mul_support_to_finset_subset
{f : α → M} (hf : (mul_support (f ∘ plift.down)).finite) {s : finset (plift α)}
(hs : hf.to_finset ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i.down | begin
rw [finprod, dif_pos],
refine finset.prod_subset hs (λ x hx hxf, _),
rwa [hf.mem_to_finset, nmem_mul_support] at hxf
end | lemma | finprod_eq_prod_plift_of_mul_support_to_finset_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod",
"finset",
"finset.prod_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_prod_plift_of_mul_support_subset
{f : α → M} {s : finset (plift α)} (hs : mul_support (f ∘ plift.down) ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i.down | finprod_eq_prod_plift_of_mul_support_to_finset_subset
(s.finite_to_set.subset hs) $ λ x hx, by { rw finite.mem_to_finset at hx, exact hs hx } | lemma | finprod_eq_prod_plift_of_mul_support_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_plift_of_mul_support_to_finset_subset",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_one : ∏ᶠ i : α, (1 : M) = 1 | begin
have : mul_support (λ x : plift α, (λ _, 1 : α → M) x.down) ⊆ (∅ : finset (plift α)),
from λ x h, h rfl,
rw [finprod_eq_prod_plift_of_mul_support_subset this, finset.prod_empty]
end | lemma | finprod_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_plift_of_mul_support_subset",
"finset",
"finset.prod_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_of_is_empty [is_empty α] (f : α → M) : ∏ᶠ i, f i = 1 | by { rw ← finprod_one, congr } | lemma | finprod_of_is_empty | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_one",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_false (f : false → M) : ∏ᶠ i, f i = 1 | finprod_of_is_empty _ | lemma | finprod_false | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_of_is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_single (f : α → M) (a : α) (ha : ∀ x ≠ a, f x = 1) :
∏ᶠ x, f x = f a | begin
have : mul_support (f ∘ plift.down) ⊆ ({plift.up a} : finset (plift α)),
{ intro x, contrapose,
simpa [plift.eq_up_iff_down_eq] using ha x.down },
rw [finprod_eq_prod_plift_of_mul_support_subset this, finset.prod_singleton],
end | lemma | finprod_eq_single | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_plift_of_mul_support_subset",
"finset",
"finset.prod_singleton",
"plift.eq_up_iff_down_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_unique [unique α] (f : α → M) : ∏ᶠ i, f i = f default | finprod_eq_single f default $ λ x hx, (hx $ unique.eq_default _).elim | lemma | finprod_unique | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_single",
"unique",
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_true (f : true → M) : ∏ᶠ i, f i = f trivial | @finprod_unique M true _ ⟨⟨trivial⟩, λ _, rfl⟩ f | lemma | finprod_true | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_dif {p : Prop} [decidable p] (f : p → M) :
∏ᶠ i, f i = if h : p then f h else 1 | begin
split_ifs,
{ haveI : unique p := ⟨⟨h⟩, λ _, rfl⟩, exact finprod_unique f },
{ haveI : is_empty p := ⟨h⟩, exact finprod_of_is_empty f }
end | lemma | finprod_eq_dif | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_of_is_empty",
"finprod_unique",
"is_empty",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_if {p : Prop} [decidable p] {x : M} :
∏ᶠ i : p, x = if p then x else 1 | finprod_eq_dif (λ _, x) | lemma | finprod_eq_if | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_dif"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_congr {f g : α → M} (h : ∀ x, f x = g x) :
finprod f = finprod g | congr_arg _ $ funext h | lemma | finprod_congr | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_congr_Prop {p q : Prop} {f : p → M} {g : q → M} (hpq : p = q)
(hfg : ∀ h : q, f (hpq.mpr h) = g h) :
finprod f = finprod g | by { subst q, exact finprod_congr hfg } | lemma | finprod_congr_Prop | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod",
"finprod_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_induction {f : α → M} (p : M → Prop) (hp₀ : p 1) (hp₁ : ∀ x y, p x → p y → p (x * y))
(hp₂ : ∀ i, p (f i)) :
p (∏ᶠ i, f i) | begin
rw finprod,
split_ifs,
exacts [finset.prod_induction _ _ hp₁ hp₀ (λ i hi, hp₂ _), hp₀]
end | lemma | finprod_induction | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod",
"finset.prod_induction"
] | To prove a property of a finite product, it suffices to prove that the property is
multiplicative and holds on the factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_nonneg {R : Type*} [ordered_comm_semiring R] {f : α → R} (hf : ∀ x, 0 ≤ f x) :
0 ≤ ∏ᶠ x, f x | finprod_induction (λ x, 0 ≤ x) zero_le_one (λ x y, mul_nonneg) hf | lemma | finprod_nonneg | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_induction",
"ordered_comm_semiring",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_finprod' {M : Type*} [ordered_comm_monoid M] {f : α → M} (hf : ∀ i, 1 ≤ f i) :
1 ≤ ∏ᶠ i, f i | finprod_induction _ le_rfl (λ _ _, one_le_mul) hf | lemma | one_le_finprod' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_induction",
"le_rfl",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_finprod_plift (f : M →* N) (g : α → M)
(h : (mul_support $ g ∘ plift.down).finite) :
f (∏ᶠ x, g x) = ∏ᶠ x, f (g x) | begin
rw [finprod_eq_prod_plift_of_mul_support_subset h.coe_to_finset.ge,
finprod_eq_prod_plift_of_mul_support_subset, f.map_prod],
rw [h.coe_to_finset],
exact mul_support_comp_subset f.map_one (g ∘ plift.down)
end | lemma | monoid_hom.map_finprod_plift | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_eq_prod_plift_of_mul_support_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_finprod_Prop {p : Prop} (f : M →* N) (g : p → M) :
f (∏ᶠ x, g x) = ∏ᶠ x, f (g x) | f.map_finprod_plift g (set.to_finite _) | lemma | monoid_hom.map_finprod_Prop | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"set.to_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_finprod_of_preimage_one (f : M →* N)
(hf : ∀ x, f x = 1 → x = 1) (g : α → M) :
f (∏ᶠ i, g i) = ∏ᶠ i, f (g i) | begin
by_cases hg : (mul_support $ g ∘ plift.down).finite, { exact f.map_finprod_plift g hg },
rw [finprod, dif_neg, f.map_one, finprod, dif_neg],
exacts [infinite.mono (λ x hx, mt (hf (g x.down)) hx) hg, hg]
end | lemma | monoid_hom.map_finprod_of_preimage_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_finprod_of_injective (g : M →* N) (hg : injective g)
(f : α → M) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) | g.map_finprod_of_preimage_one (λ x, (hg.eq_iff' g.map_one).mp) f | lemma | monoid_hom.map_finprod_of_injective | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.map_finprod (g : M ≃* N) (f : α → M) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) | g.to_monoid_hom.map_finprod_of_injective g.injective f | lemma | mul_equiv.map_finprod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsum_smul {R M : Type*} [ring R] [add_comm_group M] [module R M] [no_zero_smul_divisors R M]
(f : ι → R) (x : M) :
(∑ᶠ i, f i) • x = ∑ᶠ i, f i • x | begin
rcases eq_or_ne x 0 with rfl|hx, { simp },
exact ((smul_add_hom R M).flip x).map_finsum_of_injective (smul_left_injective R hx) _
end | lemma | finsum_smul | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"add_comm_group",
"eq_or_ne",
"module",
"no_zero_smul_divisors",
"ring",
"smul_add_hom",
"smul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_finsum {R M : Type*} [ring R] [add_comm_group M] [module R M]
[no_zero_smul_divisors R M] (c : R) (f : ι → M) :
c • (∑ᶠ i, f i) = (∑ᶠ i, c • f i) | begin
rcases eq_or_ne c 0 with rfl|hc, { simp },
exact (smul_add_hom R M c).map_finsum_of_injective (smul_right_injective M hc) _
end | lemma | smul_finsum | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"add_comm_group",
"eq_or_ne",
"module",
"no_zero_smul_divisors",
"ring",
"smul_add_hom",
"smul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_inv_distrib [division_comm_monoid G] (f : α → G) :
∏ᶠ x, (f x)⁻¹ = (∏ᶠ x, f x)⁻¹ | ((mul_equiv.inv G).map_finprod f).symm | lemma | finprod_inv_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"division_comm_monoid",
"mul_equiv.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_mul_indicator_apply (s : set α)
(f : α → M) (a : α) :
∏ᶠ (h : a ∈ s), f a = mul_indicator s f a | by convert finprod_eq_if | lemma | finprod_eq_mul_indicator_apply | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_if"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_mul_support (f : α → M) (a : α) :
∏ᶠ (h : f a ≠ 1), f a = f a | by rw [← mem_mul_support, finprod_eq_mul_indicator_apply, mul_indicator_mul_support] | lemma | finprod_mem_mul_support | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_mul_indicator_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_def (s : set α) (f : α → M) :
∏ᶠ a ∈ s, f a = ∏ᶠ a, mul_indicator s f a | finprod_congr $ finprod_eq_mul_indicator_apply s f | lemma | finprod_mem_def | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_congr",
"finprod_eq_mul_indicator_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_prod_of_mul_support_subset (f : α → M) {s : finset α}
(h : mul_support f ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i | begin
have A : mul_support (f ∘ plift.down) = equiv.plift.symm '' mul_support f,
{ rw mul_support_comp_eq_preimage,
exact (equiv.plift.symm.image_eq_preimage _).symm },
have : mul_support (f ∘ plift.down) ⊆ s.map equiv.plift.symm.to_embedding,
{ rw [A, finset.coe_map], exact image_subset _ h },
rw [finpro... | lemma | finprod_eq_prod_of_mul_support_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_plift_of_mul_support_subset",
"finset",
"finset.coe_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_prod_of_mul_support_to_finset_subset (f : α → M)
(hf : (mul_support f).finite) {s : finset α} (h : hf.to_finset ⊆ s) :
∏ᶠ i, f i = ∏ i in s, f i | finprod_eq_prod_of_mul_support_subset _ $ λ x hx, h $ hf.mem_to_finset.2 hx | lemma | finprod_eq_prod_of_mul_support_to_finset_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_eq_prod_of_mul_support_subset",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_finset_prod_of_mul_support_subset
(f : α → M) {s : finset α} (h : mul_support f ⊆ (s : set α)) :
∏ᶠ i, f i = ∏ i in s, f i | begin
have h' : (s.finite_to_set.subset h).to_finset ⊆ s,
{ simpa [← finset.coe_subset, set.coe_to_finset], },
exact finprod_eq_prod_of_mul_support_to_finset_subset _ _ h',
end | lemma | finprod_eq_finset_prod_of_mul_support_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset",
"finset.coe_subset",
"set.coe_to_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_def (f : α → M) [decidable (mul_support f).finite] :
∏ᶠ i : α, f i = if h : (mul_support f).finite then ∏ i in h.to_finset, f i else 1 | begin
split_ifs,
{ exact finprod_eq_prod_of_mul_support_to_finset_subset _ h (finset.subset.refl _) },
{ rw [finprod, dif_neg],
rw [mul_support_comp_eq_preimage],
exact mt (λ hf, hf.of_preimage equiv.plift.surjective) h}
end | lemma | finprod_def | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod",
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset.subset.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_of_infinite_mul_support {f : α → M} (hf : (mul_support f).infinite) :
∏ᶠ i, f i = 1 | by { classical, rw [finprod_def, dif_neg hf] } | lemma | finprod_of_infinite_mul_support | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_def",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_prod (f : α → M) (hf : (mul_support f).finite) :
∏ᶠ i : α, f i = ∏ i in hf.to_finset, f i | by { classical, rw [finprod_def, dif_pos hf] } | lemma | finprod_eq_prod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_prod_of_fintype [fintype α] (f : α → M) :
∏ᶠ i : α, f i = ∏ i, f i | finprod_eq_prod_of_mul_support_to_finset_subset _ (set.to_finite _) $ finset.subset_univ _ | lemma | finprod_eq_prod_of_fintype | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset.subset_univ",
"fintype",
"set.to_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_cond_eq_prod_of_cond_iff (f : α → M) {p : α → Prop} {t : finset α}
(h : ∀ {x}, f x ≠ 1 → (p x ↔ x ∈ t)) :
∏ᶠ i (hi : p i), f i = ∏ i in t, f i | begin
set s := {x | p x},
have : mul_support (s.mul_indicator f) ⊆ t,
{ rw [set.mul_support_mul_indicator], intros x hx, exact (h hx.2).1 hx.1 },
erw [finprod_mem_def, finprod_eq_prod_of_mul_support_subset _ this],
refine finset.prod_congr rfl (λ x hx, mul_indicator_apply_eq_self.2 $ λ hxs, _),
contrapose! ... | lemma | finprod_cond_eq_prod_of_cond_iff | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_eq_prod_of_mul_support_subset",
"finprod_mem_def",
"finset",
"finset.prod_congr",
"set.mul_support_mul_indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_cond_ne (f : α → M) (a : α) [decidable_eq α]
(hf : (mul_support f).finite) : (∏ᶠ i ≠ a, f i) = ∏ i in hf.to_finset.erase a, f i | begin
apply finprod_cond_eq_prod_of_cond_iff,
intros x hx,
rw [finset.mem_erase, finite.mem_to_finset, mem_mul_support],
exact ⟨λ h, and.intro h hx, λ h, h.1⟩
end | lemma | finprod_cond_ne | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_cond_eq_prod_of_cond_iff",
"finset.mem_erase"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_prod_of_inter_mul_support_eq (f : α → M) {s : set α}
{t : finset α} (h : s ∩ mul_support f = t ∩ mul_support f) :
∏ᶠ i ∈ s, f i = ∏ i in t, f i | finprod_cond_eq_prod_of_cond_iff _ $ by simpa [set.ext_iff] using h | lemma | finprod_mem_eq_prod_of_inter_mul_support_eq | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_cond_eq_prod_of_cond_iff",
"finset",
"set.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_prod_of_subset (f : α → M) {s : set α} {t : finset α}
(h₁ : s ∩ mul_support f ⊆ t) (h₂ : ↑t ⊆ s) :
∏ᶠ i ∈ s, f i = ∏ i in t, f i | finprod_cond_eq_prod_of_cond_iff _ $ λ x hx, ⟨λ h, h₁ ⟨h, hx⟩, λ h, h₂ h⟩ | lemma | finprod_mem_eq_prod_of_subset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_cond_eq_prod_of_cond_iff",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_prod (f : α → M) {s : set α}
(hf : (s ∩ mul_support f).finite) :
∏ᶠ i ∈ s, f i = ∏ i in hf.to_finset, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by simp [inter_assoc] | lemma | finprod_mem_eq_prod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_eq_prod_of_inter_mul_support_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_prod_filter (f : α → M) (s : set α) [decidable_pred (∈ s)]
(hf : (mul_support f).finite) :
∏ᶠ i ∈ s, f i = ∏ i in finset.filter (∈ s) hf.to_finset, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by simp [inter_comm, inter_left_comm] | lemma | finprod_mem_eq_prod_filter | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_eq_prod_of_inter_mul_support_eq",
"finset.filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_to_finset_prod (f : α → M) (s : set α) [fintype s] :
∏ᶠ i ∈ s, f i = ∏ i in s.to_finset, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by rw [coe_to_finset] | lemma | finprod_mem_eq_to_finset_prod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_eq_prod_of_inter_mul_support_eq",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_finite_to_finset_prod (f : α → M) {s : set α} (hs : s.finite) :
∏ᶠ i ∈ s, f i = ∏ i in hs.to_finset, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ $ by rw [hs.coe_to_finset] | lemma | finprod_mem_eq_finite_to_finset_prod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_eq_prod_of_inter_mul_support_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_finset_eq_prod (f : α → M) (s : finset α) :
∏ᶠ i ∈ s, f i = ∏ i in s, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ rfl | lemma | finprod_mem_finset_eq_prod | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_eq_prod_of_inter_mul_support_eq",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_coe_finset (f : α → M) (s : finset α) :
∏ᶠ i ∈ (s : set α), f i = ∏ i in s, f i | finprod_mem_eq_prod_of_inter_mul_support_eq _ rfl | lemma | finprod_mem_coe_finset | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_eq_prod_of_inter_mul_support_eq",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_one_of_infinite {f : α → M} {s : set α}
(hs : (s ∩ mul_support f).infinite) : ∏ᶠ i ∈ s, f i = 1 | begin
rw finprod_mem_def,
apply finprod_of_infinite_mul_support,
rwa [← mul_support_mul_indicator] at hs
end | lemma | finprod_mem_eq_one_of_infinite | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_def",
"finprod_of_infinite_mul_support",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_eq_one_of_forall_eq_one {f : α → M} {s : set α} (h : ∀ x ∈ s, f x = 1) :
∏ᶠ i ∈ s, f i = 1 | by simp [h] {contextual := tt} | lemma | finprod_mem_eq_one_of_forall_eq_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inter_mul_support (f : α → M) (s : set α) :
∏ᶠ i ∈ (s ∩ mul_support f), f i = ∏ᶠ i ∈ s, f i | by rw [finprod_mem_def, finprod_mem_def, mul_indicator_inter_mul_support] | lemma | finprod_mem_inter_mul_support | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inter_mul_support_eq (f : α → M) (s t : set α)
(h : s ∩ mul_support f = t ∩ mul_support f) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, f i | by rw [← finprod_mem_inter_mul_support, h, finprod_mem_inter_mul_support] | lemma | finprod_mem_inter_mul_support_eq | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_mem_inter_mul_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_inter_mul_support_eq' (f : α → M) (s t : set α)
(h : ∀ x ∈ mul_support f, x ∈ s ↔ x ∈ t) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, f i | begin
apply finprod_mem_inter_mul_support_eq,
ext x,
exact and_congr_left (h x)
end | lemma | finprod_mem_inter_mul_support_eq' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"and_congr_left",
"finprod_mem_inter_mul_support_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_univ (f : α → M) : ∏ᶠ i ∈ @set.univ α, f i = ∏ᶠ i : α, f i | finprod_congr $ λ i, finprod_true _ | lemma | finprod_mem_univ | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_congr",
"finprod_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mem_congr (h₀ : s = t) (h₁ : ∀ x ∈ t, f x = g x) :
∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, g i | h₀.symm ▸ (finprod_congr $ λ i, finprod_congr_Prop rfl (h₁ i)) | lemma | finprod_mem_congr | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finprod_congr",
"finprod_congr_Prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_eq_one_of_forall_eq_one {f : α → M} (h : ∀ x, f x = 1) :
∏ᶠ i, f i = 1 | by simp [h] {contextual := tt} | lemma | finprod_eq_one_of_forall_eq_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_mul_distrib (hf : (mul_support f).finite) (hg : (mul_support g).finite) :
∏ᶠ i, f i * g i = (∏ᶠ i, f i) * ∏ᶠ i, g i | begin
classical,
rw [finprod_eq_prod_of_mul_support_to_finset_subset _ hf (finset.subset_union_left _ _),
finprod_eq_prod_of_mul_support_to_finset_subset _ hg (finset.subset_union_right _ _),
← finset.prod_mul_distrib],
refine finprod_eq_prod_of_mul_support_subset _ _,
simp [mul_support_mul]
end | lemma | finprod_mul_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_eq_prod_of_mul_support_subset",
"finprod_eq_prod_of_mul_support_to_finset_subset",
"finset.prod_mul_distrib",
"finset.subset_union_left",
"finset.subset_union_right"
] | If the multiplicative supports of `f` and `g` are finite, then the product of `f i * g i` equals
the product of `f i` multiplied by the product of `g i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_div_distrib [division_comm_monoid G] {f g : α → G} (hf : (mul_support f).finite)
(hg : (mul_support g).finite) :
∏ᶠ i, f i / g i = (∏ᶠ i, f i) / ∏ᶠ i, g i | by simp only [div_eq_mul_inv, finprod_mul_distrib hf ((mul_support_inv g).symm.rec hg),
finprod_inv_distrib] | lemma | finprod_div_distrib | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"div_eq_mul_inv",
"division_comm_monoid",
"finite",
"finprod_inv_distrib",
"finprod_mul_distrib"
] | If the multiplicative supports of `f` and `g` are finite, then the product of `f i / g i`
equals the product of `f i` divided by the product of `g i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_mul_distrib' (hf : (s ∩ mul_support f).finite) (hg : (s ∩ mul_support g).finite) :
∏ᶠ i ∈ s, f i * g i = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ s, g i | begin
rw [← mul_support_mul_indicator] at hf hg,
simp only [finprod_mem_def, mul_indicator_mul, finprod_mul_distrib hf hg]
end | lemma | finprod_mem_mul_distrib' | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [
"finite",
"finprod_mem_def",
"finprod_mul_distrib"
] | A more general version of `finprod_mem_mul_distrib` that only requires `s ∩ mul_support f` and
`s ∩ mul_support g` rather than `s` to be finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_mem_one (s : set α) : ∏ᶠ i ∈ s, (1 : M) = 1 | by simp | lemma | finprod_mem_one | algebra.big_operators | src/algebra/big_operators/finprod.lean | [
"algebra.big_operators.order",
"algebra.indicator_function"
] | [] | The product of the constant function `1` over any set equals `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.