ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HVertexOperator.of_coeff_apply
@[simps]
def of_coeff (f : Γ → V →ₗ[R] W)
    (hf : ∀(x : V), (Function.support (f · x)).IsPWO) : HVertexOperator Γ R V W 
Arguments: {Γ : Type u_1} [PartialOrder Γ] {R : Type u_2} {V : Type u_3} {W : Type u_4} [CommRing R] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (f : Γ → V →ₗ[R] W) (hf : ∀ (x : V), (Function.support fun x_1 => (f x_1) x).IsPWO) (x : V)
Goal: (HVertexOperator.of_coeff f hf) x = (HahnModule.of R) { coeff := fun g => (f g) x, isPWO_support' := ⋯ }
-- HVertexOperator.compHahnSeries_coeff
@[simps]
def compHahnSeries (u : U) : HahnSeries Γ' (HahnSeries Γ W) 
Arguments: {Γ : Type u_5} {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ] [OrderedCancelAddCommMonoid Γ'] {R : Type u_7} [CommRing R] {U : Type u_8} {V : Type u_9} {W : Type u_10} [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (B : HVertexOperator Γ' R U V) (u : U) (g' : Γ')
Goal: (A.compHahnSeries B u).coeff g' = A ((B.coeff g') u)
-- HVertexOperator.compHahnSeries_add
@[simp]
theorem compHahnSeries_add (u v : U) :
    compHahnSeries A B (u + v) = compHahnSeries A B u + compHahnSeries A B v 
Arguments: {Γ : Type u_5} {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ] [OrderedCancelAddCommMonoid Γ'] {R : Type u_7} [CommRing R] {U : Type u_8} {V : Type u_9} {W : Type u_10} [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (B : HVertexOperator Γ' R U V) (u v : U)
Goal: A.compHahnSeries B (u + v) = A.compHahnSeries B u + A.compHahnSeries B v
-- HVertexOperator.compHahnSeries_smul
@[simp]
theorem compHahnSeries_smul (r : R) (u : U) :
    compHahnSeries A B (r • u) = r • compHahnSeries A B u 
Arguments: {Γ : Type u_5} {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ] [OrderedCancelAddCommMonoid Γ'] {R : Type u_7} [CommRing R] {U : Type u_8} {V : Type u_9} {W : Type u_10} [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (B : HVertexOperator Γ' R U V) (r : R) (u : U)
Goal: A.compHahnSeries B (r • u) = r • A.compHahnSeries B u
-- HVertexOperator.comp_apply
@[simps]
def comp : HVertexOperator (Γ' ×ₗ Γ) R U W 
Arguments: {Γ : Type u_5} {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ] [OrderedCancelAddCommMonoid Γ'] {R : Type u_7} [CommRing R] {U : Type u_8} {V : Type u_9} {W : Type u_10} [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (B : HVertexOperator Γ' R U V) (u : U)
Goal: (A.comp B) u = (HahnModule.of R) (A.compHahnSeries B u).ofIterate
-- HVertexOperator.comp_coeff
@[simp]
theorem comp_coeff (g : Γ' ×ₗ Γ) :
    (comp A B).coeff g = A.coeff (ofLex g).2 ∘ₗ B.coeff (ofLex g).1 
Arguments: {Γ : Type u_5} {Γ' : Type u_6} [OrderedCancelAddCommMonoid Γ] [OrderedCancelAddCommMonoid Γ'] {R : Type u_7} [CommRing R] {U : Type u_8} {V : Type u_9} {W : Type u_10} [AddCommGroup U] [Module R U] [AddCommGroup V] [Module R V] [AddCommGroup W] [Module R W] (A : HVertexOperator Γ R V W) (B : HVertexOperator Γ' R U V) (g : Lex (Γ' × Γ))
Goal: (A.comp B).coeff g = A.coeff (ofLex g).2 ∘ₗ B.coeff (ofLex g).1
-- Finset.le_sum_nonempty_of_subadditive_on_pred
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M → N) (p : M → Prop)
    (h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
    (g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i)) :
    f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [OrderedAddCommMonoid N] (f : M → N) (p : M → Prop) (h_mul : ∀ (x y : M), p x → p y → f (x + y) ≤ f x + f y) (hp_mul : ∀ (x y : M), p x → p y → p (x + y)) (g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i))
Goal: f (∑ i ∈ s, g i) ≤ ∑ i ∈ s, f (g i)
-- Finset.le_prod_nonempty_of_submultiplicative_on_pred
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M → N) (p : M → Prop)
    (h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
    (g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i)) :
    f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [CommMonoid M] [OrderedCommMonoid N] (f : M → N) (p : M → Prop) (h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ (x y : M), p x → p y → p (x * y)) (g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i))
Goal: f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i)
-- Finset.le_sum_nonempty_of_subadditive
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M → N) (h_mul : ∀ x y, f (x * y) ≤ f x * f y)
    {s : Finset ι} (hs : s.Nonempty) (g : ι → M) : f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [OrderedAddCommMonoid N] (f : M → N) (h_mul : ∀ (x y : M), f (x + y) ≤ f x + f y) {s : Finset ι} (hs : s.Nonempty) (g : ι → M)
Goal: f (∑ i ∈ s, g i) ≤ ∑ i ∈ s, f (g i)
-- Finset.le_prod_nonempty_of_submultiplicative
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M → N) (h_mul : ∀ x y, f (x * y) ≤ f x * f y)
    {s : Finset ι} (hs : s.Nonempty) (g : ι → M) : f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [CommMonoid M] [OrderedCommMonoid N] (f : M → N) (h_mul : ∀ (x y : M), f (x * y) ≤ f x * f y) {s : Finset ι} (hs : s.Nonempty) (g : ι → M)
Goal: f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i)
-- Finset.le_sum_of_subadditive_on_pred
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M → N) (p : M → Prop) (h_one : f 1 = 1)
    (h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
    (g : ι → M) {s : Finset ι} (hs : ∀ i ∈ s, p (g i)) : f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [OrderedAddCommMonoid N] (f : M → N) (p : M → Prop) (h_one : f 0 = 0) (h_mul : ∀ (x y : M), p x → p y → f (x + y) ≤ f x + f y) (hp_mul : ∀ (x y : M), p x → p y → p (x + y)) (g : ι → M) {s : Finset ι} (hs : ∀ i ∈ s, p (g i))
Goal: f (∑ i ∈ s, g i) ≤ ∑ i ∈ s, f (g i)
-- Finset.le_prod_of_submultiplicative_on_pred
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M → N) (p : M → Prop) (h_one : f 1 = 1)
    (h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
    (g : ι → M) {s : Finset ι} (hs : ∀ i ∈ s, p (g i)) : f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [CommMonoid M] [OrderedCommMonoid N] (f : M → N) (p : M → Prop) (h_one : f 1 = 1) (h_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ (x y : M), p x → p y → p (x * y)) (g : ι → M) {s : Finset ι} (hs : ∀ i ∈ s, p (g i))
Goal: f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i)
-- Finset.le_prod_of_submultiplicative
@[to_additive le_sum_of_subadditive]
theorem 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 ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [CommMonoid M] [OrderedCommMonoid N] (f : M → N) (h_one : f 1 = 1) (h_mul : ∀ (x y : M), f (x * y) ≤ f x * f y) (s : Finset ι) (g : ι → M)
Goal: f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i)
-- Finset.le_sum_of_subadditive
@[to_additive le_sum_of_subadditive]
theorem 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 ∈ s, g i) ≤ ∏ i ∈ s, f (g i) 
Arguments: {ι : Type u_1} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [OrderedAddCommMonoid N] (f : M → N) (h_one : f 0 = 0) (h_mul : ∀ (x y : M), f (x + y) ≤ f x + f y) (s : Finset ι) (g : ι → M)
Goal: f (∑ i ∈ s, g i) ≤ ∑ i ∈ s, f (g i)
-- Finset.sum_le_sum
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : ∀ i ∈ s, f i ≤ g i) : ∏ i ∈ s, f i ≤ ∏ i ∈ s, g i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f g : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ g i)
Goal: ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i
-- Finset.prod_le_prod'
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : ∀ i ∈ s, f i ≤ g i) : ∏ i ∈ s, f i ≤ ∏ i ∈ s, g i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f g : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ g i)
Goal: ∏ i ∈ s, f i ≤ ∏ i ∈ s, g i
-- GCongr.prod_le_prod'
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : ∀ i ∈ s, f i ≤ g i) : s.prod f ≤ s.prod g 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f g : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ g i)
Goal: s.prod f ≤ s.prod g
-- GCongr.sum_le_sum
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : ∀ i ∈ s, f i ≤ g i) : s.prod f ≤ s.prod g 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f g : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ g i)
Goal: s.sum f ≤ s.sum g
-- Finset.one_le_prod'
@[to_additive sum_nonneg]
theorem one_le_prod' (h : ∀ i ∈ s, 1 ≤ f i) : 1 ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ i ∈ s, 1 ≤ f i)
Goal: 1 ≤ ∏ i ∈ s, f i
-- Finset.sum_nonneg
@[to_additive sum_nonneg]
theorem one_le_prod' (h : ∀ i ∈ s, 1 ≤ f i) : 1 ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ i ∈ s, 0 ≤ f i)
Goal: 0 ≤ ∑ i ∈ s, f i
-- Finset.one_le_prod''
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : ∀ i : ι, 1 ≤ f i) : 1 ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ (i : ι), 1 ≤ f i)
Goal: 1 ≤ ∏ i ∈ s, f i
-- Finset.sum_nonneg'
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : ∀ i : ι, 1 ≤ f i) : 1 ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ (i : ι), 0 ≤ f i)
Goal: 0 ≤ ∑ i ∈ s, f i
-- Finset.sum_nonpos
@[to_additive sum_nonpos]
theorem prod_le_one' (h : ∀ i ∈ s, f i ≤ 1) : ∏ i ∈ s, f i ≤ 1 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ 0)
Goal: ∑ i ∈ s, f i ≤ 0
-- Finset.prod_le_one'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : ∀ i ∈ s, f i ≤ 1) : ∏ i ∈ s, f i ≤ 1 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι} (h : ∀ i ∈ s, f i ≤ 1)
Goal: ∏ i ∈ s, f i ≤ 1
-- Finset.prod_le_prod_of_subset_of_one_le'
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
    ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s t : Finset ι} (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i)
Goal: ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i
-- Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
    ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s t : Finset ι} (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 0 ≤ f i)
Goal: ∑ i ∈ s, f i ≤ ∑ i ∈ t, f i
-- Finset.prod_mono_set_of_one_le'
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : ∀ x, 1 ≤ f x) : Monotone fun s ↦ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} (hf : ∀ (x : ι), 1 ≤ f x)
Goal: Monotone fun s => ∏ x ∈ s, f x
-- Finset.sum_mono_set_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : ∀ x, 1 ≤ f x) : Monotone fun s ↦ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} (hf : ∀ (x : ι), 0 ≤ f x)
Goal: Monotone fun s => ∑ x ∈ s, f x
-- Finset.sum_le_univ_sum_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ι] {s : Finset ι} (w : ∀ x, 1 ≤ f x) :
    ∏ x ∈ s, f x ≤ ∏ x, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} [Fintype ι] {s : Finset ι} (w : ∀ (x : ι), 0 ≤ f x)
Goal: ∑ x ∈ s, f x ≤ ∑ x : ι, f x
-- Finset.prod_le_univ_prod_of_one_le'
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ι] {s : Finset ι} (w : ∀ x, 1 ≤ f x) :
    ∏ x ∈ s, f x ≤ ∏ x, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} [Fintype ι] {s : Finset ι} (w : ∀ (x : ι), 1 ≤ f x)
Goal: ∏ x ∈ s, f x ≤ ∏ x : ι, f x
-- Finset.sum_eq_zero_iff_of_nonneg
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
    (∀ i ∈ s, 1 ≤ f i) → ((∏ i ∈ s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι}
Goal: (∀ i ∈ s, 0 ≤ f i) → (∑ i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0)
-- Finset.prod_eq_one_iff_of_one_le'
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
    (∀ i ∈ s, 1 ≤ f i) → ((∏ i ∈ s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι}
Goal: (∀ i ∈ s, 1 ≤ f i) → (∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1)
-- Finset.sum_eq_zero_iff_of_nonpos
@[to_additive sum_eq_zero_iff_of_nonpos]
theorem prod_eq_one_iff_of_le_one' :
    (∀ i ∈ s, f i ≤ 1) → ((∏ i ∈ s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι}
Goal: (∀ i ∈ s, f i ≤ 0) → (∑ i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0)
-- Finset.prod_eq_one_iff_of_le_one'
@[to_additive sum_eq_zero_iff_of_nonpos]
theorem prod_eq_one_iff_of_le_one' :
    (∀ i ∈ s, f i ≤ 1) → ((∏ i ∈ s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι}
Goal: (∀ i ∈ s, f i ≤ 1) → (∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1)
-- Finset.single_le_prod'
@[to_additive single_le_sum]
theorem single_le_prod' (hf : ∀ i ∈ s, 1 ≤ f i) {a} (h : a ∈ s) : f a ≤ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι} (hf : ∀ i ∈ s, 1 ≤ f i) {a : ι} (h : a ∈ s)
Goal: f a ≤ ∏ x ∈ s, f x
-- Finset.single_le_sum
@[to_additive single_le_sum]
theorem single_le_prod' (hf : ∀ i ∈ s, 1 ≤ f i) {a} (h : a ∈ s) : f a ≤ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι} (hf : ∀ i ∈ s, 0 ≤ f i) {a : ι} (h : a ∈ s)
Goal: f a ≤ ∑ x ∈ s, f x
-- Finset.add_le_sum
@[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
    f i * f j ≤ ∏ k ∈ s, f k 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {f : ι → N} {s : Finset ι} {i j : ι} (hf : ∀ i ∈ s, 0 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j)
Goal: f i + f j ≤ ∑ k ∈ s, f k
-- Finset.mul_le_prod
@[to_additive]
lemma mul_le_prod {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j) :
    f i * f j ≤ ∏ k ∈ s, f k 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {f : ι → N} {s : Finset ι} {i j : ι} (hf : ∀ i ∈ s, 1 ≤ f i) (hi : i ∈ s) (hj : j ∈ s) (hne : i ≠ j)
Goal: f i * f j ≤ ∏ k ∈ s, f k
-- Finset.sum_le_card_nsmul
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
    s.prod f ≤ n ^ s.card 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n)
Goal: s.sum f ≤ s.card • n
-- Finset.prod_le_pow_card
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n) :
    s.prod f ≤ n ^ s.card 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, f x ≤ n)
Goal: s.prod f ≤ n ^ s.card
-- Finset.pow_card_le_prod
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, n ≤ f x) :
    n ^ s.card ≤ s.prod f 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, n ≤ f x)
Goal: n ^ s.card ≤ s.prod f
-- Finset.card_nsmul_le_sum
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, n ≤ f x) :
    n ^ s.card ≤ s.prod f 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] (s : Finset ι) (f : ι → N) (n : N) (h : ∀ x ∈ s, n ≤ f x)
Goal: s.card • n ≤ s.sum f
-- Finset.card_biUnion_le_card_mul
theorem card_biUnion_le_card_mul [DecidableEq β] (s : Finset ι) (f : ι → Finset β) (n : ℕ)
    (h : ∀ a ∈ s, (f a).card ≤ n) : (s.biUnion f).card ≤ s.card * n 
Arguments: {ι : Type u_1} {β : Type u_3} [DecidableEq β] (s : Finset ι) (f : ι → Finset β) (n : ℕ) (h : ∀ a ∈ s, (f a).card ≤ n)
Goal: (s.biUnion f).card ≤ s.card * n
-- Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ι'} {g : ι → ι'} {f : ι → N}
    (h : ∀ y ∉ t, (1 : N) ≤ ∏ x ∈ s.filter fun x ↦ g x = y, f x) :
    (∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x) ≤ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {s : Finset ι} {ι' : Type u_9} [DecidableEq ι'] {t : Finset ι'} {g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, 0 ≤ ∑ x ∈ Finset.filter (fun x => g x = y) s, f x)
Goal: ∑ y ∈ t, ∑ x ∈ Finset.filter (fun x => g x = y) s, f x ≤ ∑ x ∈ s, f x
-- Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ι'} {g : ι → ι'} {f : ι → N}
    (h : ∀ y ∉ t, (1 : N) ≤ ∏ x ∈ s.filter fun x ↦ g x = y, f x) :
    (∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x) ≤ ∏ x ∈ s, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {s : Finset ι} {ι' : Type u_9} [DecidableEq ι'] {t : Finset ι'} {g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, 1 ≤ ∏ x ∈ Finset.filter (fun x => g x = y) s, f x)
Goal: ∏ y ∈ t, ∏ x ∈ Finset.filter (fun x => g x = y) s, f x ≤ ∏ x ∈ s, f x
-- Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ι'} {g : ι → ι'} {f : ι → N}
    (h : ∀ y ∉ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x ≤ 1) :
    ∏ x ∈ s, f x ≤ ∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedAddCommMonoid N] {s : Finset ι} {ι' : Type u_9} [DecidableEq ι'] {t : Finset ι'} {g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, ∑ x ∈ Finset.filter (fun x => g x = y) s, f x ≤ 0)
Goal: ∑ x ∈ s, f x ≤ ∑ y ∈ t, ∑ x ∈ Finset.filter (fun x => g x = y) s, f x
-- Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ι'} {g : ι → ι'} {f : ι → N}
    (h : ∀ y ∉ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x ≤ 1) :
    ∏ x ∈ s, f x ≤ ∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f x 
Arguments: {ι : Type u_1} {N : Type u_5} [OrderedCommMonoid N] {s : Finset ι} {ι' : Type u_9} [DecidableEq ι'] {t : Finset ι'} {g : ι → ι'} {f : ι → N} (h : ∀ y ∉ t, ∏ x ∈ Finset.filter (fun x => g x = y) s, f x ≤ 1)
Goal: ∏ x ∈ s, f x ≤ ∏ y ∈ t, ∏ x ∈ Finset.filter (fun x => g x = y) s, f x
-- Finset.abs_sum_le_sum_abs
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ι → G) (s : Finset ι) :
    |∑ i ∈ s, f i| ≤ ∑ i ∈ s, |f i| 
Arguments: {ι : Type u_1} {G : Type u_9} [LinearOrderedAddCommGroup G] (f : ι → G) (s : Finset ι)
Goal: |∑ i ∈ s, f i| ≤ ∑ i ∈ s, |f i|
-- Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
    (hf : ∀ i ∈ s, 0 ≤ f i) : |∑ i ∈ s, f i| = ∑ i ∈ s, f i 
Arguments: {ι : Type u_1} {G : Type u_9} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι} (hf : ∀ i ∈ s, 0 ≤ f i)
Goal: |∑ i ∈ s, f i| = ∑ i ∈ s, f i
-- Finset.abs_sum_of_nonneg'
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
    (hf : ∀ i, 0 ≤ f i) : |∑ i ∈ s, f i| = ∑ i ∈ s, f i 
Arguments: {ι : Type u_1} {G : Type u_9} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι} (hf : ∀ (i : ι), 0 ≤ f i)
Goal: |∑ i ∈ s, f i| = ∑ i ∈ s, f i
-- Finset.card_le_mul_card_image_of_maps_to
theorem 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 fun x ↦ f x = a).card ≤ n) :
    s.card ≤ n * t.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] {f : α → β} {s : Finset α} {t : Finset β} (Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, (Finset.filter (fun x => f x = a) s).card ≤ n)
Goal: s.card ≤ n * t.card
-- Finset.card_le_mul_card_image
theorem card_le_mul_card_image {f : α → β} (s : Finset α) (n : ℕ)
    (hn : ∀ a ∈ s.image f, (s.filter fun x ↦ f x = a).card ≤ n) : s.card ≤ n * (s.image f).card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] {f : α → β} (s : Finset α) (n : ℕ) (hn : ∀ a ∈ Finset.image f s, (Finset.filter (fun x => f x = a) s).card ≤ n)
Goal: s.card ≤ n * (Finset.image f s).card
-- Finset.mul_card_image_le_card_of_maps_to
theorem 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 fun x ↦ f x = a).card) :
    n * t.card ≤ s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] {f : α → β} {s : Finset α} {t : Finset β} (Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (Finset.filter (fun x => f x = a) s).card)
Goal: n * t.card ≤ s.card
-- Finset.mul_card_image_le_card
theorem mul_card_image_le_card {f : α → β} (s : Finset α) (n : ℕ)
    (hn : ∀ a ∈ s.image f, n ≤ (s.filter fun x ↦ f x = a).card) : n * (s.image f).card ≤ s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] {f : α → β} (s : Finset α) (n : ℕ) (hn : ∀ a ∈ Finset.image f s, n ≤ (Finset.filter (fun x => f x = a) s).card)
Goal: n * (Finset.image f s).card ≤ s.card
-- Finset.sum_card_inter_le
theorem sum_card_inter_le (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card ≤ n) :
    (∑ t ∈ B, (s ∩ t).card) ≤ s.card * n 
Arguments: {α : Type u_2} [DecidableEq α] {s : Finset α} {B : Finset (Finset α)} {n : ℕ} (h : ∀ a ∈ s, (Finset.filter (fun x => a ∈ x) B).card ≤ n)
Goal: ∑ t ∈ B, (s ∩ t).card ≤ s.card * n
-- Finset.sum_card_le
theorem sum_card_le [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card ≤ n) :
    ∑ s ∈ B, s.card ≤ Fintype.card α * n 
Arguments: {α : Type u_2} [DecidableEq α] {B : Finset (Finset α)} {n : ℕ} [Fintype α] (h : ∀ (a : α), (Finset.filter (fun x => a ∈ x) B).card ≤ n)
Goal: ∑ s ∈ B, s.card ≤ Fintype.card α * n
-- Finset.le_sum_card_inter
theorem le_sum_card_inter (h : ∀ a ∈ s, n ≤ (B.filter (a ∈ ·)).card) :
    s.card * n ≤ ∑ t ∈ B, (s ∩ t).card 
Arguments: {α : Type u_2} [DecidableEq α] {s : Finset α} {B : Finset (Finset α)} {n : ℕ} (h : ∀ a ∈ s, n ≤ (Finset.filter (fun x => a ∈ x) B).card)
Goal: s.card * n ≤ ∑ t ∈ B, (s ∩ t).card
-- Finset.le_sum_card
theorem le_sum_card [Fintype α] (h : ∀ a, n ≤ (B.filter (a ∈ ·)).card) :
    Fintype.card α * n ≤ ∑ s ∈ B, s.card 
Arguments: {α : Type u_2} [DecidableEq α] {B : Finset (Finset α)} {n : ℕ} [Fintype α] (h : ∀ (a : α), n ≤ (Finset.filter (fun x => a ∈ x) B).card)
Goal: Fintype.card α * n ≤ ∑ s ∈ B, s.card
-- Finset.sum_card_inter
theorem sum_card_inter (h : ∀ a ∈ s, (B.filter (a ∈ ·)).card = n) :
    (∑ t ∈ B, (s ∩ t).card) = s.card * n 
Arguments: {α : Type u_2} [DecidableEq α] {s : Finset α} {B : Finset (Finset α)} {n : ℕ} (h : ∀ a ∈ s, (Finset.filter (fun x => a ∈ x) B).card = n)
Goal: ∑ t ∈ B, (s ∩ t).card = s.card * n
-- Finset.sum_card
theorem sum_card [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card = n) :
    ∑ s ∈ B, s.card = Fintype.card α * n 
Arguments: {α : Type u_2} [DecidableEq α] {B : Finset (Finset α)} {n : ℕ} [Fintype α] (h : ∀ (a : α), (Finset.filter (fun x => a ∈ x) B).card = n)
Goal: ∑ s ∈ B, s.card = Fintype.card α * n