ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- abs_add_three
theorem abs_add_three (a b c : α) : |a + b + c| ≤ |a| + |b| + |c| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: |a + b + c| ≤ |a| + |b| + |c|
-- dist_bdd_within_interval
theorem dist_bdd_within_interval {a b lb ub : α} (hal : lb ≤ a) (hau : a ≤ ub) (hbl : lb ≤ b)
    (hbu : b ≤ ub) : |a - b| ≤ ub - lb 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b lb ub : α} (hal : lb ≤ a) (hau : a ≤ ub) (hbl : lb ≤ b) (hbu : b ≤ ub)
Goal: |a - b| ≤ ub - lb
-- eq_of_abs_sub_nonpos
theorem eq_of_abs_sub_nonpos (h : |a - b| ≤ 0) : a = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : |a - b| ≤ 0)
Goal: a = b
-- abs_sub_nonpos
theorem abs_sub_nonpos : |a - b| ≤ 0 ↔ a = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: |a - b| ≤ 0 ↔ a = b
-- abs_sub_pos
theorem abs_sub_pos : 0 < |a - b| ↔ a ≠ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: 0 < |a - b| ↔ a ≠ b
-- abs_eq_self
@[simp]
theorem abs_eq_self : |a| = a ↔ 0 ≤ a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α}
Goal: |a| = a ↔ 0 ≤ a
-- abs_eq_neg_self
@[simp]
theorem abs_eq_neg_self : |a| = -a ↔ a ≤ 0 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α}
Goal: |a| = -a ↔ a ≤ 0
-- abs_cases
theorem abs_cases (a : α) : |a| = a ∧ 0 ≤ a ∨ |a| = -a ∧ a < 0 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α)
Goal: |a| = a ∧ 0 ≤ a ∨ |a| = -a ∧ a < 0
-- max_zero_add_max_neg_zero_eq_abs_self
@[simp]
theorem max_zero_add_max_neg_zero_eq_abs_self (a : α) : max a 0 + max (-a) 0 = |a| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α)
Goal: max a 0 + max (-a) 0 = |a|
-- Function.support_sup
@[to_additive]
lemma mulSupport_sup [SemilatticeSup M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊔ g x) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [SemilatticeSup M] (f g : α → M)
Goal: (Function.support fun x => f x ⊔ g x) ⊆ Function.support f ∪ Function.support g
-- Function.mulSupport_sup
@[to_additive]
lemma mulSupport_sup [SemilatticeSup M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊔ g x) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [One M] [SemilatticeSup M] (f g : α → M)
Goal: (Function.mulSupport fun x => f x ⊔ g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g
-- Function.mulSupport_inf
@[to_additive]
lemma mulSupport_inf [SemilatticeInf M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊓ g x) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [One M] [SemilatticeInf M] (f g : α → M)
Goal: (Function.mulSupport fun x => f x ⊓ g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g
-- Function.support_inf
@[to_additive]
lemma mulSupport_inf [SemilatticeInf M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊓ g x) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [SemilatticeInf M] (f g : α → M)
Goal: (Function.support fun x => f x ⊓ g x) ⊆ Function.support f ∪ Function.support g
-- Function.support_max
@[to_additive]
lemma mulSupport_max [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ max (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [LinearOrder M] (f g : α → M)
Goal: (Function.support fun x => max (f x) (g x)) ⊆ Function.support f ∪ Function.support g
-- Function.mulSupport_max
@[to_additive]
lemma mulSupport_max [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ max (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [One M] [LinearOrder M] (f g : α → M)
Goal: (Function.mulSupport fun x => max (f x) (g x)) ⊆ Function.mulSupport f ∪ Function.mulSupport g
-- Function.support_min
@[to_additive]
lemma mulSupport_min [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ min (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [LinearOrder M] (f g : α → M)
Goal: (Function.support fun x => min (f x) (g x)) ⊆ Function.support f ∪ Function.support g
-- Function.mulSupport_min
@[to_additive]
lemma mulSupport_min [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ min (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
Arguments: {α : Type u_2} {M : Type u_4} [One M] [LinearOrder M] (f g : α → M)
Goal: (Function.mulSupport fun x => min (f x) (g x)) ⊆ Function.mulSupport f ∪ Function.mulSupport g
-- Function.support_iSup
@[to_additive]
lemma mulSupport_iSup [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨆ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [Zero M] [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M)
Goal: (Function.support fun x => ⨆ i, f i x) ⊆ ⋃ i, Function.support (f i)
-- Function.mulSupport_iSup
@[to_additive]
lemma mulSupport_iSup [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨆ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [One M] [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M)
Goal: (Function.mulSupport fun x => ⨆ i, f i x) ⊆ ⋃ i, Function.mulSupport (f i)
-- Function.mulSupport_iInf
@[to_additive]
lemma mulSupport_iInf [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨅ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [One M] [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M)
Goal: (Function.mulSupport fun x => ⨅ i, f i x) ⊆ ⋃ i, Function.mulSupport (f i)
-- Function.support_iInf
@[to_additive]
lemma mulSupport_iInf [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨅ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [Zero M] [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M)
Goal: (Function.support fun x => ⨅ i, f i x) ⊆ ⋃ i, Function.support (f i)
-- Set.mulIndicator_apply_le'
@[to_additive]
lemma mulIndicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
    mulIndicator s f a ≤ y 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [One M] {s : Set α} {f : α → M} {a : α} {y : M} (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y)
Goal: s.mulIndicator f a ≤ y
-- Set.indicator_apply_le'
@[to_additive]
lemma mulIndicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
    mulIndicator s f a ≤ y 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [Zero M] {s : Set α} {f : α → M} {a : α} {y : M} (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 0 ≤ y)
Goal: s.indicator f a ≤ y
-- Set.mulIndicator_le'
@[to_additive]
lemma mulIndicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a, a ∉ s → 1 ≤ g a) :
    mulIndicator s f ≤ g 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [One M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a ∉ s, 1 ≤ g a)
Goal: s.mulIndicator f ≤ g
-- Set.indicator_le'
@[to_additive]
lemma mulIndicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a, a ∉ s → 1 ≤ g a) :
    mulIndicator s f ≤ g 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [Zero M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a ∉ s, 0 ≤ g a)
Goal: s.indicator f ≤ g
-- Set.le_indicator_apply
@[to_additive]
lemma le_mulIndicator_apply (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
    y ≤ mulIndicator s g a 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [Zero M] {s : Set α} {g : α → M} {a : α} {y : M} (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 0)
Goal: y ≤ s.indicator g a
-- Set.le_mulIndicator_apply
@[to_additive]
lemma le_mulIndicator_apply (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
    y ≤ mulIndicator s g a 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [One M] {s : Set α} {g : α → M} {a : α} {y : M} (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1)
Goal: y ≤ s.mulIndicator g a
-- Set.le_mulIndicator
@[to_additive]
lemma le_mulIndicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
    f ≤ mulIndicator s g 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [One M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1)
Goal: f ≤ s.mulIndicator g
-- Set.le_indicator
@[to_additive]
lemma le_mulIndicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
    f ≤ mulIndicator s g 
Arguments: {α : Type u_2} {M : Type u_4} [LE M] [Zero M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 0)
Goal: f ≤ s.indicator g
-- Set.one_le_mulIndicator_apply
@[to_additive indicator_apply_nonneg]
lemma one_le_mulIndicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mulIndicator s f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f : α → M} {a : α} (h : a ∈ s → 1 ≤ f a)
Goal: 1 ≤ s.mulIndicator f a
-- Set.indicator_apply_nonneg
@[to_additive indicator_apply_nonneg]
lemma one_le_mulIndicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mulIndicator s f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f : α → M} {a : α} (h : a ∈ s → 0 ≤ f a)
Goal: 0 ≤ s.indicator f a
-- Set.one_le_mulIndicator
@[to_additive indicator_nonneg]
lemma one_le_mulIndicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mulIndicator s f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f : α → M} (h : ∀ a ∈ s, 1 ≤ f a) (a : α)
Goal: 1 ≤ s.mulIndicator f a
-- Set.indicator_nonneg
@[to_additive indicator_nonneg]
lemma one_le_mulIndicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mulIndicator s f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f : α → M} (h : ∀ a ∈ s, 0 ≤ f a) (a : α)
Goal: 0 ≤ s.indicator f a
-- Set.mulIndicator_apply_le_one
@[to_additive]
lemma mulIndicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mulIndicator s f a ≤ 1 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f : α → M} {a : α} (h : a ∈ s → f a ≤ 1)
Goal: s.mulIndicator f a ≤ 1
-- Set.indicator_apply_nonpos
@[to_additive]
lemma mulIndicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mulIndicator s f a ≤ 1 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f : α → M} {a : α} (h : a ∈ s → f a ≤ 0)
Goal: s.indicator f a ≤ 0
-- Set.indicator_nonpos
@[to_additive]
lemma mulIndicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) : mulIndicator s f a ≤ 1 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f : α → M} (h : ∀ a ∈ s, f a ≤ 0) (a : α)
Goal: s.indicator f a ≤ 0
-- Set.mulIndicator_le_one
@[to_additive]
lemma mulIndicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) : mulIndicator s f a ≤ 1 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f : α → M} (h : ∀ a ∈ s, f a ≤ 1) (a : α)
Goal: s.mulIndicator f a ≤ 1
-- Set.indicator_le_indicator
@[to_additive]
lemma mulIndicator_le_mulIndicator (h : f a ≤ g a) : mulIndicator s f a ≤ mulIndicator s g a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f g : α → M} {a : α} (h : f a ≤ g a)
Goal: s.indicator f a ≤ s.indicator g a
-- Set.mulIndicator_le_mulIndicator
@[to_additive]
lemma mulIndicator_le_mulIndicator (h : f a ≤ g a) : mulIndicator s f a ≤ mulIndicator s g a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f g : α → M} {a : α} (h : f a ≤ g a)
Goal: s.mulIndicator f a ≤ s.mulIndicator g a
-- Set.indicator_le_indicator_of_subset
@[to_additive]
lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a) (a : α) :
    mulIndicator s f a ≤ mulIndicator t f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s t : Set α} {f : α → M} (h : s ⊆ t) (hf : ∀ (a : α), 0 ≤ f a) (a : α)
Goal: s.indicator f a ≤ t.indicator f a
-- Set.mulIndicator_le_mulIndicator_of_subset
@[to_additive]
lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a) (a : α) :
    mulIndicator s f a ≤ mulIndicator t f a 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s t : Set α} {f : α → M} (h : s ⊆ t) (hf : ∀ (a : α), 1 ≤ f a) (a : α)
Goal: s.mulIndicator f a ≤ t.mulIndicator f a
-- Set.mulIndicator_le_self'
@[to_additive]
lemma mulIndicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mulIndicator s f ≤ f 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [One M] {s : Set α} {f : α → M} (hf : ∀ x ∉ s, 1 ≤ f x)
Goal: s.mulIndicator f ≤ f
-- Set.indicator_le_self'
@[to_additive]
lemma mulIndicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mulIndicator s f ≤ f 
Arguments: {α : Type u_2} {M : Type u_4} [Preorder M] [Zero M] {s : Set α} {f : α → M} (hf : ∀ x ∉ s, 0 ≤ f x)
Goal: s.indicator f ≤ f
-- Set.indicator_le_indicator_nonneg
lemma indicator_le_indicator_nonneg (s : Set α) (f : α → M) :
    s.indicator f ≤ {a | 0 ≤ f a}.indicator f 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [LinearOrder M] (s : Set α) (f : α → M)
Goal: s.indicator f ≤ {a | 0 ≤ f a}.indicator f
-- Set.indicator_nonpos_le_indicator
lemma indicator_nonpos_le_indicator (s : Set α) (f : α → M) :
    {a | f a ≤ 0}.indicator f ≤ s.indicator f 
Arguments: {α : Type u_2} {M : Type u_4} [Zero M] [LinearOrder M] (s : Set α) (f : α → M)
Goal: {a | f a ≤ 0}.indicator f ≤ s.indicator f
-- Set.indicator_iUnion_apply
@[to_additive]
lemma mulIndicator_iUnion_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋃ i, s i) f x = ⨆ i, mulIndicator (s i) f x 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [CompleteLattice M] [Zero M] (h1 : ⊥ = 0) (s : ι → Set α) (f : α → M) (x : α)
Goal: (⋃ i, s i).indicator f x = ⨆ i, (s i).indicator f x
-- Set.mulIndicator_iUnion_apply
@[to_additive]
lemma mulIndicator_iUnion_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋃ i, s i) f x = ⨆ i, mulIndicator (s i) f x 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [CompleteLattice M] [One M] (h1 : ⊥ = 1) (s : ι → Set α) (f : α → M) (x : α)
Goal: (⋃ i, s i).mulIndicator f x = ⨆ i, (s i).mulIndicator f x
-- Set.mulIndicator_iInter_apply
@[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [CompleteLattice M] [One M] [Nonempty ι] (h1 : ⊥ = 1) (s : ι → Set α) (f : α → M) (x : α)
Goal: (⋂ i, s i).mulIndicator f x = ⨅ i, (s i).mulIndicator f x
-- Set.indicator_iInter_apply
@[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x 
Arguments: {ι : Sort u_1} {α : Type u_2} {M : Type u_4} [CompleteLattice M] [Zero M] [Nonempty ι] (h1 : ⊥ = 0) (s : ι → Set α) (f : α → M) (x : α)
Goal: (⋂ i, s i).indicator f x = ⨅ i, (s i).indicator f x
-- Set.indicator_le_self
@[to_additive]
lemma mulIndicator_le_self (s : Set α) (f : α → M) : mulIndicator s f ≤ f 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedAddCommMonoid M] (s : Set α) (f : α → M)
Goal: s.indicator f ≤ f
-- Set.mulIndicator_le_self
@[to_additive]
lemma mulIndicator_le_self (s : Set α) (f : α → M) : mulIndicator s f ≤ f 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedCommMonoid M] (s : Set α) (f : α → M)
Goal: s.mulIndicator f ≤ f
-- Set.indicator_apply_le
@[to_additive]
lemma mulIndicator_apply_le {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a) :
    mulIndicator s f a ≤ g a 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedAddCommMonoid M] {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a)
Goal: s.indicator f a ≤ g a
-- Set.mulIndicator_apply_le
@[to_additive]
lemma mulIndicator_apply_le {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a) :
    mulIndicator s f a ≤ g a 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedCommMonoid M] {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a)
Goal: s.mulIndicator f a ≤ g a
-- Set.mulIndicator_le
@[to_additive]
lemma mulIndicator_le {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
    mulIndicator s f ≤ g 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedCommMonoid M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a)
Goal: s.mulIndicator f ≤ g
-- Set.indicator_le
@[to_additive]
lemma mulIndicator_le {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
    mulIndicator s f ≤ g 
Arguments: {α : Type u_2} {M : Type u_4} [CanonicallyOrderedAddCommMonoid M] {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a)
Goal: s.indicator f ≤ g
-- Set.abs_indicator_symmDiff
@[to_additive]
lemma mabs_mulIndicator_symmDiff (s t : Set α) (f : α → M) (x : α) :
    |mulIndicator (s ∆ t) f x|ₘ = |mulIndicator s f x / mulIndicator t f x|ₘ 
Arguments: {α : Type u_2} {M : Type u_4} [LinearOrderedAddCommGroup M] (s t : Set α) (f : α → M) (x : α)
Goal: |(symmDiff s t).indicator f x| = |s.indicator f x - t.indicator f x|
-- Set.mabs_mulIndicator_symmDiff
@[to_additive]
lemma mabs_mulIndicator_symmDiff (s t : Set α) (f : α → M) (x : α) :
    |mulIndicator (s ∆ t) f x|ₘ = |mulIndicator s f x / mulIndicator t f x|ₘ 
Arguments: {α : Type u_2} {M : Type u_4} [LinearOrderedCommGroup M] (s t : Set α) (f : α → M) (x : α)
Goal: mabs ((symmDiff s t).mulIndicator f x) = mabs (s.mulIndicator f x / t.mulIndicator f x)
-- Ring.PositiveCone.one_nonneg
structure PositiveCone (α : Type*) [Ring α] extends AddCommGroup.PositiveCone α where
    one_nonneg : nonneg 1
    mul_pos : ∀ a b, pos a → pos b → pos (a * b)
Arguments: {α : Type u_2} [Ring α] (self : Ring.PositiveCone α)
Goal: self.nonneg 1
-- Ring.PositiveCone.mul_pos
structure PositiveCone (α : Type*) [Ring α] extends AddCommGroup.PositiveCone α where
    one_nonneg : nonneg 1
    mul_pos : ∀ a b, pos a → pos b → pos (a * b)
Arguments: {α : Type u_2} [Ring α] (self : Ring.PositiveCone α) (a b : α)
Goal: self.pos a → self.pos b → self.pos (a * b)
-- Ring.PositiveCone.one_pos
theorem PositiveCone.one_pos (C : PositiveCone α) : C.pos 1 
Arguments: {α : Type u_1} [Ring α] [Nontrivial α] (C : Ring.PositiveCone α)
Goal: C.pos 1
-- oneLePart_mono
@[to_additive] lemma oneLePart_mono : Monotone (oneLePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [Group α]
Goal: Monotone oneLePart
-- posPart_mono
@[to_additive] lemma oneLePart_mono : Monotone (oneLePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α]
Goal: Monotone posPart
-- oneLePart_one
@[to_additive (attr := simp)] lemma oneLePart_one : (1 : α)⁺ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [Group α]
Goal: 1⁺ᵐ = 1
-- posPart_zero
@[to_additive (attr := simp)] lemma oneLePart_one : (1 : α)⁺ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α]
Goal: 0⁺ = 0
-- negPart_zero
@[to_additive (attr := simp)] lemma leOnePart_one : (1 : α)⁻ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α]
Goal: 0⁻ = 0
-- leOnePart_one
@[to_additive (attr := simp)] lemma leOnePart_one : (1 : α)⁻ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [Group α]
Goal: 1⁻ᵐ = 1
-- one_le_oneLePart
@[to_additive posPart_nonneg] lemma one_le_oneLePart (a : α) : 1 ≤ a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: 1 ≤ a⁺ᵐ
-- posPart_nonneg
@[to_additive posPart_nonneg] lemma one_le_oneLePart (a : α) : 1 ≤ a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: 0 ≤ a⁺
-- one_le_leOnePart
@[to_additive negPart_nonneg] lemma one_le_leOnePart (a : α) : 1 ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: 1 ≤ a⁻ᵐ
-- negPart_nonneg
@[to_additive negPart_nonneg] lemma one_le_leOnePart (a : α) : 1 ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: 0 ≤ a⁻
-- le_posPart
@[to_additive le_posPart] lemma le_oneLePart (a : α) : a ≤ a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: a ≤ a⁺
-- le_oneLePart
@[to_additive le_posPart] lemma le_oneLePart (a : α) : a ≤ a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a ≤ a⁺ᵐ
-- neg_le_negPart
@[to_additive] lemma inv_le_leOnePart (a : α) : a⁻¹ ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: -a ≤ a⁻
-- inv_le_leOnePart
@[to_additive] lemma inv_le_leOnePart (a : α) : a⁻¹ ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a⁻¹ ≤ a⁻ᵐ
-- oneLePart_eq_self
@[to_additive (attr := simp)] lemma oneLePart_eq_self : a⁺ᵐ = a ↔ 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁺ᵐ = a ↔ 1 ≤ a
-- posPart_eq_self
@[to_additive (attr := simp)] lemma oneLePart_eq_self : a⁺ᵐ = a ↔ 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁺ = a ↔ 0 ≤ a
-- oneLePart_eq_one
@[to_additive] lemma oneLePart_eq_one : a⁺ᵐ = 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁺ᵐ = 1 ↔ a ≤ 1
-- posPart_eq_zero
@[to_additive] lemma oneLePart_eq_one : a⁺ᵐ = 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁺ = 0 ↔ a ≤ 0
-- leOnePart_eq_inv'
@[to_additive "See also `negPart_eq_neg`."]
lemma leOnePart_eq_inv' : a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹
-- negPart_eq_neg'
@[to_additive "See also `negPart_eq_neg`."]
lemma leOnePart_eq_inv' : a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁻ = -a ↔ 0 ≤ -a
-- leOnePart_eq_one'
@[to_additive "See also `negPart_eq_zero`."]
lemma leOnePart_eq_one' : a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1
-- negPart_eq_zero'
@[to_additive "See also `negPart_eq_zero`."]
lemma leOnePart_eq_one' : a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁻ = 0 ↔ -a ≤ 0
-- oneLePart_le_one
@[to_additive] lemma oneLePart_le_one : a⁺ᵐ ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁺ᵐ ≤ 1 ↔ a ≤ 1
-- posPart_nonpos
@[to_additive] lemma oneLePart_le_one : a⁺ᵐ ≤ 1 ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁺ ≤ 0 ↔ a ≤ 0
-- leOnePart_le_one'
@[to_additive "See also `negPart_nonpos`."]
lemma leOnePart_le_one' : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1
-- negPart_nonpos'
@[to_additive "See also `negPart_nonpos`."]
lemma leOnePart_le_one' : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁻ ≤ 0 ↔ -a ≤ 0
-- leOnePart_le_one
@[to_additive] lemma leOnePart_le_one : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α}
Goal: a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1
-- negPart_nonpos
@[to_additive] lemma leOnePart_le_one : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α}
Goal: a⁻ ≤ 0 ↔ -a ≤ 0
-- posPart_pos
@[to_additive (attr := simp) posPart_pos] lemma one_lt_oneLePart (ha : 1 < a) : 1 < a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} (ha : 0 < a)
Goal: 0 < a⁺
-- one_lt_oneLePart
@[to_additive (attr := simp) posPart_pos] lemma one_lt_oneLePart (ha : 1 < a) : 1 < a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} (ha : 1 < a)
Goal: 1 < a⁺ᵐ
-- oneLePart_inv
@[to_additive (attr := simp)] lemma oneLePart_inv (a : α) : a⁻¹⁺ᵐ = a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a⁻¹⁺ᵐ = a⁻ᵐ
-- posPart_neg
@[to_additive (attr := simp)] lemma oneLePart_inv (a : α) : a⁻¹⁺ᵐ = a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: (-a)⁺ = a⁻
-- negPart_neg
@[to_additive (attr := simp)] lemma leOnePart_inv (a : α) : a⁻¹⁻ᵐ = a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: (-a)⁻ = a⁺
-- leOnePart_inv
@[to_additive (attr := simp)] lemma leOnePart_inv (a : α) : a⁻¹⁻ᵐ = a⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a⁻¹⁻ᵐ = a⁺ᵐ
-- leOnePart_eq_inv
@[to_additive (attr := simp)] lemma leOnePart_eq_inv : a⁻ᵐ = a⁻¹ ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ᵐ = a⁻¹ ↔ a ≤ 1
-- negPart_eq_neg
@[to_additive (attr := simp)] lemma leOnePart_eq_inv : a⁻ᵐ = a⁻¹ ↔ a ≤ 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ = -a ↔ a ≤ 0
-- negPart_eq_zero
@[to_additive (attr := simp)]
lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ = 0 ↔ 0 ≤ a