File: Mathlib/Algebra/Order/Pi.lean

full_name: Pi.orderedCommMonoid
code: /-- The product of a family of ordered commutative monoids is an ordered commutative monoid. -/
@[to_additive
      "The product of a family of ordered additive commutative monoids is
an ordered additive commutative monoid."]
instance orderedCommMonoid {ι : Type*} {Z : ι → Type*} [∀ i, OrderedCommMonoid (Z i)] :
    OrderedCommMonoid (∀ i, Z i) 
goal: OrderedCommMonoid ((i : ι) → Z i)

full_name: Pi.orderedAddCommMonoid
code: /-- The product of a family of ordered commutative monoids is an ordered commutative monoid. -/
@[to_additive
      "The product of a family of ordered additive commutative monoids is
an ordered additive commutative monoid."]
instance orderedCommMonoid {ι : Type*} {Z : ι → Type*} [∀ i, OrderedCommMonoid (Z i)] :
    OrderedCommMonoid (∀ i, Z i) 
goal: OrderedAddCommMonoid ((i : ι) → Z i)

full_name: Pi.existsAddOfLe
code: @[to_additive]
instance existsMulOfLe {ι : Type*} {α : ι → Type*} [∀ i, LE (α i)] [∀ i, Mul (α i)]
    [∀ i, ExistsMulOfLE (α i)] : ExistsMulOfLE (∀ i, α i) 
goal: ExistsAddOfLE ((i : ι) → α i)

full_name: Pi.existsMulOfLe
code: @[to_additive]
instance existsMulOfLe {ι : Type*} {α : ι → Type*} [∀ i, LE (α i)] [∀ i, Mul (α i)]
    [∀ i, ExistsMulOfLE (α i)] : ExistsMulOfLE (∀ i, α i) 
goal: ExistsMulOfLE ((i : ι) → α i)

full_name: Pi.instCanonicallyOrderedAddCommMonoidForall
code: /-- The product of a family of canonically ordered monoids is a canonically ordered monoid. -/
@[to_additive
      "The product of a family of canonically ordered additive monoids is
a canonically ordered additive monoid."]
instance {ι : Type*} {Z : ι → Type*} [∀ i, CanonicallyOrderedCommMonoid (Z i)] :
    CanonicallyOrderedCommMonoid (∀ i, Z i) 
goal: CanonicallyOrderedAddCommMonoid ((i : ι) → Z i)

full_name: Pi.instCanonicallyOrderedCommMonoidForall
code: /-- The product of a family of canonically ordered monoids is a canonically ordered monoid. -/
@[to_additive
      "The product of a family of canonically ordered additive monoids is
a canonically ordered additive monoid."]
instance {ι : Type*} {Z : ι → Type*} [∀ i, CanonicallyOrderedCommMonoid (Z i)] :
    CanonicallyOrderedCommMonoid (∀ i, Z i) 
goal: CanonicallyOrderedCommMonoid ((i : ι) → Z i)

full_name: Pi.orderedCancelCommMonoid
code: @[to_additive]
instance orderedCancelCommMonoid [∀ i, OrderedCancelCommMonoid <| f i] :
    OrderedCancelCommMonoid (∀ i : I, f i) 
goal: OrderedCancelCommMonoid ((i : I) → f i)

full_name: Pi.orderedAddCancelCommMonoid
code: @[to_additive]
instance orderedCancelCommMonoid [∀ i, OrderedCancelCommMonoid <| f i] :
    OrderedCancelCommMonoid (∀ i : I, f i) 
goal: OrderedCancelAddCommMonoid ((i : I) → f i)

full_name: Pi.orderedAddCommGroup
code: @[to_additive]
instance orderedCommGroup [∀ i, OrderedCommGroup <| f i] : OrderedCommGroup (∀ i : I, f i) 
goal: OrderedAddCommGroup ((i : I) → f i)

full_name: Pi.orderedCommGroup
code: @[to_additive]
instance orderedCommGroup [∀ i, OrderedCommGroup <| f i] : OrderedCommGroup (∀ i : I, f i) 
goal: OrderedCommGroup ((i : I) → f i)

full_name: Pi.orderedSemiring
code: instance orderedSemiring [∀ i, OrderedSemiring (f i)] : OrderedSemiring (∀ i, f i) 
goal: OrderedSemiring ((i : I) → f i)

full_name: Pi.orderedCommSemiring
code: instance orderedCommSemiring [∀ i, OrderedCommSemiring (f i)] : OrderedCommSemiring (∀ i, f i) 
goal: OrderedCommSemiring ((i : I) → f i)

full_name: Pi.orderedRing
code: instance orderedRing [∀ i, OrderedRing (f i)] : OrderedRing (∀ i, f i) 
goal: OrderedRing ((i : I) → f i)

full_name: Pi.orderedCommRing
code: instance orderedCommRing [∀ i, OrderedCommRing (f i)] : OrderedCommRing (∀ i, f i) 
goal: OrderedCommRing ((i : I) → f i)

full_name: Function.one_le_const_of_one_le
code: @[to_additive const_nonneg_of_nonneg]
theorem one_le_const_of_one_le (ha : 1 ≤ a) : 1 ≤ const β a 
goal: 1 ≤ Function.const β a

full_name: Function.const_nonneg_of_nonneg
code: @[to_additive const_nonneg_of_nonneg]
theorem one_le_const_of_one_le (ha : 1 ≤ a) : 1 ≤ const β a 
goal: 0 ≤ Function.const β a

full_name: Function.const_le_one_of_le_one
code: @[to_additive]
theorem const_le_one_of_le_one (ha : a ≤ 1) : const β a ≤ 1 
goal: Function.const β a ≤ 1

full_name: Function.const_nonpos_of_nonpos
code: @[to_additive]
theorem const_le_one_of_le_one (ha : a ≤ 1) : const β a ≤ 1 
goal: Function.const β a ≤ 0

full_name: Function.one_le_const
code: @[to_additive (attr := simp) const_nonneg]
theorem one_le_const : 1 ≤ const β a ↔ 1 ≤ a 
goal: 1 ≤ Function.const β a ↔ 1 ≤ a

full_name: Function.const_nonneg
code: @[to_additive (attr := simp) const_nonneg]
theorem one_le_const : 1 ≤ const β a ↔ 1 ≤ a 
goal: 0 ≤ Function.const β a ↔ 0 ≤ a

full_name: Function.const_pos
code: @[to_additive (attr := simp) const_pos]
theorem one_lt_const : 1 < const β a ↔ 1 < a 
goal: 0 < Function.const β a ↔ 0 < a

full_name: Function.one_lt_const
code: @[to_additive (attr := simp) const_pos]
theorem one_lt_const : 1 < const β a ↔ 1 < a 
goal: 1 < Function.const β a ↔ 1 < a

full_name: Function.const_nonpos
code: @[to_additive (attr := simp)]
theorem const_le_one : const β a ≤ 1 ↔ a ≤ 1 
goal: Function.const β a ≤ 0 ↔ a ≤ 0

full_name: Function.const_le_one
code: @[to_additive (attr := simp)]
theorem const_le_one : const β a ≤ 1 ↔ a ≤ 1 
goal: Function.const β a ≤ 1 ↔ a ≤ 1

full_name: Function.const_lt_one
code: @[to_additive (attr := simp) const_neg']
theorem const_lt_one : const β a < 1 ↔ a < 1 
goal: Function.const β a < 1 ↔ a < 1

full_name: Function.const_neg'
code: @[to_additive (attr := simp) const_neg']
theorem const_lt_one : const β a < 1 ↔ a < 1 
goal: Function.const β a < 0 ↔ a < 0

full_name: Function.one_le_extend
code: @[to_additive extend_nonneg] lemma one_le_extend (hg : 1 ≤ g) (he : 1 ≤ e) : 1 ≤ extend f g e 
goal: 1 ≤ Function.extend f g e

full_name: Function.extend_nonneg
code: @[to_additive extend_nonneg] lemma one_le_extend (hg : 1 ≤ g) (he : 1 ≤ e) : 1 ≤ extend f g e 
goal: 0 ≤ Function.extend f g e

full_name: Function.extend_le_one
code: @[to_additive] lemma extend_le_one (hg : g ≤ 1) (he : e ≤ 1) : extend f g e ≤ 1 
goal: Function.extend f g e ≤ 1

full_name: Function.extend_nonpos
code: @[to_additive] lemma extend_le_one (hg : g ≤ 1) (he : e ≤ 1) : extend f g e ≤ 1 
goal: Function.extend f g e ≤ 0