File: Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean

full_name: ExistsMulOfLE.exists_mul_of_le
code: /-- An `OrderedCommMonoid` with one-sided 'division' in the sense that
if `a ≤ b`, there is some `c` for which `a * c = b`. This is a weaker version
of the condition on canonical orderings defined by `CanonicallyOrderedCommMonoid`. -/
class ExistsMulOfLE (α : Type u) [Mul α] [LE α] : Prop where
  /-- For `a ≤ b`, `a` left divides `b` -/
  exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a * c
goal: a ≤ b → ∃ c, b = a * c

full_name: ExistsMulOfLE
code: /-- An `OrderedCommMonoid` with one-sided 'division' in the sense that
if `a ≤ b`, there is some `c` for which `a * c = b`. This is a weaker version
of the condition on canonical orderings defined by `CanonicallyOrderedCommMonoid`. -/
class ExistsMulOfLE (α : Type u) [Mul α] [LE α] : Prop where
  /-- For `a ≤ b`, `a` left divides `b` -/
  exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a * c
goal: Prop

full_name: ExistsAddOfLE.exists_add_of_le
code: /-- An `OrderedAddCommMonoid` with one-sided 'subtraction' in the sense that
if `a ≤ b`, then there is some `c` for which `a + c = b`. This is a weaker version
of the condition on canonical orderings defined by `CanonicallyOrderedAddCommMonoid`. -/
class ExistsAddOfLE (α : Type u) [Add α] [LE α] : Prop where
  /-- For `a ≤ b`, there is a `c` so `b = a + c`. -/
  exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a + c
goal: a ≤ b → ∃ c, b = a + c

full_name: ExistsAddOfLE
code: /-- An `OrderedAddCommMonoid` with one-sided 'subtraction' in the sense that
if `a ≤ b`, then there is some `c` for which `a + c = b`. This is a weaker version
of the condition on canonical orderings defined by `CanonicallyOrderedAddCommMonoid`. -/
class ExistsAddOfLE (α : Type u) [Add α] [LE α] : Prop where
  /-- For `a ≤ b`, there is a `c` so `b = a + c`. -/
  exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ c : α, b = a + c
goal: Prop

full_name: Group.existsMulOfLE
code: @[to_additive]
instance (priority := 100) Group.existsMulOfLE (α : Type u) [Group α] [LE α] : ExistsMulOfLE α 
goal: ExistsMulOfLE α

full_name: AddGroup.existsAddOfLE
code: @[to_additive]
instance (priority := 100) Group.existsMulOfLE (α : Type u) [Group α] [LE α] : ExistsMulOfLE α 
goal: ExistsAddOfLE α

full_name: exists_pos_add_of_lt'
code: @[to_additive]
theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b 
goal: ∃ c, 0 < c ∧ a + c = b

full_name: exists_one_lt_mul_of_lt'
code: @[to_additive]
theorem exists_one_lt_mul_of_lt' (h : a < b) : ∃ c, 1 < c ∧ a * c = b 
goal: ∃ c, 1 < c ∧ a * c = b

full_name: le_of_forall_pos_le_add
code: @[to_additive]
theorem le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b 
goal: a ≤ b

full_name: le_of_forall_one_lt_le_mul
code: @[to_additive]
theorem le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b 
goal: a ≤ b

full_name: le_of_forall_pos_lt_add'
code: @[to_additive]
theorem le_of_forall_one_lt_lt_mul' (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
goal: a ≤ b

full_name: le_of_forall_one_lt_lt_mul'
code: @[to_additive]
theorem le_of_forall_one_lt_lt_mul' (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b 
goal: a ≤ b

full_name: le_iff_forall_pos_lt_add'
code: @[to_additive]
theorem le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
goal: a ≤ b ↔ ∀ (ε : α), 0 < ε → a < b + ε

full_name: le_iff_forall_one_lt_lt_mul'
code: @[to_additive]
theorem le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε 
goal: a ≤ b ↔ ∀ (ε : α), 1 < ε → a < b * ε