# Note about Mathlib/Init/#

The files in Mathlib/Init are leftovers from the port from Mathlib3. (They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)

We intend to move all the content of these files out into the main Mathlib directory structure. Contributions assisting with this are appreciated.

# Unbundled algebra classes #

These classes were part of an incomplete refactor described here on the github Wiki. However a subset of them are widely used in mathlib3, and it has been tricky to clean this up as this file was in core Lean 3.

@[deprecated]
class IsLeftCancel (α : Sort u) (op : ααα) :
• left_cancel : ∀ (a b c : α), op a b = op a cb = c
Instances
theorem IsLeftCancel.left_cancel {α : Sort u} {op : ααα} [self : IsLeftCancel α op] (a : α) (b : α) (c : α) :
op a b = op a cb = c
@[deprecated]
class IsRightCancel (α : Sort u) (op : ααα) :
• right_cancel : ∀ (a b c : α), op a b = op c ba = c
Instances
theorem IsRightCancel.right_cancel {α : Sort u} {op : ααα} [self : ] (a : α) (b : α) (c : α) :
op a b = op c ba = c
@[deprecated]
class IsTotalPreorder (α : Sort u) (r : ααProp) extends , :

IsTotalPreorder X r means that the binary relation r on X is total and a preorder.

Instances
@[instance 100]
instance isTotalPreorder_isPreorder (α : Sort u) (r : ααProp) [s : ] :

Every total pre-order is a pre-order.

Equations
• =
@[deprecated]
class IsIncompTrans (α : Sort u) (lt : ααProp) :

IsIncompTrans X lt means that for lt a binary relation on X, the incomparable relation fun a b => ¬ lt a b ∧ ¬ lt b a is transitive.

• incomp_trans : ∀ (a b c : α), ¬lt a b ¬lt b a¬lt b c ¬lt c b¬lt a c ¬lt c a
Instances
theorem IsIncompTrans.incomp_trans {α : Sort u} {lt : ααProp} [self : ] (a : α) (b : α) (c : α) :
¬lt a b ¬lt b a¬lt b c ¬lt c b¬lt a c ¬lt c a
@[deprecated, instance 100]
instance instIsIncompTransOfIsStrictWeakOrder (α : Sort u) (lt : ααProp) [] :
Equations
• =
@[deprecated]
theorem incomp_trans {α : Sort u} {r : ααProp} [] {a : α} {b : α} {c : α} :
¬r a b ¬r b a¬r b c ¬r c b¬r a c ¬r c a
@[elab_without_expected_type, deprecated]
theorem incomp_trans_of {α : Sort u} (r : ααProp) [] {a : α} {b : α} {c : α} :
¬r a b ¬r b a¬r b c ¬r c b¬r a c ¬r c a
@[deprecated]
def StrictWeakOrder.Equiv {α : Sort u} {r : ααProp} (a : α) (b : α) :
Equations
Instances For
@[deprecated]
theorem StrictWeakOrder.esymm {α : Sort u} {r : ααProp} {a : α} {b : α} :
@[deprecated]
theorem StrictWeakOrder.not_lt_of_equiv {α : Sort u} {r : ααProp} {a : α} {b : α} :
¬r a b
@[deprecated]
theorem StrictWeakOrder.not_lt_of_equiv' {α : Sort u} {r : ααProp} {a : α} {b : α} :
¬r b a
@[deprecated]
theorem StrictWeakOrder.erefl {α : Sort u} {r : ααProp} [] (a : α) :
@[deprecated]
theorem StrictWeakOrder.etrans {α : Sort u} {r : ααProp} [] {a : α} {b : α} {c : α} :
@[deprecated]
instance StrictWeakOrder.isEquiv {α : Sort u} {r : ααProp} [] :
IsEquiv α StrictWeakOrder.Equiv
Equations
• =

The equivalence relation induced by lt

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[deprecated]
theorem isStrictWeakOrder_of_isTotalPreorder {α : Sort u} {le : ααProp} {lt : ααProp} [] [] (h : ∀ (a b : α), lt a b ¬le b a) :
@[deprecated]
instance instIsTotalPreorderLe {α : Type u_1} [] :
IsTotalPreorder α fun (x1 x2 : α) => x1 x2
Equations
• =
@[deprecated]
instance isStrictWeakOrder_of_linearOrder {α : Type u_1} [] :
IsStrictWeakOrder α fun (x1 x2 : α) => x1 < x2
Equations
• =
@[deprecated]
theorem lt_of_lt_of_incomp {α : Sort u} {lt : ααProp} [] [] {a : α} {b : α} {c : α} :
lt a b¬lt b c ¬lt c blt a c
@[deprecated]
theorem lt_of_incomp_of_lt {α : Sort u} {lt : ααProp} [] [] {a : α} {b : α} {c : α} :
¬lt a b ¬lt b alt b clt a c
@[deprecated]
theorem eq_of_incomp {α : Sort u} {lt : ααProp} [] {a : α} {b : α} :
¬lt a b ¬lt b aa = b
@[deprecated]
theorem eq_of_eqv_lt {α : Sort u} {lt : ααProp} [] {a : α} {b : α} :
a = b
@[deprecated]
theorem incomp_iff_eq {α : Sort u} {lt : ααProp} [] [IsIrrefl α lt] (a : α) (b : α) :
¬lt a b ¬lt b a a = b
@[deprecated]
theorem eqv_lt_iff_eq {α : Sort u} {lt : ααProp} [] [IsIrrefl α lt] (a : α) (b : α) :
a = b