Tooling to make copies of lattice structures

Sometimes it is useful to make a copy of a lattice structure where one replaces the data parts with provably equal definitions that have better definitional properties.

def BoundedOrder.copy {α : Type u} {h : LE α} {h' : LE α} (c : BoundedOrder α) (top : α) (eq_top : top = ⊤) (bot : α) (eq_bot : bot = ⊥) (le_eq : ∀ (x y : α), x ≤ y ↔ x ≤ y) : BoundedOrder α

A function to create a provable equal copy of a bounded order with possibly different definitional equalities.

Equations

• c.copy top eq_top bot eq_bot le_eq = BoundedOrder.mk

def Lattice.copy {α : Type u} (c : Lattice α) (le : α → α → Prop) (eq_le : le = LE.le) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) : Lattice α

A function to create a provable equal copy of a lattice with possibly different definitional equalities.

Equations

• c.copy le eq_le sup eq_sup inf eq_inf = Lattice.mk ⋯ ⋯ ⋯

def DistribLattice.copy {α : Type u} (c : DistribLattice α) (le : α → α → Prop) (eq_le : le = LE.le) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) : DistribLattice α

A function to create a provable equal copy of a distributive lattice with possibly different definitional equalities.

Equations

• c.copy le eq_le sup eq_sup inf eq_inf = DistribLattice.mk ⋯

def CompleteLattice.copy {α : Type u} (c : CompleteLattice α) (le : α → α → Prop) (eq_le : le = LE.le) (top : α) (eq_top : top = ⊤) (bot : α) (eq_bot : bot = ⊥) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) (sSup : Set α → α) (eq_sSup : sSup = SupSet.sSup) (sInf : Set α → α) (eq_sInf : sInf = InfSet.sInf) : CompleteLattice α

A function to create a provable equal copy of a complete lattice with possibly different definitional equalities.

Equations

• c.copy le eq_le top eq_top bot eq_bot sup eq_sup inf eq_inf sSup eq_sSup sInf eq_sInf = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

def Frame.copy {α : Type u} (c : Order.Frame α) (le : α → α → Prop) (eq_le : le = LE.le) (top : α) (eq_top : top = ⊤) (bot : α) (eq_bot : bot = ⊥) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) (sSup : Set α → α) (eq_sSup : sSup = SupSet.sSup) (sInf : Set α → α) (eq_sInf : sInf = InfSet.sInf) : Order.Frame α

A function to create a provable equal copy of a frame with possibly different definitional equalities.

Equations

• Frame.copy c le eq_le top eq_top bot eq_bot sup eq_sup inf eq_inf sSup eq_sSup sInf eq_sInf = Order.Frame.mk ⋯

def Coframe.copy {α : Type u} (c : Order.Coframe α) (le : α → α → Prop) (eq_le : le = LE.le) (top : α) (eq_top : top = ⊤) (bot : α) (eq_bot : bot = ⊥) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) (sSup : Set α → α) (eq_sSup : sSup = SupSet.sSup) (sInf : Set α → α) (eq_sInf : sInf = InfSet.sInf) : Order.Coframe α

A function to create a provable equal copy of a coframe with possibly different definitional equalities.

Equations

• Coframe.copy c le eq_le top eq_top bot eq_bot sup eq_sup inf eq_inf sSup eq_sSup sInf eq_sInf = Order.Coframe.mk ⋯

def CompleteDistribLattice.copy {α : Type u} (c : CompleteDistribLattice α) (le : α → α → Prop) (eq_le : le = LE.le) (top : α) (eq_top : top = ⊤) (bot : α) (eq_bot : bot = ⊥) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) (sSup : Set α → α) (eq_sSup : sSup = SupSet.sSup) (sInf : Set α → α) (eq_sInf : sInf = InfSet.sInf) : CompleteDistribLattice α

A function to create a provable equal copy of a complete distributive lattice with possibly different definitional equalities.

Equations

• One or more equations did not get rendered due to their size.

def ConditionallyCompleteLattice.copy {α : Type u} (c : ConditionallyCompleteLattice α) (le : α → α → Prop) (eq_le : le = LE.le) (sup : α → α → α) (eq_sup : sup = Sup.sup) (inf : α → α → α) (eq_inf : inf = Inf.inf) (sSup : Set α → α) (eq_sSup : sSup = SupSet.sSup) (sInf : Set α → α) (eq_sInf : sInf = InfSet.sInf) : ConditionallyCompleteLattice α

A function to create a provable equal copy of a conditionally complete lattice with possibly different definitional equalities.

Equations

• c.copy le eq_le sup eq_sup inf eq_inf sSup eq_sSup sInf eq_sInf = ConditionallyCompleteLattice.mk ⋯ ⋯ ⋯ ⋯