The commands promote and lift are usually for when we have a natural map of rings A --> B. In this situation, promote applies the map to an element of A, and lift computes a preimage (typically some sort of normal form.)

A=QQ[x]
B=A[y]
elem1 = y^3 -- in B
C=B/(x^2-y^3)
elem2 = x^3 -- in A
methods symbol +
elem3 = elem1+elem2 -- in B
elem4 = promote(elem3,C) -- x^3+x^2 in C
elem5 = lift(elem4,B) -- in B

In the line elem3 = elem1+elem2 , new methods (+, B, A) and (+, B, B) are created behind the scenes and stashed in the youngest object B as described here. Type code (symbol +, RingElement, RingElement) to see another use of promote in this case.

Examples of natural ring maps (from aring-glue.hpp):

• ZZ --> R, for any R
• QQ --> RR --> CC
• ZZ --> ZZ/p
• ZZ/p --> GF(p^n)
• GF(p^m) --> GF(p^n), where m divides n
• A --> A[vars]/I
• A[vars]/J --> A[vars]/I (assumption: I contains J).

There are some oddities due to floating-point:

x=1e-16
y=1+x -- not representable w/ default 53 bits of precision
lift(y, ZZ) -- works
lift(x, ZZ) -- error
lift(1e-50,QQ) -- not what you might expect

Complicated due to lack of multiple dispatch in C++. Roughly speaking, the target ring controls the implementation. The meat is in aring-glue.hpp (RingPromoter namespace) and aring-translate.hpp.