linarith computes the linear form of its input expressions,
assuming (without justification) that the type of these expressions
is a commutative semiring.
It identifies atoms up to ring-equivalence: that is, (y*3)*x will be identified 3*(x*y),
where the monomial x*y is the linear atom.
Variables are represented by natural numbers.
Monomials are represented by monom := rb_map ℕ ℕ.
The monomial 1 is represented by the empty map.
Linear combinations of monomials are represented by sum := rb_map monom ℤ.
All input expressions are converted to sums, preserving the map from expressions to variables.
We then discard the monomial information, mapping each distinct monomial to a natural number.
The resulting rb_map ℕ ℤ represents the ring-normalized linear form of the expression.
This is ultimately converted into a linexp in the obvious way.
linear_forms_and_vars is the main entry point into this file. Everything else is contained.
linear_form_of_atom red map e is the atomic case for linear_form_of_expr.
If e appears with index k in map, it returns the singleton sum var k.
Otherwise it updates map, adding e with index n, and returns the singleton sum var n.
linear_form_of_expr red map e computes the linear form of e.
map is a lookup map from atomic expressions to variable numbers.
If a new atomic expression is encountered, it is added to the map with a new number.
It matches atomic expressions up to reducibility given by red.
Because it matches up to definitional equality, this function must be in the tactic monad,
and forces some functions that call it into tactic as well.
sum_to_lf s map eliminates the monomial level of the sums.
map is a lookup map from monomials to variable numbers.
The output rb_map ℕ ℤ has the same structure as sum,
but each monomial key is replaced with its index according to map.
If any new monomials are encountered, they are assigned variable numbers and map is updated.
linear_forms_and_vars red pfs is the main interface for computing the linear forms of a list
of expressions. Given a list pfs of proofs of comparisons, it produces a list c of comps of
the same length, such that c[i] represents the linear form of the type of pfs[i].
It also returns the largest variable index that appears in comparisons in c.