A weighted sum of semiring values. Mathematically, this represents an affine operation on the underlying expressions.

Runtime representation of the semiring for this sum.

Retrieve the constant addend of the weighted sum.

Create a sum from a constant coefficient value.

Create a weighted sum corresponding to the given variable.

This returns a variable times a constant.

Attempt to parse a weighted sum as a constant.

Attempt to parse a weighted sum as a single expression. asVar w = Just r when denotation(w) = r

Attempt to parse weighted sum as a single expression with a coefficient. asWeightedVar w = Just (c,r) when denotation(w) = c*r.

Attempt to parse a weighted sum as a single expression with a coefficient and offset. asAffineVar w = Just (c,r,o) when denotation(w) = c*r + o.

Return true if a weighted sum is equal to constant 0.

Traverse the expressions in a weighted sum.

Traverse the coefficients in a weighted sum.

Add two sums, collecting terms as necessary and deleting terms whose coefficients sum to 0.

Add a variable to the sum.

Create a weighted sum that represents the sum of two terms.

Add a constant to the sum.

Multiply a sum by a constant coefficient.

Evaluate a sum given interpretations of addition, scalar multiplication, and a constant rational.

Evaluate a sum given interpretations of addition, scalar multiplication, and a constant. This evaluation is threaded through a monad. The addition function is associated to the left, as in foldlM.

Given two weighted sums x and y, this returns a triple (z,x',y') where x = z + x' and y = z + y' and z contains the "common" parts of x and y. We only extract common terms when both terms occur with the same coefficient in each sum.

This is primarily used to simplify if-then-else expressions to preserve shared subterms.

Produce a weighted sum from a list of terms and an offset.

Apply update functions to the terms and coefficients of a weighted sum.

Reduce a weighted sum of integers modulo a concrete integer. This reduces each of the coefficients modulo the given integer, removing any that are congruent to 0; the offset value is also reduced.

A product of semiring values.

Traverse the expressions in a product.

Returns true if the product is trivial (contains no terms).

If the product consists of exactly on term, return it.

Runtime representation of the semiring for this product

Produce a product representing the single given term.

Multiply two products, collecting terms and adding occurrences.

Evaluate a product, given a function representing multiplication and a function to evaluate terms.

Evaluate a product, given a function representing multiplication and a function to evaluate terms, where both functions are threaded through a monad.

Returns true if the product contains at least on occurrence of the given term.