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R1cs-subset

R1cs-polynomialp

Check if a PFCS expression is an R1CS polynomial.

Signature
(r1cs-polynomialp expr) → yes/no
Arguments: expr — Guard (expressionp expr).
Returns: yes/no — Type (booleanp yes/no).

This is a linear combination, i.e. a sum of one or more monomials.

Currently the PFCS abstract syntax only has binary addition. We pick left-associated additions as linear polynomials; the base case is that of a single monomial.

Definitions and Theorems

Function: r1cs-polynomialp

(defun r1cs-polynomialp (expr)
  (declare (xargs :guard (expressionp expr)))
  (let ((__function__ 'r1cs-polynomialp))
    (declare (ignorable __function__))
    (or (r1cs-monomialp expr)
        (and (expression-case expr :add)
             (r1cs-polynomialp (expression-add->arg1 expr))
             (r1cs-monomialp (expression-add->arg2 expr))))))

Theorem: booleanp-of-r1cs-polynomialp

(defthm booleanp-of-r1cs-polynomialp
  (b* ((yes/no (r1cs-polynomialp expr)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: r1cs-polynomialp-of-expression-fix-expr

(defthm r1cs-polynomialp-of-expression-fix-expr
  (equal (r1cs-polynomialp (expression-fix expr))
         (r1cs-polynomialp expr)))

Theorem: r1cs-polynomialp-expression-equiv-congruence-on-expr

(defthm r1cs-polynomialp-expression-equiv-congruence-on-expr
  (implies (expression-equiv expr expr-equiv)
           (equal (r1cs-polynomialp expr)
                  (r1cs-polynomialp expr-equiv)))
  :rule-classes :congruence)