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R1cs-subset

R1cs-monomialp

Check if a PFCS expression is an R1CS monomial.

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

This is an addend of an R1CS polynomial (i.e. linear combination). It is either a constant, or a variable, or a negated constant or variable, or a product of a constant (natural number) by a variable, or a product of a negated constant (negative number) by a variable.

Although it could be supported, for simplicity we disallow a product of a constant (natural number) by a negated variable.

Definitions and Theorems

Function: r1cs-monomialp

(defun r1cs-monomialp (expr)
 (declare (xargs :guard (expressionp expr)))
 (let ((__function__ 'r1cs-monomialp))
  (declare (ignorable __function__))
  (or
   (expression-case expr :const)
   (expression-case expr :var)
   (and (expression-case expr :neg)
        (expression-case (expression-neg->arg expr)
                         :const))
   (and (expression-case expr :neg)
        (expression-case (expression-neg->arg expr)
                         :var))
   (and
     (expression-case expr :mul)
     (or (expression-case (expression-mul->arg1 expr)
                          :const)
         (and (expression-case (expression-mul->arg1 expr)
                               :neg)
              (expression-case
                   (expression-neg->arg (expression-mul->arg1 expr))
                   :const)))
     (expression-case (expression-mul->arg2 expr)
                      :var)))))

Theorem: booleanp-of-r1cs-monomialp

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

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

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

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

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