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R1cs-bridge

R1cs-vector-to-pfcs

Translate an R1CS (sparse) vector to a PFCS expression.

Signature
(r1cs-vector-to-pfcs vec) → expr
Arguments: vec — Guard (r1cs::sparse-vectorp vec).
Returns: expr — Type (expressionp expr).

We translate the empty vector to the PFCS constant 0. We translate a singleton vector to the element's corresponding PFCS expression. We translate vectors of two or more elements to (nested) PFCS additions; these are nested to the left, so we reverse the vector before the recursion.

Definitions and Theorems

Function: r1cs-vector-to-pfcs-aux

(defun r1cs-vector-to-pfcs-aux (rev-vec)
  (declare (xargs :guard (r1cs::sparse-vectorp rev-vec)))
  (let ((__function__ 'r1cs-vector-to-pfcs-aux))
    (declare (ignorable __function__))
    (cond ((endp rev-vec) (expression-const 0))
          ((endp (cdr rev-vec))
           (r1cs-vec-elem-to-pfcs (car rev-vec)))
          (t (make-expression-add
                  :arg1 (r1cs-vector-to-pfcs-aux (cdr rev-vec))
                  :arg2 (r1cs-vec-elem-to-pfcs (car rev-vec)))))))

Theorem: expressionp-of-r1cs-vector-to-pfcs-aux

(defthm expressionp-of-r1cs-vector-to-pfcs-aux
  (b* ((expr (r1cs-vector-to-pfcs-aux rev-vec)))
    (expressionp expr))
  :rule-classes :rewrite)

Theorem: r1cs-polynomialp-of-r1cs-vector-to-pfcs-aux

(defthm r1cs-polynomialp-of-r1cs-vector-to-pfcs-aux
  (b* ((expr (r1cs-vector-to-pfcs-aux rev-vec)))
    (r1cs-polynomialp expr))
  :rule-classes :rewrite)

Function: r1cs-vector-to-pfcs

(defun r1cs-vector-to-pfcs (vec)
  (declare (xargs :guard (r1cs::sparse-vectorp vec)))
  (let ((__function__ 'r1cs-vector-to-pfcs))
    (declare (ignorable __function__))
    (r1cs-vector-to-pfcs-aux (rev vec))))

Theorem: expressionp-of-r1cs-vector-to-pfcs

(defthm expressionp-of-r1cs-vector-to-pfcs
  (b* ((expr (r1cs-vector-to-pfcs vec)))
    (expressionp expr))
  :rule-classes :rewrite)

Theorem: r1cs-polynomialp-of-r1cs-vector-to-pfcs

(defthm r1cs-polynomialp-of-r1cs-vector-to-pfcs
  (b* ((expr (r1cs-vector-to-pfcs vec)))
    (r1cs-polynomialp expr))
  :rule-classes :rewrite)