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Datei: Sprache: Lisp

Original von: PVS^©

(defparameter *sturm-var* nil) ;; Variable

(defparameter *sturm-rews* '("polylist_const" "polylist_monom" "polylist_sum" "polylist_neg"
        "polylist_minus" "polylist_prod" "polylist_div" "polylist_pow" "polylist_sq"
        "expt_x1" "expt_x0"))

(defparameter *sturm-invalid-forms* nil)

(defun sturm-poly-exprs-rec (exprs vars)
  (when exprs
    (let* ((e    (car exprs))
    (xprs (cdr exprs))
    (pe   (catch '*sturm-error*
     (setq *sturm-var* nil)
     (sturm-poly-expr-rec e vars))))
      (cons (cons *sturm-var* pe)
     (cond (pe (sturm-poly-exprs-rec xprs vars))
    (t (push e *sturm-invalid-forms*)
       (sturm-poly-exprs-rec xprs vars)))))))

(defun sturm-poly-exprs (exprs &optional vars)
  (setq *sturm-invalid-forms* nil)
  (extra-reset-evalexprs)
  (sturm-poly-exprs-rec exprs vars))

(defun is-sturm-var (expr vars)
  (when (is-variable-expr expr vars)
    (let ((name (expr2str expr)))
      (cond ((null *sturm-var*)
      (setq *sturm-var* name))
     ((string= *sturm-var* name) t)
     (t (throw '*sturm-error*))))))

(defun sturm-var-deg (expr vars)
  (cond ((is-sturm-var expr vars) 1)
((and (is-function-expr expr "^")
       (is-sturm-var (args1 expr) vars))
  (extra-add-evalexpr (args2 expr)))))

(defun sturm-monom-expr (expr vars)
  (let ((tk-deg (sturm-var-deg expr vars)))
    (if tk-deg
(cons 1 tk-deg)
      (when (is-function-expr expr "*")
(let ((val (extra-add-evalexpr (args1 expr))))
   (when val
     (let ((tk-deg (sturm-var-deg (args2 expr) vars)))
       (when tk-deg
  (cons val tk-deg)))))))))

(defun sturm-poly-expr-rec (expr vars)
  (let ((val (extra-add-evalexpr expr)))
    (if val
(format nil "pconst(~a)" val)
      (let ((monom (sturm-monom-expr expr vars)))
(if monom (format nil "pmonom(~a,~a)" (car monom) (cdr monom))
   (cond ((and (unary-application? expr) (is-function-expr expr "-"))
   (format nil "pneg(~a)" (sturm-poly-expr-rec (args1 expr) vars)))
  ((is-function-expr expr "^")
   (let ((val (extra-add-evalexpr (args2 expr))))
     (if val
         (format nil "ppow(~a,~a)"
          (sturm-poly-expr-rec (args1 expr) vars) val)
       (throw '*sturm-error*))))
  ((is-function-expr expr "/")
   (let ((val (extra-add-evalexpr (args2 expr))))
     (if val
         (format nil "pdiv(~a,~a)"
          (sturm-poly-expr-rec (args1 expr) vars) val)
       (throw '*sturm-error*))))
  ((is-function-expr expr "+")
   (format nil "psum(~a,~a)"
    (sturm-poly-expr-rec (args1 expr) vars)
    (sturm-poly-expr-rec (args2 expr) vars)))
  ((is-function-expr expr "-")
   (format nil "pminus(~a,~a)"
    (sturm-poly-expr-rec (args1 expr) vars)
    (sturm-poly-expr-rec (args2 expr) vars)))
  ((is-function-expr expr "*")
   (format nil "pprod(~a,~a)"
    (sturm-poly-expr-rec (args1 expr) vars)
    (sturm-poly-expr-rec (args2 expr) vars)))
  ((is-function-expr expr "sq")
   (format nil "psq(~a)" (sturm-poly-expr-rec (args1 expr) vars)))
  (t (throw '*sturm-error*))))))))

(defun last-to-first (l)
  (when l
    (let ((lst (car (last l)))
   (rst (subseq l 0 (1- (length l)))))
      (cons lst rst))))

(defun sturm-relation (expr fnum)
  (let ((rel (car (is-function-expr expr '(< <= > >= = /=)))))
    (when rel
      (cond ((and (> fnum 0) (not (equal rel '=)))
      rel)
     ((and (< fnum 0) (not (equal rel '/=)))
      (not-relation rel))))))

(defun sturm-rel-str (rel)
  (if (member rel '(< <= > >=))
      (format nil "reals.~a" rel) (format nil "~a[real]" rel)))

(defun sturm-get-expr-fms (expr quant)
  (when expr
    (cond ((< quant 0)
    (last-to-first (get-ands-expr (expression expr))))
   ((> quant 0)
    (get-hypotheses (expression expr)))
   (t (list expr)))))

(defun sturm-expr-wrt0 (expr1 expr2 &optional sign)
  (let ((num (number-from-expr expr2)))
    (if (or (null num) (/= 0 num))
(mk-application (if sign '+ '-) expr1 expr2)
      expr1)))

(defun sturm-get-xval (var fms)
  (when var
    (extra-get-var-ranges fms (enlist-it var))
    (let* ((xv  (gethash var *extra-varranges*))
    (lb  (cond ((null xv) -1)
        ((null (xterval-lb xv))
         (1- (xterval-ub xv)))
        (t (xterval-lb xv))))
    (ub  (cond ((null xv) 1)
        ((null (xterval-ub xv))
         (1+ (xterval-lb xv)))
        (t (xterval-ub xv)))))
      (format nil
       "mk_realint(~a,~a,~:[FALSE~;TRUE~],~:[FALSE~;TRUE~],~:[FALSE~;TRUE~],~:[FALSE~;TRUE~])"
       lb ub (and xv (xterval-lb xv)) (and xv (xterval-ub xv))
       (and xv (xterval-lb-closed xv)) (and xv (xterval-ub-closed xv))))))

(defstep sturm (&optional (fnum 1) (preds? t) dont-fail?)
  (let ((fn       (extra-get-fnum fnum))
(expr     (extra-get-formula-from-fnum fn))
(quant    (when fn
      (cond ((forall-expr? expr) 1)
     ((exists-expr? expr) -1)
     (t 0)))) ;; forall: quant > 0, exists: quant < 0, none: 0
(exprfms  (when fn
      (sturm-get-expr-fms
       (if (= quant 0) expr
         (lift-predicates-in-quantifier expr (list *real*)))
       quant)))
(st-fms   (cdr exprfms))
(st-rel   (when fn (sturm-relation (car exprfms) (if (= quant 0) fn quant))))
(st-fm    (when st-rel (car exprfms)))
(st-expr  (when st-rel (sturm-expr-wrt0 (args1 st-fm) (args2 st-fm))))
(st-qvars (when (/= quant 0)
      (mapcar #'(lambda (x) (format nil "~a" (id x))) (bindings expr))))
(st-vpoly (when st-expr (car (sturm-poly-exprs (list st-expr) st-qvars))))
(st-var   (car st-vpoly))
(st-poly  (cdr st-vpoly))
(st-inval (car *sturm-invalid-forms*))
(msg      (cond
     ((null expr)
      (format nil "Formula ~a not found." fnum))
     ((not (check-name "SturmStrategy__"))
      (format nil "This strategy requires importing Sturm@strategies."))
     ((null st-rel)
      (format nil "Formula ~a is not supported by this strategy." fnum))
     (st-inval
      (format nil "The following expression doesn't appear to be a polynomial~@[ on variable ~a~]:~%~a."
       st-var st-inval))
     ((null st-var)
      (format nil "No variable found in formula ~a." fnum)))))
    (if msg (printf msg)
      (then (sturm__$ fn quant st-poly st-var st-qvars st-fms st-rel preds?)
     (unless dont-fail? (fail)))))
  "[Sturm] Applies decision procedure for single variable polynomials
based on Sturm's Theorem to formula FNUM. Type predicates are
introduced as hypotheses when PREDS? is t. If DONT-FAIL? is set to t,
strategy skips instead of failing when sequent cannot be discharged."
  "Applying Sturm's Theorem to formula ~a")

(defhelper sturm__ (fnum quant poly var qvars fms rel preds?)
  (with-fresh-names
   ((stu_) (lst_))
   (let ((relstr   (sturm-rel-str rel))
  (sat      (< (* fnum quant) 0))
  (degeq    (format nil "deg(~a) = deg(~a)" stu_ lst_))
  (tpreds   (and preds? (= quant 0))))
     (with-fresh-labels
      ((!stu fnum) (!stu-lst) (!stu-deg) (!stu-tccs))
      (when tpreds (discriminate (typepred var) :label !stu-tccs))
      (let ((fns  (if (= quant 0)
        (append (mapcar #'(lambda (f) (extra-get-formula-from-fnum f))
          (extra-get-fnums `(-^ ,fnum)))
         (mapcar #'(lambda (f)
       (make-negation
        (extra-get-formula-from-fnum f)))
          (extra-get-fnums `(+^ ,fnum))))
      fms))
     (xval (sturm-get-xval var fns))
     (ri   (xterval2str (gethash var *extra-varranges*))))
(then
  (name-label stu_ poly :hide? t)
  (eval-expr stu_)
  (name-label lst_ (! -1 r) -1 :hide? t)
  (label !stu-lst -1)
  (spread
   (case degeq)
   ((then (eval-expr (! -1 2))
   (replaces -1 -2)
   (label !stu-deg -1)
   (lemma "sturm_def")
   (with-fresh-labels
    ((!stlm -1))
    (spread (inst !stlm xval stu_ relstr)
     ((then (replaces !stu-deg)
     (assert !stlm)
     (expand stu_ !stlm 2)
     (replaces !stu-lst :in !stlm)
     (let ((fnlem (extra-get-fnum !stlm)))
       (if fnlem
           (then
     (eval-expr (! !stlm 1))
     (let ((b        (args2 (extra-get-formula -1)))
           (is-true  (extra-is-true b)))
       (if (iff sat is-true)
           (printf "Formula ~a doesn't hold for ~a in ~a."
            fnum var ri)
         (then
          (replaces -1)
          (flatten !stlm)
          (let ((fa  (cond (sat !stlm)
             ((> (* fnum quant) 0) !stu)))
         (ex  (if sat !stu !stlm))
         (nn (when fa (freshnames "x" (length qvars))))
         (vs (if fa
          (let ((pns (pair-lists qvars nn)))
            (cdr (assoc var pns :test #'string=)))
        var))
         (rws *sturm-rews*)
         (eqs (extra-evalexprs)))
            (if sat
         (then
          (skolem fa nn)
          (expand "contains?")
          (flatten)
          (with-fresh-labels
           (!steqs)
           (extra-evalexprs$ eqs !steqs)
           (reveal !steqs)
           (spread (inst ex vs)
            ((rewrite* rws !stlm)))
           (replaces !steqs)
           (expand* "[||]" "##" "contains?")
           (rewrite* ("abs_lt" "abs_le" "ge_abs" "gt_abs"))
           (flatten)
           (assert)))
       (then
        (when fa (skolem fa nn :skolem-typepreds? preds?))
        (inst ex vs)
        (extra-evalexprs$ eqs)
        (expand* "[||]" "##" "contains?")
        (rewrite* ("abs_lt" "abs_le" "ge_abs" "gt_abs"))
        (flatten)
        (rewrite* rws !stlm)
        (assert))))))))
         (if sat
      (printf "Domain of variable ~a is the empty interval ~a. Formula ~a doesn't hold."
       var ri fnum)
           (then
     (printf "Domain of variable ~a is the empty interval ~a. Formula ~a trivially holds."
      var ri fnum)
     (discriminate (skosimp* :preds? t) !stlm)
     (hide-all-but !stlm)
     (grind))))))
      (then (hide-all-but 1) (expand "SturmRel?"))))))
    (replaces !stu-lst)
    (eval-formula))))))))
  "[Sturm] Internal strategy." "")

(defun mono-poly-get-xval (var1 var2 rel fms)
  (extra-get-var-ranges fms (list var1 var2))
  (let* ((xv1  (or (gethash var1 *extra-varranges*)
     (make-xterval)))
  (xv2  (or (gethash var2 *extra-varranges*)
     (make-xterval)))
  (xv (if (member rel '(< <=))
   (make-xterval :lb (xterval-lb xv1)
          :lb-closed (xterval-lb-closed xv1)
          :ub (xterval-ub xv2)
          :ub-closed (xterval-ub-closed xv2))
        (make-xterval :lb (xterval-lb xv2)
        :lb-closed (xterval-lb-closed xv2)
        :ub (xterval-ub xv1)
        :ub-closed (xterval-ub-closed xv1))))
  (lb (cond ((and (null (xterval-lb xv))
    (null (xterval-ub xv))) -1)
     ((null (xterval-lb xv))
      (1- (xterval-ub xv)))
     (t (xterval-lb xv))))
  (ub  (cond ((and (null (xterval-lb xv))
     (null (xterval-ub xv))) 1)
      ((null (xterval-ub xv))
       (1+ (xterval-lb xv)))
      (t (xterval-ub xv)))))
    (format nil
     "mk_realint(~a,~a,~:[FALSE~;TRUE~],~:[FALSE~;TRUE~],~:[FALSE~;TRUE~],~:[FALSE~;TRUE~])"
     lb ub (xterval-lb xv) (xterval-ub xv) (xterval-lb-closed xv) (xterval-ub-closed xv))))

(defstep mono-poly (&optional (fnum 1) (preds? t) dont-fail?)
  (let ((fn       (extra-get-fnum fnum))
(expr     (extra-get-formula-from-fnum fn))
(quant    (when fn
      (cond ((forall-expr? expr) 1)
     ((exists-expr? expr) -1)
     (t 0)))) ;; forall: quant > 0, exists: quant < 0, none: 0
(exprfms  (when fn (sturm-get-expr-fms
       (if (= quant 0) expr
         (lift-predicates-in-quantifier expr (list *real*)))
       quant)))
(mn-qvars (when (/= quant 0)
      (mapcar #'(lambda (x) (format nil "~a" (id x))) (bindings expr))))
(mn-fms   (cdr exprfms))
(mn-rel   (when fn (sturm-relation (car exprfms) (if (= quant 0) fn quant))))
(mn-fm    (when mn-rel (car exprfms)))
(mn-vpols (when mn-fm
      (sturm-poly-exprs (list (args1 mn-fm) (args2 mn-fm)) mn-qvars)))
(mn-varl  (car (nth 0 mn-vpols)))
(mn-polyl (cdr (nth 0 mn-vpols)))
(mn-varr  (car (nth 1 mn-vpols)))
(mn-polyr (cdr (nth 1 mn-vpols)))
(msg      (cond
     ((null expr)
      (format nil "Formula ~a not found." fnum))
     ((not (check-name "SturmStrategy__"))
      (format nil "This strategy requires importing Sturm@strategies."))
     ((null mn-rel)
      (format nil "Formula ~a is not supported by this strategy." fnum))
     ((null mn-varl)
      (format nil "No variables found in left-hand side of formula ~a." fnum))
     ((null mn-varr)
      (format nil "No variables found in right-hand side of formula ~a." fnum))
     (*sturm-invalid-forms*
      (format nil "The following expressions don't appear to be univariate polynomials:~%~{~a~%~}"
       *sturm-invalid-forms*)))))
    (if msg (printf msg)
      (then (mono-poly__$ fn quant mn-polyl mn-polyr mn-varl mn-varr
     mn-qvars mn-fms mn-rel preds?)
     (printf "Formula ~a may not hold." fnum)
     (unless dont-fail? (fail)))))
  "[Sturm] Applies decision procedure for polynomial monotonicity
based on Sturm's Theorem to formula FNUM. Type predicates are
introduced as hypotheses when PREDS? is t. If DONT-FAIL? is set to t,
strategy skips instead of failing when sequent cannot be discharged."
  "Proving polynomial monotonicity in formula ~a using Sturm's Theorem")

(defhelper mono-poly__ (fnum quant poly1 poly2 var1 var2 qvars fms rel preds?)
  (with-fresh-labels
   ((!mono fnum) (!p1) (!p2) (!mono-tccs))
   (with-fresh-names
    ((p1_ poly1) (p2_ poly2))
    (eval-expr p1_)
    (relabel !p1 -1)
    (eval-expr p2_)
    (relabel !p2 -1)
    (let ((vars   (list var1 var2))
   (tpreds (and preds? (= quant 0)))
   (p1r    (extra-get-expr '(! -2 r)))
   (p2r    (extra-get-expr '(! -1 r)))
   (eq12   (extra-is-true (evalexpr (mk-equation p1r p2r)))))
      (if eq12
   (with-fresh-names
    (pl_ (! !p1 r))
    (replace !p1 !p2 rl)
    (when tpreds
      (discriminate (typepred var1) :label !mono-tccs)
      (discriminate (typepred var2) :label !mono-tccs))
    (let ((fns (if (= quant 0)
     (append (mapcar #'(lambda (f) (extra-get-formula-from-fnum f))
       (extra-get-fnums `(-^ ,fnum)))
      (mapcar #'(lambda (f)
           (make-negation
            (extra-get-formula-from-fnum f)))
       (extra-get-fnums `(+^ ,fnum))))
   fms))
   (relvar (extra-get-var-var-relation fns var1 var2)))
      (if relvar
   (let ((xval      (mono-poly-get-xval var1 var2 relvar fns))
         (lp        (format nil "list2array[rat](0)(~a)" pl_))
         (deg       (format nil "length[rat](~a)-1" pl_))
         (relstr    (sturm-rel-str rel))
         (relvarstr (sturm-rel-str relvar)))
     (with-fresh-names
      ((xval_ xval) (lp_ lp :tcc-step (eval-formula))
       (deg_ deg) (rel_ relstr) (relvar_ relvarstr))
      (lemma "mono_def")
      (branch
       (inst -1 xval_ lp_ deg_ rel_ relvar_)
       ((then
         (delete *lp_-tccs*)
         (branch
   (split -1)
   ((with-fresh-labels
     (!l -1)
     (eval-expr (! !l 2))
     (let ((b       (args2 (extra-get-formula -1)))
    (is-true (extra-is-true b))
    (nn      (when qvars (freshnames "x" (length qvars))))
    (vs      (if qvars
          (let ((pns (pair-lists qvars nn)))
            (mapcar #'(lambda (x) (cdr (assoc x pns :test #'string=)))
             vars))
        vars))
    (fall    (>= (* fnum quant) 0))
    (fa      (if fall !mono !l))
    (ex      (if fall !l !mono))
    (eqs     (extra-evalexprs))
    (holds   (iff fall is-true))
    (rws     *sturm-rews*))
       (when holds
         (replaces -1)
         (prop)
         (if fall
      (then
       (when qvars (skolem fa nn :skolem-typepreds? preds?))
       (inst ex :terms vs)
       (flatten)
       (extra-evalexprs$ eqs)
       (expand* xval_ "[||]" "##" "contains?")
       (rewrite* ("abs_lt" "abs_le" "ge_abs" "gt_abs"))
       (flatten)
       (expand* rel_ relvar_ lp_ deg_)
       (branch
        (rewrite* "polylist_eval" :dir rl)
        ((then
          (expand pl_ !l 2)
          (replace !p1 :dir rl)
          (replaces !p1 !p2)
          (expand pl_)
          (replaces !p2 :dir rl)
          (expand* p1_ p2_)
          (rewrite* rws)
          (assert))
         (eval-formula))))
    (then
     (skolem fa nn)
     (expand* xval_ "contains?" rel_ relvar_ lp_ deg_)
     (flatten)
     (branch
      (rewrite* "polylist_eval" :dir rl)
      ((then
        (expand pl_ !l 2)
        (replace !p1 :dir rl)
        (replaces !p1 !p2)
        (expand pl_)
        (replaces !p2 :dir rl)
        (expand* p1_ p2_)
        (with-fresh-labels
         (!steqs)
         (extra-evalexprs$ eqs !steqs)
         (reveal !steqs)
         (spread (inst ex :terms vs)
          ((rewrite* rws)))
         (replaces !steqs)
         (expand* "[||]" "##" "contains?")
         (rewrite* ("abs_lt" "abs_le" "ge_abs" "gt_abs"))
         (flatten)
         (assert)))
       (eval-formula))))))))
    (eval-formula))))
        (then (hide-all-but 1)
       (expand* "SturmRel?" relvar_))
        (then (hide-all-but 1)
       (expand* "SturmRel?" rel_))
        (then (hide-all-but 1)
       (branch (splash 1 :reverse? t)
        ((assert)
         (eval-formula))))))))
        (printf "Variables ~a and ~a don't appear to be related" var1 var2)))
    (when tpreds (delete !mono-tccs)))
(printf "Left- and right-hand side polynomials appear to be different")))))
  "[Sturm] Internal strategy." "")

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