(TOTAL-NEXT
 (1 1 (:TYPE-PRESCRIPTION TOTAL-NEXT))
 )
(TOTAL-RUN
 (17 9 (:REWRITE DEFAULT-+-2))
 (12 9 (:REWRITE DEFAULT-+-1))
 (8 2 (:REWRITE COMMUTATIVITY-OF-+))
 (8 1 (:DEFINITION LENGTH))
 (7 6 (:REWRITE DEFAULT-<-1))
 (6 6 (:TYPE-PRESCRIPTION TOTAL-NEXT))
 (6 6 (:REWRITE DEFAULT-<-2))
 (5 1 (:DEFINITION LEN))
 (2 2 (:REWRITE FOLD-CONSTS-IN-+))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE DEFAULT-CDR))
 (1 1 (:TYPE-PRESCRIPTION LEN))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE DEFAULT-REALPART))
 (1 1 (:REWRITE DEFAULT-NUMERATOR))
 (1 1 (:REWRITE DEFAULT-IMAGPART))
 (1 1 (:REWRITE DEFAULT-DENOMINATOR))
 (1 1 (:REWRITE DEFAULT-COERCE-2))
 (1 1 (:REWRITE DEFAULT-COERCE-1))
 (1 1 (:REWRITE DEFAULT-CAR))
 )
(TOTAL-PRE)
(TOTAL-EXTERNAL)
(TOTAL-POST)
(TOTAL-CLOCK-FN)
(TOTAL-CLOCK-FN-IS-NATP)
(TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL)
(STANDARD-THEOREM-FOR-TOTAL-CLOCKS-1)
(STANDARD-THEOREM-FOR-TOTAL-CLOCKS-2)
(STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3)
(EXISTS-TOTAL-PRE-STATE)
(EXISTS-TOTAL-PRE-STATE-SUFF)
(TOTAL-INV)
(TOTAL-RUN-FN
 (17 9 (:REWRITE DEFAULT-+-2))
 (12 9 (:REWRITE DEFAULT-+-1))
 (8 2 (:REWRITE COMMUTATIVITY-OF-+))
 (8 1 (:DEFINITION LENGTH))
 (7 6 (:REWRITE DEFAULT-<-1))
 (6 6 (:REWRITE DEFAULT-<-2))
 (5 1 (:DEFINITION LEN))
 (2 2 (:REWRITE FOLD-CONSTS-IN-+))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE DEFAULT-CDR))
 (1 1 (:TYPE-PRESCRIPTION LEN))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE DEFAULT-REALPART))
 (1 1 (:REWRITE DEFAULT-NUMERATOR))
 (1 1 (:REWRITE DEFAULT-IMAGPART))
 (1 1 (:REWRITE DEFAULT-DENOMINATOR))
 (1 1 (:REWRITE DEFAULT-COERCE-2))
 (1 1 (:REWRITE DEFAULT-COERCE-1))
 (1 1 (:REWRITE DEFAULT-CAR))
 )
(TOTAL-RUN-FN-IS-TOTAL-RUN
 (6 6 (:REWRITE ZP-OPEN))
 (4 4 (:REWRITE DEFAULT-+-2))
 (4 4 (:REWRITE DEFAULT-+-1))
 )
(TOTAL-CLOCK-FN-IS-EQUAL-IF-<)
(TOTAL-PRE-HAS-TOTAL-CLOCK->0)
(TOTAL-PRE-HAS-TOTAL-INV
 (3 2 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 )
(TOTAL-RUN-IS-SAME-FOR-NFIX
 (2 2 (:REWRITE ZP-OPEN))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (2 2 (:REWRITE SIMPLIFY-SUMS-<))
 (2 2 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (2 2 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE NORMALIZE-ADDENDS))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (2 2 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(TOTAL-RUN-NATP-TOTAL-NEXT
 (8 8 (:REWRITE ZP-OPEN))
 (7 7 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (7 7 (:REWRITE NORMALIZE-ADDENDS))
 (7 7 (:REWRITE DEFAULT-+-2))
 (7 7 (:REWRITE DEFAULT-+-1))
 (4 4 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (4 4 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (4 4 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (4 4 (:REWRITE |(equal (- x) (- y))|))
 )
(TOTAL-CLOCK-<-TOTAL-NEXT
 (12 3 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (11 2 (:DEFINITION TOTAL-RUN))
 (4 4 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (4 4 (:REWRITE NORMALIZE-ADDENDS))
 (4 4 (:REWRITE DEFAULT-+-2))
 (4 4 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-SUMS-<))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (1 1 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 (1 1 (:REWRITE |(equal (+ c x) d)|))
 (1 1 (:REWRITE |(< 0 (- x))|))
 (1 1 (:REWRITE |(< (- x) (- y))|))
 )
(TOTAL-CLOCK-FN-<-TOTAL-NEXT-CONCRETIZED)
(TOTAL-RUN-+-REDUCTION
 (80 14 (:REWRITE ZP-OPEN))
 (28 6 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (25 13 (:REWRITE |(+ c (+ d x))|))
 (20 20 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (20 20 (:REWRITE NORMALIZE-ADDENDS))
 (20 20 (:REWRITE DEFAULT-+-2))
 (20 20 (:REWRITE DEFAULT-+-1))
 (7 7 (:REWRITE FOLD-CONSTS-IN-+))
 (6 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (6 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (6 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (6 6 (:REWRITE |(< d (+ c x))|))
 (6 6 (:REWRITE |(< (- x) (- y))|))
 (4 4 (:REWRITE SIMPLIFY-SUMS-<))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE |(< 0 (- x))|))
 (3 3 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (3 3 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (3 3 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (3 3 (:REWRITE |(equal (- x) (- y))|))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (2 2 (:REWRITE |(< d (+ c x y))|))
 )
(WEIRD-TOTAL-RUN-+-REDUCTION
 (10 10 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (10 10 (:REWRITE NORMALIZE-ADDENDS))
 (10 10 (:REWRITE DEFAULT-+-2))
 (10 10 (:REWRITE DEFAULT-+-1))
 (9 9 (:REWRITE ZP-OPEN))
 (8 8 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (8 8 (:REWRITE SIMPLIFY-SUMS-<))
 (8 8 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (8 8 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (8 8 (:REWRITE DEFAULT-<-2))
 (8 8 (:REWRITE DEFAULT-<-1))
 (8 8 (:REWRITE |(< (- x) 0)|))
 (8 8 (:REWRITE |(< (- x) (- y))|))
 (4 4 (:REWRITE REDUCE-INTEGERP-+))
 (4 4 (:REWRITE INTEGERP-MINUS-X))
 (4 4 (:META META-INTEGERP-CORRECT))
 )
(TOTAL-INV-IMPLIES-TOTAL-NEXT-TOTAL-INV
 (137 9 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (79 16 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (34 34 (:REWRITE SIMPLIFY-SUMS-<))
 (34 34 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (34 34 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (34 34 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (34 34 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (34 34 (:REWRITE |(< (- x) (- y))|))
 (27 27 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (27 27 (:REWRITE NORMALIZE-ADDENDS))
 (27 27 (:REWRITE DEFAULT-<-2))
 (27 27 (:REWRITE DEFAULT-<-1))
 (27 27 (:REWRITE DEFAULT-+-2))
 (27 27 (:REWRITE DEFAULT-+-1))
 (25 25 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (25 25 (:REWRITE |(< 0 (- x))|))
 (7 7 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (7 7 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (7 7 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (7 7 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (7 7 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (7 7 (:REWRITE |(equal (- x) (- y))|))
 (3 3 (:REWRITE |(< (+ c x) d)|))
 (2 2 (:REWRITE |(< d (+ c x))|))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-INV))
 )
(TOTAL-CLOCK-FN-IS-THE-SMALLEST
 (11 2 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (2 2 (:REWRITE SIMPLIFY-SUMS-<))
 (2 2 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (2 2 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (2 2 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (1 1 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (1 1 (:REWRITE NORMALIZE-ADDENDS))
 (1 1 (:REWRITE DEFAULT-+-2))
 (1 1 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE |(< 0 (- x))|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 )
(TOTAL-RUN-1-IS-TOTAL-NEXT)
(TOTAL-INV-AND-TOTAL-EXTERNAL-IMPLIES-TOTAL-POST
 (153 10 (:REWRITE ZP-OPEN))
 (71 12 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (42 8 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (19 19 (:REWRITE SIMPLIFY-SUMS-<))
 (19 19 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (19 19 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (19 19 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (19 19 (:REWRITE |(< (- x) (- y))|))
 (17 17 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (17 17 (:REWRITE DEFAULT-<-2))
 (17 17 (:REWRITE DEFAULT-<-1))
 (15 15 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (15 15 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (15 15 (:REWRITE NORMALIZE-ADDENDS))
 (15 15 (:REWRITE DEFAULT-+-2))
 (15 15 (:REWRITE DEFAULT-+-1))
 (15 15 (:REWRITE |(< 0 (- x))|))
 (13 2 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (8 8 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (8 6 (:REWRITE REDUCE-INTEGERP-+))
 (7 7 (:META META-INTEGERP-CORRECT))
 (6 6 (:REWRITE INTEGERP-MINUS-X))
 (4 4 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (4 4 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (4 4 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (4 4 (:REWRITE |(equal (- x) (- y))|))
 (2 2 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE |(< d (+ c x))|))
 )
(FIND-TOTAL-EXTERNAL
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (6 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (6 6 (:REWRITE |(< (- x) (- y))|))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:REWRITE DEFAULT-<-1))
 (3 3 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (3 3 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:REWRITE |(< d (+ c x))|))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(M)
(FIND-TOTAL-EXTERNAL-IS-NATP)
(M-IS-AN-ORDINAL
 (83 1 (:DEFINITION FIND-TOTAL-EXTERNAL))
 (70 1 (:REWRITE ZP-OPEN))
 (35 5 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (30 4 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (28 3 (:REWRITE REDUCE-INTEGERP-+))
 (14 1 (:REWRITE |(+ y (+ x z))|))
 (12 6 (:REWRITE NORMALIZE-ADDENDS))
 (8 4 (:REWRITE INTEGERP-MINUS-X))
 (7 1 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (6 5 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (6 5 (:REWRITE DEFAULT-+-2))
 (5 5 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (5 5 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (5 5 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (5 5 (:REWRITE DEFAULT-UNARY-MINUS))
 (5 5 (:REWRITE DEFAULT-+-1))
 (5 5 (:REWRITE |(< (- x) (- y))|))
 (5 4 (:REWRITE SIMPLIFY-SUMS-<))
 (5 4 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:META META-INTEGERP-CORRECT))
 (3 3 (:REWRITE |(< 0 (- x))|))
 (3 3 (:REWRITE |(- (- x))|))
 (3 3 (:REWRITE |(+ c (+ d x))|))
 (2 2 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (2 2 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (2 2 (:REWRITE |(< d (+ c x))|))
 (2 2 (:REWRITE |(+ 0 x)|))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 (1 1 (:REWRITE |(< (- x) 0)|))
 (1 1 (:REWRITE |(+ x (- x))|))
 (1 1 (:DEFINITION NOT))
 )
(TOTAL-EXTERNAL-IMPLIES-FIND-TOTAL-EXTERNAL-TOTAL-EXTERNAL
 (93 21 (:REWRITE ZP-OPEN))
 (24 24 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (15 12 (:REWRITE DEFAULT-+-2))
 (14 10 (:REWRITE SIMPLIFY-SUMS-<))
 (14 10 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (14 10 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (14 10 (:REWRITE DEFAULT-<-2))
 (12 12 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (12 12 (:REWRITE NORMALIZE-ADDENDS))
 (12 12 (:REWRITE DEFAULT-+-1))
 (10 10 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (10 10 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (10 10 (:REWRITE DEFAULT-<-1))
 (10 10 (:REWRITE |(< (- x) (- y))|))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (5 5 (:REWRITE |(< 0 (- x))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE |(< (- x) 0)|))
 )
(TOTAL-RUN-MINUS-REDUCTION
 (47 3 (:DEFINITION TOTAL-RUN))
 (30 3 (:REWRITE ZP-OPEN))
 (6 6 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (6 6 (:REWRITE DEFAULT-+-2))
 (6 6 (:REWRITE DEFAULT-+-1))
 (4 4 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (4 4 (:REWRITE |(+ c (+ d x))|))
 (3 3 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (3 3 (:REWRITE DEFAULT-UNARY-MINUS))
 (3 3 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE SIMPLIFY-SUMS-<))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 (1 1 (:REWRITE DEFAULT-<-2))
 (1 1 (:REWRITE DEFAULT-<-1))
 (1 1 (:REWRITE |(< d (+ c x))|))
 (1 1 (:REWRITE |(< 0 (- x))|))
 (1 1 (:REWRITE |(< (- x) 0)|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 )
(TOTAL-INV-STATES-HAVE-TOTAL-EXTERNAL
 (192 25 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (121 25 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (91 9 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (59 8 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (39 39 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (39 39 (:REWRITE |(< (- x) (- y))|))
 (38 38 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (38 38 (:REWRITE DEFAULT-+-2))
 (38 38 (:REWRITE DEFAULT-+-1))
 (35 35 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (34 34 (:REWRITE SIMPLIFY-SUMS-<))
 (28 28 (:REWRITE |(< 0 (- x))|))
 (26 26 (:REWRITE DEFAULT-<-2))
 (26 26 (:REWRITE DEFAULT-<-1))
 (23 23 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (19 19 (:REWRITE |(+ c (+ d x))|))
 (19 15 (:REWRITE INTEGERP-MINUS-X))
 (15 15 (:META META-INTEGERP-CORRECT))
 (13 13 (:REWRITE DEFAULT-UNARY-MINUS))
 (11 11 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (6 6 (:REWRITE |(< d (+ c x))|))
 (5 5 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (5 5 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (5 5 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (5 5 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (5 5 (:REWRITE FOLD-CONSTS-IN-+))
 (5 5 (:REWRITE |(equal (- x) (- y))|))
 (4 4 (:REWRITE TOTAL-RUN-1-IS-TOTAL-NEXT))
 (2 2 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 )
(EXISTS-TOTAL-PRE-STATE-TO-TOTAL-CLOCK
 (48 8 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (34 2 (:DEFINITION TOTAL-RUN))
 (31 8 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (28 10 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (24 2 (:REWRITE ZP-OPEN))
 (19 2 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (10 10 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (10 10 (:REWRITE |(< (- x) (- y))|))
 (8 8 (:REWRITE SIMPLIFY-SUMS-<))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (6 6 (:REWRITE DEFAULT-<-2))
 (6 6 (:REWRITE DEFAULT-<-1))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (5 5 (:REWRITE |(< 0 (- x))|))
 (4 4 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (4 4 (:REWRITE DEFAULT-+-2))
 (4 4 (:REWRITE DEFAULT-+-1))
 (4 2 (:REWRITE |(* y x)|))
 (3 3 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (2 2 (:REWRITE DEFAULT-UNARY-/))
 (2 2 (:REWRITE |(* a (/ a))|))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE DEFAULT-UNARY-MINUS))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 (1 1 (:REWRITE |(< d (+ c x))|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 (1 1 (:REWRITE |(+ c (+ d x))|))
 )
(EXISTS-TOTAL-PRE-STATE-TO-TOTAL-CLOCK-2
 (50 10 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (46 10 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (34 2 (:DEFINITION TOTAL-RUN))
 (31 13 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (24 2 (:REWRITE ZP-OPEN))
 (20 3 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (13 13 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (13 13 (:REWRITE |(< (- x) (- y))|))
 (11 11 (:REWRITE SIMPLIFY-SUMS-<))
 (11 11 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (9 9 (:REWRITE DEFAULT-<-2))
 (9 9 (:REWRITE DEFAULT-<-1))
 (6 6 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (6 6 (:REWRITE |(< 0 (- x))|))
 (4 4 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (4 2 (:REWRITE |(* y x)|))
 (3 3 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (3 3 (:REWRITE DEFAULT-+-2))
 (3 3 (:REWRITE DEFAULT-+-1))
 (3 3 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (2 2 (:REWRITE REDUCE-INTEGERP-+))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (2 2 (:REWRITE INTEGERP-MINUS-X))
 (2 2 (:REWRITE DEFAULT-UNARY-/))
 (2 2 (:REWRITE |(* a (/ a))|))
 (2 2 (:META META-INTEGERP-CORRECT))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE DEFAULT-UNARY-MINUS))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 (1 1 (:REWRITE |(+ c (+ d x))|))
 )
(EXISTS-TOTAL-PRE-STATE-TO-TOTAL-CLOCK-3
 (315 30 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (230 30 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (186 30 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (30 30 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (30 30 (:REWRITE |(< (- x) (- y))|))
 (25 25 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (21 21 (:REWRITE SIMPLIFY-SUMS-<))
 (21 21 (:REWRITE DEFAULT-<-2))
 (21 21 (:REWRITE DEFAULT-<-1))
 (20 20 (:REWRITE DEFAULT-UNARY-MINUS))
 (20 10 (:REWRITE |(* y x)|))
 (20 2 (:REWRITE NINTP-2))
 (17 17 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (17 17 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (17 17 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (17 17 (:REWRITE |(< 0 (- x))|))
 (13 13 (:REWRITE |(- (- x))|))
 (10 10 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (10 10 (:REWRITE DEFAULT-UNARY-/))
 (10 10 (:REWRITE |(* a (/ a))|))
 (9 9 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (9 9 (:REWRITE DEFAULT-+-2))
 (9 9 (:REWRITE DEFAULT-+-1))
 (6 2 (:REWRITE NINTP-1))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (5 5 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (5 5 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (5 5 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (5 5 (:REWRITE |(equal (- x) 0)|))
 (5 5 (:REWRITE |(equal (- x) (- y))|))
 (4 4 (:REWRITE |(< (+ c x) d)|))
 (3 3 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE |(< d (+ c x))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 (1 1 (:REWRITE |(< (+ c x y) d)|))
 )
(NO-TOTAL-EXTERNAL-STATE
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (6 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (6 6 (:REWRITE |(< (- x) (- y))|))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:REWRITE DEFAULT-<-1))
 (3 3 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (3 3 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:REWRITE |(< d (+ c x))|))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(NO-TOTAL-EXTERNAL-STATE-TO<=
 (22 22 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (10 10 (:REWRITE ZP-OPEN))
 (4 4 (:REWRITE SIMPLIFY-SUMS-<))
 (4 4 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (4 4 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (4 4 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (4 4 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (4 4 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (4 4 (:REWRITE NORMALIZE-ADDENDS))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE DEFAULT-+-2))
 (4 4 (:REWRITE DEFAULT-+-1))
 (4 4 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE |(< (- x) 0)|))
 )
(NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL
 (26 26 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (9 9 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (5 5 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (5 5 (:REWRITE NORMALIZE-ADDENDS))
 (5 5 (:REWRITE DEFAULT-+-2))
 (5 5 (:REWRITE DEFAULT-+-1))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (2 2 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (1 1 (:REWRITE SIMPLIFY-SUMS-<))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (1 1 (:REWRITE DEFAULT-<-2))
 (1 1 (:REWRITE DEFAULT-<-1))
 (1 1 (:REWRITE |(< (- x) (- y))|))
 )
(NO-TOTAL-EXTERNAL-TO-<=-CONCRETIZED
 (41 4 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (22 22 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (22 10 (:REWRITE ZP-OPEN))
 (12 12 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (9 1 (:REWRITE |(< d (+ c x))|))
 (6 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (6 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (5 5 (:REWRITE SIMPLIFY-SUMS-<))
 (5 5 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (5 5 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (5 5 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (5 5 (:REWRITE NORMALIZE-ADDENDS))
 (5 5 (:REWRITE DEFAULT-<-2))
 (5 5 (:REWRITE DEFAULT-<-1))
 (5 5 (:REWRITE DEFAULT-+-2))
 (5 5 (:REWRITE DEFAULT-+-1))
 (5 5 (:REWRITE |(< (- x) (- y))|))
 (5 1 (:REWRITE |(+ c (+ d x))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE |(< (- x) 0)|))
 )
(NO-TOTAL-EXTERNAL-EXTATE-TO-FIND-TOTAL-EXTERNAL
 (77 4 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (72 72 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (54 41 (:REWRITE DEFAULT-+-2))
 (41 41 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (41 41 (:REWRITE NORMALIZE-ADDENDS))
 (41 41 (:REWRITE DEFAULT-+-1))
 (22 22 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (22 22 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (18 9 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (18 9 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (18 9 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (15 15 (:REWRITE SIMPLIFY-SUMS-<))
 (15 15 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (15 15 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (15 15 (:REWRITE DEFAULT-<-2))
 (15 15 (:REWRITE DEFAULT-<-1))
 (15 15 (:REWRITE |(< (- x) (- y))|))
 (9 9 (:REWRITE |(equal (- x) (- y))|))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (5 5 (:REWRITE |(equal (- x) 0)|))
 )
(NO-TOTAL-EXTERNAL-WITNESS
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (6 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (6 6 (:REWRITE |(< (- x) (- y))|))
 (4 4 (:REWRITE DEFAULT-<-2))
 (4 4 (:REWRITE DEFAULT-<-1))
 (3 3 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (3 3 (:REWRITE |(< (- x) 0)|))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:REWRITE |(< d (+ c x))|))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(NO-TOTAL-EXTERNAL-WITNESS-<=N
 (11 7 (:REWRITE DEFAULT-+-2))
 (11 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (10 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (9 5 (:REWRITE SIMPLIFY-SUMS-<))
 (9 5 (:REWRITE DEFAULT-<-2))
 (8 6 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (7 7 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (7 7 (:REWRITE NORMALIZE-ADDENDS))
 (7 7 (:REWRITE DEFAULT-+-1))
 (6 6 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (6 6 (:REWRITE |(< (- x) (- y))|))
 (5 5 (:REWRITE ZP-OPEN))
 (5 5 (:REWRITE DEFAULT-<-1))
 (4 2 (:REWRITE |(< d (+ c x))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE |(< (- x) 0)|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 )
(NO-TOTAL-EXTERNAL-WITNESS-IS-NATP)
(NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL
 (221 5 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (198 18 (:REWRITE ZP-OPEN))
 (52 16 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (46 12 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (31 31 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (24 19 (:REWRITE DEFAULT-+-2))
 (24 12 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (22 11 (:REWRITE SIMPLIFY-SUMS-<))
 (22 11 (:REWRITE DEFAULT-<-2))
 (19 19 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (19 19 (:REWRITE NORMALIZE-ADDENDS))
 (19 19 (:REWRITE DEFAULT-+-1))
 (12 12 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (12 12 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (12 12 (:REWRITE |(< (- x) (- y))|))
 (11 11 (:REWRITE DEFAULT-<-1))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (10 10 (:REWRITE |(< 0 (- x))|))
 (2 1 (:REWRITE |(< d (+ c x))|))
 (2 1 (:REWRITE |(+ c (+ d x))|))
 (1 1 (:REWRITE |(< (+ c x) d)|))
 (1 1 (:REWRITE |(+ 0 x)|))
 )
(TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL
 (471 10 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (359 29 (:REWRITE ZP-OPEN))
 (330 4 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (276 12 (:DEFINITION TOTAL-RUN))
 (271 13 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (56 43 (:REWRITE DEFAULT-+-2))
 (56 8 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (47 29 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (44 29 (:REWRITE SIMPLIFY-SUMS-<))
 (44 29 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (43 43 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (43 43 (:REWRITE NORMALIZE-ADDENDS))
 (43 43 (:REWRITE DEFAULT-+-1))
 (43 26 (:REWRITE DEFAULT-<-2))
 (38 8 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (36 4 (:REWRITE |(< d (+ c x))|))
 (33 33 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (31 31 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (31 31 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (29 29 (:REWRITE |(< (- x) (- y))|))
 (27 26 (:REWRITE DEFAULT-<-1))
 (20 4 (:REWRITE |(+ c (+ d x))|))
 (19 19 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (19 19 (:REWRITE |(< 0 (- x))|))
 (9 9 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (8 1 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (7 1 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (3 3 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (1 1 (:REWRITE TOTAL-RUN-1-IS-TOTAL-NEXT))
 )
(NO-TOTAL-EXTERNAL-TO-NO-TOTAL-EXTERNAL-TOTAL-RUN
 (966 21 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (186 30 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (136 110 (:REWRITE DEFAULT-+-2))
 (110 110 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (110 110 (:REWRITE DEFAULT-+-1))
 (66 66 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (65 65 (:REWRITE |(< (- x) (- y))|))
 (61 61 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (50 50 (:REWRITE |(+ c (+ d x))|))
 (50 39 (:REWRITE DEFAULT-<-2))
 (48 39 (:REWRITE SIMPLIFY-SUMS-<))
 (40 39 (:REWRITE DEFAULT-<-1))
 (38 38 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (33 33 (:REWRITE |(< 0 (- x))|))
 (27 27 (:REWRITE |(< d (+ c x))|))
 (27 26 (:REWRITE DEFAULT-UNARY-MINUS))
 (25 25 (:REWRITE FOLD-CONSTS-IN-+))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (4 4 (:REWRITE |(< (- x) 0)|))
 (3 3 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (3 2 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (3 2 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (3 2 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (2 2 (:REWRITE REDUCE-INTEGERP-+))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (2 2 (:REWRITE INTEGERP-MINUS-X))
 (2 2 (:REWRITE DEFAULT-UNARY-/))
 (2 2 (:REWRITE |(equal (- x) (- y))|))
 (2 2 (:REWRITE |(< (+ c x) d)|))
 (2 2 (:META META-INTEGERP-CORRECT))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (1 1 (:REWRITE |(equal (- x) 0)|))
 )
(TOTAL-PRE-TO-NO-TOTAL-EXTERNAL-TOTAL-RUN
 (21 3 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (13 3 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (12 2 (:DEFINITION TOTAL-RUN))
 (7 1 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (4 4 (:REWRITE SIMPLIFY-SUMS-<))
 (4 4 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (4 4 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (4 4 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (4 4 (:REWRITE |(< (- x) (- y))|))
 (3 3 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (3 3 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (3 3 (:REWRITE NORMALIZE-ADDENDS))
 (3 3 (:REWRITE DEFAULT-<-2))
 (3 3 (:REWRITE DEFAULT-<-1))
 (3 3 (:REWRITE DEFAULT-+-2))
 (3 3 (:REWRITE DEFAULT-+-1))
 (2 2 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (1 1 (:REWRITE DEFAULT-UNARY-MINUS))
 (1 1 (:REWRITE |(< 0 (- x))|))
 )
(TOTAL-INV-TO-NO-TOTAL-EXTERNAL
 (1561 26 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (1008 12 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (536 19 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (534 139 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (402 74 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (311 27 (:META META-INTEGERP-CORRECT))
 (270 57 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (231 12 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (185 185 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (139 139 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (139 139 (:REWRITE |(< (- x) (- y))|))
 (126 126 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (120 102 (:REWRITE DEFAULT-<-2))
 (110 98 (:REWRITE DEFAULT-+-2))
 (102 102 (:REWRITE DEFAULT-<-1))
 (98 98 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (98 98 (:REWRITE DEFAULT-+-1))
 (96 96 (:REWRITE |(< 0 (- x))|))
 (96 4 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (92 12 (:REWRITE REDUCE-INTEGERP-+))
 (84 84 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (47 21 (:REWRITE INTEGERP-MINUS-X))
 (44 44 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (40 20 (:REWRITE |(* y x)|))
 (27 27 (:REWRITE DEFAULT-UNARY-MINUS))
 (26 26 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (26 26 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (26 26 (:REWRITE |(< d (+ c x))|))
 (20 20 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (20 20 (:REWRITE DEFAULT-UNARY-/))
 (20 20 (:REWRITE |(* a (/ a))|))
 (20 2 (:REWRITE NINTP-2))
 (16 16 (:REWRITE FOLD-CONSTS-IN-+))
 (14 14 (:REWRITE |(< (+ c x) d)|))
 (14 3 (:REWRITE INTP-2))
 (12 12 (:REWRITE TOTAL-RUN-1-IS-TOTAL-NEXT))
 (12 12 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (12 12 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (12 12 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (12 12 (:REWRITE |(equal (- x) (- y))|))
 (12 12 (:REWRITE |(< d (+ c x y))|))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (10 10 (:REWRITE |(equal (- x) 0)|))
 (6 6 (:TYPE-PRESCRIPTION INTP-+))
 (6 2 (:REWRITE NINTP-1))
 (2 2 (:DEFINITION FIX))
 )
(NO-TOTAL-EXTERNAL-IMPLIES-+-2
 (406 61 (:REWRITE ZP-OPEN))
 (149 149 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (112 14 (:REWRITE |(< d (+ c x))|))
 (88 88 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (86 67 (:REWRITE DEFAULT-+-2))
 (67 67 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (67 67 (:REWRITE NORMALIZE-ADDENDS))
 (67 67 (:REWRITE DEFAULT-+-1))
 (48 48 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (48 48 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (45 31 (:REWRITE SIMPLIFY-SUMS-<))
 (45 31 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (45 31 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (45 31 (:REWRITE DEFAULT-<-2))
 (44 6 (:REWRITE |(< x (if a b c))|))
 (31 31 (:REWRITE DEFAULT-<-1))
 (31 31 (:REWRITE |(< (- x) (- y))|))
 (14 14 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (12 6 (:REWRITE |(+ x (if a b c))|))
 (11 11 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (10 10 (:REWRITE |(< 0 (- x))|))
 )
(NO-TOTAL-EXTERNAL-IMPLIES-+-2-CONCRETIZED
 (401 9 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (303 6 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (217 3 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (193 19 (:REWRITE ZP-OPEN))
 (190 10 (:DEFINITION TOTAL-RUN))
 (125 6 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (58 58 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (39 9 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (34 30 (:REWRITE NORMALIZE-ADDENDS))
 (34 28 (:REWRITE DEFAULT-+-2))
 (33 19 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (31 19 (:REWRITE SIMPLIFY-SUMS-<))
 (30 4 (:REWRITE |(< d (+ c x))|))
 (28 28 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (28 28 (:REWRITE DEFAULT-+-1))
 (27 9 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (25 19 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (22 22 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (20 20 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (20 20 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (19 19 (:REWRITE |(< (- x) (- y))|))
 (18 12 (:REWRITE DEFAULT-<-2))
 (15 3 (:REWRITE |(+ c (+ d x))|))
 (12 12 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (12 12 (:REWRITE DEFAULT-<-1))
 (12 12 (:REWRITE |(< 0 (- x))|))
 (9 2 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (6 6 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (6 6 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (4 2 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (3 3 (:REWRITE REDUCE-INTEGERP-+))
 (3 3 (:REWRITE INTEGERP-MINUS-X))
 (3 3 (:META META-INTEGERP-CORRECT))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 (2 2 (:REWRITE |(+ x (- x))|))
 (1 1 (:REWRITE TOTAL-CLOCK-FN-IS-NATP))
 )
(TOTAL-PRE-IMPLIES-TOTAL-CLOCK-NATP
 (66 9 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (61 9 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (48 8 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (11 6 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (8 8 (:REWRITE |(< (- x) (- y))|))
 (7 7 (:REWRITE SIMPLIFY-SUMS-<))
 (7 7 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (7 7 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (7 7 (:REWRITE DEFAULT-<-2))
 (7 7 (:REWRITE DEFAULT-<-1))
 (6 6 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (6 6 (:REWRITE |(< 0 (- x))|))
 (5 5 (:DEFINITION TOTAL-RUN))
 (4 2 (:REWRITE |(* y x)|))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (2 2 (:REWRITE DEFAULT-UNARY-/))
 (2 2 (:REWRITE |(* a (/ a))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:REWRITE |(equal (- x) 0)|))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 (1 1 (:REWRITE |(< (- x) 0)|))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(FIND-TOTAL-EXTERNAL-FOR-STATE
 (10556 2462 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (8130 549 (:REWRITE REDUCE-INTEGERP-+))
 (6068 82 (:DEFINITION FIND-TOTAL-EXTERNAL))
 (5686 229 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (2777 2465 (:REWRITE DEFAULT-+-2))
 (2676 2676 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (2615 2615 (:REWRITE |(< (- x) (- y))|))
 (2465 2465 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2465 2465 (:REWRITE DEFAULT-+-1))
 (2452 879 (:META META-INTEGERP-CORRECT))
 (2060 1907 (:REWRITE DEFAULT-<-2))
 (1953 1953 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (1937 848 (:REWRITE INTEGERP-MINUS-X))
 (1907 1907 (:REWRITE DEFAULT-<-1))
 (1805 1805 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (1805 1805 (:REWRITE |(< 0 (- x))|))
 (1804 1804 (:REWRITE DEFAULT-UNARY-MINUS))
 (1733 1080 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (1382 130 (:REWRITE |(< (- x) 0)|))
 (1229 1229 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (1130 1130 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (996 230 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (724 358 (:REWRITE |(< (+ c x) d)|))
 (687 687 (:REWRITE FOLD-CONSTS-IN-+))
 (480 319 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (469 469 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (409 254 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (405 405 (:REWRITE |(< d (+ c x))|))
 (319 319 (:REWRITE |(equal (- x) (- y))|))
 (316 237 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (312 312 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (312 312 (:REWRITE DEFAULT-UNARY-/))
 (256 98 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (255 232 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (161 161 (:REWRITE |(equal (+ c x) d)|))
 (121 121 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (121 121 (:REWRITE |(equal (- x) 0)|))
 (82 82 (:REWRITE |(< (+ d x) (+ c y))|))
 (82 82 (:REWRITE |(< (+ c x) (+ d y))|))
 (77 77 (:REWRITE |(< d (+ c x y))|))
 (73 73 (:REWRITE |(< (+ c x y) d)|))
 (2 2 (:REWRITE TOTAL-RUN-1-IS-TOTAL-NEXT))
 )
(NATP-TO-TOTAL-RUN-S-N
 (26 4 (:REWRITE ZP-OPEN))
 (10 2 (:REWRITE |(+ c (+ d x))|))
 (9 2 (:REWRITE |(< d (+ c x))|))
 (6 6 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (6 6 (:REWRITE NORMALIZE-ADDENDS))
 (6 6 (:REWRITE DEFAULT-+-2))
 (6 6 (:REWRITE DEFAULT-+-1))
 (3 3 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (3 3 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (2 2 (:REWRITE SIMPLIFY-SUMS-<))
 (2 2 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 (2 2 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 (1 1 (:REWRITE |(< 0 (- x))|))
 )
(TOTAL-INV-STATES-HAVE-TOTAL-EXTERNAL-TOTAL-NEXT
 (298 6 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (242 12 (:REWRITE ZP-OPEN))
 (228 4 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (198 6 (:DEFINITION TOTAL-RUN))
 (172 2 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (112 4 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (70 10 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (58 10 (:REWRITE REDUCE-INTEGERP-+))
 (40 40 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (26 22 (:REWRITE DEFAULT-+-2))
 (23 13 (:REWRITE |(+ c (+ d x))|))
 (22 22 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (22 22 (:REWRITE NORMALIZE-ADDENDS))
 (22 22 (:REWRITE DEFAULT-+-1))
 (21 17 (:REWRITE SIMPLIFY-SUMS-<))
 (21 17 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (21 17 (:REWRITE DEFAULT-<-2))
 (20 12 (:REWRITE INTEGERP-MINUS-X))
 (17 17 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (17 17 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (17 17 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (17 17 (:REWRITE DEFAULT-<-1))
 (17 17 (:REWRITE |(< 0 (- x))|))
 (17 17 (:REWRITE |(< (- x) (- y))|))
 (15 15 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (14 14 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-STATE))
 (12 12 (:META META-INTEGERP-CORRECT))
 (11 11 (:REWRITE FOLD-CONSTS-IN-+))
 (10 10 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (8 8 (:REWRITE |(< d (+ c x))|))
 (7 7 (:REWRITE DEFAULT-UNARY-MINUS))
 (6 6 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (6 2 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (5 5 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (5 5 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (4 4 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (4 4 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (2 2 (:REWRITE |(< d (+ c x y))|))
 (2 2 (:REWRITE |(- (- x))|))
 )
(EXISTS-TOTAL-PRE-STATE-TO-WITNESS
 (25 4 (:REWRITE REDUCE-INTEGERP-+))
 (22 16 (:REWRITE NORMALIZE-ADDENDS))
 (19 3 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (17 1 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (15 15 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (15 15 (:REWRITE DEFAULT-+-2))
 (15 15 (:REWRITE DEFAULT-+-1))
 (15 1 (:REWRITE |(+ (+ x y) z)|))
 (12 3 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (10 10 (:REWRITE SIMPLIFY-SUMS-<))
 (10 10 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (10 10 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (10 10 (:REWRITE |(< (- x) (- y))|))
 (9 9 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (9 9 (:REWRITE DEFAULT-<-2))
 (9 9 (:REWRITE DEFAULT-<-1))
 (8 8 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (8 8 (:REWRITE |(< 0 (- x))|))
 (7 1 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (4 4 (:REWRITE INTEGERP-MINUS-X))
 (4 4 (:REWRITE |(+ c (+ d x))|))
 (4 4 (:META META-INTEGERP-CORRECT))
 (3 3 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (3 3 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (3 3 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (3 3 (:REWRITE FOLD-CONSTS-IN-+))
 (3 3 (:REWRITE |(equal (- x) (- y))|))
 (3 3 (:REWRITE |(< d (+ c x))|))
 (2 2 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (1 1 (:REWRITE DEFAULT-UNARY-MINUS))
 (1 1 (:REWRITE |(+ x (- x))|))
 (1 1 (:REWRITE |(+ 0 x)|))
 )
(EXISTS-TOTAL-PRE-STATE-TO-WITNESS-2
 (302 6 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (202 4 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (196 2 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (190 6 (:REWRITE ZP-OPEN))
 (186 4 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (103 19 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (94 15 (:META META-INTEGERP-CORRECT))
 (82 19 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (57 27 (:REWRITE |(+ c (+ d x))|))
 (31 27 (:REWRITE DEFAULT-+-2))
 (27 27 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (27 27 (:REWRITE NORMALIZE-ADDENDS))
 (27 27 (:REWRITE DEFAULT-+-1))
 (24 12 (:REWRITE INTEGERP-MINUS-X))
 (21 6 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (16 16 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (16 16 (:REWRITE SIMPLIFY-SUMS-<))
 (16 16 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (16 16 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (16 16 (:REWRITE DEFAULT-<-2))
 (16 16 (:REWRITE DEFAULT-<-1))
 (16 16 (:REWRITE |(< (- x) (- y))|))
 (15 15 (:REWRITE FOLD-CONSTS-IN-+))
 (14 14 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-STATE))
 (13 13 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (13 13 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (13 13 (:REWRITE |(< 0 (- x))|))
 (12 12 (:REWRITE DEFAULT-UNARY-MINUS))
 (9 9 (:REWRITE REDUCE-INTEGERP-+))
 (7 7 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (7 7 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (6 6 (:REWRITE |(< d (+ c x))|))
 (6 6 (:REWRITE |(< d (+ c x y))|))
 (4 4 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (4 4 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (4 4 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (4 4 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (3 3 (:REWRITE |(< (+ c x) d)|))
 (3 3 (:REWRITE |(< (+ c x y) d)|))
 (3 3 (:REWRITE |(- (- x))|))
 (3 3 (:REWRITE |(+ 0 x)|))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 )
(TOTAL-CLOCK-FN-IS-LESS-1
 (1159 184 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (957 15 (:REWRITE ZP-OPEN))
 (823 184 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (731 10 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (716 5 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (691 10 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (192 99 (:REWRITE REDUCE-INTEGERP-+))
 (157 82 (:REWRITE |(+ c (+ d x))|))
 (154 154 (:REWRITE SIMPLIFY-SUMS-<))
 (154 154 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (154 154 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (154 154 (:REWRITE |(< (- x) (- y))|))
 (138 138 (:REWRITE DEFAULT-<-2))
 (138 138 (:REWRITE DEFAULT-<-1))
 (100 100 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (94 94 (:REWRITE DEFAULT-UNARY-MINUS))
 (94 84 (:REWRITE DEFAULT-+-2))
 (84 84 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (84 84 (:REWRITE NORMALIZE-ADDENDS))
 (84 84 (:REWRITE DEFAULT-+-1))
 (76 76 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (76 76 (:REWRITE |(< 0 (- x))|))
 (61 61 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (44 44 (:REWRITE FOLD-CONSTS-IN-+))
 (36 36 (:REWRITE |(- (- x))|))
 (33 33 (:REWRITE |(< (+ c x) d)|))
 (24 6 (:REWRITE NINTP-2))
 (15 15 (:REWRITE |(< d (+ c x))|))
 (15 15 (:REWRITE |(< d (+ c x y))|))
 (13 13 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (13 13 (:REWRITE |(< (- x) 0)|))
 (12 12 (:REWRITE |(< (+ c x y) d)|))
 (10 10 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (10 10 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (8 8 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (6 6 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (6 6 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (6 6 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (6 6 (:REWRITE NINTP-1))
 (6 6 (:REWRITE |(equal (- x) (- y))|))
 )
(NATP-TO-TOTAL-RUN-S-N-2
 (12 2 (:DEFINITION TOTAL-RUN))
 (3 3 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (3 3 (:REWRITE NORMALIZE-ADDENDS))
 (3 3 (:REWRITE DEFAULT-+-2))
 (3 3 (:REWRITE DEFAULT-+-1))
 (2 2 (:REWRITE ZP-OPEN))
 )
(TOTAL-INV-STATES-HAVE-TOTAL-EXTERNAL-TOTAL-PREVIOUS
 (510 66 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (258 66 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (233 89 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (233 83 (:REWRITE SIMPLIFY-SUMS-<))
 (131 3 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (129 24 (:REWRITE |(< (- x) 0)|))
 (119 7 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (101 101 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (100 100 (:REWRITE |(< (- x) (- y))|))
 (94 14 (:REWRITE REDUCE-INTEGERP-+))
 (92 2 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (78 52 (:REWRITE NORMALIZE-ADDENDS))
 (74 74 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (71 69 (:REWRITE DEFAULT-<-2))
 (70 20 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (69 69 (:REWRITE DEFAULT-<-1))
 (65 1 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (56 56 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (56 56 (:REWRITE |(< 0 (- x))|))
 (45 4 (:REWRITE NATP-TO-TOTAL-RUN-S-N-2))
 (42 6 (:REWRITE |(+ y (+ x z))|))
 (41 39 (:REWRITE DEFAULT-+-2))
 (39 39 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (39 39 (:REWRITE DEFAULT-+-1))
 (38 2 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (29 17 (:REWRITE INTEGERP-MINUS-X))
 (28 28 (:REWRITE DEFAULT-UNARY-MINUS))
 (26 13 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (25 2 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (24 24 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (24 12 (:REWRITE |(* y x)|))
 (21 11 (:REWRITE |(< d (+ c x))|))
 (17 17 (:META META-INTEGERP-CORRECT))
 (16 16 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (14 14 (:REWRITE |(- (- x))|))
 (13 13 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (13 13 (:REWRITE |(+ x (- x))|))
 (12 12 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (12 12 (:REWRITE DEFAULT-UNARY-/))
 (12 12 (:REWRITE |(* a (/ a))|))
 (10 10 (:REWRITE FOLD-CONSTS-IN-+))
 (10 10 (:REWRITE |(< (+ c x) d)|))
 (9 9 (:TYPE-PRESCRIPTION NATP))
 (7 7 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-STATE))
 (7 7 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (7 7 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (7 7 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (7 7 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (7 7 (:REWRITE |(equal (- x) (- y))|))
 (6 1 (:REWRITE |(+ (+ x y) z)|))
 (5 5 (:REWRITE |(< d (+ c x y))|))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (2 2 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (2 2 (:REWRITE |(equal (- x) 0)|))
 (1 1 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 )
(EXISTS-TOTAL-PRE-STATE-TO-WITNESS-3
 (593 21 (:REWRITE ZP-OPEN))
 (544 86 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (422 86 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (376 54 (:META META-INTEGERP-CORRECT))
 (126 18 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (103 85 (:REWRITE DEFAULT-+-2))
 (88 48 (:REWRITE INTEGERP-MINUS-X))
 (85 85 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (85 85 (:REWRITE NORMALIZE-ADDENDS))
 (85 85 (:REWRITE DEFAULT-+-1))
 (84 63 (:REWRITE |(+ c (+ d x))|))
 (77 77 (:REWRITE SIMPLIFY-SUMS-<))
 (77 77 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (77 77 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (77 77 (:REWRITE DEFAULT-<-2))
 (77 77 (:REWRITE DEFAULT-<-1))
 (77 77 (:REWRITE |(< (- x) (- y))|))
 (55 55 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (55 55 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (55 55 (:REWRITE |(< 0 (- x))|))
 (49 49 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (46 46 (:REWRITE DEFAULT-UNARY-MINUS))
 (39 39 (:REWRITE FOLD-CONSTS-IN-+))
 (38 38 (:REWRITE REDUCE-INTEGERP-+))
 (28 28 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (28 28 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (27 27 (:REWRITE |(< d (+ c x))|))
 (22 22 (:REWRITE |(< (+ c x) d)|))
 (19 19 (:REWRITE |(< d (+ c x y))|))
 (18 18 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (18 18 (:REWRITE |(< (+ c x y) d)|))
 (18 6 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (14 14 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (13 13 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (12 12 (:TYPE-PRESCRIPTION NATP))
 (12 12 (:REWRITE |(+ 0 x)|))
 (10 10 (:REWRITE |(- (- x))|))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 )
(TOTAL-CLOCK-FN-IS-LESS-2
 (1366 10 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO-NO-TOTAL-EXTERNAL))
 (693 8 (:REWRITE NO-TOTAL-EXTERNAL-WITNESS-IMPLIES-NO-TOTAL-EXTERNAL))
 (683 4 (:DEFINITION NO-TOTAL-EXTERNAL-STATE))
 (661 10 (:DEFINITION NO-TOTAL-EXTERNAL-WITNESS))
 (568 82 (:META META-INTEGERP-CORRECT))
 (458 14 (:REWRITE ZP-OPEN))
 (410 82 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (394 82 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (231 18 (:REWRITE INTEGERP-+-REDUCE-LEADING-RATIONAL-CONSTANT))
 (172 8 (:REWRITE TOTAL-PRE-IMPLIES-NO-TOTAL-EXTERNAL))
 (148 76 (:REWRITE INTEGERP-MINUS-X))
 (97 2 (:REWRITE TOTAL-RUN-+-REDUCTION))
 (74 62 (:REWRITE DEFAULT-+-2))
 (72 72 (:REWRITE SIMPLIFY-SUMS-<))
 (72 72 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (72 72 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (72 72 (:REWRITE |(< (- x) (- y))|))
 (71 71 (:REWRITE DEFAULT-<-2))
 (71 71 (:REWRITE DEFAULT-<-1))
 (65 51 (:REWRITE |(+ c (+ d x))|))
 (62 62 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (62 62 (:REWRITE NORMALIZE-ADDENDS))
 (62 62 (:REWRITE DEFAULT-+-1))
 (58 58 (:REWRITE REDUCE-INTEGERP-+))
 (55 55 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (51 51 (:REWRITE DEFAULT-UNARY-MINUS))
 (48 48 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (48 48 (:REWRITE |(< 0 (- x))|))
 (32 32 (:REWRITE FOLD-CONSTS-IN-+))
 (30 30 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (29 29 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (22 22 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-STATE))
 (18 18 (:REWRITE |(< d (+ c x))|))
 (18 18 (:REWRITE |(< d (+ c x y))|))
 (18 18 (:REWRITE |(- (- x))|))
 (12 12 (:TYPE-PRESCRIPTION NO-TOTAL-EXTERNAL-WITNESS))
 (10 10 (:REWRITE |(+ 0 x)|))
 (9 9 (:REWRITE |(< (+ c x) d)|))
 (9 9 (:REWRITE |(< (+ c x y) d)|))
 (8 8 (:REWRITE NO-TOTAL-EXTERNAL-STATE-TO<=))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (5 5 (:REWRITE |(< (- x) 0)|))
 (4 4 (:TYPE-PRESCRIPTION NATP))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (1 1 (:REWRITE |(equal (- x) (- y))|))
 )
(TOTAL-CLOCK-FN-IS-SAME
 (84 12 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (20 20 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (20 20 (:REWRITE DEFAULT-+-2))
 (20 20 (:REWRITE DEFAULT-+-1))
 (12 12 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (9 9 (:REWRITE FOLD-CONSTS-IN-+))
 (8 8 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (8 8 (:REWRITE SIMPLIFY-SUMS-<))
 (8 8 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (8 8 (:REWRITE DEFAULT-UNARY-MINUS))
 (8 8 (:REWRITE DEFAULT-<-2))
 (8 8 (:REWRITE DEFAULT-<-1))
 (8 8 (:REWRITE |(< (- x) (- y))|))
 (6 6 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (6 6 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (6 6 (:REWRITE |(< 0 (- x))|))
 (4 4 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (4 4 (:REWRITE |(equal (- x) (- y))|))
 (4 4 (:REWRITE |(equal (+ d x) (+ c y))|))
 (4 4 (:REWRITE |(equal (+ c x) d)|))
 (4 4 (:REWRITE |(equal (+ c x) (+ d y))|))
 (4 4 (:REWRITE |(equal (+ c x y) d)|))
 (4 4 (:REWRITE |(- (- x))|))
 (3 3 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (2 2 (:REWRITE |(< (+ c x) d)|))
 (2 2 (:REWRITE |(< (+ c x y) d)|))
 (1 1 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 )
(M-IS-A-NATP)
(TOTAL-INV-IMPLIES-M-TOTAL-NEXT
 (4 4 (:REWRITE TOTAL-PRE-HAS-TOTAL-INV))
 (3 3 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 )
(INTERNAL-STEPS-DECREASE-M
 (50 9 (:REWRITE TOTAL-CLOCK-FN-IS-THE-SMALLEST))
 (45 9 (:REWRITE TOTAL-CLOCK-FN-IS-EQUAL-IF-<))
 (11 11 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (11 11 (:REWRITE DEFAULT-+-2))
 (11 11 (:REWRITE DEFAULT-+-1))
 (8 8 (:REWRITE DEFAULT-UNARY-MINUS))
 (7 7 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (7 7 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (7 7 (:REWRITE |(< (- x) (- y))|))
 (5 5 (:REWRITE REMOVE-WEAK-INEQUALITIES-ONE))
 (5 5 (:REWRITE DEFAULT-<-2))
 (5 5 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE |(< (+ c x) d)|))
 (3 3 (:REWRITE TOTAL-PRE-HAS-TOTAL-CLOCK->0))
 (3 3 (:REWRITE STANDARD-THEOREM-FOR-TOTAL-CLOCKS-3))
 (3 3 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (3 3 (:REWRITE |(< 0 (- x))|))
 (2 2 (:REWRITE FOLD-CONSTS-IN-+))
 (2 2 (:REWRITE |(< d (+ c x))|))
 (2 2 (:REWRITE |(< (+ d x) (+ c y))|))
 (2 2 (:REWRITE |(< (+ c x) (+ d y))|))
 (2 2 (:REWRITE |(< (+ c x y) d)|))
 (2 2 (:REWRITE |(- (- x))|))
 (2 2 (:REWRITE |(+ 0 x)|))
 (1 1 (:REWRITE TOTAL-PRE-IS-NOT-TOTAL-EXTERNAL))
 (1 1 (:REWRITE TOTAL-PRE-HAS-TOTAL-INV))
 )
