* interp/g-opt.boot (simplifyVMForm): New.

	(optRetract): Simplify the predicate when possible.
	(optNot): New transformer.
	(optAnd): Likewise.
	(optOr): Likewise.
	(optIlt): Likewise.
	(optIle): Likewise.
	(optIgt): Likewise.
	(optIge): Likewise.

22 files changed, 203 insertions, 109 deletions

diff --git a/src/ChangeLog b/src/ChangeLog
index 953e5257..accd5363 100644
--- a/src/ChangeLog
+++ b/src/ChangeLog
@@ -1,3 +1,15 @@
+2010-07-22  Gabriel Dos Reis  <gdr@cs.tamu.edu>
+
+	* interp/g-opt.boot (simplifyVMForm): New.
+	(optRetract): Simplify the predicate when possible.
+	(optNot): New transformer.
+	(optAnd): Likewise.
+	(optOr): Likewise.
+	(optIlt): Likewise.
+	(optIle): Likewise.
+	(optIgt): Likewise.
+	(optIge): Likewise.
+
 2010-07-21  Gabriel Dos Reis  <gdr@cse.tamu.edu>
 
 	* algebra/data.spad.pamphlet (Byte) [~]: Tidy.
diff --git a/src/algebra/strap/ABELGRP-.lsp b/src/algebra/strap/ABELGRP-.lsp
index 47493f92..d436441d 100644
--- a/src/algebra/strap/ABELGRP-.lsp
+++ b/src/algebra/strap/ABELGRP-.lsp
@@ -27,7 +27,7 @@
 (DEFUN |ABELGRP-;*;I2S;4| (|n| |x| $)
   (COND
     ((ZEROP |n|) (|spadConstant| $ 19))
-    ((< 0 |n|) (SPADCALL |n| |x| (|getShellEntry| $ 24)))
+    ((PLUSP |n|) (SPADCALL |n| |x| (|getShellEntry| $ 24)))
     ('T
      (SPADCALL (- |n|) (SPADCALL |x| (|getShellEntry| $ 7))
          (|getShellEntry| $ 24))))) 
diff --git a/src/algebra/strap/CHAR.lsp b/src/algebra/strap/CHAR.lsp
index ed6ac052..869562e9 100644
--- a/src/algebra/strap/CHAR.lsp
+++ b/src/algebra/strap/CHAR.lsp
@@ -123,13 +123,13 @@
 (DEFUN |CHAR;size;Nni;6| ($) (DECLARE (IGNORE $)) 256) 
 
 (DEFUN |CHAR;index;Pi$;7| (|n| $)
-  (CODE-CHAR
-      (LET ((#0=#:G1405 (- |n| 1)))
-        (|check-subtype| (>= #0# 0) '(|NonNegativeInteger|) #0#)))) 
+  (CODE-CHAR (LET ((#0=#:G1405 (- |n| 1)))
+               (|check-subtype| (NOT (MINUSP #0#))
+                   '(|NonNegativeInteger|) #0#)))) 
 
 (DEFUN |CHAR;lookup;$Pi;8| (|c| $)
   (LET ((#0=#:G1407 (+ 1 (CHAR-CODE |c|))))
-    (|check-subtype| (< 0 #0#) '(|PositiveInteger|) #0#))) 
+    (|check-subtype| (PLUSP #0#) '(|PositiveInteger|) #0#))) 
 
 (DEFUN |CHAR;char;Nni$;9| (|n| $)
   (DECLARE (IGNORE $))
diff --git a/src/algebra/strap/DFLOAT.lsp b/src/algebra/strap/DFLOAT.lsp
index 596307d8..6f32c302 100644
--- a/src/algebra/strap/DFLOAT.lsp
+++ b/src/algebra/strap/DFLOAT.lsp
@@ -446,8 +446,8 @@
                        (|DFLOAT;log2;2$;40|
                            (FLOAT 2 |$DoubleFloatMaximum|) $)
                        (|getShellEntry| $ 32)))))
-       (|check-subtype| (AND (>= #0# 0) (< 0 #0#)) '(|PositiveInteger|)
-           #0#))))) 
+       (|check-subtype| (AND (NOT (MINUSP #0#)) (PLUSP #0#))
+           '(|PositiveInteger|) #0#))))) 
 
 (DEFUN |DFLOAT;max;$;11| ($)
   (DECLARE (IGNORE $))
@@ -636,14 +636,16 @@
 (DEFUN |DFLOAT;retract;$F;80| (|x| $)
   (|DFLOAT;rationalApproximation;$2NniF;87| |x|
       (LET ((#0=#:G1506 (- 53 1)))
-        (|check-subtype| (>= #0# 0) '(|NonNegativeInteger|) #0#))
+        (|check-subtype| (NOT (MINUSP #0#)) '(|NonNegativeInteger|)
+            #0#))
       2 $)) 
 
 (DEFUN |DFLOAT;retractIfCan;$U;81| (|x| $)
   (CONS 0
         (|DFLOAT;rationalApproximation;$2NniF;87| |x|
             (LET ((#0=#:G1514 (- 53 1)))
-              (|check-subtype| (>= #0# 0) '(|NonNegativeInteger|) #0#))
+              (|check-subtype| (NOT (MINUSP #0#))
+                  '(|NonNegativeInteger|) #0#))
             2 $))) 
 
 (DEFUN |DFLOAT;retract;$I;82| (|x| $)
@@ -706,18 +708,19 @@
         (SEQ |#G109|
              (LETT BASE 2 |DFLOAT;rationalApproximation;$2NniF;87|)
              (EXIT (COND
-                     ((>= |ex| 0)
+                     ((NOT (MINUSP |ex|))
                       (SPADCALL
                           (* |nu|
                              (EXPT BASE
-                                   (|check-subtype| (>= |ex| 0)
+                                   (|check-subtype| (NOT (MINUSP |ex|))
                                     '(|NonNegativeInteger|) |ex|)))
                           (|getShellEntry| $ 134)))
                      ('T
                       (SEQ (LETT |de|
                                  (EXPT BASE
                                        (LET ((#0=#:G1542 (- |ex|)))
-                                         (|check-subtype| (>= #0# 0)
+                                         (|check-subtype|
+                                          (NOT (MINUSP #0#))
                                           '(|NonNegativeInteger|) #0#)))
                                  |DFLOAT;rationalApproximation;$2NniF;87|)
                            (EXIT (COND
diff --git a/src/algebra/strap/FFIELDC-.lsp b/src/algebra/strap/FFIELDC-.lsp
index 9ad8712c..1c17371d 100644
--- a/src/algebra/strap/FFIELDC-.lsp
+++ b/src/algebra/strap/FFIELDC-.lsp
@@ -104,7 +104,8 @@
                    (T (SEQ (LETT |e|
                                  (SPADCALL
                                      (|check-subtype|
-                                      (AND (>= |i| 0) (< 0 |i|))
+                                      (AND (NOT (MINUSP |i|))
+                                       (PLUSP |i|))
                                       '(|PositiveInteger|) |i|)
                                      (|getShellEntry| $ 14))
                                  |FFIELDC-;createPrimitiveElement;S;8|)
diff --git a/src/algebra/strap/FPS-.lsp b/src/algebra/strap/FPS-.lsp
index 562d295c..cee87448 100644
--- a/src/algebra/strap/FPS-.lsp
+++ b/src/algebra/strap/FPS-.lsp
@@ -19,8 +19,8 @@
                          (- (SPADCALL (|getShellEntry| $ 14)) 1)
                          (|getShellEntry| $ 16))
                      13301))))
-    (|check-subtype| (AND (>= #0# 0) (< 0 #0#)) '(|PositiveInteger|)
-        #0#))) 
+    (|check-subtype| (AND (NOT (MINUSP #0#)) (PLUSP #0#))
+        '(|PositiveInteger|) #0#))) 
 
 (DEFUN |FloatingPointSystem&| (|#1|)
   (LET* ((|dv$1| (|devaluate| |#1|))
diff --git a/src/algebra/strap/GCDDOM-.lsp b/src/algebra/strap/GCDDOM-.lsp
index 92b58126..ff12789d 100644
--- a/src/algebra/strap/GCDDOM-.lsp
+++ b/src/algebra/strap/GCDDOM-.lsp
@@ -76,7 +76,7 @@
                               (SPADCALL |p1| (|getShellEntry| $ 29))
                               |GCDDOM-;gcdPolynomial;3Sup;4|)
                         (EXIT (COND
-                                ((< 0 |e1|)
+                                ((PLUSP |e1|)
                                  (SETQ |p1|
                                        (LET
                                         ((#0#
@@ -95,7 +95,7 @@
                               (SPADCALL |p2| (|getShellEntry| $ 29))
                               |GCDDOM-;gcdPolynomial;3Sup;4|)
                         (EXIT (COND
-                                ((< 0 |e2|)
+                                ((PLUSP |e2|)
                                  (SETQ |p2|
                                        (LET
                                         ((#0#
diff --git a/src/algebra/strap/ILIST.lsp b/src/algebra/strap/ILIST.lsp
index 8ba9493f..b9fcc572 100644
--- a/src/algebra/strap/ILIST.lsp
+++ b/src/algebra/strap/ILIST.lsp
@@ -387,7 +387,7 @@
               (SEQ (SETQ |p|
                          (|ILIST;rest;$Nni$;19| |p|
                              (LET ((#0=#:G1506 (- |n| 1)))
-                               (|check-subtype| (>= #0# 0)
+                               (|check-subtype| (NOT (MINUSP #0#))
                                    '(|NonNegativeInteger|) #0#))
                              $))
                    (LETT |q| (CDR |p|) |ILIST;split!;$I$;29|)
@@ -407,7 +407,7 @@
                    ('T
                     (SEQ (LETT |l|
                                (LET ((#0=#:G1511 (QUOTIENT2 |n| 2)))
-                                 (|check-subtype| (>= #0# 0)
+                                 (|check-subtype| (NOT (MINUSP #0#))
                                      '(|NonNegativeInteger|) #0#))
                                |ILIST;mergeSort|)
                          (LETT |q| (|ILIST;split!;$I$;29| |p| |l| $)
diff --git a/src/algebra/strap/INS-.lsp b/src/algebra/strap/INS-.lsp
index 794d4374..6df95b2e 100644
--- a/src/algebra/strap/INS-.lsp
+++ b/src/algebra/strap/INS-.lsp
@@ -125,10 +125,10 @@
      (|error| "euclideanSize called on zero"))
     ((SPADCALL |x| (|spadConstant| $ 10) (|getShellEntry| $ 28))
      (LET ((#0=#:G1426 (- (SPADCALL |x| (|getShellEntry| $ 30)))))
-       (|check-subtype| (>= #0# 0) '(|NonNegativeInteger|) #0#)))
+       (|check-subtype| (NOT (MINUSP #0#)) '(|NonNegativeInteger|) #0#)))
     ('T
      (LET ((#1=#:G1427 (SPADCALL |x| (|getShellEntry| $ 30))))
-       (|check-subtype| (>= #1# 0) '(|NonNegativeInteger|) #1#))))) 
+       (|check-subtype| (NOT (MINUSP #1#)) '(|NonNegativeInteger|) #1#))))) 
 
 (DEFUN |INS-;convert;SF;10| (|x| $)
   (SPADCALL (SPADCALL |x| (|getShellEntry| $ 30))
diff --git a/src/algebra/strap/INT.lsp b/src/algebra/strap/INT.lsp
index c489ab5c..28fee72a 100644
--- a/src/algebra/strap/INT.lsp
+++ b/src/algebra/strap/INT.lsp
@@ -148,17 +148,19 @@
 (DECLAIM (FTYPE (FUNCTION (|%Integer| |%Integer| |%Shell|) |%Boolean|)
                 |INT;>;2$B;36|)) 
 
-(PUT '|INT;>;2$B;36| '|SPADreplace| '|%igt|) 
+(PUT '|INT;>;2$B;36| '|SPADreplace| '(XLAM (|x| |y|) (|%ilt| |y| |x|))) 
 
 (DECLAIM (FTYPE (FUNCTION (|%Integer| |%Integer| |%Shell|) |%Boolean|)
                 |INT;<=;2$B;37|)) 
 
-(PUT '|INT;<=;2$B;37| '|SPADreplace| '|%ile|) 
+(PUT '|INT;<=;2$B;37| '|SPADreplace|
+     '(XLAM (|x| |y|) (|%not| (|%ilt| |y| |x|)))) 
 
 (DECLAIM (FTYPE (FUNCTION (|%Integer| |%Integer| |%Shell|) |%Boolean|)
                 |INT;>=;2$B;38|)) 
 
-(PUT '|INT;>=;2$B;38| '|SPADreplace| '|%ige|) 
+(PUT '|INT;>=;2$B;38| '|SPADreplace|
+     '(XLAM (|x| |y|) (|%not| (|%ilt| |x| |y|)))) 
 
 (DECLAIM (FTYPE (FUNCTION (|%Integer| |%Shell|) |%Integer|)
                 |INT;-;2$;39|)) 
@@ -338,7 +340,8 @@
   (INTEGER-LENGTH |a|)) 
 
 (DEFUN |INT;addmod;4$;20| (|a| |b| |p| $)
-  (LET ((|c| (+ |a| |b|))) (COND ((>= |c| |p|) (- |c| |p|)) ('T |c|)))) 
+  (LET ((|c| (+ |a| |b|)))
+    (COND ((NOT (< |c| |p|)) (- |c| |p|)) ('T |c|)))) 
 
 (DEFUN |INT;submod;4$;21| (|a| |b| |p| $)
   (LET ((|c| (- |a| |b|))) (COND ((MINUSP |c|) (+ |c| |p|)) ('T |c|)))) 
@@ -392,9 +395,13 @@
 
 (DEFUN |INT;>;2$B;36| (|x| |y| $) (DECLARE (IGNORE $)) (< |y| |x|)) 
 
-(DEFUN |INT;<=;2$B;37| (|x| |y| $) (DECLARE (IGNORE $)) (<= |x| |y|)) 
+(DEFUN |INT;<=;2$B;37| (|x| |y| $)
+  (DECLARE (IGNORE $))
+  (NOT (< |y| |x|))) 
 
-(DEFUN |INT;>=;2$B;38| (|x| |y| $) (DECLARE (IGNORE $)) (>= |x| |y|)) 
+(DEFUN |INT;>=;2$B;38| (|x| |y| $)
+  (DECLARE (IGNORE $))
+  (NOT (< |x| |y|))) 
 
 (DEFUN |INT;-;2$;39| (|x| $) (DECLARE (IGNORE $)) (- |x|)) 
 
@@ -491,7 +498,7 @@
 
 (DEFUN |Integer| ()
   (DECLARE (SPECIAL |$ConstructorCache|))
-  (PROG (#0=#:G1525)
+  (PROG (#0=#:G1527)
     (RETURN
       (COND
         ((SETQ #0# (HGET |$ConstructorCache| '|Integer|))
diff --git a/src/algebra/strap/ISTRING.lsp b/src/algebra/strap/ISTRING.lsp
index 55b813f7..e8a3de19 100644
--- a/src/algebra/strap/ISTRING.lsp
+++ b/src/algebra/strap/ISTRING.lsp
@@ -191,13 +191,14 @@
                      (- (SPADCALL |s| (|getShellEntry| $ 47))
                         (|getShellEntry| $ 6))))))
         (SEQ (COND
-               ((OR (OR (MINUSP |l|) (>= |h| |m|)) (< |h| (- |l| 1)))
+               ((OR (OR (MINUSP |l|) (NOT (< |h| |m|)))
+                    (< |h| (- |l| 1)))
                 (EXIT (|error| "index out of range"))))
              (LETT |r|
                    (MAKE-FULL-CVEC
                        (LET ((#0=#:G1444
                                  (+ (- |m| (+ (- |h| |l|) 1)) |n|)))
-                         (|check-subtype| (>= #0# 0)
+                         (|check-subtype| (NOT (MINUSP #0#))
                              '(|NonNegativeInteger|) #0#))
                        (|spadConstant| $ 53))
                    |ISTRING;replace;$Us2$;15|)
@@ -264,7 +265,7 @@
            (EXIT (COND
                    ((MINUSP |startpos|)
                     (|error| "index out of bounds"))
-                   ((>= |startpos| (QCSIZE |t|))
+                   ((NOT (< |startpos| (QCSIZE |t|)))
                     (- (|getShellEntry| $ 6) 1))
                    ('T
                     (SEQ (LETT |r| (STRPOS |s| |t| |startpos| NIL)
@@ -278,7 +279,7 @@
   (SEQ (SETQ |startpos| (- |startpos| (|getShellEntry| $ 6)))
        (EXIT (COND
                ((MINUSP |startpos|) (|error| "index out of bounds"))
-               ((>= |startpos| (QCSIZE |t|))
+               ((NOT (< |startpos| (QCSIZE |t|)))
                 (- (|getShellEntry| $ 6) 1))
                ('T
                 (SEQ (LET ((|r| |startpos|)
@@ -298,7 +299,7 @@
   (SEQ (SETQ |startpos| (- |startpos| (|getShellEntry| $ 6)))
        (EXIT (COND
                ((MINUSP |startpos|) (|error| "index out of bounds"))
-               ((>= |startpos| (QCSIZE |t|))
+               ((NOT (< |startpos| (QCSIZE |t|)))
                 (- (|getShellEntry| $ 6) 1))
                ('T
                 (SEQ (LET ((|r| |startpos|)
@@ -344,11 +345,11 @@
                  ((NOT (COND
                          ((< |n| |i|) NIL)
                          ('T
-                          (>= (LETT |j|
-                                    (|ISTRING;position;C$2I;19| |c| |s|
-                                     |i| $)
-                                    |ISTRING;split;$CL;22|)
-                              (|getShellEntry| $ 6)))))
+                          (NOT (< (LETT |j|
+                                        (|ISTRING;position;C$2I;19| |c|
+                                         |s| |i| $)
+                                        |ISTRING;split;$CL;22|)
+                                  (|getShellEntry| $ 6))))))
                   (RETURN NIL))
                  (T (SEQ (SETQ |l|
                                (SPADCALL
@@ -400,11 +401,11 @@
                  ((NOT (COND
                          ((< |n| |i|) NIL)
                          ('T
-                          (>= (LETT |j|
-                                    (|ISTRING;position;Cc$2I;20| |cc|
-                                     |s| |i| $)
-                                    |ISTRING;split;$CcL;23|)
-                              (|getShellEntry| $ 6)))))
+                          (NOT (< (LETT |j|
+                                        (|ISTRING;position;Cc$2I;20|
+                                         |cc| |s| |i| $)
+                                        |ISTRING;split;$CcL;23|)
+                                  (|getShellEntry| $ 6))))))
                   (RETURN NIL))
                  (T (SEQ (SETQ |l|
                                (SPADCALL
@@ -471,7 +472,7 @@
     (SEQ (LOOP
            (COND
              ((NOT (COND
-                     ((>= |j| (|getShellEntry| $ 6))
+                     ((NOT (< |j| (|getShellEntry| $ 6)))
                       (SPADCALL (|ISTRING;elt;$IC;30| |s| |j| $) |c|
                           (|getShellEntry| $ 69)))
                      ('T NIL)))
@@ -487,7 +488,7 @@
     (SEQ (LOOP
            (COND
              ((NOT (COND
-                     ((>= |j| (|getShellEntry| $ 6))
+                     ((NOT (< |j| (|getShellEntry| $ 6)))
                       (SPADCALL (|ISTRING;elt;$IC;30| |s| |j| $) |cc|
                           (|getShellEntry| $ 65)))
                      ('T NIL)))
@@ -553,7 +554,7 @@
                  (- (SPADCALL |s| (|getShellEntry| $ 47))
                     (|getShellEntry| $ 6))))))
     (SEQ (COND
-           ((OR (MINUSP |l|) (>= |h| (QCSIZE |s|)))
+           ((OR (MINUSP |l|) (NOT (< |h| (QCSIZE |s|))))
             (EXIT (|error| "index out of bound"))))
          (EXIT (SUBSTRING |s| |l| (MAX 0 (+ (- |h| |l|) 1))))))) 
 
@@ -577,7 +578,7 @@
                                         |pattern| $)
                                        |ISTRING;match?;2$CB;34|)
                                  $)))
-                     (|check-subtype| (>= #0# 0)
+                     (|check-subtype| (NOT (MINUSP #0#))
                          '(|NonNegativeInteger|) #0#))
                    |ISTRING;match?;2$CB;34|)
              (EXIT (COND
@@ -599,7 +600,7 @@
                                         (|ISTRING;position;C$2I;19|
                                          |dontcare| |pattern| (+ |p| 1)
                                          $)))
-                                   (|check-subtype| (>= #1# 0)
+                                   (|check-subtype| (NOT (MINUSP #1#))
                                     '(|NonNegativeInteger|) #1#))
                                  |ISTRING;match?;2$CB;34|)
                            (LOOP
@@ -619,7 +620,8 @@
                                          ((#2=#:G1527
                                            (|ISTRING;position;2$2I;18|
                                             |s| |target| |i| $)))
-                                          (|check-subtype| (>= #2# 0)
+                                          (|check-subtype|
+                                           (NOT (MINUSP #2#))
                                            '(|NonNegativeInteger|) #2#)))
                                        (EXIT
                                         (COND
@@ -640,7 +642,7 @@
                                                   |dontcare| |pattern|
                                                   (+ |q| 1) $)))
                                                 (|check-subtype|
-                                                 (>= #3# 0)
+                                                 (NOT (MINUSP #3#))
                                                  '(|NonNegativeInteger|)
                                                  #3#))))))))))))
                            (COND
diff --git a/src/algebra/strap/LNAGG-.lsp b/src/algebra/strap/LNAGG-.lsp
index 71915a22..e0bd3c89 100644
--- a/src/algebra/strap/LNAGG-.lsp
+++ b/src/algebra/strap/LNAGG-.lsp
@@ -32,7 +32,7 @@
 
 (DEFUN |LNAGG-;index?;IAB;2| (|i| |a| $)
   (COND
-    ((>= |i| (SPADCALL |a| (|getShellEntry| $ 9)))
+    ((NOT (< |i| (SPADCALL |a| (|getShellEntry| $ 9))))
      (NOT (< (SPADCALL |a| (|getShellEntry| $ 10)) |i|)))
     ('T NIL))) 
 
diff --git a/src/algebra/strap/LSAGG-.lsp b/src/algebra/strap/LSAGG-.lsp
index a2d94892..eac3d76a 100644
--- a/src/algebra/strap/LSAGG-.lsp
+++ b/src/algebra/strap/LSAGG-.lsp
@@ -223,7 +223,7 @@
            (SEQ (LETT |y|
                       (SPADCALL |x|
                           (LET ((#0=#:G1467 (- (- |i| 1) |m|)))
-                            (|check-subtype| (>= #0# 0)
+                            (|check-subtype| (NOT (MINUSP #0#))
                                 '(|NonNegativeInteger|) #0#))
                           (|getShellEntry| $ 39))
                       |LSAGG-;insert!;SAIA;7|)
@@ -244,7 +244,7 @@
            (SEQ (LETT |y|
                       (SPADCALL |x|
                           (LET ((#0=#:G1471 (- (- |i| 1) |m|)))
-                            (|check-subtype| (>= #0# 0)
+                            (|check-subtype| (NOT (MINUSP #0#))
                                 '(|NonNegativeInteger|) #0#))
                           (|getShellEntry| $ 39))
                       |LSAGG-;insert!;2AIA;8|)
@@ -307,7 +307,7 @@
            (SEQ (LETT |y|
                       (SPADCALL |x|
                           (LET ((#0=#:G1483 (- (- |i| 1) |m|)))
-                            (|check-subtype| (>= #0# 0)
+                            (|check-subtype| (NOT (MINUSP #0#))
                                 '(|NonNegativeInteger|) #0#))
                           (|getShellEntry| $ 39))
                       |LSAGG-;delete!;AIA;10|)
@@ -334,7 +334,7 @@
                         ((EQL |l| |m|)
                          (SPADCALL |x|
                              (LET ((#0=#:G1489 (- (+ |h| 1) |m|)))
-                               (|check-subtype| (>= #0# 0)
+                               (|check-subtype| (NOT (MINUSP #0#))
                                    '(|NonNegativeInteger|) #0#))
                              (|getShellEntry| $ 39)))
                         ('T
@@ -342,7 +342,8 @@
                                     (SPADCALL |x|
                                      (LET
                                       ((#1=#:G1490 (- (- |l| 1) |m|)))
-                                       (|check-subtype| (>= #1# 0)
+                                       (|check-subtype|
+                                        (NOT (MINUSP #1#))
                                         '(|NonNegativeInteger|) #1#))
                                      (|getShellEntry| $ 39))
                                     |LSAGG-;delete!;AUsA;11|)
@@ -350,7 +351,8 @@
                                   (SPADCALL |t|
                                       (LET
                                        ((#2=#:G1491 (+ (- |h| |l|) 2)))
-                                        (|check-subtype| (>= #2# 0)
+                                        (|check-subtype|
+                                         (NOT (MINUSP #2#))
                                          '(|NonNegativeInteger|) #2#))
                                       (|getShellEntry| $ 39))
                                   (|getShellEntry| $ 27))
@@ -405,7 +407,7 @@
                    ('T
                     (SEQ (LETT |l|
                                (LET ((#0=#:G1511 (QUOTIENT2 |n| 2)))
-                                 (|check-subtype| (>= #0# 0)
+                                 (|check-subtype| (NOT (MINUSP #0#))
                                      '(|NonNegativeInteger|) #0#))
                                |LSAGG-;mergeSort|)
                          (LETT |q|
@@ -567,7 +569,7 @@
            (SEQ (LETT |z|
                       (SPADCALL |y|
                           (LET ((#0=#:G1552 (- |s| |m|)))
-                            (|check-subtype| (>= #0# 0)
+                            (|check-subtype| (NOT (MINUSP #0#))
                                 '(|NonNegativeInteger|) #0#))
                           (|getShellEntry| $ 39))
                       |LSAGG-;copyInto!;2AIA;22|)
@@ -598,7 +600,7 @@
            (SEQ (SETQ |x|
                       (SPADCALL |x|
                           (LET ((#0=#:G1559 (- |s| |m|)))
-                            (|check-subtype| (>= #0# 0)
+                            (|check-subtype| (NOT (MINUSP #0#))
                                 '(|NonNegativeInteger|) #0#))
                           (|getShellEntry| $ 39)))
                 (LETT |k| |s| |LSAGG-;position;SA2I;23|)
diff --git a/src/algebra/strap/NNI.lsp b/src/algebra/strap/NNI.lsp
index 4347bb34..dbe1dda2 100644
--- a/src/algebra/strap/NNI.lsp
+++ b/src/algebra/strap/NNI.lsp
@@ -2,7 +2,7 @@
 (/VERSIONCHECK 2) 
 
 (|noteSubDomainInfo| '|NonNegativeInteger| '(|Integer|)
-    '(|%ige| |#1| 0)) 
+    '(|%not| (|%ilt| |#1| 0))) 
 
 (DECLAIM (FTYPE (FUNCTION
                     ((|%IntegerSection| 0) (|%IntegerSection| 0)
@@ -36,7 +36,8 @@
       ((MINUSP |c|) (CONS 1 "failed"))
       ('T
        (CONS 0
-             (|check-subtype| (>= |c| 0) '(|NonNegativeInteger|) |c|)))))) 
+             (|check-subtype| (NOT (MINUSP |c|))
+                 '(|NonNegativeInteger|) |c|)))))) 
 
 (DEFUN |NonNegativeInteger| ()
   (DECLARE (SPECIAL |$ConstructorCache|))
diff --git a/src/algebra/strap/OUTFORM.lsp b/src/algebra/strap/OUTFORM.lsp
index 64660b88..6bfe294e 100644
--- a/src/algebra/strap/OUTFORM.lsp
+++ b/src/algebra/strap/OUTFORM.lsp
@@ -616,20 +616,20 @@
 
 (DEFUN |OUTFORM;vspace;I$;28| (|n| $)
   (COND
-    ((< 0 |n|)
+    ((PLUSP |n|)
      (|OUTFORM;vconcat;3$;50| " " (|OUTFORM;vspace;I$;28| (- |n| 1) $)
          $))
     ('T (|OUTFORM;empty;$;73| $)))) 
 
 (DEFUN |OUTFORM;hspace;I$;29| (|n| $)
   (COND
-    ((< 0 |n|) (|fillerSpaces| |n|))
+    ((PLUSP |n|) (|fillerSpaces| |n|))
     ('T (|OUTFORM;empty;$;73| $)))) 
 
 (DEFUN |OUTFORM;rspace;2I$;30| (|n| |m| $)
   (SEQ (COND
-         ((< 0 |n|)
-          (COND ((NOT (< 0 |m|)) (EXIT (|OUTFORM;empty;$;73| $)))))
+         ((PLUSP |n|)
+          (COND ((NOT (PLUSP |m|)) (EXIT (|OUTFORM;empty;$;73| $)))))
          ('T (EXIT (|OUTFORM;empty;$;73| $))))
        (EXIT (|OUTFORM;vconcat;3$;50| (|OUTFORM;hspace;I$;29| |n| $)
                  (|OUTFORM;rspace;2I$;30| |n| (- |m| 1) $) $)))) 
@@ -937,7 +937,7 @@
              ('T
               (SEQ (LETT |r|
                          (SPADCALL
-                             (|check-subtype| (< 0 |nn|)
+                             (|check-subtype| (PLUSP |nn|)
                                  '(|PositiveInteger|) |nn|)
                              (|getShellEntry| $ 137))
                          |OUTFORM;differentiate;$Nni$;97|)
diff --git a/src/algebra/strap/PI.lsp b/src/algebra/strap/PI.lsp
index 3ed76477..4f7ecd8d 100644
--- a/src/algebra/strap/PI.lsp
+++ b/src/algebra/strap/PI.lsp
@@ -2,7 +2,7 @@
 (/VERSIONCHECK 2) 
 
 (|noteSubDomainInfo| '|PositiveInteger| '(|NonNegativeInteger|)
-    '(|%igt| |#1| 0)) 
+    '(|%ilt| 0 |#1|)) 
 
 (DEFUN |PositiveInteger| ()
   (DECLARE (SPECIAL |$ConstructorCache|))
diff --git a/src/algebra/strap/POLYCAT-.lsp b/src/algebra/strap/POLYCAT-.lsp
index 9426e0ee..ca0cad36 100644
--- a/src/algebra/strap/POLYCAT-.lsp
+++ b/src/algebra/strap/POLYCAT-.lsp
@@ -752,7 +752,9 @@
                                                          ((#10=#:G1612
                                                            (CDR |nd|)))
                                                           (|check-subtype|
-                                                           (>= #10# 0)
+                                                           (NOT
+                                                            (MINUSP
+                                                             #10#))
                                                            '(|NonNegativeInteger|)
                                                            #10#))))))
                                                    #9#)))))
@@ -936,7 +938,7 @@
                          |POLYCAT-;charthRootlv|)
                    (LOOP
                      (COND
-                       ((NOT (< 0 |d|)) (RETURN NIL))
+                       ((NOT (PLUSP |d|)) (RETURN NIL))
                        (T (SEQ (LETT |dd|
                                      (SPADCALL |d| |ch|
                                       (|getShellEntry| $ 173))
@@ -981,7 +983,7 @@
                                                    ((#0=#:G1640
                                                      (CDR |dd|)))
                                                     (|check-subtype|
-                                                     (>= #0# 0)
+                                                     (NOT (MINUSP #0#))
                                                      '(|NonNegativeInteger|)
                                                      #0#))
                                                   (|getShellEntry| $
diff --git a/src/algebra/strap/SINT.lsp b/src/algebra/strap/SINT.lsp
index d37ff34e..d5afc6c2 100644
--- a/src/algebra/strap/SINT.lsp
+++ b/src/algebra/strap/SINT.lsp
@@ -118,12 +118,14 @@
 (DECLAIM (FTYPE (FUNCTION (|%Short| |%Short| |%Shell|) |%Boolean|)
                 |SINT;<=;2$B;28|)) 
 
-(PUT '|SINT;<=;2$B;28| '|SPADreplace| '|%ile|) 
+(PUT '|SINT;<=;2$B;28| '|SPADreplace|
+     '(XLAM (|x| |y|) (|%not| (|%ilt| |y| |x|)))) 
 
 (DECLAIM (FTYPE (FUNCTION (|%Short| |%Short| |%Shell|) |%Boolean|)
                 |SINT;>=;2$B;29|)) 
 
-(PUT '|SINT;>=;2$B;29| '|SPADreplace| '|%ige|) 
+(PUT '|SINT;>=;2$B;29| '|SPADreplace|
+     '(XLAM (|x| |y|) (|%not| (|%ilt| |x| |y|)))) 
 
 (DECLAIM (FTYPE (FUNCTION (|%Short| |%Shell|) |%Short|)
                 |SINT;inc;2$;30|)) 
@@ -389,9 +391,13 @@
   (DECLARE (IGNORE $))
   (QSGREATERP |x| |y|)) 
 
-(DEFUN |SINT;<=;2$B;28| (|x| |y| $) (DECLARE (IGNORE $)) (<= |x| |y|)) 
+(DEFUN |SINT;<=;2$B;28| (|x| |y| $)
+  (DECLARE (IGNORE $))
+  (NOT (< |y| |x|))) 
 
-(DEFUN |SINT;>=;2$B;29| (|x| |y| $) (DECLARE (IGNORE $)) (>= |x| |y|)) 
+(DEFUN |SINT;>=;2$B;29| (|x| |y| $)
+  (DECLARE (IGNORE $))
+  (NOT (< |x| |y|))) 
 
 (DEFUN |SINT;inc;2$;30| (|x| $) (DECLARE (IGNORE $)) (QSADD1 |x|)) 
 
diff --git a/src/algebra/strap/STAGG-.lsp b/src/algebra/strap/STAGG-.lsp
index 53991265..cdcf96d8 100644
--- a/src/algebra/strap/STAGG-.lsp
+++ b/src/algebra/strap/STAGG-.lsp
@@ -78,7 +78,7 @@
               (SPADCALL
                   (SETQ |x|
                         (SPADCALL |x|
-                            (|check-subtype| (>= |i| 0)
+                            (|check-subtype| (NOT (MINUSP |i|))
                                 '(|NonNegativeInteger|) |i|)
                             (|getShellEntry| $ 25)))
                   (|getShellEntry| $ 18)))
@@ -93,11 +93,12 @@
         (COND
           ((MINUSP |l|) (|error| "index out of range"))
           ((NOT (SPADCALL |i| (|getShellEntry| $ 29)))
-           (SPADCALL (SPADCALL |x|
-                         (|check-subtype| (>= |l| 0)
-                             '(|NonNegativeInteger|) |l|)
-                         (|getShellEntry| $ 25))
-                     (|getShellEntry| $ 30)))
+           (SPADCALL
+               (SPADCALL |x|
+                   (|check-subtype| (NOT (MINUSP |l|))
+                       '(|NonNegativeInteger|) |l|)
+                   (|getShellEntry| $ 25))
+               (|getShellEntry| $ 30)))
           ('T
            (SEQ (LETT |h|
                       (- (SPADCALL |i| (|getShellEntry| $ 31))
@@ -108,11 +109,11 @@
                         ('T
                          (SPADCALL
                              (SPADCALL |x|
-                                 (|check-subtype| (>= |l| 0)
+                                 (|check-subtype| (NOT (MINUSP |l|))
                                      '(|NonNegativeInteger|) |l|)
                                  (|getShellEntry| $ 25))
                              (LET ((#0=#:G1420 (+ (- |h| |l|) 1)))
-                               (|check-subtype| (>= #0# 0)
+                               (|check-subtype| (NOT (MINUSP #0#))
                                    '(|NonNegativeInteger|) #0#))
                              (|getShellEntry| $ 35)))))))))))) 
 
@@ -160,7 +161,7 @@
               (SPADCALL
                   (SETQ |x|
                         (SPADCALL |x|
-                            (|check-subtype| (>= |i| 0)
+                            (|check-subtype| (NOT (MINUSP |i|))
                                 '(|NonNegativeInteger|) |i|)
                             (|getShellEntry| $ 25)))
                   (|getShellEntry| $ 18)))
@@ -187,7 +188,8 @@
                         ('T
                          (SEQ (LETT |y|
                                     (SPADCALL |x|
-                                     (|check-subtype| (>= |l| 0)
+                                     (|check-subtype|
+                                      (NOT (MINUSP |l|))
                                       '(|NonNegativeInteger|) |l|)
                                      (|getShellEntry| $ 25))
                                     |STAGG-;setelt;AUs2S;12|)
@@ -195,7 +197,8 @@
                                     (SPADCALL |y|
                                      (LET
                                       ((#0=#:G1443 (+ (- |h| |l|) 1)))
-                                       (|check-subtype| (>= #0# 0)
+                                       (|check-subtype|
+                                        (NOT (MINUSP #0#))
                                         '(|NonNegativeInteger|) #0#))
                                      (|getShellEntry| $ 25))
                                     |STAGG-;setelt;AUs2S;12|)
diff --git a/src/algebra/strap/SYMBOL.lsp b/src/algebra/strap/SYMBOL.lsp
index 0c928c37..190f733b 100644
--- a/src/algebra/strap/SYMBOL.lsp
+++ b/src/algebra/strap/SYMBOL.lsp
@@ -200,7 +200,8 @@
     (SEQ (LOOP
            (COND
              ((NOT (COND
-                     ((>= (LENGTH |ns|) 2) (ZEROP (|SPADfirst| |ns|)))
+                     ((NOT (< (LENGTH |ns|) 2))
+                      (ZEROP (|SPADfirst| |ns|)))
                      ('T NIL)))
               (RETURN NIL))
              (T (SETQ |ns| (CDR |ns|)))))
@@ -440,14 +441,15 @@
            (SETQ |xx| (STRCONC "%" |xx|))
            (SETQ |xx|
                  (COND
-                   ((>= (SPADCALL
-                            (SPADCALL |xx|
-                                (SPADCALL |xx| (|getShellEntry| $ 128))
-                                (|getShellEntry| $ 106))
-                            (|getShellEntry| $ 19)
-                            (|getShellEntry| $ 129))
-                        (SPADCALL (|getShellEntry| $ 19)
-                            (|getShellEntry| $ 117)))
+                   ((NOT (< (SPADCALL
+                                (SPADCALL |xx|
+                                    (SPADCALL |xx|
+                                     (|getShellEntry| $ 128))
+                                    (|getShellEntry| $ 106))
+                                (|getShellEntry| $ 19)
+                                (|getShellEntry| $ 129))
+                            (SPADCALL (|getShellEntry| $ 19)
+                                (|getShellEntry| $ 117))))
                     (STRCONC |xx|
                              (|SYMBOL;anyRadix| |n|
                                  (|getShellEntry| $ 21) $)))
@@ -546,7 +548,7 @@
                                           (|getShellEntry| $ 106))
                                          (|getShellEntry| $ 44))
                                         (|getShellEntry| $ 45))))
-                                  (|check-subtype| (>= #0# 0)
+                                  (|check-subtype| (NOT (MINUSP #0#))
                                       '(|NonNegativeInteger|) #0#))
                                 (|getShellEntry| $ 148))))
                        (SETQ |i| (+ |i| 1))
diff --git a/src/algebra/strap/URAGG-.lsp b/src/algebra/strap/URAGG-.lsp
index a9dab43a..fb03bf3a 100644
--- a/src/algebra/strap/URAGG-.lsp
+++ b/src/algebra/strap/URAGG-.lsp
@@ -168,20 +168,20 @@
     (SEQ (LOOP
            (COND
              ((NOT (COND
-                     ((< 0 |i|)
+                     ((PLUSP |i|)
                       (NOT (SPADCALL |l| (|getShellEntry| $ 20))))
                      ('T NIL)))
               (RETURN NIL))
              (T (SEQ (SETQ |l| (SPADCALL |l| (|getShellEntry| $ 14)))
                      (EXIT (SETQ |i| (- |i| 1)))))))
-         (EXIT (< 0 |i|))))) 
+         (EXIT (PLUSP |i|))))) 
 
 (DEFUN |URAGG-;more?;ANniB;13| (|l| |n| $)
   (LET ((|i| |n|))
     (SEQ (LOOP
            (COND
              ((NOT (COND
-                     ((< 0 |i|)
+                     ((PLUSP |i|)
                       (NOT (SPADCALL |l| (|getShellEntry| $ 20))))
                      ('T NIL)))
               (RETURN NIL))
@@ -198,7 +198,7 @@
            (COND
              ((NOT (COND
                      ((SPADCALL |l| (|getShellEntry| $ 20)) NIL)
-                     ('T (< 0 |i|))))
+                     ('T (PLUSP |i|))))
               (RETURN NIL))
              (T (SEQ (SETQ |l| (SPADCALL |l| (|getShellEntry| $ 14)))
                      (EXIT (SETQ |i| (- |i| 1)))))))
@@ -381,8 +381,8 @@
        (SPADCALL
            (SPADCALL |x|
                (LET ((#0=#:G1502 (- |m| |n|)))
-                 (|check-subtype| (>= #0# 0) '(|NonNegativeInteger|)
-                     #0#))
+                 (|check-subtype| (NOT (MINUSP #0#))
+                     '(|NonNegativeInteger|) #0#))
                (|getShellEntry| $ 62))
            (|getShellEntry| $ 63)))))) 
 
@@ -491,7 +491,7 @@
               (SEQ (SETQ |p|
                          (SPADCALL |p|
                              (LET ((#0=#:G1528 (- |n| 1)))
-                               (|check-subtype| (>= #0# 0)
+                               (|check-subtype| (NOT (MINUSP #0#))
                                    '(|NonNegativeInteger|) #0#))
                              (|getShellEntry| $ 62)))
                    (LETT |q| (SPADCALL |p| (|getShellEntry| $ 14))
diff --git a/src/interp/g-opt.boot b/src/interp/g-opt.boot
index b9705f6b..b29d9034 100644
--- a/src/interp/g-opt.boot
+++ b/src/interp/g-opt.boot
@@ -137,7 +137,13 @@ optimizeFunctionDef(def) ==
         replaceThrowByReturn(rest x,g)
   changeVariableDefinitionToStore(body',args)
   [name,[slamOrLam,args,groupVariableDefinitions body']]
- 
+
+++ Like `optimize', except that non-atomic form may be reduced to
+++ to atomic forms.  In particular, the address of the input may
+++ not be the same as that of the output.
+simplifyVMForm x ==
+  first optimize [x]
+
 optimize x ==
   (opt x; x) where
     opt x ==
@@ -151,8 +157,7 @@ optimize x ==
           SAY '"length mismatch in XLAM expression"
           PRETTYPRINT y
         x.first := optimize optXLAMCond SUBLIS(pairList(argl,a),body)
-      atom y =>
-        optimize rest x
+      atom y => optimize rest x
       if first y="IF" then (x.first := optIF2COND y; y:= first x)
       op:= GETL(subrname first y,"OPTIMIZE") =>
         (optimize rest x; x.first := FUNCALL(op,optimize first x))
@@ -285,7 +290,7 @@ optSpecialCall(x,y,n) ==
   fn := getFunctionReplacement compileTimeBindingOf first yval.n =>
     x.rest := CDAR x
     x.first := fn
-    if fn is ["XLAM",:.] then x:=first optimize [x]
+    if fn is ["XLAM",:.] then x := simplifyVMForm x
     x is ["EQUAL",:args] => RPLACW(x,DEF_-EQUAL args)
                 --DEF-EQUAL is really an optimiser
     x
@@ -661,11 +666,52 @@ optCollectVector form ==
 ++ Translate retraction of a value denoted by `e' to sub-domain `m'
 ++ defined by predicate `pred',
 optRetract ["%retract",e,m,pred] ==
-  atom e => ["check-subtype",substitute(e,"#1",pred),MKQ m,e]
+  atom e =>
+    cond := simplifyVMForm substitute(e,"#1",pred)
+    cond = '%true => e
+    ["check-subtype",cond,MKQ m,e]
   g := gensym()
   ["LET",[[g,e]],["check-subtype",substitute(g,"#1",pred),MKQ m,g]]
 
-lispize x == first optimize [x]
+
+--%  Boolean expression transformers
+
+optNot(x is ['%not,a]) ==
+  a = '%true => '%false
+  a = '%false => '%true
+  a is ['%not,b] => b
+  x
+
+optAnd(x is ['%and,a,b]) ==
+  a = '%true => b
+  b = '%true => a
+  a = '%false => '%false
+  x
+
+optOr(x is ['%or,a,b]) ==
+  a = '%false => b
+  b = '%false => a
+  a = '%true => '%true
+  x
+
+optIlt(x is ['%ilt,a,b]) ==
+  integer? a and integer? b =>
+    a < b => '%true
+    '%false
+  x
+
+optIle(x is ['%ile,a,b]) ==
+  optNot ['%not,optIlt ['%ilt,b,a]]
+
+optIgt x ==
+  optIlt ['%ilt,third x, second x]
+
+optIge x ==
+  optNot ['%not,optIlt ['%ilt,second x,third x]]
+
+--%  
+
+lispize x == simplifyVMForm x
  
 --% optimizer hash table
  
@@ -674,6 +720,13 @@ for x in '( (%call         optCall) _
            (LET          optLET)_
            (LET_*        optLET_*)_
            (%bind        optBind)_
+           (%not         optNot)_
+           (%and         optAnd)_
+           (%or          optOr)_
+           (%ilt         optIlt)_
+           (%ile         optIle)_
+           (%igt         optIgt)_
+           (%ige         optIge)_
            (LIST         optLIST)_
            (MINUS        optMINUS)_
            (QSMINUS      optQSMINUS)_