Running and testing a property expressing a relationship between TAKE and APPEND

Question

Basically, I need to write a what the title says, the only relationship I have been able to think of is if I take some number of elements from a list with TAKE and then take the not-as-important other half with CDR and then APPEND the two that I took to prove that it's the same as the original list. (After long painful hours of building it), the proof seems to be fine (It compiles just fine), but for some reason it fails when run. I am using Proofpad with Dracula, in case you need the information.

Here is the code:

(include-book "doublecheck" :dir :teachpacks)

(defproperty take-append-relationship-test ;not sure why this fails.
   (xs :value (random-integer-list))
   (iff (consp xs)
        (let* ((x1 (take 1 xs))
               (xs2 (cdr xs)))
            (equal (append x1 xs2)
                   xs))))

Here is the error log I get.

By the simple :definition TAKE we reduce the conjecture to

Goal'
(COND ((CONSP XS)
       (LET ((X1 (FIRST-N-AC 1 XS NIL)))
            (EQUAL (APPEND X1 (CDR XS)) XS)))
      ((LET ((X1 (FIRST-N-AC 1 XS NIL)))
            (EQUAL (APPEND X1 (CDR XS)) XS))
       NIL)
      (T T)).

This simplifies, using the :definition FIRST-N-AC, the :executable-
counterparts of BINARY-+, BINARY-APPEND, CONS, FIRST-N-AC and ZP, primitive
type reasoning and the :rewrite rules DEFAULT-CAR and DEFAULT-CDR,
to

Goal''
(IMPLIES (CONSP XS)
         (EQUAL (APPEND (FIRST-N-AC 1 XS NIL) (CDR XS))
                XS)).

The destructor terms (CAR XS) and (CDR XS) can be eliminated by using
CAR-CDR-ELIM to replace XS by (CONS XS1 XS2), (CAR XS) by XS1 and (CDR XS)
by XS2.  This produces the following goal.

Goal'''
(IMPLIES (CONSP (CONS XS1 XS2))
         (EQUAL (APPEND (FIRST-N-AC 1 (CONS XS1 XS2) NIL)
                        XS2)
                (CONS XS1 XS2))).

This simplifies, using primitive type reasoning, to

Goal'4'
(EQUAL (APPEND (FIRST-N-AC 1 (CONS XS1 XS2) NIL)
               XS2)
       (CONS XS1 XS2)).

Normally we would attempt to prove Goal'4' by induction.  However,
we prefer in this instance to focus on the original input conjecture
rather than this simplified special case.  We therefore abandon our
previous work on this conjecture and reassign the name *1 to the original
conjecture.  (See :DOC otf-flg.)

No induction schemes are suggested by *1.  Consequently, the proof
attempt has failed.

Summary
Form:  ( DEFTHM TAKE-APPEND-RELATIONSHIP-TEST ...)
Rules: ((:DEFINITION FIRST-N-AC)
        (:DEFINITION IFF)
        (:DEFINITION TAKE)
        (:ELIM CAR-CDR-ELIM)
        (:EXECUTABLE-COUNTERPART BINARY-+)
        (:EXECUTABLE-COUNTERPART BINARY-APPEND)
        (:EXECUTABLE-COUNTERPART CONS)
        (:EXECUTABLE-COUNTERPART FIRST-N-AC)
        (:EXECUTABLE-COUNTERPART ZP)
        (:FAKE-RUNE-FOR-TYPE-SET NIL)
        (:REWRITE DEFAULT-CAR)
        (:REWRITE DEFAULT-CDR))
Time:  0.00 seconds (prove: 0.00, print: 0.00, other: 0.00)
Prover steps counted:  328

---
The key checkpoint goal, below, may help you to debug this failure.
See :DOC failure and see :DOC set-checkpoint-summary-limit.
---

*** Key checkpoint before reverting to proof by induction: ***

Goal''
(IMPLIES (CONSP XS)
         (EQUAL (APPEND (FIRST-N-AC 1 XS NIL) (CDR XS))
                XS))

ACL2 Error in ( DEFTHM TAKE-APPEND-RELATIONSHIP-TEST ...):  See :DOC
failure.

******** FAILED ********

Could someone at least point me in the right direction? I have read the error log and what I get is that there is some sort of redundancy. I am not sure how to fix this at all.

---

If I do that (so like '(1 2 3 4) instead of (random-integer-list)), it throws an error:

ACL2 Error in TOP-LEVEL:  One value, '(1 2 3 4 5), is being returned
where 2 values were expected.  Note:  This error occurred in the context
(MV-LET (XS STATE)
        '(1 2 3 4 5)
        (IF T
            (MV-LET (STATE RESULT ASSIGNMENTS)
                    (EXPAND-VARS NIL (IFF # #))
                    (MV STATE RESULT (CONS # ASSIGNMENTS)))
            (MV STATE 'WHERE-NOT-MATCHED NIL))).
(See :DOC set-iprint to be able to see elided values in this message.)

---

My setup:

1. Install Racket

2. Open it and run the following code:

   (pound)lang racket (require (planet cce/dracula:8:23/lang/dracula))

3. Restart Dr Racket. Click the "Choose Language" at the bottom of the window and choose ACL2 under Dracula.

4. From the Dracula menu, choose "Change ACL2 Executable Path..." and choose "run_acl2" (or "run_acl2.exe") which is located in the installation folder of proofpad (i.e. "C:\Program Files (x86)\Proof Pad", if you have a 64 bits PC)

5. download PSP++ and run it. Make sure to unzip it in the Proofpad installation directory, otherwise it will tell you it didn't find the things it needed.