Homework Assignment 6
                               CS 389R

                        Unique Numbers: 52325
                             Spring, 2017

                       Given:  March 7, 2017
                        Due:  March 28, 2017

Prove the seven theorems from the first exam in the ACL2 proof
checker. (For the problem(s) that were not theorems, correct the
conjecture so that it is a theorem, and prove that.) The solutions
produced using the proof checker must be submitted electronically as
ACL2 DEFTHM events that can be processed by the ACL2 system.  We have
shown how to use the ACL2 proof checker in class to produce such a
log, but if you are unsure, please contact Warren or Ben, either via
email or office hours.

Email your proof checker solutions to the class TA by the beginning of
class on the due date.  The solutions should be attached as a file
named "hw6-firstname-lastname.lisp" where firstname and lastname are
your first and last name.  The file should execute from start to
finish in ACL2.  You can test this by starting ACL2 and executing the
command (ld "hw6-firstname-lastname.lisp").  In other words, all
definitions related to the problems should be included in the file.
We will inspect your definitions and your theorems with proof checker
instructions in addition to executing the file.

When you use the proof checker, limit your use of the ACL2 proof
checker to the commands listed below:

[ACL2-pc::=]
     (atomic macro) attempt an equality (or equivalence) substitution

[ACL2-pc::bk]
    (atomic macro) move backward one argument in the enclosing term

[ACL2-pc::casesplit]
    (primitive) split into two cases

[ACL2-pc::cg]
    (macro) change to another goal.

[ACL2-pc::change-goal]
    (primitive) change to another goal.

[ACL2-pc::comm]
    (macro) display instructions from the current interactive session

[ACL2-pc::commands]
    (macro) display instructions from the current interactive session

[ACL2-pc::comment]
    (primitive) insert a comment

[ACL2-pc::contradict]
    (macro) same as contrapose

[ACL2-pc::contrapose]
    (primitive) switch a hypothesis with the conclusion, negating both

[ACL2-pc::demote]
    (primitive) move top-level hypotheses to the conclusion

[ACL2-pc::dive]
    (primitive) move to the indicated subterm

[ACL2-pc::doc]
    (macro) access documentation inside the proof-checker

[ACL2-pc::drop]
    (primitive) drop top-level hypotheses

[ACL2-pc::dv]
    (atomic macro) move to the indicated subterm

[ACL2-pc::equiv]
    (primitive) attempt an equality (or congruence-based) substitution

[ACL2-pc::exit]
    (meta) exit the interactive proof-checker

[ACL2-pc::expand]
    (primitive) expand the current function call without simplification

[ACL2-pc::goals]
    (macro) list the names of goals on the stack

[ACL2-pc::help]
    (macro) proof-checker help facility

[ACL2-pc::hyps]
    (macro) print the hypotheses

[ACL2-pc::in-theory]
    (primitive) set the current proof-checker theory

[ACL2-pc::induct]
    (atomic macro) generate subgoals using induction

[ACL2-pc::nx]
   (atomic macro) move forward one argument in the enclosing term

[ACL2-pc::p]
    (macro) prettyprint the current term

[ACL2-pc::p-top]
    (macro) prettyprint the conclusion, highlighting the current term

[ACL2-pc::pp]
    (macro) prettyprint the current term

[ACL2-pc::print]
    (macro) print the result of evaluating the given form

[ACL2-pc::print-all-concs]
    (macro) print all the conclusions of (as yet unproved) goals

[ACL2-pc::print-all-goals]
    (macro) print all the (as yet unproved) goals

[ACL2-pc::print-main]
    (macro) print the original goal

[ACL2-pc::promote]
    (primitive) move antecedents of conclusion's implies term to
    top-level hypotheses

[ACL2-pc::r]
    (macro) same as rewrite

[ACL2-pc::restore]
    (meta) remove the effect of an UNDO command

[ACL2-pc::retrieve]
    (macro) re-enter the proof-checker

[ACL2-pc::rewrite]
    (primitive) apply a rewrite rule

[ACL2-pc::s]
    (primitive) simplify the current subterm

[ACL2-pc::s-prop]
    (atomic macro) simplify propositionally

[ACL2-pc::save]
    (macro) save the proof-checker state (state-stack)

[ACL2-pc::show-rewrites]
    (macro) display the applicable [rewrite] rules

[ACL2-pc::sl]
    (atomic macro) simplify with lemmas

[ACL2-pc::split]
    (atomic macro) split the current goal into cases

[ACL2-pc::sr]
    (macro) same as SHOW-REWRITES

[ACL2-pc::th]
    (macro) print the top-level hypotheses and the current subterm

[ACL2-pc::top]
    (atomic macro) move to the top of the goal

[ACL2-pc::undo]
    (meta) undo some instructions

[ACL2-pc::up]
    (primitive) move to the parent (or some ancestor) of the current
    subterm

[ACL2-pc::use]
    (atomic macro) use a lemma instance