# MATH illustrate lists and some list operators
       # to make list describable as a function, need to preceed lists
       # ... with an argument count
       # Lists keep an explicit record of their length
       # this is to avoid the need for using a special 'nil' symbol
       # ... which cannot itself be placed in the list.
       # pending: should introduce number? check function
[hear] (define list-helper /
         ? n /
         ? ret /
         if (> (n) 1)
          (? x /
           list-helper
            (- (n) 1)
            (? y /
             ? z /
             ret (+ 1 (y)) (cons (x) (z))))
          (? x /
           ret 1 (x)));

[hear] (define list /
         ? n /
         if (= (n) 0)
          (cons 0 0)
          (list-helper (n) (? y / ? z / cons (y) (z))));

[hear] (define head /
         ? lst /
         if (= (car / lst) 0)
          (undefined)
          (if (= (car / lst) 1)
            (cdr /
             lst)
            (car /
             cdr /
             lst)));

[hear] (define tail /
         ? lst /
         if (= (car / lst) 0)
          (undefined)
          (if (= (car / lst) 1)
            (cons 0 0)
            (cons (- (car / lst) 1) (cdr / cdr / lst))));

[hear] (define list-length / ? lst / car / lst);

[hear] (define list-ref /
         ? lst /
         ? n /
         if (= (list-ref / lst) 0)
          (undefined)
          (if (= (n) 0)
            (head /
             lst)
            (list-ref (tail / lst) (- (n) 1))));

[hear] (define prepend /
         ? x /
         ? lst /
         if (= (list-length / lst) 0)
          (cons 1 (x))
          (cons (+ (list-length / lst) 1)
                (cons (x) (cdr / lst))));

[hear] (define equal /
         ? x /
         ? y /
         if (= (number? (x)) (number? (y)))
          (if (number? (x)) (= (x) (y)) (list= (x) (y)))
          (false));

[hear] (define list= /
         ? x /
         ? y /
         if (= (list-length / x) (list-length / y))
          (if (> (list-length / x) 0)
            (and (equal (head / x) (head / y))
                 (list= (tail / x) (tail / y)))
            (true))
          (false));

[hear] (= (list-length / (list 0)) 0);

[hear] (= (list-length / (list 4) 6 1 0 4) 4);

[hear] (= (list-length / (list 6) 6 2 7 0 9 4) 6);

[hear] (= (list-length / (list 2) 4 9) 2);

[hear] (= (list-length / (list 3) 6 1 7) 3);

[hear] (= (head / (list 6) 12 11 10 4 1 5) 12);

[hear] (list= (tail /
                (list 6) 12 11 10 4 1 5)
               ((list 5) 11 10 4 1 5));

[hear] (= (head / (list 8) 15 13 12 7 10 11 13 18) 15);

[hear] (list= (tail /
                (list 8) 15 13 12 7 10 11 13 18)
               ((list 7) 13 12 7 10 11 13 18));

[hear] (= (head / (list 2) 11 1) 11);

[hear] (list= (tail / (list 2) 11 1) ((list 1) 1));

[hear] (= (head / (list 6) 5 19 4 16 6 11) 5);

[hear] (list= (tail /
                (list 6) 5 19 4 16 6 11)
               ((list 5) 19 4 16 6 11));

[hear] (= (head /
            (list 10) 12 18 7 4 9 18 6 16 6 18)
           12);

[hear] (list= (tail /
                (list 10) 12 18 7 4 9 18 6 16 6 18)
               ((list 9) 18 7 4 9 18 6 16 6 18));

[hear] (= (head / (list 6) 19 7 3 10 19 13) 19);

[hear] (list= (tail /
                (list 6) 19 7 3 10 19 13)
               ((list 5) 7 3 10 19 13));

[hear] (= (head / (list 6) 19 7 19 12 16 13) 19);

[hear] (list= (tail /
                (list 6) 19 7 19 12 16 13)
               ((list 5) 7 19 12 16 13));

[hear] (= (head / (list 1) 3) 3);

[hear] (list= (tail / (list 1) 3) ((list 0)));

[hear] (= (head / (list 3) 2 19 17) 2);

[hear] (list= (tail /
                (list 3) 2 19 17)
               ((list 2) 19 17));

[hear] (= (head / (list 7) 1 16 5 14 6 19 2) 1);

[hear] (list= (tail /
                (list 7) 1 16 5 14 6 19 2)
               ((list 6) 16 5 14 6 19 2));

[hear] (= (list-ref ((list 3) 18 14 17) 1) 14);

[hear] (= (list-ref ((list 3) 8 11 10) 2) 10);

[hear] (= (list-ref ((list 8) 15 0 4 9 9 2 10 17) 3) 9);

[hear] (= (list-ref ((list 7) 4 8 8 5 14 5 13) 4) 14);

[hear] (= (list-ref ((list 4) 1 4 7 18) 2) 7);

[hear] (= (list-ref ((list 3) 12 2 3) 1) 2);

[hear] (= (list-ref ((list 6) 12 5 7 15 7 16) 2) 7);

[hear] (= (list-ref ((list 8) 5 15 7 14 7 1 11 19) 0) 5);

[hear] (= (list-ref ((list 3) 19 17 8) 2) 8);

[hear] (= (list-ref ((list 4) 10 10 4 11) 1) 10);

[hear] (list= ((list 0)) ((list 0)));

[hear] (list= ((list 1) 4) ((list 1) 4));

[hear] (list= ((list 2) 7 5) ((list 2) 7 5));

[hear] (list= ((list 3) 15 13 11) ((list 3) 15 13 11));

[hear] (list= ((list 4) 2 8 0 6) ((list 4) 2 8 0 6));

       # this next batch of examples are a bit misleading, should streamline
[hear] (not / list= ((list 0)) ((list 1) 9));

[hear] (not / list= ((list 0)) ((list 1) 5));

[hear] (not / list= ((list 1) 18) ((list 2) 8 18));

[hear] (not / list= ((list 1) 18) ((list 2) 18 5));

[hear] (not /
         list= ((list 2) 11 18) ((list 3) 7 11 18));

[hear] (not /
         list= ((list 2) 11 18) ((list 3) 11 18 6));

[hear] (not /
         list= ((list 3) 7 19 17) ((list 4) 6 7 19 17));

[hear] (not /
         list= ((list 3) 7 19 17) ((list 4) 7 19 17 0));

[hear] (not /
         list= ((list 4) 10 0 11 1)
               ((list 5) 0 10 0 11 1));

[hear] (not /
         list= ((list 4) 10 0 11 1)
               ((list 5) 10 0 11 1 8));

       # some helpful functions
[hear] (list= (prepend 8 ((list 0))) ((list 1) 8));

[hear] (list= (prepend 11 ((list 1) 8)) ((list 2) 11 8));

[hear] (list= (prepend 13 ((list 2) 1 12))
               ((list 3) 13 1 12));

[hear] (list= (prepend 0 ((list 3) 7 7 5))
               ((list 4) 0 7 7 5));

[hear] (list= (prepend 16 ((list 4) 16 0 19 3))
               ((list 5) 16 16 0 19 3));

[hear] (list= (prepend 10 ((list 5) 5 6 7 9 10))
               ((list 6) 10 5 6 7 9 10));

[hear] (list= (prepend 19 ((list 6) 3 19 18 6 10 16))
               ((list 7) 19 3 19 18 6 10 16));

[hear] (list= (prepend 19 ((list 7) 17 17 10 1 18 12 14))
               ((list 8) 19 17 17 10 1 18 12 14));

[hear] (define pair /
         ? x /
         ? y /
         (list 2) (x) (y));

[hear] (define first / ? lst / head / lst);

[hear] (define second /
         ? lst /
         head /
         tail /
         lst);

[hear] (list= (pair 3 6) ((list 2) 3 6));

[hear] (= (first / pair 3 6) 3);

[hear] (= (second / pair 3 6) 6);

[hear] (list= (pair 4 9) ((list 2) 4 9));

[hear] (= (first / pair 4 9) 4);

[hear] (= (second / pair 4 9) 9);

[hear] (list= (pair 8 3) ((list 2) 8 3));

[hear] (= (first / pair 8 3) 8);

[hear] (= (second / pair 8 3) 3);

[hear] (define list-find-helper /
         ? lst /
         ? key /
         ? fail /
         ? idx /
         if (= (list-length / lst) 0)
          (fail 0)
          (if (equal (head / lst) (key))
            (idx)
            (list-find-helper
              (tail /
               lst)
              (key)
              (fail)
              (+ (idx) 1))));

[hear] (define list-find /
         ? lst /
         ? key /
         ? fail /
         list-find-helper (lst) (key) (fail) 0);

[hear] (define example-fail / ? x 100);

[hear] (= (list-find ((list 1) 13) 13 (example-fail)) 0);

[hear] (= (list-find
             ((list 10) 0 9 8 16 15 14 17 5 9 2)
             15
             (example-fail))
           4);

[hear] (= (list-find ((list 3) 7 4 10) 7 (example-fail))
           0);

[hear] (= (list-find
             ((list 6) 0 17 10 13 11 5)
             17
             (example-fail))
           1);

[hear] (= (list-find ((list 3) 12 9 6) 12 (example-fail))
           0);

[hear] (= (list-find
             ((list 7) 17 1 4 17 14 13 13)
             14
             (example-fail))
           4);

[hear] (= (list-find ((list 3) 2 15 2) 15 (example-fail))
           1);

[hear] (= (list-find
             ((list 9) 6 13 10 8 10 9 6 15 18)
             13
             (example-fail))
           1);

[hear] (= (list-find ((list 3) 12 16 0) 12 (example-fail))
           0);

[hear] (= (list-find ((list 1) 15) 15 (example-fail)) 0);

[hear] (= (list-find
             ((list 4) 2 17 11 5)
             14
             (example-fail))
           100);

[hear] (= (list-find
             ((list 6) 12 1 19 6 17 9)
             2
             (example-fail))
           100);

[hear] (= (list-find
             ((list 8) 11 6 17 8 13 10 9 16)
             19
             (example-fail))
           100);