% question-for-Yadunand-Prem.txt
% CS3234, Fri 05 Apr 2024

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This oral exam is a continuation of the midterm project. 
So, make a copy of midterm_project.v (naming it midterm_project_copy-for-oral-exam.v)
and start editing this file after

(* end of midterm_project.v *)

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The goal of this oral exam is to revisit

  Proposition list_reverse_acc_and_list_append_commute_with_each_other :
    forall (V : Type)
           (v1s v2s: list V),
      list_append V (list_reverse_acc V v2s nil) (list_reverse_acc V v1s nil) =
      list_reverse_acc V (list_append V v1s v2s) nil.

So, copy the following proposition at the very end of midterm_project_copy-for-oral-exam.v:

  Proposition a_generalized_alternative :
    forall (V : Type)
           (v1s v2s v3s : list V),
      list_reverse_acc V (list_append V v1s v2s) v3s
      = nil.

%%%%%

a. The first thing to do is to figure out what to put instead of nil on the right-hand side,
   so that the equality holds.
   We (in the sense of you) are going to do that empirically by testing.
   To this end, write:

     Proof.
       Compute (let V := nat in
                let v1s := 11 :: 12 :: nil in
                let v2s := 21 :: 22 :: nil in
                let v3s := 31 :: 32 :: nil in
                list_reverse_acc V (list_append V v1s v2s) v3s
                = nil).
      
   Then, replace nil on the right-hand-side by an expression that does not use list_append
   but that only uses two occurrences of list_reverse_acc,
   and that is such that the equality holds.
   (Let us name this expression as YOUR_EXPRESSION.)

   Hint: use "(list_reverse_acc V ... ...)" as an accumulator for list_reverse_acc.

b. To make sure, write:

       Compute (let V := nat in
                let v1s := 11 :: 12 :: nil in
                let v2s := 21 :: 22 :: nil in
                let v3s := 31 :: 32 :: nil in
                eqb_list
                  nat
                  Nat.eqb
                  (list_reverse_acc V (list_append V v1s v2s) v3s)
                  YOUR_EXPRESSION).

    Verify that the result of this computation is true.
    If it is false, then go back to a. and revise YOUR_EXPRESSION
    until the result of the computation just above is true.

c. Prove this equality by induction.

d. Time permitting, write:

       Restart.

   Prove these equalities not by induction,
   but using the many properties you have proved earlier in the file,
   e.g., about_list_reverse_acc_and_append,
   list_reverse_acc_and_list_append_commute_with_each_other,
   list_append_is_associative,
   etc.
   (Maybe you will need to state and prove more similar properties to carry our this proof,
   maybe not, who knows, this is an exploration.)

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% end of question-for-Yadunand-Prem.txt