theory example_2 imports Isac.Isac_Knowledge

begin

section\<open>Symbol Representation\<close>

ML \<open>

  (* 

   * alpha -->  "</alpha>" 

   *)

  @{term "\<alpha> "};

  @{term "\<delta> "};

  @{term "\<phi> "};

  @{term "\<rho> "};

  UnparseC.term @{term "\<rho> "};

\<close>

section\<open>Working with Rulesets\<close>

axiomatization where 

  rule1: "1 = \<delta>[n]" and

  rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and

  rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and 

  rule4: "|| z || > || \<alpha> || ==> z / (z - \<alpha>) = \<alpha>\<up>n * u [n]"

ML \<open>

  @{thm rule1};

  @{thm rule2};

  @{thm rule3};

  @{thm rule4};

\<close>

text\<open>Create Ruleset and insert rules\<close>

ML \<open>

  val transform = Rule_Set.append_rules "transform" Rule_Set.empty

    [ Thm  ("rule3",ThmC.numerals_to_Free @{thm rule3}),

      Thm  ("rule4",ThmC.numerals_to_Free @{thm rule4}),

      Thm  ("rule1",ThmC.numerals_to_Free @{thm rule1})   

    ];

\<close>

text\<open>Apply the Ruleset to a term\<close>

ML \<open>

  val t = parse_test @{context} "z / (z - 1) + z / (z - \<alpha>) + 1::real";

  val SOME (t', asm) = rewrite_set_ thy true transform t;

\<close>

ML\<open>

 UnparseC.term t';

 UnparseC.term t' = "z / (z - ?\<delta> [?n]) + z / (z - \<alpha>) + ?\<delta> [?n]";

 UnparseC.term t' = "?u [?n] + \<alpha>\<up>?n u[n] + ?\<delta> [?n]";

\<close>

text\<open>Problems: Rule~1 is applied before the expression is checked for Rule~4

      which would be the correct one. Rule~1 only matches a part of the 

      expression.\<close>

end