(*  Nominal Termination

    Author: Christian Urban

    heavily based on the code of Alexander Krauss

    (code forked on 18 July 2011)

Redefinition of the termination command

*)

signature NOMINAL_FUNCTION_TERMINATION =

sig

  include NOMINAL_FUNCTION_DATA

  val termination : bool -> term option -> local_theory -> Proof.state

  val termination_cmd : bool -> string option -> local_theory -> Proof.state

end

structure Nominal_Function_Termination : NOMINAL_FUNCTION_TERMINATION =

struct

open Function_Lib

open Function_Common

open Nominal_Function_Common

val simp_attribs = map (Attrib.internal o K)

  [Simplifier.simp_add,

   Code.add_default_eqn_attribute,

   Nitpick_Simps.add]

val eqvt_attrib =  Attrib.internal (K Nominal_ThmDecls.eqvt_add)

fun prepare_termination_proof prep_term is_eqvt raw_term_opt lthy =

  let

    val term_opt = Option.map (prep_term lthy) raw_term_opt

    val info = the (case term_opt of

                      SOME t => (import_function_data t lthy

                        handle Option.Option =>

                          error ("Not a function: " ^ quote (Syntax.string_of_term lthy t)))

                    | NONE => (import_last_function lthy handle Option.Option => error "Not a function"))

      val { termination, fs, R, add_simps, case_names, psimps,

        pinducts, defname, eqvts, ...} = info

      val domT = domain_type (fastype_of R)

      val goal = HOLogic.mk_Trueprop

                   (HOLogic.mk_all ("x", domT, mk_acc domT R $ Free ("x", domT)))

      fun afterqed [[totality]] lthy =

        let

          val totality = Thm.close_derivation totality

          val remove_domain_condition =

            full_simplify (HOL_basic_ss addsimps [totality, @{thm True_implies_equals}])

          val tsimps = map remove_domain_condition psimps

          val tinducts = map remove_domain_condition pinducts

          val teqvts = map remove_domain_condition eqvts

          fun qualify n = Binding.name n

            |> Binding.qualify true defname

        in

          lthy

          |> add_simps I "simps" I simp_attribs tsimps

          ||>> Local_Theory.note ((qualify "eqvt", if is_eqvt then [eqvt_attrib] else []), teqvts)

          ||>> Local_Theory.note

             ((qualify "induct",

               [Attrib.internal (K (Rule_Cases.case_names case_names))]),

              tinducts)

          |-> (fn ((simps, (_, eqvts)), (_, inducts)) => fn lthy =>

            let val info' = { is_partial=false, defname=defname, add_simps=add_simps,

              case_names=case_names, fs=fs, R=R, psimps=psimps, pinducts=pinducts,

              simps=SOME simps, inducts=SOME inducts, termination=termination, eqvts=teqvts }

            in

              (info',

               lthy 

               |> Local_Theory.declaration false (add_function_data o morph_function_data info')

               |> Spec_Rules.add Spec_Rules.Equational (fs, tsimps))

            end)

        end

  in

    (goal, afterqed, termination)

  end

fun gen_termination prep_term is_eqvt raw_term_opt lthy =

  let

    val (goal, afterqed, termination) = prepare_termination_proof prep_term is_eqvt raw_term_opt lthy

  in

    lthy

    |> Proof_Context.note_thmss ""

       [((Binding.empty, [Context_Rules.rule_del]), [([allI], [])])] |> snd

    |> Proof_Context.note_thmss ""

       [((Binding.empty, [Context_Rules.intro_bang (SOME 1)]), [([allI], [])])] |> snd

    |> Proof_Context.note_thmss ""

       [((Binding.name "termination", [Context_Rules.intro_bang (SOME 0)]),

         [([Goal.norm_result termination], [])])] |> snd

    |> Proof.theorem NONE (snd oo afterqed) [[(goal, [])]]

  end

val termination = gen_termination Syntax.check_term

val termination_cmd = gen_termination Syntax.read_term

(* outer syntax *)

val option_parser =

  (Scan.optional (Parse.$$$ "(" |-- Parse.!!! 

    ((Parse.reserved "eqvt" >> K (true, true)) ||

     (Parse.reserved "no_eqvt" >> K (true, false))) --| Parse.$$$ ")") (false, false))

val _ =

  Outer_Syntax.local_theory_to_proof "termination" "prove termination of a recursive function"

  Keyword.thy_goal

  (option_parser -- Scan.option Parse.term >> 

    (fn ((is_nominal, is_eqvt), opt_trm) => 

       if is_nominal then termination_cmd is_eqvt opt_trm else Function.termination_cmd opt_trm))

end