(*  Title:      nominal_eqvt.ML

    Author:     Stefan Berghofer

    Author:     Christian Urban

    Automatic proofs for equivariance of inductive predicates.

*)

signature NOMINAL_EQVT =

sig

  val eqvt_rel_tac : xstring -> Proof.context -> local_theory

end

structure Nominal_Eqvt : NOMINAL_EQVT =

struct

open Nominal_Permeq;

open Nominal_ThmDecls;

val atomize_conv = 

  MetaSimplifier.rewrite_cterm (true, false, false) (K (K NONE))

    (HOL_basic_ss addsimps @{thms induct_atomize});

val atomize_intr = Conv.fconv_rule (Conv.prems_conv ~1 atomize_conv);

fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1

  (Conv.params_conv ~1 (K (Conv.prems_conv ~1 atomize_conv)) ctxt));

fun map_term f t = 

  (case f t of

     NONE => map_term' f t 

   | x => x)

and map_term' f (t $ u) = 

    (case (map_term f t, map_term f u) of

        (NONE, NONE) => NONE

      | (SOME t'', NONE) => SOME (t'' $ u)

      | (NONE, SOME u'') => SOME (t $ u'')

      | (SOME t'', SOME u'') => SOME (t'' $ u''))

  | map_term' f (Abs (s, T, t)) = 

      (case map_term f t of

        NONE => NONE

      | SOME t'' => SOME (Abs (s, T, t'')))

  | map_term' _ _  = NONE;

fun map_thm_tac ctxt tac thm =

let

  val monos = Inductive.get_monos ctxt

in

  EVERY [cut_facts_tac [thm] 1, etac rev_mp 1,

    REPEAT_DETERM (FIRSTGOAL (resolve_tac monos)),

    REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))]

end

(* 

 proves F[f t] from F[t] where F[t] is the given theorem  

  - F needs to be monotone

  - f returns either SOME for a term it fires 

    and NONE elsewhere 

*)

fun map_thm ctxt f tac thm =

let

  val opt_goal_trm = map_term f (prop_of thm)

  fun prove goal = 

    Goal.prove ctxt [] [] goal (fn _ => map_thm_tac ctxt tac thm)

in

  case opt_goal_trm of

    NONE => thm

  | SOME goal => prove goal

end

fun transform_prem ctxt names thm =

let

  fun split_conj names (Const ("op &", _) $ p $ q) = 

      (case head_of p of

         Const (name, _) => if name mem names then SOME q else NONE

       | _ => NONE)

  | split_conj _ _ = NONE;

in

  map_thm ctxt (split_conj names) (etac conjunct2 1) thm

end

fun single_case_tac ctxt pred_names pi intro  = 

let

  val thy = ProofContext.theory_of ctxt

  val cpi = Thm.cterm_of thy (mk_minus pi)

  val rule = Drule.instantiate' [] [SOME cpi] @{thm permute_boolE}

in

  eqvt_strict_tac ctxt [] [] THEN' 

  SUBPROOF (fn {prems, context as ctxt, ...} =>

    let

      val prems' = map (transform_prem ctxt pred_names) prems

      val side_cond_tac = EVERY' 

        [ rtac rule, 

          eqvt_strict_tac ctxt @{thms permute_minus_cancel(2)} [],

          resolve_tac prems' ]

    in

      HEADGOAL (rtac intro THEN_ALL_NEW (resolve_tac prems' ORELSE' side_cond_tac)) 

    end) ctxt

end

fun prepare_pred params_no pi pred =

let

  val (c, xs) = strip_comb pred;

  val (xs1, xs2) = chop params_no xs

in

  HOLogic.mk_imp 

    (pred, list_comb (c, xs1 @ map (mk_perm pi) xs2))

end

fun note_named_thm (name, thm) ctxt = 

let

  val thm_name = Binding.qualified_name 

    (Long_Name.qualify (Long_Name.base_name name) "eqvt")

  val attr = Attrib.internal (K eqvt_add)

in

  Local_Theory.note ((thm_name, [attr]), [thm]) ctxt

end

fun eqvt_rel_tac pred_name ctxt = 

let

  val thy = ProofContext.theory_of ctxt

  val ({names, ...}, {raw_induct, intrs, ...}) =

    Inductive.the_inductive ctxt (Sign.intern_const thy pred_name)

  val raw_induct = atomize_induct ctxt raw_induct;

  val intros = map atomize_intr intrs;

  val params_no = length (Inductive.params_of raw_induct)

  val (([raw_concl], [raw_pi]), ctxt') = 

    ctxt |> Variable.import_terms false [concl_of raw_induct] 

         ||>> Variable.variant_fixes ["pi"]

  val pi = Free (raw_pi, @{typ perm})

  val preds = map (fst o HOLogic.dest_imp)

    (HOLogic.dest_conj (HOLogic.dest_Trueprop raw_concl));

  val goal = HOLogic.mk_Trueprop 

    (foldr1 HOLogic.mk_conj (map (prepare_pred params_no pi) preds))

  val thm = Goal.prove ctxt' [] [] goal (fn {context,...} => 

    HEADGOAL (EVERY' (rtac raw_induct :: map (single_case_tac context names pi) intros)))

    |> singleton (ProofContext.export ctxt' ctxt)

  val thms = map (fn th => zero_var_indexes (th RS mp)) (Datatype_Aux.split_conj_thm thm)

in

   ctxt |> fold_map note_named_thm (names ~~ thms)

        |> snd  

end

local structure P = OuterParse and K = OuterKeyword in

val _ =

  OuterSyntax.local_theory "equivariance"

    "prove equivariance for inductive predicate involving nominal datatypes" K.thy_decl

    (P.xname >> eqvt_rel_tac);

end;

end (* structure *)