(*  Title:      nominal_dt_alpha.ML

    Author:     Christian Urban

    Author:     Cezary Kaliszyk

  Deriving support propoerties for the quotient types.

*)

signature NOMINAL_DT_SUPP =

sig

end

structure Nominal_Dt_Supp: NOMINAL_DT_SUPP =

struct

(* supports lemmas for constructors *)

fun mk_supports_goal ctxt qtrm =

  let  

    val vs = fresh_args ctxt qtrm

    val rhs = list_comb (qtrm, vs)

    val lhs = fold (curry HOLogic.mk_prod) vs @{term "()"}

      |> mk_supp

  in

    mk_supports lhs rhs

    |> HOLogic.mk_Trueprop

  end

fun supports_tac ctxt perm_simps =

  let

    val ss1 = HOL_basic_ss addsimps @{thms supports_def fresh_def[symmetric]}

    val ss2 = HOL_ss addsimps @{thms swap_fresh_fresh fresh_Pair}

  in

    EVERY' [ simp_tac ss1,

             Nominal_Permeq.eqvt_strict_tac ctxt perm_simps [],

             simp_tac ss2 ]

  end

fun prove_supports_single ctxt perm_simps qtrm =

  let

    val goal = mk_supports_goal ctxt qtrm 

    val ctxt' = Variable.auto_fixes goal ctxt

  in

    Goal.prove ctxt' [] [] goal

      (K (HEADGOAL (supports_tac ctxt perm_simps)))

    |> singleton (ProofContext.export ctxt' ctxt)

  end

fun prove_supports ctxt perm_simps qtrms =

  map (prove_supports_single ctxt perm_simps) qtrms

(* finite supp lemmas for qtypes *)

fun prove_fsupp ctxt qtys qinduct qsupports_thms =

  let

    val (vs, ctxt') = Variable.variant_fixes (replicate (length qtys) "x") ctxt

    val goals = vs ~~ qtys

      |> map Free

      |> map (mk_finite o mk_supp)

      |> foldr1 (HOLogic.mk_conj)

      |> HOLogic.mk_Trueprop

    val tac = 

      EVERY' [ rtac @{thm supports_finite},

               resolve_tac qsupports_thms,

               asm_simp_tac (HOL_ss addsimps @{thms finite_supp supp_Pair finite_Un}) ]

  in

    Goal.prove ctxt' [] [] goals

      (K (HEADGOAL (rtac qinduct THEN_ALL_NEW tac)))

    |> singleton (ProofContext.export ctxt' ctxt)

    |> Datatype_Aux.split_conj_thm

    |> map zero_var_indexes

  end

(* finite supp instances *)

fun fs_instance qtys qfull_ty_names tvs qfsupp_thms lthy =

  let

    val lthy1 = 

      lthy

      |> Local_Theory.exit_global

      |> Class.instantiation (qfull_ty_names, tvs, @{sort fs}) 

    fun tac _ =

      Class.intro_classes_tac [] THEN

        (ALLGOALS (resolve_tac qfsupp_thms))

  in

    lthy1

    |> Class.prove_instantiation_exit tac 

    |> Named_Target.theory_init

  end

(* proves that fv and fv_bn equals supp *)

fun gen_mk_goals fv supp =

  let

    val arg_ty = 

      fastype_of fv

      |> domain_type

  in

    (arg_ty, fn x => HOLogic.mk_eq (fv $ x, supp x))

  end

fun mk_fvs_goals fv = gen_mk_goals fv mk_supp

fun mk_fv_bns_goals fv_bn alpha_bn = gen_mk_goals fv_bn (mk_supp_rel alpha_bn)

fun add_ss thms =

  HOL_basic_ss addsimps thms

fun symmetric thms = 

  map (fn thm => thm RS @{thm sym}) thms

val supp_Abs_set = @{thms supp_Abs(1)[symmetric]}

val supp_Abs_res = @{thms supp_Abs(2)[symmetric]}

val supp_Abs_lst = @{thms supp_Abs(3)[symmetric]}

fun mk_supp_abs ctxt (BC (Set, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_Abs_set 

  | mk_supp_abs ctxt (BC (Res, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_Abs_res

  | mk_supp_abs ctxt (BC (Lst, _, _)) = EqSubst.eqsubst_tac ctxt [1] supp_Abs_lst

fun mk_supp_abs_tac ctxt [] = []

  | mk_supp_abs_tac ctxt (BC (_, [], _)::xs) = mk_supp_abs_tac ctxt xs

  | mk_supp_abs_tac ctxt (bc::xs) = (DETERM o mk_supp_abs ctxt bc)::mk_supp_abs_tac ctxt xs

fun mk_bn_supp_abs_tac trm =

  trm

  |> fastype_of

  |> body_type

  |> (fn ty => case ty of

        @{typ "atom set"}  => simp_tac (add_ss supp_Abs_set)

      | @{typ "atom list"} => simp_tac (add_ss supp_Abs_lst)

      | _ => raise TERM ("mk_bn_supp_abs_tac", [trm]))

val thms1 = @{thms supp_Pair supp_eqvt[symmetric] Un_assoc conj_assoc}

val thms2 = @{thms de_Morgan_conj Collect_disj_eq finite_Un}

val thms3 = @{thms alphas prod_alpha_def prod_fv.simps prod_rel_def permute_prod_def 

  prod.recs prod.cases prod.inject not_True_eq_False empty_def[symmetric] finite.emptyI}

fun prove_fv_supp qtys qtrms fvs fv_bns alpha_bns fv_simps eq_iffs perm_simps 

  fv_bn_eqvts qinduct bclausess ctxt =

  let

    val goals1 = map mk_fvs_goals fvs

    val goals2 = map2 mk_fv_bns_goals fv_bns alpha_bns   

    fun tac ctxt =

      SUBGOAL (fn (goal, i) =>

        let

          val (fv_fun, arg) = 

            goal |> Envir.eta_contract

                 |> Logic.strip_assums_concl

                 |> HOLogic.dest_Trueprop

                 |> fst o HOLogic.dest_eq

                 |> dest_comb

          val supp_abs_tac = 

            case (AList.lookup (op=) (qtrms ~~ bclausess) (head_of arg)) of

              SOME bclauses => EVERY' (mk_supp_abs_tac ctxt bclauses)

            | NONE => mk_bn_supp_abs_tac fv_fun

        in

          EVERY' [ TRY o asm_full_simp_tac (add_ss (@{thm supp_Pair[symmetric]}::fv_simps)),

                   TRY o supp_abs_tac,

                   TRY o simp_tac (add_ss @{thms supp_def supp_rel_def}),

                   TRY o Nominal_Permeq.eqvt_tac ctxt (perm_simps @ fv_bn_eqvts) [], 

                   TRY o simp_tac (add_ss (@{thms Abs_eq_iff} @ eq_iffs)),

                   TRY o asm_full_simp_tac (add_ss thms3),

                   TRY o simp_tac (add_ss thms2),

                   TRY o asm_full_simp_tac (add_ss (thms1 @ (symmetric fv_bn_eqvts)))] i

        end)

  in

    induct_prove qtys (goals1 @ goals2) qinduct tac ctxt

    |> map atomize

    |> map (simplify (HOL_basic_ss addsimps @{thms fun_eq_iff[symmetric]}))

  end

fun prove_bns_finite qtys qbns qinduct qbn_simps ctxt =

  let

    fun mk_goal qbn = 

      let

        val arg_ty = domain_type (fastype_of qbn)

        val finite = @{term "finite :: atom set => bool"}

      in

        (arg_ty, fn x => finite $ (to_set (qbn $ x)))

      end

    val props = map mk_goal qbns

    val ss_tac = asm_full_simp_tac (HOL_basic_ss addsimps (qbn_simps @ 

      @{thms set.simps set_append finite_insert finite.emptyI finite_Un}))

  in

    induct_prove qtys props qinduct (K ss_tac) ctxt

  end

fun prove_perm_bn_alpha_thms qtys qperm_bns alpha_bns qinduct qperm_bn_simps qeq_iffs qalpha_refls ctxt =

  let 

    val ([p], ctxt') = Variable.variant_fixes ["p"] ctxt

    val p = Free (p, @{typ perm})

    fun mk_goal qperm_bn alpha_bn =

      let

        val arg_ty = domain_type (fastype_of alpha_bn)

      in

        (arg_ty, fn x => (mk_id (Abs ("", arg_ty, alpha_bn $ Bound 0 $ (qperm_bn $ p $ Bound 0)))) $ x)

      end

    val props = map2 mk_goal qperm_bns alpha_bns

  in

    induct_prove qtys props qinduct (K ss_tac) ctxt'

    |> ProofContext.export ctxt' ctxt

    |> map (simplify (HOL_basic_ss addsimps @{thms id_def})) 

  end

end (* structure *)