(*  Title:      nominal_inductive.ML

    Author:     Christian Urban

    Author:     Tjark Weber

    Infrastructure for proving strong induction theorems

    for inductive predicates involving nominal datatypes.

    Code based on an earlier version by Stefan Berghofer.

*)

signature NOMINAL_INDUCTIVE =

sig

  val prove_strong_inductive: string list -> string list -> term list list -> thm -> thm list ->

    Proof.context -> Proof.state

  val prove_strong_inductive_cmd: xstring * (string * string list) list -> Proof.context -> Proof.state

end

structure Nominal_Inductive : NOMINAL_INDUCTIVE =

struct

fun mk_cplus p q =

  Thm.apply (Thm.apply @{cterm "plus :: perm => perm => perm"} p) q

fun mk_cminus p =

  Thm.apply @{cterm "uminus :: perm => perm"} p

fun minus_permute_intro_tac p =

  rtac (Drule.instantiate' [] [SOME (mk_cminus p)] @{thm permute_boolE})

fun minus_permute_elim p thm =

  thm RS (Drule.instantiate' [] [NONE, SOME (mk_cminus p)] @{thm permute_boolI})

(* fixme: move to nominal_library *)

fun real_head_of (@{term Trueprop} $ t) = real_head_of t

  | real_head_of (Const (@{const_name Pure.imp}, _) $ _ $ t) = real_head_of t

  | real_head_of (Const (@{const_name Pure.all}, _) $ Abs (_, _, t)) = real_head_of t

  | real_head_of (Const (@{const_name All}, _) $ Abs (_, _, t)) = real_head_of t

  | real_head_of (Const ("HOL.induct_forall", _) $ Abs (_, _, t)) = real_head_of t

  | real_head_of t = head_of t

fun mk_vc_compat (avoid, avoid_trm) prems concl_args params =

  if null avoid then

    []

  else

    let

      val vc_goal = concl_args

        |> HOLogic.mk_tuple

        |> mk_fresh_star avoid_trm

        |> HOLogic.mk_Trueprop

        |> (curry Logic.list_implies) prems

        |> fold_rev (Logic.all o Free) params

      val finite_goal = avoid_trm

        |> mk_finite

        |> HOLogic.mk_Trueprop

        |> (curry Logic.list_implies) prems

        |> fold_rev (Logic.all o Free) params

    in

      [vc_goal, finite_goal]

    end

(* fixme: move to nominal_library *)

fun map_term prop f trm =

  if prop trm 

  then f trm

  else case trm of

    (t1 $ t2) => map_term prop f t1 $ map_term prop f t2

  | Abs (x, T, t) => Abs (x, T, map_term prop f t)

  | _ => trm

fun add_p_c p (c, c_ty) trm =

  let

    val (P, args) = strip_comb trm

    val (P_name, P_ty) = dest_Free P

    val (ty_args, bool) = strip_type P_ty

    val args' = map (mk_perm p) args

  in

    list_comb (Free (P_name, (c_ty :: ty_args) ---> bool),  c :: args')

    |> (fn t => HOLogic.all_const c_ty $ lambda c t )

    |> (fn t => HOLogic.all_const @{typ perm} $  lambda p t)

  end

fun induct_forall_const T =

  Const ("HOL.induct_forall", (T --> @{typ bool}) --> @{typ bool})

fun mk_induct_forall (a, T) t =

  induct_forall_const T $ Abs (a, T, t)

fun add_c_prop qnt Ps (c, c_name, c_ty) trm =

  let

    fun add t =

      let

        val (P, args) = strip_comb t

        val (P_name, P_ty) = dest_Free P

        val (ty_args, bool) = strip_type P_ty

        val args' = args

          |> qnt ? map (incr_boundvars 1)

      in

        list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')

          |> qnt ? mk_induct_forall (c_name, c_ty)

      end

  in

    map_term (member (op =) Ps o head_of) add trm

  end

fun prep_prem Ps c_name c_ty (avoid, avoid_trm) (params, prems, concl) =

  let

    val prems' = prems

      |> map (incr_boundvars 1)

      |> map (add_c_prop true Ps (Bound 0, c_name, c_ty))

    val avoid_trm' = avoid_trm

      |> fold_rev absfree (params @ [(c_name, c_ty)])

      |> strip_abs_body

      |> (fn t => mk_fresh_star_ty c_ty t (Bound 0))

      |> HOLogic.mk_Trueprop

    val prems'' =

      if null avoid

      then prems'

      else avoid_trm' :: prems'

    val concl' = concl

      |> incr_boundvars 1

      |> add_c_prop false Ps (Bound 0, c_name, c_ty)

  in

    mk_full_horn (params @ [(c_name, c_ty)]) prems'' concl'

  end

(* fixme: move to nominal_library *)

fun same_name (Free (a1, _), Free (a2, _)) = (a1 = a2)

  | same_name (Var (a1, _), Var (a2, _)) = (a1 = a2)

  | same_name (Const (a1, _), Const (a2, _)) = (a1 = a2)

  | same_name _ = false

(* fixme: move to nominal_library *)

fun map7 _ [] [] [] [] [] [] [] = []

  | map7 f (x :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) (r :: rs) (s :: ss) =

      f x y z u v r s :: map7 f xs ys zs us vs rs ss

(* local abbreviations *)

local

  open Nominal_Permeq

in

  (* by default eqvt_strict_config contains unwanted @{thm permute_pure} *) 

  val eqvt_sconfig = eqvt_strict_config addpres @{thms permute_minus_cancel}

  fun eqvt_stac ctxt = eqvt_tac ctxt eqvt_sconfig

  fun eqvt_srule ctxt = eqvt_rule ctxt eqvt_sconfig

end

val all_elims = 

  let

    fun spec' ct =

      Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec}

  in

    fold (fn ct => fn th => th RS spec' ct)

  end

fun helper_tac flag prm p ctxt =

  Subgoal.SUBPROOF (fn {context, prems, ...} =>

    let

      val prems' = prems

        |> map (minus_permute_elim p)

        |> map (eqvt_srule context)

      val prm' = (prems' MRS prm)

        |> flag ? (all_elims [p])

        |> flag ? (eqvt_srule context)

    in

      asm_full_simp_tac (put_simpset HOL_ss ctxt addsimps (prm' :: @{thms induct_forall_def})) 1

    end) ctxt

fun non_binder_tac prem intr_cvars Ps ctxt = 

  Subgoal.SUBPROOF (fn {context, params, prems, ...} =>

    let

      val thy = Proof_Context.theory_of context

      val (prms, p, _) = split_last2 (map snd params)

      val prm_tys = map (fastype_of o term_of) prms

      val cperms = map (cterm_of thy o perm_const) prm_tys

      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 

      val prem' = prem

        |> cterm_instantiate (intr_cvars ~~ p_prms) 

        |> eqvt_srule ctxt

      (* for inductive-premises*)

      fun tac1 prm = helper_tac true prm p context 

      (* for non-inductive premises *)   

      fun tac2 prm =  

        EVERY' [ minus_permute_intro_tac p, 

                 eqvt_stac context, 

                 helper_tac false prm p context ]

      fun select prm (t, i) =

        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i

    in

      EVERY1 [ eqvt_stac context, 

               rtac prem', 

               RANGE (map (SUBGOAL o select) prems) ]

    end) ctxt

fun fresh_thm ctxt user_thm p c concl_args avoid_trm =

  let

    val conj1 = 

      mk_fresh_star (mk_perm (Bound 0) (mk_perm p avoid_trm)) c

    val conj2 =

      mk_fresh_star_ty @{typ perm} (mk_supp (HOLogic.mk_tuple (map (mk_perm p) concl_args))) (Bound 0)

    val fresh_goal = mk_exists ("q", @{typ perm}) (HOLogic.mk_conj (conj1, conj2))

      |> HOLogic.mk_Trueprop

    val ss = @{thms finite_supp supp_Pair finite_Un permute_finite} @ 

             @{thms fresh_star_Pair fresh_star_permute_iff}

    val simp = asm_full_simp_tac (put_simpset HOL_ss ctxt addsimps ss)

  in 

    Goal.prove ctxt [] [] fresh_goal

      (K (HEADGOAL (rtac @{thm at_set_avoiding2} 

          THEN_ALL_NEW EVERY' [cut_facts_tac user_thm, REPEAT o etac @{thm conjE}, simp])))

  end

val supp_perm_eq' = @{lemma "fresh_star (supp (permute p x)) q ==> permute p x == permute (q + p) x" 

  by (simp add: supp_perm_eq)}

val fresh_star_plus = @{lemma "fresh_star (permute q (permute p x)) c ==> fresh_star (permute (q + p) x) c" 

  by (simp add: permute_plus)}

fun binder_tac prem intr_cvars param_trms Ps user_thm avoid_trm concl_args ctxt = 

  Subgoal.FOCUS (fn {context = ctxt, params, prems, concl, ...} =>

    let

      val thy = Proof_Context.theory_of ctxt

      val (prms, p, c) = split_last2 (map snd params)

      val prm_trms = map term_of prms

      val prm_tys = map fastype_of prm_trms

      val avoid_trm' = subst_free (param_trms ~~ prm_trms) avoid_trm 

      val concl_args' = map (subst_free (param_trms ~~ prm_trms)) concl_args 

      val user_thm' = map (cterm_instantiate (intr_cvars ~~ prms)) user_thm

        |> map (full_simplify (put_simpset HOL_ss ctxt addsimps (@{thm fresh_star_Pair}::prems)))

      val fthm = fresh_thm ctxt user_thm' (term_of p) (term_of c) concl_args' avoid_trm'

      val (([(_, q)], fprop :: fresh_eqs), ctxt') = Obtain.result

              (K (EVERY1 [etac @{thm exE}, 

                          full_simp_tac (put_simpset HOL_basic_ss ctxt

                            addsimps @{thms supp_Pair fresh_star_Un}),

                          REPEAT o etac @{thm conjE},

                          dtac fresh_star_plus,

                          REPEAT o dtac supp_perm_eq'])) [fthm] ctxt 

      val expand_conv = Conv.try_conv (Conv.rewrs_conv fresh_eqs)

      fun expand_conv_bot ctxt = Conv.bottom_conv (K expand_conv) ctxt

      val cperms = map (cterm_of thy o perm_const) prm_tys

      val qp_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 (mk_cplus q p) ct2) cperms prms 

      val prem' = prem

        |> cterm_instantiate (intr_cvars ~~ qp_prms)

        |> eqvt_srule ctxt

      val fprop' = eqvt_srule ctxt' fprop 

      val tac_fresh = simp_tac (put_simpset HOL_basic_ss ctxt' addsimps [fprop'])

      (* for inductive-premises*)

      fun tac1 prm = helper_tac true prm (mk_cplus q p) ctxt' 

      (* for non-inductive premises *)   

      fun tac2 prm =  

        EVERY' [ minus_permute_intro_tac (mk_cplus q p), 

                 eqvt_stac ctxt, 

                 helper_tac false prm (mk_cplus q p) ctxt' ]

      fun select prm (t, i) =

        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i

      val side_thm = Goal.prove ctxt' [] [] (term_of concl)

        (fn {context, ...} => 

           EVERY1 [ CONVERSION (expand_conv_bot context),

                    eqvt_stac context,

                    rtac prem',

                    RANGE (tac_fresh :: map (SUBGOAL o select) prems) ])

        |> singleton (Proof_Context.export ctxt' ctxt)        

    in

      rtac side_thm 1

    end) ctxt

fun case_tac ctxt Ps avoid avoid_trm intr_cvars param_trms prem user_thm concl_args =

  let

    val tac1 = non_binder_tac prem intr_cvars Ps ctxt

    val tac2 = binder_tac prem intr_cvars param_trms Ps user_thm avoid_trm concl_args ctxt

  in 

    EVERY' [ rtac @{thm allI}, rtac @{thm allI}, 

             if null avoid then tac1 else tac2 ]

  end

fun prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms concl_args 

  {prems, context} =

  let

    val cases_tac = 

      map7 (case_tac context Ps) avoids avoid_trms intr_cvars param_trms prems user_thms concl_args

  in 

    EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]

  end

val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (!!c. Q ==> P (0::perm) c)" by simp}

fun prove_strong_inductive pred_names rule_names avoids raw_induct intrs ctxt =

  let

    val thy = Proof_Context.theory_of ctxt

    val ((_, [raw_induct']), ctxt') = Variable.import true [raw_induct] ctxt

    val (ind_prems, ind_concl) = raw_induct'

      |> prop_of

      |> Logic.strip_horn

      |>> map strip_full_horn

    val params = map (fn (x, _, _) => x) ind_prems

    val param_trms = (map o map) Free params  

    val intr_vars_tys = map (fn t => rev (Term.add_vars (prop_of t) [])) intrs

    val intr_vars = (map o map) fst intr_vars_tys

    val intr_vars_substs = map2 (curry (op ~~)) intr_vars param_trms

    val intr_cvars = (map o map) (cterm_of thy o Var) intr_vars_tys      

    val (intr_prems, intr_concls) = intrs

      |> map prop_of

      |> map2 subst_Vars intr_vars_substs

      |> map Logic.strip_horn

      |> split_list

    val intr_concls_args =

      map (snd o fixed_nonfixed_args ctxt' o HOLogic.dest_Trueprop) intr_concls

    val avoid_trms = avoids

      |> (map o map) (setify ctxt') 

      |> map fold_union

    val vc_compat_goals = 

      map4 mk_vc_compat (avoids ~~ avoid_trms) intr_prems intr_concls_args params

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

    val c_ty = TFree (a, @{sort fs})

    val c = Free (c_name, c_ty)

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

    val (preconds, ind_concls) = ind_concl

      |> HOLogic.dest_Trueprop

      |> HOLogic.dest_conj 

      |> map HOLogic.dest_imp

      |> split_list

    val Ps = map (fst o strip_comb) ind_concls

    val ind_concl' = ind_concls

      |> map (add_p_c p (c, c_ty))

      |> (curry (op ~~)) preconds  

      |> map HOLogic.mk_imp

      |> fold_conj

      |> HOLogic.mk_Trueprop

    val ind_prems' = ind_prems

      |> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)   

    fun after_qed ctxt_outside user_thms ctxt = 

      let

        val strong_ind_thms = Goal.prove ctxt [] ind_prems' ind_concl' 

        (prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms intr_concls_args) 

          |> singleton (Proof_Context.export ctxt ctxt_outside)

          |> Datatype_Aux.split_conj_thm

          |> map (fn thm => thm RS normalise)

          |> map (asm_full_simplify (put_simpset HOL_basic_ss ctxt

              addsimps @{thms permute_zero induct_rulify})) 

          |> map (Drule.rotate_prems (length ind_prems'))

          |> map zero_var_indexes

        val qualified_thm_name = pred_names

          |> map Long_Name.base_name

          |> space_implode "_"

          |> (fn s => Binding.qualify false s (Binding.name "strong_induct"))

        val attrs = 

          [ Attrib.internal (K (Rule_Cases.consumes 1)),

            Attrib.internal (K (Rule_Cases.case_names rule_names)) ]

      in

        ctxt

        |> Local_Theory.note ((qualified_thm_name, attrs), strong_ind_thms)    

        |> snd   

      end

  in

    Proof.theorem NONE (after_qed ctxt) ((map o map) (rpair []) vc_compat_goals) ctxt''

  end

fun prove_strong_inductive_cmd (pred_name, avoids) ctxt =

  let

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

      Inductive.the_inductive ctxt (long_name ctxt pred_name)

    val rule_names = hd names

      |> the o Induct.lookup_inductP ctxt

      |> fst o Rule_Cases.get

      |> map (fst o fst)

    val case_names = map fst avoids

    val _ = case duplicates (op =) case_names of

        [] => ()

      | xs => error ("Duplicate case names: " ^ commas_quote xs)

    val _ = case subtract (op =) rule_names case_names of

        [] => ()

      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs)

    val avoids_ordered = order_default (op =) [] rule_names avoids

    fun read_avoids avoid_trms intr =

      let

        (* fixme hack *)

        val (((_, ctrms), _), ctxt') = Variable.import true [intr] ctxt

        val trms = map (term_of o snd) ctrms

        val ctxt'' = fold Variable.declare_term trms ctxt'

      in

        map (Syntax.read_term ctxt'') avoid_trms

      end

    val avoid_trms = map2 read_avoids avoids_ordered intrs

  in

    prove_strong_inductive names rule_names avoid_trms raw_induct intrs ctxt

  end

(* outer syntax *)

local

  val single_avoid_parser =

    Parse.name -- (@{keyword ":"} |-- Parse.and_list1 Parse.term)

  val avoids_parser =

    Scan.optional (@{keyword "avoids"} |-- Parse.enum1 "|" single_avoid_parser) []

  val main_parser = Parse.xname -- avoids_parser

in

  val _ =

    Outer_Syntax.local_theory_to_proof @{command_spec "nominal_inductive"}

      "prove strong induction theorem for inductive predicate involving nominal datatypes"

        (main_parser >> prove_strong_inductive_cmd)

end

end