(*  Title:      nominal_dt_alpha.ML

    Author:     Cezary Kaliszyk

    Author:     Christian Urban

  Definitions and proofs for the alpha-relations.

*)

signature NOMINAL_DT_ALPHA =

sig

  val define_raw_alpha: Datatype_Aux.descr -> (string * sort) list -> bn_info ->

    bclause list list list -> term list -> Proof.context -> 

    term list * term list * thm list * thm list * thm * local_theory

  val mk_alpha_distincts: Proof.context -> thm list -> thm list list -> 

    term list -> term list -> bn_info -> thm list * thm list

  val mk_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> 

    thm list -> (thm list * thm list)

  val alpha_prove: term list -> (term * ((term * term) -> term)) list -> thm -> 

    (Proof.context -> int -> tactic) -> Proof.context -> thm list

  val raw_prove_refl: term list -> term list -> thm list -> thm -> Proof.context -> thm list

  val raw_prove_sym: term list -> thm list -> thm -> Proof.context -> thm list

  val raw_prove_trans: term list -> thm list -> thm list -> thm -> thm list -> Proof.context -> thm list

  val raw_prove_equivp: term list -> thm list -> thm list -> thm list -> Proof.context -> thm list

  val raw_prove_bn_imp: term list -> term list -> thm list -> thm -> Proof.context -> thm list

  val raw_fv_bn_rsp_aux: term list -> term list -> term list -> term list -> 

    term list -> thm -> thm list -> Proof.context -> thm list

end

structure Nominal_Dt_Alpha: NOMINAL_DT_ALPHA =

struct

fun lookup xs x = the (AList.lookup (op=) xs x)

fun group xs = AList.group (op=) xs

(** definition of the inductive rules for alpha and alpha_bn **)

(* construct the compound terms for prod_fv and prod_alpha *)

fun mk_prod_fv (t1, t2) =

let

  val ty1 = fastype_of t1

  val ty2 = fastype_of t2 

  val resT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2) --> @{typ "atom set"}

in

  Const (@{const_name "prod_fv"}, [ty1, ty2] ---> resT) $ t1 $ t2

end

fun mk_prod_alpha (t1, t2) =

let

  val ty1 = fastype_of t1

  val ty2 = fastype_of t2 

  val prodT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2)

  val resT = [prodT, prodT] ---> @{typ "bool"}

in

  Const (@{const_name "prod_alpha"}, [ty1, ty2] ---> resT) $ t1 $ t2

end

(* generates the compound binder terms *)

fun mk_binders lthy bmode args bodies = 

let  

  fun bind_set lthy args (NONE, i) = setify lthy (nth args i)

    | bind_set _ args (SOME bn, i) = bn $ (nth args i)

  fun bind_lst lthy args (NONE, i) = listify lthy (nth args i)

    | bind_lst _ args (SOME bn, i) = bn $ (nth args i)

  val (combine_fn, bind_fn) =

    case bmode of

      Lst => (mk_append, bind_lst) 

    | Set => (mk_union,  bind_set)

    | Res => (mk_union,  bind_set)

in

  bodies

  |> map (bind_fn lthy args)

  |> foldl1 combine_fn 

end

(* produces the term for an alpha with abstraction *)

fun mk_alpha_term bmode fv alpha args args' binders binders' =

let

  val (alpha_name, binder_ty) = 

    case bmode of

      Lst => (@{const_name "alpha_lst"}, @{typ "atom list"})

    | Set => (@{const_name "alpha_gen"}, @{typ "atom set"})

    | Res => (@{const_name "alpha_res"}, @{typ "atom set"})

  val ty = fastype_of args

  val pair_ty = HOLogic.mk_prodT (binder_ty, ty)

  val alpha_ty = [ty, ty] ---> @{typ "bool"}

  val fv_ty = ty --> @{typ "atom set"}

  val pair_lhs = HOLogic.mk_prod (binders, args)

  val pair_rhs = HOLogic.mk_prod (binders', args')

in

  HOLogic.exists_const @{typ perm} $ Abs ("p", @{typ perm},

    Const (alpha_name, [pair_ty, alpha_ty, fv_ty, @{typ "perm"}, pair_ty] ---> @{typ bool}) 

      $ pair_lhs $ alpha $ fv $ (Bound 0) $ pair_rhs)

end

(* for non-recursive binders we have to produce alpha_bn premises *)

fun mk_alpha_bn_prem alpha_bn_map args args' bodies binder = 

  case binder of

    (NONE, _) => []

  | (SOME bn, i) =>

     if member (op=) bodies i then [] 

     else [lookup alpha_bn_map bn $ nth args i $ nth args' i]

(* generat the premises for an alpha rule; mk_frees is used

   if no binders are present *)

fun mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause =

let

  fun mk_frees i =

    let

      val arg = nth args i

      val arg' = nth args' i

      val ty = fastype_of arg

    in

      if nth is_rec i

      then fst (lookup alpha_map ty) $ arg $ arg'

      else HOLogic.mk_eq (arg, arg')

    end

  fun mk_alpha_fv i = 

    let

      val ty = fastype_of (nth args i)

    in

      case AList.lookup (op=) alpha_map ty of

        NONE => (HOLogic.eq_const ty, supp_const ty) 

      | SOME (alpha, fv) => (alpha, fv) 

    end  

in

  case bclause of

    BC (_, [], bodies) => map (HOLogic.mk_Trueprop o mk_frees) bodies 

  | BC (bmode, binders, bodies) => 

    let

      val (alphas, fvs) = split_list (map mk_alpha_fv bodies)

      val comp_fv = foldl1 mk_prod_fv fvs

      val comp_alpha = foldl1 mk_prod_alpha alphas

      val comp_args = foldl1 HOLogic.mk_prod (map (nth args) bodies)

      val comp_args' = foldl1 HOLogic.mk_prod (map (nth args') bodies)

      val comp_binders = mk_binders lthy bmode args binders

      val comp_binders' = mk_binders lthy bmode args' binders

      val alpha_prem = 

        mk_alpha_term bmode comp_fv comp_alpha comp_args comp_args' comp_binders comp_binders'

      val alpha_bn_prems = flat (map (mk_alpha_bn_prem alpha_bn_map args args' bodies) binders)

    in

      map HOLogic.mk_Trueprop (alpha_prem::alpha_bn_prems)

    end

end

(* produces the introduction rule for an alpha rule *)

fun mk_alpha_intros lthy alpha_map alpha_bn_map (constr, ty, arg_tys, is_rec) bclauses = 

let

  val arg_names = Datatype_Prop.make_tnames arg_tys

  val arg_names' = Name.variant_list arg_names arg_names

  val args = map Free (arg_names ~~ arg_tys)

  val args' = map Free (arg_names' ~~ arg_tys)

  val alpha = fst (lookup alpha_map ty)

  val concl = HOLogic.mk_Trueprop (alpha $ list_comb (constr, args) $ list_comb (constr, args'))

  val prems = map (mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args')) bclauses

in

  Library.foldr Logic.mk_implies (flat prems, concl)

end

(* produces the premise of an alpha-bn rule; we only need to

   treat the case special where the binding clause is empty;

   - if the body is not included in the bn_info, then we either

     produce an equation or an alpha-premise

   - if the body is included in the bn_info, then we create

     either a recursive call to alpha-bn, or no premise  *)

fun mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args') bclause =

let

  fun mk_alpha_bn_prem alpha_map alpha_bn_map bn_args (args, args') i = 

  let

    val arg = nth args i

    val arg' = nth args' i

    val ty = fastype_of arg

  in

    case AList.lookup (op=) bn_args i of

      NONE => (case (AList.lookup (op=) alpha_map ty) of

                 NONE =>  [HOLogic.mk_eq (arg, arg')]

               | SOME (alpha, _) => [alpha $ arg $ arg'])

    | SOME (NONE) => []

    | SOME (SOME bn) => [lookup alpha_bn_map bn $ arg $ arg']

  end  

in

  case bclause of

    BC (_, [], bodies) => 

      map HOLogic.mk_Trueprop 

        (flat (map (mk_alpha_bn_prem alpha_map alpha_bn_map bn_args (args, args')) bodies))

  | _ => mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause

end

fun mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map (bn_args, (constr, _, arg_tys, is_rec)) bclauses =

let

  val arg_names = Datatype_Prop.make_tnames arg_tys

  val arg_names' = Name.variant_list arg_names arg_names

  val args = map Free (arg_names ~~ arg_tys)

  val args' = map Free (arg_names' ~~ arg_tys)

  val alpha_bn = lookup alpha_bn_map bn_trm

  val concl = HOLogic.mk_Trueprop (alpha_bn $ list_comb (constr, args) $ list_comb (constr, args'))

  val prems = map (mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args')) bclauses

in

  Library.foldr Logic.mk_implies (flat prems, concl)

end

fun mk_alpha_bn_intros lthy alpha_map alpha_bn_map constrs_info bclausesss (bn_trm, bn_n, bn_argss) = 

let

  val nth_constrs_info = nth constrs_info bn_n

  val nth_bclausess = nth bclausesss bn_n

in

  map2 (mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map) (bn_argss ~~ nth_constrs_info) nth_bclausess

end

fun define_raw_alpha descr sorts bn_info bclausesss fvs lthy =

let

  val alpha_names = prefix_dt_names descr sorts "alpha_"

  val alpha_arg_tys = all_dtyps descr sorts

  val alpha_tys = map (fn ty => [ty, ty] ---> @{typ bool}) alpha_arg_tys

  val alpha_frees = map Free (alpha_names ~~ alpha_tys)

  val alpha_map = alpha_arg_tys ~~ (alpha_frees ~~ fvs)

  val (bns, bn_tys) = split_list (map (fn (bn, i, _) => (bn, i)) bn_info)

  val bn_names = map (fn bn => Long_Name.base_name (fst (dest_Const bn))) bns

  val alpha_bn_names = map (prefix "alpha_") bn_names

  val alpha_bn_arg_tys = map (fn i => nth_dtyp descr sorts i) bn_tys

  val alpha_bn_tys = map (fn ty => [ty, ty] ---> @{typ "bool"}) alpha_bn_arg_tys

  val alpha_bn_frees = map Free (alpha_bn_names ~~ alpha_bn_tys)

  val alpha_bn_map = bns ~~ alpha_bn_frees

  val constrs_info = all_dtyp_constrs_types descr sorts

  val alpha_intros = map2 (map2 (mk_alpha_intros lthy alpha_map alpha_bn_map)) constrs_info bclausesss 

  val alpha_bn_intros = map (mk_alpha_bn_intros lthy alpha_map alpha_bn_map constrs_info bclausesss) bn_info

  val all_alpha_names = map (fn (a, ty) => ((Binding.name a, ty), NoSyn))

    (alpha_names @ alpha_bn_names ~~ alpha_tys @ alpha_bn_tys)

  val all_alpha_intros = map (pair Attrib.empty_binding) (flat alpha_intros @ flat alpha_bn_intros)

  val (alphas, lthy') = Inductive.add_inductive_i

     {quiet_mode = true, verbose = false, alt_name = Binding.empty,

      coind = false, no_elim = false, no_ind = false, skip_mono = false, fork_mono = false}

     all_alpha_names [] all_alpha_intros [] lthy

  val all_alpha_trms_loc = #preds alphas;

  val alpha_induct_loc = #raw_induct alphas;

  val alpha_intros_loc = #intrs alphas;

  val alpha_cases_loc = #elims alphas;

  val phi = ProofContext.export_morphism lthy' lthy;

  val all_alpha_trms = map (Morphism.term phi) all_alpha_trms_loc;

  val alpha_induct = Morphism.thm phi alpha_induct_loc;

  val alpha_intros = map (Morphism.thm phi) alpha_intros_loc

  val alpha_cases = map (Morphism.thm phi) alpha_cases_loc

  val (alpha_trms, alpha_bn_trms) = chop (length fvs) all_alpha_trms

in

  (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy')

end

(** produces the distinctness theorems **)

(* transforms the distinctness theorems of the constructors 

   to "not-alphas" of the constructors *)

fun mk_alpha_distinct_goal alpha neq =

let

  val (lhs, rhs) = 

    neq

    |> HOLogic.dest_Trueprop

    |> HOLogic.dest_not

    |> HOLogic.dest_eq

in

  alpha $ lhs $ rhs

  |> HOLogic.mk_not

  |> HOLogic.mk_Trueprop

end

fun distinct_tac cases distinct_thms =

  rtac notI THEN' eresolve_tac cases 

  THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps distinct_thms)

fun mk_alpha_distinct ctxt cases_thms (distinct_thm, alpha) =

let

  val ((_, thms), ctxt') = Variable.import false distinct_thm ctxt

  val goals = map (mk_alpha_distinct_goal alpha o prop_of) thms

  val nrels = map (fn t => Goal.prove ctxt' [] [] t (K (distinct_tac cases_thms distinct_thm 1))) goals

in

  Variable.export ctxt' ctxt nrels

end

fun mk_alpha_distincts ctxt alpha_cases constrs_distinct_thms alpha_trms alpha_bn_trms bn_infos =

let

  val alpha_distincts = 

    map (mk_alpha_distinct ctxt alpha_cases) (constrs_distinct_thms ~~ alpha_trms)

  val distinc_thms = map 

  val alpha_bn_distincts_aux = map (fn (_, i, _) => nth constrs_distinct_thms i) bn_infos

  val alpha_bn_distincts =  

    map (mk_alpha_distinct ctxt alpha_cases) (alpha_bn_distincts_aux ~~ alpha_bn_trms)

in

  (flat alpha_distincts, flat alpha_bn_distincts)

end

(** produces the alpha_eq_iff simplification rules **)

(* in case a theorem is of the form (C.. = C..), it will be

   rewritten to ((C.. = C..) = True) *)

fun mk_simp_rule thm =

  case (prop_of thm) of

    @{term "Trueprop"} $ (Const (@{const_name "op ="}, _) $ _ $ _) => @{thm eqTrueI} OF [thm]

  | _ => thm

fun alpha_eq_iff_tac dist_inj intros elims =

  SOLVED' (asm_full_simp_tac (HOL_ss addsimps intros)) ORELSE'

  (rtac @{thm iffI} THEN' 

    RANGE [eresolve_tac elims THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps dist_inj),

           asm_full_simp_tac (HOL_ss addsimps intros)])

fun mk_alpha_eq_iff_goal thm =

  let

    val prop = prop_of thm;

    val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop);

    val hyps = map HOLogic.dest_Trueprop (Logic.strip_imp_prems prop);

    fun list_conj l = foldr1 HOLogic.mk_conj l;

  in

    if hyps = [] then HOLogic.mk_Trueprop concl

    else HOLogic.mk_Trueprop (HOLogic.mk_eq (concl, list_conj hyps))

  end;

fun mk_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims =

let

  val ((_, thms_imp), ctxt') = Variable.import false alpha_intros ctxt;

  val goals = map mk_alpha_eq_iff_goal thms_imp;

  val tac = alpha_eq_iff_tac (distinct_thms @ inject_thms) alpha_intros alpha_elims 1;

  val thms = map (fn goal => Goal.prove ctxt' [] [] goal (K tac)) goals;

in

  Variable.export ctxt' ctxt thms

  |> `(map mk_simp_rule)

end

(** proof by induction over the alpha-definitions **)

fun is_true @{term "Trueprop True"} = true

  | is_true _ = false 

fun alpha_prove alphas props alpha_induct_thm cases_tac ctxt =

let

  val arg_tys = map (domain_type o fastype_of) alphas

  val ((arg_names1, arg_names2), ctxt') =

    ctxt

    |> Variable.variant_fixes (replicate (length alphas) "x") 

    ||>> Variable.variant_fixes (replicate (length alphas) "y")

  val args1 = map2 (curry Free) arg_names1 arg_tys

  val args2 = map2 (curry Free) arg_names2 arg_tys

  val true_trms = replicate (length alphas) (K @{term True})

  fun apply_all x fs = map (fn f => f x) fs

    group (props @ (alphas ~~ true_trms))

    |> map snd 

    |> map2 apply_all (args1 ~~ args2)

    |> map fold_conj

  fun apply_trm_pair t (ar1, ar2) = t $ ar1 $ ar2

    |> foldr1 HOLogic.mk_conj

    |> HOLogic.mk_Trueprop

in

  |> singleton (ProofContext.export ctxt' ctxt)

  |> Datatype_Aux.split_conj_thm

  |> map (fn th => th RS mp) 

  |> map Datatype_Aux.split_conj_thm

  |> flat

  |> map zero_var_indexes

  |> filter_out (is_true o concl_of)

end

(** reflexivity proof for the alphas **)

val exi_zero = @{lemma "P (0::perm) ==> (? x. P x)" by auto}

fun cases_tac intros =

let

  val prod_simps = @{thms split_conv prod_alpha_def prod_rel.simps}

  val unbound_tac = REPEAT o (etac @{thm conjE}) THEN' atac  

  val bound_tac = 

    EVERY' [ rtac exi_zero, 

             resolve_tac @{thms alpha_refl}, 

             asm_full_simp_tac (HOL_ss addsimps prod_simps) ]

in

  REPEAT o FIRST' [rtac @{thm conjI}, 

    resolve_tac intros THEN_ALL_NEW FIRST' [rtac @{thm refl}, unbound_tac, bound_tac]]

end

fun raw_prove_refl alpha_trms alpha_bns alpha_intros raw_dt_induct ctxt =

let

  val arg_tys = 

    alpha_trms

    |> map fastype_of

    |> map domain_type

  val arg_bn_tys = 

    alpha_bns

    |> map fastype_of

    |> map domain_type

  val arg_names = Datatype_Prop.make_tnames arg_tys

  val arg_bn_names = map (lookup (arg_tys ~~ arg_names)) arg_bn_tys

  val args = map Free (arg_names ~~ arg_tys)

  val arg_bns = map Free (arg_bn_names ~~ arg_bn_tys)

  val goal = 

    group ((arg_bns ~~ alpha_bns) @ (args ~~ alpha_trms))

    |> map (fn (ar, cnsts) => map (fn c => c $ ar $ ar) cnsts) 

    |> map (foldr1 HOLogic.mk_conj)

    |> foldr1 HOLogic.mk_conj

    |> HOLogic.mk_Trueprop

in

  Goal.prove ctxt arg_names [] goal

    (fn {context, ...} => 

       HEADGOAL (DETERM o (rtac raw_dt_induct) THEN_ALL_NEW cases_tac alpha_intros)) 

  |> Datatype_Aux.split_conj_thm 

  |> map Datatype_Aux.split_conj_thm 

  |> flat

end

(** symmetry proof for the alphas **)

val exi_neg = @{lemma "(EX (p::perm). P p) ==> (!!q. P q ==> Q (- q)) ==> EX p. Q p"

  by (erule exE, rule_tac x="-p" in exI, auto)}

(* for premises that contain binders *)

fun prem_bound_tac pred_names ctxt = 

let

  fun trans_prem_tac pred_names ctxt = 

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

    let

      val prems' = map (transform_prem1 context pred_names) prems

    in

      resolve_tac prems' 1

    end) ctxt

  val prod_simps = @{thms split_conv permute_prod.simps prod_alpha_def prod_rel.simps alphas}

in

  EVERY' 

    [ etac exi_neg,

      resolve_tac @{thms alpha_sym_eqvt},

      asm_full_simp_tac (HOL_ss addsimps prod_simps),

      Nominal_Permeq.eqvt_tac ctxt [] [] THEN' rtac @{thm refl},

      trans_prem_tac pred_names ctxt ] 

end

fun raw_prove_sym alpha_trms alpha_intros alpha_induct ctxt =

let

  val props = map (fn t => fn (x, y) => t $ y $ x) alpha_trms

  fun tac ctxt = 

    let

      val alpha_names =  map (fst o dest_Const) alpha_trms   

    in

      resolve_tac alpha_intros THEN_ALL_NEW 

      FIRST' [atac, rtac @{thm sym} THEN' atac, prem_bound_tac alpha_names ctxt]

  end

in

  alpha_prove alpha_trms (alpha_trms ~~ props) alpha_induct tac ctxt 

end

(** transitivity proof for alphas **)

(* applies cases rules and resolves them with the last premise *)

fun ecases_tac cases = 

  Subgoal.FOCUS (fn {prems, ...} =>

    HEADGOAL (resolve_tac cases THEN' rtac (List.last prems)))

fun aatac pred_names = 

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

    HEADGOAL (resolve_tac (map (transform_prem1 context pred_names) prems)))

(* instantiates exI with the permutation p + q *)

val perm_inst_tac =

  Subgoal.FOCUS (fn {params, ...} => 

    let

      val (p, q) = pairself snd (last2 params)

      val pq_inst = foldl1 (uncurry Thm.capply) [@{cterm "plus::perm => perm => perm"}, p, q]

      val exi_inst = Drule.instantiate' [SOME (@{ctyp "perm"})] [NONE, SOME pq_inst] @{thm exI}