(*  Title:      HOL/Tools/SMT/smt_translate.ML

     Author:     Sascha Boehme, TU Muenchen

 Translate theorems into an SMT intermediate format and serialize them.

 *)

 signature SMT_TRANSLATE =

 sig

   (*intermediate term structure*)

   datatype squant = SForall | SExists

   datatype 'a spattern = SPat of 'a list | SNoPat of 'a list

   datatype sterm =

     SVar of int |

     SApp of string * sterm list |

     SLet of string * sterm * sterm |

     SQua of squant * string list * sterm spattern list * int option * sterm

   (*translation configuration*)

   type prefixes = {sort_prefix: string, func_prefix: string}

   type sign = {

     header: string list,

     sorts: string list,

     dtyps: (string * (string * (string * string) list) list) list list,

     funcs: (string * (string list * string)) list }

   type config = {

     prefixes: prefixes,

     header: term list -> string list,

     is_fol: bool,

     has_datatypes: bool,

     serialize: string list -> sign -> sterm list -> string }

   type recon = {

     context: Proof.context,

     typs: typ Symtab.table,

     terms: term Symtab.table,

     rewrite_rules: thm list,

     assms: (int * thm) list }

   (*translation*)

   val add_config: SMT_Utils.class * (Proof.context -> config) ->

     Context.generic -> Context.generic 

   val translate: Proof.context -> string list -> (int * thm) list ->

     string * recon

 end

 structure SMT_Translate: SMT_TRANSLATE =

 struct

 (* intermediate term structure *)

 datatype squant = SForall | SExists

 datatype 'a spattern = SPat of 'a list | SNoPat of 'a list

 datatype sterm =

   SVar of int |

   SApp of string * sterm list |

   SLet of string * sterm * sterm |

   SQua of squant * string list * sterm spattern list * int option * sterm

 (* translation configuration *)

 type prefixes = {sort_prefix: string, func_prefix: string}

 type sign = {

   header: string list,

   sorts: string list,

   dtyps: (string * (string * (string * string) list) list) list list,

   funcs: (string * (string list * string)) list }

 type config = {

   prefixes: prefixes,

   header: term list -> string list,

   is_fol: bool,

   has_datatypes: bool,

   serialize: string list -> sign -> sterm list -> string }

 type recon = {

   context: Proof.context,

   typs: typ Symtab.table,

   terms: term Symtab.table,

   rewrite_rules: thm list,

   assms: (int * thm) list }

 (* translation context *)

 fun make_tr_context {sort_prefix, func_prefix} =

   (sort_prefix, 1, Typtab.empty, func_prefix, 1, Termtab.empty)

 fun string_of_index pre i = pre ^ string_of_int i

 fun add_typ T proper (cx as (sp, Tidx, typs, fp, idx, terms)) =

   (case Typtab.lookup typs T of

     SOME (n, _) => (n, cx)

   | NONE =>

       let

         val n = string_of_index sp Tidx

         val typs' = Typtab.update (T, (n, proper)) typs

       in (n, (sp, Tidx+1, typs', fp, idx, terms)) end)

 fun add_fun t sort (cx as (sp, Tidx, typs, fp, idx, terms)) =

   (case Termtab.lookup terms t of

     SOME (n, _) => (n, cx)

   | NONE => 

       let

         val n = string_of_index fp idx

         val terms' = Termtab.update (t, (n, sort)) terms

       in (n, (sp, Tidx, typs, fp, idx+1, terms')) end)

 fun sign_of header dtyps (_, _, typs, _, _, terms) = {

   header = header,

   sorts = Typtab.fold (fn (_, (n, true)) => cons n | _ => I) typs [],

   dtyps = dtyps,

   funcs = Termtab.fold (fn (_, (n, SOME ss)) => cons (n,ss) | _ => I) terms []}

 fun recon_of ctxt rules thms ithms (_, _, typs, _, _, terms) =

   let

     fun add_typ (T, (n, _)) = Symtab.update (n, T)

     val typs' = Typtab.fold add_typ typs Symtab.empty

     fun add_fun (t, (n, _)) = Symtab.update (n, t)

     val terms' = Termtab.fold add_fun terms Symtab.empty

     val assms = map (pair ~1) thms @ ithms

   in

     {context=ctxt, typs=typs', terms=terms', rewrite_rules=rules, assms=assms}

   end

 (* preprocessing *)

 (** datatype declarations **)

 fun collect_datatypes_and_records (tr_context, ctxt) ts =

   let

     val (declss, ctxt') =

       fold (Term.fold_types SMT_Datatypes.add_decls) ts ([], ctxt)

     fun is_decl_typ T = exists (exists (equal T o fst)) declss

     fun add_typ' T proper =

       (case SMT_Builtin.dest_builtin_typ ctxt' T of

         SOME n => pair n

       | NONE => add_typ T proper)

     fun tr_select sel =

       let val T = Term.range_type (Term.fastype_of sel)

       in add_fun sel NONE ##>> add_typ' T (not (is_decl_typ T)) end

     fun tr_constr (constr, selects) =

       add_fun constr NONE ##>> fold_map tr_select selects

     fun tr_typ (T, cases) = add_typ' T false ##>> fold_map tr_constr cases

     val (declss', tr_context') = fold_map (fold_map tr_typ) declss tr_context

     fun add (constr, selects) =

       Termtab.update (constr, length selects) #>

       fold (Termtab.update o rpair 1) selects

     val funcs = fold (fold (fold add o snd)) declss Termtab.empty

   in ((funcs, declss', tr_context', ctxt'), ts) end

     (* FIXME: also return necessary datatype and record theorems *)

 (** eta-expand quantifiers, let expressions and built-ins *)

 local

   fun eta f T t = Abs (Name.uu, T, f (Term.incr_boundvars 1 t $ Bound 0))

   fun exp f T = eta f (Term.domain_type (Term.domain_type T))

   fun exp2 T q =

     let val U = Term.domain_type T

     in Abs (Name.uu, U, q $ eta I (Term.domain_type U) (Bound 0)) end

   fun exp2' T l =

     let val (U1, U2) = Term.dest_funT T ||> Term.domain_type

     in Abs (Name.uu, U1, eta I U2 (l $ Bound 0)) end

   fun expf k i T t =

     let val Ts = drop i (fst (SMT_Utils.dest_funT k T))

     in

       Term.incr_boundvars (length Ts) t

       |> fold_rev (fn i => fn u => u $ Bound i) (0 upto length Ts - 1)

       |> fold_rev (fn T => fn u => Abs (Name.uu, T, u)) Ts

     end

 in

 fun eta_expand ctxt is_fol funcs =

   let

     fun exp_func t T ts =

       (case Termtab.lookup funcs t of

         SOME k =>

           Term.list_comb (t, ts)

           |> k <> length ts ? expf k (length ts) T

       | NONE => Term.list_comb (t, ts))

     fun expand ((q as Const (@{const_name All}, _)) $ Abs a) = q $ abs_expand a

       | expand ((q as Const (@{const_name All}, T)) $ t) = q $ exp expand T t

       | expand (q as Const (@{const_name All}, T)) = exp2 T q

       | expand ((q as Const (@{const_name Ex}, _)) $ Abs a) = q $ abs_expand a

       | expand ((q as Const (@{const_name Ex}, T)) $ t) = q $ exp expand T t

       | expand (q as Const (@{const_name Ex}, T)) = exp2 T q

       | expand ((l as Const (@{const_name Let}, _)) $ t $ Abs a) =

           if is_fol then expand (Term.betapply (Abs a, t))

           else l $ expand t $ abs_expand a

       | expand ((l as Const (@{const_name Let}, T)) $ t $ u) =

           if is_fol then expand (u $ t)

           else l $ expand t $ exp expand (Term.range_type T) u

       | expand ((l as Const (@{const_name Let}, T)) $ t) =

           if is_fol then

             let val U = Term.domain_type (Term.range_type T)

             in Abs (Name.uu, U, Bound 0 $ Term.incr_boundvars 1 t) end

           else exp2 T (l $ expand t)

       | expand (l as Const (@{const_name Let}, T)) =

           if is_fol then 

             let val U = Term.domain_type (Term.range_type T)

             in

               Abs (Name.uu, Term.domain_type T, Abs (Name.uu, U,

                 Bound 0 $ Bound 1))

             end

           else exp2' T l

       | expand t =

           (case Term.strip_comb t of

             (u as Const (c as (_, T)), ts) =>

               (case SMT_Builtin.dest_builtin ctxt c ts of

                 SOME (_, k, us, mk) =>

                   if k = length us then mk (map expand us)

                   else expf k (length ts) T (mk (map expand us))

               | NONE => exp_func u T (map expand ts))

           | (u as Free (_, T), ts) => exp_func u T (map expand ts)

           | (Abs a, ts) => Term.list_comb (abs_expand a, map expand ts)

           | (u, ts) => Term.list_comb (u, map expand ts))

     and abs_expand (n, T, t) = Abs (n, T, expand t)

   in map expand end

 end

 (** introduce explicit applications **)

 local

   (*

     Make application explicit for functions with varying number of arguments.

   *)

   fun add t i = apfst (Termtab.map_default (t, i) (Integer.min i))

   fun add_type T = apsnd (Typtab.update (T, ()))

   fun min_arities t =

     (case Term.strip_comb t of

       (u as Const _, ts) => add u (length ts) #> fold min_arities ts

     | (u as Free _, ts) => add u (length ts) #> fold min_arities ts

     | (Abs (_, T, u), ts) => add_type T #> min_arities u #> fold min_arities ts

     | (_, ts) => fold min_arities ts)

   fun minimize types t i =

     let

       fun find_min j [] _ = j

         | find_min j (U :: Us) T =

             if Typtab.defined types T then j

             else find_min (j + 1) Us (U --> T)

       val (Ts, T) = Term.strip_type (Term.type_of t)

     in find_min 0 (take i (rev Ts)) T end

   fun app u (t, T) =

     (Const (@{const_name SMT.fun_app}, T --> T) $ t $ u, Term.range_type T)

   fun apply i t T ts =

     let

       val (ts1, ts2) = chop i ts

       val (_, U) = SMT_Utils.dest_funT i T

     in fst (fold app ts2 (Term.list_comb (t, ts1), U)) end

 in

 fun intro_explicit_application ctxt funcs ts =

   let

     val (arities, types) = fold min_arities ts (Termtab.empty, Typtab.empty)

     val arities' = Termtab.map (minimize types) arities

     fun app_func t T ts =

       if is_some (Termtab.lookup funcs t) then Term.list_comb (t, ts)

       else apply (the (Termtab.lookup arities' t)) t T ts

     fun in_list T f t = HOLogic.mk_list T (map f (HOLogic.dest_list t))

     fun traverse Ts t =

       (case Term.strip_comb t of

         (q as Const (@{const_name All}, _), [Abs (x, T, u)]) =>

           q $ Abs (x, T, in_trigger (T :: Ts) u)

       | (q as Const (@{const_name Ex}, _), [Abs (x, T, u)]) =>

           q $ Abs (x, T, in_trigger (T :: Ts) u)

       | (q as Const (@{const_name Let}, _), [u1, u2 as Abs _]) =>

           q $ traverse Ts u1 $ traverse Ts u2

       | (u as Const (c as (_, T)), ts) =>

           (case SMT_Builtin.dest_builtin ctxt c ts of

             SOME (_, _, us, mk) => mk (map (traverse Ts) us)

           | NONE => app_func u T (map (traverse Ts) ts))

       | (u as Free (_, T), ts) => app_func u T (map (traverse Ts) ts)

       | (u as Bound i, ts) => apply 0 u (nth Ts i) (map (traverse Ts) ts)

       | (Abs (n, T, u), ts) => traverses Ts (Abs (n, T, traverse (T::Ts) u)) ts

       | (u, ts) => traverses Ts u ts)

     and in_trigger Ts ((c as @{const SMT.trigger}) $ p $ t) =

           c $ in_pats Ts p $ in_weight Ts t

       | in_trigger Ts t = in_weight Ts t

     and in_pats Ts ps =

       in_list @{typ "SMT.pattern list"}

         (in_list @{typ SMT.pattern} (in_pat Ts)) ps

     and in_pat Ts ((p as Const (@{const_name SMT.pat}, _)) $ t) =

           p $ traverse Ts t

       | in_pat Ts ((p as Const (@{const_name SMT.nopat}, _)) $ t) =

           p $ traverse Ts t

       | in_pat _ t = raise TERM ("bad pattern", [t])

     and in_weight Ts ((c as @{const SMT.weight}) $ w $ t) =

           c $ w $ traverse Ts t

       | in_weight Ts t = traverse Ts t 

     and traverses Ts t ts = Term.list_comb (t, map (traverse Ts) ts)

   in map (traverse []) ts end

 val fun_app_eq = mk_meta_eq @{thm SMT.fun_app_def}

 end

 (** map HOL formulas to FOL formulas (i.e., separate formulas froms terms) **)

 local

   val term_bool = @{lemma "SMT.term_true ~= SMT.term_false"

     by (simp add: SMT.term_true_def SMT.term_false_def)}

   val is_quant = member (op =) [@{const_name All}, @{const_name Ex}]

   val fol_rules = [

     Let_def,

     mk_meta_eq @{thm SMT.term_true_def},

     mk_meta_eq @{thm SMT.term_false_def},

     @{lemma "P = True == P" by (rule eq_reflection) simp},

     @{lemma "if P then True else False == P" by (rule eq_reflection) simp}]

   fun as_term t = @{const HOL.eq (bool)} $ t $ @{const SMT.term_true}

   fun wrap_in_if t =

     @{const If (bool)} $ t $ @{const SMT.term_true} $ @{const SMT.term_false}

   fun is_builtin_conn_or_pred ctxt c ts =

     is_some (SMT_Builtin.dest_builtin_conn ctxt c ts) orelse

     is_some (SMT_Builtin.dest_builtin_pred ctxt c ts)

   fun builtin b ctxt c ts =

     (case (Const c, ts) of

       (@{const HOL.eq (bool)}, [t, u]) =>

         if t = @{const SMT.term_true} orelse u = @{const SMT.term_true} then

           SMT_Builtin.dest_builtin_eq ctxt t u

         else b ctxt c ts

     | _ => b ctxt c ts)

 in

 fun folify ctxt =

   let

     fun in_list T f t = HOLogic.mk_list T (map f (HOLogic.dest_list t))

     fun in_term t =

       (case Term.strip_comb t of

         (@{const True}, []) => @{const SMT.term_true}

       | (@{const False}, []) => @{const SMT.term_false}

       | (u as Const (@{const_name If}, _), [t1, t2, t3]) =>

           u $ in_form t1 $ in_term t2 $ in_term t3

       | (Const (c as (n, _)), ts) =>

           if is_builtin_conn_or_pred ctxt c ts then wrap_in_if (in_form t)

           else  if is_quant n then wrap_in_if (in_form t)

           else Term.list_comb (Const c, map in_term ts)

       | (Free c, ts) => Term.list_comb (Free c, map in_term ts)

       | _ => t)

     and in_weight ((c as @{const SMT.weight}) $ w $ t) = c $ w $ in_form t

       | in_weight t = in_form t 

     and in_pat ((p as Const (@{const_name SMT.pat}, _)) $ t) = p $ in_term t

       | in_pat ((p as Const (@{const_name SMT.nopat}, _)) $ t) = p $ in_term t

       | in_pat t = raise TERM ("bad pattern", [t])

     and in_pats ps =

       in_list @{typ "SMT.pattern list"} (in_list @{typ SMT.pattern} in_pat) ps

     and in_trigger ((c as @{const SMT.trigger}) $ p $ t) =

           c $ in_pats p $ in_weight t

       | in_trigger t = in_weight t

     and in_form t =

       (case Term.strip_comb t of

         (q as Const (qn, _), [Abs (n, T, u)]) =>

           if is_quant qn then q $ Abs (n, T, in_trigger u)

           else as_term (in_term t)

       | (Const c, ts) =>

           (case SMT_Builtin.dest_builtin_conn ctxt c ts of

             SOME (_, _, us, mk) => mk (map in_form us)

           | NONE =>

               (case SMT_Builtin.dest_builtin_pred ctxt c ts of

                 SOME (_, _, us, mk) => mk (map in_term us)

               | NONE => as_term (in_term t)))

       | _ => as_term (in_term t))

   in

     map in_form #>

     cons (SMT_Utils.prop_of term_bool) #>

     pair (fol_rules, [term_bool], builtin)

   end

 end

 (* translation into intermediate format *)

 (** utility functions **)

 val quantifier = (fn

     @{const_name All} => SOME SForall

   | @{const_name Ex} => SOME SExists

   | _ => NONE)

 fun group_quant qname Ts (t as Const (q, _) $ Abs (_, T, u)) =

       if q = qname then group_quant qname (T :: Ts) u else (Ts, t)

   | group_quant _ Ts t = (Ts, t)

 fun dest_weight (@{const SMT.weight} $ w $ t) =

       (SOME (snd (HOLogic.dest_number w)), t)

   | dest_weight t = (NONE, t)

 fun dest_pat (Const (@{const_name SMT.pat}, _) $ t) = (t, true)

   | dest_pat (Const (@{const_name SMT.nopat}, _) $ t) = (t, false)

   | dest_pat t = raise TERM ("bad pattern", [t])

 fun dest_pats [] = I

   | dest_pats ts =

       (case map dest_pat ts |> split_list ||> distinct (op =) of

         (ps, [true]) => cons (SPat ps)

       | (ps, [false]) => cons (SNoPat ps)

       | _ => raise TERM ("bad multi-pattern", ts))

 fun dest_trigger (@{const SMT.trigger} $ tl $ t) =

       (rev (fold (dest_pats o HOLogic.dest_list) (HOLogic.dest_list tl) []), t)

   | dest_trigger t = ([], t)

 fun dest_quant qn T t = quantifier qn |> Option.map (fn q =>

   let

     val (Ts, u) = group_quant qn [T] t

     val (ps, p) = dest_trigger u

     val (w, b) = dest_weight p

   in (q, rev Ts, ps, w, b) end)

 fun fold_map_pat f (SPat ts) = fold_map f ts #>> SPat

   | fold_map_pat f (SNoPat ts) = fold_map f ts #>> SNoPat

 (** translation from Isabelle terms into SMT intermediate terms **)

 fun intermediate header dtyps builtin ctxt ts trx =

   let

     fun transT (T as TFree _) = add_typ T true

       | transT (T as TVar _) = (fn _ => raise TYPE ("bad SMT type", [T], []))

       | transT (T as Type _) =

           (case SMT_Builtin.dest_builtin_typ ctxt T of

             SOME n => pair n

           | NONE => add_typ T true)

     fun app n ts = SApp (n, ts)

     fun trans t =

       (case Term.strip_comb t of

         (Const (qn, _), [Abs (_, T, t1)]) =>

           (case dest_quant qn T t1 of

             SOME (q, Ts, ps, w, b) =>

               fold_map transT Ts ##>> fold_map (fold_map_pat trans) ps ##>>

               trans b #>> (fn ((Ts', ps'), b') => SQua (q, Ts', ps', w, b'))

           | NONE => raise TERM ("unsupported quantifier", [t]))

       | (Const (@{const_name Let}, _), [t1, Abs (_, T, t2)]) =>

           transT T ##>> trans t1 ##>> trans t2 #>>

           (fn ((U, u1), u2) => SLet (U, u1, u2))

       | (u as Const (c as (_, T)), ts) =>

           (case builtin ctxt c ts of

             SOME (n, _, us, _) => fold_map trans us #>> app n

           | NONE => transs u T ts)

       | (u as Free (_, T), ts) => transs u T ts

       | (Bound i, []) => pair (SVar i)

       | _ => raise TERM ("bad SMT term", [t]))

     and transs t T ts =

       let val (Us, U) = SMT_Utils.dest_funT (length ts) T

       in

         fold_map transT Us ##>> transT U #-> (fn Up =>

         add_fun t (SOME Up) ##>> fold_map trans ts #>> SApp)

       end

     val (us, trx') = fold_map trans ts trx

   in ((sign_of (header ts) dtyps trx', us), trx') end

 (* translation *)

 structure Configs = Generic_Data

 (

   type T = (Proof.context -> config) SMT_Utils.dict

   val empty = []

   val extend = I

   fun merge data = SMT_Utils.dict_merge fst data

 )

 fun add_config (cs, cfg) = Configs.map (SMT_Utils.dict_update (cs, cfg))

 fun get_config ctxt = 

   let val cs = SMT_Config.solver_class_of ctxt

   in

     (case SMT_Utils.dict_get (Configs.get (Context.Proof ctxt)) cs of

       SOME cfg => cfg ctxt

     | NONE => error ("SMT: no translation configuration found " ^

         "for solver class " ^ quote (SMT_Utils.string_of_class cs)))

   end

 fun translate ctxt comments ithms =

   let

     val {prefixes, is_fol, header, has_datatypes, serialize} = get_config ctxt

     val with_datatypes =

       has_datatypes andalso Config.get ctxt SMT_Config.datatypes

     fun no_dtyps (tr_context, ctxt) ts =

       ((Termtab.empty, [], tr_context, ctxt), ts)

     val ts1 = map (Envir.beta_eta_contract o SMT_Utils.prop_of o snd) ithms

     val ((funcs, dtyps, tr_context, ctxt1), ts2) =

       ((make_tr_context prefixes, ctxt), ts1)

       |-> (if with_datatypes then collect_datatypes_and_records else no_dtyps)

     fun is_binder (Const (@{const_name Let}, _) $ _) = true

       | is_binder t = Lambda_Lifting.is_quantifier t

     fun mk_trigger ((q as Const (@{const_name All}, _)) $ Abs (n, T, t)) =

           q $ Abs (n, T, mk_trigger t)

       | mk_trigger (eq as (Const (@{const_name HOL.eq}, T) $ lhs $ _)) =

           Term.domain_type T --> @{typ SMT.pattern}

           |> (fn T => Const (@{const_name SMT.pat}, T) $ lhs)

           |> HOLogic.mk_list @{typ SMT.pattern} o single

           |> HOLogic.mk_list @{typ "SMT.pattern list"} o single

           |> (fn t => @{const SMT.trigger} $ t $ eq)

       | mk_trigger t = t

     val (ctxt2, ts3) =

       ts2

       |> eta_expand ctxt1 is_fol funcs

       |> rpair ctxt1

       |-> Lambda_Lifting.lift_lambdas NONE is_binder

       |-> (fn (ts', defs) => fn ctxt' =>

           map mk_trigger defs @ ts'

           |> intro_explicit_application ctxt' funcs 

           |> pair ctxt')

     val ((rewrite_rules, extra_thms, builtin), ts4) =

       (if is_fol then folify ctxt2 else pair ([], [], I)) ts3

     val rewrite_rules' = fun_app_eq :: rewrite_rules

   in

     (ts4, tr_context)

     |-> intermediate header dtyps (builtin SMT_Builtin.dest_builtin) ctxt2

     |>> uncurry (serialize comments)

     ||> recon_of ctxt2 rewrite_rules' extra_thms ithms

   end

 end