(*  Title:      Pure/drule.ML

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1993  University of Cambridge

Derived rules and other operations on theorems and theories

*)

infix 0 RS RSN RL RLN MRS MRL COMP;

signature DRULE =

  sig

  structure Thm : THM

  local open Thm  in

  val add_defs		: (string * string) list -> theory -> theory

  val add_defs_i	: (string * term) list -> theory -> theory

  val asm_rl		: thm

  val assume_ax		: theory -> string -> thm

  val COMP		: thm * thm -> thm

  val compose		: thm * int * thm -> thm list

  val cprems_of		: thm -> cterm list

  val cskip_flexpairs	: cterm -> cterm

  val cstrip_imp_prems	: cterm -> cterm list

  val cterm_instantiate	: (cterm*cterm)list -> thm -> thm

  val cut_rl		: thm

  val equal_abs_elim	: cterm  -> thm -> thm

  val equal_abs_elim_list: cterm list -> thm -> thm

  val eq_thm		: thm * thm -> bool

  val eq_thm_sg		: thm * thm -> bool

  val flexpair_abs_elim_list: cterm list -> thm -> thm

  val forall_intr_list	: cterm list -> thm -> thm

  val forall_intr_frees	: thm -> thm

  val forall_elim_list	: cterm list -> thm -> thm

  val forall_elim_var	: int -> thm -> thm

  val forall_elim_vars	: int -> thm -> thm

  val implies_elim_list	: thm -> thm list -> thm

  val implies_intr_list	: cterm list -> thm -> thm

  val MRL		: thm list list * thm list -> thm list

  val MRS		: thm list * thm -> thm

  val pprint_cterm	: cterm -> pprint_args -> unit

  val pprint_ctyp	: ctyp -> pprint_args -> unit

  val pprint_theory	: theory -> pprint_args -> unit

  val pprint_thm	: thm -> pprint_args -> unit

  val pretty_thm	: thm -> Sign.Syntax.Pretty.T

  val print_cterm	: cterm -> unit

  val print_ctyp	: ctyp -> unit

  val print_goals	: int -> thm -> unit

  val print_goals_ref	: (int -> thm -> unit) ref

  val print_syntax	: theory -> unit

  val print_sign	: theory -> unit

  val print_axioms	: theory -> unit

  val print_theory	: theory -> unit

  val print_thm		: thm -> unit

  val prth		: thm -> thm

  val prthq		: thm Sequence.seq -> thm Sequence.seq

  val prths		: thm list -> thm list

  val read_instantiate	: (string*string)list -> thm -> thm

  val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm

  val read_insts	:

          Sign.sg -> (indexname -> typ option) * (indexname -> sort option)

                  -> (indexname -> typ option) * (indexname -> sort option)

                  -> (string*string)list

                  -> (indexname*ctyp)list * (cterm*cterm)list

  val reflexive_thm	: thm

  val revcut_rl		: thm

  val rewrite_goal_rule	: bool*bool -> (meta_simpset -> thm -> thm option)

        -> meta_simpset -> int -> thm -> thm

  val rewrite_goals_rule: thm list -> thm -> thm

  val rewrite_rule	: thm list -> thm -> thm

  val RS		: thm * thm -> thm

  val RSN		: thm * (int * thm) -> thm

  val RL		: thm list * thm list -> thm list

  val RLN		: thm list * (int * thm list) -> thm list

  val show_hyps		: bool ref

  val size_of_thm	: thm -> int

  val standard		: thm -> thm

  val string_of_cterm	: cterm -> string

  val string_of_ctyp	: ctyp -> string

  val string_of_thm	: thm -> string

  val symmetric_thm	: thm

  val thin_rl		: thm

  val transitive_thm	: thm

  val triv_forall_equality: thm

  val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)

  val zero_var_indexes	: thm -> thm

  end

  end;

functor DruleFun (structure Logic: LOGIC and Thm: THM): DRULE =

struct

structure Thm = Thm;

structure Sign = Thm.Sign;

structure Type = Sign.Type;

structure Syntax = Sign.Syntax;

structure Pretty = Syntax.Pretty

structure Symtab = Sign.Symtab;

local open Thm

in

(**** Extend Theories ****)

(** add constant definitions **)

(* all_axioms_of *)

(*results may contain duplicates!*)

fun ancestry_of thy =

  thy :: flat (map ancestry_of (parents_of thy));

val all_axioms_of = flat o map axioms_of o ancestry_of;

(* clash_types, clash_consts *)

(*check if types have common instance (ignoring sorts)*)

fun clash_types ty1 ty2 =

  let

    val ty1' = Type.varifyT ty1;

    val ty2' = incr_tvar (maxidx_of_typ ty1' + 1) (Type.varifyT ty2);

  in

    Type.raw_unify (ty1', ty2')

  end;

fun clash_consts (c1, ty1) (c2, ty2) =

  c1 = c2 andalso clash_types ty1 ty2;

(* clash_defns *)

fun clash_defn c_ty (name, tm) =

  let val (c, ty') = dest_Const (head_of (fst (Logic.dest_equals tm))) in

    if clash_consts c_ty (c, ty') then Some (name, ty') else None

  end handle TERM _ => None;

fun clash_defns c_ty axms =

  distinct (mapfilter (clash_defn c_ty) axms);

(* dest_defn *)

fun dest_defn tm =

  let

    fun err msg = raise_term msg [tm];

    val (lhs, rhs) = Logic.dest_equals tm

      handle TERM _ => err "Not a meta-equality (==)";

    val (head, args) = strip_comb lhs;

    val (c, ty) = dest_Const head

      handle TERM _ => err "Head of lhs not a constant";

    fun occs_const (Const c_ty') = (c_ty' = (c, ty))

      | occs_const (Abs (_, _, t)) = occs_const t

      | occs_const (t $ u) = occs_const t orelse occs_const u

      | occs_const _ = false;

    val show_frees = commas_quote o map (fst o dest_Free);

    val show_tfrees = commas_quote o map fst;

    val lhs_dups = duplicates args;

    val rhs_extras = gen_rems (op =) (term_frees rhs, args);

    val rhs_extrasT = gen_rems (op =) (term_tfrees rhs, typ_tfrees ty);

  in

    if not (forall is_Free args) then

      err "Arguments of lhs have to be variables"

    else if not (null lhs_dups) then

      err ("Duplicate variables on lhs: " ^ show_frees lhs_dups)

    else if not (null rhs_extras) then

      err ("Extra variables on rhs: " ^ show_frees rhs_extras)

    else if not (null rhs_extrasT) then

      err ("Extra type variables on rhs: " ^ show_tfrees rhs_extrasT)

    else if occs_const rhs then

      err ("Constant to be defined occurs on rhs")

    else (c, ty)

  end;

(* check_defn *)

fun err_in_defn name msg =

  (writeln msg; error ("The error(s) above occurred in definition " ^ quote name));

fun check_defn sign (axms, (name, tm)) =

  let

    fun show_const (c, ty) = quote (Pretty.string_of (Pretty.block

      [Pretty.str (c ^ " ::"), Pretty.brk 1, Sign.pretty_typ sign ty]));

    fun show_defn c (dfn, ty') = show_const (c, ty') ^ " in " ^ dfn;

    fun show_defns c = commas o map (show_defn c);

    val (c, ty) = dest_defn tm

      handle TERM (msg, _) => err_in_defn name msg;

    val defns = clash_defns (c, ty) axms;

  in

    if not (null defns) then

      err_in_defn name ("Definition of " ^ show_const (c, ty) ^

        " clashes with " ^ show_defns c defns)

    else (name, tm) :: axms

  end;

(* add_defs *)

fun ext_defns prep_axm raw_axms thy =

  let

    val axms = map (prep_axm (sign_of thy)) raw_axms;

    val all_axms = all_axioms_of thy;

  in

    foldl (check_defn (sign_of thy)) (all_axms, axms);

    add_axioms_i axms thy

  end;

val add_defs_i = ext_defns cert_axm;

val add_defs = ext_defns read_axm;

(**** More derived rules and operations on theorems ****)

(** some cterm->cterm operations: much faster than calling cterm_of! **)

(*Discard flexflex pairs; return a cterm*)

fun cskip_flexpairs ct =

    case term_of ct of

	(Const("==>", _) $ (Const("=?=",_)$_$_) $ _) =>

	    cskip_flexpairs (#2 (dest_cimplies ct))

      | _ => ct;

(* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)

fun cstrip_imp_prems ct =

    let val (cA,cB) = dest_cimplies ct

    in  cA :: cstrip_imp_prems cB  end

    handle TERM _ => [];

(*The premises of a theorem, as a cterm list*)

val cprems_of = cstrip_imp_prems o cskip_flexpairs o cprop_of;

(** reading of instantiations **)

fun indexname cs = case Syntax.scan_varname cs of (v,[]) => v

        | _ => error("Lexical error in variable name " ^ quote (implode cs));

fun absent ixn =

  error("No such variable in term: " ^ Syntax.string_of_vname ixn);

fun inst_failure ixn =

  error("Instantiation of " ^ Syntax.string_of_vname ixn ^ " fails");

fun read_insts sign (rtypes,rsorts) (types,sorts) insts =

let val {tsig,...} = Sign.rep_sg sign

    fun split([],tvs,vs) = (tvs,vs)

      | split((sv,st)::l,tvs,vs) = (case explode sv of

                  "'"::cs => split(l,(indexname cs,st)::tvs,vs)

                | cs => split(l,tvs,(indexname cs,st)::vs));

    val (tvs,vs) = split(insts,[],[]);

    fun readT((a,i),st) =

        let val ixn = ("'" ^ a,i);

            val S = case rsorts ixn of Some S => S | None => absent ixn;

            val T = Sign.read_typ (sign,sorts) st;

        in if Type.typ_instance(tsig,T,TVar(ixn,S)) then (ixn,T)

           else inst_failure ixn

        end

    val tye = map readT tvs;

    fun add_cterm ((cts,tye), (ixn,st)) =

        let val T = case rtypes ixn of

                      Some T => typ_subst_TVars tye T

                    | None => absent ixn;

            val (ct,tye2) = read_def_cterm (sign,types,sorts) (st,T);

            val cv = cterm_of sign (Var(ixn,typ_subst_TVars tye2 T))

        in ((cv,ct)::cts,tye2 @ tye) end

    val (cterms,tye') = foldl add_cterm (([],tye), vs);

in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) tye', cterms) end;

(*** Printing of theories, theorems, etc. ***)

(*If false, hypotheses are printed as dots*)

val show_hyps = ref true;

fun pretty_thm th =

let val {sign, hyps, prop,...} = rep_thm th

    val hsymbs = if null hyps then []

                 else if !show_hyps then

                      [Pretty.brk 2,

                       Pretty.lst("[","]") (map (Sign.pretty_term sign) hyps)]

                 else Pretty.str" [" :: map (fn _ => Pretty.str".") hyps @

                      [Pretty.str"]"];

in Pretty.blk(0, Sign.pretty_term sign prop :: hsymbs) end;

val string_of_thm = Pretty.string_of o pretty_thm;

val pprint_thm = Pretty.pprint o Pretty.quote o pretty_thm;

(** Top-level commands for printing theorems **)

val print_thm = writeln o string_of_thm;

fun prth th = (print_thm th; th);

(*Print and return a sequence of theorems, separated by blank lines. *)

fun prthq thseq =

  (Sequence.prints (fn _ => print_thm) 100000 thseq; thseq);

(*Print and return a list of theorems, separated by blank lines. *)

fun prths ths = (print_list_ln print_thm ths; ths);

(* other printing commands *)

fun pprint_ctyp cT =

  let val {sign, T} = rep_ctyp cT in Sign.pprint_typ sign T end;

fun string_of_ctyp cT =

  let val {sign, T} = rep_ctyp cT in Sign.string_of_typ sign T end;

val print_ctyp = writeln o string_of_ctyp;

fun pprint_cterm ct =

  let val {sign, t, ...} = rep_cterm ct in Sign.pprint_term sign t end;

fun string_of_cterm ct =

  let val {sign, t, ...} = rep_cterm ct in Sign.string_of_term sign t end;

val print_cterm = writeln o string_of_cterm;

(* print theory *)

val pprint_theory = Sign.pprint_sg o sign_of;

val print_syntax = Syntax.print_syntax o syn_of;

val print_sign = Sign.print_sg o sign_of;

fun print_axioms thy =

  let

    val {sign, new_axioms, ...} = rep_theory thy;

    val axioms = Symtab.dest new_axioms;

    fun prt_axm (a, t) = Pretty.block [Pretty.str (a ^ ":"), Pretty.brk 1,

      Pretty.quote (Sign.pretty_term sign t)];

  in

    Pretty.writeln (Pretty.big_list "additional axioms:" (map prt_axm axioms))

  end;

fun print_theory thy = (print_sign thy; print_axioms thy);

(** Print thm A1,...,An/B in "goal style" -- premises as numbered subgoals **)

(* get type_env, sort_env of term *)

local

  open Syntax;

  fun ins_entry (x, y) [] = [(x, [y])]

    | ins_entry (x, y) ((pair as (x', ys')) :: pairs) =

        if x = x' then (x', y ins ys') :: pairs

        else pair :: ins_entry (x, y) pairs;

  fun add_type_env (Free (x, T), env) = ins_entry (T, x) env

    | add_type_env (Var (xi, T), env) = ins_entry (T, string_of_vname xi) env

    | add_type_env (Abs (_, _, t), env) = add_type_env (t, env)

    | add_type_env (t $ u, env) = add_type_env (u, add_type_env (t, env))

    | add_type_env (_, env) = env;

  fun add_sort_env (Type (_, Ts), env) = foldr add_sort_env (Ts, env)

    | add_sort_env (TFree (x, S), env) = ins_entry (S, x) env

    | add_sort_env (TVar (xi, S), env) = ins_entry (S, string_of_vname xi) env;

  val sort = map (apsnd sort_strings);

in

  fun type_env t = sort (add_type_env (t, []));

  fun sort_env t = rev (sort (it_term_types add_sort_env (t, [])));

end;

(* print_goals *)

fun print_goals maxgoals state =

  let

    open Syntax;

    val {sign, prop, ...} = rep_thm state;

    val pretty_term = Sign.pretty_term sign;

    val pretty_typ = Sign.pretty_typ sign;

    val pretty_sort = Sign.pretty_sort;

    fun pretty_vars prtf (X, vs) = Pretty.block

      [Pretty.block (Pretty.commas (map Pretty.str vs)),

        Pretty.str " ::", Pretty.brk 1, prtf X];

    fun print_list _ _ [] = ()

      | print_list name prtf lst =

          (writeln ""; Pretty.writeln (Pretty.big_list name (map prtf lst)));

    fun print_goals (_, []) = ()

      | print_goals (n, A :: As) = (Pretty.writeln (Pretty.blk (0,

          [Pretty.str (" " ^ string_of_int n ^ ". "), pretty_term A]));

            print_goals (n + 1, As));

    val print_ffpairs =

      print_list "Flex-flex pairs:" (pretty_term o Logic.mk_flexpair);

    val print_types = print_list "Types:" (pretty_vars pretty_typ) o type_env;

    val print_sorts = print_list "Sorts:" (pretty_vars pretty_sort) o sort_env;

    val (tpairs, As, B) = Logic.strip_horn prop;

    val ngoals = length As;

    val orig_no_freeTs = ! show_no_free_types;

    val orig_sorts = ! show_sorts;

    fun restore () =

      (show_no_free_types := orig_no_freeTs; show_sorts := orig_sorts);

  in

    (show_no_free_types := true; show_sorts := false;

      Pretty.writeln (pretty_term B);

      if ngoals = 0 then writeln "No subgoals!"

      else if ngoals > maxgoals then

        (print_goals (1, take (maxgoals, As));

          writeln ("A total of " ^ string_of_int ngoals ^ " subgoals..."))

      else print_goals (1, As);

      print_ffpairs tpairs;

      if orig_sorts then

        (print_types prop; print_sorts prop)

      else if ! show_types then

        print_types prop

      else ())

    handle exn => (restore (); raise exn);

    restore ()

  end;

(*"hook" for user interfaces: allows print_goals to be replaced*)

val print_goals_ref = ref print_goals;

(*** Find the type (sort) associated with a (T)Var or (T)Free in a term

     Used for establishing default types (of variables) and sorts (of

     type variables) when reading another term.

     Index -1 indicates that a (T)Free rather than a (T)Var is wanted.

***)

fun types_sorts thm =

    let val {prop,hyps,...} = rep_thm thm;

        val big = list_comb(prop,hyps); (* bogus term! *)

        val vars = map dest_Var (term_vars big);

        val frees = map dest_Free (term_frees big);

        val tvars = term_tvars big;

        val tfrees = term_tfrees big;

        fun typ(a,i) = if i<0 then assoc(frees,a) else assoc(vars,(a,i));

        fun sort(a,i) = if i<0 then assoc(tfrees,a) else assoc(tvars,(a,i));

    in (typ,sort) end;

(** Standardization of rules **)

(*Generalization over a list of variables, IGNORING bad ones*)

fun forall_intr_list [] th = th

  | forall_intr_list (y::ys) th =

        let val gth = forall_intr_list ys th

        in  forall_intr y gth   handle THM _ =>  gth  end;

(*Generalization over all suitable Free variables*)

fun forall_intr_frees th =

    let val {prop,sign,...} = rep_thm th

    in  forall_intr_list

         (map (cterm_of sign) (sort atless (term_frees prop)))

         th

    end;

(*Replace outermost quantified variable by Var of given index.

    Could clash with Vars already present.*)

fun forall_elim_var i th =

    let val {prop,sign,...} = rep_thm th

    in case prop of

          Const("all",_) $ Abs(a,T,_) =>

              forall_elim (cterm_of sign (Var((a,i), T)))  th

        | _ => raise THM("forall_elim_var", i, [th])

    end;

(*Repeat forall_elim_var until all outer quantifiers are removed*)

fun forall_elim_vars i th =

    forall_elim_vars i (forall_elim_var i th)

        handle THM _ => th;

(*Specialization over a list of cterms*)

fun forall_elim_list cts th = foldr (uncurry forall_elim) (rev cts, th);

(* maps [A1,...,An], B   to   [| A1;...;An |] ==> B  *)

fun implies_intr_list cAs th = foldr (uncurry implies_intr) (cAs,th);

(* maps [| A1;...;An |] ==> B and [A1,...,An]   to   B *)

fun implies_elim_list impth ths = foldl (uncurry implies_elim) (impth,ths);

(*Reset Var indexes to zero, renaming to preserve distinctness*)

fun zero_var_indexes th =

    let val {prop,sign,...} = rep_thm th;

        val vars = term_vars prop

        val bs = foldl add_new_id ([], map (fn Var((a,_),_)=>a) vars)

        val inrs = add_term_tvars(prop,[]);

        val nms' = rev(foldl add_new_id ([], map (#1 o #1) inrs));

        val tye = map (fn ((v,rs),a) => (v, TVar((a,0),rs))) (inrs ~~ nms')

        val ctye = map (fn (v,T) => (v,ctyp_of sign T)) tye;

        fun varpairs([],[]) = []

          | varpairs((var as Var(v,T)) :: vars, b::bs) =

                let val T' = typ_subst_TVars tye T

                in (cterm_of sign (Var(v,T')),

                    cterm_of sign (Var((b,0),T'))) :: varpairs(vars,bs)

                end

          | varpairs _ = raise TERM("varpairs", []);

    in instantiate (ctye, varpairs(vars,rev bs)) th end;

(*Standard form of object-rule: no hypotheses, Frees, or outer quantifiers;

    all generality expressed by Vars having index 0.*)

fun standard th =

    let val {maxidx,...} = rep_thm th

    in  varifyT (zero_var_indexes (forall_elim_vars(maxidx+1)

                         (forall_intr_frees(implies_intr_hyps th))))

    end;

(*Assume a new formula, read following the same conventions as axioms.

  Generalizes over Free variables,

  creates the assumption, and then strips quantifiers.

  Example is [| ALL x:?A. ?P(x) |] ==> [| ?P(?a) |]

             [ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ]    *)

fun assume_ax thy sP =

    let val sign = sign_of thy

        val prop = Logic.close_form (term_of (read_cterm sign

                         (sP, propT)))

    in forall_elim_vars 0 (assume (cterm_of sign prop))  end;

(*Resolution: exactly one resolvent must be produced.*)

fun tha RSN (i,thb) =

  case Sequence.chop (2, biresolution false [(false,tha)] i thb) of

      ([th],_) => th

    | ([],_)   => raise THM("RSN: no unifiers", i, [tha,thb])

    |      _   => raise THM("RSN: multiple unifiers", i, [tha,thb]);

(*resolution: P==>Q, Q==>R gives P==>R. *)

fun tha RS thb = tha RSN (1,thb);

(*For joining lists of rules*)

fun thas RLN (i,thbs) =

  let val resolve = biresolution false (map (pair false) thas) i

      fun resb thb = Sequence.list_of_s (resolve thb) handle THM _ => []

  in  flat (map resb thbs)  end;

fun thas RL thbs = thas RLN (1,thbs);

(*Resolve a list of rules against bottom_rl from right to left;

  makes proof trees*)

fun rls MRS bottom_rl =

  let fun rs_aux i [] = bottom_rl

        | rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)

  in  rs_aux 1 rls  end;

(*As above, but for rule lists*)

fun rlss MRL bottom_rls =

  let fun rs_aux i [] = bottom_rls

        | rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)

  in  rs_aux 1 rlss  end;

(*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R

  with no lifting or renaming!  Q may contain ==> or meta-quants

  ALWAYS deletes premise i *)

fun compose(tha,i,thb) =

    Sequence.list_of_s (bicompose false (false,tha,0) i thb);

(*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)

fun tha COMP thb =

    case compose(tha,1,thb) of

        [th] => th

      | _ =>   raise THM("COMP", 1, [tha,thb]);

(*Instantiate theorem th, reading instantiations under signature sg*)

fun read_instantiate_sg sg sinsts th =

    let val ts = types_sorts th;

    in  instantiate (read_insts sg ts ts sinsts) th  end;

(*Instantiate theorem th, reading instantiations under theory of th*)

fun read_instantiate sinsts th =

    read_instantiate_sg (#sign (rep_thm th)) sinsts th;

(*Left-to-right replacements: tpairs = [...,(vi,ti),...].

  Instantiates distinct Vars by terms, inferring type instantiations. *)

local

  fun add_types ((ct,cu), (sign,tye)) =

    let val {sign=signt, t=t, T= T, ...} = rep_cterm ct

        and {sign=signu, t=u, T= U, ...} = rep_cterm cu

        val sign' = Sign.merge(sign, Sign.merge(signt, signu))

        val tye' = Type.unify (#tsig(Sign.rep_sg sign')) ((T,U), tye)

          handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])

    in  (sign', tye')  end;

in

fun cterm_instantiate ctpairs0 th =

  let val (sign,tye) = foldr add_types (ctpairs0, (#sign(rep_thm th),[]))

      val tsig = #tsig(Sign.rep_sg sign);

      fun instT(ct,cu) = let val inst = subst_TVars tye

                         in (cterm_fun inst ct, cterm_fun inst cu) end

      fun ctyp2 (ix,T) = (ix, ctyp_of sign T)

  in  instantiate (map ctyp2 tye, map instT ctpairs0) th  end

  handle TERM _ =>

           raise THM("cterm_instantiate: incompatible signatures",0,[th])

       | TYPE _ => raise THM("cterm_instantiate: types", 0, [th])

end;

(** theorem equality test is exported and used by BEST_FIRST **)

(*equality of theorems uses equality of signatures and

  the a-convertible test for terms*)

fun eq_thm (th1,th2) =

    let val {sign=sg1, hyps=hyps1, prop=prop1, ...} = rep_thm th1

        and {sign=sg2, hyps=hyps2, prop=prop2, ...} = rep_thm th2

    in  Sign.eq_sg (sg1,sg2) andalso

        aconvs(hyps1,hyps2) andalso

        prop1 aconv prop2

    end;

(*Do the two theorems have the same signature?*)

fun eq_thm_sg (th1,th2) = Sign.eq_sg(#sign(rep_thm th1), #sign(rep_thm th2));

(*Useful "distance" function for BEST_FIRST*)

val size_of_thm = size_of_term o #prop o rep_thm;

(*** Meta-Rewriting Rules ***)

val reflexive_thm =

  let val cx = cterm_of Sign.pure (Var(("x",0),TVar(("'a",0),logicS)))

  in Thm.reflexive cx end;

val symmetric_thm =

  let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)

  in standard(Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;

val transitive_thm =

  let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)

      val yz = read_cterm Sign.pure ("y::'a::logic == z",propT)

      val xythm = Thm.assume xy and yzthm = Thm.assume yz

  in standard(Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;

(** Below, a "conversion" has type cterm -> thm **)

val refl_cimplies = reflexive (cterm_of Sign.pure implies);

(*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)

(*Do not rewrite flex-flex pairs*)

fun goals_conv pred cv =

  let fun gconv i ct =

        let val (A,B) = Thm.dest_cimplies ct

            val (thA,j) = case term_of A of

                  Const("=?=",_)$_$_ => (reflexive A, i)

                | _ => (if pred i then cv A else reflexive A, i+1)

        in  combination (combination refl_cimplies thA) (gconv j B) end

        handle TERM _ => reflexive ct

  in gconv 1 end;

(*Use a conversion to transform a theorem*)

fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th;

(*rewriting conversion*)

fun rew_conv mode prover mss = rewrite_cterm mode mss prover;

(*Rewrite a theorem*)

fun rewrite_rule thms =

  fconv_rule (rew_conv (true,false) (K(K None)) (Thm.mss_of thms));

(*Rewrite the subgoals of a proof state (represented by a theorem) *)

fun rewrite_goals_rule thms =

  fconv_rule (goals_conv (K true) (rew_conv (true,false) (K(K None))

             (Thm.mss_of thms)));

(*Rewrite the subgoal of a proof state (represented by a theorem) *)

fun rewrite_goal_rule mode prover mss i thm =

  if 0 < i  andalso  i <= nprems_of thm

  then fconv_rule (goals_conv (fn j => j=i) (rew_conv mode prover mss)) thm

  else raise THM("rewrite_goal_rule",i,[thm]);

(** Derived rules mainly for METAHYPS **)

(*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)

fun equal_abs_elim ca eqth =

  let val {sign=signa, t=a, ...} = rep_cterm ca

      and combth = combination eqth (reflexive ca)

      val {sign,prop,...} = rep_thm eqth

      val (abst,absu) = Logic.dest_equals prop

      val cterm = cterm_of (Sign.merge (sign,signa))

  in  transitive (symmetric (beta_conversion (cterm (abst$a))))

           (transitive combth (beta_conversion (cterm (absu$a))))

  end

  handle THM _ => raise THM("equal_abs_elim", 0, [eqth]);

(*Calling equal_abs_elim with multiple terms*)

fun equal_abs_elim_list cts th = foldr (uncurry equal_abs_elim) (rev cts, th);

local

  open Logic

  val alpha = TVar(("'a",0), [])     (*  type ?'a::{}  *)

  fun err th = raise THM("flexpair_inst: ", 0, [th])

  fun flexpair_inst def th =

    let val {prop = Const _ $ t $ u,  sign,...} = rep_thm th

        val cterm = cterm_of sign

        fun cvar a = cterm(Var((a,0),alpha))

        val def' = cterm_instantiate [(cvar"t", cterm t), (cvar"u", cterm u)]

                   def

    in  equal_elim def' th

    end

    handle THM _ => err th | bind => err th

in

val flexpair_intr = flexpair_inst (symmetric flexpair_def)

and flexpair_elim = flexpair_inst flexpair_def

end;

(*Version for flexflex pairs -- this supports lifting.*)

fun flexpair_abs_elim_list cts =

    flexpair_intr o equal_abs_elim_list cts o flexpair_elim;

(*** Some useful meta-theorems ***)

(*The rule V/V, obtains assumption solving for eresolve_tac*)

val asm_rl = trivial(read_cterm Sign.pure ("PROP ?psi",propT));

(*Meta-level cut rule: [| V==>W; V |] ==> W *)

val cut_rl = trivial(read_cterm Sign.pure

        ("PROP ?psi ==> PROP ?theta", propT));

(*Generalized elim rule for one conclusion; cut_rl with reversed premises:

     [| PROP V;  PROP V ==> PROP W |] ==> PROP W *)

val revcut_rl =

  let val V = read_cterm Sign.pure ("PROP V", propT)

      and VW = read_cterm Sign.pure ("PROP V ==> PROP W", propT);

  in  standard (implies_intr V

                (implies_intr VW

                 (implies_elim (assume VW) (assume V))))

  end;

(*for deleting an unwanted assumption*)

val thin_rl =

  let val V = read_cterm Sign.pure ("PROP V", propT)

      and W = read_cterm Sign.pure ("PROP W", propT);

  in  standard (implies_intr V (implies_intr W (assume W)))

  end;

(* (!!x. PROP ?V) == PROP ?V       Allows removal of redundant parameters*)

val triv_forall_equality =

  let val V  = read_cterm Sign.pure ("PROP V", propT)

      and QV = read_cterm Sign.pure ("!!x::'a. PROP V", propT)

      and x  = read_cterm Sign.pure ("x", TFree("'a",logicS));

  in  standard (equal_intr (implies_intr QV (forall_elim x (assume QV)))

                           (implies_intr V  (forall_intr x (assume V))))

  end;

end

end;