(*  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.

*)

infix 0 RS RSN RL RLN MRS MRL OF COMP INCR_COMP COMP_INCR;

signature BASIC_DRULE =

sig

  val mk_implies: cterm * cterm -> cterm

  val list_implies: cterm list * cterm -> cterm

  val dest_implies: cterm -> cterm * cterm

  val dest_equals: cterm -> cterm * cterm

  val dest_equals_lhs: cterm -> cterm

  val dest_equals_rhs: cterm -> cterm

  val strip_imp_prems: cterm -> cterm list

  val strip_imp_concl: cterm -> cterm

  val cprems_of: thm -> cterm list

  val cterm_fun: (term -> term) -> (cterm -> cterm)

  val ctyp_fun: (typ -> typ) -> (ctyp -> ctyp)

  val read_insts: theory -> (indexname -> typ option) * (indexname -> sort option) ->

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

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

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

  val forall_intr_list: cterm list -> thm -> thm

  val forall_intr_frees: thm -> thm

  val forall_intr_vars: 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 gen_all: thm -> thm

  val lift_all: cterm -> thm -> thm

  val freeze_thaw: thm -> thm * (thm -> thm)

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

  val implies_elim_list: thm -> thm list -> thm

  val implies_intr_list: cterm list -> thm -> thm

  val instantiate: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm

  val zero_var_indexes: thm -> thm

  val implies_intr_hyps: thm -> thm

  val standard: thm -> thm

  val standard': thm -> thm

  val rotate_prems: int -> thm -> thm

  val rearrange_prems: int list -> thm -> thm

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

  val RS: thm * thm -> thm

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

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

  val MRS: thm list * thm -> thm

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

  val OF: thm * thm list -> thm

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

  val COMP: thm * thm -> thm

  val INCR_COMP: thm * thm -> thm

  val COMP_INCR: thm * thm -> thm

  val read_instantiate_sg: theory -> (string*string)list -> thm -> thm

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

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

  val eq_thm_thy: thm * thm -> bool

  val eq_thm_prop: thm * thm -> bool

  val equiv_thm: thm * thm -> bool

  val size_of_thm: thm -> int

  val reflexive_thm: thm

  val symmetric_thm: thm

  val transitive_thm: thm

  val symmetric_fun: thm -> thm

  val extensional: thm -> thm

  val equals_cong: thm

  val imp_cong: thm

  val swap_prems_eq: thm

  val asm_rl: thm

  val cut_rl: thm

  val revcut_rl: thm

  val thin_rl: thm

  val triv_forall_equality: thm

  val distinct_prems_rl: thm

  val swap_prems_rl: thm

  val equal_intr_rule: thm

  val equal_elim_rule1: thm

  val equal_elim_rule2: thm

  val inst: string -> string -> thm -> thm

  val instantiate': ctyp option list -> cterm option list -> thm -> thm

end;

signature DRULE =

sig

  include BASIC_DRULE

  val generalize: string list * string list -> thm -> thm

  val list_comb: cterm * cterm list -> cterm

  val strip_comb: cterm -> cterm * cterm list

  val strip_type: ctyp -> ctyp list * ctyp

  val lhs_of: thm -> cterm

  val rhs_of: thm -> cterm

  val beta_conv: cterm -> cterm -> cterm

  val plain_prop_of: thm -> term

  val fold_terms: (term -> 'a -> 'a) -> thm -> 'a -> 'a

  val add_used: thm -> string list -> string list

  val flexflex_unique: thm -> thm

  val close_derivation: thm -> thm

  val local_standard: thm -> thm

  val store_thm: bstring -> thm -> thm

  val store_standard_thm: bstring -> thm -> thm

  val store_thm_open: bstring -> thm -> thm

  val store_standard_thm_open: bstring -> thm -> thm

  val compose_single: thm * int * thm -> thm

  val add_rule: thm -> thm list -> thm list

  val del_rule: thm -> thm list -> thm list

  val merge_rules: thm list * thm list -> thm list

  val imp_cong_rule: thm -> thm -> thm

  val beta_eta_conversion: cterm -> thm

  val eta_long_conversion: cterm -> thm

  val eta_contraction_rule: thm -> thm

  val forall_conv: int -> (cterm -> thm) -> cterm -> thm

  val concl_conv: int -> (cterm -> thm) -> cterm -> thm

  val prems_conv: int -> (int -> cterm -> thm) -> cterm -> thm

  val goals_conv: (int -> bool) -> (cterm -> thm) -> cterm -> thm

  val fconv_rule: (cterm -> thm) -> thm -> thm

  val norm_hhf_eq: thm

  val is_norm_hhf: term -> bool

  val norm_hhf: theory -> term -> term

  val norm_hhf_cterm: cterm -> cterm

  val unvarify: thm -> thm

  val protect: cterm -> cterm

  val protectI: thm

  val protectD: thm

  val protect_cong: thm

  val implies_intr_protected: cterm list -> thm -> thm

  val termI: thm

  val mk_term: cterm -> thm

  val dest_term: thm -> cterm

  val cterm_rule: (thm -> thm) -> cterm -> cterm

  val term_rule: theory -> (thm -> thm) -> term -> term

  val sort_triv: theory -> typ * sort -> thm list

  val unconstrainTs: thm -> thm

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

  val rename_bvars': string option list -> thm -> thm

  val incr_indexes: thm -> thm -> thm

  val incr_indexes2: thm -> thm -> thm -> thm

  val remdups_rl: thm

  val multi_resolve: thm list -> thm -> thm Seq.seq

  val multi_resolves: thm list -> thm list -> thm Seq.seq

  val abs_def: thm -> thm

  val read_instantiate_sg': theory -> (indexname * string) list -> thm -> thm

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

end;

structure Drule: DRULE =

struct

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

fun dest_implies ct =

  (case Thm.term_of ct of

    Const ("==>", _) $ _ $ _ => Thm.dest_binop ct

  | _ => raise TERM ("dest_implies", [Thm.term_of ct]));

fun dest_equals ct =

  (case Thm.term_of ct of

    Const ("==", _) $ _ $ _ => Thm.dest_binop ct

  | _ => raise TERM ("dest_equals", [Thm.term_of ct]));

fun dest_equals_lhs ct =

  (case Thm.term_of ct of

    Const ("==", _) $ _ $ _ => #1 (Thm.dest_binop ct)

  | _ => raise TERM ("dest_equals_lhs", [Thm.term_of ct]));

fun dest_equals_rhs ct =

  (case Thm.term_of ct of

    Const ("==", _) $ _ $ _ => Thm.dest_arg ct

  | _ => raise TERM ("dest_equals_rhs", [Thm.term_of ct]));

val lhs_of = dest_equals_lhs o Thm.cprop_of;

val rhs_of = dest_equals_rhs o Thm.cprop_of;

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

fun strip_imp_prems ct =

  let val (cA, cB) = dest_implies ct

  in cA :: strip_imp_prems cB end

  handle TERM _ => [];

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

fun strip_imp_concl ct =

  (case Thm.term_of ct of

    Const ("==>", _) $ _ $ _ => strip_imp_concl (Thm.dest_arg ct)

  | _ => ct);

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

val cprems_of = strip_imp_prems o cprop_of;

fun cterm_fun f ct =

  let val {t, thy, ...} = Thm.rep_cterm ct

  in Thm.cterm_of thy (f t) end;

fun ctyp_fun f cT =

  let val {T, thy, ...} = Thm.rep_ctyp cT

  in Thm.ctyp_of thy (f T) end;

val cert = cterm_of ProtoPure.thy;

val implies = cert Term.implies;

fun mk_implies (A, B) = Thm.capply (Thm.capply implies A) B;

(*cterm version of list_implies: [A1,...,An], B  goes to [|A1;==>;An|]==>B *)

fun list_implies([], B) = B

  | list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));

(*cterm version of list_comb: maps  (f, [t1,...,tn])  to  f(t1,...,tn) *)

fun list_comb (f, []) = f

  | list_comb (f, t::ts) = list_comb (Thm.capply f t, ts);

(*cterm version of strip_comb: maps  f(t1,...,tn)  to  (f, [t1,...,tn]) *)

fun strip_comb ct =

  let

    fun stripc (p as (ct, cts)) =

      let val (ct1, ct2) = Thm.dest_comb ct

      in stripc (ct1, ct2 :: cts) end handle CTERM _ => p

  in stripc (ct, []) end;

(* cterm version of strip_type: maps  [T1,...,Tn]--->T  to   ([T1,T2,...,Tn], T) *)

fun strip_type cT = (case Thm.typ_of cT of

    Type ("fun", _) =>

      let

        val [cT1, cT2] = Thm.dest_ctyp cT;

        val (cTs, cT') = strip_type cT2

      in (cT1 :: cTs, cT') end

  | _ => ([], cT));

(*Beta-conversion for cterms, where x is an abstraction. Simply returns the rhs

  of the meta-equality returned by the beta_conversion rule.*)

fun beta_conv x y =

  Thm.dest_arg (cprop_of (Thm.beta_conversion false (Thm.capply x y)));

fun plain_prop_of raw_thm =

  let

    val thm = Thm.strip_shyps raw_thm;

    fun err msg = raise THM ("plain_prop_of: " ^ msg, 0, [thm]);

    val {hyps, prop, tpairs, ...} = Thm.rep_thm thm;

  in

    if not (null hyps) then

      err "theorem may not contain hypotheses"

    else if not (null (Thm.extra_shyps thm)) then

      err "theorem may not contain sort hypotheses"

    else if not (null tpairs) then

      err "theorem may not contain flex-flex pairs"

    else prop

  end;

fun fold_terms f th =

  let val {tpairs, prop, hyps, ...} = Thm.rep_thm th

  in fold (fn (t, u) => f t #> f u) tpairs #> f prop #> fold f hyps end;

(** reading of instantiations **)

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 thy (rtypes,rsorts) (types,sorts) used insts =

let

    fun is_tv ((a, _), _) =

      (case Symbol.explode a of "'" :: _ => true | _ => false);

    val (tvs, vs) = List.partition is_tv insts;

    fun sort_of ixn = case rsorts ixn of SOME S => S | NONE => absent ixn;

    fun readT (ixn, st) =

        let val S = sort_of ixn;

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

        in if Sign.typ_instance thy (T, TVar(ixn,S)) then (ixn,T)

           else inst_failure ixn

        end

    val tye = map readT tvs;

    fun mkty(ixn,st) = (case rtypes ixn of

                          SOME T => (ixn,(st,typ_subst_TVars tye T))

                        | NONE => absent ixn);

    val ixnsTs = map mkty vs;

    val ixns = map fst ixnsTs

    and sTs  = map snd ixnsTs

    val (cts,tye2) = read_def_cterms(thy,types,sorts) used false sTs;

    fun mkcVar(ixn,T) =

        let val U = typ_subst_TVars tye2 T

        in cterm_of thy (Var(ixn,U)) end

    val ixnTs = ListPair.zip(ixns, map snd sTs)

in (map (fn (ixn, T) => (ctyp_of thy (TVar (ixn, sort_of ixn)),

      ctyp_of thy T)) (tye2 @ tye),

    ListPair.zip(map mkcVar ixnTs,cts))

end;

(*** 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 vars = fold_terms Term.add_vars thm [];

    val frees = fold_terms Term.add_frees thm [];

    val tvars = fold_terms Term.add_tvars thm [];

    val tfrees = fold_terms Term.add_tfrees thm [];

    fun types (a, i) =

      if i < 0 then AList.lookup (op =) frees a else AList.lookup (op =) vars (a, i);

    fun sorts (a, i) =

      if i < 0 then AList.lookup (op =) tfrees a else AList.lookup (op =) tvars (a, i);

  in (types, sorts) end;

val add_used =

  (fold_terms o fold_types o fold_atyps)

    (fn TFree (a, _) => insert (op =) a

      | TVar ((a, _), _) => insert (op =) a

      | _ => I);

(** Standardization of rules **)

(* type classes and sorts *)

fun sort_triv thy (T, S) =

  let

    val certT = Thm.ctyp_of thy;

    val cT = certT T;

    fun class_triv c =

      Thm.class_triv thy c

      |> Thm.instantiate ([(certT (TVar (("'a", 0), [c])), cT)], []);

  in map class_triv S end;

fun unconstrainTs th =

  fold (Thm.unconstrainT o Thm.ctyp_of (Thm.theory_of_thm th) o TVar)

    (fold_terms Term.add_tvars th []) th;

(*Generalization over a list of variables*)

val forall_intr_list = fold_rev forall_intr;

(*Generalization over all suitable Free variables*)

fun forall_intr_frees th =

    let

      val {prop, hyps, tpairs, thy,...} = rep_thm th;

      val fixed = fold Term.add_frees (Thm.terms_of_tpairs tpairs @ hyps) [];

      val frees = Term.fold_aterms (fn Free v =>

        if member (op =) fixed v then I else insert (op =) v | _ => I) prop [];

    in fold (forall_intr o cterm_of thy o Free) frees th end;

(*Generalization over Vars -- canonical order*)

fun forall_intr_vars th =

  fold forall_intr

    (map (Thm.cterm_of (Thm.theory_of_thm th) o Var) (fold_terms Term.add_vars th [])) th;

val forall_elim_var = PureThy.forall_elim_var;

val forall_elim_vars = PureThy.forall_elim_vars;

fun outer_params t =

  let val vs = Term.strip_all_vars t

  in Name.variant_list [] (map (Name.clean o #1) vs) ~~ map #2 vs end;

(*generalize outermost parameters*)

fun gen_all th =

  let

    val {thy, prop, maxidx, ...} = Thm.rep_thm th;

    val cert = Thm.cterm_of thy;

    fun elim (x, T) = Thm.forall_elim (cert (Var ((x, maxidx + 1), T)));

  in fold elim (outer_params prop) th end;

(*lift vars wrt. outermost goal parameters

  -- reverses the effect of gen_all modulo higher-order unification*)

fun lift_all goal th =

  let

    val thy = Theory.merge (Thm.theory_of_cterm goal, Thm.theory_of_thm th);

    val cert = Thm.cterm_of thy;

    val maxidx = Thm.maxidx_of th;

    val ps = outer_params (Thm.term_of goal)

      |> map (fn (x, T) => Var ((x, maxidx + 1), Logic.incr_tvar (maxidx + 1) T));

    val Ts = map Term.fastype_of ps;

    val inst = fold_terms Term.add_vars th [] |> map (fn (xi, T) =>

      (cert (Var (xi, T)), cert (Term.list_comb (Var (xi, Ts ---> T), ps))));

  in

    th |> Thm.instantiate ([], inst)

    |> fold_rev (Thm.forall_intr o cert) ps

  end;

(*direct generalization*)

fun generalize names th = Thm.generalize names (Thm.maxidx_of th + 1) th;

(*specialization over a list of cterms*)

val forall_elim_list = fold forall_elim;

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

val implies_intr_list = fold_rev implies_intr;

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

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

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

fun zero_var_indexes th =

  let

    val thy = Thm.theory_of_thm th;

    val certT = Thm.ctyp_of thy and cert = Thm.cterm_of thy;

    val (instT, inst) = TermSubst.zero_var_indexes_inst (Thm.full_prop_of th);

    val cinstT = map (fn (v, T) => (certT (TVar v), certT T)) instT;

    val cinst = map (fn (v, t) => (cert (Var v), cert t)) inst;

  in Thm.adjust_maxidx_thm ~1 (Thm.instantiate (cinstT, cinst) th) end;

(** Standard form of object-rule: no hypotheses, flexflex constraints,

    Frees, or outer quantifiers; all generality expressed by Vars of index 0.**)

(*Discharge all hypotheses.*)

fun implies_intr_hyps th =

  fold Thm.implies_intr (#hyps (Thm.crep_thm th)) th;

(*Squash a theorem's flexflex constraints provided it can be done uniquely.

  This step can lose information.*)

fun flexflex_unique th =

  if null (tpairs_of th) then th else

    case Seq.chop 2 (flexflex_rule th) of

      ([th],_) => th

    | ([],_)   => raise THM("flexflex_unique: impossible constraints", 0, [th])

    |      _   => raise THM("flexflex_unique: multiple unifiers", 0, [th]);

fun close_derivation thm =

  if Thm.get_name_tags thm = ("", []) then Thm.name_thm ("", thm)

  else thm;

val standard' =

  implies_intr_hyps

  #> forall_intr_frees

  #> `Thm.maxidx_of

  #-> (fn maxidx =>

    forall_elim_vars (maxidx + 1)

    #> Thm.strip_shyps

    #> zero_var_indexes

    #> Thm.varifyT

    #> Thm.compress);

val standard =

  flexflex_unique

  #> standard'

  #> close_derivation;

val local_standard =

  flexflex_unique

  #> Thm.strip_shyps

  #> zero_var_indexes

  #> Thm.compress

  #> close_derivation;

(*Convert all Vars in a theorem to Frees.  Also return a function for

  reversing that operation.  DOES NOT WORK FOR TYPE VARIABLES.

  Similar code in type/freeze_thaw*)

fun freeze_thaw_robust th =

 let val fth = Thm.freezeT th

     val {prop, tpairs, thy, ...} = rep_thm fth

 in

   case foldr add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) of

       [] => (fth, fn i => fn x => x)   (*No vars: nothing to do!*)

     | vars =>

         let fun newName (Var(ix,_)) = (ix, gensym (string_of_indexname ix))

             val alist = map newName vars

             fun mk_inst (Var(v,T)) =

                 (cterm_of thy (Var(v,T)),

                  cterm_of thy (Free(((the o AList.lookup (op =) alist) v), T)))

             val insts = map mk_inst vars

             fun thaw i th' = (*i is non-negative increment for Var indexes*)

                 th' |> forall_intr_list (map #2 insts)

                     |> forall_elim_list (map (Thm.cterm_incr_indexes i o #1) insts)

         in  (Thm.instantiate ([],insts) fth, thaw)  end

 end;

(*Basic version of the function above. No option to rename Vars apart in thaw.

  The Frees created from Vars have nice names. FIXME: does not check for

  clashes with variables in the assumptions, so delete and use freeze_thaw_robust instead?*)

fun freeze_thaw th =

 let val fth = Thm.freezeT th

     val {prop, tpairs, thy, ...} = rep_thm fth

 in

   case foldr add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) of

       [] => (fth, fn x => x)

     | vars =>

         let fun newName (Var(ix,_), (pairs,used)) =

                   let val v = Name.variant used (string_of_indexname ix)

                   in  ((ix,v)::pairs, v::used)  end;

             val (alist, _) = foldr newName ([], Library.foldr add_term_names

               (prop :: Thm.terms_of_tpairs tpairs, [])) vars

             fun mk_inst (Var(v,T)) =

                 (cterm_of thy (Var(v,T)),

                  cterm_of thy (Free(((the o AList.lookup (op =) alist) v), T)))

             val insts = map mk_inst vars

             fun thaw th' =

                 th' |> forall_intr_list (map #2 insts)

                     |> forall_elim_list (map #1 insts)

         in  (Thm.instantiate ([],insts) fth, thaw)  end

 end;

(*Rotates a rule's premises to the left by k*)

val rotate_prems = permute_prems 0;

(* permute prems, where the i-th position in the argument list (counting from 0)

   gives the position within the original thm to be transferred to position i.

   Any remaining trailing positions are left unchanged. *)

val rearrange_prems = let

  fun rearr new []      thm = thm

  |   rearr new (p::ps) thm = rearr (new+1)

     (map (fn q => if new<=q andalso q<p then q+1 else q) ps)

     (permute_prems (new+1) (new-p) (permute_prems new (p-new) thm))

  in rearr 0 end;

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

fun tha RSN (i,thb) =

  case Seq.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 = Seq.list_of (resolve thb) handle THM _ => []

  in maps 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;

(*A version of MRS with more appropriate argument order*)

fun bottom_rl OF rls = rls MRS bottom_rl;

(*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) =

    Seq.list_of (bicompose false (false,tha,0) i thb);

fun compose_single (tha,i,thb) =

  (case compose (tha,i,thb) of

    [th] => th

  | _ => raise THM ("compose: unique result expected", i, [tha,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]);

(** theorem equality **)

(*True if the two theorems have the same theory.*)

val eq_thm_thy = eq_thy o pairself Thm.theory_of_thm;

(*True if the two theorems have the same prop field, ignoring hyps, der, etc.*)

val eq_thm_prop = op aconv o pairself Thm.full_prop_of;

(*Useful "distance" function for BEST_FIRST*)

val size_of_thm = size_of_term o Thm.full_prop_of;

(*maintain lists of theorems --- preserving canonical order*)

val del_rule = remove eq_thm_prop;

fun add_rule th = cons th o del_rule th;

val merge_rules = Library.merge eq_thm_prop;

(*pattern equivalence*)

fun equiv_thm ths =

  Pattern.equiv (Theory.merge (pairself Thm.theory_of_thm ths)) (pairself Thm.full_prop_of ths);

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

fun read_prop s = read_cterm ProtoPure.thy (s, propT);

fun store_thm name thm = hd (PureThy.smart_store_thms (name, [thm]));

fun store_standard_thm name thm = store_thm name (standard thm);

fun store_thm_open name thm = hd (PureThy.smart_store_thms_open (name, [thm]));

fun store_standard_thm_open name thm = store_thm_open name (standard' thm);

val reflexive_thm =

  let val cx = cert (Var(("x",0),TVar(("'a",0),[])))

  in store_standard_thm_open "reflexive" (Thm.reflexive cx) end;

val symmetric_thm =

  let val xy = read_prop "x == y"

  in store_standard_thm_open "symmetric" (Thm.implies_intr xy (Thm.symmetric (Thm.assume xy))) end;

val transitive_thm =

  let val xy = read_prop "x == y"

      val yz = read_prop "y == z"

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

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

fun symmetric_fun thm = thm RS symmetric_thm;

fun extensional eq =

  let val eq' =

    abstract_rule "x" (Thm.dest_arg (fst (dest_equals (cprop_of eq)))) eq

  in equal_elim (eta_conversion (cprop_of eq')) eq' end;

val equals_cong =

  store_standard_thm_open "equals_cong" (Thm.reflexive (read_prop "x == y"));

val imp_cong =

  let

    val ABC = read_prop "PROP A ==> PROP B == PROP C"

    val AB = read_prop "PROP A ==> PROP B"

    val AC = read_prop "PROP A ==> PROP C"

    val A = read_prop "PROP A"

  in

    store_standard_thm_open "imp_cong" (implies_intr ABC (equal_intr

      (implies_intr AB (implies_intr A

        (equal_elim (implies_elim (assume ABC) (assume A))

          (implies_elim (assume AB) (assume A)))))

      (implies_intr AC (implies_intr A

        (equal_elim (symmetric (implies_elim (assume ABC) (assume A)))

          (implies_elim (assume AC) (assume A)))))))

  end;

val swap_prems_eq =

  let

    val ABC = read_prop "PROP A ==> PROP B ==> PROP C"

    val BAC = read_prop "PROP B ==> PROP A ==> PROP C"

    val A = read_prop "PROP A"

    val B = read_prop "PROP B"

  in

    store_standard_thm_open "swap_prems_eq" (equal_intr

      (implies_intr ABC (implies_intr B (implies_intr A

        (implies_elim (implies_elim (assume ABC) (assume A)) (assume B)))))

      (implies_intr BAC (implies_intr A (implies_intr B

        (implies_elim (implies_elim (assume BAC) (assume B)) (assume A))))))

  end;

val imp_cong_rule = combination o combination (reflexive implies);

local

  val dest_eq = dest_equals o cprop_of

  val rhs_of = snd o dest_eq

in

fun beta_eta_conversion t =

  let val thm = beta_conversion true t

  in transitive thm (eta_conversion (rhs_of thm)) end

end;

fun eta_long_conversion ct = transitive (beta_eta_conversion ct)

  (symmetric (beta_eta_conversion (cterm_fun (Pattern.eta_long []) ct)));

(*Contract all eta-redexes in the theorem, lest they give rise to needless abstractions*)

fun eta_contraction_rule th =

  equal_elim (eta_conversion (cprop_of th)) th;

val abs_def =

  let

    fun contract_lhs th =

      Thm.transitive (Thm.symmetric (beta_eta_conversion (fst (dest_equals (cprop_of th))))) th;

    fun abstract cx th = Thm.abstract_rule

        (case Thm.term_of cx of Var ((x, _), _) => x | Free (x, _) => x | _ => "x") cx th

      handle THM _ => raise THM ("Malformed definitional equation", 0, [th]);

  in

    contract_lhs

    #> `(snd o strip_comb o fst o dest_equals o cprop_of)

    #-> fold_rev abstract

    #> contract_lhs

  end;

(*rewrite B in !!x1 ... xn. B*)

fun forall_conv 0 cv ct = cv ct

  | forall_conv n cv ct =

      (case try Thm.dest_comb ct of

        NONE => cv ct

      | SOME (A, B) =>

          (case (term_of A, term_of B) of

            (Const ("all", _), Abs (x, _, _)) =>

              let val (v, B') = Thm.dest_abs (SOME (gensym "all_")) B in

                Thm.combination (Thm.reflexive A)

                  (Thm.abstract_rule x v (forall_conv (n - 1) cv B'))

              end

          | _ => cv ct));

(*rewrite B in A1 ==> ... ==> An ==> B*)

fun concl_conv 0 cv ct = cv ct

  | concl_conv n cv ct =

      (case try dest_implies ct of

        NONE => cv ct

      | SOME (A, B) => imp_cong_rule (reflexive A) (concl_conv (n - 1) cv B));

(*rewrite the A's in A1 ==> ... ==> An ==> B*)

fun prems_conv 0 _ = reflexive

  | prems_conv n cv =

      let

        fun conv i ct =

          if i = n + 1 then reflexive ct

          else

            (case try dest_implies ct of

              NONE => reflexive ct

            | SOME (A, B) => imp_cong_rule (cv i A) (conv (i + 1) B));

  in conv 1 end;

fun goals_conv pred cv = prems_conv ~1 (fn i => if pred i then cv else reflexive);

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

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

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

val asm_rl = store_standard_thm_open "asm_rl" (Thm.trivial (read_prop "PROP ?psi"));

val _ = store_thm "_" asm_rl;

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

val cut_rl =

  store_standard_thm_open "cut_rl"

    (Thm.trivial (read_prop "PROP ?psi ==> PROP ?theta"));

(*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_prop "PROP V"

      and VW = read_prop "PROP V ==> PROP W";

  in

    store_standard_thm_open "revcut_rl"

      (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_prop "PROP V"

      and W = read_prop "PROP W";

  in store_standard_thm_open "thin_rl" (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_prop "PROP V"

      and QV = read_prop "!!x::'a. PROP V"

      and x  = cert (Free ("x", Term.aT []));

  in

    store_standard_thm_open "triv_forall_equality"

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

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

  end;

(* (PROP ?Phi ==> PROP ?Phi ==> PROP ?Psi) ==>

   (PROP ?Phi ==> PROP ?Psi)

*)

val distinct_prems_rl =

  let

    val AAB = read_prop "PROP Phi ==> PROP Phi ==> PROP Psi"

    val A = read_prop "PROP Phi";

  in

    store_standard_thm_open "distinct_prems_rl"

      (implies_intr_list [AAB, A] (implies_elim_list (assume AAB) [assume A, assume A]))

  end;

(* (PROP ?PhiA ==> PROP ?PhiB ==> PROP ?Psi) ==>

   (PROP ?PhiB ==> PROP ?PhiA ==> PROP ?Psi)

   `thm COMP swap_prems_rl' swaps the first two premises of `thm'

*)

val swap_prems_rl =

  let val cmajor = read_prop "PROP PhiA ==> PROP PhiB ==> PROP Psi";

      val major = assume cmajor;

      val cminor1 = read_prop "PROP PhiA";

      val minor1 = assume cminor1;

      val cminor2 = read_prop "PROP PhiB";

      val minor2 = assume cminor2;

  in store_standard_thm_open "swap_prems_rl"

       (implies_intr cmajor (implies_intr cminor2 (implies_intr cminor1

         (implies_elim (implies_elim major minor1) minor2))))

  end;

(* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]

   ==> PROP ?phi == PROP ?psi

   Introduction rule for == as a meta-theorem.

*)

val equal_intr_rule =

  let val PQ = read_prop "PROP phi ==> PROP psi"

      and QP = read_prop "PROP psi ==> PROP phi"

  in

    store_standard_thm_open "equal_intr_rule"

      (implies_intr PQ (implies_intr QP (equal_intr (assume PQ) (assume QP))))

  end;

(* PROP ?phi == PROP ?psi ==> PROP ?phi ==> PROP ?psi *)

val equal_elim_rule1 =

  let val eq = read_prop "PROP phi == PROP psi"

      and P = read_prop "PROP phi"

  in store_standard_thm_open "equal_elim_rule1"

    (Thm.equal_elim (assume eq) (assume P) |> implies_intr_list [eq, P])

  end;

(* PROP ?psi == PROP ?phi ==> PROP ?phi ==> PROP ?psi *)

val equal_elim_rule2 =

  store_standard_thm_open "equal_elim_rule2" (symmetric_thm RS equal_elim_rule1);

(* "[| PROP ?phi; PROP ?phi; PROP ?psi |] ==> PROP ?psi" *)

val remdups_rl =

  let val P = read_prop "PROP phi" and Q = read_prop "PROP psi";

  in store_standard_thm_open "remdups_rl" (implies_intr_list [P, P, Q] (Thm.assume Q)) end;

(*(PROP ?phi ==> (!!x. PROP ?psi(x))) == (!!x. PROP ?phi ==> PROP ?psi(x))

  Rewrite rule for HHF normalization.*)

val norm_hhf_eq =

  let

    val aT = TFree ("'a", []);

    val all = Term.all aT;

    val x = Free ("x", aT);

    val phi = Free ("phi", propT);

    val psi = Free ("psi", aT --> propT);

    val cx = cert x;

    val cphi = cert phi;

    val lhs = cert (Logic.mk_implies (phi, all $ Abs ("x", aT, psi $ Bound 0)));

    val rhs = cert (all $ Abs ("x", aT, Logic.mk_implies (phi, psi $ Bound 0)));

  in

    Thm.equal_intr

      (Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)

        |> Thm.forall_elim cx

        |> Thm.implies_intr cphi

        |> Thm.forall_intr cx

        |> Thm.implies_intr lhs)

      (Thm.implies_elim

          (Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)

        |> Thm.forall_intr cx

        |> Thm.implies_intr cphi

        |> Thm.implies_intr rhs)

    |> store_standard_thm_open "norm_hhf_eq"

  end;

val norm_hhf_prop = Logic.dest_equals (Thm.prop_of norm_hhf_eq);

fun is_norm_hhf tm =

  let

    fun is_norm (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false

      | is_norm (t $ u) = is_norm t andalso is_norm u

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

      | is_norm _ = true;

  in is_norm (Envir.beta_eta_contract tm) end;

fun norm_hhf thy t =

  if is_norm_hhf t then t

  else Pattern.rewrite_term thy [norm_hhf_prop] [] t;

fun norm_hhf_cterm ct =

  if is_norm_hhf (Thm.term_of ct) then ct

  else cterm_fun (Pattern.rewrite_term (Thm.theory_of_cterm ct) [norm_hhf_prop] []) ct;

(*** Instantiate theorem th, reading instantiations in theory thy ****)

(*Version that normalizes the result: Thm.instantiate no longer does that*)

fun instantiate instpair th = Thm.instantiate instpair th  COMP   asm_rl;

fun read_instantiate_sg' thy sinsts th =

    let val ts = types_sorts th;

        val used = add_used th [];

    in  instantiate (read_insts thy ts ts used sinsts) th  end;

fun read_instantiate_sg thy sinsts th =

  read_instantiate_sg' thy (map (apfst Syntax.read_indexname) sinsts) th;

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

fun read_instantiate sinsts th =

    read_instantiate_sg (Thm.theory_of_thm th) sinsts th;

fun read_instantiate' sinsts th =

    read_instantiate_sg' (Thm.theory_of_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), (thy,tye,maxidx)) =

    let val {thy=thyt, t=t, T= T, maxidx=maxt,...} = rep_cterm ct

        and {thy=thyu, t=u, T= U, maxidx=maxu,...} = rep_cterm cu;

        val maxi = Int.max(maxidx, Int.max(maxt, maxu));

        val thy' = Theory.merge(thy, Theory.merge(thyt, thyu))

        val (tye',maxi') = Sign.typ_unify thy' (T, U) (tye, maxi)

          handle Type.TUNIFY => raise TYPE("Ill-typed instantiation", [T,U], [t,u])

    in  (thy', tye', maxi')  end;

in

fun cterm_instantiate ctpairs0 th =

  let val (thy,tye,_) = foldr add_types (Thm.theory_of_thm th, Vartab.empty, 0) ctpairs0

      fun instT(ct,cu) =

        let val inst = cterm_of thy o Envir.subst_TVars tye o term_of

        in (inst ct, inst cu) end

      fun ctyp2 (ixn, (S, T)) = (ctyp_of thy (TVar (ixn, S)), ctyp_of thy T)

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

  handle TERM _ =>

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

       | TYPE (msg, _, _) => raise THM(msg, 0, [th])

end;

(* global schematic variables *)

fun unvarify th =

  let

    val thy = Thm.theory_of_thm th;

    val cert = Thm.cterm_of thy;

    val certT = Thm.ctyp_of thy;

    val prop = Thm.full_prop_of th;

    val _ = map Logic.unvarify (prop :: Thm.hyps_of th)

      handle TERM (msg, _) => raise THM (msg, 0, [th]);

    val instT0 = rev (Term.add_tvars prop []) |> map (fn v as ((a, _), S) => (v, TFree (a, S)));

    val instT = map (fn (v, T) => (certT (TVar v), certT T)) instT0;

    val inst = rev (Term.add_vars prop []) |> map (fn ((a, i), T) =>

      let val T' = TermSubst.instantiateT instT0 T

      in (cert (Var ((a, i), T')), cert (Free ((a, T')))) end);

  in Thm.instantiate (instT, inst) th end;

(** protected propositions and embedded terms **)

local

  val A = cert (Free ("A", propT));

  val prop_def = unvarify ProtoPure.prop_def;

  val term_def = unvarify ProtoPure.term_def;

in

  val protect = Thm.capply (cert Logic.protectC);

  val protectI = store_thm "protectI" (PureThy.kind_rule Thm.internalK (standard

      (Thm.equal_elim (Thm.symmetric prop_def) (Thm.assume A))));

  val protectD = store_thm "protectD" (PureThy.kind_rule Thm.internalK (standard

      (Thm.equal_elim prop_def (Thm.assume (protect A)))));

  val protect_cong = store_standard_thm_open "protect_cong" (Thm.reflexive (protect A));

  val termI = store_thm "termI" (PureThy.kind_rule Thm.internalK (standard

      (Thm.equal_elim (Thm.symmetric term_def) (Thm.forall_intr A (Thm.trivial A)))));

end;

fun implies_intr_protected asms th =

  let val asms' = map protect asms in

    implies_elim_list

      (implies_intr_list asms th)

      (map (fn asm' => Thm.assume asm' RS protectD) asms')

    |> implies_intr_list asms'

  end;

fun mk_term ct =

  let

    val {thy, T, ...} = Thm.rep_cterm ct;

    val cert = Thm.cterm_of thy;

    val certT = Thm.ctyp_of thy;

    val a = certT (TVar (("'a", 0), []));

    val x = cert (Var (("x", 0), T));

  in Thm.instantiate ([(a, certT T)], [(x, ct)]) termI end;

fun dest_term th =

  let val cprop = strip_imp_concl (Thm.cprop_of th) in

    if can Logic.dest_term (Thm.term_of cprop) then

      Thm.dest_arg cprop

    else raise THM ("dest_term", 0, [th])

  end;

fun cterm_rule f = dest_term o f o mk_term;

fun term_rule thy f t = Thm.term_of (cterm_rule f (Thm.cterm_of thy t));

(** variations on instantiate **)

(*shorthand for instantiating just one variable in the current theory*)

fun inst x t = read_instantiate_sg (the_context()) [(x,t)];

(* instantiate by left-to-right occurrence of variables *)

fun instantiate' cTs cts thm =

  let

    fun err msg =

      raise TYPE ("instantiate': " ^ msg,

        map_filter (Option.map Thm.typ_of) cTs,

        map_filter (Option.map Thm.term_of) cts);

    fun inst_of (v, ct) =