--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/TFL/tfl.ML Wed Jan 03 21:20:40 2001 +0100
@@ -0,0 +1,1001 @@
+(* Title: TFL/tfl.ML
+ ID: $Id$
+ Author: Konrad Slind, Cambridge University Computer Laboratory
+ Copyright 1997 University of Cambridge
+
+First part of main module.
+*)
+
+signature PRIM =
+sig
+ val trace: bool ref
+ type pattern
+ val mk_functional: theory -> term list -> {functional: term, pats: pattern list}
+ val wfrec_definition0: theory -> string -> term -> term -> theory * thm
+ val post_definition: thm list -> theory * (thm * pattern list) ->
+ {theory: theory,
+ rules: thm,
+ rows: int list,
+ TCs: term list list,
+ full_pats_TCs: (term * term list) list}
+ val wfrec_eqns: theory -> xstring -> thm list -> term list ->
+ {WFR: term,
+ SV: term list,
+ proto_def: term,
+ extracta: (thm * term list) list,
+ pats: pattern list}
+ val lazyR_def: theory -> xstring -> thm list -> term list ->
+ {theory: theory,
+ rules: thm,
+ R: term,
+ SV: term list,
+ full_pats_TCs: (term * term list) list,
+ patterns : pattern list}
+ val mk_induction: theory ->
+ {fconst: term, R: term, SV: term list, pat_TCs_list: (term * term list) list} -> thm
+ val postprocess: {wf_tac: tactic, terminator: tactic, simplifier: cterm -> thm} -> theory ->
+ {rules: thm, induction: thm, TCs: term list list} ->
+ {rules: thm, induction: thm, nested_tcs: thm list}
+end;
+
+structure Prim: PRIM =
+struct
+
+val trace = ref false;
+
+open BasisLibrary;
+
+structure R = Rules;
+structure S = USyntax;
+structure U = Utils;
+
+
+fun TFL_ERR func mesg = U.ERR {module = "Tfl", func = func, mesg = mesg};
+
+val concl = #2 o R.dest_thm;
+val hyp = #1 o R.dest_thm;
+
+val list_mk_type = U.end_itlist (curry (op -->));
+
+fun enumerate xs = ListPair.zip(xs, 0 upto (length xs - 1));
+
+fun front_last [] = raise TFL_ERR "front_last" "empty list"
+ | front_last [x] = ([],x)
+ | front_last (h::t) =
+ let val (pref,x) = front_last t
+ in
+ (h::pref,x)
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * The next function is common to pattern-match translation and
+ * proof of completeness of cases for the induction theorem.
+ *
+ * The curried function "gvvariant" returns a function to generate distinct
+ * variables that are guaranteed not to be in names. The names of
+ * the variables go u, v, ..., z, aa, ..., az, ... The returned
+ * function contains embedded refs!
+ *---------------------------------------------------------------------------*)
+fun gvvariant names =
+ let val slist = ref names
+ val vname = ref "u"
+ fun new() =
+ if !vname mem_string (!slist)
+ then (vname := bump_string (!vname); new())
+ else (slist := !vname :: !slist; !vname)
+ in
+ fn ty => Free(new(), ty)
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Used in induction theorem production. This is the simple case of
+ * partitioning up pattern rows by the leading constructor.
+ *---------------------------------------------------------------------------*)
+fun ipartition gv (constructors,rows) =
+ let fun pfail s = raise TFL_ERR "partition.part" s
+ fun part {constrs = [], rows = [], A} = rev A
+ | part {constrs = [], rows = _::_, A} = pfail"extra cases in defn"
+ | part {constrs = _::_, rows = [], A} = pfail"cases missing in defn"
+ | part {constrs = c::crst, rows, A} =
+ let val (Name,Ty) = dest_Const c
+ val L = binder_types Ty
+ val (in_group, not_in_group) =
+ U.itlist (fn (row as (p::rst, rhs)) =>
+ fn (in_group,not_in_group) =>
+ let val (pc,args) = S.strip_comb p
+ in if (#1(dest_Const pc) = Name)
+ then ((args@rst, rhs)::in_group, not_in_group)
+ else (in_group, row::not_in_group)
+ end) rows ([],[])
+ val col_types = U.take type_of (length L, #1(hd in_group))
+ in
+ part{constrs = crst, rows = not_in_group,
+ A = {constructor = c,
+ new_formals = map gv col_types,
+ group = in_group}::A}
+ end
+ in part{constrs = constructors, rows = rows, A = []}
+ end;
+
+
+
+(*---------------------------------------------------------------------------
+ * Each pattern carries with it a tag (i,b) where
+ * i is the clause it came from and
+ * b=true indicates that clause was given by the user
+ * (or is an instantiation of a user supplied pattern)
+ * b=false --> i = ~1
+ *---------------------------------------------------------------------------*)
+
+type pattern = term * (int * bool)
+
+fun pattern_map f (tm,x) = (f tm, x);
+
+fun pattern_subst theta = pattern_map (subst_free theta);
+
+val pat_of = fst;
+fun row_of_pat x = fst (snd x);
+fun given x = snd (snd x);
+
+(*---------------------------------------------------------------------------
+ * Produce an instance of a constructor, plus genvars for its arguments.
+ *---------------------------------------------------------------------------*)
+fun fresh_constr ty_match colty gv c =
+ let val (_,Ty) = dest_Const c
+ val L = binder_types Ty
+ and ty = body_type Ty
+ val ty_theta = ty_match ty colty
+ val c' = S.inst ty_theta c
+ val gvars = map (S.inst ty_theta o gv) L
+ in (c', gvars)
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Goes through a list of rows and picks out the ones beginning with a
+ * pattern with constructor = Name.
+ *---------------------------------------------------------------------------*)
+fun mk_group Name rows =
+ U.itlist (fn (row as ((prfx, p::rst), rhs)) =>
+ fn (in_group,not_in_group) =>
+ let val (pc,args) = S.strip_comb p
+ in if ((#1 (Term.dest_Const pc) = Name) handle TERM _ => false)
+ then (((prfx,args@rst), rhs)::in_group, not_in_group)
+ else (in_group, row::not_in_group) end)
+ rows ([],[]);
+
+(*---------------------------------------------------------------------------
+ * Partition the rows. Not efficient: we should use hashing.
+ *---------------------------------------------------------------------------*)
+fun partition _ _ (_,_,_,[]) = raise TFL_ERR "partition" "no rows"
+ | partition gv ty_match
+ (constructors, colty, res_ty, rows as (((prfx,_),_)::_)) =
+let val fresh = fresh_constr ty_match colty gv
+ fun part {constrs = [], rows, A} = rev A
+ | part {constrs = c::crst, rows, A} =
+ let val (c',gvars) = fresh c
+ val (Name,Ty) = dest_Const c'
+ val (in_group, not_in_group) = mk_group Name rows
+ val in_group' =
+ if (null in_group) (* Constructor not given *)
+ then [((prfx, #2(fresh c)), (S.ARB res_ty, (~1,false)))]
+ else in_group
+ in
+ part{constrs = crst,
+ rows = not_in_group,
+ A = {constructor = c',
+ new_formals = gvars,
+ group = in_group'}::A}
+ end
+in part{constrs=constructors, rows=rows, A=[]}
+end;
+
+(*---------------------------------------------------------------------------
+ * Misc. routines used in mk_case
+ *---------------------------------------------------------------------------*)
+
+fun mk_pat (c,l) =
+ let val L = length (binder_types (type_of c))
+ fun build (prfx,tag,plist) =
+ let val args = take (L,plist)
+ and plist' = drop(L,plist)
+ in (prfx,tag,list_comb(c,args)::plist') end
+ in map build l end;
+
+fun v_to_prfx (prfx, v::pats) = (v::prfx,pats)
+ | v_to_prfx _ = raise TFL_ERR "mk_case" "v_to_prfx";
+
+fun v_to_pats (v::prfx,tag, pats) = (prfx, tag, v::pats)
+ | v_to_pats _ = raise TFL_ERR "mk_case" "v_to_pats";
+
+
+(*----------------------------------------------------------------------------
+ * Translation of pattern terms into nested case expressions.
+ *
+ * This performs the translation and also builds the full set of patterns.
+ * Thus it supports the construction of induction theorems even when an
+ * incomplete set of patterns is given.
+ *---------------------------------------------------------------------------*)
+
+fun mk_case ty_info ty_match usednames range_ty =
+ let
+ fun mk_case_fail s = raise TFL_ERR "mk_case" s
+ val fresh_var = gvvariant usednames
+ val divide = partition fresh_var ty_match
+ fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
+ | expand constructors ty (row as ((prfx, p::rst), rhs)) =
+ if (is_Free p)
+ then let val fresh = fresh_constr ty_match ty fresh_var
+ fun expnd (c,gvs) =
+ let val capp = list_comb(c,gvs)
+ in ((prfx, capp::rst), pattern_subst[(p,capp)] rhs)
+ end
+ in map expnd (map fresh constructors) end
+ else [row]
+ fun mk{rows=[],...} = mk_case_fail"no rows"
+ | mk{path=[], rows = ((prfx, []), (tm,tag))::_} = (* Done *)
+ ([(prfx,tag,[])], tm)
+ | mk{path=[], rows = _::_} = mk_case_fail"blunder"
+ | mk{path as u::rstp, rows as ((prfx, []), rhs)::rst} =
+ mk{path = path,
+ rows = ((prfx, [fresh_var(type_of u)]), rhs)::rst}
+ | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
+ let val (pat_rectangle,rights) = ListPair.unzip rows
+ val col0 = map(hd o #2) pat_rectangle
+ in
+ if (forall is_Free col0)
+ then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
+ (ListPair.zip (col0, rights))
+ val pat_rectangle' = map v_to_prfx pat_rectangle
+ val (pref_patl,tm) = mk{path = rstp,
+ rows = ListPair.zip (pat_rectangle',
+ rights')}
+ in (map v_to_pats pref_patl, tm)
+ end
+ else
+ let val pty as Type (ty_name,_) = type_of p
+ in
+ case (ty_info ty_name)
+ of None => mk_case_fail("Not a known datatype: "^ty_name)
+ | Some{case_const,constructors} =>
+ let
+ val case_const_name = #1(dest_Const case_const)
+ val nrows = List.concat (map (expand constructors pty) rows)
+ val subproblems = divide(constructors, pty, range_ty, nrows)
+ val groups = map #group subproblems
+ and new_formals = map #new_formals subproblems
+ and constructors' = map #constructor subproblems
+ val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
+ (ListPair.zip (new_formals, groups))
+ val rec_calls = map mk news
+ val (pat_rect,dtrees) = ListPair.unzip rec_calls
+ val case_functions = map S.list_mk_abs
+ (ListPair.zip (new_formals, dtrees))
+ val types = map type_of (case_functions@[u]) @ [range_ty]
+ val case_const' = Const(case_const_name, list_mk_type types)
+ val tree = list_comb(case_const', case_functions@[u])
+ val pat_rect1 = List.concat
+ (ListPair.map mk_pat (constructors', pat_rect))
+ in (pat_rect1,tree)
+ end
+ end end
+ in mk
+ end;
+
+
+(* Repeated variable occurrences in a pattern are not allowed. *)
+fun FV_multiset tm =
+ case (S.dest_term tm)
+ of S.VAR{Name,Ty} => [Free(Name,Ty)]
+ | S.CONST _ => []
+ | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
+ | S.LAMB _ => raise TFL_ERR "FV_multiset" "lambda";
+
+fun no_repeat_vars thy pat =
+ let fun check [] = true
+ | check (v::rst) =
+ if mem_term (v,rst) then
+ raise TFL_ERR "no_repeat_vars"
+ (quote (#1 (dest_Free v)) ^
+ " occurs repeatedly in the pattern " ^
+ quote (string_of_cterm (Thry.typecheck thy pat)))
+ else check rst
+ in check (FV_multiset pat)
+ end;
+
+fun dest_atom (Free p) = p
+ | dest_atom (Const p) = p
+ | dest_atom _ = raise TFL_ERR "dest_atom" "function name not an identifier";
+
+fun same_name (p,q) = #1(dest_atom p) = #1(dest_atom q);
+
+local fun mk_functional_err s = raise TFL_ERR "mk_functional" s
+ fun single [_$_] =
+ mk_functional_err "recdef does not allow currying"
+ | single [f] = f
+ | single fs =
+ (*multiple function names?*)
+ if length (gen_distinct same_name fs) < length fs
+ then mk_functional_err
+ "The function being declared appears with multiple types"
+ else mk_functional_err
+ (Int.toString (length fs) ^
+ " distinct function names being declared")
+in
+fun mk_functional thy clauses =
+ let val (L,R) = ListPair.unzip (map HOLogic.dest_eq clauses
+ handle TERM _ => raise TFL_ERR "mk_functional"
+ "recursion equations must use the = relation")
+ val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
+ val atom = single (gen_distinct (op aconv) funcs)
+ val (fname,ftype) = dest_atom atom
+ val dummy = map (no_repeat_vars thy) pats
+ val rows = ListPair.zip (map (fn x => ([]:term list,[x])) pats,
+ map (fn (t,i) => (t,(i,true))) (enumerate R))
+ val names = foldr add_term_names (R,[])
+ val atype = type_of(hd pats)
+ and aname = variant names "a"
+ val a = Free(aname,atype)
+ val ty_info = Thry.match_info thy
+ val ty_match = Thry.match_type thy
+ val range_ty = type_of (hd R)
+ val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty
+ {path=[a], rows=rows}
+ val patts1 = map (fn (_,tag,[pat]) => (pat,tag)) patts
+ handle Match => mk_functional_err "error in pattern-match translation"
+ val patts2 = Library.sort (Library.int_ord o Library.pairself row_of_pat) patts1
+ val finals = map row_of_pat patts2
+ val originals = map (row_of_pat o #2) rows
+ val dummy = case (originals\\finals)
+ of [] => ()
+ | L => mk_functional_err
+ ("The following clauses are redundant (covered by preceding clauses): " ^
+ commas (map (fn i => Int.toString (i + 1)) L))
+ in {functional = Abs(Sign.base_name fname, ftype,
+ abstract_over (atom,
+ absfree(aname,atype, case_tm))),
+ pats = patts2}
+end end;
+
+
+(*----------------------------------------------------------------------------
+ *
+ * PRINCIPLES OF DEFINITION
+ *
+ *---------------------------------------------------------------------------*)
+
+
+(*For Isabelle, the lhs of a definition must be a constant.*)
+fun mk_const_def sign (Name, Ty, rhs) =
+ Sign.infer_types sign (K None) (K None) [] false
+ ([Const("==",dummyT) $ Const(Name,Ty) $ rhs], propT)
+ |> #1;
+
+(*Make all TVars available for instantiation by adding a ? to the front*)
+fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
+ | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
+ | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);
+
+local val f_eq_wfrec_R_M =
+ #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
+ val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
+ val (fname,_) = dest_Free f
+ val (wfrec,_) = S.strip_comb rhs
+in
+fun wfrec_definition0 thy fid R (functional as Abs(Name, Ty, _)) =
+ let val def_name = if Name<>fid then
+ raise TFL_ERR "wfrec_definition0"
+ ("Expected a definition of " ^
+ quote fid ^ " but found one of " ^
+ quote Name)
+ else Name ^ "_def"
+ val wfrec_R_M = map_term_types poly_tvars
+ (wfrec $ map_term_types poly_tvars R)
+ $ functional
+ val def_term = mk_const_def (Theory.sign_of thy) (Name, Ty, wfrec_R_M)
+ val (thy', [def]) = PureThy.add_defs_i false [Thm.no_attributes (def_name, def_term)] thy
+ in (thy', def) end;
+end;
+
+
+
+(*---------------------------------------------------------------------------
+ * This structure keeps track of congruence rules that aren't derived
+ * from a datatype definition.
+ *---------------------------------------------------------------------------*)
+fun extraction_thms thy =
+ let val {case_rewrites,case_congs} = Thry.extract_info thy
+ in (case_rewrites, case_congs)
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Pair patterns with termination conditions. The full list of patterns for
+ * a definition is merged with the TCs arising from the user-given clauses.
+ * There can be fewer clauses than the full list, if the user omitted some
+ * cases. This routine is used to prepare input for mk_induction.
+ *---------------------------------------------------------------------------*)
+fun merge full_pats TCs =
+let fun insert (p,TCs) =
+ let fun insrt ((x as (h,[]))::rst) =
+ if (p aconv h) then (p,TCs)::rst else x::insrt rst
+ | insrt (x::rst) = x::insrt rst
+ | insrt[] = raise TFL_ERR "merge.insert" "pattern not found"
+ in insrt end
+ fun pass ([],ptcl_final) = ptcl_final
+ | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
+in
+ pass (TCs, map (fn p => (p,[])) full_pats)
+end;
+
+
+fun givens pats = map pat_of (filter given pats);
+
+fun post_definition meta_tflCongs (theory, (def, pats)) =
+ let val tych = Thry.typecheck theory
+ val f = #lhs(S.dest_eq(concl def))
+ val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
+ val pats' = filter given pats
+ val given_pats = map pat_of pats'
+ val rows = map row_of_pat pats'
+ val WFR = #ant(S.dest_imp(concl corollary))
+ val R = #Rand(S.dest_comb WFR)
+ val corollary' = R.UNDISCH corollary (* put WF R on assums *)
+ val corollaries = map (fn pat => R.SPEC (tych pat) corollary')
+ given_pats
+ val (case_rewrites,context_congs) = extraction_thms theory
+ val corollaries' = map(rewrite_rule case_rewrites) corollaries
+ val extract = R.CONTEXT_REWRITE_RULE
+ (f, [R], cut_apply, meta_tflCongs@context_congs)
+ val (rules, TCs) = ListPair.unzip (map extract corollaries')
+ val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
+ val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
+ val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
+ in
+ {theory = theory,
+ rules = rules1,
+ rows = rows,
+ full_pats_TCs = merge (map pat_of pats) (ListPair.zip (given_pats, TCs)),
+ TCs = TCs}
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Perform the extraction without making the definition. Definition and
+ * extraction commute for the non-nested case. (Deferred recdefs)
+ *
+ * The purpose of wfrec_eqns is merely to instantiate the recursion theorem
+ * and extract termination conditions: no definition is made.
+ *---------------------------------------------------------------------------*)
+
+fun wfrec_eqns thy fid tflCongs eqns =
+ let val {lhs,rhs} = S.dest_eq (hd eqns)
+ val (f,args) = S.strip_comb lhs
+ val (fname,fty) = dest_atom f
+ val (SV,a) = front_last args (* SV = schematic variables *)
+ val g = list_comb(f,SV)
+ val h = Free(fname,type_of g)
+ val eqns1 = map (subst_free[(g,h)]) eqns
+ val {functional as Abs(Name, Ty, _), pats} = mk_functional thy eqns1
+ val given_pats = givens pats
+ (* val f = Free(Name,Ty) *)
+ val Type("fun", [f_dty, f_rty]) = Ty
+ val dummy = if Name<>fid then
+ raise TFL_ERR "wfrec_eqns"
+ ("Expected a definition of " ^
+ quote fid ^ " but found one of " ^
+ quote Name)
+ else ()
+ val (case_rewrites,context_congs) = extraction_thms thy
+ val tych = Thry.typecheck thy
+ val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
+ val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0
+ val R = Free (variant (foldr add_term_names (eqns,[])) Rname,
+ Rtype)
+ val WFREC_THM = R.ISPECL [tych R, tych g] WFREC_THM0
+ val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
+ val dummy =
+ if !trace then
+ writeln ("ORIGINAL PROTO_DEF: " ^
+ Sign.string_of_term (Theory.sign_of thy) proto_def)
+ else ()
+ val R1 = S.rand WFR
+ val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
+ val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats
+ val corollaries' = map (rewrite_rule case_rewrites) corollaries
+ fun extract X = R.CONTEXT_REWRITE_RULE
+ (f, R1::SV, cut_apply, tflCongs@context_congs) X
+ in {proto_def = proto_def,
+ SV=SV,
+ WFR=WFR,
+ pats=pats,
+ extracta = map extract corollaries'}
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Define the constant after extracting the termination conditions. The
+ * wellfounded relation used in the definition is computed by using the
+ * choice operator on the extracted conditions (plus the condition that
+ * such a relation must be wellfounded).
+ *---------------------------------------------------------------------------*)
+
+fun lazyR_def thy fid tflCongs eqns =
+ let val {proto_def,WFR,pats,extracta,SV} =
+ wfrec_eqns thy fid tflCongs eqns
+ val R1 = S.rand WFR
+ val f = #lhs(S.dest_eq proto_def)
+ val (extractants,TCl) = ListPair.unzip extracta
+ val dummy = if !trace
+ then (writeln "Extractants = ";
+ prths extractants;
+ ())
+ else ()
+ val TCs = foldr (gen_union (op aconv)) (TCl, [])
+ val full_rqt = WFR::TCs
+ val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt}
+ val R'abs = S.rand R'
+ val proto_def' = subst_free[(R1,R')] proto_def
+ val dummy = if !trace then writeln ("proto_def' = " ^
+ Sign.string_of_term
+ (Theory.sign_of thy) proto_def')
+ else ()
+ val {lhs,rhs} = S.dest_eq proto_def'
+ val (c,args) = S.strip_comb lhs
+ val (Name,Ty) = dest_atom c
+ val defn = mk_const_def (Theory.sign_of thy)
+ (Name, Ty, S.list_mk_abs (args,rhs))
+ val (theory, [def0]) =
+ thy
+ |> PureThy.add_defs_i false
+ [Thm.no_attributes (fid ^ "_def", defn)]
+ val def = freezeT def0;
+ val dummy = if !trace then writeln ("DEF = " ^ string_of_thm def)
+ else ()
+ (* val fconst = #lhs(S.dest_eq(concl def)) *)
+ val tych = Thry.typecheck theory
+ val full_rqt_prop = map (Dcterm.mk_prop o tych) full_rqt
+ (*lcp: a lot of object-logic inference to remove*)
+ val baz = R.DISCH_ALL
+ (U.itlist R.DISCH full_rqt_prop
+ (R.LIST_CONJ extractants))
+ val dum = if !trace then writeln ("baz = " ^ string_of_thm baz)
+ else ()
+ val f_free = Free (fid, fastype_of f) (*'cos f is a Const*)
+ val SV' = map tych SV;
+ val SVrefls = map reflexive SV'
+ val def0 = (U.rev_itlist (fn x => fn th => R.rbeta(combination th x))
+ SVrefls def)
+ RS meta_eq_to_obj_eq
+ val def' = R.MP (R.SPEC (tych R') (R.GEN (tych R1) baz)) def0
+ val body_th = R.LIST_CONJ (map R.ASSUME full_rqt_prop)
+ val bar = R.MP (R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX)
+ body_th
+ in {theory = theory, R=R1, SV=SV,
+ rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def',
+ full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
+ patterns = pats}
+ end;
+
+
+
+(*----------------------------------------------------------------------------
+ *
+ * INDUCTION THEOREM
+ *
+ *---------------------------------------------------------------------------*)
+
+
+(*------------------------ Miscellaneous function --------------------------
+ *
+ * [x_1,...,x_n] ?v_1...v_n. M[v_1,...,v_n]
+ * -----------------------------------------------------------
+ * ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
+ * ...
+ * (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
+ *
+ * This function is totally ad hoc. Used in the production of the induction
+ * theorem. The nchotomy theorem can have clauses that look like
+ *
+ * ?v1..vn. z = C vn..v1
+ *
+ * in which the order of quantification is not the order of occurrence of the
+ * quantified variables as arguments to C. Since we have no control over this
+ * aspect of the nchotomy theorem, we make the correspondence explicit by
+ * pairing the incoming new variable with the term it gets beta-reduced into.
+ *---------------------------------------------------------------------------*)
+
+fun alpha_ex_unroll (xlist, tm) =
+ let val (qvars,body) = S.strip_exists tm
+ val vlist = #2(S.strip_comb (S.rhs body))
+ val plist = ListPair.zip (vlist, xlist)
+ val args = map (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars
+ handle Library.OPTION => sys_error
+ "TFL fault [alpha_ex_unroll]: no correspondence"
+ fun build ex [] = []
+ | build (_$rex) (v::rst) =
+ let val ex1 = betapply(rex, v)
+ in ex1 :: build ex1 rst
+ end
+ val (nex::exl) = rev (tm::build tm args)
+ in
+ (nex, ListPair.zip (args, rev exl))
+ end;
+
+
+
+(*----------------------------------------------------------------------------
+ *
+ * PROVING COMPLETENESS OF PATTERNS
+ *
+ *---------------------------------------------------------------------------*)
+
+fun mk_case ty_info usednames thy =
+ let
+ val divide = ipartition (gvvariant usednames)
+ val tych = Thry.typecheck thy
+ fun tych_binding(x,y) = (tych x, tych y)
+ fun fail s = raise TFL_ERR "mk_case" s
+ fun mk{rows=[],...} = fail"no rows"
+ | mk{path=[], rows = [([], (thm, bindings))]} =
+ R.IT_EXISTS (map tych_binding bindings) thm
+ | mk{path = u::rstp, rows as (p::_, _)::_} =
+ let val (pat_rectangle,rights) = ListPair.unzip rows
+ val col0 = map hd pat_rectangle
+ val pat_rectangle' = map tl pat_rectangle
+ in
+ if (forall is_Free col0) (* column 0 is all variables *)
+ then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
+ (ListPair.zip (rights, col0))
+ in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
+ end
+ else (* column 0 is all constructors *)
+ let val Type (ty_name,_) = type_of p
+ in
+ case (ty_info ty_name)
+ of None => fail("Not a known datatype: "^ty_name)
+ | Some{constructors,nchotomy} =>
+ let val thm' = R.ISPEC (tych u) nchotomy
+ val disjuncts = S.strip_disj (concl thm')
+ val subproblems = divide(constructors, rows)
+ val groups = map #group subproblems
+ and new_formals = map #new_formals subproblems
+ val existentials = ListPair.map alpha_ex_unroll
+ (new_formals, disjuncts)
+ val constraints = map #1 existentials
+ val vexl = map #2 existentials
+ fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
+ val news = map (fn (nf,rows,c) => {path = nf@rstp,
+ rows = map (expnd c) rows})
+ (U.zip3 new_formals groups constraints)
+ val recursive_thms = map mk news
+ val build_exists = foldr
+ (fn((x,t), th) =>
+ R.CHOOSE (tych x, R.ASSUME (tych t)) th)
+ val thms' = ListPair.map build_exists (vexl, recursive_thms)
+ val same_concls = R.EVEN_ORS thms'
+ in R.DISJ_CASESL thm' same_concls
+ end
+ end end
+ in mk
+ end;
+
+
+fun complete_cases thy =
+ let val tych = Thry.typecheck thy
+ val ty_info = Thry.induct_info thy
+ in fn pats =>
+ let val names = foldr add_term_names (pats,[])
+ val T = type_of (hd pats)
+ val aname = Term.variant names "a"
+ val vname = Term.variant (aname::names) "v"
+ val a = Free (aname, T)
+ val v = Free (vname, T)
+ val a_eq_v = HOLogic.mk_eq(a,v)
+ val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
+ (R.REFL (tych a))
+ val th0 = R.ASSUME (tych a_eq_v)
+ val rows = map (fn x => ([x], (th0,[]))) pats
+ in
+ R.GEN (tych a)
+ (R.RIGHT_ASSOC
+ (R.CHOOSE(tych v, ex_th0)
+ (mk_case ty_info (vname::aname::names)
+ thy {path=[v], rows=rows})))
+ end end;
+
+
+(*---------------------------------------------------------------------------
+ * Constructing induction hypotheses: one for each recursive call.
+ *
+ * Note. R will never occur as a variable in the ind_clause, because
+ * to do so, it would have to be from a nested definition, and we don't
+ * allow nested defns to have R variable.
+ *
+ * Note. When the context is empty, there can be no local variables.
+ *---------------------------------------------------------------------------*)
+(*
+local infix 5 ==>
+ fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
+in
+fun build_ih f P (pat,TCs) =
+ let val globals = S.free_vars_lr pat
+ fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
+ fun dest_TC tm =
+ let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
+ val (R,y,_) = S.dest_relation R_y_pat
+ val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
+ in case cntxt
+ of [] => (P_y, (tm,[]))
+ | _ => let
+ val imp = S.list_mk_conj cntxt ==> P_y
+ val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
+ val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
+ in (S.list_mk_forall(locals,imp), (tm,locals)) end
+ end
+ in case TCs
+ of [] => (S.list_mk_forall(globals, P$pat), [])
+ | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
+ val ind_clause = S.list_mk_conj ihs ==> P$pat
+ in (S.list_mk_forall(globals,ind_clause), TCs_locals)
+ end
+ end
+end;
+*)
+
+local infix 5 ==>
+ fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
+in
+fun build_ih f (P,SV) (pat,TCs) =
+ let val pat_vars = S.free_vars_lr pat
+ val globals = pat_vars@SV
+ fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
+ fun dest_TC tm =
+ let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
+ val (R,y,_) = S.dest_relation R_y_pat
+ val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
+ in case cntxt
+ of [] => (P_y, (tm,[]))
+ | _ => let
+ val imp = S.list_mk_conj cntxt ==> P_y
+ val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
+ val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
+ in (S.list_mk_forall(locals,imp), (tm,locals)) end
+ end
+ in case TCs
+ of [] => (S.list_mk_forall(pat_vars, P$pat), [])
+ | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
+ val ind_clause = S.list_mk_conj ihs ==> P$pat
+ in (S.list_mk_forall(pat_vars,ind_clause), TCs_locals)
+ end
+ end
+end;
+
+(*---------------------------------------------------------------------------
+ * This function makes good on the promise made in "build_ih".
+ *
+ * Input is tm = "(!y. R y pat ==> P y) ==> P pat",
+ * TCs = TC_1[pat] ... TC_n[pat]
+ * thm = ih1 /\ ... /\ ih_n |- ih[pat]
+ *---------------------------------------------------------------------------*)
+fun prove_case f thy (tm,TCs_locals,thm) =
+ let val tych = Thry.typecheck thy
+ val antc = tych(#ant(S.dest_imp tm))
+ val thm' = R.SPEC_ALL thm
+ fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
+ fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
+ fun mk_ih ((TC,locals),th2,nested) =
+ R.GENL (map tych locals)
+ (if nested then R.DISCH (get_cntxt TC) th2 handle U.ERR _ => th2
+ else if S.is_imp (concl TC) then R.IMP_TRANS TC th2
+ else R.MP th2 TC)
+ in
+ R.DISCH antc
+ (if S.is_imp(concl thm') (* recursive calls in this clause *)
+ then let val th1 = R.ASSUME antc
+ val TCs = map #1 TCs_locals
+ val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o
+ #2 o S.strip_forall) TCs
+ val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
+ TCs_locals
+ val th2list = map (U.C R.SPEC th1 o tych) ylist
+ val nlist = map nested TCs
+ val triples = U.zip3 TClist th2list nlist
+ val Pylist = map mk_ih triples
+ in R.MP thm' (R.LIST_CONJ Pylist) end
+ else thm')
+ end;
+
+
+(*---------------------------------------------------------------------------
+ *
+ * x = (v1,...,vn) |- M[x]
+ * ---------------------------------------------
+ * ?v1 ... vn. x = (v1,...,vn) |- M[x]
+ *
+ *---------------------------------------------------------------------------*)
+fun LEFT_ABS_VSTRUCT tych thm =
+ let fun CHOOSER v (tm,thm) =
+ let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
+ in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
+ end
+ val [veq] = filter (can S.dest_eq) (#1 (R.dest_thm thm))
+ val {lhs,rhs} = S.dest_eq veq
+ val L = S.free_vars_lr rhs
+ in #2 (U.itlist CHOOSER L (veq,thm)) end;
+
+
+(*----------------------------------------------------------------------------
+ * Input : f, R, and [(pat1,TCs1),..., (patn,TCsn)]
+ *
+ * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
+ * recursion induction (Rinduct) by proving the antecedent of Sinduct from
+ * the antecedent of Rinduct.
+ *---------------------------------------------------------------------------*)
+fun mk_induction thy {fconst, R, SV, pat_TCs_list} =
+let val tych = Thry.typecheck thy
+ val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
+ val (pats,TCsl) = ListPair.unzip pat_TCs_list
+ val case_thm = complete_cases thy pats
+ val domain = (type_of o hd) pats
+ val Pname = Term.variant (foldr (foldr add_term_names)
+ (pats::TCsl, [])) "P"
+ val P = Free(Pname, domain --> HOLogic.boolT)
+ val Sinduct = R.SPEC (tych P) Sinduction
+ val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
+ val Rassums_TCl' = map (build_ih fconst (P,SV)) pat_TCs_list
+ val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
+ val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
+ val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats
+ val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
+ val proved_cases = map (prove_case fconst thy) tasks
+ val v = Free (variant (foldr add_term_names (map concl proved_cases, []))
+ "v",
+ domain)
+ val vtyped = tych v
+ val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
+ val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th')
+ (substs, proved_cases)
+ val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
+ val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
+ val dc = R.MP Sinduct dant
+ val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
+ val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
+ val dc' = U.itlist (R.GEN o tych) vars
+ (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
+in
+ R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
+end
+handle U.ERR _ => raise TFL_ERR "mk_induction" "failed derivation";
+
+
+
+
+(*---------------------------------------------------------------------------
+ *
+ * POST PROCESSING
+ *
+ *---------------------------------------------------------------------------*)
+
+
+fun simplify_induction thy hth ind =
+ let val tych = Thry.typecheck thy
+ val (asl,_) = R.dest_thm ind
+ val (_,tc_eq_tc') = R.dest_thm hth
+ val tc = S.lhs tc_eq_tc'
+ fun loop [] = ind
+ | loop (asm::rst) =
+ if (can (Thry.match_term thy asm) tc)
+ then R.UNDISCH
+ (R.MATCH_MP
+ (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind))
+ hth)
+ else loop rst
+ in loop asl
+end;
+
+
+(*---------------------------------------------------------------------------
+ * The termination condition is an antecedent to the rule, and an
+ * assumption to the theorem.
+ *---------------------------------------------------------------------------*)
+fun elim_tc tcthm (rule,induction) =
+ (R.MP rule tcthm, R.PROVE_HYP tcthm induction)
+
+
+fun postprocess{wf_tac, terminator, simplifier} theory {rules,induction,TCs} =
+let val tych = Thry.typecheck theory
+
+ (*---------------------------------------------------------------------
+ * Attempt to eliminate WF condition. It's the only assumption of rules
+ *---------------------------------------------------------------------*)
+ val (rules1,induction1) =
+ let val thm = R.prove(tych(HOLogic.mk_Trueprop
+ (hd(#1(R.dest_thm rules)))),
+ wf_tac)
+ in (R.PROVE_HYP thm rules, R.PROVE_HYP thm induction)
+ end handle U.ERR _ => (rules,induction);
+
+ (*----------------------------------------------------------------------
+ * The termination condition (tc) is simplified to |- tc = tc' (there
+ * might not be a change!) and then 3 attempts are made:
+ *
+ * 1. if |- tc = T, then eliminate it with eqT; otherwise,
+ * 2. apply the terminator to tc'. If |- tc' = T then eliminate; else
+ * 3. replace tc by tc' in both the rules and the induction theorem.
+ *---------------------------------------------------------------------*)
+
+ fun print_thms s L =
+ if !trace then writeln (cat_lines (s :: map string_of_thm L))
+ else ();
+
+ fun print_cterms s L =
+ if !trace then writeln (cat_lines (s :: map string_of_cterm L))
+ else ();;
+
+ fun simplify_tc tc (r,ind) =
+ let val tc1 = tych tc
+ val _ = print_cterms "TC before simplification: " [tc1]
+ val tc_eq = simplifier tc1
+ val _ = print_thms "result: " [tc_eq]
+ in
+ elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
+ handle U.ERR _ =>
+ (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
+ (R.prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))),
+ terminator)))
+ (r,ind)
+ handle U.ERR _ =>
+ (R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq),
+ simplify_induction theory tc_eq ind))
+ end
+
+ (*----------------------------------------------------------------------
+ * Nested termination conditions are harder to get at, since they are
+ * left embedded in the body of the function (and in induction
+ * theorem hypotheses). Our "solution" is to simplify them, and try to
+ * prove termination, but leave the application of the resulting theorem
+ * to a higher level. So things go much as in "simplify_tc": the
+ * termination condition (tc) is simplified to |- tc = tc' (there might
+ * not be a change) and then 2 attempts are made:
+ *
+ * 1. if |- tc = T, then return |- tc; otherwise,
+ * 2. apply the terminator to tc'. If |- tc' = T then return |- tc; else
+ * 3. return |- tc = tc'
+ *---------------------------------------------------------------------*)
+ fun simplify_nested_tc tc =
+ let val tc_eq = simplifier (tych (#2 (S.strip_forall tc)))
+ in
+ R.GEN_ALL
+ (R.MATCH_MP Thms.eqT tc_eq
+ handle U.ERR _ =>
+ (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
+ (R.prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))),
+ terminator))
+ handle U.ERR _ => tc_eq))
+ end
+
+ (*-------------------------------------------------------------------
+ * Attempt to simplify the termination conditions in each rule and
+ * in the induction theorem.
+ *-------------------------------------------------------------------*)
+ fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm
+ fun loop ([],extras,R,ind) = (rev R, ind, extras)
+ | loop ((r,ftcs)::rst, nthms, R, ind) =
+ let val tcs = #1(strip_imp (concl r))
+ val extra_tcs = gen_rems (op aconv) (ftcs, tcs)
+ val extra_tc_thms = map simplify_nested_tc extra_tcs
+ val (r1,ind1) = U.rev_itlist simplify_tc tcs (r,ind)
+ val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1
+ in loop(rst, nthms@extra_tc_thms, r2::R, ind1)
+ end
+ val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs)
+ val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1)
+in
+ {induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras}
+end;
+
+
+end;