diff -r 81c8d46edfa3 -r 3902e9af752f TFL/tfl.sml --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/TFL/tfl.sml Fri Oct 18 12:41:04 1996 +0200 @@ -0,0 +1,922 @@ +functor TFL(structure Rules : Rules_sig + structure Thry : Thry_sig + structure Thms : Thms_sig + sharing type Rules.binding = Thry.binding = + Thry.USyntax.binding = Mask.binding + sharing type Rules.Type = Thry.Type = Thry.USyntax.Type + sharing type Rules.Preterm = Thry.Preterm = Thry.USyntax.Preterm + sharing type Rules.Term = Thry.Term = Thry.USyntax.Term + sharing type Thms.Thm = Rules.Thm = Thry.Thm) : TFL_sig = +struct + +(* Declarations *) +structure Thms = Thms; +structure Rules = Rules; +structure Thry = Thry; +structure USyntax = Thry.USyntax; + +type Preterm = Thry.USyntax.Preterm; +type Term = Thry.USyntax.Term; +type Thm = Thms.Thm; +type Thry = Thry.Thry; +type Tactic = Rules.Tactic; + + +(* Abbreviations *) +structure R = Rules; +structure S = USyntax; +structure U = S.Utils; + +(* Declares 'a binding datatype *) +open Mask; + +nonfix mem --> |-> ##; +val --> = S.-->; +val ## = U.##; + +infixr 3 -->; +infixr 7 |->; +infix 4 ##; + +val concl = #2 o R.dest_thm; +val hyp = #1 o R.dest_thm; + +val list_mk_type = U.end_itlist (U.curry(op -->)); + +fun flatten [] = [] + | flatten (h::t) = h@flatten t; + + +fun gtake f = + let fun grab(0,rst) = ([],rst) + | grab(n, x::rst) = + let val (taken,left) = grab(n-1,rst) + in (f x::taken, left) end + in grab + end; + +fun enumerate L = + rev(#1(U.rev_itlist (fn x => fn (alist,i) => ((x,i)::alist, i+1)) L ([],0))); + +fun stringize [] = "" + | stringize [i] = U.int_to_string i + | stringize (h::t) = (U.int_to_string h^", "^stringize t); + + +fun TFL_ERR{func,mesg} = U.ERR{module = "Tfl", func = func, mesg = mesg}; + + +(*--------------------------------------------------------------------------- + * The next function is common to pattern-match translation and + * proof of completeness of cases for the induction theorem. + * + * "gvvariant" make variables that are guaranteed not to be in vlist and + * furthermore, are guaranteed not to be equal to each other. The names of + * the variables will start with "v" and end in a number. + *---------------------------------------------------------------------------*) +local val counter = ref 0 +in +fun gvvariant vlist = + let val slist = ref (map (#Name o S.dest_var) vlist) + val mem = U.mem (U.curry (op=)) + val _ = counter := 0 + fun pass str = + if (mem str (!slist)) + then ( counter := !counter + 1; + pass (U.concat"v" (U.int_to_string(!counter)))) + else (slist := str :: !slist; str) + in + fn ty => S.mk_var{Name=pass "v", Ty=ty} + end +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{func = "partition.part", mesg = 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} = S.dest_const c + val (L,_) = S.strip_type 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 (#Name(S.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 S.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; + + + +(*--------------------------------------------------------------------------- + * This datatype carries some information about the origin of a + * clause in a function definition. + *---------------------------------------------------------------------------*) +datatype pattern = GIVEN of S.Preterm * int + | OMITTED of S.Preterm * int + +fun psubst theta (GIVEN (tm,i)) = GIVEN(S.subst theta tm, i) + | psubst theta (OMITTED (tm,i)) = OMITTED(S.subst theta tm, i); + +fun dest_pattern (GIVEN (tm,i)) = ((GIVEN,i),tm) + | dest_pattern (OMITTED (tm,i)) = ((OMITTED,i),tm); + +val pat_of = #2 o dest_pattern; +val row_of_pat = #2 o #1 o dest_pattern; + +(*--------------------------------------------------------------------------- + * Produce an instance of a constructor, plus genvars for its arguments. + *---------------------------------------------------------------------------*) +fun fresh_constr ty_match colty gv c = + let val {Ty,...} = S.dest_const c + val (L,ty) = S.strip_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 ((prefix, p::rst), rhs)) => + fn (in_group,not_in_group) => + let val (pc,args) = S.strip_comb p + in if ((#Name(S.dest_const pc) = Name) handle _ => false) + then (((prefix,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{func="partition", mesg="no rows"} + | partition gv ty_match + (constructors, colty, res_ty, rows as (((prefix,_),_)::_)) = +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} = S.dest_const c' + val (in_group, not_in_group) = mk_group Name rows + val in_group' = + if (null in_group) (* Constructor not given *) + then [((prefix, #2(fresh c)), OMITTED (S.ARB res_ty, ~1))] + 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 = + let val L = length(#1(S.strip_type(S.type_of c))) + fun build (prefix,tag,plist) = + let val (args,plist') = gtake U.I (L, plist) + in (prefix,tag,S.list_mk_comb(c,args)::plist') end + in map build + end; + +fun v_to_prefix (prefix, v::pats) = (v::prefix,pats) + | v_to_prefix _ = raise TFL_ERR{func="mk_case", mesg="v_to_prefix"}; + +fun v_to_pats (v::prefix,tag, pats) = (prefix, tag, v::pats) + | v_to_pats _ = raise TFL_ERR{func="mk_case", mesg="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 FV range_ty = + let + fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = s} + val fresh_var = gvvariant FV + val divide = partition fresh_var ty_match + fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row" + | expand constructors ty (row as ((prefix, p::rst), rhs)) = + if (S.is_var p) + then let val fresh = fresh_constr ty_match ty fresh_var + fun expnd (c,gvs) = + let val capp = S.list_mk_comb(c,gvs) + in ((prefix, capp::rst), psubst[p |-> capp] rhs) + end + in map expnd (map fresh constructors) end + else [row] + fun mk{rows=[],...} = mk_case_fail"no rows" + | mk{path=[], rows = ((prefix, []), rhs)::_} = (* Done *) + let val (tag,tm) = dest_pattern rhs + in ([(prefix,tag,[])], tm) + end + | mk{path=[], rows = _::_} = mk_case_fail"blunder" + | mk{path as u::rstp, rows as ((prefix, []), rhs)::rst} = + mk{path = path, + rows = ((prefix, [fresh_var(S.type_of u)]), rhs)::rst} + | mk{path = u::rstp, rows as ((_, p::_), _)::_} = + let val (pat_rectangle,rights) = U.unzip rows + val col0 = map(hd o #2) pat_rectangle + in + if (U.all S.is_var col0) + then let val rights' = map(fn(v,e) => psubst[v|->u] e) (U.zip col0 rights) + val pat_rectangle' = map v_to_prefix pat_rectangle + val (pref_patl,tm) = mk{path = rstp, + rows = U.zip pat_rectangle' rights'} + in (map v_to_pats pref_patl, tm) + end + else + let val pty = S.type_of p + val ty_name = (#Tyop o S.dest_type) pty + in + case (ty_info ty_name) + of U.NONE => mk_case_fail("Not a known datatype: "^ty_name) + | U.SOME{case_const,constructors} => + let val case_const_name = #Name(S.dest_const case_const) + val nrows = flatten (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}) + (U.zip new_formals groups) + val rec_calls = map mk news + val (pat_rect,dtrees) = U.unzip rec_calls + val case_functions = map S.list_mk_abs(U.zip new_formals dtrees) + val types = map S.type_of (case_functions@[u]) @ [range_ty] + val case_const' = S.mk_const{Name = case_const_name, + Ty = list_mk_type types} + val tree = S.list_mk_comb(case_const', case_functions@[u]) + val pat_rect1 = flatten(U.map2 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 v => [S.mk_var v] + | S.CONST _ => [] + | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand + | S.LAMB _ => raise TFL_ERR{func = "FV_multiset", mesg = "lambda"}; + +fun no_repeat_vars thy pat = + let fun check [] = true + | check (v::rst) = + if (U.mem S.aconv v rst) + then raise TFL_ERR{func = "no_repeat_vars", + mesg = U.concat(U.quote(#Name(S.dest_var v))) + (U.concat" occurs repeatedly in the pattern " + (U.quote(S.Term_to_string (Thry.typecheck thy pat))))} + else check rst + in check (FV_multiset pat) + end; + +local fun paired1{lhs,rhs} = (lhs,rhs) + and paired2{Rator,Rand} = (Rator,Rand) + fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s} +in +fun mk_functional thy eqs = + let val clauses = S.strip_conj eqs + val (L,R) = U.unzip (map (paired1 o S.dest_eq o U.snd o S.strip_forall) + clauses) + val (funcs,pats) = U.unzip(map (paired2 o S.dest_comb) L) + val [f] = U.mk_set (S.aconv) funcs + handle _ => mk_functional_err "function name not unique" + val _ = map (no_repeat_vars thy) pats + val rows = U.zip (map (fn x => ([],[x])) pats) (map GIVEN (enumerate R)) + val fvs = S.free_varsl R + val a = S.variant fvs (S.mk_var{Name="a", Ty = S.type_of(hd pats)}) + val FV = a::fvs + val ty_info = Thry.match_info thy + val ty_match = Thry.match_type thy + val range_ty = S.type_of (hd R) + val (patts, case_tm) = mk_case ty_info ty_match FV range_ty + {path=[a], rows=rows} + val patts1 = map (fn (_,(tag,i),[pat]) => tag (pat,i)) patts handle _ + => mk_functional_err "error in pattern-match translation" + val patts2 = U.sort(fn p1=>fn p2=> row_of_pat p1 < row_of_pat p2) patts1 + val finals = map row_of_pat patts2 + val originals = map (row_of_pat o #2) rows + fun int_eq i1 (i2:int) = (i1=i2) + val _ = case (U.set_diff int_eq originals finals) + of [] => () + | L => mk_functional_err("The following rows (counting from zero)\ + \ are inaccessible: "^stringize L) + in {functional = S.list_mk_abs ([f,a], case_tm), + pats = patts2} +end end; + + +(*---------------------------------------------------------------------------- + * + * PRINCIPLES OF DEFINITION + * + *---------------------------------------------------------------------------*) + + +(*---------------------------------------------------------------------------- + * This basic principle of definition takes a functional M and a relation R + * and specializes the following theorem + * + * |- !M R f. (f = WFREC R M) ==> WF R ==> !x. f x = M (f%R,x) x + * + * to them (getting "th1", say). Then we make the definition "f = WFREC R M" + * and instantiate "th1" to the constant "f" (getting th2). Then we use the + * definition to delete the first antecedent to th2. Hence the result in + * the "corollary" field is + * + * |- WF R ==> !x. f x = M (f%R,x) x + * + *---------------------------------------------------------------------------*) + +fun prim_wfrec_definition thy {R, functional} = + let val tych = Thry.typecheck thy + val {Bvar,...} = S.dest_abs functional + val {Name,...} = S.dest_var Bvar (* Intended name of definition *) + val cor1 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY + val cor2 = R.ISPEC (tych R) cor1 + val f_eq_WFREC_R_M = (#ant o S.dest_imp o #Body + o S.dest_forall o concl) cor2 + val {lhs,rhs} = S.dest_eq f_eq_WFREC_R_M + val {Ty, ...} = S.dest_var lhs + val def_term = S.mk_eq{lhs = S.mk_var{Name=Name,Ty=Ty}, rhs=rhs} + val (def_thm,thy1) = Thry.make_definition thy + (U.concat Name "_def") def_term + val (_,[f,_]) = (S.strip_comb o concl) def_thm + val cor3 = R.ISPEC (Thry.typecheck thy1 f) cor2 + in + {theory = thy1, def=def_thm, corollary=R.MP cor3 def_thm} + end; + + +(*--------------------------------------------------------------------------- + * This structure keeps track of congruence rules that aren't derived + * from a datatype definition. + *---------------------------------------------------------------------------*) +structure Context = +struct + val non_datatype_context = ref []:Rules.Thm list ref + fun read() = !non_datatype_context + fun write L = (non_datatype_context := L) +end; + +fun extraction_thms thy = + let val {case_rewrites,case_congs} = Thry.extract_info thy + in (case_rewrites, case_congs@Context.read()) + 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 (S.aconv p h) then (p,TCs)::rst else x::insrt rst + | insrt (x::rst) = x::insrt rst + | insrt[] = raise TFL_ERR{func="merge.insert",mesg="pat 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 not_omitted (GIVEN(tm,_)) = tm + | not_omitted (OMITTED _) = raise TFL_ERR{func="not_omitted",mesg=""} +val givens = U.mapfilter not_omitted; + + +(*-------------------------------------------------------------------------- + * This is a wrapper for "prim_wfrec_definition": it builds a functional, + * calls "prim_wfrec_definition", then specializes the result. This gives a + * list of rewrite rules where the right hand sides are quite ugly, so we + * simplify to get rid of the case statements. In essence, this function + * performs pre- and post-processing for patterns. As well, after + * simplification, termination conditions are extracted. + *-------------------------------------------------------------------------*) + +fun gen_wfrec_definition thy {R, eqs} = + let val {functional,pats} = mk_functional thy eqs + val given_pats = givens pats + val {def,corollary,theory} = prim_wfrec_definition thy + {R=R, functional=functional} + val tych = Thry.typecheck theory + val {lhs=f,...} = S.dest_eq(concl def) + val WFR = #ant(S.dest_imp(concl corollary)) + val corollary' = R.UNDISCH corollary (* put WF R on assums *) + val corollaries = map (U.C R.SPEC corollary' o tych) given_pats + val (case_rewrites,context_congs) = extraction_thms thy + val corollaries' = map(R.simplify case_rewrites) corollaries + fun xtract th = R.CONTEXT_REWRITE_RULE(f,R) + {thms = [(R.ISPECL o map tych)[f,R] Thms.CUT_LEMMA], + congs = context_congs, + th = th} + val (rules, TCs) = U.unzip (map xtract corollaries') + val rules0 = map (R.simplify [Thms.CUT_DEF]) rules + val mk_cond_rule = R.FILTER_DISCH_ALL(not o S.aconv WFR) + val rules1 = R.LIST_CONJ(map mk_cond_rule rules0) + in + {theory = theory, (* holds def, if it's needed *) + rules = rules1, + full_pats_TCs = merge (map pat_of pats) (U.zip given_pats TCs), + TCs = TCs, + patterns = pats} + end; + + +(*--------------------------------------------------------------------------- + * Perform the extraction without making the definition. Definition and + * extraction commute for the non-nested case. For hol90 users, this + * function can be invoked without being in draft mode. + *---------------------------------------------------------------------------*) +fun wfrec_eqns thy eqns = + let val {functional,pats} = mk_functional thy eqns + val given_pats = givens pats + val {Bvar = f, Body} = S.dest_abs functional + val {Bvar = x, ...} = S.dest_abs Body + val {Name,Ty = fty} = S.dest_var f + val {Tyop="fun", Args = [f_dty, f_rty]} = S.dest_type fty + val (case_rewrites,context_congs) = extraction_thms thy + val tych = Thry.typecheck thy + val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY + val R = S.variant(S.free_vars eqns) + (#Bvar(S.dest_forall(concl WFREC_THM0))) + val WFREC_THM = R.ISPECL [tych R, tych f] WFREC_THM0 + val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM) + val R1 = S.rand WFR + val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM) + val corollaries = map (U.C R.SPEC corollary' o tych) given_pats + val corollaries' = map (R.simplify case_rewrites) corollaries + fun extract th = R.CONTEXT_REWRITE_RULE(f,R1) + {thms = [(R.ISPECL o map tych)[f,R1] Thms.CUT_LEMMA], + congs = context_congs, + th = th} + in {proto_def=proto_def, + 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 eqns = + let val {proto_def,WFR,pats,extracta} = wfrec_eqns thy eqns + val R1 = S.rand WFR + val f = S.lhs proto_def + val {Name,...} = S.dest_var f + val (extractants,TCl) = U.unzip extracta + val TCs = U.Union S.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 (def,theory) = Thry.make_definition thy (U.concat Name "_def") + (S.subst[R1 |-> R'] proto_def) + val fconst = #lhs(S.dest_eq(concl def)) + val tych = Thry.typecheck theory + val baz = R.DISCH (tych proto_def) + (U.itlist (R.DISCH o tych) full_rqt (R.LIST_CONJ extractants)) + val def' = R.MP (R.SPEC (tych fconst) + (R.SPEC (tych R') (R.GENL[tych R1, tych f] baz))) + def + val body_th = R.LIST_CONJ (map (R.ASSUME o tych) full_rqt) + val bar = R.MP (R.BETA_RULE(R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX)) + body_th + in {theory = theory, R=R1, + rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def', + full_pats_TCs = merge (map pat_of pats) (U.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 = U.zip vlist xlist + val args = map (U.C (U.assoc1 (U.uncurry S.aconv)) plist) qvars + val args' = map (fn U.SOME(_,v) => v + | U.NONE => raise TFL_ERR{func = "alpha_ex_unroll", + mesg = "no correspondence"}) args + fun build ex [] = [] + | build ex (v::rst) = + let val ex1 = S.beta_conv(S.mk_comb{Rator=S.rand ex, Rand=v}) + in ex1::build ex1 rst + end + val (nex::exl) = rev (tm::build tm args') + in + (nex, U.zip args' (rev exl)) + end; + + + +(*---------------------------------------------------------------------------- + * + * PROVING COMPLETENESS OF PATTERNS + * + *---------------------------------------------------------------------------*) + +fun mk_case ty_info FV thy = + let + val divide = ipartition (gvvariant FV) + val tych = Thry.typecheck thy + fun tych_binding(x|->y) = (tych x |-> tych y) + fun fail s = raise TFL_ERR{func = "mk_case", mesg = 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) = U.unzip rows + val col0 = map hd pat_rectangle + val pat_rectangle' = map tl pat_rectangle + in + if (U.all S.is_var col0) (* column 0 is all variables *) + then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[u|->v])) + (U.zip rights col0) + in mk{path = rstp, rows = U.zip pat_rectangle' rights'} + end + else (* column 0 is all constructors *) + let val ty_name = (#Tyop o S.dest_type o S.type_of) p + in + case (ty_info ty_name) + of U.NONE => fail("Not a known datatype: "^ty_name) + | U.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 = U.map2 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 = U.itlist(R.CHOOSE o (tych##(R.ASSUME o tych))) + val thms' = U.map2 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 + fun pmk_var n ty = S.mk_var{Name = n,Ty = ty} + val ty_info = Thry.induct_info thy + in fn pats => + let val FV0 = S.free_varsl pats + val a = S.variant FV0 (pmk_var "a" (S.type_of(hd pats))) + val v = S.variant (a::FV0) (pmk_var "v" (S.type_of a)) + val FV = a::v::FV0 + val a_eq_v = S.mk_eq{lhs = a, rhs = v} + val ex_th0 = R.EXISTS ((tych##tych) (S.mk_exists{Bvar=v,Body=a_eq_v},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 FV 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 nonfix ^ ; infix 9 ^ ; infix 5 ==> + fun (tm1 ^ tm2) = S.mk_comb{Rator = tm1, Rand = tm2} + 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 = U.can(S.find_term (S.aconv f)) tm handle _ => false + 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 = U.set_diff S.aconv (S.free_vars_lr imp) globals + val locals = #2(U.pluck (S.aconv P) lvs) handle _ => 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) = U.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; + + + +(*--------------------------------------------------------------------------- + * This function makes good on the promise made in "build_ih: we prove + * . + * + * 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 = U.can(S.find_term (S.aconv f)) tm handle _ => false + 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 _ => 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] = U.filter (U.can S.dest_eq) (#1 (R.dest_thm thm)) + val {lhs,rhs} = S.dest_eq veq + val L = S.free_vars_lr rhs + in U.snd(U.itlist CHOOSER L (veq,thm)) + end; + + +fun combize M N = S.mk_comb{Rator=M,Rand=N}; +fun eq v tm = S.mk_eq{lhs=v,rhs=tm}; + + +(*---------------------------------------------------------------------------- + * 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 f R pat_TCs_list = +let val tych = Thry.typecheck thy + val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM) + val (pats,TCsl) = U.unzip pat_TCs_list + val case_thm = complete_cases thy pats + val domain = (S.type_of o hd) pats + val P = S.variant (S.all_varsl (pats@flatten TCsl)) + (S.mk_var{Name="P", Ty=domain --> S.bool}) + 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 f P) pat_TCs_list + val (Rassums,TCl') = U.unzip Rassums_TCl' + val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums)) + val cases = map (S.beta_conv o combize Sinduct_assumf) pats + val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum) + val proved_cases = map (prove_case f thy) tasks + val v = S.variant (S.free_varsl (map concl proved_cases)) + (S.mk_var{Name="v", Ty=domain}) + val vtyped = tych v + val substs = map (R.SYM o R.ASSUME o tych o eq v) pats + val proved_cases1 = U.map2 (fn th => R.SUBS[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 = S.type_of(#Bvar(S.dest_forall(concl dc))) + val vars = map (gvvariant[P]) (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 _ => raise TFL_ERR{func = "mk_induction", mesg = "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 (U.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{WFtac, 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(hd(#1(R.dest_thm rules))),WFtac) + in (R.PROVE_HYP thm rules, R.PROVE_HYP thm induction) + end handle _ => (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 simplify_tc tc (r,ind) = + let val tc_eq = simplifier (tych tc) + in + elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind) + handle _ => + (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq) + (R.prove(tych(S.rhs(concl tc_eq)),terminator))) + (r,ind) + handle _ => + (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 _ + => (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq) + (R.prove(tych(S.rhs(concl tc_eq)),terminator)) + handle _ => 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 = U.set_diff S.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 = U.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; + + +(*--------------------------------------------------------------------------- + * Extract termination goals so that they can be put it into a goalstack, or + * have a tactic directly applied to them. + *--------------------------------------------------------------------------*) +local exception IS_NEG + fun strip_imp tm = if S.is_neg tm then raise IS_NEG else S.strip_imp tm +in +fun termination_goals rules = + U.itlist (fn th => fn A => + let val tcl = (#1 o S.strip_imp o #2 o S.strip_forall o concl) th + in tcl@A + end handle _ => A) (R.CONJUNCTS rules) (hyp rules) +end; + +end; (* TFL *)