author  paulson 
Thu, 25 Sep 1997 12:24:53 +0200  
changeset 3712  242546f35f8e 
parent 3459  112cbb8301dc 
child 3768  67f4ac759100 
permissions  rwrr 
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(* Title: TFL/tfl 
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ID: $Id$ 
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Author: Konrad Slind, Cambridge University Computer Laboratory 
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Copyright 1997 University of Cambridge 
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Main module 
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*) 
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structure Prim : TFL_sig = 
2112  10 
struct 
11 

12 
(* Abbreviations *) 

13 
structure R = Rules; 

14 
structure S = USyntax; 

15 
structure U = S.Utils; 

16 

17 
val concl = #2 o R.dest_thm; 

18 
val hyp = #1 o R.dest_thm; 

19 

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val list_mk_type = U.end_itlist (curry(op >)); 
2112  21 

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fun enumerate l = 
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rev(#1(foldl (fn ((alist,i), x) => ((x,i)::alist, i+1)) (([],0), l))); 
2112  24 

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fun stringize [] = "" 

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 stringize [i] = Int.toString i 
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 stringize (h::t) = (Int.toString h^", "^stringize t); 
2112  28 

29 

30 
fun TFL_ERR{func,mesg} = U.ERR{module = "Tfl", func = func, mesg = mesg}; 

31 

32 

33 
(* 

34 
* The next function is common to patternmatch translation and 

35 
* proof of completeness of cases for the induction theorem. 

36 
* 

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* The curried function "gvvariant" returns a function to generate distinct 
3405  38 
* variables that are guaranteed not to be in names. The names of 
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* the variables go u, v, ..., z, aa, ..., az, ... The returned 
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* function contains embedded refs! 
2112  41 
**) 
3405  42 
fun gvvariant names = 
43 
let val slist = ref names 

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val vname = ref "u" 
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fun new() = 
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if !vname mem_string (!slist) 
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then (vname := bump_string (!vname); new()) 
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else (slist := !vname :: !slist; !vname) 
2112  49 
in 
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fn ty => Free(new(), ty) 
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end; 
2112  52 

53 

54 
(* 

55 
* Used in induction theorem production. This is the simple case of 

56 
* partitioning up pattern rows by the leading constructor. 

57 
**) 

58 
fun ipartition gv (constructors,rows) = 

59 
let fun pfail s = raise TFL_ERR{func = "partition.part", mesg = s} 

60 
fun part {constrs = [], rows = [], A} = rev A 

61 
 part {constrs = [], rows = _::_, A} = pfail"extra cases in defn" 

62 
 part {constrs = _::_, rows = [], A} = pfail"cases missing in defn" 

63 
 part {constrs = c::crst, rows, A} = 

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let val (Name,Ty) = dest_Const c 
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val L = binder_types Ty 
2112  66 
val (in_group, not_in_group) = 
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U.itlist (fn (row as (p::rst, rhs)) => 

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fn (in_group,not_in_group) => 

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let val (pc,args) = S.strip_comb p 

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in if (#1(dest_Const pc) = Name) 
2112  71 
then ((args@rst, rhs)::in_group, not_in_group) 
72 
else (in_group, row::not_in_group) 

73 
end) rows ([],[]) 

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val col_types = U.take type_of (length L, #1(hd in_group)) 
2112  75 
in 
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part{constrs = crst, rows = not_in_group, 

77 
A = {constructor = c, 

78 
new_formals = map gv col_types, 

79 
group = in_group}::A} 

80 
end 

81 
in part{constrs = constructors, rows = rows, A = []} 

82 
end; 

83 

84 

85 

86 
(* 

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* This datatype carries some information about the origin of a 

88 
* clause in a function definition. 

89 
**) 

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datatype pattern = GIVEN of term * int 
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 OMITTED of term * int 
2112  92 

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fun pattern_map f (GIVEN (tm,i)) = GIVEN(f tm, i) 
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 pattern_map f (OMITTED (tm,i)) = OMITTED(f tm, i); 
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fun pattern_subst theta = pattern_map (subst_free theta); 
2112  97 

98 
fun dest_pattern (GIVEN (tm,i)) = ((GIVEN,i),tm) 

99 
 dest_pattern (OMITTED (tm,i)) = ((OMITTED,i),tm); 

100 

101 
val pat_of = #2 o dest_pattern; 

102 
val row_of_pat = #2 o #1 o dest_pattern; 

103 

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(* 

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* Produce an instance of a constructor, plus genvars for its arguments. 

106 
**) 

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fun fresh_constr ty_match colty gv c = 

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let val (_,Ty) = dest_Const c 
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val L = binder_types Ty 
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and ty = body_type Ty 
2112  111 
val ty_theta = ty_match ty colty 
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val c' = S.inst ty_theta c 

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val gvars = map (S.inst ty_theta o gv) L 

114 
in (c', gvars) 

115 
end; 

116 

117 

118 
(* 

119 
* Goes through a list of rows and picks out the ones beginning with a 

120 
* pattern with constructor = Name. 

121 
**) 

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fun mk_group Name rows = 

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U.itlist (fn (row as ((prefix, p::rst), rhs)) => 

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fn (in_group,not_in_group) => 

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let val (pc,args) = S.strip_comb p 

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in if ((#1(dest_Const pc) = Name) handle _ => false) 
2112  127 
then (((prefix,args@rst), rhs)::in_group, not_in_group) 
128 
else (in_group, row::not_in_group) end) 

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rows ([],[]); 

130 

131 
(* 

132 
* Partition the rows. Not efficient: we should use hashing. 

133 
**) 

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fun partition _ _ (_,_,_,[]) = raise TFL_ERR{func="partition", mesg="no rows"} 

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 partition gv ty_match 

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(constructors, colty, res_ty, rows as (((prefix,_),_)::_)) = 

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let val fresh = fresh_constr ty_match colty gv 

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fun part {constrs = [], rows, A} = rev A 

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 part {constrs = c::crst, rows, A} = 

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let val (c',gvars) = fresh c 

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val (Name,Ty) = dest_Const c' 
2112  142 
val (in_group, not_in_group) = mk_group Name rows 
143 
val in_group' = 

144 
if (null in_group) (* Constructor not given *) 

145 
then [((prefix, #2(fresh c)), OMITTED (S.ARB res_ty, ~1))] 

146 
else in_group 

147 
in 

148 
part{constrs = crst, 

149 
rows = not_in_group, 

150 
A = {constructor = c', 

151 
new_formals = gvars, 

152 
group = in_group'}::A} 

153 
end 

154 
in part{constrs=constructors, rows=rows, A=[]} 

155 
end; 

156 

157 
(* 

158 
* Misc. routines used in mk_case 

159 
**) 

160 

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fun mk_pat (c,l) = 
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let val L = length (binder_types (type_of c)) 
2112  163 
fun build (prefix,tag,plist) = 
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let val args = take (L,plist) 
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and plist' = drop(L,plist) 
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in (prefix,tag,list_comb(c,args)::plist') end 
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in map build l end; 
2112  168 

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fun v_to_prefix (prefix, v::pats) = (v::prefix,pats) 

170 
 v_to_prefix _ = raise TFL_ERR{func="mk_case", mesg="v_to_prefix"}; 

171 

172 
fun v_to_pats (v::prefix,tag, pats) = (prefix, tag, v::pats) 

173 
 v_to_pats _ = raise TFL_ERR{func="mk_case", mesg="v_to_pats"}; 

174 

175 

176 
(* 

177 
* Translation of pattern terms into nested case expressions. 

178 
* 

179 
* This performs the translation and also builds the full set of patterns. 

180 
* Thus it supports the construction of induction theorems even when an 

181 
* incomplete set of patterns is given. 

182 
**) 

183 

3405  184 
fun mk_case ty_info ty_match usednames range_ty = 
2112  185 
let 
186 
fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = s} 

3405  187 
val fresh_var = gvvariant usednames 
2112  188 
val divide = partition fresh_var ty_match 
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fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row" 

190 
 expand constructors ty (row as ((prefix, p::rst), rhs)) = 

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if (is_Free p) 
2112  192 
then let val fresh = fresh_constr ty_match ty fresh_var 
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fun expnd (c,gvs) = 

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let val capp = list_comb(c,gvs) 
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in ((prefix, capp::rst), pattern_subst[(p,capp)] rhs) 
2112  196 
end 
197 
in map expnd (map fresh constructors) end 

198 
else [row] 

199 
fun mk{rows=[],...} = mk_case_fail"no rows" 

200 
 mk{path=[], rows = ((prefix, []), rhs)::_} = (* Done *) 

201 
let val (tag,tm) = dest_pattern rhs 

202 
in ([(prefix,tag,[])], tm) 

203 
end 

204 
 mk{path=[], rows = _::_} = mk_case_fail"blunder" 

205 
 mk{path as u::rstp, rows as ((prefix, []), rhs)::rst} = 

206 
mk{path = path, 

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rows = ((prefix, [fresh_var(type_of u)]), rhs)::rst} 
2112  208 
 mk{path = u::rstp, rows as ((_, p::_), _)::_} = 
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let val (pat_rectangle,rights) = ListPair.unzip rows 
2112  210 
val col0 = map(hd o #2) pat_rectangle 
211 
in 

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if (forall is_Free col0) 
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then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e) 
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(ListPair.zip (col0, rights)) 
2112  215 
val pat_rectangle' = map v_to_prefix pat_rectangle 
216 
val (pref_patl,tm) = mk{path = rstp, 

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rows = ListPair.zip (pat_rectangle', 
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rights')} 
2112  219 
in (map v_to_pats pref_patl, tm) 
220 
end 

221 
else 

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let val pty as Type (ty_name,_) = type_of p 
2112  223 
in 
224 
case (ty_info ty_name) 

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of None => mk_case_fail("Not a known datatype: "^ty_name) 
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 Some{case_const,constructors} => 
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let val case_const_name = #1(dest_Const case_const) 
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val nrows = List_.concat (map (expand constructors pty) rows) 
2112  229 
val subproblems = divide(constructors, pty, range_ty, nrows) 
230 
val groups = map #group subproblems 

231 
and new_formals = map #new_formals subproblems 

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and constructors' = map #constructor subproblems 

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val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows}) 

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(ListPair.zip (new_formals, groups)) 
2112  235 
val rec_calls = map mk news 
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val (pat_rect,dtrees) = ListPair.unzip rec_calls 
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val case_functions = map S.list_mk_abs 
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(ListPair.zip (new_formals, dtrees)) 
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val types = map type_of (case_functions@[u]) @ [range_ty] 
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val case_const' = Const(case_const_name, list_mk_type types) 
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val tree = list_comb(case_const', case_functions@[u]) 
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val pat_rect1 = List_.concat 
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(ListPair.map mk_pat (constructors', pat_rect)) 
2112  244 
in (pat_rect1,tree) 
245 
end 

246 
end end 

247 
in mk 

248 
end; 

249 

250 

251 
(* Repeated variable occurrences in a pattern are not allowed. *) 

252 
fun FV_multiset tm = 

253 
case (S.dest_term tm) 

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of S.VAR{Name,Ty} => [Free(Name,Ty)] 
2112  255 
 S.CONST _ => [] 
256 
 S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand 

257 
 S.LAMB _ => raise TFL_ERR{func = "FV_multiset", mesg = "lambda"}; 

258 

259 
fun no_repeat_vars thy pat = 

260 
let fun check [] = true 

261 
 check (v::rst) = 

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if mem_term (v,rst) then 
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raise TFL_ERR{func = "no_repeat_vars", 
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mesg = quote(#1(dest_Free v)) ^ 
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" occurs repeatedly in the pattern " ^ 
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quote (string_of_cterm (Thry.typecheck thy pat))} 
2112  267 
else check rst 
268 
in check (FV_multiset pat) 

269 
end; 

270 

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local fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s} 
3191  272 
fun single [f] = f 
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 single fs = mk_functional_err (Int.toString (length fs) ^ 
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" distinct function names!") 
2112  275 
in 
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276 
fun mk_functional thy clauses = 
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277 
let val (L,R) = ListPair.unzip 
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278 
(map (fn (Const("op =",_) $ t $ u) => (t,u)) clauses) 
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279 
val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L) 
3405  280 
val fcon as Const (fname, ftype) = single (gen_distinct (op aconv) funcs) 
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281 
val dummy = map (no_repeat_vars thy) pats 
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282 
val rows = ListPair.zip (map (fn x => ([],[x])) pats, 
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283 
map GIVEN (enumerate R)) 
3405  284 
val names = foldr add_term_names (R,[]) 
285 
val atype = type_of(hd pats) 

286 
and aname = variant names "a" 

287 
val a = Free(aname,atype) 

2112  288 
val ty_info = Thry.match_info thy 
289 
val ty_match = Thry.match_type thy 

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290 
val range_ty = type_of (hd R) 
3405  291 
val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty 
2112  292 
{path=[a], rows=rows} 
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293 
val patts1 = map (fn (_,(tag,i),[pat]) => tag (pat,i)) patts 
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294 
handle _ => mk_functional_err "error in patternmatch translation" 
2112  295 
val patts2 = U.sort(fn p1=>fn p2=> row_of_pat p1 < row_of_pat p2) patts1 
296 
val finals = map row_of_pat patts2 

297 
val originals = map (row_of_pat o #2) rows 

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298 
val dummy = case (originals\\finals) 
2112  299 
of [] => () 
300 
 L => mk_functional_err("The following rows (counting from zero)\ 

301 
\ are inaccessible: "^stringize L) 

3405  302 
in {functional = Abs(fname, ftype, 
303 
abstract_over (fcon, 

304 
absfree(aname,atype, case_tm))), 

2112  305 
pats = patts2} 
306 
end end; 

307 

308 

309 
(* 

310 
* 

311 
* PRINCIPLES OF DEFINITION 

312 
* 

313 
**) 

314 

315 

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316 
(*Make all TVars available for instantiation by adding a ? to the front*) 
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317 
fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts) 
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318 
 poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort) 
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319 
 poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort); 
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320 

3191  321 
local val f_eq_wfrec_R_M = 
322 
#ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY)))) 

323 
val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M 

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324 
val (fname,_) = dest_Free f 
3191  325 
val (wfrec,_) = S.strip_comb rhs 
326 
in 

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327 
fun wfrec_definition0 thy fid R (functional as Abs(Name, Ty, _)) = 
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328 
let val def_name = if Name<>fid then 
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329 
raise TFL_ERR{func = "wfrec_definition0", 
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330 
mesg = "Expected a definition of " ^ 
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331 
quote fid ^ " but found one of " ^ 
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332 
quote Name} 
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333 
else Name ^ "_def" 
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334 
val wfrec_R_M = map_term_types poly_tvars 
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335 
(wfrec $ map_term_types poly_tvars R) 
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336 
$ functional 
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337 
val (_, def_term, _) = 
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338 
Sign.infer_types (sign_of thy) (K None) (K None) [] false 
3405  339 
([Const("==",dummyT) $ Const(Name,Ty) $ wfrec_R_M], 
340 
propT) 

341 
in add_defs_i [(def_name, def_term)] thy end 

3191  342 
end; 
2112  343 

344 

345 

346 
(* 

347 
* This structure keeps track of congruence rules that aren't derived 

348 
* from a datatype definition. 

349 
**) 

350 
fun extraction_thms thy = 

351 
let val {case_rewrites,case_congs} = Thry.extract_info thy 

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352 
in (case_rewrites, case_congs) 
2112  353 
end; 
354 

355 

356 
(* 

357 
* Pair patterns with termination conditions. The full list of patterns for 

358 
* a definition is merged with the TCs arising from the usergiven clauses. 

359 
* There can be fewer clauses than the full list, if the user omitted some 

360 
* cases. This routine is used to prepare input for mk_induction. 

361 
**) 

362 
fun merge full_pats TCs = 

363 
let fun insert (p,TCs) = 

364 
let fun insrt ((x as (h,[]))::rst) = 

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365 
if (p aconv h) then (p,TCs)::rst else x::insrt rst 
2112  366 
 insrt (x::rst) = x::insrt rst 
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367 
 insrt[] = raise TFL_ERR{func="merge.insert", 
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368 
mesg="pattern not found"} 
2112  369 
in insrt end 
370 
fun pass ([],ptcl_final) = ptcl_final 

371 
 pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl) 

372 
in 

373 
pass (TCs, map (fn p => (p,[])) full_pats) 

374 
end; 

375 

376 

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377 
fun givens [] = [] 
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378 
 givens (GIVEN(tm,_)::pats) = tm :: givens pats 
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379 
 givens (OMITTED _::pats) = givens pats; 
2112  380 

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381 
fun post_definition (ss, tflCongs) (theory, (def, pats)) = 
3191  382 
let val tych = Thry.typecheck theory 
383 
val f = #lhs(S.dest_eq(concl def)) 

384 
val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def 

2112  385 
val given_pats = givens pats 
386 
val WFR = #ant(S.dest_imp(concl corollary)) 

3191  387 
val R = #Rand(S.dest_comb WFR) 
2112  388 
val corollary' = R.UNDISCH corollary (* put WF R on assums *) 
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389 
val corollaries = map (fn pat => R.SPEC (tych pat) corollary') 
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390 
given_pats 
3191  391 
val (case_rewrites,context_congs) = extraction_thms theory 
3405  392 
val corollaries' = map(rewrite_rule case_rewrites) corollaries 
393 
val extract = R.CONTEXT_REWRITE_RULE 

394 
(ss, f, R, 

395 
R.ISPECL (map tych [f,R]) Thms.CUT_LEMMA, 

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396 
tflCongs@context_congs) 
3405  397 
val (rules, TCs) = ListPair.unzip (map extract corollaries') 
398 
val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules 

399 
val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR) 

2112  400 
val rules1 = R.LIST_CONJ(map mk_cond_rule rules0) 
401 
in 

402 
{theory = theory, (* holds def, if it's needed *) 

403 
rules = rules1, 

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404 
full_pats_TCs = merge (map pat_of pats) 
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405 
(ListPair.zip (given_pats, TCs)), 
2112  406 
TCs = TCs, 
407 
patterns = pats} 

408 
end; 

409 

410 
(* 

411 
* Perform the extraction without making the definition. Definition and 

412 
* extraction commute for the nonnested case. For hol90 users, this 

413 
* function can be invoked without being in draft mode. 

3405  414 
* CURRENTLY UNUSED 
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415 
fun wfrec_eqns (ss, tflCongs) thy eqns = 
2112  416 
let val {functional,pats} = mk_functional thy eqns 
417 
val given_pats = givens pats 

418 
val {Bvar = f, Body} = S.dest_abs functional 

419 
val {Bvar = x, ...} = S.dest_abs Body 

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420 
val (Name, Type("fun", [f_dty, f_rty])) = dest_Free f 
2112  421 
val (case_rewrites,context_congs) = extraction_thms thy 
422 
val tych = Thry.typecheck thy 

423 
val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY 

3405  424 
val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0 
425 
val R = Free (variant (foldr add_term_names (eqns,[])) Rname, 

426 
Rtype) 

2112  427 
val WFREC_THM = R.ISPECL [tych R, tych f] WFREC_THM0 
428 
val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM) 

429 
val R1 = S.rand WFR 

430 
val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM) 

3405  431 
val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats 
432 
val corollaries' = map (rewrite_rule case_rewrites) corollaries 

433 
val extract = R.CONTEXT_REWRITE_RULE 

434 
(ss, f, R1, 

435 
R.ISPECL (map tych [f,R1]) Thms.CUT_LEMMA, 

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changeset

436 
tflCongs@context_congs) 
2112  437 
in {proto_def=proto_def, 
438 
WFR=WFR, 

439 
pats=pats, 

440 
extracta = map extract corollaries'} 

441 
end; 

3405  442 
**) 
2112  443 

444 

445 
(* 

446 
* Define the constant after extracting the termination conditions. The 

447 
* wellfounded relation used in the definition is computed by using the 

448 
* choice operator on the extracted conditions (plus the condition that 

449 
* such a relation must be wellfounded). 

3405  450 
* CURRENTLY UNUSED 
451 
fun lazyR_def ss thy eqns = 

452 
let val {proto_def,WFR,pats,extracta} = wfrec_eqns ss thy eqns 

2112  453 
val R1 = S.rand WFR 
454 
val f = S.lhs proto_def 

3333
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455 
val (Name,_) = dest_Free f 
3245
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changeset

456 
val (extractants,TCl) = ListPair.unzip extracta 
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3191
diff
changeset

457 
val TCs = foldr (gen_union (op aconv)) (TCl, []) 
2112  458 
val full_rqt = WFR::TCs 
459 
val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt} 

460 
val R'abs = S.rand R' 

3405  461 
val theory = add_defs_i [(Name ^ "_def", subst_free[(R1,R')] proto_def)] 
462 
thy 

463 
val def = freezeT((get_axiom theory (Name ^ "_def")) RS meta_eq_to_obj_eq) 

2112  464 
val fconst = #lhs(S.dest_eq(concl def)) 
465 
val tych = Thry.typecheck theory 

466 
val baz = R.DISCH (tych proto_def) 

467 
(U.itlist (R.DISCH o tych) full_rqt (R.LIST_CONJ extractants)) 

468 
val def' = R.MP (R.SPEC (tych fconst) 

469 
(R.SPEC (tych R') (R.GENL[tych R1, tych f] baz))) 

470 
def 

471 
val body_th = R.LIST_CONJ (map (R.ASSUME o tych) full_rqt) 

3245
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changeset

472 
val bar = R.MP (R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX) 
3191  473 
body_th 
2112  474 
in {theory = theory, R=R1, 
475 
rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def', 

3245
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changeset

476 
full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)), 
2112  477 
patterns = pats} 
478 
end; 

3405  479 
**) 
2112  480 

481 

482 

483 
(* 

484 
* 

485 
* INDUCTION THEOREM 

486 
* 

487 
**) 

488 

489 

490 
(* Miscellaneous function  

491 
* 

492 
* [x_1,...,x_n] ?v_1...v_n. M[v_1,...,v_n] 

493 
*  

494 
* ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]), 

495 
* ... 

496 
* (x_j,?v_n. M[x_1,...,x_(n1),v_n])] ) 

497 
* 

498 
* This function is totally ad hoc. Used in the production of the induction 

499 
* theorem. The nchotomy theorem can have clauses that look like 

500 
* 

501 
* ?v1..vn. z = C vn..v1 

502 
* 

503 
* in which the order of quantification is not the order of occurrence of the 

504 
* quantified variables as arguments to C. Since we have no control over this 

505 
* aspect of the nchotomy theorem, we make the correspondence explicit by 

506 
* pairing the incoming new variable with the term it gets betareduced into. 

507 
**) 

508 

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changeset

509 
fun alpha_ex_unroll (xlist, tm) = 
2112  510 
let val (qvars,body) = S.strip_exists tm 
511 
val vlist = #2(S.strip_comb (S.rhs body)) 

3245
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512 
val plist = ListPair.zip (vlist, xlist) 
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changeset

513 
val args = map (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars 
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changeset

514 
handle OPTION _ => error 
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diff
changeset

515 
"TFL fault [alpha_ex_unroll]: no correspondence" 
3405  516 
fun build ex [] = [] 
517 
 build (_$rex) (v::rst) = 

518 
let val ex1 = betapply(rex, v) 

519 
in ex1 :: build ex1 rst 

2112  520 
end 
3245
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changeset

521 
val (nex::exl) = rev (tm::build tm args) 
2112  522 
in 
3245
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changeset

523 
(nex, ListPair.zip (args, rev exl)) 
2112  524 
end; 
525 

526 

527 

528 
(* 

529 
* 

530 
* PROVING COMPLETENESS OF PATTERNS 

531 
* 

532 
**) 

533 

3405  534 
fun mk_case ty_info usednames thy = 
2112  535 
let 
3405  536 
val divide = ipartition (gvvariant usednames) 
2112  537 
val tych = Thry.typecheck thy 
3353
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Removal of module Mask and datatype binding with its constructor >
paulson
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3333
diff
changeset

538 
fun tych_binding(x,y) = (tych x, tych y) 
2112  539 
fun fail s = raise TFL_ERR{func = "mk_case", mesg = s} 
540 
fun mk{rows=[],...} = fail"no rows" 

541 
 mk{path=[], rows = [([], (thm, bindings))]} = 

542 
R.IT_EXISTS (map tych_binding bindings) thm 

543 
 mk{path = u::rstp, rows as (p::_, _)::_} = 

3245
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3191
diff
changeset

544 
let val (pat_rectangle,rights) = ListPair.unzip rows 
2112  545 
val col0 = map hd pat_rectangle 
546 
val pat_rectangle' = map tl pat_rectangle 

547 
in 

3333
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3301
diff
changeset

548 
if (forall is_Free col0) (* column 0 is all variables *) 
3353
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paulson
parents:
3333
diff
changeset

549 
then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)])) 
3245
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changeset

550 
(ListPair.zip (rights, col0)) 
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diff
changeset

551 
in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')} 
2112  552 
end 
553 
else (* column 0 is all constructors *) 

3245
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Removal of redundant code (unused or already present in Isabelle.
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diff
changeset

554 
let val Type (ty_name,_) = type_of p 
2112  555 
in 
556 
case (ty_info ty_name) 

3245
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diff
changeset

557 
of None => fail("Not a known datatype: "^ty_name) 
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diff
changeset

558 
 Some{constructors,nchotomy} => 
2112  559 
let val thm' = R.ISPEC (tych u) nchotomy 
560 
val disjuncts = S.strip_disj (concl thm') 

561 
val subproblems = divide(constructors, rows) 

562 
val groups = map #group subproblems 

563 
and new_formals = map #new_formals subproblems 

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Removal of redundant code (unused or already present in Isabelle.
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changeset

564 
val existentials = ListPair.map alpha_ex_unroll 
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Removal of redundant code (unused or already present in Isabelle.
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changeset

565 
(new_formals, disjuncts) 
2112  566 
val constraints = map #1 existentials 
567 
val vexl = map #2 existentials 

568 
fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b)) 

569 
val news = map (fn (nf,rows,c) => {path = nf@rstp, 

570 
rows = map (expnd c) rows}) 

571 
(U.zip3 new_formals groups constraints) 

572 
val recursive_thms = map mk news 

3245
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Removal of redundant code (unused or already present in Isabelle.
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changeset

573 
val build_exists = foldr 
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Removal of redundant code (unused or already present in Isabelle.
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changeset

574 
(fn((x,t), th) => 
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changeset

575 
R.CHOOSE (tych x, R.ASSUME (tych t)) th) 
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changeset

576 
val thms' = ListPair.map build_exists (vexl, recursive_thms) 
2112  577 
val same_concls = R.EVEN_ORS thms' 
578 
in R.DISJ_CASESL thm' same_concls 

579 
end 

580 
end end 

581 
in mk 

582 
end; 

583 

584 

585 
fun complete_cases thy = 

586 
let val tych = Thry.typecheck thy 

587 
val ty_info = Thry.induct_info thy 

588 
in fn pats => 

3405  589 
let val names = foldr add_term_names (pats,[]) 
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More deHOLification: using Free, Const, etc. instead of mk_var, mk_const
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changeset

590 
val T = type_of (hd pats) 
3405  591 
val aname = Term.variant names "a" 
592 
val vname = Term.variant (aname::names) "v" 

593 
val a = Free (aname, T) 

594 
val v = Free (vname, T) 

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Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
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diff
changeset

595 
val a_eq_v = HOLogic.mk_eq(a,v) 
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Removal of redundant code (unused or already present in Isabelle.
paulson
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diff
changeset

596 
val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a) 
2112  597 
(R.REFL (tych a)) 
598 
val th0 = R.ASSUME (tych a_eq_v) 

599 
val rows = map (fn x => ([x], (th0,[]))) pats 

600 
in 

601 
R.GEN (tych a) 

602 
(R.RIGHT_ASSOC 

603 
(R.CHOOSE(tych v, ex_th0) 

3405  604 
(mk_case ty_info (vname::aname::names) 
605 
thy {path=[v], rows=rows}))) 

2112  606 
end end; 
607 

608 

609 
(* 

610 
* Constructing induction hypotheses: one for each recursive call. 

611 
* 

612 
* Note. R will never occur as a variable in the ind_clause, because 

613 
* to do so, it would have to be from a nested definition, and we don't 

614 
* allow nested defns to have R variable. 

615 
* 

616 
* Note. When the context is empty, there can be no local variables. 

617 
**) 

618 

3405  619 
local infix 5 ==> 
2112  620 
fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2} 
621 
in 

622 
fun build_ih f P (pat,TCs) = 

623 
let val globals = S.free_vars_lr pat 

3405  624 
fun nested tm = is_some (S.find_term (curry (op aconv) f) tm) 
2112  625 
fun dest_TC tm = 
626 
let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm)) 

627 
val (R,y,_) = S.dest_relation R_y_pat 

3405  628 
val P_y = if (nested tm) then R_y_pat ==> P$y else P$y 
2112  629 
in case cntxt 
630 
of [] => (P_y, (tm,[])) 

631 
 _ => let 

632 
val imp = S.list_mk_conj cntxt ==> P_y 

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More deHOLification: using Free, Const, etc. instead of mk_var, mk_const
paulson
parents:
3388
diff
changeset

633 
val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals) 
3405  634 
val locals = #2(U.pluck (curry (op aconv) P) lvs) handle _ => lvs 
2112  635 
in (S.list_mk_forall(locals,imp), (tm,locals)) end 
636 
end 

637 
in case TCs 

3405  638 
of [] => (S.list_mk_forall(globals, P$pat), []) 
3245
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Removal of redundant code (unused or already present in Isabelle.
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parents:
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diff
changeset

639 
 _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs) 
3405  640 
val ind_clause = S.list_mk_conj ihs ==> P$pat 
2112  641 
in (S.list_mk_forall(globals,ind_clause), TCs_locals) 
642 
end 

643 
end 

644 
end; 

645 

646 

647 

648 
(* 

649 
* This function makes good on the promise made in "build_ih: we prove 

650 
* <something>. 

651 
* 

652 
* Input is tm = "(!y. R y pat ==> P y) ==> P pat", 

653 
* TCs = TC_1[pat] ... TC_n[pat] 

654 
* thm = ih1 /\ ... /\ ih_n  ih[pat] 

655 
**) 

656 
fun prove_case f thy (tm,TCs_locals,thm) = 

657 
let val tych = Thry.typecheck thy 

658 
val antc = tych(#ant(S.dest_imp tm)) 

659 
val thm' = R.SPEC_ALL thm 

3405  660 
fun nested tm = is_some (S.find_term (curry (op aconv) f) tm) 
2112  661 
fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC))))) 
662 
fun mk_ih ((TC,locals),th2,nested) = 

663 
R.GENL (map tych locals) 

664 
(if nested 

665 
then R.DISCH (get_cntxt TC) th2 handle _ => th2 

666 
else if S.is_imp(concl TC) 

667 
then R.IMP_TRANS TC th2 

668 
else R.MP th2 TC) 

669 
in 

670 
R.DISCH antc 

671 
(if S.is_imp(concl thm') (* recursive calls in this clause *) 

672 
then let val th1 = R.ASSUME antc 

673 
val TCs = map #1 TCs_locals 

674 
val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o 

675 
#2 o S.strip_forall) TCs 

676 
val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs)) 

677 
TCs_locals 

678 
val th2list = map (U.C R.SPEC th1 o tych) ylist 

679 
val nlist = map nested TCs 

680 
val triples = U.zip3 TClist th2list nlist 

681 
val Pylist = map mk_ih triples 

682 
in R.MP thm' (R.LIST_CONJ Pylist) end 

683 
else thm') 

684 
end; 

685 

686 

687 
(* 

688 
* 

689 
* x = (v1,...,vn)  M[x] 

690 
*  

691 
* ?v1 ... vn. x = (v1,...,vn)  M[x] 

692 
* 

693 
**) 

694 
fun LEFT_ABS_VSTRUCT tych thm = 

695 
let fun CHOOSER v (tm,thm) = 

696 
let val ex_tm = S.mk_exists{Bvar=v,Body=tm} 

697 
in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm) 

698 
end 

3245
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Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

699 
val [veq] = filter (U.can S.dest_eq) (#1 (R.dest_thm thm)) 
2112  700 
val {lhs,rhs} = S.dest_eq veq 
701 
val L = S.free_vars_lr rhs 

3245
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Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

702 
in #2 (U.itlist CHOOSER L (veq,thm)) end; 
2112  703 

704 

705 
(* 

706 
* Input : f, R, and [(pat1,TCs1),..., (patn,TCsn)] 

707 
* 

708 
* Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove 

709 
* recursion induction (Rinduct) by proving the antecedent of Sinduct from 

710 
* the antecedent of Rinduct. 

711 
**) 

712 
fun mk_induction thy f R pat_TCs_list = 

713 
let val tych = Thry.typecheck thy 

714 
val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM) 

3245
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Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

715 
val (pats,TCsl) = ListPair.unzip pat_TCs_list 
2112  716 
val case_thm = complete_cases thy pats 
3245
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Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

717 
val domain = (type_of o hd) pats 
3405  718 
val Pname = Term.variant (foldr (foldr add_term_names) 
719 
(pats::TCsl, [])) "P" 

720 
val P = Free(Pname, domain > HOLogic.boolT) 

2112  721 
val Sinduct = R.SPEC (tych P) Sinduction 
722 
val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct) 

723 
val Rassums_TCl' = map (build_ih f P) pat_TCs_list 

3245
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

724 
val (Rassums,TCl') = ListPair.unzip Rassums_TCl' 
2112  725 
val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums)) 
3405  726 
val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats 
2112  727 
val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum) 
728 
val proved_cases = map (prove_case f thy) tasks 

3405  729 
val v = Free (variant (foldr add_term_names (map concl proved_cases, [])) 
730 
"v", 

731 
domain) 

2112  732 
val vtyped = tych v 
3245
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

733 
val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats 
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

734 
val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th') 
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

735 
(substs, proved_cases) 
2112  736 
val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1 
737 
val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases) 

738 
val dc = R.MP Sinduct dant 

3245
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

739 
val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc))) 
3405  740 
val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty) 
2112  741 
val dc' = U.itlist (R.GEN o tych) vars 
742 
(R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc) 

743 
in 

744 
R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc') 

3391
5e45dd3b64e9
More deHOLification: using Free, Const, etc. instead of mk_var, mk_const
paulson
parents:
3388
diff
changeset

745 
end 
2112  746 
handle _ => raise TFL_ERR{func = "mk_induction", mesg = "failed derivation"}; 
747 

748 

749 

3391
5e45dd3b64e9
More deHOLification: using Free, Const, etc. instead of mk_var, mk_const
paulson
parents:
3388
diff
changeset

750 

2112  751 
(* 
752 
* 

753 
* POST PROCESSING 

754 
* 

755 
**) 

756 

757 

758 
fun simplify_induction thy hth ind = 

759 
let val tych = Thry.typecheck thy 

760 
val (asl,_) = R.dest_thm ind 

761 
val (_,tc_eq_tc') = R.dest_thm hth 

762 
val tc = S.lhs tc_eq_tc' 

763 
fun loop [] = ind 

764 
 loop (asm::rst) = 

765 
if (U.can (Thry.match_term thy asm) tc) 

766 
then R.UNDISCH 

767 
(R.MATCH_MP 

768 
(R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind)) 

769 
hth) 

770 
else loop rst 

771 
in loop asl 

772 
end; 

773 

774 

775 
(* 

776 
* The termination condition is an antecedent to the rule, and an 

777 
* assumption to the theorem. 

778 
**) 

779 
fun elim_tc tcthm (rule,induction) = 

780 
(R.MP rule tcthm, R.PROVE_HYP tcthm induction) 

781 

782 

783 
fun postprocess{WFtac, terminator, simplifier} theory {rules,induction,TCs} = 

784 
let val tych = Thry.typecheck theory 

785 

786 
(* 

787 
* Attempt to eliminate WF condition. It's the only assumption of rules 

788 
**) 

789 
val (rules1,induction1) = 

3405  790 
let val thm = R.prove(tych(HOLogic.mk_Trueprop 
791 
(hd(#1(R.dest_thm rules)))), 

792 
WFtac) 

2112  793 
in (R.PROVE_HYP thm rules, R.PROVE_HYP thm induction) 
794 
end handle _ => (rules,induction) 

795 

796 
(* 

797 
* The termination condition (tc) is simplified to  tc = tc' (there 

798 
* might not be a change!) and then 3 attempts are made: 

799 
* 

800 
* 1. if  tc = T, then eliminate it with eqT; otherwise, 

801 
* 2. apply the terminator to tc'. If  tc' = T then eliminate; else 

802 
* 3. replace tc by tc' in both the rules and the induction theorem. 

803 
**) 

804 
fun simplify_tc tc (r,ind) = 

805 
let val tc_eq = simplifier (tych tc) 

806 
in 

807 
elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind) 

808 
handle _ => 

809 
(elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq) 

3405  810 
(R.prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))), 
811 
terminator))) 

2112  812 
(r,ind) 
813 
handle _ => 

814 
(R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq), 

815 
simplify_induction theory tc_eq ind)) 

816 
end 

817 

818 
(* 

819 
* Nested termination conditions are harder to get at, since they are 

820 
* left embedded in the body of the function (and in induction 

821 
* theorem hypotheses). Our "solution" is to simplify them, and try to 

822 
* prove termination, but leave the application of the resulting theorem 

823 
* to a higher level. So things go much as in "simplify_tc": the 

824 
* termination condition (tc) is simplified to  tc = tc' (there might 

825 
* not be a change) and then 2 attempts are made: 

826 
* 

827 
* 1. if  tc = T, then return  tc; otherwise, 

828 
* 2. apply the terminator to tc'. If  tc' = T then return  tc; else 

829 
* 3. return  tc = tc' 

830 
**) 

831 
fun simplify_nested_tc tc = 

832 
let val tc_eq = simplifier (tych (#2 (S.strip_forall tc))) 

833 
in 

834 
R.GEN_ALL 

835 
(R.MATCH_MP Thms.eqT tc_eq 

836 
handle _ 

837 
=> (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq) 

3405  838 
(R.prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))), 
839 
terminator)) 

2112  840 
handle _ => tc_eq)) 
841 
end 

842 

843 
(* 

844 
* Attempt to simplify the termination conditions in each rule and 

845 
* in the induction theorem. 

846 
**) 

847 
fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm 

848 
fun loop ([],extras,R,ind) = (rev R, ind, extras) 

849 
 loop ((r,ftcs)::rst, nthms, R, ind) = 

850 
let val tcs = #1(strip_imp (concl r)) 

3391
5e45dd3b64e9
More deHOLification: using Free, Const, etc. instead of mk_var, mk_const
paulson
parents:
3388
diff
changeset

851 
val extra_tcs = gen_rems (op aconv) (ftcs, tcs) 
2112  852 
val extra_tc_thms = map simplify_nested_tc extra_tcs 
853 
val (r1,ind1) = U.rev_itlist simplify_tc tcs (r,ind) 

854 
val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1 

855 
in loop(rst, nthms@extra_tc_thms, r2::R, ind1) 

856 
end 

3245
241838c01caf
Removal of redundant code (unused or already present in Isabelle.
paulson
parents:
3191
diff
changeset

857 
val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs) 
2112  858 
val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1) 
859 
in 

860 
{induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras} 

861 
end; 

862 

863 
end; (* TFL *) 