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
* <something>.
*
* 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 *)