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% FILE : unwinding.ml %
% DESCRIPTION : Rules for unfolding, unwinding, pruning, etc. %
% %
% READS FILES : <none> %
% WRITES FILES : <none> %
% %
% AUTHOR : Originally written for LCF-LSM by Mike Gordon (MJCG). %
% 21.May.1985 : Additions by Tom Melham (TFM). %
% 10.Mar.1988 : Modifications by David Shepherd (DES) of INMOS. %
% 24.Mar.1988 : Bug fixes by David Shepherd (DES). %
% 23.Apr.1990 : Modifications by Tom Melham (TFM). %
% 22.Aug.1991 : Additions and tidying-up by Richard Boulton (RJB). %
% %
% LAST MODIFIED : R.J.Boulton %
% DATE : 21st July 1992 %
%****************************************************************************%
%============================================================================%
% Tools for manipulating device implementations `by hand' %
%============================================================================%
%----------------------------------------------------------------------------%
% DEPTH_FORALL_CONV : conv -> conv %
% %
% DEPTH_FORALL_CONV conv "!x1 ... xn. body" applies conv to "body" and %
% returns a theorem of the form: %
% %
% |- (!x1 ... xn. body) = (!x1 ... xn. body') %
%----------------------------------------------------------------------------%
letrec DEPTH_FORALL_CONV conv tm =
if (is_forall tm)
then RAND_CONV (ABS_CONV (DEPTH_FORALL_CONV conv)) tm
else conv tm;;
%----------------------------------------------------------------------------%
% DEPTH_EXISTS_CONV : conv -> conv %
% %
% DEPTH_EXISTS_CONV conv "?x1 ... xn. body" applies conv to "body" and %
% returns a theorem of the form: %
% %
% |- (?x1 ... xn. body) = (?x1 ... xn. body') %
%----------------------------------------------------------------------------%
letrec DEPTH_EXISTS_CONV conv tm =
if (is_exists tm)
then RAND_CONV (ABS_CONV (DEPTH_EXISTS_CONV conv)) tm
else conv tm;;
%----------------------------------------------------------------------------%
% FLATTEN_CONJ_CONV : conv %
% %
% "t1 /\ ... /\ tn" %
% ----> %
% |- t1 /\ ... /\ tn = u1 /\ ... /\ un %
% %
% where the RHS of the equation is a flattened version of the LHS. %
%----------------------------------------------------------------------------%
let FLATTEN_CONJ_CONV t =
CONJUNCTS_CONV (t,list_mk_conj (conjuncts t));;
%============================================================================%
% Moving universal quantifiers in and out of conjunctions %
%============================================================================%
%----------------------------------------------------------------------------%
% CONJ_FORALL_ONCE_CONV : conv %
% %
% "(!x. t1) /\ ... /\ (!x. tn)" %
% ----> %
% |- (!x. t1) /\ ... /\ (!x. tn) = !x. t1 /\ ... /\ tn %
% %
% where the original term can be an arbitrary tree of /\s, not just linear. %
% The structure of the tree is retained in both sides of the equation. %
% %
% To avoid deriving incompatible theorems for IMP_ANTISYM_RULE when one or %
% more of the ti's is a conjunction, the original term is broken up as well %
% as the theorem. If this were not done, the conversion would fail in such %
% cases. %
%----------------------------------------------------------------------------%
let CONJ_FORALL_ONCE_CONV t =
letrec conj_tree_map f t th =
(let t1,t2 = dest_conj t
and th1,th2 = CONJ_PAIR th
in CONJ (conj_tree_map f t1 th1) (conj_tree_map f t2 th2)
) ? (f th)
in
(let conjs = conjuncts t
in
if (length conjs = 1) then REFL t
else
let var = case (setify (map (fst o dest_forall) conjs))
of [x] . x
| (_) . fail
in
let th = GEN var (conj_tree_map (SPEC var) t (ASSUME t))
in
let th1 = DISCH t th
and th2 = DISCH (concl th)
(conj_tree_map (GEN var) t (SPEC var (ASSUME (concl th))))
in IMP_ANTISYM_RULE th1 th2
) ? failwith `CONJ_FORALL_ONCE_CONV`;;
%----------------------------------------------------------------------------%
% FORALL_CONJ_ONCE_CONV : conv %
% %
% "!x. t1 /\ ... /\ tn" %
% ----> %
% |- !x. t1 /\ ... /\ tn = (!x. t1) /\ ... /\ (!x. tn) %
% %
% where the original term can be an arbitrary tree of /\s, not just linear. %
% The structure of the tree is retained in both sides of the equation. %
%----------------------------------------------------------------------------%
let FORALL_CONJ_ONCE_CONV t =
letrec conj_tree_map f th =
(let th1,th2 = CONJ_PAIR th
in CONJ (conj_tree_map f th1) (conj_tree_map f th2)
) ? (f th)
in
(let var = fst (dest_forall t)
in
let th = conj_tree_map (GEN var) (SPEC var (ASSUME t))
in
let th1 = DISCH t th
and th2 = DISCH (concl th)
(GEN var (conj_tree_map (SPEC var) (ASSUME (concl th))))
in IMP_ANTISYM_RULE th1 th2
) ? failwith `FORALL_CONJ_ONCE_CONV`;;
%----------------------------------------------------------------------------%
% CONJ_FORALL_CONV : conv %
% %
% "(!x1 ... xm. t1) /\ ... /\ (!x1 ... xm. tn)" %
% ----> %
% |- (!x1 ... xm. t1) /\ ... /\ (!x1 ... xm. tn) = %
% !x1 ... xm. t1 /\ ... /\ tn %
% %
% where the original term can be an arbitrary tree of /\s, not just linear. %
% The structure of the tree is retained in both sides of the equation. %
%----------------------------------------------------------------------------%
letrec CONJ_FORALL_CONV tm =
(if (length (conjuncts tm) = 1)
then fail
else (CONJ_FORALL_ONCE_CONV THENC (RAND_CONV (ABS_CONV CONJ_FORALL_CONV))) tm
) ? REFL tm;;
%----------------------------------------------------------------------------%
% FORALL_CONJ_CONV : conv %
% %
% "!x1 ... xm. t1 /\ ... /\ tn" %
% ----> %
% |- !x1 ... xm. t1 /\ ... /\ tn = %
% (!x1 ... xm. t1) /\ ... /\ (!x1 ... xm. tn) %
% %
% where the original term can be an arbitrary tree of /\s, not just linear. %
% The structure of the tree is retained in both sides of the equation. %
%----------------------------------------------------------------------------%
letrec FORALL_CONJ_CONV tm =
if (is_forall tm)
then (RAND_CONV (ABS_CONV FORALL_CONJ_CONV) THENC FORALL_CONJ_ONCE_CONV) tm
else REFL tm;;
%----------------------------------------------------------------------------%
% CONJ_FORALL_RIGHT_RULE : thm -> thm %
% %
% A |- !z1 ... zr. %
% t = ?y1 ... yp. (!x1 ... xm. t1) /\ ... /\ (!x1 ... xm. tn) %
% ------------------------------------------------------------------- %
% A |- !z1 ... zr. t = ?y1 ... yp. !x1 ... xm. t1 /\ ... /\ tn %
% %
%----------------------------------------------------------------------------%
let CONJ_FORALL_RIGHT_RULE th =
CONV_RULE
(DEPTH_FORALL_CONV (RAND_CONV (DEPTH_EXISTS_CONV CONJ_FORALL_CONV))) th
? failwith `CONJ_FORALL_RIGHT_RULE`;;
%----------------------------------------------------------------------------%
% FORALL_CONJ_RIGHT_RULE : thm -> thm %
% %
% A |- !z1 ... zr. t = ?y1 ... yp. !x1 ... xm. t1 /\ ... /\ tn %
% ------------------------------------------------------------------- %
% A |- !z1 ... zr. %
% t = ?y1 ... yp. (!x1 ... xm. t1) /\ ... /\ (!x1 ... xm. tn) %
% %
%----------------------------------------------------------------------------%
let FORALL_CONJ_RIGHT_RULE th =
CONV_RULE
(DEPTH_FORALL_CONV (RAND_CONV (DEPTH_EXISTS_CONV FORALL_CONJ_CONV))) th
? failwith `FORALL_CONJ_RIGHT_RULE`;;
%============================================================================%
% Rules for unfolding %
%============================================================================%
%----------------------------------------------------------------------------%
% UNFOLD_CONV : thm list -> conv %
% %
% UNFOLD_CONV thl %
% %
% "t1 /\ ... /\ tn" %
% ----> %
% B |- t1 /\ ... /\ tn = t1' /\ ... /\ tn' %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% ti, it is unchanged. %
%----------------------------------------------------------------------------%
let UNFOLD_CONV thl =
let net = mk_conv_net thl
in
let REWRITES_CONV net = \tm. FIRST_CONV (lookup_term net tm) tm
in
let THENQC conv1 conv2 tm =
(let th1 = conv1 tm
in ((th1 TRANS (conv2 (rhs (concl th1)))) ? th1))
? (conv2 tm)
in
letrec CONJ_TREE_CONV conv tm =
if (is_conj tm)
then THENQC (RATOR_CONV (RAND_CONV (CONJ_TREE_CONV conv)))
(RAND_CONV (CONJ_TREE_CONV conv))
tm
else conv tm
in
\t. if (null thl)
then REFL t
else CONJ_TREE_CONV (REWRITES_CONV net) t ? REFL t;;
%----------------------------------------------------------------------------%
% UNFOLD_RIGHT_RULE : thm list -> thm -> thm %
% %
% UNFOLD_RIGHT_RULE thl %
% %
% A |- !z1 ... zr. t = ?y1 ... yp. t1 /\ ... /\ tn %
% -------------------------------------------------------- %
% B u A |- !z1 ... zr. t = ?y1 ... yp. t1' /\ ... /\ tn' %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% ti, it is unchanged. %
%----------------------------------------------------------------------------%
let UNFOLD_RIGHT_RULE thl th =
CONV_RULE
(DEPTH_FORALL_CONV (RAND_CONV (DEPTH_EXISTS_CONV (UNFOLD_CONV thl))))
th
? failwith `UNFOLD_RIGHT_RULE`;;
%============================================================================%
% Rules for unwinding device implementations %
%============================================================================%
%----------------------------------------------------------------------------%
% line_var : term -> term %
% %
% line_var "!y1 ... ym. f x1 ... xn = t" ----> "f" %
%----------------------------------------------------------------------------%
let line_var tm =
(fst o strip_comb o lhs o snd o strip_forall) tm ? failwith `line_var`;;
%----------------------------------------------------------------------------%
% line_name : term -> string %
% %
% line_name "!y1 ... ym. f x1 ... xn = t" ----> `f` %
%----------------------------------------------------------------------------%
let line_name tm = (fst o dest_var o line_var) tm ? failwith `line_name`;;
%----------------------------------------------------------------------------%
% UNWIND_ONCE_CONV : (term -> bool) -> conv %
% %
% Basic conversion for parallel unwinding of equations defining wire values %
% in a standard device specification. %
% %
% USAGE: UNWIND_ONCE_CONV p tm %
% %
% DESCRIPTION: tm should be a conjunction of terms, equivalent under %
% associative-commutative reordering to: %
% %
% t1 /\ t2 /\ ... /\ tn. %
% %
% The function p:term->bool is a predicate which will be %
% used to partition the terms ti for 1 <= i <= n into two %
% disjoint sets: %
% %
% REW = {ti | p ti} and OBJ = {ti | ~p ti} %
% %
% The terms ti for which p is true are then used as a set %
% of rewrite rules (thus they should be equations) to do a %
% single top-down parallel rewrite of the remaining terms. %
% The rewritten terms take the place of the original terms %
% in the input conjunction. %
% %
% For example, if tm is: %
% %
% t1 /\ t2 /\ t3 /\ t4 %
% %
% and REW = {t1,t3} then the result is: %
% %
% |- t1 /\ t2 /\ t3 /\ t4 = t1 /\ t2' /\ t3 /\ t4' %
% %
% where ti' is ti rewritten with the equations REW. %
% %
% IMPLEMENTATION NOTE: %
% %
% The assignment: %
% %
% let pf,fn,eqns = trav p tm [] %
% %
% makes %
% %
% eqns = a list of theorems constructed by assuming each term for %
% which p is true, i.e., eqns = the list of rewrites. %
% %
% fn = a function which, when applied to a rewriting conversion %
% will reconstruct the original term in the original structure, %
% but with the subterms for which p is not true rewritten %
% using the supplied conversion. %
% %
% pf = a function which maps |- tm to [|- t1;...;|- tj] where each %
% ti is a term for which p is true. %
%----------------------------------------------------------------------------%
let UNWIND_ONCE_CONV =
let REWRITES_CONV net = \tm. FIRST_CONV (lookup_term net tm) tm
in
let AND = mk_const (`/\\`,":bool->bool->bool")
in
letrec trav p tm rl =
(let (l,r) = dest_conj tm
in
let (pf2,fn2,pf1,fn1,rew) = (I # (I # trav p l)) (trav p r rl)
in
let pf = $@ o (pf1 # pf2) o CONJ_PAIR
in (pf,(\rf. MK_COMB ((AP_TERM AND (fn1 rf)),(fn2 rf))),rew)
) ? if ((p tm) ? false)
then ((\th.[th]),(\rf. REFL tm),(ASSUME tm . rl))
else ((\th.[]),(\rf. rf tm),rl)
in
(\p tm.
let (pf,fn,eqns) = trav p tm []
in
let rconv = ONCE_DEPTH_CONV (REWRITES_CONV (mk_conv_net eqns))
in
let th = fn rconv
in
let l,r = (dest_eq (concl th))
in
let lth = ASSUME l
and rth = ASSUME r
in
let imp1 = DISCH l (EQ_MP (itlist PROVE_HYP (pf lth) th) lth)
and imp2 = DISCH r (EQ_MP (SYM (itlist PROVE_HYP (pf rth) th)) rth)
in IMP_ANTISYM_RULE imp1 imp2
) ? failwith `UNWIND_ONCE_CONV`;;
%----------------------------------------------------------------------------%
% UNWIND_CONV : (term -> bool) -> conv %
% %
% Unwind device behaviour using line equations eqnx selected by p until no %
% change. %
% %
% WARNING -- MAY LOOP! %
% %
% UNWIND_CONV p %
% %
% "t1 /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn" %
% ----> %
% |- t1 /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn = %
% t1' /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn' %
% %
% where ti' (for 1 <= i <= n) is ti rewritten with the equations %
% eqni (1 <= i <= m). These equations are the conjuncts for which the %
% predicate p is true. The ti terms are the conjuncts for which p is false. %
% The rewriting is repeated until no changes take place. %
%----------------------------------------------------------------------------%
letrec UNWIND_CONV p tm =
let th = UNWIND_ONCE_CONV p tm
in if lhs (concl th) = rhs (concl th)
then th
else th TRANS (UNWIND_CONV p (rhs (concl th)));;
%----------------------------------------------------------------------------%
% UNWIND_ALL_BUT_CONV : string list -> conv %
% %
% Unwind all lines (except those in the list l) as much as possible. %
% %
% WARNING -- MAY LOOP! %
% %
% UNWIND_ALL_BUT_CONV l %
% %
% "t1 /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn" %
% ----> %
% |- t1 /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn = %
% t1' /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn' %
% %
% where ti' (for 1 <= i <= n) is ti rewritten with the equations %
% eqni (1 <= i <= m). These equations are those conjuncts with line name not %
% in l (and which are equations). %
%----------------------------------------------------------------------------%
let UNWIND_ALL_BUT_CONV l tm =
(let line_names = subtract (mapfilter line_name (conjuncts tm)) l
in
let p line = \tm. (line_name tm) = line
in
let itfn line = \th. th TRANS (UNWIND_CONV (p line) (rhs (concl th)))
in
itlist itfn line_names (REFL tm)
) ? failwith `UNWIND_ALL_BUT_CONV`;;
%----------------------------------------------------------------------------%
% UNWIND_AUTO_CONV : conv %
% %
% "?l1 ... lm. t1 /\ ... /\ tn" %
% ----> %
% |- (?l1 ... lm. t1 /\ ... /\ tn) = (?l1 ... lm. t1' /\ ... /\ tn') %
% %
% where tj' is tj rewritten with equations selected from the ti's. The %
% function decides which equations to use by performing a loop analysis on %
% the graph representing the dependencies of the lines. By this means the %
% term can be unwound as much as possible without the risk of looping. The %
% user is left to deal with the recursive equations. %
% %
% There is some inefficiency in that certain lines may be used in unwinding %
% even though they do not appear in any RHS's. %
% %
% The algorithms used for loop analysis in this function have exponential %
% complexity in the number of lines. However, steps are taken to limit this %
% as much as possible. The internal function `next_combination' computes %
% combinations of a list, but only as they are required. This puts the %
% burden on time resources rather than space resources. The computation time %
% would be less if a complete list of combinations were computed in one go, %
% but the list generated might exhaust memory. The first argument to %
% `next_combination' is the list to find the combinations of. The second %
% argument is the reverse of the previous combination. Initially this should %
% be the empty list. The result of a call is the reverse of the next %
% combination. The function assumes that the source list is a set. %
% %
% Amongst other things, the internal function `graph_of_term' rearranges the %
% lines in the graph representation so that external lines come first. It is %
% important that the number of internal lines left unwound because of loops %
% is minimised since it is the internal lines that are existentially %
% quantified. Further manipulations by the user should be easier if any %
% loops remaining do not involve existentially quantified variables. %
% %
% The algorithm for breaking loops is: %
% %
% 1. Isolate any loops of length one. Each such loop involves only one line %
% so must be broken at that line. %
% %
% 2. Eliminate from consideration the single element loops and any other %
% loop that will be broken by the elements in those loops. %
% %
% 3. Determine those loops that consist entirely of internal lines. All %
% other loops can be broken at an external line. A minimal set of %
% internal lines that break all the exclusively internal loops is then %
% computed. This may not be the only minimal set. %
% %
% 4. Any loop broken by the minimal set of internal lines is eliminated from %
% consideration. A list of external lines appearing in these remaining %
% loops is computed. From these external lines a minimal set that breaks %
% all the remaining loops is computed. This set will only be minimal %
% relative to the choice of minimal set of internals. A different choice %
% for the latter might have produced a smaller set of external lines. %
% %
% 5. The three lists of lines computed in 1-4 are concatenated and returned. %
%----------------------------------------------------------------------------%
let UNWIND_AUTO_CONV =
let graph_of_term tm =
(let internals,t = strip_exists tm
in let (lines,rhs_frees) =
split (mapfilter ((((assert is_var) o fst o strip_comb) # frees) o
dest_eq o snd o strip_forall) (conjuncts t))
in
if (distinct lines)
then let graph = combine (lines,map (intersect lines) rhs_frees)
in let (intern,extern) = partition (\p. mem (fst p) internals) graph
in extern @ intern
else fail)
in
letrec loops_containing_line line graph chain =
(let successors =
map fst (filter (\(_,predecs). mem (hd chain) predecs) graph)
in let not_in_chain = filter (\line. not (mem line chain)) successors
in let new_chains = map (\line. line.chain) not_in_chain
in let new_loops = flat (map (loops_containing_line line graph) new_chains)
in if (mem line successors)
then (rev chain).new_loops
else new_loops)
in
letrec chop_at x l =
(if (hd l = x)
then ([],l)
else let (l1,l2) = chop_at x (tl l)
in ((hd l).l1,l2))
in
let equal_loops lp1 lp2 =
(if (set_equal lp1 lp2)
then let (before,rest) = chop_at (hd lp1) lp2
in lp1 = (rest @ before)
else false)
in
letrec distinct_loops lps =
(if (null lps)
then []
else let (h.t) = lps
in if (exists (\lp. equal_loops lp h) t)
then distinct_loops t
else h.(distinct_loops t))
in
let loops_of_graph graph =
(distinct_loops
(flat
(map (\line. loops_containing_line line graph [line]) (map fst graph))))
in
letrec list_after x l =
(if (x = hd l)
then tl l
else list_after x (tl l))
in
letrec rev_front_of l n front =
(if (n < 0) then fail
if (n = 0) then front
else rev_front_of (tl l) (n - 1) ((hd l).front))
in
letrec next_comb_at_this_level source n (h.t) =
(let l = list_after h source
in if (length l > n)
then (rev_front_of l (n + 1) []) @ t
else next_comb_at_this_level source (n + 1) t)
in
let next_combination source previous =
((next_comb_at_this_level source 0 previous) ?
(rev_front_of source ((length previous) + 1) []))
in
letrec break_all_loops lps lines previous =
(let comb = next_combination lines previous
in if (forall (\lp. not (null (intersect lp comb))) lps)
then comb
else break_all_loops lps lines comb)
in
let breaks internals graph =
(let loops = loops_of_graph graph
in let single_el_loops = filter (\l. length l = 1) loops
in let single_breaks = flat single_el_loops
in let loops' = filter (null o (intersect single_breaks)) loops
in let only_internal_loops =
filter (\l. null (subtract l internals)) loops'
in let only_internal_lines = end_itlist union only_internal_loops ? []
in let internal_breaks =
break_all_loops only_internal_loops only_internal_lines [] ? []
in let external_loops = filter (null o (intersect internal_breaks)) loops'
in let external_lines =
subtract (end_itlist union external_loops ? []) internals
in let external_breaks =
break_all_loops external_loops external_lines [] ? []
in single_breaks @ (rev internal_breaks) @ (rev external_breaks))
in
letrec conv dependencies used t =
(let vars =
map fst (filter ((\l. null (subtract l used)) o snd) dependencies)
in if (null vars)
then REFL t
else ((UNWIND_ONCE_CONV (\tm. mem (line_var tm) vars)) THENC
(conv (filter (\p. not (mem (fst p) vars)) dependencies)
(used @ vars))) t)
in
\tm.
(let internals = fst (strip_exists tm)
and graph = graph_of_term tm
in
let brks = breaks internals graph
in
let dependencies =
map (I # (\l. subtract l brks)) (filter (\p. not (mem (fst p) brks)) graph)
in
DEPTH_EXISTS_CONV (conv dependencies []) tm
) ? failwith `UNWIND_AUTO_CONV`;;
%----------------------------------------------------------------------------%
% UNWIND_ALL_BUT_RIGHT_RULE : string list -> thm -> thm %
% %
% Unwind all lines (except those in the list l) as much as possible. %
% %
% WARNING -- MAY LOOP! %
% %
% UNWIND_ALL_BUT_RIGHT_RULE l %
% %
% A |- !z1 ... zr. %
% t = %
% (?l1 ... lp. t1 /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn) %
% --------------------------------------------------------------------- %
% A |- !z1 ... zr. %
% t = %
% (?l1 ... lp. t1' /\ ... /\ eqn1 /\ ... /\ eqnm /\ ... /\ tn') %
% %
% where ti' (for 1 <= i <= n) is ti rewritten with the equations %
% eqni (1 <= i <= m). These equations are those conjuncts with line name not %
% in l (and which are equations). %
%----------------------------------------------------------------------------%
let UNWIND_ALL_BUT_RIGHT_RULE l th =
CONV_RULE
(DEPTH_FORALL_CONV (RAND_CONV (DEPTH_EXISTS_CONV (UNWIND_ALL_BUT_CONV l))))
th
? failwith `UNWIND_ALL_BUT_RIGHT_RULE`;;
%----------------------------------------------------------------------------%
% UNWIND_AUTO_RIGHT_RULE : thm -> thm %
% %
% A |- !z1 ... zr. t = ?l1 ... lm. t1 /\ ... /\ tn %
% ---------------------------------------------------- %
% A |- !z1 ... zr. t = ?l1 ... lm. t1' /\ ... /\ tn' %
% %
% where tj' is tj rewritten with equations selected from the ti's. The %
% function decides which equations to use by performing a loop analysis on %
% the graph representing the dependencies of the lines. By this means the %
% equations can be unwound as much as possible without the risk of looping. %
% The user is left to deal with the recursive equations. %
%----------------------------------------------------------------------------%
let UNWIND_AUTO_RIGHT_RULE th =
CONV_RULE (DEPTH_FORALL_CONV (RAND_CONV UNWIND_AUTO_CONV)) th
? failwith `UNWIND_AUTO_RIGHT_RULE`;;
%============================================================================%
% Rules for pruning %
%============================================================================%
%----------------------------------------------------------------------------%
% EXISTS_DEL1_CONV : conv %
% %
% "?x. t" %
% ----> %
% |- (?x. t) = t %
% %
% provided x is not free in t. %
% %
% Deletes one existential quantifier. %
%----------------------------------------------------------------------------%
let EXISTS_DEL1_CONV tm =
(let x,t = dest_exists tm
in
let th = ASSUME t
in
let th1 = DISCH tm (CHOOSE (x, ASSUME tm) th)
and th2 = DISCH t (EXISTS (tm,x) th)
in
IMP_ANTISYM_RULE th1 th2
) ? failwith `EXISTS_DEL1_CONV`;;
%----------------------------------------------------------------------------%
% EXISTS_DEL_CONV : conv %
% %
% "?x1 ... xn. t" %
% ----> %
% |- (?x1 ... xn. t) = t %
% %
% provided x1,...,xn are not free in t. %
% %
% Deletes existential quantifiers. The conversion fails if any of the x's %
% appear free in t. It does not perform a partial deletion; for example, if %
% x1 and x2 do not appear free in t but x3 does, the function will fail; it %
% will not return |- ?x1 ... xn. t = ?x3 ... xn. t. %
%----------------------------------------------------------------------------%
let EXISTS_DEL_CONV tm =
letrec terms_and_vars tm =
(let x,tm' = dest_exists tm
in (tm,x).(terms_and_vars tm')
) ? []
in
(let txs = terms_and_vars tm
in
let t = snd (dest_exists (fst (last txs))) ? tm
in
let th = ASSUME t
in
let th1 = DISCH tm (itlist (\(tm,x). CHOOSE (x, ASSUME tm)) txs th)
and th2 = DISCH t (itlist EXISTS txs th)
in
IMP_ANTISYM_RULE th1 th2
) ? failwith `EXISTS_DEL_CONV`;;
%----------------------------------------------------------------------------%
% EXISTS_EQN_CONV : conv %
% %
% "?l. !y1 ... ym. l x1 ... xn = t" %
% ----> %
% |- (?l. !y1 ... ym. l x1 ... xn = t) = T %
% %
% provided l is not free in t. Both m and n may be zero. %
%----------------------------------------------------------------------------%
let EXISTS_EQN_CONV t =
(let l,ys,t1,t2 = (I # ((I # dest_eq) o strip_forall)) (dest_exists t)
in
let xs = snd ((assert (curry $= l) # I) (strip_comb t1))
in
let t3 = list_mk_abs (xs,t2)
in
let th1 =
GENL ys (RIGHT_CONV_RULE LIST_BETA_CONV (REFL (list_mk_comb (t3,xs))))
in
EQT_INTRO (EXISTS (t,t3) th1)
) ? failwith `EXISTS_EQN_CONV`;;
%----------------------------------------------------------------------------%
% PRUNE_ONCE_CONV : conv %
% %
% Prunes one hidden variable. %
% %
% "?l. t1 /\ ... /\ ti /\ eq /\ t(i+1) /\ ... /\ tp" %
% ----> %
% |- (?l. t1 /\ ... /\ ti /\ eq /\ t(i+1) /\ ... /\ tp) = %
% (t1 /\ ... /\ ti /\ t(i+1) /\ ... /\ tp) %
% %
% where eq has the form "!y1 ... ym. l x1 ... xn = b" and l does not appear %
% free in the ti's or in b. The conversion works if eq is not present, %
% i.e. if l is not free in any of the conjuncts, but does not work if l %
% appears free in more than one of the conjuncts. Each of m, n and p may be %
% zero. %
%----------------------------------------------------------------------------%
let PRUNE_ONCE_CONV tm =
(let x,t = dest_exists tm
in
let l1,l2 = partition (free_in x) (conjuncts t)
in
if (null l1) then EXISTS_DEL1_CONV tm
else
let [eq] = l1
in
let th1 = EXISTS_EQN_CONV (mk_exists (x,eq))
in
if (null l2) then th1
else
let conj = list_mk_conj l2
in
let th2 = AP_THM (AP_TERM "$/\" th1) conj
in
let th3 = EXISTS_EQ x (CONJUNCTS_CONV (t,mk_conj(eq,conj)))
in
let th4 = RIGHT_CONV_RULE EXISTS_AND_CONV th3
in
th4 TRANS th2 TRANS (CONJUNCT1 (SPEC conj AND_CLAUSES))
) ? failwith `PRUNE_ONCE_CONV`;;
%----------------------------------------------------------------------------%
% PRUNE_ONE_CONV : string -> conv %
% %
% Prunes one hidden variable. %
% %
% PRUNE_ONE_CONV `lj` %
% %
% "?l1 ... lj ... lr. t1 /\ ... /\ ti /\ eq /\ t(i+1) /\ ... /\ tp" %
% ----> %
% |- (?l1 ... lj ... lr. t1 /\ ... /\ ti /\ eq /\ t(i+1) /\ ... /\ tp) = %
% (?l1 ... l(j-1) l(j+1) ... lr. %
% t1 /\ ... /\ ti /\ t(i+1) /\ ... /\ tp) %
% %
% where eq has the form "!y1 ... ym. lj x1 ... xn = b" and lj does not %
% appear free in the ti's or in b. The conversion works if eq is not %
% present, i.e. if lj is not free in any of the conjuncts, but does not work %
% if lj appears free in more than one of the conjuncts. Each of m, n and p %
% may be zero. %
% %
% If there is more than one line with the specified name (but with different %
% types), the one that appears outermost in the existential quantifications %
% is pruned. %
%----------------------------------------------------------------------------%
letrec PRUNE_ONE_CONV v tm =
(let x,tm' = dest_exists tm
in if (fst (dest_var x) = v)
then if (is_exists tm')
then (SWAP_EXISTS_CONV THENC
(RAND_CONV (ABS_CONV (PRUNE_ONE_CONV v)))) tm
else PRUNE_ONCE_CONV tm
else RAND_CONV (ABS_CONV (PRUNE_ONE_CONV v)) tm
) ? failwith `PRUNE_ONE_CONV`;;
%----------------------------------------------------------------------------%
% PRUNE_SOME_CONV : string list -> conv %
% %
% Prunes several hidden variables. %
% %
% PRUNE_SOME_CONV [`li1`;...;`lik`] %
% %
% "?l1 ... lr. t1 /\ ... /\ eqni1 /\ ... /\ eqnik /\ ... /\ tp" %
% ----> %
% |- (?l1 ... lr. t1 /\ ... /\ eqni1 /\ ... /\ eqnik /\ ... /\ tp) = %
% (?li(k+1) ... lir. t1 /\ ... /\ tp) %
% %
% where for 1 <= j <= k, each eqnij has the form: %
% %
% "!y1 ... ym. lij x1 ... xn = b" %
% %
% and lij does not appear free in any of the other conjuncts or in b. %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lir} = {l1,...,lr} %
% %
% The conversion works if one or more of the eqnij's are not present, %
% i.e. if lij is not free in any of the conjuncts, but does not work if lij %
% appears free in more than one of the conjuncts. p may be zero, that is, %
% all the conjuncts may be eqnij's. In this case the body of the result will %
% be T (true). Also, for each eqnij, m and n may be zero. %
% %
% If there is more than one line with a specified name (but with different %
% types), the one that appears outermost in the existential quantifications %
% is pruned. If such a line name is mentioned twice in the list, the two %
% outermost occurrences of lines with that name will be pruned, and so on. %
%----------------------------------------------------------------------------%
letrec PRUNE_SOME_CONV vs tm =
(if (null vs)
then REFL tm
else (PRUNE_SOME_CONV (tl vs) THENC PRUNE_ONE_CONV (hd vs)) tm
) ? failwith `PRUNE_SOME_CONV`;;
%----------------------------------------------------------------------------%
% PRUNE_CONV : conv %
% %
% Prunes all hidden variables. %
% %
% "?l1 ... lr. t1 /\ ... /\ eqn1 /\ ... /\ eqnr /\ ... /\ tp" %
% ----> %
% |- (?l1 ... lr. t1 /\ ... /\ eqn1 /\ ... /\ eqnr /\ ... /\ tp) = %
% (t1 /\ ... /\ tp) %
% %
% where each eqni has the form "!y1 ... ym. li x1 ... xn = b" and li does %
% not appear free in any of the other conjuncts or in b. The conversion %
% works if one or more of the eqni's are not present, i.e. if li is not free %
% in any of the conjuncts, but does not work if li appears free in more than %
% one of the conjuncts. p may be zero, that is, all the conjuncts may be %
% eqni's. In this case the result will be simply T (true). Also, for each %
% eqni, m and n may be zero. %
%----------------------------------------------------------------------------%
letrec PRUNE_CONV tm =
(if (is_exists tm)
then (RAND_CONV (ABS_CONV PRUNE_CONV) THENC PRUNE_ONCE_CONV) tm
else REFL tm
) ? failwith `PRUNE_CONV`;;
%----------------------------------------------------------------------------%
% PRUNE_SOME_RIGHT_RULE : string list -> thm -> thm %
% %
% Prunes several hidden variables. %
% %
% PRUNE_SOME_RIGHT_RULE [`li1`;...;`lik`] %
% %
% A |- !z1 ... zr. %
% t = ?l1 ... lr. t1 /\ ... /\ eqni1 /\ ... /\ eqnik /\ ... /\ tp %
% ----------------------------------------------------------------------- %
% A |- !z1 ... zr. t = ?li(k+1) ... lir. t1 /\ ... /\ tp %
% %
% where for 1 <= j <= k, each eqnij has the form: %
% %
% "!y1 ... ym. lij x1 ... xn = b" %
% %
% and lij does not appear free in any of the other conjuncts or in b. %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lir} = {l1,...,lr} %
% %
% The rule works if one or more of the eqnij's are not present, i.e. if lij %
% is not free in any of the conjuncts, but does not work if lij appears free %
% in more than one of the conjuncts. p may be zero, that is, all the %
% conjuncts may be eqnij's. In this case the conjunction will be transformed %
% to T (true). Also, for each eqnij, m and n may be zero. %
% %
% If there is more than one line with a specified name (but with different %
% types), the one that appears outermost in the existential quantifications %
% is pruned. If such a line name is mentioned twice in the list, the two %
% outermost occurrences of lines with that name will be pruned, and so on. %
%----------------------------------------------------------------------------%
let PRUNE_SOME_RIGHT_RULE vs th =
CONV_RULE (DEPTH_FORALL_CONV (RAND_CONV (PRUNE_SOME_CONV vs))) th
? failwith `PRUNE_SOME_RIGHT_RULE`;;
%----------------------------------------------------------------------------%
% PRUNE_RIGHT_RULE : thm -> thm %
% %
% Prunes all hidden variables. %
% %
% A |- !z1 ... zr. %
% t = ?l1 ... lr. t1 /\ ... /\ eqn1 /\ ... /\ eqnr /\ ... /\ tp %
% --------------------------------------------------------------------- %
% A |- !z1 ... zr. t = t1 /\ ... /\ tp %
% %
% where each eqni has the form "!y1 ... ym. li x1 ... xn = b" and li does %
% not appear free in any of the other conjuncts or in b. The rule works if %
% one or more of the eqni's are not present, i.e. if li is not free in any %
% of the conjuncts, but does not work if li appears free in more than one of %
% the conjuncts. p may be zero, that is, all the conjuncts may be eqni's. In %
% this case the result will be simply T (true). Also, for each eqni, m and n %
% may be zero. %
%----------------------------------------------------------------------------%
let PRUNE_RIGHT_RULE th =
CONV_RULE (DEPTH_FORALL_CONV (RAND_CONV PRUNE_CONV)) th
? failwith `PRUNE_RIGHT_RULE`;;
%============================================================================%
% Functions which do unfolding, unwinding and pruning %
%============================================================================%
%----------------------------------------------------------------------------%
% EXPAND_ALL_BUT_CONV : string list -> thm list -> conv %
% %
% Unfold with the theorems thl, then unwind all lines (except those in the %
% list) as much as possible and prune the unwound lines. %
% %
% WARNING -- MAY LOOP! %
% %
% EXPAND_ALL_BUT_CONV [`li(k+1)`;...;`lim`] thl %
% %
% "?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn" %
% ----> %
% B |- (?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn) = %
% (?li(k+1) ... lim. t1' /\ ... /\ tn') %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% conjunct, it is unchanged. Those conjuncts that after rewriting are %
% equations for the lines li1,...,lik (they are denoted by ui1,...,uik) are %
% used to unwind and the lines li1,...,lik are then pruned. If this is not %
% possible the function will fail. It is also possible for the function to %
% attempt unwinding indefinitely (to loop). %
% %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lim} = {l1,...,lm} %
%----------------------------------------------------------------------------%
let EXPAND_ALL_BUT_CONV l thl tm =
(DEPTH_EXISTS_CONV ((UNFOLD_CONV thl) THENC (UNWIND_ALL_BUT_CONV l)) THENC
(\tm. let var_names = map (fst o dest_var) (fst (strip_exists tm))
in PRUNE_SOME_CONV (subtract var_names l) tm))
tm
? failwith `EXPAND_ALL_BUT_CONV`;;
%----------------------------------------------------------------------------%
% EXPAND_AUTO_CONV : thm list -> conv %
% %
% Unfold with the theorems thl, then unwind as much as possible and prune %
% the unwound lines. %
% %
% EXPAND_AUTO_CONV thl %
% %
% "?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn" %
% ----> %
% B |- (?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn) = %
% (?li(k+1) ... lim. t1' /\ ... /\ tn') %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% conjunct, it is unchanged. After rewriting the function decides which of %
% the resulting terms to use for unwinding by performing a loop analysis on %
% the graph representing the dependencies of the lines. %
% %
% Suppose the function decides to unwind the lines li1,...,lik using the %
% terms ui1',...,uik' respectively. Then after unwinding the lines %
% li1,...,lik are pruned (provided they have been eliminated from the RHS's %
% of the conjuncts that are equations, and from the whole of any other %
% conjuncts) resulting in the elimination of ui1',...,uik'. %
% %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lim} = {l1,...,lm} %
% %
% The loop analysis allows the term to be unwound as much as possible %
% without the risk of looping. The user is left to deal with the recursive %
% equations. %
%----------------------------------------------------------------------------%
let EXPAND_AUTO_CONV thl tm =
(DEPTH_EXISTS_CONV (UNFOLD_CONV thl) THENC
UNWIND_AUTO_CONV THENC
(\tm. let internals,conjs = (I # conjuncts) (strip_exists tm)
in
let vars =
flat (map (frees o (\tm. rhs tm ? tm) o snd o strip_forall) conjs)
in
PRUNE_SOME_CONV (map (fst o dest_var) (subtract internals vars)) tm))
tm
? failwith `EXPAND_AUTO_CONV`;;
%----------------------------------------------------------------------------%
% EXPAND_ALL_BUT_RIGHT_RULE : string list -> thm list -> thm -> thm %
% %
% Unfold with the theorems thl, then unwind all lines (except those in the %
% list) as much as possible and prune the unwound lines. %
% %
% WARNING -- MAY LOOP! %
% %
% EXPAND_ALL_BUT_RIGHT_RULE [`li(k+1)`;...;`lim`] thl %
% %
% A |- !z1 ... zr. %
% t = ?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn %
% ----------------------------------------------------------------------- %
% B u A |- !z1 ... zr. t = ?li(k+1) ... lim. t1' /\ ... /\ tn' %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% conjunct, it is unchanged. Those conjuncts that after rewriting are %
% equations for the lines li1,...,lik (they are denoted by ui1,...,uik) are %
% used to unwind and the lines li1,...,lik are then pruned. If this is not %
% possible the function will fail. It is also possible for the function to %
% attempt unwinding indefinitely (to loop). %
% %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lim} = {l1,...,lm} %
%----------------------------------------------------------------------------%
let EXPAND_ALL_BUT_RIGHT_RULE l thl th =
CONV_RULE (DEPTH_FORALL_CONV (RAND_CONV (EXPAND_ALL_BUT_CONV l thl))) th
? failwith `EXPAND_ALL_BUT_RIGHT_RULE`;;
%----------------------------------------------------------------------------%
% EXPAND_AUTO_RIGHT_RULE : thm list -> thm -> thm %
% %
% Unfold with the theorems thl, then unwind as much as possible and prune %
% the unwound lines. %
% %
% EXPAND_AUTO_RIGHT_RULE thl %
% %
% A |- !z1 ... zr. %
% t = ?l1 ... lm. t1 /\ ... /\ ui1 /\ ... /\ uik /\ ... /\ tn %
% ----------------------------------------------------------------------- %
% B u A |- !z1 ... zr. t = ?li(k+1) ... lim. t1' /\ ... /\ tn' %
% %
% where each ti' is the result of rewriting ti with the theorems in thl. The %
% set of assumptions B is the union of the instantiated assumptions of the %
% theorems used for rewriting. If none of the rewrites are applicable to a %
% conjunct, it is unchanged. After rewriting the function decides which of %
% the resulting terms to use for unwinding by performing a loop analysis on %
% the graph representing the dependencies of the lines. %
% %
% Suppose the function decides to unwind the lines li1,...,lik using the %
% terms ui1',...,uik' respectively. Then after unwinding the lines %
% li1,...,lik are pruned (provided they have been eliminated from the RHS's %
% of the conjuncts that are equations, and from the whole of any other %
% conjuncts) resulting in the elimination of ui1',...,uik'. %
% %
% The li's are related by the equation: %
% %
% {li1,...,lik} u {li(k+1),...,lim} = {l1,...,lm} %
% %
% The loop analysis allows the term to be unwound as much as possible %
% without the risk of looping. The user is left to deal with the recursive %
% equations. %
%----------------------------------------------------------------------------%
let EXPAND_AUTO_RIGHT_RULE thl th =
CONV_RULE (DEPTH_FORALL_CONV (RAND_CONV (EXPAND_AUTO_CONV thl))) th
? failwith `EXPAND_AUTO_RIGHT_RULE`;;
|