--- a/src/HOL/IsaMakefile Tue Sep 05 10:14:36 2000 +0200
+++ b/src/HOL/IsaMakefile Tue Sep 05 10:15:23 2000 +0200
@@ -65,7 +65,7 @@
SVC_Oracle.ML SVC_Oracle.thy Sum.ML Sum.thy Tools/datatype_aux.ML \
Tools/datatype_abs_proofs.ML Tools/datatype_package.ML Tools/datatype_prop.ML \
Tools/datatype_rep_proofs.ML Tools/induct_method.ML \
- Tools/inductive_package.ML Tools/numeral_syntax.ML \
+ Tools/inductive_package.ML Tools/meson.ML Tools/numeral_syntax.ML \
Tools/primrec_package.ML Tools/recdef_package.ML \
Tools/record_package.ML Tools/svc_funcs.ML Tools/typedef_package.ML \
Trancl.ML Trancl.thy Univ.ML Univ.thy Vimage.ML Vimage.thy WF.ML \
@@ -436,7 +436,7 @@
ex/Factorization.ML ex/Factorization.thy \
ex/Primrec.ML ex/Primrec.thy \
ex/Puzzle.ML ex/Puzzle.thy ex/Qsort.ML ex/Qsort.thy \
- ex/ROOT.ML ex/Recdefs.ML ex/Recdefs.thy ex/cla.ML ex/meson.ML \
+ ex/ROOT.ML ex/Recdefs.ML ex/Recdefs.thy ex/cla.ML \
ex/mesontest.ML ex/mesontest2.ML ex/set.thy ex/set.ML \
ex/Group.ML ex/Group.thy ex/IntRing.ML ex/IntRing.thy \
ex/Lagrange.ML ex/Lagrange.thy ex/Ring.ML ex/Ring.thy ex/StringEx.ML \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/meson.ML Tue Sep 05 10:15:23 2000 +0200
@@ -0,0 +1,467 @@
+(* Title: HOL/ex/meson
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+The MESON resolution proof procedure for HOL
+
+When making clauses, avoids using the rewriter -- instead uses RS recursively
+
+NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR
+FUNCTION nodups -- if done to goal clauses too!
+*)
+
+
+(**** LEMMAS : outside the "local" block ****)
+
+(** "Axiom" of Choice, proved using the description operator **)
+
+Goal "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)";
+by (fast_tac (claset() addEs [selectI]) 1);
+qed "choice";
+
+(*** Generation of contrapositives ***)
+
+(*Inserts negated disjunct after removing the negation; P is a literal*)
+val [major,minor] = Goal "~P|Q ==> ((~P==>P) ==> Q)";
+by (rtac (major RS disjE) 1);
+by (rtac notE 1);
+by (etac minor 2);
+by (ALLGOALS assume_tac);
+qed "make_neg_rule";
+
+(*For Plaisted's "Postive refinement" of the MESON procedure*)
+Goal "~P|Q ==> (P ==> Q)";
+by (Blast_tac 1);
+qed "make_refined_neg_rule";
+
+(*P should be a literal*)
+val [major,minor] = Goal "P|Q ==> ((P==>~P) ==> Q)";
+by (rtac (major RS disjE) 1);
+by (rtac notE 1);
+by (etac minor 1);
+by (ALLGOALS assume_tac);
+qed "make_pos_rule";
+
+(*** Generation of a goal clause -- put away the final literal ***)
+
+val [major,minor] = Goal "~P ==> ((~P==>P) ==> False)";
+by (rtac notE 1);
+by (rtac minor 2);
+by (ALLGOALS (rtac major));
+qed "make_neg_goal";
+
+val [major,minor] = Goal "P ==> ((P==>~P) ==> False)";
+by (rtac notE 1);
+by (rtac minor 1);
+by (ALLGOALS (rtac major));
+qed "make_pos_goal";
+
+
+(**** Lemmas for forward proof (like congruence rules) ****)
+
+(*NOTE: could handle conjunctions (faster?) by
+ nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
+val major::prems = Goal
+ "[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q";
+by (rtac (major RS conjE) 1);
+by (rtac conjI 1);
+by (ALLGOALS (eresolve_tac prems));
+qed "conj_forward";
+
+val major::prems = Goal
+ "[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q";
+by (rtac (major RS disjE) 1);
+by (ALLGOALS (dresolve_tac prems));
+by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
+qed "disj_forward";
+
+(*Version for removal of duplicate literals*)
+val major::prems = Goal
+ "[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q";
+by (cut_facts_tac [major] 1);
+by (blast_tac (claset() addIs prems) 1);
+qed "disj_forward2";
+
+val major::prems = Goal
+ "[| ALL x. P'(x); !!x. P'(x) ==> P(x) |] ==> ALL x. P(x)";
+by (rtac allI 1);
+by (resolve_tac prems 1);
+by (rtac (major RS spec) 1);
+qed "all_forward";
+
+val major::prems = Goal
+ "[| EX x. P'(x); !!x. P'(x) ==> P(x) |] ==> EX x. P(x)";
+by (rtac (major RS exE) 1);
+by (rtac exI 1);
+by (eresolve_tac prems 1);
+qed "ex_forward";
+
+(**** END OF LEMMAS ****)
+
+local
+
+ (*Prove theorems using fast_tac*)
+ fun prove_fun s =
+ prove_goal (the_context ()) s
+ (fn prems => [ cut_facts_tac prems 1, Fast_tac 1 ]);
+
+ (**** Negation Normal Form ****)
+
+ (*** de Morgan laws ***)
+
+ val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
+ val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
+ val not_notD = prove_fun "~~P ==> P";
+ val not_allD = prove_fun "~(ALL x. P(x)) ==> EX x. ~P(x)";
+ val not_exD = prove_fun "~(EX x. P(x)) ==> ALL x. ~P(x)";
+
+
+ (*** Removal of --> and <-> (positive and negative occurrences) ***)
+
+ val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
+ val not_impD = prove_fun "~(P-->Q) ==> P & ~Q";
+
+ val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
+
+ (*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
+ val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
+
+
+ (**** Pulling out the existential quantifiers ****)
+
+ (*** Conjunction ***)
+
+ val conj_exD1 = prove_fun "(EX x. P(x)) & Q ==> EX x. P(x) & Q";
+ val conj_exD2 = prove_fun "P & (EX x. Q(x)) ==> EX x. P & Q(x)";
+
+ (*** Disjunction ***)
+
+ (*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
+ With ex-Skolemization, makes fewer Skolem constants*)
+ val disj_exD = prove_fun "(EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)";
+
+ val disj_exD1 = prove_fun "(EX x. P(x)) | Q ==> EX x. P(x) | Q";
+ val disj_exD2 = prove_fun "P | (EX x. Q(x)) ==> EX x. P | Q(x)";
+
+
+
+ (***** Generating clauses for the Meson Proof Procedure *****)
+
+ (*** Disjunctions ***)
+
+ val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
+
+ val disj_comm = prove_fun "P|Q ==> Q|P";
+
+ val disj_FalseD1 = prove_fun "False|P ==> P";
+ val disj_FalseD2 = prove_fun "P|False ==> P";
+
+
+ (**** Operators for forward proof ****)
+
+ (*raises exception if no rules apply -- unlike RL*)
+ fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
+ | tryres (th, []) = raise THM("tryres", 0, [th]);
+
+ val prop_of = #prop o rep_thm;
+
+ (*Permits forward proof from rules that discharge assumptions*)
+ fun forward_res nf st =
+ case Seq.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
+ of Some(th,_) => th
+ | None => raise THM("forward_res", 0, [st]);
+
+
+ (*Are any of the constants in "bs" present in the term?*)
+ fun has_consts bs =
+ let fun has (Const(a,_)) = a mem bs
+ | has (f$u) = has f orelse has u
+ | has (Abs(_,_,t)) = has t
+ | has _ = false
+ in has end;
+
+
+ (**** Clause handling ****)
+
+ fun literals (Const("Trueprop",_) $ P) = literals P
+ | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
+ | literals (Const("Not",_) $ P) = [(false,P)]
+ | literals P = [(true,P)];
+
+ (*number of literals in a term*)
+ val nliterals = length o literals;
+
+ (*to detect, and remove, tautologous clauses*)
+ fun taut_lits [] = false
+ | taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
+
+ (*Include False as a literal: an occurrence of ~False is a tautology*)
+ fun is_taut th = taut_lits ((true, HOLogic.false_const) ::
+ literals (prop_of th));
+
+ (*Generation of unique names -- maxidx cannot be relied upon to increase!
+ Cannot rely on "variant", since variables might coincide when literals
+ are joined to make a clause...
+ 19 chooses "U" as the first variable name*)
+ val name_ref = ref 19;
+
+ (*Replaces universally quantified variables by FREE variables -- because
+ assumptions may not contain scheme variables. Later, call "generalize". *)
+ fun freeze_spec th =
+ let val sth = th RS spec
+ val newname = (name_ref := !name_ref + 1;
+ radixstring(26, "A", !name_ref))
+ in read_instantiate [("x", newname)] sth end;
+
+ fun resop nf [prem] = resolve_tac (nf prem) 1;
+
+ (*Conjunctive normal form, detecting tautologies early.
+ Strips universal quantifiers and breaks up conjunctions. *)
+ fun cnf_aux seen (th,ths) =
+ if taut_lits (literals(prop_of th) @ seen) then ths
+ else if not (has_consts ["All","op &"] (prop_of th)) then th::ths
+ else (*conjunction?*)
+ cnf_aux seen (th RS conjunct1,
+ cnf_aux seen (th RS conjunct2, ths))
+ handle THM _ => (*universal quant?*)
+ cnf_aux seen (freeze_spec th, ths)
+ handle THM _ => (*disjunction?*)
+ let val tac =
+ (METAHYPS (resop (cnf_nil seen)) 1) THEN
+ (fn st' => st' |>
+ METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1)
+ in Seq.list_of (tac (th RS disj_forward)) @ ths end
+ and cnf_nil seen th = cnf_aux seen (th,[]);
+
+ (*Top-level call to cnf -- it's safe to reset name_ref*)
+ fun cnf (th,ths) =
+ (name_ref := 19; cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
+ handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
+
+ (**** Removal of duplicate literals ****)
+
+ (*Forward proof, passing extra assumptions as theorems to the tactic*)
+ fun forward_res2 nf hyps st =
+ case Seq.pull
+ (REPEAT
+ (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
+ st)
+ of Some(th,_) => th
+ | None => raise THM("forward_res2", 0, [st]);
+
+ (*Remove duplicates in P|Q by assuming ~P in Q
+ rls (initially []) accumulates assumptions of the form P==>False*)
+ fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
+ handle THM _ => tryres(th,rls)
+ handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
+ [disj_FalseD1, disj_FalseD2, asm_rl])
+ handle THM _ => th;
+
+ (*Remove duplicate literals, if there are any*)
+ fun nodups th =
+ if null(findrep(literals(prop_of th))) then th
+ else nodups_aux [] th;
+
+
+ (**** Generation of contrapositives ****)
+
+ (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
+ fun assoc_right th = assoc_right (th RS disj_assoc)
+ handle THM _ => th;
+
+ (*Must check for negative literal first!*)
+ val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
+
+ (*For Plaisted's postive refinement. [currently unused] *)
+ val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
+
+ (*Create a goal or support clause, conclusing False*)
+ fun make_goal th = (*Must check for negative literal first!*)
+ make_goal (tryres(th, clause_rules))
+ handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
+
+ (*Sort clauses by number of literals*)
+ fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
+
+ (*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
+ fun sort_clauses ths = sort (make_ord fewerlits) (filter (not o is_taut) ths);
+
+ (*Convert all suitable free variables to schematic variables*)
+ fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
+
+ (*Create a meta-level Horn clause*)
+ fun make_horn crules th = make_horn crules (tryres(th,crules))
+ handle THM _ => th;
+
+ (*Generate Horn clauses for all contrapositives of a clause*)
+ fun add_contras crules (th,hcs) =
+ let fun rots (0,th) = hcs
+ | rots (k,th) = zero_var_indexes (make_horn crules th) ::
+ rots(k-1, assoc_right (th RS disj_comm))
+ in case nliterals(prop_of th) of
+ 1 => th::hcs
+ | n => rots(n, assoc_right th)
+ end;
+
+ (*Use "theorem naming" to label the clauses*)
+ fun name_thms label =
+ let fun name1 (th, (k,ths)) =
+ (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
+
+ in fn ths => #2 (foldr name1 (ths, (length ths, []))) end;
+
+ (*Find an all-negative support clause*)
+ fun is_negative th = forall (not o #1) (literals (prop_of th));
+
+ val neg_clauses = filter is_negative;
+
+
+ (***** MESON PROOF PROCEDURE *****)
+
+ fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
+ As) = rhyps(phi, A::As)
+ | rhyps (_, As) = As;
+
+ (** Detecting repeated assumptions in a subgoal **)
+
+ (*The stringtree detects repeated assumptions.*)
+ fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
+
+ (*detects repetitions in a list of terms*)
+ fun has_reps [] = false
+ | has_reps [_] = false
+ | has_reps [t,u] = (t aconv u)
+ | has_reps ts = (foldl ins_term (Net.empty, ts); false)
+ handle INSERT => true;
+
+ (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
+ fun TRYALL_eq_assume_tac 0 st = Seq.single st
+ | TRYALL_eq_assume_tac i st =
+ TRYALL_eq_assume_tac (i-1) (eq_assumption i st)
+ handle THM _ => TRYALL_eq_assume_tac (i-1) st;
+
+ (*Loop checking: FAIL if trying to prove the same thing twice
+ -- if *ANY* subgoal has repeated literals*)
+ fun check_tac st =
+ if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
+ then Seq.empty else Seq.single st;
+
+
+ (* net_resolve_tac actually made it slower... *)
+ fun prolog_step_tac horns i =
+ (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
+ TRYALL eq_assume_tac;
+
+
+in
+
+
+(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
+local fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
+in
+fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
+end;
+
+(*Negation Normal Form*)
+val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
+ not_impD, not_iffD, not_allD, not_exD, not_notD];
+fun make_nnf th = make_nnf (tryres(th, nnf_rls))
+ handle THM _ =>
+ forward_res make_nnf
+ (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
+ handle THM _ => th;
+
+(*Pull existential quantifiers (Skolemization)*)
+fun skolemize th =
+ if not (has_consts ["Ex"] (prop_of th)) then th
+ else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
+ disj_exD, disj_exD1, disj_exD2]))
+ handle THM _ =>
+ skolemize (forward_res skolemize
+ (tryres (th, [conj_forward, disj_forward, all_forward])))
+ handle THM _ => forward_res skolemize (th RS ex_forward);
+
+
+(*Make clauses from a list of theorems, previously Skolemized and put into nnf.
+ The resulting clauses are HOL disjunctions.*)
+fun make_clauses ths =
+ sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
+
+(*Convert a list of clauses to (contrapositive) Horn clauses*)
+fun make_horns ths =
+ name_thms "Horn#"
+ (gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[])));
+
+(*Could simply use nprems_of, which would count remaining subgoals -- no
+ discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
+
+fun best_prolog_tac sizef horns =
+ BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
+
+fun depth_prolog_tac horns =
+ DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
+
+(*Return all negative clauses, as possible goal clauses*)
+fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
+
+
+fun skolemize_tac prems =
+ cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
+ REPEAT o (etac exE);
+
+(*Shell of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
+fun MESON cltac = SELECT_GOAL
+ (EVERY1 [rtac ccontr,
+ METAHYPS (fn negs =>
+ EVERY1 [skolemize_tac negs,
+ METAHYPS (cltac o make_clauses)])]);
+
+(** Best-first search versions **)
+
+fun best_meson_tac sizef =
+ MESON (fn cls =>
+ THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
+ (has_fewer_prems 1, sizef)
+ (prolog_step_tac (make_horns cls) 1));
+
+(*First, breaks the goal into independent units*)
+val safe_best_meson_tac =
+ SELECT_GOAL (TRY Safe_tac THEN
+ TRYALL (best_meson_tac size_of_subgoals));
+
+(** Depth-first search version **)
+
+val depth_meson_tac =
+ MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
+ depth_prolog_tac (make_horns cls)]);
+
+
+
+(** Iterative deepening version **)
+
+(*This version does only one inference per call;
+ having only one eq_assume_tac speeds it up!*)
+fun prolog_step_tac' horns =
+ let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
+ take_prefix Thm.no_prems horns
+ val nrtac = net_resolve_tac horns
+ in fn i => eq_assume_tac i ORELSE
+ match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
+ ((assume_tac i APPEND nrtac i) THEN check_tac)
+ end;
+
+fun iter_deepen_prolog_tac horns =
+ ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
+
+val iter_deepen_meson_tac =
+ MESON (fn cls =>
+ (THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
+ (has_fewer_prems 1)
+ (prolog_step_tac' (make_horns cls))));
+
+val meson_tac =
+ SELECT_GOAL (TRY Safe_tac THEN
+ TRYALL (iter_deepen_meson_tac));
+
+end;
--- a/src/HOL/ex/ROOT.ML Tue Sep 05 10:14:36 2000 +0200
+++ b/src/HOL/ex/ROOT.ML Tue Sep 05 10:15:23 2000 +0200
@@ -13,7 +13,6 @@
time_use_thy "NatSum";
time_use "cla.ML";
-time_use "meson.ML";
time_use "mesontest.ML";
time_use "mesontest2.ML";
time_use_thy "BT";
--- a/src/HOL/ex/meson.ML Tue Sep 05 10:14:36 2000 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,461 +0,0 @@
-(* Title: HOL/ex/meson
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1992 University of Cambridge
-
-The MESON resolution proof procedure for HOL
-
-When making clauses, avoids using the rewriter -- instead uses RS recursively
-
-NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR
-FUNCTION nodups -- if done to goal clauses too!
-*)
-
-writeln"File HOL/ex/meson.";
-
-context HOL.thy;
-
-(*Prove theorems using fast_tac*)
-fun prove_fun s =
- prove_goal HOL.thy s
- (fn prems => [ cut_facts_tac prems 1, Fast_tac 1 ]);
-
-(**** Negation Normal Form ****)
-
-(*** de Morgan laws ***)
-
-val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
-val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
-val not_notD = prove_fun "~~P ==> P";
-val not_allD = prove_fun "~(! x. P(x)) ==> ? x. ~P(x)";
-val not_exD = prove_fun "~(? x. P(x)) ==> ! x. ~P(x)";
-
-
-(*** Removal of --> and <-> (positive and negative occurrences) ***)
-
-val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
-val not_impD = prove_fun "~(P-->Q) ==> P & ~Q";
-
-val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
-
-(*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
-val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
-
-
-(**** Pulling out the existential quantifiers ****)
-
-(*** Conjunction ***)
-
-val conj_exD1 = prove_fun "(? x. P(x)) & Q ==> ? x. P(x) & Q";
-val conj_exD2 = prove_fun "P & (? x. Q(x)) ==> ? x. P & Q(x)";
-
-(*** Disjunction ***)
-
-(*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
- With ex-Skolemization, makes fewer Skolem constants*)
-val disj_exD = prove_fun "(? x. P(x)) | (? x. Q(x)) ==> ? x. P(x) | Q(x)";
-
-val disj_exD1 = prove_fun "(? x. P(x)) | Q ==> ? x. P(x) | Q";
-val disj_exD2 = prove_fun "P | (? x. Q(x)) ==> ? x. P | Q(x)";
-
-
-
-(***** Generating clauses for the Meson Proof Procedure *****)
-
-(*** Disjunctions ***)
-
-val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
-
-val disj_comm = prove_fun "P|Q ==> Q|P";
-
-val disj_FalseD1 = prove_fun "False|P ==> P";
-val disj_FalseD2 = prove_fun "P|False ==> P";
-
-(*** Generation of contrapositives ***)
-
-(*Inserts negated disjunct after removing the negation; P is a literal*)
-val [major,minor] = Goal "~P|Q ==> ((~P==>P) ==> Q)";
-by (rtac (major RS disjE) 1);
-by (rtac notE 1);
-by (etac minor 2);
-by (ALLGOALS assume_tac);
-qed "make_neg_rule";
-
-(*For Plaisted's "Postive refinement" of the MESON procedure*)
-Goal "~P|Q ==> (P ==> Q)";
-by (Blast_tac 1);
-qed "make_refined_neg_rule";
-
-(*P should be a literal*)
-val [major,minor] = Goal "P|Q ==> ((P==>~P) ==> Q)";
-by (rtac (major RS disjE) 1);
-by (rtac notE 1);
-by (etac minor 1);
-by (ALLGOALS assume_tac);
-qed "make_pos_rule";
-
-(*** Generation of a goal clause -- put away the final literal ***)
-
-val [major,minor] = Goal "~P ==> ((~P==>P) ==> False)";
-by (rtac notE 1);
-by (rtac minor 2);
-by (ALLGOALS (rtac major));
-qed "make_neg_goal";
-
-val [major,minor] = Goal "P ==> ((P==>~P) ==> False)";
-by (rtac notE 1);
-by (rtac minor 1);
-by (ALLGOALS (rtac major));
-qed "make_pos_goal";
-
-
-(**** Lemmas for forward proof (like congruence rules) ****)
-
-(*NOTE: could handle conjunctions (faster?) by
- nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
-val major::prems = Goal
- "[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q";
-by (rtac (major RS conjE) 1);
-by (rtac conjI 1);
-by (ALLGOALS (eresolve_tac prems));
-qed "conj_forward";
-
-val major::prems = Goal
- "[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q";
-by (rtac (major RS disjE) 1);
-by (ALLGOALS (dresolve_tac prems));
-by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
-qed "disj_forward";
-
-val major::prems = Goal
- "[| ! x. P'(x); !!x. P'(x) ==> P(x) |] ==> ! x. P(x)";
-by (rtac allI 1);
-by (resolve_tac prems 1);
-by (rtac (major RS spec) 1);
-qed "all_forward";
-
-val major::prems = Goal
- "[| ? x. P'(x); !!x. P'(x) ==> P(x) |] ==> ? x. P(x)";
-by (rtac (major RS exE) 1);
-by (rtac exI 1);
-by (eresolve_tac prems 1);
-qed "ex_forward";
-
-
-(**** Operators for forward proof ****)
-
-(*raises exception if no rules apply -- unlike RL*)
-fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
- | tryres (th, []) = raise THM("tryres", 0, [th]);
-
-val prop_of = #prop o rep_thm;
-
-(*Permits forward proof from rules that discharge assumptions*)
-fun forward_res nf st =
- case Seq.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
- of Some(th,_) => th
- | None => raise THM("forward_res", 0, [st]);
-
-
-(*Negation Normal Form*)
-val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
- not_impD, not_iffD, not_allD, not_exD, not_notD];
-fun make_nnf th = make_nnf (tryres(th, nnf_rls))
- handle THM _ =>
- forward_res make_nnf
- (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
- handle THM _ => th;
-
-
-(*Are any of the constants in "bs" present in the term?*)
-fun has_consts bs =
- let fun has (Const(a,_)) = a mem bs
- | has (f$u) = has f orelse has u
- | has (Abs(_,_,t)) = has t
- | has _ = false
- in has end;
-
-(*Pull existential quantifiers (Skolemization)*)
-fun skolemize th =
- if not (has_consts ["Ex"] (prop_of th)) then th
- else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
- disj_exD, disj_exD1, disj_exD2]))
- handle THM _ =>
- skolemize (forward_res skolemize
- (tryres (th, [conj_forward, disj_forward, all_forward])))
- handle THM _ => forward_res skolemize (th RS ex_forward);
-
-
-(**** Clause handling ****)
-
-fun literals (Const("Trueprop",_) $ P) = literals P
- | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
- | literals (Const("Not",_) $ P) = [(false,P)]
- | literals P = [(true,P)];
-
-(*number of literals in a term*)
-val nliterals = length o literals;
-
-(*to detect, and remove, tautologous clauses*)
-fun taut_lits [] = false
- | taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
-
-val term_False = term_of (read_cterm (sign_of HOL.thy)
- ("False", Type("bool",[])));
-
-(*Include False as a literal: an occurrence of ~False is a tautology*)
-fun is_taut th = taut_lits ((true,term_False) :: literals (prop_of th));
-
-(*Generation of unique names -- maxidx cannot be relied upon to increase!
- Cannot rely on "variant", since variables might coincide when literals
- are joined to make a clause...
- 19 chooses "U" as the first variable name*)
-val name_ref = ref 19;
-
-(*Replaces universally quantified variables by FREE variables -- because
- assumptions may not contain scheme variables. Later, call "generalize". *)
-fun freeze_spec th =
- let val sth = th RS spec
- val newname = (name_ref := !name_ref + 1;
- radixstring(26, "A", !name_ref))
- in read_instantiate [("x", newname)] sth end;
-
-fun resop nf [prem] = resolve_tac (nf prem) 1;
-
-(*Conjunctive normal form, detecting tautologies early.
- Strips universal quantifiers and breaks up conjunctions. *)
-fun cnf_aux seen (th,ths) =
- if taut_lits (literals(prop_of th) @ seen) then ths
- else if not (has_consts ["All","op &"] (prop_of th)) then th::ths
- else (*conjunction?*)
- cnf_aux seen (th RS conjunct1,
- cnf_aux seen (th RS conjunct2, ths))
- handle THM _ => (*universal quant?*)
- cnf_aux seen (freeze_spec th, ths)
- handle THM _ => (*disjunction?*)
- let val tac =
- (METAHYPS (resop (cnf_nil seen)) 1) THEN
- (fn st' => st' |>
- METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1)
- in Seq.list_of (tac (th RS disj_forward)) @ ths end
-and cnf_nil seen th = cnf_aux seen (th,[]);
-
-(*Top-level call to cnf -- it's safe to reset name_ref*)
-fun cnf (th,ths) =
- (name_ref := 19; cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
- handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
-
-(**** Removal of duplicate literals ****)
-
-(*Version for removal of duplicate literals*)
-val major::prems = Goal
- "[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q";
-by (rtac (major RS disjE) 1);
-by (rtac disjI1 1);
-by (rtac (disjCI RS disj_comm) 2);
-by (ALLGOALS (eresolve_tac prems));
-by (etac notE 1);
-by (assume_tac 1);
-qed "disj_forward2";
-
-(*Forward proof, passing extra assumptions as theorems to the tactic*)
-fun forward_res2 nf hyps st =
- case Seq.pull
- (REPEAT
- (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
- st)
- of Some(th,_) => th
- | None => raise THM("forward_res2", 0, [st]);
-
-(*Remove duplicates in P|Q by assuming ~P in Q
- rls (initially []) accumulates assumptions of the form P==>False*)
-fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
- handle THM _ => tryres(th,rls)
- handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
- [disj_FalseD1, disj_FalseD2, asm_rl])
- handle THM _ => th;
-
-(*Remove duplicate literals, if there are any*)
-fun nodups th =
- if null(findrep(literals(prop_of th))) then th
- else nodups_aux [] th;
-
-
-(**** Generation of contrapositives ****)
-
-(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
-fun assoc_right th = assoc_right (th RS disj_assoc)
- handle THM _ => th;
-
-(*Must check for negative literal first!*)
-val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
-
-(*For Plaisted's postive refinement. [currently unused] *)
-val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
-
-(*Create a goal or support clause, conclusing False*)
-fun make_goal th = (*Must check for negative literal first!*)
- make_goal (tryres(th, clause_rules))
- handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
-
-(*Sort clauses by number of literals*)
-fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
-
-(*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
-fun sort_clauses ths = sort (make_ord fewerlits) (filter (not o is_taut) ths);
-
-(*Convert all suitable free variables to schematic variables*)
-fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
-
-(*Make clauses from a list of theorems, previously Skolemized and put into nnf.
- The resulting clauses are HOL disjunctions.*)
-fun make_clauses ths =
- sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
-
-(*Create a meta-level Horn clause*)
-fun make_horn crules th = make_horn crules (tryres(th,crules))
- handle THM _ => th;
-
-(*Generate Horn clauses for all contrapositives of a clause*)
-fun add_contras crules (th,hcs) =
- let fun rots (0,th) = hcs
- | rots (k,th) = zero_var_indexes (make_horn crules th) ::
- rots(k-1, assoc_right (th RS disj_comm))
- in case nliterals(prop_of th) of
- 1 => th::hcs
- | n => rots(n, assoc_right th)
- end;
-
-(*Use "theorem naming" to label the clauses*)
-fun name_thms label =
- let fun name1 (th, (k,ths)) =
- (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
-
- in fn ths => #2 (foldr name1 (ths, (length ths, []))) end;
-
-(*Convert a list of clauses to (contrapositive) Horn clauses*)
-fun make_horns ths =
- name_thms "Horn#"
- (gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[])));
-
-(*Find an all-negative support clause*)
-fun is_negative th = forall (not o #1) (literals (prop_of th));
-
-val neg_clauses = filter is_negative;
-
-
-(***** MESON PROOF PROCEDURE *****)
-
-fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
- As) = rhyps(phi, A::As)
- | rhyps (_, As) = As;
-
-(** Detecting repeated assumptions in a subgoal **)
-
-(*The stringtree detects repeated assumptions.*)
-fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
-
-(*detects repetitions in a list of terms*)
-fun has_reps [] = false
- | has_reps [_] = false
- | has_reps [t,u] = (t aconv u)
- | has_reps ts = (foldl ins_term (Net.empty, ts); false)
- handle INSERT => true;
-
-(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
-fun TRYALL_eq_assume_tac 0 st = Seq.single st
- | TRYALL_eq_assume_tac i st = TRYALL_eq_assume_tac (i-1) (eq_assumption i st)
- handle THM _ => TRYALL_eq_assume_tac (i-1) st;
-
-(*Loop checking: FAIL if trying to prove the same thing twice
- -- if *ANY* subgoal has repeated literals*)
-fun check_tac st =
- if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
- then Seq.empty else Seq.single st;
-
-
-(* net_resolve_tac actually made it slower... *)
-fun prolog_step_tac horns i =
- (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
- TRYALL eq_assume_tac;
-
-
-(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
-local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz
-in
-fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
-end;
-
-(*Could simply use nprems_of, which would count remaining subgoals -- no
- discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
-
-fun best_prolog_tac sizef horns =
- BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
-
-fun depth_prolog_tac horns =
- DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
-
-(*Return all negative clauses, as possible goal clauses*)
-fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
-
-
-fun skolemize_tac prems =
- cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
- REPEAT o (etac exE);
-
-(*Shell of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
-fun MESON cltac = SELECT_GOAL
- (EVERY1 [rtac ccontr,
- METAHYPS (fn negs =>
- EVERY1 [skolemize_tac negs,
- METAHYPS (cltac o make_clauses)])]);
-
-(** Best-first search versions **)
-
-fun best_meson_tac sizef =
- MESON (fn cls =>
- THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
- (has_fewer_prems 1, sizef)
- (prolog_step_tac (make_horns cls) 1));
-
-(*First, breaks the goal into independent units*)
-val safe_best_meson_tac =
- SELECT_GOAL (TRY Safe_tac THEN
- TRYALL (best_meson_tac size_of_subgoals));
-
-(** Depth-first search version **)
-
-val depth_meson_tac =
- MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
- depth_prolog_tac (make_horns cls)]);
-
-
-
-(** Iterative deepening version **)
-
-(*This version does only one inference per call;
- having only one eq_assume_tac speeds it up!*)
-fun prolog_step_tac' horns =
- let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
- take_prefix Thm.no_prems horns
- val nrtac = net_resolve_tac horns
- in fn i => eq_assume_tac i ORELSE
- match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
- ((assume_tac i APPEND nrtac i) THEN check_tac)
- end;
-
-fun iter_deepen_prolog_tac horns =
- ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
-
-val iter_deepen_meson_tac =
- MESON (fn cls =>
- (THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
- (has_fewer_prems 1)
- (prolog_step_tac' (make_horns cls))));
-
-val safe_meson_tac =
- SELECT_GOAL (TRY Safe_tac THEN
- TRYALL (iter_deepen_meson_tac));
-
-
-writeln"Reached end of file.";