--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Provers/classical.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,206 @@
+(* Title: Provers/classical
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+Theorem prover for classical reasoning, including predicate calculus, set
+theory, etc.
+
+Rules must be classified as intr, elim, safe, hazardous.
+
+A rule is unsafe unless it can be applied blindly without harmful results.
+For a rule to be safe, its premises and conclusion should be logically
+equivalent. There should be no variables in the premises that are not in
+the conclusion.
+*)
+
+signature CLASSICAL_DATA =
+ sig
+ val mp: thm (* [| P-->Q; P |] ==> Q *)
+ val not_elim: thm (* [| ~P; P |] ==> R *)
+ val swap: thm (* ~P ==> (~Q ==> P) ==> Q *)
+ val sizef : thm -> int (* size function for BEST_FIRST *)
+ val hyp_subst_tacs: (int -> tactic) list
+ end;
+
+(*Higher precedence than := facilitates use of references*)
+infix 4 addSIs addSEs addSDs addIs addEs addDs;
+
+
+signature CLASSICAL =
+ sig
+ type claset
+ val empty_cs: claset
+ val addDs : claset * thm list -> claset
+ val addEs : claset * thm list -> claset
+ val addIs : claset * thm list -> claset
+ val addSDs: claset * thm list -> claset
+ val addSEs: claset * thm list -> claset
+ val addSIs: claset * thm list -> claset
+ val print_cs: claset -> unit
+ val rep_claset: claset ->
+ {safeIs: thm list, safeEs: thm list, hazIs: thm list, hazEs: thm list}
+ val best_tac : claset -> int -> tactic
+ val chain_tac : int -> tactic
+ val contr_tac : int -> tactic
+ val eq_mp_tac: int -> tactic
+ val fast_tac : claset -> int -> tactic
+ val joinrules : thm list * thm list -> (bool * thm) list
+ val mp_tac: int -> tactic
+ val safe_tac : claset -> tactic
+ val safe_step_tac : claset -> int -> tactic
+ val slow_step_tac : claset -> int -> tactic
+ val slow_best_tac : claset -> int -> tactic
+ val slow_tac : claset -> int -> tactic
+ val step_tac : claset -> int -> tactic
+ val swapify : thm list -> thm list
+ val swap_res_tac : thm list -> int -> tactic
+ val inst_step_tac : claset -> int -> tactic
+ end;
+
+
+functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL =
+struct
+
+local open Data in
+
+(** Useful tactics for classical reasoning **)
+
+val imp_elim = make_elim mp;
+
+(*Solve goal that assumes both P and ~P. *)
+val contr_tac = eresolve_tac [not_elim] THEN' assume_tac;
+
+(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
+fun mp_tac i = eresolve_tac ([not_elim,imp_elim]) i THEN assume_tac i;
+
+(*Like mp_tac but instantiates no variables*)
+fun eq_mp_tac i = ematch_tac ([not_elim,imp_elim]) i THEN eq_assume_tac i;
+
+(*Creates rules to eliminate ~A, from rules to introduce A*)
+fun swapify intrs = intrs RLN (2, [swap]);
+
+(*Uses introduction rules in the normal way, or on negated assumptions,
+ trying rules in order. *)
+fun swap_res_tac rls =
+ let fun tacf rl = rtac rl ORELSE' etac (rl RSN (2,swap))
+ in assume_tac ORELSE' contr_tac ORELSE' FIRST' (map tacf rls)
+ end;
+
+(*Given assumption P-->Q, reduces subgoal Q to P [deletes the implication!] *)
+fun chain_tac i =
+ eresolve_tac [imp_elim] i THEN
+ (assume_tac (i+1) ORELSE contr_tac (i+1));
+
+(*** Classical rule sets ***)
+
+type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;
+
+datatype claset =
+ CS of {safeIs : thm list,
+ safeEs : thm list,
+ hazIs : thm list,
+ hazEs : thm list,
+ safe0_netpair : netpair,
+ safep_netpair : netpair,
+ haz_netpair : netpair};
+
+fun rep_claset (CS{safeIs,safeEs,hazIs,hazEs,...}) =
+ {safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs};
+
+(*For use with biresolve_tac. Combines intrs with swap to catch negated
+ assumptions; pairs elims with true; sorts. *)
+fun joinrules (intrs,elims) =
+ sort lessb
+ (map (pair true) (elims @ swapify intrs) @
+ map (pair false) intrs);
+
+(*Make a claset from the four kinds of rules*)
+fun make_cs {safeIs,safeEs,hazIs,hazEs} =
+ let val (safe0_brls, safep_brls) = (*0 subgoals vs 1 or more*)
+ take_prefix (fn brl => subgoals_of_brl brl=0)
+ (joinrules(safeIs, safeEs))
+ in CS{safeIs = safeIs,
+ safeEs = safeEs,
+ hazIs = hazIs,
+ hazEs = hazEs,
+ safe0_netpair = build_netpair safe0_brls,
+ safep_netpair = build_netpair safep_brls,
+ haz_netpair = build_netpair (joinrules(hazIs, hazEs))}
+ end;
+
+(*** Manipulation of clasets ***)
+
+val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[]};
+
+fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) =
+ (writeln"Introduction rules"; prths hazIs;
+ writeln"Safe introduction rules"; prths safeIs;
+ writeln"Elimination rules"; prths hazEs;
+ writeln"Safe elimination rules"; prths safeEs;
+ ());
+
+fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSIs ths =
+ make_cs {safeIs=ths@safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs};
+
+fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSEs ths =
+ make_cs {safeIs=safeIs, safeEs=ths@safeEs, hazIs=hazIs, hazEs=hazEs};
+
+fun cs addSDs ths = cs addSEs (map make_elim ths);
+
+fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addIs ths =
+ make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=ths@hazIs, hazEs=hazEs};
+
+fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addEs ths =
+ make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=ths@hazEs};
+
+fun cs addDs ths = cs addEs (map make_elim ths);
+
+(*** Simple tactics for theorem proving ***)
+
+(*Attack subgoals using safe inferences -- matching, not resolution*)
+fun safe_step_tac (CS{safe0_netpair,safep_netpair,...}) =
+ FIRST' [eq_assume_tac,
+ eq_mp_tac,
+ bimatch_from_nets_tac safe0_netpair,
+ FIRST' hyp_subst_tacs,
+ bimatch_from_nets_tac safep_netpair] ;
+
+(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
+fun safe_tac cs = DETERM (REPEAT_FIRST (safe_step_tac cs));
+
+(*These steps could instantiate variables and are therefore unsafe.*)
+fun inst_step_tac (CS{safe0_netpair,safep_netpair,...}) =
+ assume_tac APPEND'
+ contr_tac APPEND'
+ biresolve_from_nets_tac safe0_netpair APPEND'
+ biresolve_from_nets_tac safep_netpair;
+
+(*Single step for the prover. FAILS unless it makes progress. *)
+fun step_tac (cs as (CS{haz_netpair,...})) i =
+ FIRST [safe_tac cs,
+ inst_step_tac cs i,
+ biresolve_from_nets_tac haz_netpair i];
+
+(*Using a "safe" rule to instantiate variables is unsafe. This tactic
+ allows backtracking from "safe" rules to "unsafe" rules here.*)
+fun slow_step_tac (cs as (CS{haz_netpair,...})) i =
+ safe_tac cs ORELSE
+ (inst_step_tac cs i APPEND biresolve_from_nets_tac haz_netpair i);
+
+(*** The following tactics all fail unless they solve one goal ***)
+
+(*Dumb but fast*)
+fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1));
+
+(*Slower but smarter than fast_tac*)
+fun best_tac cs =
+ SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1));
+
+fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1));
+
+fun slow_best_tac cs =
+ SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1));
+
+end;
+end;