|
1 (* Title: FOLP/classical |
|
2 ID: $Id$ |
|
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 Copyright 1992 University of Cambridge |
|
5 |
|
6 Like Provers/classical but modified because match_tac is unsuitable for |
|
7 proof objects. |
|
8 |
|
9 Theorem prover for classical reasoning, including predicate calculus, set |
|
10 theory, etc. |
|
11 |
|
12 Rules must be classified as intr, elim, safe, hazardous. |
|
13 |
|
14 A rule is unsafe unless it can be applied blindly without harmful results. |
|
15 For a rule to be safe, its premises and conclusion should be logically |
|
16 equivalent. There should be no variables in the premises that are not in |
|
17 the conclusion. |
|
18 *) |
|
19 |
|
20 signature CLASSICAL_DATA = |
|
21 sig |
|
22 val mp: thm (* [| P-->Q; P |] ==> Q *) |
|
23 val not_elim: thm (* [| ~P; P |] ==> R *) |
|
24 val swap: thm (* ~P ==> (~Q ==> P) ==> Q *) |
|
25 val sizef : thm -> int (* size function for BEST_FIRST *) |
|
26 val hyp_subst_tacs: (int -> tactic) list |
|
27 end; |
|
28 |
|
29 (*Higher precedence than := facilitates use of references*) |
|
30 infix 4 addSIs addSEs addSDs addIs addEs addDs; |
|
31 |
|
32 |
|
33 signature CLASSICAL = |
|
34 sig |
|
35 type claset |
|
36 val empty_cs: claset |
|
37 val addDs : claset * thm list -> claset |
|
38 val addEs : claset * thm list -> claset |
|
39 val addIs : claset * thm list -> claset |
|
40 val addSDs: claset * thm list -> claset |
|
41 val addSEs: claset * thm list -> claset |
|
42 val addSIs: claset * thm list -> claset |
|
43 val print_cs: claset -> unit |
|
44 val rep_claset: claset -> |
|
45 {safeIs: thm list, safeEs: thm list, hazIs: thm list, hazEs: thm list, |
|
46 safe0_brls:(bool*thm)list, safep_brls: (bool*thm)list, |
|
47 haz_brls: (bool*thm)list} |
|
48 val best_tac : claset -> int -> tactic |
|
49 val chain_tac : int -> tactic |
|
50 val contr_tac : int -> tactic |
|
51 val fast_tac : claset -> int -> tactic |
|
52 val inst_step_tac : int -> tactic |
|
53 val joinrules : thm list * thm list -> (bool * thm) list |
|
54 val mp_tac: int -> tactic |
|
55 val safe_tac : claset -> tactic |
|
56 val safe_step_tac : claset -> int -> tactic |
|
57 val slow_step_tac : claset -> int -> tactic |
|
58 val step_tac : claset -> int -> tactic |
|
59 val swapify : thm list -> thm list |
|
60 val swap_res_tac : thm list -> int -> tactic |
|
61 val uniq_mp_tac: int -> tactic |
|
62 end; |
|
63 |
|
64 |
|
65 functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL = |
|
66 struct |
|
67 |
|
68 local open Data in |
|
69 |
|
70 (** Useful tactics for classical reasoning **) |
|
71 |
|
72 val imp_elim = make_elim mp; |
|
73 |
|
74 (*Solve goal that assumes both P and ~P. *) |
|
75 val contr_tac = eresolve_tac [not_elim] THEN' assume_tac; |
|
76 |
|
77 (*Finds P-->Q and P in the assumptions, replaces implication by Q *) |
|
78 fun mp_tac i = eresolve_tac ([not_elim,imp_elim]) i THEN assume_tac i; |
|
79 |
|
80 (*Like mp_tac but instantiates no variables*) |
|
81 fun uniq_mp_tac i = ematch_tac ([not_elim,imp_elim]) i THEN uniq_assume_tac i; |
|
82 |
|
83 (*Creates rules to eliminate ~A, from rules to introduce A*) |
|
84 fun swapify intrs = intrs RLN (2, [swap]); |
|
85 |
|
86 (*Uses introduction rules in the normal way, or on negated assumptions, |
|
87 trying rules in order. *) |
|
88 fun swap_res_tac rls = |
|
89 let fun tacf rl = rtac rl ORELSE' etac (rl RSN (2,swap)) |
|
90 in assume_tac ORELSE' contr_tac ORELSE' FIRST' (map tacf rls) |
|
91 end; |
|
92 |
|
93 (*Given assumption P-->Q, reduces subgoal Q to P [deletes the implication!] *) |
|
94 fun chain_tac i = |
|
95 eresolve_tac [imp_elim] i THEN |
|
96 (assume_tac (i+1) ORELSE contr_tac (i+1)); |
|
97 |
|
98 (*** Classical rule sets ***) |
|
99 |
|
100 datatype claset = |
|
101 CS of {safeIs: thm list, |
|
102 safeEs: thm list, |
|
103 hazIs: thm list, |
|
104 hazEs: thm list, |
|
105 (*the following are computed from the above*) |
|
106 safe0_brls: (bool*thm)list, |
|
107 safep_brls: (bool*thm)list, |
|
108 haz_brls: (bool*thm)list}; |
|
109 |
|
110 fun rep_claset (CS x) = x; |
|
111 |
|
112 (*For use with biresolve_tac. Combines intrs with swap to catch negated |
|
113 assumptions. Also pairs elims with true. *) |
|
114 fun joinrules (intrs,elims) = |
|
115 map (pair true) (elims @ swapify intrs) @ map (pair false) intrs; |
|
116 |
|
117 (*Note that allE precedes exI in haz_brls*) |
|
118 fun make_cs {safeIs,safeEs,hazIs,hazEs} = |
|
119 let val (safe0_brls, safep_brls) = (*0 subgoals vs 1 or more*) |
|
120 partition (apl(0,op=) o subgoals_of_brl) |
|
121 (sort lessb (joinrules(safeIs, safeEs))) |
|
122 in CS{safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs, |
|
123 safe0_brls=safe0_brls, safep_brls=safep_brls, |
|
124 haz_brls = sort lessb (joinrules(hazIs, hazEs))} |
|
125 end; |
|
126 |
|
127 (*** Manipulation of clasets ***) |
|
128 |
|
129 val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[]}; |
|
130 |
|
131 fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) = |
|
132 (writeln"Introduction rules"; prths hazIs; |
|
133 writeln"Safe introduction rules"; prths safeIs; |
|
134 writeln"Elimination rules"; prths hazEs; |
|
135 writeln"Safe elimination rules"; prths safeEs; |
|
136 ()); |
|
137 |
|
138 fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSIs ths = |
|
139 make_cs {safeIs=ths@safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs}; |
|
140 |
|
141 fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSEs ths = |
|
142 make_cs {safeIs=safeIs, safeEs=ths@safeEs, hazIs=hazIs, hazEs=hazEs}; |
|
143 |
|
144 fun cs addSDs ths = cs addSEs (map make_elim ths); |
|
145 |
|
146 fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addIs ths = |
|
147 make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=ths@hazIs, hazEs=hazEs}; |
|
148 |
|
149 fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addEs ths = |
|
150 make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=ths@hazEs}; |
|
151 |
|
152 fun cs addDs ths = cs addEs (map make_elim ths); |
|
153 |
|
154 (*** Simple tactics for theorem proving ***) |
|
155 |
|
156 (*Attack subgoals using safe inferences*) |
|
157 fun safe_step_tac (CS{safe0_brls,safep_brls,...}) = |
|
158 FIRST' [uniq_assume_tac, |
|
159 uniq_mp_tac, |
|
160 biresolve_tac safe0_brls, |
|
161 FIRST' hyp_subst_tacs, |
|
162 biresolve_tac safep_brls] ; |
|
163 |
|
164 (*Repeatedly attack subgoals using safe inferences*) |
|
165 fun safe_tac cs = DETERM (REPEAT_FIRST (safe_step_tac cs)); |
|
166 |
|
167 (*These steps could instantiate variables and are therefore unsafe.*) |
|
168 val inst_step_tac = assume_tac APPEND' contr_tac; |
|
169 |
|
170 (*Single step for the prover. FAILS unless it makes progress. *) |
|
171 fun step_tac (cs as (CS{haz_brls,...})) i = |
|
172 FIRST [safe_tac cs, |
|
173 inst_step_tac i, |
|
174 biresolve_tac haz_brls i]; |
|
175 |
|
176 (*** The following tactics all fail unless they solve one goal ***) |
|
177 |
|
178 (*Dumb but fast*) |
|
179 fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1)); |
|
180 |
|
181 (*Slower but smarter than fast_tac*) |
|
182 fun best_tac cs = |
|
183 SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1)); |
|
184 |
|
185 (*Using a "safe" rule to instantiate variables is unsafe. This tactic |
|
186 allows backtracking from "safe" rules to "unsafe" rules here.*) |
|
187 fun slow_step_tac (cs as (CS{haz_brls,...})) i = |
|
188 safe_tac cs ORELSE (assume_tac i APPEND biresolve_tac haz_brls i); |
|
189 |
|
190 end; |
|
191 end; |