src/FOLP/ex/Classical.thy
author wenzelm
Sat Jul 25 10:31:27 2009 +0200 (2009-07-25)
changeset 32187 cca43ca13f4f
parent 26322 eaf634e975fa
child 35762 af3ff2ba4c54
permissions -rw-r--r--
renamed structure Display_Goal to Goal_Display;
     1 (*  Title:      FOLP/ex/Classical.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Classical First-Order Logic.
     7 *)
     8 
     9 theory Classical
    10 imports FOLP
    11 begin
    12 
    13 lemma "?p : (P --> Q | R) --> (P-->Q) | (P-->R)"
    14   by (tactic "fast_tac FOLP_cs 1")
    15 
    16 (*If and only if*)
    17 lemma "?p : (P<->Q) <-> (Q<->P)"
    18   by (tactic "fast_tac FOLP_cs 1")
    19 
    20 lemma "?p : ~ (P <-> ~P)"
    21   by (tactic "fast_tac FOLP_cs 1")
    22 
    23 
    24 (*Sample problems from 
    25   F. J. Pelletier, 
    26   Seventy-Five Problems for Testing Automatic Theorem Provers,
    27   J. Automated Reasoning 2 (1986), 191-216.
    28   Errata, JAR 4 (1988), 236-236.
    29 
    30 The hardest problems -- judging by experience with several theorem provers,
    31 including matrix ones -- are 34 and 43.
    32 *)
    33 
    34 text "Pelletier's examples"
    35 (*1*)
    36 lemma "?p : (P-->Q)  <->  (~Q --> ~P)"
    37   by (tactic "fast_tac FOLP_cs 1")
    38 
    39 (*2*)
    40 lemma "?p : ~ ~ P  <->  P"
    41   by (tactic "fast_tac FOLP_cs 1")
    42 
    43 (*3*)
    44 lemma "?p : ~(P-->Q) --> (Q-->P)"
    45   by (tactic "fast_tac FOLP_cs 1")
    46 
    47 (*4*)
    48 lemma "?p : (~P-->Q)  <->  (~Q --> P)"
    49   by (tactic "fast_tac FOLP_cs 1")
    50 
    51 (*5*)
    52 lemma "?p : ((P|Q)-->(P|R)) --> (P|(Q-->R))"
    53   by (tactic "fast_tac FOLP_cs 1")
    54 
    55 (*6*)
    56 lemma "?p : P | ~ P"
    57   by (tactic "fast_tac FOLP_cs 1")
    58 
    59 (*7*)
    60 lemma "?p : P | ~ ~ ~ P"
    61   by (tactic "fast_tac FOLP_cs 1")
    62 
    63 (*8.  Peirce's law*)
    64 lemma "?p : ((P-->Q) --> P)  -->  P"
    65   by (tactic "fast_tac FOLP_cs 1")
    66 
    67 (*9*)
    68 lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
    69   by (tactic "fast_tac FOLP_cs 1")
    70 
    71 (*10*)
    72 lemma "?p : (Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"
    73   by (tactic "fast_tac FOLP_cs 1")
    74 
    75 (*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
    76 lemma "?p : P<->P"
    77   by (tactic "fast_tac FOLP_cs 1")
    78 
    79 (*12.  "Dijkstra's law"*)
    80 lemma "?p : ((P <-> Q) <-> R)  <->  (P <-> (Q <-> R))"
    81   by (tactic "fast_tac FOLP_cs 1")
    82 
    83 (*13.  Distributive law*)
    84 lemma "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
    85   by (tactic "fast_tac FOLP_cs 1")
    86 
    87 (*14*)
    88 lemma "?p : (P <-> Q) <-> ((Q | ~P) & (~Q|P))"
    89   by (tactic "fast_tac FOLP_cs 1")
    90 
    91 (*15*)
    92 lemma "?p : (P --> Q) <-> (~P | Q)"
    93   by (tactic "fast_tac FOLP_cs 1")
    94 
    95 (*16*)
    96 lemma "?p : (P-->Q) | (Q-->P)"
    97   by (tactic "fast_tac FOLP_cs 1")
    98 
    99 (*17*)
   100 lemma "?p : ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
   101   by (tactic "fast_tac FOLP_cs 1")
   102 
   103 
   104 text "Classical Logic: examples with quantifiers"
   105 
   106 lemma "?p : (ALL x. P(x) & Q(x)) <-> (ALL x. P(x))  &  (ALL x. Q(x))"
   107   by (tactic "fast_tac FOLP_cs 1")
   108 
   109 lemma "?p : (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))"
   110   by (tactic "fast_tac FOLP_cs 1")
   111 
   112 lemma "?p : (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q"
   113   by (tactic "fast_tac FOLP_cs 1")
   114 
   115 lemma "?p : (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)"
   116   by (tactic "fast_tac FOLP_cs 1")
   117 
   118 
   119 text "Problems requiring quantifier duplication"
   120 
   121 (*Needs multiple instantiation of ALL.*)
   122 lemma "?p : (ALL x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))"
   123   by (tactic "best_tac FOLP_dup_cs 1")
   124 
   125 (*Needs double instantiation of the quantifier*)
   126 lemma "?p : EX x. P(x) --> P(a) & P(b)"
   127   by (tactic "best_tac FOLP_dup_cs 1")
   128 
   129 lemma "?p : EX z. P(z) --> (ALL x. P(x))"
   130   by (tactic "best_tac FOLP_dup_cs 1")
   131 
   132 
   133 text "Hard examples with quantifiers"
   134 
   135 text "Problem 18"
   136 lemma "?p : EX y. ALL x. P(y)-->P(x)"
   137   by (tactic "best_tac FOLP_dup_cs 1")
   138 
   139 text "Problem 19"
   140 lemma "?p : EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
   141   by (tactic "best_tac FOLP_dup_cs 1")
   142 
   143 text "Problem 20"
   144 lemma "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))      
   145     --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
   146   by (tactic "fast_tac FOLP_cs 1")
   147 
   148 text "Problem 21"
   149 lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
   150   by (tactic "best_tac FOLP_dup_cs 1")
   151 
   152 text "Problem 22"
   153 lemma "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
   154   by (tactic "fast_tac FOLP_cs 1")
   155 
   156 text "Problem 23"
   157 lemma "?p : (ALL x. P | Q(x))  <->  (P | (ALL x. Q(x)))"
   158   by (tactic "best_tac FOLP_dup_cs 1")
   159 
   160 text "Problem 24"
   161 lemma "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &   
   162      (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))   
   163     --> (EX x. P(x)&R(x))"
   164   by (tactic "fast_tac FOLP_cs 1")
   165 
   166 text "Problem 25"
   167 lemma "?p : (EX x. P(x)) &  
   168        (ALL x. L(x) --> ~ (M(x) & R(x))) &  
   169        (ALL x. P(x) --> (M(x) & L(x))) &   
   170        ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))  
   171    --> (EX x. Q(x)&P(x))"
   172   oops
   173 
   174 text "Problem 26"
   175 lemma "?u : ((EX x. p(x)) <-> (EX x. q(x))) &   
   176      (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y)))   
   177   --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))";
   178   by (tactic "fast_tac FOLP_cs 1")
   179 
   180 text "Problem 27"
   181 lemma "?p : (EX x. P(x) & ~Q(x)) &    
   182               (ALL x. P(x) --> R(x)) &    
   183               (ALL x. M(x) & L(x) --> P(x)) &    
   184               ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))   
   185           --> (ALL x. M(x) --> ~L(x))"
   186   by (tactic "fast_tac FOLP_cs 1")
   187 
   188 text "Problem 28.  AMENDED"
   189 lemma "?p : (ALL x. P(x) --> (ALL x. Q(x))) &    
   190         ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &   
   191         ((EX x. S(x)) --> (ALL x. L(x) --> M(x)))   
   192     --> (ALL x. P(x) & L(x) --> M(x))"
   193   by (tactic "fast_tac FOLP_cs 1")
   194 
   195 text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
   196 lemma "?p : (EX x. P(x)) & (EX y. Q(y))   
   197     --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->      
   198          (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
   199   by (tactic "fast_tac FOLP_cs 1")
   200 
   201 text "Problem 30"
   202 lemma "?p : (ALL x. P(x) | Q(x) --> ~ R(x)) &  
   203         (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))   
   204     --> (ALL x. S(x))"
   205   by (tactic "fast_tac FOLP_cs 1")
   206 
   207 text "Problem 31"
   208 lemma "?p : ~(EX x. P(x) & (Q(x) | R(x))) &  
   209         (EX x. L(x) & P(x)) &  
   210         (ALL x. ~ R(x) --> M(x))   
   211     --> (EX x. L(x) & M(x))"
   212   by (tactic "fast_tac FOLP_cs 1")
   213 
   214 text "Problem 32"
   215 lemma "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &  
   216         (ALL x. S(x) & R(x) --> L(x)) &  
   217         (ALL x. M(x) --> R(x))   
   218     --> (ALL x. P(x) & M(x) --> L(x))"
   219   by (tactic "best_tac FOLP_dup_cs 1")
   220 
   221 text "Problem 33"
   222 lemma "?p : (ALL x. P(a) & (P(x)-->P(b))-->P(c))  <->     
   223      (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
   224   by (tactic "best_tac FOLP_dup_cs 1")
   225 
   226 text "Problem 35"
   227 lemma "?p : EX x y. P(x,y) -->  (ALL u v. P(u,v))"
   228   by (tactic "best_tac FOLP_dup_cs 1")
   229 
   230 text "Problem 36"
   231 lemma
   232 "?p : (ALL x. EX y. J(x,y)) &  
   233       (ALL x. EX y. G(x,y)) &  
   234       (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z)))    
   235   --> (ALL x. EX y. H(x,y))"
   236   by (tactic "fast_tac FOLP_cs 1")
   237 
   238 text "Problem 37"
   239 lemma "?p : (ALL z. EX w. ALL x. EX y.  
   240            (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) &  
   241         (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) &  
   242         ((EX x y. Q(x,y)) --> (ALL x. R(x,x)))   
   243     --> (ALL x. EX y. R(x,y))"
   244   by (tactic "fast_tac FOLP_cs 1")
   245 
   246 text "Problem 39"
   247 lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
   248   by (tactic "fast_tac FOLP_cs 1")
   249 
   250 text "Problem 40.  AMENDED"
   251 lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->   
   252               ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
   253   by (tactic "fast_tac FOLP_cs 1")
   254 
   255 text "Problem 41"
   256 lemma "?p : (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x))   
   257           --> ~ (EX z. ALL x. f(x,z))"
   258   by (tactic "best_tac FOLP_dup_cs 1")
   259 
   260 text "Problem 44"
   261 lemma "?p : (ALL x. f(x) -->                                     
   262               (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &        
   263               (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                    
   264               --> (EX x. j(x) & ~f(x))"
   265   by (tactic "fast_tac FOLP_cs 1")
   266 
   267 text "Problems (mainly) involving equality or functions"
   268 
   269 text "Problem 48"
   270 lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
   271   by (tactic "fast_tac FOLP_cs 1")
   272 
   273 text "Problem 50"
   274 (*What has this to do with equality?*)
   275 lemma "?p : (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))"
   276   by (tactic "best_tac FOLP_dup_cs 1")
   277 
   278 text "Problem 56"
   279 lemma
   280  "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
   281   by (tactic "fast_tac FOLP_cs 1")
   282 
   283 text "Problem 57"
   284 lemma
   285 "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &  
   286       (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
   287   by (tactic "fast_tac FOLP_cs 1")
   288 
   289 text "Problem 58  NOT PROVED AUTOMATICALLY"
   290 lemma
   291   notes f_cong = subst_context [where t = f]
   292   shows "?p : (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"
   293   by (tactic {* fast_tac (FOLP_cs addSIs [@{thm f_cong}]) 1 *})
   294 
   295 text "Problem 59"
   296 lemma "?p : (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"
   297   by (tactic "best_tac FOLP_dup_cs 1")
   298 
   299 text "Problem 60"
   300 lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
   301   by (tactic "fast_tac FOLP_cs 1")
   302 
   303 end