src/FOLP/FOLP.thy
author haftmann
Thu Nov 23 17:03:27 2017 +0000 (20 months ago)
changeset 67087 733017b19de9
parent 61337 4645502c3c64
child 69593 3dda49e08b9d
permissions -rw-r--r--
generalized more lemmas
     1 (*  Title:      FOLP/FOLP.thy
     2     Author:     Martin D Coen, Cambridge University Computer Laboratory
     3     Copyright   1992  University of Cambridge
     4 *)
     5 
     6 section \<open>Classical First-Order Logic with Proofs\<close>
     7 
     8 theory FOLP
     9 imports IFOLP
    10 begin
    11 
    12 axiomatization cla :: "[p=>p]=>p"
    13   where classical: "(!!x. x:~P ==> f(x):P) ==> cla(f):P"
    14 
    15 
    16 (*** Classical introduction rules for | and EX ***)
    17 
    18 schematic_goal disjCI:
    19   assumes "!!x. x:~Q ==> f(x):P"
    20   shows "?p : P|Q"
    21   apply (rule classical)
    22   apply (assumption | rule assms disjI1 notI)+
    23   apply (assumption | rule disjI2 notE)+
    24   done
    25 
    26 (*introduction rule involving only EX*)
    27 schematic_goal ex_classical:
    28   assumes "!!u. u:~(EX x. P(x)) ==> f(u):P(a)"
    29   shows "?p : EX x. P(x)"
    30   apply (rule classical)
    31   apply (rule exI, rule assms, assumption)
    32   done
    33 
    34 (*version of above, simplifying ~EX to ALL~ *)
    35 schematic_goal exCI:
    36   assumes "!!u. u:ALL x. ~P(x) ==> f(u):P(a)"
    37   shows "?p : EX x. P(x)"
    38   apply (rule ex_classical)
    39   apply (rule notI [THEN allI, THEN assms])
    40   apply (erule notE)
    41   apply (erule exI)
    42   done
    43 
    44 schematic_goal excluded_middle: "?p : ~P | P"
    45   apply (rule disjCI)
    46   apply assumption
    47   done
    48 
    49 
    50 (*** Special elimination rules *)
    51 
    52 (*Classical implies (-->) elimination. *)
    53 schematic_goal impCE:
    54   assumes major: "p:P-->Q"
    55     and r1: "!!x. x:~P ==> f(x):R"
    56     and r2: "!!y. y:Q ==> g(y):R"
    57   shows "?p : R"
    58   apply (rule excluded_middle [THEN disjE])
    59    apply (tactic \<open>DEPTH_SOLVE (assume_tac @{context} 1 ORELSE
    60        resolve_tac @{context} [@{thm r1}, @{thm r2}, @{thm major} RS @{thm mp}] 1)\<close>)
    61   done
    62 
    63 (*Double negation law*)
    64 schematic_goal notnotD: "p:~~P ==> ?p : P"
    65   apply (rule classical)
    66   apply (erule notE)
    67   apply assumption
    68   done
    69 
    70 
    71 (*** Tactics for implication and contradiction ***)
    72 
    73 (*Classical <-> elimination.  Proof substitutes P=Q in
    74     ~P ==> ~Q    and    P ==> Q  *)
    75 schematic_goal iffCE:
    76   assumes major: "p:P<->Q"
    77     and r1: "!!x y.[| x:P; y:Q |] ==> f(x,y):R"
    78     and r2: "!!x y.[| x:~P; y:~Q |] ==> g(x,y):R"
    79   shows "?p : R"
    80   apply (insert major)
    81   apply (unfold iff_def)
    82   apply (rule conjE)
    83   apply (tactic \<open>DEPTH_SOLVE_1 (eresolve_tac @{context} @{thms impCE} 1 ORELSE
    84       eresolve_tac @{context} [@{thm notE}, @{thm impE}] 1 THEN assume_tac @{context} 1 ORELSE
    85       assume_tac @{context} 1 ORELSE
    86       resolve_tac @{context} [@{thm r1}, @{thm r2}] 1)\<close>)+
    87   done
    88 
    89 
    90 (*Should be used as swap since ~P becomes redundant*)
    91 schematic_goal swap:
    92   assumes major: "p:~P"
    93     and r: "!!x. x:~Q ==> f(x):P"
    94   shows "?p : Q"
    95   apply (rule classical)
    96   apply (rule major [THEN notE])
    97   apply (rule r)
    98   apply assumption
    99   done
   100 
   101 ML_file "classical.ML"      (* Patched because matching won't instantiate proof *)
   102 ML_file "simp.ML"           (* Patched because matching won't instantiate proof *)
   103 
   104 ML \<open>
   105 structure Cla = Classical
   106 (
   107   val sizef = size_of_thm
   108   val mp = @{thm mp}
   109   val not_elim = @{thm notE}
   110   val swap = @{thm swap}
   111   fun hyp_subst_tacs ctxt = [hyp_subst_tac ctxt]
   112 );
   113 open Cla;
   114 
   115 (*Propositional rules
   116   -- iffCE might seem better, but in the examples in ex/cla
   117      run about 7% slower than with iffE*)
   118 val prop_cs =
   119   empty_cs addSIs [@{thm refl}, @{thm TrueI}, @{thm conjI}, @{thm disjCI},
   120       @{thm impI}, @{thm notI}, @{thm iffI}]
   121     addSEs [@{thm conjE}, @{thm disjE}, @{thm impCE}, @{thm FalseE}, @{thm iffE}];
   122 
   123 (*Quantifier rules*)
   124 val FOLP_cs =
   125   prop_cs addSIs [@{thm allI}] addIs [@{thm exI}, @{thm ex1I}]
   126     addSEs [@{thm exE}, @{thm ex1E}] addEs [@{thm allE}];
   127 
   128 val FOLP_dup_cs =
   129   prop_cs addSIs [@{thm allI}] addIs [@{thm exCI}, @{thm ex1I}]
   130     addSEs [@{thm exE}, @{thm ex1E}] addEs [@{thm all_dupE}];
   131 \<close>
   132 
   133 schematic_goal cla_rews:
   134   "?p1 : P | ~P"
   135   "?p2 : ~P | P"
   136   "?p3 : ~ ~ P <-> P"
   137   "?p4 : (~P --> P) <-> P"
   138   apply (tactic \<open>ALLGOALS (Cla.fast_tac @{context} FOLP_cs)\<close>)
   139   done
   140 
   141 ML_file "simpdata.ML"
   142 
   143 end