src/HOL/subset.ML
author wenzelm
Thu Jan 23 14:19:16 1997 +0100 (1997-01-23)
changeset 2545 d10abc8c11fb
parent 2515 6ff9bd353121
child 2893 2ee005e46d6d
permissions -rw-r--r--
added AxClasses test;
     1 (*  Title:      HOL/subset
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 Derived rules involving subsets
     7 Union and Intersection as lattice operations
     8 *)
     9 
    10 (*** insert ***)
    11 
    12 qed_goal "subset_insertI" Set.thy "B <= insert a B"
    13  (fn _=> [ (rtac subsetI 1), (etac insertI2 1) ]);
    14 
    15 goal Set.thy "!!x. x ~: A ==> (A <= insert x B) = (A <= B)";
    16 by (Fast_tac 1);
    17 qed "subset_insert";
    18 
    19 (*** Big Union -- least upper bound of a set  ***)
    20 
    21 val prems = goal Set.thy
    22     "B:A ==> B <= Union(A)";
    23 by (REPEAT (ares_tac (prems@[subsetI,UnionI]) 1));
    24 qed "Union_upper";
    25 
    26 val [prem] = goal Set.thy
    27     "[| !!X. X:A ==> X<=C |] ==> Union(A) <= C";
    28 by (rtac subsetI 1);
    29 by (REPEAT (eresolve_tac [asm_rl, UnionE, prem RS subsetD] 1));
    30 qed "Union_least";
    31 
    32 (** General union **)
    33 
    34 val prems = goal Set.thy
    35     "a:A ==> B(a) <= (UN x:A. B(x))";
    36 by (REPEAT (ares_tac (prems@[UN_I RS subsetI]) 1));
    37 qed "UN_upper";
    38 
    39 val [prem] = goal Set.thy
    40     "[| !!x. x:A ==> B(x)<=C |] ==> (UN x:A. B(x)) <= C";
    41 by (rtac subsetI 1);
    42 by (REPEAT (eresolve_tac [asm_rl, UN_E, prem RS subsetD] 1));
    43 qed "UN_least";
    44 
    45 goal Set.thy "B(a) <= (UN x. B(x))";
    46 by (REPEAT (ares_tac [UN1_I RS subsetI] 1));
    47 qed "UN1_upper";
    48 
    49 val [prem] = goal Set.thy "[| !!x. B(x)<=C |] ==> (UN x. B(x)) <= C";
    50 by (rtac subsetI 1);
    51 by (REPEAT (eresolve_tac [asm_rl, UN1_E, prem RS subsetD] 1));
    52 qed "UN1_least";
    53 
    54 
    55 (*** Big Intersection -- greatest lower bound of a set ***)
    56 
    57 val prems = goal Set.thy "B:A ==> Inter(A) <= B";
    58 by (rtac subsetI 1);
    59 by (REPEAT (resolve_tac prems 1 ORELSE etac InterD 1));
    60 qed "Inter_lower";
    61 
    62 val [prem] = goal Set.thy
    63     "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)";
    64 by (rtac (InterI RS subsetI) 1);
    65 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    66 qed "Inter_greatest";
    67 
    68 val prems = goal Set.thy "a:A ==> (INT x:A. B(x)) <= B(a)";
    69 by (rtac subsetI 1);
    70 by (REPEAT (resolve_tac prems 1 ORELSE etac INT_D 1));
    71 qed "INT_lower";
    72 
    73 val [prem] = goal Set.thy
    74     "[| !!x. x:A ==> C<=B(x) |] ==> C <= (INT x:A. B(x))";
    75 by (rtac (INT_I RS subsetI) 1);
    76 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    77 qed "INT_greatest";
    78 
    79 goal Set.thy "(INT x. B(x)) <= B(a)";
    80 by (rtac subsetI 1);
    81 by (REPEAT (resolve_tac prems 1 ORELSE etac INT1_D 1));
    82 qed "INT1_lower";
    83 
    84 val [prem] = goal Set.thy
    85     "[| !!x. C<=B(x) |] ==> C <= (INT x. B(x))";
    86 by (rtac (INT1_I RS subsetI) 1);
    87 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    88 qed "INT1_greatest";
    89 
    90 (*** Finite Union -- the least upper bound of 2 sets ***)
    91 
    92 goal Set.thy "A <= A Un B";
    93 by (REPEAT (ares_tac [subsetI,UnI1] 1));
    94 qed "Un_upper1";
    95 
    96 goal Set.thy "B <= A Un B";
    97 by (REPEAT (ares_tac [subsetI,UnI2] 1));
    98 qed "Un_upper2";
    99 
   100 val prems = goal Set.thy "[| A<=C;  B<=C |] ==> A Un B <= C";
   101 by (cut_facts_tac prems 1);
   102 by (DEPTH_SOLVE (ares_tac [subsetI] 1 
   103           ORELSE eresolve_tac [UnE,subsetD] 1));
   104 qed "Un_least";
   105 
   106 (*** Finite Intersection -- the greatest lower bound of 2 sets *)
   107 
   108 goal Set.thy "A Int B <= A";
   109 by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
   110 qed "Int_lower1";
   111 
   112 goal Set.thy "A Int B <= B";
   113 by (REPEAT (ares_tac [subsetI] 1 ORELSE etac IntE 1));
   114 qed "Int_lower2";
   115 
   116 val prems = goal Set.thy "[| C<=A;  C<=B |] ==> C <= A Int B";
   117 by (cut_facts_tac prems 1);
   118 by (REPEAT (ares_tac [subsetI,IntI] 1
   119      ORELSE etac subsetD 1));
   120 qed "Int_greatest";
   121 
   122 (*** Set difference ***)
   123 
   124 qed_goal "Diff_subset" Set.thy "A-B <= (A::'a set)"
   125  (fn _ => [ (REPEAT (ares_tac [subsetI] 1 ORELSE etac DiffE 1)) ]);
   126 
   127 (*** Monotonicity ***)
   128 
   129 val [prem] = goal Set.thy "mono(f) ==> f(A) Un f(B) <= f(A Un B)";
   130 by (rtac Un_least 1);
   131 by (rtac (Un_upper1 RS (prem RS monoD)) 1);
   132 by (rtac (Un_upper2 RS (prem RS monoD)) 1);
   133 qed "mono_Un";
   134 
   135 val [prem] = goal Set.thy "mono(f) ==> f(A Int B) <= f(A) Int f(B)";
   136 by (rtac Int_greatest 1);
   137 by (rtac (Int_lower1 RS (prem RS monoD)) 1);
   138 by (rtac (Int_lower2 RS (prem RS monoD)) 1);
   139 qed "mono_Int";