src/HOL/subset.ML
author nipkow
Tue Apr 08 10:48:42 1997 +0200 (1997-04-08)
changeset 2919 953a47dc0519
parent 2893 2ee005e46d6d
child 4159 4aff9b7e5597
permissions -rw-r--r--
Dep. on Provers/nat_transitive
     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 (Blast_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 goal Set.thy "!!B. B:A ==> Inter(A) <= B";
    58 by (Blast_tac 1);
    59 qed "Inter_lower";
    60 
    61 val [prem] = goal Set.thy
    62     "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)";
    63 by (rtac (InterI RS subsetI) 1);
    64 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    65 qed "Inter_greatest";
    66 
    67 val prems = goal Set.thy "a:A ==> (INT x:A. B(x)) <= B(a)";
    68 by (rtac subsetI 1);
    69 by (REPEAT (resolve_tac prems 1 ORELSE etac INT_D 1));
    70 qed "INT_lower";
    71 
    72 val [prem] = goal Set.thy
    73     "[| !!x. x:A ==> C<=B(x) |] ==> C <= (INT x:A. B(x))";
    74 by (rtac (INT_I RS subsetI) 1);
    75 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    76 qed "INT_greatest";
    77 
    78 goal Set.thy "(INT x. B(x)) <= B(a)";
    79 by (Blast_tac 1);
    80 qed "INT1_lower";
    81 
    82 val [prem] = goal Set.thy
    83     "[| !!x. C<=B(x) |] ==> C <= (INT x. B(x))";
    84 by (rtac (INT1_I RS subsetI) 1);
    85 by (REPEAT (eresolve_tac [asm_rl, prem RS subsetD] 1));
    86 qed "INT1_greatest";
    87 
    88 (*** Finite Union -- the least upper bound of 2 sets ***)
    89 
    90 goal Set.thy "A <= A Un B";
    91 by (Blast_tac 1);
    92 qed "Un_upper1";
    93 
    94 goal Set.thy "B <= A Un B";
    95 by (Blast_tac 1);
    96 qed "Un_upper2";
    97 
    98 goal Set.thy "!!C. [| A<=C;  B<=C |] ==> A Un B <= C";
    99 by (Blast_tac 1);
   100 qed "Un_least";
   101 
   102 (*** Finite Intersection -- the greatest lower bound of 2 sets *)
   103 
   104 goal Set.thy "A Int B <= A";
   105 by (Blast_tac 1);
   106 qed "Int_lower1";
   107 
   108 goal Set.thy "A Int B <= B";
   109 by (Blast_tac 1);
   110 qed "Int_lower2";
   111 
   112 goal Set.thy "!!C. [| C<=A;  C<=B |] ==> C <= A Int B";
   113 by (Blast_tac 1);
   114 qed "Int_greatest";
   115 
   116 (*** Set difference ***)
   117 
   118 qed_goal "Diff_subset" Set.thy "A-B <= (A::'a set)"
   119  (fn _ => [ (Blast_tac 1) ]);
   120 
   121 (*** Monotonicity ***)
   122 
   123 val [prem] = goal Set.thy "mono(f) ==> f(A) Un f(B) <= f(A Un B)";
   124 by (rtac Un_least 1);
   125 by (rtac (Un_upper1 RS (prem RS monoD)) 1);
   126 by (rtac (Un_upper2 RS (prem RS monoD)) 1);
   127 qed "mono_Un";
   128 
   129 val [prem] = goal Set.thy "mono(f) ==> f(A Int B) <= f(A) Int f(B)";
   130 by (rtac Int_greatest 1);
   131 by (rtac (Int_lower1 RS (prem RS monoD)) 1);
   132 by (rtac (Int_lower2 RS (prem RS monoD)) 1);
   133 qed "mono_Int";