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--
```     1 (*  Title:      HOL/subset
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1991  University of Cambridge
```
```     5
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```     6 Derived rules involving subsets
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```     7 Union and Intersection as lattice operations
```
```     8 *)
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```     9
```
```    10 (*** insert ***)
```
```    11
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```    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";
```