src/HOL/Fun.ML
 author paulson Mon Oct 07 10:28:44 1996 +0200 (1996-10-07) changeset 2056 93c093620c28 parent 2031 03a843f0f447 child 2499 0bc87b063447 permissions -rw-r--r--
Removed commands made redundant by new one-point rules
```     1 (*  Title:      HOL/Fun
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
```
```     4     Copyright   1993  University of Cambridge
```
```     5
```
```     6 Lemmas about functions.
```
```     7 *)
```
```     8
```
```     9 goal Fun.thy "(f = g) = (!x. f(x)=g(x))";
```
```    10 by (rtac iffI 1);
```
```    11 by (Asm_simp_tac 1);
```
```    12 by (rtac ext 1 THEN Asm_simp_tac 1);
```
```    13 qed "expand_fun_eq";
```
```    14
```
```    15 val prems = goal Fun.thy
```
```    16     "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)";
```
```    17 by (rtac (arg_cong RS box_equals) 1);
```
```    18 by (REPEAT (resolve_tac (prems@[refl]) 1));
```
```    19 qed "apply_inverse";
```
```    20
```
```    21
```
```    22 (*** Image of a set under a function ***)
```
```    23
```
```    24 (*Frequently b does not have the syntactic form of f(x).*)
```
```    25 val prems = goalw Fun.thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
```
```    26 by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
```
```    27 qed "image_eqI";
```
```    28
```
```    29 bind_thm ("imageI", refl RS image_eqI);
```
```    30
```
```    31 (*The eta-expansion gives variable-name preservation.*)
```
```    32 val major::prems = goalw Fun.thy [image_def]
```
```    33     "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P";
```
```    34 by (rtac (major RS CollectD RS bexE) 1);
```
```    35 by (REPEAT (ares_tac prems 1));
```
```    36 qed "imageE";
```
```    37
```
```    38 goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)";
```
```    39 by (rtac set_ext 1);
```
```    40 by (fast_tac (!claset addIs [imageI] addSEs [imageE]) 1);
```
```    41 qed "image_compose";
```
```    42
```
```    43 goal Fun.thy "f``(A Un B) = f``A Un f``B";
```
```    44 by (rtac set_ext 1);
```
```    45 by (fast_tac (!claset addIs [imageI,UnCI] addSEs [imageE,UnE]) 1);
```
```    46 qed "image_Un";
```
```    47
```
```    48 (*** Range of a function -- just a translation for image! ***)
```
```    49
```
```    50 goal Fun.thy "!!b. b=f(x) ==> b : range(f)";
```
```    51 by (EVERY1 [etac image_eqI, rtac UNIV_I]);
```
```    52 bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
```
```    53
```
```    54 bind_thm ("rangeI", UNIV_I RS imageI);
```
```    55
```
```    56 val [major,minor] = goal Fun.thy
```
```    57     "[| b : range(%x.f(x));  !!x. b=f(x) ==> P |] ==> P";
```
```    58 by (rtac (major RS imageE) 1);
```
```    59 by (etac minor 1);
```
```    60 qed "rangeE";
```
```    61 (*** inj(f): f is a one-to-one function ***)
```
```    62
```
```    63 val prems = goalw Fun.thy [inj_def]
```
```    64     "[| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f)";
```
```    65 by (fast_tac (!claset addIs prems) 1);
```
```    66 qed "injI";
```
```    67
```
```    68 val [major] = goal Fun.thy "(!!x. g(f(x)) = x) ==> inj(f)";
```
```    69 by (rtac injI 1);
```
```    70 by (etac (arg_cong RS box_equals) 1);
```
```    71 by (rtac major 1);
```
```    72 by (rtac major 1);
```
```    73 qed "inj_inverseI";
```
```    74
```
```    75 val [major,minor] = goalw Fun.thy [inj_def]
```
```    76     "[| inj(f); f(x) = f(y) |] ==> x=y";
```
```    77 by (rtac (major RS spec RS spec RS mp) 1);
```
```    78 by (rtac minor 1);
```
```    79 qed "injD";
```
```    80
```
```    81 (*Useful with the simplifier*)
```
```    82 val [major] = goal Fun.thy "inj(f) ==> (f(x) = f(y)) = (x=y)";
```
```    83 by (rtac iffI 1);
```
```    84 by (etac (major RS injD) 1);
```
```    85 by (etac arg_cong 1);
```
```    86 qed "inj_eq";
```
```    87
```
```    88 val [major] = goal Fun.thy "inj(f) ==> (@x.f(x)=f(y)) = y";
```
```    89 by (rtac (major RS injD) 1);
```
```    90 by (rtac selectI 1);
```
```    91 by (rtac refl 1);
```
```    92 qed "inj_select";
```
```    93
```
```    94 (*A one-to-one function has an inverse (given using select).*)
```
```    95 val [major] = goalw Fun.thy [Inv_def] "inj(f) ==> Inv f (f x) = x";
```
```    96 by (EVERY1 [rtac (major RS inj_select)]);
```
```    97 qed "Inv_f_f";
```
```    98
```
```    99 (* Useful??? *)
```
```   100 val [oneone,minor] = goal Fun.thy
```
```   101     "[| inj(f); !!y. y: range(f) ==> P(Inv f y) |] ==> P(x)";
```
```   102 by (res_inst_tac [("t", "x")] (oneone RS (Inv_f_f RS subst)) 1);
```
```   103 by (rtac (rangeI RS minor) 1);
```
```   104 qed "inj_transfer";
```
```   105
```
```   106
```
```   107 (*** inj_onto f A: f is one-to-one over A ***)
```
```   108
```
```   109 val prems = goalw Fun.thy [inj_onto_def]
```
```   110     "(!! x y. [| f(x) = f(y);  x:A;  y:A |] ==> x=y) ==> inj_onto f A";
```
```   111 by (fast_tac (!claset addIs prems addSIs [ballI]) 1);
```
```   112 qed "inj_ontoI";
```
```   113
```
```   114 val [major] = goal Fun.thy
```
```   115     "(!!x. x:A ==> g(f(x)) = x) ==> inj_onto f A";
```
```   116 by (rtac inj_ontoI 1);
```
```   117 by (etac (apply_inverse RS trans) 1);
```
```   118 by (REPEAT (eresolve_tac [asm_rl,major] 1));
```
```   119 qed "inj_onto_inverseI";
```
```   120
```
```   121 val major::prems = goalw Fun.thy [inj_onto_def]
```
```   122     "[| inj_onto f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y";
```
```   123 by (rtac (major RS bspec RS bspec RS mp) 1);
```
```   124 by (REPEAT (resolve_tac prems 1));
```
```   125 qed "inj_ontoD";
```
```   126
```
```   127 goal Fun.thy "!!x y.[| inj_onto f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)";
```
```   128 by (fast_tac (!claset addSEs [inj_ontoD]) 1);
```
```   129 qed "inj_onto_iff";
```
```   130
```
```   131 val major::prems = goal Fun.thy
```
```   132     "[| inj_onto f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)";
```
```   133 by (rtac contrapos 1);
```
```   134 by (etac (major RS inj_ontoD) 2);
```
```   135 by (REPEAT (resolve_tac prems 1));
```
```   136 qed "inj_onto_contraD";
```
```   137
```
```   138
```
```   139 (*** Lemmas about inj ***)
```
```   140
```
```   141 val prems = goalw Fun.thy [o_def]
```
```   142     "[| inj(f);  inj_onto g (range f) |] ==> inj(g o f)";
```
```   143 by (cut_facts_tac prems 1);
```
```   144 by (fast_tac (!claset addIs [injI,rangeI]
```
```   145                      addEs [injD,inj_ontoD]) 1);
```
```   146 qed "comp_inj";
```
```   147
```
```   148 val [prem] = goal Fun.thy "inj(f) ==> inj_onto f A";
```
```   149 by (fast_tac (!claset addIs [prem RS injD, inj_ontoI]) 1);
```
```   150 qed "inj_imp";
```
```   151
```
```   152 val [prem] = goalw Fun.thy [Inv_def] "y : range(f) ==> f(Inv f y) = y";
```
```   153 by (EVERY1 [rtac (prem RS rangeE), rtac selectI, etac sym]);
```
```   154 qed "f_Inv_f";
```
```   155
```
```   156 val prems = goal Fun.thy
```
```   157     "[| Inv f x=Inv f y; x: range(f);  y: range(f) |] ==> x=y";
```
```   158 by (rtac (arg_cong RS box_equals) 1);
```
```   159 by (REPEAT (resolve_tac (prems @ [f_Inv_f]) 1));
```
```   160 qed "Inv_injective";
```
```   161
```
```   162 val prems = goal Fun.thy
```
```   163     "[| inj(f);  A<=range(f) |] ==> inj_onto (Inv f) A";
```
```   164 by (cut_facts_tac prems 1);
```
```   165 by (fast_tac (!claset addIs [inj_ontoI]
```
```   166                      addEs [Inv_injective,injD,subsetD]) 1);
```
```   167 qed "inj_onto_Inv";
```
```   168
```
```   169
```
```   170 (*** Set reasoning tools ***)
```
```   171
```
```   172 AddSIs [ballI, PowI, subsetI, InterI, INT_I, INT1_I, CollectI,
```
```   173             ComplI, IntI, DiffI, UnCI, insertCI];
```
```   174 AddIs  [bexI, UnionI, UN_I, UN1_I, imageI, rangeI];
```
```   175 AddSEs [bexE, make_elim PowD, UnionE, UN_E, UN1_E, DiffE,
```
```   176             make_elim singleton_inject,
```
```   177             CollectE, ComplE, IntE, UnE, insertE, imageE, rangeE, emptyE];
```
```   178 AddEs  [ballE, InterD, InterE, INT_D, INT_E, make_elim INT1_D,
```
```   179             subsetD, subsetCE];
```
```   180
```
```   181 val set_cs = !claset delrules [equalityI];
```
```   182
```
```   183
```