src/HOL/Integ/Equiv.ML
author berghofe
Tue Jul 30 17:33:26 1996 +0200 (1996-07-30)
changeset 1894 c2c8279d40f0
parent 1844 791e79473966
child 1978 e7df069acb74
permissions -rw-r--r--
Classical tactics now use default claset.
     1 (*  Title:      Equiv.ML
     2     ID:         $Id$
     3     Authors:    Riccardo Mattolini, Dip. Sistemi e Informatica
     4                 Lawrence C Paulson, Cambridge University Computer Laboratory
     5     Copyright   1994 Universita' di Firenze
     6     Copyright   1993  University of Cambridge
     7 
     8 Equivalence relations in HOL Set Theory 
     9 *)
    10 
    11 open Equiv;
    12 
    13 Delrules [equalityI];
    14 
    15 (*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
    16 
    17 (** first half: equiv A r ==> converse(r) O r = r **)
    18 
    19 goalw Equiv.thy [trans_def,sym_def,converse_def]
    20     "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
    21 by (fast_tac (!claset addSEs [converseD]) 1);
    22 qed "sym_trans_comp_subset";
    23 
    24 goalw Equiv.thy [refl_def]
    25     "!!A r. refl A r ==> r <= converse(r) O r";
    26 by (fast_tac (!claset addIs [compI]) 1);
    27 qed "refl_comp_subset";
    28 
    29 goalw Equiv.thy [equiv_def]
    30     "!!A r. equiv A r ==> converse(r) O r = r";
    31 by (rtac equalityI 1);
    32 by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
    33      ORELSE etac conjE 1));
    34 qed "equiv_comp_eq";
    35 
    36 (*second half*)
    37 goalw Equiv.thy [equiv_def,refl_def,sym_def,trans_def]
    38     "!!A r. [| converse(r) O r = r;  Domain(r) = A |] ==> equiv A r";
    39 by (etac equalityE 1);
    40 by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
    41 by (safe_tac (!claset));
    42 by (fast_tac (!claset addSIs [converseI] addIs [compI]) 3);
    43 by (ALLGOALS (fast_tac (!claset addIs [compI] addSEs [compE])));
    44 qed "comp_equivI";
    45 
    46 (** Equivalence classes **)
    47 
    48 (*Lemma for the next result*)
    49 goalw Equiv.thy [equiv_def,trans_def,sym_def]
    50     "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} <= r^^{b}";
    51 by (safe_tac (!claset));
    52 by (rtac ImageI 1);
    53 by (Fast_tac 2);
    54 by (Fast_tac 1);
    55 qed "equiv_class_subset";
    56 
    57 goal Equiv.thy "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} = r^^{b}";
    58 by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
    59 by (rewrite_goals_tac [equiv_def,sym_def]);
    60 by (Fast_tac 1);
    61 qed "equiv_class_eq";
    62 
    63 val prems = goalw Equiv.thy [equiv_def,refl_def]
    64     "[| equiv A r;  a: A |] ==> a: r^^{a}";
    65 by (cut_facts_tac prems 1);
    66 by (Fast_tac 1);
    67 qed "equiv_class_self";
    68 
    69 (*Lemma for the next result*)
    70 goalw Equiv.thy [equiv_def,refl_def]
    71     "!!A r. [| equiv A r;  r^^{b} <= r^^{a};  b: A |] ==> (a,b): r";
    72 by (Fast_tac 1);
    73 qed "subset_equiv_class";
    74 
    75 val prems = goal Equiv.thy
    76     "[| r^^{a} = r^^{b};  equiv A r;  b: A |] ==> (a,b): r";
    77 by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
    78 qed "eq_equiv_class";
    79 
    80 (*thus r^^{a} = r^^{b} as well*)
    81 goalw Equiv.thy [equiv_def,trans_def,sym_def]
    82     "!!A r. [| equiv A r;  x: (r^^{a} Int r^^{b}) |] ==> (a,b): r";
    83 by (Fast_tac 1);
    84 qed "equiv_class_nondisjoint";
    85 
    86 val [major] = goalw Equiv.thy [equiv_def,refl_def]
    87     "equiv A r ==> r <= A Times A";
    88 by (rtac (major RS conjunct1 RS conjunct1) 1);
    89 qed "equiv_type";
    90 
    91 goal Equiv.thy
    92     "!!A r. equiv A r ==> ((x,y): r) = (r^^{x} = r^^{y} & x:A & y:A)";
    93 by (safe_tac (!claset));
    94 by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
    95 by ((rtac eq_equiv_class 3) THEN 
    96     (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
    97 by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
    98     (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1));
    99 by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
   100     (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1));
   101 qed "equiv_class_eq_iff";
   102 
   103 goal Equiv.thy
   104     "!!A r. [| equiv A r;  x: A;  y: A |] ==> (r^^{x} = r^^{y}) = ((x,y): r)";
   105 by (safe_tac (!claset));
   106 by ((rtac eq_equiv_class 1) THEN 
   107     (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
   108 by ((rtac equiv_class_eq 1) THEN 
   109     (assume_tac 1) THEN (assume_tac 1));
   110 qed "eq_equiv_class_iff";
   111 
   112 (*** Quotients ***)
   113 
   114 (** Introduction/elimination rules -- needed? **)
   115 
   116 val prems = goalw Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r";
   117 by (rtac UN_I 1);
   118 by (resolve_tac prems 1);
   119 by (rtac singletonI 1);
   120 qed "quotientI";
   121 
   122 val [major,minor] = goalw Equiv.thy [quotient_def]
   123     "[| X:(A/r);  !!x. [| X = r^^{x};  x:A |] ==> P |]  \
   124 \    ==> P";
   125 by (resolve_tac [major RS UN_E] 1);
   126 by (rtac minor 1);
   127 by (assume_tac 2);
   128 by (Fast_tac 1);
   129 qed "quotientE";
   130 
   131 (** Not needed by Theory Integ --> bypassed **)
   132 (**goalw Equiv.thy [equiv_def,refl_def,quotient_def]
   133     "!!A r. equiv A r ==> Union(A/r) = A";
   134 by (Fast_tac 1);
   135 qed "Union_quotient";
   136 **)
   137 
   138 (** Not needed by Theory Integ --> bypassed **)
   139 (*goalw Equiv.thy [quotient_def]
   140     "!!A r. [| equiv A r;  X: A/r;  Y: A/r |] ==> X=Y | (X Int Y <= 0)";
   141 by (safe_tac (!claset addSIs [equiv_class_eq]));
   142 by (assume_tac 1);
   143 by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
   144 by (Fast_tac 1);
   145 qed "quotient_disj";
   146 **)
   147 
   148 (**** Defining unary operations upon equivalence classes ****)
   149 
   150 (* theorem needed to prove UN_equiv_class *)
   151 goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
   152 by (fast_tac (!claset addSEs [equalityE] addSIs [equalityI]) 1);
   153 qed "UN_singleton_lemma";
   154 val UN_singleton = ballI RSN (2,UN_singleton_lemma);
   155 
   156 
   157 (** These proofs really require as local premises
   158      equiv A r;  congruent r b
   159 **)
   160 
   161 (*Conversion rule*)
   162 val prems as [equivA,bcong,_] = goal Equiv.thy
   163     "[| equiv A r;  congruent r b;  a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)";
   164 by (cut_facts_tac prems 1);
   165 by (rtac UN_singleton 1);
   166 by (rtac equiv_class_self 1);
   167 by (assume_tac 1);
   168 by (assume_tac 1);
   169 by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
   170 by (Fast_tac 1);
   171 qed "UN_equiv_class";
   172 
   173 (*Resolve th against the "local" premises*)
   174 val localize = RSLIST [equivA,bcong];
   175 
   176 (*type checking of  UN x:r``{a}. b(x) *)
   177 val _::_::prems = goalw Equiv.thy [quotient_def]
   178     "[| equiv A r;  congruent r b;  X: A/r;     \
   179 \       !!x.  x : A ==> b(x) : B |]     \
   180 \    ==> (UN x:X. b(x)) : B";
   181 by (cut_facts_tac prems 1);
   182 by (safe_tac (!claset));
   183 by (rtac (localize UN_equiv_class RS ssubst) 1);
   184 by (REPEAT (ares_tac prems 1));
   185 qed "UN_equiv_class_type";
   186 
   187 (*Sufficient conditions for injectiveness.  Could weaken premises!
   188   major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
   189 *)
   190 val _::_::prems = goalw Equiv.thy [quotient_def]
   191     "[| equiv A r;   congruent r b;  \
   192 \       (UN x:X. b(x))=(UN y:Y. b(y));  X: A/r;  Y: A/r;  \
   193 \       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
   194 \    ==> X=Y";
   195 by (cut_facts_tac prems 1);
   196 by (safe_tac ((!claset) delrules [equalityI]));
   197 by (rtac (equivA RS equiv_class_eq) 1);
   198 by (REPEAT (ares_tac prems 1));
   199 by (etac box_equals 1);
   200 by (REPEAT (ares_tac [localize UN_equiv_class] 1));
   201 qed "UN_equiv_class_inject";
   202 
   203 
   204 (**** Defining binary operations upon equivalence classes ****)
   205 
   206 
   207 goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
   208     "!!A r. [| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
   209 by (Fast_tac 1);
   210 qed "congruent2_implies_congruent";
   211 
   212 val equivA::prems = goalw Equiv.thy [congruent_def]
   213     "[| equiv A r;  congruent2 r b;  a: A |] ==> \
   214 \    congruent r (%x1. UN x2:r^^{a}. b x1 x2)";
   215 by (cut_facts_tac (equivA::prems) 1);
   216 by (safe_tac (!claset));
   217 by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
   218 by (assume_tac 1);
   219 by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
   220                                      congruent2_implies_congruent]) 1);
   221 by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
   222 by (Fast_tac 1);
   223 qed "congruent2_implies_congruent_UN";
   224 
   225 val prems as equivA::_ = goal Equiv.thy
   226     "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
   227 \    ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b x1 x2) = b a1 a2";
   228 by (cut_facts_tac prems 1);
   229 by (asm_simp_tac (!simpset addsimps [equivA RS UN_equiv_class,
   230                                      congruent2_implies_congruent,
   231                                      congruent2_implies_congruent_UN]) 1);
   232 qed "UN_equiv_class2";
   233 
   234 (*type checking*)
   235 val prems = goalw Equiv.thy [quotient_def]
   236     "[| equiv A r;  congruent2 r b;  \
   237 \       X1: A/r;  X2: A/r;      \
   238 \       !!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
   239 \    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
   240 by (cut_facts_tac prems 1);
   241 by (safe_tac (!claset));
   242 by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
   243                              congruent2_implies_congruent_UN,
   244                              congruent2_implies_congruent, quotientI]) 1));
   245 qed "UN_equiv_class_type2";
   246 
   247 
   248 (*Suggested by John Harrison -- the two subproofs may be MUCH simpler
   249   than the direct proof*)
   250 val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def]
   251     "[| equiv A r;      \
   252 \       !! y z w. [| w: A;  (y,z) : r |] ==> b y w = b z w;      \
   253 \       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
   254 \    |] ==> congruent2 r b";
   255 by (cut_facts_tac prems 1);
   256 by (safe_tac (!claset));
   257 by (rtac trans 1);
   258 by (REPEAT (ares_tac prems 1
   259      ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
   260 qed "congruent2I";
   261 
   262 val [equivA,commute,congt] = goal Equiv.thy
   263     "[| equiv A r;      \
   264 \       !! y z. [| y: A;  z: A |] ==> b y z = b z y;        \
   265 \       !! y z w. [| w: A;  (y,z): r |] ==> b w y = b w z       \
   266 \    |] ==> congruent2 r b";
   267 by (resolve_tac [equivA RS congruent2I] 1);
   268 by (rtac (commute RS trans) 1);
   269 by (rtac (commute RS trans RS sym) 3);
   270 by (rtac sym 5);
   271 by (REPEAT (ares_tac [congt] 1
   272      ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
   273 qed "congruent2_commuteI";
   274