src/HOL/Integ/Equiv.ML
 author clasohm Fri Mar 24 12:30:35 1995 +0100 (1995-03-24) changeset 972 e61b058d58d2 parent 925 15539deb6863 child 1045 0cdf86973c49 permissions -rw-r--r--
changed syntax of tuples from <..., ...> to (..., ...)
```     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 (*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
```
```    14
```
```    15 (** first half: equiv A r ==> converse(r) O r = r **)
```
```    16
```
```    17 goalw Equiv.thy [trans_def,sym_def,converse_def]
```
```    18     "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
```
```    19 by (fast_tac (comp_cs addSEs [converseD]) 1);
```
```    20 qed "sym_trans_comp_subset";
```
```    21
```
```    22 val [major,minor]=goal Equiv.thy "[|(x,y):r; z=(x,y)|] ==>  z:r";
```
```    23 by (simp_tac (prod_ss addsimps [minor]) 1);
```
```    24 by (rtac major 1);
```
```    25 qed "BreakPair";
```
```    26
```
```    27 val [major]=goal Equiv.thy "[|? x y. (x,y):r & z=(x,y)|] ==>  z:r";
```
```    28 by (resolve_tac [major RS exE] 1);
```
```    29 by (etac exE 1);
```
```    30 by (etac conjE 1);
```
```    31 by (asm_simp_tac (prod_ss addsimps [minor]) 1);
```
```    32 qed "BreakPair1";
```
```    33
```
```    34 val [major,minor]=goal Equiv.thy "[|z:r; z=(x,y)|] ==> (x,y):r";
```
```    35 by (simp_tac (prod_ss addsimps [minor RS sym]) 1);
```
```    36 by (rtac major 1);
```
```    37 qed "BuildPair";
```
```    38
```
```    39 val [major]=goal Equiv.thy "[|? z:r. (x,y)=z|] ==> (x,y):r";
```
```    40 by (resolve_tac [major RS bexE] 1);
```
```    41 by (asm_simp_tac (prod_ss addsimps []) 1);
```
```    42 qed "BuildPair1";
```
```    43
```
```    44 val rel_pair_cs = rel_cs addIs [BuildPair1]
```
```    45                          addEs [BreakPair1];
```
```    46
```
```    47 goalw Equiv.thy [refl_def,converse_def]
```
```    48     "!!A r. refl A r ==> r <= converse(r) O r";
```
```    49 by (step_tac comp_cs 1);
```
```    50 by (dtac subsetD 1);
```
```    51 by (assume_tac 1);
```
```    52 by (etac SigmaE 1);
```
```    53 by (rtac BreakPair1 1);
```
```    54 by (fast_tac comp_cs 1);
```
```    55 qed "refl_comp_subset";
```
```    56
```
```    57 goalw Equiv.thy [equiv_def]
```
```    58     "!!A r. equiv A r ==> converse(r) O r = r";
```
```    59 by (rtac equalityI 1);
```
```    60 by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
```
```    61      ORELSE etac conjE 1));
```
```    62 qed "equiv_comp_eq";
```
```    63
```
```    64 (*second half*)
```
```    65 goalw Equiv.thy [equiv_def,refl_def,sym_def,trans_def]
```
```    66     "!!A r. [| converse(r) O r = r;  Domain(r) = A |] ==> equiv A r";
```
```    67 by (etac equalityE 1);
```
```    68 by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
```
```    69 by (safe_tac set_cs);
```
```    70 by (fast_tac (set_cs addSIs [converseI] addIs [compI]) 3);
```
```    71 by (fast_tac (set_cs addSIs [converseI] addIs [compI] addSEs [DomainE]) 2);
```
```    72 by (fast_tac (rel_pair_cs addSEs [SigmaE] addSIs [SigmaI]) 1);
```
```    73 by (dtac subsetD 1);
```
```    74 by (dtac subsetD 1);
```
```    75 by (fast_tac rel_cs 1);
```
```    76 by (fast_tac rel_cs 1);
```
```    77 by flexflex_tac;
```
```    78 by (dtac subsetD 1);
```
```    79 by (fast_tac converse_cs 2);
```
```    80 by (fast_tac converse_cs 1);
```
```    81 qed "comp_equivI";
```
```    82
```
```    83 (** Equivalence classes **)
```
```    84
```
```    85 (*Lemma for the next result*)
```
```    86 goalw Equiv.thy [equiv_def,trans_def,sym_def]
```
```    87     "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} <= r^^{b}";
```
```    88 by (safe_tac rel_cs);
```
```    89 by (rtac ImageI 1);
```
```    90 by (fast_tac rel_cs 2);
```
```    91 by (fast_tac rel_cs 1);
```
```    92 qed "equiv_class_subset";
```
```    93
```
```    94 goal Equiv.thy "!!A r. [| equiv A r;  (a,b): r |] ==> r^^{a} = r^^{b}";
```
```    95 by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
```
```    96 by (rewrite_goals_tac [equiv_def,sym_def]);
```
```    97 by (fast_tac rel_cs 1);
```
```    98 qed "equiv_class_eq";
```
```    99
```
```   100 val prems = goalw Equiv.thy [equiv_def,refl_def]
```
```   101     "[| equiv A r;  a: A |] ==> a: r^^{a}";
```
```   102 by (cut_facts_tac prems 1);
```
```   103 by (fast_tac rel_cs 1);
```
```   104 qed "equiv_class_self";
```
```   105
```
```   106 (*Lemma for the next result*)
```
```   107 goalw Equiv.thy [equiv_def,refl_def]
```
```   108     "!!A r. [| equiv A r;  r^^{b} <= r^^{a};  b: A |] ==> (a,b): r";
```
```   109 by (fast_tac rel_cs 1);
```
```   110 qed "subset_equiv_class";
```
```   111
```
```   112 val prems = goal Equiv.thy
```
```   113     "[| r^^{a} = r^^{b};  equiv A r;  b: A |] ==> (a,b): r";
```
```   114 by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
```
```   115 qed "eq_equiv_class";
```
```   116
```
```   117 (*thus r^^{a} = r^^{b} as well*)
```
```   118 goalw Equiv.thy [equiv_def,trans_def,sym_def]
```
```   119     "!!A r. [| equiv A r;  x: (r^^{a} Int r^^{b}) |] ==> (a,b): r";
```
```   120 by (fast_tac rel_cs 1);
```
```   121 qed "equiv_class_nondisjoint";
```
```   122
```
```   123 val [major] = goalw Equiv.thy [equiv_def,refl_def]
```
```   124     "equiv A r ==> r <= Sigma A (%x.A)";
```
```   125 by (rtac (major RS conjunct1 RS conjunct1) 1);
```
```   126 qed "equiv_type";
```
```   127
```
```   128 goal Equiv.thy
```
```   129     "!!A r. equiv A r ==> ((x,y): r) = (r^^{x} = r^^{y} & x:A & y:A)";
```
```   130 by (safe_tac rel_cs);
```
```   131 by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
```
```   132 by ((rtac eq_equiv_class 3) THEN
```
```   133     (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
```
```   134 by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
```
```   135     (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1));
```
```   136 by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
```
```   137     (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1));
```
```   138 qed "equiv_class_eq_iff";
```
```   139
```
```   140 goal Equiv.thy
```
```   141     "!!A r. [| equiv A r;  x: A;  y: A |] ==> (r^^{x} = r^^{y}) = ((x,y): r)";
```
```   142 by (safe_tac rel_cs);
```
```   143 by ((rtac eq_equiv_class 1) THEN
```
```   144     (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
```
```   145 by ((rtac equiv_class_eq 1) THEN
```
```   146     (assume_tac 1) THEN (assume_tac 1));
```
```   147 qed "eq_equiv_class_iff";
```
```   148
```
```   149 (*** Quotients ***)
```
```   150
```
```   151 (** Introduction/elimination rules -- needed? **)
```
```   152
```
```   153 val prems = goalw Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r";
```
```   154 by (rtac UN_I 1);
```
```   155 by (resolve_tac prems 1);
```
```   156 by (rtac singletonI 1);
```
```   157 qed "quotientI";
```
```   158
```
```   159 val [major,minor] = goalw Equiv.thy [quotient_def]
```
```   160     "[| X:(A/r);  !!x. [| X = r^^{x};  x:A |] ==> P |] 	\
```
```   161 \    ==> P";
```
```   162 by (resolve_tac [major RS UN_E] 1);
```
```   163 by (rtac minor 1);
```
```   164 by (assume_tac 2);
```
```   165 by (fast_tac rel_cs 1);
```
```   166 qed "quotientE";
```
```   167
```
```   168 (** Not needed by Theory Integ --> bypassed **)
```
```   169 (**goalw Equiv.thy [equiv_def,refl_def,quotient_def]
```
```   170     "!!A r. equiv A r ==> Union(A/r) = A";
```
```   171 by (fast_tac eq_cs 1);
```
```   172 qed "Union_quotient";
```
```   173 **)
```
```   174
```
```   175 (** Not needed by Theory Integ --> bypassed **)
```
```   176 (*goalw Equiv.thy [quotient_def]
```
```   177     "!!A r. [| equiv A r;  X: A/r;  Y: A/r |] ==> X=Y | (X Int Y <= 0)";
```
```   178 by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
```
```   179 by (assume_tac 1);
```
```   180 by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
```
```   181 by (fast_tac ZF_cs 1);
```
```   182 qed "quotient_disj";
```
```   183 **)
```
```   184
```
```   185 (**** Defining unary operations upon equivalence classes ****)
```
```   186
```
```   187 (* theorem needed to prove UN_equiv_class *)
```
```   188 goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
```
```   189 by (fast_tac (eq_cs addSEs [equalityE]) 1);
```
```   190 qed "UN_singleton_lemma";
```
```   191 val UN_singleton = ballI RSN (2,UN_singleton_lemma);
```
```   192
```
```   193
```
```   194 (** These proofs really require as local premises
```
```   195      equiv A r;  congruent r b
```
```   196 **)
```
```   197
```
```   198 (*Conversion rule*)
```
```   199 val prems as [equivA,bcong,_] = goal Equiv.thy
```
```   200     "[| equiv A r;  congruent r b;  a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)";
```
```   201 by (cut_facts_tac prems 1);
```
```   202 by (rtac UN_singleton 1);
```
```   203 by (rtac equiv_class_self 1);
```
```   204 by (assume_tac 1);
```
```   205 by (assume_tac 1);
```
```   206 by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
```
```   207 by (fast_tac rel_cs 1);
```
```   208 qed "UN_equiv_class";
```
```   209
```
```   210 (*Resolve th against the "local" premises*)
```
```   211 val localize = RSLIST [equivA,bcong];
```
```   212
```
```   213 (*type checking of  UN x:r``{a}. b(x) *)
```
```   214 val _::_::prems = goalw Equiv.thy [quotient_def]
```
```   215     "[| equiv A r;  congruent r b;  X: A/r;	\
```
```   216 \	!!x.  x : A ==> b(x) : B |] 	\
```
```   217 \    ==> (UN x:X. b(x)) : B";
```
```   218 by (cut_facts_tac prems 1);
```
```   219 by (safe_tac rel_cs);
```
```   220 by (rtac (localize UN_equiv_class RS ssubst) 1);
```
```   221 by (REPEAT (ares_tac prems 1));
```
```   222 qed "UN_equiv_class_type";
```
```   223
```
```   224 (*Sufficient conditions for injectiveness.  Could weaken premises!
```
```   225   major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
```
```   226 *)
```
```   227 val _::_::prems = goalw Equiv.thy [quotient_def]
```
```   228     "[| equiv A r;   congruent r b;  \
```
```   229 \       (UN x:X. b(x))=(UN y:Y. b(y));  X: A/r;  Y: A/r;  \
```
```   230 \       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |] 	\
```
```   231 \    ==> X=Y";
```
```   232 by (cut_facts_tac prems 1);
```
```   233 by (safe_tac rel_cs);
```
```   234 by (rtac (equivA RS equiv_class_eq) 1);
```
```   235 by (REPEAT (ares_tac prems 1));
```
```   236 by (etac box_equals 1);
```
```   237 by (REPEAT (ares_tac [localize UN_equiv_class] 1));
```
```   238 qed "UN_equiv_class_inject";
```
```   239
```
```   240
```
```   241 (**** Defining binary operations upon equivalence classes ****)
```
```   242
```
```   243
```
```   244 goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
```
```   245     "!!A r. [| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
```
```   246 by (fast_tac rel_cs 1);
```
```   247 qed "congruent2_implies_congruent";
```
```   248
```
```   249 val equivA::prems = goalw Equiv.thy [congruent_def]
```
```   250     "[| equiv A r;  congruent2 r b;  a: A |] ==> \
```
```   251 \    congruent r (%x1. UN x2:r^^{a}. b x1 x2)";
```
```   252 by (cut_facts_tac (equivA::prems) 1);
```
```   253 by (safe_tac rel_cs);
```
```   254 by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
```
```   255 by (assume_tac 1);
```
```   256 by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class,
```
```   257 				 congruent2_implies_congruent]) 1);
```
```   258 by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
```
```   259 by (fast_tac rel_cs 1);
```
```   260 qed "congruent2_implies_congruent_UN";
```
```   261
```
```   262 val prems as equivA::_ = goal Equiv.thy
```
```   263     "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
```
```   264 \    ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b x1 x2) = b a1 a2";
```
```   265 by (cut_facts_tac prems 1);
```
```   266 by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class,
```
```   267 				    congruent2_implies_congruent,
```
```   268 				    congruent2_implies_congruent_UN]) 1);
```
```   269 qed "UN_equiv_class2";
```
```   270
```
```   271 (*type checking*)
```
```   272 val prems = goalw Equiv.thy [quotient_def]
```
```   273     "[| equiv A r;  congruent2 r b;  \
```
```   274 \       X1: A/r;  X2: A/r;	\
```
```   275 \	!!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
```
```   276 \    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
```
```   277 by (cut_facts_tac prems 1);
```
```   278 by (safe_tac rel_cs);
```
```   279 by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
```
```   280 			     congruent2_implies_congruent_UN,
```
```   281 			     congruent2_implies_congruent, quotientI]) 1));
```
```   282 qed "UN_equiv_class_type2";
```
```   283
```
```   284
```
```   285 (*Suggested by John Harrison -- the two subproofs may be MUCH simpler
```
```   286   than the direct proof*)
```
```   287 val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def]
```
```   288     "[| equiv A r;	\
```
```   289 \       !! y z w. [| w: A;  (y,z) : r |] ==> b y w = b z w;      \
```
```   290 \       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
```
```   291 \    |] ==> congruent2 r b";
```
```   292 by (cut_facts_tac prems 1);
```
```   293 by (safe_tac rel_cs);
```
```   294 by (rtac trans 1);
```
```   295 by (REPEAT (ares_tac prems 1
```
```   296      ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
```
```   297 qed "congruent2I";
```
```   298
```
```   299 val [equivA,commute,congt] = goal Equiv.thy
```
```   300     "[| equiv A r;	\
```
```   301 \       !! y z. [| y: A;  z: A |] ==> b y z = b z y;        \
```
```   302 \       !! y z w. [| w: A;  (y,z): r |] ==> b w y = b w z	\
```
```   303 \    |] ==> congruent2 r b";
```
```   304 by (resolve_tac [equivA RS congruent2I] 1);
```
```   305 by (rtac (commute RS trans) 1);
```
```   306 by (rtac (commute RS trans RS sym) 3);
```
```   307 by (rtac sym 5);
```
```   308 by (REPEAT (ares_tac [congt] 1
```
```   309      ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
```
```   310 qed "congruent2_commuteI";
```
```   311
```