src/HOL/Algebra/Congruence.thy
 author ballarin Tue Dec 16 21:10:53 2008 +0100 (2008-12-16) changeset 29237 e90d9d51106b parent 27717 21bbd410ba04 child 35355 613e133966ea permissions -rw-r--r--
More porting to new locales.
```     1 (*
```
```     2   Title:  Algebra/Congruence.thy
```
```     3   Author: Clemens Ballarin, started 3 January 2008
```
```     4   Copyright: Clemens Ballarin
```
```     5 *)
```
```     6
```
```     7 theory Congruence imports Main begin
```
```     8
```
```     9 section {* Objects *}
```
```    10
```
```    11 subsection {* Structure with Carrier Set. *}
```
```    12
```
```    13 record 'a partial_object =
```
```    14   carrier :: "'a set"
```
```    15
```
```    16
```
```    17 subsection {* Structure with Carrier and Equivalence Relation @{text eq} *}
```
```    18
```
```    19 record 'a eq_object = "'a partial_object" +
```
```    20   eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".=\<index>" 50)
```
```    21
```
```    22 constdefs (structure S)
```
```    23   elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<in>\<index>" 50)
```
```    24   "x .\<in> A \<equiv> (\<exists>y \<in> A. x .= y)"
```
```    25
```
```    26   set_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.=}\<index>" 50)
```
```    27   "A {.=} B == ((\<forall>x \<in> A. x .\<in> B) \<and> (\<forall>x \<in> B. x .\<in> A))"
```
```    28
```
```    29   eq_class_of :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set" ("class'_of\<index> _")
```
```    30   "class_of x == {y \<in> carrier S. x .= y}"
```
```    31
```
```    32   eq_closure_of :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set" ("closure'_of\<index> _")
```
```    33   "closure_of A == {y \<in> carrier S. y .\<in> A}"
```
```    34
```
```    35   eq_is_closed :: "_ \<Rightarrow> 'a set \<Rightarrow> bool" ("is'_closed\<index> _")
```
```    36   "is_closed A == (A \<subseteq> carrier S \<and> closure_of A = A)"
```
```    37
```
```    38 syntax
```
```    39   not_eq :: "_ \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".\<noteq>\<index>" 50)
```
```    40   not_elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<notin>\<index>" 50)
```
```    41   set_not_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.\<noteq>}\<index>" 50)
```
```    42
```
```    43 translations
```
```    44   "x .\<noteq>\<index> y" == "~(x .=\<index> y)"
```
```    45   "x .\<notin>\<index> A" == "~(x .\<in>\<index> A)"
```
```    46   "A {.\<noteq>}\<index> B" == "~(A {.=}\<index> B)"
```
```    47
```
```    48 locale equivalence =
```
```    49   fixes S (structure)
```
```    50   assumes refl [simp, intro]: "x \<in> carrier S \<Longrightarrow> x .= x"
```
```    51     and sym [sym]: "\<lbrakk> x .= y; x \<in> carrier S; y \<in> carrier S \<rbrakk> \<Longrightarrow> y .= x"
```
```    52     and trans [trans]: "\<lbrakk> x .= y; y .= z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S \<rbrakk> \<Longrightarrow> x .= z"
```
```    53
```
```    54 (* Lemmas by Stephan Hohe *)
```
```    55
```
```    56 lemma elemI:
```
```    57   fixes R (structure)
```
```    58   assumes "a' \<in> A" and "a .= a'"
```
```    59   shows "a .\<in> A"
```
```    60 unfolding elem_def
```
```    61 using assms
```
```    62 by fast
```
```    63
```
```    64 lemma (in equivalence) elem_exact:
```
```    65   assumes "a \<in> carrier S" and "a \<in> A"
```
```    66   shows "a .\<in> A"
```
```    67 using assms
```
```    68 by (fast intro: elemI)
```
```    69
```
```    70 lemma elemE:
```
```    71   fixes S (structure)
```
```    72   assumes "a .\<in> A"
```
```    73     and "\<And>a'. \<lbrakk>a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```    74   shows "P"
```
```    75 using assms
```
```    76 unfolding elem_def
```
```    77 by fast
```
```    78
```
```    79 lemma (in equivalence) elem_cong_l [trans]:
```
```    80   assumes cong: "a' .= a"
```
```    81     and a: "a .\<in> A"
```
```    82     and carr: "a \<in> carrier S"  "a' \<in> carrier S"
```
```    83     and Acarr: "A \<subseteq> carrier S"
```
```    84   shows "a' .\<in> A"
```
```    85 using a
```
```    86 apply (elim elemE, intro elemI)
```
```    87 proof assumption
```
```    88   fix b
```
```    89   assume bA: "b \<in> A"
```
```    90   note [simp] = carr bA[THEN subsetD[OF Acarr]]
```
```    91   note cong
```
```    92   also assume "a .= b"
```
```    93   finally show "a' .= b" by simp
```
```    94 qed
```
```    95
```
```    96 lemma (in equivalence) elem_subsetD:
```
```    97   assumes "A \<subseteq> B"
```
```    98     and aA: "a .\<in> A"
```
```    99   shows "a .\<in> B"
```
```   100 using assms
```
```   101 by (fast intro: elemI elim: elemE dest: subsetD)
```
```   102
```
```   103 lemma (in equivalence) mem_imp_elem [simp, intro]:
```
```   104   "[| x \<in> A; x \<in> carrier S |] ==> x .\<in> A"
```
```   105   unfolding elem_def by blast
```
```   106
```
```   107 lemma set_eqI:
```
```   108   fixes R (structure)
```
```   109   assumes ltr: "\<And>a. a \<in> A \<Longrightarrow> a .\<in> B"
```
```   110     and rtl: "\<And>b. b \<in> B \<Longrightarrow> b .\<in> A"
```
```   111   shows "A {.=} B"
```
```   112 unfolding set_eq_def
```
```   113 by (fast intro: ltr rtl)
```
```   114
```
```   115 lemma set_eqI2:
```
```   116   fixes R (structure)
```
```   117   assumes ltr: "\<And>a b. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .= b"
```
```   118     and rtl: "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .= a"
```
```   119   shows "A {.=} B"
```
```   120   by (intro set_eqI, unfold elem_def) (fast intro: ltr rtl)+
```
```   121
```
```   122 lemma set_eqD1:
```
```   123   fixes R (structure)
```
```   124   assumes AA': "A {.=} A'"
```
```   125     and "a \<in> A"
```
```   126   shows "\<exists>a'\<in>A'. a .= a'"
```
```   127 using assms
```
```   128 unfolding set_eq_def elem_def
```
```   129 by fast
```
```   130
```
```   131 lemma set_eqD2:
```
```   132   fixes R (structure)
```
```   133   assumes AA': "A {.=} A'"
```
```   134     and "a' \<in> A'"
```
```   135   shows "\<exists>a\<in>A. a' .= a"
```
```   136 using assms
```
```   137 unfolding set_eq_def elem_def
```
```   138 by fast
```
```   139
```
```   140 lemma set_eqE:
```
```   141   fixes R (structure)
```
```   142   assumes AB: "A {.=} B"
```
```   143     and r: "\<lbrakk>\<forall>a\<in>A. a .\<in> B; \<forall>b\<in>B. b .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   144   shows "P"
```
```   145 using AB
```
```   146 unfolding set_eq_def
```
```   147 by (blast dest: r)
```
```   148
```
```   149 lemma set_eqE2:
```
```   150   fixes R (structure)
```
```   151   assumes AB: "A {.=} B"
```
```   152     and r: "\<lbrakk>\<forall>a\<in>A. (\<exists>b\<in>B. a .= b); \<forall>b\<in>B. (\<exists>a\<in>A. b .= a)\<rbrakk> \<Longrightarrow> P"
```
```   153   shows "P"
```
```   154 using AB
```
```   155 unfolding set_eq_def elem_def
```
```   156 by (blast dest: r)
```
```   157
```
```   158 lemma set_eqE':
```
```   159   fixes R (structure)
```
```   160   assumes AB: "A {.=} B"
```
```   161     and aA: "a \<in> A" and bB: "b \<in> B"
```
```   162     and r: "\<And>a' b'. \<lbrakk>a' \<in> A; b .= a'; b' \<in> B; a .= b'\<rbrakk> \<Longrightarrow> P"
```
```   163   shows "P"
```
```   164 proof -
```
```   165   from AB aA
```
```   166       have "\<exists>b'\<in>B. a .= b'" by (rule set_eqD1)
```
```   167   from this obtain b'
```
```   168       where b': "b' \<in> B" "a .= b'" by auto
```
```   169
```
```   170   from AB bB
```
```   171       have "\<exists>a'\<in>A. b .= a'" by (rule set_eqD2)
```
```   172   from this obtain a'
```
```   173       where a': "a' \<in> A" "b .= a'" by auto
```
```   174
```
```   175   from a' b'
```
```   176       show "P" by (rule r)
```
```   177 qed
```
```   178
```
```   179 lemma (in equivalence) eq_elem_cong_r [trans]:
```
```   180   assumes a: "a .\<in> A"
```
```   181     and cong: "A {.=} A'"
```
```   182     and carr: "a \<in> carrier S"
```
```   183     and Carr: "A \<subseteq> carrier S" "A' \<subseteq> carrier S"
```
```   184   shows "a .\<in> A'"
```
```   185 using a cong
```
```   186 proof (elim elemE set_eqE)
```
```   187   fix b
```
```   188   assume bA: "b \<in> A"
```
```   189      and inA': "\<forall>b\<in>A. b .\<in> A'"
```
```   190   note [simp] = carr Carr Carr[THEN subsetD] bA
```
```   191   assume "a .= b"
```
```   192   also from bA inA'
```
```   193        have "b .\<in> A'" by fast
```
```   194   finally
```
```   195        show "a .\<in> A'" by simp
```
```   196 qed
```
```   197
```
```   198 lemma (in equivalence) set_eq_sym [sym]:
```
```   199   assumes "A {.=} B"
```
```   200     and "A \<subseteq> carrier S" "B \<subseteq> carrier S"
```
```   201   shows "B {.=} A"
```
```   202 using assms
```
```   203 unfolding set_eq_def elem_def
```
```   204 by fast
```
```   205
```
```   206 (* FIXME: the following two required in Isabelle 2008, not Isabelle 2007 *)
```
```   207 (* alternatively, could declare lemmas [trans] = ssubst [where 'a = "'a set"] *)
```
```   208
```
```   209 lemma (in equivalence) equal_set_eq_trans [trans]:
```
```   210   assumes AB: "A = B" and BC: "B {.=} C"
```
```   211   shows "A {.=} C"
```
```   212   using AB BC by simp
```
```   213
```
```   214 lemma (in equivalence) set_eq_equal_trans [trans]:
```
```   215   assumes AB: "A {.=} B" and BC: "B = C"
```
```   216   shows "A {.=} C"
```
```   217   using AB BC by simp
```
```   218
```
```   219
```
```   220 lemma (in equivalence) set_eq_trans [trans]:
```
```   221   assumes AB: "A {.=} B" and BC: "B {.=} C"
```
```   222     and carr: "A \<subseteq> carrier S"  "B \<subseteq> carrier S"  "C \<subseteq> carrier S"
```
```   223   shows "A {.=} C"
```
```   224 proof (intro set_eqI)
```
```   225   fix a
```
```   226   assume aA: "a \<in> A"
```
```   227   with carr have "a \<in> carrier S" by fast
```
```   228   note [simp] = carr this
```
```   229
```
```   230   from aA
```
```   231        have "a .\<in> A" by (simp add: elem_exact)
```
```   232   also note AB
```
```   233   also note BC
```
```   234   finally
```
```   235        show "a .\<in> C" by simp
```
```   236 next
```
```   237   fix c
```
```   238   assume cC: "c \<in> C"
```
```   239   with carr have "c \<in> carrier S" by fast
```
```   240   note [simp] = carr this
```
```   241
```
```   242   from cC
```
```   243        have "c .\<in> C" by (simp add: elem_exact)
```
```   244   also note BC[symmetric]
```
```   245   also note AB[symmetric]
```
```   246   finally
```
```   247        show "c .\<in> A" by simp
```
```   248 qed
```
```   249
```
```   250 (* FIXME: generalise for insert *)
```
```   251
```
```   252 (*
```
```   253 lemma (in equivalence) set_eq_insert:
```
```   254   assumes x: "x .= x'"
```
```   255     and carr: "x \<in> carrier S" "x' \<in> carrier S" "A \<subseteq> carrier S"
```
```   256   shows "insert x A {.=} insert x' A"
```
```   257   unfolding set_eq_def elem_def
```
```   258 apply rule
```
```   259 apply rule
```
```   260 apply (case_tac "xa = x")
```
```   261 using x apply fast
```
```   262 apply (subgoal_tac "xa \<in> A") prefer 2 apply fast
```
```   263 apply (rule_tac x=xa in bexI)
```
```   264 using carr apply (rule_tac refl) apply auto 
```
```   265 apply safe
```
```   266 *)
```
```   267
```
```   268 lemma (in equivalence) set_eq_pairI:
```
```   269   assumes xx': "x .= x'"
```
```   270     and carr: "x \<in> carrier S" "x' \<in> carrier S" "y \<in> carrier S"
```
```   271   shows "{x, y} {.=} {x', y}"
```
```   272 unfolding set_eq_def elem_def
```
```   273 proof safe
```
```   274   have "x' \<in> {x', y}" by fast
```
```   275   with xx' show "\<exists>b\<in>{x', y}. x .= b" by fast
```
```   276 next
```
```   277   have "y \<in> {x', y}" by fast
```
```   278   with carr show "\<exists>b\<in>{x', y}. y .= b" by fast
```
```   279 next
```
```   280   have "x \<in> {x, y}" by fast
```
```   281   with xx'[symmetric] carr
```
```   282   show "\<exists>a\<in>{x, y}. x' .= a" by fast
```
```   283 next
```
```   284   have "y \<in> {x, y}" by fast
```
```   285   with carr show "\<exists>a\<in>{x, y}. y .= a" by fast
```
```   286 qed
```
```   287
```
```   288 lemma (in equivalence) is_closedI:
```
```   289   assumes closed: "!!x y. [| x .= y; x \<in> A; y \<in> carrier S |] ==> y \<in> A"
```
```   290     and S: "A \<subseteq> carrier S"
```
```   291   shows "is_closed A"
```
```   292   unfolding eq_is_closed_def eq_closure_of_def elem_def
```
```   293   using S
```
```   294   by (blast dest: closed sym)
```
```   295
```
```   296 lemma (in equivalence) closure_of_eq:
```
```   297   "[| x .= x'; A \<subseteq> carrier S; x \<in> closure_of A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> closure_of A"
```
```   298   unfolding eq_closure_of_def elem_def
```
```   299   by (blast intro: trans sym)
```
```   300
```
```   301 lemma (in equivalence) is_closed_eq [dest]:
```
```   302   "[| x .= x'; x \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> A"
```
```   303   unfolding eq_is_closed_def
```
```   304   using closure_of_eq [where A = A]
```
```   305   by simp
```
```   306
```
```   307 lemma (in equivalence) is_closed_eq_rev [dest]:
```
```   308   "[| x .= x'; x' \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x \<in> A"
```
```   309   by (drule sym) (simp_all add: is_closed_eq)
```
```   310
```
```   311 lemma closure_of_closed [simp, intro]:
```
```   312   fixes S (structure)
```
```   313   shows "closure_of A \<subseteq> carrier S"
```
```   314 unfolding eq_closure_of_def
```
```   315 by fast
```
```   316
```
```   317 lemma closure_of_memI:
```
```   318   fixes S (structure)
```
```   319   assumes "a .\<in> A"
```
```   320     and "a \<in> carrier S"
```
```   321   shows "a \<in> closure_of A"
```
```   322 unfolding eq_closure_of_def
```
```   323 using assms
```
```   324 by fast
```
```   325
```
```   326 lemma closure_ofI2:
```
```   327   fixes S (structure)
```
```   328   assumes "a .= a'"
```
```   329     and "a' \<in> A"
```
```   330     and "a \<in> carrier S"
```
```   331   shows "a \<in> closure_of A"
```
```   332 unfolding eq_closure_of_def elem_def
```
```   333 using assms
```
```   334 by fast
```
```   335
```
```   336 lemma closure_of_memE:
```
```   337   fixes S (structure)
```
```   338   assumes p: "a \<in> closure_of A"
```
```   339     and r: "\<lbrakk>a \<in> carrier S; a .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   340   shows "P"
```
```   341 proof -
```
```   342   from p
```
```   343       have acarr: "a \<in> carrier S"
```
```   344       and "a .\<in> A"
```
```   345       by (simp add: eq_closure_of_def)+
```
```   346   thus "P" by (rule r)
```
```   347 qed
```
```   348
```
```   349 lemma closure_ofE2:
```
```   350   fixes S (structure)
```
```   351   assumes p: "a \<in> closure_of A"
```
```   352     and r: "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```   353   shows "P"
```
```   354 proof -
```
```   355   from p have acarr: "a \<in> carrier S" by (simp add: eq_closure_of_def)
```
```   356
```
```   357   from p have "\<exists>a'\<in>A. a .= a'" by (simp add: eq_closure_of_def elem_def)
```
```   358   from this obtain a'
```
```   359       where "a' \<in> A" and "a .= a'" by auto
```
```   360
```
```   361   from acarr and this
```
```   362       show "P" by (rule r)
```
```   363 qed
```
```   364
```
```   365 (*
```
```   366 lemma (in equivalence) classes_consistent:
```
```   367   assumes Acarr: "A \<subseteq> carrier S"
```
```   368   shows "is_closed (closure_of A)"
```
```   369 apply (blast intro: elemI elim elemE)
```
```   370 using assms
```
```   371 apply (intro is_closedI closure_of_memI, simp)
```
```   372  apply (elim elemE closure_of_memE)
```
```   373 proof -
```
```   374   fix x a' a''
```
```   375   assume carr: "x \<in> carrier S" "a' \<in> carrier S"
```
```   376   assume a''A: "a'' \<in> A"
```
```   377   with Acarr have "a'' \<in> carrier S" by fast
```
```   378   note [simp] = carr this Acarr
```
```   379
```
```   380   assume "x .= a'"
```
```   381   also assume "a' .= a''"
```
```   382   also from a''A
```
```   383        have "a'' .\<in> A" by (simp add: elem_exact)
```
```   384   finally show "x .\<in> A" by simp
```
```   385 qed
```
```   386 *)
```
```   387 (*
```
```   388 lemma (in equivalence) classes_small:
```
```   389   assumes "is_closed B"
```
```   390     and "A \<subseteq> B"
```
```   391   shows "closure_of A \<subseteq> B"
```
```   392 using assms
```
```   393 by (blast dest: is_closedD2 elem_subsetD elim: closure_of_memE)
```
```   394
```
```   395 lemma (in equivalence) classes_eq:
```
```   396   assumes "A \<subseteq> carrier S"
```
```   397   shows "A {.=} closure_of A"
```
```   398 using assms
```
```   399 by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE)
```
```   400
```
```   401 lemma (in equivalence) complete_classes:
```
```   402   assumes c: "is_closed A"
```
```   403   shows "A = closure_of A"
```
```   404 using assms
```
```   405 by (blast intro: closure_of_memI elem_exact dest: is_closedD1 is_closedD2 closure_of_memE)
```
```   406 *)
```
```   407
```
```   408 end
```