src/HOL/Algebra/Congruence.thy
 author wenzelm Sun Mar 21 15:57:40 2010 +0100 (2010-03-21) changeset 35847 19f1f7066917 parent 35355 613e133966ea child 35848 5443079512ea permissions -rw-r--r--
eliminated old constdefs;
```     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 definition
```
```    23   elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<in>\<index>" 50)
```
```    24   where "x .\<in>\<^bsub>S\<^esub> A \<equiv> (\<exists>y \<in> A. x .=\<^bsub>S\<^esub> y)"
```
```    25
```
```    26 definition
```
```    27   set_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.=}\<index>" 50)
```
```    28   where "A {.=}\<^bsub>S\<^esub> B == ((\<forall>x \<in> A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x \<in> B. x .\<in>\<^bsub>S\<^esub> A))"
```
```    29
```
```    30 definition
```
```    31   eq_class_of :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set" ("class'_of\<index> _")
```
```    32   where "class_of\<^bsub>S\<^esub> x == {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}"
```
```    33
```
```    34 definition
```
```    35   eq_closure_of :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set" ("closure'_of\<index> _")
```
```    36   where "closure_of\<^bsub>S\<^esub> A == {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}"
```
```    37
```
```    38 definition
```
```    39   eq_is_closed :: "_ \<Rightarrow> 'a set \<Rightarrow> bool" ("is'_closed\<index> _")
```
```    40   where "is_closed\<^bsub>S\<^esub> A == (A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A)"
```
```    41
```
```    42 abbreviation
```
```    43   not_eq :: "_ \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".\<noteq>\<index>" 50)
```
```    44   where "x .\<noteq>\<^bsub>S\<^esub> y == ~(x .=\<^bsub>S\<^esub> y)"
```
```    45
```
```    46 abbreviation
```
```    47   not_elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<notin>\<index>" 50)
```
```    48   where "x .\<notin>\<^bsub>S\<^esub> A == ~(x .\<in>\<^bsub>S\<^esub> A)"
```
```    49
```
```    50 abbreviation
```
```    51   set_not_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.\<noteq>}\<index>" 50)
```
```    52   where "A {.\<noteq>}\<^bsub>S\<^esub> B == ~(A {.=}\<^bsub>S\<^esub> B)"
```
```    53
```
```    54 locale equivalence =
```
```    55   fixes S (structure)
```
```    56   assumes refl [simp, intro]: "x \<in> carrier S \<Longrightarrow> x .= x"
```
```    57     and sym [sym]: "\<lbrakk> x .= y; x \<in> carrier S; y \<in> carrier S \<rbrakk> \<Longrightarrow> y .= x"
```
```    58     and trans [trans]: "\<lbrakk> x .= y; y .= z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S \<rbrakk> \<Longrightarrow> x .= z"
```
```    59
```
```    60 (* Lemmas by Stephan Hohe *)
```
```    61
```
```    62 lemma elemI:
```
```    63   fixes R (structure)
```
```    64   assumes "a' \<in> A" and "a .= a'"
```
```    65   shows "a .\<in> A"
```
```    66 unfolding elem_def
```
```    67 using assms
```
```    68 by fast
```
```    69
```
```    70 lemma (in equivalence) elem_exact:
```
```    71   assumes "a \<in> carrier S" and "a \<in> A"
```
```    72   shows "a .\<in> A"
```
```    73 using assms
```
```    74 by (fast intro: elemI)
```
```    75
```
```    76 lemma elemE:
```
```    77   fixes S (structure)
```
```    78   assumes "a .\<in> A"
```
```    79     and "\<And>a'. \<lbrakk>a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```    80   shows "P"
```
```    81 using assms
```
```    82 unfolding elem_def
```
```    83 by fast
```
```    84
```
```    85 lemma (in equivalence) elem_cong_l [trans]:
```
```    86   assumes cong: "a' .= a"
```
```    87     and a: "a .\<in> A"
```
```    88     and carr: "a \<in> carrier S"  "a' \<in> carrier S"
```
```    89     and Acarr: "A \<subseteq> carrier S"
```
```    90   shows "a' .\<in> A"
```
```    91 using a
```
```    92 apply (elim elemE, intro elemI)
```
```    93 proof assumption
```
```    94   fix b
```
```    95   assume bA: "b \<in> A"
```
```    96   note [simp] = carr bA[THEN subsetD[OF Acarr]]
```
```    97   note cong
```
```    98   also assume "a .= b"
```
```    99   finally show "a' .= b" by simp
```
```   100 qed
```
```   101
```
```   102 lemma (in equivalence) elem_subsetD:
```
```   103   assumes "A \<subseteq> B"
```
```   104     and aA: "a .\<in> A"
```
```   105   shows "a .\<in> B"
```
```   106 using assms
```
```   107 by (fast intro: elemI elim: elemE dest: subsetD)
```
```   108
```
```   109 lemma (in equivalence) mem_imp_elem [simp, intro]:
```
```   110   "[| x \<in> A; x \<in> carrier S |] ==> x .\<in> A"
```
```   111   unfolding elem_def by blast
```
```   112
```
```   113 lemma set_eqI:
```
```   114   fixes R (structure)
```
```   115   assumes ltr: "\<And>a. a \<in> A \<Longrightarrow> a .\<in> B"
```
```   116     and rtl: "\<And>b. b \<in> B \<Longrightarrow> b .\<in> A"
```
```   117   shows "A {.=} B"
```
```   118 unfolding set_eq_def
```
```   119 by (fast intro: ltr rtl)
```
```   120
```
```   121 lemma set_eqI2:
```
```   122   fixes R (structure)
```
```   123   assumes ltr: "\<And>a b. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .= b"
```
```   124     and rtl: "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .= a"
```
```   125   shows "A {.=} B"
```
```   126   by (intro set_eqI, unfold elem_def) (fast intro: ltr rtl)+
```
```   127
```
```   128 lemma set_eqD1:
```
```   129   fixes R (structure)
```
```   130   assumes AA': "A {.=} A'"
```
```   131     and "a \<in> A"
```
```   132   shows "\<exists>a'\<in>A'. a .= a'"
```
```   133 using assms
```
```   134 unfolding set_eq_def elem_def
```
```   135 by fast
```
```   136
```
```   137 lemma set_eqD2:
```
```   138   fixes R (structure)
```
```   139   assumes AA': "A {.=} A'"
```
```   140     and "a' \<in> A'"
```
```   141   shows "\<exists>a\<in>A. a' .= a"
```
```   142 using assms
```
```   143 unfolding set_eq_def elem_def
```
```   144 by fast
```
```   145
```
```   146 lemma set_eqE:
```
```   147   fixes R (structure)
```
```   148   assumes AB: "A {.=} B"
```
```   149     and r: "\<lbrakk>\<forall>a\<in>A. a .\<in> B; \<forall>b\<in>B. b .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   150   shows "P"
```
```   151 using AB
```
```   152 unfolding set_eq_def
```
```   153 by (blast dest: r)
```
```   154
```
```   155 lemma set_eqE2:
```
```   156   fixes R (structure)
```
```   157   assumes AB: "A {.=} B"
```
```   158     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"
```
```   159   shows "P"
```
```   160 using AB
```
```   161 unfolding set_eq_def elem_def
```
```   162 by (blast dest: r)
```
```   163
```
```   164 lemma set_eqE':
```
```   165   fixes R (structure)
```
```   166   assumes AB: "A {.=} B"
```
```   167     and aA: "a \<in> A" and bB: "b \<in> B"
```
```   168     and r: "\<And>a' b'. \<lbrakk>a' \<in> A; b .= a'; b' \<in> B; a .= b'\<rbrakk> \<Longrightarrow> P"
```
```   169   shows "P"
```
```   170 proof -
```
```   171   from AB aA
```
```   172       have "\<exists>b'\<in>B. a .= b'" by (rule set_eqD1)
```
```   173   from this obtain b'
```
```   174       where b': "b' \<in> B" "a .= b'" by auto
```
```   175
```
```   176   from AB bB
```
```   177       have "\<exists>a'\<in>A. b .= a'" by (rule set_eqD2)
```
```   178   from this obtain a'
```
```   179       where a': "a' \<in> A" "b .= a'" by auto
```
```   180
```
```   181   from a' b'
```
```   182       show "P" by (rule r)
```
```   183 qed
```
```   184
```
```   185 lemma (in equivalence) eq_elem_cong_r [trans]:
```
```   186   assumes a: "a .\<in> A"
```
```   187     and cong: "A {.=} A'"
```
```   188     and carr: "a \<in> carrier S"
```
```   189     and Carr: "A \<subseteq> carrier S" "A' \<subseteq> carrier S"
```
```   190   shows "a .\<in> A'"
```
```   191 using a cong
```
```   192 proof (elim elemE set_eqE)
```
```   193   fix b
```
```   194   assume bA: "b \<in> A"
```
```   195      and inA': "\<forall>b\<in>A. b .\<in> A'"
```
```   196   note [simp] = carr Carr Carr[THEN subsetD] bA
```
```   197   assume "a .= b"
```
```   198   also from bA inA'
```
```   199        have "b .\<in> A'" by fast
```
```   200   finally
```
```   201        show "a .\<in> A'" by simp
```
```   202 qed
```
```   203
```
```   204 lemma (in equivalence) set_eq_sym [sym]:
```
```   205   assumes "A {.=} B"
```
```   206     and "A \<subseteq> carrier S" "B \<subseteq> carrier S"
```
```   207   shows "B {.=} A"
```
```   208 using assms
```
```   209 unfolding set_eq_def elem_def
```
```   210 by fast
```
```   211
```
```   212 (* FIXME: the following two required in Isabelle 2008, not Isabelle 2007 *)
```
```   213 (* alternatively, could declare lemmas [trans] = ssubst [where 'a = "'a set"] *)
```
```   214
```
```   215 lemma (in equivalence) equal_set_eq_trans [trans]:
```
```   216   assumes AB: "A = B" and BC: "B {.=} C"
```
```   217   shows "A {.=} C"
```
```   218   using AB BC by simp
```
```   219
```
```   220 lemma (in equivalence) set_eq_equal_trans [trans]:
```
```   221   assumes AB: "A {.=} B" and BC: "B = C"
```
```   222   shows "A {.=} C"
```
```   223   using AB BC by simp
```
```   224
```
```   225
```
```   226 lemma (in equivalence) set_eq_trans [trans]:
```
```   227   assumes AB: "A {.=} B" and BC: "B {.=} C"
```
```   228     and carr: "A \<subseteq> carrier S"  "B \<subseteq> carrier S"  "C \<subseteq> carrier S"
```
```   229   shows "A {.=} C"
```
```   230 proof (intro set_eqI)
```
```   231   fix a
```
```   232   assume aA: "a \<in> A"
```
```   233   with carr have "a \<in> carrier S" by fast
```
```   234   note [simp] = carr this
```
```   235
```
```   236   from aA
```
```   237        have "a .\<in> A" by (simp add: elem_exact)
```
```   238   also note AB
```
```   239   also note BC
```
```   240   finally
```
```   241        show "a .\<in> C" by simp
```
```   242 next
```
```   243   fix c
```
```   244   assume cC: "c \<in> C"
```
```   245   with carr have "c \<in> carrier S" by fast
```
```   246   note [simp] = carr this
```
```   247
```
```   248   from cC
```
```   249        have "c .\<in> C" by (simp add: elem_exact)
```
```   250   also note BC[symmetric]
```
```   251   also note AB[symmetric]
```
```   252   finally
```
```   253        show "c .\<in> A" by simp
```
```   254 qed
```
```   255
```
```   256 (* FIXME: generalise for insert *)
```
```   257
```
```   258 (*
```
```   259 lemma (in equivalence) set_eq_insert:
```
```   260   assumes x: "x .= x'"
```
```   261     and carr: "x \<in> carrier S" "x' \<in> carrier S" "A \<subseteq> carrier S"
```
```   262   shows "insert x A {.=} insert x' A"
```
```   263   unfolding set_eq_def elem_def
```
```   264 apply rule
```
```   265 apply rule
```
```   266 apply (case_tac "xa = x")
```
```   267 using x apply fast
```
```   268 apply (subgoal_tac "xa \<in> A") prefer 2 apply fast
```
```   269 apply (rule_tac x=xa in bexI)
```
```   270 using carr apply (rule_tac refl) apply auto 
```
```   271 apply safe
```
```   272 *)
```
```   273
```
```   274 lemma (in equivalence) set_eq_pairI:
```
```   275   assumes xx': "x .= x'"
```
```   276     and carr: "x \<in> carrier S" "x' \<in> carrier S" "y \<in> carrier S"
```
```   277   shows "{x, y} {.=} {x', y}"
```
```   278 unfolding set_eq_def elem_def
```
```   279 proof safe
```
```   280   have "x' \<in> {x', y}" by fast
```
```   281   with xx' show "\<exists>b\<in>{x', y}. x .= b" by fast
```
```   282 next
```
```   283   have "y \<in> {x', y}" by fast
```
```   284   with carr show "\<exists>b\<in>{x', y}. y .= b" by fast
```
```   285 next
```
```   286   have "x \<in> {x, y}" by fast
```
```   287   with xx'[symmetric] carr
```
```   288   show "\<exists>a\<in>{x, y}. x' .= a" by fast
```
```   289 next
```
```   290   have "y \<in> {x, y}" by fast
```
```   291   with carr show "\<exists>a\<in>{x, y}. y .= a" by fast
```
```   292 qed
```
```   293
```
```   294 lemma (in equivalence) is_closedI:
```
```   295   assumes closed: "!!x y. [| x .= y; x \<in> A; y \<in> carrier S |] ==> y \<in> A"
```
```   296     and S: "A \<subseteq> carrier S"
```
```   297   shows "is_closed A"
```
```   298   unfolding eq_is_closed_def eq_closure_of_def elem_def
```
```   299   using S
```
```   300   by (blast dest: closed sym)
```
```   301
```
```   302 lemma (in equivalence) closure_of_eq:
```
```   303   "[| x .= x'; A \<subseteq> carrier S; x \<in> closure_of A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> closure_of A"
```
```   304   unfolding eq_closure_of_def elem_def
```
```   305   by (blast intro: trans sym)
```
```   306
```
```   307 lemma (in equivalence) is_closed_eq [dest]:
```
```   308   "[| x .= x'; x \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> A"
```
```   309   unfolding eq_is_closed_def
```
```   310   using closure_of_eq [where A = A]
```
```   311   by simp
```
```   312
```
```   313 lemma (in equivalence) is_closed_eq_rev [dest]:
```
```   314   "[| x .= x'; x' \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x \<in> A"
```
```   315   by (drule sym) (simp_all add: is_closed_eq)
```
```   316
```
```   317 lemma closure_of_closed [simp, intro]:
```
```   318   fixes S (structure)
```
```   319   shows "closure_of A \<subseteq> carrier S"
```
```   320 unfolding eq_closure_of_def
```
```   321 by fast
```
```   322
```
```   323 lemma closure_of_memI:
```
```   324   fixes S (structure)
```
```   325   assumes "a .\<in> A"
```
```   326     and "a \<in> carrier S"
```
```   327   shows "a \<in> closure_of A"
```
```   328 unfolding eq_closure_of_def
```
```   329 using assms
```
```   330 by fast
```
```   331
```
```   332 lemma closure_ofI2:
```
```   333   fixes S (structure)
```
```   334   assumes "a .= a'"
```
```   335     and "a' \<in> A"
```
```   336     and "a \<in> carrier S"
```
```   337   shows "a \<in> closure_of A"
```
```   338 unfolding eq_closure_of_def elem_def
```
```   339 using assms
```
```   340 by fast
```
```   341
```
```   342 lemma closure_of_memE:
```
```   343   fixes S (structure)
```
```   344   assumes p: "a \<in> closure_of A"
```
```   345     and r: "\<lbrakk>a \<in> carrier S; a .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   346   shows "P"
```
```   347 proof -
```
```   348   from p
```
```   349       have acarr: "a \<in> carrier S"
```
```   350       and "a .\<in> A"
```
```   351       by (simp add: eq_closure_of_def)+
```
```   352   thus "P" by (rule r)
```
```   353 qed
```
```   354
```
```   355 lemma closure_ofE2:
```
```   356   fixes S (structure)
```
```   357   assumes p: "a \<in> closure_of A"
```
```   358     and r: "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```   359   shows "P"
```
```   360 proof -
```
```   361   from p have acarr: "a \<in> carrier S" by (simp add: eq_closure_of_def)
```
```   362
```
```   363   from p have "\<exists>a'\<in>A. a .= a'" by (simp add: eq_closure_of_def elem_def)
```
```   364   from this obtain a'
```
```   365       where "a' \<in> A" and "a .= a'" by auto
```
```   366
```
```   367   from acarr and this
```
```   368       show "P" by (rule r)
```
```   369 qed
```
```   370
```
```   371 (*
```
```   372 lemma (in equivalence) classes_consistent:
```
```   373   assumes Acarr: "A \<subseteq> carrier S"
```
```   374   shows "is_closed (closure_of A)"
```
```   375 apply (blast intro: elemI elim elemE)
```
```   376 using assms
```
```   377 apply (intro is_closedI closure_of_memI, simp)
```
```   378  apply (elim elemE closure_of_memE)
```
```   379 proof -
```
```   380   fix x a' a''
```
```   381   assume carr: "x \<in> carrier S" "a' \<in> carrier S"
```
```   382   assume a''A: "a'' \<in> A"
```
```   383   with Acarr have "a'' \<in> carrier S" by fast
```
```   384   note [simp] = carr this Acarr
```
```   385
```
```   386   assume "x .= a'"
```
```   387   also assume "a' .= a''"
```
```   388   also from a''A
```
```   389        have "a'' .\<in> A" by (simp add: elem_exact)
```
```   390   finally show "x .\<in> A" by simp
```
```   391 qed
```
```   392 *)
```
```   393 (*
```
```   394 lemma (in equivalence) classes_small:
```
```   395   assumes "is_closed B"
```
```   396     and "A \<subseteq> B"
```
```   397   shows "closure_of A \<subseteq> B"
```
```   398 using assms
```
```   399 by (blast dest: is_closedD2 elem_subsetD elim: closure_of_memE)
```
```   400
```
```   401 lemma (in equivalence) classes_eq:
```
```   402   assumes "A \<subseteq> carrier S"
```
```   403   shows "A {.=} closure_of A"
```
```   404 using assms
```
```   405 by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE)
```
```   406
```
```   407 lemma (in equivalence) complete_classes:
```
```   408   assumes c: "is_closed A"
```
```   409   shows "A = closure_of A"
```
```   410 using assms
```
```   411 by (blast intro: closure_of_memI elem_exact dest: is_closedD1 is_closedD2 closure_of_memE)
```
```   412 *)
```
```   413
```
```   414 end
```