src/HOL/Algebra/Congruence.thy
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more symbols;
```     1 (*  Title:      HOL/Algebra/Congruence.thy
```
```     2     Author:     Clemens Ballarin, started 3 January 2008
```
```     3     Copyright:  Clemens Ballarin
```
```     4 *)
```
```     5
```
```     6 theory Congruence
```
```     7 imports
```
```     8   Main
```
```     9   "HOL-Library.FuncSet"
```
```    10 begin
```
```    11
```
```    12 section \<open>Objects\<close>
```
```    13
```
```    14 subsection \<open>Structure with Carrier Set.\<close>
```
```    15
```
```    16 record 'a partial_object =
```
```    17   carrier :: "'a set"
```
```    18
```
```    19 lemma funcset_carrier:
```
```    20   "\<lbrakk> f \<in> carrier X \<rightarrow> carrier Y; x \<in> carrier X \<rbrakk> \<Longrightarrow> f x \<in> carrier Y"
```
```    21   by (fact funcset_mem)
```
```    22
```
```    23 lemma funcset_carrier':
```
```    24   "\<lbrakk> f \<in> carrier A \<rightarrow> carrier A; x \<in> carrier A \<rbrakk> \<Longrightarrow> f x \<in> carrier A"
```
```    25   by (fact funcset_mem)
```
```    26
```
```    27
```
```    28 subsection \<open>Structure with Carrier and Equivalence Relation \<open>eq\<close>\<close>
```
```    29
```
```    30 record 'a eq_object = "'a partial_object" +
```
```    31   eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".=\<index>" 50)
```
```    32
```
```    33 definition
```
```    34   elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<in>\<index>" 50)
```
```    35   where "x .\<in>\<^bsub>S\<^esub> A \<longleftrightarrow> (\<exists>y \<in> A. x .=\<^bsub>S\<^esub> y)"
```
```    36
```
```    37 definition
```
```    38   set_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.=}\<index>" 50)
```
```    39   where "A {.=}\<^bsub>S\<^esub> B \<longleftrightarrow> ((\<forall>x \<in> A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x \<in> B. x .\<in>\<^bsub>S\<^esub> A))"
```
```    40
```
```    41 definition
```
```    42   eq_class_of :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set" ("class'_of\<index>")
```
```    43   where "class_of\<^bsub>S\<^esub> x = {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}"
```
```    44
```
```    45 definition
```
```    46   eq_closure_of :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set" ("closure'_of\<index>")
```
```    47   where "closure_of\<^bsub>S\<^esub> A = {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}"
```
```    48
```
```    49 definition
```
```    50   eq_is_closed :: "_ \<Rightarrow> 'a set \<Rightarrow> bool" ("is'_closed\<index>")
```
```    51   where "is_closed\<^bsub>S\<^esub> A \<longleftrightarrow> A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A"
```
```    52
```
```    53 abbreviation
```
```    54   not_eq :: "_ \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".\<noteq>\<index>" 50)
```
```    55   where "x .\<noteq>\<^bsub>S\<^esub> y \<equiv> \<not>(x .=\<^bsub>S\<^esub> y)"
```
```    56
```
```    57 abbreviation
```
```    58   not_elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<notin>\<index>" 50)
```
```    59   where "x .\<notin>\<^bsub>S\<^esub> A \<equiv> \<not>(x .\<in>\<^bsub>S\<^esub> A)"
```
```    60
```
```    61 abbreviation
```
```    62   set_not_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.\<noteq>}\<index>" 50)
```
```    63   where "A {.\<noteq>}\<^bsub>S\<^esub> B \<equiv> \<not>(A {.=}\<^bsub>S\<^esub> B)"
```
```    64
```
```    65 locale equivalence =
```
```    66   fixes S (structure)
```
```    67   assumes refl [simp, intro]: "x \<in> carrier S \<Longrightarrow> x .= x"
```
```    68     and sym [sym]: "\<lbrakk> x .= y; x \<in> carrier S; y \<in> carrier S \<rbrakk> \<Longrightarrow> y .= x"
```
```    69     and trans [trans]:
```
```    70       "\<lbrakk> x .= y; y .= z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S \<rbrakk> \<Longrightarrow> x .= z"
```
```    71
```
```    72 (* Lemmas by Stephan Hohe *)
```
```    73
```
```    74 lemma elemI:
```
```    75   fixes R (structure)
```
```    76   assumes "a' \<in> A" and "a .= a'"
```
```    77   shows "a .\<in> A"
```
```    78 unfolding elem_def
```
```    79 using assms
```
```    80 by fast
```
```    81
```
```    82 lemma (in equivalence) elem_exact:
```
```    83   assumes "a \<in> carrier S" and "a \<in> A"
```
```    84   shows "a .\<in> A"
```
```    85 using assms
```
```    86 by (fast intro: elemI)
```
```    87
```
```    88 lemma elemE:
```
```    89   fixes S (structure)
```
```    90   assumes "a .\<in> A"
```
```    91     and "\<And>a'. \<lbrakk>a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```    92   shows "P"
```
```    93 using assms
```
```    94 unfolding elem_def
```
```    95 by fast
```
```    96
```
```    97 lemma (in equivalence) elem_cong_l [trans]:
```
```    98   assumes cong: "a' .= a"
```
```    99     and a: "a .\<in> A"
```
```   100     and carr: "a \<in> carrier S"  "a' \<in> carrier S"
```
```   101     and Acarr: "A \<subseteq> carrier S"
```
```   102   shows "a' .\<in> A"
```
```   103 using a
```
```   104 apply (elim elemE, intro elemI)
```
```   105 proof assumption
```
```   106   fix b
```
```   107   assume bA: "b \<in> A"
```
```   108   note [simp] = carr bA[THEN subsetD[OF Acarr]]
```
```   109   note cong
```
```   110   also assume "a .= b"
```
```   111   finally show "a' .= b" by simp
```
```   112 qed
```
```   113
```
```   114 lemma (in equivalence) elem_subsetD:
```
```   115   assumes "A \<subseteq> B"
```
```   116     and aA: "a .\<in> A"
```
```   117   shows "a .\<in> B"
```
```   118 using assms
```
```   119 by (fast intro: elemI elim: elemE dest: subsetD)
```
```   120
```
```   121 lemma (in equivalence) mem_imp_elem [simp, intro]:
```
```   122   "[| x \<in> A; x \<in> carrier S |] ==> x .\<in> A"
```
```   123   unfolding elem_def by blast
```
```   124
```
```   125 lemma set_eqI:
```
```   126   fixes R (structure)
```
```   127   assumes ltr: "\<And>a. a \<in> A \<Longrightarrow> a .\<in> B"
```
```   128     and rtl: "\<And>b. b \<in> B \<Longrightarrow> b .\<in> A"
```
```   129   shows "A {.=} B"
```
```   130 unfolding set_eq_def
```
```   131 by (fast intro: ltr rtl)
```
```   132
```
```   133 lemma set_eqI2:
```
```   134   fixes R (structure)
```
```   135   assumes ltr: "\<And>a b. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .= b"
```
```   136     and rtl: "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .= a"
```
```   137   shows "A {.=} B"
```
```   138   by (intro set_eqI, unfold elem_def) (fast intro: ltr rtl)+
```
```   139
```
```   140 lemma set_eqD1:
```
```   141   fixes R (structure)
```
```   142   assumes AA': "A {.=} A'"
```
```   143     and "a \<in> A"
```
```   144   shows "\<exists>a'\<in>A'. a .= a'"
```
```   145 using assms
```
```   146 unfolding set_eq_def elem_def
```
```   147 by fast
```
```   148
```
```   149 lemma set_eqD2:
```
```   150   fixes R (structure)
```
```   151   assumes AA': "A {.=} A'"
```
```   152     and "a' \<in> A'"
```
```   153   shows "\<exists>a\<in>A. a' .= a"
```
```   154 using assms
```
```   155 unfolding set_eq_def elem_def
```
```   156 by fast
```
```   157
```
```   158 lemma set_eqE:
```
```   159   fixes R (structure)
```
```   160   assumes AB: "A {.=} B"
```
```   161     and r: "\<lbrakk>\<forall>a\<in>A. a .\<in> B; \<forall>b\<in>B. b .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   162   shows "P"
```
```   163 using AB
```
```   164 unfolding set_eq_def
```
```   165 by (blast dest: r)
```
```   166
```
```   167 lemma set_eqE2:
```
```   168   fixes R (structure)
```
```   169   assumes AB: "A {.=} B"
```
```   170     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"
```
```   171   shows "P"
```
```   172 using AB
```
```   173 unfolding set_eq_def elem_def
```
```   174 by (blast dest: r)
```
```   175
```
```   176 lemma set_eqE':
```
```   177   fixes R (structure)
```
```   178   assumes AB: "A {.=} B"
```
```   179     and aA: "a \<in> A" and bB: "b \<in> B"
```
```   180     and r: "\<And>a' b'. \<lbrakk>a' \<in> A; b .= a'; b' \<in> B; a .= b'\<rbrakk> \<Longrightarrow> P"
```
```   181   shows "P"
```
```   182 proof -
```
```   183   from AB aA
```
```   184       have "\<exists>b'\<in>B. a .= b'" by (rule set_eqD1)
```
```   185   from this obtain b'
```
```   186       where b': "b' \<in> B" "a .= b'" by auto
```
```   187
```
```   188   from AB bB
```
```   189       have "\<exists>a'\<in>A. b .= a'" by (rule set_eqD2)
```
```   190   from this obtain a'
```
```   191       where a': "a' \<in> A" "b .= a'" by auto
```
```   192
```
```   193   from a' b'
```
```   194       show "P" by (rule r)
```
```   195 qed
```
```   196
```
```   197 lemma (in equivalence) eq_elem_cong_r [trans]:
```
```   198   assumes a: "a .\<in> A"
```
```   199     and cong: "A {.=} A'"
```
```   200     and carr: "a \<in> carrier S"
```
```   201     and Carr: "A \<subseteq> carrier S" "A' \<subseteq> carrier S"
```
```   202   shows "a .\<in> A'"
```
```   203 using a cong
```
```   204 proof (elim elemE set_eqE)
```
```   205   fix b
```
```   206   assume bA: "b \<in> A"
```
```   207      and inA': "\<forall>b\<in>A. b .\<in> A'"
```
```   208   note [simp] = carr Carr Carr[THEN subsetD] bA
```
```   209   assume "a .= b"
```
```   210   also from bA inA'
```
```   211        have "b .\<in> A'" by fast
```
```   212   finally
```
```   213        show "a .\<in> A'" by simp
```
```   214 qed
```
```   215
```
```   216 lemma (in equivalence) set_eq_sym [sym]:
```
```   217   assumes "A {.=} B"
```
```   218     and "A \<subseteq> carrier S" "B \<subseteq> carrier S"
```
```   219   shows "B {.=} A"
```
```   220 using assms
```
```   221 unfolding set_eq_def elem_def
```
```   222 by fast
```
```   223
```
```   224 (* FIXME: the following two required in Isabelle 2008, not Isabelle 2007 *)
```
```   225 (* alternatively, could declare lemmas [trans] = ssubst [where 'a = "'a set"] *)
```
```   226
```
```   227 lemma (in equivalence) equal_set_eq_trans [trans]:
```
```   228   assumes AB: "A = B" and BC: "B {.=} C"
```
```   229   shows "A {.=} C"
```
```   230   using AB BC by simp
```
```   231
```
```   232 lemma (in equivalence) set_eq_equal_trans [trans]:
```
```   233   assumes AB: "A {.=} B" and BC: "B = C"
```
```   234   shows "A {.=} C"
```
```   235   using AB BC by simp
```
```   236
```
```   237
```
```   238 lemma (in equivalence) set_eq_trans [trans]:
```
```   239   assumes AB: "A {.=} B" and BC: "B {.=} C"
```
```   240     and carr: "A \<subseteq> carrier S"  "B \<subseteq> carrier S"  "C \<subseteq> carrier S"
```
```   241   shows "A {.=} C"
```
```   242 proof (intro set_eqI)
```
```   243   fix a
```
```   244   assume aA: "a \<in> A"
```
```   245   with carr have "a \<in> carrier S" by fast
```
```   246   note [simp] = carr this
```
```   247
```
```   248   from aA
```
```   249        have "a .\<in> A" by (simp add: elem_exact)
```
```   250   also note AB
```
```   251   also note BC
```
```   252   finally
```
```   253        show "a .\<in> C" by simp
```
```   254 next
```
```   255   fix c
```
```   256   assume cC: "c \<in> C"
```
```   257   with carr have "c \<in> carrier S" by fast
```
```   258   note [simp] = carr this
```
```   259
```
```   260   from cC
```
```   261        have "c .\<in> C" by (simp add: elem_exact)
```
```   262   also note BC[symmetric]
```
```   263   also note AB[symmetric]
```
```   264   finally
```
```   265        show "c .\<in> A" by simp
```
```   266 qed
```
```   267
```
```   268 (* FIXME: generalise for insert *)
```
```   269
```
```   270 (*
```
```   271 lemma (in equivalence) set_eq_insert:
```
```   272   assumes x: "x .= x'"
```
```   273     and carr: "x \<in> carrier S" "x' \<in> carrier S" "A \<subseteq> carrier S"
```
```   274   shows "insert x A {.=} insert x' A"
```
```   275   unfolding set_eq_def elem_def
```
```   276 apply rule
```
```   277 apply rule
```
```   278 apply (case_tac "xa = x")
```
```   279 using x apply fast
```
```   280 apply (subgoal_tac "xa \<in> A") prefer 2 apply fast
```
```   281 apply (rule_tac x=xa in bexI)
```
```   282 using carr apply (rule_tac refl) apply auto 
```
```   283 apply safe
```
```   284 *)
```
```   285
```
```   286 lemma (in equivalence) set_eq_pairI:
```
```   287   assumes xx': "x .= x'"
```
```   288     and carr: "x \<in> carrier S" "x' \<in> carrier S" "y \<in> carrier S"
```
```   289   shows "{x, y} {.=} {x', y}"
```
```   290 unfolding set_eq_def elem_def
```
```   291 proof safe
```
```   292   have "x' \<in> {x', y}" by fast
```
```   293   with xx' show "\<exists>b\<in>{x', y}. x .= b" by fast
```
```   294 next
```
```   295   have "y \<in> {x', y}" by fast
```
```   296   with carr show "\<exists>b\<in>{x', y}. y .= b" by fast
```
```   297 next
```
```   298   have "x \<in> {x, y}" by fast
```
```   299   with xx'[symmetric] carr
```
```   300   show "\<exists>a\<in>{x, y}. x' .= a" by fast
```
```   301 next
```
```   302   have "y \<in> {x, y}" by fast
```
```   303   with carr show "\<exists>a\<in>{x, y}. y .= a" by fast
```
```   304 qed
```
```   305
```
```   306 lemma (in equivalence) is_closedI:
```
```   307   assumes closed: "!!x y. [| x .= y; x \<in> A; y \<in> carrier S |] ==> y \<in> A"
```
```   308     and S: "A \<subseteq> carrier S"
```
```   309   shows "is_closed A"
```
```   310   unfolding eq_is_closed_def eq_closure_of_def elem_def
```
```   311   using S
```
```   312   by (blast dest: closed sym)
```
```   313
```
```   314 lemma (in equivalence) closure_of_eq:
```
```   315   "[| x .= x'; A \<subseteq> carrier S; x \<in> closure_of A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> closure_of A"
```
```   316   unfolding eq_closure_of_def elem_def
```
```   317   by (blast intro: trans sym)
```
```   318
```
```   319 lemma (in equivalence) is_closed_eq [dest]:
```
```   320   "[| x .= x'; x \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> A"
```
```   321   unfolding eq_is_closed_def
```
```   322   using closure_of_eq [where A = A]
```
```   323   by simp
```
```   324
```
```   325 lemma (in equivalence) is_closed_eq_rev [dest]:
```
```   326   "[| x .= x'; x' \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x \<in> A"
```
```   327   by (drule sym) (simp_all add: is_closed_eq)
```
```   328
```
```   329 lemma closure_of_closed [simp, intro]:
```
```   330   fixes S (structure)
```
```   331   shows "closure_of A \<subseteq> carrier S"
```
```   332 unfolding eq_closure_of_def
```
```   333 by fast
```
```   334
```
```   335 lemma closure_of_memI:
```
```   336   fixes S (structure)
```
```   337   assumes "a .\<in> A"
```
```   338     and "a \<in> carrier S"
```
```   339   shows "a \<in> closure_of A"
```
```   340 unfolding eq_closure_of_def
```
```   341 using assms
```
```   342 by fast
```
```   343
```
```   344 lemma closure_ofI2:
```
```   345   fixes S (structure)
```
```   346   assumes "a .= a'"
```
```   347     and "a' \<in> A"
```
```   348     and "a \<in> carrier S"
```
```   349   shows "a \<in> closure_of A"
```
```   350 unfolding eq_closure_of_def elem_def
```
```   351 using assms
```
```   352 by fast
```
```   353
```
```   354 lemma closure_of_memE:
```
```   355   fixes S (structure)
```
```   356   assumes p: "a \<in> closure_of A"
```
```   357     and r: "\<lbrakk>a \<in> carrier S; a .\<in> A\<rbrakk> \<Longrightarrow> P"
```
```   358   shows "P"
```
```   359 proof -
```
```   360   from p
```
```   361       have acarr: "a \<in> carrier S"
```
```   362       and "a .\<in> A"
```
```   363       by (simp add: eq_closure_of_def)+
```
```   364   thus "P" by (rule r)
```
```   365 qed
```
```   366
```
```   367 lemma closure_ofE2:
```
```   368   fixes S (structure)
```
```   369   assumes p: "a \<in> closure_of A"
```
```   370     and r: "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
```
```   371   shows "P"
```
```   372 proof -
```
```   373   from p have acarr: "a \<in> carrier S" by (simp add: eq_closure_of_def)
```
```   374
```
```   375   from p have "\<exists>a'\<in>A. a .= a'" by (simp add: eq_closure_of_def elem_def)
```
```   376   from this obtain a'
```
```   377       where "a' \<in> A" and "a .= a'" by auto
```
```   378
```
```   379   from acarr and this
```
```   380       show "P" by (rule r)
```
```   381 qed
```
```   382
```
```   383 (*
```
```   384 lemma (in equivalence) classes_consistent:
```
```   385   assumes Acarr: "A \<subseteq> carrier S"
```
```   386   shows "is_closed (closure_of A)"
```
```   387 apply (blast intro: elemI elim elemE)
```
```   388 using assms
```
```   389 apply (intro is_closedI closure_of_memI, simp)
```
```   390  apply (elim elemE closure_of_memE)
```
```   391 proof -
```
```   392   fix x a' a''
```
```   393   assume carr: "x \<in> carrier S" "a' \<in> carrier S"
```
```   394   assume a''A: "a'' \<in> A"
```
```   395   with Acarr have "a'' \<in> carrier S" by fast
```
```   396   note [simp] = carr this Acarr
```
```   397
```
```   398   assume "x .= a'"
```
```   399   also assume "a' .= a''"
```
```   400   also from a''A
```
```   401        have "a'' .\<in> A" by (simp add: elem_exact)
```
```   402   finally show "x .\<in> A" by simp
```
```   403 qed
```
```   404 *)
```
```   405 (*
```
```   406 lemma (in equivalence) classes_small:
```
```   407   assumes "is_closed B"
```
```   408     and "A \<subseteq> B"
```
```   409   shows "closure_of A \<subseteq> B"
```
```   410 using assms
```
```   411 by (blast dest: is_closedD2 elem_subsetD elim: closure_of_memE)
```
```   412
```
```   413 lemma (in equivalence) classes_eq:
```
```   414   assumes "A \<subseteq> carrier S"
```
```   415   shows "A {.=} closure_of A"
```
```   416 using assms
```
```   417 by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE)
```
```   418
```
```   419 lemma (in equivalence) complete_classes:
```
```   420   assumes c: "is_closed A"
```
```   421   shows "A = closure_of A"
```
```   422 using assms
```
```   423 by (blast intro: closure_of_memI elem_exact dest: is_closedD1 is_closedD2 closure_of_memE)
```
```   424 *)
```
```   425
```
```   426 lemma equivalence_subset:
```
```   427   assumes "equivalence L" "A \<subseteq> carrier L"
```
```   428   shows "equivalence (L\<lparr> carrier := A \<rparr>)"
```
```   429 proof -
```
```   430   interpret L: equivalence L
```
```   431     by (simp add: assms)
```
```   432   show ?thesis
```
```   433     by (unfold_locales, simp_all add: L.sym assms rev_subsetD, meson L.trans assms(2) contra_subsetD)
```
```   434 qed
```
```   435
```
```   436 end
```