src/HOL/Library/Order_Union.thy
 author haftmann Fri Nov 01 18:51:14 2013 +0100 (2013-11-01) changeset 54230 b1d955791529 parent 52199 d25fc4c0ff62 child 54473 8bee5ca99e63 permissions -rw-r--r--
more simplification rules on unary and binary minus
```     1 (*  Title:      HOL/Library/Order_Union.thy
```
```     2     Author:     Andrei Popescu, TU Muenchen
```
```     3
```
```     4 The ordinal-like sum of two orders with disjoint fields
```
```     5 *)
```
```     6
```
```     7 header {* Order Union *}
```
```     8
```
```     9 theory Order_Union
```
```    10 imports "~~/src/HOL/Cardinals/Wellfounded_More_Base"
```
```    11 begin
```
```    12
```
```    13 definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel"  (infix "Osum" 60) where
```
```    14   "r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
```
```    15
```
```    16 notation Osum  (infix "\<union>o" 60)
```
```    17
```
```    18 lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"
```
```    19   unfolding Osum_def Field_def by blast
```
```    20
```
```    21 lemma Osum_wf:
```
```    22 assumes FLD: "Field r Int Field r' = {}" and
```
```    23         WF: "wf r" and WF': "wf r'"
```
```    24 shows "wf (r Osum r')"
```
```    25 unfolding wf_eq_minimal2 unfolding Field_Osum
```
```    26 proof(intro allI impI, elim conjE)
```
```    27   fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
```
```    28   obtain B where B_def: "B = A Int Field r" by blast
```
```    29   show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
```
```    30   proof(cases "B = {}")
```
```    31     assume Case1: "B \<noteq> {}"
```
```    32     hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
```
```    33     then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
```
```    34     using WF  unfolding wf_eq_minimal2 by blast
```
```    35     hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
```
```    36     (*  *)
```
```    37     have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
```
```    38     proof(intro ballI)
```
```    39       fix a1 assume **: "a1 \<in> A"
```
```    40       {assume Case11: "a1 \<in> Field r"
```
```    41        hence "(a1,a) \<notin> r" using B_def ** 2 by auto
```
```    42        moreover
```
```    43        have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
```
```    44        ultimately have "(a1,a) \<notin> r Osum r'"
```
```    45        using 3 unfolding Osum_def by auto
```
```    46       }
```
```    47       moreover
```
```    48       {assume Case12: "a1 \<notin> Field r"
```
```    49        hence "(a1,a) \<notin> r" unfolding Field_def by auto
```
```    50        moreover
```
```    51        have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
```
```    52        ultimately have "(a1,a) \<notin> r Osum r'"
```
```    53        using 3 unfolding Osum_def by auto
```
```    54       }
```
```    55       ultimately show "(a1,a) \<notin> r Osum r'" by blast
```
```    56     qed
```
```    57     thus ?thesis using 1 B_def by auto
```
```    58   next
```
```    59     assume Case2: "B = {}"
```
```    60     hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
```
```    61     then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
```
```    62     using WF' unfolding wf_eq_minimal2 by blast
```
```    63     hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
```
```    64     (*  *)
```
```    65     have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
```
```    66     proof(unfold Osum_def, auto simp add: 3)
```
```    67       fix a1' assume "(a1', a') \<in> r"
```
```    68       thus False using 4 unfolding Field_def by blast
```
```    69     next
```
```    70       fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
```
```    71       thus False using Case2 B_def by auto
```
```    72     qed
```
```    73     thus ?thesis using 2 by blast
```
```    74   qed
```
```    75 qed
```
```    76
```
```    77 lemma Osum_Refl:
```
```    78 assumes FLD: "Field r Int Field r' = {}" and
```
```    79         REFL: "Refl r" and REFL': "Refl r'"
```
```    80 shows "Refl (r Osum r')"
```
```    81 using assms
```
```    82 unfolding refl_on_def Field_Osum unfolding Osum_def by blast
```
```    83
```
```    84 lemma Osum_trans:
```
```    85 assumes FLD: "Field r Int Field r' = {}" and
```
```    86         TRANS: "trans r" and TRANS': "trans r'"
```
```    87 shows "trans (r Osum r')"
```
```    88 proof(unfold trans_def, auto)
```
```    89   fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
```
```    90   show  "(x, z) \<in> r \<union>o r'"
```
```    91   proof-
```
```    92     {assume Case1: "(x,y) \<in> r"
```
```    93      hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
```
```    94      have ?thesis
```
```    95      proof-
```
```    96        {assume Case11: "(y,z) \<in> r"
```
```    97         hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
```
```    98         hence ?thesis unfolding Osum_def by auto
```
```    99        }
```
```   100        moreover
```
```   101        {assume Case12: "(y,z) \<in> r'"
```
```   102         hence "y \<in> Field r'" unfolding Field_def by auto
```
```   103         hence False using FLD 1 by auto
```
```   104        }
```
```   105        moreover
```
```   106        {assume Case13: "z \<in> Field r'"
```
```   107         hence ?thesis using 1 unfolding Osum_def by auto
```
```   108        }
```
```   109        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   110      qed
```
```   111     }
```
```   112     moreover
```
```   113     {assume Case2: "(x,y) \<in> r'"
```
```   114      hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
```
```   115      have ?thesis
```
```   116      proof-
```
```   117        {assume Case21: "(y,z) \<in> r"
```
```   118         hence "y \<in> Field r" unfolding Field_def by auto
```
```   119         hence False using FLD 2 by auto
```
```   120        }
```
```   121        moreover
```
```   122        {assume Case22: "(y,z) \<in> r'"
```
```   123         hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
```
```   124         hence ?thesis unfolding Osum_def by auto
```
```   125        }
```
```   126        moreover
```
```   127        {assume Case23: "y \<in> Field r"
```
```   128         hence False using FLD 2 by auto
```
```   129        }
```
```   130        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   131      qed
```
```   132     }
```
```   133     moreover
```
```   134     {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
```
```   135      have ?thesis
```
```   136      proof-
```
```   137        {assume Case31: "(y,z) \<in> r"
```
```   138         hence "y \<in> Field r" unfolding Field_def by auto
```
```   139         hence False using FLD Case3 by auto
```
```   140        }
```
```   141        moreover
```
```   142        {assume Case32: "(y,z) \<in> r'"
```
```   143         hence "z \<in> Field r'" unfolding Field_def by blast
```
```   144         hence ?thesis unfolding Osum_def using Case3 by auto
```
```   145        }
```
```   146        moreover
```
```   147        {assume Case33: "y \<in> Field r"
```
```   148         hence False using FLD Case3 by auto
```
```   149        }
```
```   150        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   151      qed
```
```   152     }
```
```   153     ultimately show ?thesis using * unfolding Osum_def by blast
```
```   154   qed
```
```   155 qed
```
```   156
```
```   157 lemma Osum_Preorder:
```
```   158 "\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
```
```   159 unfolding preorder_on_def using Osum_Refl Osum_trans by blast
```
```   160
```
```   161 lemma Osum_antisym:
```
```   162 assumes FLD: "Field r Int Field r' = {}" and
```
```   163         AN: "antisym r" and AN': "antisym r'"
```
```   164 shows "antisym (r Osum r')"
```
```   165 proof(unfold antisym_def, auto)
```
```   166   fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
```
```   167   show  "x = y"
```
```   168   proof-
```
```   169     {assume Case1: "(x,y) \<in> r"
```
```   170      hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
```
```   171      have ?thesis
```
```   172      proof-
```
```   173        have "(y,x) \<in> r \<Longrightarrow> ?thesis"
```
```   174        using Case1 AN antisym_def[of r] by blast
```
```   175        moreover
```
```   176        {assume "(y,x) \<in> r'"
```
```   177         hence "y \<in> Field r'" unfolding Field_def by auto
```
```   178         hence False using FLD 1 by auto
```
```   179        }
```
```   180        moreover
```
```   181        have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
```
```   182        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   183      qed
```
```   184     }
```
```   185     moreover
```
```   186     {assume Case2: "(x,y) \<in> r'"
```
```   187      hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
```
```   188      have ?thesis
```
```   189      proof-
```
```   190        {assume "(y,x) \<in> r"
```
```   191         hence "y \<in> Field r" unfolding Field_def by auto
```
```   192         hence False using FLD 2 by auto
```
```   193        }
```
```   194        moreover
```
```   195        have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
```
```   196        using Case2 AN' antisym_def[of r'] by blast
```
```   197        moreover
```
```   198        {assume "y \<in> Field r"
```
```   199         hence False using FLD 2 by auto
```
```   200        }
```
```   201        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   202      qed
```
```   203     }
```
```   204     moreover
```
```   205     {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
```
```   206      have ?thesis
```
```   207      proof-
```
```   208        {assume "(y,x) \<in> r"
```
```   209         hence "y \<in> Field r" unfolding Field_def by auto
```
```   210         hence False using FLD Case3 by auto
```
```   211        }
```
```   212        moreover
```
```   213        {assume Case32: "(y,x) \<in> r'"
```
```   214         hence "x \<in> Field r'" unfolding Field_def by blast
```
```   215         hence False using FLD Case3 by auto
```
```   216        }
```
```   217        moreover
```
```   218        have "\<not> y \<in> Field r" using FLD Case3 by auto
```
```   219        ultimately show ?thesis using ** unfolding Osum_def by blast
```
```   220      qed
```
```   221     }
```
```   222     ultimately show ?thesis using * unfolding Osum_def by blast
```
```   223   qed
```
```   224 qed
```
```   225
```
```   226 lemma Osum_Partial_order:
```
```   227 "\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
```
```   228  Partial_order (r Osum r')"
```
```   229 unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
```
```   230
```
```   231 lemma Osum_Total:
```
```   232 assumes FLD: "Field r Int Field r' = {}" and
```
```   233         TOT: "Total r" and TOT': "Total r'"
```
```   234 shows "Total (r Osum r')"
```
```   235 using assms
```
```   236 unfolding total_on_def  Field_Osum unfolding Osum_def by blast
```
```   237
```
```   238 lemma Osum_Linear_order:
```
```   239 "\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
```
```   240  Linear_order (r Osum r')"
```
```   241 unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
```
```   242
```
```   243 lemma Osum_minus_Id1:
```
```   244 assumes "r \<le> Id"
```
```   245 shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
```
```   246 proof-
```
```   247   let ?Left = "(r Osum r') - Id"
```
```   248   let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
```
```   249   {fix a::'a and b assume *: "(a,b) \<notin> Id"
```
```   250    {assume "(a,b) \<in> r"
```
```   251     with * have False using assms by auto
```
```   252    }
```
```   253    moreover
```
```   254    {assume "(a,b) \<in> r'"
```
```   255     with * have "(a,b) \<in> r' - Id" by auto
```
```   256    }
```
```   257    ultimately
```
```   258    have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
```
```   259    unfolding Osum_def by auto
```
```   260   }
```
```   261   thus ?thesis by auto
```
```   262 qed
```
```   263
```
```   264 lemma Osum_minus_Id2:
```
```   265 assumes "r' \<le> Id"
```
```   266 shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
```
```   267 proof-
```
```   268   let ?Left = "(r Osum r') - Id"
```
```   269   let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
```
```   270   {fix a::'a and b assume *: "(a,b) \<notin> Id"
```
```   271    {assume "(a,b) \<in> r'"
```
```   272     with * have False using assms by auto
```
```   273    }
```
```   274    moreover
```
```   275    {assume "(a,b) \<in> r"
```
```   276     with * have "(a,b) \<in> r - Id" by auto
```
```   277    }
```
```   278    ultimately
```
```   279    have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
```
```   280    unfolding Osum_def by auto
```
```   281   }
```
```   282   thus ?thesis by auto
```
```   283 qed
```
```   284
```
```   285 lemma Osum_minus_Id:
```
```   286 assumes TOT: "Total r" and TOT': "Total r'" and
```
```   287         NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
```
```   288 shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
```
```   289 proof-
```
```   290   {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
```
```   291    have "(a,a') \<in> (r - Id) Osum (r' - Id)"
```
```   292    proof-
```
```   293      {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
```
```   294       with ** have ?thesis unfolding Osum_def by auto
```
```   295      }
```
```   296      moreover
```
```   297      {assume "a \<in> Field r \<and> a' \<in> Field r'"
```
```   298       hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
```
```   299       using assms Total_Id_Field by blast
```
```   300       hence ?thesis unfolding Osum_def by auto
```
```   301      }
```
```   302      ultimately show ?thesis using * unfolding Osum_def by blast
```
```   303    qed
```
```   304   }
```
```   305   thus ?thesis by(auto simp add: Osum_def)
```
```   306 qed
```
```   307
```
```   308 lemma wf_Int_Times:
```
```   309 assumes "A Int B = {}"
```
```   310 shows "wf(A \<times> B)"
```
```   311 proof(unfold wf_def, auto)
```
```   312   fix P x
```
```   313   assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
```
```   314   moreover have "\<forall>y \<in> A. P y" using assms * by blast
```
```   315   ultimately show "P x" using * by (case_tac "x \<in> B", auto)
```
```   316 qed
```
```   317
```
```   318 lemma Osum_wf_Id:
```
```   319 assumes TOT: "Total r" and TOT': "Total r'" and
```
```   320         FLD: "Field r Int Field r' = {}" and
```
```   321         WF: "wf(r - Id)" and WF': "wf(r' - Id)"
```
```   322 shows "wf ((r Osum r') - Id)"
```
```   323 proof(cases "r \<le> Id \<or> r' \<le> Id")
```
```   324   assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
```
```   325   have "Field(r - Id) Int Field(r' - Id) = {}"
```
```   326   using FLD mono_Field[of "r - Id" r]  mono_Field[of "r' - Id" r']
```
```   327             Diff_subset[of r Id] Diff_subset[of r' Id] by blast
```
```   328   thus ?thesis
```
```   329   using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
```
```   330         wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
```
```   331 next
```
```   332   have 1: "wf(Field r \<times> Field r')"
```
```   333   using FLD by (auto simp add: wf_Int_Times)
```
```   334   assume Case2: "r \<le> Id \<or> r' \<le> Id"
```
```   335   moreover
```
```   336   {assume Case21: "r \<le> Id"
```
```   337    hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
```
```   338    using Osum_minus_Id1[of r r'] by simp
```
```   339    moreover
```
```   340    {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
```
```   341     using FLD unfolding Field_def by blast
```
```   342     hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
```
```   343     using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
```
```   344     by (auto simp add: Un_commute)
```
```   345    }
```
```   346    ultimately have ?thesis by (auto simp add: wf_subset)
```
```   347   }
```
```   348   moreover
```
```   349   {assume Case22: "r' \<le> Id"
```
```   350    hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
```
```   351    using Osum_minus_Id2[of r' r] by simp
```
```   352    moreover
```
```   353    {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
```
```   354     using FLD unfolding Field_def by blast
```
```   355     hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
```
```   356     using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
```
```   357     by (auto simp add: Un_commute)
```
```   358    }
```
```   359    ultimately have ?thesis by (auto simp add: wf_subset)
```
```   360   }
```
```   361   ultimately show ?thesis by blast
```
```   362 qed
```
```   363
```
```   364 lemma Osum_Well_order:
```
```   365 assumes FLD: "Field r Int Field r' = {}" and
```
```   366         WELL: "Well_order r" and WELL': "Well_order r'"
```
```   367 shows "Well_order (r Osum r')"
```
```   368 proof-
```
```   369   have "Total r \<and> Total r'" using WELL WELL'
```
```   370   by (auto simp add: order_on_defs)
```
```   371   thus ?thesis using assms unfolding well_order_on_def
```
```   372   using Osum_Linear_order Osum_wf_Id by blast
```
```   373 qed
```
```   374
```
```   375 end
```
```   376
```