src/HOL/Library/Bourbaki_Witt_Fixpoint.thy
 author Andreas Lochbihler Tue Jan 12 15:23:54 2016 +0100 (2016-01-12) changeset 62141 00bfdf4bf237 parent 62058 1cfd5d604937 child 62390 842917225d56 permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Bourbaki_Witt_Fixpoint.thy
```
```     2     Author:     Andreas Lochbihler, ETH Zurich
```
```     3
```
```     4   Follows G. Smolka, S. SchÃ¤fer and C. Doczkal: Transfinite Constructions in
```
```     5   Classical Type Theory. ITP 2015
```
```     6 *)
```
```     7
```
```     8 section \<open>The Bourbaki-Witt tower construction for transfinite iteration\<close>
```
```     9
```
```    10 theory Bourbaki_Witt_Fixpoint imports Main begin
```
```    11
```
```    12 lemma ChainsI [intro?]:
```
```    13   "(\<And>a b. \<lbrakk> a \<in> Y; b \<in> Y \<rbrakk> \<Longrightarrow> (a, b) \<in> r \<or> (b, a) \<in> r) \<Longrightarrow> Y \<in> Chains r"
```
```    14 unfolding Chains_def by blast
```
```    15
```
```    16 lemma in_Chains_subset: "\<lbrakk> M \<in> Chains r; M' \<subseteq> M \<rbrakk> \<Longrightarrow> M' \<in> Chains r"
```
```    17 by(auto simp add: Chains_def)
```
```    18
```
```    19 lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
```
```    20   unfolding Field_def by auto
```
```    21
```
```    22 lemma Chains_FieldD: "\<lbrakk> M \<in> Chains r; x \<in> M \<rbrakk> \<Longrightarrow> x \<in> Field r"
```
```    23 by(auto simp add: Chains_def intro: FieldI1 FieldI2)
```
```    24
```
```    25 lemma partial_order_on_trans:
```
```    26   "\<lbrakk> partial_order_on A r; (x, y) \<in> r; (y, z) \<in> r \<rbrakk> \<Longrightarrow> (x, z) \<in> r"
```
```    27 by(auto simp add: order_on_defs dest: transD)
```
```    28
```
```    29 locale bourbaki_witt_fixpoint =
```
```    30   fixes lub :: "'a set \<Rightarrow> 'a"
```
```    31   and leq :: "('a \<times> 'a) set"
```
```    32   and f :: "'a \<Rightarrow> 'a"
```
```    33   assumes po: "Partial_order leq"
```
```    34   and lub_least: "\<lbrakk> M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> (x, z) \<in> leq \<rbrakk> \<Longrightarrow> (lub M, z) \<in> leq"
```
```    35   and lub_upper: "\<lbrakk> M \<in> Chains leq; x \<in> M \<rbrakk> \<Longrightarrow> (x, lub M) \<in> leq"
```
```    36   and lub_in_Field: "\<lbrakk> M \<in> Chains leq; M \<noteq> {} \<rbrakk> \<Longrightarrow> lub M \<in> Field leq"
```
```    37   and increasing: "\<And>x. x \<in> Field leq \<Longrightarrow> (x, f x) \<in> leq"
```
```    38 begin
```
```    39
```
```    40 lemma leq_trans: "\<lbrakk> (x, y) \<in> leq; (y, z) \<in> leq \<rbrakk> \<Longrightarrow> (x, z) \<in> leq"
```
```    41 by(rule partial_order_on_trans[OF po])
```
```    42
```
```    43 lemma leq_refl: "x \<in> Field leq \<Longrightarrow> (x, x) \<in> leq"
```
```    44 using po by(simp add: order_on_defs refl_on_def)
```
```    45
```
```    46 lemma leq_antisym: "\<lbrakk> (x, y) \<in> leq; (y, x) \<in> leq \<rbrakk> \<Longrightarrow> x = y"
```
```    47 using po by(simp add: order_on_defs antisym_def)
```
```    48
```
```    49 inductive_set iterates_above :: "'a \<Rightarrow> 'a set"
```
```    50   for a
```
```    51 where
```
```    52   base: "a \<in> iterates_above a"
```
```    53 | step: "x \<in> iterates_above a \<Longrightarrow> f x \<in> iterates_above a"
```
```    54 | Sup: "\<lbrakk> M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above a \<rbrakk> \<Longrightarrow> lub M \<in> iterates_above a"
```
```    55
```
```    56 definition fixp_above :: "'a \<Rightarrow> 'a"
```
```    57 where "fixp_above a = lub (iterates_above a)"
```
```    58
```
```    59 context
```
```    60   notes leq_refl [intro!, simp]
```
```    61   and base [intro]
```
```    62   and step [intro]
```
```    63   and Sup [intro]
```
```    64   and leq_trans [trans]
```
```    65 begin
```
```    66
```
```    67 lemma iterates_above_le_f: "\<lbrakk> x \<in> iterates_above a; a \<in> Field leq \<rbrakk> \<Longrightarrow> (x, f x) \<in> leq"
```
```    68 by(induction x rule: iterates_above.induct)(blast intro: increasing FieldI2 lub_in_Field)+
```
```    69
```
```    70 lemma iterates_above_Field: "\<lbrakk> x \<in> iterates_above a; a \<in> Field leq \<rbrakk> \<Longrightarrow> x \<in> Field leq"
```
```    71 by(drule (1) iterates_above_le_f)(rule FieldI1)
```
```    72
```
```    73 lemma iterates_above_ge:
```
```    74   assumes y: "y \<in> iterates_above a"
```
```    75   and a: "a \<in> Field leq"
```
```    76   shows "(a, y) \<in> leq"
```
```    77 using y by(induction)(auto intro: a increasing iterates_above_le_f leq_trans leq_trans[OF _ lub_upper])
```
```    78
```
```    79 lemma iterates_above_lub:
```
```    80   assumes M: "M \<in> Chains leq"
```
```    81   and nempty: "M \<noteq> {}"
```
```    82   and upper: "\<And>y. y \<in> M \<Longrightarrow> \<exists>z \<in> M. (y, z) \<in> leq \<and> z \<in> iterates_above a"
```
```    83   shows "lub M \<in> iterates_above a"
```
```    84 proof -
```
```    85   let ?M = "M \<inter> iterates_above a"
```
```    86   from M have M': "?M \<in> Chains leq" by(rule in_Chains_subset)simp
```
```    87   have "?M \<noteq> {}" using nempty by(auto dest: upper)
```
```    88   with M' have "lub ?M \<in> iterates_above a" by(rule Sup) blast
```
```    89   also have "lub ?M = lub M" using nempty
```
```    90     by(intro leq_antisym)(blast intro!: lub_least[OF M] lub_least[OF M'] intro: lub_upper[OF M'] lub_upper[OF M] leq_trans dest: upper)+
```
```    91   finally show ?thesis .
```
```    92 qed
```
```    93
```
```    94 lemma iterates_above_successor:
```
```    95   assumes y: "y \<in> iterates_above a"
```
```    96   and a: "a \<in> Field leq"
```
```    97   shows "y = a \<or> y \<in> iterates_above (f a)"
```
```    98 using y
```
```    99 proof induction
```
```   100   case base thus ?case by simp
```
```   101 next
```
```   102   case (step x) thus ?case by auto
```
```   103 next
```
```   104   case (Sup M)
```
```   105   show ?case
```
```   106   proof(cases "\<exists>x. M \<subseteq> {x}")
```
```   107     case True
```
```   108     with \<open>M \<noteq> {}\<close> obtain y where M: "M = {y}" by auto
```
```   109     have "lub M = y"
```
```   110       by(rule leq_antisym)(auto intro!: lub_upper Sup lub_least ChainsI simp add: a M Sup.hyps(3)[of y, THEN iterates_above_Field] dest: iterates_above_Field)
```
```   111     with Sup.IH[of y] M show ?thesis by simp
```
```   112   next
```
```   113     case False
```
```   114     from Sup(1-2) have "lub M \<in> iterates_above (f a)"
```
```   115     proof(rule iterates_above_lub)
```
```   116       fix y
```
```   117       assume y: "y \<in> M"
```
```   118       from Sup.IH[OF this] show "\<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above (f a)"
```
```   119       proof
```
```   120         assume "y = a"
```
```   121         from y False obtain z where z: "z \<in> M" and neq: "y \<noteq> z" by (metis insertI1 subsetI)
```
```   122         with Sup.IH[OF z] \<open>y = a\<close> Sup.hyps(3)[OF z]
```
```   123         show ?thesis by(auto dest: iterates_above_ge intro: a)
```
```   124       next
```
```   125         assume "y \<in> iterates_above (f a)"
```
```   126         moreover with increasing[OF a] have "y \<in> Field leq"
```
```   127           by(auto dest!: iterates_above_Field intro: FieldI2)
```
```   128         ultimately show ?thesis using y by(auto)
```
```   129       qed
```
```   130     qed
```
```   131     thus ?thesis by simp
```
```   132   qed
```
```   133 qed
```
```   134
```
```   135 lemma iterates_above_Sup_aux:
```
```   136   assumes M: "M \<in> Chains leq" "M \<noteq> {}"
```
```   137   and M': "M' \<in> Chains leq" "M' \<noteq> {}"
```
```   138   and comp: "\<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x"
```
```   139   shows "(lub M, lub M') \<in> leq \<or> lub M \<in> iterates_above (lub M')"
```
```   140 proof(cases "\<exists>x \<in> M. x \<in> iterates_above (lub M')")
```
```   141   case True
```
```   142   then obtain x where x: "x \<in> M" "x \<in> iterates_above (lub M')" by blast
```
```   143   have lub_M': "lub M' \<in> Field leq" using M' by(rule lub_in_Field)
```
```   144   have "lub M \<in> iterates_above (lub M')" using M
```
```   145   proof(rule iterates_above_lub)
```
```   146     fix y
```
```   147     assume y: "y \<in> M"
```
```   148     from comp[OF y] show "\<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above (lub M')"
```
```   149     proof
```
```   150       assume "y \<in> iterates_above (lub M')"
```
```   151       from this iterates_above_Field[OF this] y lub_M' show ?thesis by blast
```
```   152     next
```
```   153       assume "lub M' \<in> iterates_above y"
```
```   154       hence "(y, lub M') \<in> leq" using Chains_FieldD[OF M(1) y] by(rule iterates_above_ge)
```
```   155       also have "(lub M', x) \<in> leq" using x(2) lub_M' by(rule iterates_above_ge)
```
```   156       finally show ?thesis using x by blast
```
```   157     qed
```
```   158   qed
```
```   159   thus ?thesis ..
```
```   160 next
```
```   161   case False
```
```   162   have "(lub M, lub M') \<in> leq" using M
```
```   163   proof(rule lub_least)
```
```   164     fix x
```
```   165     assume x: "x \<in> M"
```
```   166     from comp[OF x] x False have "lub M' \<in> iterates_above x" by auto
```
```   167     moreover from M(1) x have "x \<in> Field leq" by(rule Chains_FieldD)
```
```   168     ultimately show "(x, lub M') \<in> leq" by(rule iterates_above_ge)
```
```   169   qed
```
```   170   thus ?thesis ..
```
```   171 qed
```
```   172
```
```   173 lemma iterates_above_triangle:
```
```   174   assumes x: "x \<in> iterates_above a"
```
```   175   and y: "y \<in> iterates_above a"
```
```   176   and a: "a \<in> Field leq"
```
```   177   shows "x \<in> iterates_above y \<or> y \<in> iterates_above x"
```
```   178 using x y
```
```   179 proof(induction arbitrary: y)
```
```   180   case base then show ?case by simp
```
```   181 next
```
```   182   case (step x) thus ?case using a
```
```   183     by(auto dest: iterates_above_successor intro: iterates_above_Field)
```
```   184 next
```
```   185   case x: (Sup M)
```
```   186   hence lub: "lub M \<in> iterates_above a" by blast
```
```   187   from \<open>y \<in> iterates_above a\<close> show ?case
```
```   188   proof(induction)
```
```   189     case base show ?case using lub by simp
```
```   190   next
```
```   191     case (step y) thus ?case using a
```
```   192       by(auto dest: iterates_above_successor intro: iterates_above_Field)
```
```   193   next
```
```   194     case y: (Sup M')
```
```   195     hence lub': "lub M' \<in> iterates_above a" by blast
```
```   196     have *: "x \<in> iterates_above (lub M') \<or> lub M' \<in> iterates_above x" if "x \<in> M" for x
```
```   197       using that lub' by(rule x.IH)
```
```   198     with x(1-2) y(1-2) have "(lub M, lub M') \<in> leq \<or> lub M \<in> iterates_above (lub M')"
```
```   199       by(rule iterates_above_Sup_aux)
```
```   200     moreover from y(1-2) x(1-2) have "(lub M', lub M) \<in> leq \<or> lub M' \<in> iterates_above (lub M)"
```
```   201       by(rule iterates_above_Sup_aux)(blast dest: y.IH)
```
```   202     ultimately show ?case by(auto 4 3 dest: leq_antisym)
```
```   203   qed
```
```   204 qed
```
```   205
```
```   206 lemma chain_iterates_above:
```
```   207   assumes a: "a \<in> Field leq"
```
```   208   shows "iterates_above a \<in> Chains leq" (is "?C \<in> _")
```
```   209 proof (rule ChainsI)
```
```   210   fix x y
```
```   211   assume "x \<in> ?C" "y \<in> ?C"
```
```   212   hence "x \<in> iterates_above y \<or> y \<in> iterates_above x" using a by(rule iterates_above_triangle)
```
```   213   moreover from \<open>x \<in> ?C\<close> a have "x \<in> Field leq" by(rule iterates_above_Field)
```
```   214   moreover from \<open>y \<in> ?C\<close> a have "y \<in> Field leq" by(rule iterates_above_Field)
```
```   215   ultimately show "(x, y) \<in> leq \<or> (y, x) \<in> leq" by(auto dest: iterates_above_ge)
```
```   216 qed
```
```   217
```
```   218 lemma fixp_iterates_above: "a \<in> Field leq \<Longrightarrow> fixp_above a \<in> iterates_above a"
```
```   219 unfolding fixp_above_def by(rule iterates_above.Sup)(blast intro: chain_iterates_above)+
```
```   220
```
```   221 lemma
```
```   222   assumes b: "b \<in> iterates_above a"
```
```   223   and fb: "f b = b"
```
```   224   and x: "x \<in> iterates_above a"
```
```   225   and a: "a \<in> Field leq"
```
```   226   shows "b \<in> iterates_above x"
```
```   227 using x
```
```   228 proof(induction)
```
```   229   case base show ?case using b by simp
```
```   230 next
```
```   231   case (step x)
```
```   232   from step.hyps a have "x \<in> Field leq" by(rule iterates_above_Field)
```
```   233   from iterates_above_successor[OF step.IH this] fb
```
```   234   show ?case by(auto)
```
```   235 next
```
```   236   case (Sup M)
```
```   237   oops
```
```   238
```
```   239 lemma fixp_above_Field: "a \<in> Field leq \<Longrightarrow> fixp_above a \<in> Field leq"
```
```   240 using fixp_iterates_above by(rule iterates_above_Field)
```
```   241
```
```   242 lemma fixp_above_unfold:
```
```   243   assumes a: "a \<in> Field leq"
```
```   244   shows "fixp_above a = f (fixp_above a)" (is "?a = f ?a")
```
```   245 proof(rule leq_antisym)
```
```   246   show "(?a, f ?a) \<in> leq" using fixp_above_Field[OF a] by(rule increasing)
```
```   247
```
```   248   have "f ?a \<in> iterates_above a" using fixp_iterates_above[OF a] by(rule iterates_above.step)
```
```   249   with chain_iterates_above[OF a] show "(f ?a, ?a) \<in> leq" unfolding fixp_above_def by(rule lub_upper)
```
```   250 qed
```
```   251
```
```   252 end
```
```   253
```
```   254 end
```
```   255
```
`   256 end`