src/HOL/Wfrec.thy
 author wenzelm Tue Sep 26 20:54:40 2017 +0200 (22 months ago) changeset 66695 91500c024c7f parent 63572 c0cbfd2b5a45 child 69593 3dda49e08b9d permissions -rw-r--r--
tuned;
```     1 (*  Title:      HOL/Wfrec.thy
```
```     2     Author:     Tobias Nipkow
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Konrad Slind
```
```     5 *)
```
```     6
```
```     7 section \<open>Well-Founded Recursion Combinator\<close>
```
```     8
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```     9 theory Wfrec
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```    10   imports Wellfounded
```
```    11 begin
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```    12
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```    13 inductive wfrec_rel :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" for R F
```
```    14   where wfrecI: "(\<And>z. (z, x) \<in> R \<Longrightarrow> wfrec_rel R F z (g z)) \<Longrightarrow> wfrec_rel R F x (F g x)"
```
```    15
```
```    16 definition cut :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    17   where "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
```
```    18
```
```    19 definition adm_wf :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> bool"
```
```    20   where "adm_wf R F \<longleftrightarrow> (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
```
```    21
```
```    22 definition wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```    23   where "wfrec R F = (\<lambda>x. THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y)"
```
```    24
```
```    25 lemma cuts_eq: "(cut f R x = cut g R x) \<longleftrightarrow> (\<forall>y. (y, x) \<in> R \<longrightarrow> f y = g y)"
```
```    26   by (simp add: fun_eq_iff cut_def)
```
```    27
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```    28 lemma cut_apply: "(x, a) \<in> R \<Longrightarrow> cut f R a x = f x"
```
```    29   by (simp add: cut_def)
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```    30
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```    31 text \<open>
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```    32   Inductive characterization of \<open>wfrec\<close> combinator; for details see:
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```    33   John Harrison, "Inductive definitions: automation and application".
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```    34 \<close>
```
```    35
```
```    36 lemma theI_unique: "\<exists>!x. P x \<Longrightarrow> P x \<longleftrightarrow> x = The P"
```
```    37   by (auto intro: the_equality[symmetric] theI)
```
```    38
```
```    39 lemma wfrec_unique:
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```    40   assumes "adm_wf R F" "wf R"
```
```    41   shows "\<exists>!y. wfrec_rel R F x y"
```
```    42   using \<open>wf R\<close>
```
```    43 proof induct
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```    44   define f where "f y = (THE z. wfrec_rel R F y z)" for y
```
```    45   case (less x)
```
```    46   then have "\<And>y z. (y, x) \<in> R \<Longrightarrow> wfrec_rel R F y z \<longleftrightarrow> z = f y"
```
```    47     unfolding f_def by (rule theI_unique)
```
```    48   with \<open>adm_wf R F\<close> show ?case
```
```    49     by (subst wfrec_rel.simps) (auto simp: adm_wf_def)
```
```    50 qed
```
```    51
```
```    52 lemma adm_lemma: "adm_wf R (\<lambda>f x. F (cut f R x) x)"
```
```    53   by (auto simp: adm_wf_def intro!: arg_cong[where f="\<lambda>x. F x y" for y] cuts_eq[THEN iffD2])
```
```    54
```
```    55 lemma wfrec: "wf R \<Longrightarrow> wfrec R F a = F (cut (wfrec R F) R a) a"
```
```    56   apply (simp add: wfrec_def)
```
```    57   apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality])
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```    58    apply assumption
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```    59   apply (rule wfrec_rel.wfrecI)
```
```    60   apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
```
```    61   done
```
```    62
```
```    63
```
```    64 text \<open>This form avoids giant explosions in proofs.  NOTE USE OF \<open>\<equiv>\<close>.\<close>
```
```    65 lemma def_wfrec: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> f a = F (cut f R a) a"
```
```    66   by (auto intro: wfrec)
```
```    67
```
```    68
```
```    69 subsubsection \<open>Well-founded recursion via genuine fixpoints\<close>
```
```    70
```
```    71 lemma wfrec_fixpoint:
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```    72   assumes wf: "wf R"
```
```    73     and adm: "adm_wf R F"
```
```    74   shows "wfrec R F = F (wfrec R F)"
```
```    75 proof (rule ext)
```
```    76   fix x
```
```    77   have "wfrec R F x = F (cut (wfrec R F) R x) x"
```
```    78     using wfrec[of R F] wf by simp
```
```    79   also
```
```    80   have "\<And>y. (y, x) \<in> R \<Longrightarrow> cut (wfrec R F) R x y = wfrec R F y"
```
```    81     by (auto simp add: cut_apply)
```
```    82   then have "F (cut (wfrec R F) R x) x = F (wfrec R F) x"
```
```    83     using adm adm_wf_def[of R F] by auto
```
```    84   finally show "wfrec R F x = F (wfrec R F) x" .
```
```    85 qed
```
```    86
```
```    87
```
```    88 subsection \<open>Wellfoundedness of \<open>same_fst\<close>\<close>
```
```    89
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```    90 definition same_fst :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set"
```
```    91   where "same_fst P R = {((x', y'), (x, y)) . x' = x \<and> P x \<and> (y',y) \<in> R x}"
```
```    92    \<comment> \<open>For @{const wfrec} declarations where the first n parameters
```
```    93        stay unchanged in the recursive call.\<close>
```
```    94
```
```    95 lemma same_fstI [intro!]: "P x \<Longrightarrow> (y', y) \<in> R x \<Longrightarrow> ((x, y'), (x, y)) \<in> same_fst P R"
```
```    96   by (simp add: same_fst_def)
```
```    97
```
```    98 lemma wf_same_fst:
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```    99   assumes prem: "\<And>x. P x \<Longrightarrow> wf (R x)"
```
```   100   shows "wf (same_fst P R)"
```
```   101   apply (simp cong del: imp_cong add: wf_def same_fst_def)
```
```   102   apply (intro strip)
```
```   103   apply (rename_tac a b)
```
```   104   apply (case_tac "wf (R a)")
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```   105    apply (erule_tac a = b in wf_induct)
```
```   106    apply blast
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```   107   apply (blast intro: prem)
```
```   108   done
```
```   109
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```   110 end
```