(*  ID:         $Id$

     Author:     Tobias Nipkow

     Copyright   1992  University of Cambridge

 *)

 header {*Well-founded Recursion*}

 theory Wellfounded_Recursion

 imports Transitive_Closure

 begin

 consts

   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => ('a * 'b) set"

 inductive "wfrec_rel R F"

 intros

   wfrecI: "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>

             (x, F g x) : wfrec_rel R F"

 constdefs

   wf         :: "('a * 'a)set => bool"

   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"

   acyclic :: "('a*'a)set => bool"

   "acyclic r == !x. (x,x) ~: r^+"

   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"

   "cut f r x == (%y. if (y,x):r then f y else arbitrary)"

   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"

   "adm_wf R F == ALL f g x.

      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"

   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"

   "wfrec R F == %x. THE y. (x, y) : wfrec_rel R (%f x. F (cut f R x) x)"

 axclass wellorder \<subseteq> linorder

   wf: "wf {(x,y::'a::ord). x<y}"

 lemma wfUNIVI: 

    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"

 by (unfold wf_def, blast)

 text{*Restriction to domain @{term A}.  

   If @{term r} is well-founded over @{term A} then @{term "wf r"}*}

 lemma wfI: 

  "[| r <= A <*> A;   

      !!x P. [| ALL x. (ALL y. (y,x) : r --> P y) --> P x;  x:A |] ==> P x |]   

   ==>  wf r"

 by (unfold wf_def, blast)

 lemma wf_induct: 

     "[| wf(r);           

         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  

      |]  ==>  P(a)"

 by (unfold wf_def, blast)

 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]

 lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r"

 by (erule_tac a=a in wf_induct, blast)

 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)

 lemmas wf_asym = wf_not_sym [elim_format]

 lemma wf_not_refl [simp]: "wf(r) ==> (a,a) ~: r"

 by (blast elim: wf_asym)

 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)

 lemmas wf_irrefl = wf_not_refl [elim_format]

 text{*transitive closure of a well-founded relation is well-founded! *}

 lemma wf_trancl: "wf(r) ==> wf(r^+)"

 apply (subst wf_def, clarify)

 apply (rule allE, assumption)

   --{*Retains the universal formula for later use!*}

 apply (erule mp)

 apply (erule_tac a = x in wf_induct)

 apply (blast elim: tranclE)

 done

 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"

 apply (subst trancl_converse [symmetric])

 apply (erule wf_trancl)

 done

 subsubsection{*Minimal-element characterization of well-foundedness*}

 lemma lemma1: "wf r ==> x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)"

 apply (unfold wf_def)

 apply (drule spec)

 apply (erule mp [THEN spec], blast)

 done

 lemma lemma2: "(ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)) ==> wf r"

 apply (unfold wf_def, clarify)

 apply (drule_tac x = "{x. ~ P x}" in spec, blast)

 done

 lemma wf_eq_minimal: "wf r = (ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q))"

 by (blast intro!: lemma1 lemma2)

 subsubsection{*Other simple well-foundedness results*}

 text{*Well-foundedness of subsets*}

 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"

 apply (simp (no_asm_use) add: wf_eq_minimal)

 apply fast

 done

 text{*Well-foundedness of the empty relation*}

 lemma wf_empty [iff]: "wf({})"

 by (simp add: wf_def)

 text{*Well-foundedness of insert*}

 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"

 apply (rule iffI)

  apply (blast elim: wf_trancl [THEN wf_irrefl]

               intro: rtrancl_into_trancl1 wf_subset 

                      rtrancl_mono [THEN [2] rev_subsetD])

 apply (simp add: wf_eq_minimal, safe)

 apply (rule allE, assumption, erule impE, blast) 

 apply (erule bexE)

 apply (rename_tac "a", case_tac "a = x")

  prefer 2

 apply blast 

 apply (case_tac "y:Q")

  prefer 2 apply blast

 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)

  apply assumption

 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 

   --{*essential for speed*}

 txt{*Blast with new substOccur fails*}

 apply (fast intro: converse_rtrancl_into_rtrancl)

 done

 text{*Well-foundedness of image*}

 lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"

 apply (simp only: wf_eq_minimal, clarify)

 apply (case_tac "EX p. f p : Q")

 apply (erule_tac x = "{p. f p : Q}" in allE)

 apply (fast dest: inj_onD, blast)

 done

 subsubsection{*Well-Foundedness Results for Unions*}

 text{*Well-foundedness of indexed union with disjoint domains and ranges*}

 lemma wf_UN: "[| ALL i:I. wf(r i);  

          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  

       |] ==> wf(UN i:I. r i)"

 apply (simp only: wf_eq_minimal, clarify)

 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")

  prefer 2

  apply force 

 apply clarify

 apply (drule bspec, assumption)  

 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)

 apply (blast elim!: allE)  

 done

 lemma wf_Union: 

  "[| ALL r:R. wf r;  

      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  

   |] ==> wf(Union R)"

 apply (simp add: Union_def)

 apply (blast intro: wf_UN)

 done

 (*Intuition: we find an (R u S)-min element of a nonempty subset A

              by case distinction.

   1. There is a step a -R-> b with a,b : A.

      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.

      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the

      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot

      have an S-successor and is thus S-min in A as well.

   2. There is no such step.

      Pick an S-min element of A. In this case it must be an R-min

      element of A as well.

 *)

 lemma wf_Un:

      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"

 apply (simp only: wf_eq_minimal, clarify) 

 apply (rename_tac A a)

 apply (case_tac "EX a:A. EX b:A. (b,a) : r") 

  prefer 2

  apply simp

  apply (drule_tac x=A in spec)+

  apply blast 

 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r) }" in allE)+

 apply (blast elim!: allE)  

 done

 subsubsection {*acyclic*}

 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"

 by (simp add: acyclic_def)

 lemma wf_acyclic: "wf r ==> acyclic r"

 apply (simp add: acyclic_def)

 apply (blast elim: wf_trancl [THEN wf_irrefl])

 done

 lemma acyclic_insert [iff]:

      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"

 apply (simp add: acyclic_def trancl_insert)

 apply (blast intro: rtrancl_trans)

 done

 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"

 by (simp add: acyclic_def trancl_converse)

 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"

 apply (simp add: acyclic_def antisym_def)

 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)

 done

 (* Other direction:

 acyclic = no loops

 antisym = only self loops

 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)

 ==> antisym( r^* ) = acyclic(r - Id)";

 *)

 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"

 apply (simp add: acyclic_def)

 apply (blast intro: trancl_mono)

 done

 subsection{*Well-Founded Recursion*}

 text{*cut*}

 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"

 by (simp add: expand_fun_eq cut_def)

 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"

 by (simp add: cut_def)

 text{*Inductive characterization of wfrec combinator; for details see:  

 John Harrison, "Inductive definitions: automation and application"*}

 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F"

 apply (simp add: adm_wf_def)

 apply (erule_tac a=x in wf_induct) 

 apply (rule ex1I)

 apply (rule_tac g = "%x. THE y. (x, y) : wfrec_rel R F" in wfrec_rel.wfrecI)

 apply (fast dest!: theI')

 apply (erule wfrec_rel.cases, simp)

 apply (erule allE, erule allE, erule allE, erule mp)

 apply (fast intro: the_equality [symmetric])

 done

 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"

 apply (simp add: adm_wf_def)

 apply (intro strip)

 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)

 apply (rule refl)

 done

 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"

 apply (simp add: wfrec_def)

 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)

 apply (rule wfrec_rel.wfrecI)

 apply (intro strip)

 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])

 done

 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}

 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"

 apply auto

 apply (blast intro: wfrec)

 done

 subsection {* Code generator setup *}

 consts_code

   "wfrec"   ("\<module>wfrec?")

 attach {*

 fun wfrec f x = f (wfrec f) x;

 *}

 code_primconst wfrec

 ml {*

 fun wfrec f x = f (wfrec f) x;

 *}

 haskell {*

 wfrec f x = f (wfrec f) x

 *}

 (* code_syntax_const

   wfrec

     ml ("{*wfrec*}?")

     haskell ("{*wfrec*}?") *)

 subsection{*Variants for TFL: the Recdef Package*}

 lemma tfl_wf_induct: "ALL R. wf R -->  

        (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"

 apply clarify

 apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)

 done

 lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"

 apply clarify

 apply (rule cut_apply, assumption)

 done

 lemma tfl_wfrec:

      "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"

 apply clarify

 apply (erule wfrec)

 done

 subsection {*LEAST and wellorderings*}

 text{* See also @{text wf_linord_ex_has_least} and its consequences in

  @{text Wellfounded_Relations.ML}*}

 lemma wellorder_Least_lemma [rule_format]:

      "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"

 apply (rule_tac a = k in wf [THEN wf_induct])

 apply (rule impI)

 apply (rule classical)

 apply (rule_tac s = x in Least_equality [THEN ssubst], auto)

 apply (auto simp add: linorder_not_less [symmetric])

 done

 lemmas LeastI   = wellorder_Least_lemma [THEN conjunct1, standard]

 lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]

 -- "The following 3 lemmas are due to Brian Huffman"

 lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"

 apply (erule exE)

 apply (erule LeastI)

 done

 lemma LeastI2:

   "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"

 by (blast intro: LeastI)

 lemma LeastI2_ex:

   "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"

 by (blast intro: LeastI_ex)

 lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"

 apply (simp (no_asm_use) add: linorder_not_le [symmetric])

 apply (erule contrapos_nn)

 apply (erule Least_le)

 done

 ML

 {*

 val wf_def = thm "wf_def";

 val wfUNIVI = thm "wfUNIVI";

 val wfI = thm "wfI";

 val wf_induct = thm "wf_induct";

 val wf_not_sym = thm "wf_not_sym";

 val wf_asym = thm "wf_asym";

 val wf_not_refl = thm "wf_not_refl";

 val wf_irrefl = thm "wf_irrefl";

 val wf_trancl = thm "wf_trancl";

 val wf_converse_trancl = thm "wf_converse_trancl";

 val wf_eq_minimal = thm "wf_eq_minimal";

 val wf_subset = thm "wf_subset";

 val wf_empty = thm "wf_empty";

 val wf_insert = thm "wf_insert";

 val wf_UN = thm "wf_UN";

 val wf_Union = thm "wf_Union";

 val wf_Un = thm "wf_Un";

 val wf_prod_fun_image = thm "wf_prod_fun_image";

 val acyclicI = thm "acyclicI";

 val wf_acyclic = thm "wf_acyclic";

 val acyclic_insert = thm "acyclic_insert";

 val acyclic_converse = thm "acyclic_converse";

 val acyclic_impl_antisym_rtrancl = thm "acyclic_impl_antisym_rtrancl";

 val acyclic_subset = thm "acyclic_subset";

 val cuts_eq = thm "cuts_eq";

 val cut_apply = thm "cut_apply";

 val wfrec_unique = thm "wfrec_unique";

 val wfrec = thm "wfrec";

 val def_wfrec = thm "def_wfrec";

 val tfl_wf_induct = thm "tfl_wf_induct";

 val tfl_cut_apply = thm "tfl_cut_apply";

 val tfl_wfrec = thm "tfl_wfrec";

 val LeastI = thm "LeastI";

 val Least_le = thm "Least_le";

 val not_less_Least = thm "not_less_Least";

 *}

 end