src/CCL/Fix.thy
author wenzelm
Sat May 15 22:15:57 2010 +0200 (2010-05-15)
changeset 36948 d2cdad45fd14
parent 36319 8feb2c4bef1a
child 42156 df219e736a5d
permissions -rw-r--r--
renamed Outer_Parse to Parse (in Scala);
     1 (*  Title:      CCL/Fix.thy
     2     Author:     Martin Coen
     3     Copyright   1993  University of Cambridge
     4 *)
     5 
     6 header {* Tentative attempt at including fixed point induction; justified by Smith *}
     7 
     8 theory Fix
     9 imports Type
    10 begin
    11 
    12 consts
    13   idgen      ::       "[i]=>i"
    14   INCL      :: "[i=>o]=>o"
    15 
    16 defs
    17   idgen_def:
    18   "idgen(f) == lam t. case(t,true,false,%x y.<f`x, f`y>,%u. lam x. f ` u(x))"
    19 
    20 axioms
    21   INCL_def:   "INCL(%x. P(x)) == (ALL f.(ALL n:Nat. P(f^n`bot)) --> P(fix(f)))"
    22   po_INCL:    "INCL(%x. a(x) [= b(x))"
    23   INCL_subst: "INCL(P) ==> INCL(%x. P((g::i=>i)(x)))"
    24 
    25 
    26 subsection {* Fixed Point Induction *}
    27 
    28 lemma fix_ind:
    29   assumes base: "P(bot)"
    30     and step: "!!x. P(x) ==> P(f(x))"
    31     and incl: "INCL(P)"
    32   shows "P(fix(f))"
    33   apply (rule incl [unfolded INCL_def, rule_format])
    34   apply (rule Nat_ind [THEN ballI], assumption)
    35    apply simp_all
    36    apply (rule base)
    37   apply (erule step)
    38   done
    39 
    40 
    41 subsection {* Inclusive Predicates *}
    42 
    43 lemma inclXH: "INCL(P) <-> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) --> P(fix(f)))"
    44   by (simp add: INCL_def)
    45 
    46 lemma inclI: "[| !!f. ALL n:Nat. P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x. P(x))"
    47   unfolding inclXH by blast
    48 
    49 lemma inclD: "[| INCL(P);  !!n. n:Nat ==> P(f^n`bot) |] ==> P(fix(f))"
    50   unfolding inclXH by blast
    51 
    52 lemma inclE: "[| INCL(P);  (ALL n:Nat. P(f^n`bot))-->P(fix(f)) ==> R |] ==> R"
    53   by (blast dest: inclD)
    54 
    55 
    56 subsection {* Lemmas for Inclusive Predicates *}
    57 
    58 lemma npo_INCL: "INCL(%x.~ a(x) [= t)"
    59   apply (rule inclI)
    60   apply (drule bspec)
    61    apply (rule zeroT)
    62   apply (erule contrapos)
    63   apply (rule po_trans)
    64    prefer 2
    65    apply assumption
    66   apply (subst napplyBzero)
    67   apply (rule po_cong, rule po_bot)
    68   done
    69 
    70 lemma conj_INCL: "[| INCL(P);  INCL(Q) |] ==> INCL(%x. P(x) & Q(x))"
    71   by (blast intro!: inclI dest!: inclD)
    72 
    73 lemma all_INCL: "[| !!a. INCL(P(a)) |] ==> INCL(%x. ALL a. P(a,x))"
    74   by (blast intro!: inclI dest!: inclD)
    75 
    76 lemma ball_INCL: "[| !!a. a:A ==> INCL(P(a)) |] ==> INCL(%x. ALL a:A. P(a,x))"
    77   by (blast intro!: inclI dest!: inclD)
    78 
    79 lemma eq_INCL: "INCL(%x. a(x) = (b(x)::'a::prog))"
    80   apply (simp add: eq_iff)
    81   apply (rule conj_INCL po_INCL)+
    82   done
    83 
    84 
    85 subsection {* Derivation of Reachability Condition *}
    86 
    87 (* Fixed points of idgen *)
    88 
    89 lemma fix_idgenfp: "idgen(fix(idgen)) = fix(idgen)"
    90   apply (rule fixB [symmetric])
    91   done
    92 
    93 lemma id_idgenfp: "idgen(lam x. x) = lam x. x"
    94   apply (simp add: idgen_def)
    95   apply (rule term_case [THEN allI])
    96       apply simp_all
    97   done
    98 
    99 (* All fixed points are lam-expressions *)
   100 
   101 schematic_lemma idgenfp_lam: "idgen(d) = d ==> d = lam x. ?f(x)"
   102   apply (unfold idgen_def)
   103   apply (erule ssubst)
   104   apply (rule refl)
   105   done
   106 
   107 (* Lemmas for rewriting fixed points of idgen *)
   108 
   109 lemma l_lemma: "[| a = b;  a ` t = u |] ==> b ` t = u"
   110   by (simp add: idgen_def)
   111 
   112 lemma idgen_lemmas:
   113   "idgen(d) = d ==> d ` bot = bot"
   114   "idgen(d) = d ==> d ` true = true"
   115   "idgen(d) = d ==> d ` false = false"
   116   "idgen(d) = d ==> d ` <a,b> = <d ` a,d ` b>"
   117   "idgen(d) = d ==> d ` (lam x. f(x)) = lam x. d ` f(x)"
   118   by (erule l_lemma, simp add: idgen_def)+
   119 
   120 
   121 (* Proof of Reachability law - show that fix and lam x.x both give LEAST fixed points
   122   of idgen and hence are they same *)
   123 
   124 lemma po_eta:
   125   "[| ALL x. t ` x [= u ` x;  EX f. t=lam x. f(x);  EX f. u=lam x. f(x) |] ==> t [= u"
   126   apply (drule cond_eta)+
   127   apply (erule ssubst)
   128   apply (erule ssubst)
   129   apply (rule po_lam [THEN iffD2])
   130   apply simp
   131   done
   132 
   133 schematic_lemma po_eta_lemma: "idgen(d) = d ==> d = lam x. ?f(x)"
   134   apply (unfold idgen_def)
   135   apply (erule sym)
   136   done
   137 
   138 lemma lemma1:
   139   "idgen(d) = d ==>
   140     {p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t & b = d ` t)} <=
   141       POgen({p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t  & b = d ` t)})"
   142   apply clarify
   143   apply (rule_tac t = t in term_case)
   144       apply (simp_all add: POgenXH idgen_lemmas idgen_lemmas [OF fix_idgenfp])
   145    apply blast
   146   apply fast
   147   done
   148 
   149 lemma fix_least_idgen: "idgen(d) = d ==> fix(idgen) [= d"
   150   apply (rule allI [THEN po_eta])
   151     apply (rule lemma1 [THEN [2] po_coinduct])
   152      apply (blast intro: po_eta_lemma fix_idgenfp)+
   153   done
   154 
   155 lemma lemma2:
   156   "idgen(d) = d ==>
   157     {p. EX a b. p=<a,b> & b = d ` a} <= POgen({p. EX a b. p=<a,b> & b = d ` a})"
   158   apply clarify
   159   apply (rule_tac t = a in term_case)
   160       apply (simp_all add: POgenXH idgen_lemmas)
   161   apply fast
   162   done
   163 
   164 lemma id_least_idgen: "idgen(d) = d ==> lam x. x [= d"
   165   apply (rule allI [THEN po_eta])
   166     apply (rule lemma2 [THEN [2] po_coinduct])
   167      apply simp
   168     apply (fast intro: po_eta_lemma fix_idgenfp)+
   169   done
   170 
   171 lemma reachability: "fix(idgen) = lam x. x"
   172   apply (fast intro: eq_iff [THEN iffD2]
   173     id_idgenfp [THEN fix_least_idgen] fix_idgenfp [THEN id_least_idgen])
   174   done
   175 
   176 (********)
   177 
   178 lemma id_apply: "f = lam x. x ==> f`t = t"
   179   apply (erule ssubst)
   180   apply (rule applyB)
   181   done
   182 
   183 lemma term_ind:
   184   assumes 1: "P(bot)" and 2: "P(true)" and 3: "P(false)"
   185     and 4: "!!x y.[| P(x);  P(y) |] ==> P(<x,y>)"
   186     and 5: "!!u.(!!x. P(u(x))) ==> P(lam x. u(x))"
   187     and 6: "INCL(P)"
   188   shows "P(t)"
   189   apply (rule reachability [THEN id_apply, THEN subst])
   190   apply (rule_tac x = t in spec)
   191   apply (rule fix_ind)
   192     apply (unfold idgen_def)
   193     apply (rule allI)
   194     apply (subst applyBbot)
   195     apply (rule 1)
   196    apply (rule allI)
   197    apply (rule applyB [THEN ssubst])
   198     apply (rule_tac t = "xa" in term_case)
   199        apply simp_all
   200        apply (fast intro: assms INCL_subst all_INCL)+
   201   done
   202 
   203 end