src/ZF/Main.thy
author wenzelm
Tue Jul 31 19:40:22 2007 +0200 (2007-07-31)
changeset 24091 109f19a13872
parent 22814 4cd25f1706bb
child 24893 b8ef7afe3a6b
permissions -rw-r--r--
added Tools/lin_arith.ML;
     1 (*$Id$*)
     2 
     3 header{*Theory Main: Everything Except AC*}
     4 
     5 theory Main imports List IntDiv CardinalArith begin
     6 
     7 (*The theory of "iterates" logically belongs to Nat, but can't go there because
     8   primrec isn't available into after Datatype.*)
     9 
    10 subsection{* Iteration of the function @{term F} *}
    11 
    12 consts  iterates :: "[i=>i,i,i] => i"   ("(_^_ '(_'))" [60,1000,1000] 60)
    13 
    14 primrec
    15     "F^0 (x) = x"
    16     "F^(succ(n)) (x) = F(F^n (x))"
    17 
    18 constdefs
    19   iterates_omega :: "[i=>i,i] => i"
    20     "iterates_omega(F,x) == \<Union>n\<in>nat. F^n (x)"
    21 
    22 syntax (xsymbols)
    23   iterates_omega :: "[i=>i,i] => i"   ("(_^\<omega> '(_'))" [60,1000] 60)
    24 syntax (HTML output)
    25   iterates_omega :: "[i=>i,i] => i"   ("(_^\<omega> '(_'))" [60,1000] 60)
    26 
    27 lemma iterates_triv:
    28      "[| n\<in>nat;  F(x) = x |] ==> F^n (x) = x"  
    29 by (induct n rule: nat_induct, simp_all)
    30 
    31 lemma iterates_type [TC]:
    32      "[| n:nat;  a: A; !!x. x:A ==> F(x) : A |] 
    33       ==> F^n (a) : A"  
    34 by (induct n rule: nat_induct, simp_all)
    35 
    36 lemma iterates_omega_triv:
    37     "F(x) = x ==> F^\<omega> (x) = x" 
    38 by (simp add: iterates_omega_def iterates_triv) 
    39 
    40 lemma Ord_iterates [simp]:
    41      "[| n\<in>nat;  !!i. Ord(i) ==> Ord(F(i));  Ord(x) |] 
    42       ==> Ord(F^n (x))"  
    43 by (induct n rule: nat_induct, simp_all)
    44 
    45 lemma iterates_commute: "n \<in> nat ==> F(F^n (x)) = F^n (F(x))"
    46 by (induct_tac n, simp_all)
    47 
    48 
    49 subsection{* Transfinite Recursion *}
    50 
    51 text{*Transfinite recursion for definitions based on the 
    52     three cases of ordinals*}
    53 
    54 constdefs
    55   transrec3 :: "[i, i, [i,i]=>i, [i,i]=>i] =>i"
    56     "transrec3(k, a, b, c) ==                     
    57        transrec(k, \<lambda>x r.
    58          if x=0 then a
    59          else if Limit(x) then c(x, \<lambda>y\<in>x. r`y)
    60          else b(Arith.pred(x), r ` Arith.pred(x)))"
    61 
    62 lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a"
    63 by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
    64 
    65 lemma transrec3_succ [simp]:
    66      "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))"
    67 by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
    68 
    69 lemma transrec3_Limit:
    70      "Limit(i) ==> 
    71       transrec3(i,a,b,c) = c(i, \<lambda>j\<in>i. transrec3(j,a,b,c))"
    72 by (rule transrec3_def [THEN def_transrec, THEN trans], force)
    73 
    74 
    75 ML_setup {*
    76   change_simpset (fn ss => ss setmksimps (map mk_eq o Ord_atomize o gen_all));
    77 *}
    78 
    79 end