author | krauss |
Mon, 11 Feb 2008 15:40:21 +0100 | |
changeset 26056 | 6a0801279f4c |
child 26339 | 7825c83c9eff |
permissions | -rw-r--r-- |
(*$Id$*) header{*Theory Main: Everything Except AC*} theory Main_ZF imports List_ZF IntDiv_ZF CardinalArith begin (*The theory of "iterates" logically belongs to Nat, but can't go there because primrec isn't available into after Datatype.*) subsection{* Iteration of the function @{term F} *} consts iterates :: "[i=>i,i,i] => i" ("(_^_ '(_'))" [60,1000,1000] 60) primrec "F^0 (x) = x" "F^(succ(n)) (x) = F(F^n (x))" definition iterates_omega :: "[i=>i,i] => i" where "iterates_omega(F,x) == \<Union>n\<in>nat. F^n (x)" notation (xsymbols) iterates_omega ("(_^\<omega> '(_'))" [60,1000] 60) notation (HTML output) iterates_omega ("(_^\<omega> '(_'))" [60,1000] 60) lemma iterates_triv: "[| n\<in>nat; F(x) = x |] ==> F^n (x) = x" by (induct n rule: nat_induct, simp_all) lemma iterates_type [TC]: "[| n:nat; a: A; !!x. x:A ==> F(x) : A |] ==> F^n (a) : A" by (induct n rule: nat_induct, simp_all) lemma iterates_omega_triv: "F(x) = x ==> F^\<omega> (x) = x" by (simp add: iterates_omega_def iterates_triv) lemma Ord_iterates [simp]: "[| n\<in>nat; !!i. Ord(i) ==> Ord(F(i)); Ord(x) |] ==> Ord(F^n (x))" by (induct n rule: nat_induct, simp_all) lemma iterates_commute: "n \<in> nat ==> F(F^n (x)) = F^n (F(x))" by (induct_tac n, simp_all) subsection{* Transfinite Recursion *} text{*Transfinite recursion for definitions based on the three cases of ordinals*} definition transrec3 :: "[i, i, [i,i]=>i, [i,i]=>i] =>i" where "transrec3(k, a, b, c) == transrec(k, \<lambda>x r. if x=0 then a else if Limit(x) then c(x, \<lambda>y\<in>x. r`y) else b(Arith.pred(x), r ` Arith.pred(x)))" lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a" by (rule transrec3_def [THEN def_transrec, THEN trans], simp) lemma transrec3_succ [simp]: "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))" by (rule transrec3_def [THEN def_transrec, THEN trans], simp) lemma transrec3_Limit: "Limit(i) ==> transrec3(i,a,b,c) = c(i, \<lambda>j\<in>i. transrec3(j,a,b,c))" by (rule transrec3_def [THEN def_transrec, THEN trans], force) ML_setup {* change_simpset (fn ss => ss setmksimps (map mk_eq o Ord_atomize o gen_all)); *} end