(*$Id: Main.thy,v 1.21 2007/10/07 19:19:32 wenzelm Exp $*) header{*Theory Main: Everything Except AC*} theory Main imports List IntDiv 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∈nat. F^n (x)" notation (xsymbols) iterates_omega ("(_^ω '(_'))" [60,1000] 60) notation (HTML output) iterates_omega ("(_^ω '(_'))" [60,1000] 60) lemma iterates_triv: "[| n∈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^ω (x) = x" by (simp add: iterates_omega_def iterates_triv) lemma Ord_iterates [simp]: "[| n∈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 ∈ 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, λx r. if x=0 then a else if Limit(x) then c(x, λy∈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, λj∈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
lemma iterates_triv:
[| n ∈ nat; F(x) = x |] ==> F^n (x) = x
lemma iterates_type:
[| n ∈ nat; a ∈ A; !!x. x ∈ A ==> F(x) ∈ A |] ==> F^n (a) ∈ A
lemma iterates_omega_triv:
F(x) = x ==> F^ω (x) = x
lemma Ord_iterates:
[| n ∈ nat; !!i. Ord(i) ==> Ord(F(i)); Ord(x) |] ==> Ord(F^n (x))
lemma iterates_commute:
n ∈ nat ==> F(F^n (x)) = F^n (F(x))
lemma transrec3_0:
transrec3(0, a, b, c) = a
lemma transrec3_succ:
transrec3(succ(i), a, b, c) = b(i, transrec3(i, a, b, c))
lemma transrec3_Limit:
Limit(i) ==> transrec3(i, a, b, c) = c(i, λj∈i. transrec3(j, a, b, c))