(* Title: CCL/ex/List.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section \<open>Programs defined over lists\<close>
theory List
imports Nat
begin
definition map :: "[i\<Rightarrow>i,i]\<Rightarrow>i"
where "map(f,l) == lrec(l, [], \<lambda>x xs g. f(x)$g)"
definition comp :: "[i\<Rightarrow>i,i\<Rightarrow>i]\<Rightarrow>i\<Rightarrow>i" (infixr "\<circ>" 55)
where "f \<circ> g == (\<lambda>x. f(g(x)))"
definition append :: "[i,i]\<Rightarrow>i" (infixr "@" 55)
where "l @ m == lrec(l, m, \<lambda>x xs g. x$g)"
axiomatization member :: "[i,i]\<Rightarrow>i" (infixr "mem" 55) (* FIXME dangling eq *)
where member_ax: "a mem l == lrec(l, false, \<lambda>h t g. if eq(a,h) then true else g)"
definition filter :: "[i,i]\<Rightarrow>i"
where "filter(f,l) == lrec(l, [], \<lambda>x xs g. if f`x then x$g else g)"
definition flat :: "i\<Rightarrow>i"
where "flat(l) == lrec(l, [], \<lambda>h t g. h @ g)"
definition partition :: "[i,i]\<Rightarrow>i" where
"partition(f,l) == letrec part l a b be lcase(l, <a,b>, \<lambda>x xs.
if f`x then part(xs,x$a,b) else part(xs,a,x$b))
in part(l,[],[])"
definition insert :: "[i,i,i]\<Rightarrow>i"
where "insert(f,a,l) == lrec(l, a$[], \<lambda>h t g. if f`a`h then a$h$t else h$g)"
definition isort :: "i\<Rightarrow>i"
where "isort(f) == lam l. lrec(l, [], \<lambda>h t g. insert(f,h,g))"
definition qsort :: "i\<Rightarrow>i" where
"qsort(f) == lam l. letrec qsortx l be lcase(l, [], \<lambda>h t.
let p be partition(f`h,t)
in split(p, \<lambda>x y. qsortx(x) @ h$qsortx(y)))
in qsortx(l)"
lemmas list_defs = map_def comp_def append_def filter_def flat_def
insert_def isort_def partition_def qsort_def
lemma listBs [simp]:
"\<And>f g. (f \<circ> g) = (\<lambda>a. f(g(a)))"
"\<And>a f g. (f \<circ> g)(a) = f(g(a))"
"\<And>f. map(f,[]) = []"
"\<And>f x xs. map(f,x$xs) = f(x)$map(f,xs)"
"\<And>m. [] @ m = m"
"\<And>x xs m. x$xs @ m = x$(xs @ m)"
"\<And>f. filter(f,[]) = []"
"\<And>f x xs. filter(f,x$xs) = if f`x then x$filter(f,xs) else filter(f,xs)"
"flat([]) = []"
"\<And>x xs. flat(x$xs) = x @ flat(xs)"
"\<And>a f. insert(f,a,[]) = a$[]"
"\<And>a f xs. insert(f,a,x$xs) = if f`a`x then a$x$xs else x$insert(f,a,xs)"
by (simp_all add: list_defs)
lemma nmapBnil: "n:Nat \<Longrightarrow> map(f) ^ n ` [] = []"
apply (erule Nat_ind)
apply simp_all
done
lemma nmapBcons: "n:Nat \<Longrightarrow> map(f)^n`(x$xs) = (f^n`x)$(map(f)^n`xs)"
apply (erule Nat_ind)
apply simp_all
done
lemma mapT: "\<lbrakk>\<And>x. x:A \<Longrightarrow> f(x):B; l : List(A)\<rbrakk> \<Longrightarrow> map(f,l) : List(B)"
apply (unfold map_def)
apply typechk
apply blast
done
lemma appendT: "\<lbrakk>l : List(A); m : List(A)\<rbrakk> \<Longrightarrow> l @ m : List(A)"
apply (unfold append_def)
apply typechk
done
lemma appendTS:
"\<lbrakk>l : {l:List(A). m : {m:List(A).P(l @ m)}}\<rbrakk> \<Longrightarrow> l @ m : {x:List(A). P(x)}"
by (blast intro!: appendT)
lemma filterT: "\<lbrakk>f:A->Bool; l : List(A)\<rbrakk> \<Longrightarrow> filter(f,l) : List(A)"
apply (unfold filter_def)
apply typechk
done
lemma flatT: "l : List(List(A)) \<Longrightarrow> flat(l) : List(A)"
apply (unfold flat_def)
apply (typechk appendT)
done
lemma insertT: "\<lbrakk>f : A->A->Bool; a:A; l : List(A)\<rbrakk> \<Longrightarrow> insert(f,a,l) : List(A)"
apply (unfold insert_def)
apply typechk
done
lemma insertTS:
"\<lbrakk>f : {f:A->A->Bool. a : {a:A. l : {l:List(A).P(insert(f,a,l))}}} \<rbrakk> \<Longrightarrow>
insert(f,a,l) : {x:List(A). P(x)}"
by (blast intro!: insertT)
lemma partitionT: "\<lbrakk>f:A->Bool; l : List(A)\<rbrakk> \<Longrightarrow> partition(f,l) : List(A)*List(A)"
apply (unfold partition_def)
apply typechk
apply clean_ccs
apply (rule ListPRI [THEN wfstI, THEN ListPR_wf [THEN wmap_wf, THEN wfI]])
apply assumption+
apply (rule ListPRI [THEN wfstI, THEN ListPR_wf [THEN wmap_wf, THEN wfI]])
apply assumption+
done
end