author  wenzelm 
Sat, 25 May 2013 17:40:44 +0200  
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permissions  rwrr 
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(* Title: CCL/ex/List.thy 
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Author: Martin Coen, Cambridge University Computer Laboratory 
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Copyright 1993 University of Cambridge 
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*) 

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header {* Programs defined over lists *} 
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theory List 

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imports Nat 

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begin 

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definition map :: "[i=>i,i]=>i" 
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where "map(f,l) == lrec(l,[],%x xs g. f(x)$g)" 
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definition comp :: "[i=>i,i=>i]=>i=>i" (infixr "\<circ>" 55) 
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where "f \<circ> g == (%x. f(g(x)))" 
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definition append :: "[i,i]=>i" (infixr "@" 55) 
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where "l @ m == lrec(l,m,%x xs g. x$g)" 
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axiomatization member :: "[i,i]=>i" (infixr "mem" 55) (* FIXME dangling eq *) 
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where member_ax: "a mem l == lrec(l,false,%h t g. if eq(a,h) then true else g)" 
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definition filter :: "[i,i]=>i" 
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where "filter(f,l) == lrec(l,[],%x xs g. if f`x then x$g else g)" 
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definition flat :: "i=>i" 
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where "flat(l) == lrec(l,[],%h t g. h @ g)" 
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definition partition :: "[i,i]=>i" where 
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"partition(f,l) == letrec part l a b be lcase(l,<a,b>,%x xs. 
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if f`x then part(xs,x$a,b) else part(xs,a,x$b)) 
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in part(l,[],[])" 
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definition insert :: "[i,i,i]=>i" 
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where "insert(f,a,l) == lrec(l,a$[],%h t g. if f`a`h then a$h$t else h$g)" 
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definition isort :: "i=>i" 
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where "isort(f) == lam l. lrec(l,[],%h t g. insert(f,h,g))" 
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definition qsort :: "i=>i" where 
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"qsort(f) == lam l. letrec qsortx l be lcase(l,[],%h t. 
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let p be partition(f`h,t) 
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in split(p,%x y. qsortx(x) @ h$qsortx(y))) 

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in qsortx(l)" 
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lemmas list_defs = map_def comp_def append_def filter_def flat_def 
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insert_def isort_def partition_def qsort_def 

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lemma listBs [simp]: 

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"!!f g. (f \<circ> g) = (%a. f(g(a)))" 
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"!!a f g. (f \<circ> g)(a) = f(g(a))" 
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"!!f. map(f,[]) = []" 
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"!!f x xs. map(f,x$xs) = f(x)$map(f,xs)" 

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"!!m. [] @ m = m" 

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"!!x xs m. x$xs @ m = x$(xs @ m)" 

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"!!f. filter(f,[]) = []" 

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"!!f x xs. filter(f,x$xs) = if f`x then x$filter(f,xs) else filter(f,xs)" 

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"flat([]) = []" 

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"!!x xs. flat(x$xs) = x @ flat(xs)" 

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"!!a f. insert(f,a,[]) = a$[]" 

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"!!a f xs. insert(f,a,x$xs) = if f`a`x then a$x$xs else x$insert(f,a,xs)" 

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by (simp_all add: list_defs) 

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lemma nmapBnil: "n:Nat ==> map(f) ^ n ` [] = []" 

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apply (erule Nat_ind) 

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apply simp_all 

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done 

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lemma nmapBcons: "n:Nat ==> map(f)^n`(x$xs) = (f^n`x)$(map(f)^n`xs)" 

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apply (erule Nat_ind) 

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apply simp_all 

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done 

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lemma mapT: "[ !!x. x:A==>f(x):B; l : List(A) ] ==> map(f,l) : List(B)" 

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apply (unfold map_def) 

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apply (tactic "typechk_tac @{context} [] 1") 
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apply blast 
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done 

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lemma appendT: "[ l : List(A); m : List(A) ] ==> l @ m : List(A)" 

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apply (unfold append_def) 

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apply (tactic "typechk_tac @{context} [] 1") 
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done 
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lemma appendTS: 

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"[ l : {l:List(A). m : {m:List(A).P(l @ m)}} ] ==> l @ m : {x:List(A). P(x)}" 

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by (blast intro!: appendT) 
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lemma filterT: "[ f:A>Bool; l : List(A) ] ==> filter(f,l) : List(A)" 

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apply (unfold filter_def) 

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apply (tactic "typechk_tac @{context} [] 1") 
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done 
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lemma flatT: "l : List(List(A)) ==> flat(l) : List(A)" 

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apply (unfold flat_def) 

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apply (tactic {* typechk_tac @{context} @{thms appendT} 1 *}) 
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done 
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lemma insertT: "[ f : A>A>Bool; a:A; l : List(A) ] ==> insert(f,a,l) : List(A)" 

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apply (unfold insert_def) 

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apply (tactic "typechk_tac @{context} [] 1") 
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done 
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lemma insertTS: 

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"[ f : {f:A>A>Bool. a : {a:A. l : {l:List(A).P(insert(f,a,l))}}} ] ==> 

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insert(f,a,l) : {x:List(A). P(x)}" 

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by (blast intro!: insertT) 
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lemma partitionT: 

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"[ f:A>Bool; l : List(A) ] ==> partition(f,l) : List(A)*List(A)" 

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apply (unfold partition_def) 

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apply (tactic "typechk_tac @{context} [] 1") 
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apply (tactic "clean_ccs_tac @{context}") 
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apply (rule ListPRI [THEN wfstI, THEN ListPR_wf [THEN wmap_wf, THEN wfI]]) 
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apply assumption+ 

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apply (rule ListPRI [THEN wfstI, THEN ListPR_wf [THEN wmap_wf, THEN wfI]]) 

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apply assumption+ 

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done 

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end 