author  wenzelm 
Sun, 30 Nov 2008 14:43:29 +0100  
changeset 28917  20f43e0e0958 
parent 26480  544cef16045b 
child 32010  cb1a1c94b4cd 
permissions  rwrr 
17456  1 
(* Title: CCL/Term.thy 
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ID: $Id$ 
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Author: Martin Coen 
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Copyright 1993 University of Cambridge 
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*) 

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header {* Definitions of usual program constructs in CCL *} 
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theory Term 

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imports CCL 

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begin 

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consts 

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one :: "i" 
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"if" :: "[i,i,i]=>i" ("(3if _/ then _/ else _)" [0,0,60] 60) 
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inl :: "i=>i" 
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inr :: "i=>i" 

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when :: "[i,i=>i,i=>i]=>i" 

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split :: "[i,[i,i]=>i]=>i" 
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fst :: "i=>i" 
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snd :: "i=>i" 

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thd :: "i=>i" 
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zero :: "i" 
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succ :: "i=>i" 
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ncase :: "[i,i,i=>i]=>i" 
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nrec :: "[i,i,[i,i]=>i]=>i" 
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nil :: "i" ("([])") 
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cons :: "[i,i]=>i" (infixr "$" 80) 
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lcase :: "[i,i,[i,i]=>i]=>i" 
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lrec :: "[i,i,[i,i,i]=>i]=>i" 
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"let" :: "[i,i=>i]=>i" 
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letrec :: "[[i,i=>i]=>i,(i=>i)=>i]=>i" 
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letrec2 :: "[[i,i,i=>i=>i]=>i,(i=>i=>i)=>i]=>i" 
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letrec3 :: "[[i,i,i,i=>i=>i=>i]=>i,(i=>i=>i=>i)=>i]=>i" 
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syntax 
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"@let" :: "[idt,i,i]=>i" ("(3let _ be _/ in _)" 
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[0,0,60] 60) 
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"@letrec" :: "[idt,idt,i,i]=>i" ("(3letrec _ _ be _/ in _)" 
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[0,0,0,60] 60) 
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"@letrec2" :: "[idt,idt,idt,i,i]=>i" ("(3letrec _ _ _ be _/ in _)" 
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[0,0,0,0,60] 60) 
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"@letrec3" :: "[idt,idt,idt,idt,i,i]=>i" ("(3letrec _ _ _ _ be _/ in _)" 
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[0,0,0,0,0,60] 60) 
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ML {* 
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(** Quantifier translations: variable binding **) 
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(* FIXME does not handle "_idtdummy" *) 
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(* FIXME should use Syntax.mark_bound(T), Syntax.variant_abs' *) 
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0  62 
fun let_tr [Free(id,T),a,b] = Const("let",dummyT) $ a $ absfree(id,T,b); 
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fun let_tr' [a,Abs(id,T,b)] = 

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let val (id',b') = variant_abs(id,T,b) 

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in Const("@let",dummyT) $ Free(id',T) $ a $ b' end; 

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fun letrec_tr [Free(f,S),Free(x,T),a,b] = 
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Const("letrec",dummyT) $ absfree(x,T,absfree(f,S,a)) $ absfree(f,S,b); 
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fun letrec2_tr [Free(f,S),Free(x,T),Free(y,U),a,b] = 
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Const("letrec2",dummyT) $ absfree(x,T,absfree(y,U,absfree(f,S,a))) $ absfree(f,S,b); 
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fun letrec3_tr [Free(f,S),Free(x,T),Free(y,U),Free(z,V),a,b] = 
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Const("letrec3",dummyT) $ absfree(x,T,absfree(y,U,absfree(z,U,absfree(f,S,a)))) $ absfree(f,S,b); 
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fun letrec_tr' [Abs(x,T,Abs(f,S,a)),Abs(ff,SS,b)] = 

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let val (f',b') = variant_abs(ff,SS,b) 

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val (_,a'') = variant_abs(f,S,a) 

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val (x',a') = variant_abs(x,T,a'') 

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in Const("@letrec",dummyT) $ Free(f',SS) $ Free(x',T) $ a' $ b' end; 

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fun letrec2_tr' [Abs(x,T,Abs(y,U,Abs(f,S,a))),Abs(ff,SS,b)] = 

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let val (f',b') = variant_abs(ff,SS,b) 

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val ( _,a1) = variant_abs(f,S,a) 

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val (y',a2) = variant_abs(y,U,a1) 

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val (x',a') = variant_abs(x,T,a2) 

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in Const("@letrec2",dummyT) $ Free(f',SS) $ Free(x',T) $ Free(y',U) $ a' $ b' 

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end; 

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fun letrec3_tr' [Abs(x,T,Abs(y,U,Abs(z,V,Abs(f,S,a)))),Abs(ff,SS,b)] = 

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let val (f',b') = variant_abs(ff,SS,b) 

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val ( _,a1) = variant_abs(f,S,a) 

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val (z',a2) = variant_abs(z,V,a1) 

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val (y',a3) = variant_abs(y,U,a2) 

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val (x',a') = variant_abs(x,T,a3) 

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in Const("@letrec3",dummyT) $ Free(f',SS) $ Free(x',T) $ Free(y',U) $ Free(z',V) $ a' $ b' 

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end; 

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*} 
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parse_translation {* 

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[("@let", let_tr), 

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("@letrec", letrec_tr), 

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("@letrec2", letrec2_tr), 

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("@letrec3", letrec3_tr)] *} 

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print_translation {* 

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[("let", let_tr'), 

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("letrec", letrec_tr'), 

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("letrec2", letrec2_tr'), 

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("letrec3", letrec3_tr')] *} 

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consts 

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napply :: "[i=>i,i,i]=>i" ("(_ ^ _ ` _)" [56,56,56] 56) 

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axioms 

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one_def: "one == true" 

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if_def: "if b then t else u == case(b,t,u,% x y. bot,%v. bot)" 

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inl_def: "inl(a) == <true,a>" 

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inr_def: "inr(b) == <false,b>" 

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when_def: "when(t,f,g) == split(t,%b x. if b then f(x) else g(x))" 

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split_def: "split(t,f) == case(t,bot,bot,f,%u. bot)" 

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fst_def: "fst(t) == split(t,%x y. x)" 

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snd_def: "snd(t) == split(t,%x y. y)" 

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thd_def: "thd(t) == split(t,%x p. split(p,%y z. z))" 

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zero_def: "zero == inl(one)" 

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succ_def: "succ(n) == inr(n)" 

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ncase_def: "ncase(n,b,c) == when(n,%x. b,%y. c(y))" 

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nrec_def: " nrec(n,b,c) == letrec g x be ncase(x,b,%y. c(y,g(y))) in g(n)" 

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nil_def: "[] == inl(one)" 

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cons_def: "h$t == inr(<h,t>)" 

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lcase_def: "lcase(l,b,c) == when(l,%x. b,%y. split(y,c))" 

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lrec_def: "lrec(l,b,c) == letrec g x be lcase(x,b,%h t. c(h,t,g(t))) in g(l)" 

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let_def: "let x be t in f(x) == case(t,f(true),f(false),%x y. f(<x,y>),%u. f(lam x. u(x)))" 

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letrec_def: 

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"letrec g x be h(x,g) in b(g) == b(%x. fix(%f. lam x. h(x,%y. f`y))`x)" 

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letrec2_def: "letrec g x y be h(x,y,g) in f(g)== 

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letrec g' p be split(p,%x y. h(x,y,%u v. g'(<u,v>))) 

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in f(%x y. g'(<x,y>))" 

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letrec3_def: "letrec g x y z be h(x,y,z,g) in f(g) == 

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letrec g' p be split(p,%x xs. split(xs,%y z. h(x,y,z,%u v w. g'(<u,<v,w>>)))) 

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in f(%x y z. g'(<x,<y,z>>))" 

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napply_def: "f ^n` a == nrec(n,a,%x g. f(g))" 

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lemmas simp_can_defs = one_def inl_def inr_def 

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and simp_ncan_defs = if_def when_def split_def fst_def snd_def thd_def 

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lemmas simp_defs = simp_can_defs simp_ncan_defs 

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lemmas ind_can_defs = zero_def succ_def nil_def cons_def 

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and ind_ncan_defs = ncase_def nrec_def lcase_def lrec_def 

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lemmas ind_defs = ind_can_defs ind_ncan_defs 

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lemmas data_defs = simp_defs ind_defs napply_def 

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and genrec_defs = letrec_def letrec2_def letrec3_def 

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subsection {* Beta Rules, including strictness *} 

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lemma letB: "~ t=bot ==> let x be t in f(x) = f(t)" 

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apply (unfold let_def) 

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apply (erule rev_mp) 

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apply (rule_tac t = "t" in term_case) 

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apply (simp_all add: caseBtrue caseBfalse caseBpair caseBlam) 

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done 

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lemma letBabot: "let x be bot in f(x) = bot" 

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apply (unfold let_def) 

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apply (rule caseBbot) 

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done 

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lemma letBbbot: "let x be t in bot = bot" 

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apply (unfold let_def) 

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apply (rule_tac t = t in term_case) 

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apply (rule caseBbot) 

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apply (simp_all add: caseBtrue caseBfalse caseBpair caseBlam) 

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done 

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lemma applyB: "(lam x. b(x)) ` a = b(a)" 

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apply (unfold apply_def) 

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apply (simp add: caseBtrue caseBfalse caseBpair caseBlam) 

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done 

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lemma applyBbot: "bot ` a = bot" 

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apply (unfold apply_def) 

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apply (rule caseBbot) 

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done 

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lemma fixB: "fix(f) = f(fix(f))" 

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apply (unfold fix_def) 

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apply (rule applyB [THEN ssubst], rule refl) 

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done 

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lemma letrecB: 

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"letrec g x be h(x,g) in g(a) = h(a,%y. letrec g x be h(x,g) in g(y))" 

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apply (unfold letrec_def) 

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apply (rule fixB [THEN ssubst], rule applyB [THEN ssubst], rule refl) 

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done 

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lemmas rawBs = caseBs applyB applyBbot 

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ML {* 

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local 

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val letrecB = thm "letrecB" 

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val rawBs = thms "rawBs" 

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val data_defs = thms "data_defs" 

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in 

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fun raw_mk_beta_rl defs s = prove_goalw (the_context ()) defs s 

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(fn _ => [stac letrecB 1, 

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simp_tac (@{simpset} addsimps rawBs) 1]); 
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fun mk_beta_rl s = raw_mk_beta_rl data_defs s; 
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fun raw_mk_beta_rl defs s = prove_goalw (the_context ()) defs s 

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(fn _ => [simp_tac (@{simpset} addsimps rawBs 
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setloop (stac letrecB)) 1]); 
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fun mk_beta_rl s = raw_mk_beta_rl data_defs s; 

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end 

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*} 
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ML {* 
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bind_thm ("ifBtrue", mk_beta_rl "if true then t else u = t"); 
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bind_thm ("ifBfalse", mk_beta_rl "if false then t else u = u"); 

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bind_thm ("ifBbot", mk_beta_rl "if bot then t else u = bot"); 

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bind_thm ("whenBinl", mk_beta_rl "when(inl(a),t,u) = t(a)"); 

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bind_thm ("whenBinr", mk_beta_rl "when(inr(a),t,u) = u(a)"); 

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bind_thm ("whenBbot", mk_beta_rl "when(bot,t,u) = bot"); 

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bind_thm ("splitB", mk_beta_rl "split(<a,b>,h) = h(a,b)"); 

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bind_thm ("splitBbot", mk_beta_rl "split(bot,h) = bot"); 

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bind_thm ("fstB", mk_beta_rl "fst(<a,b>) = a"); 

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bind_thm ("fstBbot", mk_beta_rl "fst(bot) = bot"); 

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bind_thm ("sndB", mk_beta_rl "snd(<a,b>) = b"); 

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bind_thm ("sndBbot", mk_beta_rl "snd(bot) = bot"); 

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bind_thm ("thdB", mk_beta_rl "thd(<a,<b,c>>) = c"); 

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bind_thm ("thdBbot", mk_beta_rl "thd(bot) = bot"); 

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bind_thm ("ncaseBzero", mk_beta_rl "ncase(zero,t,u) = t"); 

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bind_thm ("ncaseBsucc", mk_beta_rl "ncase(succ(n),t,u) = u(n)"); 

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bind_thm ("ncaseBbot", mk_beta_rl "ncase(bot,t,u) = bot"); 

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bind_thm ("nrecBzero", mk_beta_rl "nrec(zero,t,u) = t"); 

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bind_thm ("nrecBsucc", mk_beta_rl "nrec(succ(n),t,u) = u(n,nrec(n,t,u))"); 

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bind_thm ("nrecBbot", mk_beta_rl "nrec(bot,t,u) = bot"); 

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bind_thm ("lcaseBnil", mk_beta_rl "lcase([],t,u) = t"); 

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bind_thm ("lcaseBcons", mk_beta_rl "lcase(x$xs,t,u) = u(x,xs)"); 

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bind_thm ("lcaseBbot", mk_beta_rl "lcase(bot,t,u) = bot"); 

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bind_thm ("lrecBnil", mk_beta_rl "lrec([],t,u) = t"); 

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bind_thm ("lrecBcons", mk_beta_rl "lrec(x$xs,t,u) = u(x,xs,lrec(xs,t,u))"); 

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bind_thm ("lrecBbot", mk_beta_rl "lrec(bot,t,u) = bot"); 

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bind_thm ("letrec2B", raw_mk_beta_rl (thms "data_defs" @ [thm "letrec2_def"]) 

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"letrec g x y be h(x,y,g) in g(p,q) = h(p,q,%u v. letrec g x y be h(x,y,g) in g(u,v))"); 

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bind_thm ("letrec3B", raw_mk_beta_rl (thms "data_defs" @ [thm "letrec3_def"]) 

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"letrec g x y z be h(x,y,z,g) in g(p,q,r) = h(p,q,r,%u v w. letrec g x y z be h(x,y,z,g) in g(u,v,w))"); 

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bind_thm ("napplyBzero", mk_beta_rl "f^zero`a = a"); 

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bind_thm ("napplyBsucc", mk_beta_rl "f^succ(n)`a = f(f^n`a)"); 

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bind_thms ("termBs", [thm "letB", thm "applyB", thm "applyBbot", splitB,splitBbot, 

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fstB,fstBbot,sndB,sndBbot,thdB,thdBbot, 

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ifBtrue,ifBfalse,ifBbot,whenBinl,whenBinr,whenBbot, 

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ncaseBzero,ncaseBsucc,ncaseBbot,nrecBzero,nrecBsucc,nrecBbot, 

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lcaseBnil,lcaseBcons,lcaseBbot,lrecBnil,lrecBcons,lrecBbot, 

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napplyBzero,napplyBsucc]); 

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*} 

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subsection {* Constructors are injective *} 

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ML {* 
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bind_thms ("term_injs", map (mk_inj_rl (the_context ()) 

278 
[thm "applyB", thm "splitB", thm "whenBinl", thm "whenBinr", thm "ncaseBsucc", thm "lcaseBcons"]) 

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["(inl(a) = inl(a')) <> (a=a')", 

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"(inr(a) = inr(a')) <> (a=a')", 

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"(succ(a) = succ(a')) <> (a=a')", 

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"(a$b = a'$b') <> (a=a' & b=b')"]) 

283 
*} 

284 

285 

286 
subsection {* Constructors are distinct *} 

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ML {* 
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bind_thms ("term_dstncts", 
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mkall_dstnct_thms (the_context ()) (thms "data_defs") (thms "ccl_injs" @ thms "term_injs") 

24825  291 
[["bot","inl","inr"], ["bot","zero","succ"], ["bot","nil","cons"]]); 
20140  292 
*} 
293 

294 

295 
subsection {* Rules for preorder @{text "[="} *} 

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ML {* 
20140  298 

299 
local 

300 
fun mk_thm s = prove_goalw (the_context ()) (thms "data_defs") s (fn _ => 

26342  301 
[simp_tac (@{simpset} addsimps (thms "ccl_porews")) 1]) 
20140  302 
in 
303 
val term_porews = map mk_thm ["inl(a) [= inl(a') <> a [= a'", 

304 
"inr(b) [= inr(b') <> b [= b'", 

305 
"succ(n) [= succ(n') <> n [= n'", 

306 
"x$xs [= x'$xs' <> x [= x' & xs [= xs'"] 

307 
end; 

308 

309 
bind_thms ("term_porews", term_porews); 

310 
*} 

311 

312 
subsection {* Rewriting and Proving *} 

313 

26480  314 
ML {* 
24790  315 
bind_thms ("term_injDs", XH_to_Ds @{thms term_injs}); 
20140  316 
*} 
317 

20917  318 
lemmas term_rews = termBs term_injs term_dstncts ccl_porews term_porews 
319 

20140  320 
lemmas [simp] = term_rews 
20917  321 
lemmas [elim!] = term_dstncts [THEN notE] 
322 
lemmas [dest!] = term_injDs 

20140  323 

324 
end 