Reimplemented algebra method; now controlled by attribute.
(* Title: CCL/Term.thy
ID: $Id$
Author: Martin Coen
Copyright 1993 University of Cambridge
*)
header {* Definitions of usual program constructs in CCL *}
theory Term
imports CCL
begin
consts
one :: "i"
"if" :: "[i,i,i]=>i" ("(3if _/ then _/ else _)" [0,0,60] 60)
inl :: "i=>i"
inr :: "i=>i"
when :: "[i,i=>i,i=>i]=>i"
split :: "[i,[i,i]=>i]=>i"
fst :: "i=>i"
snd :: "i=>i"
thd :: "i=>i"
zero :: "i"
succ :: "i=>i"
ncase :: "[i,i,i=>i]=>i"
nrec :: "[i,i,[i,i]=>i]=>i"
nil :: "i" ("([])")
"$" :: "[i,i]=>i" (infixr 80)
lcase :: "[i,i,[i,i]=>i]=>i"
lrec :: "[i,i,[i,i,i]=>i]=>i"
"let" :: "[i,i=>i]=>i"
letrec :: "[[i,i=>i]=>i,(i=>i)=>i]=>i"
letrec2 :: "[[i,i,i=>i=>i]=>i,(i=>i=>i)=>i]=>i"
letrec3 :: "[[i,i,i,i=>i=>i=>i]=>i,(i=>i=>i=>i)=>i]=>i"
syntax
"@let" :: "[idt,i,i]=>i" ("(3let _ be _/ in _)"
[0,0,60] 60)
"@letrec" :: "[idt,idt,i,i]=>i" ("(3letrec _ _ be _/ in _)"
[0,0,0,60] 60)
"@letrec2" :: "[idt,idt,idt,i,i]=>i" ("(3letrec _ _ _ be _/ in _)"
[0,0,0,0,60] 60)
"@letrec3" :: "[idt,idt,idt,idt,i,i]=>i" ("(3letrec _ _ _ _ be _/ in _)"
[0,0,0,0,0,60] 60)
ML {*
(** Quantifier translations: variable binding **)
(* FIXME does not handle "_idtdummy" *)
(* FIXME should use Syntax.mark_bound(T), Syntax.variant_abs' *)
fun let_tr [Free(id,T),a,b] = Const("let",dummyT) $ a $ absfree(id,T,b);
fun let_tr' [a,Abs(id,T,b)] =
let val (id',b') = variant_abs(id,T,b)
in Const("@let",dummyT) $ Free(id',T) $ a $ b' end;
fun letrec_tr [Free(f,S),Free(x,T),a,b] =
Const("letrec",dummyT) $ absfree(x,T,absfree(f,S,a)) $ absfree(f,S,b);
fun letrec2_tr [Free(f,S),Free(x,T),Free(y,U),a,b] =
Const("letrec2",dummyT) $ absfree(x,T,absfree(y,U,absfree(f,S,a))) $ absfree(f,S,b);
fun letrec3_tr [Free(f,S),Free(x,T),Free(y,U),Free(z,V),a,b] =
Const("letrec3",dummyT) $ absfree(x,T,absfree(y,U,absfree(z,U,absfree(f,S,a)))) $ absfree(f,S,b);
fun letrec_tr' [Abs(x,T,Abs(f,S,a)),Abs(ff,SS,b)] =
let val (f',b') = variant_abs(ff,SS,b)
val (_,a'') = variant_abs(f,S,a)
val (x',a') = variant_abs(x,T,a'')
in Const("@letrec",dummyT) $ Free(f',SS) $ Free(x',T) $ a' $ b' end;
fun letrec2_tr' [Abs(x,T,Abs(y,U,Abs(f,S,a))),Abs(ff,SS,b)] =
let val (f',b') = variant_abs(ff,SS,b)
val ( _,a1) = variant_abs(f,S,a)
val (y',a2) = variant_abs(y,U,a1)
val (x',a') = variant_abs(x,T,a2)
in Const("@letrec2",dummyT) $ Free(f',SS) $ Free(x',T) $ Free(y',U) $ a' $ b'
end;
fun letrec3_tr' [Abs(x,T,Abs(y,U,Abs(z,V,Abs(f,S,a)))),Abs(ff,SS,b)] =
let val (f',b') = variant_abs(ff,SS,b)
val ( _,a1) = variant_abs(f,S,a)
val (z',a2) = variant_abs(z,V,a1)
val (y',a3) = variant_abs(y,U,a2)
val (x',a') = variant_abs(x,T,a3)
in Const("@letrec3",dummyT) $ Free(f',SS) $ Free(x',T) $ Free(y',U) $ Free(z',V) $ a' $ b'
end;
*}
parse_translation {*
[("@let", let_tr),
("@letrec", letrec_tr),
("@letrec2", letrec2_tr),
("@letrec3", letrec3_tr)] *}
print_translation {*
[("let", let_tr'),
("letrec", letrec_tr'),
("letrec2", letrec2_tr'),
("letrec3", letrec3_tr')] *}
consts
napply :: "[i=>i,i,i]=>i" ("(_ ^ _ ` _)" [56,56,56] 56)
axioms
one_def: "one == true"
if_def: "if b then t else u == case(b,t,u,% x y. bot,%v. bot)"
inl_def: "inl(a) == <true,a>"
inr_def: "inr(b) == <false,b>"
when_def: "when(t,f,g) == split(t,%b x. if b then f(x) else g(x))"
split_def: "split(t,f) == case(t,bot,bot,f,%u. bot)"
fst_def: "fst(t) == split(t,%x y. x)"
snd_def: "snd(t) == split(t,%x y. y)"
thd_def: "thd(t) == split(t,%x p. split(p,%y z. z))"
zero_def: "zero == inl(one)"
succ_def: "succ(n) == inr(n)"
ncase_def: "ncase(n,b,c) == when(n,%x. b,%y. c(y))"
nrec_def: " nrec(n,b,c) == letrec g x be ncase(x,b,%y. c(y,g(y))) in g(n)"
nil_def: "[] == inl(one)"
cons_def: "h$t == inr(<h,t>)"
lcase_def: "lcase(l,b,c) == when(l,%x. b,%y. split(y,c))"
lrec_def: "lrec(l,b,c) == letrec g x be lcase(x,b,%h t. c(h,t,g(t))) in g(l)"
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)))"
letrec_def:
"letrec g x be h(x,g) in b(g) == b(%x. fix(%f. lam x. h(x,%y. f`y))`x)"
letrec2_def: "letrec g x y be h(x,y,g) in f(g)==
letrec g' p be split(p,%x y. h(x,y,%u v. g'(<u,v>)))
in f(%x y. g'(<x,y>))"
letrec3_def: "letrec g x y z be h(x,y,z,g) in f(g) ==
letrec g' p be split(p,%x xs. split(xs,%y z. h(x,y,z,%u v w. g'(<u,<v,w>>))))
in f(%x y z. g'(<x,<y,z>>))"
napply_def: "f ^n` a == nrec(n,a,%x g. f(g))"
lemmas simp_can_defs = one_def inl_def inr_def
and simp_ncan_defs = if_def when_def split_def fst_def snd_def thd_def
lemmas simp_defs = simp_can_defs simp_ncan_defs
lemmas ind_can_defs = zero_def succ_def nil_def cons_def
and ind_ncan_defs = ncase_def nrec_def lcase_def lrec_def
lemmas ind_defs = ind_can_defs ind_ncan_defs
lemmas data_defs = simp_defs ind_defs napply_def
and genrec_defs = letrec_def letrec2_def letrec3_def
subsection {* Beta Rules, including strictness *}
lemma letB: "~ t=bot ==> let x be t in f(x) = f(t)"
apply (unfold let_def)
apply (erule rev_mp)
apply (rule_tac t = "t" in term_case)
apply (simp_all add: caseBtrue caseBfalse caseBpair caseBlam)
done
lemma letBabot: "let x be bot in f(x) = bot"
apply (unfold let_def)
apply (rule caseBbot)
done
lemma letBbbot: "let x be t in bot = bot"
apply (unfold let_def)
apply (rule_tac t = t in term_case)
apply (rule caseBbot)
apply (simp_all add: caseBtrue caseBfalse caseBpair caseBlam)
done
lemma applyB: "(lam x. b(x)) ` a = b(a)"
apply (unfold apply_def)
apply (simp add: caseBtrue caseBfalse caseBpair caseBlam)
done
lemma applyBbot: "bot ` a = bot"
apply (unfold apply_def)
apply (rule caseBbot)
done
lemma fixB: "fix(f) = f(fix(f))"
apply (unfold fix_def)
apply (rule applyB [THEN ssubst], rule refl)
done
lemma letrecB:
"letrec g x be h(x,g) in g(a) = h(a,%y. letrec g x be h(x,g) in g(y))"
apply (unfold letrec_def)
apply (rule fixB [THEN ssubst], rule applyB [THEN ssubst], rule refl)
done
lemmas rawBs = caseBs applyB applyBbot
ML {*
local
val letrecB = thm "letrecB"
val rawBs = thms "rawBs"
val data_defs = thms "data_defs"
in
fun raw_mk_beta_rl defs s = prove_goalw (the_context ()) defs s
(fn _ => [stac letrecB 1,
simp_tac (simpset () addsimps rawBs) 1]);
fun mk_beta_rl s = raw_mk_beta_rl data_defs s;
fun raw_mk_beta_rl defs s = prove_goalw (the_context ()) defs s
(fn _ => [simp_tac (simpset () addsimps rawBs
setloop (stac letrecB)) 1]);
fun mk_beta_rl s = raw_mk_beta_rl data_defs s;
end
*}
ML {*
bind_thm ("ifBtrue", mk_beta_rl "if true then t else u = t");
bind_thm ("ifBfalse", mk_beta_rl "if false then t else u = u");
bind_thm ("ifBbot", mk_beta_rl "if bot then t else u = bot");
bind_thm ("whenBinl", mk_beta_rl "when(inl(a),t,u) = t(a)");
bind_thm ("whenBinr", mk_beta_rl "when(inr(a),t,u) = u(a)");
bind_thm ("whenBbot", mk_beta_rl "when(bot,t,u) = bot");
bind_thm ("splitB", mk_beta_rl "split(<a,b>,h) = h(a,b)");
bind_thm ("splitBbot", mk_beta_rl "split(bot,h) = bot");
bind_thm ("fstB", mk_beta_rl "fst(<a,b>) = a");
bind_thm ("fstBbot", mk_beta_rl "fst(bot) = bot");
bind_thm ("sndB", mk_beta_rl "snd(<a,b>) = b");
bind_thm ("sndBbot", mk_beta_rl "snd(bot) = bot");
bind_thm ("thdB", mk_beta_rl "thd(<a,<b,c>>) = c");
bind_thm ("thdBbot", mk_beta_rl "thd(bot) = bot");
bind_thm ("ncaseBzero", mk_beta_rl "ncase(zero,t,u) = t");
bind_thm ("ncaseBsucc", mk_beta_rl "ncase(succ(n),t,u) = u(n)");
bind_thm ("ncaseBbot", mk_beta_rl "ncase(bot,t,u) = bot");
bind_thm ("nrecBzero", mk_beta_rl "nrec(zero,t,u) = t");
bind_thm ("nrecBsucc", mk_beta_rl "nrec(succ(n),t,u) = u(n,nrec(n,t,u))");
bind_thm ("nrecBbot", mk_beta_rl "nrec(bot,t,u) = bot");
bind_thm ("lcaseBnil", mk_beta_rl "lcase([],t,u) = t");
bind_thm ("lcaseBcons", mk_beta_rl "lcase(x$xs,t,u) = u(x,xs)");
bind_thm ("lcaseBbot", mk_beta_rl "lcase(bot,t,u) = bot");
bind_thm ("lrecBnil", mk_beta_rl "lrec([],t,u) = t");
bind_thm ("lrecBcons", mk_beta_rl "lrec(x$xs,t,u) = u(x,xs,lrec(xs,t,u))");
bind_thm ("lrecBbot", mk_beta_rl "lrec(bot,t,u) = bot");
bind_thm ("letrec2B", raw_mk_beta_rl (thms "data_defs" @ [thm "letrec2_def"])
"letrec g x y be h(x,y,g) in g(p,q) = h(p,q,%u v. letrec g x y be h(x,y,g) in g(u,v))");
bind_thm ("letrec3B", raw_mk_beta_rl (thms "data_defs" @ [thm "letrec3_def"])
"letrec g x y z be h(x,y,z,g) in g(p,q,r) = h(p,q,r,%u v w. letrec g x y z be h(x,y,z,g) in g(u,v,w))");
bind_thm ("napplyBzero", mk_beta_rl "f^zero`a = a");
bind_thm ("napplyBsucc", mk_beta_rl "f^succ(n)`a = f(f^n`a)");
bind_thms ("termBs", [thm "letB", thm "applyB", thm "applyBbot", splitB,splitBbot,
fstB,fstBbot,sndB,sndBbot,thdB,thdBbot,
ifBtrue,ifBfalse,ifBbot,whenBinl,whenBinr,whenBbot,
ncaseBzero,ncaseBsucc,ncaseBbot,nrecBzero,nrecBsucc,nrecBbot,
lcaseBnil,lcaseBcons,lcaseBbot,lrecBnil,lrecBcons,lrecBbot,
napplyBzero,napplyBsucc]);
*}
subsection {* Constructors are injective *}
ML {*
bind_thms ("term_injs", map (mk_inj_rl (the_context ())
[thm "applyB", thm "splitB", thm "whenBinl", thm "whenBinr", thm "ncaseBsucc", thm "lcaseBcons"])
["(inl(a) = inl(a')) <-> (a=a')",
"(inr(a) = inr(a')) <-> (a=a')",
"(succ(a) = succ(a')) <-> (a=a')",
"(a$b = a'$b') <-> (a=a' & b=b')"])
*}
subsection {* Constructors are distinct *}
ML {*
bind_thms ("term_dstncts",
mkall_dstnct_thms (the_context ()) (thms "data_defs") (thms "ccl_injs" @ thms "term_injs")
[["bot","inl","inr"], ["bot","zero","succ"], ["bot","nil","op $"]]);
*}
subsection {* Rules for pre-order @{text "[="} *}
ML {*
local
fun mk_thm s = prove_goalw (the_context ()) (thms "data_defs") s (fn _ =>
[simp_tac (simpset () addsimps (thms "ccl_porews")) 1])
in
val term_porews = map mk_thm ["inl(a) [= inl(a') <-> a [= a'",
"inr(b) [= inr(b') <-> b [= b'",
"succ(n) [= succ(n') <-> n [= n'",
"x$xs [= x'$xs' <-> x [= x' & xs [= xs'"]
end;
bind_thms ("term_porews", term_porews);
*}
subsection {* Rewriting and Proving *}
lemmas term_rews = termBs term_injs term_dstncts ccl_porews term_porews
ML {*
bind_thms ("term_injDs", XH_to_Ds term_injs);
*}
lemmas [simp] = term_rews
and [elim!] = term_dstncts [THEN notE]
and [dest!] = term_injDs
end