src/CCL/Term.thy

PRIVATE_FILE: slightly more robust way to create and dispose;

(*  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