(* Title: CCL/Wfd.thy
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header {* Well-founded relations in CCL *}
theory Wfd
imports Trancl Type Hered
begin
consts
(*** Predicates ***)
Wfd :: "[i set] => o"
(*** Relations ***)
wf :: "[i set] => i set"
wmap :: "[i=>i,i set] => i set"
lex :: "[i set,i set] => i set" (infixl "**" 70)
NatPR :: "i set"
ListPR :: "i set => i set"
defs
Wfd_def:
"Wfd(R) == ALL P.(ALL x.(ALL y.<y,x> : R --> y:P) --> x:P) --> (ALL a. a:P)"
wf_def: "wf(R) == {x. x:R & Wfd(R)}"
wmap_def: "wmap(f,R) == {p. EX x y. p=<x,y> & <f(x),f(y)> : R}"
lex_def:
"ra**rb == {p. EX a a' b b'. p = <<a,b>,<a',b'>> & (<a,a'> : ra | (a=a' & <b,b'> : rb))}"
NatPR_def: "NatPR == {p. EX x:Nat. p=<x,succ(x)>}"
ListPR_def: "ListPR(A) == {p. EX h:A. EX t:List(A). p=<t,h$t>}"
lemma wfd_induct:
assumes 1: "Wfd(R)"
and 2: "!!x.[| ALL y. <y,x>: R --> P(y) |] ==> P(x)"
shows "P(a)"
apply (rule 1 [unfolded Wfd_def, rule_format, THEN CollectD])
using 2 apply blast
done
lemma wfd_strengthen_lemma:
assumes 1: "!!x y.<x,y> : R ==> Q(x)"
and 2: "ALL x. (ALL y. <y,x> : R --> y : P) --> x : P"
and 3: "!!x. Q(x) ==> x:P"
shows "a:P"
apply (rule 2 [rule_format])
using 1 3
apply blast
done
ML {*
fun wfd_strengthen_tac ctxt s i =
RuleInsts.res_inst_tac ctxt [(("Q", 0), s)] @{thm wfd_strengthen_lemma} i THEN assume_tac (i+1)
*}
lemma wf_anti_sym: "[| Wfd(r); <a,x>:r; <x,a>:r |] ==> P"
apply (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P")
apply blast
apply (erule wfd_induct)
apply blast
done
lemma wf_anti_refl: "[| Wfd(r); <a,a>: r |] ==> P"
apply (rule wf_anti_sym)
apply assumption+
done
subsection {* Irreflexive transitive closure *}
lemma trancl_wf:
assumes 1: "Wfd(R)"
shows "Wfd(R^+)"
apply (unfold Wfd_def)
apply (rule allI ballI impI)+
(*must retain the universal formula for later use!*)
apply (rule allE, assumption)
apply (erule mp)
apply (rule 1 [THEN wfd_induct])
apply (rule impI [THEN allI])
apply (erule tranclE)
apply blast
apply (erule spec [THEN mp, THEN spec, THEN mp])
apply assumption+
done
subsection {* Lexicographic Ordering *}
lemma lexXH:
"p : ra**rb <-> (EX a a' b b'. p = <<a,b>,<a',b'>> & (<a,a'> : ra | a=a' & <b,b'> : rb))"
unfolding lex_def by blast
lemma lexI1: "<a,a'> : ra ==> <<a,b>,<a',b'>> : ra**rb"
by (blast intro!: lexXH [THEN iffD2])
lemma lexI2: "<b,b'> : rb ==> <<a,b>,<a,b'>> : ra**rb"
by (blast intro!: lexXH [THEN iffD2])
lemma lexE:
assumes 1: "p : ra**rb"
and 2: "!!a a' b b'.[| <a,a'> : ra; p=<<a,b>,<a',b'>> |] ==> R"
and 3: "!!a b b'.[| <b,b'> : rb; p = <<a,b>,<a,b'>> |] ==> R"
shows R
apply (rule 1 [THEN lexXH [THEN iffD1], THEN exE])
using 2 3
apply blast
done
lemma lex_pair: "[| p : r**s; !!a a' b b'. p = <<a,b>,<a',b'>> ==> P |] ==>P"
apply (erule lexE)
apply blast+
done
lemma lex_wf:
assumes 1: "Wfd(R)"
and 2: "Wfd(S)"
shows "Wfd(R**S)"
apply (unfold Wfd_def)
apply safe
apply (tactic {* wfd_strengthen_tac @{context} "%x. EX a b. x=<a,b>" 1 *})
apply (blast elim!: lex_pair)
apply (subgoal_tac "ALL a b.<a,b>:P")
apply blast
apply (rule 1 [THEN wfd_induct, THEN allI])
apply (rule 2 [THEN wfd_induct, THEN allI]) back
apply (fast elim!: lexE)
done
subsection {* Mapping *}
lemma wmapXH: "p : wmap(f,r) <-> (EX x y. p=<x,y> & <f(x),f(y)> : r)"
unfolding wmap_def by blast
lemma wmapI: "<f(a),f(b)> : r ==> <a,b> : wmap(f,r)"
by (blast intro!: wmapXH [THEN iffD2])
lemma wmapE: "[| p : wmap(f,r); !!a b.[| <f(a),f(b)> : r; p=<a,b> |] ==> R |] ==> R"
by (blast dest!: wmapXH [THEN iffD1])
lemma wmap_wf:
assumes 1: "Wfd(r)"
shows "Wfd(wmap(f,r))"
apply (unfold Wfd_def)
apply clarify
apply (subgoal_tac "ALL b. ALL a. f (a) =b-->a:P")
apply blast
apply (rule 1 [THEN wfd_induct, THEN allI])
apply clarify
apply (erule spec [THEN mp])
apply (safe elim!: wmapE)
apply (erule spec [THEN mp, THEN spec, THEN mp])
apply assumption
apply (rule refl)
done
subsection {* Projections *}
lemma wfstI: "<xa,ya> : r ==> <<xa,xb>,<ya,yb>> : wmap(fst,r)"
apply (rule wmapI)
apply simp
done
lemma wsndI: "<xb,yb> : r ==> <<xa,xb>,<ya,yb>> : wmap(snd,r)"
apply (rule wmapI)
apply simp
done
lemma wthdI: "<xc,yc> : r ==> <<xa,<xb,xc>>,<ya,<yb,yc>>> : wmap(thd,r)"
apply (rule wmapI)
apply simp
done
subsection {* Ground well-founded relations *}
lemma wfI: "[| Wfd(r); a : r |] ==> a : wf(r)"
unfolding wf_def by blast
lemma Empty_wf: "Wfd({})"
unfolding Wfd_def by (blast elim: EmptyXH [THEN iffD1, THEN FalseE])
lemma wf_wf: "Wfd(wf(R))"
unfolding wf_def
apply (rule_tac Q = "Wfd(R)" in excluded_middle [THEN disjE])
apply simp_all
apply (rule Empty_wf)
done
lemma NatPRXH: "p : NatPR <-> (EX x:Nat. p=<x,succ(x)>)"
unfolding NatPR_def by blast
lemma ListPRXH: "p : ListPR(A) <-> (EX h:A. EX t:List(A).p=<t,h$t>)"
unfolding ListPR_def by blast
lemma NatPRI: "x : Nat ==> <x,succ(x)> : NatPR"
by (auto simp: NatPRXH)
lemma ListPRI: "[| t : List(A); h : A |] ==> <t,h $ t> : ListPR(A)"
by (auto simp: ListPRXH)
lemma NatPR_wf: "Wfd(NatPR)"
apply (unfold Wfd_def)
apply clarify
apply (tactic {* wfd_strengthen_tac @{context} "%x. x:Nat" 1 *})
apply (fastsimp iff: NatPRXH)
apply (erule Nat_ind)
apply (fastsimp iff: NatPRXH)+
done
lemma ListPR_wf: "Wfd(ListPR(A))"
apply (unfold Wfd_def)
apply clarify
apply (tactic {* wfd_strengthen_tac @{context} "%x. x:List (A)" 1 *})
apply (fastsimp iff: ListPRXH)
apply (erule List_ind)
apply (fastsimp iff: ListPRXH)+
done
subsection {* General Recursive Functions *}
lemma letrecT:
assumes 1: "a : A"
and 2: "!!p g.[| p:A; ALL x:{x: A. <x,p>:wf(R)}. g(x) : D(x) |] ==> h(p,g) : D(p)"
shows "letrec g x be h(x,g) in g(a) : D(a)"
apply (rule 1 [THEN rev_mp])
apply (rule wf_wf [THEN wfd_induct])
apply (subst letrecB)
apply (rule impI)
apply (erule 2)
apply blast
done
lemma SPLITB: "SPLIT(<a,b>,B) = B(a,b)"
unfolding SPLIT_def
apply (rule set_ext)
apply blast
done
lemma letrec2T:
assumes "a : A"
and "b : B"
and "!!p q g.[| p:A; q:B;
ALL x:A. ALL y:{y: B. <<x,y>,<p,q>>:wf(R)}. g(x,y) : D(x,y) |] ==>
h(p,q,g) : D(p,q)"
shows "letrec g x y be h(x,y,g) in g(a,b) : D(a,b)"
apply (unfold letrec2_def)
apply (rule SPLITB [THEN subst])
apply (assumption | rule letrecT pairT splitT prems)+
apply (subst SPLITB)
apply (assumption | rule ballI SubtypeI prems)+
apply (rule SPLITB [THEN subst])
apply (assumption | rule letrecT SubtypeI pairT splitT prems |
erule bspec SubtypeE sym [THEN subst])+
done
lemma lem: "SPLIT(<a,<b,c>>,%x xs. SPLIT(xs,%y z. B(x,y,z))) = B(a,b,c)"
by (simp add: SPLITB)
lemma letrec3T:
assumes "a : A"
and "b : B"
and "c : C"
and "!!p q r g.[| p:A; q:B; r:C;
ALL x:A. ALL y:B. ALL z:{z:C. <<x,<y,z>>,<p,<q,r>>> : wf(R)}.
g(x,y,z) : D(x,y,z) |] ==>
h(p,q,r,g) : D(p,q,r)"
shows "letrec g x y z be h(x,y,z,g) in g(a,b,c) : D(a,b,c)"
apply (unfold letrec3_def)
apply (rule lem [THEN subst])
apply (assumption | rule letrecT pairT splitT prems)+
apply (simp add: SPLITB)
apply (assumption | rule ballI SubtypeI prems)+
apply (rule lem [THEN subst])
apply (assumption | rule letrecT SubtypeI pairT splitT prems |
erule bspec SubtypeE sym [THEN subst])+
done
lemmas letrecTs = letrecT letrec2T letrec3T
subsection {* Type Checking for Recursive Calls *}
lemma rcallT:
"[| ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x);
g(a) : D(a) ==> g(a) : E; a:A; <a,p>:wf(R) |] ==>
g(a) : E"
by blast
lemma rcall2T:
"[| ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y);
g(a,b) : D(a,b) ==> g(a,b) : E; a:A; b:B; <<a,b>,<p,q>>:wf(R) |] ==>
g(a,b) : E"
by blast
lemma rcall3T:
"[| ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}. g(x,y,z):D(x,y,z);
g(a,b,c) : D(a,b,c) ==> g(a,b,c) : E;
a:A; b:B; c:C; <<a,<b,c>>,<p,<q,r>>> : wf(R) |] ==>
g(a,b,c) : E"
by blast
lemmas rcallTs = rcallT rcall2T rcall3T
subsection {* Instantiating an induction hypothesis with an equality assumption *}
lemma hyprcallT:
assumes 1: "g(a) = b"
and 2: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x)"
and 3: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> b=g(a) ==> g(a) : D(a) ==> P"
and 4: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> a:A"
and 5: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> <a,p>:wf(R)"
shows P
apply (rule 3 [OF 2, OF 1 [symmetric]])
apply (rule rcallT [OF 2])
apply assumption
apply (rule 4 [OF 2])
apply (rule 5 [OF 2])
done
lemma hyprcall2T:
assumes 1: "g(a,b) = c"
and 2: "ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y)"
and 3: "[| c=g(a,b); g(a,b) : D(a,b) |] ==> P"
and 4: "a:A"
and 5: "b:B"
and 6: "<<a,b>,<p,q>>:wf(R)"
shows P
apply (rule 3)
apply (rule 1 [symmetric])
apply (rule rcall2T)
apply (rule 2)
apply assumption
apply (rule 4)
apply (rule 5)
apply (rule 6)
done
lemma hyprcall3T:
assumes 1: "g(a,b,c) = d"
and 2: "ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z)"
and 3: "[| d=g(a,b,c); g(a,b,c) : D(a,b,c) |] ==> P"
and 4: "a:A"
and 5: "b:B"
and 6: "c:C"
and 7: "<<a,<b,c>>,<p,<q,r>>> : wf(R)"
shows P
apply (rule 3)
apply (rule 1 [symmetric])
apply (rule rcall3T)
apply (rule 2)
apply assumption
apply (rule 4)
apply (rule 5)
apply (rule 6)
apply (rule 7)
done
lemmas hyprcallTs = hyprcallT hyprcall2T hyprcall3T
subsection {* Rules to Remove Induction Hypotheses after Type Checking *}
lemma rmIH1: "[| ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x); P |] ==> P" .
lemma rmIH2: "[| ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y); P |] ==> P" .
lemma rmIH3:
"[| ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z);
P |] ==>
P" .
lemmas rmIHs = rmIH1 rmIH2 rmIH3
subsection {* Lemmas for constructors and subtypes *}
(* 0-ary constructors do not need additional rules as they are handled *)
(* correctly by applying SubtypeI *)
lemma Subtype_canTs:
"!!a b A B P. a : {x:A. b:{y:B(a).P(<x,y>)}} ==> <a,b> : {x:Sigma(A,B).P(x)}"
"!!a A B P. a : {x:A. P(inl(x))} ==> inl(a) : {x:A+B. P(x)}"
"!!b A B P. b : {x:B. P(inr(x))} ==> inr(b) : {x:A+B. P(x)}"
"!!a P. a : {x:Nat. P(succ(x))} ==> succ(a) : {x:Nat. P(x)}"
"!!h t A P. h : {x:A. t : {y:List(A).P(x$y)}} ==> h$t : {x:List(A).P(x)}"
by (assumption | rule SubtypeI canTs icanTs | erule SubtypeE)+
lemma letT: "[| f(t):B; ~t=bot |] ==> let x be t in f(x) : B"
apply (erule letB [THEN ssubst])
apply assumption
done
lemma applyT2: "[| a:A; f : Pi(A,B) |] ==> f ` a : B(a)"
apply (erule applyT)
apply assumption
done
lemma rcall_lemma1: "[| a:A; a:A ==> P(a) |] ==> a : {x:A. P(x)}"
by blast
lemma rcall_lemma2: "[| a:{x:A. Q(x)}; [| a:A; Q(a) |] ==> P(a) |] ==> a : {x:A. P(x)}"
by blast
lemmas rcall_lemmas = asm_rl rcall_lemma1 SubtypeD1 rcall_lemma2
subsection {* Typechecking *}
ML {*
local
val type_rls =
thms "canTs" @ thms "icanTs" @ thms "applyT2" @ thms "ncanTs" @ thms "incanTs" @
thms "precTs" @ thms "letrecTs" @ thms "letT" @ thms "Subtype_canTs";
val rcallT = thm "rcallT";
val rcall2T = thm "rcall2T";
val rcall3T = thm "rcall3T";
val rcallTs = thms "rcallTs";
val rcall_lemmas = thms "rcall_lemmas";
val SubtypeE = thm "SubtypeE";
val SubtypeI = thm "SubtypeI";
val rmIHs = thms "rmIHs";
val hyprcallTs = thms "hyprcallTs";
fun bvars (Const("all",_) $ Abs(s,_,t)) l = bvars t (s::l)
| bvars _ l = l
fun get_bno l n (Const("all",_) $ Abs(s,_,t)) = get_bno (s::l) n t
| get_bno l n (Const("Trueprop",_) $ t) = get_bno l n t
| get_bno l n (Const("Ball",_) $ _ $ Abs(s,_,t)) = get_bno (s::l) (n+1) t
| get_bno l n (Const("mem",_) $ t $ _) = get_bno l n t
| get_bno l n (t $ s) = get_bno l n t
| get_bno l n (Bound m) = (m-length(l),n)
(* Not a great way of identifying induction hypothesis! *)
fun could_IH x = could_unify(x,hd (prems_of rcallT)) orelse
could_unify(x,hd (prems_of rcall2T)) orelse
could_unify(x,hd (prems_of rcall3T))
fun IHinst tac rls = SUBGOAL (fn (Bi,i) =>
let val bvs = bvars Bi []
val ihs = List.filter could_IH (Logic.strip_assums_hyp Bi)
val rnames = map (fn x=>
let val (a,b) = get_bno [] 0 x
in (List.nth(bvs,a),b) end) ihs
fun try_IHs [] = no_tac
| try_IHs ((x,y)::xs) = tac [("g",x)] (List.nth(rls,y-1)) i ORELSE (try_IHs xs)
in try_IHs rnames end)
fun is_rigid_prog t =
case (Logic.strip_assums_concl t) of
(Const("Trueprop",_) $ (Const("mem",_) $ a $ _)) => (term_vars a = [])
| _ => false
in
fun rcall_tac i = let fun tac ps rl i = Tactic.res_inst_tac ps rl i THEN atac i
in IHinst tac rcallTs i end THEN
eresolve_tac rcall_lemmas i
fun raw_step_tac prems i = ares_tac (prems@type_rls) i ORELSE
rcall_tac i ORELSE
ematch_tac [SubtypeE] i ORELSE
match_tac [SubtypeI] i
fun tc_step_tac prems = SUBGOAL (fn (Bi,i) =>
if is_rigid_prog Bi then raw_step_tac prems i else no_tac)
fun typechk_tac rls i = SELECT_GOAL (REPEAT_FIRST (tc_step_tac rls)) i
val tac = typechk_tac [] 1
(*** Clean up Correctness Condictions ***)
val clean_ccs_tac = REPEAT_FIRST (eresolve_tac ([SubtypeE]@rmIHs) ORELSE'
hyp_subst_tac)
val clean_ccs_tac =
let fun tac ps rl i = Tactic.eres_inst_tac ps rl i THEN atac i
in TRY (REPEAT_FIRST (IHinst tac hyprcallTs ORELSE'
eresolve_tac ([asm_rl,SubtypeE]@rmIHs) ORELSE'
hyp_subst_tac)) end
fun gen_ccs_tac rls i = SELECT_GOAL (REPEAT_FIRST (tc_step_tac rls) THEN
clean_ccs_tac) i
end
*}
subsection {* Evaluation *}
ML {*
local
structure Data = NamedThmsFun(val name = "eval" val description = "evaluation rules");
in
fun eval_tac ctxt ths =
METAHYPS (fn prems =>
DEPTH_SOLVE_1 (resolve_tac (ths @ prems @ Data.get ctxt) 1)) 1;
val eval_setup =
Data.setup #>
Method.add_methods [("eval", Method.thms_ctxt_args (fn ths => fn ctxt =>
Method.SIMPLE_METHOD (CHANGED (eval_tac ctxt ths))), "evaluation")];
end;
*}
setup eval_setup
lemmas eval_rls [eval] = trueV falseV pairV lamV caseVtrue caseVfalse caseVpair caseVlam
lemma applyV [eval]:
assumes "f ---> lam x. b(x)"
and "b(a) ---> c"
shows "f ` a ---> c"
unfolding apply_def by (eval prems)
lemma letV:
assumes 1: "t ---> a"
and 2: "f(a) ---> c"
shows "let x be t in f(x) ---> c"
apply (unfold let_def)
apply (rule 1 [THEN canonical])
apply (tactic {*
REPEAT (DEPTH_SOLVE_1 (resolve_tac (@{thms assms} @ @{thms eval_rls}) 1 ORELSE
etac @{thm substitute} 1)) *})
done
lemma fixV: "f(fix(f)) ---> c ==> fix(f) ---> c"
apply (unfold fix_def)
apply (rule applyV)
apply (rule lamV)
apply assumption
done
lemma letrecV:
"h(t,%y. letrec g x be h(x,g) in g(y)) ---> c ==>
letrec g x be h(x,g) in g(t) ---> c"
apply (unfold letrec_def)
apply (assumption | rule fixV applyV lamV)+
done
lemmas [eval] = letV letrecV fixV
lemma V_rls [eval]:
"true ---> true"
"false ---> false"
"!!b c t u. [| b--->true; t--->c |] ==> if b then t else u ---> c"
"!!b c t u. [| b--->false; u--->c |] ==> if b then t else u ---> c"
"!!a b. <a,b> ---> <a,b>"
"!!a b c t h. [| t ---> <a,b>; h(a,b) ---> c |] ==> split(t,h) ---> c"
"zero ---> zero"
"!!n. succ(n) ---> succ(n)"
"!!c n t u. [| n ---> zero; t ---> c |] ==> ncase(n,t,u) ---> c"
"!!c n t u x. [| n ---> succ(x); u(x) ---> c |] ==> ncase(n,t,u) ---> c"
"!!c n t u. [| n ---> zero; t ---> c |] ==> nrec(n,t,u) ---> c"
"!!c n t u x. [| n--->succ(x); u(x,nrec(x,t,u))--->c |] ==> nrec(n,t,u)--->c"
"[] ---> []"
"!!h t. h$t ---> h$t"
"!!c l t u. [| l ---> []; t ---> c |] ==> lcase(l,t,u) ---> c"
"!!c l t u x xs. [| l ---> x$xs; u(x,xs) ---> c |] ==> lcase(l,t,u) ---> c"
"!!c l t u. [| l ---> []; t ---> c |] ==> lrec(l,t,u) ---> c"
"!!c l t u x xs. [| l--->x$xs; u(x,xs,lrec(xs,t,u))--->c |] ==> lrec(l,t,u)--->c"
unfolding data_defs by eval+
subsection {* Factorial *}
lemma
"letrec f n be ncase(n,succ(zero),%x. nrec(n,zero,%y g. nrec(f(x),g,%z h. succ(h))))
in f(succ(succ(zero))) ---> ?a"
by eval
lemma
"letrec f n be ncase(n,succ(zero),%x. nrec(n,zero,%y g. nrec(f(x),g,%z h. succ(h))))
in f(succ(succ(succ(zero)))) ---> ?a"
by eval
subsection {* Less Than Or Equal *}
lemma
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>))))
in f(<succ(zero), succ(zero)>) ---> ?a"
by eval
lemma
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>))))
in f(<succ(zero), succ(succ(succ(succ(zero))))>) ---> ?a"
by eval
lemma
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>))))
in f(<succ(succ(succ(succ(succ(zero))))), succ(succ(succ(succ(zero))))>) ---> ?a"
by eval
subsection {* Reverse *}
lemma
"letrec id l be lcase(l,[],%x xs. x$id(xs))
in id(zero$succ(zero)$[]) ---> ?a"
by eval
lemma
"letrec rev l be lcase(l,[],%x xs. lrec(rev(xs),x$[],%y ys g. y$g))
in rev(zero$succ(zero)$(succ((lam x. x)`succ(zero)))$([])) ---> ?a"
by eval
end