src/CCL/CCL.thy
 author wenzelm Thu, 23 Jul 2009 21:59:56 +0200 changeset 32153 a0e57fb1b930 parent 32149 ef59550a55d3 child 32154 9721e8e4d48c permissions -rw-r--r--
```
(*  Title:      CCL/CCL.thy
Author:     Martin Coen
*)

header {* Classical Computational Logic for Untyped Lambda Calculus
with reduction to weak head-normal form *}

theory CCL
imports Gfp
begin

text {*
Based on FOL extended with set collection, a primitive higher-order
logic.  HOL is too strong - descriptions prevent a type of programs
being defined which contains only executable terms.
*}

classes prog < "term"
defaultsort prog

arities "fun" :: (prog, prog) prog

typedecl i
arities i :: prog

consts
(*** Evaluation Judgement ***)
Eval      ::       "[i,i]=>prop"          (infixl "--->" 20)

(*** Bisimulations for pre-order and equality ***)
po          ::       "['a,'a]=>o"           (infixl "[=" 50)
SIM         ::       "[i,i,i set]=>o"
POgen       ::       "i set => i set"
EQgen       ::       "i set => i set"
PO          ::       "i set"
EQ          ::       "i set"

(*** Term Formers ***)
true        ::       "i"
false       ::       "i"
pair        ::       "[i,i]=>i"             ("(1<_,/_>)")
lambda      ::       "(i=>i)=>i"            (binder "lam " 55)
"case"      ::       "[i,i,i,[i,i]=>i,(i=>i)=>i]=>i"
"apply"     ::       "[i,i]=>i"             (infixl "`" 56)
bot         ::       "i"
"fix"       ::       "(i=>i)=>i"

(*** Defined Predicates ***)
Trm         ::       "i => o"
Dvg         ::       "i => o"

axioms

(******* EVALUATION SEMANTICS *******)

(**  This is the evaluation semantics from which the axioms below were derived.  **)
(**  It is included here just as an evaluator for FUN and has no influence on    **)
(**  inference in the theory CCL.                                                **)

trueV:       "true ---> true"
falseV:      "false ---> false"
pairV:       "<a,b> ---> <a,b>"
lamV:        "lam x. b(x) ---> lam x. b(x)"
caseVtrue:   "[| t ---> true;  d ---> c |] ==> case(t,d,e,f,g) ---> c"
caseVfalse:  "[| t ---> false;  e ---> c |] ==> case(t,d,e,f,g) ---> c"
caseVpair:   "[| t ---> <a,b>;  f(a,b) ---> c |] ==> case(t,d,e,f,g) ---> c"
caseVlam:    "[| t ---> lam x. b(x);  g(b) ---> c |] ==> case(t,d,e,f,g) ---> c"

(*** Properties of evaluation: note that "t ---> c" impies that c is canonical ***)

canonical:  "[| t ---> c; c==true ==> u--->v;
c==false ==> u--->v;
!!a b. c==<a,b> ==> u--->v;
!!f. c==lam x. f(x) ==> u--->v |] ==>
u--->v"

(* Should be derivable - but probably a bitch! *)
substitute: "[| a==a'; t(a)--->c(a) |] ==> t(a')--->c(a')"

(************** LOGIC ***************)

(*** Definitions used in the following rules ***)

apply_def:     "f ` t == case(f,bot,bot,%x y. bot,%u. u(t))"
bot_def:         "bot == (lam x. x`x)`(lam x. x`x)"
fix_def:      "fix(f) == (lam x. f(x`x))`(lam x. f(x`x))"

(*  The pre-order ([=) is defined as a simulation, and behavioural equivalence (=) *)
(*  as a bisimulation.  They can both be expressed as (bi)simulations up to        *)
(*  behavioural equivalence (ie the relations PO and EQ defined below).            *)

SIM_def:
"SIM(t,t',R) ==  (t=true & t'=true) | (t=false & t'=false) |
(EX a a' b b'. t=<a,b> & t'=<a',b'> & <a,a'> : R & <b,b'> : R) |
(EX f f'. t=lam x. f(x) & t'=lam x. f'(x) & (ALL x.<f(x),f'(x)> : R))"

POgen_def:  "POgen(R) == {p. EX t t'. p=<t,t'> & (t = bot | SIM(t,t',R))}"
EQgen_def:  "EQgen(R) == {p. EX t t'. p=<t,t'> & (t = bot & t' = bot | SIM(t,t',R))}"

PO_def:    "PO == gfp(POgen)"
EQ_def:    "EQ == gfp(EQgen)"

(*** Rules ***)

(** Partial Order **)

po_refl:        "a [= a"
po_trans:       "[| a [= b;  b [= c |] ==> a [= c"
po_cong:        "a [= b ==> f(a) [= f(b)"

(* Extend definition of [= to program fragments of higher type *)
po_abstractn:   "(!!x. f(x) [= g(x)) ==> (%x. f(x)) [= (%x. g(x))"

(** Equality - equivalence axioms inherited from FOL.thy   **)
(**          - congruence of "=" is axiomatised implicitly **)

eq_iff:         "t = t' <-> t [= t' & t' [= t"

(** Properties of canonical values given by greatest fixed point definitions **)

PO_iff:         "t [= t' <-> <t,t'> : PO"
EQ_iff:         "t =  t' <-> <t,t'> : EQ"

(** Behaviour of non-canonical terms (ie case) given by the following beta-rules **)

caseBtrue:            "case(true,d,e,f,g) = d"
caseBfalse:          "case(false,d,e,f,g) = e"
caseBpair:           "case(<a,b>,d,e,f,g) = f(a,b)"
caseBlam:       "case(lam x. b(x),d,e,f,g) = g(b)"
caseBbot:              "case(bot,d,e,f,g) = bot"            (* strictness *)

(** The theory is non-trivial **)
distinctness:   "~ lam x. b(x) = bot"

(*** Definitions of Termination and Divergence ***)

Dvg_def:  "Dvg(t) == t = bot"
Trm_def:  "Trm(t) == ~ Dvg(t)"

text {*
Would be interesting to build a similar theory for a typed programming language:
ie.     true :: bool,      fix :: ('a=>'a)=>'a  etc......

This is starting to look like LCF.
What are the advantages of this approach?
- less axiomatic
- wfd induction / coinduction and fixed point induction available
*}

lemmas ccl_data_defs = apply_def fix_def

declare po_refl [simp]

subsection {* Congruence Rules *}

(*similar to AP_THM in Gordon's HOL*)
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
by simp

(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
lemma arg_cong: "x=y ==> f(x)=f(y)"
by simp

lemma abstractn: "(!!x. f(x) = g(x)) ==> f = g"
apply (blast intro: po_abstractn)
done

lemmas caseBs = caseBtrue caseBfalse caseBpair caseBlam caseBbot

subsection {* Termination and Divergence *}

lemma Trm_iff: "Trm(t) <-> ~ t = bot"

lemma Dvg_iff: "Dvg(t) <-> t = bot"

subsection {* Constructors are injective *}

lemma eq_lemma: "[| x=a;  y=b;  x=y |] ==> a=b"
by simp

ML {*
fun mk_inj_rl thy rews s =
let
fun mk_inj_lemmas r = [@{thm arg_cong}] RL [r RS (r RS @{thm eq_lemma})]
val inj_lemmas = maps mk_inj_lemmas rews
val tac = REPEAT (ares_tac [iffI, allI, conjI] 1 ORELSE
eresolve_tac inj_lemmas 1 ORELSE
asm_simp_tac (Simplifier.theory_context thy @{simpset} addsimps rews) 1)
in prove_goal thy s (fn _ => [tac]) end
*}

ML {*
bind_thms ("ccl_injs",
map (mk_inj_rl @{theory} @{thms caseBs})
["<a,b> = <a',b'> <-> (a=a' & b=b')",
"(lam x. b(x) = lam x. b'(x)) <-> ((ALL z. b(z)=b'(z)))"])
*}

lemma pair_inject: "<a,b> = <a',b'> \<Longrightarrow> (a = a' \<Longrightarrow> b = b' \<Longrightarrow> R) \<Longrightarrow> R"

subsection {* Constructors are distinct *}

lemma lem: "t=t' ==> case(t,b,c,d,e) = case(t',b,c,d,e)"
by simp

ML {*

local
fun pairs_of f x [] = []
| pairs_of f x (y::ys) = (f x y) :: (f y x) :: (pairs_of f x ys)

fun mk_combs ff [] = []
| mk_combs ff (x::xs) = (pairs_of ff x xs) @ mk_combs ff xs

(* Doesn't handle binder types correctly *)
fun saturate thy sy name =
let fun arg_str 0 a s = s
| arg_str 1 a s = "(" ^ a ^ "a" ^ s ^ ")"
| arg_str n a s = arg_str (n-1) a ("," ^ a ^ (chr((ord "a")+n-1)) ^ s)
val T = Sign.the_const_type thy (Sign.intern_const thy sy);
val arity = length (fst (strip_type T))
in sy ^ (arg_str arity name "") end

fun mk_thm_str thy a b = "~ " ^ (saturate thy a "a") ^ " = " ^ (saturate thy b "b")

val lemma = thm "lem";
val eq_lemma = thm "eq_lemma";
val distinctness = thm "distinctness";
fun mk_lemma (ra,rb) = [lemma] RL [ra RS (rb RS eq_lemma)] RL
[distinctness RS notE,sym RS (distinctness RS notE)]
in
fun mk_lemmas rls = maps mk_lemma (mk_combs pair rls)
fun mk_dstnct_rls thy xs = mk_combs (mk_thm_str thy) xs
end

*}

ML {*

val caseB_lemmas = mk_lemmas @{thms caseBs}

val ccl_dstncts =
let fun mk_raw_dstnct_thm rls s =
prove_goal @{theory} s (fn _=> [rtac notI 1,eresolve_tac rls 1])
in map (mk_raw_dstnct_thm caseB_lemmas)
(mk_dstnct_rls @{theory} ["bot","true","false","pair","lambda"]) end

fun mk_dstnct_thms thy defs inj_rls xs =
let fun mk_dstnct_thm rls s = prove_goalw thy defs s
(fn _ => [simp_tac (global_simpset_of thy addsimps (rls@inj_rls)) 1])
in map (mk_dstnct_thm ccl_dstncts) (mk_dstnct_rls thy xs) end

fun mkall_dstnct_thms thy defs i_rls xss = maps (mk_dstnct_thms thy defs i_rls) xss

(*** Rewriting and Proving ***)

fun XH_to_I rl = rl RS iffD2
fun XH_to_D rl = rl RS iffD1
val XH_to_E = make_elim o XH_to_D
val XH_to_Is = map XH_to_I
val XH_to_Ds = map XH_to_D
val XH_to_Es = map XH_to_E;

bind_thms ("ccl_rews", @{thms caseBs} @ ccl_injs @ ccl_dstncts);
bind_thms ("ccl_dstnctsEs", ccl_dstncts RL [notE]);
bind_thms ("ccl_injDs", XH_to_Ds @{thms ccl_injs});
*}

lemmas [simp] = ccl_rews
and [elim!] = pair_inject ccl_dstnctsEs
and [dest!] = ccl_injDs

subsection {* Facts from gfp Definition of @{text "[="} and @{text "="} *}

lemma XHlemma1: "[| A=B;  a:B <-> P |] ==> a:A <-> P"
by simp

lemma XHlemma2: "(P(t,t') <-> Q) ==> (<t,t'> : {p. EX t t'. p=<t,t'> &  P(t,t')} <-> Q)"
by blast

subsection {* Pre-Order *}

lemma POgen_mono: "mono(%X. POgen(X))"
apply (unfold POgen_def SIM_def)
apply (rule monoI)
apply blast
done

lemma POgenXH:
"<t,t'> : POgen(R) <-> t= bot | (t=true & t'=true)  | (t=false & t'=false) |
(EX a a' b b'. t=<a,b> &  t'=<a',b'>  & <a,a'> : R & <b,b'> : R) |
(EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. <f(x),f'(x)> : R))"
apply (unfold POgen_def SIM_def)
apply (rule iff_refl [THEN XHlemma2])
done

lemma poXH:
"t [= t' <-> t=bot | (t=true & t'=true) | (t=false & t'=false) |
(EX a a' b b'. t=<a,b> &  t'=<a',b'>  & a [= a' & b [= b') |
(EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. f(x) [= f'(x)))"
apply (simp add: PO_iff del: ex_simps)
apply (rule POgen_mono
[THEN PO_def [THEN def_gfp_Tarski], THEN XHlemma1, unfolded POgen_def SIM_def])
apply (rule iff_refl [THEN XHlemma2])
done

lemma po_bot: "bot [= b"
apply (rule poXH [THEN iffD2])
apply simp
done

lemma bot_poleast: "a [= bot ==> a=bot"
apply (drule poXH [THEN iffD1])
apply simp
done

lemma po_pair: "<a,b> [= <a',b'> <->  a [= a' & b [= b'"
apply (rule poXH [THEN iff_trans])
apply simp
done

lemma po_lam: "lam x. f(x) [= lam x. f'(x) <-> (ALL x. f(x) [= f'(x))"
apply (rule poXH [THEN iff_trans])
apply fastsimp
done

lemmas ccl_porews = po_bot po_pair po_lam

lemma case_pocong:
assumes 1: "t [= t'"
and 2: "a [= a'"
and 3: "b [= b'"
and 4: "!!x y. c(x,y) [= c'(x,y)"
and 5: "!!u. d(u) [= d'(u)"
shows "case(t,a,b,c,d) [= case(t',a',b',c',d')"
apply (rule 1 [THEN po_cong, THEN po_trans])
apply (rule 2 [THEN po_cong, THEN po_trans])
apply (rule 3 [THEN po_cong, THEN po_trans])
apply (rule 4 [THEN po_abstractn, THEN po_abstractn, THEN po_cong, THEN po_trans])
apply (rule_tac f1 = "%d. case (t',a',b',c',d)" in
5 [THEN po_abstractn, THEN po_cong, THEN po_trans])
apply (rule po_refl)
done

lemma apply_pocong: "[| f [= f';  a [= a' |] ==> f ` a [= f' ` a'"
unfolding ccl_data_defs
apply (rule case_pocong, (rule po_refl | assumption)+)
apply (erule po_cong)
done

lemma npo_lam_bot: "~ lam x. b(x) [= bot"
apply (rule notI)
apply (drule bot_poleast)
apply (erule distinctness [THEN notE])
done

lemma po_lemma: "[| x=a;  y=b;  x[=y |] ==> a[=b"
by simp

lemma npo_pair_lam: "~ <a,b> [= lam x. f(x)"
apply (rule notI)
apply (rule npo_lam_bot [THEN notE])
apply (erule case_pocong [THEN caseBlam [THEN caseBpair [THEN po_lemma]]])
apply (rule po_refl npo_lam_bot)+
done

lemma npo_lam_pair: "~ lam x. f(x) [= <a,b>"
apply (rule notI)
apply (rule npo_lam_bot [THEN notE])
apply (erule case_pocong [THEN caseBpair [THEN caseBlam [THEN po_lemma]]])
apply (rule po_refl npo_lam_bot)+
done

lemma npo_rls1:
"~ true [= false"
"~ false [= true"
"~ true [= <a,b>"
"~ <a,b> [= true"
"~ true [= lam x. f(x)"
"~ lam x. f(x) [= true"
"~ false [= <a,b>"
"~ <a,b> [= false"
"~ false [= lam x. f(x)"
"~ lam x. f(x) [= false"
by (tactic {*
REPEAT (rtac notI 1 THEN
dtac @{thm case_pocong} 1 THEN
etac rev_mp 5 THEN
ALLGOALS (simp_tac @{simpset}) THEN
REPEAT (resolve_tac [@{thm po_refl}, @{thm npo_lam_bot}] 1)) *})

lemmas npo_rls = npo_pair_lam npo_lam_pair npo_rls1

subsection {* Coinduction for @{text "[="} *}

lemma po_coinduct: "[|  <t,u> : R;  R <= POgen(R) |] ==> t [= u"
apply (rule PO_def [THEN def_coinduct, THEN PO_iff [THEN iffD2]])
apply assumption+
done

ML {*
fun po_coinduct_tac ctxt s i =
res_inst_tac ctxt [(("R", 0), s)] @{thm po_coinduct} i
*}

subsection {* Equality *}

lemma EQgen_mono: "mono(%X. EQgen(X))"
apply (unfold EQgen_def SIM_def)
apply (rule monoI)
apply blast
done

lemma EQgenXH:
"<t,t'> : EQgen(R) <-> (t=bot & t'=bot)  | (t=true & t'=true)  |
(t=false & t'=false) |
(EX a a' b b'. t=<a,b> &  t'=<a',b'>  & <a,a'> : R & <b,b'> : R) |
(EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x.<f(x),f'(x)> : R))"
apply (unfold EQgen_def SIM_def)
apply (rule iff_refl [THEN XHlemma2])
done

lemma eqXH:
"t=t' <-> (t=bot & t'=bot)  | (t=true & t'=true)  | (t=false & t'=false) |
(EX a a' b b'. t=<a,b> &  t'=<a',b'>  & a=a' & b=b') |
(EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. f(x)=f'(x)))"
apply (subgoal_tac "<t,t'> : EQ <-> (t=bot & t'=bot) | (t=true & t'=true) | (t=false & t'=false) | (EX a a' b b'. t=<a,b> & t'=<a',b'> & <a,a'> : EQ & <b,b'> : EQ) | (EX f f'. t=lam x. f (x) & t'=lam x. f' (x) & (ALL x. <f (x) ,f' (x) > : EQ))")
apply (erule rev_mp)
apply (simp add: EQ_iff [THEN iff_sym])
apply (rule EQgen_mono [THEN EQ_def [THEN def_gfp_Tarski], THEN XHlemma1,
unfolded EQgen_def SIM_def])
apply (rule iff_refl [THEN XHlemma2])
done

lemma eq_coinduct: "[|  <t,u> : R;  R <= EQgen(R) |] ==> t = u"
apply (rule EQ_def [THEN def_coinduct, THEN EQ_iff [THEN iffD2]])
apply assumption+
done

lemma eq_coinduct3:
"[|  <t,u> : R;  R <= EQgen(lfp(%x. EQgen(x) Un R Un EQ)) |] ==> t = u"
apply (rule EQ_def [THEN def_coinduct3, THEN EQ_iff [THEN iffD2]])
apply (rule EQgen_mono | assumption)+
done

ML {*
fun eq_coinduct_tac ctxt s i = res_inst_tac ctxt [(("R", 0), s)] @{thm eq_coinduct} i
fun eq_coinduct3_tac ctxt s i = res_inst_tac ctxt [(("R", 0), s)] @{thm eq_coinduct3} i
*}

subsection {* Untyped Case Analysis and Other Facts *}

lemma cond_eta: "(EX f. t=lam x. f(x)) ==> t = lam x.(t ` x)"
by (auto simp: apply_def)

lemma exhaustion: "(t=bot) | (t=true) | (t=false) | (EX a b. t=<a,b>) | (EX f. t=lam x. f(x))"
apply (cut_tac refl [THEN eqXH [THEN iffD1]])
apply blast
done

lemma term_case:
"[| P(bot);  P(true);  P(false);  !!x y. P(<x,y>);  !!b. P(lam x. b(x)) |] ==> P(t)"
using exhaustion [of t] by blast

end
```