(* Title: CCL/ccl.thy
ID: $Id$
Author: Martin Coen
Copyright 1993 University of Cambridge
Classical Computational Logic for Untyped Lambda Calculus with reduction to
weak head-normal form.
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.
*)
CCL = Gfp +
classes prog < term
default prog
types i
arities
i :: prog
fun :: (prog,prog)prog
consts
(*** Evaluation Judgement ***)
"--->" :: "[i,i]=>prop" (infixl 20)
(*** Bisimulations for pre-order and equality ***)
"[=" :: "['a,'a]=>o" (infixl 50)
SIM :: "[i,i,i set]=>o"
POgen,EQgen :: "i set => i set"
PO,EQ :: "i set"
(*** Term Formers ***)
true,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"
"`" :: "[i,i]=>i" (infixl 56)
bot :: "i"
fix :: "(i=>i)=>i"
(*** Defined Predicates ***)
Trm,Dvg :: "i => o"
rules
(******* 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)"
end
(*
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
*)