Theory CCL

```(*  Title:      CCL/CCL.thy
Author:     Martin Coen
*)

section ‹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.
›

class prog = "term"
default_sort prog

instance "fun" :: (prog, prog) prog ..

typedecl i
instance i :: prog ..

consts
(*** Evaluation Judgement ***)
Eval      ::       "[i,i]⇒prop"          (infixl "⤏" 20)

(*** Bisimulations for pre-order and equality ***)
po          ::       "['a,'a]⇒o"           (infixl "[=" 50)

(*** 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"

(******* 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.                                                **)

axiomatization where
trueV:       "true ⤏ true" and
falseV:      "false ⤏ false" and
pairV:       "<a,b> ⤏ <a,b>" and
lamV:        "⋀b. lam x. b(x) ⤏ lam x. b(x)" and

caseVtrue:   "⟦t ⤏ true; d ⤏ c⟧ ⟹ case(t,d,e,f,g) ⤏ c" and
caseVfalse:  "⟦t ⤏ false; e ⤏ c⟧ ⟹ case(t,d,e,f,g) ⤏ c" and
caseVpair:   "⟦t ⤏ <a,b>; f(a,b) ⤏ c⟧ ⟹ case(t,d,e,f,g) ⤏ c" and
caseVlam:    "⋀b. ⟦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 ***)

axiomatization where
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! *)
axiomatization where
substitute: "⟦a==a'; t(a)⤏c(a)⟧ ⟹ t(a')⤏c(a')"

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

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

axiomatization where
bot_def:         "bot == (lam x. x`x)`(lam x. x`x)" and
apply_def:     "f ` t == case(f, bot, bot, λx y. bot, λu. u(t))"

definition "fix" :: "(i⇒i)⇒i"
where "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).            *)

definition SIM :: "[i,i,i set]⇒o"
where
"SIM(t,t',R) ==  (t=true ∧ t'=true) | (t=false ∧ t'=false) |
(∃a a' b b'. t=<a,b> ∧ t'=<a',b'> ∧ <a,a'> : R ∧ <b,b'> : R) |
(∃f f'. t=lam x. f(x) ∧ t'=lam x. f'(x) ∧ (ALL x.<f(x),f'(x)> : R))"

definition POgen :: "i set ⇒ i set"
where "POgen(R) == {p. ∃t t'. p=<t,t'> ∧ (t = bot | SIM(t,t',R))}"

definition EQgen :: "i set ⇒ i set"
where "EQgen(R) == {p. ∃t t'. p=<t,t'> ∧ (t = bot ∧ t' = bot | SIM(t,t',R))}"

definition PO :: "i set"
where "PO == gfp(POgen)"

definition EQ :: "i set"
where "EQ == gfp(EQgen)"

(*** Rules ***)

(** Partial Order **)

axiomatization where
po_refl:        "a [= a" and
po_trans:       "⟦a [= b;  b [= c⟧ ⟹ a [= c" and
po_cong:        "a [= b ⟹ f(a) [= f(b)" and

(* Extend definition of [= to program fragments of higher type *)
po_abstractn:   "(⋀x. f(x) [= g(x)) ⟹ (λx. f(x)) [= (λx. g(x))"

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

axiomatization where
eq_iff:         "t = t' ⟷ t [= t' ∧ t' [= t"

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

axiomatization where
PO_iff:         "t [= t' ⟷ <t,t'> : PO" and
EQ_iff:         "t =  t' ⟷ <t,t'> : EQ"

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

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

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

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

definition Dvg :: "i ⇒ o"
where "Dvg(t) == t = bot"

definition Trm :: "i ⇒ o"
where "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 inj_rl_tac ctxt rews i =
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
in
CHANGED (REPEAT (assume_tac ctxt i ORELSE
resolve_tac ctxt @{thms iffI allI conjI} i ORELSE
eresolve_tac ctxt inj_lemmas i ORELSE
end;
›

method_setup inj_rl = ‹
Attrib.thms >> (fn rews => fn ctxt => SIMPLE_METHOD' (inj_rl_tac ctxt rews))
›

lemma ccl_injs:
"<a,b> = <a',b'> ⟷ (a=a' ∧ b=b')"
"⋀b b'. (lam x. b(x) = lam x. b'(x)) ⟷ ((ALL z. b(z)=b'(z)))"
by (inj_rl caseBs)

lemma pair_inject: "<a,b> = <a',b'> ⟹ (a = a' ⟹ b = b' ⟹ R) ⟹ 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 _ _ [] = []
| pairs_of f x (y::ys) = (f x y) :: (f y x) :: (pairs_of f x ys)

fun mk_combs _ [] = []
| 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 _ 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 (binder_types 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 @{thm notE}, @{thm sym} RS (distinctness RS @{thm 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 =
Goal.prove_global \<^theory> [] [] (Syntax.read_prop_global \<^theory> s)
(fn {context = ctxt, ...} => resolve_tac ctxt @{thms notI} 1 THEN eresolve_tac ctxt rls 1)
in map (mk_raw_dstnct_thm caseB_lemmas)
(mk_dstnct_rls \<^theory> ["bot","true","false","pair","lambda"]) end

fun mk_dstnct_thms ctxt defs inj_rls xs =
let
val thy = Proof_Context.theory_of ctxt;
fun mk_dstnct_thm rls s =
Goal.prove_global thy [] [] (Syntax.read_prop ctxt s)
(fn _ =>
rewrite_goals_tac ctxt defs THEN
simp_tac (ctxt addsimps (rls @ inj_rls)) 1)
in map (mk_dstnct_thm ccl_dstncts) (mk_dstnct_rls thy xs) end

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

(*** Rewriting and Proving ***)

fun XH_to_I rl = rl RS @{thm iffD2}
fun XH_to_D rl = rl RS @{thm 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;

ML_Thms.bind_thms ("ccl_rews", @{thms caseBs} @ @{thms ccl_injs} @ ccl_dstncts);
ML_Thms.bind_thms ("ccl_dstnctsEs", ccl_dstncts RL [@{thm notE}]);
ML_Thms.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 ‹[=› and ‹=››

lemma XHlemma1: "⟦A=B; a:B ⟷ P⟧ ⟹ a:A ⟷ P"
by simp

lemma XHlemma2: "(P(t,t') ⟷ Q) ⟹ (<t,t'> : {p. ∃t t'. p=<t,t'> ∧  P(t,t')} ⟷ Q)"
by blast

subsection ‹Pre-Order›

lemma POgen_mono: "mono(λX. 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 fastforce
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 = "λd. 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 (rule notI, drule case_pocong, erule_tac [5] rev_mp, simp_all,
(rule po_refl npo_lam_bot)+)+

lemmas npo_rls = npo_pair_lam npo_lam_pair npo_rls1

subsection ‹Coinduction for ‹[=››

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

subsection ‹Equality›

lemma EQgen_mono: "mono(λX. 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(λx. 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

method_setup eq_coinduct3 = ‹
Scan.lift Parse.embedded_inner_syntax >> (fn s => fn ctxt =>
SIMPLE_METHOD'
(Rule_Insts.res_inst_tac ctxt [((("R", 0), Position.none), s)] [] @{thm eq_coinduct3}))
›

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
```