(* Title: CCL/CCL.thy ID: $Id$ Author: Martin Coen Copyright 1993 University of Cambridge *) 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: " ---> " 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 ---> ; 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== ==> 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= & t'= & : R & : R) | (EX f f'. t=lam x. f(x) & t'=lam x. f'(x) & (ALL x. : R))" POgen_def: "POgen(R) == {p. EX t t'. p= & (t = bot | SIM(t,t',R))}" EQgen_def: "EQgen(R) == {p. EX t t'. p= & (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' <-> : PO" EQ_iff: "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(,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 and [simp] = po_refl 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 (simp add: eq_iff) apply (blast intro: po_abstractn) done lemmas caseBs = caseBtrue caseBfalse caseBpair caseBlam caseBbot subsection {* Termination and Divergence *} lemma Trm_iff: "Trm(t) <-> ~ t = bot" by (simp add: Trm_def Dvg_def) lemma Dvg_iff: "Dvg(t) <-> t = bot" by (simp add: Trm_def Dvg_def) 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 = List.concat (map 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=a' & b=b')", "(lam x. b(x) = lam x. b'(x)) <-> ((ALL z. b(z)=b'(z)))"]) *} lemma pair_inject: " = \ (a = a' \ b = b' \ R) \ R" by (simp add: ccl_injs) 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 = List.concat (map 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 (the_context ()) s (fn _=> [rtac notI 1,eresolve_tac rls 1]) in map (mk_raw_dstnct_thm caseB_lemmas) (mk_dstnct_rls (the_context ()) ["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 (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 = List.concat (map (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) ==> ( : {p. EX t t'. p= & 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: " : POgen(R) <-> t= bot | (t=true & t'=true) | (t=false & t'=false) | (EX a a' b b'. t= & t'= & : R & : R) | (EX f f'. t=lam x. f(x) & t'=lam x. f'(x) & (ALL 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= & t'= & 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 [= 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: "~ [= 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) [= " 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 ML {* local fun mk_thm s = prove_goal (the_context ()) s (fn _ => [rtac notI 1,dtac (thm "case_pocong") 1,etac rev_mp 5, ALLGOALS (simp_tac (simpset ())), REPEAT (resolve_tac [thm "po_refl", thm "npo_lam_bot"] 1)]) in val npo_rls = [thm "npo_pair_lam", thm "npo_lam_pair"] @ map mk_thm ["~ true [= false", "~ false [= true", "~ true [= ", "~ [= true", "~ true [= lam x. f(x)","~ lam x. f(x) [= true", "~ false [= ", "~ [= false", "~ false [= lam x. f(x)","~ lam x. f(x) [= false"] end; bind_thms ("npo_rls", npo_rls); *} subsection {* Coinduction for @{text "[="} *} lemma po_coinduct: "[| : R; R <= POgen(R) |] ==> t [= u" apply (rule PO_def [THEN def_coinduct, THEN PO_iff [THEN iffD2]]) apply assumption+ done ML {* local val po_coinduct = thm "po_coinduct" in fun po_coinduct_tac s i = res_inst_tac [("R",s)] po_coinduct i end *} subsection {* Equality *} lemma EQgen_mono: "mono(%X. EQgen(X))" apply (unfold EQgen_def SIM_def) apply (rule monoI) apply blast done lemma EQgenXH: " : EQgen(R) <-> (t=bot & t'=bot) | (t=true & t'=true) | (t=false & t'=false) | (EX a a' b b'. t= & t'= & : R & : R) | (EX f f'. t=lam x. f(x) & t'=lam x. f'(x) & (ALL 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= & t'= & 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 " : EQ <-> (t=bot & t'=bot) | (t=true & t'=true) | (t=false & t'=false) | (EX a a' b b'. t= & t'= & : EQ & : EQ) | (EX f f'. t=lam x. f (x) & t'=lam x. f' (x) & (ALL 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: "[| : R; R <= EQgen(R) |] ==> t = u" apply (rule EQ_def [THEN def_coinduct, THEN EQ_iff [THEN iffD2]]) apply assumption+ done lemma eq_coinduct3: "[| : 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 {* local val eq_coinduct = thm "eq_coinduct" val eq_coinduct3 = thm "eq_coinduct3" in fun eq_coinduct_tac s i = res_inst_tac [("R",s)] eq_coinduct i fun eq_coinduct3_tac s i = res_inst_tac [("R",s)] eq_coinduct3 i end *} 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=) | (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(); !!b. P(lam x. b(x)) |] ==> P(t)" using exhaustion [of t] by blast end