(* Title: CCL/Type.ML ID: $Id: Type.ML,v 1.11 2005/09/17 15:35:29 wenzelm Exp $ Author: Martin Coen, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) val simp_type_defs = [Subtype_def,Unit_def,Bool_def,Plus_def,Sigma_def,Pi_def, Lift_def,Tall_def,Tex_def]; val ind_type_defs = [Nat_def,List_def]; val simp_data_defs = [one_def,inl_def,inr_def]; val ind_data_defs = [zero_def,succ_def,nil_def,cons_def]; goal (the_context ()) "A <= B <-> (ALL x. x:A --> x:B)"; by (fast_tac set_cs 1); qed "subsetXH"; (*** Exhaustion Rules ***) fun mk_XH_tac thy defs rls s = prove_goalw thy defs s (fn _ => [cfast_tac rls 1]); val XH_tac = mk_XH_tac (the_context ()) simp_type_defs []; val EmptyXH = XH_tac "a : {} <-> False"; val SubtypeXH = XH_tac "a : {x:A. P(x)} <-> (a:A & P(a))"; val UnitXH = XH_tac "a : Unit <-> a=one"; val BoolXH = XH_tac "a : Bool <-> a=true | a=false"; val PlusXH = XH_tac "a : A+B <-> (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))"; val PiXH = XH_tac "a : PROD x:A. B(x) <-> (EX b. a=lam x. b(x) & (ALL x:A. b(x):B(x)))"; val SgXH = XH_tac "a : SUM x:A. B(x) <-> (EX x:A. EX y:B(x).a=<x,y>)"; val XHs = [EmptyXH,SubtypeXH,UnitXH,BoolXH,PlusXH,PiXH,SgXH]; val LiftXH = XH_tac "a : [A] <-> (a=bot | a:A)"; val TallXH = XH_tac "a : TALL X. B(X) <-> (ALL X. a:B(X))"; val TexXH = XH_tac "a : TEX X. B(X) <-> (EX X. a:B(X))"; val case_rls = XH_to_Es XHs; (*** Canonical Type Rules ***) fun mk_canT_tac thy xhs s = prove_goal thy s (fn prems => [fast_tac (set_cs addIs (prems @ (xhs RL [iffD2]))) 1]); val canT_tac = mk_canT_tac (the_context ()) XHs; val oneT = canT_tac "one : Unit"; val trueT = canT_tac "true : Bool"; val falseT = canT_tac "false : Bool"; val lamT = canT_tac "[| !!x. x:A ==> b(x):B(x) |] ==> lam x. b(x) : Pi(A,B)"; val pairT = canT_tac "[| a:A; b:B(a) |] ==> <a,b>:Sigma(A,B)"; val inlT = canT_tac "a:A ==> inl(a) : A+B"; val inrT = canT_tac "b:B ==> inr(b) : A+B"; val canTs = [oneT,trueT,falseT,pairT,lamT,inlT,inrT]; (*** Non-Canonical Type Rules ***) local val lemma = prove_goal (the_context ()) "[| a:B(u); u=v |] ==> a : B(v)" (fn prems => [cfast_tac prems 1]); in fun mk_ncanT_tac thy defs top_crls crls s = prove_goalw thy defs s (fn major::prems => [(resolve_tac ([major] RL top_crls) 1), (REPEAT_SOME (eresolve_tac (crls @ [exE,bexE,conjE,disjE]))), (ALLGOALS (asm_simp_tac term_ss)), (ALLGOALS (ares_tac (prems RL [lemma]) ORELSE' etac bspec )), (safe_tac (ccl_cs addSIs prems))]); end; val ncanT_tac = mk_ncanT_tac (the_context ()) [] case_rls case_rls; val ifT = ncanT_tac "[| b:Bool; b=true ==> t:A(true); b=false ==> u:A(false) |] ==> \ \ if b then t else u : A(b)"; val applyT = ncanT_tac "[| f : Pi(A,B); a:A |] ==> f ` a : B(a)"; val splitT = ncanT_tac "[| p:Sigma(A,B); !!x y. [| x:A; y:B(x); p=<x,y> |] ==> c(x,y):C(<x,y>) |] ==> \ \ split(p,c):C(p)"; val whenT = ncanT_tac "[| p:A+B; !!x.[| x:A; p=inl(x) |] ==> a(x):C(inl(x)); \ \ !!y.[| y:B; p=inr(y) |] ==> b(y):C(inr(y)) |] ==> \ \ when(p,a,b) : C(p)"; val ncanTs = [ifT,applyT,splitT,whenT]; (*** Subtypes ***) val SubtypeD1 = standard ((SubtypeXH RS iffD1) RS conjunct1); val SubtypeD2 = standard ((SubtypeXH RS iffD1) RS conjunct2); val prems = goal (the_context ()) "[| a:A; P(a) |] ==> a : {x:A. P(x)}"; by (REPEAT (resolve_tac (prems@[SubtypeXH RS iffD2,conjI]) 1)); qed "SubtypeI"; val prems = goal (the_context ()) "[| a : {x:A. P(x)}; [| a:A; P(a) |] ==> Q |] ==> Q"; by (REPEAT (resolve_tac (prems@[SubtypeD1,SubtypeD2]) 1)); qed "SubtypeE"; (*** Monotonicity ***) Goal "mono (%X. X)"; by (REPEAT (ares_tac [monoI] 1)); qed "idM"; Goal "mono(%X. A)"; by (REPEAT (ares_tac [monoI,subset_refl] 1)); qed "constM"; val major::prems = goal (the_context ()) "mono(%X. A(X)) ==> mono(%X.[A(X)])"; by (rtac (subsetI RS monoI) 1); by (dtac (LiftXH RS iffD1) 1); by (etac disjE 1); by (etac (disjI1 RS (LiftXH RS iffD2)) 1); by (rtac (disjI2 RS (LiftXH RS iffD2)) 1); by (etac (major RS monoD RS subsetD) 1); by (assume_tac 1); qed "LiftM"; val prems = goal (the_context ()) "[| mono(%X. A(X)); !!x X. x:A(X) ==> mono(%X. B(X,x)) |] ==> \ \ mono(%X. Sigma(A(X),B(X)))"; by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE hyp_subst_tac 1)); qed "SgM"; val prems = goal (the_context ()) "[| !!x. x:A ==> mono(%X. B(X,x)) |] ==> mono(%X. Pi(A,B(X)))"; by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE hyp_subst_tac 1)); qed "PiM"; val prems = goal (the_context ()) "[| mono(%X. A(X)); mono(%X. B(X)) |] ==> mono(%X. A(X)+B(X))"; by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE hyp_subst_tac 1)); qed "PlusM"; (**************** RECURSIVE TYPES ******************) (*** Conversion Rules for Fixed Points via monotonicity and Tarski ***) Goal "mono(%X. Unit+X)"; by (REPEAT (ares_tac [PlusM,constM,idM] 1)); qed "NatM"; bind_thm("def_NatB", result() RS (Nat_def RS def_lfp_Tarski)); Goal "mono(%X.(Unit+Sigma(A,%y. X)))"; by (REPEAT (ares_tac [PlusM,SgM,constM,idM] 1)); qed "ListM"; bind_thm("def_ListB", result() RS (List_def RS def_lfp_Tarski)); bind_thm("def_ListsB", result() RS (Lists_def RS def_gfp_Tarski)); Goal "mono(%X.({} + Sigma(A,%y. X)))"; by (REPEAT (ares_tac [PlusM,SgM,constM,idM] 1)); qed "IListsM"; bind_thm("def_IListsB", result() RS (ILists_def RS def_gfp_Tarski)); val ind_type_eqs = [def_NatB,def_ListB,def_ListsB,def_IListsB]; (*** Exhaustion Rules ***) fun mk_iXH_tac teqs ddefs rls s = prove_goalw (the_context ()) ddefs s (fn _ => [resolve_tac (teqs RL [XHlemma1]) 1, fast_tac (set_cs addSIs canTs addSEs case_rls) 1]); val iXH_tac = mk_iXH_tac ind_type_eqs ind_data_defs []; val NatXH = iXH_tac "a : Nat <-> (a=zero | (EX x:Nat. a=succ(x)))"; val ListXH = iXH_tac "a : List(A) <-> (a=[] | (EX x:A. EX xs:List(A).a=x$xs))"; val ListsXH = iXH_tac "a : Lists(A) <-> (a=[] | (EX x:A. EX xs:Lists(A).a=x$xs))"; val IListsXH = iXH_tac "a : ILists(A) <-> (EX x:A. EX xs:ILists(A).a=x$xs)"; val iXHs = [NatXH,ListXH]; val icase_rls = XH_to_Es iXHs; (*** Type Rules ***) val icanT_tac = mk_canT_tac (the_context ()) iXHs; val incanT_tac = mk_ncanT_tac (the_context ()) [] icase_rls case_rls; val zeroT = icanT_tac "zero : Nat"; val succT = icanT_tac "n:Nat ==> succ(n) : Nat"; val nilT = icanT_tac "[] : List(A)"; val consT = icanT_tac "[| h:A; t:List(A) |] ==> h$t : List(A)"; val icanTs = [zeroT,succT,nilT,consT]; val ncaseT = incanT_tac "[| n:Nat; n=zero ==> b:C(zero); \ \ !!x.[| x:Nat; n=succ(x) |] ==> c(x):C(succ(x)) |] ==> \ \ ncase(n,b,c) : C(n)"; val lcaseT = incanT_tac "[| l:List(A); l=[] ==> b:C([]); \ \ !!h t.[| h:A; t:List(A); l=h$t |] ==> c(h,t):C(h$t) |] ==> \ \ lcase(l,b,c) : C(l)"; val incanTs = [ncaseT,lcaseT]; (*** Induction Rules ***) val ind_Ms = [NatM,ListM]; fun mk_ind_tac ddefs tdefs Ms canTs case_rls s = prove_goalw (the_context ()) ddefs s (fn major::prems => [resolve_tac (Ms RL ([major] RL (tdefs RL [def_induct]))) 1, fast_tac (set_cs addSIs (prems @ canTs) addSEs case_rls) 1]); val ind_tac = mk_ind_tac ind_data_defs ind_type_defs ind_Ms canTs case_rls; val Nat_ind = ind_tac "[| n:Nat; P(zero); !!x.[| x:Nat; P(x) |] ==> P(succ(x)) |] ==> \ \ P(n)"; val List_ind = ind_tac "[| l:List(A); P([]); \ \ !!x xs.[| x:A; xs:List(A); P(xs) |] ==> P(x$xs) |] ==> \ \ P(l)"; val inds = [Nat_ind,List_ind]; (*** Primitive Recursive Rules ***) fun mk_prec_tac inds s = prove_goal (the_context ()) s (fn major::prems => [resolve_tac ([major] RL inds) 1, ALLGOALS (simp_tac term_ss THEN' fast_tac (set_cs addSIs prems))]); val prec_tac = mk_prec_tac inds; val nrecT = prec_tac "[| n:Nat; b:C(zero); \ \ !!x g.[| x:Nat; g:C(x) |] ==> c(x,g):C(succ(x)) |] ==> \ \ nrec(n,b,c) : C(n)"; val lrecT = prec_tac "[| l:List(A); b:C([]); \ \ !!x xs g.[| x:A; xs:List(A); g:C(xs) |] ==> c(x,xs,g):C(x$xs) |] ==> \ \ lrec(l,b,c) : C(l)"; val precTs = [nrecT,lrecT]; (*** Theorem proving ***) val [major,minor] = goal (the_context ()) "[| <a,b> : Sigma(A,B); [| a:A; b:B(a) |] ==> P \ \ |] ==> P"; by (rtac (major RS (XH_to_E SgXH)) 1); by (rtac minor 1); by (ALLGOALS (fast_tac term_cs)); qed "SgE2"; (* General theorem proving ignores non-canonical term-formers, *) (* - intro rules are type rules for canonical terms *) (* - elim rules are case rules (no non-canonical terms appear) *) val type_cs = term_cs addSIs (SubtypeI::(canTs @ icanTs)) addSEs (SubtypeE::(XH_to_Es XHs)); (*** Infinite Data Types ***) val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) <= gfp(f)"; by (rtac (lfp_lowerbound RS subset_trans) 1); by (rtac (mono RS gfp_lemma3) 1); by (rtac subset_refl 1); qed "lfp_subset_gfp"; val prems = goal (the_context ()) "[| a:A; !!x X.[| x:A; ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==> \ \ t(a) : gfp(B)"; by (rtac coinduct 1); by (res_inst_tac [("P","%x. EX y:A. x=t(y)")] CollectI 1); by (ALLGOALS (fast_tac (ccl_cs addSIs prems))); qed "gfpI"; val rew::prem::prems = goal (the_context ()) "[| C==gfp(B); a:A; !!x X.[| x:A; ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==> \ \ t(a) : C"; by (rewtac rew); by (REPEAT (ares_tac ((prem RS gfpI)::prems) 1)); qed "def_gfpI"; (* EG *) val prems = goal (the_context ()) "letrec g x be zero$g(x) in g(bot) : Lists(Nat)"; by (rtac (refl RS (XH_to_I UnitXH) RS (Lists_def RS def_gfpI)) 1); by (stac letrecB 1); by (rewtac cons_def); by (fast_tac type_cs 1); result();