simple cases for strong inducts done; infrastructure for the difficult ones is there
(* Title: nominal_library.ML Author: Christian Urban Basic functions for nominal.*)signature NOMINAL_LIBRARY =sig val last2: 'a list -> 'a * 'a val order: ('a * 'a -> bool) -> 'a list -> ('a * 'b) list -> 'b list val remove_dups: ('a * 'a -> bool) -> 'a list -> 'a list val is_true: term -> bool val dest_listT: typ -> typ val dest_fsetT: typ -> typ val mk_id: term -> term val mk_all: (string * typ) -> term -> term val sum_case_const: typ -> typ -> typ -> term val mk_sum_case: term -> term -> term val mk_minus: term -> term val mk_plus: term -> term -> term val perm_ty: typ -> typ val mk_perm_ty: typ -> term -> term -> term val mk_perm: term -> term -> term val dest_perm: term -> term * term val mk_sort_of: term -> term val atom_ty: typ -> typ val atom_const: typ -> term val mk_atom_ty: typ -> term -> term val mk_atom: term -> term val mk_atom_set_ty: typ -> term -> term val mk_atom_set: term -> term val mk_atom_fset_ty: typ -> term -> term val mk_atom_fset: term -> term val mk_atom_list_ty: typ -> term -> term val mk_atom_list: term -> term val is_atom: Proof.context -> typ -> bool val is_atom_set: Proof.context -> typ -> bool val is_atom_fset: Proof.context -> typ -> bool val is_atom_list: Proof.context -> typ -> bool val to_set_ty: typ -> term -> term val to_set: term -> term val atomify_ty: Proof.context -> typ -> term -> term val atomify: Proof.context -> term -> term val setify_ty: Proof.context -> typ -> term -> term val setify: Proof.context -> term -> term val listify_ty: Proof.context -> typ -> term -> term val listify: Proof.context -> term -> term val fresh_star_ty: typ -> typ val fresh_star_const: typ -> term val mk_fresh_star_ty: typ -> term -> term -> term val mk_fresh_star: term -> term -> term val supp_ty: typ -> typ val supp_const: typ -> term val mk_supp_ty: typ -> term -> term val mk_supp: term -> term val supp_rel_ty: typ -> typ val supp_rel_const: typ -> term val mk_supp_rel_ty: typ -> term -> term -> term val mk_supp_rel: term -> term -> term val supports_const: typ -> term val mk_supports_ty: typ -> term -> term -> term val mk_supports: term -> term -> term val finite_const: typ -> term val mk_finite_ty: typ -> term -> term val mk_finite: term -> term val mk_equiv: thm -> thm val safe_mk_equiv: thm -> thm val mk_diff: term * term -> term val mk_append: term * term -> term val mk_union: term * term -> term val fold_union: term list -> term val fold_append: term list -> term val mk_conj: term * term -> term val fold_conj: term list -> term (* fresh arguments for a term *) val fresh_args: Proof.context -> term -> term list (* datatype operations *) type cns_info = (term * typ * typ list * bool list) list val all_dtyps: Datatype_Aux.descr -> (string * sort) list -> typ list val nth_dtyp: Datatype_Aux.descr -> (string * sort) list -> int -> typ val all_dtyp_constrs_types: Datatype_Aux.descr -> (string * sort) list -> cns_info list val nth_dtyp_constrs_types: Datatype_Aux.descr -> (string * sort) list -> int -> cns_info val prefix_dt_names: Datatype_Aux.descr -> (string * sort) list -> string -> string list (* tactics for function package *) val pat_completeness_simp: thm list -> Proof.context -> tactic val prove_termination: thm list -> Proof.context -> Function.info * local_theory (* transformations of premises in inductions *) val transform_prem1: Proof.context -> string list -> thm -> thm val transform_prem2: Proof.context -> string list -> thm -> thm (* transformation into the object logic *) val atomize: thm -> thmendstructure Nominal_Library: NOMINAL_LIBRARY =struct(* orders an AList according to keys *)fun order eq keys list = map (the o AList.lookup eq list) keys(* remove duplicates *)fun remove_dups eq [] = [] | remove_dups eq (x :: xs) = if member eq xs x then remove_dups eq xs else x :: remove_dups eq xsfun last2 [] = raise Empty | last2 [_] = raise Empty | last2 [x, y] = (x, y) | last2 (_ :: xs) = last2 xsfun is_true @{term "Trueprop True"} = true | is_true _ = false fun dest_listT (Type (@{type_name list}, [T])) = T | dest_listT T = raise TYPE ("dest_listT: list type expected", [T], [])fun dest_fsetT (Type (@{type_name fset}, [T])) = T | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);fun mk_id trm = let val ty = fastype_of trm in Const (@{const_name id}, ty --> ty) $ trm endfun mk_all (a, T) t = Term.all T $ Abs (a, T, t)fun sum_case_const ty1 ty2 ty3 = Const (@{const_name sum_case}, [ty1 --> ty3, ty2 --> ty3, Type (@{type_name sum}, [ty1, ty2])] ---> ty3)fun mk_sum_case trm1 trm2 = let val ([ty1], ty3) = strip_type (fastype_of trm1) val ty2 = domain_type (fastype_of trm2) in sum_case_const ty1 ty2 ty3 $ trm1 $ trm2 end fun mk_minus p = @{term "uminus::perm => perm"} $ pfun mk_plus p q = @{term "plus::perm => perm => perm"} $ p $ qfun perm_ty ty = @{typ "perm"} --> ty --> tyfun mk_perm_ty ty p trm = Const (@{const_name "permute"}, perm_ty ty) $ p $ trmfun mk_perm p trm = mk_perm_ty (fastype_of trm) p trmfun dest_perm (Const (@{const_name "permute"}, _) $ p $ t) = (p, t) | dest_perm t = raise TERM ("dest_perm", [t]);fun mk_sort_of t = @{term "sort_of"} $ t;fun atom_ty ty = ty --> @{typ "atom"};fun atom_const ty = Const (@{const_name "atom"}, atom_ty ty)fun mk_atom_ty ty t = atom_const ty $ t;fun mk_atom t = mk_atom_ty (fastype_of t) t;fun mk_atom_set_ty ty t = let val atom_ty = HOLogic.dest_setT ty val img_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom set"}; in Const (@{const_name image}, img_ty) $ atom_const atom_ty $ t endfun mk_atom_fset_ty ty t = let val atom_ty = dest_fsetT ty val fmap_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom fset"}; in Const (@{const_name map_fset}, fmap_ty) $ atom_const atom_ty $ t endfun mk_atom_list_ty ty t = let val atom_ty = dest_listT ty val map_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom list"} in Const (@{const_name map}, map_ty) $ atom_const atom_ty $ t endfun mk_atom_set t = mk_atom_set_ty (fastype_of t) tfun mk_atom_fset t = mk_atom_fset_ty (fastype_of t) tfun mk_atom_list t = mk_atom_list_ty (fastype_of t) t(* coerces a list into a set *)fun to_set_ty ty t = case ty of @{typ "atom list"} => @{term "set :: atom list => atom set"} $ t | @{typ "atom fset"} => @{term "fset :: atom fset => atom set"} $ t | _ => tfun to_set t = to_set_ty (fastype_of t) t(* testing for concrete atom types *)fun is_atom ctxt ty = Sign.of_sort (ProofContext.theory_of ctxt) (ty, @{sort at_base})fun is_atom_set ctxt (Type ("fun", [ty, @{typ bool}])) = is_atom ctxt ty | is_atom_set _ _ = false;fun is_atom_fset ctxt (Type (@{type_name "fset"}, [ty])) = is_atom ctxt ty | is_atom_fset _ _ = false;fun is_atom_list ctxt (Type (@{type_name "list"}, [ty])) = is_atom ctxt ty | is_atom_list _ _ = false(* functions that coerce singletons, sets, fsets and lists of concrete atoms into general atoms sets / lists *)fun atomify_ty ctxt ty t = if is_atom ctxt ty then mk_atom_ty ty t else if is_atom_set ctxt ty then mk_atom_set_ty ty t else if is_atom_fset ctxt ty then mk_atom_fset_ty ty t else if is_atom_list ctxt ty then mk_atom_list_ty ty t else raise TERM ("atomify", [t])fun setify_ty ctxt ty t = if is_atom ctxt ty then HOLogic.mk_set @{typ atom} [mk_atom_ty ty t] else if is_atom_set ctxt ty then mk_atom_set_ty ty t else if is_atom_fset ctxt ty then @{term "fset :: atom fset => atom set"} $ mk_atom_fset_ty ty t else if is_atom_list ctxt ty then @{term "set :: atom list => atom set"} $ mk_atom_list_ty ty t else raise TERM ("setify", [t])fun listify_ty ctxt ty t = if is_atom ctxt ty then HOLogic.mk_list @{typ atom} [mk_atom_ty ty t] else if is_atom_list ctxt ty then mk_atom_list_ty ty t else raise TERM ("listify", [t])fun atomify ctxt t = atomify_ty ctxt (fastype_of t) tfun setify ctxt t = setify_ty ctxt (fastype_of t) tfun listify ctxt t = listify_ty ctxt (fastype_of t) tfun fresh_star_ty ty = [@{typ "atom set"}, ty] ---> @{typ bool}fun fresh_star_const ty = Const (@{const_name fresh_star}, fresh_star_ty ty)fun mk_fresh_star_ty ty t1 t2 = fresh_star_const ty $ t1 $ t2fun mk_fresh_star t1 t2 = mk_fresh_star_ty (fastype_of t2) t1 t2fun supp_ty ty = ty --> @{typ "atom set"};fun supp_const ty = Const (@{const_name supp}, supp_ty ty)fun mk_supp_ty ty t = supp_const ty $ tfun mk_supp t = mk_supp_ty (fastype_of t) tfun supp_rel_ty ty = ([ty, ty] ---> @{typ bool}) --> ty --> @{typ "atom set"};fun supp_rel_const ty = Const (@{const_name supp_rel}, supp_rel_ty ty)fun mk_supp_rel_ty ty r t = supp_rel_const ty $ r $ tfun mk_supp_rel r t = mk_supp_rel_ty (fastype_of t) r tfun supports_const ty = Const (@{const_name supports}, [@{typ "atom set"}, ty] ---> @{typ bool});fun mk_supports_ty ty t1 t2 = supports_const ty $ t1 $ t2;fun mk_supports t1 t2 = mk_supports_ty (fastype_of t2) t1 t2;fun finite_const ty = Const (@{const_name finite}, ty --> @{typ bool})fun mk_finite_ty ty t = finite_const ty $ tfun mk_finite t = mk_finite_ty (fastype_of t) tfun mk_equiv r = r RS @{thm eq_reflection};fun safe_mk_equiv r = mk_equiv r handle Thm.THM _ => r;(* functions that construct differences, appends and unions but avoid producing empty atom sets or empty atom lists *)fun mk_diff (@{term "{}::atom set"}, _) = @{term "{}::atom set"} | mk_diff (t1, @{term "{}::atom set"}) = t1 | mk_diff (@{term "set ([]::atom list)"}, _) = @{term "set ([]::atom list)"} | mk_diff (t1, @{term "set ([]::atom list)"}) = t1 | mk_diff (t1, t2) = HOLogic.mk_binop @{const_name minus} (t1, t2)fun mk_append (t1, @{term "[]::atom list"}) = t1 | mk_append (@{term "[]::atom list"}, t2) = t2 | mk_append (t1, t2) = HOLogic.mk_binop @{const_name "append"} (t1, t2) fun mk_union (t1, @{term "{}::atom set"}) = t1 | mk_union (@{term "{}::atom set"}, t2) = t2 | mk_union (t1, @{term "set ([]::atom list)"}) = t1 | mk_union (@{term "set ([]::atom list)"}, t2) = t2 | mk_union (t1, t2) = HOLogic.mk_binop @{const_name "sup"} (t1, t2) fun fold_union trms = fold_rev (curry mk_union) trms @{term "{}::atom set"}fun fold_append trms = fold_rev (curry mk_append) trms @{term "[]::atom list"}fun mk_conj (t1, @{term "True"}) = t1 | mk_conj (@{term "True"}, t2) = t2 | mk_conj (t1, t2) = HOLogic.mk_conj (t1, t2)fun fold_conj trms = fold_rev (curry mk_conj) trms @{term "True"}(* produces fresh arguments for a term *)fun fresh_args ctxt f = f |> fastype_of |> binder_types |> map (pair "z") |> Variable.variant_frees ctxt [f] |> map Free(** datatypes **)(* constructor infos *)type cns_info = (term * typ * typ list * bool list) list(* - term for constructor constant - type of the constructor - types of the arguments - flags indicating whether the argument is recursive*)(* returns the type of the nth datatype *)fun all_dtyps descr sorts = map (fn n => Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec n)) (0 upto (length descr - 1))fun nth_dtyp descr sorts n = Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec n);(* returns info about constructors in a datatype *)fun all_dtyp_constrs_info descr = map (fn (_, (ty, vs, constrs)) => map (pair (ty, vs)) constrs) descr(* returns the constants of the constructors plus the corresponding type and types of arguments *)fun all_dtyp_constrs_types descr sorts = let fun aux ((ty_name, vs), (cname, args)) = let val vs_tys = map (Datatype_Aux.typ_of_dtyp descr sorts) vs val ty = Type (ty_name, vs_tys) val arg_tys = map (Datatype_Aux.typ_of_dtyp descr sorts) args val is_rec = map Datatype_Aux.is_rec_type args in (Const (cname, arg_tys ---> ty), ty, arg_tys, is_rec) end in map (map aux) (all_dtyp_constrs_info descr) endfun nth_dtyp_constrs_types descr sorts n = nth (all_dtyp_constrs_types descr sorts) n(* generates for every datatype a name str ^ dt_name plus and index for multiple occurences of a string *)fun prefix_dt_names descr sorts str = let fun get_nth_name (i, _) = Datatype_Aux.name_of_typ (nth_dtyp descr sorts i) in Datatype_Prop.indexify_names (map (prefix str o get_nth_name) descr) end(** function package tactics **)fun pat_completeness_simp simps lthy = let val simp_set = HOL_basic_ss addsimps (@{thms sum.inject sum.distinct} @ simps) in Pat_Completeness.pat_completeness_tac lthy 1 THEN ALLGOALS (asm_full_simp_tac simp_set) endfun prove_termination_tac size_simps ctxt = let val natT = @{typ nat} fun prod_size_const fT sT = let val fT_fun = fT --> natT val sT_fun = sT --> natT val prodT = Type (@{type_name prod}, [fT, sT]) in Const (@{const_name prod_size}, [fT_fun, sT_fun, prodT] ---> natT) end fun mk_size_measure T = case T of (Type (@{type_name Sum_Type.sum}, [fT, sT])) => SumTree.mk_sumcase fT sT natT (mk_size_measure fT) (mk_size_measure sT) | (Type (@{type_name Product_Type.prod}, [fT, sT])) => prod_size_const fT sT $ (mk_size_measure fT) $ (mk_size_measure sT) | _ => HOLogic.size_const T fun mk_measure_trm T = HOLogic.dest_setT T |> fst o HOLogic.dest_prodT |> mk_size_measure |> curry (op $) (Const (@{const_name "measure"}, dummyT)) |> Syntax.check_term ctxt val ss = HOL_ss addsimps @{thms in_measure wf_measure sum.cases add_Suc_right add.right_neutral zero_less_Suc prod.size(1) mult_Suc_right} @ size_simps val ss' = ss addsimprocs Nat_Numeral_Simprocs.cancel_numerals in Function_Relation.relation_tac ctxt mk_measure_trm THEN_ALL_NEW simp_tac ss' endfun prove_termination size_simps ctxt = Function.prove_termination NONE (HEADGOAL (prove_termination_tac size_simps ctxt)) ctxt(** transformations of premises (in inductive proofs) **)(* given the theorem F[t]; proves the theorem F[f t] - F needs to be monotone - f returns either SOME for a term it fires on and NONE elsewhere *)fun map_term f t = (case f t of NONE => map_term' f t | x => x)and map_term' f (t $ u) = (case (map_term f t, map_term f u) of (NONE, NONE) => NONE | (SOME t'', NONE) => SOME (t'' $ u) | (NONE, SOME u'') => SOME (t $ u'') | (SOME t'', SOME u'') => SOME (t'' $ u'')) | map_term' f (Abs (s, T, t)) = (case map_term f t of NONE => NONE | SOME t'' => SOME (Abs (s, T, t''))) | map_term' _ _ = NONE;fun map_thm_tac ctxt tac thm = let val monos = Inductive.get_monos ctxt val simps = HOL_basic_ss addsimps @{thms split_def} in EVERY [cut_facts_tac [thm] 1, etac rev_mp 1, REPEAT_DETERM (FIRSTGOAL (simp_tac simps THEN' resolve_tac monos)), REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))] endfun map_thm ctxt f tac thm = let val opt_goal_trm = map_term f (prop_of thm) in case opt_goal_trm of NONE => thm | SOME goal => Goal.prove ctxt [] [] goal (fn _ => map_thm_tac ctxt tac thm) end(* inductive premises can be of the form R ... /\ P ...; split_conj_i picks out the part R or P part*)fun split_conj1 names (Const (@{const_name "conj"}, _) $ f1 $ _) = (case head_of f1 of Const (name, _) => if member (op =) names name then SOME f1 else NONE | _ => NONE)| split_conj1 _ _ = NONE;fun split_conj2 names (Const (@{const_name "conj"}, _) $ f1 $ f2) = (case head_of f1 of Const (name, _) => if member (op =) names name then SOME f2 else NONE | _ => NONE)| split_conj2 _ _ = NONE;fun transform_prem1 ctxt names thm = map_thm ctxt (split_conj1 names) (etac conjunct1 1) thmfun transform_prem2 ctxt names thm = map_thm ctxt (split_conj2 names) (etac conjunct2 1) thm(* transformes a theorem into one of the object logic *)val atomize = Conv.fconv_rule Object_Logic.atomize o forall_intr_varsend (* structure *)open Nominal_Library;