--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Fv.thy Thu Feb 25 07:48:33 2010 +0100
@@ -0,0 +1,399 @@
+theory Fv
+imports "Nominal2_Atoms" "Abs"
+begin
+
+(* Bindings are given as a list which has a length being equal
+ to the length of the number of constructors.
+
+ Each element is a list whose length is equal to the number
+ of arguents.
+
+ Every element specifies bindings of this argument given as
+ a tuple: function, bound argument.
+
+ Eg:
+nominal_datatype
+
+ C1
+ | C2 x y z bind x in z
+ | C3 x y z bind f x in z bind g y in z
+
+yields:
+[
+ [],
+ [[], [], [(NONE, 0)]],
+ [[], [], [(SOME (Const f), 0), (Some (Const g), 1)]]]
+
+A SOME binding has to have a function returning an atom set,
+and a NONE binding has to be on an argument that is an atom
+or an atom set.
+
+How the procedure works:
+ For each of the defined datatypes,
+ For each of the constructors,
+ It creates a union of free variables for each argument.
+
+ For an argument the free variables are the variables minus
+ bound variables.
+
+ The variables are:
+ For an atom, a singleton set with the atom itself.
+ For an atom set, the atom set itself.
+ For a recursive argument, the appropriate fv function applied to it.
+ (* TODO: This one is not implemented *)
+ For other arguments it should be an appropriate fv function stored
+ in the database.
+ The bound variables are a union of results of all bindings that
+ involve the given argument. For a paricular binding the result is:
+ For a function applied to an argument this function with the argument.
+ For an atom, a singleton set with the atom itself.
+ For an atom set, the atom set itself.
+ For a recursive argument, the appropriate fv function applied to it.
+ (* TODO: This one is not implemented *)
+ For other arguments it should be an appropriate fv function stored
+ in the database.
+*)
+
+ML {*
+ open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);
+ (* TODO: It is the same as one in 'nominal_atoms' *)
+ fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});
+ val noatoms = @{term "{} :: atom set"};
+ fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];
+ fun mk_union sets =
+ fold (fn a => fn b =>
+ if a = noatoms then b else
+ if b = noatoms then a else
+ HOLogic.mk_binop @{const_name union} (a, b)) (rev sets) noatoms;
+ fun mk_diff a b =
+ if b = noatoms then a else
+ if b = a then noatoms else
+ HOLogic.mk_binop @{const_name minus} (a, b);
+ fun mk_atoms t =
+ let
+ val ty = fastype_of t;
+ val atom_ty = HOLogic.dest_setT ty --> @{typ atom};
+ val img_ty = atom_ty --> ty --> @{typ "atom set"};
+ in
+ (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)
+ end;
+ (* Copy from Term *)
+ fun is_funtype (Type ("fun", [_, _])) = true
+ | is_funtype _ = false;
+ (* Similar to one in USyntax *)
+ fun mk_pair (fst, snd) =
+ let val ty1 = fastype_of fst
+ val ty2 = fastype_of snd
+ val c = HOLogic.pair_const ty1 ty2
+ in c $ fst $ snd
+ end;
+
+*}
+
+(* TODO: Notice datatypes without bindings and replace alpha with equality *)
+ML {*
+(* Currently needs just one full_tname to access Datatype *)
+fun define_fv_alpha full_tname bindsall lthy =
+let
+ val thy = ProofContext.theory_of lthy;
+ val {descr, ...} = Datatype.the_info thy full_tname;
+ val sorts = []; (* TODO *)
+ fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
+ val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>
+ "fv_" ^ name_of_typ (nth_dtyp i)) descr);
+ val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;
+ val fv_frees = map Free (fv_names ~~ fv_types);
+ val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>
+ "alpha_" ^ name_of_typ (nth_dtyp i)) descr);
+ val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;
+ val alpha_frees = map Free (alpha_names ~~ alpha_types);
+ fun fv_alpha_constr i (cname, dts) bindcs =
+ let
+ val Ts = map (typ_of_dtyp descr sorts) dts;
+ val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
+ val args = map Free (names ~~ Ts);
+ val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);
+ val args2 = map Free (names2 ~~ Ts);
+ val c = Const (cname, Ts ---> (nth_dtyp i));
+ val fv_c = nth fv_frees i;
+ val alpha = nth alpha_frees i;
+ fun fv_bind args (NONE, i) =
+ if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else
+ (* TODO we assume that all can be 'atomized' *)
+ if (is_funtype o fastype_of) (nth args i) then mk_atoms (nth args i) else
+ mk_single_atom (nth args i)
+ | fv_bind args (SOME f, i) = f $ (nth args i);
+ fun fv_arg ((dt, x), bindxs) =
+ let
+ val arg =
+ if is_rec_type dt then nth fv_frees (body_index dt) $ x else
+ (* TODO: we just assume everything can be 'atomized' *)
+ if (is_funtype o fastype_of) x then mk_atoms x else
+ HOLogic.mk_set @{typ atom} [mk_atom (fastype_of x) $ x]
+ val sub = mk_union (map (fv_bind args) bindxs)
+ in
+ mk_diff arg sub
+ end;
+ val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ bindcs))))
+ val alpha_rhs =
+ HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));
+ fun alpha_arg ((dt, bindxs), (arg, arg2)) =
+ if bindxs = [] then (
+ if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)
+ else (HOLogic.mk_eq (arg, arg2)))
+ else
+ if is_rec_type dt then let
+ (* THE HARD CASE *)
+ val lhs_binds = mk_union (map (fv_bind args) bindxs);
+ val lhs = mk_pair (lhs_binds, arg);
+ val rhs_binds = mk_union (map (fv_bind args2) bindxs);
+ val rhs = mk_pair (rhs_binds, arg2);
+ val alpha = nth alpha_frees (body_index dt);
+ val fv = nth fv_frees (body_index dt);
+ val alpha_gen_pre = Const (@{const_name alpha_gen}, dummyT) $ lhs $ alpha $ fv $ (Free ("pi", @{typ perm})) $ rhs;
+ val alpha_gen_t = Syntax.check_term lthy alpha_gen_pre
+ in
+ HOLogic.mk_exists ("pi", @{typ perm}, alpha_gen_t)
+ (* TODO Add some test that is makes sense *)
+ end else @{term "True"}
+ val alpha_lhss = map (HOLogic.mk_Trueprop o alpha_arg) (dts ~~ bindcs ~~ (args ~~ args2))
+ val alpha_eq = Logic.list_implies (alpha_lhss, alpha_rhs)
+ in
+ (fv_eq, alpha_eq)
+ end;
+ fun fv_alpha_eq (i, (_, _, constrs)) binds = map2 (fv_alpha_constr i) constrs binds;
+ val (fv_eqs, alpha_eqs) = split_list (flat (map2 fv_alpha_eq descr bindsall))
+ val add_binds = map (fn x => (Attrib.empty_binding, x))
+ val (fvs, lthy') = (Primrec.add_primrec
+ (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) (add_binds fv_eqs) lthy)
+ val (alphas, lthy'') = (Inductive.add_inductive_i
+ {quiet_mode = false, verbose = true, alt_name = Binding.empty,
+ coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}
+ (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) alpha_names alpha_types) []
+ (add_binds alpha_eqs) [] lthy')
+in
+ ((fvs, alphas), lthy'')
+end
+*}
+
+(* tests
+atom_decl name
+
+datatype ty =
+ Var "name set"
+
+ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *}
+
+local_setup {* define_fv_alpha "Fv.ty" [[[[]]]] *}
+print_theorems
+
+
+datatype rtrm1 =
+ rVr1 "name"
+| rAp1 "rtrm1" "rtrm1"
+| rLm1 "name" "rtrm1" --"name is bound in trm1"
+| rLt1 "bp" "rtrm1" "rtrm1" --"all variables in bp are bound in the 2nd trm1"
+and bp =
+ BUnit
+| BVr "name"
+| BPr "bp" "bp"
+
+(* to be given by the user *)
+
+primrec
+ bv1
+where
+ "bv1 (BUnit) = {}"
+| "bv1 (BVr x) = {atom x}"
+| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"
+
+setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Fv.rtrm1", "Fv.bp"] *}
+
+local_setup {* define_fv_alpha "Fv.rtrm1"
+ [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],
+ [[], [[]], [[], []]]] *}
+print_theorems
+*)
+
+
+ML {*
+fun alpha_inj_tac dist_inj intrs elims =
+ SOLVED' (asm_full_simp_tac (HOL_ss addsimps intrs)) ORELSE'
+ (rtac @{thm iffI} THEN' RANGE [
+ (eresolve_tac elims THEN_ALL_NEW
+ asm_full_simp_tac (HOL_ss addsimps dist_inj)
+ ),
+ asm_full_simp_tac (HOL_ss addsimps intrs)])
+*}
+
+ML {*
+fun build_alpha_inj_gl thm =
+ let
+ val prop = prop_of thm;
+ val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop);
+ val hyps = map HOLogic.dest_Trueprop (Logic.strip_imp_prems prop);
+ fun list_conj l = foldr1 HOLogic.mk_conj l;
+ in
+ if hyps = [] then concl
+ else HOLogic.mk_eq (concl, list_conj hyps)
+ end;
+*}
+
+ML {*
+fun build_alpha_inj intrs dist_inj elims ctxt =
+let
+ val ((_, thms_imp), ctxt') = Variable.import false intrs ctxt;
+ val gls = map (HOLogic.mk_Trueprop o build_alpha_inj_gl) thms_imp;
+ fun tac _ = alpha_inj_tac dist_inj intrs elims 1;
+ val thms = map (fn gl => Goal.prove ctxt' [] [] gl tac) gls;
+in
+ Variable.export ctxt' ctxt thms
+end
+*}
+
+ML {*
+fun build_alpha_refl_gl alphas (x, y, z) =
+let
+ fun build_alpha alpha =
+ let
+ val ty = domain_type (fastype_of alpha);
+ val var = Free(x, ty);
+ val var2 = Free(y, ty);
+ val var3 = Free(z, ty);
+ val symp = HOLogic.mk_imp (alpha $ var $ var2, alpha $ var2 $ var);
+ val transp = HOLogic.mk_imp (alpha $ var $ var2,
+ HOLogic.mk_all (z, ty,
+ HOLogic.mk_imp (alpha $ var2 $ var3, alpha $ var $ var3)))
+ in
+ ((alpha $ var $ var), (symp, transp))
+ end;
+ val (refl_eqs, eqs) = split_list (map build_alpha alphas)
+ val (sym_eqs, trans_eqs) = split_list eqs
+ fun conj l = @{term Trueprop} $ foldr1 HOLogic.mk_conj l
+in
+ (conj refl_eqs, (conj sym_eqs, conj trans_eqs))
+end
+*}
+
+ML {*
+fun reflp_tac induct inj =
+ rtac induct THEN_ALL_NEW
+ asm_full_simp_tac (HOL_ss addsimps inj) THEN_ALL_NEW
+ TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI) THEN_ALL_NEW
+ (rtac @{thm exI[of _ "0 :: perm"]} THEN'
+ asm_full_simp_tac (HOL_ss addsimps
+ @{thms alpha_gen fresh_star_def fresh_zero_perm permute_zero ball_triv}))
+*}
+
+ML {*
+fun symp_tac induct inj eqvt =
+ ((rtac @{thm impI} THEN' etac induct) ORELSE' rtac induct) THEN_ALL_NEW
+ asm_full_simp_tac (HOL_ss addsimps inj) THEN_ALL_NEW
+ TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI) THEN_ALL_NEW
+ (etac @{thm alpha_gen_compose_sym} THEN' eresolve_tac eqvt)
+*}
+
+ML {*
+fun imp_elim_tac case_rules =
+ Subgoal.FOCUS (fn {concl, context, ...} =>
+ case term_of concl of
+ _ $ (_ $ asm $ _) =>
+ let
+ fun filter_fn case_rule = (
+ case Logic.strip_assums_hyp (prop_of case_rule) of
+ ((_ $ asmc) :: _) =>
+ let
+ val thy = ProofContext.theory_of context
+ in
+ Pattern.matches thy (asmc, asm)
+ end
+ | _ => false)
+ val matching_rules = filter filter_fn case_rules
+ in
+ (rtac impI THEN' rotate_tac (~1) THEN' eresolve_tac matching_rules) 1
+ end
+ | _ => no_tac
+ )
+*}
+
+ML {*
+fun transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt =
+ ((rtac impI THEN' etac induct) ORELSE' rtac induct) THEN_ALL_NEW
+ (TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW
+ (
+ asm_full_simp_tac (HOL_ss addsimps alpha_inj @ term_inj @ distinct) THEN'
+ TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI) THEN_ALL_NEW
+ (etac @{thm alpha_gen_compose_trans} THEN' RANGE [atac, eresolve_tac eqvt])
+ )
+*}
+
+lemma transp_aux:
+ "(\<And>xa ya. R xa ya \<longrightarrow> (\<forall>z. R ya z \<longrightarrow> R xa z)) \<Longrightarrow> transp R"
+ unfolding transp_def
+ by blast
+
+ML {*
+fun equivp_tac reflps symps transps =
+ simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def})
+ THEN' rtac conjI THEN' rtac allI THEN'
+ resolve_tac reflps THEN'
+ rtac conjI THEN' rtac allI THEN' rtac allI THEN'
+ resolve_tac symps THEN'
+ rtac @{thm transp_aux} THEN' resolve_tac transps
+*}
+
+ML {*
+fun build_equivps alphas term_induct alpha_induct term_inj alpha_inj distinct cases eqvt ctxt =
+let
+ val ([x, y, z], ctxt') = Variable.variant_fixes ["x","y","z"] ctxt;
+ val (reflg, (symg, transg)) = build_alpha_refl_gl alphas (x, y, z)
+ fun reflp_tac' _ = reflp_tac term_induct alpha_inj 1;
+ fun symp_tac' _ = symp_tac alpha_induct alpha_inj eqvt 1;
+ fun transp_tac' _ = transp_tac ctxt alpha_induct alpha_inj term_inj distinct cases eqvt 1;
+ val reflt = Goal.prove ctxt' [] [] reflg reflp_tac';
+ val symt = Goal.prove ctxt' [] [] symg symp_tac';
+ val transt = Goal.prove ctxt' [] [] transg transp_tac';
+ val [refltg, symtg, transtg] = Variable.export ctxt' ctxt [reflt, symt, transt]
+ val reflts = HOLogic.conj_elims refltg
+ val symts = HOLogic.conj_elims symtg
+ val transts = HOLogic.conj_elims transtg
+ fun equivp alpha =
+ let
+ val equivp = Const (@{const_name equivp}, fastype_of alpha --> @{typ bool})
+ val goal = @{term Trueprop} $ (equivp $ alpha)
+ fun tac _ = equivp_tac reflts symts transts 1
+ in
+ Goal.prove ctxt [] [] goal tac
+ end
+in
+ map equivp alphas
+end
+*}
+
+(*
+Tests:
+prove alpha1_reflp_aux: {* fst (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}
+by (tactic {* reflp_tac @{thm rtrm1_bp.induct} @{thms alpha1_inj} 1 *})
+
+prove alpha1_symp_aux: {* (fst o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}
+by (tactic {* symp_tac @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms alpha1_eqvt} 1 *})
+
+prove alpha1_transp_aux: {* (snd o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}
+by (tactic {* transp_tac @{context} @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms rtrm1.inject bp.inject} @{thms rtrm1.distinct bp.distinct} @{thms alpha_rtrm1.cases alpha_bp.cases} @{thms alpha1_eqvt} 1 *})
+
+lemma alpha1_equivp:
+ "equivp alpha_rtrm1"
+ "equivp alpha_bp"
+apply (tactic {*
+ (simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def})
+ THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN'
+ resolve_tac (HOLogic.conj_elims @{thm alpha1_reflp_aux})
+ THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN'
+ resolve_tac (HOLogic.conj_elims @{thm alpha1_symp_aux}) THEN' rtac @{thm transp_aux}
+ THEN' resolve_tac (HOLogic.conj_elims @{thm alpha1_transp_aux})
+)
+1 *})
+done*)
+
+end