--- a/Quot/Nominal/Fv.thy Wed Feb 24 17:32:43 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,399 +0,0 @@
-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