
diff -r 4b0563bc4b03 -r 7d8949da7d99 Quot/Nominal/Fv.thy
--- 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




