
diff -r e1e2ca92760b -r a8fc346deda3 Nominal/Ex/Typing.thy
--- a/Nominal/Ex/Typing.thy	Wed Jan 05 16:51:27 2011 +0000
+++ b/Nominal/Ex/Typing.thy	Wed Jan 05 17:33:43 2011 +0000
@@ -20,498 +20,6 @@
 thm lam.fv_bn_eqvt
 thm lam.size_eqvt
 
-ML {*
-fun mk_cplus p q = Thm.capply (Thm.capply @{cterm "plus::perm \<Rightarrow> perm \<Rightarrow> perm"} p) q 
-
-fun mk_cminus p = Thm.capply @{cterm "uminus::perm \<Rightarrow> perm"} p 
-
-
-fun minus_permute_intro_tac p = 
-  rtac (Drule.instantiate' [] [SOME (mk_cminus p)] @{thm permute_boolE})
-
-fun minus_permute_elim p thm = 
-  thm RS (Drule.instantiate' [] [NONE, SOME (mk_cminus p)] @{thm permute_boolI})
-*}
-
-ML {*
-fun real_head_of (@{term Trueprop} $ t) = real_head_of t
-  | real_head_of (Const ("==>", _) $ _ $ t) = real_head_of t
-  | real_head_of (Const (@{const_name all}, _) $ Abs (_, _, t)) = real_head_of t
-  | real_head_of (Const (@{const_name All}, _) $ Abs (_, _, t)) = real_head_of t
-  | real_head_of (Const ("HOL.induct_forall", _) $ Abs (_, _, t)) = real_head_of t
-  | real_head_of t = head_of t  
-*}
-
-ML {* 
-fun mk_vc_compat (avoid, avoid_trm) prems concl_args params = 
-  let
-    val vc_goal = concl_args
-      |> HOLogic.mk_tuple
-      |> mk_fresh_star avoid_trm 
-      |> HOLogic.mk_Trueprop
-      |> (curry Logic.list_implies) prems
-      |> (curry list_all_free) params
-    val finite_goal = avoid_trm
-      |> mk_finite
-      |> HOLogic.mk_Trueprop
-      |> (curry Logic.list_implies) prems
-      |> (curry list_all_free) params
-  in 
-    if null avoid then [] else [vc_goal, finite_goal]
-  end
-*}
-
-ML {*
-fun map_term prop f trm =
-  if prop trm 
-  then f trm
-  else case trm of
-    (t1 $ t2) => map_term prop f t1 $ map_term prop f t2
-  | Abs (x, T, t) => Abs (x, T, map_term prop f t)
-  | _ => trm
-*}
-
-ML {*
-fun add_p_c p (c, c_ty) trm =
-  let
-    val (P, args) = strip_comb trm
-    val (P_name, P_ty) = dest_Free P
-    val (ty_args, bool) = strip_type P_ty
-    val args' = map (mk_perm p) args
-  in
-    list_comb (Free (P_name, (c_ty :: ty_args) ---> bool),  c :: args')
-    |> (fn t => HOLogic.all_const c_ty $ lambda c t )
-    |> (fn t => HOLogic.all_const @{typ perm} $  lambda p t)
-  end
-*}
-
-ML {*
-fun induct_forall_const T = Const ("HOL.induct_forall", (T --> @{typ bool}) --> @{typ bool})
-fun mk_induct_forall (a, T) t =  induct_forall_const T $ Abs (a, T, t)
-*}
-
-ML {*
-fun add_c_prop qnt Ps (c, c_name, c_ty) trm =
-  let
-    fun add t = 
-      let
-        val (P, args) = strip_comb t
-        val (P_name, P_ty) = dest_Free P
-        val (ty_args, bool) = strip_type P_ty
-        val args' = args
-          |> qnt ? map (incr_boundvars 1)
-      in
-        list_comb (Free (P_name, (c_ty :: ty_args) ---> bool), c :: args')
-        |> qnt ? mk_induct_forall (c_name, c_ty)
-      end
-  in
-    map_term (member (op =) Ps o head_of) add trm
-  end
-*}
-
-ML {*
-fun prep_prem Ps c_name c_ty (avoid, avoid_trm) (params, prems, concl) =
-  let
-    val prems' = prems
-      |> map (incr_boundvars 1) 
-      |> map (add_c_prop true Ps (Bound 0, c_name, c_ty))
-
-    val avoid_trm' = avoid_trm
-      |> (curry list_abs_free) (params @ [(c_name, c_ty)])
-      |> strip_abs_body
-      |> (fn t => mk_fresh_star_ty c_ty t (Bound 0))
-      |> HOLogic.mk_Trueprop
-
-    val prems'' = 
-      if null avoid 
-      then prems' 
-      else avoid_trm' :: prems'
-
-    val concl' = concl
-      |> incr_boundvars 1 
-      |> add_c_prop false Ps (Bound 0, c_name, c_ty)  
-  in
-    mk_full_horn (params @ [(c_name, c_ty)]) prems'' concl'
-  end
-*}
-
-
-ML {*
-fun same_name (Free (a1, _), Free (a2, _)) = (a1 = a2)
-  | same_name (Var (a1, _), Var (a2, _)) = (a1 = a2)
-  | same_name (Const (a1, _), Const (a2, _)) = (a1 = a2)
-  | same_name _ = false
-*}
-
-ML {*
-fun map7 _ [] [] [] [] [] [] [] = []
-  | map7 f (x :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) (r :: rs) (s :: ss) = 
-      f x y z u v r s :: map7 f xs ys zs us vs rs ss
-*}
-
-ML {*
-(* local abbreviations *)
-fun eqvt_stac ctxt = Nominal_Permeq.eqvt_strict_tac ctxt @{thms permute_minus_cancel} []  
-fun eqvt_srule ctxt = Nominal_Permeq.eqvt_strict_rule ctxt @{thms permute_minus_cancel} []  
-*}
-
-ML {*
-val all_elims = 
-  let
-     fun spec' ct = Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec}
-  in
-    fold (fn ct => fn th => th RS spec' ct)
-  end
-*}
-
-ML {*
-fun helper_tac flag prm p ctxt =
-  Subgoal.SUBPROOF (fn {context, prems, ...} =>
-    let
-      val prems' = prems
-        |> map (minus_permute_elim p)
-        |> map (eqvt_srule context)
-
-      val prm' = (prems' MRS prm)
-        |> flag ? (all_elims [p])
-        |> flag ? (eqvt_srule context)
-
-      val _ = tracing ("prm':" ^ @{make_string} prm')
-    in
-      print_tac "start helper"
-      THEN asm_full_simp_tac (HOL_ss addsimps (prm' :: @{thms induct_forall_def})) 1
-      THEN print_tac "final helper"
-    end) ctxt
-*}
-
-ML {*
-fun non_binder_tac prem intr_cvars Ps ctxt = 
-  Subgoal.SUBPROOF (fn {context, params, prems, ...} =>
-    let
-      val thy = ProofContext.theory_of context
-      val (prms, p, _) = split_last2 (map snd params)
-      val prm_tys = map (fastype_of o term_of) prms
-      val cperms = map (cterm_of thy o perm_const) prm_tys
-      val p_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 p ct2) cperms prms 
-      val prem' = cterm_instantiate (intr_cvars ~~ p_prms) prem
-
-      (* for inductive-premises*)
-      fun tac1 prm = helper_tac true prm p context 
-
-      (* for non-inductive premises *)   
-      fun tac2 prm =  
-        EVERY' [ minus_permute_intro_tac p, 
-                 eqvt_stac context, 
-                 helper_tac false prm p context ]
-
-      fun select prm (t, i) =
-        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
-    in
-      EVERY1 [eqvt_stac ctxt, rtac prem', RANGE (map (SUBGOAL o select) prems) ]
-    end) ctxt
-*}
-
-
-ML {*
-fun fresh_thm ctxt user_thm p c concl_args avoid_trm =
-  let
-    val conj1 = 
-      mk_fresh_star (mk_perm (Bound 0) (mk_perm p avoid_trm)) c
-    val conj2 =
-      mk_fresh_star_ty @{typ perm} (mk_supp (HOLogic.mk_tuple (map (mk_perm p) concl_args))) (Bound 0)
-    val fresh_goal = mk_exists ("q", @{typ perm}) (HOLogic.mk_conj (conj1, conj2))
-      |> HOLogic.mk_Trueprop
-
-    val ss = @{thms finite_supp supp_Pair finite_Un permute_finite} @ 
-             @{thms fresh_star_Pair fresh_star_permute_iff}
-    val simp = asm_full_simp_tac (HOL_ss addsimps ss)
-  in 
-    Goal.prove ctxt [] [] fresh_goal
-      (K (HEADGOAL (rtac @{thm at_set_avoiding2} 
-          THEN_ALL_NEW EVERY' [cut_facts_tac user_thm, REPEAT o etac @{thm conjE}, simp])))
-  end
-*}
-
-ML {* 
-val supp_perm_eq' = 
-  @{lemma "supp (p \<bullet> x) \<sharp>* q ==> p \<bullet> x == (q + p) \<bullet> x" by (simp add: supp_perm_eq)}
-val fresh_star_plus =
-  @{lemma "(q \<bullet> (p \<bullet> x)) \<sharp>* c ==> ((q + p) \<bullet> x) \<sharp>* c" by (simp add: permute_plus)}
-*}
-
-ML {*
-fun binder_tac prem intr_cvars param_trms Ps user_thm avoid avoid_trm concl_args ctxt = 
-  Subgoal.FOCUS (fn {context = ctxt, params, prems, concl, ...} =>
-    let
-      val thy = ProofContext.theory_of ctxt
-      val (prms, p, c) = split_last2 (map snd params)
-      val prm_trms = map term_of prms
-      val prm_tys = map fastype_of prm_trms
-
-      val avoid_trm' = subst_free (param_trms ~~ prm_trms) avoid_trm 
-      val concl_args' = map (subst_free (param_trms ~~ prm_trms)) concl_args 
-      
-      val user_thm' = map (cterm_instantiate (intr_cvars ~~ prms)) user_thm
-        |> map (full_simplify (HOL_ss addsimps (@{thm fresh_star_Pair}::prems)))
-      
-      val fthm = fresh_thm ctxt user_thm' (term_of p) (term_of c) concl_args' avoid_trm'
-
-      val (([(_, q)], fprop :: fresh_eqs), ctxt') = Obtain.result
-              (K (EVERY1 [etac @{thm exE}, 
-                          full_simp_tac (HOL_basic_ss addsimps @{thms supp_Pair fresh_star_Un}), 
-                          REPEAT o etac @{thm conjE},
-                          dtac fresh_star_plus,
-                          REPEAT o dtac supp_perm_eq'])) [fthm] ctxt 
-
-      val expand_conv = Conv.try_conv (Conv.rewrs_conv fresh_eqs)
-      fun expand_conv_bot ctxt = Conv.bottom_conv (K expand_conv) ctxt
-
-      val cperms = map (cterm_of thy o perm_const) prm_tys
-      val qp_prms = map2 (fn ct1 => fn ct2 => Thm.mk_binop ct1 (mk_cplus q p) ct2) cperms prms 
-      val prem' = cterm_instantiate (intr_cvars ~~ qp_prms) prem
-
-      val fprop' = eqvt_srule ctxt' fprop 
-      val tac_fresh = simp_tac (HOL_basic_ss addsimps [fprop'])
-
-      (* for inductive-premises*)
-      fun tac1 prm = helper_tac true prm (mk_cplus q p) ctxt' 
-
-      (* for non-inductive premises *)   
-      fun tac2 prm =  
-        EVERY' [ minus_permute_intro_tac (mk_cplus q p), 
-                 eqvt_stac ctxt, 
-                 helper_tac false prm (mk_cplus q p) ctxt' ]
-
-      fun select prm (t, i) =
-        (if member same_name Ps (real_head_of t) then tac1 prm else tac2 prm) i
-
-      val _ = tracing ("fthm:\n" ^ @{make_string} fthm)
-      val _ = tracing ("fr_eqs:\n" ^ cat_lines (map @{make_string} fresh_eqs))
-      val _ = tracing ("fprop:\n" ^ @{make_string} fprop)
-      val _ = tracing ("fprop':\n" ^ @{make_string} fprop')
-      val _ = tracing ("fperm:\n" ^ @{make_string} q)
-      val _ = tracing ("prem':\n" ^ @{make_string} prem')
-
-      val side_thm = Goal.prove ctxt' [] [] (term_of concl)
-        (fn {context, ...} => 
-           EVERY1 [ CONVERSION (expand_conv_bot context),
-                    eqvt_stac context,
-                    rtac prem',
-                    RANGE (tac_fresh :: map (SUBGOAL o select) prems),
-                    K (print_tac "GOAL") ])
-        |> singleton (ProofContext.export ctxt' ctxt)        
-    in
-      rtac side_thm 1
-    end) ctxt
-*}
-
-ML {*
-fun case_tac ctxt Ps avoid avoid_trm intr_cvars param_trms prem user_thm concl_args =
-  let
-    val tac1 = non_binder_tac prem intr_cvars Ps ctxt
-    val tac2 = binder_tac prem intr_cvars param_trms Ps user_thm avoid avoid_trm concl_args ctxt
-  in 
-    EVERY' [ rtac @{thm allI}, rtac @{thm allI}, if null avoid then tac1 else tac2 ]
-  end
-*}
-
-ML {*
-fun prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms concl_args 
-  {prems, context} =
-  let
-    val cases_tac = 
-      map7 (case_tac context Ps) avoids avoid_trms intr_cvars param_trms prems user_thms concl_args
-  in 
-    EVERY1 [ DETERM o rtac raw_induct, RANGE cases_tac ]
-  end
-*}
-
-ML {*
-val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (!!c. Q ==> P (0::perm) c)" by simp}
-*}
-
-ML {* Local_Theory.note *}
-
-ML {*
-fun prove_strong_inductive pred_names rule_names avoids raw_induct intrs ctxt =
-  let
-    val thy = ProofContext.theory_of ctxt
-    val ((_, [raw_induct']), ctxt') = Variable.import true [raw_induct] ctxt
-
-    val (ind_prems, ind_concl) = raw_induct'
-      |> prop_of
-      |> Logic.strip_horn
-      |>> map strip_full_horn
-    val params = map (fn (x, _, _) => x) ind_prems
-    val param_trms = (map o map) Free params  
-
-    val intr_vars_tys = map (fn t => rev (Term.add_vars (prop_of t) [])) intrs
-    val intr_vars = (map o map) fst intr_vars_tys
-    val intr_vars_substs = map2 (curry (op ~~)) intr_vars param_trms
-    val intr_cvars = (map o map) (cterm_of thy o Var) intr_vars_tys      
-
-    val (intr_prems, intr_concls) = intrs
-      |> map prop_of
-      |> map2 subst_Vars intr_vars_substs
-      |> map Logic.strip_horn
-      |> split_list
-
-    val intr_concls_args = map (snd o strip_comb o HOLogic.dest_Trueprop) intr_concls 
-      
-    val avoid_trms = avoids
-      |> (map o map) (setify ctxt') 
-      |> map fold_union
-
-    val vc_compat_goals = 
-      map4 mk_vc_compat (avoids ~~ avoid_trms) intr_prems intr_concls_args params
-
-    val ([c_name, a, p], ctxt'') = Variable.variant_fixes ["c", "'a", "p"] ctxt'
-    val c_ty = TFree (a, @{sort fs})
-    val c = Free (c_name, c_ty)
-    val p = Free (p, @{typ perm})
-
-    val (preconds, ind_concls) = ind_concl
-      |> HOLogic.dest_Trueprop
-      |> HOLogic.dest_conj 
-      |> map HOLogic.dest_imp
-      |> split_list
-
-    val Ps = map (fst o strip_comb) ind_concls
-
-    val ind_concl' = ind_concls
-      |> map (add_p_c p (c, c_ty))
-      |> (curry (op ~~)) preconds  
-      |> map HOLogic.mk_imp
-      |> fold_conj
-      |> HOLogic.mk_Trueprop
-
-    val ind_prems' = ind_prems
-      |> map2 (prep_prem Ps c_name c_ty) (avoids ~~ avoid_trms)   
-
-    fun after_qed ctxt_outside user_thms ctxt = 
-      let
-        val strong_ind_thms = Goal.prove ctxt [] ind_prems' ind_concl' 
-        (prove_sinduct_tac raw_induct user_thms Ps avoids avoid_trms intr_cvars param_trms intr_concls_args) 
-          |> singleton (ProofContext.export ctxt ctxt_outside)
-          |> Datatype_Aux.split_conj_thm
-          |> map (fn thm => thm RS normalise)
-          |> map (asm_full_simplify (HOL_basic_ss addsimps @{thms permute_zero induct_rulify})) 
-          |> map (Drule.rotate_prems (length ind_prems'))
-          |> map zero_var_indexes
-
-        val qualified_thm_name = pred_names
-          |> map Long_Name.base_name
-          |> space_implode "_"
-          |> (fn s => Binding.qualify false s (Binding.name "strong_induct"))
-
-        val attrs = 
-          [ Attrib.internal (K (Rule_Cases.consumes 1)),
-            Attrib.internal (K (Rule_Cases.case_names rule_names)) ]
-        val _ = tracing ("RESULTS\n" ^ cat_lines (map (Syntax.string_of_term ctxt o prop_of) strong_ind_thms))
-        val _ = tracing ("rule_names: " ^ commas rule_names)
-        val _ = tracing ("pred_names: " ^ commas pred_names)
-      in
-        ctxt
-        |> Local_Theory.note ((qualified_thm_name, attrs), strong_ind_thms)    
-        |> snd   
-      end
-  in
-    Proof.theorem NONE (after_qed ctxt) ((map o map) (rpair []) vc_compat_goals) ctxt''
-  end
-*}
-
-ML {*
-fun prove_strong_inductive_cmd (pred_name, avoids) ctxt =
-  let
-    val thy = ProofContext.theory_of ctxt;
-    val ({names, ...}, {raw_induct, intrs, ...}) =
-      Inductive.the_inductive ctxt (Sign.intern_const thy pred_name);
-
-    val rule_names = 
-      hd names
-      |> the o Induct.lookup_inductP ctxt
-      |> fst o Rule_Cases.get
-      |> map fst
-
-    val _ = (case duplicates (op = o pairself fst) avoids of
-        [] => ()
-      | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)))
-
-    val _ = (case subtract (op =) rule_names (map fst avoids) of
-        [] => ()
-      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs))
-
-    val avoids_ordered = order_default (op =) [] rule_names avoids
-      
-    fun read_avoids avoid_trms intr =
-      let
-        (* fixme hack *)
-        val (((_, ctrms), _), ctxt') = Variable.import true [intr] ctxt
-        val trms = map (term_of o snd) ctrms
-        val ctxt'' = fold Variable.declare_term trms ctxt' 
-      in
-        map (Syntax.read_term ctxt'') avoid_trms 
-      end 
-
-    val avoid_trms = map2 read_avoids avoids_ordered intrs
-  in
-    prove_strong_inductive names rule_names avoid_trms raw_induct intrs ctxt
-  end
-*}
-
-ML {*
-(* outer syntax *)
-local
-  structure P = Parse;
-  structure S = Scan
-  
-  val _ = Keyword.keyword "avoids"
-
-  val single_avoid_parser = 
-    P.name -- (P.$$$ ":" |-- P.and_list1 P.term)
-
-  val avoids_parser = 
-    S.optional (P.$$$ "avoids" |-- P.enum1 "|" single_avoid_parser) []
-
-  val main_parser = P.xname -- avoids_parser
-in
-  val _ =
-  Outer_Syntax.local_theory_to_proof "nominal_inductive"
-    "prove strong induction theorem for inductive predicate involving nominal datatypes"
-      Keyword.thy_goal (main_parser >> prove_strong_inductive_cmd)
-end
-*}
-
-inductive
-  Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
-where
-  AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"
-
-(*
-equivariance Acc
-*)
-
-lemma Acc_eqvt [eqvt]:
-  fixes p::"perm"
-  assumes a: "Acc R x"
-  shows "Acc (p \<bullet> R) (p \<bullet> x)"
-using a
-apply(induct)
-apply(rule AccI)
-apply(rotate_tac 1)
-apply(drule_tac x="-p \<bullet> y" in meta_spec)
-apply(simp)
-apply(drule meta_mp)
-apply(rule_tac p="p" in permute_boolE)
-apply(perm_simp add: permute_minus_cancel)
-apply(assumption)
-apply(assumption)
-done
- 
-
-nominal_inductive Acc .
-
-thm Acc.strong_induct
 
 section {* Typing *}
 
@@ -527,8 +35,6 @@
 apply (simp_all add: ty.fresh pure_fresh)
 done
 
-
-
 inductive
   valid :: "(name \<times> ty) list \<Rightarrow> bool"
 where
@@ -545,16 +51,13 @@
 thm typing.intros
 thm typing.induct
 
-
-
 equivariance valid
 equivariance typing
 
 nominal_inductive typing
   avoids t_Lam: "x"
-  apply -
-  apply(simp_all add: fresh_star_def ty_fresh lam.fresh)?
-  done
+  by (simp_all add: fresh_star_def ty_fresh lam.fresh)
+
 
 thm typing.strong_induct
 
@@ -611,6 +114,34 @@
 done
 
 
+inductive
+  Acc :: "('a::pt \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+where
+  AccI: "(\<And>y. R y x \<Longrightarrow> Acc R y) \<Longrightarrow> Acc R x"
+
+
+lemma Acc_eqvt [eqvt]:
+  fixes p::"perm"
+  assumes a: "Acc R x"
+  shows "Acc (p \<bullet> R) (p \<bullet> x)"
+using a
+apply(induct)
+apply(rule AccI)
+apply(rotate_tac 1)
+apply(drule_tac x="-p \<bullet> y" in meta_spec)
+apply(simp)
+apply(drule meta_mp)
+apply(rule_tac p="p" in permute_boolE)
+apply(perm_simp add: permute_minus_cancel)
+apply(assumption)
+apply(assumption)
+done
+ 
+
+nominal_inductive Acc .
+
+thm Acc.strong_induct
+
 
 end
 




