diff -r e1e2ca92760b -r a8fc346deda3 Nominal/nominal_inductive.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/nominal_inductive.ML	Wed Jan 05 17:33:43 2011 +0000
@@ -0,0 +1,437 @@
+(*  Title:      nominal_inductive.ML
+    Author:     Christian Urban
+
+    Infrastructure for proving strong induction theorems
+    for inductive predicates involving nominal datatypes.
+
+    Code based on an earlier version by Stefan Berghofer.
+*)
+
+
+signature NOMINAL_INDUCTIVE =
+sig
+  val prove_strong_inductive: string list -> string list -> term list list -> thm -> thm list -> 
+    Proof.context -> Proof.state
+
+  val prove_strong_inductive_cmd: xstring * (string * string list) list -> Proof.context -> Proof.state
+end
+
+structure Nominal_Inductive : NOMINAL_INDUCTIVE =
+struct
+
+
+fun mk_cplus p q = Thm.capply (Thm.capply @{cterm "plus :: perm => perm => perm"} p) q 
+
+fun mk_cminus p = Thm.capply @{cterm "uminus :: perm => 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})
+
+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  
+
+
+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
+
+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
+
+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
+
+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)
+
+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
+
+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
+
+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
+
+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
+
+(* 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} []  
+
+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
+
+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
+
+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
+
+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
+
+val supp_perm_eq' = @{lemma "fresh_star (supp (permute p x)) q ==> permute p x == permute (q + p) x" 
+  by (simp add: supp_perm_eq)}
+val fresh_star_plus = @{lemma "fresh_star (permute q (permute p x)) c ==> fresh_star (permute (q + p) x) c" 
+  by (simp add: permute_plus)}
+
+
+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
+
+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
+
+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
+
+val normalise = @{lemma "(Q --> (!p c. P p c)) ==> (!!c. Q ==> P (0::perm) c)" by simp}
+
+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
+
+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
+
+(* 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
+
+end
\ No newline at end of file