src/HOL/Nominal/nominal_permeq.ML

modified the perm_compose rule such that it
is applied as simplification rule (as simproc)
in the restricted case where the first
permutation is a swapping coming from a supports
problem

also deleted the perm_compose' rule from the set
of rules that are automatically tried

(* $Id$ *)

(* METHOD FOR ANALYSING EQUATION INVOLVING PERMUTATION *)

local

(* pulls out dynamically a thm via the simpset *)
fun dynamic_thms ss name = 
    ProofContext.get_thms (Simplifier.the_context ss) (Name name);
fun dynamic_thm ss name = 
    ProofContext.get_thm (Simplifier.the_context ss) (Name name);

(* initial simplification step in the tactic *)
fun perm_eval_tac ss i =
    let
        fun perm_eval_simproc sg ss redex =
        let 
	   
           (* the "application" case below is only applicable when the head   *)
           (* of f is not a constant  or when it is a permuattion with two or *) 
           (* more arguments                                                  *)
           fun applicable t = 
	       (case (strip_comb t) of
		  (Const ("nominal.perm",_),ts) => (length ts) >= 2
		| (Const _,_) => false
		| _ => true)

	in
        (case redex of 
        (* case pi o (f x) == (pi o f) (pi o x)          *)
        (* special treatment according to the head of f  *)
        (Const("nominal.perm",
          Type("fun",[Type("List.list",[Type("*",[Type(n,_),_])]),_])) $ pi $ (f $ x)) => 
	   (case (applicable f) of
                false => NONE  
              | _ => 
		let
		    val name = Sign.base_name n
		    val at_inst     = dynamic_thm ss ("at_"^name^"_inst")
		    val pt_inst     = dynamic_thm ss ("pt_"^name^"_inst")  
		    val perm_eq_app = thm "nominal.pt_fun_app_eq"	  
		in SOME ((at_inst RS (pt_inst RS perm_eq_app)) RS eq_reflection) end)

        (* case pi o (%x. f x) == (%x. pi o (f ((rev pi)o x))) *)
        | (Const("nominal.perm",_) $ pi $ (Abs _)) => 
           let 
               val perm_fun_def = thm "nominal.perm_fun_def"
           in SOME (perm_fun_def) end

        (* no redex otherwise *) 
        | _ => NONE) end

	val perm_eval =
	    Simplifier.simproc (Theory.sign_of (the_context ())) "perm_eval" 
	    ["nominal.perm pi x"] perm_eval_simproc;

      (* applies the pt_perm_compose lemma              *)
      (*                                                *)
      (*     pi1 o (pi2 o x) = (pi1 o pi2) o (pi1 o x)  *)
      (*                                                *)
      (* in the restricted case where pi1 is a swapping *)
      (* (a b) coming from a "supports problem"; in     *)
      (* this rule would cause loops in the simplifier  *) 
      val pt_perm_compose = thm "pt_perm_compose";
	  
      fun perm_compose_simproc i sg ss redex =
      (case redex of
        (Const ("nominal.perm", _) $ (pi1 as Const ("List.list.Cons", _) $
         (Const ("Pair", _) $ Free (a as (_, T as Type (tname, _))) $ Free b) $ 
           Const ("List.list.Nil", _)) $ (Const ("nominal.perm", 
            Type ("fun", [Type ("List.list", [Type ("*", [U, _])]), _])) $ pi2 $ t)) =>
        let
            val ({bounds = (_, xs), ...}, _) = rep_ss ss
            val ai = find_index (fn (x, _) => x = a) xs
	    val bi = find_index (fn (x, _) => x = b) xs
	    val tname' = Sign.base_name tname
        in
            if ai = length xs - i - 1 andalso 
               bi = length xs - i - 2 andalso 
               T = U andalso pi1 <> pi2 then
                SOME (Drule.instantiate'
	              [SOME (ctyp_of sg (fastype_of t))]
		      [SOME (cterm_of sg pi1), SOME (cterm_of sg pi2), SOME (cterm_of sg t)]
		      (mk_meta_eq ([PureThy.get_thm sg (Name ("pt_"^tname'^"_inst")),
	               PureThy.get_thm sg (Name ("at_"^tname'^"_inst"))] MRS pt_perm_compose)))
            else NONE
        end
       | _ => NONE);

      fun perm_compose i =
	Simplifier.simproc (the_context()) "perm_compose" 
	["nominal.perm [(a, b)] (nominal.perm pi t)"] (perm_compose_simproc i);
         
      (* these lemmas are created dynamically according to the atom types *) 
      val perm_swap     = dynamic_thms ss "perm_swap"
      val perm_fresh    = dynamic_thms ss "perm_fresh_fresh"
      val perm_bij      = dynamic_thms ss "perm_bij"
      val perm_pi_simp  = dynamic_thms ss "perm_pi_simp"
      val ss' = ss addsimps (perm_swap@perm_fresh@perm_bij@perm_pi_simp)
    in
      ("general simplification step", 
        FIRST [rtac conjI i, 
               SUBGOAL (fn (g, i) => asm_full_simp_tac 
                 (ss' addsimprocs [perm_eval,perm_compose (length (Logic.strip_params g)-2)]) i) i])
    end;

(* applies the perm_compose rule - this rule would loop in the simplifier     *)
(* in case there are more atom-types we have to check every possible instance *)
(* of perm_compose                                                            *)
fun apply_perm_compose_tac ss i = 
    let
	val perm_compose = dynamic_thms ss "perm_compose"; 
        val tacs = map (fn thm => (rtac (thm RS trans) i)) perm_compose
    in
	("analysing permutation compositions on the lhs",FIRST (tacs))
    end

(* applying Stefan's smart congruence tac *)
fun apply_cong_tac i = 
    ("application of congruence",
     (fn st => DatatypeAux.cong_tac  i st handle Subscript => no_tac st));

(* unfolds the definition of permutations applied to functions *)
fun unfold_perm_fun_def_tac i = 
    let
	val perm_fun_def = thm "nominal.perm_fun_def"
    in
	("unfolding of permutations on functions", 
	 simp_tac (HOL_basic_ss addsimps [perm_fun_def]) i)
    end

(* applies the expand_fun_eq rule to the first goal and strips off all universal quantifiers *)
fun expand_fun_eq_tac i =    
    ("extensionality expansion of functions",
    EVERY [simp_tac (HOL_basic_ss addsimps [expand_fun_eq]) i, REPEAT_DETERM (rtac allI i)]);

(* debugging *)
fun DEBUG_tac (msg,tac) = 
    CHANGED (EVERY [tac, print_tac ("after "^msg)]); 
fun NO_DEBUG_tac (_,tac) = CHANGED tac; 

(* Main Tactic *)

fun perm_simp_tac tactical ss i = 
    DETERM (tactical (perm_eval_tac ss i));


(* perm_simp_tac plus additional tactics to decide            *)
(* support problems                                           *)
(* the "recursion"-depth is set to 10 - this seems sufficient *)
fun perm_supports_tac tactical ss n = 
    if n=0 then K all_tac
    else DETERM o 
         (FIRST'[fn i => tactical (perm_eval_tac ss i),
                 (*fn i => tactical (apply_perm_compose_tac ss i),*)
		 fn i => tactical (apply_cong_tac i), 
                 fn i => tactical (unfold_perm_fun_def_tac i),
		 fn i => tactical (expand_fun_eq_tac i)]
         THEN_ALL_NEW (TRY o (perm_supports_tac tactical ss (n-1))));

(* tactic that first unfolds the support definition          *)
(* and strips off the intros, then applies perm_supports_tac *)
fun supports_tac tactical ss i =
  let 
      val supports_def = thm "nominal.op supports_def";
      val fresh_def    = thm "nominal.fresh_def";
      val fresh_prod   = thm "nominal.fresh_prod";
      val simps        = [supports_def,symmetric fresh_def,fresh_prod]
  in
      EVERY [tactical ("unfolding of supports ", simp_tac (HOL_basic_ss addsimps simps) i),
             tactical ("stripping of foralls  ", REPEAT_DETERM (rtac allI i)),
             tactical ("geting rid of the imps", rtac impI i),
             tactical ("eliminating conjuncts ", REPEAT_DETERM (etac  conjE i)),
             tactical ("applying perm_simp    ", perm_supports_tac tactical ss 10 i)]
  end;


(* tactic that guesses the finite-support of a goal *)

fun collect_vars i (Bound j) vs = if j < i then vs else Bound (j - i) ins vs
  | collect_vars i (v as Free _) vs = v ins vs
  | collect_vars i (v as Var _) vs = v ins vs
  | collect_vars i (Const _) vs = vs
  | collect_vars i (Abs (_, _, t)) vs = collect_vars (i+1) t vs
  | collect_vars i (t $ u) vs = collect_vars i u (collect_vars i t vs);

val supports_rule = thm "supports_finite";

fun finite_guess_tac tactical ss i st =
    let val goal = List.nth(cprems_of st, i-1)
    in
      case Logic.strip_assums_concl (term_of goal) of
          _ $ (Const ("op :", _) $ (Const ("nominal.supp", T) $ x) $
            Const ("Finite_Set.Finites", _)) =>
          let
            val cert = Thm.cterm_of (sign_of_thm st);
            val ps = Logic.strip_params (term_of goal);
            val Ts = rev (map snd ps);
            val vs = collect_vars 0 x [];
            val s = foldr (fn (v, s) =>
                HOLogic.pair_const (fastype_of1 (Ts, v)) (fastype_of1 (Ts, s)) $ v $ s)
              HOLogic.unit vs;
            val s' = list_abs (ps,
              Const ("nominal.supp", fastype_of1 (Ts, s) --> body_type T) $ s);
            val supports_rule' = Thm.lift_rule goal supports_rule;
            val _ $ (_ $ S $ _) =
              Logic.strip_assums_concl (hd (prems_of supports_rule'));
            val supports_rule'' = Drule.cterm_instantiate
              [(cert (head_of S), cert s')] supports_rule'
          in
            (tactical ("guessing of the right supports-set",
                      EVERY [compose_tac (false, supports_rule'', 2) i,
                             asm_full_simp_tac ss (i+1),
                             supports_tac tactical ss i])) st
          end
        | _ => Seq.empty
    end
    handle Subscript => Seq.empty

in             


fun simp_meth_setup tac =
  Method.only_sectioned_args (Simplifier.simp_modifiers' @ Splitter.split_modifiers)
  (Method.SIMPLE_METHOD' HEADGOAL o tac o local_simpset_of);

val perm_eq_meth         = simp_meth_setup (perm_simp_tac NO_DEBUG_tac);
val perm_eq_meth_debug   = simp_meth_setup (perm_simp_tac DEBUG_tac);
val supports_meth        = simp_meth_setup (supports_tac NO_DEBUG_tac);
val supports_meth_debug  = simp_meth_setup (supports_tac DEBUG_tac);
val finite_gs_meth       = simp_meth_setup (finite_guess_tac NO_DEBUG_tac);
val finite_gs_meth_debug = simp_meth_setup (finite_guess_tac DEBUG_tac);

end