src/HOL/Statespace/distinct_tree_prover.ML

prefix for equations in primrec specifications

(*  Title:      distinct_tree_prover.thy
    ID:         $Id$
    Author:     Norbert Schirmer, TU Muenchen
*)

signature DISTINCT_TREE_PROVER =
sig
  datatype Direction = Left | Right
  val mk_tree : ('a -> Term.term) -> Term.typ -> 'a list -> Term.term
  val dest_tree : Term.term -> Term.term list
  val find_tree :
       Term.term -> Term.term -> Direction list option 

  val neq_to_eq_False : Thm.thm
  val distinctTreeProver : Thm.thm -> Direction list -> Direction list -> Thm.thm
  val neq_x_y :
       Proof.context -> Term.term -> Term.term -> string -> Thm.thm option   
  val distinctFieldSolver : string list -> MetaSimplifier.solver
  val distinctTree_tac :
       string list -> Proof.context -> Term.term * int -> Tactical.tactic
  val distinct_implProver : Thm.thm -> Thm.cterm -> Thm.thm
  val subtractProver : Term.term -> Thm.cterm -> Thm.thm -> Thm.thm
  val distinct_simproc : string list -> MetaSimplifier.simproc
  
  val discharge : Thm.thm list -> Thm.thm -> Thm.thm
end;

structure DistinctTreeProver : DISTINCT_TREE_PROVER =
struct
val all_distinct_left = thm "DistinctTreeProver.all_distinct_left";
val all_distinct_right = thm "DistinctTreeProver.all_distinct_right";

val distinct_left = thm "DistinctTreeProver.distinct_left";
val distinct_right = thm "DistinctTreeProver.distinct_right";
val distinct_left_right = thm "DistinctTreeProver.distinct_left_right";
val in_set_root = thm "DistinctTreeProver.in_set_root";
val in_set_left = thm "DistinctTreeProver.in_set_left";
val in_set_right = thm "DistinctTreeProver.in_set_right";

val swap_neq = thm "DistinctTreeProver.swap_neq";
val neq_to_eq_False = thm "DistinctTreeProver.neq_to_eq_False"

datatype Direction = Left | Right 

fun treeT T = Type ("DistinctTreeProver.tree",[T]);
fun mk_tree' e T n []     = Const ("DistinctTreeProver.tree.Tip",treeT T)
  | mk_tree' e T n xs =
     let
       val m = (n - 1) div 2;
       val (xsl,x::xsr) = chop m xs;
       val l = mk_tree' e T m xsl;
       val r = mk_tree' e T (n-(m+1)) xsr;
     in Const ("DistinctTreeProver.tree.Node",
                treeT T --> T --> HOLogic.boolT--> treeT T --> treeT T) $ 
          l$ e x $ HOLogic.false_const $ r 
     end

fun mk_tree e T xs = mk_tree' e T (length xs) xs;         

fun dest_tree (Const ("DistinctTreeProver.tree.Tip",_)) = []
  | dest_tree (Const ("DistinctTreeProver.tree.Node",_)$l$e$_$r) = dest_tree l @ e :: dest_tree r
  | dest_tree t = raise TERM ("DistinctTreeProver.dest_tree",[t]);



fun lin_find_tree e (Const ("DistinctTreeProver.tree.Tip",_)) = NONE
  | lin_find_tree e (Const ("DistinctTreeProver.tree.Node",_) $ l$ x $ _ $ r) =
      if e aconv x 
      then SOME []
      else (case lin_find_tree e l of
              SOME path => SOME (Left::path)
            | NONE => (case lin_find_tree e r of
                        SOME path => SOME (Right::path)
                       | NONE => NONE))
  | lin_find_tree e t = raise TERM ("find_tree: input not a tree",[t])

fun bin_find_tree order e (Const ("DistinctTreeProver.tree.Tip",_)) = NONE
  | bin_find_tree order e (Const ("DistinctTreeProver.tree.Node",_) $ l$ x $ _ $ r) =
      (case order (e,x) of
         EQUAL => SOME []
       | LESS => Option.map (cons Left) (bin_find_tree order e l)
       | GREATER => Option.map (cons Right) (bin_find_tree order e r))
  | bin_find_tree order e t = raise TERM ("find_tree: input not a tree",[t])

fun find_tree e t =
  (case bin_find_tree Term.fast_term_ord e t of
     NONE => lin_find_tree e t
   | x => x);

 
fun index_tree (Const ("DistinctTreeProver.tree.Tip",_)) path tab = tab
  | index_tree (Const ("DistinctTreeProver.tree.Node",_) $ l$ x $ _ $ r) path tab =
      tab 
      |> Termtab.update_new (x,path) 
      |> index_tree l (path@[Left])
      |> index_tree r (path@[Right])
  | index_tree t _ _ = raise TERM ("index_tree: input not a tree",[t])

fun split_common_prefix xs [] = ([],xs,[])
  | split_common_prefix [] ys = ([],[],ys)
  | split_common_prefix (xs as (x::xs')) (ys as (y::ys')) =
     if x=y 
     then let val (ps,xs'',ys'') = split_common_prefix xs' ys' in (x::ps,xs'',ys'') end
     else ([],xs,ys)


(* Wrapper around Thm.instantiate. The type instiations of instTs are applied to
 * the right hand sides of insts
 *)
fun instantiate instTs insts =
  let
    val instTs' = map (fn (T,U) => (dest_TVar (typ_of T),typ_of U)) instTs;
    fun substT x = (case AList.lookup (op =) instTs' x of NONE => TVar x | SOME T' => T');
    fun mapT_and_recertify ct =
      let
        val thy = theory_of_cterm ct;
      in (cterm_of thy (Term.map_types (Term.map_type_tvar substT) (term_of ct))) end;
    val insts' = map (apfst mapT_and_recertify) insts;
  in Thm.instantiate (instTs,insts') end;

fun tvar_clash ixn S S' = raise TYPE ("Type variable " ^
  quote (Term.string_of_vname ixn) ^ " has two distinct sorts",
  [TVar (ixn, S), TVar (ixn, S')], []);

fun lookup (tye, (ixn, S)) =
  (case AList.lookup (op =) tye ixn of
    NONE => NONE
  | SOME (S', T) => if S = S' then SOME T else tvar_clash ixn S S');

val naive_typ_match =
  let
    fun match (TVar (v, S), T) subs =
          (case lookup (subs, (v, S)) of
            NONE => ((v, (S, T))::subs)
          | SOME _ => subs)
      | match (Type (a, Ts), Type (b, Us)) subs =
          if a <> b then raise Type.TYPE_MATCH
          else matches (Ts, Us) subs
      | match (TFree x, TFree y) subs =
          if x = y then subs else raise Type.TYPE_MATCH
      | match _ _ = raise Type.TYPE_MATCH
    and matches (T :: Ts, U :: Us) subs = matches (Ts, Us) (match (T, U) subs)
      | matches _ subs = subs;
  in match end;


(* expects that relevant type variables are already contained in 
 * term variables. First instantiation of variables is returned without further
 * checking.
 *)
fun naive_cterm_first_order_match (t,ct) env =
  let
    val thy = (theory_of_cterm ct);
    fun mtch (env as (tyinsts,insts)) = fn
         (Var(ixn,T),ct) =>
           (case AList.lookup (op =) insts ixn of
             NONE => (naive_typ_match (T,typ_of (ctyp_of_term ct)) tyinsts,
                      (ixn, ct)::insts)
            | SOME _ => env)
        | (f$t,ct) => let val (cf,ct') = Thm.dest_comb ct;
                      in mtch (mtch env (f,cf)) (t,ct') end
        | _ => env
  in mtch env (t,ct) end;


fun mp prem rule = implies_elim rule prem;

fun discharge prems rule =
  let
     val thy = theory_of_thm (hd prems);
     val (tyinsts,insts) =  
           fold naive_cterm_first_order_match (prems_of rule ~~ map cprop_of prems) ([],[]);

     val tyinsts' = map (fn (v,(S,U)) => (ctyp_of thy (TVar (v,S)),ctyp_of thy U)) 
                     tyinsts;
     val insts' = map (fn (idxn,ct) => (cterm_of thy (Var (idxn,typ_of (ctyp_of_term ct))),ct))  
                     insts;
     val rule' = Thm.instantiate (tyinsts',insts') rule;
   in fold mp prems rule' end;

local

val (l_in_set_root,x_in_set_root,r_in_set_root) =
  let val (Node_l_x_d,r) = (cprop_of in_set_root) 
                         |> Thm.dest_comb |> #2 
                         |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2 |> Thm.dest_comb;
      val (Node_l,x) = Node_l_x_d |> Thm.dest_comb |> #1 |> Thm.dest_comb;
      val l = Node_l |> Thm.dest_comb |> #2;
  in (l,x,r) end
val (x_in_set_left,r_in_set_left) = 
  let val (Node_l_x_d,r) = (cprop_of in_set_left) 
                         |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2
                         |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2 |> Thm.dest_comb;
      val x = Node_l_x_d |> Thm.dest_comb |> #1 |> Thm.dest_comb |> #2;
  in (x,r) end

val (x_in_set_right,l_in_set_right) = 
  let val (Node_l,x) = (cprop_of in_set_right) 
                         |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2
                         |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2 
                         |> Thm.dest_comb |> #1 |> Thm.dest_comb |> #1 
                         |> Thm.dest_comb
      val l = Node_l |> Thm.dest_comb |> #2;
  in (x,l) end

in
(*
1. First get paths x_path y_path of x and y in the tree.
2. For the common prefix descend into the tree according to the path
   and lemmas all_distinct_left/right
3. If one restpath is empty use distinct_left/right,
   otherwise all_distinct_left_right
*)

fun distinctTreeProver dist_thm x_path y_path =
  let
    fun dist_subtree [] thm = thm
      | dist_subtree (p::ps) thm =
         let 
           val rule = (case p of Left => all_distinct_left | Right => all_distinct_right)
         in dist_subtree ps (discharge [thm] rule) end;

    val (ps,x_rest,y_rest) = split_common_prefix x_path y_path;
    val dist_subtree_thm = dist_subtree ps dist_thm;
    val subtree = cprop_of dist_subtree_thm |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2;
    val (_,[l,_,_,r]) = Drule.strip_comb subtree;
    
    fun in_set ps tree =
      let
        val (_,[l,x,_,r]) = Drule.strip_comb tree;
        val xT = ctyp_of_term x;
      in (case ps of
            [] => instantiate 
                    [(ctyp_of_term x_in_set_root,xT)]
                    [(l_in_set_root,l),(x_in_set_root,x),(r_in_set_root,r)] in_set_root
          | (Left::ps') => 
               let
                  val in_set_l = in_set ps' l;
                  val in_set_left' = instantiate [(ctyp_of_term x_in_set_left,xT)]
                                      [(x_in_set_left,x),(r_in_set_left,r)] in_set_left;
               in discharge [in_set_l] in_set_left' end
          | (Right::ps') => 
               let
                  val in_set_r = in_set ps' r;
                  val in_set_right' = instantiate [(ctyp_of_term x_in_set_right,xT)] 
                                      [(x_in_set_right,x),(l_in_set_right,l)] in_set_right;
               in discharge [in_set_r] in_set_right' end)
      end 
       
  fun in_set' [] = raise TERM ("distinctTreeProver",[])
    | in_set' (Left::ps) = in_set ps l
    | in_set' (Right::ps) = in_set ps r;

  fun distinct_lr node_in_set Left  = discharge [dist_subtree_thm,node_in_set] distinct_left 
    | distinct_lr node_in_set Right = discharge [dist_subtree_thm,node_in_set] distinct_right 

  val (swap,neq) = 
       (case x_rest of
         [] => let 
                 val y_in_set = in_set' y_rest;
               in (false,distinct_lr y_in_set (hd y_rest)) end
       | (xr::xrs) => 
           (case y_rest of
             [] => let 
                     val x_in_set = in_set' x_rest;
               in (true,distinct_lr x_in_set (hd x_rest)) end
           | (yr::yrs) =>
               let
                 val x_in_set = in_set' x_rest;
                 val y_in_set = in_set' y_rest;
               in (case xr of
                    Left => (false,
                             discharge [dist_subtree_thm,x_in_set,y_in_set] distinct_left_right)
                   |Right => (true,
                             discharge [dist_subtree_thm,y_in_set,x_in_set] distinct_left_right))
               end
        ))
  in if swap then discharge [neq] swap_neq else neq
  end  


val delete_root = thm "DistinctTreeProver.delete_root";
val delete_left = thm "DistinctTreeProver.delete_left";
val delete_right = thm "DistinctTreeProver.delete_right";

fun deleteProver dist_thm [] = delete_root OF [dist_thm]
  | deleteProver dist_thm (p::ps) =
     let
       val dist_rule = (case p of Left => all_distinct_left | Right => all_distinct_right);
       val dist_thm' = discharge [dist_thm] dist_rule 
       val del_rule = (case p of Left => delete_left | Right => delete_right)
       val del = deleteProver dist_thm' ps;
     in discharge [dist_thm, del] del_rule end;

val subtract_Tip = thm "DistinctTreeProver.subtract_Tip";
val subtract_Node = thm "DistinctTreeProver.subtract_Node";
val delete_Some_all_distinct = thm "DistinctTreeProver.delete_Some_all_distinct";
val subtract_Some_all_distinct_res = thm "DistinctTreeProver.subtract_Some_all_distinct_res";

local
  val (alpha,v) = 
    let
      val ct = subtract_Tip |> Thm.cprop_of |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2 
               |> Thm.dest_comb |> #2
      val [alpha] = ct |> Thm.ctyp_of_term |> Thm.dest_ctyp;
    in (alpha, #1 (dest_Var (term_of ct))) end;
in
fun subtractProver (Const ("DistinctTreeProver.tree.Tip",T)) ct dist_thm =
    let 
      val ct' = dist_thm |> Thm.cprop_of |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2;
      val thy = theory_of_cterm ct;
      val [alphaI] = #2 (dest_Type T);
    in Thm.instantiate ([(alpha,ctyp_of thy alphaI)],
                        [(cterm_of thy (Var (v,treeT alphaI)),ct')]) subtract_Tip
    end
  | subtractProver (Const ("DistinctTreeProver.tree.Node",nT)$l$x$d$r) ct dist_thm =
    let
      val ct' = dist_thm |> Thm.cprop_of |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2;
      val (_,[cl,_,_,cr]) = Drule.strip_comb ct;
      val ps = the (find_tree x (term_of ct'));
      val del_tree = deleteProver dist_thm ps;
      val dist_thm' = discharge [del_tree, dist_thm] delete_Some_all_distinct; 
      val sub_l = subtractProver (term_of cl) cl (dist_thm');
      val sub_r = subtractProver (term_of cr) cr 
                    (discharge [sub_l, dist_thm'] subtract_Some_all_distinct_res);
    in discharge [del_tree, sub_l, sub_r] subtract_Node end
end

val subtract_Some_all_distinct = thm "DistinctTreeProver.subtract_Some_all_distinct";
fun distinct_implProver dist_thm ct =
  let
    val ctree = ct |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2;
    val sub = subtractProver (term_of ctree) ctree dist_thm;
  in subtract_Some_all_distinct OF [sub, dist_thm] end;

fun get_fst_success f [] = NONE
  | get_fst_success f (x::xs) = (case f x of NONE => get_fst_success f xs 
                                 | SOME v => SOME v);

fun neq_x_y ctxt x y name =
  (let
    val dist_thm = the (try (ProofContext.get_thm ctxt) name);
    val ctree = cprop_of dist_thm |> Thm.dest_comb |> #2 |> Thm.dest_comb |> #2;
    val tree = term_of ctree;
    val x_path = the (find_tree x tree);
    val y_path = the (find_tree y tree);
    val thm = distinctTreeProver dist_thm x_path y_path;
  in SOME thm  
  end handle Option => NONE)

fun distinctTree_tac names ctxt 
      (Const ("Trueprop",_) $ (Const ("Not", _) $ (Const ("op =", _) $ x $ y)), i) =
  (case get_fst_success (neq_x_y ctxt x y) names of
     SOME neq => rtac neq i
   | NONE => no_tac)
  | distinctTree_tac _ _ _ = no_tac;

fun distinctFieldSolver names = mk_solver' "distinctFieldSolver"
     (fn ss => case #context (#1 (rep_ss ss)) of
                 SOME ctxt => SUBGOAL (distinctTree_tac names ctxt)
                | NONE => fn i => no_tac)

fun distinct_simproc names =
  Simplifier.simproc HOL.thy "DistinctTreeProver.distinct_simproc" ["x = y"]
    (fn thy => fn ss => fn (Const ("op =",_)$x$y) =>
        case #context (#1 (rep_ss ss)) of
        SOME ctxt => Option.map (fn neq => neq_to_eq_False OF [neq]) 
                      (get_fst_success (neq_x_y ctxt x y) names)
       | NONE => NONE
    )

end
end;