(*

  Title:	Transitivity reasoner for transitive closures of relations

  Id:		$Id$

  Author:	Oliver Kutter

  Copyright:	TU Muenchen

*)

(*

The packages provides tactics trancl_tac and rtrancl_tac that prove

goals of the form

   (x,y) : r^+     and     (x,y) : r^* (rtrancl_tac only)

from premises of the form

   (x,y) : r,     (x,y) : r^+     and     (x,y) : r^* (rtrancl_tac only)

by reflexivity and transitivity.  The relation r is determined by inspecting

the conclusion.

The package is implemented as an ML functor and thus not limited to

particular constructs for transitive and reflexive-transitive

closures, neither need relations be represented as sets of pairs.  In

order to instantiate the package for transitive closure only, supply

dummy theorems to the additional rules for reflexive-transitive

closures, and don't use rtrancl_tac!

*)

signature TRANCL_ARITH = 

sig

  (* theorems for transitive closure *)

  val r_into_trancl : thm

      (* (a,b) : r ==> (a,b) : r^+ *)

  val trancl_trans : thm

      (* [| (a,b) : r^+ ; (b,c) : r^+ |] ==> (a,c) : r^+ *)

  (* additional theorems for reflexive-transitive closure *)

  val rtrancl_refl : thm

      (* (a,a): r^* *)

  val r_into_rtrancl : thm

      (* (a,b) : r ==> (a,b) : r^* *)

  val trancl_into_rtrancl : thm

      (* (a,b) : r^+ ==> (a,b) : r^* *)

  val rtrancl_trancl_trancl : thm

      (* [| (a,b) : r^* ; (b,c) : r^+ |] ==> (a,c) : r^+ *)

  val trancl_rtrancl_trancl : thm

      (* [| (a,b) : r^+ ; (b,c) : r^* |] ==> (a,c) : r^+ *)

  val rtrancl_trans : thm

      (* [| (a,b) : r^* ; (b,c) : r^* |] ==> (a,c) : r^* *)

  (* decomp: decompose a premise or conclusion

     Returns one of the following:

     NONE if not an instance of a relation,

     SOME (x, y, r, s) if instance of a relation, where

       x: left hand side argument, y: right hand side argument,

       r: the relation,

       s: the kind of closure, one of

            "r":   the relation itself,

            "r^+": transitive closure of the relation,

            "r^*": reflexive-transitive closure of the relation

  *)  

  val decomp: term ->  (term * term * term * string) option 

end;

signature TRANCL_TAC = 

sig

  val trancl_tac: int -> tactic;

  val rtrancl_tac: int -> tactic;

end;

functor Trancl_Tac_Fun (Cls : TRANCL_ARITH): TRANCL_TAC = 

struct

 datatype proof

  = Asm of int 

  | Thm of proof list * thm; 

 exception Cannot; (* internal exception: raised if no proof can be found *)

 fun prove asms = 

  let fun pr (Asm i) = List.nth (asms, i)

  |       pr (Thm (prfs, thm)) = (map pr prfs) MRS thm

  in pr end;

(* Internal datatype for inequalities *)

datatype rel 

   = Trans  of term * term * proof  (* R^+ *)

   | RTrans of term * term * proof; (* R^* *)

 (* Misc functions for datatype rel *)

fun lower (Trans (x, _, _)) = x

  | lower (RTrans (x,_,_)) = x;

fun upper (Trans (_, y, _)) = y

  | upper (RTrans (_,y,_)) = y;

fun getprf   (Trans   (_, _, p)) = p

|   getprf   (RTrans (_,_, p)) = p; 

(* ************************************************************************ *)

(*                                                                          *)

(*  mkasm_trancl Rel (t,n): term -> (term , int) -> rel list                *)

(*                                                                          *)

(*  Analyse assumption t with index n with respect to relation Rel:         *)

(*  If t is of the form "(x, y) : Rel" (or Rel^+), translate to             *)

(*  an object (singleton list) of internal datatype rel.                    *)

(*  Otherwise return empty list.                                            *)

(*                                                                          *)

(* ************************************************************************ *)

fun mkasm_trancl  Rel  (t, n) =

  case Cls.decomp t of

    SOME (x, y, rel,r) => if rel aconv Rel then  

    (case r of

      "r"   => [Trans (x,y, Thm([Asm n], Cls.r_into_trancl))]

    | "r+"  => [Trans (x,y, Asm n)]

    | "r*"  => []

    | _     => error ("trancl_tac: unknown relation symbol"))

    else [] 

  | NONE => [];

(* ************************************************************************ *)

(*                                                                          *)

(*  mkasm_rtrancl Rel (t,n): term -> (term , int) -> rel list               *)

(*                                                                          *)

(*  Analyse assumption t with index n with respect to relation Rel:         *)

(*  If t is of the form "(x, y) : Rel" (or Rel^+ or Rel^* ), translate to   *)

(*  an object (singleton list) of internal datatype rel.                    *)

(*  Otherwise return empty list.                                            *)

(*                                                                          *)

(* ************************************************************************ *)

fun mkasm_rtrancl Rel (t, n) =

  case Cls.decomp t of

   SOME (x, y, rel, r) => if rel aconv Rel then

    (case r of

      "r"   => [ Trans (x,y, Thm([Asm n], Cls.r_into_trancl))]

    | "r+"  => [ Trans (x,y, Asm n)]

    | "r*"  => [ RTrans(x,y, Asm n)]

    | _     => error ("rtrancl_tac: unknown relation symbol" ))

   else [] 

  | NONE => [];

(* ************************************************************************ *)

(*                                                                          *)

(*  mkconcl_trancl t: term -> (term, rel, proof)                            *)

(*  mkconcl_rtrancl t: term -> (term, rel, proof)                           *)

(*                                                                          *)

(*  Analyse conclusion t:                                                   *)

(*    - must be of form "(x, y) : r^+ (or r^* for rtrancl)                  *)

(*    - returns r                                                           *)

(*    - conclusion in internal form                                         *)

(*    - proof object                                                        *)

(*                                                                          *)

(* ************************************************************************ *)

fun mkconcl_trancl  t =

  case Cls.decomp t of

    SOME (x, y, rel, r) => (case r of

      "r+"  => (rel, Trans (x,y, Asm ~1), Asm 0)

    | _     => raise Cannot)

  | NONE => raise Cannot;

fun mkconcl_rtrancl  t =

  case Cls.decomp t of

    SOME (x,  y, rel,r ) => (case r of

      "r+"  => (rel, Trans (x,y, Asm ~1),  Asm 0)

    | "r*"  => (rel, RTrans (x,y, Asm ~1), Asm 0)

    | _     => raise Cannot)

  | NONE => raise Cannot;

(* ************************************************************************ *)

(*                                                                          *)

(*  makeStep (r1, r2): rel * rel -> rel                                     *)

(*                                                                          *)

(*  Apply transitivity to r1 and r2, obtaining a new element of r^+ or r^*, *)

(*  according the following rules:                                          *)

(*                                                                          *)

(* ( (a, b) : r^+ , (b,c) : r^+ ) --> (a,c) : r^+                           *)

(* ( (a, b) : r^* , (b,c) : r^+ ) --> (a,c) : r^+                           *)

(* ( (a, b) : r^+ , (b,c) : r^* ) --> (a,c) : r^+                           *)

(* ( (a, b) : r^* , (b,c) : r^* ) --> (a,c) : r^*                           *)

(*                                                                          *)

(* ************************************************************************ *)

fun makeStep (Trans (a,_,p), Trans(_,c,q))  = Trans (a,c, Thm ([p,q], Cls.trancl_trans))

(* refl. + trans. cls. rules *)

|   makeStep (RTrans (a,_,p), Trans(_,c,q))  = Trans (a,c, Thm ([p,q], Cls.rtrancl_trancl_trancl))

|   makeStep (Trans (a,_,p), RTrans(_,c,q))  = Trans (a,c, Thm ([p,q], Cls.trancl_rtrancl_trancl))   

|   makeStep (RTrans (a,_,p), RTrans(_,c,q))  = RTrans (a,c, Thm ([p,q], Cls.rtrancl_trans));

(* ******************************************************************* *)

(*                                                                     *)

(* transPath (Clslist, Cls): (rel  list * rel) -> rel                  *)

(*                                                                     *)

(* If a path represented by a list of elements of type rel is found,   *)

(* this needs to be contracted to a single element of type rel.        *)

(* Prior to each transitivity step it is checked whether the step is   *)

(* valid.                                                              *)

(*                                                                     *)

(* ******************************************************************* *)

fun transPath ([],acc) = acc

|   transPath (x::xs,acc) = transPath (xs, makeStep(acc,x))

(* ********************************************************************* *)

(* Graph functions                                                       *)

(* ********************************************************************* *)

(* *********************************************************** *)

(* Functions for constructing graphs                           *)

(* *********************************************************** *)

fun addEdge (v,d,[]) = [(v,d)]

|   addEdge (v,d,((u,dl)::el)) = if v aconv u then ((v,d@dl)::el)

    else (u,dl):: (addEdge(v,d,el));

(* ********************************************************************** *)

(*                                                                        *)

(* mkGraph constructs from a list of objects of type rel  a graph g       *)

(* and a list of all edges with label r+.                                 *)

(*                                                                        *)

(* ********************************************************************** *)

fun mkGraph [] = ([],[])

|   mkGraph ys =  

 let

  fun buildGraph ([],g,zs) = (g,zs)

  |   buildGraph (x::xs, g, zs) = 

        case x of (Trans (_,_,_)) => 

	       buildGraph (xs, addEdge((upper x), [],(addEdge ((lower x),[((upper x),x)],g))), x::zs) 

	| _ => buildGraph (xs, addEdge((upper x), [],(addEdge ((lower x),[((upper x),x)],g))), zs)

in buildGraph (ys, [], []) end;

(* *********************************************************************** *)

(*                                                                         *)

(* adjacent g u : (''a * 'b list ) list -> ''a -> 'b list                  *)

(*                                                                         *)

(* List of successors of u in graph g                                      *)

(*                                                                         *)

(* *********************************************************************** *)

fun adjacent eq_comp ((v,adj)::el) u = 

    if eq_comp (u, v) then adj else adjacent eq_comp el u

|   adjacent _  []  _ = []  

(* *********************************************************************** *)

(*                                                                         *)

(* dfs eq_comp g u v:                                                      *)

(* ('a * 'a -> bool) -> ('a  *( 'a * rel) list) list ->                    *)

(* 'a -> 'a -> (bool * ('a * rel) list)                                    *) 

(*                                                                         *)

(* Depth first search of v from u.                                         *)

(* Returns (true, path(u, v)) if successful, otherwise (false, []).        *)

(*                                                                         *)

(* *********************************************************************** *)

fun dfs eq_comp g u v = 

 let 

    val pred = ref nil;

    val visited = ref nil;

    fun been_visited v = exists (fn w => eq_comp (w, v)) (!visited)

    fun dfs_visit u' = 

    let val _ = visited := u' :: (!visited)

    fun update (x,l) = let val _ = pred := (x,l) ::(!pred) in () end;

    in if been_visited v then () 

    else (app (fn (v',l) => if been_visited v' then () else (

       update (v',l); 

       dfs_visit v'; ()) )) (adjacent eq_comp g u')

     end

  in 

    dfs_visit u; 

    if (been_visited v) then (true, (!pred)) else (false , [])   

  end;

(* *********************************************************************** *)

(*                                                                         *)

(* transpose g:                                                            *)

(* (''a * ''a list) list -> (''a * ''a list) list                          *)

(*                                                                         *)

(* Computes transposed graph g' from g                                     *)

(* by reversing all edges u -> v to v -> u                                 *)

(*                                                                         *)

(* *********************************************************************** *)

fun transpose eq_comp g =

  let

   (* Compute list of reversed edges for each adjacency list *)

   fun flip (u,(v,l)::el) = (v,(u,l)) :: flip (u,el)

     | flip (_,nil) = nil

   (* Compute adjacency list for node u from the list of edges

      and return a likewise reduced list of edges.  The list of edges

      is searches for edges starting from u, and these edges are removed. *)

   fun gather (u,(v,w)::el) =

    let

     val (adj,edges) = gather (u,el)

    in

     if eq_comp (u, v) then (w::adj,edges)

     else (adj,(v,w)::edges)

    end

   | gather (_,nil) = (nil,nil)

   (* For every node in the input graph, call gather to find all reachable

      nodes in the list of edges *)

   fun assemble ((u,_)::el) edges =

       let val (adj,edges) = gather (u,edges)

       in (u,adj) :: assemble el edges

       end

     | assemble nil _ = nil

   (* Compute, for each adjacency list, the list with reversed edges,

      and concatenate these lists. *)

   val flipped = Library.foldr (op @) ((map flip g),nil)

 in assemble g flipped end    

(* *********************************************************************** *)

(*                                                                         *)

(* dfs_reachable eq_comp g u:                                              *)

(* (int * int list) list -> int -> int list                                *) 

(*                                                                         *)

(* Computes list of all nodes reachable from u in g.                       *)

(*                                                                         *)

(* *********************************************************************** *)

fun dfs_reachable eq_comp g u = 

 let

  (* List of vertices which have been visited. *)

  val visited  = ref nil;

  fun been_visited v = exists (fn w => eq_comp (w, v)) (!visited)

  fun dfs_visit g u  =

      let

   val _ = visited := u :: !visited

   val descendents =

       Library.foldr (fn ((v,l),ds) => if been_visited v then ds

            else v :: dfs_visit g v @ ds)

        ( ((adjacent eq_comp g u)) ,nil)

   in  descendents end

 in u :: dfs_visit g u end;

(* *********************************************************************** *)

(*                                                                         *)

(* dfs_term_reachable g u:                                                  *)

(* (term * term list) list -> term -> term list                            *) 

(*                                                                         *)

(* Computes list of all nodes reachable from u in g.                       *)

(*                                                                         *)

(* *********************************************************************** *)

fun dfs_term_reachable g u = dfs_reachable (op aconv) g u;

(* ************************************************************************ *) 

(*                                                                          *)

(* findPath x y g: Term.term -> Term.term ->                                *)

(*                  (Term.term * (Term.term * rel list) list) ->            *) 

(*                  (bool, rel list)                                        *)

(*                                                                          *)

(*  Searches a path from vertex x to vertex y in Graph g, returns true and  *)

(*  the list of edges if path is found, otherwise false and nil.            *)

(*                                                                          *)

(* ************************************************************************ *) 

fun findPath x y g = 

  let 

   val (found, tmp) =  dfs (op aconv) g x y ;

   val pred = map snd tmp;

   fun path x y  =

    let

	 (* find predecessor u of node v and the edge u -> v *)

      fun lookup v [] = raise Cannot

      |   lookup v (e::es) = if (upper e) aconv v then e else lookup v es;

      (* traverse path backwards and return list of visited edges *)   

      fun rev_path v = 

	let val l = lookup v pred

	    val u = lower l;

	in

	  if u aconv x then [l] else (rev_path u) @ [l] 

	end

    in rev_path y end;

   in 

      if found then ( (found, (path x y) )) else (found,[])

   end;

(* ************************************************************************ *)

(*                                                                          *)

(* findRtranclProof g tranclEdges subgoal:                                  *)

(* (Term.term * (Term.term * rel list) list) -> rel -> proof list           *)

(*                                                                          *)

(* Searches in graph g a proof for subgoal.                                 *)

(*                                                                          *)

(* ************************************************************************ *)

fun findRtranclProof g tranclEdges subgoal = 

   case subgoal of (RTrans (x,y,_)) => if x aconv y then [Thm ([], Cls.rtrancl_refl)] else (

     let val (found, path) = findPath (lower subgoal) (upper subgoal) g

     in 

       if found then (

          let val path' = (transPath (tl path, hd path))

	  in 

	    case path' of (Trans (_,_,p)) => [Thm ([p], Cls.trancl_into_rtrancl )] 

	    | _ => [getprf path']

	  end

       )

       else raise Cannot

     end

   )

| (Trans (x,y,_)) => (

  let

   val Vx = dfs_term_reachable g x;

   val g' = transpose (op aconv) g;

   val Vy = dfs_term_reachable g' y;

   fun processTranclEdges [] = raise Cannot

   |   processTranclEdges (e::es) = 

          if (upper e) mem Vx andalso (lower e) mem Vx

	  andalso (upper e) mem Vy andalso (lower e) mem Vy

	  then (

	    if (lower e) aconv x then (

	      if (upper e) aconv y then (

	          [(getprf e)] 

	      )

	      else (

	          let 

		    val (found,path) = findPath (upper e) y g

		  in

		   if found then ( 

		       [getprf (transPath (path, e))]

		      ) else processTranclEdges es

		  end 

	      )   

	    )

	    else if (upper e) aconv y then (

	       let val (xufound,xupath) = findPath x (lower e) g

	       in 

	          if xufound then (

		    let val xuRTranclEdge = transPath (tl xupath, hd xupath)

			    val xyTranclEdge = makeStep(xuRTranclEdge,e)

				in [getprf xyTranclEdge] end

	         ) else processTranclEdges es

	       end

	    )

	    else ( 

	        let val (xufound,xupath) = findPath x (lower e) g

		    val (vyfound,vypath) = findPath (upper e) y g

		 in 

		    if xufound then (

		         if vyfound then ( 

			    let val xuRTranclEdge = transPath (tl xupath, hd xupath)

			        val vyRTranclEdge = transPath (tl vypath, hd vypath)

				val xyTranclEdge = makeStep (makeStep(xuRTranclEdge,e),vyRTranclEdge)

				in [getprf xyTranclEdge] end

			 ) else processTranclEdges es

		    ) 

		    else processTranclEdges es

		 end

	    )

	  )

	  else processTranclEdges es;

   in processTranclEdges tranclEdges end )

| _ => raise Cannot

fun solveTrancl (asms, concl) = 

 let val (g,_) = mkGraph asms

 in

  let val (_, subgoal, _) = mkconcl_trancl concl

      val (found, path) = findPath (lower subgoal) (upper subgoal) g

  in

    if found then  [getprf (transPath (tl path, hd path))]

    else raise Cannot 

  end

 end;

fun solveRtrancl (asms, concl) = 

 let val (g,tranclEdges) = mkGraph asms

     val (_, subgoal, _) = mkconcl_rtrancl concl

in

  findRtranclProof g tranclEdges subgoal

end;

val trancl_tac =   SUBGOAL (fn (A, n) =>

 let

  val Hs = Logic.strip_assums_hyp A;

  val C = Logic.strip_assums_concl A;

  val (rel,subgoals, prf) = mkconcl_trancl C;

  val prems = List.concat (ListPair.map (mkasm_trancl rel) (Hs, 0 upto (length Hs - 1)))

  val prfs = solveTrancl (prems, C);

 in

  METAHYPS (fn asms =>

    let val thms = map (prove asms) prfs

    in rtac (prove thms prf) 1 end) n

 end

handle  Cannot  => no_tac);

val rtrancl_tac =   SUBGOAL (fn (A, n) =>

 let

  val Hs = Logic.strip_assums_hyp A;

  val C = Logic.strip_assums_concl A;

  val (rel,subgoals, prf) = mkconcl_rtrancl C;

  val prems = List.concat (ListPair.map (mkasm_rtrancl rel) (Hs, 0 upto (length Hs - 1)))

  val prfs = solveRtrancl (prems, C);

 in

  METAHYPS (fn asms =>

    let val thms = map (prove asms) prfs

    in rtac (prove thms prf) 1 end) n

 end

handle  Cannot  => no_tac);

end;