(*  Title:      Provers/trancl.ML

     Author:     Oliver Kutter, TU Muenchen

 Transitivity reasoner for transitive closures of relations

 *)

 (*

 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: Proof.context -> int -> tactic

   val rtrancl_tac: Proof.context -> int -> tactic

 end;

 functor Trancl_Tac(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 decomp t = Option.map (fn (x, y, rel, r) =>

   (Envir.beta_eta_contract x, Envir.beta_eta_contract y,

    Envir.beta_eta_contract rel, r)) (Cls.decomp t);

 fun prove thy r asms =

   let

     fun inst thm =

       let val SOME (_, _, r', _) = decomp (concl_of thm)

       in Drule.cterm_instantiate [(cterm_of thy r', cterm_of thy r)] thm end;

     fun pr (Asm i) = nth asms i

       | pr (Thm (prfs, thm)) = map pr prfs MRS inst 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 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 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 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 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 = Unsynchronized.ref [];

     val visited = Unsynchronized.ref [];

     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 (_,[]) = []

    (* 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 (_,[]) = ([],[])

    (* 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 [] _ = []

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

       and concatenate these lists. *)

    val flipped = maps flip g

  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  = Unsynchronized.ref [];

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

   fun dfs_visit g u  =

       let

    val _ = visited := u :: !visited

    val descendents =

        List.foldr (fn ((v,_),ds) => if been_visited v then ds

             else v :: dfs_visit g v @ ds)

         [] (adjacent eq_comp g u)

    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 member (op =) Vx (upper e) andalso member (op =) Vx (lower e)

           andalso member (op =) Vy (upper e) andalso member (op =) Vy (lower e)

           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 )

 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;

 fun trancl_tac ctxt = SUBGOAL (fn (A, n) => fn st =>

  let

   val thy = Proof_Context.theory_of ctxt;

   val Hs = Logic.strip_assums_hyp A;

   val C = Logic.strip_assums_concl A;

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

   val prems = flat (map_index (mkasm_trancl rel o swap) Hs);

   val prfs = solveTrancl (prems, C);

  in

   Subgoal.FOCUS (fn {prems, concl, ...} =>

     let

       val SOME (_, _, rel', _) = decomp (term_of concl);

       val thms = map (prove thy rel' prems) prfs

     in rtac (prove thy rel' thms prf) 1 end) ctxt n st

  end

  handle Cannot => Seq.empty);

 fun rtrancl_tac ctxt = SUBGOAL (fn (A, n) => fn st =>

  let

   val thy = Proof_Context.theory_of ctxt;

   val Hs = Logic.strip_assums_hyp A;

   val C = Logic.strip_assums_concl A;

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

   val prems = flat (map_index (mkasm_rtrancl rel o swap) Hs);

   val prfs = solveRtrancl (prems, C);

  in

   Subgoal.FOCUS (fn {prems, concl, ...} =>

     let

       val SOME (_, _, rel', _) = decomp (term_of concl);

       val thms = map (prove thy rel' prems) prfs

     in rtac (prove thy rel' thms prf) 1 end) ctxt n st

  end

  handle Cannot => Seq.empty | General.Subscript => Seq.empty);

 end;