Some debug info; if this occurs here we should deal with it, but it seems to occur elsewhere (in KB?)

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()

option package;

#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".

global Piece;

local
   diff_,

   Umap := proc(f,x,$)
       f(op(0,x))( map( p -> Piece(f(condOf(p)),f(valOf(p)))
                      , op(1,x)
                      )
                 )
   end proc,

   # could also be named
   # simplifyPartition_by_the_removal_of_particular_singular_points
   simplifyPartition_mergeSinglePts := module()
       local canSimp := c -> valOf(c) :: identical('undefined') and condOf(c) :: `=`
                                  and (lhs(condOf(c)) :: name or rhs(condOf(c)) :: name);

       local mentions_t_hi :=
                t -> bnd -> cl -> has(cl, t<bnd) or has(cl, bnd>t);

       local mentions_t_lo :=
                t -> bnd -> cl -> has(cl, bnd<t) or has(cl, t>bnd);

       local replace_with := t -> bnd -> x ->
                if mentions_t_hi(t)(bnd)(x) then
                    t <= bnd
                elif mentions_t_lo(t)(bnd)(x) then
                    t >= bnd
                else
                    x
                end if;

       local set_xor := proc(a,b,$) (a union b) intersect (a minus b) end proc;

       local tryReplacePieces :=
               proc(replPieces, otherPieces,cmp,$)
                 local rpp := replPieces, otp := otherPieces, nm, val, rp, rpv;

                 for rp in rpp do
                     rp, rpv := op(rp);

                     nm   := `if`(lhs(rp)::name, lhs(rp), rhs(rp));
                     val  := `if`(lhs(rp)::name, rhs(rp), lhs(rp));

                     otp := tryReplacePiece( nm, val, rpv, otp, cmp )
                 end do;

                 otp;

               end proc;

       local the_val := cmp -> xs ->
                evalb( foldl( proc(x,y) if cmp(x,y) then x else NULL end if; end proc, op(xs)) = NULL ) ;

       local tryReplacePiece :=
               proc(vrNm, vrVal, pc0val, pcs, cmp,$)
                 local pcs0 := pcs, pcs1, qs0, qs1, qs2, vrEq := vrNm=vrVal ;
                 pcs1 := subsindets(pcs0, relation, replace_with(vrNm)(vrVal));

                 qs0, qs1 := seq({op(qs)},qs=(pcs0,pcs1));

                 qs2 := set_xor(qs1, qs0);

                 if nops(qs2) = 2 then
                     qs2 := map(condOf, qs2);

                     if not pc0val :: identical('undefined') then
                         qs2 := { pc0val, op(qs2) };
                     end if;

                     qs2 := map(q -> subs(vrEq, q), qs2);

                     if the_val(cmp)(qs2) then
                         pcs1;
                     else
                         [ Piece(vrEq, pc0val), op(pcs0) ] ;
                     end if
                 else
                     [ Piece(vrEq, pc0val), op(pcs0) ] ;
                 end if;

               end proc;

       export ModuleApply := proc(p_,{cmp:=`=`},$)

          local p := p_, r := p, uc, oc;

          # if the partition contains case of the form `x = t', where `t' is a
          # constant (or term??) and `x' is a variable, and the value of that
          # case is `undefined', then we may be able to eliminate it (if another
          # case includes that point)
          r := op(1,r);
          uc, oc := selectremove(canSimp, r);

          # special case: if there is exactly one `undefined' case, and one
          # other case, just get rid of the `undefined' piece
          if nops(uc)=1 and nops(oc)=1 then
              p := PARTITION(oc);
          else
              p := PARTITION(tryReplacePieces(uc, oc, cmp));
          end if;

          return p;

       end proc;

   end module,

   simplifyPartitionCtx := module()

       local is_extra_sol := x -> (x :: `=` and rhs(x)=lhs(x) and lhs(x) :: name);
                               # or (x :: name);

       local preproc_for_solve := proc(ctx, $)
                 ctx;
             end proc;

       local postproc_for_solve := proc(ctx, ctxSlv, $)::{identical(false), list({boolean,relation})};
                 local ctxC := ctxSlv;

                 if ctxC = [] then
                     ctxC := false ;

                 elif nops(ctxC)> 1 then
                     ctxC := map(x -> postproc_for_solve(ctx, [x])[], ctxC);

                 elif nops(ctxC) = 1 then
                     ctxC := op(1,ctxC);

                     if ctxC :: set then
                         ctxC := remove(is_extra_sol, ctxC);

                         if ctxC :: identical('{}') then
                             ctxC := eval(ctx, [`And`=`and`, `Not`=`not`]);
                         else
                             ctxC := `and`(op(ctxC));
                         end if ;

                         ctxC := [ctxC];

                     elif ctxC :: specfunc('piecewise') then
                         ctxC := PWToPartition(ctxC);

                         ctxC := [ seq( map( o -> condOf(c) and o
                                           , postproc_for_solve(ctx, valOf(c)))[]
                                      , c=op(1, ctxC)
                                      )
                                 ] ;
                     else
                         error "simplifyPartitionCtx: don't know what to do with %1", ctxC;
                     end if;
                 else
                     error "simplifyPartitionCtx: don't know what to do with %1", ctxC;

                 end if;

                 ctxC;

             end proc;

       export ModuleApply := proc(ctx, $)
       local ctxC := ctx;

           if ctx :: identical(true) then
               error "simplifyPartitionCtx: don't know what to do with %1", ctxC;
           end if;

           ctxC := solve({ctxC});
           ctxC := postproc_for_solve(ctx, [ctxC]);

           if indets(ctxC, specfunc({`Or`, `or`})) <> {} then
               userinfo(10, 'simplifyPartitionCtx',
                   printf("output: \n"
                          "  %a\n\n"
                          , ctxC )
                  );
           end if;

           ctxC;

       end proc;

   end module,

   ModuleLoad::static:= proc()
      local prev;

      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([ condOf(eval(branch)), valOf(eval(branch))][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(Partition, specfunc(PARTITION));

      # global extensions to maple functionality

      :-`eval/PARTITION` :=
      proc(p, eqs, $)
          local q, r;
          q := Umap(x->eval(x,eqs), p);

          r := eval( PartitionToPW(q), eqs );
          if r :: specfunc('piecewise') then
              q := PWToPartition( r );
          else
              q := r;
          end if;

          q;

      end proc;

      :-`depends/PARTITION` :=
      proc(parts, nms, $)
         local dp := (x -> depends(x, nms));
         `or`(op ( map(p-> dp(condOf(p)) or dp(valOf(p)), parts) ));
      end proc;

      :-`diff/PARTITION` :=
      proc(parts, wrt, $)
          local pw  := PartitionToPW(PARTITION(parts))
              , dpw := diff(pw, wrt)
              , r   := PWToPartition(dpw)
              , r0, r1;
            ;
          r0 := simplifyPartition_mergeSinglePts(r);

          # probably a better way to do this; we really only want to simplify
          # sums and products of integrals and summations
          r1 := subsindets(r0, algebraic, `simplify`);

          userinfo(10, 'disint_trace',
                   printf("  input        : \n\t%a\n\n"
                          "  diff         : \n\t%a\n\n"
                          "  singular pts : \n\t%a\n\n"
                          "  simplified   : \n\t%a\n\n\n"
                          , parts, r, r0, r1 ));

          r1;
      end proc;

      NULL
   end proc,

   ModuleUnload::static:= proc()
      TypeTools:-RemoveType(Partition);
      NULL
   end proc,

   # abstract out all argument checking for map-like functions
   map_check := proc(p)
      local pos, err;
      if p::indexed then
         pos:= op(p);
         if not [pos]::[posint] then
            err := sprintf("Expected positive integer index; received %a", [pos]);
            return err;
         end if
      else
         pos:= 1
      end if;
      if nargs-1 <= pos then
         err := sprintf("Expected at least %d arguments; received %d", pos+1, nargs-1);
         return err;
      end if;
      if not args[pos+2]::Partition then
         err := sprintf("Expected a Partition; received %a", args[pos+2]);
         return err;
      end if;
      return pos;
   end proc
;
export

   condOf := proc(x::specfunc(`Piece`),$)
       op(1,x);
   end proc,

   valOf := proc(x::specfunc(`Piece`),$)
       op(2,x);
   end proc,

   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= [ seq(Piece( Terms[k], Terms[k+1]), k= 1..nargs-1, 2) ] ;
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(condOf(p), p= op(1,P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, anything]);
   local p;
      {seq([condOf(p), valOf(p)], p= op(1,P)) }
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos,res;
      res := map_check(procname, args);
      if res::string then error res else pos := res; end if;
      PARTITION(
         [seq(
            Piece(
               condOf(pair),
               f(args[2..pos], valOf(pair), args[pos+2..])
            ),
            pair= op(1, args[pos+1])
         )]
      )
   end proc,

   # a more complex mapping combinator which works on all 3 parts
   # not fully general, but made to work with KB
   # also, does not handle extra arguments (on purpose!)
   Amap::static:= proc(
      funcs::[anything, anything, anything], #`appliable` not inclusive enough.
      part::Partition
   )::Partition;
   local pair,pos,f,g,h,doIt;
      (f,g,h) := op(funcs);
      #sigh, we don't have a decent 'let', need to use a local proc
      doIt := proc(pair)
        local kb0 := h(condOf(pair));
        Piece( f(condOf(pair), kb0),
               g(valOf(pair), kb0));
      end proc;
      PARTITION(map(doIt,op(1,part)));
   end proc,

   PartitionToPW := proc(x::Partition)::specfunc(piecewise);
       # piecewise can reduce immediately to not-piecewise which makes the type
       # test fail
       'piecewise'( op( ListTools[Flatten]( [ seq([condOf(p), valOf(p)], p= op(1,x)) ] ) ) );
   end proc,

   # convert a piecewise to a partition, which is straightforward except:
   # - if any of the branches are unreachable, they are removed
   # - if the last clause is (implicitly) `otherwise`, that clause is filled in
   #     appropriately

   # note that if the piecewise does not cover the entire domain,
   #   then this Partition will be 'invalid' (in the sense that it also
   #   will not cover the entire domain) - the 'correct' thing to do would
   #   probably be to add a new clause whose value is 'undefined'

   # the logic of this function is already essentially implemented, by KB
   # in fact, kb_piecewise does something extremely similar to this
   PWToPartition := proc(x::specfunc(piecewise))::Partition;

       # each clause evaluated under the context so far,
       # which is the conjunction of the negations of all clauses
       # so far
       local ctx := true, n := nops(x), cls := [], cnd, ncnd, i, q
           , ctxC, cl
         ;

       userinfo(5, PWToPartition
               , printf("PWToPartition: found %d ops in %a \n ", n, x) );

       # handles all but the `otherwise` case if there is such a case
       for i in seq(q, q = 1 .. iquo(n, 2)) do

           userinfo(3, PWToPartition
                    , printf("PWToPartition: looking at clause %d (op %d) \n ", i, 2*i-1));

           cnd := op(2*i-1,x); # the clause as given

           # if this clause is unreachable, then every subsequent clause will be as well
           if ctx :: identical(false) then
               return PARTITION( cls );
           else
               ctxC := `And`(cnd, ctx);              # the condition, along with the context (which is implicit in pw)
               ctxC := simplifyPartitionCtx(ctxC);

               if cnd :: `=` then
                   ncnd := lhs(cnd) <> rhs(cnd);
               else
                   ncnd := Not(cnd);
               end if;

               ctx  := `And`(ncnd, ctx); # the context for the next clause

               userinfo(3, PWToPartition, printf("PWToPartition: ctx after %d clauses "
                                                 "is %a\n", i, ctx));

               if ctxC :: identical(false) then # this clause is actually unreachable
                   return(PARTITION(cls));
               else
                   cls := [ op(cls)
                          , seq(Piece(cl
                                     ,op(2*i ,x)
                                     )
                               ,cl=ctxC
                               )
                          ];

               end if;

           end if;
       end do;

       # if there is an otherwise case, handle that.
       if n::odd then
           ctx := simplifyPartitionCtx(ctx);
           if not ctx :: identical(false) then
               cls := [ op(cls)
                      , seq(Piece(cl
                                 ,op(n,x)
                                 )
                           ,cl=ctx
                           )
                      ];
           end if;
       end if;

       PARTITION( cls );
   end proc,

   MbAppPartOrPw := proc(chk,f,x::Or(Partition,specfunc(piecewise)))
       local isPr := evalb(x::Partition), pr := `if`(isPr, x, PWToPartition(x));
       if chk(pr) then
           (`if`(isPr, a->a, PartitionToPW))(f(pr));
       else
           FAIL
       end if;
   end proc,

   # applies a function to the arg if arg::Partition,
   # and if arg::piecewise, then converts the piecewise to a partition,
   # applies the function, then converts back to piecewise
   # this mainly acts as a sanity check

   AppPartOrPw := proc(f::anything # TODO better type
                      ,x::Or(Partition,specfunc(piecewise))
                      )
       if x::Partition then
           f(x);
       else
           PartitionToPW(f(PWToPartition(x)));
       end if;
   end proc,

   #Check whether the conditions of a Partition depend on any of a set of names.
   ConditionsDepend:= proc(P::Partition, V::{name, list(name), set(name)}, $)
   local p;
      for p in op(1,P) do if depends(condOf(p), V) then return true end if end do;
      false
   end proc
;

uses Hakaru;

   ModuleLoad()
end module:

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()

option package;

#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".

global Piece;

local
   diff_,

   Umap := proc(f,x,$)
       f(op(0,x))( map( p -> Piece(f(condOf(p)),f(valOf(p)))
                      , op(1,x)
                      )
                 )
   end proc,

   # could also be named
   # simplifyPartition_by_the_removal_of_particular_singular_points
   simplifyPartition_mergeSinglePts := module()
       local canSimp := c -> valOf(c) :: identical('undefined') and condOf(c) :: `=`
                                  and (lhs(condOf(c)) :: name or rhs(condOf(c)) :: name);

       local mentions_t_hi :=
                t -> bnd -> cl -> has(cl, t<bnd) or has(cl, bnd>t);

       local mentions_t_lo :=
                t -> bnd -> cl -> has(cl, bnd<t) or has(cl, t>bnd);

       local replace_with := t -> bnd -> x ->
                if mentions_t_hi(t)(bnd)(x) then
                    t <= bnd
                elif mentions_t_lo(t)(bnd)(x) then
                    t >= bnd
                else
                    x
                end if;

       local set_xor := proc(a,b,$) (a union b) intersect (a minus b) end proc;

       local tryReplacePieces :=
               proc(replPieces, otherPieces,cmp,$)
                 local rpp := replPieces, otp := otherPieces, nm, val, rp, rpv;

                 for rp in rpp do
                     rp, rpv := op(rp);

                     nm   := `if`(lhs(rp)::name, lhs(rp), rhs(rp));
                     val  := `if`(lhs(rp)::name, rhs(rp), lhs(rp));

                     otp := tryReplacePiece( nm, val, rpv, otp, cmp )
                 end do;

                 otp;

               end proc;

       local the_val := cmp -> xs ->
                evalb( foldl( proc(x,y) if cmp(x,y) then x else NULL end if; end proc, op(xs)) = NULL ) ;

       local tryReplacePiece :=
               proc(vrNm, vrVal, pc0val, pcs, cmp,$)
                 local pcs0 := pcs, pcs1, qs0, qs1, qs2, vrEq := vrNm=vrVal ;
                 pcs1 := subsindets(pcs0, relation, replace_with(vrNm)(vrVal));

                 qs0, qs1 := seq({op(qs)},qs=(pcs0,pcs1));

                 qs2 := set_xor(qs1, qs0);

                 if nops(qs2) = 2 then
                     qs2 := map(condOf, qs2);

                     if not pc0val :: identical('undefined') then
                         qs2 := { pc0val, op(qs2) };
                     end if;

                     qs2 := map(q -> subs(vrEq, q), qs2);

                     if the_val(cmp)(qs2) then
                         pcs1;
                     else
                         [ Piece(vrEq, pc0val), op(pcs0) ] ;
                     end if
                 else
                     [ Piece(vrEq, pc0val), op(pcs0) ] ;
                 end if;

               end proc;

       export ModuleApply := proc(p_,{cmp:=`=`},$)

          local p := p_, r := p, uc, oc;

          # if the partition contains case of the form `x = t', where `t' is a
          # constant (or term??) and `x' is a variable, and the value of that
          # case is `undefined', then we may be able to eliminate it (if another
          # case includes that point)
          r := op(1,r);
          uc, oc := selectremove(canSimp, r);

          # special case: if there is exactly one `undefined' case, and one
          # other case, just get rid of the `undefined' piece
          if nops(uc)=1 and nops(oc)=1 then
              p := PARTITION(oc);
          else
              p := PARTITION(tryReplacePieces(uc, oc, cmp));
          end if;

          return p;

       end proc;

   end module,

   simplifyPartitionCtx := module()

       local is_extra_sol := x -> (x :: `=` and rhs(x)=lhs(x) and lhs(x) :: name);
                               # or (x :: name);

       local preproc_for_solve := proc(ctx, $)
                 ctx;
             end proc;

       local postproc_for_solve := proc(ctx, ctxSlv, $)::{identical(false), list({boolean,relation})};
                 local ctxC := ctxSlv;

                 if ctxC = [] then
                     ctxC := false ;

                 elif nops(ctxC)> 1 then
                     ctxC := map(x -> postproc_for_solve(ctx, [x])[], ctxC);

                 elif nops(ctxC) = 1 then
                     ctxC := op(1,ctxC);

                     if ctxC :: set then
                         ctxC := remove(is_extra_sol, ctxC);

                         if ctxC :: identical('{}') then
                             ctxC := eval(ctx, [`And`=`and`, `Not`=`not`]);
                         else
                             ctxC := `and`(op(ctxC));
                         end if ;

                         ctxC := [ctxC];

                     elif ctxC :: specfunc('piecewise') then
                         ctxC := PWToPartition(ctxC);

                         ctxC := [ seq( map( o -> condOf(c) and o
                                           , postproc_for_solve(ctx, valOf(c)))[]
                                      , c=op(1, ctxC)
                                      )
                                 ] ;
                     else
                         error "simplifyPartitionCtx: don't know what to do with %1", ctxC;
                     end if;
                 else
                     error "simplifyPartitionCtx: don't know what to do with %1", ctxC;

                 end if;

                 ctxC;

             end proc;

       export ModuleApply := proc(ctx, $)
       local ctxC := ctx;

           if ctx :: identical(true) then
               error "simplifyPartitionCtx: don't know what to do with %1", ctxC;
           end if;

           ctxC := solve({ctxC});
           postproc_for_solve(ctx, [ctxC]);

       end proc;

   end module,

   ModuleLoad::static:= proc()
      local prev;

      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([ condOf(eval(branch)), valOf(eval(branch))][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(Partition, specfunc(PARTITION));

      # global extensions to maple functionality

      :-`eval/PARTITION` :=
      proc(p, eqs, $)
          local q, r;
          q := Umap(x->eval(x,eqs), p);

          r := eval( PartitionToPW(q), eqs );
          if r :: specfunc('piecewise') then
              q := PWToPartition( r );
          else
              q := r;
          end if;

          q;

      end proc;

      :-`depends/PARTITION` :=
      proc(parts, nms, $)
         local dp := (x -> depends(x, nms));
         `or`(op ( map(p-> dp(condOf(p)) or dp(valOf(p)), parts) ));
      end proc;

      :-`diff/PARTITION` :=
      proc(parts, wrt, $)
          local pw  := PartitionToPW(PARTITION(parts))
              , dpw := diff(pw, wrt)
              , r   := PWToPartition(dpw)
              , r0, r1;
            ;
          r0 := simplifyPartition_mergeSinglePts(r);

          # probably a better way to do this; we really only want to simplify
          # sums and products of integrals and summations
          r1 := subsindets(r0, algebraic, `simplify`);

          userinfo(10, 'disint_trace',
                   printf("  input        : \n\t%a\n\n"
                          "  diff         : \n\t%a\n\n"
                          "  singular pts : \n\t%a\n\n"
                          "  simplified   : \n\t%a\n\n\n"
                          , parts, r, r0, r1 ));

          r1;
      end proc;

      NULL
   end proc,

   ModuleUnload::static:= proc()
      TypeTools:-RemoveType(Partition);
      NULL
   end proc,

   # abstract out all argument checking for map-like functions
   map_check := proc(p)
      local pos, err;
      if p::indexed then
         pos:= op(p);
         if not [pos]::[posint] then
            err := sprintf("Expected positive integer index; received %a", [pos]);
            return err;
         end if
      else
         pos:= 1
      end if;
      if nargs-1 <= pos then
         err := sprintf("Expected at least %d arguments; received %d", pos+1, nargs-1);
         return err;
      end if;
      if not args[pos+2]::Partition then
         err := sprintf("Expected a Partition; received %a", args[pos+2]);
         return err;
      end if;
      return pos;
   end proc
;
export

   condOf := proc(x::specfunc(`Piece`),$)
       op(1,x);
   end proc,

   valOf := proc(x::specfunc(`Piece`),$)
       op(2,x);
   end proc,

   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= [ seq(Piece( Terms[k], Terms[k+1]), k= 1..nargs-1, 2) ] ;
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(condOf(p), p= op(1,P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, anything]);
   local p;
      {seq([condOf(p), valOf(p)], p= op(1,P)) }
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos,res;
      res := map_check(procname, args);
      if res::string then error res else pos := res; end if;
      PARTITION(
         [seq(
            Piece(
               condOf(pair),
               f(args[2..pos], valOf(pair), args[pos+2..])
            ),
            pair= op(1, args[pos+1])
         )]
      )
   end proc,

   # a more complex mapping combinator which works on all 3 parts
   # not fully general, but made to work with KB
   # also, does not handle extra arguments (on purpose!)
   Amap::static:= proc(
      funcs::[anything, anything, anything], #`appliable` not inclusive enough.
      part::Partition
   )::Partition;
   local pair,pos,f,g,h,doIt;
      (f,g,h) := op(funcs);
      #sigh, we don't have a decent 'let', need to use a local proc
      doIt := proc(pair)
        local kb0 := h(condOf(pair));
        Piece( f(condOf(pair), kb0),
               g(valOf(pair), kb0));
      end proc;
      PARTITION(map(doIt,op(1,part)));
   end proc,

   PartitionToPW := proc(x::Partition)::specfunc(piecewise);
       # piecewise can reduce immediately to not-piecewise which makes the type
       # test fail
       'piecewise'( op( ListTools[Flatten]( [ seq([condOf(p), valOf(p)], p= op(1,x)) ] ) ) );
   end proc,

   # convert a piecewise to a partition, which is straightforward except:
   # - if any of the branches are unreachable, they are removed
   # - if the last clause is (implicitly) `otherwise`, that clause is filled in
   #     appropriately

   # note that if the piecewise does not cover the entire domain,
   #   then this Partition will be 'invalid' (in the sense that it also
   #   will not cover the entire domain) - the 'correct' thing to do would
   #   probably be to add a new clause whose value is 'undefined'

   # the logic of this function is already essentially implemented, by KB
   # in fact, kb_piecewise does something extremely similar to this
   PWToPartition := proc(x::specfunc(piecewise))::Partition;

       # each clause evaluated under the context so far,
       # which is the conjunction of the negations of all clauses
       # so far
       local ctx := true, n := nops(x), cls := [], cnd, ncnd, i, q
           , ctxC, cl
         ;

       userinfo(5, PWToPartition
               , printf("PWToPartition: found %d ops in %a \n ", n, x) );

       # handles all but the `otherwise` case if there is such a case
       for i in seq(q, q = 1 .. iquo(n, 2)) do

           userinfo(3, PWToPartition
                    , printf("PWToPartition: looking at clause %d (op %d) \n ", i, 2*i-1));

           cnd := op(2*i-1,x); # the clause as given

           # if this clause is unreachable, then every subsequent clause will be as well
           if ctx :: identical(false) then
               return PARTITION( cls );
           else
               ctxC := `And`(cnd, ctx);              # the condition, along with the context (which is implicit in pw)
               ctxC := simplifyPartitionCtx(ctxC);

               if cnd :: `=` then
                   ncnd := lhs(cnd) <> rhs(cnd);
               else
                   ncnd := Not(cnd);
               end if;

               ctx  := `And`(ncnd, ctx); # the context for the next clause

               userinfo(3, PWToPartition, printf("PWToPartition: ctx after %d clauses "
                                                 "is %a\n", i, ctx));

               if ctxC :: identical(false) then # this clause is actually unreachable
                   return(PARTITION(cls));
               else
                   cls := [ op(cls)
                          , seq(Piece(cl
                                     ,op(2*i ,x)
                                     )
                               ,cl=ctxC
                               )
                          ];

               end if;

           end if;
       end do;

       # if there is an otherwise case, handle that.
       if n::odd then
           ctx := simplifyPartitionCtx(ctx);
           if not ctx :: identical(false) then
               cls := [ op(cls)
                      , seq(Piece(cl
                                 ,op(n,x)
                                 )
                           ,cl=ctx
                           )
                      ];
           end if;
       end if;

       PARTITION( cls );
   end proc,

   MbAppPartOrPw := proc(chk,f,x::Or(Partition,specfunc(piecewise)))
       local isPr := evalb(x::Partition), pr := `if`(isPr, x, PWToPartition(x));
       if chk(pr) then
           (`if`(isPr, a->a, PartitionToPW))(f(pr));
       else
           FAIL
       end if;
   end proc,

   # applies a function to the arg if arg::Partition,
   # and if arg::piecewise, then converts the piecewise to a partition,
   # applies the function, then converts back to piecewise
   # this mainly acts as a sanity check

   AppPartOrPw := proc(f::anything # TODO better type
                      ,x::Or(Partition,specfunc(piecewise))
                      )
       if x::Partition then
           f(x);
       else
           PartitionToPW(f(PWToPartition(x)));
       end if;
   end proc,

   #Check whether the conditions of a Partition depend on any of a set of names.
   ConditionsDepend:= proc(P::Partition, V::{name, list(name), set(name)}, $)
   local p;
      for p in op(1,P) do if depends(condOf(p), V) then return true end if end do;
      false
   end proc
;

uses Hakaru;

   ModuleLoad()
end module:


abstract out argument checking.

abstract out argument checking.

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()
#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".
local
   #The object's (internal) constructor. This just checks the argument types and
   #returns unevaluated.
   PARTITION::static:= proc(
      Pairs::set(
         record(
            #The type `anything` below should be some boolean type, but we'll
            #need to write our own as neither Maple's 'boolean' nor
            #'boolean &under (convert, boolean_operator)' is inclusive enough.
            cond::anything,
            val::t_Hakaru
            #Is that inclusive enough?
         )
      ),
      $ #no optional arguments, for now at least
   )::Partition;
     'procname'(_passed)
   end proc,

   ModuleLoad::static:= proc()
      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([eval(branch):-cond, eval(branch):-val][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(Partition, specfunc(PARTITION));
      NULL
   end proc,

   ModuleUnload:= proc()
      TypeTools:-RemoveType(Partition);
      NULL
   end proc,

   # abstract out all argument checking for map-like functions
   map_check := proc(p)
      local pos, err;
      if p::indexed then
         pos:= op(p);
         if not [pos]::[posint] then
            err := sprintf("Expected positive integer index; received %1", [pos]);
            return err;
         end if
      else
         pos:= 1
      end if;
      if nargs-1 <= pos then
         err := sprintf("Expected at least %1 arguments; received %2", pos+1, nargs-1);
         return err;
      end if;
      if not args[pos+2]::Partition then
         err := sprintf("Expected a Partition; received %1", args[pos+2]);
         return err;
      end if;
      return pos;
   end proc
;
export
   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= {seq(Record('cond'= Terms[k], 'val'= Terms[k+1]), k= 1..nargs-1, 2)};
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(p:-cond, p= op(P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, t_Hakaru]);
   local p;
      {seq([p:-cond, p:-val], p= op(P))}
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos;
      res := map_check(procname, args);
      if res::string then error res else pos := res; end if;
      PARTITION(
         {seq(
            Record(
               'cond'= pair:-cond,
               'val'= f(args[2..pos], pair:-val, args[pos+2..])
            ),
            pair= op(args[pos+1])
         )}
      )
   end proc

;
   ModuleLoad()
end module:

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()
#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".
local
   #The object's (internal) constructor. This just checks the argument types and
   #returns unevaluated.
   PARTITION::static:= proc(
      Pairs::set(
         record(
            #The type `anything` below should be some boolean type, but we'll
            #need to write our own as neither Maple's 'boolean' nor
            #'boolean &under (convert, boolean_operator)' is inclusive enough.
            cond::anything,
            val::t_Hakaru
            #Is that inclusive enough?
         )
      ),
      $ #no optional arguments, for now at least
   )::Partition;
     'procname'(_passed)
   end proc,

   ModuleLoad::static:= proc()
      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([eval(branch):-cond, eval(branch):-val][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(Partition, specfunc(PARTITION));
      NULL
   end proc,

   ModuleUnload:= proc()
      TypeTools:-RemoveType(Partition);
      NULL
   end proc
;
export
   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= {seq(Record('cond'= Terms[k], 'val'= Terms[k+1]), k= 1..nargs-1, 2)};
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(p:-cond, p= op(P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, t_Hakaru]);
   local p;
      {seq([p:-cond, p:-val], p= op(P))}
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos;
      if procname::indexed then
         pos:= op(procname);
         if not [pos]::[posint] then
            error "Expected positive integer index; received %1", [pos]
         end if
      else
         pos:= 1
      end if;
      if nargs <= pos then
         error "Expected at least %1 arguments; received %2", pos+1, nargs
      end if;
      if not args[pos+1]::Partition then
         error "Expected a Partition; received %1", args[pos+1]
      end if;
      PARTITION(
         {seq(
            Record(
               'cond'= pair:-cond,
               'val'= f(args[2..pos], pair:-val, args[pos+2..])
            ),
            pair= op(args[pos+1])
         )}
      )
   end proc
;
   ModuleLoad()
end module:


Not needed here

Not needed here

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()

option package;

#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".

global Piece;

local
   Umap := proc(f,x,$)
       f(op(0,x))( map( p -> Piece(f(condOf(p)),f(valOf(p)))
                      , op(1,x) ))
   end proc,

   isPartitionPieceOf := proc( p, elem_t := anything )
      type(p, 'Piece(PartitionCond, elem_t)');
   end proc,

   isPartitionOf := proc( e, elem_t := anything )
      type(e, 'PARTITION(list(PartitionPiece(elem_t)))' );
   end proc,

   ModuleLoad::static:= proc()
      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([ condOf(eval(branch)), valOf(eval(branch))][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(PartitionCond, {relation, boolean, `::`, specfunc({`And`,`Or`,`Not`}), `and`, `or`, `not`});
      TypeTools:-AddType(PartitionPiece, isPartitionPieceOf);
      TypeTools:-AddType(Partition, isPartitionOf);

      # global extensions to maple functionality
      :-`eval/PARTITION` :=
      proc(p, eqs, $)
          local q, r;
          q := Umap(x->eval(x,eqs), p);
          r := eval( PartitionToPW(q), eqs );
          if r :: specfunc('piecewise') then
              q := PWToPartition( r , 'do_solve' );
          else
              q := r;
          end if;
          q;
      end proc;

      :-`depends/PARTITION` :=
      proc(parts, nms, $)
         local dp := (x -> depends(x, nms));
         `or`(op ( map(p-> dp(condOf(p)) or dp(valOf(p)), parts) ));
      end proc;

      :-`diff/PARTITION` :=
      proc(parts, wrt, $)
          local pw  := PartitionToPW(PARTITION(parts))
              , dpw := diff(pw, wrt)
              , r   := PWToPartition(dpw, 'do_solve' )
              , r0, r1;
            ;
          r0 := Simpl:-singular_pts(r);

          # probably a better way to do this; we really only want to simplify
          # sums and products of integrals and summations
          r1 := subsindets(r0, algebraic, `simplify`);

          userinfo(10, 'disint_trace',
                   printf("  input        : \n\t%a\n\n"
                          "  diff         : \n\t%a\n\n"
                          "  singular pts : \n\t%a\n\n"
                          "  simplified   : \n\t%a\n\n\n"
                          , parts, r, r0, r1 ));

          r1;
      end proc;

      :-`simplify/PARTITION` := Simpl;

      NULL
   end proc,

   ModuleUnload::static:= proc()
      TypeTools:-RemoveType(Partition);
      TypeTools:-RemoveType(PartitionPiece);
      NULL
   end proc,

   # abstract out all argument checking for map-like functions
   map_check := proc(p)
      local pos, err;
      if p::indexed then
         pos:= op(p);
         if not [pos]::[posint] then
            err := sprintf("Expected positive integer index; received %a", [pos]);
            return err;
         end if
      else
         pos:= 1
      end if;
      if nargs-1 <= pos then
         err := sprintf("Expected at least %d arguments; received %d", pos+1, nargs-1);
         return err;
      end if;
      if not args[pos+2]::Partition then
         err := sprintf("Expected a Partition; received %a", args[pos+2]);
         return err;
      end if;
      return pos;
   end proc,

   bool_And := proc()
       if nargs=0 then true
       elif nargs=1 then args[1]
       else `And`(args)
       end if;
   end proc,

   bool_Or := proc()
       if nargs=0 then false
       elif nargs=1 then args[1]
       else `Or`(args)
       end if;
   end proc,

   extr_conjs := proc(x,$)
       if x::{specfunc(`And`), `and`} then
           map(extr_conjs, [op(x)])[];
       else
           x
       end if
   end proc,

   pw_cond_ctx := proc(ctx_, p, $)
       local ps, ctx, ctxN, cond, ncond;
       ps, ctx := op(ctx_); cond := condOf(p);
       cond := {extr_conjs(cond)};
       cond := remove(x->x in ctx, cond);
       ncond := `if`(nops(cond)=1
                     , KB:-negate_rel(op(1,cond))
                     , `Not`(bool_And(op(cond))) );
       cond := bool_And(op(cond));
       ctx := ctx union { ncond };
       [ [ op(ps), Piece(cond, valOf(p)) ], ctx ]
   end proc
;
export

   condOf := proc(x::specfunc(`Piece`),$)
       op(1,x);
   end proc,

   valOf := proc(x::specfunc(`Piece`),$)
       op(2,x);
   end proc,

   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= [ seq(Piece( Terms[k], Terms[k+1]), k= 1..nargs-1, 2) ] ;
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(condOf(p), p= op(1,P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, anything]);
   local p;
      {seq([condOf(p), valOf(p)], p= op(1,P)) }
   end proc,

   Pieces := proc(cs0,es0)::list(PartitionPiece);
       local es, cs;
       if es0 :: {set,list} then
           es := es0;
       else
           es := {es0};
       end if;
       if cs0 :: {set,list,specfunc(`And`),`and`} then
           cs := cs0;
       else
           cs := {cs0};
       end if;
       [seq(seq(Piece(c,e),c=cs),e=es)];
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos,res;
      res := map_check(procname, args);
      if res::string then error res else pos := res; end if;
      PARTITION(
         [seq(
            Piece(
               condOf(pair),
               f(args[2..pos], valOf(pair), args[pos+2..])
            ),
            pair= op(1, args[pos+1])
         )]
      )
   end proc,

   # a more complex mapping combinator which works on all 3 parts
   # not fully general, but made to work with KB
   # also, does not handle extra arguments (on purpose!)
   Amap::static:= proc(
      funcs::[anything, anything, anything], #`appliable` not inclusive enough.
      part::Partition
   )::Partition;
   local pair,pos,f,g,h,doIt;
      (f,g,h) := op(funcs);
      #sigh, we don't have a decent 'let', need to use a local proc
      doIt := proc(pair)
        local kb0 := h(condOf(pair));
        Piece( f(condOf(pair), kb0),
               g(valOf(pair), kb0));
      end proc;
      PARTITION(map(doIt,op(1,part)));
   end proc,

   Foldr := proc( cons, nil, prt :: Partition, $ )
       foldr( proc(p, x) cons(condOf(p), valOf(p), x); end proc
            , nil
            , op(op(1, prt))
            );
   end proc,

   Foldr_mb := proc( cons, nil, prt, $ )
       if prt :: Partition then
           Foldr(_passed);
       else
           cons( true, prt, nil )
       end if;
   end proc,

   PartitionToPW := proc(x::Partition)#::specfunc(piecewise);
       local parts := op(1,x);
       parts := foldl(pw_cond_ctx, [ [], {} ], op(parts) );
       parts := [seq([condOf(p), valOf(p)][], p=op(1,parts))];
       if op(-2, parts) :: identical(true) then
           parts := subsop(-2=NULL, parts);
       end if;
       piecewise(op(parts));
   end proc,

   # convert a piecewise to a partition, which is straightforward except:
   # - if any of the branches are unreachable, they are removed
   # - if the last clause is (implicitly) `otherwise`, that clause is filled in
   #     appropriately

   # note that if the piecewise does not cover the entire domain,
   #   then this Partition will be 'invalid' (in the sense that it also
   #   will not cover the entire domain) - the 'correct' thing to do would
   #   probably be to add a new clause whose value is 'undefined'

   # the logic of this function is already essentially implemented, by KB
   # in fact, kb_piecewise does something extremely similar to this
   PWToPartition := proc(x::specfunc(piecewise))::Partition;

       # each clause evaluated under the context so far,
       # which is the conjunction of the negations of all clauses
       # so far
       local ctx := true, n := nops(x), cls := [], cnd, ncnd, i, q
           , ctxC, cl;
       userinfo(5, PWToPartition
               , printf("PWToPartition: found %d ops in %a \n ", n, x) );
       # handles all but the `otherwise` case if there is such a case
       for i in seq(q, q = 1 .. iquo(n, 2)) do
           userinfo(3, PWToPartition
                    , printf("PWToPartition: looking at clause %d (op %d) \n ", i, 2*i-1));

           cnd := op(2*i-1,x); # the clause as given

           # if this clause is unreachable, then every subsequent clause will be as well
           if ctx :: identical(false) then
               return PARTITION( cls );
           else
               ctxC := `And`(cnd, ctx);              # the condition, along with the context (which is implicit in pw)
               ctxC := Simpl:-condition(ctxC, _rest);

               if cnd :: `=` then
                   ncnd := lhs(cnd) <> rhs(cnd);
               else
                   ncnd := Not(cnd);
               end if;

               ctx  := `And`(ncnd, ctx); # the context for the next clause

               userinfo(3, PWToPartition, printf("PWToPartition: ctx after %d clauses "
                                                 "is %a\n", i, ctx));

               if ctxC :: identical(false,[]) then # this clause is actually unreachable
                   return(PARTITION(cls));
               else
                   cls := [ op(cls), op(Pieces(ctxC,[op(2*i,x)])) ];
               end if;
           end if;

       end do;

       # if there is an otherwise case, handle that.
       if n::odd then
           ctx := Simpl:-condition(ctx, _rest);
           if not ctx :: identical(false,[]) then
               cls := [ op(cls), op(Pieces(ctx,[op(n,x)])) ];
           end if;
       end if;

       if nops(cls) = 0 then
           error "PWToPartition: the piecewise  %1  produced an empty partition\n", x;
       end if;

       PARTITION( cls );
   end proc,

   # applies a function to the arg if arg::Partition,
   # and if arg::piecewise, then converts the piecewise to a partition,
   # applies the function, then converts back to piecewise
   # this mainly acts as a sanity check
   AppPartOrPw := proc(f::anything # TODO better type
                      ,x::Or(Partition,specfunc(piecewise))
                      )
       if x::Partition then
           f(x);
       else
           PartitionToPW(f(PWToPartition(x)));
       end if;
   end proc,

   #Check whether the conditions of a Partition depend on any of a set of names.
   ConditionsDepend:= proc(P::Partition, V::{name, list(name), set(name)}, $)
   local p;
      for p in op(1,P) do if depends(condOf(p), V) then return true end if end do;
      false
   end proc,

   Simpl := module()
       export ModuleApply := (single_branch@remove_false_pieces);

       export single_nonzero_piece := proc(e, $)
           local zs, nzs;
           if e :: Partition then
               zs, nzs := selectremove(p -> Testzero(valOf(p)), op(1, e));
               if nops(nzs) = 1 then
                   return condOf(op(1,nzs)) , valOf(op(1,nzs))
               end if;
           end if;
           true, e
       end proc;

       export remove_false_pieces := proc(e::Partition, $)
           PARTITION(remove(p -> not(coulditbe(condOf(p))), op(1, e)));
       end proc;

       export single_branch := proc(e::Partition, $)
           if nops(op(1,e)) = 1 then
               op([1,1,2], e)
           else
               e
           end if;
       end proc;

       # Removal of singular points from partitions
       export singular_pts := module()
           # we can simplify the pieces which are equalities and whose LHS or RHS is
           # a name.
           local canSimp := c -> #valOf(c) :: identical('undefined') and
                                 condOf(c) :: `=` and (lhs(condOf(c)) :: name or rhs(condOf(c)) :: name);

           # determines if a given variable `t' has the given upper/lower `bnd'.
           local mentions_t_hi :=
                    t -> bnd -> cl -> has(cl, t<bnd) or has(cl, bnd>t);

           local mentions_t_lo :=
                    t -> bnd -> cl -> has(cl, bnd<t) or has(cl, t>bnd);

           # replace bounds with what we would get if the equality can be
           # integrated into other pieces
           local replace_with := t -> bnd -> x ->
                    if mentions_t_hi(t)(bnd)(x) then
                        t <= bnd
                    elif mentions_t_lo(t)(bnd)(x) then
                        t >= bnd
                    else
                        x
                    end if;

           local set_xor := proc(a,b,$) (a union b) intersect (a minus b) end proc;

           # this loops over the pieces to replace, keeping a state consisting of
           # the "rest" of the pieces
           local tryReplacePieces :=
                   proc(replPieces, otherPieces,cmp,$)
                     local rpp := replPieces, otp := otherPieces, nm, val, rp, rpv;
                     for rp in rpp do
                         rp, rpv := op(rp);
                         nm   := `if`(lhs(rp)::name, lhs(rp), rhs(rp));
                         val  := `if`(lhs(rp)::name, rhs(rp), lhs(rp));
                         otp := tryReplacePiece( nm, val, rpv, otp, cmp )
                     end do;
                     otp;
                   end proc;

           local do_eval_for_cmp := proc(ev, x, $)
              try
                  return ev(x);
              catch "numeric exception: division by zero":
                  # a cheap and dirty way of discovering we have
                  # an expression which is singular at the point
              end try;
              x
           end proc;

           local tryReplacePiece :=
               proc(vrNm, vrVal, pc0val, pcs, eval_cmp,$)
                 local pcs0 := pcs, pcs1, qs0, qs1, qs2, vrEq := vrNm=vrVal, vs2, ret;
                 ret := [ Piece(vrEq, pc0val), op(pcs0) ] ;
                 # speculatively replace the conditions
                 pcs1 := subsindets(pcs0, relation, replace_with(vrNm)(vrVal));
                 # convert to sets and take the "set xor", which will contain
                 # only those elements which are not common to both sets.
                 qs0, qs1 := seq({op(qs)},qs=(pcs0,pcs1));
                 qs2 := set_xor(qs1, qs0);
                 # if we have updated precisely two pieces (an upper and lower bound)
                 if nops(qs2) = 2 then
                     # get the values of those pieces, and the value of the
                     # piece to be replaced, if that isn't undefined
                     vs2 := map(valOf, qs2);
                     if not pc0val :: identical('undefined') then
                         vs2 := { pc0val, op(vs2) };
                     end if;
                     # substitute the equality over the piece values
                     vs2 := map(q -> do_eval_for_cmp(r -> eval_cmp(subs(vrEq, r)), q), vs2);
                     # if they are identically equal, return the original
                     # "guess"
                     if nops(vs2) = 1 then ret := pcs1; end if;
                 end if;
                 ret;
               end proc;

           export ModuleApply := proc(p_,{eval_cmp:='value'},$)
              local p := p_, r := p, uc, oc;
              # if the partition contains case of the form `x = t', where `t' is a
              # constant (or term??) and `x' is a variable, and the value of that
              # case is `undefined', then we may be able to eliminate it (if another
              # case includes that point)
              r := op(1,r);
              uc, oc := selectremove(canSimp, r);
              PARTITION(tryReplacePieces(uc, oc, eval_cmp));
           end proc;
       end module; # singular_pts

       export condition := module()
           local is_extra_sol := x -> (x :: `=` and rhs(x)=lhs(x) and lhs(x) :: name);
           local eval_ctrs := ctx -> eval(ctx, [`And`=`and`, `Not`=`not`]);

           local postproc_for_solve := proc(ctx, ctxSlv)::{identical(false), list({boolean,relation})};
               local ctxC := ctxSlv;
               if ctxC = [] then
                   ctxC := false ;
               elif nops(ctxC)> 1 then
                   ctxC := map(x -> postproc_for_solve(ctx, [x], _rest)[], ctxC);
               elif nops(ctxC) = 1 then
                   ctxC := op(1,ctxC);
                   if ctxC :: set then
                       ctxC := remove(is_extra_sol, ctxC);
                       if ctxC :: identical('{}') then
                           ctxC := eval_ctrs(ctx);
                       else
                           ctxC := `and`(op(ctxC));
                       end if ;
                       ctxC := [ctxC];
                   elif ctxC :: specfunc('piecewise') then
                       ctxC := PWToPartition(ctxC, _rest);
                       ctxC := [ seq( map( o -> condOf(c) and o
                                         , postproc_for_solve(ctx, valOf(c), _rest))[]
                                    , c=op(1, ctxC) )] ;
                   else
                       ctxC := FAIL;
                   end if;
               else
                   ctxC := FAIL;
               end if;
               if ctxC = FAIL then
                   error "don't know what to do with %1", ctxSlv;
               else
                   ctxC;
               end if;
           end proc;

           export ModuleApply := proc(ctx)::list;
               local ctxC := ctx;

               if ctx :: identical(true) then
                   error "Simpl:-condition: don't know what to do with %1", ctxC;
               end if;

               if 'do_solve' in {_rest} then
                   ctxC := solve({ctxC}, 'useassumptions'=true);
                   if ctxC = NULL and indets(ctxC, specfunc(`exp`)) <> {} then
                       return [ctx];
                   end if;
                   ctxC := postproc_for_solve(ctx, [ctxC], _rest);
                   if indets(ctxC, specfunc({`Or`, `or`})) <> {} then
                       userinfo(10, 'Simpl:-condition',
                           printf("output: \n"
                                  "  %a\n\n"
                                  , ctxC )
                          );
                   end if;
               else
                   ctxC := Domain:-simpl_relation(ctxC, norty='DNF');
                   ctxC := subsindets(ctxC, list, `and`@op);
                   if ctxC :: set then
                       ctxC := [op(ctxC)]
                   else
                       ctxC := [ctxC];
                   end if;
               end if;
               ctxC;
           end proc;
       end module; #Simpl:-condition
   end module, #Simpl

   SamePartition := proc(eqCond, eqPart, $) proc(p0 :: Partition, p1 :: Partition, $)
     local ps0, ps1, pc0, the;
     ps0, ps1 := seq(op(1,p),p=(p0,p1));

     if nops(ps0) <> nops(ps1) then
         return false;
     end if;

     for pc0 in ps0 do
         the, ps1 :=
           selectremove(pc1 -> eqCond( condOf(pc1), condOf(pc0) ) and
                               eqPart( valOf(pc1) , valOf(pc0)  )
                       ,ps1);
         if nops(the) <> 1 then
             return false;
         end if;
     end do;

     true;
   end proc; end proc

;

uses Hakaru;

   ModuleLoad()
end module:

#This module implements `Partition`---Hakaru's replacement for Maple's
#endogenous and unwieldy `piecewise`.

#The outer data structure for a Partition is a function, PARTITION(...), (just like it
#is for piecewise.

Partition:= module()

option package;

#This module is essentially an object, but we decided, for now at least, to not
#use Maple's "option object".

global Piece;

local
   Umap := proc(f,x,$)
       f(op(0,x))( map( p -> Piece(f(condOf(p)),f(valOf(p)))
                      , op(1,x) ))
   end proc,

   isPartitionPieceOf := proc( p, elem_t := anything )
      type(p, 'Piece(PartitionCond, elem_t)');
   end proc,

   isPartitionOf := proc( e, elem_t := anything )
      type(e, 'PARTITION(list(PartitionPiece(elem_t)))' );
   end proc,

   ModuleLoad::static:= proc()
      :-`print/PARTITION`:= proc(SetOfRecords)
      local branch;
         `print/%piecewise`(
            seq([ condOf(eval(branch)), valOf(eval(branch))][], branch= SetOfRecords)
         )
      end proc;

      TypeTools:-AddType(PartitionCond, {relation, boolean, `::`, specfunc({`And`,`Or`,`Not`}), `and`, `or`, `not`});
      TypeTools:-AddType(PartitionPiece, isPartitionPieceOf);
      TypeTools:-AddType(Partition, isPartitionOf);

      # global extensions to maple functionality
      :-`eval/PARTITION` :=
      proc(p, eqs, $)
          local q, r;
          q := Umap(x->eval(x,eqs), p);
          r := eval( PartitionToPW(q), eqs );
          if r :: specfunc('piecewise') then
              q := PWToPartition( r , 'do_solve' );
          else
              q := r;
          end if;
          q;
      end proc;

      :-`depends/PARTITION` :=
      proc(parts, nms, $)
         local dp := (x -> depends(x, nms));
         `or`(op ( map(p-> dp(condOf(p)) or dp(valOf(p)), parts) ));
      end proc;

      :-`diff/PARTITION` :=
      proc(parts, wrt, $)
          local pw  := PartitionToPW(PARTITION(parts))
              , dpw := diff(pw, wrt)
              , r   := PWToPartition(dpw, 'do_solve' )
              , r0, r1;
            ;
          r0 := Simpl:-singular_pts(r);

          # probably a better way to do this; we really only want to simplify
          # sums and products of integrals and summations
          r1 := subsindets(r0, algebraic, `simplify`);

          userinfo(10, 'disint_trace',
                   printf("  input        : \n\t%a\n\n"
                          "  diff         : \n\t%a\n\n"
                          "  singular pts : \n\t%a\n\n"
                          "  simplified   : \n\t%a\n\n\n"
                          , parts, r, r0, r1 ));

          r1;
      end proc;

      :-`simplify/PARTITION` := Simpl;

      NULL
   end proc,

   ModuleUnload::static:= proc()
      TypeTools:-RemoveType(Partition);
      TypeTools:-RemoveType(PartitionPiece);
      NULL
   end proc,

   # abstract out all argument checking for map-like functions
   map_check := proc(p)
      local pos, err;
      if p::indexed then
         pos:= op(p);
         if not [pos]::[posint] then
            err := sprintf("Expected positive integer index; received %a", [pos]);
            return err;
         end if
      else
         pos:= 1
      end if;
      if nargs-1 <= pos then
         err := sprintf("Expected at least %d arguments; received %d", pos+1, nargs-1);
         return err;
      end if;
      if not args[pos+2]::Partition then
         err := sprintf("Expected a Partition; received %a", args[pos+2]);
         return err;
      end if;
      return pos;
   end proc,

   bool_And := proc()
       if nargs=0 then true
       elif nargs=1 then args[1]
       else `And`(map(a->if a::specfunc(`And`)then op(a) else a end if
                     ,a=[args])[])
       end if;
   end proc,

   bool_Or := proc()
       if nargs=0 then false
       elif nargs=1 then args[1]
       else `Or`(map(a->if a::specfunc(`Or`)then op(a) else a end if
                     ,a=[args])[])
       end if;
   end proc,

   extr_conjs := proc(x,$)
       if x::{specfunc(`And`), `and`} then
           map(extr_conjs, [op(x)])[];
       else
           x
       end if
   end proc,

   pw_cond_ctx := proc(ctx_, p, $)
       local ps, ctx, ctxN, cond, ncond;
       ps, ctx := op(ctx_); cond := condOf(p);
       cond := {extr_conjs(cond)};
       cond := remove(x->x in ctx, cond);
       ncond := `if`(nops(cond)=1
                     , KB:-negate_rel(op(1,cond))
                     , `Not`(bool_And(op(cond))) );
       cond := bool_And(op(cond));
       ctx := ctx union { ncond };
       [ [ op(ps), Piece(cond, valOf(p)) ], ctx ]
   end proc
;
export

   condOf := proc(x::specfunc(`Piece`),$)
       op(1,x);
   end proc,

   valOf := proc(x::specfunc(`Piece`),$)
       op(2,x);
   end proc,

   #This is the exported lazy-syntax constructor. The syntax is like piecewise except
   #that there can be no 'otherwise'.
   ModuleApply::static:= proc(Terms::seq(anything))::Partition;
   local pair, s, Pairs, k;
      if nargs::odd then
         error "Expected an even number of arguments"
      end if;
      s:= [ seq(Piece( Terms[k], Terms[k+1]), k= 1..nargs-1, 2) ] ;
      userinfo(3, PARTITION, s);
      PARTITION(s)
   end proc,

   #Deconstructor that returns just the conditions as a set
   Conditions::static:= proc(P::Partition)::set;
   local p;
      {seq(condOf(p), p= op(1,P))}
   end proc,

   #Deconstructor that returns a set of [cond, val] pairs
   Pairs:= proc(P::Partition)::set([anything, anything]);
   local p;
      {seq([condOf(p), valOf(p)], p= op(1,P)) }
   end proc,

   Pieces := proc(cs0,es0)::list(PartitionPiece);
       local es, cs;
       if es0 :: {set,list} then
           es := es0;
       else
           es := {es0};
       end if;
       if cs0 :: {set,list,specfunc(`And`),`and`} then
           cs := cs0;
       else
           cs := {cs0};
       end if;
       [seq(seq(Piece(c,e),c=cs),e=es)];
   end proc,

   #This is just `map` for Partitions.
   Pmap::static:= proc(
      f::anything #`appliable` not inclusive enough.
      #Allow additional args, just like `map`
   )::Partition;
   local pair,pos,res;
      res := map_check(procname, args);
      if res::string then error res else pos := res; end if;
      PARTITION(
         [seq(
            Piece(
               condOf(pair),
               f(args[2..pos], valOf(pair), args[pos+2..])
            ),
            pair= op(1, args[pos+1])
         )]
      )
   end proc,

   # a more complex mapping combinator which works on all 3 parts
   # not fully general, but made to work with KB
   # also, does not handle extra arguments (on purpose!)
   Amap::static:= proc(
      funcs::[anything, anything, anything], #`appliable` not inclusive enough.
      part::Partition
   )::Partition;
   local pair,pos,f,g,h,doIt;
      (f,g,h) := op(funcs);
      #sigh, we don't have a decent 'let', need to use a local proc
      doIt := proc(pair)
        local kb0 := h(condOf(pair));
        Piece( f(condOf(pair), kb0),
               g(valOf(pair), kb0));
      end proc;
      PARTITION(map(doIt,op(1,part)));
   end proc,

   Foldr := proc( cons, nil, prt :: Partition, $ )
       foldr( proc(p, x) cons(condOf(p), valOf(p), x); end proc
            , nil
            , op(op(1, prt))
            );
   end proc,

   Foldr_mb := proc( cons, nil, prt, $ )
       if prt :: Partition then
           Foldr(_passed);
       else
           cons( true, prt, nil )
       end if;
   end proc,

   PartitionToPW := proc(x::Partition)#::specfunc(piecewise);
       local parts := op(1,x);
       parts := foldl(pw_cond_ctx, [ [], {} ], op(parts) );
       parts := [seq([condOf(p), valOf(p)][], p=op(1,parts))];
       if op(-2, parts) :: identical(true) then
           parts := subsop(-2=NULL, parts);
       end if;
       piecewise(op(parts));
   end proc,

   # convert a piecewise to a partition, which is straightforward except:
   # - if any of the branches are unreachable, they are removed
   # - if the last clause is (implicitly) `otherwise`, that clause is filled in
   #     appropriately

   # note that if the piecewise does not cover the entire domain,
   #   then this Partition will be 'invalid' (in the sense that it also
   #   will not cover the entire domain) - the 'correct' thing to do would
   #   probably be to add a new clause whose value is 'undefined'

   # the logic of this function is already essentially implemented, by KB
   # in fact, kb_piecewise does something extremely similar to this
   PWToPartition := proc(x::specfunc(piecewise))::Partition;

       # each clause evaluated under the context so far,
       # which is the conjunction of the negations of all clauses
       # so far
       local ctx := true, n := nops(x), cls := [], cnd, ncnd, i, q
           , ctxC, cl;
       userinfo(5, PWToPartition
               , printf("PWToPartition: found %d ops in %a \n ", n, x) );
       # handles all but the `otherwise` case if there is such a case
       for i in seq(q, q = 1 .. iquo(n, 2)) do
           userinfo(3, PWToPartition
                    , printf("PWToPartition: looking at clause %d (op %d) \n ", i, 2*i-1));

           cnd := op(2*i-1,x); # the clause as given

           # if this clause is unreachable, then every subsequent clause will be as well
           if ctx :: identical(false) then
               return PARTITION( cls );
           else
               ctxC := `And`(cnd, ctx);              # the condition, along with the context (which is implicit in pw)
               ctxC := Simpl:-condition(ctxC, _rest);

               if cnd :: `=` then
                   ncnd := lhs(cnd) <> rhs(cnd);
               else
                   ncnd := Not(cnd);
               end if;

               ctx  := `And`(ncnd, ctx); # the context for the next clause

               userinfo(3, PWToPartition, printf("PWToPartition: ctx after %d clauses "
                                                 "is %a\n", i, ctx));

               if ctxC :: identical(false,[]) then # this clause is actually unreachable
                   return(PARTITION(cls));
               else
                   cls := [ op(cls), op(Pieces(ctxC,[op(2*i,x)])) ];
               end if;
           end if;

       end do;

       # if there is an otherwise case, handle that.
       if n::odd then
           ctx := Simpl:-condition(ctx, _rest);
           if not ctx :: identical(false,[]) then
               cls := [ op(cls), op(Pieces(ctx,[op(n,x)])) ];
           end if;
       end if;

       if nops(cls) = 0 then
           error "PWToPartition: the piecewise  %1  produced an empty partition\n", x;
       end if;

       PARTITION( cls );
   end proc,

   # applies a function to the arg if arg::Partition,
   # and if arg::piecewise, then converts the piecewise to a partition,
   # applies the function, then converts back to piecewise
   # this mainly acts as a sanity check
   AppPartOrPw := proc(f::anything # TODO better type
                      ,x::Or(Partition,specfunc(piecewise))
                      )
       if x::Partition then
           f(x);
       else
           PartitionToPW(f(PWToPartition(x)));
       end if;
   end proc,

   #Check whether the conditions of a Partition depend on any of a set of names.
   ConditionsDepend:= proc(P::Partition, V::{name, list(name), set(name)}, $)
   local p;
      for p in op(1,P) do if depends(condOf(p), V) then return true end if end do;
      false
   end proc,

   Simpl := module()
       export ModuleApply := (single_branch@remove_false_pieces);

       export single_nonzero_piece := proc(e, $)
           local zs, nzs;
           if e :: Partition then
               zs, nzs := selectremove(p -> Testzero(valOf(p)), op(1, e));
               if nops(nzs) = 1 then
                   return condOf(op(1,nzs)) , valOf(op(1,nzs))
               end if;
           end if;
           true, e
       end proc;

       export remove_false_pieces := proc(e::Partition, $)
           PARTITION(remove(p -> not(coulditbe(condOf(p))), op(1, e)));
       end proc;

       export single_branch := proc(e::Partition, $)
           if nops(op(1,e)) = 1 then
               op([1,1,2], e)
           else
               e
           end if;
       end proc;

       # Removal of singular points from partitions
       export singular_pts := module()
           # we can simplify the pieces which are equalities and whose LHS or RHS is
           # a name.
           local canSimp := c -> #valOf(c) :: identical('undefined') and
                                 condOf(c) :: `=` and (lhs(condOf(c)) :: name or rhs(condOf(c)) :: name);

           # determines if a given variable `t' has the given upper/lower `bnd'.
           local mentions_t_hi :=
                    t -> bnd -> cl -> has(cl, t<bnd) or has(cl, bnd>t);

           local mentions_t_lo :=
                    t -> bnd -> cl -> has(cl, bnd<t) or has(cl, t>bnd);

           # replace bounds with what we would get if the equality can be
           # integrated into other pieces
           local replace_with := t -> bnd -> x ->
                    if mentions_t_hi(t)(bnd)(x) then
                        t <= bnd
                    elif mentions_t_lo(t)(bnd)(x) then
                        t >= bnd
                    else
                        x
                    end if;

           local set_xor := proc(a,b,$) (a union b) intersect (a minus b) end proc;

           # this loops over the pieces to replace, keeping a state consisting of
           # the "rest" of the pieces
           local tryReplacePieces :=
                   proc(replPieces, otherPieces,cmp,$)
                     local rpp := replPieces, otp := otherPieces, nm, val, rp, rpv;
                     for rp in rpp do
                         rp, rpv := op(rp);
                         nm   := `if`(lhs(rp)::name, lhs(rp), rhs(rp));
                         val  := `if`(lhs(rp)::name, rhs(rp), lhs(rp));
                         otp := tryReplacePiece( nm, val, rpv, otp, cmp )
                     end do;
                     otp;
                   end proc;

           local do_eval_for_cmp := proc(ev, x, $)
              try
                  return ev(x);
              catch "numeric exception: division by zero":
                  # a cheap and dirty way of discovering we have
                  # an expression which is singular at the point
              end try;
              x
           end proc;

           local tryReplacePiece :=
               proc(vrNm, vrVal, pc0val, pcs, eval_cmp,$)
                 local pcs0 := pcs, pcs1, qs0, qs1, qs2, vrEq := vrNm=vrVal, vs2, ret;
                 ret := [ Piece(vrEq, pc0val), op(pcs0) ] ;
                 # speculatively replace the conditions
                 pcs1 := subsindets(pcs0, relation, replace_with(vrNm)(vrVal));
                 # convert to sets and take the "set xor", which will contain
                 # only those elements which are not common to both sets.
                 qs0, qs1 := seq({op(qs)},qs=(pcs0,pcs1));
                 qs2 := set_xor(qs1, qs0);
                 # if we have updated precisely two pieces (an upper and lower bound)
                 if nops(qs2) = 2 then
                     # get the values of those pieces, and the value of the
                     # piece to be replaced, if that isn't undefined
                     vs2 := map(valOf, qs2);
                     if not pc0val :: identical('undefined') then
                         vs2 := { pc0val, op(vs2) };
                     end if;
                     # substitute the equality over the piece values
                     vs2 := map(q -> do_eval_for_cmp(r -> eval_cmp(subs(vrEq, r)), q), vs2);
                     # if they are identically equal, return the original
                     # "guess"
                     if nops(vs2) = 1 then ret := pcs1; end if;
                 end if;
                 ret;
               end proc;

           export ModuleApply := proc(p_,{eval_cmp:='value'},$)
              local p := p_, r := p, uc, oc;
              # if the partition contains case of the form `x = t', where `t' is a
              # constant (or term??) and `x' is a variable, and the value of that
              # case is `undefined', then we may be able to eliminate it (if another
              # case includes that point)
              r := op(1,r);
              uc, oc := selectremove(canSimp, r);
              PARTITION(tryReplacePieces(uc, oc, eval_cmp));
           end proc;
       end module; # singular_pts

       export condition := module()
           local is_extra_sol := x -> (x :: `=` and rhs(x)=lhs(x) and lhs(x) :: name);
           local eval_ctrs := ctx -> eval(ctx, [`And`=`and`, `Not`=`not`]);

           local postproc_for_solve := proc(ctx, ctxSlv)::{identical(false), list({boolean,relation})};
               local ctxC := ctxSlv;
               if ctxC = [] then
                   ctxC := false ;
               elif nops(ctxC)> 1 then
                   ctxC := map(x -> postproc_for_solve(ctx, [x], _rest)[], ctxC);
               elif nops(ctxC) = 1 then
                   ctxC := op(1,ctxC);
                   if ctxC :: set then
                       ctxC := remove(is_extra_sol, ctxC);
                       if ctxC :: identical('{}') then
                           ctxC := eval_ctrs(ctx);
                       else
                           ctxC := `and`(op(ctxC));
                       end if ;
                       ctxC := [ctxC];
                   elif ctxC :: specfunc('piecewise') then
                       ctxC := PWToPartition(ctxC, _rest);
                       ctxC := [ seq( map( o -> condOf(c) and o
                                         , postproc_for_solve(ctx, valOf(c), _rest))[]
                                    , c=op(1, ctxC) )] ;
                   else
                       ctxC := FAIL;
                   end if;
               else
                   ctxC := FAIL;
               end if;
               if ctxC = FAIL then
                   error "don't know what to do with %1", ctxSlv;
               else
                   ctxC;
               end if;
           end proc;

           export ModuleApply := proc(ctx)::list;
               local ctxC := ctx;

               if ctx :: identical(true) then
                   error "Simpl:-condition: don't know what to do with %1", ctxC;
               end if;

               if 'do_solve' in {_rest} then
                   ctxC := solve({ctxC}, 'useassumptions'=true);
                   if ctxC = NULL and indets(ctxC, specfunc(`exp`)) <> {} then
                       return [ctx];
                   end if;
                   ctxC := postproc_for_solve(ctx, [ctxC], _rest);
                   if indets(ctxC, specfunc({`Or`, `or`})) <> {} then
                       userinfo(10, 'Simpl:-condition',
                           printf("output: \n"
                                  "  %a\n\n"
                                  , ctxC )
                          );
                   end if;
               else
                   ctxC := Domain:-simpl_relation(ctxC, norty='DNF');
                   ctxC := subsindets(ctxC, list, `and`@op);
                   if ctxC :: set then
                       ctxC := [op(ctxC)]
                   else
                       ctxC := [ctxC];
                   end if;
               end if;
               ctxC;
           end proc;
       end module; #Simpl:-condition
   end module, #Simpl

   SamePartition := proc(eqCond, eqPart, $) proc(p0 :: Partition, p1 :: Partition, $)
     local ps0, ps1, pc0, the;
     ps0, ps1 := seq(op(1,p),p=(p0,p1));

     if nops(ps0) <> nops(ps1) then
         return false;
     end if;

     for pc0 in ps0 do
         the, ps1 :=
           selectremove(pc1 -> eqCond( condOf(pc1), condOf(pc0) ) and
                               eqPart( valOf(pc1) , valOf(pc0)  )
                       ,ps1);
         if nops(the) <> 1 then
             return false;
         end if;
     end do;

     true;
   end proc; end proc

;

uses Hakaru;

   ModuleLoad()
end module: