src/Provers/quasi.ML
 author wenzelm Sun Mar 07 12:19:47 2010 +0100 (2010-03-07) changeset 35625 9c818cab0dd0 parent 33063 4d462963a7db child 37744 3daaf23b9ab4 permissions -rw-r--r--
modernized structure Object_Logic;
```     1 (*  Author:     Oliver Kutter, TU Muenchen
```
```     2
```
```     3 Reasoner for simple transitivity and quasi orders.
```
```     4 *)
```
```     5
```
```     6 (*
```
```     7
```
```     8 The package provides tactics trans_tac and quasi_tac that use
```
```     9 premises of the form
```
```    10
```
```    11   t = u, t ~= u, t < u and t <= u
```
```    12
```
```    13 to
```
```    14 - either derive a contradiction, in which case the conclusion can be
```
```    15   any term,
```
```    16 - or prove the concluson, which must be of the form t ~= u, t < u or
```
```    17   t <= u.
```
```    18
```
```    19 Details:
```
```    20
```
```    21 1. trans_tac:
```
```    22    Only premises of form t <= u are used and the conclusion must be of
```
```    23    the same form.  The conclusion is proved, if possible, by a chain of
```
```    24    transitivity from the assumptions.
```
```    25
```
```    26 2. quasi_tac:
```
```    27    <= is assumed to be a quasi order and < its strict relative, defined
```
```    28    as t < u == t <= u & t ~= u.  Again, the conclusion is proved from
```
```    29    the assumptions.
```
```    30    Note that the presence of a strict relation is not necessary for
```
```    31    quasi_tac.  Configure decomp_quasi to ignore < and ~=.  A list of
```
```    32    required theorems for both situations is given below.
```
```    33 *)
```
```    34
```
```    35 signature LESS_ARITH =
```
```    36 sig
```
```    37   (* Transitivity of <=
```
```    38      Note that transitivities for < hold for partial orders only. *)
```
```    39   val le_trans: thm  (* [| x <= y; y <= z |] ==> x <= z *)
```
```    40
```
```    41   (* Additional theorem for quasi orders *)
```
```    42   val le_refl: thm  (* x <= x *)
```
```    43   val eqD1: thm (* x = y ==> x <= y *)
```
```    44   val eqD2: thm (* x = y ==> y <= x *)
```
```    45
```
```    46   (* Additional theorems for premises of the form x < y *)
```
```    47   val less_reflE: thm  (* x < x ==> P *)
```
```    48   val less_imp_le : thm (* x < y ==> x <= y *)
```
```    49
```
```    50   (* Additional theorems for premises of the form x ~= y *)
```
```    51   val le_neq_trans : thm (* [| x <= y ; x ~= y |] ==> x < y *)
```
```    52   val neq_le_trans : thm (* [| x ~= y ; x <= y |] ==> x < y *)
```
```    53
```
```    54   (* Additional theorem for goals of form x ~= y *)
```
```    55   val less_imp_neq : thm (* x < y ==> x ~= y *)
```
```    56
```
```    57   (* Analysis of premises and conclusion *)
```
```    58   (* decomp_x (`x Rel y') should yield SOME (x, Rel, y)
```
```    59        where Rel is one of "<", "<=", "=" and "~=",
```
```    60        other relation symbols cause an error message *)
```
```    61   (* decomp_trans is used by trans_tac, it may only return Rel = "<=" *)
```
```    62   val decomp_trans: theory -> term -> (term * string * term) option
```
```    63   (* decomp_quasi is used by quasi_tac *)
```
```    64   val decomp_quasi: theory -> term -> (term * string * term) option
```
```    65 end;
```
```    66
```
```    67 signature QUASI_TAC =
```
```    68 sig
```
```    69   val trans_tac: Proof.context -> int -> tactic
```
```    70   val quasi_tac: Proof.context -> int -> tactic
```
```    71 end;
```
```    72
```
```    73 functor Quasi_Tac(Less: LESS_ARITH): QUASI_TAC =
```
```    74 struct
```
```    75
```
```    76 (* Internal datatype for the proof *)
```
```    77 datatype proof
```
```    78   = Asm of int
```
```    79   | Thm of proof list * thm;
```
```    80
```
```    81 exception Cannot;
```
```    82  (* Internal exception, raised if conclusion cannot be derived from
```
```    83      assumptions. *)
```
```    84 exception Contr of proof;
```
```    85   (* Internal exception, raised if contradiction ( x < x ) was derived *)
```
```    86
```
```    87 fun prove asms =
```
```    88   let fun pr (Asm i) = List.nth (asms, i)
```
```    89   |       pr (Thm (prfs, thm)) = (map pr prfs) MRS thm
```
```    90   in pr end;
```
```    91
```
```    92 (* Internal datatype for inequalities *)
```
```    93 datatype less
```
```    94    = Less  of term * term * proof
```
```    95    | Le    of term * term * proof
```
```    96    | NotEq of term * term * proof;
```
```    97
```
```    98  (* Misc functions for datatype less *)
```
```    99 fun lower (Less (x, _, _)) = x
```
```   100   | lower (Le (x, _, _)) = x
```
```   101   | lower (NotEq (x,_,_)) = x;
```
```   102
```
```   103 fun upper (Less (_, y, _)) = y
```
```   104   | upper (Le (_, y, _)) = y
```
```   105   | upper (NotEq (_,y,_)) = y;
```
```   106
```
```   107 fun getprf   (Less (_, _, p)) = p
```
```   108 |   getprf   (Le   (_, _, p)) = p
```
```   109 |   getprf   (NotEq (_,_, p)) = p;
```
```   110
```
```   111 (* ************************************************************************ *)
```
```   112 (*                                                                          *)
```
```   113 (* mkasm_trans sign (t, n) :  theory -> (Term.term * int)  -> less          *)
```
```   114 (*                                                                          *)
```
```   115 (* Tuple (t, n) (t an assumption, n its index in the assumptions) is        *)
```
```   116 (* translated to an element of type less.                                   *)
```
```   117 (* Only assumptions of form x <= y are used, all others are ignored         *)
```
```   118 (*                                                                          *)
```
```   119 (* ************************************************************************ *)
```
```   120
```
```   121 fun mkasm_trans thy (t, n) =
```
```   122   case Less.decomp_trans thy t of
```
```   123     SOME (x, rel, y) =>
```
```   124     (case rel of
```
```   125      "<="  =>  [Le (x, y, Asm n)]
```
```   126     | _     => error ("trans_tac: unknown relation symbol ``" ^ rel ^
```
```   127                  "''returned by decomp_trans."))
```
```   128   | NONE => [];
```
```   129
```
```   130 (* ************************************************************************ *)
```
```   131 (*                                                                          *)
```
```   132 (* mkasm_quasi sign (t, n) : theory -> (Term.term * int) -> less            *)
```
```   133 (*                                                                          *)
```
```   134 (* Tuple (t, n) (t an assumption, n its index in the assumptions) is        *)
```
```   135 (* translated to an element of type less.                                   *)
```
```   136 (* Quasi orders only.                                                       *)
```
```   137 (*                                                                          *)
```
```   138 (* ************************************************************************ *)
```
```   139
```
```   140 fun mkasm_quasi thy (t, n) =
```
```   141   case Less.decomp_quasi thy t of
```
```   142     SOME (x, rel, y) => (case rel of
```
```   143       "<"   => if (x aconv y) then raise Contr (Thm ([Asm n], Less.less_reflE))
```
```   144                else [Less (x, y, Asm n)]
```
```   145     | "<="  => [Le (x, y, Asm n)]
```
```   146     | "="   => [Le (x, y, Thm ([Asm n], Less.eqD1)),
```
```   147                 Le (y, x, Thm ([Asm n], Less.eqD2))]
```
```   148     | "~="  => if (x aconv y) then
```
```   149                   raise Contr (Thm ([(Thm ([(Thm ([], Less.le_refl)) ,(Asm n)], Less.le_neq_trans))], Less.less_reflE))
```
```   150                else [ NotEq (x, y, Asm n),
```
```   151                       NotEq (y, x,Thm ( [Asm n], thm "not_sym"))]
```
```   152     | _     => error ("quasi_tac: unknown relation symbol ``" ^ rel ^
```
```   153                  "''returned by decomp_quasi."))
```
```   154   | NONE => [];
```
```   155
```
```   156
```
```   157 (* ************************************************************************ *)
```
```   158 (*                                                                          *)
```
```   159 (* mkconcl_trans sign t : theory -> Term.term -> less                       *)
```
```   160 (*                                                                          *)
```
```   161 (* Translates conclusion t to an element of type less.                      *)
```
```   162 (* Only for Conclusions of form x <= y or x < y.                            *)
```
```   163 (*                                                                          *)
```
```   164 (* ************************************************************************ *)
```
```   165
```
```   166
```
```   167 fun mkconcl_trans thy t =
```
```   168   case Less.decomp_trans thy t of
```
```   169     SOME (x, rel, y) => (case rel of
```
```   170      "<="  => (Le (x, y, Asm ~1), Asm 0)
```
```   171     | _  => raise Cannot)
```
```   172   | NONE => raise Cannot;
```
```   173
```
```   174
```
```   175 (* ************************************************************************ *)
```
```   176 (*                                                                          *)
```
```   177 (* mkconcl_quasi sign t : theory -> Term.term -> less                       *)
```
```   178 (*                                                                          *)
```
```   179 (* Translates conclusion t to an element of type less.                      *)
```
```   180 (* Quasi orders only.                                                       *)
```
```   181 (*                                                                          *)
```
```   182 (* ************************************************************************ *)
```
```   183
```
```   184 fun mkconcl_quasi thy t =
```
```   185   case Less.decomp_quasi thy t of
```
```   186     SOME (x, rel, y) => (case rel of
```
```   187       "<"   => ([Less (x, y, Asm ~1)], Asm 0)
```
```   188     | "<="  => ([Le (x, y, Asm ~1)], Asm 0)
```
```   189     | "~="  => ([NotEq (x,y, Asm ~1)], Asm 0)
```
```   190     | _  => raise Cannot)
```
```   191 | NONE => raise Cannot;
```
```   192
```
```   193
```
```   194 (* ******************************************************************* *)
```
```   195 (*                                                                     *)
```
```   196 (* mergeLess (less1,less2):  less * less -> less                       *)
```
```   197 (*                                                                     *)
```
```   198 (* Merge to elements of type less according to the following rules     *)
```
```   199 (*                                                                     *)
```
```   200 (* x <= y && y <= z ==> x <= z                                         *)
```
```   201 (* x <= y && x ~= y ==> x < y                                          *)
```
```   202 (* x ~= y && x <= y ==> x < y                                          *)
```
```   203 (*                                                                     *)
```
```   204 (* ******************************************************************* *)
```
```   205
```
```   206 fun mergeLess (Le (x, _, p) , Le (_ , z, q)) =
```
```   207       Le (x, z, Thm ([p,q] , Less.le_trans))
```
```   208 |   mergeLess (Le (x, z, p) , NotEq (x', z', q)) =
```
```   209       if (x aconv x' andalso z aconv z' )
```
```   210        then Less (x, z, Thm ([p,q] , Less.le_neq_trans))
```
```   211         else error "quasi_tac: internal error le_neq_trans"
```
```   212 |   mergeLess (NotEq (x, z, p) , Le (x' , z', q)) =
```
```   213       if (x aconv x' andalso z aconv z')
```
```   214       then Less (x, z, Thm ([p,q] , Less.neq_le_trans))
```
```   215       else error "quasi_tac: internal error neq_le_trans"
```
```   216 |   mergeLess (_, _) =
```
```   217       error "quasi_tac: internal error: undefined case";
```
```   218
```
```   219
```
```   220 (* ******************************************************************** *)
```
```   221 (* tr checks for valid transitivity step                                *)
```
```   222 (* ******************************************************************** *)
```
```   223
```
```   224 infix tr;
```
```   225 fun (Le (_, y, _))   tr (Le (x', _, _))   = ( y aconv x' )
```
```   226   | _ tr _ = false;
```
```   227
```
```   228 (* ******************************************************************* *)
```
```   229 (*                                                                     *)
```
```   230 (* transPath (Lesslist, Less): (less list * less) -> less              *)
```
```   231 (*                                                                     *)
```
```   232 (* If a path represented by a list of elements of type less is found,  *)
```
```   233 (* this needs to be contracted to a single element of type less.       *)
```
```   234 (* Prior to each transitivity step it is checked whether the step is   *)
```
```   235 (* valid.                                                              *)
```
```   236 (*                                                                     *)
```
```   237 (* ******************************************************************* *)
```
```   238
```
```   239 fun transPath ([],lesss) = lesss
```
```   240 |   transPath (x::xs,lesss) =
```
```   241       if lesss tr x then transPath (xs, mergeLess(lesss,x))
```
```   242       else error "trans/quasi_tac: internal error transpath";
```
```   243
```
```   244 (* ******************************************************************* *)
```
```   245 (*                                                                     *)
```
```   246 (* less1 subsumes less2 : less -> less -> bool                         *)
```
```   247 (*                                                                     *)
```
```   248 (* subsumes checks whether less1 implies less2                         *)
```
```   249 (*                                                                     *)
```
```   250 (* ******************************************************************* *)
```
```   251
```
```   252 infix subsumes;
```
```   253 fun (Le (x, y, _)) subsumes (Le (x', y', _)) =
```
```   254       (x aconv x' andalso y aconv y')
```
```   255   | (Le _) subsumes (Less _) =
```
```   256       error "trans/quasi_tac: internal error: Le cannot subsume Less"
```
```   257   | (NotEq(x,y,_)) subsumes (NotEq(x',y',_)) = x aconv x' andalso y aconv y' orelse x aconv y' andalso y aconv x'
```
```   258   | _ subsumes _ = false;
```
```   259
```
```   260 (* ******************************************************************* *)
```
```   261 (*                                                                     *)
```
```   262 (* triv_solv less1 : less ->  proof option                     *)
```
```   263 (*                                                                     *)
```
```   264 (* Solves trivial goal x <= x.                                         *)
```
```   265 (*                                                                     *)
```
```   266 (* ******************************************************************* *)
```
```   267
```
```   268 fun triv_solv (Le (x, x', _)) =
```
```   269     if x aconv x' then  SOME (Thm ([], Less.le_refl))
```
```   270     else NONE
```
```   271 |   triv_solv _ = NONE;
```
```   272
```
```   273 (* ********************************************************************* *)
```
```   274 (* Graph functions                                                       *)
```
```   275 (* ********************************************************************* *)
```
```   276
```
```   277 (* *********************************************************** *)
```
```   278 (* Functions for constructing graphs                           *)
```
```   279 (* *********************************************************** *)
```
```   280
```
```   281 fun addEdge (v,d,[]) = [(v,d)]
```
```   282 |   addEdge (v,d,((u,dl)::el)) = if v aconv u then ((v,d@dl)::el)
```
```   283     else (u,dl):: (addEdge(v,d,el));
```
```   284
```
```   285 (* ********************************************************************** *)
```
```   286 (*                                                                        *)
```
```   287 (* mkQuasiGraph constructs from a list of objects of type less a graph g, *)
```
```   288 (* by taking all edges that are candidate for a <=, and a list neqE, by   *)
```
```   289 (* taking all edges that are candiate for a ~=                            *)
```
```   290 (*                                                                        *)
```
```   291 (* ********************************************************************** *)
```
```   292
```
```   293 fun mkQuasiGraph [] = ([],[])
```
```   294 |   mkQuasiGraph lessList =
```
```   295  let
```
```   296  fun buildGraphs ([],leG, neqE) = (leG,  neqE)
```
```   297   |   buildGraphs (l::ls, leG,  neqE) = case l of
```
```   298        (Less (x,y,p)) =>
```
```   299          let
```
```   300           val leEdge  = Le (x,y, Thm ([p], Less.less_imp_le))
```
```   301           val neqEdges = [ NotEq (x,y, Thm ([p], Less.less_imp_neq)),
```
```   302                            NotEq (y,x, Thm ( [Thm ([p], Less.less_imp_neq)], thm "not_sym"))]
```
```   303          in
```
```   304            buildGraphs (ls, addEdge(y,[],(addEdge (x,[(y,leEdge)],leG))), neqEdges@neqE)
```
```   305          end
```
```   306      |  (Le (x,y,p))   => buildGraphs (ls, addEdge(y,[],(addEdge (x,[(y,l)],leG))), neqE)
```
```   307      | _ =>  buildGraphs (ls, leG,  l::neqE) ;
```
```   308
```
```   309 in buildGraphs (lessList, [],  []) end;
```
```   310
```
```   311 (* ********************************************************************** *)
```
```   312 (*                                                                        *)
```
```   313 (* mkGraph constructs from a list of objects of type less a graph g       *)
```
```   314 (* Used for plain transitivity chain reasoning.                           *)
```
```   315 (*                                                                        *)
```
```   316 (* ********************************************************************** *)
```
```   317
```
```   318 fun mkGraph [] = []
```
```   319 |   mkGraph lessList =
```
```   320  let
```
```   321   fun buildGraph ([],g) = g
```
```   322   |   buildGraph (l::ls, g) =  buildGraph (ls, (addEdge ((lower l),[((upper l),l)],g)))
```
```   323
```
```   324 in buildGraph (lessList, []) end;
```
```   325
```
```   326 (* *********************************************************************** *)
```
```   327 (*                                                                         *)
```
```   328 (* adjacent g u : (''a * 'b list ) list -> ''a -> 'b list                  *)
```
```   329 (*                                                                         *)
```
```   330 (* List of successors of u in graph g                                      *)
```
```   331 (*                                                                         *)
```
```   332 (* *********************************************************************** *)
```
```   333
```
```   334 fun adjacent eq_comp ((v,adj)::el) u =
```
```   335     if eq_comp (u, v) then adj else adjacent eq_comp el u
```
```   336 |   adjacent _  []  _ = []
```
```   337
```
```   338 (* *********************************************************************** *)
```
```   339 (*                                                                         *)
```
```   340 (* dfs eq_comp g u v:                                                      *)
```
```   341 (* ('a * 'a -> bool) -> ('a  *( 'a * less) list) list ->                   *)
```
```   342 (* 'a -> 'a -> (bool * ('a * less) list)                                   *)
```
```   343 (*                                                                         *)
```
```   344 (* Depth first search of v from u.                                         *)
```
```   345 (* Returns (true, path(u, v)) if successful, otherwise (false, []).        *)
```
```   346 (*                                                                         *)
```
```   347 (* *********************************************************************** *)
```
```   348
```
```   349 fun dfs eq_comp g u v =
```
```   350  let
```
```   351     val pred = Unsynchronized.ref [];
```
```   352     val visited = Unsynchronized.ref [];
```
```   353
```
```   354     fun been_visited v = exists (fn w => eq_comp (w, v)) (!visited)
```
```   355
```
```   356     fun dfs_visit u' =
```
```   357     let val _ = visited := u' :: (!visited)
```
```   358
```
```   359     fun update (x,l) = let val _ = pred := (x,l) ::(!pred) in () end;
```
```   360
```
```   361     in if been_visited v then ()
```
```   362     else (app (fn (v',l) => if been_visited v' then () else (
```
```   363        update (v',l);
```
```   364        dfs_visit v'; ()) )) (adjacent eq_comp g u')
```
```   365      end
```
```   366   in
```
```   367     dfs_visit u;
```
```   368     if (been_visited v) then (true, (!pred)) else (false , [])
```
```   369   end;
```
```   370
```
```   371 (* ************************************************************************ *)
```
```   372 (*                                                                          *)
```
```   373 (* Begin: Quasi Order relevant functions                                    *)
```
```   374 (*                                                                          *)
```
```   375 (*                                                                          *)
```
```   376 (* ************************************************************************ *)
```
```   377
```
```   378 (* ************************************************************************ *)
```
```   379 (*                                                                          *)
```
```   380 (* findPath x y g: Term.term -> Term.term ->                                *)
```
```   381 (*                  (Term.term * (Term.term * less list) list) ->           *)
```
```   382 (*                  (bool, less list)                                       *)
```
```   383 (*                                                                          *)
```
```   384 (*  Searches a path from vertex x to vertex y in Graph g, returns true and  *)
```
```   385 (*  the list of edges forming the path, if a path is found, otherwise false *)
```
```   386 (*  and nil.                                                                *)
```
```   387 (*                                                                          *)
```
```   388 (* ************************************************************************ *)
```
```   389
```
```   390
```
```   391  fun findPath x y g =
```
```   392   let
```
```   393     val (found, tmp) =  dfs (op aconv) g x y ;
```
```   394     val pred = map snd tmp;
```
```   395
```
```   396     fun path x y  =
```
```   397       let
```
```   398        (* find predecessor u of node v and the edge u -> v *)
```
```   399        fun lookup v [] = raise Cannot
```
```   400        |   lookup v (e::es) = if (upper e) aconv v then e else lookup v es;
```
```   401
```
```   402        (* traverse path backwards and return list of visited edges *)
```
```   403        fun rev_path v =
```
```   404         let val l = lookup v pred
```
```   405             val u = lower l;
```
```   406         in
```
```   407            if u aconv x then [l] else (rev_path u) @ [l]
```
```   408         end
```
```   409       in rev_path y end;
```
```   410
```
```   411   in
```
```   412    if found then (
```
```   413     if x aconv y then (found,[(Le (x, y, (Thm ([], Less.le_refl))))])
```
```   414     else (found, (path x y) ))
```
```   415    else (found,[])
```
```   416   end;
```
```   417
```
```   418
```
```   419 (* ************************************************************************ *)
```
```   420 (*                                                                          *)
```
```   421 (* findQuasiProof (leqG, neqE) subgoal:                                     *)
```
```   422 (* (Term.term * (Term.term * less list) list) * less list  -> less -> proof *)
```
```   423 (*                                                                          *)
```
```   424 (* Constructs a proof for subgoal by searching a special path in leqG and   *)
```
```   425 (* neqE. Raises Cannot if construction of the proof fails.                  *)
```
```   426 (*                                                                          *)
```
```   427 (* ************************************************************************ *)
```
```   428
```
```   429
```
```   430 (* As the conlusion can be either of form x <= y, x < y or x ~= y we have        *)
```
```   431 (* three cases to deal with. Finding a transitivity path from x to y with label  *)
```
```   432 (* 1. <=                                                                         *)
```
```   433 (*    This is simply done by searching any path from x to y in the graph leG.    *)
```
```   434 (*    The graph leG contains only edges with label <=.                           *)
```
```   435 (*                                                                               *)
```
```   436 (* 2. <                                                                          *)
```
```   437 (*    A path from x to y with label < can be found by searching a path with      *)
```
```   438 (*    label <= from x to y in the graph leG and merging the path x <= y with     *)
```
```   439 (*    a parallel edge x ~= y resp. y ~= x to x < y.                              *)
```
```   440 (*                                                                               *)
```
```   441 (* 3. ~=                                                                         *)
```
```   442 (*   If the conclusion is of form x ~= y, we can find a proof either directly,   *)
```
```   443 (*   if x ~= y or y ~= x are among the assumptions, or by constructing x ~= y if *)
```
```   444 (*   x < y or y < x follows from the assumptions.                                *)
```
```   445
```
```   446 fun findQuasiProof (leG, neqE) subgoal =
```
```   447   case subgoal of (Le (x,y, _)) => (
```
```   448    let
```
```   449     val (xyLefound,xyLePath) = findPath x y leG
```
```   450    in
```
```   451     if xyLefound then (
```
```   452      let
```
```   453       val Le_x_y = (transPath (tl xyLePath, hd xyLePath))
```
```   454      in getprf Le_x_y end )
```
```   455     else raise Cannot
```
```   456    end )
```
```   457   | (Less (x,y,_))  => (
```
```   458    let
```
```   459     fun findParallelNeq []  = NONE
```
```   460     |   findParallelNeq (e::es)  =
```
```   461      if      (x aconv (lower e) andalso y aconv (upper e)) then SOME e
```
```   462      else if (y aconv (lower e) andalso x aconv (upper e)) then SOME (NotEq (x,y, (Thm ([getprf e], thm "not_sym"))))
```
```   463      else findParallelNeq es ;
```
```   464    in
```
```   465    (* test if there is a edge x ~= y respectivly  y ~= x and     *)
```
```   466    (* if it possible to find a path x <= y in leG, thus we can conclude x < y *)
```
```   467     (case findParallelNeq neqE of (SOME e) =>
```
```   468       let
```
```   469        val (xyLeFound,xyLePath) = findPath x y leG
```
```   470       in
```
```   471        if xyLeFound then (
```
```   472         let
```
```   473          val Le_x_y = (transPath (tl xyLePath, hd xyLePath))
```
```   474          val Less_x_y = mergeLess (e, Le_x_y)
```
```   475         in getprf Less_x_y end
```
```   476        ) else raise Cannot
```
```   477       end
```
```   478     | _ => raise Cannot)
```
```   479    end )
```
```   480  | (NotEq (x,y,_)) =>
```
```   481   (* First check if a single premiss is sufficient *)
```
```   482   (case (Library.find_first (fn fact => fact subsumes subgoal) neqE, subgoal) of
```
```   483     (SOME (NotEq (x, y, p)), NotEq (x', y', _)) =>
```
```   484       if  (x aconv x' andalso y aconv y') then p
```
```   485       else Thm ([p], thm "not_sym")
```
```   486     | _  => raise Cannot
```
```   487   )
```
```   488
```
```   489
```
```   490 (* ************************************************************************ *)
```
```   491 (*                                                                          *)
```
```   492 (* End: Quasi Order relevant functions                                      *)
```
```   493 (*                                                                          *)
```
```   494 (*                                                                          *)
```
```   495 (* ************************************************************************ *)
```
```   496
```
```   497 (* *********************************************************************** *)
```
```   498 (*                                                                         *)
```
```   499 (* solveLeTrans sign (asms,concl) :                                        *)
```
```   500 (* theory -> less list * Term.term -> proof list                           *)
```
```   501 (*                                                                         *)
```
```   502 (* Solves                                                                  *)
```
```   503 (*                                                                         *)
```
```   504 (* *********************************************************************** *)
```
```   505
```
```   506 fun solveLeTrans thy (asms, concl) =
```
```   507  let
```
```   508   val g = mkGraph asms
```
```   509  in
```
```   510    let
```
```   511     val (subgoal, prf) = mkconcl_trans thy concl
```
```   512     val (found, path) = findPath (lower subgoal) (upper subgoal) g
```
```   513    in
```
```   514     if found then [getprf (transPath (tl path, hd path))]
```
```   515     else raise Cannot
```
```   516   end
```
```   517  end;
```
```   518
```
```   519
```
```   520 (* *********************************************************************** *)
```
```   521 (*                                                                         *)
```
```   522 (* solveQuasiOrder sign (asms,concl) :                                     *)
```
```   523 (* theory -> less list * Term.term -> proof list                           *)
```
```   524 (*                                                                         *)
```
```   525 (* Find proof if possible for quasi order.                                 *)
```
```   526 (*                                                                         *)
```
```   527 (* *********************************************************************** *)
```
```   528
```
```   529 fun solveQuasiOrder thy (asms, concl) =
```
```   530  let
```
```   531   val (leG, neqE) = mkQuasiGraph asms
```
```   532  in
```
```   533    let
```
```   534    val (subgoals, prf) = mkconcl_quasi thy concl
```
```   535    fun solve facts less =
```
```   536        (case triv_solv less of NONE => findQuasiProof (leG, neqE) less
```
```   537        | SOME prf => prf )
```
```   538   in   map (solve asms) subgoals end
```
```   539  end;
```
```   540
```
```   541 (* ************************************************************************ *)
```
```   542 (*                                                                          *)
```
```   543 (* Tactics                                                                  *)
```
```   544 (*                                                                          *)
```
```   545 (*  - trans_tac                                                          *)
```
```   546 (*  - quasi_tac, solves quasi orders                                        *)
```
```   547 (* ************************************************************************ *)
```
```   548
```
```   549
```
```   550 (* trans_tac - solves transitivity chains over <= *)
```
```   551
```
```   552 fun trans_tac ctxt = SUBGOAL (fn (A, n) => fn st =>
```
```   553  let
```
```   554   val thy = ProofContext.theory_of ctxt;
```
```   555   val rfrees = map Free (Term.rename_wrt_term A (Logic.strip_params A));
```
```   556   val Hs = map (fn H => subst_bounds (rfrees, H)) (Logic.strip_assums_hyp A);
```
```   557   val C = subst_bounds (rfrees, Logic.strip_assums_concl A);
```
```   558   val lesss = flat (map_index (mkasm_trans thy o swap) Hs);
```
```   559   val prfs = solveLeTrans thy (lesss, C);
```
```   560
```
```   561   val (subgoal, prf) = mkconcl_trans thy C;
```
```   562  in
```
```   563   Subgoal.FOCUS (fn {prems, ...} =>
```
```   564     let val thms = map (prove prems) prfs
```
```   565     in rtac (prove thms prf) 1 end) ctxt n st
```
```   566  end
```
```   567  handle Contr p => Subgoal.FOCUS (fn {prems, ...} => rtac (prove prems p) 1) ctxt n st
```
```   568   | Cannot  => Seq.empty);
```
```   569
```
```   570
```
```   571 (* quasi_tac - solves quasi orders *)
```
```   572
```
```   573 fun quasi_tac ctxt = SUBGOAL (fn (A, n) => fn st =>
```
```   574  let
```
```   575   val thy = ProofContext.theory_of ctxt
```
```   576   val rfrees = map Free (Term.rename_wrt_term A (Logic.strip_params A));
```
```   577   val Hs = map (fn H => subst_bounds (rfrees, H)) (Logic.strip_assums_hyp A);
```
```   578   val C = subst_bounds (rfrees, Logic.strip_assums_concl A);
```
```   579   val lesss = flat (map_index (mkasm_quasi thy o swap) Hs);
```
```   580   val prfs = solveQuasiOrder thy (lesss, C);
```
```   581   val (subgoals, prf) = mkconcl_quasi thy C;
```
```   582  in
```
```   583   Subgoal.FOCUS (fn {prems, ...} =>
```
```   584     let val thms = map (prove prems) prfs
```
```   585     in rtac (prove thms prf) 1 end) ctxt n st
```
```   586  end
```
```   587  handle Contr p =>
```
```   588     (Subgoal.FOCUS (fn {prems, ...} => rtac (prove prems p) 1) ctxt n st
```
```   589       handle Subscript => Seq.empty)
```
```   590   | Cannot => Seq.empty
```
```   591   | Subscript => Seq.empty);
```
```   592
```
```   593 end;
```