************************************* Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES f (true, x, y, z) -> f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) -> f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) -> n plus (n, s (m)) -> s (plus (n, m)) gt (0, v) -> false gt (s (u), 0) -> true gt (s (u), s (v)) -> gt (u, v)) , property = Termination , truth = Nothing , transform = Ignore_Strategy , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES f (true, x, y, z) -> f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) -> f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) -> n plus (n, s (m)) -> s (plus (n, m)) gt (0, v) -> false gt (s (u), 0) -> true gt (s (u), s (v)) -> gt (u, v)) , property = Termination , truth = Nothing , transform = Dependency_Pair_Transformation , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) fP (true, x, y, z) -> gtP (x, plus (y, z)) fP (true, x, y, z) -> plusP (y, z) fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) fP (true, x, y, z) -> gtP (x, plus (y, z)) fP (true, x, y, z) -> plusP (y, z) plusP (n, s (m)) -> plusP (n, m) gtP (s (u), s (v)) -> gtP (u, v) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Nothing , transform = Remove { interpretation = Interpretation gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2) plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0) 0 |-> () |-> E^0x1 f |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1 s |-> (x) |-> E^0x0 * x + E^0x1 false |-> () |-> E^0x1 fP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (3) gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 true |-> () |-> E^0x1 plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 , removes_rules = [ fP (true, x, y, z) -> gtP (x, plus (y, z)) , fP (true, x, y, z) -> plusP (y, z) , fP (true, x, y, z) -> gtP (x, plus (y, z)) , fP (true, x, y, z) -> plusP (y, z) ] , comment = size 0 , heights ( 3 , 7 ) , time } , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) plusP (n, s (m)) -> plusP (n, m) gtP (s (u), s (v)) -> gtP (u, v) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Nothing , transform = Split { interpretation = Interpretation gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2) plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0) 0 |-> () |-> E^0x1 f |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1 s |-> (x) |-> E^0x0 * x + E^0x1 false |-> () |-> E^0x1 fP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (3) gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 true |-> () |-> E^0x1 plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 , clusters = [ [ plusP (n, s (m)) -> plusP (n, m) ] , [ gtP (s (u), s (v)) -> gtP (u, v) ] , [ fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) , fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) ] ] , comment = size 0 , heights ( 3 , 7 ) , time split_dimension: 0 } , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES plusP (n, s (m)) -> plusP (n, m) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Just True , transform = Remove { interpretation = Interpretation plusP |-> (x , y) |-> (1) * x + (2) * y + (0) 0 |-> () |-> (0) f |-> (p , q , r , s) |-> (0) * p + (0) * q + (0) * r + (0) * s + (0) s |-> (x) |-> (1) * x + (1) false |-> () |-> (0) gt |-> (x , y) |-> (0) * x + (0) * y + (0) true |-> () |-> (0) plus |-> (x , y) |-> (1) * x + (1) * y + (0) , removes_rules = [ plusP (n, s (m)) -> plusP (n, m) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec } , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Just True , transform = No_Strict_Rules , to = [ ] } ] } , Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES gtP (s (u), s (v)) -> gtP (u, v) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Just True , transform = Remove { interpretation = Interpretation gtP |-> (x , y) |-> (0) * x + (1) * y + (0) 0 |-> () |-> (0) f |-> (p , q , r , s) |-> (0) * p + (0) * q + (0) * r + (0) * s + (0) s |-> (x) |-> (1) * x + (1) false |-> () |-> (0) gt |-> (x , y) |-> (0) * x + (0) * y + (0) true |-> () |-> (0) plus |-> (x , y) |-> (4) * x + (1) * y + (0) , removes_rules = [ gtP (s (u), s (v)) -> gtP (u, v) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec } , to = [ Proof { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination , truth = Just True , transform = No_Strict_Rules , to = [ ] } ] } , Claim { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination } ] } ] } ] } ] } Proof summary: value Nothing for property Termination for system with 7 strict rules and 0 non-strict rules follows by transformation Ignore_Strategy from value Nothing for property Termination for system with 7 strict rules and 0 non-strict rules follows by transformation Dependency_Pair_Transformation from value Nothing for property Top_Termination for system with 8 strict rules and 7 non-strict rules follows by transformation Remove { interpretation = Interpretation gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2) plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0) 0 |-> () |-> E^0x1 f |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1 s |-> (x) |-> E^0x0 * x + E^0x1 false |-> () |-> E^0x1 fP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (3) gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 true |-> () |-> E^0x1 plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 , removes_rules = [ fP (true, x, y, z) -> gtP (x, plus (y, z)) , fP (true, x, y, z) -> plusP (y, z) , fP (true, x, y, z) -> gtP (x, plus (y, z)) , fP (true, x, y, z) -> plusP (y, z) ] , comment = size 0 , heights ( 3 , 7 ) , time } from value Nothing for property Top_Termination for system with 4 strict rules and 7 non-strict rules follows by transformation Split { interpretation = Interpretation gtP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (2) plusP |-> (x , y) |-> E^1x0 * x + E^1x0 * y + (0) 0 |-> () |-> E^0x1 f |-> (p , q , r , s) |-> E^0x0 * p + E^0x0 * q + E^0x0 * r + E^0x0 * s + E^0x1 s |-> (x) |-> E^0x0 * x + E^0x1 false |-> () |-> E^0x1 fP |-> (p , q , r , s) |-> E^1x0 * p + E^1x0 * q + E^1x0 * r + E^1x0 * s + (3) gt |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 true |-> () |-> E^0x1 plus |-> (x , y) |-> E^0x0 * x + E^0x0 * y + E^0x1 , clusters = [ [ plusP (n, s (m)) -> plusP (n, m) ] , [ gtP (s (u), s (v)) -> gtP (u, v) ] , [ fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) , fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) ] ] , comment = size 0 , heights ( 3 , 7 ) , time split_dimension: 0 } from value Just True for property Top_Termination for system with 1 strict rules and 7 non-strict rules follows by transformation Remove { interpretation = Interpretation plusP |-> (x , y) |-> (1) * x + (2) * y + (0) 0 |-> () |-> (0) f |-> (p , q , r , s) |-> (0) * p + (0) * q + (0) * r + (0) * s + (0) s |-> (x) |-> (1) * x + (1) false |-> () |-> (0) gt |-> (x , y) |-> (0) * x + (0) * y + (0) true |-> () |-> (0) plus |-> (x , y) |-> (1) * x + (1) * y + (0) , removes_rules = [ plusP (n, s (m)) -> plusP (n, m) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec } from value Just True for property Top_Termination for system with 0 strict rules and 7 non-strict rules follows by transformation No_Strict_Rules from value Just True for property Top_Termination for system with 1 strict rules and 7 non-strict rules follows by transformation Remove { interpretation = Interpretation gtP |-> (x , y) |-> (0) * x + (1) * y + (0) 0 |-> () |-> (0) f |-> (p , q , r , s) |-> (0) * p + (0) * q + (0) * r + (0) * s + (0) s |-> (x) |-> (1) * x + (1) false |-> () |-> (0) gt |-> (x , y) |-> (0) * x + (0) * y + (0) true |-> () |-> (0) plus |-> (x , y) |-> (4) * x + (1) * y + (0) , removes_rules = [ gtP (s (u), s (v)) -> gtP (u, v) ] , comment = size 1 , heights ( 7 , 15 ) , time 1 sec } from value Just True for property Top_Termination for system with 0 strict rules and 7 non-strict rules follows by transformation No_Strict_Rules from Claim { system = (VAR x y z n m u v) (STRATEGY INNERMOST) (RULES fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, s (y), z) fP (true, x, y, z) -> fP (gt (x, plus (y, z)), x, y, s (z)) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, s (y), z) f (true, x, y, z) ->= f (gt (x, plus (y, z)), x, y, s (z)) plus (n, 0) ->= n plus (n, s (m)) ->= s (plus (n, m)) gt (0, v) ->= false gt (s (u), 0) ->= true gt (s (u), s (v)) ->= gt (u, v)) , property = Top_Termination } ------------------------------------------------------------------ matchbox general information (including details on proof methods): http://dfa.imn.htwk-leipzig.de/matchbox/ this matchbox implementation uses the SAT solver SatELite/MiniSat by Niklas Een and Niklas Sörensson http://www.cs.chalmers.se/Cs/Research/FormalMethods/MiniSat/ matchbox process information arguments : --solver=/home/nowonder/forschung/increasing/wst06/matchbox/SatELiteGTI --timeout-command=/home/nowonder/forschung/increasing/wst06/matchbox/timeout --tmpdir=/home/nowonder/forschung/increasing/wst06/matchbox --timeout=60 /tmp/tmp6Tjvxx/ex06.trs started : Thu Feb 22 16:37:25 CET 2007 finished : Thu Feb 22 16:38:29 CET 2007 run system : Linux aprove 2.6.14-gentoo-r5 #1 SMP Sun Dec 25 15:42:02 CET 2005 x86_64 release date : Thu Jun 8 23:18:07 CEST 2006 build date : Thu Jun 8 23:18:07 CEST 2006 build system : Linux dfa 2.6.8-2-k7 #1 Tue Aug 16 14:00:15 UTC 2005 i686 GNU/Linux used clock time: 64 secs