(ignored inputs)COMMENT R3 of \cite{AT2012}
Rewrite Rules:
[ +(0,?y) -> ?y,
+(?x,0) -> ?x,
+(s(?x),?y) -> s(+(?x,?y)),
+(?x,s(?y)) -> s(+(?x,?y)),
+(?x,+(?y,?z)) -> +(+(?x,?y),?z),
+(?x,?y) -> +(?y,?x) ]
Apply Direct Methods...
Inner CPs:
[ +(?x_4,?y) = +(+(?x_4,0),?y),
+(?x_4,?x_1) = +(+(?x_4,?x_1),0),
+(?x_4,s(+(?x_2,?y_2))) = +(+(?x_4,s(?x_2)),?y_2),
+(?x_4,s(+(?x_3,?y_3))) = +(+(?x_4,?x_3),s(?y_3)),
+(?x_4,+(?y_5,?x_5)) = +(+(?x_4,?x_5),?y_5),
+(?x_1,+(+(?x,?y),?z)) = +(+(?x_1,?x),+(?y,?z)) ]
Outer CPs:
[ 0 = 0,
s(?y_3) = s(+(0,?y_3)),
+(?y_4,?z_4) = +(+(0,?y_4),?z_4),
?y = +(?y,0),
s(?x_2) = s(+(?x_2,0)),
?x_1 = +(0,?x_1),
s(+(?x_2,s(?y_3))) = s(+(s(?x_2),?y_3)),
s(+(?x_2,+(?y_4,?z_4))) = +(+(s(?x_2),?y_4),?z_4),
s(+(?x_2,?y_2)) = +(?y_2,s(?x_2)),
s(+(?x_3,?y_3)) = +(s(?y_3),?x_3),
+(+(?x_4,?y_4),?z_4) = +(+(?y_4,?z_4),?x_4) ]
not Overlay, check Termination...
unknown/not Terminating
unknown Knuth&Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ +(+(?x_6,?x),?y) = +(?x_6,+(?y,?x)),
+(+(?x,?y_5),?z_5) = +(+(?y_5,?z_5),?x),
s(+(?x,?y_4)) = +(s(?y_4),?x),
s(+(?x_3,?y)) = +(?y,s(?x_3)),
?x = +(0,?x),
?y = +(?y,0),
+(+(?x_1,?x),+(?y,?z)) = +(?x_1,+(+(?x,?y),?z)),
+(+(?x_1,?x),+(?z,?y)) = +(?x_1,+(+(?x,?y),?z)),
+(+(?x_1,?x),s(+(?y,?y_6))) = +(?x_1,+(+(?x,?y),s(?y_6))),
+(+(?x_1,?x),s(+(?x_5,?z))) = +(?x_1,+(+(?x,s(?x_5)),?z)),
+(+(?x_1,?x),?y) = +(?x_1,+(+(?x,?y),0)),
+(+(?x_1,?x),?z) = +(?x_1,+(+(?x,0),?z)),
+(+(?x_1,?x),+(+(?y,?y_2),?z_2)) = +(?x_1,+(+(?x,?y),+(?y_2,?z_2))),
+(?x,+(?z,?y)) = +(+(?x,?y),?z),
+(?x,s(+(?y,?y_5))) = +(+(?x,?y),s(?y_5)),
+(?x,s(+(?x_4,?z))) = +(+(?x,s(?x_4)),?z),
+(?x,?y) = +(+(?x,?y),0),
+(?x,?z) = +(+(?x,0),?z),
+(?x,+(+(?y,?y_1),?z_1)) = +(+(?x,?y),+(?y_1,?z_1)),
+(+(?y,?z),?x) = +(+(?x,?y),?z),
s(+(?x_3,+(?y,?z))) = +(+(s(?x_3),?y),?z),
+(?y,?z) = +(+(0,?y),?z),
+(+(?z,?y),?x) = +(+(?x,?y),?z),
+(s(+(?y,?y_5)),?x) = +(+(?x,?y),s(?y_5)),
+(s(+(?x_4,?z)),?x) = +(+(?x,s(?x_4)),?z),
+(?y,?x) = +(+(?x,?y),0),
+(?z,?x) = +(+(?x,0),?z),
+(+(+(?y,?y_1),?z_1),?x) = +(+(?x,?y),+(?y_1,?z_1)),
s(+(?x_3,+(?z,?y))) = +(+(s(?x_3),?y),?z),
s(+(?x_3,s(+(?y,?y_8)))) = +(+(s(?x_3),?y),s(?y_8)),
s(+(?x_3,s(+(?x_7,?z)))) = +(+(s(?x_3),s(?x_7)),?z),
s(+(?x_3,?y)) = +(+(s(?x_3),?y),0),
s(+(?x_3,?z)) = +(+(s(?x_3),0),?z),
s(+(?x_3,+(+(?y,?y_4),?z_4))) = +(+(s(?x_3),?y),+(?y_4,?z_4)),
+(?z,?y) = +(+(0,?y),?z),
s(+(?y,?y_5)) = +(+(0,?y),s(?y_5)),
s(+(?x_4,?z)) = +(+(0,s(?x_4)),?z),
?y = +(+(0,?y),0),
?z = +(+(0,0),?z),
+(+(?y,?y_1),?z_1) = +(+(0,?y),+(?y_1,?z_1)),
+(+(?x_5,?x),s(?y)) = +(?x_5,s(+(?x,?y))),
+(s(?y),?x) = s(+(?x,?y)),
s(+(?x_3,s(?y))) = s(+(s(?x_3),?y)),
s(?y) = s(+(0,?y)),
+(+(?x_5,s(?x)),?y) = +(?x_5,s(+(?x,?y))),
+(?y,s(?x)) = s(+(?x,?y)),
+(+(s(?x),?y_4),?z_4) = s(+(?x,+(?y_4,?z_4))),
s(+(s(?x),?y_3)) = s(+(?x,s(?y_3))),
s(?x) = s(+(?x,0)),
+(+(?x_5,?x),0) = +(?x_5,?x),
+(0,?x) = ?x,
s(+(?x_2,0)) = s(?x_2),
+(+(?x_5,0),?y) = +(?x_5,?y),
+(?y,0) = ?y,
+(+(0,?y_4),?z_4) = +(?y_4,?z_4),
s(+(0,?y_3)) = s(?y_3),
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <+(?x_4,?y), +(+(?x_4,0),?y)> by Rules <0, 4> preceded by [(+,2)]
joinable by a reduction of rules <[], [([(+,1)],1)]>
Critical Pair <+(?x_4,?x_1), +(+(?x_4,?x_1),0)> by Rules <1, 4> preceded by [(+,2)]
joinable by a reduction of rules <[], [([],1)]>
Critical Pair <+(?x_4,s(+(?x_2,?y_2))), +(+(?x_4,s(?x_2)),?y_2)> by Rules <2, 4> preceded by [(+,2)]
joinable by a reduction of rules <[([],3),([(s,1)],4)], [([(+,1)],3),([],2)]>
Critical Pair <+(?x_4,s(+(?x_3,?y_3))), +(+(?x_4,?x_3),s(?y_3))> by Rules <3, 4> preceded by [(+,2)]
joinable by a reduction of rules <[([],3),([(s,1)],4)], [([],3)]>
Critical Pair <+(?x_4,+(?y_5,?x_5)), +(+(?x_4,?x_5),?y_5)> by Rules <5, 4> preceded by [(+,2)]
joinable by a reduction of rules <[([(+,2)],5),([],4)], []>
joinable by a reduction of rules <[([],4),([(+,1)],5)], [([],5),([],4)]>
Critical Pair <+(?x_1,+(+(?x,?y),?z)), +(+(?x_1,?x),+(?y,?z))> by Rules <4, 4> preceded by [(+,2)]
joinable by a reduction of rules <[([],4),([(+,1)],4)], [([],4)]>
Critical Pair <0, 0> by Rules <1, 0> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair ~~ by Rules <3, 0> preceded by []
joinable by a reduction of rules <[([(s,1)],0)], []>
Critical Pair <+(+(0,?y_4),?z_4), +(?y_4,?z_4)> by Rules <4, 0> preceded by []
joinable by a reduction of rules <[([(+,1)],0)], []>
Critical Pair <+(?y_5,0), ?y_5> by Rules <5, 0> preceded by []
joinable by a reduction of rules <[([],1)], []>
Critical Pair ~~~~ by Rules <2, 1> preceded by []
joinable by a reduction of rules <[([(s,1)],1)], []>
Critical Pair <+(0,?x_5), ?x_5> by Rules <5, 1> preceded by []
joinable by a reduction of rules <[([],0)], []>
Critical Pair ~~~~ by Rules <3, 2> preceded by []
joinable by a reduction of rules <[([(s,1)],2)], [([(s,1)],3)]>
Critical Pair <+(+(s(?x_2),?y_4),?z_4), s(+(?x_2,+(?y_4,?z_4)))> by Rules <4, 2> preceded by []
joinable by a reduction of rules <[([(+,1)],2),([],2)], [([(s,1)],4)]>
Critical Pair <+(?y_5,s(?x_2)), s(+(?x_2,?y_5))> by Rules <5, 2> preceded by []
joinable by a reduction of rules <[([],3)], [([(s,1)],5)]>
Critical Pair <+(s(?y_3),?x_5), s(+(?x_5,?y_3))> by Rules <5, 3> preceded by []
joinable by a reduction of rules <[([],2)], [([(s,1)],5)]>
Critical Pair <+(+(?y_4,?z_4),?x_5), +(+(?x_5,?y_4),?z_4)> by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([],5),([],4)], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
[ +(0,?y) -> ?y,
+(?x,0) -> ?x,
+(s(?x),?y) -> s(+(?x,?y)),
+(?x,s(?y)) -> s(+(?x,?y)) ]
P:
[ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
+(?x,?y) -> +(?y,?x) ]
S: terminating
CP(S,S):
<0, 0> --> <0, 0> => yes
~~~~ --> ~~~~ => yes
~~~~ --> ~~~~ => yes
~~~~ --> ~~~~ => yes
PCP_in(symP,S):
CP(S,symP):
<+(?y_1,?z_1), +(+(0,?y_1),?z_1)> --> <+(?y_1,?z_1), +(?y_1,?z_1)> => yes
<+(?x_1,?y), +(+(?x_1,0),?y)> --> <+(?x_1,?y), +(?x_1,?y)> => yes
<+(?y,?z_1), +(0,+(?y,?z_1))> --> <+(?y,?z_1), +(?y,?z_1)> => yes
--> => yes
<+(?x_1,?x), +(+(?x_1,?x),0)> --> <+(?x_1,?x), +(?x_1,?x)> => yes
<+(?x_1,?y_1), +(?x_1,+(?y_1,0))> --> <+(?x_1,?y_1), +(?x_1,?y_1)> => yes
<+(?x,?z_1), +(?x,+(0,?z_1))> --> <+(?x,?z_1), +(?x,?z_1)> => yes
--> => yes
~~~~ --> ~~~~ => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,s(?x)),?y)> --> ~~~~ => yes
<+(s(+(?x,?y)),?z_1), +(s(?x),+(?y,?z_1))> --> ~~~~ => yes
~~~~ --> ~~~~ => yes
<+(?x_1,s(+(?x,?y))), +(+(?x_1,?x),s(?y))> --> ~~~~ => yes
~~~~ --> ~~~~ => yes
<+(s(+(?x,?y)),?z_1), +(?x,+(s(?y),?z_1))> --> ~~~~ => yes
~~~~ --> ~~~~ => yes
S:
[ +(0,?y) -> ?y,
+(?x,0) -> ?x,
+(s(?x),?y) -> s(+(?x,?y)),
+(?x,s(?y)) -> s(+(?x,?y)) ]
P:
[ +(?x,+(?y,?z)) -> +(+(?x,?y),?z),
+(?x,?y) -> +(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR
Final result: CR
/local-scratch/hzankl/2012/cops2012/125.trs: Success(CR)
(23 msec.)
~~