(ignored inputs)COMMENT associative law + commutative law
Rewrite Rules:
[ f(f(?x,?y),?z) -> f(?x,f(?y,?z)),
f(?x,?y) -> f(?y,?x) ]
Apply Direct Methods...
Inner CPs:
[ f(f(?y_1,?x_1),?z) = f(?x_1,f(?y_1,?z)),
f(f(?x,f(?y,?z)),?z_1) = f(f(?x,?y),f(?z,?z_1)) ]
Outer CPs:
[ f(?x,f(?y,?z)) = f(?z,f(?x,?y)) ]
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:
[ f(?x,f(?y,?z_2)) = f(f(?y,?x),?z_2),
f(?x_1,f(?y_1,?y)) = f(?y,f(?x_1,?y_1)),
f(f(?x,?y),f(?z,?z_1)) = f(f(?x,f(?y,?z)),?z_1),
f(f(?y,?x),f(?z,?z_1)) = f(f(?x,f(?y,?z)),?z_1),
f(f(?x_2,f(?y_2,?y)),f(?z,?z_1)) = f(f(f(?x_2,?y_2),f(?y,?z)),?z_1),
f(f(?y,?x),?z) = f(?x,f(?y,?z)),
f(f(?x_1,f(?y_1,?y)),?z) = f(f(?x_1,?y_1),f(?y,?z)),
f(?z,f(?x,?y)) = f(?x,f(?y,?z)),
f(?z,f(?y,?x)) = f(?x,f(?y,?z)),
f(?z,f(?x_1,f(?y_1,?y))) = f(f(?x_1,?y_1),f(?y,?z)) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <1, 0> preceded by [(f,1)]
joinable by a reduction of rules <[([(f,1)],1),([],0)], []>
joinable by a reduction of rules <[([],0),([(f,2)],1)], [([],1),([],0)]>
Critical Pair by Rules <0, 0> preceded by [(f,1)]
joinable by a reduction of rules <[([],0),([(f,2)],0)], [([],0)]>
Critical Pair by Rules <1, 0> preceded by []
joinable by a reduction of rules <[([],1),([],0)], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
STEP: 1 (parallel)
S:
[ ]
P:
[ f(f(?x,?y),?z) -> f(?x,f(?y,?z)),
f(?x,?y) -> f(?y,?x) ]
S: terminating
CP(S,S):
PCP_in(symP,S):
CP(S,symP):
S:
[ ]
P:
[ f(f(?x,?y),?z) -> f(?x,f(?y,?z)),
f(?x,?y) -> f(?y,?x) ]
Success
Reduction-Preserving Completion
Direct Methods: CR
Final result: CR
/local-scratch/hzankl/2012/cops2012/103.trs: Success(CR)
(5 msec.)