(ignored inputs)COMMENT from Example 3 of \cite{OO03}
Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))),
0 = 0,
gcd(0,mod(?x_11,0)) = ?x_11,
gcd(?y_11,mod(0,?y_11)) = ?y_11,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[([(if,1)],6),([],0)], []>
Critical Pair <0, 0> by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],10),([],13)]>
joinable by a reduction of rules <[], [([],13),([],10)]>
Critical Pair by Rules <13, 11> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],12)]>
Critical Pair <0, 0> by Rules <13, 12> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 42*42=>42
0 : =>42
< : 42*42=>48
s : 42=>42
if : 48*42*42=>42
gcd : 42*42=>42
mod : 42*42=>42
true : =>48
false : =>48
maximal types: {42,48}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
Try Commutative Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Outside Critical Pair: <0, 0> by Rules <4, 3>
develop reducts from lhs term...
<{}, 0>
develop reducts from rhs term...
<{}, 0>
Outside Critical Pair: by Rules <9, 8>
develop reducts from lhs term...
<{4,6,9}, if(true,0,if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))))>
<{6,9}, if(true,0,if(<(-(0,s(?y_9)),s(?y_9)),-(0,s(?y_9)),mod(-(-(0,s(?y_9)),s(?y_9)),s(?y_9))))>
<{4,6}, if(true,0,mod(0,s(?y_9)))>
<{6}, if(true,0,mod(-(0,s(?y_9)),s(?y_9)))>
<{4,9}, if(<(0,s(?y_9)),0,if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))))>
<{9}, if(<(0,s(?y_9)),0,if(<(-(0,s(?y_9)),s(?y_9)),-(0,s(?y_9)),mod(-(-(0,s(?y_9)),s(?y_9)),s(?y_9))))>
<{4}, if(<(0,s(?y_9)),0,mod(0,s(?y_9)))>
<{}, if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9)))>
develop reducts from rhs term...
<{}, 0>
Outside Critical Pair: <0, 0> by Rules <10, 8>
develop reducts from lhs term...
<{}, 0>
develop reducts from rhs term...
<{}, 0>
Outside Critical Pair: by Rules <12, 11>
develop reducts from lhs term...
<{}, ?x_12>
develop reducts from rhs term...
<{10,13}, ?x_12>
<{13}, mod(?x_12,0)>
<{10,11}, gcd(?x_12,mod(0,?x_12))>
<{11}, gcd(mod(?x_12,0),mod(0,mod(?x_12,0)))>
<{10}, gcd(0,?x_12)>
<{}, gcd(0,mod(?x_12,0))>
Outside Critical Pair: by Rules <13, 11>
develop reducts from lhs term...
<{}, ?x_13>
develop reducts from rhs term...
<{8,11}, gcd(0,mod(?x_13,0))>
<{11}, gcd(mod(0,?x_13),mod(?x_13,mod(0,?x_13)))>
<{8}, gcd(?x_13,0)>
<{}, gcd(?x_13,mod(0,?x_13))>
Outside Critical Pair: <0, 0> by Rules <13, 12>
develop reducts from lhs term...
<{}, 0>
develop reducts from rhs term...
<{}, 0>
Try A Minimal Decomposition {8,9}{13,11,12}{0}{1}{2}{3}{4}{5}{6}{7}{10}
{8,9}
(cm)Rewrite Rules:
[ mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))) ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_1)),0,mod(-(0,s(?y_1)),s(?y_1))) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ 0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_1)),0,mod(-(0,s(?y_1)),s(?y_1))) = 0 ]
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 []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_1)),0,if(<(-(0,s(?y_1)),s(?y_1)),-(0,s(?y_1)),if(<(-(-(0,s(?y_1)),s(?y_1)),s(?y_1)),-(-(0,s(?y_1)),s(?y_1)),mod(-(-(-(0,s(?y_1)),s(?y_1)),s(?y_1)),s(?y_1)))))> from mod(0,s(?y_1))
Not Confluent
Direct Methods: not CR
{13,11,12}
(cm)Rewrite Rules:
[ gcd(0,?x) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ?x = gcd(?x,mod(0,?x)),
0 = 0,
gcd(0,mod(?x_1,0)) = ?x_1 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(0,mod(?x,0)) = ?x,
?x = gcd(0,mod(?x,0)),
?y = gcd(?y,mod(0,?y)),
0 = 0,
gcd(?x,mod(0,?x)) = ?x ]
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 []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_1,0)
Not Confluent
Direct Methods: not CR
{0}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{1}
(cm)Rewrite Rules:
[ if(false,?x,?y) -> ?y ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{2}
(cm)Rewrite Rules:
[ -(s(?x),s(?y)) -> -(?x,?y) ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{3}
(cm)Rewrite Rules:
[ -(?x,0) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{4}
(cm)Rewrite Rules:
[ -(0,?x) -> 0 ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{5}
(cm)Rewrite Rules:
[ <(s(?x),s(?y)) -> <(?x,?y) ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{6}
(cm)Rewrite Rules:
[ <(0,s(?x)) -> true ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{7}
(cm)Rewrite Rules:
[ <(?x,0) -> false ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
{10}
(cm)Rewrite Rules:
[ mod(?x,0) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ ]
Overlay, check Innermost Termination...
Innermost Terminating (hence Terminating), WCR
Knuth&Bendix
Direct Methods: CR
Try to add other components to {8,9}
Add {13,11,12}
{8,9,11,12,13}
(cm)Rewrite Rules:
[ mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_1)),0,mod(-(0,s(?y_1)),s(?y_1))),
gcd(0,mod(?x_2,0)) = ?x_2,
gcd(?y_2,mod(0,?y_2)) = ?y_2,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_1)),0,mod(-(0,s(?y_1)),s(?y_1))) = 0 ]
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 []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_2,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}
{0,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_2)),0,mod(-(0,s(?y_2)),s(?y_2))),
gcd(0,mod(?x_3,0)) = ?x_3,
gcd(?y_3,mod(0,?y_3)) = ?y_3,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_2)),0,mod(-(0,s(?y_2)),s(?y_2))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <2, 1> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_3,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}
{0,1,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))),
gcd(0,mod(?x_4,0)) = ?x_4,
gcd(?y_4,mod(0,?y_4)) = ?y_4,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <3, 2> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_4,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}
{0,1,2,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}
{0,1,2,3,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}
{0,1,2,3,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}
{0,1,2,3,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}{6}
{0,1,2,3,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],6),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}{6}{7}
{0,1,2,3,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))),
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_9)),0,mod(-(0,s(?y_9)),s(?y_9))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[([(if,1)],6),([],0)], []>
Critical Pair by Rules <11, 10> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_10,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}{6}{10}
{0,1,2,3,4,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],6),([],0)], []>
Critical Pair <0, 0> by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],9),([],12)]>
joinable by a reduction of rules <[], [([],12),([],9)]>
Critical Pair by Rules <12, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],11)]>
Critical Pair <0, 0> by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 41*41=>41
0 : =>41
< : 41*41=>47
s : 41=>41
if : 47*41*41=>41
gcd : 41*41=>41
mod : 41*41=>41
true : =>47
false : =>47
maximal types: {41,47}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{4}{5}{7}
{0,1,2,3,4,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}{10}
{0,1,2,3,4,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_7)),0,0)> from mod(0,s(?y_7))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{5}{7}{10}
{0,1,2,3,4,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_8)),0,0)> from mod(0,s(?y_8))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{6}
{0,1,2,3,4,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{6}{7}
{0,1,2,3,4,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{6}{10}
{0,1,2,3,4,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{4}{7}
{0,1,2,3,4,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{10}
{0,1,2,3,4,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{7}{10}
{0,1,2,3,4,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_7)),0,0)> from mod(0,s(?y_7))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{4}{6}{7}{10}
{0,1,2,3,4,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <4, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],9),([],12)]>
joinable by a reduction of rules <[], [([],12),([],9)]>
Critical Pair by Rules <12, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],11)]>
Critical Pair <0, 0> by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{5}
{0,1,2,3,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{5}{6}
{0,1,2,3,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{5}{6}{7}
{0,1,2,3,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{5}{6}{10}
{0,1,2,3,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{5}{7}
{0,1,2,3,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{5}{10}
{0,1,2,3,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>27
s : 39=>39
if : 27*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>27
false : =>27
maximal types: {27,39}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{5}{7}{10}
{0,1,2,3,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{6}
{0,1,2,3,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{6}{7}
{0,1,2,3,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{6}{10}
{0,1,2,3,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{7}
{0,1,2,3,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{3}{10}
{0,1,2,3,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>25
s : 37=>37
if : 25*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>25
false : =>25
maximal types: {25,37}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{7}{10}
{0,1,2,3,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{6}{7}{10}
{0,1,2,3,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{3}{5}{6}{7}{10}
{0,1,2,3,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],9),([],12)]>
joinable by a reduction of rules <[], [([],12),([],9)]>
Critical Pair by Rules <12, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],11)]>
Critical Pair <0, 0> by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 41*41=>41
0 : =>41
< : 41*41=>47
s : 41=>41
if : 47*41*41=>41
gcd : 41*41=>41
mod : 41*41=>41
true : =>47
false : =>47
maximal types: {41,47}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{4}
{0,1,2,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}
{0,1,2,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}{6}
{0,1,2,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}{6}{7}
{0,1,2,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}{6}{10}
{0,1,2,4,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{4}{5}{7}
{0,1,2,4,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}{10}
{0,1,2,4,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{5}{7}{10}
{0,1,2,4,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_7)),0,0)> from mod(0,s(?y_7))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{6}
{0,1,2,4,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{6}{7}
{0,1,2,4,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{6}{10}
{0,1,2,4,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{4}{7}
{0,1,2,4,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{10}
{0,1,2,4,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_5)),0,0)> from mod(0,s(?y_5))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{7}{10}
{0,1,2,4,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{4}{6}{7}{10}
{0,1,2,4,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{5}
{0,1,2,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{5}{6}
{0,1,2,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{5}{6}{7}
{0,1,2,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{5}{6}{10}
{0,1,2,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{5}{7}
{0,1,2,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{5}{10}
{0,1,2,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>26
s : 38=>38
if : 26*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>26
false : =>26
maximal types: {26,38}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{5}{7}{10}
{0,1,2,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{6}
{0,1,2,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{6}{7}
{0,1,2,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{6}{10}
{0,1,2,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{7}
{0,1,2,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{2}{10}
{0,1,2,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>24
s : 36=>36
if : 24*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>24
false : =>24
maximal types: {24,36}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{7}{10}
{0,1,2,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{6}{7}{10}
{0,1,2,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{5}{6}{7}{10}
{0,1,2,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{2}{4}{5}{6}{7}{10}
{0,1,2,4,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],9),([],12)]>
joinable by a reduction of rules <[], [([],12),([],9)]>
Critical Pair by Rules <12, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],11)]>
Critical Pair <0, 0> by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 41*41=>41
0 : =>41
< : 41*41=>47
s : 41=>41
if : 47*41*41=>41
gcd : 41*41=>41
mod : 41*41=>41
true : =>47
false : =>47
maximal types: {41,47}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}
{0,1,3,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}
{0,1,3,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}
{0,1,3,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}{6}
{0,1,3,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}{6}{7}
{0,1,3,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}{6}{10}
{0,1,3,4,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{4}{5}{7}
{0,1,3,4,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}{10}
{0,1,3,4,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{5}{7}{10}
{0,1,3,4,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_7)),0,0)> from mod(0,s(?y_7))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{6}
{0,1,3,4,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{6}{7}
{0,1,3,4,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{6}{10}
{0,1,3,4,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 23=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {23,37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{4}{7}
{0,1,3,4,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{10}
{0,1,3,4,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_5)),0,0)> from mod(0,s(?y_5))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{7}{10}
{0,1,3,4,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{4}{6}{7}{10}
{0,1,3,4,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 24=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {24,38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{5}
{0,1,3,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{5}{6}
{0,1,3,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{5}{6}{7}
{0,1,3,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{5}{6}{10}
{0,1,3,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{5}{7}
{0,1,3,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{5}{10}
{0,1,3,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>25
s : 37=>37
if : 25*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>25
false : =>25
maximal types: {25,37}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{5}{7}{10}
{0,1,3,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{6}
{0,1,3,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{6}{7}
{0,1,3,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{6}{10}
{0,1,3,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>42
s : 22=>36
if : 42*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>42
false : =>42
maximal types: {22,36,42}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{7}
{0,1,3,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{3}{10}
{0,1,3,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 35*35=>35
0 : =>35
< : 35*35=>23
s : 21=>35
if : 23*35*35=>35
gcd : 35*35=>35
mod : 35*35=>35
true : =>23
false : =>23
maximal types: {21,23,35}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{7}{10}
{0,1,3,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>42
s : 22=>36
if : 42*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>42
false : =>42
maximal types: {22,36,42}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{6}{7}{10}
{0,1,3,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 23=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {23,37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{5}{6}{7}{10}
{0,1,3,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{4}
{0,1,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}
{0,1,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}{6}
{0,1,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}{6}{7}
{0,1,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}{6}{10}
{0,1,4,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{4}{5}{7}
{0,1,4,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}{10}
{0,1,4,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_5)),0,0)> from mod(0,s(?y_5))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{5}{7}{10}
{0,1,4,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{6}
{0,1,4,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{6}{7}
{0,1,4,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{6}{10}
{0,1,4,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>42
s : 22=>36
if : 42*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>42
false : =>42
maximal types: {22,36,42}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{4}{7}
{0,1,4,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{10}
{0,1,4,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_4)),0,0)> from mod(0,s(?y_4))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{7}{10}
{0,1,4,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_5)),0,0)> from mod(0,s(?y_5))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{4}{6}{7}{10}
{0,1,4,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 23=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {23,37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{5}
{0,1,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{5}{6}
{0,1,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{5}{6}{7}
{0,1,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{5}{6}{10}
{0,1,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{5}{7}
{0,1,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{5}{10}
{0,1,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>24
s : 36=>36
if : 24*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>24
false : =>24
maximal types: {24,36}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{5}{7}{10}
{0,1,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{6}
{0,1,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
joinable by a reduction of rules <[([(if,1)],2),([],0)], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{6}{7}
{0,1,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],2),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{6}{10}
{0,1,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
joinable by a reduction of rules <[([(if,1)],2),([],0)], []>
Critical Pair <0, 0> by Rules <5, 3> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],8)]>
joinable by a reduction of rules <[], [([],8),([],5)]>
Critical Pair by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],3),([],7)]>
Critical Pair <0, 0> by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 35*35=>35
0 : =>35
< : 35*35=>41
s : 21=>35
if : 41*35*35=>35
gcd : 35*35=>35
mod : 35*35=>35
true : =>41
false : =>41
maximal types: {21,35,41}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{7}
{0,1,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{1}{10}
{0,1,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))),
0 = 0,
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <3, 2> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 34*34=>34
0 : =>34
< : 34*34=>22
s : 20=>34
if : 22*34*34=>34
gcd : 34*34=>34
mod : 34*34=>34
true : =>22
false : =>22
maximal types: {20,22,34}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{7}{10}
{0,1,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 35*35=>35
0 : =>35
< : 35*35=>41
s : 21=>35
if : 41*35*35=>35
gcd : 35*35=>35
mod : 35*35=>35
true : =>41
false : =>41
maximal types: {21,35,41}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{6}{7}{10}
{0,1,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],2),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>42
s : 22=>36
if : 42*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>42
false : =>42
maximal types: {22,36,42}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{5}{6}{7}{10}
{0,1,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{4}{5}{6}{7}{10}
{0,1,4,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{1}{3}{4}{5}{6}{7}{10}
{0,1,3,4,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
0 = 0,
gcd(0,mod(?x_10,0)) = ?x_10,
gcd(?y_10,mod(0,?y_10)) = ?y_10,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],9),([],12)]>
joinable by a reduction of rules <[], [([],12),([],9)]>
Critical Pair by Rules <12, 10> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],11)]>
Critical Pair <0, 0> by Rules <12, 11> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 40*40=>40
0 : =>40
< : 40*40=>46
s : 40=>40
if : 46*40*40=>40
gcd : 40*40=>40
mod : 40*40=>40
true : =>46
false : =>46
maximal types: {40,46}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
if(false,?x,?y) -> ?y,
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}
{0,2,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))),
gcd(0,mod(?x_4,0)) = ?x_4,
gcd(?y_4,mod(0,?y_4)) = ?y_4,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_3)),0,mod(-(0,s(?y_3)),s(?y_3))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <3, 2> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_4,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}
{0,2,3,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}
{0,2,3,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}
{0,2,3,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}{6}
{0,2,3,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}{6}{7}
{0,2,3,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))),
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_8)),0,mod(-(0,s(?y_8)),s(?y_8))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair by Rules <10, 9> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_9,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}{6}{10}
{0,2,3,4,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],5),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{4}{5}{7}
{0,2,3,4,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}{10}
{0,2,3,4,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{5}{7}{10}
{0,2,3,4,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_7)),0,0)> from mod(0,s(?y_7))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{6}
{0,2,3,4,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{6}{7}
{0,2,3,4,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{6}{10}
{0,2,3,4,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>42
s : 36=>36
if : 42*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>42
maximal types: {36,42}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{4}{7}
{0,2,3,4,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{10}
{0,2,3,4,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_5)),0,0)> from mod(0,s(?y_5))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{7}{10}
{0,2,3,4,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: <0, if(<(0,s(?y_6)),0,0)> from mod(0,s(?y_6))
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{4}{6}{7}{10}
{0,2,3,4,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = 0,
0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0,
0 = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <0, 0> by Rules <3, 2> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>44
s : 38=>38
if : 44*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>44
false : =>44
maximal types: {38,44}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
-(0,?x) -> 0,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{5}
{0,2,3,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{5}{6}
{0,2,3,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{5}{6}{7}
{0,2,3,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <9, 8> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_8,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{5}{6}{10}
{0,2,3,5,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{5}{7}
{0,2,3,5,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{5}{10}
{0,2,3,5,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>25
s : 36=>36
if : 25*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>25
maximal types: {25,36}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{5}{7}{10}
{0,2,3,5,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 38*38=>38
0 : =>38
< : 38*38=>26
s : 38=>38
if : 26*38*38=>38
gcd : 38*38=>38
mod : 38*38=>38
true : =>26
false : =>26
maximal types: {26,38}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{6}
{0,2,3,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <7, 6> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{6}{7}
{0,2,3,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{6}{10}
{0,2,3,6,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <6, 4> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <8, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],9)]>
joinable by a reduction of rules <[], [([],9),([],6)]>
Critical Pair by Rules <9, 7> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],4),([],8)]>
Critical Pair <0, 0> by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 35*35=>35
0 : =>35
< : 35*35=>41
s : 35=>35
if : 41*35*35=>35
gcd : 35*35=>35
mod : 35*35=>35
true : =>41
maximal types: {35,41}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{7}
{0,2,3,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{3}{10}
{0,2,3,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
0 = 0,
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 34*34=>34
0 : =>34
< : 34*34=>23
s : 34=>34
if : 23*34*34=>34
gcd : 34*34=>34
mod : 34*34=>34
true : =>23
maximal types: {23,34}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{7}{10}
{0,2,3,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
0 = 0,
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 36*36=>36
0 : =>36
< : 36*36=>24
s : 36=>36
if : 24*36*36=>36
gcd : 36*36=>36
mod : 36*36=>36
true : =>24
false : =>24
maximal types: {24,36}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{6}{7}{10}
{0,2,3,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
0 = 0,
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],3),([],0)], []>
Critical Pair <0, 0> by Rules <7, 5> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <9, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],7),([],10)]>
joinable by a reduction of rules <[], [([],10),([],7)]>
Critical Pair by Rules <10, 8> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],5),([],9)]>
Critical Pair <0, 0> by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 37*37=>37
0 : =>37
< : 37*37=>43
s : 37=>37
if : 43*37*37=>37
gcd : 37*37=>37
mod : 37*37=>37
true : =>43
false : =>43
maximal types: {37,43}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{3}{5}{6}{7}{10}
{0,2,3,5,6,7,8,9,10,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
0 = 0,
gcd(0,mod(?x_9,0)) = ?x_9,
gcd(?y_9,mod(0,?y_9)) = ?y_9,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
0 = 0,
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <7, 6> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair <0, 0> by Rules <8, 6> preceded by []
joinable by a reduction of rules <[], []>
Critical Pair by Rules <10, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],8),([],11)]>
joinable by a reduction of rules <[], [([],11),([],8)]>
Critical Pair by Rules <11, 9> preceded by []
joinable by a reduction of rules <[], [([(gcd,2)],6),([],10)]>
Critical Pair <0, 0> by Rules <11, 10> preceded by []
joinable by a reduction of rules <[], []>
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Unknown
Direct Methods: Can't judge
Try Persistent Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Sort Assignment:
- : 39*39=>39
0 : =>39
< : 39*39=>45
s : 39=>39
if : 45*39*39=>39
gcd : 39*39=>39
mod : 39*39=>39
true : =>45
false : =>45
maximal types: {39,45}
Persistent Decomposition failed: Can't judge
Try Layer Preserving Decomposition for...
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(?x,0) -> ?x,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
mod(?x,0) -> ?x,
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Layer Preserving Decomposition failed: Can't judge
No further decomposition possible
Add {13,11,12}{0}{2}{4}
{0,2,4,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))),
gcd(0,mod(?x_5,0)) = ?x_5,
gcd(?y_5,mod(0,?y_5)) = ?y_5,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_4)),0,mod(-(0,s(?y_4)),s(?y_4))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <4, 3> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_5,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{4}{5}
{0,2,4,5,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))),
gcd(0,mod(?x_6,0)) = ?x_6,
gcd(?y_6,mod(0,?y_6)) = ?y_6,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_5)),0,mod(-(0,s(?y_5)),s(?y_5))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <5, 4> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_6,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{4}{5}{6}
{0,2,4,5,6,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))),
gcd(0,mod(?x_7,0)) = ?x_7,
gcd(?y_7,mod(0,?y_7)) = ?y_7,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_6)),0,mod(-(0,s(?y_6)),s(?y_6))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair by Rules <6, 5> preceded by []
joinable by a reduction of rules <[([(if,1)],4),([],0)], []>
Critical Pair by Rules <8, 7> preceded by []
unknown Diagram Decreasing
check by Reduction-Preserving Completion...
failure(empty P)
unknown Reduction-Preserving Completion
check Non-Confluence...Find Non-Joinable CP reducts: from gcd(?x_7,0)
Not Confluent
Direct Methods: not CR
Add {13,11,12}{0}{2}{4}{5}{6}{7}
{0,2,4,5,6,7,8,9,11,12,13}
(cm)Rewrite Rules:
[ if(true,?x,?y) -> ?x,
-(s(?x),s(?y)) -> -(?x,?y),
-(0,?x) -> 0,
<(s(?x),s(?y)) -> <(?x,?y),
<(0,s(?x)) -> true,
<(?x,0) -> false,
mod(0,?y) -> 0,
mod(?x,s(?y)) -> if(<(?x,s(?y)),?x,mod(-(?x,s(?y)),s(?y))),
gcd(?x,?y) -> gcd(?y,mod(?x,?y)),
gcd(?x,0) -> ?x,
gcd(0,?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
[ ]
Outer CPs:
[ 0 = if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))),
gcd(0,mod(?x_8,0)) = ?x_8,
gcd(?y_8,mod(0,?y_8)) = ?y_8,
0 = 0 ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth&Bendix
Left-Linear, not Right-Linear
unknown Development Closed
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
[ gcd(?x,mod(0,?x)) = ?x,
0 = 0,
gcd(0,mod(?x,0)) = ?x,
?y = gcd(?y,mod(0,?y)),
?x = gcd(0,mod(?x,0)),
0 = if(<(0,s(?y)),0,mod(-(0,s(?y)),s(?y))),
if(<(0,s(?y_7)),0,mod(-(0,s(?y_7)),s(?y_7))) = 0 ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair