0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 PisEmptyProof (⇔)
↳9 TRUE
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 TRUE
↳15 QDP
↳16 QDPOrderProof (⇔)
↳17 QDP
↳18 PisEmptyProof (⇔)
↳19 TRUE
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
+1(a, b) → +1(b, a)
+1(a, +(b, z)) → +1(b, +(a, z))
+1(a, +(b, z)) → +1(a, z)
+1(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)
F(+(x, y), z) → +1(f(x, z), f(y, z))
F(+(x, y), z) → F(x, z)
F(+(x, y), z) → F(y, z)
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
+1(a, +(b, z)) → +1(a, z)
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(a, +(b, z)) → +1(a, z)
[a, b] > [+^11, +1]
+^11: multiset
a: multiset
+1: multiset
b: multiset
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
[+^12, +2] > [a, b]
+^12: [1,2]
+2: [1,2]
a: multiset
b: multiset
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
F(+(x, y), z) → F(y, z)
F(+(x, y), z) → F(x, z)
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
F(+(x, y), z) → F(y, z)
F(+(x, y), z) → F(x, z)
trivial
F2: [2,1]
+2: multiset
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))