+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
*12(*2(x, y), z) -> *12(y, z)
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
*12(s1(x), s1(y)) -> +12(x, y)
*12(s1(x), s1(y)) -> *12(x, y)
SUM1(cons2(x, l)) -> SUM1(l)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(+2(x, y), z) -> +12(x, +2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
*12(*2(x, y), z) -> *12(y, z)
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
*12(s1(x), s1(y)) -> +12(x, y)
*12(s1(x), s1(y)) -> *12(x, y)
SUM1(cons2(x, l)) -> SUM1(l)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(+2(x, y), z) -> +12(x, +2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
+12(s1(x), s1(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(s1(x), s1(y)) -> +12(x, y)
Used ordering: Polynomial Order [17,21] with Interpretation:
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
POL( s1(x1) ) = x1 + 1
POL( 0 ) = 1
POL( +12(x1, x2) ) = max{0, x1 + x2 - 1}
POL( +2(x1, x2) ) = x1 + x2 + 1
+2(0, x) -> x
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(x, 0) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(+2(x, y), z) -> +12(y, z)
Used ordering: Polynomial Order [17,21] with Interpretation:
+12(+2(x, y), z) -> +12(x, +2(y, z))
POL( s1(x1) ) = max{0, -1}
POL( 0 ) = max{0, -1}
POL( +12(x1, x2) ) = x1 + x2
POL( +2(x1, x2) ) = x1 + x2 + 1
+2(0, x) -> x
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(x, 0) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
+12(+2(x, y), z) -> +12(x, +2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
+12(+2(x, y), z) -> +12(x, +2(y, z))
POL( s1(x1) ) = max{0, -1}
POL( 0 ) = 1
POL( +12(x1, x2) ) = x1 + 1
POL( +2(x1, x2) ) = x1 + 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
SUM1(cons2(x, l)) -> SUM1(l)
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SUM1(cons2(x, l)) -> SUM1(l)
POL( cons2(x1, x2) ) = x2 + 1
POL( SUM1(x1) ) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
*12(*2(x, y), z) -> *12(y, z)
*12(s1(x), s1(y)) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
Used ordering: Polynomial Order [17,21] with Interpretation:
*12(s1(x), s1(y)) -> *12(x, y)
POL( s1(x1) ) = x1
POL( 0 ) = max{0, -1}
POL( *12(x1, x2) ) = x1 + 1
POL( +2(x1, x2) ) = 1
POL( *2(x1, x2) ) = x1 + x2 + 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
*12(s1(x), s1(y)) -> *12(x, y)
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
*12(s1(x), s1(y)) -> *12(x, y)
POL( *12(x1, x2) ) = x1
POL( s1(x1) ) = x1 + 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
PROD1(cons2(x, l)) -> PROD1(l)
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROD1(cons2(x, l)) -> PROD1(l)
POL( PROD1(x1) ) = x1
POL( cons2(x1, x2) ) = x2 + 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))