0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 QReductionProof (⇔)
↳11 QDP
↳12 QDPSizeChangeProof (⇔)
↳13 TRUE
↳14 QDP
↳15 QDPOrderProof (⇔)
↳16 QDP
↳17 PisEmptyProof (⇔)
↳18 TRUE
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
F(s(x)) → F(x)
G(s(x), s(y)) → IF(f(x), s(x), s(y))
G(s(x), s(y)) → F(x)
G(x, c(y)) → G(x, g(s(c(y)), y))
G(x, c(y)) → G(s(c(y)), y)
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
F(s(x)) → F(x)
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
F(s(x)) → F(x)
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
F(s(x)) → F(x)
From the DPs we obtained the following set of size-change graphs:
G(x, c(y)) → G(s(c(y)), y)
G(x, c(y)) → G(x, g(s(c(y)), y))
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
G(x, c(y)) → G(s(c(y)), y)
G(x, c(y)) → G(x, g(s(c(y)), y))
POL(0) = 0
POL(1) = 0
POL(G(x1, x2)) = x2
POL(c(x1)) = 1 + x1
POL(f(x1)) = 0
POL(false) = 0
POL(g(x1, x2)) = 0
POL(if(x1, x2, x3)) = x2 + x3
POL(s(x1)) = 0
POL(true) = 0
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0) → true
f(1) → false
f(s(x)) → f(x)
if(true, x, y) → x
if(false, x, y) → y
g(s(x), s(y)) → if(f(x), s(x), s(y))
g(x, c(y)) → g(x, g(s(c(y)), y))
f(0)
f(1)
f(s(x0))
if(true, x0, x1)
if(false, x0, x1)
g(s(x0), s(x1))
g(x0, c(x1))