YES
0 QTRS
↳1 Overlay + Local Confluence (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 0 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 QDPQMonotonicMRRProof (⇔, 39 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 QDP
↳11 UsableRulesProof (⇔, 0 ms)
↳12 QDP
↳13 QReductionProof (⇔, 0 ms)
↳14 QDP
↳15 UsableRulesReductionPairsProof (⇔, 0 ms)
↳16 QDP
↳17 QDPSizeChangeProof (⇔, 0 ms)
↳18 YES
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
DIN(der(plus(X, Y))) → U21(din(der(X)), X, Y)
DIN(der(plus(X, Y))) → DIN(der(X))
U21(dout(DX), X, Y) → U22(din(der(Y)), X, Y, DX)
U21(dout(DX), X, Y) → DIN(der(Y))
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
DIN(der(times(X, Y))) → DIN(der(X))
U31(dout(DX), X, Y) → U32(din(der(Y)), X, Y, DX)
U31(dout(DX), X, Y) → DIN(der(Y))
DIN(der(der(X))) → U41(din(der(X)), X)
DIN(der(der(X))) → DIN(der(X))
U41(dout(DX), X) → U42(din(der(DX)), X, DX)
U41(dout(DX), X) → DIN(der(DX))
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
U21(dout(DX), X, Y) → DIN(der(Y))
DIN(der(plus(X, Y))) → U21(din(der(X)), X, Y)
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
U31(dout(DX), X, Y) → DIN(der(Y))
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(der(X))) → U41(din(der(X)), X)
U41(dout(DX), X) → DIN(der(DX))
DIN(der(der(X))) → DIN(der(X))
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) → u41(din(der(X)), X)
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
U21(dout(DX), X, Y) → DIN(der(Y))
DIN(der(plus(X, Y))) → U21(din(der(X)), X, Y)
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
U31(dout(DX), X, Y) → DIN(der(Y))
DIN(der(der(X))) → U41(din(der(X)), X)
u21(dout(DX), X, Y) → u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) → dout(plus(DX, DY))
u31(dout(DX), X, Y) → u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) → dout(plus(times(X, DY), times(Y, DX)))
u41(dout(DX), X) → u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) → dout(DDX)
POL(DIN(x1)) = 2
POL(U21(x1, x2, x3)) = 2·x1
POL(U31(x1, x2, x3)) = 2·x1
POL(U41(x1, x2)) = x1
POL(der(x1)) = 2·x1
POL(din(x1)) = 0
POL(dout(x1)) = 2 + 2·x1
POL(plus(x1, x2)) = 0
POL(times(x1, x2)) = x1
POL(u21(x1, x2, x3)) = x1
POL(u22(x1, x2, x3, x4)) = 2·x1 + 2·x4
POL(u31(x1, x2, x3)) = x1
POL(u32(x1, x2, x3, x4)) = 2·x1
POL(u41(x1, x2)) = x1
POL(u42(x1, x2, x3)) = 2·x1
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → DIN(der(X))
U41(dout(DX), X) → DIN(der(DX))
DIN(der(der(X))) → DIN(der(X))
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
din(der(der(X))) → u41(din(der(X)), X)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(der(X))) → DIN(der(X))
din(der(plus(X, Y))) → u21(din(der(X)), X, Y)
din(der(times(X, Y))) → u31(din(der(X)), X, Y)
din(der(der(X))) → u41(din(der(X)), X)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(der(X))) → DIN(der(X))
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(der(X))) → DIN(der(X))
No rules are removed from R.
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
POL(DIN(x1)) = 2·x1
POL(der(x1)) = x1
POL(plus(x1, x2)) = 2·x1 + x2
POL(times(x1, x2)) = 2·x1 + x2
DIN(der(der(X))) → DIN(der(X))
From the DPs we obtained the following set of size-change graphs: