YES
0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 0 ms)
↳4 AND
↳5 QDP
↳6 UsableRulesProof (⇔, 0 ms)
↳7 QDP
↳8 MRRProof (⇔, 0 ms)
↳9 QDP
↳10 PisEmptyProof (⇔, 0 ms)
↳11 YES
↳12 QDP
↳13 UsableRulesProof (⇔, 0 ms)
↳14 QDP
↳15 MRRProof (⇔, 0 ms)
↳16 QDP
↳17 PisEmptyProof (⇔, 0 ms)
↳18 YES
↳19 QDP
↳20 UsableRulesProof (⇔, 0 ms)
↳21 QDP
↳22 QDPSizeChangeProof (⇔, 0 ms)
↳23 YES
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
+1(O(x), O(y)) → O1(+(x, y))
+1(O(x), O(y)) → +1(x, y)
+1(O(x), I(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → O1(+(+(x, y), I(0)))
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
*1(O(x), y) → O1(*(x, y))
*1(O(x), y) → *1(x, y)
*1(I(x), y) → +1(O(*(x, y)), y)
*1(I(x), y) → O1(*(x, y))
*1(I(x), y) → *1(x, y)
-1(O(x), O(y)) → O1(-(x, y))
-1(O(x), O(y)) → -1(x, y)
-1(O(x), I(y)) → -1(-(x, y), I(1))
-1(O(x), I(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(I(x), I(y)) → O1(-(x, y))
-1(I(x), I(y)) → -1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
-1(O(x), I(y)) → -1(-(x, y), I(1))
-1(O(x), I(y)) → -1(x, y)
-1(O(x), O(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(I(x), I(y)) → -1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
-1(O(x), I(y)) → -1(-(x, y), I(1))
-1(O(x), I(y)) → -1(x, y)
-1(O(x), O(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(I(x), I(y)) → -1(x, y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
O(0) → 0
-1(O(x), I(y)) → -1(-(x, y), I(1))
-1(O(x), I(y)) → -1(x, y)
-1(O(x), O(y)) → -1(x, y)
-1(I(x), O(y)) → -1(x, y)
-1(I(x), I(y)) → -1(x, y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
O(0) → 0
-^12 > 0 > -2 > 1 > O1 > I1
0=1
1=2
O_1=5
I_1=3
-_2=0
-^1_2=0
+1(O(x), I(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
+1(O(x), I(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
O(0) → 0
+1(O(x), I(y)) → +1(x, y)
+1(O(x), O(y)) → +1(x, y)
+1(I(x), O(y)) → +1(x, y)
+1(I(x), I(y)) → +1(+(x, y), I(0))
+1(I(x), I(y)) → +1(x, y)
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
O(0) → 0
0 > +^12 > +2 > I1 > O1
0=2
O_1=1
I_1=3
+_2=0
+^1_2=0
*1(I(x), y) → *1(x, y)
*1(O(x), y) → *1(x, y)
O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))
*1(I(x), y) → *1(x, y)
*1(O(x), y) → *1(x, y)
From the DPs we obtained the following set of size-change graphs: