2.60/2.68	YES
2.60/2.68	
2.60/2.68	DP problem for innermost termination.
2.60/2.68	P =
2.60/2.68	  init#(x1, x2) -> f1#(rnd1, rnd2) 
2.60/2.68	  f3#(I0, I1) -> f2#(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2#(I5, I6) -> f3#(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  f1#(I7, I8) -> f2#(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	R = 
2.60/2.68	  init(x1, x2) -> f1(rnd1, rnd2) 
2.60/2.68	  f3(I0, I1) -> f2(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3(I3, I4) -> f3(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2(I5, I6) -> f3(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  f1(I7, I8) -> f2(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	
2.60/2.68	The dependency graph for this problem is:
2.60/2.68	  0 -> 4
2.60/2.68	  1 -> 3
2.60/2.68	  2 -> 1, 2
2.60/2.68	  3 -> 2
2.60/2.68	  4 -> 3
2.60/2.68	Where:
2.60/2.68	  0) init#(x1, x2) -> f1#(rnd1, rnd2) 
2.60/2.68	  1) f3#(I0, I1) -> f2#(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  2) f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  3) f2#(I5, I6) -> f3#(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  4) f1#(I7, I8) -> f2#(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	
2.60/2.68	We have the following SCCs.
2.60/2.68	  { 1, 2, 3 }
2.60/2.68	
2.60/2.68	DP problem for innermost termination.
2.60/2.68	P =
2.60/2.68	  f3#(I0, I1) -> f2#(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2#(I5, I6) -> f3#(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	R = 
2.60/2.68	  init(x1, x2) -> f1(rnd1, rnd2) 
2.60/2.68	  f3(I0, I1) -> f2(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3(I3, I4) -> f3(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2(I5, I6) -> f3(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  f1(I7, I8) -> f2(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	
2.60/2.68	We use the reverse value criterion with the projection function NU:
2.60/2.68	  NU[f2#(z1,z2)] = z1 - 1 + -1 * -1
2.60/2.68	  NU[f3#(z1,z2)] = z1 - 1 + -1 * 0
2.60/2.68	
2.60/2.68	This gives the following inequalities:
2.60/2.68	  I0 <= I1 - 1 /\ 0 <= I0 - 1  ==>  I0 - 1 + -1 * 0 > I0 - 2 - 1 + -1 * -1  with  I0 - 1 + -1 * 0 >= 0
2.60/2.68	  I4 <= I3  ==>  I3 - 1 + -1 * 0 >= I3 - 1 + -1 * 0
2.60/2.68	  -1 <= I5 - 1  ==>  I5 - 1 + -1 * -1 >= I5 + 1 - 1 + -1 * 0
2.60/2.68	
2.60/2.68	We remove all the strictly oriented dependency pairs.
2.60/2.68	
2.60/2.68	DP problem for innermost termination.
2.60/2.68	P =
2.60/2.68	  f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2#(I5, I6) -> f3#(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	R = 
2.60/2.68	  init(x1, x2) -> f1(rnd1, rnd2) 
2.60/2.68	  f3(I0, I1) -> f2(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3(I3, I4) -> f3(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2(I5, I6) -> f3(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  f1(I7, I8) -> f2(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	
2.60/2.68	The dependency graph for this problem is:
2.60/2.68	  2 -> 2
2.60/2.68	  3 -> 2
2.60/2.68	Where:
2.60/2.68	  2) f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  3) f2#(I5, I6) -> f3#(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	
2.60/2.68	We have the following SCCs.
2.60/2.68	  { 2 }
2.60/2.68	
2.60/2.68	DP problem for innermost termination.
2.60/2.68	P =
2.60/2.68	  f3#(I3, I4) -> f3#(I3, I4 + 1) [I4 <= I3]
2.60/2.68	R = 
2.60/2.68	  init(x1, x2) -> f1(rnd1, rnd2) 
2.60/2.68	  f3(I0, I1) -> f2(I0 - 2, I2) [I0 <= I1 - 1 /\ 0 <= I0 - 1]
2.60/2.68	  f3(I3, I4) -> f3(I3, I4 + 1) [I4 <= I3]
2.60/2.68	  f2(I5, I6) -> f3(I5 + 1, 1) [-1 <= I5 - 1]
2.60/2.68	  f1(I7, I8) -> f2(I9, I10) [0 <= I7 - 1 /\ -1 <= I9 - 1 /\ -1 <= I8 - 1]
2.60/2.68	
2.60/2.68	We use the reverse value criterion with the projection function NU:
2.60/2.68	  NU[f3#(z1,z2)] = z1 + -1 * z2
2.60/2.68	
2.60/2.68	This gives the following inequalities:
2.60/2.68	  I4 <= I3  ==>  I3 + -1 * I4 > I3 + -1 * (I4 + 1)  with  I3 + -1 * I4 >= 0
2.60/2.68	
2.60/2.68	All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed.
2.60/5.67	EOF