1.09/1.55 YES 1.09/1.55 1.09/1.55 DP problem for innermost termination. 1.09/1.55 P = 1.09/1.55 f4#(x1, x2) -> f3#(x1, x2) 1.09/1.55 f3#(I0, I1) -> f1#(I0, I1) 1.09/1.55 f2#(I2, I3) -> f1#(I2, I3) 1.09/1.55 f1#(I4, I5) -> f2#(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 R = 1.09/1.55 f4(x1, x2) -> f3(x1, x2) 1.09/1.55 f3(I0, I1) -> f1(I0, I1) 1.09/1.55 f2(I2, I3) -> f1(I2, I3) 1.09/1.55 f1(I4, I5) -> f2(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 1.09/1.55 The dependency graph for this problem is: 1.09/1.55 0 -> 1 1.09/1.55 1 -> 3 1.09/1.55 2 -> 3 1.09/1.55 3 -> 2 1.09/1.55 Where: 1.09/1.55 0) f4#(x1, x2) -> f3#(x1, x2) 1.09/1.55 1) f3#(I0, I1) -> f1#(I0, I1) 1.09/1.55 2) f2#(I2, I3) -> f1#(I2, I3) 1.09/1.55 3) f1#(I4, I5) -> f2#(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 1.09/1.55 We have the following SCCs. 1.09/1.55 { 2, 3 } 1.09/1.55 1.09/1.55 DP problem for innermost termination. 1.09/1.55 P = 1.09/1.55 f2#(I2, I3) -> f1#(I2, I3) 1.09/1.55 f1#(I4, I5) -> f2#(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 R = 1.09/1.55 f4(x1, x2) -> f3(x1, x2) 1.09/1.55 f3(I0, I1) -> f1(I0, I1) 1.09/1.55 f2(I2, I3) -> f1(I2, I3) 1.09/1.55 f1(I4, I5) -> f2(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 1.09/1.55 We use the reverse value criterion with the projection function NU: 1.09/1.55 NU[f1#(z1,z2)] = 0 + -1 * z2 1.09/1.55 NU[f2#(z1,z2)] = 0 + -1 * z2 1.09/1.55 1.09/1.55 This gives the following inequalities: 1.09/1.55 ==> 0 + -1 * I3 >= 0 + -1 * I3 1.09/1.55 I5 <= 0 /\ 1 <= I4 ==> 0 + -1 * I5 > 0 + -1 * (1 + I5) with 0 + -1 * I5 >= 0 1.09/1.55 1.09/1.55 We remove all the strictly oriented dependency pairs. 1.09/1.55 1.09/1.55 DP problem for innermost termination. 1.09/1.55 P = 1.09/1.55 f2#(I2, I3) -> f1#(I2, I3) 1.09/1.55 R = 1.09/1.55 f4(x1, x2) -> f3(x1, x2) 1.09/1.55 f3(I0, I1) -> f1(I0, I1) 1.09/1.55 f2(I2, I3) -> f1(I2, I3) 1.09/1.55 f1(I4, I5) -> f2(I4, 1 + I5) [I5 <= 0 /\ 1 <= I4] 1.09/1.55 1.09/1.55 The dependency graph for this problem is: 1.09/1.55 2 -> 1.09/1.55 Where: 1.09/1.55 2) f2#(I2, I3) -> f1#(I2, I3) 1.09/1.55 1.09/1.55 We have the following SCCs. 1.09/1.55 1.09/4.52 EOF