YES(O(1), O(n^1)) 4.69/1.67 YES(O(1), O(n^1)) 5.09/1.73 5.09/1.73 5.09/1.73
5.09/1.73 5.09/1.730 CpxTRS5.09/1.73
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))5.09/1.73
↳2 CdtProblem5.09/1.73
↳3 CdtUnreachableProof (⇔)5.09/1.73
↳4 CdtProblem5.09/1.73
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))5.09/1.73
↳6 CdtProblem5.09/1.73
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)5.09/1.73
↳8 CdtProblem5.09/1.73
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))5.09/1.73
↳10 CdtProblem5.09/1.73
↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))5.09/1.73
↳12 CdtProblem5.09/1.73
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))5.09/1.73
↳14 CdtProblem5.09/1.73
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))5.09/1.73
↳16 CdtProblem5.09/1.73
↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID))5.09/1.73
↳18 BOUNDS(O(1), O(1))5.09/1.73
p(0) → 0 5.09/1.73
p(s(X)) → X 5.09/1.73
leq(0, Y) → true 5.09/1.73
leq(s(X), 0) → false 5.09/1.73
leq(s(X), s(Y)) → leq(X, Y) 5.09/1.73
if(true, X, Y) → activate(X) 5.09/1.73
if(false, X, Y) → activate(Y) 5.09/1.73
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 5.09/1.73
0 → n__0 5.09/1.73
s(X) → n__s(X) 5.09/1.73
diff(X1, X2) → n__diff(X1, X2) 5.09/1.73
p(X) → n__p(X) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(X)) → s(activate(X)) 5.09/1.73
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2)) 5.09/1.73
activate(n__p(X)) → p(activate(X)) 5.09/1.73
activate(X) → X
Tuples:
p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0 → n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
S tuples:
P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1)) 5.09/1.73
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
K tuples:none
P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1)) 5.09/1.73
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
P, LEQ, IF, DIFF, ACTIVATE
c, c5, c6, c7, c8, c12, c13, c14, c15
P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1))
Tuples:
p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0 → n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
S tuples:
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
K tuples:none
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
IF, DIFF, ACTIVATE
c6, c7, c8, c12, c13, c14, c15
Tuples:
p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0 → n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
S tuples:
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:none
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
IF, ACTIVATE, DIFF
c6, c7, c14, c8, c12, c13, c15
Removed 2 trailing nodes:
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1))
ACTIVATE(n__0) → c12 5.09/1.73
DIFF(z0, z1) → c8
Tuples:
p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0 → n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
S tuples:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:none
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
ACTIVATE, DIFF
c14, c8, c12, c13, c15
We considered the (Usable) Rules:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__0) → c12
And the Tuples:
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
0 → n__0
The order we found is given by the following interpretation:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
POL(0) = [3] 5.09/1.73
POL(ACTIVATE(x1)) = [4]x1 5.09/1.73
POL(DIFF(x1, x2)) = 0 5.09/1.73
POL(activate(x1)) = 0 5.09/1.73
POL(c12) = 0 5.09/1.73
POL(c13(x1)) = x1 5.09/1.73
POL(c14(x1, x2, x3)) = x1 + x2 + x3 5.09/1.73
POL(c15(x1)) = x1 5.09/1.73
POL(c8) = 0 5.09/1.73
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.73
POL(if(x1, x2, x3)) = [3] + x2 5.09/1.73
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.73
POL(n__0) = [1] 5.09/1.73
POL(n__diff(x1, x2)) = [3] + x1 + x2 5.09/1.73
POL(n__p(x1)) = x1 5.09/1.73
POL(n__s(x1)) = x1 5.09/1.73
POL(p(x1)) = [3] 5.09/1.73
POL(s(x1)) = [3] + [3]x1
Tuples:
p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0 → n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
S tuples:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12
p, leq, if, diff, 0, s, activate
ACTIVATE, DIFF
c14, c8, c12, c13, c15
DIFF(z0, z1) → c8
Tuples:
p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0 → n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
S tuples:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8
p, leq, if, diff, 0, s, activate
ACTIVATE, DIFF
c14, c8, c12, c13, c15
We considered the (Usable) Rules:
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
And the Tuples:
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
0 → n__0
The order we found is given by the following interpretation:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
POL(0) = [3] 5.09/1.74
POL(ACTIVATE(x1)) = [2]x1 5.09/1.74
POL(DIFF(x1, x2)) = 0 5.09/1.74
POL(activate(x1)) = 0 5.09/1.74
POL(c12) = 0 5.09/1.74
POL(c13(x1)) = x1 5.09/1.74
POL(c14(x1, x2, x3)) = x1 + x2 + x3 5.09/1.74
POL(c15(x1)) = x1 5.09/1.74
POL(c8) = 0 5.09/1.74
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.74
POL(if(x1, x2, x3)) = [3] + x2 5.09/1.74
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.74
POL(n__0) = 0 5.09/1.74
POL(n__diff(x1, x2)) = x1 + x2 5.09/1.74
POL(n__p(x1)) = [1] + x1 5.09/1.74
POL(n__s(x1)) = x1 5.09/1.74
POL(p(x1)) = [3] 5.09/1.74
POL(s(x1)) = [3] + [3]x1
Tuples:
p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0 → n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
S tuples:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
ACTIVATE, DIFF
c14, c8, c12, c13, c15
We considered the (Usable) Rules:
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
And the Tuples:
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
0 → n__0
The order we found is given by the following interpretation:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
POL(0) = [3] 5.09/1.74
POL(ACTIVATE(x1)) = [1] + [4]x1 5.09/1.74
POL(DIFF(x1, x2)) = [5] 5.09/1.74
POL(activate(x1)) = 0 5.09/1.74
POL(c12) = 0 5.09/1.74
POL(c13(x1)) = x1 5.09/1.74
POL(c14(x1, x2, x3)) = x1 + x2 + x3 5.09/1.74
POL(c15(x1)) = x1 5.09/1.74
POL(c8) = 0 5.09/1.74
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.74
POL(if(x1, x2, x3)) = [3] + x2 5.09/1.74
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2 5.09/1.74
POL(n__0) = 0 5.09/1.74
POL(n__diff(x1, x2)) = [4] + x1 + x2 5.09/1.74
POL(n__p(x1)) = x1 5.09/1.74
POL(n__s(x1)) = [1] + x1 5.09/1.74
POL(p(x1)) = [3] 5.09/1.74
POL(s(x1)) = [3] + [3]x1
Tuples:
p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0 → n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
S tuples:none
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
p, leq, if, diff, 0, s, activate
ACTIVATE, DIFF
c14, c8, c12, c13, c15