YES(O(1), O(n^2)) 5.88/1.95 YES(O(1), O(n^2)) 6.31/2.00 6.31/2.00 6.31/2.00
6.31/2.00 6.31/2.000 CpxTRS6.31/2.00
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))6.31/2.00
↳2 CdtProblem6.31/2.00
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))6.31/2.00
↳4 CdtProblem6.31/2.00
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))6.31/2.00
↳6 CdtProblem6.31/2.00
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))6.31/2.00
↳8 CdtProblem6.31/2.00
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))6.31/2.00
↳10 CdtProblem6.31/2.00
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))6.31/2.00
↳12 CdtProblem6.31/2.00
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))6.31/2.00
↳14 BOUNDS(O(1), O(1))6.31/2.00
le(0, y) → true 6.31/2.00
le(s(x), 0) → false 6.31/2.00
le(s(x), s(y)) → le(x, y) 6.31/2.00
pred(s(x)) → x 6.31/2.00
minus(x, 0) → x 6.31/2.00
minus(x, s(y)) → pred(minus(x, y)) 6.31/2.00
mod(0, y) → 0 6.31/2.00
mod(s(x), 0) → 0 6.31/2.00
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y)) 6.31/2.00
if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y)) 6.31/2.00
if_mod(false, s(x), s(y)) → s(x)
Tuples:
le(0, z0) → true 6.31/2.00
le(s(z0), 0) → false 6.31/2.00
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.00
pred(s(z0)) → z0 6.31/2.00
minus(z0, 0) → z0 6.31/2.00
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.00
mod(0, z0) → 0 6.31/2.00
mod(s(z0), 0) → 0 6.31/2.00
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.00
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.00
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.00
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1)) 6.31/2.00
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.00
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.00
MINUS(z0, s(z1)) → c5(PRED(minus(z0, z1)), MINUS(z0, z1)) 6.31/2.00
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.00
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
le, pred, minus, mod, if_mod
LE, MINUS, MOD, IF_MOD
c2, c5, c8, c9
Tuples:
le(0, z0) → true 6.31/2.00
le(s(z0), 0) → false 6.31/2.00
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.00
pred(s(z0)) → z0 6.31/2.00
minus(z0, 0) → z0 6.31/2.00
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.00
mod(0, z0) → 0 6.31/2.00
mod(s(z0), 0) → 0 6.31/2.00
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.00
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.00
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.00
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.01
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.01
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:none
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.01
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.01
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.01
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
le, pred, minus, mod, if_mod
LE, MOD, IF_MOD, MINUS
c2, c8, c9, c5
We considered the (Usable) Rules:
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
And the Tuples:
minus(z0, 0) → z0 6.31/2.01
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.01
pred(s(z0)) → z0 6.31/2.01
le(0, z0) → true 6.31/2.01
le(s(z0), 0) → false 6.31/2.01
le(s(z0), s(z1)) → le(z0, z1)
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.01
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.01
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.01
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
POL(0) = [1] 6.31/2.01
POL(IF_MOD(x1, x2, x3)) = [4]x2 6.31/2.01
POL(LE(x1, x2)) = 0 6.31/2.01
POL(MINUS(x1, x2)) = 0 6.31/2.01
POL(MOD(x1, x2)) = [4]x1 6.31/2.02
POL(c2(x1)) = x1 6.31/2.02
POL(c5(x1)) = x1 6.31/2.02
POL(c8(x1, x2)) = x1 + x2 6.31/2.02
POL(c9(x1, x2)) = x1 + x2 6.31/2.02
POL(false) = [3] 6.31/2.02
POL(le(x1, x2)) = 0 6.31/2.02
POL(minus(x1, x2)) = x1 6.31/2.02
POL(pred(x1)) = x1 6.31/2.02
POL(s(x1)) = [2] + x1 6.31/2.02
POL(true) = 0
Tuples:
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
mod(0, z0) → 0 6.31/2.02
mod(s(z0), 0) → 0 6.31/2.02
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.02
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.02
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
le, pred, minus, mod, if_mod
LE, MOD, IF_MOD, MINUS
c2, c8, c9, c5
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
Tuples:
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
mod(0, z0) → 0 6.31/2.02
mod(s(z0), 0) → 0 6.31/2.02
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.02
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.02
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
le, pred, minus, mod, if_mod
LE, MOD, IF_MOD, MINUS
c2, c8, c9, c5
We considered the (Usable) Rules:
LE(s(z0), s(z1)) → c2(LE(z0, z1))
And the Tuples:
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1)
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
POL(0) = 0 6.31/2.02
POL(IF_MOD(x1, x2, x3)) = [2]x22 6.31/2.02
POL(LE(x1, x2)) = x2 6.31/2.02
POL(MINUS(x1, x2)) = [3]x1 6.31/2.02
POL(MOD(x1, x2)) = [1] + x1 + [2]x12 6.31/2.02
POL(c2(x1)) = x1 6.31/2.02
POL(c5(x1)) = x1 6.31/2.02
POL(c8(x1, x2)) = x1 + x2 6.31/2.02
POL(c9(x1, x2)) = x1 + x2 6.31/2.02
POL(false) = [3] 6.31/2.02
POL(le(x1, x2)) = 0 6.31/2.02
POL(minus(x1, x2)) = x1 6.31/2.02
POL(pred(x1)) = x1 6.31/2.02
POL(s(x1)) = [1] + x1 6.31/2.02
POL(true) = 0
Tuples:
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
mod(0, z0) → 0 6.31/2.02
mod(s(z0), 0) → 0 6.31/2.02
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.02
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.02
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
K tuples:
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
LE(s(z0), s(z1)) → c2(LE(z0, z1))
le, pred, minus, mod, if_mod
LE, MOD, IF_MOD, MINUS
c2, c8, c9, c5
We considered the (Usable) Rules:
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
And the Tuples:
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1)
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
POL(0) = 0 6.31/2.02
POL(IF_MOD(x1, x2, x3)) = [2]x2·x3 6.31/2.02
POL(LE(x1, x2)) = 0 6.31/2.02
POL(MINUS(x1, x2)) = x2 6.31/2.02
POL(MOD(x1, x2)) = [2]x1·x2 6.31/2.02
POL(c2(x1)) = x1 6.31/2.02
POL(c5(x1)) = x1 6.31/2.02
POL(c8(x1, x2)) = x1 + x2 6.31/2.02
POL(c9(x1, x2)) = x1 + x2 6.31/2.02
POL(false) = [3] 6.31/2.02
POL(le(x1, x2)) = 0 6.31/2.02
POL(minus(x1, x2)) = x1 6.31/2.02
POL(pred(x1)) = x1 6.31/2.02
POL(s(x1)) = [2] + x1 6.31/2.02
POL(true) = 0
Tuples:
le(0, z0) → true 6.31/2.02
le(s(z0), 0) → false 6.31/2.02
le(s(z0), s(z1)) → le(z0, z1) 6.31/2.02
pred(s(z0)) → z0 6.31/2.02
minus(z0, 0) → z0 6.31/2.02
minus(z0, s(z1)) → pred(minus(z0, z1)) 6.31/2.02
mod(0, z0) → 0 6.31/2.02
mod(s(z0), 0) → 0 6.31/2.02
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 6.31/2.02
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 6.31/2.02
if_mod(false, s(z0), s(z1)) → s(z0)
S tuples:none
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
Defined Rule Symbols:
IF_MOD(true, s(z0), s(z1)) → c9(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 6.31/2.02
MOD(s(z0), s(z1)) → c8(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 6.31/2.02
LE(s(z0), s(z1)) → c2(LE(z0, z1)) 6.31/2.02
MINUS(z0, s(z1)) → c5(MINUS(z0, z1))
le, pred, minus, mod, if_mod
LE, MOD, IF_MOD, MINUS
c2, c8, c9, c5