YES(O(1), O(n^2)) 7.09/2.27 YES(O(1), O(n^2)) 7.51/2.33 7.51/2.33 7.51/2.33
7.51/2.33 7.51/2.330 CpxTRS7.51/2.33
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))7.51/2.33
↳2 CdtProblem7.51/2.33
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))7.51/2.33
↳4 CdtProblem7.51/2.33
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))7.51/2.33
↳6 CdtProblem7.51/2.33
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))7.51/2.33
↳8 CdtProblem7.51/2.33
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))7.51/2.33
↳10 CdtProblem7.51/2.33
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))7.51/2.33
↳12 CdtProblem7.51/2.33
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))7.51/2.33
↳14 BOUNDS(O(1), O(1))7.51/2.33
minus(X, s(Y)) → pred(minus(X, Y)) 7.51/2.33
minus(X, 0) → X 7.51/2.33
pred(s(X)) → X 7.51/2.33
le(s(X), s(Y)) → le(X, Y) 7.51/2.33
le(s(X), 0) → false 7.51/2.33
le(0, Y) → true 7.51/2.33
gcd(0, Y) → 0 7.51/2.33
gcd(s(X), 0) → s(X) 7.51/2.33
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y)) 7.51/2.33
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y)) 7.51/2.33
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.33
minus(z0, 0) → z0 7.51/2.33
pred(s(z0)) → z0 7.51/2.33
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.33
le(s(z0), 0) → false 7.51/2.33
le(0, z0) → true 7.51/2.33
gcd(0, z0) → 0 7.51/2.33
gcd(s(z0), 0) → s(z0) 7.51/2.33
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.33
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.33
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:
MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1)) 7.51/2.33
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.33
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.33
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.33
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
K tuples:none
MINUS(z0, s(z1)) → c(PRED(minus(z0, z1)), MINUS(z0, z1)) 7.51/2.33
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.33
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.33
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.33
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
minus, pred, le, gcd, if
MINUS, LE, GCD, IF
c, c3, c8, c9, c10
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.33
minus(z0, 0) → z0 7.51/2.33
pred(s(z0)) → z0 7.51/2.33
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.33
le(s(z0), 0) → false 7.51/2.33
le(0, z0) → true 7.51/2.33
gcd(0, z0) → 0 7.51/2.33
gcd(s(z0), 0) → s(z0) 7.51/2.33
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:none
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
minus, pred, le, gcd, if
LE, GCD, IF, MINUS
c3, c8, c9, c10, c
We considered the (Usable) Rules:
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
And the Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
POL(0) = [3] 7.51/2.34
POL(GCD(x1, x2)) = [4]x1 + [4]x2 7.51/2.34
POL(IF(x1, x2, x3)) = [4]x2 + [4]x3 7.51/2.34
POL(LE(x1, x2)) = 0 7.51/2.34
POL(MINUS(x1, x2)) = 0 7.51/2.34
POL(c(x1)) = x1 7.51/2.34
POL(c10(x1, x2)) = x1 + x2 7.51/2.34
POL(c3(x1)) = x1 7.51/2.34
POL(c8(x1, x2)) = x1 + x2 7.51/2.34
POL(c9(x1, x2)) = x1 + x2 7.51/2.34
POL(false) = 0 7.51/2.34
POL(le(x1, x2)) = 0 7.51/2.34
POL(minus(x1, x2)) = x1 7.51/2.34
POL(pred(x1)) = x1 7.51/2.34
POL(s(x1)) = [4] + x1 7.51/2.34
POL(true) = 0
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
Defined Rule Symbols:
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
minus, pred, le, gcd, if
LE, GCD, IF, MINUS
c3, c8, c9, c10, c
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0))
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
Defined Rule Symbols:
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
minus, pred, le, gcd, if
LE, GCD, IF, MINUS
c3, c8, c9, c10, c
We considered the (Usable) Rules:
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
And the Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
POL(0) = 0 7.51/2.34
POL(GCD(x1, x2)) = [2] + x1·x2 7.51/2.34
POL(IF(x1, x2, x3)) = x2·x3 7.51/2.34
POL(LE(x1, x2)) = 0 7.51/2.34
POL(MINUS(x1, x2)) = [2]x2 7.51/2.34
POL(c(x1)) = x1 7.51/2.34
POL(c10(x1, x2)) = x1 + x2 7.51/2.34
POL(c3(x1)) = x1 7.51/2.34
POL(c8(x1, x2)) = x1 + x2 7.51/2.34
POL(c9(x1, x2)) = x1 + x2 7.51/2.34
POL(false) = 0 7.51/2.34
POL(le(x1, x2)) = 0 7.51/2.34
POL(minus(x1, x2)) = x1 7.51/2.34
POL(pred(x1)) = x1 7.51/2.34
POL(s(x1)) = [2] + x1 7.51/2.34
POL(true) = 0
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
K tuples:
LE(s(z0), s(z1)) → c3(LE(z0, z1))
Defined Rule Symbols:
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
minus, pred, le, gcd, if
LE, GCD, IF, MINUS
c3, c8, c9, c10, c
We considered the (Usable) Rules:
LE(s(z0), s(z1)) → c3(LE(z0, z1))
And the Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true
The order we found is given by the following interpretation:
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
POL(0) = 0 7.51/2.34
POL(GCD(x1, x2)) = [2]x1 + x22 + x12 7.51/2.34
POL(IF(x1, x2, x3)) = [1] + x32 + x22 7.51/2.34
POL(LE(x1, x2)) = [3] + x2 7.51/2.34
POL(MINUS(x1, x2)) = 0 7.51/2.34
POL(c(x1)) = x1 7.51/2.34
POL(c10(x1, x2)) = x1 + x2 7.51/2.34
POL(c3(x1)) = x1 7.51/2.34
POL(c8(x1, x2)) = x1 + x2 7.51/2.34
POL(c9(x1, x2)) = x1 + x2 7.51/2.34
POL(false) = 0 7.51/2.34
POL(le(x1, x2)) = 0 7.51/2.34
POL(minus(x1, x2)) = x1 7.51/2.34
POL(pred(x1)) = x1 7.51/2.34
POL(s(x1)) = [2] + x1 7.51/2.34
POL(true) = 0
Tuples:
minus(z0, s(z1)) → pred(minus(z0, z1)) 7.51/2.34
minus(z0, 0) → z0 7.51/2.34
pred(s(z0)) → z0 7.51/2.34
le(s(z0), s(z1)) → le(z0, z1) 7.51/2.34
le(s(z0), 0) → false 7.51/2.34
le(0, z0) → true 7.51/2.34
gcd(0, z0) → 0 7.51/2.34
gcd(s(z0), 0) → s(z0) 7.51/2.34
gcd(s(z0), s(z1)) → if(le(z1, z0), s(z0), s(z1)) 7.51/2.34
if(true, s(z0), s(z1)) → gcd(minus(z0, z1), s(z1)) 7.51/2.34
if(false, s(z0), s(z1)) → gcd(minus(z1, z0), s(z0))
S tuples:none
LE(s(z0), s(z1)) → c3(LE(z0, z1)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1))
Defined Rule Symbols:
IF(true, s(z0), s(z1)) → c9(GCD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 7.51/2.34
IF(false, s(z0), s(z1)) → c10(GCD(minus(z1, z0), s(z0)), MINUS(z1, z0)) 7.51/2.34
GCD(s(z0), s(z1)) → c8(IF(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 7.51/2.34
MINUS(z0, s(z1)) → c(MINUS(z0, z1)) 7.51/2.34
LE(s(z0), s(z1)) → c3(LE(z0, z1))
minus, pred, le, gcd, if
LE, GCD, IF, MINUS
c3, c8, c9, c10, c