Termination of the given ITRSProblem could successfully be proven:
↳ ITRS
↳ ITRStoIDPProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
Cond_gcd1(TRUE, x, y) → gcd(-@z(y, x), x)
Cond_gcd(TRUE, x, y) → gcd(-@z(x, y), y)
gcd(0@z, y) → y
gcd(x, 0@z) → x
gcd(x, y) → Cond_gcd(&&(>=@z(x, y), >@z(y, 0@z)), x, y)
gcd(x, y) → Cond_gcd1(&&(>@z(y, x), >@z(x, 0@z)), x, y)
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
Added dependency pairs
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
I DP problem:
The following domains are used:
z
The ITRS R consists of the following rules:
Cond_gcd1(TRUE, x, y) → gcd(-@z(y, x), x)
Cond_gcd(TRUE, x, y) → gcd(-@z(x, y), y)
gcd(0@z, y) → y
gcd(x, 0@z) → x
gcd(x, y) → Cond_gcd(&&(>=@z(x, y), >@z(y, 0@z)), x, y)
gcd(x, y) → Cond_gcd1(&&(>@z(y, x), >@z(x, 0@z)), x, y)
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(1): GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(3): COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3])
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
(1) -> (3), if ((x[1] →* x[3])∧(y[1] →* y[3])∧(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)) →* TRUE))
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(2) -> (1), if ((x[2] →* y[1])∧(-@z(y[2], x[2]) →* x[1]))
(3) -> (0), if ((y[3] →* y[0])∧(-@z(x[3], y[3]) →* x[0]))
(3) -> (1), if ((y[3] →* y[1])∧(-@z(x[3], y[3]) →* x[1]))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(1): GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(3): COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3])
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
(1) -> (3), if ((x[1] →* x[3])∧(y[1] →* y[3])∧(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)) →* TRUE))
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(2) -> (1), if ((x[2] →* y[1])∧(-@z(y[2], x[2]) →* x[1]))
(3) -> (0), if ((y[3] →* y[0])∧(-@z(x[3], y[3]) →* x[0]))
(3) -> (1), if ((y[3] →* y[1])∧(-@z(x[3], y[3]) →* x[1]))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair GCD(x, y) → COND_GCD1(&&(>@z(y, x), >@z(x, 0@z)), x, y) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(1) (GCD(x[0], y[0])≥NonInfC∧GCD(x[0], y[0])≥COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair GCD(x, y) → COND_GCD(&&(>=@z(x, y), >@z(y, 0@z)), x, y) the following chains were created:
- We consider the chain GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1]) which results in the following constraint:
(6) (GCD(x[1], y[1])≥NonInfC∧GCD(x[1], y[1])≥COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])∧(UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) (0 ≥ 0∧(UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥)∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) (0 = 0∧(UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
For Pair COND_GCD1(TRUE, x, y) → GCD(-@z(y, x), x) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]), COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]), GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(11) (x[0]=x[2]∧y[0]=y[2]∧&&(>@z(y[0], x[0]), >@z(x[0], 0@z))=TRUE∧x[2]=y[0]1∧-@z(y[2], x[2])=x[0]1 ⇒ COND_GCD1(TRUE, x[2], y[2])≥NonInfC∧COND_GCD1(TRUE, x[2], y[2])≥GCD(-@z(y[2], x[2]), x[2])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12) (>@z(y[0], x[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_GCD1(TRUE, x[0], y[0])≥NonInfC∧COND_GCD1(TRUE, x[0], y[0])≥GCD(-@z(y[0], x[0]), x[0])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧0 ≥ 0∧x[0] ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧0 ≥ 0∧x[0] ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16) (y[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17) (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + x[0] ≥ 0∧0 ≥ 0)
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]), COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]), GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1]) which results in the following constraint:
(18) (x[0]=x[2]∧y[0]=y[2]∧&&(>@z(y[0], x[0]), >@z(x[0], 0@z))=TRUE∧x[2]=y[1]∧-@z(y[2], x[2])=x[1] ⇒ COND_GCD1(TRUE, x[2], y[2])≥NonInfC∧COND_GCD1(TRUE, x[2], y[2])≥GCD(-@z(y[2], x[2]), x[2])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (18) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(19) (>@z(y[0], x[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_GCD1(TRUE, x[0], y[0])≥NonInfC∧COND_GCD1(TRUE, x[0], y[0])≥GCD(-@z(y[0], x[0]), x[0])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧0 ≥ 0∧x[0] ≥ 0)
We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(21) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧0 ≥ 0∧x[0] ≥ 0)
We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(22) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(23) (y[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧x[0] ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24) (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + x[0] ≥ 0∧0 ≥ 0)
For Pair COND_GCD(TRUE, x, y) → GCD(-@z(x, y), y) the following chains were created:
- We consider the chain GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1]), COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3]), GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1]) which results in the following constraint:
(25) (y[1]=y[3]∧&&(>=@z(x[1], y[1]), >@z(y[1], 0@z))=TRUE∧-@z(x[3], y[3])=x[1]1∧x[1]=x[3]∧y[3]=y[1]1 ⇒ COND_GCD(TRUE, x[3], y[3])≥NonInfC∧COND_GCD(TRUE, x[3], y[3])≥GCD(-@z(x[3], y[3]), y[3])∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26) (>=@z(x[1], y[1])=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_GCD(TRUE, x[1], y[1])≥NonInfC∧COND_GCD(TRUE, x[1], y[1])≥GCD(-@z(x[1], y[1]), y[1])∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) (x[1] + (-1)y[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-2 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) (x[1] + (-1)y[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-2 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) (-1 + y[1] ≥ 0∧x[1] + (-1)y[1] ≥ 0 ⇒ -2 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (-1 + y[1] ≥ 0∧x[1] ≥ 0 ⇒ -2 + (-1)Bound + (2)y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31) (y[1] ≥ 0∧x[1] ≥ 0 ⇒ (-1)Bound + (2)y[1] + x[1] ≥ 0∧y[1] ≥ 0∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
- We consider the chain GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1]), COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3]), GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(32) (y[1]=y[3]∧y[3]=y[0]∧-@z(x[3], y[3])=x[0]∧&&(>=@z(x[1], y[1]), >@z(y[1], 0@z))=TRUE∧x[1]=x[3] ⇒ COND_GCD(TRUE, x[3], y[3])≥NonInfC∧COND_GCD(TRUE, x[3], y[3])≥GCD(-@z(x[3], y[3]), y[3])∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>=@z(x[1], y[1])=TRUE∧>@z(y[1], 0@z)=TRUE ⇒ COND_GCD(TRUE, x[1], y[1])≥NonInfC∧COND_GCD(TRUE, x[1], y[1])≥GCD(-@z(x[1], y[1]), y[1])∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (x[1] + (-1)y[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-2 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (x[1] + (-1)y[1] ≥ 0∧-1 + y[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-2 + (-1)Bound + y[1] + x[1] ≥ 0∧-1 + y[1] ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + y[1] ≥ 0∧x[1] + (-1)y[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-1 + y[1] ≥ 0∧-2 + (-1)Bound + y[1] + x[1] ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (-1 + y[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧-1 + y[1] ≥ 0∧-2 + (-1)Bound + (2)y[1] + x[1] ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧y[1] ≥ 0∧(-1)Bound + (2)y[1] + x[1] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- GCD(x, y) → COND_GCD1(&&(>@z(y, x), >@z(x, 0@z)), x, y)
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- GCD(x, y) → COND_GCD(&&(>=@z(x, y), >@z(y, 0@z)), x, y)
- (0 = 0∧(UIncreasing(COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
- COND_GCD1(TRUE, x, y) → GCD(-@z(y, x), x)
- (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + x[0] ≥ 0∧0 ≥ 0)
- (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + x[0] ≥ 0∧0 ≥ 0)
- COND_GCD(TRUE, x, y) → GCD(-@z(x, y), y)
- (y[1] ≥ 0∧x[1] ≥ 0 ⇒ (-1)Bound + (2)y[1] + x[1] ≥ 0∧y[1] ≥ 0∧(UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥))
- (y[1] ≥ 0∧x[1] ≥ 0 ⇒ (UIncreasing(GCD(-@z(x[3], y[3]), y[3])), ≥)∧y[1] ≥ 0∧(-1)Bound + (2)y[1] + x[1] ≥ 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(COND_GCD1(x1, x2, x3)) = -1 + x3 + x2
POL(>=@z(x1, x2)) = -1
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_GCD(x1, x2, x3)) = -1 + x3 + x2 + x1
POL(GCD(x1, x2)) = -1 + x2 + x1
POL(FALSE) = -1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])
The following pairs are in Pbound:
COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3])
The following pairs are in P≥:
GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3])
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 → FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 → TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 → FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(1): GCD(x[1], y[1]) → COND_GCD(&&(>=@z(x[1], y[1]), >@z(y[1], 0@z)), x[1], y[1])
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(2) -> (1), if ((x[2] →* y[1])∧(-@z(y[2], x[2]) →* x[1]))
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]), COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]), GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(1) (x[0]=x[2]∧y[0]=y[2]∧&&(>@z(y[0], x[0]), >@z(x[0], 0@z))=TRUE∧x[2]=y[0]1∧-@z(y[2], x[2])=x[0]1 ⇒ COND_GCD1(TRUE, x[2], y[2])≥NonInfC∧COND_GCD1(TRUE, x[2], y[2])≥GCD(-@z(y[2], x[2]), x[2])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(y[0], x[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_GCD1(TRUE, x[0], y[0])≥NonInfC∧COND_GCD1(TRUE, x[0], y[0])≥GCD(-@z(y[0], x[0]), x[0])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x[0] + -1 ≥ 0∧-1 + y[0] + (-1)x[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x[0] + -1 ≥ 0∧y[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[0] ≥ 0∧y[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧5 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0∧(2)x[0] ≥ 0)
For Pair GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(8) (GCD(x[0], y[0])≥NonInfC∧GCD(x[0], y[0])≥COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (0 ≥ 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 ≥ 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
- (x[0] ≥ 0∧y[0] ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧5 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0∧(2)x[0] ≥ 0)
- GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
- (0 ≥ 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(COND_GCD1(x1, x2, x3)) = -1 + (2)x3 + (2)x2
POL(0@z) = 0
POL(TRUE) = 2
POL(&&(x1, x2)) = 0
POL(GCD(x1, x2)) = (2)x2 + (2)x1
POL(FALSE) = 0
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
The following pairs are in Pbound:
COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
The following pairs are in P≥:
none
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
TRUE1 → &&(TRUE, TRUE)1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(TRUE, FALSE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:none
R is empty.
The integer pair graph is empty.
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(3): COND_GCD(TRUE, x[3], y[3]) → GCD(-@z(x[3], y[3]), y[3])
(3) -> (0), if ((y[3] →* y[0])∧(-@z(x[3], y[3]) →* x[0]))
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(2): COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
(2) -> (0), if ((x[2] →* y[0])∧(-@z(y[2], x[2]) →* x[0]))
(0) -> (2), if ((x[0] →* x[2])∧(y[0] →* y[2])∧(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.
For Pair COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]), COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2]), GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(1) (x[0]=x[2]∧y[0]=y[2]∧&&(>@z(y[0], x[0]), >@z(x[0], 0@z))=TRUE∧x[2]=y[0]1∧-@z(y[2], x[2])=x[0]1 ⇒ COND_GCD1(TRUE, x[2], y[2])≥NonInfC∧COND_GCD1(TRUE, x[2], y[2])≥GCD(-@z(y[2], x[2]), x[2])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(y[0], x[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_GCD1(TRUE, x[0], y[0])≥NonInfC∧COND_GCD1(TRUE, x[0], y[0])≥GCD(-@z(y[0], x[0]), x[0])∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ (UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0∧-2 + (2)x[0] ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ -2 + (2)x[0] ≥ 0∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧-1 + (-1)Bound + (2)y[0] + (2)x[0] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (y[0] ≥ 0∧x[0] + -1 ≥ 0 ⇒ -2 + (2)x[0] ≥ 0∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧1 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (2)x[0] ≥ 0∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧5 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0)
For Pair GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) the following chains were created:
- We consider the chain GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0]) which results in the following constraint:
(8) (GCD(x[0], y[0])≥NonInfC∧GCD(x[0], y[0])≥COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (0 ≥ 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 = 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
- (y[0] ≥ 0∧x[0] ≥ 0 ⇒ (2)x[0] ≥ 0∧(UIncreasing(GCD(-@z(y[2], x[2]), x[2])), ≥)∧5 + (-1)Bound + (4)x[0] + (2)y[0] ≥ 0)
- GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
- (0 = 0∧(UIncreasing(COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(COND_GCD1(x1, x2, x3)) = -1 + (2)x3 + (2)x2
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = 2
POL(GCD(x1, x2)) = (2)x2 + (2)x1
POL(FALSE) = -1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
The following pairs are in Pbound:
COND_GCD1(TRUE, x[2], y[2]) → GCD(-@z(y[2], x[2]), x[2])
The following pairs are in P≥:
none
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 → FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 → TRUE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:none
R is empty.
The integer pair graph is empty.
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): GCD(x[0], y[0]) → COND_GCD1(&&(>@z(y[0], x[0]), >@z(x[0], 0@z)), x[0], y[0])
The set Q consists of the following terms:
Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.