Termination proof of /tmp/tmpYBUPPG/csharp1.itrs

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:

b14(sv14_14, sv23_37, sv24_38) → Cond_b14(&&(>=@z(sv23_37, sv14_14), &&(<@z(1@z, sv14_14), &&(TRUE, &&(<@z(0@z, sv24_38), <@z(1@z, sv23_37))))), sv14_14, sv23_37, sv24_38)
Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, -@z(sv23_37, sv14_14), +@z(sv24_38, 1@z))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)

The set Q consists of the following terms:

b14(x0, x1, x2)
Cond_b14(TRUE, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

b14(sv14_14, sv23_37, sv24_38) → Cond_b14(&&(>=@z(sv23_37, sv14_14), &&(<@z(1@z, sv14_14), &&(TRUE, &&(<@z(0@z, sv24_38), <@z(1@z, sv23_37))))), sv14_14, sv23_37, sv24_38)
Cond_b14(TRUE, sv14_14, sv23_37, sv24_38) → b15(sv14_14, sv23_37, sv24_38)
b15(sv14_14, sv23_37, sv24_38) → b10(sv14_14, -@z(sv23_37, sv14_14), +@z(sv24_38, 1@z))
b10(sv14_14, sv23_37, sv24_38) → b14(sv14_14, sv23_37, sv24_38)

The integer pair graph contains the following rules and edges:

(0): B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0])
(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1])
(2): B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])
(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))

(0) -> (2), if ((sv23_37[0] →^* sv23_37[2])∧(sv24_38[0] →^* sv24_38[2])∧(sv14_14[0] →^* sv14_14[2]))

(1) -> (3), if ((sv23_37[1] →^* sv23_37[3])∧(sv24_38[1] →^* sv24_38[3])∧(sv14_14[1] →^* sv14_14[3]))

(2) -> (1), if ((sv24_38[2] →^* sv24_38[1])∧(sv14_14[2] →^* sv14_14[1])∧(sv23_37[2] →^* sv23_37[1])∧(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))) →^* TRUE))

(3) -> (0), if ((-@z(sv23_37[3], sv14_14[3]) →^* sv23_37[0])∧(+@z(sv24_38[3], 1@z) →^* sv24_38[0])∧(sv14_14[3] →^* sv14_14[0]))

The set Q consists of the following terms:

b14(x0, x1, x2)
Cond_b14(TRUE, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)

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): B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0])
(1): COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1])
(2): B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])
(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))

(0) -> (2), if ((sv23_37[0] →^* sv23_37[2])∧(sv24_38[0] →^* sv24_38[2])∧(sv14_14[0] →^* sv14_14[2]))

(1) -> (3), if ((sv23_37[1] →^* sv23_37[3])∧(sv24_38[1] →^* sv24_38[3])∧(sv14_14[1] →^* sv14_14[3]))

(3) -> (0), if ((-@z(sv23_37[3], sv14_14[3]) →^* sv23_37[0])∧(+@z(sv24_38[3], 1@z) →^* sv24_38[0])∧(sv14_14[3] →^* sv14_14[0]))

The set Q consists of the following terms:

b14(x0, x1, x2)
Cond_b14(TRUE, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)

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 B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38) the following chains were created:

We consider the chain B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z)), B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0]), B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2]) which results in the following constraint:
(1)    (-@z(sv23_37[3], sv14_14[3])=sv23_37[0]∧+@z(sv24_38[3], 1@z)=sv24_38[0]∧sv24_38[0]=sv24_38[2]∧sv14_14[0]=sv14_14[2]∧sv14_14[3]=sv14_14[0]∧sv23_37[0]=sv23_37[2] ⇒ B10(sv14_14[0], sv23_37[0], sv24_38[0])≥NonInfC∧B10(sv14_14[0], sv23_37[0], sv24_38[0])≥B14(sv14_14[0], sv23_37[0], sv24_38[0])∧(U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))≥NonInfC∧B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))≥B14(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))∧(U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    ((U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥))

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥)∧0 = 0)

For Pair COND_B14(TRUE, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38) the following chains were created:

We consider the chain B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2]), COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z)) which results in the following constraint:
(7)    (sv14_14[1]=sv14_14[3]∧sv23_37[1]=sv23_37[3]∧sv24_38[1]=sv24_38[3]∧sv23_37[2]=sv23_37[1]∧sv14_14[2]=sv14_14[1]∧sv24_38[2]=sv24_38[1]∧&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2])))))=TRUE ⇒ COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])≥NonInfC∧COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1])≥B15(sv14_14[1], sv23_37[1], sv24_38[1])∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥))

We simplified constraint (7) using rules (III), (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
(8)    (>=@z(sv23_37[2], sv14_14[2])=TRUE∧<@z(1@z, sv14_14[2])=TRUE∧<@z(0@z, sv24_38[2])=TRUE∧<@z(1@z, sv23_37[2])=TRUE ⇒ COND_B14(TRUE, sv14_14[2], sv23_37[2], sv24_38[2])≥NonInfC∧COND_B14(TRUE, sv14_14[2], sv23_37[2], sv24_38[2])≥B15(sv14_14[2], sv23_37[2], sv24_38[2])∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (sv23_37[2] + (-1)sv14_14[2] ≥ 0∧sv14_14[2] + -2 ≥ 0∧sv24_38[2] + -1 ≥ 0∧-2 + sv23_37[2] ≥ 0 ⇒ (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧-1 + (-1)Bound + sv23_37[2] ≥ 0∧-2 + sv14_14[2] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (sv23_37[2] + (-1)sv14_14[2] ≥ 0∧sv14_14[2] + -2 ≥ 0∧sv24_38[2] + -1 ≥ 0∧-2 + sv23_37[2] ≥ 0 ⇒ (U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧-1 + (-1)Bound + sv23_37[2] ≥ 0∧-2 + sv14_14[2] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (sv23_37[2] + (-1)sv14_14[2] ≥ 0∧sv24_38[2] + -1 ≥ 0∧sv14_14[2] + -2 ≥ 0∧-2 + sv23_37[2] ≥ 0 ⇒ -2 + sv14_14[2] ≥ 0∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧-1 + (-1)Bound + sv23_37[2] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (sv23_37[2] ≥ 0∧sv24_38[2] + -1 ≥ 0∧sv14_14[2] + -2 ≥ 0∧-2 + sv14_14[2] + sv23_37[2] ≥ 0 ⇒ -2 + sv14_14[2] ≥ 0∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧-1 + (-1)Bound + sv14_14[2] + sv23_37[2] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (sv23_37[2] ≥ 0∧sv24_38[2] + -1 ≥ 0∧sv14_14[2] ≥ 0∧sv14_14[2] + sv23_37[2] ≥ 0 ⇒ sv14_14[2] ≥ 0∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧1 + (-1)Bound + sv14_14[2] + sv23_37[2] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (sv23_37[2] ≥ 0∧sv24_38[2] ≥ 0∧sv14_14[2] ≥ 0∧sv14_14[2] + sv23_37[2] ≥ 0 ⇒ sv14_14[2] ≥ 0∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧1 + (-1)Bound + sv14_14[2] + sv23_37[2] ≥ 0)

For Pair B14(sv14_14, sv23_37, sv24_38) → COND_B14(&&(>=@z(sv23_37, sv14_14), &&(<@z(1@z, sv14_14), &&(TRUE, &&(<@z(0@z, sv24_38), <@z(1@z, sv23_37))))), sv14_14, sv23_37, sv24_38) the following chains were created:

We consider the chain B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2]) which results in the following constraint:
(15)    (B14(sv14_14[2], sv23_37[2], sv24_38[2])≥NonInfC∧B14(sv14_14[2], sv23_37[2], sv24_38[2])≥COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])∧(U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    ((U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    ((U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥))

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥)∧0 = 0∧0 = 0)

For Pair B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, -@z(sv23_37, sv14_14), +@z(sv24_38, 1@z)) the following chains were created:

We consider the chain COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1]), B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z)), B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0]) which results in the following constraint:
(20)    (sv14_14[1]=sv14_14[3]∧sv23_37[1]=sv23_37[3]∧-@z(sv23_37[3], sv14_14[3])=sv23_37[0]∧sv24_38[1]=sv24_38[3]∧+@z(sv24_38[3], 1@z)=sv24_38[0]∧sv14_14[3]=sv14_14[0] ⇒ B15(sv14_14[3], sv23_37[3], sv24_38[3])≥NonInfC∧B15(sv14_14[3], sv23_37[3], sv24_38[3])≥B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))∧(U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥))

We simplified constraint (20) using rules (III), (IV) which results in the following new constraint:
(21)    (B15(sv14_14[1], sv23_37[1], sv24_38[1])≥NonInfC∧B15(sv14_14[1], sv23_37[1], sv24_38[1])≥B10(sv14_14[1], -@z(sv23_37[1], sv14_14[1]), +@z(sv24_38[1], 1@z))∧(U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    (0 = 0∧(U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)

To summarize, we get the following constraints P_≥ for the following pairs.

B10(sv14_14, sv23_37, sv24_38) → B14(sv14_14, sv23_37, sv24_38)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(B14(sv14_14[0], sv23_37[0], sv24_38[0])), ≥)∧0 = 0)
COND_B14(TRUE, sv14_14, sv23_37, sv24_38) → B15(sv14_14, sv23_37, sv24_38)
- (sv23_37[2] ≥ 0∧sv24_38[2] ≥ 0∧sv14_14[2] ≥ 0∧sv14_14[2] + sv23_37[2] ≥ 0 ⇒ sv14_14[2] ≥ 0∧(U^Increasing(B15(sv14_14[1], sv23_37[1], sv24_38[1])), ≥)∧1 + (-1)Bound + sv14_14[2] + sv23_37[2] ≥ 0)
B14(sv14_14, sv23_37, sv24_38) → COND_B14(&&(>=@z(sv23_37, sv14_14), &&(<@z(1@z, sv14_14), &&(TRUE, &&(<@z(0@z, sv24_38), <@z(1@z, sv23_37))))), sv14_14, sv23_37, sv24_38)
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])), ≥)∧0 = 0∧0 = 0)
B15(sv14_14, sv23_37, sv24_38) → B10(sv14_14, -@z(sv23_37, sv14_14), +@z(sv24_38, 1@z))
- (0 = 0∧(U^Increasing(B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)

The constraints for P_> respective P_bound 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 P_bound.
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(B15(x₁, x₂, x₃)) = x₂ + (-1)x₁
POL(0@z) = 0
POL(TRUE) = 1
POL(&&(x₁, x₂)) = 0
POL(FALSE) = 0
POL(<@z(x₁, x₂)) = -1
POL(COND_B14(x₁, x₂, x₃, x₄)) = x₃ + (-1)x₁
POL(>=@z(x₁, x₂)) = -1
POL(B14(x₁, x₂, x₃)) = x₂
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(B10(x₁, x₂, x₃)) = x₂
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1])

The following pairs are in P_bound:

COND_B14(TRUE, sv14_14[1], sv23_37[1], sv24_38[1]) → B15(sv14_14[1], sv23_37[1], sv24_38[1])

The following pairs are in P_≥:

B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0])
B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])
B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
-@z¹ ↔
TRUE¹ → &&(TRUE, TRUE)¹
+@z¹ ↔
&&(TRUE, FALSE)¹ ↔ FALSE¹
FALSE¹ → &&(FALSE, TRUE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ 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): B10(sv14_14[0], sv23_37[0], sv24_38[0]) → B14(sv14_14[0], sv23_37[0], sv24_38[0])
(2): B14(sv14_14[2], sv23_37[2], sv24_38[2]) → COND_B14(&&(>=@z(sv23_37[2], sv14_14[2]), &&(<@z(1@z, sv14_14[2]), &&(TRUE, &&(<@z(0@z, sv24_38[2]), <@z(1@z, sv23_37[2]))))), sv14_14[2], sv23_37[2], sv24_38[2])
(3): B15(sv14_14[3], sv23_37[3], sv24_38[3]) → B10(sv14_14[3], -@z(sv23_37[3], sv14_14[3]), +@z(sv24_38[3], 1@z))

(0) -> (2), if ((sv23_37[0] →^* sv23_37[2])∧(sv24_38[0] →^* sv24_38[2])∧(sv14_14[0] →^* sv14_14[2]))

(3) -> (0), if ((-@z(sv23_37[3], sv14_14[3]) →^* sv23_37[0])∧(+@z(sv24_38[3], 1@z) →^* sv24_38[0])∧(sv14_14[3] →^* sv14_14[0]))

The set Q consists of the following terms:

b14(x0, x1, x2)
Cond_b14(TRUE, x0, x1, x2)
b15(x0, x1, x2)
b10(x0, x1, x2)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.