6.98/2.70 YES 7.57/2.87 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 7.57/2.87 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.57/2.87 7.57/2.87 7.57/2.87 Termination of the given ITRS could be proven: 7.57/2.87 7.57/2.87 (0) ITRS 7.57/2.87 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 7.57/2.87 (2) IDP 7.57/2.87 (3) UsableRulesProof [EQUIVALENT, 0 ms] 7.57/2.87 (4) IDP 7.57/2.87 (5) IDPNonInfProof [SOUND, 413 ms] 7.57/2.87 (6) IDP 7.57/2.87 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.57/2.87 (8) IDP 7.57/2.87 (9) IDPNonInfProof [SOUND, 168 ms] 7.57/2.87 (10) IDP 7.57/2.87 (11) IDependencyGraphProof [EQUIVALENT, 0 ms] 7.57/2.87 (12) TRUE 7.57/2.87 7.57/2.87 7.57/2.87 ---------------------------------------- 7.57/2.87 7.57/2.87 (0) 7.57/2.87 Obligation: 7.57/2.87 ITRS problem: 7.57/2.87 7.57/2.87 The following function symbols are pre-defined: 7.57/2.87 <<< 7.57/2.87 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.87 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.87 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.87 / ~ Div: (Integer, Integer) -> Integer 7.57/2.87 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.87 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.87 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.87 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.87 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.87 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 The TRS R consists of the following rules: 7.57/2.88 eval(x, y, z) -> Cond_eval(x > z, x, y, z) 7.57/2.88 Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval1(y > z && x > z, x, y, z) 7.57/2.88 Cond_eval1(TRUE, x, y, z) -> eval(x - 1, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval2(x > z && z >= x && y > z, x, y, z) 7.57/2.88 Cond_eval2(TRUE, x, y, z) -> eval(x, y - 1, z) 7.57/2.88 eval(x, y, z) -> Cond_eval3(y > z && z >= x, x, y, z) 7.57/2.88 Cond_eval3(TRUE, x, y, z) -> eval(x, y - 1, z) 7.57/2.88 eval(x, y, z) -> Cond_eval4(x > z && z >= x && z >= y, x, y, z) 7.57/2.88 Cond_eval4(TRUE, x, y, z) -> eval(x, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval5(y > z && z >= x && z >= y, x, y, z) 7.57/2.88 Cond_eval5(TRUE, x, y, z) -> eval(x, y, z) 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (1) ITRStoIDPProof (EQUIVALENT) 7.57/2.88 Added dependency pairs 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (2) 7.57/2.88 Obligation: 7.57/2.88 IDP problem: 7.57/2.88 The following function symbols are pre-defined: 7.57/2.88 <<< 7.57/2.88 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.88 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.88 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.88 / ~ Div: (Integer, Integer) -> Integer 7.57/2.88 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.88 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.88 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.88 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.88 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.88 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 7.57/2.88 The following domains are used: 7.57/2.88 Integer, Boolean 7.57/2.88 7.57/2.88 The ITRS R consists of the following rules: 7.57/2.88 eval(x, y, z) -> Cond_eval(x > z, x, y, z) 7.57/2.88 Cond_eval(TRUE, x, y, z) -> eval(x - 1, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval1(y > z && x > z, x, y, z) 7.57/2.88 Cond_eval1(TRUE, x, y, z) -> eval(x - 1, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval2(x > z && z >= x && y > z, x, y, z) 7.57/2.88 Cond_eval2(TRUE, x, y, z) -> eval(x, y - 1, z) 7.57/2.88 eval(x, y, z) -> Cond_eval3(y > z && z >= x, x, y, z) 7.57/2.88 Cond_eval3(TRUE, x, y, z) -> eval(x, y - 1, z) 7.57/2.88 eval(x, y, z) -> Cond_eval4(x > z && z >= x && z >= y, x, y, z) 7.57/2.88 Cond_eval4(TRUE, x, y, z) -> eval(x, y, z) 7.57/2.88 eval(x, y, z) -> Cond_eval5(y > z && z >= x && z >= y, x, y, z) 7.57/2.88 Cond_eval5(TRUE, x, y, z) -> eval(x, y, z) 7.57/2.88 7.57/2.88 The integer pair graph contains the following rules and edges: 7.57/2.88 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) 7.57/2.88 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.57/2.88 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) 7.57/2.88 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) 7.57/2.88 (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] > z[4] && z[4] >= x[4] && y[4] > z[4], x[4], y[4], z[4]) 7.57/2.88 (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5] - 1, z[5]) 7.57/2.88 (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) 7.57/2.88 (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) 7.57/2.88 (8): EVAL(x[8], y[8], z[8]) -> COND_EVAL4(x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8], x[8], y[8], z[8]) 7.57/2.88 (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) 7.57/2.88 (10): EVAL(x[10], y[10], z[10]) -> COND_EVAL5(y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10], x[10], y[10], z[10]) 7.57/2.88 (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) 7.57/2.88 7.57/2.88 (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.57/2.88 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.57/2.88 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.57/2.88 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 7.57/2.88 (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 7.57/2.88 (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8] & z[1] ->^* z[8]) 7.57/2.88 (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10] & z[1] ->^* z[10]) 7.57/2.88 (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.57/2.88 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) 7.57/2.88 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) 7.57/2.88 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) 7.57/2.88 (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) 7.57/2.88 (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8] & z[3] ->^* z[8]) 7.57/2.88 (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10] & z[3] ->^* z[10]) 7.57/2.88 (4) -> (5), if (x[4] > z[4] && z[4] >= x[4] && y[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 7.57/2.88 (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0] & z[5] ->^* z[0]) 7.57/2.88 (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2] & z[5] ->^* z[2]) 7.57/2.88 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4] & z[5] ->^* z[4]) 7.57/2.88 (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6] & z[5] ->^* z[6]) 7.57/2.88 (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8] & z[5] ->^* z[8]) 7.57/2.88 (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10] & z[5] ->^* z[10]) 7.57/2.88 (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 7.57/2.88 (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) 7.57/2.88 (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) 7.57/2.88 (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4] & z[7] ->^* z[4]) 7.57/2.88 (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) 7.57/2.88 (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8] & z[7] ->^* z[8]) 7.57/2.88 (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10] & z[7] ->^* z[10]) 7.57/2.88 (8) -> (9), if (x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9] & z[8] ->^* z[9]) 7.57/2.88 (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) 7.57/2.88 (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) 7.57/2.88 (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4] & z[9] ->^* z[4]) 7.57/2.88 (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6] & z[9] ->^* z[6]) 7.57/2.88 (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8] & z[9] ->^* z[8]) 7.57/2.88 (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10] & z[9] ->^* z[10]) 7.57/2.88 (10) -> (11), if (y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11] & z[10] ->^* z[11]) 7.57/2.88 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) 7.57/2.88 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) 7.57/2.88 (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4] & z[11] ->^* z[4]) 7.57/2.88 (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6] & z[11] ->^* z[6]) 7.57/2.88 (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8] & z[11] ->^* z[8]) 7.57/2.88 (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10] & z[11] ->^* z[10]) 7.57/2.88 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (3) UsableRulesProof (EQUIVALENT) 7.57/2.88 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. 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (4) 7.57/2.88 Obligation: 7.57/2.88 IDP problem: 7.57/2.88 The following function symbols are pre-defined: 7.57/2.88 <<< 7.57/2.88 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.88 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.88 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.88 / ~ Div: (Integer, Integer) -> Integer 7.57/2.88 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.88 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.88 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.88 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.88 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.88 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 7.57/2.88 The following domains are used: 7.57/2.88 Integer, Boolean 7.57/2.88 7.57/2.88 R is empty. 7.57/2.88 7.57/2.88 The integer pair graph contains the following rules and edges: 7.57/2.88 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) 7.57/2.88 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.57/2.88 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) 7.57/2.88 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) 7.57/2.88 (4): EVAL(x[4], y[4], z[4]) -> COND_EVAL2(x[4] > z[4] && z[4] >= x[4] && y[4] > z[4], x[4], y[4], z[4]) 7.57/2.88 (5): COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], y[5] - 1, z[5]) 7.57/2.88 (6): EVAL(x[6], y[6], z[6]) -> COND_EVAL3(y[6] > z[6] && z[6] >= x[6], x[6], y[6], z[6]) 7.57/2.88 (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) 7.57/2.88 (8): EVAL(x[8], y[8], z[8]) -> COND_EVAL4(x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8], x[8], y[8], z[8]) 7.57/2.88 (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) 7.57/2.88 (10): EVAL(x[10], y[10], z[10]) -> COND_EVAL5(y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10], x[10], y[10], z[10]) 7.57/2.88 (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) 7.57/2.88 7.57/2.88 (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.57/2.88 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.57/2.88 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.57/2.88 (1) -> (4), if (x[1] - 1 ->^* x[4] & y[1] ->^* y[4] & z[1] ->^* z[4]) 7.57/2.88 (1) -> (6), if (x[1] - 1 ->^* x[6] & y[1] ->^* y[6] & z[1] ->^* z[6]) 7.57/2.88 (1) -> (8), if (x[1] - 1 ->^* x[8] & y[1] ->^* y[8] & z[1] ->^* z[8]) 7.57/2.88 (1) -> (10), if (x[1] - 1 ->^* x[10] & y[1] ->^* y[10] & z[1] ->^* z[10]) 7.57/2.88 (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.57/2.88 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) 7.57/2.88 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) 7.57/2.88 (3) -> (4), if (x[3] - 1 ->^* x[4] & y[3] ->^* y[4] & z[3] ->^* z[4]) 7.57/2.88 (3) -> (6), if (x[3] - 1 ->^* x[6] & y[3] ->^* y[6] & z[3] ->^* z[6]) 7.57/2.88 (3) -> (8), if (x[3] - 1 ->^* x[8] & y[3] ->^* y[8] & z[3] ->^* z[8]) 7.57/2.88 (3) -> (10), if (x[3] - 1 ->^* x[10] & y[3] ->^* y[10] & z[3] ->^* z[10]) 7.57/2.88 (4) -> (5), if (x[4] > z[4] && z[4] >= x[4] && y[4] > z[4] & x[4] ->^* x[5] & y[4] ->^* y[5] & z[4] ->^* z[5]) 7.57/2.88 (5) -> (0), if (x[5] ->^* x[0] & y[5] - 1 ->^* y[0] & z[5] ->^* z[0]) 7.57/2.88 (5) -> (2), if (x[5] ->^* x[2] & y[5] - 1 ->^* y[2] & z[5] ->^* z[2]) 7.57/2.88 (5) -> (4), if (x[5] ->^* x[4] & y[5] - 1 ->^* y[4] & z[5] ->^* z[4]) 7.57/2.88 (5) -> (6), if (x[5] ->^* x[6] & y[5] - 1 ->^* y[6] & z[5] ->^* z[6]) 7.57/2.88 (5) -> (8), if (x[5] ->^* x[8] & y[5] - 1 ->^* y[8] & z[5] ->^* z[8]) 7.57/2.88 (5) -> (10), if (x[5] ->^* x[10] & y[5] - 1 ->^* y[10] & z[5] ->^* z[10]) 7.57/2.88 (6) -> (7), if (y[6] > z[6] && z[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7] & z[6] ->^* z[7]) 7.57/2.88 (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) 7.57/2.88 (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) 7.57/2.88 (7) -> (4), if (x[7] ->^* x[4] & y[7] - 1 ->^* y[4] & z[7] ->^* z[4]) 7.57/2.88 (7) -> (6), if (x[7] ->^* x[6] & y[7] - 1 ->^* y[6] & z[7] ->^* z[6]) 7.57/2.88 (7) -> (8), if (x[7] ->^* x[8] & y[7] - 1 ->^* y[8] & z[7] ->^* z[8]) 7.57/2.88 (7) -> (10), if (x[7] ->^* x[10] & y[7] - 1 ->^* y[10] & z[7] ->^* z[10]) 7.57/2.88 (8) -> (9), if (x[8] > z[8] && z[8] >= x[8] && z[8] >= y[8] & x[8] ->^* x[9] & y[8] ->^* y[9] & z[8] ->^* z[9]) 7.57/2.88 (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) 7.57/2.88 (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) 7.57/2.88 (9) -> (4), if (x[9] ->^* x[4] & y[9] ->^* y[4] & z[9] ->^* z[4]) 7.57/2.88 (9) -> (6), if (x[9] ->^* x[6] & y[9] ->^* y[6] & z[9] ->^* z[6]) 7.57/2.88 (9) -> (8), if (x[9] ->^* x[8] & y[9] ->^* y[8] & z[9] ->^* z[8]) 7.57/2.88 (9) -> (10), if (x[9] ->^* x[10] & y[9] ->^* y[10] & z[9] ->^* z[10]) 7.57/2.88 (10) -> (11), if (y[10] > z[10] && z[10] >= x[10] && z[10] >= y[10] & x[10] ->^* x[11] & y[10] ->^* y[11] & z[10] ->^* z[11]) 7.57/2.88 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) 7.57/2.88 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) 7.57/2.88 (11) -> (4), if (x[11] ->^* x[4] & y[11] ->^* y[4] & z[11] ->^* z[4]) 7.57/2.88 (11) -> (6), if (x[11] ->^* x[6] & y[11] ->^* y[6] & z[11] ->^* z[6]) 7.57/2.88 (11) -> (8), if (x[11] ->^* x[8] & y[11] ->^* y[8] & z[11] ->^* z[8]) 7.57/2.88 (11) -> (10), if (x[11] ->^* x[10] & y[11] ->^* y[10] & z[11] ->^* z[10]) 7.57/2.88 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (5) IDPNonInfProof (SOUND) 7.57/2.88 Used the following options for this NonInfProof: 7.57/2.88 7.57/2.88 IDPGPoloSolver: 7.57/2.88 Range: [(-1,2)] 7.57/2.88 IsNat: false 7.57/2.88 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@21d7b6ed 7.57/2.88 Constraint Generator: NonInfConstraintGenerator: 7.57/2.88 PathGenerator: MetricPathGenerator: 7.57/2.88 Max Left Steps: 1 7.57/2.88 Max Right Steps: 1 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 The constraints were generated the following way: 7.57/2.88 7.57/2.88 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 7.57/2.88 7.57/2.88 Note that final constraints are written in bold face. 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL(>(x, z), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] + [bni_40]y[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] + [bni_40]y[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] + [bni_40]y[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (III) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] + [bni_42]y[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] + [bni_42]y[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] + [bni_42]y[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL1(&&(>(y, z), >(x, z)), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]z[2] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL1(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]z[2] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL2(&&(&&(>(x, z), >=(z, x)), >(y, z)), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[4], z[4])=TRUE & >(x[4], z[4])=TRUE & >=(z[4], x[4])=TRUE ==> EVAL(x[4], y[4], z[4])_>=_NonInfC & EVAL(x[4], y[4], z[4])_>=_COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) & (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] + [bni_48]y[4] >= 0 & [1 + (-1)bso_49] + [2]y[4] + x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] + [bni_48]y[4] >= 0 & [1 + (-1)bso_49] + [2]y[4] + x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4])), >=) & [(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]z[4] + [bni_48]y[4] >= 0 & [1 + (-1)bso_49] + [2]y[4] + x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We solved constraint (5) using rule (IDP_SMT_SPLIT). 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL2(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]), COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4]))=TRUE & x[4]=x[5] & y[4]=y[5] & z[4]=z[5] ==> COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5], z[5])_>=_EVAL(x[5], -(y[5], 1), z[5]) & (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[4], z[4])=TRUE & >(x[4], z[4])=TRUE & >=(z[4], x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4], z[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4], z[4])_>=_EVAL(x[4], -(y[4], 1), z[4]) & (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-2)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-2]y[4] + [-1]x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-2)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-2]y[4] + [-1]x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[4] + [-1] + [-1]z[4] >= 0 & x[4] + [-1] + [-1]z[4] >= 0 & z[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(x[5], -(y[5], 1), z[5])), >=) & [(-2)bni_50 + (-1)Bound*bni_50] + [(-1)bni_50]z[4] + [(-1)bni_50]y[4] + [(-1)bni_50]x[4] >= 0 & [(-1)bso_51] + [-2]y[4] + [-1]x[4] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We solved constraint (5) using rule (IDP_SMT_SPLIT). 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL3(&&(>(y, z), >=(z, x)), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> EVAL(x[6], y[6], z[6])_>=_NonInfC & EVAL(x[6], y[6], z[6])_>=_COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)bni_52 + (-1)Bound*bni_52] + [(-1)bni_52]z[6] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL3(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]), COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[6], z[6]), >=(z[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] & z[6]=z[7] ==> COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7], z[7])_>=_EVAL(x[7], -(y[7], 1), z[7]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[6], z[6])=TRUE & >=(z[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6], z[6])_>=_EVAL(x[6], -(y[6], 1), z[6]) & (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-2)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-2)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[6] + [-1] + [-1]z[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-2)bni_54 + (-1)Bound*bni_54] + [(-1)bni_54]z[6] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[6] >= 0 & z[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[6] >= 0 & z[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 (9) (y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL4(&&(&&(>(x, z), >=(z, x)), >=(z, y)), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]), COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8]))=TRUE & x[8]=x[9] & y[8]=y[9] & z[8]=z[9] ==> EVAL(x[8], y[8], z[8])_>=_NonInfC & EVAL(x[8], y[8], z[8])_>=_COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>=(z[8], y[8])=TRUE & >(x[8], z[8])=TRUE & >=(z[8], x[8])=TRUE ==> EVAL(x[8], y[8], z[8])_>=_NonInfC & EVAL(x[8], y[8], z[8])_>=_COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) & (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] + [bni_56]y[8] >= 0 & [(-1)bso_57] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] + [bni_56]y[8] >= 0 & [(-1)bso_57] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (z[8] + [-1]y[8] >= 0 & x[8] + [-1] + [-1]z[8] >= 0 & z[8] + [-1]x[8] >= 0 ==> (U^Increasing(COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8])), >=) & [(-1)bni_56 + (-1)Bound*bni_56] + [(-1)bni_56]z[8] + [bni_56]y[8] >= 0 & [(-1)bso_57] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We solved constraint (5) using rule (IDP_SMT_SPLIT). 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL4(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[0] & y[9]=y[0] & z[9]=z[0] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[2] & y[9]=y[2] & z[9]=z[2] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[4] & y[9]=y[4] & z[9]=z[4] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[6] & y[9]=y[6] & z[9]=z[6] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[8] & y[9]=y[8] & z[9]=z[8] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]), EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[9]=x[10] & y[9]=y[10] & z[9]=z[10] ==> COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_NonInfC & COND_EVAL4(TRUE, x[9], y[9], z[9])_>=_EVAL(x[9], y[9], z[9]) & (U^Increasing(EVAL(x[9], y[9], z[9])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x, y, z) -> COND_EVAL5(&&(&&(>(y, z), >=(z, x)), >=(z, y)), x, y, z) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]), COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10]))=TRUE & x[10]=x[11] & y[10]=y[11] & z[10]=z[11] ==> EVAL(x[10], y[10], z[10])_>=_NonInfC & EVAL(x[10], y[10], z[10])_>=_COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>=(z[10], y[10])=TRUE & >(y[10], z[10])=TRUE & >=(z[10], x[10])=TRUE ==> EVAL(x[10], y[10], z[10])_>=_NonInfC & EVAL(x[10], y[10], z[10])_>=_COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) & (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] + [bni_60]y[10] >= 0 & [(-1)bso_61] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] + [bni_60]y[10] >= 0 & [(-1)bso_61] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (z[10] + [-1]y[10] >= 0 & y[10] + [-1] + [-1]z[10] >= 0 & z[10] + [-1]x[10] >= 0 ==> (U^Increasing(COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10])), >=) & [(-1)bni_60 + (-1)Bound*bni_60] + [(-1)bni_60]z[10] + [bni_60]y[10] >= 0 & [(-1)bso_61] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We solved constraint (5) using rule (IDP_SMT_SPLIT). 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL5(TRUE, x, y, z) -> EVAL(x, y, z) the following chains were created: 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[0] & y[11]=y[0] & z[11]=z[0] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[2] & y[11]=y[2] & z[11]=z[2] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[4] & y[11]=y[4] & z[11]=z[4] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[6] & y[11]=y[6] & z[11]=z[6] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[8] & y[11]=y[8] & z[11]=z[8] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]), EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (x[11]=x[10] & y[11]=y[10] & z[11]=z[10] ==> COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_NonInfC & COND_EVAL5(TRUE, x[11], y[11], z[11])_>=_EVAL(x[11], y[11], z[11]) & (U^Increasing(EVAL(x[11], y[11], z[11])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) ((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 To summarize, we get the following constraints P__>=_ for the following pairs. 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL(>(x, z), x, y, z) 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [bni_40] = 0 & [(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]z[0] >= 0 & [(-1)bso_41] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [bni_42] = 0 & [(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]z[0] >= 0 & [(-1)bso_43] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL1(&&(>(y, z), >(x, z)), x, y, z) 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_44] + [bni_44]y[2] >= 0 & [(-1)bso_45] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL1(TRUE, x, y, z) -> EVAL(-(x, 1), y, z) 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_46] + [bni_46]y[2] >= 0 & [(-1)bso_47] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL2(&&(&&(>(x, z), >=(z, x)), >(y, z)), x, y, z) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL2(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL3(&&(>(y, z), >=(z, x)), x, y, z) 7.57/2.88 7.57/2.88 *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6])), >=) & [(-1)Bound*bni_52] + [bni_52]y[6] >= 0 & [1 + (-1)bso_53] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL3(TRUE, x, y, z) -> EVAL(x, -(y, 1), z) 7.57/2.88 7.57/2.88 *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[6] >= 0 & z[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(x[7], -(y[7], 1), z[7])), >=) & [(-1)bni_54 + (-1)Bound*bni_54] + [bni_54]y[6] >= 0 & [(-1)bso_55] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL4(&&(&&(>(x, z), >=(z, x)), >=(z, y)), x, y, z) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL4(TRUE, x, y, z) -> EVAL(x, y, z) 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[9], y[9], z[9])), >=) & [bni_58] = 0 & [(-1)bso_59] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x, y, z) -> COND_EVAL5(&&(&&(>(y, z), >=(z, x)), >=(z, y)), x, y, z) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL5(TRUE, x, y, z) -> EVAL(x, y, z) 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *((U^Increasing(EVAL(x[11], y[11], z[11])), >=) & [bni_62] = 0 & [(-1)bso_63] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 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. 7.57/2.88 7.57/2.88 Using the following integer polynomial ordering the resulting constraints can be solved 7.57/2.88 7.57/2.88 Polynomial interpretation over integers[POLO]: 7.57/2.88 7.57/2.88 POL(TRUE) = [1] 7.57/2.88 POL(FALSE) = [3] 7.57/2.88 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 7.57/2.88 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 7.57/2.88 POL(>(x_1, x_2)) = [-1] 7.57/2.88 POL(-(x_1, x_2)) = x_1 + [-1]x_2 7.57/2.88 POL(1) = [1] 7.57/2.88 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 7.57/2.88 POL(&&(x_1, x_2)) = [1] 7.57/2.88 POL(COND_EVAL2(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + [-1]x_3 + [-1]x_2 + [-1]x_1 7.57/2.88 POL(>=(x_1, x_2)) = [-1] 7.57/2.88 POL(COND_EVAL3(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 + [-1]x_1 7.57/2.88 POL(COND_EVAL4(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 7.57/2.88 POL(COND_EVAL5(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_3 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_>: 7.57/2.88 7.57/2.88 7.57/2.88 EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) 7.57/2.88 COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) 7.57/2.88 EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) 7.57/2.88 EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) 7.57/2.88 EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_bound: 7.57/2.88 7.57/2.88 7.57/2.88 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) 7.57/2.88 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) 7.57/2.88 EVAL(x[4], y[4], z[4]) -> COND_EVAL2(&&(&&(>(x[4], z[4]), >=(z[4], x[4])), >(y[4], z[4])), x[4], y[4], z[4]) 7.57/2.88 COND_EVAL2(TRUE, x[5], y[5], z[5]) -> EVAL(x[5], -(y[5], 1), z[5]) 7.57/2.88 EVAL(x[6], y[6], z[6]) -> COND_EVAL3(&&(>(y[6], z[6]), >=(z[6], x[6])), x[6], y[6], z[6]) 7.57/2.88 COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) 7.57/2.88 EVAL(x[8], y[8], z[8]) -> COND_EVAL4(&&(&&(>(x[8], z[8]), >=(z[8], x[8])), >=(z[8], y[8])), x[8], y[8], z[8]) 7.57/2.88 EVAL(x[10], y[10], z[10]) -> COND_EVAL5(&&(&&(>(y[10], z[10]), >=(z[10], x[10])), >=(z[10], y[10])), x[10], y[10], z[10]) 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_>=: 7.57/2.88 7.57/2.88 7.57/2.88 EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 7.57/2.88 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.57/2.88 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) 7.57/2.88 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) 7.57/2.88 COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], -(y[7], 1), z[7]) 7.57/2.88 COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) 7.57/2.88 COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) 7.57/2.88 7.57/2.88 7.57/2.88 At least the following rules have been oriented under context sensitive arithmetic replacement: 7.57/2.88 7.57/2.88 &&(TRUE, TRUE)^1 <-> TRUE^1 7.57/2.88 FALSE^1 -> &&(TRUE, FALSE)^1 7.57/2.88 FALSE^1 -> &&(FALSE, TRUE)^1 7.57/2.88 FALSE^1 -> &&(FALSE, FALSE)^1 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (6) 7.57/2.88 Obligation: 7.57/2.88 IDP problem: 7.57/2.88 The following function symbols are pre-defined: 7.57/2.88 <<< 7.57/2.88 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.88 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.88 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.88 / ~ Div: (Integer, Integer) -> Integer 7.57/2.88 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.88 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.88 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.88 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.88 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.88 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 7.57/2.88 The following domains are used: 7.57/2.88 Integer, Boolean 7.57/2.88 7.57/2.88 R is empty. 7.57/2.88 7.57/2.88 The integer pair graph contains the following rules and edges: 7.57/2.88 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) 7.57/2.88 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.57/2.88 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) 7.57/2.88 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) 7.57/2.88 (7): COND_EVAL3(TRUE, x[7], y[7], z[7]) -> EVAL(x[7], y[7] - 1, z[7]) 7.57/2.88 (9): COND_EVAL4(TRUE, x[9], y[9], z[9]) -> EVAL(x[9], y[9], z[9]) 7.57/2.88 (11): COND_EVAL5(TRUE, x[11], y[11], z[11]) -> EVAL(x[11], y[11], z[11]) 7.57/2.88 7.57/2.88 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.57/2.88 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) 7.57/2.88 (7) -> (0), if (x[7] ->^* x[0] & y[7] - 1 ->^* y[0] & z[7] ->^* z[0]) 7.57/2.88 (9) -> (0), if (x[9] ->^* x[0] & y[9] ->^* y[0] & z[9] ->^* z[0]) 7.57/2.88 (11) -> (0), if (x[11] ->^* x[0] & y[11] ->^* y[0] & z[11] ->^* z[0]) 7.57/2.88 (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.57/2.88 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.57/2.88 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) 7.57/2.88 (7) -> (2), if (x[7] ->^* x[2] & y[7] - 1 ->^* y[2] & z[7] ->^* z[2]) 7.57/2.88 (9) -> (2), if (x[9] ->^* x[2] & y[9] ->^* y[2] & z[9] ->^* z[2]) 7.57/2.88 (11) -> (2), if (x[11] ->^* x[2] & y[11] ->^* y[2] & z[11] ->^* z[2]) 7.57/2.88 (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.57/2.88 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (7) IDependencyGraphProof (EQUIVALENT) 7.57/2.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (8) 7.57/2.88 Obligation: 7.57/2.88 IDP problem: 7.57/2.88 The following function symbols are pre-defined: 7.57/2.88 <<< 7.57/2.88 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.88 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.88 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.88 / ~ Div: (Integer, Integer) -> Integer 7.57/2.88 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.88 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.88 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.88 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.88 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.88 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 7.57/2.88 The following domains are used: 7.57/2.88 Integer, Boolean 7.57/2.88 7.57/2.88 R is empty. 7.57/2.88 7.57/2.88 The integer pair graph contains the following rules and edges: 7.57/2.88 (3): COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(x[3] - 1, y[3], z[3]) 7.57/2.88 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) 7.57/2.88 (1): COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(x[1] - 1, y[1], z[1]) 7.57/2.88 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) 7.57/2.88 7.57/2.88 (1) -> (0), if (x[1] - 1 ->^* x[0] & y[1] ->^* y[0] & z[1] ->^* z[0]) 7.57/2.88 (3) -> (0), if (x[3] - 1 ->^* x[0] & y[3] ->^* y[0] & z[3] ->^* z[0]) 7.57/2.88 (0) -> (1), if (x[0] > z[0] & x[0] ->^* x[1] & y[0] ->^* y[1] & z[0] ->^* z[1]) 7.57/2.88 (1) -> (2), if (x[1] - 1 ->^* x[2] & y[1] ->^* y[2] & z[1] ->^* z[2]) 7.57/2.88 (3) -> (2), if (x[3] - 1 ->^* x[2] & y[3] ->^* y[2] & z[3] ->^* z[2]) 7.57/2.88 (2) -> (3), if (y[2] > z[2] && x[2] > z[2] & x[2] ->^* x[3] & y[2] ->^* y[3] & z[2] ->^* z[3]) 7.57/2.88 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (9) IDPNonInfProof (SOUND) 7.57/2.88 Used the following options for this NonInfProof: 7.57/2.88 7.57/2.88 IDPGPoloSolver: 7.57/2.88 Range: [(-1,2)] 7.57/2.88 IsNat: false 7.57/2.88 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@21d7b6ed 7.57/2.88 Constraint Generator: NonInfConstraintGenerator: 7.57/2.88 PathGenerator: MetricPathGenerator: 7.57/2.88 Max Left Steps: 1 7.57/2.88 Max Right Steps: 1 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 The constraints were generated the following way: 7.57/2.88 7.57/2.88 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 7.57/2.88 7.57/2.88 Note that final constraints are written in bold face. 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[0] & y[3]=y[0] & z[3]=z[0] ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] & -(x[3], 1)=x[2]1 & y[3]=y[2]1 & z[3]=z[2]1 ==> COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3], z[3])_>=_EVAL(-(x[3], 1), y[3], z[3]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2], z[2])_>=_EVAL(-(x[2], 1), y[2], z[2]) & (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]z[2] + [bni_19]x[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]), COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (&&(>(y[2], z[2]), >(x[2], z[2]))=TRUE & x[2]=x[3] & y[2]=y[3] & z[2]=z[3] ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(y[2], z[2])=TRUE & >(x[2], z[2])=TRUE ==> EVAL(x[2], y[2], z[2])_>=_NonInfC & EVAL(x[2], y[2], z[2])_>=_COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (y[2] + [-1] + [-1]z[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (y[2] >= 0 & x[2] + [-1] + [-1]z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]z[2] + [bni_21]x[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (y[2] >= 0 & z[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_21] + [bni_21]z[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_21] + [bni_21]z[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 (9) (y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_21] + [bni_21]z[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[0]1 & y[1]=y[0]1 & z[1]=z[0]1 ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]), EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] & -(x[1], 1)=x[2] & y[1]=y[2] & z[1]=z[2] ==> COND_EVAL(TRUE, x[1], y[1], z[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1], z[1])_>=_EVAL(-(x[1], 1), y[1], z[1]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(x[0], z[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0], z[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0], z[0])_>=_EVAL(-(x[0], 1), y[0], z[0]) & (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]z[0] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (x[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 For Pair EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) the following chains were created: 7.57/2.88 *We consider the chain EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]), COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) which results in the following constraint: 7.57/2.88 7.57/2.88 (1) (>(x[0], z[0])=TRUE & x[0]=x[1] & y[0]=y[1] & z[0]=z[1] ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (1) using rule (IV) which results in the following new constraint: 7.57/2.88 7.57/2.88 (2) (>(x[0], z[0])=TRUE ==> EVAL(x[0], y[0], z[0])_>=_NonInfC & EVAL(x[0], y[0], z[0])_>=_COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) & (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=)) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (3) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 7.57/2.88 7.57/2.88 (4) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 7.57/2.88 7.57/2.88 (5) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 7.57/2.88 7.57/2.88 (6) (x[0] + [-1] + [-1]z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]z[0] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 7.57/2.88 7.57/2.88 (7) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 7.57/2.88 7.57/2.88 (8) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 (9) (x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 To summarize, we get the following constraints P__>=_ for the following pairs. 7.57/2.88 7.57/2.88 *COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(EVAL(-(x[3], 1), y[3], z[3])), >=) & [(-1)Bound*bni_19] + [bni_19]z[2] >= 0 & [1 + (-1)bso_20] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_21] + [bni_21]z[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(y[2] >= 0 & z[2] >= 0 & x[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2])), >=) & [(-1)Bound*bni_21] + [bni_21]z[2] >= 0 & [(-1)bso_22] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(EVAL(-(x[1], 1), y[1], z[1])), >=) & 0 = 0 & [(-1)Bound*bni_23] + [bni_23]x[0] >= 0 & [1 + (-1)bso_24] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 *EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 *(x[0] >= 0 & z[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0])), >=) & 0 = 0 & [(-1)Bound*bni_25] + [bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 7.57/2.88 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. 7.57/2.88 7.57/2.88 Using the following integer polynomial ordering the resulting constraints can be solved 7.57/2.88 7.57/2.88 Polynomial interpretation over integers[POLO]: 7.57/2.88 7.57/2.88 POL(TRUE) = 0 7.57/2.88 POL(FALSE) = [1] 7.57/2.88 POL(COND_EVAL1(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 + [-1]x_1 7.57/2.88 POL(EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_1 7.57/2.88 POL(-(x_1, x_2)) = x_1 + [-1]x_2 7.57/2.88 POL(1) = [1] 7.57/2.88 POL(&&(x_1, x_2)) = 0 7.57/2.88 POL(>(x_1, x_2)) = [-1] 7.57/2.88 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + [-1]x_4 + x_2 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_>: 7.57/2.88 7.57/2.88 7.57/2.88 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) 7.57/2.88 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_bound: 7.57/2.88 7.57/2.88 7.57/2.88 COND_EVAL1(TRUE, x[3], y[3], z[3]) -> EVAL(-(x[3], 1), y[3], z[3]) 7.57/2.88 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) 7.57/2.88 COND_EVAL(TRUE, x[1], y[1], z[1]) -> EVAL(-(x[1], 1), y[1], z[1]) 7.57/2.88 EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 7.57/2.88 7.57/2.88 7.57/2.88 The following pairs are in P_>=: 7.57/2.88 7.57/2.88 7.57/2.88 EVAL(x[2], y[2], z[2]) -> COND_EVAL1(&&(>(y[2], z[2]), >(x[2], z[2])), x[2], y[2], z[2]) 7.57/2.88 EVAL(x[0], y[0], z[0]) -> COND_EVAL(>(x[0], z[0]), x[0], y[0], z[0]) 7.57/2.88 7.57/2.88 7.57/2.88 At least the following rules have been oriented under context sensitive arithmetic replacement: 7.57/2.88 7.57/2.88 TRUE^1 -> &&(TRUE, TRUE)^1 7.57/2.88 FALSE^1 -> &&(TRUE, FALSE)^1 7.57/2.88 FALSE^1 -> &&(FALSE, TRUE)^1 7.57/2.88 FALSE^1 -> &&(FALSE, FALSE)^1 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (10) 7.57/2.88 Obligation: 7.57/2.88 IDP problem: 7.57/2.88 The following function symbols are pre-defined: 7.57/2.88 <<< 7.57/2.88 & ~ Bwand: (Integer, Integer) -> Integer 7.57/2.88 >= ~ Ge: (Integer, Integer) -> Boolean 7.57/2.88 | ~ Bwor: (Integer, Integer) -> Integer 7.57/2.88 / ~ Div: (Integer, Integer) -> Integer 7.57/2.88 != ~ Neq: (Integer, Integer) -> Boolean 7.57/2.88 && ~ Land: (Boolean, Boolean) -> Boolean 7.57/2.88 ! ~ Lnot: (Boolean) -> Boolean 7.57/2.88 = ~ Eq: (Integer, Integer) -> Boolean 7.57/2.88 <= ~ Le: (Integer, Integer) -> Boolean 7.57/2.88 ^ ~ Bwxor: (Integer, Integer) -> Integer 7.57/2.88 % ~ Mod: (Integer, Integer) -> Integer 7.57/2.88 > ~ Gt: (Integer, Integer) -> Boolean 7.57/2.88 + ~ Add: (Integer, Integer) -> Integer 7.57/2.88 -1 ~ UnaryMinus: (Integer) -> Integer 7.57/2.88 < ~ Lt: (Integer, Integer) -> Boolean 7.57/2.88 || ~ Lor: (Boolean, Boolean) -> Boolean 7.57/2.88 - ~ Sub: (Integer, Integer) -> Integer 7.57/2.88 ~ ~ Bwnot: (Integer) -> Integer 7.57/2.88 * ~ Mul: (Integer, Integer) -> Integer 7.57/2.88 >>> 7.57/2.88 7.57/2.88 7.57/2.88 The following domains are used: 7.57/2.88 Boolean, Integer 7.57/2.88 7.57/2.88 R is empty. 7.57/2.88 7.57/2.88 The integer pair graph contains the following rules and edges: 7.57/2.88 (2): EVAL(x[2], y[2], z[2]) -> COND_EVAL1(y[2] > z[2] && x[2] > z[2], x[2], y[2], z[2]) 7.57/2.88 (0): EVAL(x[0], y[0], z[0]) -> COND_EVAL(x[0] > z[0], x[0], y[0], z[0]) 7.57/2.88 7.57/2.88 7.57/2.88 The set Q consists of the following terms: 7.57/2.88 eval(x0, x1, x2) 7.57/2.88 Cond_eval(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval1(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval2(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval3(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval4(TRUE, x0, x1, x2) 7.57/2.88 Cond_eval5(TRUE, x0, x1, x2) 7.57/2.88 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (11) IDependencyGraphProof (EQUIVALENT) 7.57/2.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 7.57/2.88 ---------------------------------------- 7.57/2.88 7.57/2.88 (12) 7.57/2.88 TRUE 7.57/2.91 EOF