4.00/2.00 YES 4.00/2.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.00/2.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.00/2.01 4.00/2.01 4.00/2.01 Termination of the given ITRS could be proven: 4.00/2.01 4.00/2.01 (0) ITRS 4.00/2.01 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.00/2.01 (2) IDP 4.00/2.01 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.00/2.01 (4) IDP 4.00/2.01 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.00/2.01 (6) IDP 4.00/2.01 (7) IDPNonInfProof [SOUND, 175 ms] 4.00/2.01 (8) IDP 4.00/2.01 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.00/2.01 (10) TRUE 4.00/2.01 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (0) 4.00/2.01 Obligation: 4.00/2.01 ITRS problem: 4.00/2.01 4.00/2.01 The following function symbols are pre-defined: 4.00/2.01 <<< 4.00/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.00/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.00/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.00/2.01 / ~ Div: (Integer, Integer) -> Integer 4.00/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.00/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.00/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.00/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.00/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.00/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.00/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.00/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.00/2.01 + ~ Add: (Integer, Integer) -> Integer 4.00/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.00/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.00/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.00/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.00/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.00/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.00/2.01 >>> 4.00/2.01 4.00/2.01 The TRS R consists of the following rules: 4.00/2.01 random(x) -> Cond_random(x >= 0, x) 4.00/2.01 Cond_random(TRUE, x) -> rand(x, w(0)) 4.00/2.01 rand(x, y) -> Cond_rand(x = 0, x, y) 4.00/2.01 Cond_rand(TRUE, x, y) -> y 4.00/2.01 rand(x, y) -> Cond_rand1(x > 0, x, y) 4.00/2.01 Cond_rand1(TRUE, x, y) -> rand(x - 1, id_inc(y)) 4.00/2.01 id_inc(w(x)) -> w(x) 4.00/2.01 id_inc(w(x)) -> w(x + 1) 4.00/2.01 The set Q consists of the following terms: 4.00/2.01 random(x0) 4.00/2.01 Cond_random(TRUE, x0) 4.00/2.01 rand(x0, x1) 4.00/2.01 Cond_rand(TRUE, x0, x1) 4.00/2.01 Cond_rand1(TRUE, x0, x1) 4.00/2.01 id_inc(w(x0)) 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (1) ITRStoIDPProof (EQUIVALENT) 4.00/2.01 Added dependency pairs 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (2) 4.00/2.01 Obligation: 4.00/2.01 IDP problem: 4.00/2.01 The following function symbols are pre-defined: 4.00/2.01 <<< 4.00/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.00/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.00/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.00/2.01 / ~ Div: (Integer, Integer) -> Integer 4.00/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.00/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.00/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.00/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.00/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.00/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.00/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.00/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.00/2.01 + ~ Add: (Integer, Integer) -> Integer 4.00/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.00/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.00/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.00/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.00/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.00/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.00/2.01 >>> 4.00/2.01 4.00/2.01 4.00/2.01 The following domains are used: 4.00/2.01 Integer 4.00/2.01 4.00/2.01 The ITRS R consists of the following rules: 4.00/2.01 random(x) -> Cond_random(x >= 0, x) 4.00/2.01 Cond_random(TRUE, x) -> rand(x, w(0)) 4.00/2.01 rand(x, y) -> Cond_rand(x = 0, x, y) 4.00/2.01 Cond_rand(TRUE, x, y) -> y 4.00/2.01 rand(x, y) -> Cond_rand1(x > 0, x, y) 4.00/2.01 Cond_rand1(TRUE, x, y) -> rand(x - 1, id_inc(y)) 4.00/2.01 id_inc(w(x)) -> w(x) 4.00/2.01 id_inc(w(x)) -> w(x + 1) 4.00/2.01 4.00/2.01 The integer pair graph contains the following rules and edges: 4.00/2.01 (0): RANDOM(x[0]) -> COND_RANDOM(x[0] >= 0, x[0]) 4.00/2.01 (1): COND_RANDOM(TRUE, x[1]) -> RAND(x[1], w(0)) 4.00/2.01 (2): RAND(x[2], y[2]) -> COND_RAND(x[2] = 0, x[2], y[2]) 4.00/2.01 (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) 4.00/2.01 (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) 4.00/2.01 (5): COND_RAND1(TRUE, x[5], y[5]) -> ID_INC(y[5]) 4.00/2.01 4.00/2.01 (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) 4.00/2.01 (1) -> (2), if (x[1] ->^* x[2] & w(0) ->^* y[2]) 4.00/2.01 (1) -> (3), if (x[1] ->^* x[3] & w(0) ->^* y[3]) 4.00/2.01 (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) 4.00/2.01 (3) -> (5), if (x[3] > 0 & x[3] ->^* x[5] & y[3] ->^* y[5]) 4.00/2.01 (4) -> (2), if (x[4] - 1 ->^* x[2] & id_inc(y[4]) ->^* y[2]) 4.00/2.01 (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) 4.00/2.01 4.00/2.01 The set Q consists of the following terms: 4.00/2.01 random(x0) 4.00/2.01 Cond_random(TRUE, x0) 4.00/2.01 rand(x0, x1) 4.00/2.01 Cond_rand(TRUE, x0, x1) 4.00/2.01 Cond_rand1(TRUE, x0, x1) 4.00/2.01 id_inc(w(x0)) 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (3) UsableRulesProof (EQUIVALENT) 4.00/2.01 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. 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (4) 4.00/2.01 Obligation: 4.00/2.01 IDP problem: 4.00/2.01 The following function symbols are pre-defined: 4.00/2.01 <<< 4.00/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.00/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.00/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.00/2.01 / ~ Div: (Integer, Integer) -> Integer 4.00/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.00/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.00/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.00/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.00/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.00/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.00/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.00/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.00/2.01 + ~ Add: (Integer, Integer) -> Integer 4.00/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.00/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.00/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.00/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.00/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.00/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.00/2.01 >>> 4.00/2.01 4.00/2.01 4.00/2.01 The following domains are used: 4.00/2.01 Integer 4.00/2.01 4.00/2.01 The ITRS R consists of the following rules: 4.00/2.01 id_inc(w(x)) -> w(x) 4.00/2.01 id_inc(w(x)) -> w(x + 1) 4.00/2.01 4.00/2.01 The integer pair graph contains the following rules and edges: 4.00/2.01 (0): RANDOM(x[0]) -> COND_RANDOM(x[0] >= 0, x[0]) 4.00/2.01 (1): COND_RANDOM(TRUE, x[1]) -> RAND(x[1], w(0)) 4.00/2.01 (2): RAND(x[2], y[2]) -> COND_RAND(x[2] = 0, x[2], y[2]) 4.00/2.01 (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) 4.00/2.01 (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) 4.00/2.01 (5): COND_RAND1(TRUE, x[5], y[5]) -> ID_INC(y[5]) 4.00/2.01 4.00/2.01 (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) 4.00/2.01 (1) -> (2), if (x[1] ->^* x[2] & w(0) ->^* y[2]) 4.00/2.01 (1) -> (3), if (x[1] ->^* x[3] & w(0) ->^* y[3]) 4.00/2.01 (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) 4.00/2.01 (3) -> (5), if (x[3] > 0 & x[3] ->^* x[5] & y[3] ->^* y[5]) 4.00/2.01 (4) -> (2), if (x[4] - 1 ->^* x[2] & id_inc(y[4]) ->^* y[2]) 4.00/2.01 (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) 4.00/2.01 4.00/2.01 The set Q consists of the following terms: 4.00/2.01 random(x0) 4.00/2.01 Cond_random(TRUE, x0) 4.00/2.01 rand(x0, x1) 4.00/2.01 Cond_rand(TRUE, x0, x1) 4.00/2.01 Cond_rand1(TRUE, x0, x1) 4.00/2.01 id_inc(w(x0)) 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (5) IDependencyGraphProof (EQUIVALENT) 4.00/2.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (6) 4.00/2.01 Obligation: 4.00/2.01 IDP problem: 4.00/2.01 The following function symbols are pre-defined: 4.00/2.01 <<< 4.00/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.00/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.00/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.00/2.01 / ~ Div: (Integer, Integer) -> Integer 4.00/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.00/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.00/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.00/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.00/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.00/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.00/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.00/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.00/2.01 + ~ Add: (Integer, Integer) -> Integer 4.00/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.00/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.00/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.00/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.00/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.00/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.00/2.01 >>> 4.00/2.01 4.00/2.01 4.00/2.01 The following domains are used: 4.00/2.01 Integer 4.00/2.01 4.00/2.01 The ITRS R consists of the following rules: 4.00/2.01 id_inc(w(x)) -> w(x) 4.00/2.01 id_inc(w(x)) -> w(x + 1) 4.00/2.01 4.00/2.01 The integer pair graph contains the following rules and edges: 4.00/2.01 (4): COND_RAND1(TRUE, x[4], y[4]) -> RAND(x[4] - 1, id_inc(y[4])) 4.00/2.01 (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) 4.00/2.01 4.00/2.01 (4) -> (3), if (x[4] - 1 ->^* x[3] & id_inc(y[4]) ->^* y[3]) 4.00/2.01 (3) -> (4), if (x[3] > 0 & x[3] ->^* x[4] & y[3] ->^* y[4]) 4.00/2.01 4.00/2.01 The set Q consists of the following terms: 4.00/2.01 random(x0) 4.00/2.01 Cond_random(TRUE, x0) 4.00/2.01 rand(x0, x1) 4.00/2.01 Cond_rand(TRUE, x0, x1) 4.00/2.01 Cond_rand1(TRUE, x0, x1) 4.00/2.01 id_inc(w(x0)) 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (7) IDPNonInfProof (SOUND) 4.00/2.01 Used the following options for this NonInfProof: 4.00/2.01 4.00/2.01 IDPGPoloSolver: 4.00/2.01 Range: [(-1,2)] 4.00/2.01 IsNat: false 4.00/2.01 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5597d194 4.00/2.01 Constraint Generator: NonInfConstraintGenerator: 4.00/2.01 PathGenerator: MetricPathGenerator: 4.00/2.01 Max Left Steps: 1 4.00/2.01 Max Right Steps: 1 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 The constraints were generated the following way: 4.00/2.01 4.00/2.01 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.00/2.01 4.00/2.01 Note that final constraints are written in bold face. 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 For Pair COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) the following chains were created: 4.00/2.01 *We consider the chain RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]), COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])), RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) which results in the following constraint: 4.00/2.01 4.00/2.01 (1) (>(x[3], 0)=TRUE & x[3]=x[4] & y[3]=y[4] & -(x[4], 1)=x[3]1 & id_inc(y[4])=y[3]1 ==> COND_RAND1(TRUE, x[4], y[4])_>=_NonInfC & COND_RAND1(TRUE, x[4], y[4])_>=_RAND(-(x[4], 1), id_inc(y[4])) & (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=)) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (1) using rules (III), (IV) which results in the following new constraint: 4.00/2.01 4.00/2.01 (2) (>(x[3], 0)=TRUE ==> COND_RAND1(TRUE, x[3], y[3])_>=_NonInfC & COND_RAND1(TRUE, x[3], y[3])_>=_RAND(-(x[3], 1), id_inc(y[3])) & (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=)) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.00/2.01 4.00/2.01 (3) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[3] >= 0 & [2 + (-1)bso_16] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.00/2.01 4.00/2.01 (4) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[3] >= 0 & [2 + (-1)bso_16] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.00/2.01 4.00/2.01 (5) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[3] >= 0 & [2 + (-1)bso_16] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.00/2.01 4.00/2.01 (6) (x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & 0 = 0 & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[3] >= 0 & 0 = 0 & [2 + (-1)bso_16] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 For Pair RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) the following chains were created: 4.00/2.01 *We consider the chain RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]), COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) which results in the following constraint: 4.00/2.01 4.00/2.01 (1) (>(x[3], 0)=TRUE & x[3]=x[4] & y[3]=y[4] ==> RAND(x[3], y[3])_>=_NonInfC & RAND(x[3], y[3])_>=_COND_RAND1(>(x[3], 0), x[3], y[3]) & (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=)) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (1) using rule (IV) which results in the following new constraint: 4.00/2.01 4.00/2.01 (2) (>(x[3], 0)=TRUE ==> RAND(x[3], y[3])_>=_NonInfC & RAND(x[3], y[3])_>=_COND_RAND1(>(x[3], 0), x[3], y[3]) & (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=)) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.00/2.01 4.00/2.01 (3) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[3] >= 0 & [(-1)bso_18] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.00/2.01 4.00/2.01 (4) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[3] >= 0 & [(-1)bso_18] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.00/2.01 4.00/2.01 (5) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[3] >= 0 & [(-1)bso_18] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.00/2.01 4.00/2.01 (6) (x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & 0 = 0 & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[3] >= 0 & [(-1)bso_18] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 To summarize, we get the following constraints P__>=_ for the following pairs. 4.00/2.01 4.00/2.01 *COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) 4.00/2.01 4.00/2.01 *(x[3] + [-1] >= 0 ==> (U^Increasing(RAND(-(x[4], 1), id_inc(y[4]))), >=) & 0 = 0 & [(-1)bni_15 + (-1)Bound*bni_15] + [(2)bni_15]x[3] >= 0 & 0 = 0 & [2 + (-1)bso_16] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 *RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) 4.00/2.01 4.00/2.01 *(x[3] + [-1] >= 0 ==> (U^Increasing(COND_RAND1(>(x[3], 0), x[3], y[3])), >=) & 0 = 0 & [(-1)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]x[3] >= 0 & [(-1)bso_18] >= 0) 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 4.00/2.01 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. 4.00/2.01 4.00/2.01 Using the following integer polynomial ordering the resulting constraints can be solved 4.00/2.01 4.00/2.01 Polynomial interpretation over integers[POLO]: 4.00/2.01 4.00/2.01 POL(TRUE) = [2] 4.00/2.01 POL(FALSE) = 0 4.00/2.01 POL(id_inc(x_1)) = [1] + [-1]x_1 4.00/2.01 POL(w(x_1)) = [2] + [2]x_1 4.00/2.01 POL(+(x_1, x_2)) = x_1 + x_2 4.00/2.01 POL(1) = [1] 4.00/2.01 POL(COND_RAND1(x_1, x_2, x_3)) = [-1] + [2]x_2 4.00/2.01 POL(RAND(x_1, x_2)) = [-1] + [2]x_1 4.00/2.01 POL(-(x_1, x_2)) = x_1 + [-1]x_2 4.00/2.01 POL(>(x_1, x_2)) = [-1] 4.00/2.01 POL(0) = 0 4.00/2.01 4.00/2.01 4.00/2.01 The following pairs are in P_>: 4.00/2.01 4.00/2.01 4.00/2.01 COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) 4.00/2.01 4.00/2.01 4.00/2.01 The following pairs are in P_bound: 4.00/2.01 4.00/2.01 4.00/2.01 COND_RAND1(TRUE, x[4], y[4]) -> RAND(-(x[4], 1), id_inc(y[4])) 4.00/2.01 RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) 4.00/2.01 4.00/2.01 4.00/2.01 The following pairs are in P_>=: 4.00/2.01 4.00/2.01 4.00/2.01 RAND(x[3], y[3]) -> COND_RAND1(>(x[3], 0), x[3], y[3]) 4.00/2.01 4.00/2.01 4.00/2.01 There are no usable rules. 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (8) 4.00/2.01 Obligation: 4.00/2.01 IDP problem: 4.00/2.01 The following function symbols are pre-defined: 4.00/2.01 <<< 4.00/2.01 & ~ Bwand: (Integer, Integer) -> Integer 4.00/2.01 >= ~ Ge: (Integer, Integer) -> Boolean 4.00/2.01 | ~ Bwor: (Integer, Integer) -> Integer 4.00/2.01 / ~ Div: (Integer, Integer) -> Integer 4.00/2.01 != ~ Neq: (Integer, Integer) -> Boolean 4.00/2.01 && ~ Land: (Boolean, Boolean) -> Boolean 4.00/2.01 ! ~ Lnot: (Boolean) -> Boolean 4.00/2.01 = ~ Eq: (Integer, Integer) -> Boolean 4.00/2.01 <= ~ Le: (Integer, Integer) -> Boolean 4.00/2.01 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.00/2.01 % ~ Mod: (Integer, Integer) -> Integer 4.00/2.01 > ~ Gt: (Integer, Integer) -> Boolean 4.00/2.01 + ~ Add: (Integer, Integer) -> Integer 4.00/2.01 -1 ~ UnaryMinus: (Integer) -> Integer 4.00/2.01 < ~ Lt: (Integer, Integer) -> Boolean 4.00/2.01 || ~ Lor: (Boolean, Boolean) -> Boolean 4.00/2.01 - ~ Sub: (Integer, Integer) -> Integer 4.00/2.01 ~ ~ Bwnot: (Integer) -> Integer 4.00/2.01 * ~ Mul: (Integer, Integer) -> Integer 4.00/2.01 >>> 4.00/2.01 4.00/2.01 4.00/2.01 The following domains are used: 4.00/2.01 Integer 4.00/2.01 4.00/2.01 The ITRS R consists of the following rules: 4.00/2.01 id_inc(w(x)) -> w(x) 4.00/2.01 id_inc(w(x)) -> w(x + 1) 4.00/2.01 4.00/2.01 The integer pair graph contains the following rules and edges: 4.00/2.01 (3): RAND(x[3], y[3]) -> COND_RAND1(x[3] > 0, x[3], y[3]) 4.00/2.01 4.00/2.01 4.00/2.01 The set Q consists of the following terms: 4.00/2.01 random(x0) 4.00/2.01 Cond_random(TRUE, x0) 4.00/2.01 rand(x0, x1) 4.00/2.01 Cond_rand(TRUE, x0, x1) 4.00/2.01 Cond_rand1(TRUE, x0, x1) 4.00/2.01 id_inc(w(x0)) 4.00/2.01 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (9) IDependencyGraphProof (EQUIVALENT) 4.00/2.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.00/2.01 ---------------------------------------- 4.00/2.01 4.00/2.01 (10) 4.00/2.01 TRUE 4.00/2.03 EOF