4.11/2.67 YES 4.21/2.68 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.21/2.68 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.21/2.68 4.21/2.68 4.21/2.68 Termination of the given ITRS could be proven: 4.21/2.68 4.21/2.68 (0) ITRS 4.21/2.68 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.21/2.68 (2) IDP 4.21/2.68 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.21/2.68 (4) IDP 4.21/2.68 (5) IDPNonInfProof [SOUND, 201 ms] 4.21/2.68 (6) IDP 4.21/2.68 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.21/2.68 (8) TRUE 4.21/2.68 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (0) 4.21/2.68 Obligation: 4.21/2.68 ITRS problem: 4.21/2.68 4.21/2.68 The following function symbols are pre-defined: 4.21/2.68 <<< 4.21/2.68 & ~ Bwand: (Integer, Integer) -> Integer 4.21/2.68 >= ~ Ge: (Integer, Integer) -> Boolean 4.21/2.68 | ~ Bwor: (Integer, Integer) -> Integer 4.21/2.68 / ~ Div: (Integer, Integer) -> Integer 4.21/2.68 != ~ Neq: (Integer, Integer) -> Boolean 4.21/2.68 && ~ Land: (Boolean, Boolean) -> Boolean 4.21/2.68 ! ~ Lnot: (Boolean) -> Boolean 4.21/2.68 = ~ Eq: (Integer, Integer) -> Boolean 4.21/2.68 <= ~ Le: (Integer, Integer) -> Boolean 4.21/2.68 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.21/2.68 % ~ Mod: (Integer, Integer) -> Integer 4.21/2.68 + ~ Add: (Integer, Integer) -> Integer 4.21/2.68 > ~ Gt: (Integer, Integer) -> Boolean 4.21/2.68 -1 ~ UnaryMinus: (Integer) -> Integer 4.21/2.68 < ~ Lt: (Integer, Integer) -> Boolean 4.21/2.68 || ~ Lor: (Boolean, Boolean) -> Boolean 4.21/2.68 - ~ Sub: (Integer, Integer) -> Integer 4.21/2.68 ~ ~ Bwnot: (Integer) -> Integer 4.21/2.68 * ~ Mul: (Integer, Integer) -> Integer 4.21/2.68 >>> 4.21/2.68 4.21/2.68 The TRS R consists of the following rules: 4.21/2.68 eval(x, y) -> Cond_eval(x > 0 && x > y, x, y) 4.21/2.68 Cond_eval(TRUE, x, y) -> eval(x, x + y) 4.21/2.68 The set Q consists of the following terms: 4.21/2.68 eval(x0, x1) 4.21/2.68 Cond_eval(TRUE, x0, x1) 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (1) ITRStoIDPProof (EQUIVALENT) 4.21/2.68 Added dependency pairs 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (2) 4.21/2.68 Obligation: 4.21/2.68 IDP problem: 4.21/2.68 The following function symbols are pre-defined: 4.21/2.68 <<< 4.21/2.68 & ~ Bwand: (Integer, Integer) -> Integer 4.21/2.68 >= ~ Ge: (Integer, Integer) -> Boolean 4.21/2.68 | ~ Bwor: (Integer, Integer) -> Integer 4.21/2.68 / ~ Div: (Integer, Integer) -> Integer 4.21/2.68 != ~ Neq: (Integer, Integer) -> Boolean 4.21/2.68 && ~ Land: (Boolean, Boolean) -> Boolean 4.21/2.68 ! ~ Lnot: (Boolean) -> Boolean 4.21/2.68 = ~ Eq: (Integer, Integer) -> Boolean 4.21/2.68 <= ~ Le: (Integer, Integer) -> Boolean 4.21/2.68 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.21/2.68 % ~ Mod: (Integer, Integer) -> Integer 4.21/2.68 + ~ Add: (Integer, Integer) -> Integer 4.21/2.68 > ~ Gt: (Integer, Integer) -> Boolean 4.21/2.68 -1 ~ UnaryMinus: (Integer) -> Integer 4.21/2.68 < ~ Lt: (Integer, Integer) -> Boolean 4.21/2.68 || ~ Lor: (Boolean, Boolean) -> Boolean 4.21/2.68 - ~ Sub: (Integer, Integer) -> Integer 4.21/2.68 ~ ~ Bwnot: (Integer) -> Integer 4.21/2.68 * ~ Mul: (Integer, Integer) -> Integer 4.21/2.68 >>> 4.21/2.68 4.21/2.68 4.21/2.68 The following domains are used: 4.21/2.68 Boolean, Integer 4.21/2.68 4.21/2.68 The ITRS R consists of the following rules: 4.21/2.68 eval(x, y) -> Cond_eval(x > 0 && x > y, x, y) 4.21/2.68 Cond_eval(TRUE, x, y) -> eval(x, x + y) 4.21/2.68 4.21/2.68 The integer pair graph contains the following rules and edges: 4.21/2.68 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && x[0] > y[0], x[0], y[0]) 4.21/2.68 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], x[1] + y[1]) 4.21/2.68 4.21/2.68 (0) -> (1), if (x[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 4.21/2.68 (1) -> (0), if (x[1] ->^* x[0] & x[1] + y[1] ->^* y[0]) 4.21/2.68 4.21/2.68 The set Q consists of the following terms: 4.21/2.68 eval(x0, x1) 4.21/2.68 Cond_eval(TRUE, x0, x1) 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (3) UsableRulesProof (EQUIVALENT) 4.21/2.68 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. 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (4) 4.21/2.68 Obligation: 4.21/2.68 IDP problem: 4.21/2.68 The following function symbols are pre-defined: 4.21/2.68 <<< 4.21/2.68 & ~ Bwand: (Integer, Integer) -> Integer 4.21/2.68 >= ~ Ge: (Integer, Integer) -> Boolean 4.21/2.68 | ~ Bwor: (Integer, Integer) -> Integer 4.21/2.68 / ~ Div: (Integer, Integer) -> Integer 4.21/2.68 != ~ Neq: (Integer, Integer) -> Boolean 4.21/2.68 && ~ Land: (Boolean, Boolean) -> Boolean 4.21/2.68 ! ~ Lnot: (Boolean) -> Boolean 4.21/2.68 = ~ Eq: (Integer, Integer) -> Boolean 4.21/2.68 <= ~ Le: (Integer, Integer) -> Boolean 4.21/2.68 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.21/2.68 % ~ Mod: (Integer, Integer) -> Integer 4.21/2.68 + ~ Add: (Integer, Integer) -> Integer 4.21/2.68 > ~ Gt: (Integer, Integer) -> Boolean 4.21/2.68 -1 ~ UnaryMinus: (Integer) -> Integer 4.21/2.68 < ~ Lt: (Integer, Integer) -> Boolean 4.21/2.68 || ~ Lor: (Boolean, Boolean) -> Boolean 4.21/2.68 - ~ Sub: (Integer, Integer) -> Integer 4.21/2.68 ~ ~ Bwnot: (Integer) -> Integer 4.21/2.68 * ~ Mul: (Integer, Integer) -> Integer 4.21/2.68 >>> 4.21/2.68 4.21/2.68 4.21/2.68 The following domains are used: 4.21/2.68 Boolean, Integer 4.21/2.68 4.21/2.68 R is empty. 4.21/2.68 4.21/2.68 The integer pair graph contains the following rules and edges: 4.21/2.68 (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > 0 && x[0] > y[0], x[0], y[0]) 4.21/2.68 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], x[1] + y[1]) 4.21/2.68 4.21/2.68 (0) -> (1), if (x[0] > 0 && x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) 4.21/2.68 (1) -> (0), if (x[1] ->^* x[0] & x[1] + y[1] ->^* y[0]) 4.21/2.68 4.21/2.68 The set Q consists of the following terms: 4.21/2.68 eval(x0, x1) 4.21/2.68 Cond_eval(TRUE, x0, x1) 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (5) IDPNonInfProof (SOUND) 4.21/2.68 Used the following options for this NonInfProof: 4.21/2.68 4.21/2.68 IDPGPoloSolver: 4.21/2.68 Range: [(-1,2)] 4.21/2.68 IsNat: false 4.21/2.68 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@162d394 4.21/2.68 Constraint Generator: NonInfConstraintGenerator: 4.21/2.68 PathGenerator: MetricPathGenerator: 4.21/2.68 Max Left Steps: 1 4.21/2.68 Max Right Steps: 1 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 The constraints were generated the following way: 4.21/2.68 4.21/2.68 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.21/2.68 4.21/2.68 Note that final constraints are written in bold face. 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 For Pair EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >(x, y)), x, y) the following chains were created: 4.21/2.68 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])) which results in the following constraint: 4.21/2.68 4.21/2.68 (1) (&&(>(x[0], 0), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=)) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.21/2.68 4.21/2.68 (2) (>(x[0], 0)=TRUE & >(x[0], y[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) & (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=)) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.21/2.68 4.21/2.68 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.21/2.68 4.21/2.68 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.21/2.68 4.21/2.68 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.21/2.68 4.21/2.68 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 For Pair COND_EVAL(TRUE, x, y) -> EVAL(x, +(x, y)) the following chains were created: 4.21/2.68 *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])), EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) which results in the following constraint: 4.21/2.68 4.21/2.68 (1) (&&(>(x[0], 0), >(x[0], y[0]))=TRUE & x[0]=x[1] & y[0]=y[1] & x[1]=x[0]1 & +(x[1], y[1])=y[0]1 ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(x[1], +(x[1], y[1])) & (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=)) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 4.21/2.68 4.21/2.68 (2) (>(x[0], 0)=TRUE & >(x[0], y[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(x[0], +(x[0], y[0])) & (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=)) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.21/2.68 4.21/2.68 (3) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.21/2.68 4.21/2.68 (4) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.21/2.68 4.21/2.68 (5) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.21/2.68 4.21/2.68 (6) (x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 (7) (x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 To summarize, we get the following constraints P__>=_ for the following pairs. 4.21/2.68 4.21/2.68 *EVAL(x, y) -> COND_EVAL(&&(>(x, 0), >(x, y)), x, y) 4.21/2.68 4.21/2.68 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0])), >=) & [(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]y[0] + [bni_12]x[0] >= 0 & [(-1)bso_13] + x[0] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 *COND_EVAL(TRUE, x, y) -> EVAL(x, +(x, y)) 4.21/2.68 4.21/2.68 *(x[0] + [-1] >= 0 & x[0] + [-1] + y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 *(x[0] + [-1] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(x[1], y[1]))), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] >= 0 & [(-1)bso_15] >= 0) 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 4.21/2.68 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. 4.21/2.68 4.21/2.68 Using the following integer polynomial ordering the resulting constraints can be solved 4.21/2.68 4.21/2.68 Polynomial interpretation over integers[POLO]: 4.21/2.68 4.21/2.68 POL(TRUE) = 0 4.21/2.68 POL(FALSE) = [2] 4.21/2.68 POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 4.21/2.68 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_1 4.21/2.68 POL(&&(x_1, x_2)) = 0 4.21/2.68 POL(>(x_1, x_2)) = [-1] 4.21/2.68 POL(0) = 0 4.21/2.68 POL(+(x_1, x_2)) = x_1 + x_2 4.21/2.68 4.21/2.68 4.21/2.68 The following pairs are in P_>: 4.21/2.68 4.21/2.68 4.21/2.68 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) 4.21/2.68 4.21/2.68 4.21/2.68 The following pairs are in P_bound: 4.21/2.68 4.21/2.68 4.21/2.68 EVAL(x[0], y[0]) -> COND_EVAL(&&(>(x[0], 0), >(x[0], y[0])), x[0], y[0]) 4.21/2.68 4.21/2.68 4.21/2.68 The following pairs are in P_>=: 4.21/2.68 4.21/2.68 4.21/2.68 COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(x[1], y[1])) 4.21/2.68 4.21/2.68 4.21/2.68 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.21/2.68 4.21/2.68 TRUE^1 -> &&(TRUE, TRUE)^1 4.21/2.68 FALSE^1 -> &&(TRUE, FALSE)^1 4.21/2.68 FALSE^1 -> &&(FALSE, TRUE)^1 4.21/2.68 FALSE^1 -> &&(FALSE, FALSE)^1 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (6) 4.21/2.68 Obligation: 4.21/2.68 IDP problem: 4.21/2.68 The following function symbols are pre-defined: 4.21/2.68 <<< 4.21/2.68 & ~ Bwand: (Integer, Integer) -> Integer 4.21/2.68 >= ~ Ge: (Integer, Integer) -> Boolean 4.21/2.68 | ~ Bwor: (Integer, Integer) -> Integer 4.21/2.68 / ~ Div: (Integer, Integer) -> Integer 4.21/2.68 != ~ Neq: (Integer, Integer) -> Boolean 4.21/2.68 && ~ Land: (Boolean, Boolean) -> Boolean 4.21/2.68 ! ~ Lnot: (Boolean) -> Boolean 4.21/2.68 = ~ Eq: (Integer, Integer) -> Boolean 4.21/2.68 <= ~ Le: (Integer, Integer) -> Boolean 4.21/2.68 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.21/2.68 % ~ Mod: (Integer, Integer) -> Integer 4.21/2.68 + ~ Add: (Integer, Integer) -> Integer 4.21/2.68 > ~ Gt: (Integer, Integer) -> Boolean 4.21/2.68 -1 ~ UnaryMinus: (Integer) -> Integer 4.21/2.68 < ~ Lt: (Integer, Integer) -> Boolean 4.21/2.68 || ~ Lor: (Boolean, Boolean) -> Boolean 4.21/2.68 - ~ Sub: (Integer, Integer) -> Integer 4.21/2.68 ~ ~ Bwnot: (Integer) -> Integer 4.21/2.68 * ~ Mul: (Integer, Integer) -> Integer 4.21/2.68 >>> 4.21/2.68 4.21/2.68 4.21/2.68 The following domains are used: 4.21/2.68 Integer 4.21/2.68 4.21/2.68 R is empty. 4.21/2.68 4.21/2.68 The integer pair graph contains the following rules and edges: 4.21/2.68 (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], x[1] + y[1]) 4.21/2.68 4.21/2.68 4.21/2.68 The set Q consists of the following terms: 4.21/2.68 eval(x0, x1) 4.21/2.68 Cond_eval(TRUE, x0, x1) 4.21/2.68 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (7) IDependencyGraphProof (EQUIVALENT) 4.21/2.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.21/2.68 ---------------------------------------- 4.21/2.68 4.21/2.68 (8) 4.21/2.68 TRUE 4.21/2.70 EOF