/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.itrs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 33 ms] (6) IDP (7) PisEmptyProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ Sub: (Integer, Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval(x) -> Cond_eval(x >= 0, x) Cond_eval(TRUE, x) -> eval(-(2) * x + 10) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ Sub: (Integer, Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer The ITRS R consists of the following rules: eval(x) -> Cond_eval(x >= 0, x) Cond_eval(TRUE, x) -> eval(-(2) * x + 10) The integer pair graph contains the following rules and edges: (0): EVAL(x[0]) -> COND_EVAL(x[0] >= 0, x[0]) (1): COND_EVAL(TRUE, x[1]) -> EVAL(-(2) * x[1] + 10) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) (1) -> (0), if (-(2) * x[1] + 10 ->^* x[0]) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) 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) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ Sub: (Integer, Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0]) -> COND_EVAL(x[0] >= 0, x[0]) (1): COND_EVAL(TRUE, x[1]) -> EVAL(-(2) * x[1] + 10) (0) -> (1), if (x[0] >= 0 & x[0] ->^* x[1]) (1) -> (0), if (-(2) * x[1] + 10 ->^* x[0]) The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@670440af Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 8 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x) -> COND_EVAL(>=(x, 0), x) the following chains were created: *We consider the chain EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)) which results in the following constraint: (1) (>=(x[0], 0)=TRUE & x[0]=x[1] ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(>=(x[0], 0), x[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>=(x[0], 0)=TRUE ==> EVAL(x[0])_>=_NonInfC & EVAL(x[0])_>=_COND_EVAL(>=(x[0], 0), x[0]) & (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=) & [bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x[0] >= 0 & [2 + (-1)bso_10] + [2]x[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=) & [bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x[0] >= 0 & [2 + (-1)bso_10] + [2]x[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=) & [bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x[0] >= 0 & [2 + (-1)bso_10] + [2]x[0] >= 0) For Pair COND_EVAL(TRUE, x) -> EVAL(+(*(-(2), x), 10)) the following chains were created: *We consider the chain COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)), EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)), EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)), EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)), EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]), COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)), EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]) which results in the following constraint: (1) (+(*(-(2), x[1]), 10)=x[0] & >=(x[0], 0)=TRUE & x[0]=x[1]1 & +(*(-(2), x[1]1), 10)=x[0]1 & >=(x[0]1, 0)=TRUE & x[0]1=x[1]2 & +(*(-(2), x[1]2), 10)=x[0]2 & >=(x[0]2, 0)=TRUE & x[0]2=x[1]3 & +(*(-(2), x[1]3), 10)=x[0]3 & >=(x[0]3, 0)=TRUE & x[0]3=x[1]4 & +(*(-(2), x[1]4), 10)=x[0]4 ==> COND_EVAL(TRUE, x[1]4)_>=_NonInfC & COND_EVAL(TRUE, x[1]4)_>=_EVAL(+(*(-(2), x[1]4), 10)) & (U^Increasing(EVAL(+(*(-(2), x[1]4), 10))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_CONSTANT_FOLD) which results in the following new constraint: (2) (>=(+(*(-2, x[1]), 10), 0)=TRUE & >=(+(*(-2, +(*(-2, x[1]), 10)), 10), 0)=TRUE & >=(+(*(-2, +(*(-2, +(*(-2, x[1]), 10)), 10)), 10), 0)=TRUE & >=(+(*(-2, +(*(-2, +(*(-2, +(*(-2, x[1]), 10)), 10)), 10)), 10), 0)=TRUE ==> COND_EVAL(TRUE, +(*(-2, +(*(-2, +(*(-2, +(*(-2, x[1]), 10)), 10)), 10)), 10))_>=_NonInfC & COND_EVAL(TRUE, +(*(-2, +(*(-2, +(*(-2, +(*(-2, x[1]), 10)), 10)), 10)), 10))_>=_EVAL(+(*(-(2), +(*(-2, +(*(-2, +(*(-2, +(*(-2, x[1]), 10)), 10)), 10)), 10)), 10)) & (U^Increasing(EVAL(+(*(-(2), x[1]4), 10))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-2]x[1] + [10] >= 0 & [4]x[1] + [-10] >= 0 & [-8]x[1] + [30] >= 0 & [16]x[1] + [-50] >= 0 ==> (U^Increasing(EVAL(+(*(-(2), x[1]4), 10))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] >= 0 & [-222 + (-1)bso_12] + [64]x[1] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-2]x[1] + [10] >= 0 & [4]x[1] + [-10] >= 0 & [-8]x[1] + [30] >= 0 & [16]x[1] + [-50] >= 0 ==> (U^Increasing(EVAL(+(*(-(2), x[1]4), 10))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] >= 0 & [-222 + (-1)bso_12] + [64]x[1] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-2]x[1] + [10] >= 0 & [4]x[1] + [-10] >= 0 & [-8]x[1] + [30] >= 0 & [16]x[1] + [-50] >= 0 ==> (U^Increasing(EVAL(+(*(-(2), x[1]4), 10))), >=) & [(-1)bni_11 + (-1)Bound*bni_11] >= 0 & [-222 + (-1)bso_12] + [64]x[1] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x) -> COND_EVAL(>=(x, 0), x) *(x[0] >= 0 ==> (U^Increasing(COND_EVAL(>=(x[0], 0), x[0])), >=) & [bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x[0] >= 0 & [2 + (-1)bso_10] + [2]x[0] >= 0) *COND_EVAL(TRUE, x) -> EVAL(+(*(-(2), x), 10)) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL(x_1)) = [1] + [2]x_1 POL(COND_EVAL(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [2] POL(0) = 0 POL(+(x_1, x_2)) = x_1 + x_2 POL(*(x_1, x_2)) = x_1*x_2 POL(-(x_1)) = [-1]x_1 POL(2) = [2] POL(10) = [10] POL(-2) = [-2] The following pairs are in P_>: EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]) COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)) The following pairs are in P_bound: EVAL(x[0]) -> COND_EVAL(>=(x[0], 0), x[0]) COND_EVAL(TRUE, x[1]) -> EVAL(+(*(-(2), x[1]), 10)) The following pairs are in P_>=: none There are no usable rules. ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ Sub: (Integer, Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ UnaryMinus: (Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: none R is empty. The integer pair graph is empty. The set Q consists of the following terms: eval(x0) Cond_eval(TRUE, x0) ---------------------------------------- (7) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (8) YES