/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 270 ms] (6) IDP (7) IDependencyGraphProof [EQUIVALENT, 0 ms] (8) TRUE ---------------------------------------- (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 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval(i, j, k) -> Cond_eval(i <= 100 && j <= k, i, j, k) Cond_eval(TRUE, i, j, k) -> eval(j, i + 1, k - 1) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) ---------------------------------------- (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 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer The ITRS R consists of the following rules: eval(i, j, k) -> Cond_eval(i <= 100 && j <= k, i, j, k) Cond_eval(TRUE, i, j, k) -> eval(j, i + 1, k - 1) The integer pair graph contains the following rules and edges: (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) (1): COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], i[1] + 1, k[1] - 1) (0) -> (1), if (i[0] <= 100 && j[0] <= k[0] & i[0] ->^* i[1] & j[0] ->^* j[1] & k[0] ->^* k[1]) (1) -> (0), if (j[1] ->^* i[0] & i[1] + 1 ->^* j[0] & k[1] - 1 ->^* k[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) ---------------------------------------- (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 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) (1): COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], i[1] + 1, k[1] - 1) (0) -> (1), if (i[0] <= 100 && j[0] <= k[0] & i[0] ->^* i[1] & j[0] ->^* j[1] & k[0] ->^* k[1]) (1) -> (0), if (j[1] ->^* i[0] & i[1] + 1 ->^* j[0] & k[1] - 1 ->^* k[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) ---------------------------------------- (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@11353166 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 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(i, j, k) -> COND_EVAL(&&(<=(i, 100), <=(j, k)), i, j, k) the following chains were created: *We consider the chain EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]), COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) which results in the following constraint: (1) (&&(<=(i[0], 100), <=(j[0], k[0]))=TRUE & i[0]=i[1] & j[0]=j[1] & k[0]=k[1] ==> EVAL(i[0], j[0], k[0])_>=_NonInfC & EVAL(i[0], j[0], k[0])_>=_COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) & (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (<=(i[0], 100)=TRUE & <=(j[0], k[0])=TRUE ==> EVAL(i[0], j[0], k[0])_>=_NonInfC & EVAL(i[0], j[0], k[0])_>=_COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) & (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]j[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) ([100] + [-1]i[0] >= 0 & k[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) (8) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) (10) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (11) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) (12) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) For Pair COND_EVAL(TRUE, i, j, k) -> EVAL(j, +(i, 1), -(k, 1)) the following chains were created: *We consider the chain EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]), COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)), EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) which results in the following constraint: (1) (&&(<=(i[0], 100), <=(j[0], k[0]))=TRUE & i[0]=i[1] & j[0]=j[1] & k[0]=k[1] & j[1]=i[0]1 & +(i[1], 1)=j[0]1 & -(k[1], 1)=k[0]1 ==> COND_EVAL(TRUE, i[1], j[1], k[1])_>=_NonInfC & COND_EVAL(TRUE, i[1], j[1], k[1])_>=_EVAL(j[1], +(i[1], 1), -(k[1], 1)) & (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (<=(i[0], 100)=TRUE & <=(j[0], k[0])=TRUE ==> COND_EVAL(TRUE, i[0], j[0], k[0])_>=_NonInfC & COND_EVAL(TRUE, i[0], j[0], k[0])_>=_EVAL(j[0], +(i[0], 1), -(k[0], 1)) & (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([100] + [-1]i[0] >= 0 & k[0] + [-1]j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]j[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) ([100] + [-1]i[0] >= 0 & k[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) (8) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) (10) ([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (11) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) (12) ([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(i, j, k) -> COND_EVAL(&&(<=(i, 100), <=(j, k)), i, j, k) *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [(-1)bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0])), >=) & [(-1)Bound*bni_14] + [bni_14]k[0] + [bni_14]i[0] >= 0 & [(-1)bso_15] >= 0) *COND_EVAL(TRUE, i, j, k) -> EVAL(j, +(i, 1), -(k, 1)) *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) *([100] + [-1]i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [(-1)bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) *([100] + i[0] >= 0 & k[0] >= 0 & i[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(j[1], +(i[1], 1), -(k[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]k[0] + [bni_16]i[0] >= 0 & [1 + (-1)bso_17] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(EVAL(x_1, x_2, x_3)) = x_3 + [-1]x_2 + [-1]x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4)) = [-1] + x_4 + [-1]x_3 + [-1]x_2 + x_1 POL(&&(x_1, x_2)) = [1] POL(<=(x_1, x_2)) = [-1] POL(100) = [100] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 The following pairs are in P_>: COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) The following pairs are in P_bound: EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) COND_EVAL(TRUE, i[1], j[1], k[1]) -> EVAL(j[1], +(i[1], 1), -(k[1], 1)) The following pairs are in P_>=: EVAL(i[0], j[0], k[0]) -> COND_EVAL(&&(<=(i[0], 100), <=(j[0], k[0])), i[0], j[0], k[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 -> TRUE^1 &&(TRUE, FALSE)^1 -> FALSE^1 &&(FALSE, TRUE)^1 <-> FALSE^1 &&(FALSE, FALSE)^1 -> FALSE^1 ---------------------------------------- (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 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Boolean, Integer R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(i[0], j[0], k[0]) -> COND_EVAL(i[0] <= 100 && j[0] <= k[0], i[0], j[0], k[0]) The set Q consists of the following terms: eval(x0, x1, x2) Cond_eval(TRUE, x0, x1, x2) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (8) TRUE