/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDependencyGraphProof [EQUIVALENT, 0 ms] (6) IDP (7) IDPNonInfProof [SOUND, 155 ms] (8) IDP (9) IDependencyGraphProof [EQUIVALENT, 0 ms] (10) 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: minus(x, x) -> 0 minus(x, y) -> cond(min(x, y), x, y) cond(y, x, y) -> 1 + minus(x, y + 1) min(u, v) -> if(u < v, u, v) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The set Q consists of the following terms: minus(x0, x1) cond(x0, x1, x0) min(x0, x1) if(TRUE, x0, x1) if(FALSE, x0, x1) ---------------------------------------- (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: Integer The ITRS R consists of the following rules: minus(x, x) -> 0 minus(x, y) -> cond(min(x, y), x, y) cond(y, x, y) -> 1 + minus(x, y + 1) min(u, v) -> if(u < v, u, v) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The integer pair graph contains the following rules and edges: (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) (3): MIN(u[3], v[3]) -> IF(u[3] < v[3], u[3], v[3]) (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) (1) -> (3), if (x[1] ->^* u[3] & y[1] ->^* v[3]) (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) The set Q consists of the following terms: minus(x0, x1) cond(x0, x1, x0) min(x0, x1) if(TRUE, x0, x1) if(FALSE, x0, x1) ---------------------------------------- (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: Integer The ITRS R consists of the following rules: min(u, v) -> if(u < v, u, v) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The integer pair graph contains the following rules and edges: (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) (3): MIN(u[3], v[3]) -> IF(u[3] < v[3], u[3], v[3]) (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) (1) -> (3), if (x[1] ->^* u[3] & y[1] ->^* v[3]) (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) The set Q consists of the following terms: minus(x0, x1) cond(x0, x1, x0) min(x0, x1) if(TRUE, x0, x1) if(FALSE, x0, x1) ---------------------------------------- (5) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (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: Integer The ITRS R consists of the following rules: min(u, v) -> if(u < v, u, v) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The integer pair graph contains the following rules and edges: (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) The set Q consists of the following terms: minus(x0, x1) cond(x0, x1, x0) min(x0, x1) if(TRUE, x0, x1) if(FALSE, x0, x1) ---------------------------------------- (7) 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@e2449a2 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 COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) the following chains were created: *We consider the chain MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]), COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)), MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) which results in the following constraint: (1) (min(x[0], y[0])=y[2] & x[0]=x[2] & y[0]=y[2] & x[2]=x[0]1 & +(y[2], 1)=y[0]1 ==> COND(y[2], x[2], y[2])_>=_NonInfC & COND(y[2], x[2], y[2])_>=_MINUS(x[2], +(y[2], 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (1) using rules (III), (IV), (VII), (REWRITING) which results in the following new constraint: (2) (<(x[0], y[0])=x0 & if(x0, x[0], y[0])=y[0] ==> COND(y[0], x[0], y[0])_>=_NonInfC & COND(y[0], x[0], y[0])_>=_MINUS(x[0], +(y[0], 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on if(x0, x[0], y[0])=y[0] which results in the following new constraints: (3) (x2=x1 & <(x2, x1)=TRUE ==> COND(x1, x2, x1)_>=_NonInfC & COND(x1, x2, x1)_>=_MINUS(x2, +(x1, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) (4) (x3=x3 & <(x4, x3)=FALSE ==> COND(x3, x4, x3)_>=_NonInfC & COND(x3, x4, x3)_>=_MINUS(x4, +(x3, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (3) using rule (III) which results in the following new constraint: (5) (<(x1, x1)=TRUE ==> COND(x1, x1, x1)_>=_NonInfC & COND(x1, x1, x1)_>=_MINUS(x1, +(x1, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (4) using rule (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint: (6) (<(x4, x3)=FALSE ==> COND(x3, x4, x3)_>=_NonInfC & COND(x3, x4, x3)_>=_MINUS(x4, +(x3, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (5) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (7) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (8) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (9) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (10) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (11) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (12) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (13) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & 0 = 0 & [(2)bni_18 + (-1)Bound*bni_18] >= 0 & 0 = 0 & [1 + (-1)bso_19] >= 0) We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (14) (x4 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) We solved constraint (13) using rule (IDP_SMT_SPLIT).We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (15) (x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) (16) (x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) For Pair MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) the following chains were created: *We consider the chain COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)), MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]), COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) which results in the following constraint: (1) (x[2]=x[0] & +(y[2], 1)=y[0] & min(x[0], y[0])=y[2]1 & x[0]=x[2]1 & y[0]=y[2]1 ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_COND(min(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (1) using rules (III), (IV), (VII), (REWRITING) which results in the following new constraint: (2) (<(x[0], +(y[2], 1))=x5 & +(y[2], 1)=x6 & if(x5, x[0], x6)=+(y[2], 1) ==> MINUS(x[0], +(y[2], 1))_>=_NonInfC & MINUS(x[0], +(y[2], 1))_>=_COND(min(x[0], +(y[2], 1)), x[0], +(y[2], 1)) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on if(x5, x[0], x6)=+(y[2], 1) which results in the following new constraints: (3) (x8=+(y[2], 1) & <(x8, +(y[2], 1))=TRUE & +(y[2], 1)=x7 ==> MINUS(x8, +(y[2], 1))_>=_NonInfC & MINUS(x8, +(y[2], 1))_>=_COND(min(x8, +(y[2], 1)), x8, +(y[2], 1)) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) (4) (x9=+(y[2], 1) & <(x10, +(y[2], 1))=FALSE & +(y[2], 1)=x9 ==> MINUS(x10, +(y[2], 1))_>=_NonInfC & MINUS(x10, +(y[2], 1))_>=_COND(min(x10, +(y[2], 1)), x10, +(y[2], 1)) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (3) using rules (III), (IV) which results in the following new constraint: (5) (<(+(y[2], 1), +(y[2], 1))=TRUE ==> MINUS(+(y[2], 1), +(y[2], 1))_>=_NonInfC & MINUS(+(y[2], 1), +(y[2], 1))_>=_COND(min(+(y[2], 1), +(y[2], 1)), +(y[2], 1), +(y[2], 1)) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (4) using rules (III), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint: (6) (<(x10, +(y[2], 1))=FALSE ==> MINUS(x10, +(y[2], 1))_>=_NonInfC & MINUS(x10, +(y[2], 1))_>=_COND(min(x10, +(y[2], 1)), x10, +(y[2], 1)) & (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (5) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (7) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (8) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (9) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (10) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (11) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (12) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (13) (x10 >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: (14) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & 0 = 0 & [(2)bni_20 + (-1)Bound*bni_20] >= 0 & 0 = 0 & [(-1)bso_21] >= 0) We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (15) (x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) (16) (x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) We solved constraint (14) using rule (IDP_SMT_SPLIT). To summarize, we get the following constraints P__>=_ for the following pairs. *COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) *(x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) *(x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) *MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) *(x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) *(x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 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) = [3] POL(FALSE) = [3] POL(min(x_1, x_2)) = [1] + [-1]x_2 + [-1]x_1 POL(if(x_1, x_2, x_3)) = [2] + [-1]x_3 + [-1]x_2 POL(<(x_1, x_2)) = [2] POL(COND(x_1, x_2, x_3)) = [2] + [-1]x_3 + x_2 POL(MINUS(x_1, x_2)) = [2] + [-1]x_2 + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] The following pairs are in P_>: COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) The following pairs are in P_bound: COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) The following pairs are in P_>=: MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: if(<(u, v), u, v)^1 -> min(u, v)^1 ---------------------------------------- (8) 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: Integer The ITRS R consists of the following rules: min(u, v) -> if(u < v, u, v) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The integer pair graph contains the following rules and edges: (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) The set Q consists of the following terms: minus(x0, x1) cond(x0, x1, x0) min(x0, x1) if(TRUE, x0, x1) if(FALSE, x0, x1) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (10) TRUE