/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 229 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, y) -> minusNat(y >= 0 && x = y + 1, x, y) minusNat(TRUE, x, y) -> minus(x, round(x)) round(x) -> if(x % 2 = 0, x, x + 1) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The set Q consists of the following terms: minus(x0, x1) minusNat(TRUE, x0, x1) round(x0) 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: Boolean, Integer The ITRS R consists of the following rules: minus(x, y) -> minusNat(y >= 0 && x = y + 1, x, y) minusNat(TRUE, x, y) -> minus(x, round(x)) round(x) -> if(x % 2 = 0, x, x + 1) 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]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) (2): MINUSNAT(TRUE, x[2], y[2]) -> ROUND(x[2]) (3): ROUND(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] + 1) (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) (0) -> (2), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[2] & y[0] ->^* y[2]) (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) (2) -> (3), if (x[2] ->^* x[3]) The set Q consists of the following terms: minus(x0, x1) minusNat(TRUE, x0, x1) round(x0) 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, Boolean The ITRS R consists of the following rules: round(x) -> if(x % 2 = 0, x, x + 1) 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]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) (2): MINUSNAT(TRUE, x[2], y[2]) -> ROUND(x[2]) (3): ROUND(x[3]) -> IF(x[3] % 2 = 0, x[3], x[3] + 1) (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) (0) -> (2), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[2] & y[0] ->^* y[2]) (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) (2) -> (3), if (x[2] ->^* x[3]) The set Q consists of the following terms: minus(x0, x1) minusNat(TRUE, x0, x1) round(x0) 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, Boolean The ITRS R consists of the following rules: round(x) -> if(x % 2 = 0, x, x + 1) if(TRUE, u, v) -> u if(FALSE, u, v) -> v The integer pair graph contains the following rules and edges: (1): MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) (0): MINUS(x[0], y[0]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) (1) -> (0), if (x[1] ->^* x[0] & round(x[1]) ->^* y[0]) (0) -> (1), if (y[0] >= 0 && x[0] = y[0] + 1 & x[0] ->^* x[1] & y[0] ->^* y[1]) The set Q consists of the following terms: minus(x0, x1) minusNat(TRUE, x0, x1) round(x0) 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.IdpCand1ShapeHeuristic@4cf92d53 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 2 Max Right Steps: 0 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 MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) the following chains were created: *We consider the chain MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])), MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]), MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) which results in the following constraint: (1) (x[1]=x[0] & round(x[1])=y[0] & &&(>=(y[0], 0), =(x[0], +(y[0], 1)))=TRUE & x[0]=x[1]1 & y[0]=y[1]1 ==> MINUSNAT(TRUE, x[1]1, y[1]1)_>=_NonInfC & MINUSNAT(TRUE, x[1]1, y[1]1)_>=_MINUS(x[1]1, round(x[1]1)) & (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=)) We simplified constraint (1) using rules (III), (VII), (IDP_BOOLEAN), (REWRITING) which results in the following new constraint: (2) (=(%(x[1], 2), 0)=x0 & +(x[1], 1)=x1 & if(x0, x[1], x1)=y[0] & >=(y[0], 0)=TRUE & >=(x[1], +(y[0], 1))=TRUE & <=(x[1], +(y[0], 1))=TRUE ==> MINUSNAT(TRUE, x[1], y[0])_>=_NonInfC & MINUSNAT(TRUE, x[1], y[0])_>=_MINUS(x[1], round(x[1])) & (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) ([-1] + [-1]x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) ([-1] + x0 >= 0 & x[1] + [1] + [-1]x1 >= 0 & y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) For Pair MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) the following chains were created: *We consider the chain MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]), MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) which results in the following constraint: (1) (&&(>=(y[0], 0), =(x[0], +(y[0], 1)))=TRUE & x[0]=x[1] & y[0]=y[1] ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) & (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>=(y[0], 0)=TRUE & >=(x[0], +(y[0], 1))=TRUE & <=(x[0], +(y[0], 1))=TRUE ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) & (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) *(y[0] >= 0 & x[1] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[1] >= 0 ==> (U^Increasing(MINUS(x[1]1, round(x[1]1))), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]y[0] + [bni_20]x[1] >= 0 & [(-1)bso_21] + [-1]y[0] + x[1] >= 0) *MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) *(y[0] >= 0 & x[0] + [-1] + [-1]y[0] >= 0 & y[0] + [1] + [-1]x[0] >= 0 ==> (U^Increasing(MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0])), >=) & [(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]y[0] + [bni_22]x[0] >= 0 & [(-1)bso_23] >= 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) = 0 POL(round(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = [-1]max{[-1]x_3, [-1]x_2} POL(=(x_1, x_2)) = [-1] POL(2) = [2] POL(0) = 0 POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(MINUSNAT(x_1, x_2, x_3)) = [-1] + [-1]x_3 + x_2 POL(MINUS(x_1, x_2)) = [-1] + [-1]x_2 + x_1 POL(&&(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) The following pairs are in P_bound: MINUSNAT(TRUE, x[1], y[1]) -> MINUS(x[1], round(x[1])) MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) The following pairs are in P_>=: MINUS(x[0], y[0]) -> MINUSNAT(&&(>=(y[0], 0), =(x[0], +(y[0], 1))), x[0], y[0]) At least the following rules have been oriented under context sensitive arithmetic replacement: if(=(%(x, 2), 0), x, +(x, 1))^1 -> round(x)^1 u^1 -> if(TRUE, u, v)^1 v^1 -> if(FALSE, 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, Boolean The ITRS R consists of the following rules: round(x) -> if(x % 2 = 0, x, x + 1) 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]) -> MINUSNAT(y[0] >= 0 && x[0] = y[0] + 1, x[0], y[0]) The set Q consists of the following terms: minus(x0, x1) minusNat(TRUE, x0, x1) round(x0) 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