/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5)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, 400 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) -> Cond_eval(i - j >= 1 && nat >= 0 && pos > 0, i, j, nat, pos) Cond_eval(TRUE, i, j, nat, pos) -> eval(i - nat, j + pos) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2, x3) ---------------------------------------- (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) -> Cond_eval(i - j >= 1 && nat >= 0 && pos > 0, i, j, nat, pos) Cond_eval(TRUE, i, j, nat, pos) -> eval(i - nat, j + pos) The integer pair graph contains the following rules and edges: (0): EVAL(i[0], j[0]) -> COND_EVAL(i[0] - j[0] >= 1 && nat[0] >= 0 && pos[0] > 0, i[0], j[0], nat[0], pos[0]) (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) (0) -> (1), if (i[0] - j[0] >= 1 && nat[0] >= 0 && pos[0] > 0 & i[0] ->^* i[1] & j[0] ->^* j[1] & nat[0] ->^* nat[1] & pos[0] ->^* pos[1]) (1) -> (0), if (i[1] - nat[1] ->^* i[0] & j[1] + pos[1] ->^* j[0]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2, x3) ---------------------------------------- (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]) -> COND_EVAL(i[0] - j[0] >= 1 && nat[0] >= 0 && pos[0] > 0, i[0], j[0], nat[0], pos[0]) (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) (0) -> (1), if (i[0] - j[0] >= 1 && nat[0] >= 0 && pos[0] > 0 & i[0] ->^* i[1] & j[0] ->^* j[1] & nat[0] ->^* nat[1] & pos[0] ->^* pos[1]) (1) -> (0), if (i[1] - nat[1] ->^* i[0] & j[1] + pos[1] ->^* j[0]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2, x3) ---------------------------------------- (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@3e1b73b6 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) -> COND_EVAL(&&(&&(>=(-(i, j), 1), >=(nat, 0)), >(pos, 0)), i, j, nat, pos) the following chains were created: *We consider the chain COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(-(i[1], nat[1]), +(j[1], pos[1])), EVAL(i[0], j[0]) -> COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]), COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(-(i[1], nat[1]), +(j[1], pos[1])) which results in the following constraint: (1) (-(i[1], nat[1])=i[0] & +(j[1], pos[1])=j[0] & &&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0))=TRUE & i[0]=i[1]1 & j[0]=j[1]1 & nat[0]=nat[1]1 & pos[0]=pos[1]1 ==> EVAL(i[0], j[0])_>=_NonInfC & EVAL(i[0], j[0])_>=_COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]) & (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(pos[0], 0)=TRUE & >=(-(-(i[1], nat[1]), +(j[1], pos[1])), 1)=TRUE & >=(nat[0], 0)=TRUE ==> EVAL(-(i[1], nat[1]), +(j[1], pos[1]))_>=_NonInfC & EVAL(-(i[1], nat[1]), +(j[1], pos[1]))_>=_COND_EVAL(&&(&&(>=(-(-(i[1], nat[1]), +(j[1], pos[1])), 1), >=(nat[0], 0)), >(pos[0], 0)), -(i[1], nat[1]), +(j[1], pos[1]), nat[0], pos[0]) & (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (pos[0] + [-1] >= 0 & i[1] + [-1] + [-1]nat[1] + [-1]j[1] + [-1]pos[1] >= 0 & nat[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]j[1] + [(-1)bni_15]pos[1] + [bni_15]i[1] + [(-1)bni_15]nat[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (pos[0] + [-1] >= 0 & i[1] + [-1] + [-1]nat[1] + [-1]j[1] + [-1]pos[1] >= 0 & nat[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]j[1] + [(-1)bni_15]pos[1] + [bni_15]i[1] + [(-1)bni_15]nat[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (pos[0] + [-1] >= 0 & i[1] + [-1] + [-1]nat[1] + [-1]j[1] + [-1]pos[1] >= 0 & nat[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]j[1] + [(-1)bni_15]pos[1] + [bni_15]i[1] + [(-1)bni_15]nat[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (8) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (9) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (10) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (11) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (12) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (13) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (14) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (15) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (16) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (17) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (18) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (19) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) (20) (pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) For Pair COND_EVAL(TRUE, i, j, nat, pos) -> EVAL(-(i, nat), +(j, pos)) the following chains were created: *We consider the chain EVAL(i[0], j[0]) -> COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]), COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(-(i[1], nat[1]), +(j[1], pos[1])), EVAL(i[0], j[0]) -> COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]) which results in the following constraint: (1) (&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0))=TRUE & i[0]=i[1] & j[0]=j[1] & nat[0]=nat[1] & pos[0]=pos[1] & -(i[1], nat[1])=i[0]1 & +(j[1], pos[1])=j[0]1 ==> COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1])_>=_NonInfC & COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1])_>=_EVAL(-(i[1], nat[1]), +(j[1], pos[1])) & (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=)) We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(pos[0], 0)=TRUE & >=(-(i[0], j[0]), 1)=TRUE & >=(nat[0], 0)=TRUE ==> COND_EVAL(TRUE, i[0], j[0], nat[0], pos[0])_>=_NonInfC & COND_EVAL(TRUE, i[0], j[0], nat[0], pos[0])_>=_EVAL(-(i[0], nat[0]), +(j[0], pos[0])) & (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (pos[0] + [-1] >= 0 & i[0] + [-1] + [-1]j[0] >= 0 & nat[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [(-1)bni_17]j[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (pos[0] + [-1] >= 0 & i[0] + [-1] + [-1]j[0] >= 0 & nat[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [(-1)bni_17]j[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (pos[0] + [-1] >= 0 & i[0] + [-1] + [-1]j[0] >= 0 & nat[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [(-1)bni_17]j[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (pos[0] + [-1] >= 0 & i[0] >= 0 & nat[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (pos[0] + [-1] >= 0 & i[0] >= 0 & nat[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) (8) (pos[0] + [-1] >= 0 & i[0] >= 0 & nat[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(i, j) -> COND_EVAL(&&(&&(>=(-(i, j), 1), >=(nat, 0)), >(pos, 0)), i, j, nat, pos) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *(pos[0] + [-1] >= 0 & i[1] >= 0 & nat[0] >= 0 & nat[1] >= 0 & j[1] >= 0 & pos[1] >= 0 ==> (U^Increasing(COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0])), >=) & [(-1)Bound*bni_15] + [bni_15]i[1] >= 0 & [1 + (-1)bso_16] + nat[0] >= 0) *COND_EVAL(TRUE, i, j, nat, pos) -> EVAL(-(i, nat), +(j, pos)) *(pos[0] + [-1] >= 0 & i[0] >= 0 & nat[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 0) *(pos[0] + [-1] >= 0 & i[0] >= 0 & nat[0] >= 0 & j[0] >= 0 ==> (U^Increasing(EVAL(-(i[1], nat[1]), +(j[1], pos[1]))), >=) & [(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]nat[0] + [bni_17]i[0] >= 0 & [-1 + (-1)bso_18] + pos[0] >= 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) = [1] POL(FALSE) = [2] POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 POL(COND_EVAL(x_1, x_2, x_3, x_4, x_5)) = [-1] + [-1]x_4 + [-1]x_3 + x_2 + [-1]x_1 POL(&&(x_1, x_2)) = [1] POL(>=(x_1, x_2)) = [-1] POL(-(x_1, x_2)) = x_1 + [-1]x_2 POL(1) = [1] POL(0) = 0 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 The following pairs are in P_>: EVAL(i[0], j[0]) -> COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]) The following pairs are in P_bound: EVAL(i[0], j[0]) -> COND_EVAL(&&(&&(>=(-(i[0], j[0]), 1), >=(nat[0], 0)), >(pos[0], 0)), i[0], j[0], nat[0], pos[0]) The following pairs are in P_>=: COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(-(i[1], nat[1]), +(j[1], pos[1])) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 <-> TRUE^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, 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: Integer R is empty. The integer pair graph contains the following rules and edges: (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1, x2, x3) ---------------------------------------- (7) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (8) TRUE