/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDependencyGraphProof [EQUIVALENT, 0 ms] (6) IDP (7) IDPNonInfProof [SOUND, 228 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) = y, x, y) cond(TRUE, 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(TRUE, x0, x1) 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) = y, x, y) cond(TRUE, 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]) = y[0], x[0], y[0]) (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) (2): COND(TRUE, 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[0] & 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(TRUE, x0, x1) 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]) = y[0], x[0], y[0]) (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) (2): COND(TRUE, 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[0] & 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(TRUE, x0, x1) 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(TRUE, x[2], y[2]) -> MINUS(x[2], y[2] + 1) (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[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[0] & x[0] ->^* x[2] & y[0] ->^* y[2]) The set Q consists of the following terms: minus(x0, x1) cond(TRUE, x0, x1) 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@2986bf54 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(TRUE, 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]), y[0]), x[0], y[0]), COND(TRUE, x[2], y[2]) -> MINUS(x[2], +(y[2], 1)), MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) which results in the following constraint: (1) (=(min(x[0], y[0]), y[0])=TRUE & x[0]=x[2] & y[0]=y[2] & x[2]=x[0]1 & +(y[2], 1)=y[0]1 ==> COND(TRUE, x[2], y[2])_>=_NonInfC & COND(TRUE, 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), (IDP_BOOLEAN), (REWRITING) which results in the following new constraint: (2) (<(x[0], y[0])=x1 & if(x1, x[0], y[0])=x0 @ Ge: (Integer, Integer) -> Boolean & >=(x0, y[0])=TRUE & <(x[0], y[0])=x3 & if(x3, x[0], y[0])=x2 @ Le: (Integer, Integer) -> Boolean & <=(x2, y[0])=TRUE ==> COND(TRUE, x[0], y[0])_>=_NonInfC & COND(TRUE, 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(x1, x[0], y[0])=x0 @ Ge: (Integer, Integer) -> Boolean which results in the following new constraints: (3) (x5=x0 @ Ge: (Integer, Integer) -> Boolean & <(x5, x4)=TRUE & >=(x0, x4)=TRUE & <(x5, x4)=x3 & if(x3, x5, x4)=x2 @ Le: (Integer, Integer) -> Boolean & <=(x2, x4)=TRUE ==> COND(TRUE, x5, x4)_>=_NonInfC & COND(TRUE, x5, x4)_>=_MINUS(x5, +(x4, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) (4) (x6=x0 @ Ge: (Integer, Integer) -> Boolean & <(x7, x6)=FALSE & >=(x0, x6)=TRUE & <(x7, x6)=x3 & if(x3, x7, x6)=x2 @ Le: (Integer, Integer) -> Boolean & <=(x2, x6)=TRUE ==> COND(TRUE, x7, x6)_>=_NonInfC & COND(TRUE, x7, x6)_>=_MINUS(x7, +(x6, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on if(x3, x5, x4)=x2 @ Le: (Integer, Integer) -> Boolean which results in the following new constraints: (5) (x9=x2 @ Le: (Integer, Integer) -> Boolean & x9=x0 @ Ge: (Integer, Integer) -> Boolean & <(x9, x8)=TRUE & >=(x0, x8)=TRUE & <=(x2, x8)=TRUE ==> COND(TRUE, x9, x8)_>=_NonInfC & COND(TRUE, x9, x8)_>=_MINUS(x9, +(x8, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) (6) (x10=x2 @ Le: (Integer, Integer) -> Boolean & x11=x0 @ Ge: (Integer, Integer) -> Boolean & <(x11, x10)=TRUE & >=(x0, x10)=TRUE & <(x11, x10)=FALSE & <=(x2, x10)=TRUE ==> COND(TRUE, x11, x10)_>=_NonInfC & COND(TRUE, x11, x10)_>=_MINUS(x11, +(x10, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on if(x3, x7, x6)=x2 @ Le: (Integer, Integer) -> Boolean which results in the following new constraints: (7) (x13=x2 @ Le: (Integer, Integer) -> Boolean & x12=x0 @ Ge: (Integer, Integer) -> Boolean & <(x13, x12)=FALSE & >=(x0, x12)=TRUE & <(x13, x12)=TRUE & <=(x2, x12)=TRUE ==> COND(TRUE, x13, x12)_>=_NonInfC & COND(TRUE, x13, x12)_>=_MINUS(x13, +(x12, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) (8) (x14=x2 @ Le: (Integer, Integer) -> Boolean & x14=x0 @ Ge: (Integer, Integer) -> Boolean & <(x15, x14)=FALSE & >=(x0, x14)=TRUE & <=(x2, x14)=TRUE ==> COND(TRUE, x15, x14)_>=_NonInfC & COND(TRUE, x15, x14)_>=_MINUS(x15, +(x14, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (5) using rule (III) which results in the following new constraint: (9) (<(x0, x8)=TRUE & >=(x0, x8)=TRUE & <=(x0, x8)=TRUE ==> COND(TRUE, x0, x8)_>=_NonInfC & COND(TRUE, x0, x8)_>=_MINUS(x0, +(x8, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (6) using rule (III) which results in the following new constraint: (10) (<(x0, x2)=TRUE & >=(x0, x2)=TRUE & <(x0, x2)=FALSE & <=(x2, x2)=TRUE ==> COND(TRUE, x0, x2)_>=_NonInfC & COND(TRUE, x0, x2)_>=_MINUS(x0, +(x2, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (7) using rule (III) which results in the following new constraint: (11) (<(x2, x0)=FALSE & >=(x0, x0)=TRUE & <(x2, x0)=TRUE & <=(x2, x0)=TRUE ==> COND(TRUE, x2, x0)_>=_NonInfC & COND(TRUE, x2, x0)_>=_MINUS(x2, +(x0, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (8) using rule (III) which results in the following new constraint: (12) (<(x15, x0)=FALSE & >=(x0, x0)=TRUE & <=(x0, x0)=TRUE ==> COND(TRUE, x15, x0)_>=_NonInfC & COND(TRUE, x15, x0)_>=_MINUS(x15, +(x0, 1)) & (U^Increasing(MINUS(x[2], +(y[2], 1))), >=)) We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (13) (x8 + [-1] + [-1]x0 >= 0 & x0 + [-1]x8 >= 0 & x8 + [-1]x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x8 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (14) (x2 + [-1] + [-1]x0 >= 0 & x0 + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x2 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (15) (x2 + [-1]x0 >= 0 & 0 >= 0 & x0 + [-1] + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x2 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (16) (x15 + [-1]x0 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (17) (x8 + [-1] + [-1]x0 >= 0 & x0 + [-1]x8 >= 0 & x8 + [-1]x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x8 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (18) (x15 + [-1]x0 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (19) (x2 + [-1] + [-1]x0 >= 0 & x0 + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x2 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (20) (x2 + [-1]x0 >= 0 & 0 >= 0 & x0 + [-1] + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x2 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (21) (x8 + [-1] + [-1]x0 >= 0 & x0 + [-1]x8 >= 0 & x8 + [-1]x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x8 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (22) (x15 + [-1]x0 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (23) (x2 + [-1] + [-1]x0 >= 0 & x0 + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x2 + [bni_19]x0 >= 0 & [1 + (-1)bso_20] >= 0) We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (24) (x2 + [-1]x0 >= 0 & 0 >= 0 & x0 + [-1] + [-1]x2 >= 0 & x0 + [-1]x2 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0 + [bni_19]x2 >= 0 & [1 + (-1)bso_20] >= 0) We solved constraint (21) using rule (IDP_SMT_SPLIT).We simplified constraint (22) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (25) (x15 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) We solved constraint (23) using rule (IDP_SMT_SPLIT).We solved constraint (24) using rule (IDP_SMT_SPLIT).We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (26) (x15 >= 0 & 0 >= 0 & 0 >= 0 & x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) (27) (x15 >= 0 & 0 >= 0 & 0 >= 0 & x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) For Pair MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) the following chains were created: *We consider the chain MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]), COND(TRUE, x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) which results in the following constraint: (1) (=(min(x[0], y[0]), y[0])=TRUE & x[0]=x[2] & y[0]=y[2] ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (1) using rules (IV), (VII), (IDP_BOOLEAN), (REWRITING) which results in the following new constraint: (2) (<(x[0], y[0])=x17 & if(x17, x[0], y[0])=x16 @ Ge: (Integer, Integer) -> Boolean & >=(x16, y[0])=TRUE & <(x[0], y[0])=x19 & if(x19, x[0], y[0])=x18 @ Le: (Integer, Integer) -> Boolean & <=(x18, y[0])=TRUE ==> MINUS(x[0], y[0])_>=_NonInfC & MINUS(x[0], y[0])_>=_COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on if(x17, x[0], y[0])=x16 @ Ge: (Integer, Integer) -> Boolean which results in the following new constraints: (3) (x21=x16 @ Ge: (Integer, Integer) -> Boolean & <(x21, x20)=TRUE & >=(x16, x20)=TRUE & <(x21, x20)=x19 & if(x19, x21, x20)=x18 @ Le: (Integer, Integer) -> Boolean & <=(x18, x20)=TRUE ==> MINUS(x21, x20)_>=_NonInfC & MINUS(x21, x20)_>=_COND(=(min(x21, x20), x20), x21, x20) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) (4) (x22=x16 @ Ge: (Integer, Integer) -> Boolean & <(x23, x22)=FALSE & >=(x16, x22)=TRUE & <(x23, x22)=x19 & if(x19, x23, x22)=x18 @ Le: (Integer, Integer) -> Boolean & <=(x18, x22)=TRUE ==> MINUS(x23, x22)_>=_NonInfC & MINUS(x23, x22)_>=_COND(=(min(x23, x22), x22), x23, x22) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on if(x19, x21, x20)=x18 @ Le: (Integer, Integer) -> Boolean which results in the following new constraints: (5) (x25=x18 @ Le: (Integer, Integer) -> Boolean & x25=x16 @ Ge: (Integer, Integer) -> Boolean & <(x25, x24)=TRUE & >=(x16, x24)=TRUE & <=(x18, x24)=TRUE ==> MINUS(x25, x24)_>=_NonInfC & MINUS(x25, x24)_>=_COND(=(min(x25, x24), x24), x25, x24) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) (6) (x26=x18 @ Le: (Integer, Integer) -> Boolean & x27=x16 @ Ge: (Integer, Integer) -> Boolean & <(x27, x26)=TRUE & >=(x16, x26)=TRUE & <(x27, x26)=FALSE & <=(x18, x26)=TRUE ==> MINUS(x27, x26)_>=_NonInfC & MINUS(x27, x26)_>=_COND(=(min(x27, x26), x26), x27, x26) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on if(x19, x23, x22)=x18 @ Le: (Integer, Integer) -> Boolean which results in the following new constraints: (7) (x29=x18 @ Le: (Integer, Integer) -> Boolean & x28=x16 @ Ge: (Integer, Integer) -> Boolean & <(x29, x28)=FALSE & >=(x16, x28)=TRUE & <(x29, x28)=TRUE & <=(x18, x28)=TRUE ==> MINUS(x29, x28)_>=_NonInfC & MINUS(x29, x28)_>=_COND(=(min(x29, x28), x28), x29, x28) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) (8) (x30=x18 @ Le: (Integer, Integer) -> Boolean & x30=x16 @ Ge: (Integer, Integer) -> Boolean & <(x31, x30)=FALSE & >=(x16, x30)=TRUE & <=(x18, x30)=TRUE ==> MINUS(x31, x30)_>=_NonInfC & MINUS(x31, x30)_>=_COND(=(min(x31, x30), x30), x31, x30) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (5) using rule (III) which results in the following new constraint: (9) (<(x16, x24)=TRUE & >=(x16, x24)=TRUE & <=(x16, x24)=TRUE ==> MINUS(x16, x24)_>=_NonInfC & MINUS(x16, x24)_>=_COND(=(min(x16, x24), x24), x16, x24) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (6) using rule (III) which results in the following new constraint: (10) (<(x16, x18)=TRUE & >=(x16, x18)=TRUE & <(x16, x18)=FALSE & <=(x18, x18)=TRUE ==> MINUS(x16, x18)_>=_NonInfC & MINUS(x16, x18)_>=_COND(=(min(x16, x18), x18), x16, x18) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (7) using rule (III) which results in the following new constraint: (11) (<(x18, x16)=FALSE & >=(x16, x16)=TRUE & <(x18, x16)=TRUE & <=(x18, x16)=TRUE ==> MINUS(x18, x16)_>=_NonInfC & MINUS(x18, x16)_>=_COND(=(min(x18, x16), x16), x18, x16) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (8) using rule (III) which results in the following new constraint: (12) (<(x31, x16)=FALSE & >=(x16, x16)=TRUE & <=(x16, x16)=TRUE ==> MINUS(x31, x16)_>=_NonInfC & MINUS(x31, x16)_>=_COND(=(min(x31, x16), x16), x31, x16) & (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=)) We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (13) (x24 + [-1] + [-1]x16 >= 0 & x16 + [-1]x24 >= 0 & x24 + [-1]x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x24 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (14) (x18 + [-1] + [-1]x16 >= 0 & x16 + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x18 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (15) (x18 + [-1]x16 >= 0 & 0 >= 0 & x16 + [-1] + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x18 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (16) (x31 + [-1]x16 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (17) (x24 + [-1] + [-1]x16 >= 0 & x16 + [-1]x24 >= 0 & x24 + [-1]x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x24 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (18) (x31 + [-1]x16 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (19) (x18 + [-1] + [-1]x16 >= 0 & x16 + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x18 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (20) (x18 + [-1]x16 >= 0 & 0 >= 0 & x16 + [-1] + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x18 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (21) (x24 + [-1] + [-1]x16 >= 0 & x16 + [-1]x24 >= 0 & x24 + [-1]x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x24 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (22) (x31 + [-1]x16 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (23) (x18 + [-1] + [-1]x16 >= 0 & x16 + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x18 + [bni_21]x16 >= 0 & [(-1)bso_22] >= 0) We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (24) (x18 + [-1]x16 >= 0 & 0 >= 0 & x16 + [-1] + [-1]x18 >= 0 & x16 + [-1]x18 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x16 + [bni_21]x18 >= 0 & [(-1)bso_22] >= 0) We solved constraint (21) using rule (IDP_SMT_SPLIT).We simplified constraint (22) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (25) (x31 >= 0 & 0 >= 0 & 0 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) We solved constraint (23) using rule (IDP_SMT_SPLIT).We solved constraint (24) using rule (IDP_SMT_SPLIT).We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (26) (x31 >= 0 & 0 >= 0 & 0 >= 0 & x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) (27) (x31 >= 0 & 0 >= 0 & 0 >= 0 & x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *COND(TRUE, x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) *(x15 >= 0 & 0 >= 0 & 0 >= 0 & x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) *(x15 >= 0 & 0 >= 0 & 0 >= 0 & x0 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x15 >= 0 & [1 + (-1)bso_20] >= 0) *MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) *(x31 >= 0 & 0 >= 0 & 0 >= 0 & x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x31 >= 0 & [(-1)bso_22] >= 0) *(x31 >= 0 & 0 >= 0 & 0 >= 0 & x16 >= 0 ==> (U^Increasing(COND(=(min(x[0], y[0]), y[0]), x[0], y[0])), >=) & [(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x31 >= 0 & [(-1)bso_22] >= 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(min(x_1, x_2)) = [2] + [2]x_2 + [2]x_1 POL(if(x_1, x_2, x_3)) = [2] + x_3 + x_2 + [2]x_1 POL(<(x_1, x_2)) = 0 POL(COND(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)) = x_1 + x_2 POL(1) = [1] POL(=(x_1, x_2)) = [-1] The following pairs are in P_>: COND(TRUE, x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) The following pairs are in P_bound: COND(TRUE, x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) The following pairs are in P_>=: MINUS(x[0], y[0]) -> COND(=(min(x[0], y[0]), y[0]), x[0], y[0]) There are no usable rules. ---------------------------------------- (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]) = y[0], x[0], y[0]) The set Q consists of the following terms: minus(x0, x1) cond(TRUE, x0, x1) 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