4.03/1.91 YES 4.03/1.92 proof of /export/starexec/sandbox/benchmark/theBenchmark.itrs 4.03/1.92 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.03/1.92 4.03/1.92 4.03/1.92 Termination of the given ITRS could be proven: 4.03/1.92 4.03/1.92 (0) ITRS 4.03/1.92 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 4.03/1.92 (2) IDP 4.03/1.92 (3) UsableRulesProof [EQUIVALENT, 0 ms] 4.03/1.92 (4) IDP 4.03/1.92 (5) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.03/1.92 (6) IDP 4.03/1.92 (7) IDPNonInfProof [SOUND, 159 ms] 4.03/1.92 (8) IDP 4.03/1.92 (9) IDependencyGraphProof [EQUIVALENT, 0 ms] 4.03/1.92 (10) TRUE 4.03/1.92 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (0) 4.03/1.92 Obligation: 4.03/1.92 ITRS problem: 4.03/1.92 4.03/1.92 The following function symbols are pre-defined: 4.03/1.92 <<< 4.03/1.92 & ~ Bwand: (Integer, Integer) -> Integer 4.03/1.92 >= ~ Ge: (Integer, Integer) -> Boolean 4.03/1.92 | ~ Bwor: (Integer, Integer) -> Integer 4.03/1.92 / ~ Div: (Integer, Integer) -> Integer 4.03/1.92 != ~ Neq: (Integer, Integer) -> Boolean 4.03/1.92 && ~ Land: (Boolean, Boolean) -> Boolean 4.03/1.92 ! ~ Lnot: (Boolean) -> Boolean 4.03/1.92 = ~ Eq: (Integer, Integer) -> Boolean 4.03/1.92 <= ~ Le: (Integer, Integer) -> Boolean 4.03/1.92 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.03/1.92 % ~ Mod: (Integer, Integer) -> Integer 4.03/1.92 + ~ Add: (Integer, Integer) -> Integer 4.03/1.92 > ~ Gt: (Integer, Integer) -> Boolean 4.03/1.92 -1 ~ UnaryMinus: (Integer) -> Integer 4.03/1.92 < ~ Lt: (Integer, Integer) -> Boolean 4.03/1.92 || ~ Lor: (Boolean, Boolean) -> Boolean 4.03/1.92 - ~ Sub: (Integer, Integer) -> Integer 4.03/1.92 ~ ~ Bwnot: (Integer) -> Integer 4.03/1.92 * ~ Mul: (Integer, Integer) -> Integer 4.03/1.92 >>> 4.03/1.92 4.03/1.92 The TRS R consists of the following rules: 4.03/1.92 minus(x, x) -> 0 4.03/1.92 minus(x, y) -> cond(min(x, y), x, y) 4.03/1.92 cond(y, x, y) -> 1 + minus(x, y + 1) 4.03/1.92 min(u, v) -> if(u < v, u, v) 4.03/1.92 if(TRUE, u, v) -> u 4.03/1.92 if(FALSE, u, v) -> v 4.03/1.92 The set Q consists of the following terms: 4.03/1.92 minus(x0, x1) 4.03/1.92 cond(x0, x1, x0) 4.03/1.92 min(x0, x1) 4.03/1.92 if(TRUE, x0, x1) 4.03/1.92 if(FALSE, x0, x1) 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (1) ITRStoIDPProof (EQUIVALENT) 4.03/1.92 Added dependency pairs 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (2) 4.03/1.92 Obligation: 4.03/1.92 IDP problem: 4.03/1.92 The following function symbols are pre-defined: 4.03/1.92 <<< 4.03/1.92 & ~ Bwand: (Integer, Integer) -> Integer 4.03/1.92 >= ~ Ge: (Integer, Integer) -> Boolean 4.03/1.92 | ~ Bwor: (Integer, Integer) -> Integer 4.03/1.92 / ~ Div: (Integer, Integer) -> Integer 4.03/1.92 != ~ Neq: (Integer, Integer) -> Boolean 4.03/1.92 && ~ Land: (Boolean, Boolean) -> Boolean 4.03/1.92 ! ~ Lnot: (Boolean) -> Boolean 4.03/1.92 = ~ Eq: (Integer, Integer) -> Boolean 4.03/1.92 <= ~ Le: (Integer, Integer) -> Boolean 4.03/1.92 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.03/1.92 % ~ Mod: (Integer, Integer) -> Integer 4.03/1.92 + ~ Add: (Integer, Integer) -> Integer 4.03/1.92 > ~ Gt: (Integer, Integer) -> Boolean 4.03/1.92 -1 ~ UnaryMinus: (Integer) -> Integer 4.03/1.92 < ~ Lt: (Integer, Integer) -> Boolean 4.03/1.92 || ~ Lor: (Boolean, Boolean) -> Boolean 4.03/1.92 - ~ Sub: (Integer, Integer) -> Integer 4.03/1.92 ~ ~ Bwnot: (Integer) -> Integer 4.03/1.92 * ~ Mul: (Integer, Integer) -> Integer 4.03/1.92 >>> 4.03/1.92 4.03/1.92 4.03/1.92 The following domains are used: 4.03/1.92 Integer 4.03/1.92 4.03/1.92 The ITRS R consists of the following rules: 4.03/1.92 minus(x, x) -> 0 4.03/1.92 minus(x, y) -> cond(min(x, y), x, y) 4.03/1.92 cond(y, x, y) -> 1 + minus(x, y + 1) 4.03/1.92 min(u, v) -> if(u < v, u, v) 4.03/1.92 if(TRUE, u, v) -> u 4.03/1.92 if(FALSE, u, v) -> v 4.03/1.92 4.03/1.92 The integer pair graph contains the following rules and edges: 4.03/1.92 (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) 4.03/1.92 (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) 4.03/1.92 (3): MIN(u[3], v[3]) -> IF(u[3] < v[3], u[3], v[3]) 4.03/1.92 4.03/1.92 (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) 4.03/1.92 (1) -> (3), if (x[1] ->^* u[3] & y[1] ->^* v[3]) 4.03/1.92 (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) 4.03/1.92 (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) 4.03/1.92 4.03/1.92 The set Q consists of the following terms: 4.03/1.92 minus(x0, x1) 4.03/1.92 cond(x0, x1, x0) 4.03/1.92 min(x0, x1) 4.03/1.92 if(TRUE, x0, x1) 4.03/1.92 if(FALSE, x0, x1) 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (3) UsableRulesProof (EQUIVALENT) 4.03/1.92 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. 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (4) 4.03/1.92 Obligation: 4.03/1.92 IDP problem: 4.03/1.92 The following function symbols are pre-defined: 4.03/1.92 <<< 4.03/1.92 & ~ Bwand: (Integer, Integer) -> Integer 4.03/1.92 >= ~ Ge: (Integer, Integer) -> Boolean 4.03/1.92 | ~ Bwor: (Integer, Integer) -> Integer 4.03/1.92 / ~ Div: (Integer, Integer) -> Integer 4.03/1.92 != ~ Neq: (Integer, Integer) -> Boolean 4.03/1.92 && ~ Land: (Boolean, Boolean) -> Boolean 4.03/1.92 ! ~ Lnot: (Boolean) -> Boolean 4.03/1.92 = ~ Eq: (Integer, Integer) -> Boolean 4.03/1.92 <= ~ Le: (Integer, Integer) -> Boolean 4.03/1.92 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.03/1.92 % ~ Mod: (Integer, Integer) -> Integer 4.03/1.92 + ~ Add: (Integer, Integer) -> Integer 4.03/1.92 > ~ Gt: (Integer, Integer) -> Boolean 4.03/1.92 -1 ~ UnaryMinus: (Integer) -> Integer 4.03/1.92 < ~ Lt: (Integer, Integer) -> Boolean 4.03/1.92 || ~ Lor: (Boolean, Boolean) -> Boolean 4.03/1.92 - ~ Sub: (Integer, Integer) -> Integer 4.03/1.92 ~ ~ Bwnot: (Integer) -> Integer 4.03/1.92 * ~ Mul: (Integer, Integer) -> Integer 4.03/1.92 >>> 4.03/1.92 4.03/1.92 4.03/1.92 The following domains are used: 4.03/1.92 Integer 4.03/1.92 4.03/1.92 The ITRS R consists of the following rules: 4.03/1.92 min(u, v) -> if(u < v, u, v) 4.03/1.92 if(TRUE, u, v) -> u 4.03/1.92 if(FALSE, u, v) -> v 4.03/1.92 4.03/1.92 The integer pair graph contains the following rules and edges: 4.03/1.92 (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 (1): MINUS(x[1], y[1]) -> MIN(x[1], y[1]) 4.03/1.92 (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) 4.03/1.92 (3): MIN(u[3], v[3]) -> IF(u[3] < v[3], u[3], v[3]) 4.03/1.92 4.03/1.92 (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) 4.03/1.92 (1) -> (3), if (x[1] ->^* u[3] & y[1] ->^* v[3]) 4.03/1.92 (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) 4.03/1.92 (2) -> (1), if (x[2] ->^* x[1] & y[2] + 1 ->^* y[1]) 4.03/1.92 4.03/1.92 The set Q consists of the following terms: 4.03/1.92 minus(x0, x1) 4.03/1.92 cond(x0, x1, x0) 4.03/1.92 min(x0, x1) 4.03/1.92 if(TRUE, x0, x1) 4.03/1.92 if(FALSE, x0, x1) 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (5) IDependencyGraphProof (EQUIVALENT) 4.03/1.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (6) 4.03/1.92 Obligation: 4.03/1.92 IDP problem: 4.03/1.92 The following function symbols are pre-defined: 4.03/1.92 <<< 4.03/1.92 & ~ Bwand: (Integer, Integer) -> Integer 4.03/1.92 >= ~ Ge: (Integer, Integer) -> Boolean 4.03/1.92 | ~ Bwor: (Integer, Integer) -> Integer 4.03/1.92 / ~ Div: (Integer, Integer) -> Integer 4.03/1.92 != ~ Neq: (Integer, Integer) -> Boolean 4.03/1.92 && ~ Land: (Boolean, Boolean) -> Boolean 4.03/1.92 ! ~ Lnot: (Boolean) -> Boolean 4.03/1.92 = ~ Eq: (Integer, Integer) -> Boolean 4.03/1.92 <= ~ Le: (Integer, Integer) -> Boolean 4.03/1.92 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.03/1.92 % ~ Mod: (Integer, Integer) -> Integer 4.03/1.92 + ~ Add: (Integer, Integer) -> Integer 4.03/1.92 > ~ Gt: (Integer, Integer) -> Boolean 4.03/1.92 -1 ~ UnaryMinus: (Integer) -> Integer 4.03/1.92 < ~ Lt: (Integer, Integer) -> Boolean 4.03/1.92 || ~ Lor: (Boolean, Boolean) -> Boolean 4.03/1.92 - ~ Sub: (Integer, Integer) -> Integer 4.03/1.92 ~ ~ Bwnot: (Integer) -> Integer 4.03/1.92 * ~ Mul: (Integer, Integer) -> Integer 4.03/1.92 >>> 4.03/1.92 4.03/1.92 4.03/1.92 The following domains are used: 4.03/1.92 Integer 4.03/1.92 4.03/1.92 The ITRS R consists of the following rules: 4.03/1.92 min(u, v) -> if(u < v, u, v) 4.03/1.92 if(TRUE, u, v) -> u 4.03/1.92 if(FALSE, u, v) -> v 4.03/1.92 4.03/1.92 The integer pair graph contains the following rules and edges: 4.03/1.92 (2): COND(y[2], x[2], y[2]) -> MINUS(x[2], y[2] + 1) 4.03/1.92 (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 4.03/1.92 (2) -> (0), if (x[2] ->^* x[0] & y[2] + 1 ->^* y[0]) 4.03/1.92 (0) -> (2), if (min(x[0], y[0]) ->^* y[2] & x[0] ->^* x[2] & y[0] ->^* y[2]) 4.03/1.92 4.03/1.92 The set Q consists of the following terms: 4.03/1.92 minus(x0, x1) 4.03/1.92 cond(x0, x1, x0) 4.03/1.92 min(x0, x1) 4.03/1.92 if(TRUE, x0, x1) 4.03/1.92 if(FALSE, x0, x1) 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (7) IDPNonInfProof (SOUND) 4.03/1.92 Used the following options for this NonInfProof: 4.03/1.92 4.03/1.92 IDPGPoloSolver: 4.03/1.92 Range: [(-1,2)] 4.03/1.92 IsNat: false 4.03/1.92 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@1a63f1f9 4.03/1.92 Constraint Generator: NonInfConstraintGenerator: 4.03/1.92 PathGenerator: MetricPathGenerator: 4.03/1.92 Max Left Steps: 1 4.03/1.92 Max Right Steps: 1 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 The constraints were generated the following way: 4.03/1.92 4.03/1.92 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 4.03/1.92 4.03/1.92 Note that final constraints are written in bold face. 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 For Pair COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) the following chains were created: 4.03/1.92 *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: 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (1) using rules (III), (IV), (VII), (REWRITING) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 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: 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (3) using rule (III) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (4) using rule (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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))), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (5) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (7) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (8) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.03/1.92 4.03/1.92 (9) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.03/1.92 4.03/1.92 (10) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.03/1.92 4.03/1.92 (11) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.03/1.92 4.03/1.92 (12) (x4 + [-1]x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x3 + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (13) ([-1] >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & 0 = 0 & [bni_18 + (-1)Bound*bni_18] >= 0 & 0 = 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.03/1.92 4.03/1.92 (14) (x4 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 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: 4.03/1.92 4.03/1.92 (15) (x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 (16) (x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 For Pair MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) the following chains were created: 4.03/1.92 *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: 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (1) using rules (III), (IV), (VII), (REWRITING) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 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: 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (3) using rules (III), (IV) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (4) using rules (III), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint: 4.03/1.92 4.03/1.92 (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])), >=)) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (5) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (7) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (8) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.03/1.92 4.03/1.92 (9) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 4.03/1.92 4.03/1.92 (10) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.03/1.92 4.03/1.92 (11) (x10 + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20] + [(-1)bni_20]y[2] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 4.03/1.92 4.03/1.92 (12) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [bni_20 + (-1)Bound*bni_20] >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 4.03/1.92 4.03/1.92 (13) (x10 >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20 + bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint: 4.03/1.92 4.03/1.92 (14) ([-1] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & 0 = 0 & [bni_20 + (-1)Bound*bni_20] >= 0 & 0 = 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 4.03/1.92 4.03/1.92 (15) (x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20 + bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 (16) (x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20 + bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 We solved constraint (14) using rule (IDP_SMT_SPLIT). 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 To summarize, we get the following constraints P__>=_ for the following pairs. 4.03/1.92 4.03/1.92 *COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) 4.03/1.92 4.03/1.92 *(x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 *(x4 >= 0 & x3 >= 0 ==> (U^Increasing(MINUS(x[2], +(y[2], 1))), >=) & [bni_18 + (-1)Bound*bni_18] + [bni_18]x4 >= 0 & [1 + (-1)bso_19] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 *MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 4.03/1.92 *(x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20 + bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 *(x10 >= 0 & y[2] >= 0 ==> (U^Increasing(COND(min(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_20 + bni_20] + [bni_20]x10 >= 0 & [(-1)bso_21] >= 0) 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 4.03/1.92 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. 4.03/1.92 4.03/1.92 Using the following integer polynomial ordering the resulting constraints can be solved 4.03/1.92 4.03/1.92 Polynomial interpretation over integers[POLO]: 4.03/1.92 4.03/1.92 POL(TRUE) = [1] 4.03/1.92 POL(FALSE) = [1] 4.03/1.92 POL(min(x_1, x_2)) = [2] + [-1]x_2 + [-1]x_1 4.03/1.92 POL(if(x_1, x_2, x_3)) = [1] + [-1]x_3 + [-1]x_2 + [2]x_1 4.03/1.92 POL(<(x_1, x_2)) = [2] 4.03/1.92 POL(COND(x_1, x_2, x_3)) = [1] + [-1]x_3 + x_2 4.03/1.92 POL(MINUS(x_1, x_2)) = [1] + [-1]x_2 + x_1 4.03/1.92 POL(+(x_1, x_2)) = x_1 + x_2 4.03/1.92 POL(1) = [1] 4.03/1.92 4.03/1.92 4.03/1.92 The following pairs are in P_>: 4.03/1.92 4.03/1.92 4.03/1.92 COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) 4.03/1.92 4.03/1.92 4.03/1.92 The following pairs are in P_bound: 4.03/1.92 4.03/1.92 4.03/1.92 COND(y[2], x[2], y[2]) -> MINUS(x[2], +(y[2], 1)) 4.03/1.92 MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 4.03/1.92 4.03/1.92 The following pairs are in P_>=: 4.03/1.92 4.03/1.92 4.03/1.92 MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 4.03/1.92 4.03/1.92 At least the following rules have been oriented under context sensitive arithmetic replacement: 4.03/1.92 4.03/1.92 if(<(u, v), u, v)^1 -> min(u, v)^1 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (8) 4.03/1.92 Obligation: 4.03/1.92 IDP problem: 4.03/1.92 The following function symbols are pre-defined: 4.03/1.92 <<< 4.03/1.92 & ~ Bwand: (Integer, Integer) -> Integer 4.03/1.92 >= ~ Ge: (Integer, Integer) -> Boolean 4.03/1.92 | ~ Bwor: (Integer, Integer) -> Integer 4.03/1.92 / ~ Div: (Integer, Integer) -> Integer 4.03/1.92 != ~ Neq: (Integer, Integer) -> Boolean 4.03/1.92 && ~ Land: (Boolean, Boolean) -> Boolean 4.03/1.92 ! ~ Lnot: (Boolean) -> Boolean 4.03/1.92 = ~ Eq: (Integer, Integer) -> Boolean 4.03/1.92 <= ~ Le: (Integer, Integer) -> Boolean 4.03/1.92 ^ ~ Bwxor: (Integer, Integer) -> Integer 4.03/1.92 % ~ Mod: (Integer, Integer) -> Integer 4.03/1.92 + ~ Add: (Integer, Integer) -> Integer 4.03/1.92 > ~ Gt: (Integer, Integer) -> Boolean 4.03/1.92 -1 ~ UnaryMinus: (Integer) -> Integer 4.03/1.92 < ~ Lt: (Integer, Integer) -> Boolean 4.03/1.92 || ~ Lor: (Boolean, Boolean) -> Boolean 4.03/1.92 - ~ Sub: (Integer, Integer) -> Integer 4.03/1.92 ~ ~ Bwnot: (Integer) -> Integer 4.03/1.92 * ~ Mul: (Integer, Integer) -> Integer 4.03/1.92 >>> 4.03/1.92 4.03/1.92 4.03/1.92 The following domains are used: 4.03/1.92 Integer 4.03/1.92 4.03/1.92 The ITRS R consists of the following rules: 4.03/1.92 min(u, v) -> if(u < v, u, v) 4.03/1.92 if(TRUE, u, v) -> u 4.03/1.92 if(FALSE, u, v) -> v 4.03/1.92 4.03/1.92 The integer pair graph contains the following rules and edges: 4.03/1.92 (0): MINUS(x[0], y[0]) -> COND(min(x[0], y[0]), x[0], y[0]) 4.03/1.92 4.03/1.92 4.03/1.92 The set Q consists of the following terms: 4.03/1.92 minus(x0, x1) 4.03/1.92 cond(x0, x1, x0) 4.03/1.92 min(x0, x1) 4.03/1.92 if(TRUE, x0, x1) 4.03/1.92 if(FALSE, x0, x1) 4.03/1.92 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (9) IDependencyGraphProof (EQUIVALENT) 4.03/1.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.03/1.92 ---------------------------------------- 4.03/1.92 4.03/1.92 (10) 4.03/1.92 TRUE 4.30/1.95 EOF