5.28/2.23 YES 5.28/2.25 proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs 5.28/2.25 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.28/2.25 5.28/2.25 5.28/2.25 Termination of the given ITRS could be proven: 5.28/2.25 5.28/2.25 (0) ITRS 5.28/2.25 (1) ITRStoIDPProof [EQUIVALENT, 0 ms] 5.28/2.25 (2) IDP 5.28/2.25 (3) UsableRulesProof [EQUIVALENT, 0 ms] 5.28/2.25 (4) IDP 5.28/2.25 (5) IDPNonInfProof [SOUND, 474 ms] 5.28/2.25 (6) IDP 5.28/2.25 (7) IDependencyGraphProof [EQUIVALENT, 0 ms] 5.28/2.25 (8) TRUE 5.28/2.25 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (0) 5.28/2.25 Obligation: 5.28/2.25 ITRS problem: 5.28/2.25 5.28/2.25 The following function symbols are pre-defined: 5.28/2.25 <<< 5.28/2.25 & ~ Bwand: (Integer, Integer) -> Integer 5.28/2.25 >= ~ Ge: (Integer, Integer) -> Boolean 5.28/2.25 | ~ Bwor: (Integer, Integer) -> Integer 5.28/2.25 / ~ Div: (Integer, Integer) -> Integer 5.28/2.25 != ~ Neq: (Integer, Integer) -> Boolean 5.28/2.25 && ~ Land: (Boolean, Boolean) -> Boolean 5.28/2.25 ! ~ Lnot: (Boolean) -> Boolean 5.28/2.25 = ~ Eq: (Integer, Integer) -> Boolean 5.28/2.25 <= ~ Le: (Integer, Integer) -> Boolean 5.28/2.25 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.28/2.25 % ~ Mod: (Integer, Integer) -> Integer 5.28/2.25 + ~ Add: (Integer, Integer) -> Integer 5.28/2.25 > ~ Gt: (Integer, Integer) -> Boolean 5.28/2.25 -1 ~ UnaryMinus: (Integer) -> Integer 5.28/2.25 < ~ Lt: (Integer, Integer) -> Boolean 5.28/2.25 || ~ Lor: (Boolean, Boolean) -> Boolean 5.28/2.25 - ~ Sub: (Integer, Integer) -> Integer 5.28/2.25 ~ ~ Bwnot: (Integer) -> Integer 5.28/2.25 * ~ Mul: (Integer, Integer) -> Integer 5.28/2.25 >>> 5.28/2.25 5.28/2.25 The TRS R consists of the following rules: 5.28/2.25 eval(i, j) -> Cond_eval(i - j >= 1 && nat >= 0 && pos > 0, i, j, nat, pos) 5.28/2.25 Cond_eval(TRUE, i, j, nat, pos) -> eval(i - nat, j + pos) 5.28/2.25 The set Q consists of the following terms: 5.28/2.25 eval(x0, x1) 5.28/2.25 Cond_eval(TRUE, x0, x1, x2, x3) 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (1) ITRStoIDPProof (EQUIVALENT) 5.28/2.25 Added dependency pairs 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (2) 5.28/2.25 Obligation: 5.28/2.25 IDP problem: 5.28/2.25 The following function symbols are pre-defined: 5.28/2.25 <<< 5.28/2.25 & ~ Bwand: (Integer, Integer) -> Integer 5.28/2.25 >= ~ Ge: (Integer, Integer) -> Boolean 5.28/2.25 | ~ Bwor: (Integer, Integer) -> Integer 5.28/2.25 / ~ Div: (Integer, Integer) -> Integer 5.28/2.25 != ~ Neq: (Integer, Integer) -> Boolean 5.28/2.25 && ~ Land: (Boolean, Boolean) -> Boolean 5.28/2.25 ! ~ Lnot: (Boolean) -> Boolean 5.28/2.25 = ~ Eq: (Integer, Integer) -> Boolean 5.28/2.25 <= ~ Le: (Integer, Integer) -> Boolean 5.28/2.25 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.28/2.25 % ~ Mod: (Integer, Integer) -> Integer 5.28/2.25 + ~ Add: (Integer, Integer) -> Integer 5.28/2.25 > ~ Gt: (Integer, Integer) -> Boolean 5.28/2.25 -1 ~ UnaryMinus: (Integer) -> Integer 5.28/2.25 < ~ Lt: (Integer, Integer) -> Boolean 5.28/2.25 || ~ Lor: (Boolean, Boolean) -> Boolean 5.28/2.25 - ~ Sub: (Integer, Integer) -> Integer 5.28/2.25 ~ ~ Bwnot: (Integer) -> Integer 5.28/2.25 * ~ Mul: (Integer, Integer) -> Integer 5.28/2.25 >>> 5.28/2.25 5.28/2.25 5.28/2.25 The following domains are used: 5.28/2.25 Boolean, Integer 5.28/2.25 5.28/2.25 The ITRS R consists of the following rules: 5.28/2.25 eval(i, j) -> Cond_eval(i - j >= 1 && nat >= 0 && pos > 0, i, j, nat, pos) 5.28/2.25 Cond_eval(TRUE, i, j, nat, pos) -> eval(i - nat, j + pos) 5.28/2.25 5.28/2.25 The integer pair graph contains the following rules and edges: 5.28/2.25 (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]) 5.28/2.25 (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) 5.28/2.25 5.28/2.25 (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]) 5.28/2.25 (1) -> (0), if (i[1] - nat[1] ->^* i[0] & j[1] + pos[1] ->^* j[0]) 5.28/2.25 5.28/2.25 The set Q consists of the following terms: 5.28/2.25 eval(x0, x1) 5.28/2.25 Cond_eval(TRUE, x0, x1, x2, x3) 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (3) UsableRulesProof (EQUIVALENT) 5.28/2.25 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. 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (4) 5.28/2.25 Obligation: 5.28/2.25 IDP problem: 5.28/2.25 The following function symbols are pre-defined: 5.28/2.25 <<< 5.28/2.25 & ~ Bwand: (Integer, Integer) -> Integer 5.28/2.25 >= ~ Ge: (Integer, Integer) -> Boolean 5.28/2.25 | ~ Bwor: (Integer, Integer) -> Integer 5.28/2.25 / ~ Div: (Integer, Integer) -> Integer 5.28/2.25 != ~ Neq: (Integer, Integer) -> Boolean 5.28/2.25 && ~ Land: (Boolean, Boolean) -> Boolean 5.28/2.25 ! ~ Lnot: (Boolean) -> Boolean 5.28/2.25 = ~ Eq: (Integer, Integer) -> Boolean 5.28/2.25 <= ~ Le: (Integer, Integer) -> Boolean 5.28/2.25 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.28/2.25 % ~ Mod: (Integer, Integer) -> Integer 5.28/2.25 + ~ Add: (Integer, Integer) -> Integer 5.28/2.25 > ~ Gt: (Integer, Integer) -> Boolean 5.28/2.25 -1 ~ UnaryMinus: (Integer) -> Integer 5.28/2.25 < ~ Lt: (Integer, Integer) -> Boolean 5.28/2.25 || ~ Lor: (Boolean, Boolean) -> Boolean 5.28/2.25 - ~ Sub: (Integer, Integer) -> Integer 5.28/2.25 ~ ~ Bwnot: (Integer) -> Integer 5.28/2.25 * ~ Mul: (Integer, Integer) -> Integer 5.28/2.25 >>> 5.28/2.25 5.28/2.25 5.28/2.25 The following domains are used: 5.28/2.25 Boolean, Integer 5.28/2.25 5.28/2.25 R is empty. 5.28/2.25 5.28/2.25 The integer pair graph contains the following rules and edges: 5.28/2.25 (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]) 5.28/2.25 (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) 5.28/2.25 5.28/2.25 (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]) 5.28/2.25 (1) -> (0), if (i[1] - nat[1] ->^* i[0] & j[1] + pos[1] ->^* j[0]) 5.28/2.25 5.28/2.25 The set Q consists of the following terms: 5.28/2.25 eval(x0, x1) 5.28/2.25 Cond_eval(TRUE, x0, x1, x2, x3) 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (5) IDPNonInfProof (SOUND) 5.28/2.25 Used the following options for this NonInfProof: 5.28/2.25 5.28/2.25 IDPGPoloSolver: 5.28/2.25 Range: [(-1,2)] 5.28/2.25 IsNat: false 5.28/2.25 Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@1ccfbc04 5.28/2.25 Constraint Generator: NonInfConstraintGenerator: 5.28/2.25 PathGenerator: MetricPathGenerator: 5.28/2.25 Max Left Steps: 1 5.28/2.25 Max Right Steps: 1 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 The constraints were generated the following way: 5.28/2.25 5.28/2.25 The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: 5.28/2.25 5.28/2.25 Note that final constraints are written in bold face. 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 For Pair EVAL(i, j) -> COND_EVAL(&&(&&(>=(-(i, j), 1), >=(nat, 0)), >(pos, 0)), i, j, nat, pos) the following chains were created: 5.28/2.25 *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: 5.28/2.25 5.28/2.25 (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])), >=)) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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])), >=)) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 For Pair COND_EVAL(TRUE, i, j, nat, pos) -> EVAL(-(i, nat), +(j, pos)) the following chains were created: 5.28/2.25 *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: 5.28/2.25 5.28/2.25 (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]))), >=)) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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]))), >=)) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 (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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 To summarize, we get the following constraints P__>=_ for the following pairs. 5.28/2.25 5.28/2.25 *EVAL(i, j) -> COND_EVAL(&&(&&(>=(-(i, j), 1), >=(nat, 0)), >(pos, 0)), i, j, nat, pos) 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 *COND_EVAL(TRUE, i, j, nat, pos) -> EVAL(-(i, nat), +(j, pos)) 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 *(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) 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 5.28/2.25 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. 5.28/2.25 5.28/2.25 Using the following integer polynomial ordering the resulting constraints can be solved 5.28/2.25 5.28/2.25 Polynomial interpretation over integers[POLO]: 5.28/2.25 5.28/2.25 POL(TRUE) = [1] 5.28/2.25 POL(FALSE) = [3] 5.28/2.25 POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + x_1 5.28/2.25 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 5.28/2.25 POL(&&(x_1, x_2)) = [1] 5.28/2.25 POL(>=(x_1, x_2)) = [-1] 5.28/2.25 POL(-(x_1, x_2)) = x_1 + [-1]x_2 5.28/2.25 POL(1) = [1] 5.28/2.25 POL(0) = 0 5.28/2.25 POL(>(x_1, x_2)) = [-1] 5.28/2.25 POL(+(x_1, x_2)) = x_1 + x_2 5.28/2.25 5.28/2.25 5.28/2.25 The following pairs are in P_>: 5.28/2.25 5.28/2.25 5.28/2.25 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]) 5.28/2.25 5.28/2.25 5.28/2.25 The following pairs are in P_bound: 5.28/2.25 5.28/2.25 5.28/2.25 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]) 5.28/2.25 5.28/2.25 5.28/2.25 The following pairs are in P_>=: 5.28/2.25 5.28/2.25 5.28/2.25 COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(-(i[1], nat[1]), +(j[1], pos[1])) 5.28/2.25 5.28/2.25 5.28/2.25 At least the following rules have been oriented under context sensitive arithmetic replacement: 5.28/2.25 5.28/2.25 TRUE^1 -> &&(TRUE, TRUE)^1 5.28/2.25 FALSE^1 -> &&(TRUE, FALSE)^1 5.28/2.25 FALSE^1 -> &&(FALSE, TRUE)^1 5.28/2.25 FALSE^1 -> &&(FALSE, FALSE)^1 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (6) 5.28/2.25 Obligation: 5.28/2.25 IDP problem: 5.28/2.25 The following function symbols are pre-defined: 5.28/2.25 <<< 5.28/2.25 & ~ Bwand: (Integer, Integer) -> Integer 5.28/2.25 >= ~ Ge: (Integer, Integer) -> Boolean 5.28/2.25 | ~ Bwor: (Integer, Integer) -> Integer 5.28/2.25 / ~ Div: (Integer, Integer) -> Integer 5.28/2.25 != ~ Neq: (Integer, Integer) -> Boolean 5.28/2.25 && ~ Land: (Boolean, Boolean) -> Boolean 5.28/2.25 ! ~ Lnot: (Boolean) -> Boolean 5.28/2.25 = ~ Eq: (Integer, Integer) -> Boolean 5.28/2.25 <= ~ Le: (Integer, Integer) -> Boolean 5.28/2.25 ^ ~ Bwxor: (Integer, Integer) -> Integer 5.28/2.25 % ~ Mod: (Integer, Integer) -> Integer 5.28/2.25 + ~ Add: (Integer, Integer) -> Integer 5.28/2.25 > ~ Gt: (Integer, Integer) -> Boolean 5.28/2.25 -1 ~ UnaryMinus: (Integer) -> Integer 5.28/2.25 < ~ Lt: (Integer, Integer) -> Boolean 5.28/2.25 || ~ Lor: (Boolean, Boolean) -> Boolean 5.28/2.25 - ~ Sub: (Integer, Integer) -> Integer 5.28/2.25 ~ ~ Bwnot: (Integer) -> Integer 5.28/2.25 * ~ Mul: (Integer, Integer) -> Integer 5.28/2.25 >>> 5.28/2.25 5.28/2.25 5.28/2.25 The following domains are used: 5.28/2.25 Integer 5.28/2.25 5.28/2.25 R is empty. 5.28/2.25 5.28/2.25 The integer pair graph contains the following rules and edges: 5.28/2.25 (1): COND_EVAL(TRUE, i[1], j[1], nat[1], pos[1]) -> EVAL(i[1] - nat[1], j[1] + pos[1]) 5.28/2.25 5.28/2.25 5.28/2.25 The set Q consists of the following terms: 5.28/2.25 eval(x0, x1) 5.28/2.25 Cond_eval(TRUE, x0, x1, x2, x3) 5.28/2.25 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (7) IDependencyGraphProof (EQUIVALENT) 5.28/2.25 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 5.28/2.25 ---------------------------------------- 5.28/2.25 5.28/2.25 (8) 5.28/2.25 TRUE 5.28/2.30 EOF