WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 217 ms] (4) CpxRelTRS (5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (6) CpxRelTRS (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxWeightedTrs (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedTrs (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 2272 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 1242 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 281 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 321 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1863 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 298 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 289 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 23 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 430 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 339 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 12.4 s] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 650 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 727 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 688 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 726 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 696 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (102) CpxRNTS (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 1206 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 249 ms] (108) CpxRNTS (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 888 ms] (112) CpxRNTS (113) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (114) CpxRNTS (115) FinalProof [FINISHED, 0 ms] (116) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) and(x, y) -> if(x, y, false) or(x, y) -> if(x, true, y) implies(x, y) -> if(x, y, true) =(x, x) -> true =(x, y) -> if(x, y, not(y)) if(true, x, y) -> x if(false, x, y) -> y if(x, x, if(x, false, true)) -> true =(x, y) -> if(x, y, if(y, false, true)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(false) -> false encArg(true) -> true encArg(cons_not(x_1)) -> not(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) encArg(cons_=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_not(x_1) -> not(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_true -> true encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) and(x, y) -> if(x, y, false) or(x, y) -> if(x, true, y) implies(x, y) -> if(x, y, true) =(x, x) -> true =(x, y) -> if(x, y, not(y)) if(true, x, y) -> x if(false, x, y) -> y if(x, x, if(x, false, true)) -> true =(x, y) -> if(x, y, if(y, false, true)) The (relative) TRS S consists of the following rules: encArg(false) -> false encArg(true) -> true encArg(cons_not(x_1)) -> not(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) encArg(cons_=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_not(x_1) -> not(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_true -> true encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) and(x, y) -> if(x, y, false) or(x, y) -> if(x, true, y) implies(x, y) -> if(x, y, true) =(x, x) -> true =(x, y) -> if(x, y, not(y)) if(true, x, y) -> x if(false, x, y) -> y if(x, x, if(x, false, true)) -> true =(x, y) -> if(x, y, if(y, false, true)) The (relative) TRS S consists of the following rules: encArg(false) -> false encArg(true) -> true encArg(cons_not(x_1)) -> not(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) encArg(cons_=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_not(x_1) -> not(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_true -> true encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) and(x, y) -> if(x, y, false) or(x, y) -> if(x, true, y) implies(x, y) -> if(x, y, true) =(x, x) -> true =(x, y) -> if(x, y, not(y)) if(true, x, y) -> x if(false, x, y) -> y =(x, y) -> if(x, y, if(y, false, true)) if(x, x, c_if(x, false, true)) -> true The (relative) TRS S consists of the following rules: encArg(false) -> false encArg(true) -> true encArg(cons_not(x_1)) -> not(encArg(x_1)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) encArg(cons_=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_not(x_1) -> not(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_false -> false encode_true -> true encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) if(x0, x1, x2) -> c_if(x0, x1, x2) Rewrite Strategy: INNERMOST ---------------------------------------- (7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) [1] and(x, y) -> if(x, y, false) [1] or(x, y) -> if(x, true, y) [1] implies(x, y) -> if(x, y, true) [1] =(x, x) -> true [1] =(x, y) -> if(x, y, not(y)) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] =(x, y) -> if(x, y, if(y, false, true)) [1] if(x, x, c_if(x, false, true)) -> true [1] encArg(false) -> false [0] encArg(true) -> true [0] encArg(cons_not(x_1)) -> not(encArg(x_1)) [0] encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0] encArg(cons_=(x_1, x_2)) -> =(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_not(x_1) -> not(encArg(x_1)) [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_false -> false [0] encode_true -> true [0] encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0] encode_=(x_1, x_2) -> =(encArg(x_1), encArg(x_2)) [0] if(x0, x1, x2) -> c_if(x0, x1, x2) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: = => eq ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: not(x) -> if(x, false, true) [1] and(x, y) -> if(x, y, false) [1] or(x, y) -> if(x, true, y) [1] implies(x, y) -> if(x, y, true) [1] eq(x, x) -> true [1] eq(x, y) -> if(x, y, not(y)) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] eq(x, y) -> if(x, y, if(y, false, true)) [1] if(x, x, c_if(x, false, true)) -> true [1] encArg(false) -> false [0] encArg(true) -> true [0] encArg(cons_not(x_1)) -> not(encArg(x_1)) [0] encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0] encArg(cons_=(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_not(x_1) -> not(encArg(x_1)) [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_false -> false [0] encode_true -> true [0] encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0] encode_=(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) [0] if(x0, x1, x2) -> c_if(x0, x1, x2) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: not(x) -> if(x, false, true) [1] and(x, y) -> if(x, y, false) [1] or(x, y) -> if(x, true, y) [1] implies(x, y) -> if(x, y, true) [1] eq(x, x) -> true [1] eq(x, y) -> if(x, y, not(y)) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] eq(x, y) -> if(x, y, if(y, false, true)) [1] if(x, x, c_if(x, false, true)) -> true [1] encArg(false) -> false [0] encArg(true) -> true [0] encArg(cons_not(x_1)) -> not(encArg(x_1)) [0] encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0] encArg(cons_=(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_not(x_1) -> not(encArg(x_1)) [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_false -> false [0] encode_true -> true [0] encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0] encode_=(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) [0] if(x0, x1, x2) -> c_if(x0, x1, x2) [0] The TRS has the following type information: not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if eq :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if c_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encArg :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if cons_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if encode_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if Rewrite Strategy: INNERMOST ---------------------------------------- (13) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: not_1 and_2 eq_2 implies_2 or_2 encArg_1 encode_not_1 encode_if_3 encode_false encode_true encode_and_2 encode_or_2 encode_implies_2 encode_=_2 if_3 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_not(v0) -> null_encode_not [0] encode_if(v0, v1, v2) -> null_encode_if [0] encode_false -> null_encode_false [0] encode_true -> null_encode_true [0] encode_and(v0, v1) -> null_encode_and [0] encode_or(v0, v1) -> null_encode_or [0] encode_implies(v0, v1) -> null_encode_implies [0] encode_=(v0, v1) -> null_encode_= [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_encArg, null_encode_not, null_encode_if, null_encode_false, null_encode_true, null_encode_and, null_encode_or, null_encode_implies, null_encode_=, null_if ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: not(x) -> if(x, false, true) [1] and(x, y) -> if(x, y, false) [1] or(x, y) -> if(x, true, y) [1] implies(x, y) -> if(x, y, true) [1] eq(x, x) -> true [1] eq(x, y) -> if(x, y, not(y)) [1] if(true, x, y) -> x [1] if(false, x, y) -> y [1] eq(x, y) -> if(x, y, if(y, false, true)) [1] if(x, x, c_if(x, false, true)) -> true [1] encArg(false) -> false [0] encArg(true) -> true [0] encArg(cons_not(x_1)) -> not(encArg(x_1)) [0] encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0] encArg(cons_=(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_not(x_1) -> not(encArg(x_1)) [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_false -> false [0] encode_true -> true [0] encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0] encode_=(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) [0] if(x0, x1, x2) -> c_if(x0, x1, x2) [0] encArg(v0) -> null_encArg [0] encode_not(v0) -> null_encode_not [0] encode_if(v0, v1, v2) -> null_encode_if [0] encode_false -> null_encode_false [0] encode_true -> null_encode_true [0] encode_and(v0, v1) -> null_encode_and [0] encode_or(v0, v1) -> null_encode_or [0] encode_implies(v0, v1) -> null_encode_implies [0] encode_=(v0, v1) -> null_encode_= [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if eq :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if c_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encArg :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encArg :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: not(x) -> if(x, false, true) [1] and(x, y) -> if(x, y, false) [1] or(x, y) -> if(x, true, y) [1] implies(x, y) -> if(x, y, true) [1] eq(x, x) -> true [1] eq(x, y) -> if(x, y, if(y, false, true)) [2] if(true, x, y) -> x [1] if(false, x, y) -> y [1] eq(x, true) -> if(x, true, false) [2] eq(x, false) -> if(x, false, true) [2] eq(x, y) -> if(x, y, c_if(y, false, true)) [1] eq(x, y) -> if(x, y, null_if) [1] if(x, x, c_if(x, false, true)) -> true [1] encArg(false) -> false [0] encArg(true) -> true [0] encArg(cons_not(false)) -> not(false) [0] encArg(cons_not(true)) -> not(true) [0] encArg(cons_not(cons_not(x_1'))) -> not(not(encArg(x_1'))) [0] encArg(cons_not(cons_and(x_1'', x_2'))) -> not(and(encArg(x_1''), encArg(x_2'))) [0] encArg(cons_not(cons_or(x_11, x_2''))) -> not(or(encArg(x_11), encArg(x_2''))) [0] encArg(cons_not(cons_implies(x_12, x_21))) -> not(implies(encArg(x_12), encArg(x_21))) [0] encArg(cons_not(cons_=(x_13, x_22))) -> not(eq(encArg(x_13), encArg(x_22))) [0] encArg(cons_not(cons_if(x_14, x_23, x_3'))) -> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) [0] encArg(cons_not(x_1)) -> not(null_encArg) [0] encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0] encArg(cons_=(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_not(false) -> not(false) [0] encode_not(true) -> not(true) [0] encode_not(cons_not(x_1791)) -> not(not(encArg(x_1791))) [0] encode_not(cons_and(x_1792, x_2659)) -> not(and(encArg(x_1792), encArg(x_2659))) [0] encode_not(cons_or(x_1793, x_2660)) -> not(or(encArg(x_1793), encArg(x_2660))) [0] encode_not(cons_implies(x_1794, x_2661)) -> not(implies(encArg(x_1794), encArg(x_2661))) [0] encode_not(cons_=(x_1795, x_2662)) -> not(eq(encArg(x_1795), encArg(x_2662))) [0] encode_not(cons_if(x_1796, x_2663, x_3131)) -> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) [0] encode_not(x_1) -> not(null_encArg) [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_false -> false [0] encode_true -> true [0] encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0] encode_=(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) [0] if(x0, x1, x2) -> c_if(x0, x1, x2) [0] encArg(v0) -> null_encArg [0] encode_not(v0) -> null_encode_not [0] encode_if(v0, v1, v2) -> null_encode_if [0] encode_false -> null_encode_false [0] encode_true -> null_encode_true [0] encode_and(v0, v1) -> null_encode_and [0] encode_or(v0, v1) -> null_encode_or [0] encode_implies(v0, v1) -> null_encode_implies [0] encode_=(v0, v1) -> null_encode_= [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if eq :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if c_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encArg :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if cons_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if encode_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if -> false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encArg :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_not :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_false :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_true :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_and :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_or :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_implies :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_encode_= :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if null_if :: false:true:c_if:cons_not:cons_and:cons_or:cons_implies:cons_=:cons_if:null_encArg:null_encode_not:null_encode_if:null_encode_false:null_encode_true:null_encode_and:null_encode_or:null_encode_implies:null_encode_=:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: false => 0 true => 1 null_encArg => 0 null_encode_not => 0 null_encode_if => 0 null_encode_false => 0 null_encode_true => 0 null_encode_and => 0 null_encode_or => 0 null_encode_implies => 0 null_encode_= => 0 null_if => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 1 }-> if(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> not(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> not(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> not(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_=(z, z') -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_=(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_and(z, z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_implies(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(x_1791))) :|: z = 1 + x_1791, x_1791 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> not(1) :|: z = 1 encode_not(z) -{ 0 }-> not(0) :|: z = 0 encode_not(z) -{ 0 }-> not(0) :|: x_1 >= 0, z = x_1 encode_not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_or(z, z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 2 }-> if(x, y, if(y, 0, 1)) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 1 }-> if(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 1 }-> if(x, y, 1 + y + 0 + 1) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 2 }-> if(x, 1, 0) :|: x >= 0, z' = 1, z = x eq(z, z') -{ 2 }-> if(x, 0, 1) :|: x >= 0, z = x, z' = 0 eq(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = x if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + x + 0 + 1, z = x if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'') -{ 0 }-> 1 + x0 + x1 + x2 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1, z'' = x2, x2 >= 0 implies(z, z') -{ 1 }-> if(x, y, 1) :|: x >= 0, y >= 0, z = x, z' = y not(z) -{ 1 }-> if(x, 0, 1) :|: x >= 0, z = x or(z, z') -{ 1 }-> if(x, 1, y) :|: x >= 0, y >= 0, z = x, z' = y ---------------------------------------- (19) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if(z, z', z'') -{ 0 }-> 1 + x0 + x1 + x2 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1, z'' = x2, x2 >= 0 if(z, z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + x + 0 + 1, z = x if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 not(z) -{ 1 }-> if(x, 0, 1) :|: x >= 0, z = x and(z, z') -{ 1 }-> if(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y or(z, z') -{ 1 }-> if(x, 1, y) :|: x >= 0, y >= 0, z = x, z' = y implies(z, z') -{ 1 }-> if(x, y, 1) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = x eq(z, z') -{ 1 }-> if(x, y, 1 + y + 0 + 1) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 2 }-> if(x, y, if(y, 0, 1)) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 1 }-> if(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y eq(z, z') -{ 2 }-> if(x, 1, 0) :|: x >= 0, z' = 1, z = x eq(z, z') -{ 2 }-> if(x, 0, 1) :|: x >= 0, z = x, z' = 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 0 = y', x = 1, x' >= 0, y' >= 0 and(z, z') -{ 2 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 0 = y', x' >= 0, y' >= 0, x = 0 and(z, z') -{ 1 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 0 = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 and(z, z') -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, x = x0, x0 >= 0, x1 >= 0, y = x1, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + x_1, x_1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_=(z, z') -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_=(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_and(z, z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_implies(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(x_1791))) :|: z = 1 + x_1791, x_1791 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: x_1 >= 0, z = x_1, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_or(z, z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, y = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, y = x', 1 + x0 + x1 + x2 = y', x = 1, x' >= 0, y' >= 0 eq(z, z') -{ 3 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 = v2, v1 >= 0, y = v0, 0 = v1, v2 >= 0, y = x', 0 = y', x = 1, x' >= 0, y' >= 0 eq(z, z') -{ 3 }-> x' :|: x >= 0, z' = 1, z = x, 1 = x', 0 = y, x = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: x >= 0, z = x, z' = 0, 0 = x', 1 = y, x = 1, x' >= 0, y >= 0 eq(z, z') -{ 2 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 1 + y + 0 + 1 = y', x = 1, x' >= 0, y' >= 0 eq(z, z') -{ 2 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 0 = y', x = 1, x' >= 0, y' >= 0 eq(z, z') -{ 4 }-> x'' :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', x' >= 0, y' >= 0, y = 0, y = x'', y' = y'', x = 1, x'' >= 0, y'' >= 0 eq(z, z') -{ 4 }-> x'' :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', y = 1, x' >= 0, y' >= 0, y = x'', x' = y'', x = 1, x'' >= 0, y'' >= 0 eq(z, z') -{ 3 }-> y :|: x >= 0, z' = 1, z = x, 1 = x', 0 = y, x' >= 0, y >= 0, x = 0 eq(z, z') -{ 3 }-> y :|: x >= 0, z = x, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, x = 0 eq(z, z') -{ 3 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, y = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, y = x', 1 + x0 + x1 + x2 = y', x' >= 0, y' >= 0, x = 0 eq(z, z') -{ 3 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 = v2, v1 >= 0, y = v0, 0 = v1, v2 >= 0, y = x', 0 = y', x' >= 0, y' >= 0, x = 0 eq(z, z') -{ 2 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 1 + y + 0 + 1 = y', x' >= 0, y' >= 0, x = 0 eq(z, z') -{ 2 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 0 = y', x' >= 0, y' >= 0, x = 0 eq(z, z') -{ 4 }-> y'' :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', x' >= 0, y' >= 0, y = 0, y = x'', y' = y'', x'' >= 0, y'' >= 0, x = 0 eq(z, z') -{ 4 }-> y'' :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', y = 1, x' >= 0, y' >= 0, y = x'', x' = y'', x'' >= 0, y'' >= 0, x = 0 eq(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = x eq(z, z') -{ 3 }-> 1 :|: x >= 0, y >= 0, z = x, z' = y, y = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, y = x', x' >= 0, 1 + x0 + x1 + x2 = 1 + x' + 0 + 1, x = x' eq(z, z') -{ 2 }-> 1 :|: x >= 0, y >= 0, z = x, z' = y, y = x', x' >= 0, 1 + y + 0 + 1 = 1 + x' + 0 + 1, x = x' eq(z, z') -{ 3 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', x' >= 0, y' >= 0, y = 0, v0 >= 0, y' = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, y = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, v0 >= 0, 1 + x0 + x1 + x2 = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', y = 1, x' >= 0, y' >= 0, v0 >= 0, x' = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 = v2, v1 >= 0, y = v0, 0 = v1, v2 >= 0, v0' >= 0, 0 = v2', v1' >= 0, x = v0', y = v1', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: x >= 0, z' = 1, z = x, v0 >= 0, 0 = v2, v1 >= 0, x = v0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: x >= 0, z = x, z' = 0, v0 >= 0, 1 = v2, v1 >= 0, x = v0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 + y + 0 + 1 = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 0 = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 eq(z, z') -{ 3 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', x' >= 0, y' >= 0, y = 0, x = x0, x0 >= 0, x1 >= 0, y = x1, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, 0 = x', 1 = y', y = 1, x' >= 0, y' >= 0, x = x0, x0 >= 0, x1 >= 0, y = x1, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 = v2, v1 >= 0, y = v0, 0 = v1, v2 >= 0, x = x0, x0 >= 0, x1 >= 0, y = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + x0 + x1 + x2 :|: x >= 0, z' = 1, z = x, x = x0, x0 >= 0, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + x0 + x1 + x2 :|: x >= 0, z = x, z' = 0, x = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, x = x0, x0 >= 0, x1 >= 0, y = x1, 1 + y + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, x = x0, x0 >= 0, x1 >= 0, y = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + x0' + x1' + x2' :|: x >= 0, y >= 0, z = x, z' = y, y = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, x = x0', x0' >= 0, x1' >= 0, y = x1', 1 + x0 + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + x + 0 + 1, z = x if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'') -{ 0 }-> 1 + x0 + x1 + x2 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1, z'' = x2, x2 >= 0 implies(z, z') -{ 2 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 1 = y', x = 1, x' >= 0, y' >= 0 implies(z, z') -{ 2 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, y = x', 1 = y', x' >= 0, y' >= 0, x = 0 implies(z, z') -{ 1 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, 1 = v2, v1 >= 0, x = v0, y = v1, v2 >= 0 implies(z, z') -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, x = x0, x0 >= 0, x1 >= 0, y = x1, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: x >= 0, z = x, 0 = x', 1 = y, x = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: x >= 0, z = x, 0 = x', 1 = y, x' >= 0, y >= 0, x = 0 not(z) -{ 1 }-> 0 :|: x >= 0, z = x, v0 >= 0, 1 = v2, v1 >= 0, x = v0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, z = x, x = x0, x0 >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: x >= 0, y >= 0, z = x, z' = y, 1 = x', y = y', x = 1, x' >= 0, y' >= 0 or(z, z') -{ 2 }-> y' :|: x >= 0, y >= 0, z = x, z' = y, 1 = x', y = y', x' >= 0, y' >= 0, x = 0 or(z, z') -{ 2 }-> 1 :|: x >= 0, y >= 0, z = x, z' = y, 1 = x', x' >= 0, y = 1 + x' + 0 + 1, x = x' or(z, z') -{ 1 }-> 0 :|: x >= 0, y >= 0, z = x, z' = y, v0 >= 0, y = v2, v1 >= 0, x = v0, 1 = v1, v2 >= 0 or(z, z') -{ 1 }-> 1 + x0 + x1 + x2 :|: x >= 0, y >= 0, z = x, z' = y, x = x0, x0 >= 0, x1 >= 0, 1 = x1, y = x2, x2 >= 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { not } { implies } { encode_false } { eq } { and } { encode_true } { if } { or } { encArg } { encode_= } { encode_implies } { encode_or } { encode_and } { encode_not } { encode_if } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {not}, {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {not}, {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: not after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {not}, {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: not after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: implies after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {implies}, {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: ?, size: O(n^1) [2 + z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: implies after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_false after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_false}, {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: ?, size: O(1) [0] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_false after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + 2*z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {eq}, {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: ?, size: O(n^1) [3 + z + 2*z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {and}, {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_true}, {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: ?, size: O(1) [1] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' + z'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {if}, {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: ?, size: O(n^1) [1 + z + z' + z''] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 1 }-> if(x, 0, 1) :|: z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 0, x >= 0, 0 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z = 1, x >= 0, 1 = x encode_not(z) -{ 1 }-> if(x, 0, 1) :|: z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {or}, {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: ?, size: O(n^1) [2 + z + z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encArg}, {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: ?, size: EXP ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 6*z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> not(or(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> not(implies(encArg(x_12), encArg(x_21))) :|: z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> not(if(encArg(x_14), encArg(x_23), encArg(x_3'))) :|: x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 0 }-> not(eq(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> not(and(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> eq(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 0 }-> eq(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> and(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 0 }-> implies(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 0 }-> not(or(encArg(x_1793), encArg(x_2660))) :|: x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0 encode_not(z) -{ 0 }-> not(implies(encArg(x_1794), encArg(x_2661))) :|: x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 0 }-> not(if(encArg(x_1796), encArg(x_2663), encArg(x_3131))) :|: z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 0 }-> not(eq(encArg(x_1795), encArg(x_2662))) :|: z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> not(and(encArg(x_1792), encArg(x_2659))) :|: x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_= after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_=}, {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: ?, size: INF ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_= after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 8 + 6*z + 6*z' ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_implies after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_implies}, {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: ?, size: INF ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_implies after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 6*z + 6*z' ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_or after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_or}, {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: ?, size: INF ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_or after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 6*z + 6*z' ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (99) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_and after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_and}, {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: ?, size: INF ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_and after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 6*z + 6*z' ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (103) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF ---------------------------------------- (105) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_not after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_not}, {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_not: runtime: ?, size: INF ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_not after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 6*z ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_not: runtime: O(n^1) [4 + 6*z], size: INF ---------------------------------------- (109) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_not: runtime: O(n^1) [4 + 6*z], size: INF ---------------------------------------- (111) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: {encode_if} Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_not: runtime: O(n^1) [4 + 6*z], size: INF encode_if: runtime: ?, size: INF ---------------------------------------- (113) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 7 + 6*z + 6*z' + 6*z'' ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: and(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 and(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 and(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 encArg(z) -{ 2 }-> s :|: s >= 0, s <= x + 0 + 1 + 1, z = 1 + 0, x >= 0, 0 = x encArg(z) -{ 2 }-> s' :|: s' >= 0, s' <= x + 0 + 1 + 1, z = 1 + 1, x >= 0, 1 = x encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 0 + 1 + 1, z - 1 >= 0, x >= 0, 0 = x encArg(z) -{ -6 + 6*z }-> s14 :|: s12 >= 0, s12 <= inf4, s13 >= 0, s13 <= s12 + 2, s14 >= 0, s14 <= s13 + 2, z - 2 >= 0 encArg(z) -{ 9 + 6*x_14 + 6*x_23 + 6*x_3' }-> s22 :|: s18 >= 0, s18 <= inf6, s19 >= 0, s19 <= inf7, s20 >= 0, s20 <= inf8, s21 >= 0, s21 <= s18 + s19 + s20 + 1, s22 >= 0, s22 <= s21 + 2, x_14 >= 0, x_3' >= 0, x_23 >= 0, z = 1 + (1 + x_14 + x_23 + x_3') encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s30 :|: s28 >= 0, s28 <= inf12, s29 >= 0, s29 <= inf13, s30 >= 0, s30 <= s28 + s29 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_1'' + 6*x_2' }-> s37 :|: s34 >= 0, s34 <= inf16, s35 >= 0, s35 <= inf17, s36 >= 0, s36 <= s34 + s35 + 1, s37 >= 0, s37 <= s36 + 2, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'), x_2' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s44 :|: s42 >= 0, s42 <= inf20, s43 >= 0, s43 <= inf21, s44 >= 0, s44 <= s42 + s43 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_11 + 6*x_2'' }-> s51 :|: s48 >= 0, s48 <= inf24, s49 >= 0, s49 <= inf25, s50 >= 0, s50 <= s48 + s49 + 2, s51 >= 0, s51 <= s50 + 2, x_11 >= 0, z = 1 + (1 + x_11 + x_2''), x_2'' >= 0 encArg(z) -{ 6 + 6*x_1 + 6*x_2 }-> s58 :|: s56 >= 0, s56 <= inf28, s57 >= 0, s57 <= inf29, s58 >= 0, s58 <= s56 + s57 + 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 8 + 6*x_12 + 6*x_21 }-> s65 :|: s62 >= 0, s62 <= inf32, s63 >= 0, s63 <= inf33, s64 >= 0, s64 <= s62 + s63 + 2, s65 >= 0, s65 <= s64 + 2, z = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0 encArg(z) -{ 7 + 6*x_1 + 6*x_2 + 6*x_3 }-> s7 :|: s4 >= 0, s4 <= inf, s5 >= 0, s5 <= inf', s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= s4 + s5 + s6 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 8 + 6*x_1 + 6*x_2 }-> s72 :|: s70 >= 0, s70 <= inf36, s71 >= 0, s71 <= inf37, s72 >= 0, s72 <= s70 + 2 * s71 + 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 10 + 6*x_13 + 6*x_22 }-> s79 :|: s76 >= 0, s76 <= inf40, s77 >= 0, s77 <= inf41, s78 >= 0, s78 <= s76 + 2 * s77 + 3, s79 >= 0, s79 <= s78 + 2, x_13 >= 0, z = 1 + (1 + x_13 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_=(z, z') -{ 8 + 6*z + 6*z' }-> s75 :|: s73 >= 0, s73 <= inf38, s74 >= 0, s74 <= inf39, s75 >= 0, s75 <= s73 + 2 * s74 + 3, z >= 0, z' >= 0 encode_=(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_and(z, z') -{ 6 + 6*z + 6*z' }-> s33 :|: s31 >= 0, s31 <= inf14, s32 >= 0, s32 <= inf15, s33 >= 0, s33 <= s31 + s32 + 1, z >= 0, z' >= 0 encode_and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 7 + 6*z + 6*z' + 6*z'' }-> s11 :|: s8 >= 0, s8 <= inf1, s9 >= 0, s9 <= inf2, s10 >= 0, s10 <= inf3, s11 >= 0, s11 <= s8 + s9 + s10 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_implies(z, z') -{ 6 + 6*z + 6*z' }-> s61 :|: s59 >= 0, s59 <= inf30, s60 >= 0, s60 <= inf31, s61 >= 0, s61 <= s59 + s60 + 2, z >= 0, z' >= 0 encode_implies(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_not(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 0 + 1 + 1, z = 0, x >= 0, 0 = x encode_not(z) -{ 6*z }-> s17 :|: s15 >= 0, s15 <= inf5, s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 2, z - 1 >= 0 encode_not(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 0 + 1 + 1, z = 1, x >= 0, 1 = x encode_not(z) -{ 9 + 6*x_1796 + 6*x_2663 + 6*x_3131 }-> s27 :|: s23 >= 0, s23 <= inf9, s24 >= 0, s24 <= inf10, s25 >= 0, s25 <= inf11, s26 >= 0, s26 <= s23 + s24 + s25 + 1, s27 >= 0, s27 <= s26 + 2, z = 1 + x_1796 + x_2663 + x_3131, x_2663 >= 0, x_3131 >= 0, x_1796 >= 0 encode_not(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 0 + 1 + 1, z >= 0, x >= 0, 0 = x encode_not(z) -{ 8 + 6*x_1792 + 6*x_2659 }-> s41 :|: s38 >= 0, s38 <= inf18, s39 >= 0, s39 <= inf19, s40 >= 0, s40 <= s38 + s39 + 1, s41 >= 0, s41 <= s40 + 2, x_1792 >= 0, x_2659 >= 0, z = 1 + x_1792 + x_2659 encode_not(z) -{ 8 + 6*x_1793 + 6*x_2660 }-> s55 :|: s52 >= 0, s52 <= inf26, s53 >= 0, s53 <= inf27, s54 >= 0, s54 <= s52 + s53 + 2, s55 >= 0, s55 <= s54 + 2, x_1793 >= 0, x_2660 >= 0, z = 1 + x_1793 + x_2660 encode_not(z) -{ 8 + 6*x_1794 + 6*x_2661 }-> s69 :|: s66 >= 0, s66 <= inf34, s67 >= 0, s67 <= inf35, s68 >= 0, s68 <= s66 + s67 + 2, s69 >= 0, s69 <= s68 + 2, x_2661 >= 0, z = 1 + x_1794 + x_2661, x_1794 >= 0 encode_not(z) -{ 10 + 6*x_1795 + 6*x_2662 }-> s83 :|: s80 >= 0, s80 <= inf42, s81 >= 0, s81 <= inf43, s82 >= 0, s82 <= s80 + 2 * s81 + 3, s83 >= 0, s83 <= s82 + 2, z = 1 + x_1795 + x_2662, x_1795 >= 0, x_2662 >= 0 encode_not(z) -{ 0 }-> 0 :|: z >= 0 encode_or(z, z') -{ 6 + 6*z + 6*z' }-> s47 :|: s45 >= 0, s45 <= inf22, s46 >= 0, s46 <= inf23, s47 >= 0, s47 <= s45 + s46 + 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_true -{ 0 }-> 1 :|: encode_true -{ 0 }-> 0 :|: eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 1, 1 = x', 0 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> x' :|: z >= 0, z' = 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 1, 1 = x', 0 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y :|: z >= 0, z' = 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', y' >= 0, z = 0 eq(z, z') -{ 3 }-> y' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', y' >= 0, z = 0 eq(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 0 = y', y' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> y'' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', y'' >= 0, z = 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = y', z = 1, y' >= 0 eq(z, z') -{ 4 }-> z' :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = y'', z = 1, y'' >= 0 eq(z, z') -{ 3 }-> z' :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = y', z = 1, y' >= 0 eq(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 0 = y', z = 1, y' >= 0 eq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = z' eq(z, z') -{ 3 }-> 1 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = 1 + z' + 0 + 1, z = z' eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = v2, v2 >= 0 eq(z, z') -{ 3 }-> 0 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = v2, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = v2', v2' >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1, 0 = v2, v1 >= 0, 1 = v1, v2 >= 0 eq(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = v2, v2 >= 0 eq(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 0 = v2, v2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 1, x1 >= 0, 1 = x1, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + x1 + x2 :|: z >= 0, z' = 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', x' >= 0, y' >= 0, z' = 0, y' = x2, x2 >= 0 eq(z, z') -{ 3 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x', 1 = y', z' = 1, x' >= 0, y' >= 0, x' = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 + z' + 0 + 1 = x2, x2 >= 0 eq(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 0 = x2, x2 >= 0 eq(z, z') -{ 2 }-> 1 + z + z' + x2' :|: z >= 0, z' >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0, 1 + z' + x1 + x2 = x2', x2' >= 0 if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0 if(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + z' + 0 + 1, z = z' if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z >= 0, z' >= 0, z'' >= 0 implies(z, z') -{ 2 }-> y' :|: z >= 0, z' >= 0, 1 = y', y' >= 0, z = 0 implies(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = y', z = 1, y' >= 0 implies(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, 1 = v2, v2 >= 0 implies(z, z') -{ 1 }-> 1 + z + z' + x2 :|: z >= 0, z' >= 0, 1 = x2, x2 >= 0 not(z) -{ 2 }-> x' :|: z >= 0, 0 = x', 1 = y, z = 1, x' >= 0, y >= 0 not(z) -{ 2 }-> y :|: z >= 0, 0 = x', 1 = y, x' >= 0, y >= 0, z = 0 not(z) -{ 1 }-> 0 :|: z >= 0, 1 = v2, v1 >= 0, 0 = v1, v2 >= 0 not(z) -{ 1 }-> 1 + z + x1 + x2 :|: z >= 0, x1 >= 0, 0 = x1, 1 = x2, x2 >= 0 or(z, z') -{ 2 }-> x' :|: z >= 0, z' >= 0, 1 = x', z = 1, x' >= 0 or(z, z') -{ 2 }-> z' :|: z >= 0, z' >= 0, 1 = x', x' >= 0, z = 0 or(z, z') -{ 2 }-> 1 :|: z >= 0, z' >= 0, 1 = z' - 2, z' - 2 >= 0, z = z' - 2 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v1 >= 0, 1 = v1 or(z, z') -{ 1 }-> 1 + z + x1 + z' :|: z >= 0, z' >= 0, x1 >= 0, 1 = x1 Function symbols to be analyzed: Previous analysis results are: not: runtime: O(1) [2], size: O(n^1) [2 + z] implies: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encode_false: runtime: O(1) [0], size: O(1) [0] eq: runtime: O(1) [4], size: O(n^1) [3 + z + 2*z'] and: runtime: O(1) [2], size: O(n^1) [1 + z + z'] encode_true: runtime: O(1) [0], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [1 + z + z' + z''] or: runtime: O(1) [2], size: O(n^1) [2 + z + z'] encArg: runtime: O(n^1) [2 + 6*z], size: EXP encode_=: runtime: O(n^1) [8 + 6*z + 6*z'], size: INF encode_implies: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_or: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_and: runtime: O(n^1) [6 + 6*z + 6*z'], size: INF encode_not: runtime: O(n^1) [4 + 6*z], size: INF encode_if: runtime: O(n^1) [7 + 6*z + 6*z' + 6*z''], size: INF ---------------------------------------- (115) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (116) BOUNDS(1, n^1)