WORST_CASE(?,O(n^1)) proof of input_lmGH4pRUVq.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 110 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 1017 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 264 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 59 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 130 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 254 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1048 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 1475 ms] (48) CpxRNTS (49) FinalProof [FINISHED, 0 ms] (50) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) a(Z, y) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) a(Z, y) -> Z eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) eqZList(C(x1, x2), Z) -> False eqZList(Z, C(y1, y2)) -> False eqZList(Z, Z) -> True second(C(x1, x2)) -> x2 first(C(x1, x2)) -> x1 The (relative) TRS S consists of the following rules: and(False, False) -> False and(True, False) -> False and(False, True) -> False and(True, True) -> True Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) 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: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1] a(Z, y) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1] a(Z, y) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] The TRS has the following type information: a :: C:Z -> C:Z -> C:Z C :: C:Z -> C:Z -> C:Z Z :: C:Z eqZList :: C:Z -> C:Z -> False:True and :: False:True -> False:True -> False:True False :: False:True True :: False:True second :: C:Z -> C:Z first :: C:Z -> C:Z Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: a_2 second_1 first_1 (c) The following functions are completely defined: eqZList_2 and_2 Due to the following rules being added: and(v0, v1) -> null_and [0] And the following fresh constants: null_and ---------------------------------------- (8) 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: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1] a(Z, y) -> Z [1] eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] and(v0, v1) -> null_and [0] The TRS has the following type information: a :: C:Z -> C:Z -> C:Z C :: C:Z -> C:Z -> C:Z Z :: C:Z eqZList :: C:Z -> C:Z -> False:True:null_and and :: False:True:null_and -> False:True:null_and -> False:True:null_and False :: False:True:null_and True :: False:True:null_and second :: C:Z -> C:Z first :: C:Z -> C:Z null_and :: False:True:null_and Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) 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: a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1] a(Z, y) -> Z [1] eqZList(C(C(x1', x2'), C(x11, x21)), C(C(y1', y2'), C(y11, y21))) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) [3] eqZList(C(C(x1', x2'), C(x12, x22)), C(C(y1', y2'), Z)) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), False) [3] eqZList(C(C(x1', x2'), Z), C(C(y1', y2'), C(y12, y22))) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), False) [3] eqZList(C(C(x1', x2'), Z), C(C(y1', y2'), Z)) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), True) [3] eqZList(C(C(x1'', x2''), C(x13, x23)), C(Z, C(y13, y23))) -> and(False, and(eqZList(x13, y13), eqZList(x23, y23))) [3] eqZList(C(C(x1'', x2''), C(x14, x24)), C(Z, Z)) -> and(False, False) [3] eqZList(C(C(x1'', x2''), Z), C(Z, C(y14, y24))) -> and(False, False) [3] eqZList(C(C(x1'', x2''), Z), C(Z, Z)) -> and(False, True) [3] eqZList(C(Z, C(x15, x25)), C(C(y1'', y2''), C(y15, y25))) -> and(False, and(eqZList(x15, y15), eqZList(x25, y25))) [3] eqZList(C(Z, C(x16, x26)), C(C(y1'', y2''), Z)) -> and(False, False) [3] eqZList(C(Z, Z), C(C(y1'', y2''), C(y16, y26))) -> and(False, False) [3] eqZList(C(Z, Z), C(C(y1'', y2''), Z)) -> and(False, True) [3] eqZList(C(Z, C(x17, x27)), C(Z, C(y17, y27))) -> and(True, and(eqZList(x17, y17), eqZList(x27, y27))) [3] eqZList(C(Z, C(x18, x28)), C(Z, Z)) -> and(True, False) [3] eqZList(C(Z, Z), C(Z, C(y18, y28))) -> and(True, False) [3] eqZList(C(Z, Z), C(Z, Z)) -> and(True, True) [3] eqZList(C(x1, x2), Z) -> False [1] eqZList(Z, C(y1, y2)) -> False [1] eqZList(Z, Z) -> True [1] second(C(x1, x2)) -> x2 [1] first(C(x1, x2)) -> x1 [1] and(False, False) -> False [0] and(True, False) -> False [0] and(False, True) -> False [0] and(True, True) -> True [0] and(v0, v1) -> null_and [0] The TRS has the following type information: a :: C:Z -> C:Z -> C:Z C :: C:Z -> C:Z -> C:Z Z :: C:Z eqZList :: C:Z -> C:Z -> False:True:null_and and :: False:True:null_and -> False:True:null_and -> False:True:null_and False :: False:True:null_and True :: False:True:null_and second :: C:Z -> C:Z first :: C:Z -> C:Z null_and :: False:True:null_and Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Z => 0 False => 1 True => 2 null_and => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y a(z, z') -{ 1 }-> 1 + a(x1, y) + a(x2, 1 + x1 + x2) :|: x1 >= 0, y >= 0, z = 1 + x1 + x2, z' = y, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(2, 2) :|: z = 1 + 0 + 0, z' = 1 + 0 + 0 eqZList(z, z') -{ 3 }-> and(2, 1) :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0 eqZList(z, z') -{ 3 }-> and(2, 1) :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 3 }-> and(1, 2) :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, 2) :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0 eqZList(z, z') -{ 3 }-> and(1, 1) :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, 1) :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0 eqZList(z, z') -{ 3 }-> and(1, 1) :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0 eqZList(z, z') -{ 3 }-> and(1, 1) :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y a(z, z') -{ 1 }-> 1 + a(x1, y) + a(x2, 1 + x1 + x2) :|: x1 >= 0, y >= 0, z = 1 + x1 + x2, z' = y, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { and } { first } { second } { a } { eqZList } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList} Previous analysis results are: and: runtime: ?, size: O(1) [2] ---------------------------------------- (23) 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: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {first}, {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (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: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {first}, {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {first}, {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: first after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [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: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: second after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {second}, {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: ?, size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: second after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [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: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: a after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {a}, {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] a: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: a after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z] ---------------------------------------- (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: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eqZList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: {eqZList} Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z] eqZList: runtime: ?, size: O(1) [2] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eqZList after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 78 + 156*z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0 and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0 eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22) eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22) eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27) eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0 eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0 eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0 eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2 eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0 eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0 Function symbols to be analyzed: Previous analysis results are: and: runtime: O(1) [0], size: O(1) [2] first: runtime: O(1) [1], size: O(n^1) [z] second: runtime: O(1) [1], size: O(n^1) [z] a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z] eqZList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2] ---------------------------------------- (49) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (50) BOUNDS(1, n^1)