WORST_CASE(Omega(n^1),O(n^3)) proof of input_Gpip4l5sQT.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRNTS (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) IntTrsBoundProof [UPPER BOUND(ID), 179 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 68 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 373 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 415 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 111 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 28 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 866 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 263 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 1757 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 534 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 4706 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 1360 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 447 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (68) CpxRNTS (69) FinalProof [FINISHED, 0 ms] (70) BOUNDS(1, n^3) (71) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 46 ms] (72) CdtProblem (73) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CpxRelTRS (75) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CpxRelTRS (77) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (78) typed CpxTrs (79) OrderProof [LOWER BOUND(ID), 45 ms] (80) typed CpxTrs (81) RewriteLemmaProof [LOWER BOUND(ID), 380 ms] (82) BEST (83) proven lower bound (84) LowerBoundPropagationProof [FINISHED, 0 ms] (85) BOUNDS(n^1, INF) (86) typed CpxTrs (87) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (88) typed CpxTrs (89) RewriteLemmaProof [LOWER BOUND(ID), 20 ms] (90) typed CpxTrs (91) RewriteLemmaProof [LOWER BOUND(ID), 650 ms] (92) typed CpxTrs (93) RewriteLemmaProof [LOWER BOUND(ID), 43 ms] (94) typed CpxTrs (95) RewriteLemmaProof [LOWER BOUND(ID), 51 ms] (96) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(m)) -> false eq(s(n), 0) -> false eq(s(n), s(m)) -> eq(n, m) le(0, m) -> true le(s(n), 0) -> false le(s(n), s(m)) -> le(n, m) min(cons(x, nil)) -> x min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) replace(n, m, nil) -> nil replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) if_replace(true, n, m, cons(k, x)) -> cons(m, x) if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) empty(nil) -> true empty(cons(n, x)) -> false head(cons(n, x)) -> n tail(nil) -> nil tail(cons(n, x)) -> x sort(x) -> sortIter(x, nil) sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) if(true, x, y, z) -> y if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(m)) -> false [1] eq(s(n), 0) -> false [1] eq(s(n), s(m)) -> eq(n, m) [1] le(0, m) -> true [1] le(s(n), 0) -> false [1] le(s(n), s(m)) -> le(n, m) [1] min(cons(x, nil)) -> x [1] min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] replace(n, m, nil) -> nil [1] replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] empty(nil) -> true [1] empty(cons(n, x)) -> false [1] head(cons(n, x)) -> n [1] tail(nil) -> nil [1] tail(cons(n, x)) -> x [1] sort(x) -> sortIter(x, nil) [1] sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: eq(0, 0) -> true [1] eq(0, s(m)) -> false [1] eq(s(n), 0) -> false [1] eq(s(n), s(m)) -> eq(n, m) [1] le(0, m) -> true [1] le(s(n), 0) -> false [1] le(s(n), s(m)) -> le(n, m) [1] min(cons(x, nil)) -> x [1] min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] replace(n, m, nil) -> nil [1] replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] empty(nil) -> true [1] empty(cons(n, x)) -> false [1] head(cons(n, x)) -> n [1] tail(nil) -> nil [1] tail(cons(n, x)) -> x [1] sort(x) -> sortIter(x, nil) [1] sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false le :: 0:s -> 0:s -> true:false min :: nil:cons:append -> 0:s cons :: 0:s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append if_min :: true:false -> nil:cons:append -> 0:s replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append if_replace :: true:false -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append empty :: nil:cons:append -> true:false head :: nil:cons:append -> 0:s tail :: nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append append :: nil:cons:append -> nil:cons:append -> nil:cons:append Rewrite Strategy: INNERMOST ---------------------------------------- (5) 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: sort_1 sortIter_2 if_4 (c) The following functions are completely defined: empty_1 min_1 replace_3 head_1 tail_1 eq_2 le_2 if_replace_4 if_min_2 Due to the following rules being added: empty(v0) -> null_empty [0] min(v0) -> 0 [0] replace(v0, v1, v2) -> nil [0] head(v0) -> 0 [0] tail(v0) -> nil [0] if_replace(v0, v1, v2, v3) -> nil [0] if_min(v0, v1) -> 0 [0] And the following fresh constants: null_empty ---------------------------------------- (6) 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: eq(0, 0) -> true [1] eq(0, s(m)) -> false [1] eq(s(n), 0) -> false [1] eq(s(n), s(m)) -> eq(n, m) [1] le(0, m) -> true [1] le(s(n), 0) -> false [1] le(s(n), s(m)) -> le(n, m) [1] min(cons(x, nil)) -> x [1] min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] replace(n, m, nil) -> nil [1] replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] empty(nil) -> true [1] empty(cons(n, x)) -> false [1] head(cons(n, x)) -> n [1] tail(nil) -> nil [1] tail(cons(n, x)) -> x [1] sort(x) -> sortIter(x, nil) [1] sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] if(true, x, y, z) -> y [1] if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] empty(v0) -> null_empty [0] min(v0) -> 0 [0] replace(v0, v1, v2) -> nil [0] head(v0) -> 0 [0] tail(v0) -> nil [0] if_replace(v0, v1, v2, v3) -> nil [0] if_min(v0, v1) -> 0 [0] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false:null_empty 0 :: 0:s true :: true:false:null_empty s :: 0:s -> 0:s false :: true:false:null_empty le :: 0:s -> 0:s -> true:false:null_empty min :: nil:cons:append -> 0:s cons :: 0:s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append if_min :: true:false:null_empty -> nil:cons:append -> 0:s replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append empty :: nil:cons:append -> true:false:null_empty head :: nil:cons:append -> 0:s tail :: nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append append :: nil:cons:append -> nil:cons:append -> nil:cons:append null_empty :: true:false:null_empty Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: eq(0, 0) -> true [1] eq(0, s(m)) -> false [1] eq(s(n), 0) -> false [1] eq(s(n), s(m)) -> eq(n, m) [1] le(0, m) -> true [1] le(s(n), 0) -> false [1] le(s(n), s(m)) -> le(n, m) [1] min(cons(x, nil)) -> x [1] min(cons(0, cons(m, x))) -> if_min(true, cons(0, cons(m, x))) [2] min(cons(s(n'), cons(0, x))) -> if_min(false, cons(s(n'), cons(0, x))) [2] min(cons(s(n''), cons(s(m'), x))) -> if_min(le(n'', m'), cons(s(n''), cons(s(m'), x))) [2] if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] replace(n, m, nil) -> nil [1] replace(0, m, cons(0, x)) -> if_replace(true, 0, m, cons(0, x)) [2] replace(0, m, cons(s(m''), x)) -> if_replace(false, 0, m, cons(s(m''), x)) [2] replace(s(n1), m, cons(0, x)) -> if_replace(false, s(n1), m, cons(0, x)) [2] replace(s(n2), m, cons(s(m1), x)) -> if_replace(eq(n2, m1), s(n2), m, cons(s(m1), x)) [2] if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] empty(nil) -> true [1] empty(cons(n, x)) -> false [1] head(cons(n, x)) -> n [1] tail(nil) -> nil [1] tail(cons(n, x)) -> x [1] sort(x) -> sortIter(x, nil) [1] sortIter(nil, y) -> if(true, nil, y, append(y, cons(0, nil))) [2] sortIter(cons(n3, nil), y) -> if(false, cons(n3, nil), y, append(y, cons(n3, nil))) [3] sortIter(cons(n3, cons(m2, x'')), y) -> if(false, cons(n3, cons(m2, x'')), y, append(y, cons(if_min(le(n3, m2), cons(n3, cons(m2, x''))), nil))) [3] sortIter(cons(n3, x'), y) -> if(false, cons(n3, x'), y, append(y, cons(0, nil))) [2] sortIter(cons(x1, nil), y) -> if(null_empty, cons(x1, nil), y, append(y, cons(x1, nil))) [2] sortIter(cons(n4, cons(m3, x2)), y) -> if(null_empty, cons(n4, cons(m3, x2)), y, append(y, cons(if_min(le(n4, m3), cons(n4, cons(m3, x2))), nil))) [2] sortIter(x, y) -> if(null_empty, x, y, append(y, cons(0, nil))) [1] if(true, x, y, z) -> y [1] if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [4] if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [3] if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [3] if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [2] if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), n5, cons(m4, x4)), z) [4] if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), n5, nil), z) [3] if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), 0, cons(m4, x4)), z) [3] if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), 0, nil), z) [2] if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, x5), z) [3] if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, nil), z) [2] if(false, nil, y, z) -> sortIter(replace(0, 0, nil), z) [2] if(false, cons(n7, x6), y, z) -> sortIter(replace(0, 0, x6), z) [2] if(false, x, y, z) -> sortIter(replace(0, 0, nil), z) [1] empty(v0) -> null_empty [0] min(v0) -> 0 [0] replace(v0, v1, v2) -> nil [0] head(v0) -> 0 [0] tail(v0) -> nil [0] if_replace(v0, v1, v2, v3) -> nil [0] if_min(v0, v1) -> 0 [0] The TRS has the following type information: eq :: 0:s -> 0:s -> true:false:null_empty 0 :: 0:s true :: true:false:null_empty s :: 0:s -> 0:s false :: true:false:null_empty le :: 0:s -> 0:s -> true:false:null_empty min :: nil:cons:append -> 0:s cons :: 0:s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append if_min :: true:false:null_empty -> nil:cons:append -> 0:s replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append empty :: nil:cons:append -> true:false:null_empty head :: nil:cons:append -> 0:s tail :: nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append append :: nil:cons:append -> nil:cons:append -> nil:cons:append null_empty :: true:false:null_empty Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 nil => 0 null_empty => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 eq(z', z'') -{ 1 }-> eq(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 1 + m, z' = 0, m >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(x3, x3, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(x3, x3, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(x3, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(x3, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z) :|: z1 = y, x5 >= 0, z >= 0, z2 = z, n6 >= 0, y >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z) :|: z1 = y, x5 >= 0, z >= 0, z2 = z, n6 >= 0, y >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z) :|: z1 = y, z >= 0, z'' = 1 + n7 + x6, z2 = z, n7 >= 0, y >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(n, m, x) :|: n >= 0, z'' = n, z1 = m, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, m >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + m + x :|: n >= 0, z'' = n, z' = 2, z1 = m, z2 = 1 + k + x, x >= 0, k >= 0, m >= 0 le(z', z'') -{ 1 }-> le(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' = m, z' = 0, m >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n min(z') -{ 1 }-> x :|: x >= 0, z' = 1 + x + 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(n2, m1), 1 + n2, m, 1 + (1 + m1) + x) :|: x >= 0, n2 >= 0, m1 >= 0, z'' = m, z' = 1 + n2, z1 = 1 + (1 + m1) + x, m >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, m, 1 + 0 + x) :|: z1 = 1 + 0 + x, x >= 0, z'' = m, z' = 0, m >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, m, 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z'' = m, z' = 0, m >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + n1, m, 1 + 0 + x) :|: z1 = 1 + 0 + x, x >= 0, n1 >= 0, z' = 1 + n1, z'' = m, m >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: n >= 0, z1 = 0, z' = n, z'' = m, m >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 sort(z') -{ 1 }-> sortIter(x, 0) :|: z' = x, x >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, y, 1 + y + (1 + 0 + 0)) :|: z'' = y, y >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', y, 1 + y + (1 + 0 + 0)) :|: z' = 1 + n3 + x', z'' = y, x' >= 0, y >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + 0, y, 1 + y + (1 + n3 + 0)) :|: z'' = y, y >= 0, z' = 1 + n3 + 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), y, 1 + y + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' = y, y >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 1 }-> if(0, x, y, 1 + y + (1 + 0 + 0)) :|: z' = x, z'' = y, x >= 0, y >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), y, 1 + y + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' = y, y >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + x1 + 0, y, 1 + y + (1 + x1 + 0)) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1 + 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 ---------------------------------------- (11) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 ---------------------------------------- (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { empty } { le } { eq } { tail } { head } { min, if_min } { replace, if_replace } { sortIter, if } { sort } ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} ---------------------------------------- (15) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} ---------------------------------------- (17) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: ?, size: O(1) [2] ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: ?, size: O(1) [2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z'' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: ?, size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: eq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using KoAT for: if_min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: ?, size: O(n^1) [z'] if_min: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 5 + 4*z' + z'^2 Computed RUNTIME bound using KoAT for: if_min after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 22 + 24*z'' + 8*z''^2 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: replace after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' + z1 Computed SIZE bound using CoFloCo for: if_replace after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z1 + z2 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: ?, size: O(n^1) [z'' + z1] if_replace: runtime: ?, size: O(n^1) [z1 + z2] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: replace after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 6 + z' + z'*z1 + 5*z1 Computed RUNTIME bound using KoAT for: if_replace after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + z'' + z''*z2 + 5*z2 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sortIter after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z' + z'^2 + z'' Computed SIZE bound using KoAT for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 16*z'' + 8*z''^2 + z1 + 13*z2 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sortIter,if}, {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] sortIter: runtime: ?, size: O(n^2) [2*z' + z'^2 + z''] if: runtime: ?, size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sortIter after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 254 + 2902*z' + 3584*z'^2 + 1452*z'^3 Computed RUNTIME bound using KoAT for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z''] if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0 sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z''] if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z' + z'^2 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0 sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: {sort} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z''] if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] sort: runtime: ?, size: O(n^2) [2*z' + z'^2] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sort after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: empty(z') -{ 1 }-> 2 :|: z' = 0 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 empty(z') -{ 0 }-> 0 :|: z' >= 0 eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 head(z') -{ 0 }-> 0 :|: z' >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0 if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0 min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 min(z') -{ 0 }-> 0 :|: z' >= 0 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0 sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'') sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0 sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0 sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0 sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 tail(z') -{ 1 }-> 0 :|: z' = 0 tail(z') -{ 0 }-> 0 :|: z' >= 0 Function symbols to be analyzed: Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [2] le: runtime: O(n^1) [2 + z''], size: O(1) [2] eq: runtime: O(n^1) [3 + z'], size: O(1) [2] tail: runtime: O(1) [1], size: O(n^1) [z'] head: runtime: O(1) [1], size: O(n^1) [z'] min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1] if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2] sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z''] if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] sort: runtime: O(n^3) [255 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2] ---------------------------------------- (69) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (70) BOUNDS(1, n^3) ---------------------------------------- (71) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) S tuples: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) K tuples:none Defined Rule Symbols: eq_2, le_2, min_1, if_min_2, replace_3, if_replace_4, empty_1, head_1, tail_1, sort_1, sortIter_2, if_4 Defined Pair Symbols: EQ_2, LE_2, MIN_1, IF_MIN_2, REPLACE_3, IF_REPLACE_4, EMPTY_1, HEAD_1, TAIL_1, SORT_1, SORTITER_2, IF_4 Compound Symbols: c, c1, c2, c3_1, c4, c5, c6_1, c7, c8_2, c9_1, c10_1, c11, c12_2, c13, c14_1, c15, c16, c17, c18, c19, c20_1, c21_2, c22_2, c23, c24_3, c25_3, c26_3 ---------------------------------------- (73) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (74) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: EQ(0, 0) -> c EQ(0, s(z0)) -> c1 EQ(s(z0), 0) -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0, z0) -> c4 LE(s(z0), 0) -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) The (relative) TRS S consists of the following rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Rewrite Strategy: INNERMOST ---------------------------------------- (75) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (76) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) The (relative) TRS S consists of the following rules: eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Rewrite Strategy: INNERMOST ---------------------------------------- (77) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (78) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append ---------------------------------------- (79) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: EQ, LE, MIN, le, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: EQ < REPLACE LE < MIN le < MIN MIN < SORTITER le < min eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (80) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: EQ, LE, MIN, le, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: EQ < REPLACE LE < MIN le < MIN MIN < SORTITER le < min eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (81) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) Induction Base: EQ(gen_0':s17_27(0), gen_0':s17_27(0)) ->_R^Omega(1) c Induction Step: EQ(gen_0':s17_27(+(n21_27, 1)), gen_0':s17_27(+(n21_27, 1))) ->_R^Omega(1) c3(EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27))) ->_IH c3(gen_c:c1:c2:c316_27(c22_27)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (82) Complex Obligation (BEST) ---------------------------------------- (83) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: EQ, LE, MIN, le, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: EQ < REPLACE LE < MIN le < MIN MIN < SORTITER le < min eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (84) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (85) BOUNDS(n^1, INF) ---------------------------------------- (86) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: LE, MIN, le, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: LE < MIN le < MIN MIN < SORTITER le < min eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (87) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) Induction Base: LE(gen_0':s17_27(0), gen_0':s17_27(0)) ->_R^Omega(1) c4 Induction Step: LE(gen_0':s17_27(+(n1115_27, 1)), gen_0':s17_27(+(n1115_27, 1))) ->_R^Omega(1) c6(LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27))) ->_IH c6(gen_c4:c5:c618_27(c1116_27)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (88) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: le, MIN, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: le < MIN MIN < SORTITER le < min eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (89) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) -> true, rt in Omega(0) Induction Base: le(gen_0':s17_27(0), gen_0':s17_27(0)) ->_R^Omega(0) true Induction Step: le(gen_0':s17_27(+(n1983_27, 1)), gen_0':s17_27(+(n1983_27, 1))) ->_R^Omega(0) le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) ->_IH true We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (90) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) -> true, rt in Omega(0) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: MIN, REPLACE, eq, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: MIN < SORTITER eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (91) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MIN(gen_nil:cons:append19_27(+(1, n2530_27))) -> *20_27, rt in Omega(n2530_27) Induction Base: MIN(gen_nil:cons:append19_27(+(1, 0))) Induction Step: MIN(gen_nil:cons:append19_27(+(1, +(n2530_27, 1)))) ->_R^Omega(1) c8(IF_MIN(le(0', 0'), cons(0', cons(0', gen_nil:cons:append19_27(n2530_27)))), LE(0', 0')) ->_L^Omega(0) c8(IF_MIN(true, cons(0', cons(0', gen_nil:cons:append19_27(n2530_27)))), LE(0', 0')) ->_R^Omega(1) c8(c9(MIN(cons(0', gen_nil:cons:append19_27(n2530_27)))), LE(0', 0')) ->_IH c8(c9(*20_27), LE(0', 0')) ->_L^Omega(1) c8(c9(*20_27), gen_c4:c5:c618_27(0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (92) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) -> true, rt in Omega(0) MIN(gen_nil:cons:append19_27(+(1, n2530_27))) -> *20_27, rt in Omega(n2530_27) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: eq, REPLACE, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: eq < REPLACE REPLACE < SORTITER eq < replace min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (93) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: eq(gen_0':s17_27(n6879_27), gen_0':s17_27(n6879_27)) -> true, rt in Omega(0) Induction Base: eq(gen_0':s17_27(0), gen_0':s17_27(0)) ->_R^Omega(0) true Induction Step: eq(gen_0':s17_27(+(n6879_27, 1)), gen_0':s17_27(+(n6879_27, 1))) ->_R^Omega(0) eq(gen_0':s17_27(n6879_27), gen_0':s17_27(n6879_27)) ->_IH true We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (94) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) -> true, rt in Omega(0) MIN(gen_nil:cons:append19_27(+(1, n2530_27))) -> *20_27, rt in Omega(n2530_27) eq(gen_0':s17_27(n6879_27), gen_0':s17_27(n6879_27)) -> true, rt in Omega(0) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: REPLACE, SORTITER, min, replace, sortIter They will be analysed ascendingly in the following order: REPLACE < SORTITER min < SORTITER replace < SORTITER min < sortIter replace < sortIter ---------------------------------------- (95) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: min(gen_nil:cons:append19_27(+(1, n8298_27))) -> gen_0':s17_27(0), rt in Omega(0) Induction Base: min(gen_nil:cons:append19_27(+(1, 0))) ->_R^Omega(0) 0' Induction Step: min(gen_nil:cons:append19_27(+(1, +(n8298_27, 1)))) ->_R^Omega(0) if_min(le(0', 0'), cons(0', cons(0', gen_nil:cons:append19_27(n8298_27)))) ->_L^Omega(0) if_min(true, cons(0', cons(0', gen_nil:cons:append19_27(n8298_27)))) ->_R^Omega(0) min(cons(0', gen_nil:cons:append19_27(n8298_27))) ->_IH gen_0':s17_27(0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (96) Obligation: Innermost TRS: Rules: EQ(0', 0') -> c EQ(0', s(z0)) -> c1 EQ(s(z0), 0') -> c2 EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) LE(0', z0) -> c4 LE(s(z0), 0') -> c5 LE(s(z0), s(z1)) -> c6(LE(z0, z1)) MIN(cons(z0, nil)) -> c7 MIN(cons(z0, cons(z1, z2))) -> c8(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) IF_MIN(true, cons(z0, cons(z1, z2))) -> c9(MIN(cons(z0, z2))) IF_MIN(false, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z1, z2))) REPLACE(z0, z1, nil) -> c11 REPLACE(z0, z1, cons(z2, z3)) -> c12(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c13 IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c14(REPLACE(z0, z1, z3)) EMPTY(nil) -> c15 EMPTY(cons(z0, z1)) -> c16 HEAD(cons(z0, z1)) -> c17 TAIL(nil) -> c18 TAIL(cons(z0, z1)) -> c19 SORT(z0) -> c20(SORTITER(z0, nil)) SORTITER(z0, z1) -> c21(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), EMPTY(z0)) SORTITER(z0, z1) -> c22(IF(empty(z0), z0, z1, append(z1, cons(min(z0), nil))), MIN(z0)) IF(true, z0, z1, z2) -> c23 IF(false, z0, z1, z2) -> c24(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), MIN(z0)) IF(false, z0, z1, z2) -> c25(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), HEAD(z0)) IF(false, z0, z1, z2) -> c26(SORTITER(replace(min(z0), head(z0), tail(z0)), z2), REPLACE(min(z0), head(z0), tail(z0)), TAIL(z0)) eq(0', 0') -> true eq(0', s(z0)) -> false eq(s(z0), 0') -> false eq(s(z0), s(z1)) -> eq(z0, z1) le(0', z0) -> true le(s(z0), 0') -> false le(s(z0), s(z1)) -> le(z0, z1) min(cons(z0, nil)) -> z0 min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) replace(z0, z1, nil) -> nil replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) empty(nil) -> true empty(cons(z0, z1)) -> false head(cons(z0, z1)) -> z0 tail(nil) -> nil tail(cons(z0, z1)) -> z1 sort(z0) -> sortIter(z0, nil) sortIter(z0, z1) -> if(empty(z0), z0, z1, append(z1, cons(min(z0), nil))) if(true, z0, z1, z2) -> z1 if(false, z0, z1, z2) -> sortIter(replace(min(z0), head(z0), tail(z0)), z2) Types: EQ :: 0':s -> 0':s -> c:c1:c2:c3 0' :: 0':s c :: c:c1:c2:c3 s :: 0':s -> 0':s c1 :: c:c1:c2:c3 c2 :: c:c1:c2:c3 c3 :: c:c1:c2:c3 -> c:c1:c2:c3 LE :: 0':s -> 0':s -> c4:c5:c6 c4 :: c4:c5:c6 c5 :: c4:c5:c6 c6 :: c4:c5:c6 -> c4:c5:c6 MIN :: nil:cons:append -> c7:c8 cons :: 0':s -> nil:cons:append -> nil:cons:append nil :: nil:cons:append c7 :: c7:c8 c8 :: c9:c10 -> c4:c5:c6 -> c7:c8 IF_MIN :: true:false -> nil:cons:append -> c9:c10 le :: 0':s -> 0':s -> true:false true :: true:false c9 :: c7:c8 -> c9:c10 false :: true:false c10 :: c7:c8 -> c9:c10 REPLACE :: 0':s -> 0':s -> nil:cons:append -> c11:c12 c11 :: c11:c12 c12 :: c13:c14 -> c:c1:c2:c3 -> c11:c12 IF_REPLACE :: true:false -> 0':s -> 0':s -> nil:cons:append -> c13:c14 eq :: 0':s -> 0':s -> true:false c13 :: c13:c14 c14 :: c11:c12 -> c13:c14 EMPTY :: nil:cons:append -> c15:c16 c15 :: c15:c16 c16 :: c15:c16 HEAD :: nil:cons:append -> c17 c17 :: c17 TAIL :: nil:cons:append -> c18:c19 c18 :: c18:c19 c19 :: c18:c19 SORT :: nil:cons:append -> c20 c20 :: c21:c22 -> c20 SORTITER :: nil:cons:append -> nil:cons:append -> c21:c22 c21 :: c23:c24:c25:c26 -> c15:c16 -> c21:c22 IF :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> c23:c24:c25:c26 empty :: nil:cons:append -> true:false append :: nil:cons:append -> nil:cons:append -> nil:cons:append min :: nil:cons:append -> 0':s c22 :: c23:c24:c25:c26 -> c7:c8 -> c21:c22 c23 :: c23:c24:c25:c26 c24 :: c21:c22 -> c11:c12 -> c7:c8 -> c23:c24:c25:c26 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append head :: nil:cons:append -> 0':s tail :: nil:cons:append -> nil:cons:append c25 :: c21:c22 -> c11:c12 -> c17 -> c23:c24:c25:c26 c26 :: c21:c22 -> c11:c12 -> c18:c19 -> c23:c24:c25:c26 if_min :: true:false -> nil:cons:append -> 0':s if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append sort :: nil:cons:append -> nil:cons:append sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append hole_c:c1:c2:c31_27 :: c:c1:c2:c3 hole_0':s2_27 :: 0':s hole_c4:c5:c63_27 :: c4:c5:c6 hole_c7:c84_27 :: c7:c8 hole_nil:cons:append5_27 :: nil:cons:append hole_c9:c106_27 :: c9:c10 hole_true:false7_27 :: true:false hole_c11:c128_27 :: c11:c12 hole_c13:c149_27 :: c13:c14 hole_c15:c1610_27 :: c15:c16 hole_c1711_27 :: c17 hole_c18:c1912_27 :: c18:c19 hole_c2013_27 :: c20 hole_c21:c2214_27 :: c21:c22 hole_c23:c24:c25:c2615_27 :: c23:c24:c25:c26 gen_c:c1:c2:c316_27 :: Nat -> c:c1:c2:c3 gen_0':s17_27 :: Nat -> 0':s gen_c4:c5:c618_27 :: Nat -> c4:c5:c6 gen_nil:cons:append19_27 :: Nat -> nil:cons:append Lemmas: EQ(gen_0':s17_27(n21_27), gen_0':s17_27(n21_27)) -> gen_c:c1:c2:c316_27(n21_27), rt in Omega(1 + n21_27) LE(gen_0':s17_27(n1115_27), gen_0':s17_27(n1115_27)) -> gen_c4:c5:c618_27(n1115_27), rt in Omega(1 + n1115_27) le(gen_0':s17_27(n1983_27), gen_0':s17_27(n1983_27)) -> true, rt in Omega(0) MIN(gen_nil:cons:append19_27(+(1, n2530_27))) -> *20_27, rt in Omega(n2530_27) eq(gen_0':s17_27(n6879_27), gen_0':s17_27(n6879_27)) -> true, rt in Omega(0) min(gen_nil:cons:append19_27(+(1, n8298_27))) -> gen_0':s17_27(0), rt in Omega(0) Generator Equations: gen_c:c1:c2:c316_27(0) <=> c gen_c:c1:c2:c316_27(+(x, 1)) <=> c3(gen_c:c1:c2:c316_27(x)) gen_0':s17_27(0) <=> 0' gen_0':s17_27(+(x, 1)) <=> s(gen_0':s17_27(x)) gen_c4:c5:c618_27(0) <=> c4 gen_c4:c5:c618_27(+(x, 1)) <=> c6(gen_c4:c5:c618_27(x)) gen_nil:cons:append19_27(0) <=> nil gen_nil:cons:append19_27(+(x, 1)) <=> cons(0', gen_nil:cons:append19_27(x)) The following defined symbols remain to be analysed: replace, SORTITER, sortIter They will be analysed ascendingly in the following order: replace < SORTITER replace < sortIter