1123.63/291.55 WORST_CASE(Omega(n^1), O(n^3)) 1123.63/291.57 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1123.63/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.63/291.57 1123.63/291.57 1123.63/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). 1123.63/291.57 1123.63/291.57 (0) CpxTRS 1123.63/291.57 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (2) CpxWeightedTrs 1123.63/291.57 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (4) CpxTypedWeightedTrs 1123.63/291.57 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (6) CpxTypedWeightedCompleteTrs 1123.63/291.57 (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (8) CpxTypedWeightedCompleteTrs 1123.63/291.57 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms] 1123.63/291.57 (10) CpxRNTS 1123.63/291.57 (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (12) CpxRNTS 1123.63/291.57 (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (14) CpxRNTS 1123.63/291.57 (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (16) CpxRNTS 1123.63/291.57 (17) IntTrsBoundProof [UPPER BOUND(ID), 183 ms] 1123.63/291.57 (18) CpxRNTS 1123.63/291.57 (19) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] 1123.63/291.57 (20) CpxRNTS 1123.63/291.57 (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (22) CpxRNTS 1123.63/291.57 (23) IntTrsBoundProof [UPPER BOUND(ID), 293 ms] 1123.63/291.57 (24) CpxRNTS 1123.63/291.57 (25) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] 1123.63/291.57 (26) CpxRNTS 1123.63/291.57 (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (28) CpxRNTS 1123.63/291.57 (29) IntTrsBoundProof [UPPER BOUND(ID), 503 ms] 1123.63/291.57 (30) CpxRNTS 1123.63/291.57 (31) IntTrsBoundProof [UPPER BOUND(ID), 186 ms] 1123.63/291.57 (32) CpxRNTS 1123.63/291.57 (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (34) CpxRNTS 1123.63/291.57 (35) IntTrsBoundProof [UPPER BOUND(ID), 160 ms] 1123.63/291.57 (36) CpxRNTS 1123.63/291.57 (37) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] 1123.63/291.57 (38) CpxRNTS 1123.63/291.57 (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (40) CpxRNTS 1123.63/291.57 (41) IntTrsBoundProof [UPPER BOUND(ID), 160 ms] 1123.63/291.57 (42) CpxRNTS 1123.63/291.57 (43) IntTrsBoundProof [UPPER BOUND(ID), 34 ms] 1123.63/291.57 (44) CpxRNTS 1123.63/291.57 (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (46) CpxRNTS 1123.63/291.57 (47) IntTrsBoundProof [UPPER BOUND(ID), 1044 ms] 1123.63/291.57 (48) CpxRNTS 1123.63/291.57 (49) IntTrsBoundProof [UPPER BOUND(ID), 335 ms] 1123.63/291.57 (50) CpxRNTS 1123.63/291.57 (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (52) CpxRNTS 1123.63/291.57 (53) IntTrsBoundProof [UPPER BOUND(ID), 2077 ms] 1123.63/291.57 (54) CpxRNTS 1123.63/291.57 (55) IntTrsBoundProof [UPPER BOUND(ID), 522 ms] 1123.63/291.57 (56) CpxRNTS 1123.63/291.57 (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (58) CpxRNTS 1123.63/291.57 (59) IntTrsBoundProof [UPPER BOUND(ID), 5926 ms] 1123.63/291.57 (60) CpxRNTS 1123.63/291.57 (61) IntTrsBoundProof [UPPER BOUND(ID), 1671 ms] 1123.63/291.57 (62) CpxRNTS 1123.63/291.57 (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (64) CpxRNTS 1123.63/291.57 (65) IntTrsBoundProof [UPPER BOUND(ID), 669 ms] 1123.63/291.57 (66) CpxRNTS 1123.63/291.57 (67) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 1123.63/291.57 (68) CpxRNTS 1123.63/291.57 (69) FinalProof [FINISHED, 0 ms] 1123.63/291.57 (70) BOUNDS(1, n^3) 1123.63/291.57 (71) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (72) CpxTRS 1123.63/291.57 (73) SlicingProof [LOWER BOUND(ID), 0 ms] 1123.63/291.57 (74) CpxTRS 1123.63/291.57 (75) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 1123.63/291.57 (76) typed CpxTrs 1123.63/291.57 (77) OrderProof [LOWER BOUND(ID), 5 ms] 1123.63/291.57 (78) typed CpxTrs 1123.63/291.57 (79) RewriteLemmaProof [LOWER BOUND(ID), 332 ms] 1123.63/291.57 (80) BEST 1123.63/291.57 (81) proven lower bound 1123.63/291.57 (82) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.63/291.57 (83) BOUNDS(n^1, INF) 1123.63/291.57 (84) typed CpxTrs 1123.63/291.57 (85) RewriteLemmaProof [LOWER BOUND(ID), 31 ms] 1123.63/291.57 (86) typed CpxTrs 1123.63/291.57 (87) RewriteLemmaProof [LOWER BOUND(ID), 52 ms] 1123.63/291.57 (88) typed CpxTrs 1123.63/291.57 1123.63/291.57 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (0) 1123.63/291.57 Obligation: 1123.63/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). 1123.63/291.57 1123.63/291.57 1123.63/291.57 The TRS R consists of the following rules: 1123.63/291.57 1123.63/291.57 eq(0, 0) -> true 1123.63/291.57 eq(0, s(m)) -> false 1123.63/291.57 eq(s(n), 0) -> false 1123.63/291.57 eq(s(n), s(m)) -> eq(n, m) 1123.63/291.57 le(0, m) -> true 1123.63/291.57 le(s(n), 0) -> false 1123.63/291.57 le(s(n), s(m)) -> le(n, m) 1123.63/291.57 min(cons(x, nil)) -> x 1123.63/291.57 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.63/291.57 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.63/291.57 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.63/291.57 replace(n, m, nil) -> nil 1123.63/291.57 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.63/291.57 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.63/291.57 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.63/291.57 empty(nil) -> true 1123.63/291.57 empty(cons(n, x)) -> false 1123.63/291.57 head(cons(n, x)) -> n 1123.63/291.57 tail(nil) -> nil 1123.63/291.57 tail(cons(n, x)) -> x 1123.63/291.57 sort(x) -> sortIter(x, nil) 1123.63/291.57 sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) 1123.63/291.57 if(true, x, y, z) -> y 1123.63/291.57 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.63/291.57 1123.63/291.57 S is empty. 1123.63/291.57 Rewrite Strategy: INNERMOST 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1123.63/291.57 Transformed relative TRS to weighted TRS 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (2) 1123.63/291.57 Obligation: 1123.63/291.57 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). 1123.63/291.57 1123.63/291.57 1123.63/291.57 The TRS R consists of the following rules: 1123.63/291.57 1123.63/291.57 eq(0, 0) -> true [1] 1123.63/291.57 eq(0, s(m)) -> false [1] 1123.63/291.57 eq(s(n), 0) -> false [1] 1123.63/291.57 eq(s(n), s(m)) -> eq(n, m) [1] 1123.63/291.57 le(0, m) -> true [1] 1123.63/291.57 le(s(n), 0) -> false [1] 1123.63/291.57 le(s(n), s(m)) -> le(n, m) [1] 1123.63/291.57 min(cons(x, nil)) -> x [1] 1123.63/291.57 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] 1123.63/291.57 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] 1123.63/291.57 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] 1123.63/291.57 replace(n, m, nil) -> nil [1] 1123.63/291.57 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] 1123.63/291.57 if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] 1123.63/291.57 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] 1123.63/291.57 empty(nil) -> true [1] 1123.63/291.57 empty(cons(n, x)) -> false [1] 1123.63/291.57 head(cons(n, x)) -> n [1] 1123.63/291.57 tail(nil) -> nil [1] 1123.63/291.57 tail(cons(n, x)) -> x [1] 1123.63/291.57 sort(x) -> sortIter(x, nil) [1] 1123.63/291.57 sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] 1123.63/291.57 if(true, x, y, z) -> y [1] 1123.63/291.57 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] 1123.63/291.57 1123.63/291.57 Rewrite Strategy: INNERMOST 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1123.63/291.57 Infered types. 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (4) 1123.63/291.57 Obligation: 1123.63/291.57 Runtime Complexity Weighted TRS with Types. 1123.63/291.57 The TRS R consists of the following rules: 1123.63/291.57 1123.63/291.57 eq(0, 0) -> true [1] 1123.63/291.57 eq(0, s(m)) -> false [1] 1123.63/291.57 eq(s(n), 0) -> false [1] 1123.63/291.57 eq(s(n), s(m)) -> eq(n, m) [1] 1123.63/291.57 le(0, m) -> true [1] 1123.63/291.57 le(s(n), 0) -> false [1] 1123.63/291.57 le(s(n), s(m)) -> le(n, m) [1] 1123.63/291.57 min(cons(x, nil)) -> x [1] 1123.63/291.57 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] 1123.63/291.57 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] 1123.63/291.57 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] 1123.63/291.57 replace(n, m, nil) -> nil [1] 1123.63/291.57 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] 1123.63/291.57 if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] 1123.63/291.57 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] 1123.63/291.57 empty(nil) -> true [1] 1123.63/291.57 empty(cons(n, x)) -> false [1] 1123.63/291.57 head(cons(n, x)) -> n [1] 1123.63/291.57 tail(nil) -> nil [1] 1123.63/291.57 tail(cons(n, x)) -> x [1] 1123.63/291.57 sort(x) -> sortIter(x, nil) [1] 1123.63/291.57 sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] 1123.63/291.57 if(true, x, y, z) -> y [1] 1123.63/291.57 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] 1123.63/291.57 1123.63/291.57 The TRS has the following type information: 1123.63/291.57 eq :: 0:s -> 0:s -> true:false 1123.63/291.57 0 :: 0:s 1123.63/291.57 true :: true:false 1123.63/291.57 s :: 0:s -> 0:s 1123.63/291.57 false :: true:false 1123.63/291.57 le :: 0:s -> 0:s -> true:false 1123.63/291.57 min :: nil:cons:append -> 0:s 1123.63/291.57 cons :: 0:s -> nil:cons:append -> nil:cons:append 1123.63/291.57 nil :: nil:cons:append 1123.63/291.57 if_min :: true:false -> nil:cons:append -> 0:s 1123.63/291.57 replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.63/291.57 if_replace :: true:false -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.63/291.57 empty :: nil:cons:append -> true:false 1123.63/291.57 head :: nil:cons:append -> 0:s 1123.63/291.57 tail :: nil:cons:append -> nil:cons:append 1123.63/291.57 sort :: nil:cons:append -> nil:cons:append 1123.63/291.57 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.63/291.57 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.63/291.57 append :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.63/291.57 1123.63/291.57 Rewrite Strategy: INNERMOST 1123.63/291.57 ---------------------------------------- 1123.63/291.57 1123.63/291.57 (5) CompletionProof (UPPER BOUND(ID)) 1123.84/291.59 The transformation into a RNTS is sound, since: 1123.84/291.59 1123.84/291.59 (a) The obligation is a constructor system where every type has a constant constructor, 1123.84/291.59 1123.84/291.59 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 1123.84/291.59 1123.84/291.59 sort_1 1123.84/291.59 sortIter_2 1123.84/291.59 if_4 1123.84/291.59 1123.84/291.59 (c) The following functions are completely defined: 1123.84/291.59 1123.84/291.59 empty_1 1123.84/291.59 min_1 1123.84/291.59 replace_3 1123.84/291.59 head_1 1123.84/291.59 tail_1 1123.84/291.59 eq_2 1123.84/291.59 le_2 1123.84/291.59 if_replace_4 1123.84/291.59 if_min_2 1123.84/291.59 1123.84/291.59 Due to the following rules being added: 1123.84/291.59 1123.84/291.59 empty(v0) -> null_empty [0] 1123.84/291.59 min(v0) -> 0 [0] 1123.84/291.59 replace(v0, v1, v2) -> nil [0] 1123.84/291.59 head(v0) -> 0 [0] 1123.84/291.59 tail(v0) -> nil [0] 1123.84/291.59 if_replace(v0, v1, v2, v3) -> nil [0] 1123.84/291.59 if_min(v0, v1) -> 0 [0] 1123.84/291.59 1123.84/291.59 And the following fresh constants: null_empty 1123.84/291.59 1123.84/291.59 ---------------------------------------- 1123.84/291.59 1123.84/291.59 (6) 1123.84/291.59 Obligation: 1123.84/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1123.84/291.59 1123.84/291.59 Runtime Complexity Weighted TRS with Types. 1123.84/291.59 The TRS R consists of the following rules: 1123.84/291.59 1123.84/291.59 eq(0, 0) -> true [1] 1123.84/291.59 eq(0, s(m)) -> false [1] 1123.84/291.59 eq(s(n), 0) -> false [1] 1123.84/291.59 eq(s(n), s(m)) -> eq(n, m) [1] 1123.84/291.59 le(0, m) -> true [1] 1123.84/291.59 le(s(n), 0) -> false [1] 1123.84/291.59 le(s(n), s(m)) -> le(n, m) [1] 1123.84/291.59 min(cons(x, nil)) -> x [1] 1123.84/291.59 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1] 1123.84/291.59 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] 1123.84/291.59 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] 1123.84/291.59 replace(n, m, nil) -> nil [1] 1123.84/291.59 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1] 1123.84/291.59 if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] 1123.84/291.59 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] 1123.84/291.59 empty(nil) -> true [1] 1123.84/291.59 empty(cons(n, x)) -> false [1] 1123.84/291.59 head(cons(n, x)) -> n [1] 1123.84/291.59 tail(nil) -> nil [1] 1123.84/291.59 tail(cons(n, x)) -> x [1] 1123.84/291.59 sort(x) -> sortIter(x, nil) [1] 1123.84/291.59 sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1] 1123.84/291.59 if(true, x, y, z) -> y [1] 1123.84/291.59 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1] 1123.84/291.59 empty(v0) -> null_empty [0] 1123.84/291.59 min(v0) -> 0 [0] 1123.84/291.59 replace(v0, v1, v2) -> nil [0] 1123.84/291.59 head(v0) -> 0 [0] 1123.84/291.59 tail(v0) -> nil [0] 1123.84/291.59 if_replace(v0, v1, v2, v3) -> nil [0] 1123.84/291.59 if_min(v0, v1) -> 0 [0] 1123.84/291.59 1123.84/291.59 The TRS has the following type information: 1123.84/291.59 eq :: 0:s -> 0:s -> true:false:null_empty 1123.84/291.59 0 :: 0:s 1123.84/291.59 true :: true:false:null_empty 1123.84/291.59 s :: 0:s -> 0:s 1123.84/291.59 false :: true:false:null_empty 1123.84/291.59 le :: 0:s -> 0:s -> true:false:null_empty 1123.84/291.59 min :: nil:cons:append -> 0:s 1123.84/291.59 cons :: 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 nil :: nil:cons:append 1123.84/291.59 if_min :: true:false:null_empty -> nil:cons:append -> 0:s 1123.84/291.59 replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 empty :: nil:cons:append -> true:false:null_empty 1123.84/291.59 head :: nil:cons:append -> 0:s 1123.84/291.59 tail :: nil:cons:append -> nil:cons:append 1123.84/291.59 sort :: nil:cons:append -> nil:cons:append 1123.84/291.59 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 append :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 null_empty :: true:false:null_empty 1123.84/291.59 1123.84/291.59 Rewrite Strategy: INNERMOST 1123.84/291.59 ---------------------------------------- 1123.84/291.59 1123.84/291.59 (7) NarrowingProof (BOTH BOUNDS(ID, ID)) 1123.84/291.59 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 1123.84/291.59 ---------------------------------------- 1123.84/291.59 1123.84/291.59 (8) 1123.84/291.59 Obligation: 1123.84/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1123.84/291.59 1123.84/291.59 Runtime Complexity Weighted TRS with Types. 1123.84/291.59 The TRS R consists of the following rules: 1123.84/291.59 1123.84/291.59 eq(0, 0) -> true [1] 1123.84/291.59 eq(0, s(m)) -> false [1] 1123.84/291.59 eq(s(n), 0) -> false [1] 1123.84/291.59 eq(s(n), s(m)) -> eq(n, m) [1] 1123.84/291.59 le(0, m) -> true [1] 1123.84/291.59 le(s(n), 0) -> false [1] 1123.84/291.59 le(s(n), s(m)) -> le(n, m) [1] 1123.84/291.59 min(cons(x, nil)) -> x [1] 1123.84/291.59 min(cons(0, cons(m, x))) -> if_min(true, cons(0, cons(m, x))) [2] 1123.84/291.59 min(cons(s(n'), cons(0, x))) -> if_min(false, cons(s(n'), cons(0, x))) [2] 1123.84/291.59 min(cons(s(n''), cons(s(m'), x))) -> if_min(le(n'', m'), cons(s(n''), cons(s(m'), x))) [2] 1123.84/291.59 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1] 1123.84/291.59 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1] 1123.84/291.59 replace(n, m, nil) -> nil [1] 1123.84/291.59 replace(0, m, cons(0, x)) -> if_replace(true, 0, m, cons(0, x)) [2] 1123.84/291.59 replace(0, m, cons(s(m''), x)) -> if_replace(false, 0, m, cons(s(m''), x)) [2] 1123.84/291.59 replace(s(n1), m, cons(0, x)) -> if_replace(false, s(n1), m, cons(0, x)) [2] 1123.84/291.59 replace(s(n2), m, cons(s(m1), x)) -> if_replace(eq(n2, m1), s(n2), m, cons(s(m1), x)) [2] 1123.84/291.59 if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1] 1123.84/291.59 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1] 1123.84/291.59 empty(nil) -> true [1] 1123.84/291.59 empty(cons(n, x)) -> false [1] 1123.84/291.59 head(cons(n, x)) -> n [1] 1123.84/291.59 tail(nil) -> nil [1] 1123.84/291.59 tail(cons(n, x)) -> x [1] 1123.84/291.59 sort(x) -> sortIter(x, nil) [1] 1123.84/291.59 sortIter(nil, y) -> if(true, nil, y, append(y, cons(0, nil))) [2] 1123.84/291.59 sortIter(cons(n3, nil), y) -> if(false, cons(n3, nil), y, append(y, cons(n3, nil))) [3] 1123.84/291.59 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] 1123.84/291.59 sortIter(cons(n3, x'), y) -> if(false, cons(n3, x'), y, append(y, cons(0, nil))) [2] 1123.84/291.59 sortIter(cons(x1, nil), y) -> if(null_empty, cons(x1, nil), y, append(y, cons(x1, nil))) [2] 1123.84/291.59 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] 1123.84/291.59 sortIter(x, y) -> if(null_empty, x, y, append(y, cons(0, nil))) [1] 1123.84/291.59 if(true, x, y, z) -> y [1] 1123.84/291.59 if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [4] 1123.84/291.59 if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [3] 1123.84/291.59 if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [3] 1123.84/291.59 if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [2] 1123.84/291.59 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] 1123.84/291.59 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] 1123.84/291.59 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] 1123.84/291.59 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] 1123.84/291.59 if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, x5), z) [3] 1123.84/291.59 if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, nil), z) [2] 1123.84/291.59 if(false, nil, y, z) -> sortIter(replace(0, 0, nil), z) [2] 1123.84/291.59 if(false, cons(n7, x6), y, z) -> sortIter(replace(0, 0, x6), z) [2] 1123.84/291.59 if(false, x, y, z) -> sortIter(replace(0, 0, nil), z) [1] 1123.84/291.59 empty(v0) -> null_empty [0] 1123.84/291.59 min(v0) -> 0 [0] 1123.84/291.59 replace(v0, v1, v2) -> nil [0] 1123.84/291.59 head(v0) -> 0 [0] 1123.84/291.59 tail(v0) -> nil [0] 1123.84/291.59 if_replace(v0, v1, v2, v3) -> nil [0] 1123.84/291.59 if_min(v0, v1) -> 0 [0] 1123.84/291.59 1123.84/291.59 The TRS has the following type information: 1123.84/291.59 eq :: 0:s -> 0:s -> true:false:null_empty 1123.84/291.59 0 :: 0:s 1123.84/291.59 true :: true:false:null_empty 1123.84/291.59 s :: 0:s -> 0:s 1123.84/291.59 false :: true:false:null_empty 1123.84/291.59 le :: 0:s -> 0:s -> true:false:null_empty 1123.84/291.59 min :: nil:cons:append -> 0:s 1123.84/291.59 cons :: 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 nil :: nil:cons:append 1123.84/291.59 if_min :: true:false:null_empty -> nil:cons:append -> 0:s 1123.84/291.59 replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append 1123.84/291.59 empty :: nil:cons:append -> true:false:null_empty 1123.84/291.59 head :: nil:cons:append -> 0:s 1123.84/291.59 tail :: nil:cons:append -> nil:cons:append 1123.84/291.59 sort :: nil:cons:append -> nil:cons:append 1123.84/291.59 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 append :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.59 null_empty :: true:false:null_empty 1123.84/291.59 1123.84/291.59 Rewrite Strategy: INNERMOST 1123.84/291.59 ---------------------------------------- 1123.84/291.59 1123.84/291.59 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1123.84/291.59 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1123.84/291.59 The constant constructors are abstracted as follows: 1123.84/291.62 1123.84/291.62 0 => 0 1123.84/291.62 true => 2 1123.84/291.62 false => 1 1123.84/291.62 nil => 0 1123.84/291.62 null_empty => 0 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (10) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 1 + m, z' = 0, m >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1 1123.84/291.62 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 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 le(z', z'') -{ 1 }-> le(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z'' = m, z' = 0, m >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n 1123.84/291.62 min(z') -{ 1 }-> x :|: x >= 0, z' = 1 + x + 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: n >= 0, z1 = 0, z' = n, z'' = m, m >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(x, 0) :|: z' = x, x >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, y, 1 + y + (1 + 0 + 0)) :|: z'' = y, y >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, x, y, 1 + y + (1 + 0 + 0)) :|: z' = x, z'' = y, x >= 0, y >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 1123.84/291.62 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (11) SimplificationProof (BOTH BOUNDS(ID, ID)) 1123.84/291.62 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (12) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 1123.84/291.62 Found the following analysis order by SCC decomposition: 1123.84/291.62 1123.84/291.62 { empty } 1123.84/291.62 { le } 1123.84/291.62 { eq } 1123.84/291.62 { tail } 1123.84/291.62 { head } 1123.84/291.62 { min, if_min } 1123.84/291.62 { replace, if_replace } 1123.84/291.62 { sortIter, if } 1123.84/291.62 { sort } 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (14) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (15) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.62 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (16) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (17) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.62 1123.84/291.62 Computed SIZE bound using CoFloCo for: empty 1123.84/291.62 after applying outer abstraction to obtain an ITS, 1123.84/291.62 resulting in: O(1) with polynomial bound: 2 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (18) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 Previous analysis results are: 1123.84/291.62 empty: runtime: ?, size: O(1) [2] 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (19) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.62 1123.84/291.62 Computed RUNTIME bound using CoFloCo for: empty 1123.84/291.62 after applying outer abstraction to obtain an ITS, 1123.84/291.62 resulting in: O(1) with polynomial bound: 1 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (20) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 Previous analysis results are: 1123.84/291.62 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (21) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.62 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (22) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 Previous analysis results are: 1123.84/291.62 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (23) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.62 1123.84/291.62 Computed SIZE bound using CoFloCo for: le 1123.84/291.62 after applying outer abstraction to obtain an ITS, 1123.84/291.62 resulting in: O(1) with polynomial bound: 2 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (24) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.62 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.62 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.62 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 1123.84/291.62 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.62 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.62 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.62 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.62 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.62 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.62 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 1123.84/291.62 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'') 1123.84/291.62 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.62 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.62 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 1123.84/291.62 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.62 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.62 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 1123.84/291.62 Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.62 Previous analysis results are: 1123.84/291.62 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.62 le: runtime: ?, size: O(1) [2] 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (25) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.62 1123.84/291.62 Computed RUNTIME bound using KoAT for: le 1123.84/291.62 after applying outer abstraction to obtain an ITS, 1123.84/291.62 resulting in: O(n^1) with polynomial bound: 2 + z'' 1123.84/291.62 1123.84/291.62 ---------------------------------------- 1123.84/291.62 1123.84/291.62 (26) 1123.84/291.62 Obligation: 1123.84/291.62 Complexity RNTS consisting of the following rules: 1123.84/291.62 1123.84/291.62 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.62 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.62 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.62 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.62 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 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 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.62 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.64 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.64 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.64 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.64 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.64 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 1123.84/291.64 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.64 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.64 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.64 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.64 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 1123.84/291.64 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.64 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.64 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.64 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 1123.84/291.64 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.64 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 1123.84/291.64 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.64 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.64 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.64 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.64 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.64 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 1123.84/291.64 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'') 1123.84/291.64 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.64 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.64 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 1123.84/291.64 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.64 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.64 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 1123.84/291.64 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.64 Previous analysis results are: 1123.84/291.64 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.64 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.64 1123.84/291.64 ---------------------------------------- 1123.84/291.64 1123.84/291.64 (27) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.64 ---------------------------------------- 1123.84/291.64 1123.84/291.64 (28) 1123.84/291.64 Obligation: 1123.84/291.64 Complexity RNTS consisting of the following rules: 1123.84/291.64 1123.84/291.64 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.64 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.64 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.64 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.64 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.64 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.64 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.64 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.64 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.64 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.64 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.64 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.64 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 1123.84/291.64 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.64 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.64 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.64 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.64 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 1123.84/291.64 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.64 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.64 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.64 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 1123.84/291.64 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.64 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 1123.84/291.64 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.64 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.64 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.64 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.64 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.64 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 1123.84/291.64 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'') 1123.84/291.64 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.64 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.64 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 1123.84/291.64 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.64 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.64 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 1123.84/291.64 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.64 Previous analysis results are: 1123.84/291.64 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.64 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.64 1123.84/291.64 ---------------------------------------- 1123.84/291.64 1123.84/291.64 (29) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.64 1123.84/291.64 Computed SIZE bound using CoFloCo for: eq 1123.84/291.64 after applying outer abstraction to obtain an ITS, 1123.84/291.64 resulting in: O(1) with polynomial bound: 2 1123.84/291.64 1123.84/291.64 ---------------------------------------- 1123.84/291.64 1123.84/291.64 (30) 1123.84/291.64 Obligation: 1123.84/291.64 Complexity RNTS consisting of the following rules: 1123.84/291.64 1123.84/291.64 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.64 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.64 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.64 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.64 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.64 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.64 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: ?, size: O(1) [2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (31) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: eq 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: 3 + z'' 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (32) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (33) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (34) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (35) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: tail 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z' 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (36) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: ?, size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (37) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using CoFloCo for: tail 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(1) with polynomial bound: 1 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (38) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (39) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (40) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (41) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: head 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z' 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (42) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: ?, size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (43) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using CoFloCo for: head 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(1) with polynomial bound: 1 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (44) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (45) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (46) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (47) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: min 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z' 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: if_min 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z'' 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (48) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: ?, size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: ?, size: O(n^1) [z''] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (49) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using CoFloCo for: min 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 5 + 4*z' + z'^2 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: if_min 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 22 + 24*z'' + 8*z''^2 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (50) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0 1123.84/291.65 if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0 1123.84/291.65 min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (51) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (52) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (53) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: replace 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z'' + z1 1123.84/291.65 1123.84/291.65 Computed SIZE bound using CoFloCo for: if_replace 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^1) with polynomial bound: z1 + z2 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (54) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: ?, size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: ?, size: O(n^1) [z1 + z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (55) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using CoFloCo for: replace 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 6 + 5*z1 + z1^2 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: if_replace 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 8 + 5*z2 + z2^2 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (56) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 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 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 5 + m1 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (57) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (58) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 10 }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 120 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 119 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 9 + 5*x5 + x5^2 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 8 + 5*x6 + x6^2 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (59) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using CoFloCo for: sortIter 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 2*z' + z'^2 + z'' 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: if 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 16*z'' + 8*z''^2 + z1 + 13*z2 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (60) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 10 }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 120 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 119 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 9 + 5*x5 + x5^2 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 8 + 5*x6 + x6^2 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sortIter,if}, {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 sortIter: runtime: ?, size: O(n^2) [2*z' + z'^2 + z''] 1123.84/291.65 if: runtime: ?, size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (61) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: sortIter 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^3) with polynomial bound: 254 + 2896*z' + 3562*z'^2 + 1450*z'^3 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: if 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^3) with polynomial bound: 19132 + 87288*z'' + 128088*z''^2 + 69600*z''^3 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (62) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 10 }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 9 }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 120 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 119 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 63*x4 + 9*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 1123.84/291.65 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 + 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 1123.84/291.65 if(z', z'', z1, z2) -{ 9 + 5*x5 + x5^2 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1 1123.84/291.65 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 1123.84/291.65 if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 8 + 5*x6 + x6^2 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0 1123.84/291.65 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 1123.84/291.65 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'') 1123.84/291.65 sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0 1123.84/291.65 sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0 1123.84/291.65 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 1123.84/291.65 sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 sortIter: runtime: O(n^3) [254 + 2896*z' + 3562*z'^2 + 1450*z'^3], size: O(n^2) [2*z' + z'^2 + z''] 1123.84/291.65 if: runtime: O(n^3) [19132 + 87288*z'' + 128088*z''^2 + 69600*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (63) ResultPropagationProof (UPPER BOUND(ID)) 1123.84/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (64) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 264 + 2896*s24 + 3562*s24^2 + 1450*s24^3 }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s25 + 3562*s25^2 + 1450*s25^3 }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s26 + 3562*s26^2 + 1450*s26^3 }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s27 + 3562*s27^2 + 1450*s27^3 }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 374 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s28 + 3562*s28^2 + 1450*s28^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s29 + 3562*s29^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 373 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s30 + 3562*s30^2 + 1450*s30^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s31 + 3562*s31^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s32 + 3562*s32^2 + 1450*s32^3 + 5*x5 + x5^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s33 + 3562*s33^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s34 + 3562*s34^2 + 1450*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s35 + 3562*s35^2 + 1450*s35^3 + 5*x6 + x6^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 261 + 2896*s36 + 3562*s36^2 + 1450*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 255 + 2896*z' + 3562*z'^2 + 1450*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 19134 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 1123.84/291.65 sortIter(z', z'') -{ 19135 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262967 + 1434897*m2 + 1091392*m2*n3 + 417600*m2*n3*x'' + 208800*m2*n3^2 + 1091392*m2*x'' + 208800*m2*x''^2 + 545696*m2^2 + 208800*m2^2*n3 + 208800*m2^2*x'' + 69600*m2^3 + 1434896*n3 + 1091392*n3*x'' + 208800*n3*x''^2 + 545696*n3^2 + 208800*n3^2*x'' + 69600*n3^3 + 1434896*x'' + 545696*x''^2 + 69600*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'') 1123.84/291.65 sortIter(z', z'') -{ 304110 + 552264*n3 + 673776*n3*x' + 208800*n3*x'^2 + 336888*n3^2 + 208800*n3^2*x' + 69600*n3^3 + 552264*x' + 336888*x'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19134 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262966 + 1434897*m3 + 1091392*m3*n4 + 417600*m3*n4*x2 + 208800*m3*n4^2 + 1091392*m3*x2 + 208800*m3*x2^2 + 545696*m3^2 + 208800*m3^2*n4 + 208800*m3^2*x2 + 69600*m3^3 + 1434896*n4 + 1091392*n4*x2 + 208800*n4*x2^2 + 545696*n4^2 + 208800*n4^2*x2 + 69600*n4^3 + 1434896*x2 + 545696*x2^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19133 + 87288*z' + 128088*z'^2 + 69600*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 sortIter: runtime: O(n^3) [254 + 2896*z' + 3562*z'^2 + 1450*z'^3], size: O(n^2) [2*z' + z'^2 + z''] 1123.84/291.65 if: runtime: O(n^3) [19132 + 87288*z'' + 128088*z''^2 + 69600*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (65) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed SIZE bound using KoAT for: sort 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^2) with polynomial bound: 2*z' + z'^2 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (66) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 264 + 2896*s24 + 3562*s24^2 + 1450*s24^3 }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s25 + 3562*s25^2 + 1450*s25^3 }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s26 + 3562*s26^2 + 1450*s26^3 }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s27 + 3562*s27^2 + 1450*s27^3 }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 374 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s28 + 3562*s28^2 + 1450*s28^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s29 + 3562*s29^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 373 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s30 + 3562*s30^2 + 1450*s30^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s31 + 3562*s31^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s32 + 3562*s32^2 + 1450*s32^3 + 5*x5 + x5^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s33 + 3562*s33^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s34 + 3562*s34^2 + 1450*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s35 + 3562*s35^2 + 1450*s35^3 + 5*x6 + x6^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 261 + 2896*s36 + 3562*s36^2 + 1450*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 255 + 2896*z' + 3562*z'^2 + 1450*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 19134 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 1123.84/291.65 sortIter(z', z'') -{ 19135 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262967 + 1434897*m2 + 1091392*m2*n3 + 417600*m2*n3*x'' + 208800*m2*n3^2 + 1091392*m2*x'' + 208800*m2*x''^2 + 545696*m2^2 + 208800*m2^2*n3 + 208800*m2^2*x'' + 69600*m2^3 + 1434896*n3 + 1091392*n3*x'' + 208800*n3*x''^2 + 545696*n3^2 + 208800*n3^2*x'' + 69600*n3^3 + 1434896*x'' + 545696*x''^2 + 69600*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'') 1123.84/291.65 sortIter(z', z'') -{ 304110 + 552264*n3 + 673776*n3*x' + 208800*n3*x'^2 + 336888*n3^2 + 208800*n3^2*x' + 69600*n3^3 + 552264*x' + 336888*x'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19134 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262966 + 1434897*m3 + 1091392*m3*n4 + 417600*m3*n4*x2 + 208800*m3*n4^2 + 1091392*m3*x2 + 208800*m3*x2^2 + 545696*m3^2 + 208800*m3^2*n4 + 208800*m3^2*x2 + 69600*m3^3 + 1434896*n4 + 1091392*n4*x2 + 208800*n4*x2^2 + 545696*n4^2 + 208800*n4^2*x2 + 69600*n4^3 + 1434896*x2 + 545696*x2^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19133 + 87288*z' + 128088*z'^2 + 69600*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: {sort} 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.65 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.65 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.65 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.65 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.65 sortIter: runtime: O(n^3) [254 + 2896*z' + 3562*z'^2 + 1450*z'^3], size: O(n^2) [2*z' + z'^2 + z''] 1123.84/291.65 if: runtime: O(n^3) [19132 + 87288*z'' + 128088*z''^2 + 69600*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] 1123.84/291.65 sort: runtime: ?, size: O(n^2) [2*z' + z'^2] 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (67) IntTrsBoundProof (UPPER BOUND(ID)) 1123.84/291.65 1123.84/291.65 Computed RUNTIME bound using KoAT for: sort 1123.84/291.65 after applying outer abstraction to obtain an ITS, 1123.84/291.65 resulting in: O(n^3) with polynomial bound: 255 + 2896*z' + 3562*z'^2 + 1450*z'^3 1123.84/291.65 1123.84/291.65 ---------------------------------------- 1123.84/291.65 1123.84/291.65 (68) 1123.84/291.65 Obligation: 1123.84/291.65 Complexity RNTS consisting of the following rules: 1123.84/291.65 1123.84/291.65 empty(z') -{ 1 }-> 2 :|: z' = 0 1123.84/291.65 empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 empty(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 eq(z', z'') -{ 3 + z'' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0 1123.84/291.65 eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 head(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 264 + 2896*s24 + 3562*s24^2 + 1450*s24^3 }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s25 + 3562*s25^2 + 1450*s25^3 }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s26 + 3562*s26^2 + 1450*s26^3 }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s27 + 3562*s27^2 + 1450*s27^3 }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0 1123.84/291.65 if(z', z'', z1, z2) -{ 374 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s28 + 3562*s28^2 + 1450*s28^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s29 + 3562*s29^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 373 + 64*m4 + 16*m4*n5 + 18*m4*x4 + 9*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2896*s30 + 3562*s30^2 + 1450*s30^3 + 63*x4 + 9*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 1123.84/291.65 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 + 2896*s31 + 3562*s31^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 263 + 2896*s32 + 3562*s32^2 + 1450*s32^3 + 5*x5 + x5^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s33 + 3562*s33^2 + 1450*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 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s34 + 3562*s34^2 + 1450*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 262 + 2896*s35 + 3562*s35^2 + 1450*s35^3 + 5*x6 + x6^2 }-> 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 1123.84/291.65 if(z', z'', z1, z2) -{ 261 + 2896*s36 + 3562*s36^2 + 1450*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 1123.84/291.65 if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 7 + 5*x + x^2 }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0 1123.84/291.65 if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0 1123.84/291.65 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0 1123.84/291.65 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 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 1123.84/291.65 min(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 24 + 9*m'' + 2*m''*x + m''^2 + 9*x + x^2 }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 10 + 5*z1 + z1^2 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 27 + 10*m1 + 2*m1*x + m1^2 + 9*x + x^2 }-> 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 1123.84/291.65 replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0 1123.84/291.65 replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 1123.84/291.65 sort(z') -{ 255 + 2896*z' + 3562*z'^2 + 1450*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0 1123.84/291.65 sortIter(z', z'') -{ 19134 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0 1123.84/291.65 sortIter(z', z'') -{ 19135 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262967 + 1434897*m2 + 1091392*m2*n3 + 417600*m2*n3*x'' + 208800*m2*n3^2 + 1091392*m2*x'' + 208800*m2*x''^2 + 545696*m2^2 + 208800*m2^2*n3 + 208800*m2^2*x'' + 69600*m2^3 + 1434896*n3 + 1091392*n3*x'' + 208800*n3*x''^2 + 545696*n3^2 + 208800*n3^2*x'' + 69600*n3^3 + 1434896*x'' + 545696*x''^2 + 69600*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'') 1123.84/291.65 sortIter(z', z'') -{ 304110 + 552264*n3 + 673776*n3*x' + 208800*n3*x'^2 + 336888*n3^2 + 208800*n3^2*x' + 69600*n3^3 + 552264*x' + 336888*x'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19134 + 87288*z' + 128088*z'^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 1262966 + 1434897*m3 + 1091392*m3*n4 + 417600*m3*n4*x2 + 208800*m3*n4^2 + 1091392*m3*x2 + 208800*m3*x2^2 + 545696*m3^2 + 208800*m3^2*n4 + 208800*m3^2*x2 + 69600*m3^3 + 1434896*n4 + 1091392*n4*x2 + 208800*n4*x2^2 + 545696*n4^2 + 208800*n4^2*x2 + 69600*n4^3 + 1434896*x2 + 545696*x2^2 + 69600*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 1123.84/291.65 sortIter(z', z'') -{ 19133 + 87288*z' + 128088*z'^2 + 69600*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0 1123.84/291.65 tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0 1123.84/291.65 tail(z') -{ 1 }-> 0 :|: z' = 0 1123.84/291.65 tail(z') -{ 0 }-> 0 :|: z' >= 0 1123.84/291.65 1123.84/291.65 Function symbols to be analyzed: 1123.84/291.65 Previous analysis results are: 1123.84/291.65 empty: runtime: O(1) [1], size: O(1) [2] 1123.84/291.65 le: runtime: O(n^1) [2 + z''], size: O(1) [2] 1123.84/291.65 eq: runtime: O(n^1) [3 + z''], size: O(1) [2] 1123.84/291.65 tail: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.66 head: runtime: O(1) [1], size: O(n^1) [z'] 1123.84/291.66 min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z'] 1123.84/291.66 if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z''] 1123.84/291.66 replace: runtime: O(n^2) [6 + 5*z1 + z1^2], size: O(n^1) [z'' + z1] 1123.84/291.66 if_replace: runtime: O(n^2) [8 + 5*z2 + z2^2], size: O(n^1) [z1 + z2] 1123.84/291.66 sortIter: runtime: O(n^3) [254 + 2896*z' + 3562*z'^2 + 1450*z'^3], size: O(n^2) [2*z' + z'^2 + z''] 1123.84/291.66 if: runtime: O(n^3) [19132 + 87288*z'' + 128088*z''^2 + 69600*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2] 1123.84/291.66 sort: runtime: O(n^3) [255 + 2896*z' + 3562*z'^2 + 1450*z'^3], size: O(n^2) [2*z' + z'^2] 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (69) FinalProof (FINISHED) 1123.84/291.66 Computed overall runtime complexity 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (70) 1123.84/291.66 BOUNDS(1, n^3) 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (71) RenamingProof (BOTH BOUNDS(ID, ID)) 1123.84/291.66 Renamed function symbols to avoid clashes with predefined symbol. 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (72) 1123.84/291.66 Obligation: 1123.84/291.66 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.84/291.66 1123.84/291.66 1123.84/291.66 The TRS R consists of the following rules: 1123.84/291.66 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 S is empty. 1123.84/291.66 Rewrite Strategy: INNERMOST 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (73) SlicingProof (LOWER BOUND(ID)) 1123.84/291.66 Sliced the following arguments: 1123.84/291.66 append/0 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (74) 1123.84/291.66 Obligation: 1123.84/291.66 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.84/291.66 1123.84/291.66 1123.84/291.66 The TRS R consists of the following rules: 1123.84/291.66 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 S is empty. 1123.84/291.66 Rewrite Strategy: INNERMOST 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (75) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1123.84/291.66 Infered types. 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (76) 1123.84/291.66 Obligation: 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (77) OrderProof (LOWER BOUND(ID)) 1123.84/291.66 Heuristically decided to analyse the following defined symbols: 1123.84/291.66 eq, le, min, replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 eq < replace 1123.84/291.66 le < min 1123.84/291.66 min < sortIter 1123.84/291.66 replace < sortIter 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (78) 1123.84/291.66 Obligation: 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 1123.84/291.66 Generator Equations: 1123.84/291.66 gen_0':s4_0(0) <=> 0' 1123.84/291.66 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 1123.84/291.66 gen_nil:cons:append5_0(0) <=> nil 1123.84/291.66 gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x)) 1123.84/291.66 1123.84/291.66 1123.84/291.66 The following defined symbols remain to be analysed: 1123.84/291.66 eq, le, min, replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 eq < replace 1123.84/291.66 le < min 1123.84/291.66 min < sortIter 1123.84/291.66 replace < sortIter 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (79) RewriteLemmaProof (LOWER BOUND(ID)) 1123.84/291.66 Proved the following rewrite lemma: 1123.84/291.66 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 1123.84/291.66 1123.84/291.66 Induction Base: 1123.84/291.66 eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 1123.84/291.66 true 1123.84/291.66 1123.84/291.66 Induction Step: 1123.84/291.66 eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1) 1123.84/291.66 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH 1123.84/291.66 true 1123.84/291.66 1123.84/291.66 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (80) 1123.84/291.66 Complex Obligation (BEST) 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (81) 1123.84/291.66 Obligation: 1123.84/291.66 Proved the lower bound n^1 for the following obligation: 1123.84/291.66 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 1123.84/291.66 Generator Equations: 1123.84/291.66 gen_0':s4_0(0) <=> 0' 1123.84/291.66 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 1123.84/291.66 gen_nil:cons:append5_0(0) <=> nil 1123.84/291.66 gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x)) 1123.84/291.66 1123.84/291.66 1123.84/291.66 The following defined symbols remain to be analysed: 1123.84/291.66 eq, le, min, replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 eq < replace 1123.84/291.66 le < min 1123.84/291.66 min < sortIter 1123.84/291.66 replace < sortIter 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (82) LowerBoundPropagationProof (FINISHED) 1123.84/291.66 Propagated lower bound. 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (83) 1123.84/291.66 BOUNDS(n^1, INF) 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (84) 1123.84/291.66 Obligation: 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 1123.84/291.66 Lemmas: 1123.84/291.66 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 1123.84/291.66 1123.84/291.66 1123.84/291.66 Generator Equations: 1123.84/291.66 gen_0':s4_0(0) <=> 0' 1123.84/291.66 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 1123.84/291.66 gen_nil:cons:append5_0(0) <=> nil 1123.84/291.66 gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x)) 1123.84/291.66 1123.84/291.66 1123.84/291.66 The following defined symbols remain to be analysed: 1123.84/291.66 le, min, replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 le < min 1123.84/291.66 min < sortIter 1123.84/291.66 replace < sortIter 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (85) RewriteLemmaProof (LOWER BOUND(ID)) 1123.84/291.66 Proved the following rewrite lemma: 1123.84/291.66 le(gen_0':s4_0(n584_0), gen_0':s4_0(n584_0)) -> true, rt in Omega(1 + n584_0) 1123.84/291.66 1123.84/291.66 Induction Base: 1123.84/291.66 le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 1123.84/291.66 true 1123.84/291.66 1123.84/291.66 Induction Step: 1123.84/291.66 le(gen_0':s4_0(+(n584_0, 1)), gen_0':s4_0(+(n584_0, 1))) ->_R^Omega(1) 1123.84/291.66 le(gen_0':s4_0(n584_0), gen_0':s4_0(n584_0)) ->_IH 1123.84/291.66 true 1123.84/291.66 1123.84/291.66 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (86) 1123.84/291.66 Obligation: 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 1123.84/291.66 Lemmas: 1123.84/291.66 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 1123.84/291.66 le(gen_0':s4_0(n584_0), gen_0':s4_0(n584_0)) -> true, rt in Omega(1 + n584_0) 1123.84/291.66 1123.84/291.66 1123.84/291.66 Generator Equations: 1123.84/291.66 gen_0':s4_0(0) <=> 0' 1123.84/291.66 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 1123.84/291.66 gen_nil:cons:append5_0(0) <=> nil 1123.84/291.66 gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x)) 1123.84/291.66 1123.84/291.66 1123.84/291.66 The following defined symbols remain to be analysed: 1123.84/291.66 min, replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 min < sortIter 1123.84/291.66 replace < sortIter 1123.84/291.66 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (87) RewriteLemmaProof (LOWER BOUND(ID)) 1123.84/291.66 Proved the following rewrite lemma: 1123.84/291.66 min(gen_nil:cons:append5_0(+(1, n955_0))) -> gen_0':s4_0(0), rt in Omega(1 + n955_0) 1123.84/291.66 1123.84/291.66 Induction Base: 1123.84/291.66 min(gen_nil:cons:append5_0(+(1, 0))) ->_R^Omega(1) 1123.84/291.66 0' 1123.84/291.66 1123.84/291.66 Induction Step: 1123.84/291.66 min(gen_nil:cons:append5_0(+(1, +(n955_0, 1)))) ->_R^Omega(1) 1123.84/291.66 if_min(le(0', 0'), cons(0', cons(0', gen_nil:cons:append5_0(n955_0)))) ->_L^Omega(1) 1123.84/291.66 if_min(true, cons(0', cons(0', gen_nil:cons:append5_0(n955_0)))) ->_R^Omega(1) 1123.84/291.66 min(cons(0', gen_nil:cons:append5_0(n955_0))) ->_IH 1123.84/291.66 gen_0':s4_0(0) 1123.84/291.66 1123.84/291.66 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 1123.84/291.66 ---------------------------------------- 1123.84/291.66 1123.84/291.66 (88) 1123.84/291.66 Obligation: 1123.84/291.66 Innermost TRS: 1123.84/291.66 Rules: 1123.84/291.66 eq(0', 0') -> true 1123.84/291.66 eq(0', s(m)) -> false 1123.84/291.66 eq(s(n), 0') -> false 1123.84/291.66 eq(s(n), s(m)) -> eq(n, m) 1123.84/291.66 le(0', m) -> true 1123.84/291.66 le(s(n), 0') -> false 1123.84/291.66 le(s(n), s(m)) -> le(n, m) 1123.84/291.66 min(cons(x, nil)) -> x 1123.84/291.66 min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) 1123.84/291.66 if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) 1123.84/291.66 if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) 1123.84/291.66 replace(n, m, nil) -> nil 1123.84/291.66 replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) 1123.84/291.66 if_replace(true, n, m, cons(k, x)) -> cons(m, x) 1123.84/291.66 if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) 1123.84/291.66 empty(nil) -> true 1123.84/291.66 empty(cons(n, x)) -> false 1123.84/291.66 head(cons(n, x)) -> n 1123.84/291.66 tail(nil) -> nil 1123.84/291.66 tail(cons(n, x)) -> x 1123.84/291.66 sort(x) -> sortIter(x, nil) 1123.84/291.66 sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil))) 1123.84/291.66 if(true, x, y, z) -> y 1123.84/291.66 if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) 1123.84/291.66 1123.84/291.66 Types: 1123.84/291.66 eq :: 0':s -> 0':s -> true:false 1123.84/291.66 0' :: 0':s 1123.84/291.66 true :: true:false 1123.84/291.66 s :: 0':s -> 0':s 1123.84/291.66 false :: true:false 1123.84/291.66 le :: 0':s -> 0':s -> true:false 1123.84/291.66 min :: nil:cons:append -> 0':s 1123.84/291.66 cons :: 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 nil :: nil:cons:append 1123.84/291.66 if_min :: true:false -> nil:cons:append -> 0':s 1123.84/291.66 replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append 1123.84/291.66 empty :: nil:cons:append -> true:false 1123.84/291.66 head :: nil:cons:append -> 0':s 1123.84/291.66 tail :: nil:cons:append -> nil:cons:append 1123.84/291.66 sort :: nil:cons:append -> nil:cons:append 1123.84/291.66 sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append 1123.84/291.66 append :: nil:cons:append -> nil:cons:append 1123.84/291.66 hole_true:false1_0 :: true:false 1123.84/291.66 hole_0':s2_0 :: 0':s 1123.84/291.66 hole_nil:cons:append3_0 :: nil:cons:append 1123.84/291.66 gen_0':s4_0 :: Nat -> 0':s 1123.84/291.66 gen_nil:cons:append5_0 :: Nat -> nil:cons:append 1123.84/291.66 1123.84/291.66 1123.84/291.66 Lemmas: 1123.84/291.66 eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0) 1123.84/291.66 le(gen_0':s4_0(n584_0), gen_0':s4_0(n584_0)) -> true, rt in Omega(1 + n584_0) 1123.84/291.66 min(gen_nil:cons:append5_0(+(1, n955_0))) -> gen_0':s4_0(0), rt in Omega(1 + n955_0) 1123.84/291.66 1123.84/291.66 1123.84/291.66 Generator Equations: 1123.84/291.66 gen_0':s4_0(0) <=> 0' 1123.84/291.66 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 1123.84/291.66 gen_nil:cons:append5_0(0) <=> nil 1123.84/291.66 gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x)) 1123.84/291.66 1123.84/291.66 1123.84/291.66 The following defined symbols remain to be analysed: 1123.84/291.66 replace, sortIter 1123.84/291.66 1123.84/291.66 They will be analysed ascendingly in the following order: 1123.84/291.66 replace < sortIter 1124.15/291.73 EOF