/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 184 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 8 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 975 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 1035 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 158 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 301 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 189 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 552 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 101 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 219 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 176 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 396 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 146 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 51 ms] (66) CpxRNTS (67) FinalProof [FINISHED, 0 ms] (68) BOUNDS(1, n^2) (69) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CpxRelTRS (71) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (72) typed CpxTrs (73) OrderProof [LOWER BOUND(ID), 0 ms] (74) typed CpxTrs (75) RewriteLemmaProof [LOWER BOUND(ID), 1916 ms] (76) typed CpxTrs (77) RewriteLemmaProof [LOWER BOUND(ID), 355 ms] (78) BEST (79) proven lower bound (80) LowerBoundPropagationProof [FINISHED, 0 ms] (81) BOUNDS(n^1, INF) (82) typed CpxTrs (83) RewriteLemmaProof [LOWER BOUND(ID), 343 ms] (84) typed CpxTrs (85) RewriteLemmaProof [LOWER BOUND(ID), 360 ms] (86) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] findMin(@l) -> findMin#1(@l) [1] findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) [1] findMin#1(nil) -> nil [1] findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) [1] findMin#2(nil, @x) -> ::(@x, nil) [1] findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) [1] findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] minSort(@l) -> minSort#1(findMin(@l)) [1] minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) [1] minSort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] findMin(@l) -> findMin#1(@l) [1] findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) [1] findMin#1(nil) -> nil [1] findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) [1] findMin#2(nil, @x) -> ::(@x, nil) [1] findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) [1] findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] minSort(@l) -> minSort#1(findMin(@l)) [1] minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) [1] minSort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: minSort_1 minSort#1_1 (c) The following functions are completely defined: #less_2 findMin_1 findMin#1_1 findMin#2_2 findMin#3_4 #cklt_1 #compare_2 Due to the following rules being added: #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] findMin#3(v0, v1, v2, v3) -> nil [0] And the following fresh constants: null_#cklt, null_#compare ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] findMin(@l) -> findMin#1(@l) [1] findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) [1] findMin#1(nil) -> nil [1] findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) [1] findMin#2(nil, @x) -> ::(@x, nil) [1] findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) [1] findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] minSort(@l) -> minSort#1(findMin(@l)) [1] minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) [1] minSort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] findMin#3(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #less(#0, #0) -> #cklt(#EQ) [1] #less(#0, #neg(@y')) -> #cklt(#GT) [1] #less(#0, #pos(@y'')) -> #cklt(#LT) [1] #less(#0, #s(@y1)) -> #cklt(#LT) [1] #less(#neg(@x'), #0) -> #cklt(#LT) [1] #less(#neg(@x''), #neg(@y2)) -> #cklt(#compare(@y2, @x'')) [1] #less(#neg(@x1), #pos(@y3)) -> #cklt(#LT) [1] #less(#pos(@x2), #0) -> #cklt(#GT) [1] #less(#pos(@x3), #neg(@y4)) -> #cklt(#GT) [1] #less(#pos(@x4), #pos(@y5)) -> #cklt(#compare(@x4, @y5)) [1] #less(#s(@x5), #0) -> #cklt(#GT) [1] #less(#s(@x6), #s(@y6)) -> #cklt(#compare(@x6, @y6)) [1] #less(@x, @y) -> #cklt(null_#compare) [1] findMin(@l) -> findMin#1(@l) [1] findMin#1(::(@x, @xs)) -> findMin#2(findMin#1(@xs), @x) [2] findMin#1(nil) -> nil [1] findMin#2(::(@y, @ys), @x) -> findMin#3(#cklt(#compare(@x, @y)), @x, @y, @ys) [2] findMin#2(nil, @x) -> ::(@x, nil) [1] findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) [1] findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] minSort(@l) -> minSort#1(findMin#1(@l)) [2] minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) [1] minSort#1(nil) -> nil [1] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] findMin#3(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 #0 => 0 null_#cklt => 0 null_#compare => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 #less(z, z') -{ 1 }-> #cklt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 #less(z, z') -{ 1 }-> #cklt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(1) :|: z = 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 findMin(z) -{ 1 }-> findMin#1(@l) :|: z = @l, @l >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(@x, @y)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @y + (1 + @x + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y minSort(z) -{ 2 }-> minSort#1(findMin#1(@l)) :|: z = @l, @l >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @y + (1 + @x + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 2 :|: z = 3 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 findMin(z) -{ 1 }-> findMin#1(@l) :|: z = @l, @l >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(@x, @y)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y findMin#3(z, z', z'', z1) -{ 1 }-> 1 + @y + (1 + @x + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y minSort(z) -{ 2 }-> minSort#1(findMin#1(@l)) :|: z = @l, @l >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(z', @y)), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #compare } { findMin#3 } { #cklt } { #less } { findMin#2 } { findMin#1 } { minSort#1, minSort } { findMin } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(z', @y)), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#compare}, {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(z', @y)), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#compare}, {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(z', @y)), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#compare}, {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(#compare(z', @y)), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: findMin#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#3}, {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: findMin#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#cklt}, {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: ?, size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 2 }-> findMin#3(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {#less}, {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: ?, size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: findMin#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#2}, {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: findMin#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: findMin#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin#1}, {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: findMin#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 5*z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 1 }-> findMin#1(z) :|: z >= 0 findMin#1(z) -{ 2 }-> findMin#2(findMin#1(@xs), @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 2 }-> minSort#1(findMin#1(z)) :|: z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 3 + 5*z }-> minSort#1(s10) :|: s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minSort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: minSort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 3 + 5*z }-> minSort#1(s10) :|: s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {minSort#1,minSort}, {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] minSort#1: runtime: ?, size: O(n^1) [z] minSort: runtime: ?, size: O(n^1) [z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minSort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 8*z + 5*z^2 Computed RUNTIME bound using KoAT for: minSort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 7 + 13*z + 5*z^2 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 3 + 5*z }-> minSort#1(s10) :|: s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 1 }-> 1 + @x + minSort(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] minSort#1: runtime: O(n^2) [4 + 8*z + 5*z^2], size: O(n^1) [z] minSort: runtime: O(n^2) [7 + 13*z + 5*z^2], size: O(n^1) [z] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 7 + 8*s10 + 5*s10^2 + 5*z }-> s11 :|: s11 >= 0, s11 <= s10, s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 8 + 13*@xs + 5*@xs^2 }-> 1 + @x + s12 :|: s12 >= 0, s12 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] minSort#1: runtime: O(n^2) [4 + 8*z + 5*z^2], size: O(n^1) [z] minSort: runtime: O(n^2) [7 + 13*z + 5*z^2], size: O(n^1) [z] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: findMin after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 7 + 8*s10 + 5*s10^2 + 5*z }-> s11 :|: s11 >= 0, s11 <= s10, s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 8 + 13*@xs + 5*@xs^2 }-> 1 + @x + s12 :|: s12 >= 0, s12 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: {findMin} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] minSort#1: runtime: O(n^2) [4 + 8*z + 5*z^2], size: O(n^1) [z] minSort: runtime: O(n^2) [7 + 13*z + 5*z^2], size: O(n^1) [z] findMin: runtime: ?, size: O(n^1) [z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: findMin after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 5*z ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 findMin(z) -{ 2 + 5*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 findMin#1(z) -{ 6 + 5*@xs }-> s9 :|: s8 >= 0, s8 <= @xs, s9 >= 0, s9 <= s8 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 findMin#1(z) -{ 1 }-> 0 :|: z = 0 findMin#2(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + @y + @ys + 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 findMin#2(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 findMin#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 findMin#3(z, z', z'', z1) -{ 1 }-> 1 + z'' + (1 + z' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 minSort(z) -{ 7 + 8*s10 + 5*s10^2 + 5*z }-> s11 :|: s11 >= 0, s11 <= s10, s10 >= 0, s10 <= z, z >= 0 minSort#1(z) -{ 1 }-> 0 :|: z = 0 minSort#1(z) -{ 8 + 13*@xs + 5*@xs^2 }-> 1 + @x + s12 :|: s12 >= 0, s12 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0 Function symbols to be analyzed: Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] findMin#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] #cklt: runtime: O(1) [0], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] findMin#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'] findMin#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [z] minSort#1: runtime: O(n^2) [4 + 8*z + 5*z^2], size: O(n^1) [z] minSort: runtime: O(n^2) [7 + 13*z + 5*z^2], size: O(n^1) [z] findMin: runtime: O(n^1) [2 + 5*z], size: O(n^1) [z] ---------------------------------------- (67) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (68) BOUNDS(1, n^2) ---------------------------------------- (69) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (70) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil The (relative) TRS S consists of the following rules: #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: INNERMOST ---------------------------------------- (71) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (72) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil ---------------------------------------- (73) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: #compare, findMin, findMin#1, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (74) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: #compare, findMin, findMin#1, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (75) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) Induction Base: #compare(gen_#0:#neg:#pos:#s5_4(0), gen_#0:#neg:#pos:#s5_4(0)) ->_R^Omega(0) #EQ Induction Step: #compare(gen_#0:#neg:#pos:#s5_4(+(n8_4, 1)), gen_#0:#neg:#pos:#s5_4(+(n8_4, 1))) ->_R^Omega(0) #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) ->_IH #EQ We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (76) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: findMin#1, findMin, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (77) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: findMin#1(gen_:::nil6_4(n318649_4)) -> *7_4, rt in Omega(n318649_4) Induction Base: findMin#1(gen_:::nil6_4(0)) Induction Step: findMin#1(gen_:::nil6_4(+(n318649_4, 1))) ->_R^Omega(1) findMin#2(findMin(gen_:::nil6_4(n318649_4)), #0) ->_R^Omega(1) findMin#2(findMin#1(gen_:::nil6_4(n318649_4)), #0) ->_IH findMin#2(*7_4, #0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (78) Complex Obligation (BEST) ---------------------------------------- (79) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: findMin#1, findMin, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (80) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (81) BOUNDS(n^1, INF) ---------------------------------------- (82) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) findMin#1(gen_:::nil6_4(n318649_4)) -> *7_4, rt in Omega(n318649_4) Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: findMin, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (83) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: findMin(gen_:::nil6_4(n321788_4)) -> *7_4, rt in Omega(n321788_4) Induction Base: findMin(gen_:::nil6_4(0)) Induction Step: findMin(gen_:::nil6_4(+(n321788_4, 1))) ->_R^Omega(1) findMin#1(gen_:::nil6_4(+(n321788_4, 1))) ->_R^Omega(1) findMin#2(findMin(gen_:::nil6_4(n321788_4)), #0) ->_IH findMin#2(*7_4, #0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (84) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) findMin#1(gen_:::nil6_4(n318649_4)) -> *7_4, rt in Omega(n318649_4) findMin(gen_:::nil6_4(n321788_4)) -> *7_4, rt in Omega(n321788_4) Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: findMin#1, minSort, minSort#1 They will be analysed ascendingly in the following order: findMin = findMin#1 findMin < minSort minSort = minSort#1 ---------------------------------------- (85) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: findMin#1(gen_:::nil6_4(n326813_4)) -> *7_4, rt in Omega(n326813_4) Induction Base: findMin#1(gen_:::nil6_4(0)) Induction Step: findMin#1(gen_:::nil6_4(+(n326813_4, 1))) ->_R^Omega(1) findMin#2(findMin(gen_:::nil6_4(n326813_4)), #0) ->_R^Omega(1) findMin#2(findMin#1(gen_:::nil6_4(n326813_4)), #0) ->_IH findMin#2(*7_4, #0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (86) Obligation: Innermost TRS: Rules: #less(@x, @y) -> #cklt(#compare(@x, @y)) findMin(@l) -> findMin#1(@l) findMin#1(::(@x, @xs)) -> findMin#2(findMin(@xs), @x) findMin#1(nil) -> nil findMin#2(::(@y, @ys), @x) -> findMin#3(#less(@x, @y), @x, @y, @ys) findMin#2(nil, @x) -> ::(@x, nil) findMin#3(#false, @x, @y, @ys) -> ::(@y, ::(@x, @ys)) findMin#3(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) minSort(@l) -> minSort#1(findMin(@l)) minSort#1(::(@x, @xs)) -> ::(@x, minSort(@xs)) minSort#1(nil) -> nil #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Types: #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT findMin :: :::nil -> :::nil findMin#1 :: :::nil -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil findMin#2 :: :::nil -> #0:#neg:#pos:#s -> :::nil nil :: :::nil findMin#3 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil #false :: #false:#true #true :: #false:#true minSort :: :::nil -> :::nil minSort#1 :: :::nil -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s hole_#false:#true1_4 :: #false:#true hole_#0:#neg:#pos:#s2_4 :: #0:#neg:#pos:#s hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT hole_:::nil4_4 :: :::nil gen_#0:#neg:#pos:#s5_4 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_4 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_4(n8_4), gen_#0:#neg:#pos:#s5_4(n8_4)) -> #EQ, rt in Omega(0) findMin#1(gen_:::nil6_4(n326813_4)) -> *7_4, rt in Omega(n326813_4) findMin(gen_:::nil6_4(n321788_4)) -> *7_4, rt in Omega(n321788_4) Generator Equations: gen_#0:#neg:#pos:#s5_4(0) <=> #0 gen_#0:#neg:#pos:#s5_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_4(x)) gen_:::nil6_4(0) <=> nil gen_:::nil6_4(+(x, 1)) <=> ::(#0, gen_:::nil6_4(x)) The following defined symbols remain to be analysed: minSort#1, minSort They will be analysed ascendingly in the following order: minSort = minSort#1