323.44/291.49 WORST_CASE(?, O(n^1)) 323.44/291.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 323.44/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 323.44/291.50 323.44/291.50 323.44/291.50 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). 323.44/291.50 323.44/291.50 (0) CpxRelTRS 323.44/291.50 (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 166 ms] 323.44/291.50 (2) CpxRelTRS 323.44/291.50 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (4) CpxWeightedTrs 323.44/291.50 (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (6) CpxWeightedTrs 323.44/291.50 (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (8) CpxTypedWeightedTrs 323.44/291.50 (9) CompletionProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (10) CpxTypedWeightedCompleteTrs 323.44/291.50 (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (12) CpxTypedWeightedCompleteTrs 323.44/291.50 (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (14) CpxRNTS 323.44/291.50 (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (16) CpxRNTS 323.44/291.50 (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 323.44/291.50 (18) CpxRNTS 323.44/291.50 (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (20) CpxRNTS 323.44/291.50 (21) IntTrsBoundProof [UPPER BOUND(ID), 368 ms] 323.44/291.50 (22) CpxRNTS 323.44/291.50 (23) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] 323.44/291.50 (24) CpxRNTS 323.44/291.50 (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (26) CpxRNTS 323.44/291.50 (27) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] 323.44/291.50 (28) CpxRNTS 323.44/291.50 (29) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] 323.44/291.50 (30) CpxRNTS 323.44/291.50 (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (32) CpxRNTS 323.44/291.50 (33) IntTrsBoundProof [UPPER BOUND(ID), 1222 ms] 323.44/291.50 (34) CpxRNTS 323.44/291.50 (35) IntTrsBoundProof [UPPER BOUND(ID), 520 ms] 323.44/291.50 (36) CpxRNTS 323.44/291.50 (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 323.44/291.50 (38) CpxRNTS 323.44/291.50 (39) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] 323.44/291.50 (40) CpxRNTS 323.44/291.50 (41) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] 323.44/291.50 (42) CpxRNTS 323.44/291.50 (43) FinalProof [FINISHED, 0 ms] 323.44/291.50 (44) BOUNDS(1, n^1) 323.44/291.50 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (0) 323.44/291.50 Obligation: 323.44/291.50 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). 323.44/291.50 323.44/291.50 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs))) 323.44/291.50 ordered(Cons(x, Nil)) -> True 323.44/291.50 ordered(Nil) -> True 323.44/291.50 notEmpty(Cons(x, xs)) -> True 323.44/291.50 notEmpty(Nil) -> False 323.44/291.50 goal(xs) -> ordered(xs) 323.44/291.50 323.44/291.50 The (relative) TRS S consists of the following rules: 323.44/291.50 323.44/291.50 <(S(x), S(y)) -> <(x, y) 323.44/291.50 <(0, S(y)) -> True 323.44/291.50 <(x, 0) -> False 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) 323.44/291.50 ordered[Ite](False, xs) -> False 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) 323.44/291.50 proved termination of relative rules 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (2) 323.44/291.50 Obligation: 323.44/291.50 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). 323.44/291.50 323.44/291.50 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs))) 323.44/291.50 ordered(Cons(x, Nil)) -> True 323.44/291.50 ordered(Nil) -> True 323.44/291.50 notEmpty(Cons(x, xs)) -> True 323.44/291.50 notEmpty(Nil) -> False 323.44/291.50 goal(xs) -> ordered(xs) 323.44/291.50 323.44/291.50 The (relative) TRS S consists of the following rules: 323.44/291.50 323.44/291.50 <(S(x), S(y)) -> <(x, y) 323.44/291.50 <(0, S(y)) -> True 323.44/291.50 <(x, 0) -> False 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) 323.44/291.50 ordered[Ite](False, xs) -> False 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Transformed relative TRS to weighted TRS 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (4) 323.44/291.50 Obligation: 323.44/291.50 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 323.44/291.50 323.44/291.50 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](<(x', x), Cons(x', Cons(x, xs))) [1] 323.44/291.50 ordered(Cons(x, Nil)) -> True [1] 323.44/291.50 ordered(Nil) -> True [1] 323.44/291.50 notEmpty(Cons(x, xs)) -> True [1] 323.44/291.50 notEmpty(Nil) -> False [1] 323.44/291.50 goal(xs) -> ordered(xs) [1] 323.44/291.50 <(S(x), S(y)) -> <(x, y) [0] 323.44/291.50 <(0, S(y)) -> True [0] 323.44/291.50 <(x, 0) -> False [0] 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0] 323.44/291.50 ordered[Ite](False, xs) -> False [0] 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Renamed defined symbols to avoid conflicts with arithmetic symbols: 323.44/291.50 323.44/291.50 < => lt 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (6) 323.44/291.50 Obligation: 323.44/291.50 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 323.44/291.50 323.44/291.50 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1] 323.44/291.50 ordered(Cons(x, Nil)) -> True [1] 323.44/291.50 ordered(Nil) -> True [1] 323.44/291.50 notEmpty(Cons(x, xs)) -> True [1] 323.44/291.50 notEmpty(Nil) -> False [1] 323.44/291.50 goal(xs) -> ordered(xs) [1] 323.44/291.50 lt(S(x), S(y)) -> lt(x, y) [0] 323.44/291.50 lt(0, S(y)) -> True [0] 323.44/291.50 lt(x, 0) -> False [0] 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0] 323.44/291.50 ordered[Ite](False, xs) -> False [0] 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Infered types. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (8) 323.44/291.50 Obligation: 323.44/291.50 Runtime Complexity Weighted TRS with Types. 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1] 323.44/291.50 ordered(Cons(x, Nil)) -> True [1] 323.44/291.50 ordered(Nil) -> True [1] 323.44/291.50 notEmpty(Cons(x, xs)) -> True [1] 323.44/291.50 notEmpty(Nil) -> False [1] 323.44/291.50 goal(xs) -> ordered(xs) [1] 323.44/291.50 lt(S(x), S(y)) -> lt(x, y) [0] 323.44/291.50 lt(0, S(y)) -> True [0] 323.44/291.50 lt(x, 0) -> False [0] 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0] 323.44/291.50 ordered[Ite](False, xs) -> False [0] 323.44/291.50 323.44/291.50 The TRS has the following type information: 323.44/291.50 ordered :: Cons:Nil -> True:False 323.44/291.50 Cons :: S:0 -> Cons:Nil -> Cons:Nil 323.44/291.50 ordered[Ite] :: True:False -> Cons:Nil -> True:False 323.44/291.50 lt :: S:0 -> S:0 -> True:False 323.44/291.50 Nil :: Cons:Nil 323.44/291.50 True :: True:False 323.44/291.50 notEmpty :: Cons:Nil -> True:False 323.44/291.50 False :: True:False 323.44/291.50 goal :: Cons:Nil -> True:False 323.44/291.50 S :: S:0 -> S:0 323.44/291.50 0 :: S:0 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (9) CompletionProof (UPPER BOUND(ID)) 323.44/291.50 The transformation into a RNTS is sound, since: 323.44/291.50 323.44/291.50 (a) The obligation is a constructor system where every type has a constant constructor, 323.44/291.50 323.44/291.50 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 323.44/291.50 323.44/291.50 ordered_1 323.44/291.50 notEmpty_1 323.44/291.50 goal_1 323.44/291.50 323.44/291.50 (c) The following functions are completely defined: 323.44/291.50 323.44/291.50 lt_2 323.44/291.50 ordered[Ite]_2 323.44/291.50 323.44/291.50 Due to the following rules being added: 323.44/291.50 323.44/291.50 lt(v0, v1) -> null_lt [0] 323.44/291.50 ordered[Ite](v0, v1) -> null_ordered[Ite] [0] 323.44/291.50 323.44/291.50 And the following fresh constants: null_lt, null_ordered[Ite] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (10) 323.44/291.50 Obligation: 323.44/291.50 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 323.44/291.50 323.44/291.50 Runtime Complexity Weighted TRS with Types. 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](lt(x', x), Cons(x', Cons(x, xs))) [1] 323.44/291.50 ordered(Cons(x, Nil)) -> True [1] 323.44/291.50 ordered(Nil) -> True [1] 323.44/291.50 notEmpty(Cons(x, xs)) -> True [1] 323.44/291.50 notEmpty(Nil) -> False [1] 323.44/291.50 goal(xs) -> ordered(xs) [1] 323.44/291.50 lt(S(x), S(y)) -> lt(x, y) [0] 323.44/291.50 lt(0, S(y)) -> True [0] 323.44/291.50 lt(x, 0) -> False [0] 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0] 323.44/291.50 ordered[Ite](False, xs) -> False [0] 323.44/291.50 lt(v0, v1) -> null_lt [0] 323.44/291.50 ordered[Ite](v0, v1) -> null_ordered[Ite] [0] 323.44/291.50 323.44/291.50 The TRS has the following type information: 323.44/291.50 ordered :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 Cons :: S:0 -> Cons:Nil -> Cons:Nil 323.44/291.50 ordered[Ite] :: True:False:null_lt:null_ordered[Ite] -> Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 lt :: S:0 -> S:0 -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 Nil :: Cons:Nil 323.44/291.50 True :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 notEmpty :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 False :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 goal :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 S :: S:0 -> S:0 323.44/291.50 0 :: S:0 323.44/291.50 null_lt :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 null_ordered[Ite] :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (11) NarrowingProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (12) 323.44/291.50 Obligation: 323.44/291.50 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 323.44/291.50 323.44/291.50 Runtime Complexity Weighted TRS with Types. 323.44/291.50 The TRS R consists of the following rules: 323.44/291.50 323.44/291.50 ordered(Cons(S(x''), Cons(S(y'), xs))) -> ordered[Ite](lt(x'', y'), Cons(S(x''), Cons(S(y'), xs))) [1] 323.44/291.50 ordered(Cons(0, Cons(S(y''), xs))) -> ordered[Ite](True, Cons(0, Cons(S(y''), xs))) [1] 323.44/291.50 ordered(Cons(x', Cons(0, xs))) -> ordered[Ite](False, Cons(x', Cons(0, xs))) [1] 323.44/291.50 ordered(Cons(x', Cons(x, xs))) -> ordered[Ite](null_lt, Cons(x', Cons(x, xs))) [1] 323.44/291.50 ordered(Cons(x, Nil)) -> True [1] 323.44/291.50 ordered(Nil) -> True [1] 323.44/291.50 notEmpty(Cons(x, xs)) -> True [1] 323.44/291.50 notEmpty(Nil) -> False [1] 323.44/291.50 goal(xs) -> ordered(xs) [1] 323.44/291.50 lt(S(x), S(y)) -> lt(x, y) [0] 323.44/291.50 lt(0, S(y)) -> True [0] 323.44/291.50 lt(x, 0) -> False [0] 323.44/291.50 ordered[Ite](True, Cons(x, xs)) -> ordered(xs) [0] 323.44/291.50 ordered[Ite](False, xs) -> False [0] 323.44/291.50 lt(v0, v1) -> null_lt [0] 323.44/291.50 ordered[Ite](v0, v1) -> null_ordered[Ite] [0] 323.44/291.50 323.44/291.50 The TRS has the following type information: 323.44/291.50 ordered :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 Cons :: S:0 -> Cons:Nil -> Cons:Nil 323.44/291.50 ordered[Ite] :: True:False:null_lt:null_ordered[Ite] -> Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 lt :: S:0 -> S:0 -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 Nil :: Cons:Nil 323.44/291.50 True :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 notEmpty :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 False :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 goal :: Cons:Nil -> True:False:null_lt:null_ordered[Ite] 323.44/291.50 S :: S:0 -> S:0 323.44/291.50 0 :: S:0 323.44/291.50 null_lt :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 null_ordered[Ite] :: True:False:null_lt:null_ordered[Ite] 323.44/291.50 323.44/291.50 Rewrite Strategy: INNERMOST 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 323.44/291.50 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 323.44/291.50 The constant constructors are abstracted as follows: 323.44/291.50 323.44/291.50 Nil => 0 323.44/291.50 True => 2 323.44/291.50 False => 1 323.44/291.50 0 => 0 323.44/291.50 null_lt => 0 323.44/291.50 null_ordered[Ite] => 0 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (14) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(xs) :|: xs >= 0, z = xs 323.44/291.50 lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: x >= 0, z = 1 + x + 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: xs >= 0, z = 1, z' = xs 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 323.44/291.50 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (15) SimplificationProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (16) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 323.44/291.50 Found the following analysis order by SCC decomposition: 323.44/291.50 323.44/291.50 { lt } 323.44/291.50 { notEmpty } 323.44/291.50 { ordered[Ite], ordered } 323.44/291.50 { goal } 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (18) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (19) ResultPropagationProof (UPPER BOUND(ID)) 323.44/291.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (20) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (21) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed SIZE bound using CoFloCo for: lt 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 2 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (22) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {lt}, {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: ?, size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (23) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed RUNTIME bound using CoFloCo for: lt 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 0 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (24) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](lt(x'', y'), 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (25) ResultPropagationProof (UPPER BOUND(ID)) 323.44/291.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (26) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (27) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed SIZE bound using CoFloCo for: notEmpty 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 2 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (28) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {notEmpty}, {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: ?, size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (29) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed RUNTIME bound using CoFloCo for: notEmpty 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 1 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (30) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (31) ResultPropagationProof (UPPER BOUND(ID)) 323.44/291.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (32) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (33) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed SIZE bound using CoFloCo for: ordered[Ite] 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 2 323.44/291.50 323.44/291.50 Computed SIZE bound using CoFloCo for: ordered 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 2 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (34) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {ordered[Ite],ordered}, {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 ordered[Ite]: runtime: ?, size: O(1) [2] 323.44/291.50 ordered: runtime: ?, size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (35) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed RUNTIME bound using CoFloCo for: ordered[Ite] 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(n^1) with polynomial bound: 4 + z' 323.44/291.50 323.44/291.50 Computed RUNTIME bound using CoFloCo for: ordered 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(n^1) with polynomial bound: 5 + z 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (36) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 1 }-> ordered(z) :|: z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](s, 1 + (1 + x'') + (1 + (1 + y') + xs)) :|: s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](2, 1 + 0 + (1 + (1 + y'') + xs)) :|: xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](1, 1 + x' + (1 + 0 + xs)) :|: xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 1 }-> ordered[Ite](0, 1 + x' + (1 + x + xs)) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> ordered(xs) :|: z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2] 323.44/291.50 ordered: runtime: O(n^1) [5 + z], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (37) ResultPropagationProof (UPPER BOUND(ID)) 323.44/291.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (38) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2] 323.44/291.50 ordered: runtime: O(n^1) [5 + z], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (39) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed SIZE bound using CoFloCo for: goal 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(1) with polynomial bound: 2 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (40) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: {goal} 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2] 323.44/291.50 ordered: runtime: O(n^1) [5 + z], size: O(1) [2] 323.44/291.50 goal: runtime: ?, size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (41) IntTrsBoundProof (UPPER BOUND(ID)) 323.44/291.50 323.44/291.50 Computed RUNTIME bound using CoFloCo for: goal 323.44/291.50 after applying outer abstraction to obtain an ITS, 323.44/291.50 resulting in: O(n^1) with polynomial bound: 6 + z 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (42) 323.44/291.50 Obligation: 323.44/291.50 Complexity RNTS consisting of the following rules: 323.44/291.50 323.44/291.50 goal(z) -{ 6 + z }-> s4 :|: s4 >= 0, s4 <= 2, z >= 0 323.44/291.50 lt(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 323.44/291.50 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 323.44/291.50 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 323.44/291.50 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0 323.44/291.50 notEmpty(z) -{ 1 }-> 1 :|: z = 0 323.44/291.50 ordered(z) -{ 9 + x'' + xs + y' }-> s'' :|: s'' >= 0, s'' <= 2, s >= 0, s <= 2, xs >= 0, z = 1 + (1 + x'') + (1 + (1 + y') + xs), y' >= 0, x'' >= 0 323.44/291.50 ordered(z) -{ 8 + xs + y'' }-> s1 :|: s1 >= 0, s1 <= 2, xs >= 0, z = 1 + 0 + (1 + (1 + y'') + xs), y'' >= 0 323.44/291.50 ordered(z) -{ 7 + x' + xs }-> s2 :|: s2 >= 0, s2 <= 2, xs >= 0, x' >= 0, z = 1 + x' + (1 + 0 + xs) 323.44/291.50 ordered(z) -{ 7 + x + x' + xs }-> s3 :|: s3 >= 0, s3 <= 2, xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs) 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z - 1 >= 0 323.44/291.50 ordered(z) -{ 1 }-> 2 :|: z = 0 323.44/291.50 ordered[Ite](z, z') -{ 5 + xs }-> s5 :|: s5 >= 0, s5 <= 2, z = 2, xs >= 0, z' = 1 + x + xs, x >= 0 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 1 :|: z' >= 0, z = 1 323.44/291.50 ordered[Ite](z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 323.44/291.50 323.44/291.50 Function symbols to be analyzed: 323.44/291.50 Previous analysis results are: 323.44/291.50 lt: runtime: O(1) [0], size: O(1) [2] 323.44/291.50 notEmpty: runtime: O(1) [1], size: O(1) [2] 323.44/291.50 ordered[Ite]: runtime: O(n^1) [4 + z'], size: O(1) [2] 323.44/291.50 ordered: runtime: O(n^1) [5 + z], size: O(1) [2] 323.44/291.50 goal: runtime: O(n^1) [6 + z], size: O(1) [2] 323.44/291.50 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (43) FinalProof (FINISHED) 323.44/291.50 Computed overall runtime complexity 323.44/291.50 ---------------------------------------- 323.44/291.50 323.44/291.50 (44) 323.44/291.50 BOUNDS(1, n^1) 323.44/291.54 EOF