334.58/291.55	WORST_CASE(Omega(n^1), O(n^2))
334.58/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
334.58/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
334.58/291.57	
334.58/291.57	
334.58/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
334.58/291.57	
334.58/291.57	(0) CpxTRS
334.58/291.57	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
334.58/291.57	(2) CpxWeightedTrs
334.58/291.57	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
334.58/291.57	    (4) CpxTypedWeightedTrs
334.58/291.57	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (6) CpxTypedWeightedCompleteTrs
334.58/291.57	    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
334.58/291.57	    (8) CpxTypedWeightedCompleteTrs
334.58/291.57	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (10) CpxRNTS
334.58/291.57	    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 5 ms]
334.58/291.57	    (12) CpxRNTS
334.58/291.57	    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
334.58/291.57	    (14) CpxRNTS
334.58/291.57	    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (16) CpxRNTS
334.58/291.57	    (17) IntTrsBoundProof [UPPER BOUND(ID), 416 ms]
334.58/291.57	    (18) CpxRNTS
334.58/291.57	    (19) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
334.58/291.57	    (20) CpxRNTS
334.58/291.57	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (22) CpxRNTS
334.58/291.57	    (23) IntTrsBoundProof [UPPER BOUND(ID), 1912 ms]
334.58/291.57	    (24) CpxRNTS
334.58/291.57	    (25) IntTrsBoundProof [UPPER BOUND(ID), 554 ms]
334.58/291.57	    (26) CpxRNTS
334.58/291.57	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (28) CpxRNTS
334.58/291.57	    (29) IntTrsBoundProof [UPPER BOUND(ID), 421 ms]
334.58/291.57	    (30) CpxRNTS
334.58/291.57	    (31) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
334.58/291.57	    (32) CpxRNTS
334.58/291.57	    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (34) CpxRNTS
334.58/291.57	    (35) IntTrsBoundProof [UPPER BOUND(ID), 196 ms]
334.58/291.57	    (36) CpxRNTS
334.58/291.57	    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
334.58/291.57	    (38) CpxRNTS
334.58/291.57	    (39) FinalProof [FINISHED, 0 ms]
334.58/291.57	    (40) BOUNDS(1, n^2)
334.58/291.57	(41) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
334.58/291.57	(42) TRS for Loop Detection
334.58/291.57	    (43) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
334.58/291.57	    (44) BEST
334.58/291.57	        (45) proven lower bound
334.58/291.57	            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
334.58/291.57	            (47) BOUNDS(n^1, INF)
334.58/291.57	        (48) TRS for Loop Detection
334.58/291.57	
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(0)
334.58/291.57	Obligation:
334.58/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
334.58/291.57	
334.58/291.57	
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil)
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
334.58/291.57	   leq#2(0, x8) -> True
334.58/291.57	   leq#2(S(x12), 0) -> False
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
334.58/291.57	   main(x1) -> sort#2(x1)
334.58/291.57	
334.58/291.57	S is empty.
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
334.58/291.57	Transformed relative TRS to weighted TRS
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(2)
334.58/291.57	Obligation:
334.58/291.57	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
334.58/291.57	
334.58/291.57	
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil [1]
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
334.58/291.57	   leq#2(0, x8) -> True [1]
334.58/291.57	   leq#2(S(x12), 0) -> False [1]
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
334.58/291.57	   main(x1) -> sort#2(x1) [1]
334.58/291.57	
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
334.58/291.57	Infered types.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(4)
334.58/291.57	Obligation:
334.58/291.57	Runtime Complexity Weighted TRS with Types.
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil [1]
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
334.58/291.57	   leq#2(0, x8) -> True [1]
334.58/291.57	   leq#2(S(x12), 0) -> False [1]
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
334.58/291.57	   main(x1) -> sort#2(x1) [1]
334.58/291.57	
334.58/291.57	The TRS has the following type information:
334.58/291.57	sort#2 :: Nil:Cons -> Nil:Cons
334.58/291.57	Nil :: Nil:Cons
334.58/291.57	Cons :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	insert#3 :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	True :: True:False
334.58/291.57	False :: True:False
334.58/291.57	leq#2 :: 0:S -> 0:S -> True:False
334.58/291.57	0 :: 0:S
334.58/291.57	S :: 0:S -> 0:S
334.58/291.57	main :: Nil:Cons -> Nil:Cons
334.58/291.57	
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(5) CompletionProof (UPPER BOUND(ID))
334.58/291.57	The transformation into a RNTS is sound, since:
334.58/291.57	
334.58/291.57	(a) The obligation is a constructor system where every type has a constant constructor,
334.58/291.57	
334.58/291.57	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
334.58/291.57	
334.58/291.57	   main_1
334.58/291.57	
334.58/291.57	(c) The following functions are completely defined:
334.58/291.57	
334.58/291.57	   sort#2_1
334.58/291.57	   leq#2_2
334.58/291.57	   insert#3_2
334.58/291.57	   cond_insert_ord_x_ys_1_4
334.58/291.57	
334.58/291.57	Due to the following rules being added:
334.58/291.57	none
334.58/291.57	
334.58/291.57	And the following fresh constants: none
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(6)
334.58/291.57	Obligation:
334.58/291.57	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
334.58/291.57	
334.58/291.57	Runtime Complexity Weighted TRS with Types.
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil [1]
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
334.58/291.57	   leq#2(0, x8) -> True [1]
334.58/291.57	   leq#2(S(x12), 0) -> False [1]
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
334.58/291.57	   main(x1) -> sort#2(x1) [1]
334.58/291.57	
334.58/291.57	The TRS has the following type information:
334.58/291.57	sort#2 :: Nil:Cons -> Nil:Cons
334.58/291.57	Nil :: Nil:Cons
334.58/291.57	Cons :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	insert#3 :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	True :: True:False
334.58/291.57	False :: True:False
334.58/291.57	leq#2 :: 0:S -> 0:S -> True:False
334.58/291.57	0 :: 0:S
334.58/291.57	S :: 0:S -> 0:S
334.58/291.57	main :: Nil:Cons -> Nil:Cons
334.58/291.57	
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(7) NarrowingProof (BOTH BOUNDS(ID, ID))
334.58/291.57	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(8)
334.58/291.57	Obligation:
334.58/291.57	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
334.58/291.57	
334.58/291.57	Runtime Complexity Weighted TRS with Types.
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil [1]
334.58/291.57	   sort#2(Cons(x4, Nil)) -> insert#3(x4, Nil) [2]
334.58/291.57	   sort#2(Cons(x4, Cons(x4', x2'))) -> insert#3(x4, insert#3(x4', sort#2(x2'))) [2]
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1)) [1]
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil) [1]
334.58/291.57	   insert#3(0, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(True, 0, x4, x2) [2]
334.58/291.57	   insert#3(S(x12'), Cons(0, x2)) -> cond_insert_ord_x_ys_1(False, S(x12'), 0, x2) [2]
334.58/291.57	   insert#3(S(x4''), Cons(S(x2''), x2)) -> cond_insert_ord_x_ys_1(leq#2(x4'', x2''), S(x4''), S(x2''), x2) [2]
334.58/291.57	   leq#2(0, x8) -> True [1]
334.58/291.57	   leq#2(S(x12), 0) -> False [1]
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
334.58/291.57	   main(x1) -> sort#2(x1) [1]
334.58/291.57	
334.58/291.57	The TRS has the following type information:
334.58/291.57	sort#2 :: Nil:Cons -> Nil:Cons
334.58/291.57	Nil :: Nil:Cons
334.58/291.57	Cons :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	insert#3 :: 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
334.58/291.57	True :: True:False
334.58/291.57	False :: True:False
334.58/291.57	leq#2 :: 0:S -> 0:S -> True:False
334.58/291.57	0 :: 0:S
334.58/291.57	S :: 0:S -> 0:S
334.58/291.57	main :: Nil:Cons -> Nil:Cons
334.58/291.57	
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
334.58/291.57	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
334.58/291.57	The constant constructors are abstracted as follows:
334.58/291.57	
334.58/291.57	   Nil => 0
334.58/291.57	   True => 1
334.58/291.57	   False => 0
334.58/291.57	   0 => 0
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(10)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x2 + insert#3(x3, x1) :|: x1 >= 0, z' = x3, z1 = x1, z = 0, z'' = x2, x3 >= 0, x2 >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x3 + (1 + x2 + x1) :|: x1 >= 0, z = 1, z' = x3, z1 = x1, z'' = x2, x3 >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(x4'', x2''), 1 + x4'', 1 + x2'', x2) :|: x4'' >= 0, z = 1 + x4'', z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + x12', 0, x2) :|: z = 1 + x12', z' = 1 + 0 + x2, x12' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + x2 + 0 :|: z = x2, x2 >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(x4, x2) :|: x4 >= 0, z' = 1 + x2, z = 1 + x4, x2 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: x8 >= 0, z = 0, z' = x8
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z = 1 + x12, x12 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(x1) :|: x1 >= 0, z = x1
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, 0) :|: z = 1 + x4 + 0, x4 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(11) SimplificationProof (BOTH BOUNDS(ID, ID))
334.58/291.57	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(12)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z - 1, x2''), 1 + (z - 1), 1 + x2'', x2) :|: z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
334.58/291.57	Found the following analysis order by SCC decomposition:
334.58/291.57	
334.58/291.57	   { leq#2 }
334.58/291.57	   { cond_insert_ord_x_ys_1, insert#3 }
334.58/291.57	   { sort#2 }
334.58/291.57	   { main }
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(14)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z - 1, x2''), 1 + (z - 1), 1 + x2'', x2) :|: z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(15) ResultPropagationProof (UPPER BOUND(ID))
334.58/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(16)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z - 1, x2''), 1 + (z - 1), 1 + x2'', x2) :|: z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(17) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed SIZE bound using CoFloCo for: leq#2
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(1) with polynomial bound: 1
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(18)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z - 1, x2''), 1 + (z - 1), 1 + x2'', x2) :|: z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {leq#2}, {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: ?, size: O(1) [1]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(19) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed RUNTIME bound using KoAT for: leq#2
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: 2 + z'
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(20)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(leq#2(z - 1, x2''), 1 + (z - 1), 1 + x2'', x2) :|: z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> leq#2(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(21) ResultPropagationProof (UPPER BOUND(ID))
334.58/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(22)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(23) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed SIZE bound using CoFloCo for: cond_insert_ord_x_ys_1
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
334.58/291.57	
334.58/291.57	Computed SIZE bound using CoFloCo for: insert#3
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: 1 + z + z'
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(24)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {cond_insert_ord_x_ys_1,insert#3}, {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: ?, size: O(n^1) [1 + z + z']
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(25) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed RUNTIME bound using CoFloCo for: cond_insert_ord_x_ys_1
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: 4 + 7*z1
334.58/291.57	
334.58/291.57	Computed RUNTIME bound using CoFloCo for: insert#3
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: 6 + 7*z'
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(26)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert#3(z', z1) :|: z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 4 + x2'' }-> cond_insert_ord_x_ys_1(s, 1 + (z - 1), 1 + x2'', x2) :|: s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(1, 0, x4, x2) :|: x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 2 }-> cond_insert_ord_x_ys_1(0, 1 + (z - 1), 0, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(z - 1, 0) :|: z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(27) ResultPropagationProof (UPPER BOUND(ID))
334.58/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(28)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(29) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed SIZE bound using CoFloCo for: sort#2
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: z
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(30)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {sort#2}, {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	  sort#2: runtime: ?, size: O(n^1) [z]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(31) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed RUNTIME bound using CoFloCo for: sort#2
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^2) with polynomial bound: 15 + 21*z + 14*z^2
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(32)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 1 }-> sort#2(z) :|: z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 2 }-> insert#3(x4, insert#3(x4', sort#2(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	  sort#2: runtime: O(n^2) [15 + 21*z + 14*z^2], size: O(n^1) [z]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(33) ResultPropagationProof (UPPER BOUND(ID))
334.58/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(34)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 16 + 21*z + 14*z^2 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	  sort#2: runtime: O(n^2) [15 + 21*z + 14*z^2], size: O(n^1) [z]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(35) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed SIZE bound using CoFloCo for: main
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^1) with polynomial bound: z
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(36)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 16 + 21*z + 14*z^2 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: {main}
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	  sort#2: runtime: O(n^2) [15 + 21*z + 14*z^2], size: O(n^1) [z]
334.58/291.57	  main: runtime: ?, size: O(n^1) [z]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(37) IntTrsBoundProof (UPPER BOUND(ID))
334.58/291.57	
334.58/291.57	Computed RUNTIME bound using KoAT for: main
334.58/291.57	after applying outer abstraction to obtain an ITS,
334.58/291.57	resulting in: O(n^2) with polynomial bound: 16 + 21*z + 14*z^2
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(38)
334.58/291.57	Obligation:
334.58/291.57	Complexity RNTS consisting of the following rules:
334.58/291.57	
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
334.58/291.57	   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 7 + 7*z1 }-> 1 + z'' + s1 :|: s1 >= 0, s1 <= z' + z1 + 1, z1 >= 0, z = 0, z' >= 0, z'' >= 0
334.58/291.57	   insert#3(z, z') -{ 6 + 7*x2 }-> s2 :|: s2 >= 0, s2 <= 0 + x4 + x2 + 2, x4 >= 0, z' = 1 + x4 + x2, z = 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ -1 + 7*z' }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1) + 0 + (z' - 1) + 2, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   insert#3(z, z') -{ 8 + 7*x2 + x2'' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1) + (1 + x2'') + x2 + 2, s >= 0, s <= 1, z - 1 >= 0, z' = 1 + (1 + x2'') + x2, x2'' >= 0, x2 >= 0
334.58/291.57	   insert#3(z, z') -{ 1 }-> 1 + z + 0 :|: z >= 0, z' = 0
334.58/291.57	   leq#2(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
334.58/291.57	   leq#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
334.58/291.57	   main(z) -{ 16 + 21*z + 14*z^2 }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
334.58/291.57	   sort#2(z) -{ 8 }-> s'' :|: s'' >= 0, s'' <= z - 1 + 0 + 1, z - 1 >= 0
334.58/291.57	   sort#2(z) -{ 29 + 7*s5 + 7*s6 + 21*x2' + 14*x2'^2 }-> s7 :|: s5 >= 0, s5 <= x2', s6 >= 0, s6 <= x4' + s5 + 1, s7 >= 0, s7 <= x4 + s6 + 1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
334.58/291.57	   sort#2(z) -{ 1 }-> 0 :|: z = 0
334.58/291.57	
334.58/291.57	Function symbols to be analyzed: 
334.58/291.57	Previous analysis results are:
334.58/291.57	  leq#2: runtime: O(n^1) [2 + z'], size: O(1) [1]
334.58/291.57	  cond_insert_ord_x_ys_1: runtime: O(n^1) [4 + 7*z1], size: O(n^1) [2 + z' + z'' + z1]
334.58/291.57	  insert#3: runtime: O(n^1) [6 + 7*z'], size: O(n^1) [1 + z + z']
334.58/291.57	  sort#2: runtime: O(n^2) [15 + 21*z + 14*z^2], size: O(n^1) [z]
334.58/291.57	  main: runtime: O(n^2) [16 + 21*z + 14*z^2], size: O(n^1) [z]
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(39) FinalProof (FINISHED)
334.58/291.57	Computed overall runtime complexity
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(40)
334.58/291.57	BOUNDS(1, n^2)
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(41) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
334.58/291.57	Transformed a relative TRS into a decreasing-loop problem.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(42)
334.58/291.57	Obligation:
334.58/291.57	Analyzing the following TRS for decreasing loops:
334.58/291.57	
334.58/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
334.58/291.57	
334.58/291.57	
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil)
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
334.58/291.57	   leq#2(0, x8) -> True
334.58/291.57	   leq#2(S(x12), 0) -> False
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
334.58/291.57	   main(x1) -> sort#2(x1)
334.58/291.57	
334.58/291.57	S is empty.
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(43) DecreasingLoopProof (LOWER BOUND(ID))
334.58/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
334.58/291.57	
334.58/291.57	The rewrite sequence
334.58/291.57	
334.58/291.57	sort#2(Cons(x4, x2)) ->^+ insert#3(x4, sort#2(x2))
334.58/291.57	
334.58/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
334.58/291.57	
334.58/291.57	The pumping substitution is [x2 / Cons(x4, x2)].
334.58/291.57	
334.58/291.57	The result substitution is [ ].
334.58/291.57	
334.58/291.57	
334.58/291.57	
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(44)
334.58/291.57	Complex Obligation (BEST)
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(45)
334.58/291.57	Obligation:
334.58/291.57	Proved the lower bound n^1 for the following obligation:
334.58/291.57	
334.58/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
334.58/291.57	
334.58/291.57	
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil)
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
334.58/291.57	   leq#2(0, x8) -> True
334.58/291.57	   leq#2(S(x12), 0) -> False
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
334.58/291.57	   main(x1) -> sort#2(x1)
334.58/291.57	
334.58/291.57	S is empty.
334.58/291.57	Rewrite Strategy: INNERMOST
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(46) LowerBoundPropagationProof (FINISHED)
334.58/291.57	Propagated lower bound.
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(47)
334.58/291.57	BOUNDS(n^1, INF)
334.58/291.57	
334.58/291.57	----------------------------------------
334.58/291.57	
334.58/291.57	(48)
334.58/291.57	Obligation:
334.58/291.57	Analyzing the following TRS for decreasing loops:
334.58/291.57	
334.58/291.57	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
334.58/291.57	
334.58/291.57	
334.58/291.57	The TRS R consists of the following rules:
334.58/291.57	
334.58/291.57	   sort#2(Nil) -> Nil
334.58/291.57	   sort#2(Cons(x4, x2)) -> insert#3(x4, sort#2(x2))
334.58/291.57	   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
334.58/291.57	   cond_insert_ord_x_ys_1(False, x3, x2, x1) -> Cons(x2, insert#3(x3, x1))
334.58/291.57	   insert#3(x2, Nil) -> Cons(x2, Nil)
334.58/291.57	   insert#3(x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
334.58/291.57	   leq#2(0, x8) -> True
334.58/291.57	   leq#2(S(x12), 0) -> False
334.58/291.57	   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
334.58/291.57	   main(x1) -> sort#2(x1)
334.58/291.57	
334.58/291.57	S is empty.
334.58/291.57	Rewrite Strategy: INNERMOST
334.74/291.61	EOF