1090.44/291.55	WORST_CASE(Omega(n^1), O(n^2))
1090.44/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1090.44/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1090.44/291.58	
1090.44/291.58	(0) CpxTRS
1090.44/291.58	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
1090.44/291.58	(2) CpxTRS
1090.44/291.58	    (3) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
1090.44/291.58	    (4) CpxRelTRS
1090.44/291.58	    (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	    (6) CpxRelTRS
1090.44/291.58	    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	    (8) CpxWeightedTrs
1090.44/291.58	    (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	    (10) CpxWeightedTrs
1090.44/291.58	    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	    (12) CpxTypedWeightedTrs
1090.44/291.58	    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
1090.44/291.58	    (14) CpxTypedWeightedCompleteTrs
1090.44/291.58	    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1090.44/291.58	    (16) CpxRNTS
1090.44/291.58	    (17) CompleteCoflocoProof [FINISHED, 305 ms]
1090.44/291.58	    (18) BOUNDS(1, n^2)
1090.44/291.58	(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	(20) CpxTRS
1090.44/291.58	    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1090.44/291.58	    (22) typed CpxTrs
1090.44/291.58	    (23) OrderProof [LOWER BOUND(ID), 0 ms]
1090.44/291.58	    (24) typed CpxTrs
1090.44/291.58	    (25) RewriteLemmaProof [LOWER BOUND(ID), 1048 ms]
1090.44/291.58	    (26) BEST
1090.44/291.58	        (27) proven lower bound
1090.44/291.58	            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
1090.44/291.58	            (29) BOUNDS(n^1, INF)
1090.44/291.58	        (30) typed CpxTrs
1090.44/291.58	            (31) RewriteLemmaProof [LOWER BOUND(ID), 25 ms]
1090.44/291.58	            (32) BOUNDS(1, INF)
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(0)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0
1090.44/291.58	   +(0, x) -> x
1090.44/291.58	   +(x, 0) -> x
1090.44/291.58	   +(O(x), O(y)) -> O(+(x, y))
1090.44/291.58	   +(O(x), I(y)) -> I(+(x, y))
1090.44/291.58	   +(I(x), O(y)) -> I(+(x, y))
1090.44/291.58	   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
1090.44/291.58	   *(0, x) -> 0
1090.44/291.58	   *(x, 0) -> 0
1090.44/291.58	   *(O(x), y) -> O(*(x, y))
1090.44/291.58	   *(I(x), y) -> +(O(*(x, y)), y)
1090.44/291.58	
1090.44/291.58	S is empty.
1090.44/291.58	Rewrite Strategy: FULL
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
1090.44/291.58	The following defined symbols can occur below the 0th argument of +: O, +, *
1090.44/291.58	The following defined symbols can occur below the 0th argument of O: O, +, *
1090.44/291.58	
1090.44/291.58	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
1090.44/291.58	+(O(x), O(y)) -> O(+(x, y))
1090.44/291.58	+(I(x), O(y)) -> I(+(x, y))
1090.44/291.58	*(O(x), y) -> O(*(x, y))
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(2)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0
1090.44/291.58	   +(0, x) -> x
1090.44/291.58	   +(x, 0) -> x
1090.44/291.58	   +(O(x), I(y)) -> I(+(x, y))
1090.44/291.58	   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
1090.44/291.58	   *(0, x) -> 0
1090.44/291.58	   *(x, 0) -> 0
1090.44/291.58	   *(I(x), y) -> +(O(*(x, y)), y)
1090.44/291.58	
1090.44/291.58	S is empty.
1090.44/291.58	Rewrite Strategy: FULL
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(3) NonCtorToCtorProof (UPPER BOUND(ID))
1090.44/291.58	transformed non-ctor to ctor-system
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(4)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0
1090.44/291.58	   +(0, x) -> x
1090.44/291.58	   +(x, 0) -> x
1090.44/291.58	   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
1090.44/291.58	   *(0, x) -> 0
1090.44/291.58	   *(x, 0) -> 0
1090.44/291.58	   *(I(x), y) -> +(O(*(x, y)), y)
1090.44/291.58	   +(c_O(x), I(y)) -> I(+(x, y))
1090.44/291.58	
1090.44/291.58	The (relative) TRS S consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(x0) -> c_O(x0)
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: FULL
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Converted rc-obligation to irc-obligation.
1090.44/291.58	
1090.44/291.58	The duplicating contexts are:
1090.44/291.58	*(I(x), [])
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The defined contexts are:
1090.44/291.58	+([], x1)
1090.44/291.58	O([])
1090.44/291.58	+([], I(0))
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(6)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0
1090.44/291.58	   +(0, x) -> x
1090.44/291.58	   +(x, 0) -> x
1090.44/291.58	   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
1090.44/291.58	   *(0, x) -> 0
1090.44/291.58	   *(x, 0) -> 0
1090.44/291.58	   *(I(x), y) -> +(O(*(x, y)), y)
1090.44/291.58	   +(c_O(x), I(y)) -> I(+(x, y))
1090.44/291.58	
1090.44/291.58	The (relative) TRS S consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(x0) -> c_O(x0)
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: INNERMOST
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Transformed relative TRS to weighted TRS
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(8)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0 [1]
1090.44/291.58	   +(0, x) -> x [1]
1090.44/291.58	   +(x, 0) -> x [1]
1090.44/291.58	   +(I(x), I(y)) -> O(+(+(x, y), I(0))) [1]
1090.44/291.58	   *(0, x) -> 0 [1]
1090.44/291.58	   *(x, 0) -> 0 [1]
1090.44/291.58	   *(I(x), y) -> +(O(*(x, y)), y) [1]
1090.44/291.58	   +(c_O(x), I(y)) -> I(+(x, y)) [1]
1090.44/291.58	   O(x0) -> c_O(x0) [0]
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: INNERMOST
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Renamed defined symbols to avoid conflicts with arithmetic symbols:
1090.44/291.58	
1090.44/291.58	+ => plus
1090.44/291.58	* => times
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(10)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0 [1]
1090.44/291.58	   plus(0, x) -> x [1]
1090.44/291.58	   plus(x, 0) -> x [1]
1090.44/291.58	   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
1090.44/291.58	   times(0, x) -> 0 [1]
1090.44/291.58	   times(x, 0) -> 0 [1]
1090.44/291.58	   times(I(x), y) -> plus(O(times(x, y)), y) [1]
1090.44/291.58	   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
1090.44/291.58	   O(x0) -> c_O(x0) [0]
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: INNERMOST
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Infered types.
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(12)
1090.44/291.58	Obligation:
1090.44/291.58	Runtime Complexity Weighted TRS with Types.
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0 [1]
1090.44/291.58	   plus(0, x) -> x [1]
1090.44/291.58	   plus(x, 0) -> x [1]
1090.44/291.58	   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
1090.44/291.58	   times(0, x) -> 0 [1]
1090.44/291.58	   times(x, 0) -> 0 [1]
1090.44/291.58	   times(I(x), y) -> plus(O(times(x, y)), y) [1]
1090.44/291.58	   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
1090.44/291.58	   O(x0) -> c_O(x0) [0]
1090.44/291.58	
1090.44/291.58	The TRS has the following type information:
1090.44/291.58	O :: 0:I:c_O -> 0:I:c_O
1090.44/291.58	0 :: 0:I:c_O
1090.44/291.58	plus :: 0:I:c_O -> 0:I:c_O -> 0:I:c_O
1090.44/291.58	I :: 0:I:c_O -> 0:I:c_O
1090.44/291.58	times :: 0:I:c_O -> 0:I:c_O -> 0:I:c_O
1090.44/291.58	c_O :: 0:I:c_O -> 0:I:c_O
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: INNERMOST
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(13) CompletionProof (UPPER BOUND(ID))
1090.44/291.58	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1090.44/291.58	
1090.44/291.58	   O(v0) -> null_O [0]
1090.44/291.58	   plus(v0, v1) -> null_plus [0]
1090.44/291.58	   times(v0, v1) -> null_times [0]
1090.44/291.58	
1090.44/291.58	And the following fresh constants:    null_O, null_plus, null_times
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(14)
1090.44/291.58	Obligation:
1090.44/291.58	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1090.44/291.58	
1090.44/291.58	Runtime Complexity Weighted TRS with Types.
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0) -> 0 [1]
1090.44/291.58	   plus(0, x) -> x [1]
1090.44/291.58	   plus(x, 0) -> x [1]
1090.44/291.58	   plus(I(x), I(y)) -> O(plus(plus(x, y), I(0))) [1]
1090.44/291.58	   times(0, x) -> 0 [1]
1090.44/291.58	   times(x, 0) -> 0 [1]
1090.44/291.58	   times(I(x), y) -> plus(O(times(x, y)), y) [1]
1090.44/291.58	   plus(c_O(x), I(y)) -> I(plus(x, y)) [1]
1090.44/291.58	   O(x0) -> c_O(x0) [0]
1090.44/291.58	   O(v0) -> null_O [0]
1090.44/291.58	   plus(v0, v1) -> null_plus [0]
1090.44/291.58	   times(v0, v1) -> null_times [0]
1090.44/291.58	
1090.44/291.58	The TRS has the following type information:
1090.44/291.58	O :: 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	0 :: 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	plus :: 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	I :: 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	times :: 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	c_O :: 0:I:c_O:null_O:null_plus:null_times -> 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	null_O :: 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	null_plus :: 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	null_times :: 0:I:c_O:null_O:null_plus:null_times
1090.44/291.58	
1090.44/291.58	Rewrite Strategy: INNERMOST
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1090.44/291.58	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1090.44/291.58	The constant constructors are abstracted as follows:
1090.44/291.58	
1090.44/291.58	   0 => 0
1090.44/291.58	   null_O => 0
1090.44/291.58	   null_plus => 0
1090.44/291.58	   null_times => 0
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(16)
1090.44/291.58	Obligation:
1090.44/291.58	Complexity RNTS consisting of the following rules:
1090.44/291.58	
1090.44/291.58	   O(z) -{ 1 }-> 0 :|: z = 0
1090.44/291.58	   O(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1090.44/291.58	   O(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
1090.44/291.58	   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
1090.44/291.58	   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
1090.44/291.58	   plus(z, z') -{ 1 }-> O(plus(plus(x, y), 1 + 0)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1090.44/291.58	   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1090.44/291.58	   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1090.44/291.58	   times(z, z') -{ 1 }-> plus(O(times(x, y)), y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
1090.44/291.58	   times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
1090.44/291.58	   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
1090.44/291.58	   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1090.44/291.58	
1090.44/291.58	Only complete derivations are relevant for the runtime complexity.
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(17) CompleteCoflocoProof (FINISHED)
1090.44/291.58	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1090.44/291.58	
1090.44/291.58	eq(start(V, V1),0,[fun(V, Out)],[V >= 0]).
1090.44/291.58	eq(start(V, V1),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]).
1090.44/291.58	eq(start(V, V1),0,[times(V, V1, Out)],[V >= 0,V1 >= 0]).
1090.44/291.58	eq(fun(V, Out),1,[],[Out = 0,V = 0]).
1090.44/291.58	eq(plus(V, V1, Out),1,[],[Out = V2,V1 = V2,V2 >= 0,V = 0]).
1090.44/291.58	eq(plus(V, V1, Out),1,[],[Out = V3,V3 >= 0,V = V3,V1 = 0]).
1090.44/291.58	eq(plus(V, V1, Out),1,[plus(V4, V5, Ret00),plus(Ret00, 1 + 0, Ret0),fun(Ret0, Ret)],[Out = Ret,V1 = 1 + V5,V4 >= 0,V5 >= 0,V = 1 + V4]).
1090.44/291.58	eq(times(V, V1, Out),1,[],[Out = 0,V1 = V6,V6 >= 0,V = 0]).
1090.44/291.58	eq(times(V, V1, Out),1,[],[Out = 0,V7 >= 0,V = V7,V1 = 0]).
1090.44/291.58	eq(times(V, V1, Out),1,[times(V8, V9, Ret001),fun(Ret001, Ret01),plus(Ret01, V9, Ret1)],[Out = Ret1,V8 >= 0,V9 >= 0,V = 1 + V8,V1 = V9]).
1090.44/291.58	eq(plus(V, V1, Out),1,[plus(V11, V10, Ret11)],[Out = 1 + Ret11,V1 = 1 + V10,V11 >= 0,V10 >= 0,V = 1 + V11]).
1090.44/291.58	eq(fun(V, Out),0,[],[Out = 1 + V12,V = V12,V12 >= 0]).
1090.44/291.58	eq(fun(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]).
1090.44/291.58	eq(plus(V, V1, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V = V15,V1 = V14]).
1090.44/291.58	eq(times(V, V1, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V = V17,V1 = V16]).
1090.44/291.58	input_output_vars(fun(V,Out),[V],[Out]).
1090.44/291.58	input_output_vars(plus(V,V1,Out),[V,V1],[Out]).
1090.44/291.58	input_output_vars(times(V,V1,Out),[V,V1],[Out]).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	CoFloCo proof output:
1090.44/291.58	Preprocessing Cost Relations
1090.44/291.58	=====================================
1090.44/291.58	
1090.44/291.58	#### Computed strongly connected components 
1090.44/291.58	0. non_recursive  : [fun/2]
1090.44/291.58	1. recursive [non_tail,multiple] : [plus/3]
1090.44/291.58	2. recursive [non_tail] : [times/3]
1090.44/291.58	3. non_recursive  : [start/2]
1090.44/291.58	
1090.44/291.58	#### Obtained direct recursion through partial evaluation 
1090.44/291.58	0. SCC is partially evaluated into fun/2
1090.44/291.58	1. SCC is partially evaluated into plus/3
1090.44/291.58	2. SCC is partially evaluated into times/3
1090.44/291.58	3. SCC is partially evaluated into start/2
1090.44/291.58	
1090.44/291.58	Control-Flow Refinement of Cost Relations
1090.44/291.58	=====================================
1090.44/291.58	
1090.44/291.58	### Specialization of cost equations fun/2 
1090.44/291.58	* CE 5 is refined into CE [16] 
1090.44/291.58	* CE 4 is refined into CE [17] 
1090.44/291.58	* CE 6 is refined into CE [18] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Cost equations --> "Loop" of fun/2 
1090.44/291.58	* CEs [16] --> Loop 12 
1090.44/291.58	* CEs [17,18] --> Loop 13 
1090.44/291.58	
1090.44/291.58	### Ranking functions of CR fun(V,Out) 
1090.44/291.58	
1090.44/291.58	#### Partial ranking functions of CR fun(V,Out) 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Specialization of cost equations plus/3 
1090.44/291.58	* CE 11 is refined into CE [19] 
1090.44/291.58	* CE 8 is refined into CE [20] 
1090.44/291.58	* CE 7 is refined into CE [21] 
1090.44/291.58	* CE 10 is refined into CE [22] 
1090.44/291.58	* CE 9 is refined into CE [23,24] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Cost equations --> "Loop" of plus/3 
1090.44/291.58	* CEs [24] --> Loop 14 
1090.44/291.58	* CEs [23] --> Loop 15 
1090.44/291.58	* CEs [22] --> Loop 16 
1090.44/291.58	* CEs [19] --> Loop 17 
1090.44/291.58	* CEs [20] --> Loop 18 
1090.44/291.58	* CEs [21] --> Loop 19 
1090.44/291.58	
1090.44/291.58	### Ranking functions of CR plus(V,V1,Out) 
1090.44/291.58	
1090.44/291.58	#### Partial ranking functions of CR plus(V,V1,Out) 
1090.44/291.58	* Partial RF of phase [14,15,16]:
1090.44/291.58	  - RF of loop [14:1,15:1,16:1]:
1090.44/291.58	    V depends on loops [14:2,15:2] 
1090.44/291.58	    V1
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Specialization of cost equations times/3 
1090.44/291.58	* CE 13 is refined into CE [25] 
1090.44/291.58	* CE 12 is refined into CE [26] 
1090.44/291.58	* CE 15 is refined into CE [27] 
1090.44/291.58	* CE 14 is refined into CE [28,29,30,31,32,33] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Cost equations --> "Loop" of times/3 
1090.44/291.58	* CEs [33] --> Loop 20 
1090.44/291.58	* CEs [28] --> Loop 21 
1090.44/291.58	* CEs [31] --> Loop 22 
1090.44/291.58	* CEs [29,30,32] --> Loop 23 
1090.44/291.58	* CEs [25] --> Loop 24 
1090.44/291.58	* CEs [26,27] --> Loop 25 
1090.44/291.58	
1090.44/291.58	### Ranking functions of CR times(V,V1,Out) 
1090.44/291.58	* RF of phase [20,21,22,23]: [V]
1090.44/291.58	
1090.44/291.58	#### Partial ranking functions of CR times(V,V1,Out) 
1090.44/291.58	* Partial RF of phase [20,21,22,23]:
1090.44/291.58	  - RF of loop [20:1,21:1,22:1,23:1]:
1090.44/291.58	    V
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Specialization of cost equations start/2 
1090.44/291.58	* CE 1 is refined into CE [34,35] 
1090.44/291.58	* CE 2 is refined into CE [36,37,38,39] 
1090.44/291.58	* CE 3 is refined into CE [40,41] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	### Cost equations --> "Loop" of start/2 
1090.44/291.58	* CEs [37] --> Loop 26 
1090.44/291.58	* CEs [34,35,36,38,40,41] --> Loop 27 
1090.44/291.58	* CEs [39] --> Loop 28 
1090.44/291.58	
1090.44/291.58	### Ranking functions of CR start(V,V1) 
1090.44/291.58	
1090.44/291.58	#### Partial ranking functions of CR start(V,V1) 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Computing Bounds
1090.44/291.58	=====================================
1090.44/291.58	
1090.44/291.58	#### Cost of chains of fun(V,Out):
1090.44/291.58	* Chain [13]: 1
1090.44/291.58	  with precondition: [Out=0,V>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [12]: 0
1090.44/291.58	  with precondition: [V+1=Out,V>=0] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	#### Cost of chains of plus(V,V1,Out):
1090.44/291.58	* Chain [multiple([14,15,16],[[],[19],[18],[17]])]...: 4*it(14)+2*it([18])+0
1090.44/291.58	  Such that:aux(1) =< 1
1090.44/291.58	aux(2) =< 2*V1
1090.44/291.58	it(14) =< aux(2)
1090.44/291.58	it([18]) =< it(14)+it(14)+aux(1)
1090.44/291.58	
1090.44/291.58	  with precondition: [V>=1,V1>=1] 
1090.44/291.58	
1090.44/291.58	* Chain [19]: 1
1090.44/291.58	  with precondition: [V=0,V1=Out,V1>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [18]: 1
1090.44/291.58	  with precondition: [V1=0,V=Out,V>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [17]: 0
1090.44/291.58	  with precondition: [Out=0,V>=0,V1>=0] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	#### Cost of chains of times(V,V1,Out):
1090.44/291.58	* Chain [[20,21,22,23],25]: 9*it(20)+4*s(9)+2*s(10)+1
1090.44/291.58	  Such that:aux(4) =< 2*V1
1090.44/291.58	aux(7) =< V
1090.44/291.58	it(20) =< aux(7)
1090.44/291.58	s(12) =< it(20)*aux(4)
1090.44/291.58	s(9) =< s(12)
1090.44/291.58	s(10) =< s(9)+s(9)+aux(7)
1090.44/291.58	
1090.44/291.58	  with precondition: [V>=1,V1>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [[20,21,22,23],24]: 9*it(20)+2*s(10)+1
1090.44/291.58	  Such that:aux(8) =< V
1090.44/291.58	it(20) =< aux(8)
1090.44/291.58	s(10) =< aux(8)
1090.44/291.58	
1090.44/291.58	  with precondition: [V1=0,V>=1,Out>=0,V>=Out] 
1090.44/291.58	
1090.44/291.58	* Chain [25]: 1
1090.44/291.58	  with precondition: [Out=0,V>=0,V1>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [24]: 1
1090.44/291.58	  with precondition: [V1=0,Out=0,V>=0] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	#### Cost of chains of start(V,V1):
1090.44/291.58	* Chain [28]...: 4*s(24)+2*s(25)+0
1090.44/291.58	  Such that:s(22) =< 1
1090.44/291.58	s(23) =< 2*V1
1090.44/291.58	s(24) =< s(23)
1090.44/291.58	s(25) =< s(24)+s(24)+s(22)
1090.44/291.58	
1090.44/291.58	  with precondition: [V>=1,V1>=1] 
1090.44/291.58	
1090.44/291.58	* Chain [27]: 20*s(28)+4*s(30)+2*s(31)+1
1090.44/291.58	  Such that:s(27) =< V
1090.44/291.58	s(26) =< 2*V1
1090.44/291.58	s(28) =< s(27)
1090.44/291.58	s(29) =< s(28)*s(26)
1090.44/291.58	s(30) =< s(29)
1090.44/291.58	s(31) =< s(30)+s(30)+s(27)
1090.44/291.58	
1090.44/291.58	  with precondition: [V>=0] 
1090.44/291.58	
1090.44/291.58	* Chain [26]: 1
1090.44/291.58	  with precondition: [V1=0,V>=0] 
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Closed-form bounds of start(V,V1): 
1090.44/291.58	-------------------------------------
1090.44/291.58	* Chain [28]... with precondition: [V>=1,V1>=1] 
1090.44/291.58	    - Upper bound: 16*V1+2 
1090.44/291.58	    - Complexity: n 
1090.44/291.58	* Chain [27] with precondition: [V>=0] 
1090.44/291.58	    - Upper bound: 22*V+1+8*V*nat(2*V1) 
1090.44/291.58	    - Complexity: n^2 
1090.44/291.58	* Chain [26] with precondition: [V1=0,V>=0] 
1090.44/291.58	    - Upper bound: 1 
1090.44/291.58	    - Complexity: constant 
1090.44/291.58	
1090.44/291.58	### Maximum cost of start(V,V1): max([nat(2*V1)*8+1,8*V*nat(2*V1)+22*V])+1 
1090.44/291.58	Asymptotic class: n^2 
1090.44/291.58	* Total analysis performed in 221 ms.
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(18)
1090.44/291.58	BOUNDS(1, n^2)
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(19) RenamingProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Renamed function symbols to avoid clashes with predefined symbol.
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(20)
1090.44/291.58	Obligation:
1090.44/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The TRS R consists of the following rules:
1090.44/291.58	
1090.44/291.58	   O(0') -> 0'
1090.44/291.58	   +'(0', x) -> x
1090.44/291.58	   +'(x, 0') -> x
1090.44/291.58	   +'(O(x), O(y)) -> O(+'(x, y))
1090.44/291.58	   +'(O(x), I(y)) -> I(+'(x, y))
1090.44/291.58	   +'(I(x), O(y)) -> I(+'(x, y))
1090.44/291.58	   +'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
1090.44/291.58	   *'(0', x) -> 0'
1090.44/291.58	   *'(x, 0') -> 0'
1090.44/291.58	   *'(O(x), y) -> O(*'(x, y))
1090.44/291.58	   *'(I(x), y) -> +'(O(*'(x, y)), y)
1090.44/291.58	
1090.44/291.58	S is empty.
1090.44/291.58	Rewrite Strategy: FULL
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(21) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1090.44/291.58	Infered types.
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(22)
1090.44/291.58	Obligation:
1090.44/291.58	TRS:
1090.44/291.58	Rules:
1090.44/291.58	O(0') -> 0'
1090.44/291.58	+'(0', x) -> x
1090.44/291.58	+'(x, 0') -> x
1090.44/291.58	+'(O(x), O(y)) -> O(+'(x, y))
1090.44/291.58	+'(O(x), I(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), O(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
1090.44/291.58	*'(0', x) -> 0'
1090.44/291.58	*'(x, 0') -> 0'
1090.44/291.58	*'(O(x), y) -> O(*'(x, y))
1090.44/291.58	*'(I(x), y) -> +'(O(*'(x, y)), y)
1090.44/291.58	
1090.44/291.58	Types:
1090.44/291.58	O :: 0':I -> 0':I
1090.44/291.58	0' :: 0':I
1090.44/291.58	+' :: 0':I -> 0':I -> 0':I
1090.44/291.58	I :: 0':I -> 0':I
1090.44/291.58	*' :: 0':I -> 0':I -> 0':I
1090.44/291.58	hole_0':I1_0 :: 0':I
1090.44/291.58	gen_0':I2_0 :: Nat -> 0':I
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(23) OrderProof (LOWER BOUND(ID))
1090.44/291.58	Heuristically decided to analyse the following defined symbols:
1090.44/291.58	+', *'
1090.44/291.58	
1090.44/291.58	They will be analysed ascendingly in the following order:
1090.44/291.58	+' < *'
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(24)
1090.44/291.58	Obligation:
1090.44/291.58	TRS:
1090.44/291.58	Rules:
1090.44/291.58	O(0') -> 0'
1090.44/291.58	+'(0', x) -> x
1090.44/291.58	+'(x, 0') -> x
1090.44/291.58	+'(O(x), O(y)) -> O(+'(x, y))
1090.44/291.58	+'(O(x), I(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), O(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
1090.44/291.58	*'(0', x) -> 0'
1090.44/291.58	*'(x, 0') -> 0'
1090.44/291.58	*'(O(x), y) -> O(*'(x, y))
1090.44/291.58	*'(I(x), y) -> +'(O(*'(x, y)), y)
1090.44/291.58	
1090.44/291.58	Types:
1090.44/291.58	O :: 0':I -> 0':I
1090.44/291.58	0' :: 0':I
1090.44/291.58	+' :: 0':I -> 0':I -> 0':I
1090.44/291.58	I :: 0':I -> 0':I
1090.44/291.58	*' :: 0':I -> 0':I -> 0':I
1090.44/291.58	hole_0':I1_0 :: 0':I
1090.44/291.58	gen_0':I2_0 :: Nat -> 0':I
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Generator Equations:
1090.44/291.58	gen_0':I2_0(0) <=> 0'
1090.44/291.58	gen_0':I2_0(+(x, 1)) <=> I(gen_0':I2_0(x))
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The following defined symbols remain to be analysed:
1090.44/291.58	+', *'
1090.44/291.58	
1090.44/291.58	They will be analysed ascendingly in the following order:
1090.44/291.58	+' < *'
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(25) RewriteLemmaProof (LOWER BOUND(ID))
1090.44/291.58	Proved the following rewrite lemma:
1090.44/291.58	+'(gen_0':I2_0(n4_0), gen_0':I2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
1090.44/291.58	
1090.44/291.58	Induction Base:
1090.44/291.58	+'(gen_0':I2_0(0), gen_0':I2_0(0))
1090.44/291.58	
1090.44/291.58	Induction Step:
1090.44/291.58	+'(gen_0':I2_0(+(n4_0, 1)), gen_0':I2_0(+(n4_0, 1))) ->_R^Omega(1)
1090.44/291.58	O(+'(+'(gen_0':I2_0(n4_0), gen_0':I2_0(n4_0)), I(0'))) ->_IH
1090.44/291.58	O(+'(*3_0, I(0')))
1090.44/291.58	
1090.44/291.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(26)
1090.44/291.58	Complex Obligation (BEST)
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(27)
1090.44/291.58	Obligation:
1090.44/291.58	Proved the lower bound n^1 for the following obligation:
1090.44/291.58	
1090.44/291.58	TRS:
1090.44/291.58	Rules:
1090.44/291.58	O(0') -> 0'
1090.44/291.58	+'(0', x) -> x
1090.44/291.58	+'(x, 0') -> x
1090.44/291.58	+'(O(x), O(y)) -> O(+'(x, y))
1090.44/291.58	+'(O(x), I(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), O(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
1090.44/291.58	*'(0', x) -> 0'
1090.44/291.58	*'(x, 0') -> 0'
1090.44/291.58	*'(O(x), y) -> O(*'(x, y))
1090.44/291.58	*'(I(x), y) -> +'(O(*'(x, y)), y)
1090.44/291.58	
1090.44/291.58	Types:
1090.44/291.58	O :: 0':I -> 0':I
1090.44/291.58	0' :: 0':I
1090.44/291.58	+' :: 0':I -> 0':I -> 0':I
1090.44/291.58	I :: 0':I -> 0':I
1090.44/291.58	*' :: 0':I -> 0':I -> 0':I
1090.44/291.58	hole_0':I1_0 :: 0':I
1090.44/291.58	gen_0':I2_0 :: Nat -> 0':I
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Generator Equations:
1090.44/291.58	gen_0':I2_0(0) <=> 0'
1090.44/291.58	gen_0':I2_0(+(x, 1)) <=> I(gen_0':I2_0(x))
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The following defined symbols remain to be analysed:
1090.44/291.58	+', *'
1090.44/291.58	
1090.44/291.58	They will be analysed ascendingly in the following order:
1090.44/291.58	+' < *'
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(28) LowerBoundPropagationProof (FINISHED)
1090.44/291.58	Propagated lower bound.
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(29)
1090.44/291.58	BOUNDS(n^1, INF)
1090.44/291.58	
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(30)
1090.44/291.58	Obligation:
1090.44/291.58	TRS:
1090.44/291.58	Rules:
1090.44/291.58	O(0') -> 0'
1090.44/291.58	+'(0', x) -> x
1090.44/291.58	+'(x, 0') -> x
1090.44/291.58	+'(O(x), O(y)) -> O(+'(x, y))
1090.44/291.58	+'(O(x), I(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), O(y)) -> I(+'(x, y))
1090.44/291.58	+'(I(x), I(y)) -> O(+'(+'(x, y), I(0')))
1090.44/291.58	*'(0', x) -> 0'
1090.44/291.58	*'(x, 0') -> 0'
1090.44/291.58	*'(O(x), y) -> O(*'(x, y))
1090.44/291.58	*'(I(x), y) -> +'(O(*'(x, y)), y)
1090.44/291.58	
1090.44/291.58	Types:
1090.44/291.58	O :: 0':I -> 0':I
1090.44/291.58	0' :: 0':I
1090.44/291.58	+' :: 0':I -> 0':I -> 0':I
1090.44/291.58	I :: 0':I -> 0':I
1090.44/291.58	*' :: 0':I -> 0':I -> 0':I
1090.44/291.58	hole_0':I1_0 :: 0':I
1090.44/291.58	gen_0':I2_0 :: Nat -> 0':I
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Lemmas:
1090.44/291.58	+'(gen_0':I2_0(n4_0), gen_0':I2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	Generator Equations:
1090.44/291.58	gen_0':I2_0(0) <=> 0'
1090.44/291.58	gen_0':I2_0(+(x, 1)) <=> I(gen_0':I2_0(x))
1090.44/291.58	
1090.44/291.58	
1090.44/291.58	The following defined symbols remain to be analysed:
1090.44/291.58	*'
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(31) RewriteLemmaProof (LOWER BOUND(ID))
1090.44/291.58	Proved the following rewrite lemma:
1090.44/291.58	*'(gen_0':I2_0(n40075_0), gen_0':I2_0(0)) -> gen_0':I2_0(0), rt in Omega(1 + n40075_0)
1090.44/291.58	
1090.44/291.58	Induction Base:
1090.44/291.58	*'(gen_0':I2_0(0), gen_0':I2_0(0)) ->_R^Omega(1)
1090.44/291.58	0'
1090.44/291.58	
1090.44/291.58	Induction Step:
1090.44/291.58	*'(gen_0':I2_0(+(n40075_0, 1)), gen_0':I2_0(0)) ->_R^Omega(1)
1090.44/291.58	+'(O(*'(gen_0':I2_0(n40075_0), gen_0':I2_0(0))), gen_0':I2_0(0)) ->_IH
1090.44/291.58	+'(O(gen_0':I2_0(0)), gen_0':I2_0(0)) ->_R^Omega(1)
1090.44/291.58	+'(0', gen_0':I2_0(0)) ->_R^Omega(1)
1090.44/291.58	gen_0':I2_0(0)
1090.44/291.58	
1090.44/291.58	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1090.44/291.58	----------------------------------------
1090.44/291.58	
1090.44/291.58	(32)
1090.44/291.58	BOUNDS(1, INF)
1090.85/291.69	EOF