1107.65/291.57	WORST_CASE(Omega(n^1), O(n^3))
1107.65/291.59	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1107.65/291.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1107.65/291.59	
1107.65/291.59	(0) CpxTRS
1107.65/291.59	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
1107.65/291.59	(2) CpxTRS
1107.65/291.59	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 9 ms]
1107.65/291.59	    (4) CpxTRS
1107.65/291.59	    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1107.65/291.59	    (6) CpxWeightedTrs
1107.65/291.59	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1107.65/291.59	    (8) CpxTypedWeightedTrs
1107.65/291.59	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
1107.65/291.59	    (10) CpxTypedWeightedCompleteTrs
1107.65/291.59	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms]
1107.65/291.59	    (12) CpxRNTS
1107.65/291.59	    (13) CompleteCoflocoProof [FINISHED, 550 ms]
1107.65/291.59	    (14) BOUNDS(1, n^3)
1107.65/291.59	(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1107.65/291.59	(16) TRS for Loop Detection
1107.65/291.59	    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1107.65/291.59	    (18) BEST
1107.65/291.59	        (19) proven lower bound
1107.65/291.59	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
1107.65/291.59	            (21) BOUNDS(n^1, INF)
1107.65/291.59	        (22) TRS for Loop Detection
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(0)
1107.65/291.59	Obligation:
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   minus(x, plus(y, z)) -> minus(minus(x, y), z)
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: FULL
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
1107.65/291.59	The following defined symbols can occur below the 0th argument of minus: p, minus
1107.65/291.59	The following defined symbols can occur below the 1th argument of minus: p
1107.65/291.59	The following defined symbols can occur below the 0th argument of p: p
1107.65/291.59	The following defined symbols can occur below the 0th argument of plus: p, minus
1107.65/291.59	The following defined symbols can occur below the 1th argument of plus: p, minus
1107.65/291.59	The following defined symbols can occur below the 0th argument of div: p, minus
1107.65/291.59	
1107.65/291.59	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
1107.65/291.59	minus(x, plus(y, z)) -> minus(minus(x, y), z)
1107.65/291.59	div(plus(x, y), z) -> plus(div(x, z), div(y, z))
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(2)
1107.65/291.59	Obligation:
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: FULL
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
1107.65/291.59	Converted rc-obligation to irc-obligation.
1107.65/291.59	
1107.65/291.59	The duplicating contexts are:
1107.65/291.59	div(s(x), s([]))
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The defined contexts are:
1107.65/291.59	minus([], x1)
1107.65/291.59	minus(x0, [])
1107.65/291.59	plus(x0, [])
1107.65/291.59	div([], s(x1))
1107.65/291.59	p(s([]))
1107.65/291.59	plus([], x1)
1107.65/291.59	minus(s([]), s(0))
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(4)
1107.65/291.59	Obligation:
1107.65/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: INNERMOST
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1107.65/291.59	Transformed relative TRS to weighted TRS
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(6)
1107.65/291.59	Obligation:
1107.65/291.59	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x [1]
1107.65/291.59	   minus(0, y) -> 0 [1]
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
1107.65/291.59	   p(s(s(x))) -> s(p(s(x))) [1]
1107.65/291.59	   p(0) -> s(s(0)) [1]
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
1107.65/291.59	   plus(0, y) -> y [1]
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]
1107.65/291.59	
1107.65/291.59	Rewrite Strategy: INNERMOST
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1107.65/291.59	Infered types.
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(8)
1107.65/291.59	Obligation:
1107.65/291.59	Runtime Complexity Weighted TRS with Types.
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x [1]
1107.65/291.59	   minus(0, y) -> 0 [1]
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
1107.65/291.59	   p(s(s(x))) -> s(p(s(x))) [1]
1107.65/291.59	   p(0) -> s(s(0)) [1]
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
1107.65/291.59	   plus(0, y) -> y [1]
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]
1107.65/291.59	
1107.65/291.59	The TRS has the following type information:
1107.65/291.59	minus :: 0:s -> 0:s -> 0:s
1107.65/291.59	0 :: 0:s
1107.65/291.59	s :: 0:s -> 0:s
1107.65/291.59	p :: 0:s -> 0:s
1107.65/291.59	div :: 0:s -> 0:s -> 0:s
1107.65/291.59	plus :: 0:s -> 0:s -> 0:s
1107.65/291.59	
1107.65/291.59	Rewrite Strategy: INNERMOST
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(9) CompletionProof (UPPER BOUND(ID))
1107.65/291.59	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1107.65/291.59	
1107.65/291.59	   p(v0) -> null_p [0]
1107.65/291.59	   div(v0, v1) -> null_div [0]
1107.65/291.59	   minus(v0, v1) -> null_minus [0]
1107.65/291.59	   plus(v0, v1) -> null_plus [0]
1107.65/291.59	
1107.65/291.59	And the following fresh constants:    null_p, null_div, null_minus, null_plus
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(10)
1107.65/291.59	Obligation:
1107.65/291.59	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1107.65/291.59	
1107.65/291.59	Runtime Complexity Weighted TRS with Types.
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x [1]
1107.65/291.59	   minus(0, y) -> 0 [1]
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
1107.65/291.59	   p(s(s(x))) -> s(p(s(x))) [1]
1107.65/291.59	   p(0) -> s(s(0)) [1]
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
1107.65/291.59	   plus(0, y) -> y [1]
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]
1107.65/291.59	   p(v0) -> null_p [0]
1107.65/291.59	   div(v0, v1) -> null_div [0]
1107.65/291.59	   minus(v0, v1) -> null_minus [0]
1107.65/291.59	   plus(v0, v1) -> null_plus [0]
1107.65/291.59	
1107.65/291.59	The TRS has the following type information:
1107.65/291.59	minus :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	0 :: 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	s :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	p :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	div :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	plus :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	null_p :: 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	null_div :: 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	null_minus :: 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	null_plus :: 0:s:null_p:null_div:null_minus:null_plus
1107.65/291.59	
1107.65/291.59	Rewrite Strategy: INNERMOST
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1107.65/291.59	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1107.65/291.59	The constant constructors are abstracted as follows:
1107.65/291.59	
1107.65/291.59	   0 => 0
1107.65/291.59	   null_p => 0
1107.65/291.59	   null_div => 0
1107.65/291.59	   null_minus => 0
1107.65/291.59	   null_plus => 0
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(12)
1107.65/291.59	Obligation:
1107.65/291.59	Complexity RNTS consisting of the following rules:
1107.65/291.59	
1107.65/291.59	   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1107.65/291.59	   div(z, z') -{ 1 }-> 1 + div(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1107.65/291.59	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
1107.65/291.59	   minus(z, z') -{ 1 }-> minus(p(1 + x), p(1 + y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1107.65/291.59	   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
1107.65/291.59	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1107.65/291.59	   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1107.65/291.59	   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
1107.65/291.59	   p(z) -{ 1 }-> 1 + (1 + 0) :|: z = 0
1107.65/291.59	   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
1107.65/291.59	   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1107.65/291.59	   plus(z, z') -{ 1 }-> 1 + plus(y, minus(1 + x, 1 + 0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
1107.65/291.59	
1107.65/291.59	Only complete derivations are relevant for the runtime complexity.
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(13) CompleteCoflocoProof (FINISHED)
1107.65/291.59	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1107.65/291.59	
1107.65/291.59	eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
1107.65/291.59	eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
1107.65/291.59	eq(start(V1, V),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
1107.65/291.59	eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
1107.65/291.59	eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
1107.65/291.59	eq(minus(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 0,V = V3]).
1107.65/291.59	eq(minus(V1, V, Out),1,[p(1 + V4, Ret0),p(1 + V5, Ret1),minus(Ret0, Ret1, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
1107.65/291.59	eq(p(V1, Out),1,[p(1 + V6, Ret11)],[Out = 1 + Ret11,V6 >= 0,V1 = 2 + V6]).
1107.65/291.59	eq(p(V1, Out),1,[],[Out = 2,V1 = 0]).
1107.65/291.59	eq(div(V1, V, Out),1,[minus(V7, V8, Ret10),div(Ret10, 1 + V8, Ret12)],[Out = 1 + Ret12,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
1107.65/291.59	eq(plus(V1, V, Out),1,[],[Out = V9,V9 >= 0,V1 = 0,V = V9]).
1107.65/291.59	eq(plus(V1, V, Out),1,[minus(1 + V11, 1 + 0, Ret111),plus(V10, Ret111, Ret13)],[Out = 1 + Ret13,V11 >= 0,V10 >= 0,V1 = 1 + V11,V = V10]).
1107.65/291.59	eq(p(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
1107.65/291.59	eq(div(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
1107.65/291.59	eq(minus(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
1107.65/291.59	eq(plus(V1, V, Out),0,[],[Out = 0,V17 >= 0,V18 >= 0,V1 = V17,V = V18]).
1107.65/291.59	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
1107.65/291.59	input_output_vars(p(V1,Out),[V1],[Out]).
1107.65/291.59	input_output_vars(div(V1,V,Out),[V1,V],[Out]).
1107.65/291.59	input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	CoFloCo proof output:
1107.65/291.59	Preprocessing Cost Relations
1107.65/291.59	=====================================
1107.65/291.59	
1107.65/291.59	#### Computed strongly connected components 
1107.65/291.59	0. recursive  : [p/2]
1107.65/291.59	1. recursive  : [minus/3]
1107.65/291.59	2. recursive  : [(div)/3]
1107.65/291.59	3. recursive  : [plus/3]
1107.65/291.59	4. non_recursive  : [start/2]
1107.65/291.59	
1107.65/291.59	#### Obtained direct recursion through partial evaluation 
1107.65/291.59	0. SCC is partially evaluated into p/2
1107.65/291.59	1. SCC is partially evaluated into minus/3
1107.65/291.59	2. SCC is partially evaluated into (div)/3
1107.65/291.59	3. SCC is partially evaluated into plus/3
1107.65/291.59	4. SCC is partially evaluated into start/2
1107.65/291.59	
1107.65/291.59	Control-Flow Refinement of Cost Relations
1107.65/291.59	=====================================
1107.65/291.59	
1107.65/291.59	### Specialization of cost equations p/2 
1107.65/291.59	* CE 11 is refined into CE [17] 
1107.65/291.59	* CE 10 is refined into CE [18] 
1107.65/291.59	* CE 9 is refined into CE [19] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Cost equations --> "Loop" of p/2 
1107.65/291.59	* CEs [19] --> Loop 13 
1107.65/291.59	* CEs [17] --> Loop 14 
1107.65/291.59	* CEs [18] --> Loop 15 
1107.65/291.59	
1107.65/291.59	### Ranking functions of CR p(V1,Out) 
1107.65/291.59	* RF of phase [13]: [V1-1]
1107.65/291.59	
1107.65/291.59	#### Partial ranking functions of CR p(V1,Out) 
1107.65/291.59	* Partial RF of phase [13]:
1107.65/291.59	  - RF of loop [13:1]:
1107.65/291.59	    V1-1
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Specialization of cost equations minus/3 
1107.65/291.59	* CE 5 is refined into CE [20] 
1107.65/291.59	* CE 6 is refined into CE [21] 
1107.65/291.59	* CE 8 is refined into CE [22] 
1107.65/291.59	* CE 7 is refined into CE [23,24,25,26] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Cost equations --> "Loop" of minus/3 
1107.65/291.59	* CEs [26] --> Loop 16 
1107.65/291.59	* CEs [25] --> Loop 17 
1107.65/291.59	* CEs [24] --> Loop 18 
1107.65/291.59	* CEs [23] --> Loop 19 
1107.65/291.59	* CEs [20] --> Loop 20 
1107.65/291.59	* CEs [21,22] --> Loop 21 
1107.65/291.59	
1107.65/291.59	### Ranking functions of CR minus(V1,V,Out) 
1107.65/291.59	* RF of phase [16]: [V-1,V1-1]
1107.65/291.59	
1107.65/291.59	#### Partial ranking functions of CR minus(V1,V,Out) 
1107.65/291.59	* Partial RF of phase [16]:
1107.65/291.59	  - RF of loop [16:1]:
1107.65/291.59	    V-1
1107.65/291.59	    V1-1
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Specialization of cost equations (div)/3 
1107.65/291.59	* CE 13 is refined into CE [27] 
1107.65/291.59	* CE 12 is refined into CE [28,29,30] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Cost equations --> "Loop" of (div)/3 
1107.65/291.59	* CEs [30] --> Loop 22 
1107.65/291.59	* CEs [29] --> Loop 23 
1107.65/291.59	* CEs [28] --> Loop 24 
1107.65/291.59	* CEs [27] --> Loop 25 
1107.65/291.59	
1107.65/291.59	### Ranking functions of CR div(V1,V,Out) 
1107.65/291.59	* RF of phase [22]: [V1/2-1]
1107.65/291.59	* RF of phase [24]: [V1]
1107.65/291.59	
1107.65/291.59	#### Partial ranking functions of CR div(V1,V,Out) 
1107.65/291.59	* Partial RF of phase [22]:
1107.65/291.59	  - RF of loop [22:1]:
1107.65/291.59	    V1/2-1
1107.65/291.59	* Partial RF of phase [24]:
1107.65/291.59	  - RF of loop [24:1]:
1107.65/291.59	    V1
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Specialization of cost equations plus/3 
1107.65/291.59	* CE 16 is refined into CE [31] 
1107.65/291.59	* CE 14 is refined into CE [32] 
1107.65/291.59	* CE 15 is refined into CE [33,34] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Cost equations --> "Loop" of plus/3 
1107.65/291.59	* CEs [34] --> Loop 26 
1107.65/291.59	* CEs [33] --> Loop 27 
1107.65/291.59	* CEs [31] --> Loop 28 
1107.65/291.59	* CEs [32] --> Loop 29 
1107.65/291.59	
1107.65/291.59	### Ranking functions of CR plus(V1,V,Out) 
1107.65/291.59	* RF of phase [26,27]: [V1+V]
1107.65/291.59	
1107.65/291.59	#### Partial ranking functions of CR plus(V1,V,Out) 
1107.65/291.59	* Partial RF of phase [26,27]:
1107.65/291.59	  - RF of loop [26:1]:
1107.65/291.59	    V1+V-1
1107.65/291.59	  - RF of loop [27:1]:
1107.65/291.59	    V1+V
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Specialization of cost equations start/2 
1107.65/291.59	* CE 1 is refined into CE [35,36,37] 
1107.65/291.59	* CE 2 is refined into CE [38,39,40] 
1107.65/291.59	* CE 3 is refined into CE [41,42,43,44,45] 
1107.65/291.59	* CE 4 is refined into CE [46,47,48] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	### Cost equations --> "Loop" of start/2 
1107.65/291.59	* CEs [41] --> Loop 30 
1107.65/291.59	* CEs [35] --> Loop 31 
1107.65/291.59	* CEs [36,37,38,39,40,42,43,44,45,46,47,48] --> Loop 32 
1107.65/291.59	
1107.65/291.59	### Ranking functions of CR start(V1,V) 
1107.65/291.59	
1107.65/291.59	#### Partial ranking functions of CR start(V1,V) 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	Computing Bounds
1107.65/291.59	=====================================
1107.65/291.59	
1107.65/291.59	#### Cost of chains of p(V1,Out):
1107.65/291.59	* Chain [[13],14]: 1*it(13)+0
1107.65/291.59	  Such that:it(13) =< Out
1107.65/291.59	
1107.65/291.59	  with precondition: [Out>=1,V1>=Out+1] 
1107.65/291.59	
1107.65/291.59	* Chain [15]: 1
1107.65/291.59	  with precondition: [V1=0,Out=2] 
1107.65/291.59	
1107.65/291.59	* Chain [14]: 0
1107.65/291.59	  with precondition: [Out=0,V1>=0] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	#### Cost of chains of minus(V1,V,Out):
1107.65/291.59	* Chain [[16],21]: 1*it(16)+1*s(5)+1*s(6)+1
1107.65/291.59	  Such that:aux(3) =< V1
1107.65/291.59	aux(5) =< V
1107.65/291.59	it(16) =< aux(5)
1107.65/291.59	it(16) =< aux(3)
1107.65/291.59	s(6) =< it(16)*aux(5)
1107.65/291.59	s(5) =< it(16)*aux(3)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=2,V>=2] 
1107.65/291.59	
1107.65/291.59	* Chain [[16],19,21]: 1*it(16)+1*s(5)+1*s(6)+2
1107.65/291.59	  Such that:aux(3) =< V1
1107.65/291.59	aux(6) =< V
1107.65/291.59	it(16) =< aux(6)
1107.65/291.59	it(16) =< aux(3)
1107.65/291.59	s(6) =< it(16)*aux(6)
1107.65/291.59	s(5) =< it(16)*aux(3)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=2,V>=2] 
1107.65/291.59	
1107.65/291.59	* Chain [[16],19,20]: 1*it(16)+1*s(5)+1*s(6)+2
1107.65/291.59	  Such that:aux(3) =< V1
1107.65/291.59	aux(7) =< V
1107.65/291.59	it(16) =< aux(7)
1107.65/291.59	it(16) =< aux(3)
1107.65/291.59	s(6) =< it(16)*aux(7)
1107.65/291.59	s(5) =< it(16)*aux(3)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=2,V>=2] 
1107.65/291.59	
1107.65/291.59	* Chain [[16],18,21]: 1*it(16)+1*s(5)+1*s(6)+1*s(7)+2
1107.65/291.59	  Such that:aux(3) =< V1
1107.65/291.59	aux(8) =< V
1107.65/291.59	it(16) =< aux(8)
1107.65/291.59	s(7) =< aux(8)
1107.65/291.59	it(16) =< aux(3)
1107.65/291.59	s(6) =< it(16)*aux(8)
1107.65/291.59	s(5) =< it(16)*aux(3)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=2,V>=3] 
1107.65/291.59	
1107.65/291.59	* Chain [[16],17,21]: 1*it(16)+1*s(5)+1*s(6)+1*s(8)+2
1107.65/291.59	  Such that:aux(4) =< V
1107.65/291.59	aux(9) =< V1
1107.65/291.59	it(16) =< aux(9)
1107.65/291.59	s(8) =< aux(9)
1107.65/291.59	it(16) =< aux(4)
1107.65/291.59	s(6) =< it(16)*aux(4)
1107.65/291.59	s(5) =< it(16)*aux(9)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=3,V>=2] 
1107.65/291.59	
1107.65/291.59	* Chain [[16],17,20]: 1*it(16)+1*s(5)+1*s(6)+1*s(8)+2
1107.65/291.59	  Such that:aux(3) =< V1
1107.65/291.59	s(8) =< Out
1107.65/291.59	aux(10) =< V
1107.65/291.59	it(16) =< aux(10)
1107.65/291.59	it(16) =< aux(3)
1107.65/291.59	s(6) =< it(16)*aux(10)
1107.65/291.59	s(5) =< it(16)*aux(3)
1107.65/291.59	
1107.65/291.59	  with precondition: [V>=2,Out>=1,V1>=Out+2] 
1107.65/291.59	
1107.65/291.59	* Chain [21]: 1
1107.65/291.59	  with precondition: [Out=0,V1>=0,V>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [20]: 1
1107.65/291.59	  with precondition: [V=0,V1=Out,V1>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [19,21]: 2
1107.65/291.59	  with precondition: [Out=0,V1>=1,V>=1] 
1107.65/291.59	
1107.65/291.59	* Chain [19,20]: 2
1107.65/291.59	  with precondition: [Out=0,V1>=1,V>=1] 
1107.65/291.59	
1107.65/291.59	* Chain [18,21]: 1*s(7)+2
1107.65/291.59	  Such that:s(7) =< V
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=1,V>=2] 
1107.65/291.59	
1107.65/291.59	* Chain [17,21]: 1*s(8)+2
1107.65/291.59	  Such that:s(8) =< V1
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=0,V1>=2,V>=1] 
1107.65/291.59	
1107.65/291.59	* Chain [17,20]: 1*s(8)+2
1107.65/291.59	  Such that:s(8) =< Out
1107.65/291.59	
1107.65/291.59	  with precondition: [V>=1,Out>=1,V1>=Out+1] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	#### Cost of chains of div(V1,V,Out):
1107.65/291.59	* Chain [[24],25]: 2*it(24)+0
1107.65/291.59	  Such that:it(24) =< Out
1107.65/291.59	
1107.65/291.59	  with precondition: [V=1,Out>=1,V1>=Out] 
1107.65/291.59	
1107.65/291.59	* Chain [[24],23,25]: 2*it(24)+2*s(47)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+3
1107.65/291.59	  Such that:s(46) =< 1
1107.65/291.59	s(45) =< V1-Out+1
1107.65/291.59	it(24) =< Out
1107.65/291.59	s(47) =< s(45)
1107.65/291.59	s(48) =< s(46)
1107.65/291.59	s(49) =< s(46)
1107.65/291.59	s(49) =< s(45)
1107.65/291.59	s(50) =< s(49)*s(46)
1107.65/291.59	s(51) =< s(49)*s(45)
1107.65/291.59	
1107.65/291.59	  with precondition: [V=1,Out>=2,V1>=Out] 
1107.65/291.59	
1107.65/291.59	* Chain [[22],25]: 3*it(22)+2*s(66)+1*s(67)+1*s(68)+1*s(69)+0
1107.65/291.59	  Such that:s(59) =< V1
1107.65/291.59	it(22) =< V1/2
1107.65/291.59	s(61) =< V
1107.65/291.59	aux(16) =< s(61)
1107.65/291.59	aux(15) =< s(59)-2
1107.65/291.59	aux(14) =< s(59)
1107.65/291.59	s(71) =< it(22)*aux(16)
1107.65/291.59	s(72) =< it(22)*aux(15)
1107.65/291.59	s(70) =< it(22)*aux(14)
1107.65/291.59	s(66) =< s(72)
1107.65/291.59	s(67) =< s(71)
1107.65/291.59	s(67) =< s(70)
1107.65/291.59	s(68) =< s(67)*s(61)
1107.65/291.59	s(69) =< s(67)*s(59)
1107.65/291.59	
1107.65/291.59	  with precondition: [V>=2,Out>=1,V1>=2*Out+1] 
1107.65/291.59	
1107.65/291.59	* Chain [[22],23,25]: 5*it(22)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+2*s(66)+1*s(67)+1*s(68)+1*s(69)+3
1107.65/291.59	  Such that:aux(17) =< V1
1107.65/291.59	aux(18) =< V
1107.65/291.59	it(22) =< aux(17)
1107.65/291.59	s(48) =< aux(18)
1107.65/291.59	s(49) =< aux(18)
1107.65/291.59	s(49) =< aux(17)
1107.65/291.59	s(50) =< s(49)*aux(18)
1107.65/291.59	s(51) =< s(49)*aux(17)
1107.65/291.59	aux(16) =< aux(18)
1107.65/291.59	aux(15) =< aux(17)-2
1107.65/291.59	aux(14) =< aux(17)
1107.65/291.59	s(71) =< it(22)*aux(16)
1107.65/291.59	s(72) =< it(22)*aux(15)
1107.65/291.59	s(70) =< it(22)*aux(14)
1107.65/291.59	s(66) =< s(72)
1107.65/291.59	s(67) =< s(71)
1107.65/291.59	s(67) =< s(70)
1107.65/291.59	s(68) =< s(67)*aux(18)
1107.65/291.59	s(69) =< s(67)*aux(17)
1107.65/291.59	
1107.65/291.59	  with precondition: [V>=2,Out>=2,V1+1>=2*Out] 
1107.65/291.59	
1107.65/291.59	* Chain [25]: 0
1107.65/291.59	  with precondition: [Out=0,V1>=0,V>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [23,25]: 2*s(47)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+3
1107.65/291.59	  Such that:s(45) =< V1
1107.65/291.59	s(46) =< V
1107.65/291.59	s(47) =< s(45)
1107.65/291.59	s(48) =< s(46)
1107.65/291.59	s(49) =< s(46)
1107.65/291.59	s(49) =< s(45)
1107.65/291.59	s(50) =< s(49)*s(46)
1107.65/291.59	s(51) =< s(49)*s(45)
1107.65/291.59	
1107.65/291.59	  with precondition: [Out=1,V1>=1,V>=1] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	#### Cost of chains of plus(V1,V,Out):
1107.65/291.59	* Chain [[26,27],29]: 15*it(26)+2*s(110)+1*s(111)+1*s(112)+1*s(113)+5*s(120)+5*s(121)+1
1107.65/291.59	  Such that:aux(24) =< 1
1107.65/291.59	aux(27) =< V1+V
1107.65/291.59	it(26) =< aux(27)
1107.65/291.59	aux(22) =< aux(27)-1
1107.65/291.59	aux(21) =< aux(27)
1107.65/291.59	s(116) =< it(26)*aux(22)
1107.65/291.59	s(114) =< it(26)*aux(21)
1107.65/291.59	s(120) =< it(26)*aux(24)
1107.65/291.59	s(121) =< it(26)*aux(21)
1107.65/291.59	s(110) =< s(116)
1107.65/291.59	s(111) =< aux(27)
1107.65/291.59	s(111) =< s(114)
1107.65/291.59	s(112) =< s(111)*aux(24)
1107.65/291.59	s(113) =< s(111)*aux(27)
1107.65/291.59	
1107.65/291.59	  with precondition: [V1>=1,V>=0,Out>=1,V+V1>=Out] 
1107.65/291.59	
1107.65/291.59	* Chain [[26,27],28]: 15*it(26)+2*s(110)+1*s(111)+1*s(112)+1*s(113)+5*s(120)+5*s(121)+0
1107.65/291.59	  Such that:aux(24) =< 1
1107.65/291.59	aux(28) =< V1+V
1107.65/291.59	it(26) =< aux(28)
1107.65/291.59	aux(22) =< aux(28)-1
1107.65/291.59	aux(21) =< aux(28)
1107.65/291.59	s(116) =< it(26)*aux(22)
1107.65/291.59	s(114) =< it(26)*aux(21)
1107.65/291.59	s(120) =< it(26)*aux(24)
1107.65/291.59	s(121) =< it(26)*aux(21)
1107.65/291.59	s(110) =< s(116)
1107.65/291.59	s(111) =< aux(28)
1107.65/291.59	s(111) =< s(114)
1107.65/291.59	s(112) =< s(111)*aux(24)
1107.65/291.59	s(113) =< s(111)*aux(28)
1107.65/291.59	
1107.65/291.59	  with precondition: [V1>=1,V>=0,Out>=1] 
1107.65/291.59	
1107.65/291.59	* Chain [29]: 1
1107.65/291.59	  with precondition: [V1=0,V=Out,V>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [28]: 0
1107.65/291.59	  with precondition: [Out=0,V1>=0,V>=0] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	#### Cost of chains of start(V1,V):
1107.65/291.59	* Chain [32]: 12*s(152)+6*s(153)+16*s(154)+16*s(155)+16*s(156)+3*s(173)+2*s(181)+1*s(182)+1*s(183)+1*s(184)+2*s(198)+1*s(199)+1*s(200)+1*s(201)+30*s(204)+10*s(209)+10*s(210)+4*s(211)+2*s(212)+2*s(213)+2*s(214)+3
1107.65/291.59	  Such that:s(202) =< 1
1107.65/291.59	s(203) =< V1+V
1107.65/291.59	s(173) =< V1/2
1107.65/291.59	aux(32) =< V1
1107.65/291.59	aux(33) =< V
1107.65/291.59	s(152) =< aux(32)
1107.65/291.59	s(204) =< s(203)
1107.65/291.59	s(205) =< s(203)-1
1107.65/291.59	s(206) =< s(203)
1107.65/291.59	s(207) =< s(204)*s(205)
1107.65/291.59	s(208) =< s(204)*s(206)
1107.65/291.59	s(209) =< s(204)*s(202)
1107.65/291.59	s(210) =< s(204)*s(206)
1107.65/291.59	s(211) =< s(207)
1107.65/291.59	s(212) =< s(203)
1107.65/291.59	s(212) =< s(208)
1107.65/291.59	s(213) =< s(212)*s(202)
1107.65/291.59	s(214) =< s(212)*s(203)
1107.65/291.59	s(153) =< aux(33)
1107.65/291.59	s(154) =< aux(33)
1107.65/291.59	s(154) =< aux(32)
1107.65/291.59	s(155) =< s(154)*aux(33)
1107.65/291.59	s(156) =< s(154)*aux(32)
1107.65/291.59	s(175) =< aux(33)
1107.65/291.59	s(176) =< aux(32)-2
1107.65/291.59	s(177) =< aux(32)
1107.65/291.59	s(178) =< s(173)*s(175)
1107.65/291.59	s(179) =< s(173)*s(176)
1107.65/291.59	s(180) =< s(173)*s(177)
1107.65/291.59	s(181) =< s(179)
1107.65/291.59	s(182) =< s(178)
1107.65/291.59	s(182) =< s(180)
1107.65/291.59	s(183) =< s(182)*aux(33)
1107.65/291.59	s(184) =< s(182)*aux(32)
1107.65/291.59	s(195) =< s(152)*s(175)
1107.65/291.59	s(196) =< s(152)*s(176)
1107.65/291.59	s(197) =< s(152)*s(177)
1107.65/291.59	s(198) =< s(196)
1107.65/291.59	s(199) =< s(195)
1107.65/291.59	s(199) =< s(197)
1107.65/291.59	s(200) =< s(199)*aux(33)
1107.65/291.59	s(201) =< s(199)*aux(32)
1107.65/291.59	
1107.65/291.59	  with precondition: [V1>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [31]: 1
1107.65/291.59	  with precondition: [V=0,V1>=0] 
1107.65/291.59	
1107.65/291.59	* Chain [30]: 6*s(218)+2*s(220)+5*s(221)+5*s(222)+5*s(223)+3
1107.65/291.59	  Such that:s(215) =< 1
1107.65/291.59	aux(34) =< V1+1
1107.65/291.59	s(218) =< aux(34)
1107.65/291.59	s(220) =< s(215)
1107.65/291.59	s(221) =< s(215)
1107.65/291.59	s(221) =< aux(34)
1107.65/291.59	s(222) =< s(221)*s(215)
1107.65/291.59	s(223) =< s(221)*aux(34)
1107.65/291.59	
1107.65/291.59	  with precondition: [V=1,V1>=1] 
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	Closed-form bounds of start(V1,V): 
1107.65/291.59	-------------------------------------
1107.65/291.59	* Chain [32] with precondition: [V1>=0] 
1107.65/291.59	    - Upper bound: 12*V1+3+V1*V1*nat(V)+17*V1*nat(V)+nat(V)*V1*nat(V)+V1/2*(nat(V)*V1)+2*V1*nat(V1-2)+nat(V)*22+nat(V)*16*nat(V)+V1/2*(nat(V)*nat(V))+V1/2*nat(V)+V1/2*(nat(V1-2)*2)+nat(nat(V1+V)+ -1)*4*nat(V1+V)+nat(V1+V)*44+nat(V1+V)*12*nat(V1+V)+3/2*V1 
1107.65/291.59	    - Complexity: n^3 
1107.65/291.59	* Chain [31] with precondition: [V=0,V1>=0] 
1107.65/291.59	    - Upper bound: 1 
1107.65/291.59	    - Complexity: constant 
1107.65/291.59	* Chain [30] with precondition: [V=1,V1>=1] 
1107.65/291.59	    - Upper bound: 11*V1+26 
1107.65/291.59	    - Complexity: n 
1107.65/291.59	
1107.65/291.59	### Maximum cost of start(V1,V): max([11*V1+25,12*V1+2+V1*V1*nat(V)+17*V1*nat(V)+nat(V)*V1*nat(V)+V1/2*(nat(V)*V1)+2*V1*nat(V1-2)+nat(V)*22+nat(V)*16*nat(V)+V1/2*(nat(V)*nat(V))+V1/2*nat(V)+V1/2*(nat(V1-2)*2)+nat(nat(V1+V)+ -1)*4*nat(V1+V)+nat(V1+V)*44+nat(V1+V)*12*nat(V1+V)+3/2*V1])+1 
1107.65/291.59	Asymptotic class: n^3 
1107.65/291.59	* Total analysis performed in 442 ms.
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(14)
1107.65/291.59	BOUNDS(1, n^3)
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1107.65/291.59	Transformed a relative TRS into a decreasing-loop problem.
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(16)
1107.65/291.59	Obligation:
1107.65/291.59	Analyzing the following TRS for decreasing loops:
1107.65/291.59	
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   minus(x, plus(y, z)) -> minus(minus(x, y), z)
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: FULL
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(17) DecreasingLoopProof (LOWER BOUND(ID))
1107.65/291.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1107.65/291.59	
1107.65/291.59	The rewrite sequence
1107.65/291.59	
1107.65/291.59	p(s(s(x))) ->^+ s(p(s(x)))
1107.65/291.59	
1107.65/291.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1107.65/291.59	
1107.65/291.59	The pumping substitution is [x / s(x)].
1107.65/291.59	
1107.65/291.59	The result substitution is [ ].
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(18)
1107.65/291.59	Complex Obligation (BEST)
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(19)
1107.65/291.59	Obligation:
1107.65/291.59	Proved the lower bound n^1 for the following obligation:
1107.65/291.59	
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   minus(x, plus(y, z)) -> minus(minus(x, y), z)
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: FULL
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(20) LowerBoundPropagationProof (FINISHED)
1107.65/291.59	Propagated lower bound.
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(21)
1107.65/291.59	BOUNDS(n^1, INF)
1107.65/291.59	
1107.65/291.59	----------------------------------------
1107.65/291.59	
1107.65/291.59	(22)
1107.65/291.59	Obligation:
1107.65/291.59	Analyzing the following TRS for decreasing loops:
1107.65/291.59	
1107.65/291.59	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).
1107.65/291.59	
1107.65/291.59	
1107.65/291.59	The TRS R consists of the following rules:
1107.65/291.59	
1107.65/291.59	   minus(x, 0) -> x
1107.65/291.59	   minus(0, y) -> 0
1107.65/291.59	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
1107.65/291.59	   minus(x, plus(y, z)) -> minus(minus(x, y), z)
1107.65/291.59	   p(s(s(x))) -> s(p(s(x)))
1107.65/291.59	   p(0) -> s(s(0))
1107.65/291.59	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
1107.65/291.59	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
1107.65/291.59	   plus(0, y) -> y
1107.65/291.59	   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))
1107.65/291.59	
1107.65/291.59	S is empty.
1107.65/291.59	Rewrite Strategy: FULL
1107.89/291.66	EOF