1129.68/291.83	WORST_CASE(Omega(n^1), O(n^2))
1140.58/294.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1140.58/294.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1140.58/294.56	
1140.58/294.56	(0) CpxTRS
1140.58/294.56	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
1140.58/294.56	(2) CpxTRS
1140.58/294.56	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1140.58/294.56	    (4) CpxWeightedTrs
1140.58/294.56	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1140.58/294.56	    (6) CpxTypedWeightedTrs
1140.58/294.56	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
1140.58/294.56	    (8) CpxTypedWeightedCompleteTrs
1140.58/294.56	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1140.58/294.56	    (10) CpxRNTS
1140.58/294.56	    (11) CompleteCoflocoProof [FINISHED, 361 ms]
1140.58/294.56	    (12) BOUNDS(1, n^2)
1140.58/294.56	(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1140.58/294.56	(14) TRS for Loop Detection
1140.58/294.56	    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1140.58/294.56	    (16) BEST
1140.58/294.56	        (17) proven lower bound
1140.58/294.56	            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
1140.58/294.56	            (19) BOUNDS(n^1, INF)
1140.58/294.56	        (20) TRS for Loop Detection
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(0)
1140.58/294.56	Obligation:
1140.58/294.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y)
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y)))
1140.58/294.56	   min(0, v) -> 0
1140.58/294.56	   min(u, 0) -> 0
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v))
1140.58/294.56	   equal(0, 0) -> true
1140.58/294.56	   equal(s(x), 0) -> false
1140.58/294.56	   equal(0, s(y)) -> false
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y)
1140.58/294.56	
1140.58/294.56	S is empty.
1140.58/294.56	Rewrite Strategy: FULL
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
1140.58/294.56	Converted rc-obligation to irc-obligation.
1140.58/294.56	
1140.58/294.56	The duplicating contexts are:
1140.58/294.56	minus([], y)
1140.58/294.56	minus(x, [])
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The defined contexts are:
1140.58/294.56	cond([], x1, x2)
1140.58/294.56	equal([], x1)
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(2)
1140.58/294.56	Obligation:
1140.58/294.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y)
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y)))
1140.58/294.56	   min(0, v) -> 0
1140.58/294.56	   min(u, 0) -> 0
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v))
1140.58/294.56	   equal(0, 0) -> true
1140.58/294.56	   equal(s(x), 0) -> false
1140.58/294.56	   equal(0, s(y)) -> false
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y)
1140.58/294.56	
1140.58/294.56	S is empty.
1140.58/294.56	Rewrite Strategy: INNERMOST
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1140.58/294.56	Transformed relative TRS to weighted TRS
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(4)
1140.58/294.56	Obligation:
1140.58/294.56	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y) [1]
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y))) [1]
1140.58/294.56	   min(0, v) -> 0 [1]
1140.58/294.56	   min(u, 0) -> 0 [1]
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v)) [1]
1140.58/294.56	   equal(0, 0) -> true [1]
1140.58/294.56	   equal(s(x), 0) -> false [1]
1140.58/294.56	   equal(0, s(y)) -> false [1]
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y) [1]
1140.58/294.56	
1140.58/294.56	Rewrite Strategy: INNERMOST
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1140.58/294.56	Infered types.
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(6)
1140.58/294.56	Obligation:
1140.58/294.56	Runtime Complexity Weighted TRS with Types.
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y) [1]
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y))) [1]
1140.58/294.56	   min(0, v) -> 0 [1]
1140.58/294.56	   min(u, 0) -> 0 [1]
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v)) [1]
1140.58/294.56	   equal(0, 0) -> true [1]
1140.58/294.56	   equal(s(x), 0) -> false [1]
1140.58/294.56	   equal(0, s(y)) -> false [1]
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y) [1]
1140.58/294.56	
1140.58/294.56	The TRS has the following type information:
1140.58/294.56	minus :: s:0 -> s:0 -> s:0
1140.58/294.56	cond :: true:false -> s:0 -> s:0 -> s:0
1140.58/294.56	equal :: s:0 -> s:0 -> true:false
1140.58/294.56	min :: s:0 -> s:0 -> s:0
1140.58/294.56	true :: true:false
1140.58/294.56	s :: s:0 -> s:0
1140.58/294.56	0 :: s:0
1140.58/294.56	false :: true:false
1140.58/294.56	
1140.58/294.56	Rewrite Strategy: INNERMOST
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(7) CompletionProof (UPPER BOUND(ID))
1140.58/294.56	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
1140.58/294.56	
1140.58/294.56	   cond(v0, v1, v2) -> null_cond [0]
1140.58/294.56	   min(v0, v1) -> null_min [0]
1140.58/294.56	   equal(v0, v1) -> null_equal [0]
1140.58/294.56	
1140.58/294.56	And the following fresh constants:    null_cond, null_min, null_equal
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(8)
1140.58/294.56	Obligation:
1140.58/294.56	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
1140.58/294.56	
1140.58/294.56	Runtime Complexity Weighted TRS with Types.
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y) [1]
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y))) [1]
1140.58/294.56	   min(0, v) -> 0 [1]
1140.58/294.56	   min(u, 0) -> 0 [1]
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v)) [1]
1140.58/294.56	   equal(0, 0) -> true [1]
1140.58/294.56	   equal(s(x), 0) -> false [1]
1140.58/294.56	   equal(0, s(y)) -> false [1]
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y) [1]
1140.58/294.56	   cond(v0, v1, v2) -> null_cond [0]
1140.58/294.56	   min(v0, v1) -> null_min [0]
1140.58/294.56	   equal(v0, v1) -> null_equal [0]
1140.58/294.56	
1140.58/294.56	The TRS has the following type information:
1140.58/294.56	minus :: s:0:null_cond:null_min -> s:0:null_cond:null_min -> s:0:null_cond:null_min
1140.58/294.56	cond :: true:false:null_equal -> s:0:null_cond:null_min -> s:0:null_cond:null_min -> s:0:null_cond:null_min
1140.58/294.56	equal :: s:0:null_cond:null_min -> s:0:null_cond:null_min -> true:false:null_equal
1140.58/294.56	min :: s:0:null_cond:null_min -> s:0:null_cond:null_min -> s:0:null_cond:null_min
1140.58/294.56	true :: true:false:null_equal
1140.58/294.56	s :: s:0:null_cond:null_min -> s:0:null_cond:null_min
1140.58/294.56	0 :: s:0:null_cond:null_min
1140.58/294.56	false :: true:false:null_equal
1140.58/294.56	null_cond :: s:0:null_cond:null_min
1140.58/294.56	null_min :: s:0:null_cond:null_min
1140.58/294.56	null_equal :: true:false:null_equal
1140.58/294.56	
1140.58/294.56	Rewrite Strategy: INNERMOST
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1140.58/294.56	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1140.58/294.56	The constant constructors are abstracted as follows:
1140.58/294.56	
1140.58/294.56	   true => 2
1140.58/294.56	   0 => 0
1140.58/294.56	   false => 1
1140.58/294.56	   null_cond => 0
1140.58/294.56	   null_min => 0
1140.58/294.56	   null_equal => 0
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(10)
1140.58/294.56	Obligation:
1140.58/294.56	Complexity RNTS consisting of the following rules:
1140.58/294.56	
1140.58/294.56	   cond(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1140.58/294.56	   cond(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
1140.58/294.56	   equal(z, z') -{ 1 }-> equal(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
1140.58/294.56	   equal(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
1140.58/294.56	   equal(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
1140.58/294.56	   equal(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
1140.58/294.56	   equal(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1140.58/294.56	   min(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
1140.58/294.56	   min(z, z') -{ 1 }-> 0 :|: z = u, z' = 0, u >= 0
1140.58/294.56	   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1140.58/294.56	   min(z, z') -{ 1 }-> 1 + min(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
1140.58/294.56	   minus(z, z') -{ 1 }-> cond(equal(min(x, y), y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
1140.58/294.56	
1140.58/294.56	Only complete derivations are relevant for the runtime complexity.
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(11) CompleteCoflocoProof (FINISHED)
1140.58/294.56	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
1140.58/294.56	
1140.58/294.56	eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
1140.58/294.56	eq(start(V1, V, V5),0,[cond(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
1140.58/294.56	eq(start(V1, V, V5),0,[min(V1, V, Out)],[V1 >= 0,V >= 0]).
1140.58/294.56	eq(start(V1, V, V5),0,[equal(V1, V, Out)],[V1 >= 0,V >= 0]).
1140.58/294.56	eq(minus(V1, V, Out),1,[min(V3, V2, Ret00),equal(Ret00, V2, Ret0),cond(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
1140.58/294.56	eq(cond(V1, V, V5, Out),1,[minus(V4, 1 + V6, Ret1)],[Out = 1 + Ret1,V1 = 2,V = V4,V5 = V6,V4 >= 0,V6 >= 0]).
1140.58/294.56	eq(min(V1, V, Out),1,[],[Out = 0,V7 >= 0,V = V7,V1 = 0]).
1140.58/294.56	eq(min(V1, V, Out),1,[],[Out = 0,V1 = V8,V = 0,V8 >= 0]).
1140.58/294.56	eq(min(V1, V, Out),1,[min(V9, V10, Ret11)],[Out = 1 + Ret11,V10 >= 0,V = 1 + V10,V1 = 1 + V9,V9 >= 0]).
1140.58/294.56	eq(equal(V1, V, Out),1,[],[Out = 2,V1 = 0,V = 0]).
1140.58/294.56	eq(equal(V1, V, Out),1,[],[Out = 1,V11 >= 0,V1 = 1 + V11,V = 0]).
1140.58/294.56	eq(equal(V1, V, Out),1,[],[Out = 1,V = 1 + V12,V12 >= 0,V1 = 0]).
1140.58/294.56	eq(equal(V1, V, Out),1,[equal(V13, V14, Ret2)],[Out = Ret2,V = 1 + V14,V13 >= 0,V14 >= 0,V1 = 1 + V13]).
1140.58/294.56	eq(cond(V1, V, V5, Out),0,[],[Out = 0,V16 >= 0,V5 = V17,V15 >= 0,V1 = V16,V = V15,V17 >= 0]).
1140.58/294.56	eq(min(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
1140.58/294.56	eq(equal(V1, V, Out),0,[],[Out = 0,V21 >= 0,V20 >= 0,V1 = V21,V = V20]).
1140.58/294.56	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
1140.58/294.56	input_output_vars(cond(V1,V,V5,Out),[V1,V,V5],[Out]).
1140.58/294.56	input_output_vars(min(V1,V,Out),[V1,V],[Out]).
1140.58/294.56	input_output_vars(equal(V1,V,Out),[V1,V],[Out]).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	CoFloCo proof output:
1140.58/294.56	Preprocessing Cost Relations
1140.58/294.56	=====================================
1140.58/294.56	
1140.58/294.56	#### Computed strongly connected components 
1140.58/294.56	0. recursive  : [equal/3]
1140.58/294.56	1. recursive  : [min/3]
1140.58/294.56	2. recursive  : [cond/4,minus/3]
1140.58/294.56	3. non_recursive  : [start/3]
1140.58/294.56	
1140.58/294.56	#### Obtained direct recursion through partial evaluation 
1140.58/294.56	0. SCC is partially evaluated into equal/3
1140.58/294.56	1. SCC is partially evaluated into min/3
1140.58/294.56	2. SCC is partially evaluated into minus/3
1140.58/294.56	3. SCC is partially evaluated into start/3
1140.58/294.56	
1140.58/294.56	Control-Flow Refinement of Cost Relations
1140.58/294.56	=====================================
1140.58/294.56	
1140.58/294.56	### Specialization of cost equations equal/3 
1140.58/294.56	* CE 16 is refined into CE [17] 
1140.58/294.56	* CE 13 is refined into CE [18] 
1140.58/294.56	* CE 14 is refined into CE [19] 
1140.58/294.56	* CE 12 is refined into CE [20] 
1140.58/294.56	* CE 15 is refined into CE [21] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Cost equations --> "Loop" of equal/3 
1140.58/294.56	* CEs [21] --> Loop 12 
1140.58/294.56	* CEs [17] --> Loop 13 
1140.58/294.56	* CEs [18] --> Loop 14 
1140.58/294.56	* CEs [19] --> Loop 15 
1140.58/294.56	* CEs [20] --> Loop 16 
1140.58/294.56	
1140.58/294.56	### Ranking functions of CR equal(V1,V,Out) 
1140.58/294.56	* RF of phase [12]: [V,V1]
1140.58/294.56	
1140.58/294.56	#### Partial ranking functions of CR equal(V1,V,Out) 
1140.58/294.56	* Partial RF of phase [12]:
1140.58/294.56	  - RF of loop [12:1]:
1140.58/294.56	    V
1140.58/294.56	    V1
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Specialization of cost equations min/3 
1140.58/294.56	* CE 9 is refined into CE [22] 
1140.58/294.56	* CE 8 is refined into CE [23] 
1140.58/294.56	* CE 11 is refined into CE [24] 
1140.58/294.56	* CE 10 is refined into CE [25] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Cost equations --> "Loop" of min/3 
1140.58/294.56	* CEs [25] --> Loop 17 
1140.58/294.56	* CEs [22] --> Loop 18 
1140.58/294.56	* CEs [23,24] --> Loop 19 
1140.58/294.56	
1140.58/294.56	### Ranking functions of CR min(V1,V,Out) 
1140.58/294.56	* RF of phase [17]: [V,V1]
1140.58/294.56	
1140.58/294.56	#### Partial ranking functions of CR min(V1,V,Out) 
1140.58/294.56	* Partial RF of phase [17]:
1140.58/294.56	  - RF of loop [17:1]:
1140.58/294.56	    V
1140.58/294.56	    V1
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Specialization of cost equations minus/3 
1140.58/294.56	* CE 7 is refined into CE [26,27] 
1140.58/294.56	* CE 6 is refined into CE [28,29,30,31,32,33] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Cost equations --> "Loop" of minus/3 
1140.58/294.56	* CEs [28,29,30,31,32,33] --> Loop 20 
1140.58/294.56	* CEs [27] --> Loop 21 
1140.58/294.56	* CEs [26] --> Loop 22 
1140.58/294.56	
1140.58/294.56	### Ranking functions of CR minus(V1,V,Out) 
1140.58/294.56	* RF of phase [21]: [V1-V+1]
1140.58/294.56	
1140.58/294.56	#### Partial ranking functions of CR minus(V1,V,Out) 
1140.58/294.56	* Partial RF of phase [21]:
1140.58/294.56	  - RF of loop [21:1]:
1140.58/294.56	    V1-V+1
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Specialization of cost equations start/3 
1140.58/294.56	* CE 1 is refined into CE [34] 
1140.58/294.56	* CE 2 is refined into CE [35,36] 
1140.58/294.56	* CE 3 is refined into CE [37,38,39,40] 
1140.58/294.56	* CE 4 is refined into CE [41,42] 
1140.58/294.56	* CE 5 is refined into CE [43,44,45,46,47,48,49] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	### Cost equations --> "Loop" of start/3 
1140.58/294.56	* CEs [49] --> Loop 23 
1140.58/294.56	* CEs [35,36] --> Loop 24 
1140.58/294.56	* CEs [44] --> Loop 25 
1140.58/294.56	* CEs [34,37,38,39,40,41,42,43,45,46,47,48] --> Loop 26 
1140.58/294.56	
1140.58/294.56	### Ranking functions of CR start(V1,V,V5) 
1140.58/294.56	
1140.58/294.56	#### Partial ranking functions of CR start(V1,V,V5) 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	Computing Bounds
1140.58/294.56	=====================================
1140.58/294.56	
1140.58/294.56	#### Cost of chains of equal(V1,V,Out):
1140.58/294.56	* Chain [[12],16]: 1*it(12)+1
1140.58/294.56	  Such that:it(12) =< V1
1140.58/294.56	
1140.58/294.56	  with precondition: [Out=2,V1=V,V1>=1] 
1140.58/294.56	
1140.58/294.56	* Chain [[12],15]: 1*it(12)+1
1140.58/294.56	  Such that:it(12) =< V1
1140.58/294.56	
1140.58/294.56	  with precondition: [Out=1,V1>=1,V>=V1+1] 
1140.58/294.56	
1140.58/294.56	* Chain [[12],14]: 1*it(12)+1
1140.58/294.56	  Such that:it(12) =< V
1140.58/294.56	
1140.58/294.56	  with precondition: [Out=1,V>=1,V1>=V+1] 
1140.58/294.56	
1140.58/294.56	* Chain [[12],13]: 1*it(12)+0
1140.58/294.56	  Such that:it(12) =< V
1140.58/294.56	
1140.58/294.56	  with precondition: [Out=0,V1>=1,V>=1] 
1140.58/294.56	
1140.58/294.56	* Chain [16]: 1
1140.58/294.56	  with precondition: [V1=0,V=0,Out=2] 
1140.58/294.56	
1140.58/294.56	* Chain [15]: 1
1140.58/294.56	  with precondition: [V1=0,Out=1,V>=1] 
1140.58/294.56	
1140.58/294.56	* Chain [14]: 1
1140.58/294.56	  with precondition: [V=0,Out=1,V1>=1] 
1140.58/294.56	
1140.58/294.56	* Chain [13]: 0
1140.58/294.56	  with precondition: [Out=0,V1>=0,V>=0] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	#### Cost of chains of min(V1,V,Out):
1140.58/294.56	* Chain [[17],19]: 1*it(17)+1
1140.58/294.56	  Such that:it(17) =< Out
1140.58/294.56	
1140.58/294.56	  with precondition: [Out>=1,V1>=Out,V>=Out] 
1140.58/294.56	
1140.58/294.56	* Chain [[17],18]: 1*it(17)+1
1140.58/294.56	  Such that:it(17) =< Out
1140.58/294.56	
1140.58/294.56	  with precondition: [V=Out,V>=1,V1>=V] 
1140.58/294.56	
1140.58/294.56	* Chain [19]: 1
1140.58/294.56	  with precondition: [Out=0,V1>=0,V>=0] 
1140.58/294.56	
1140.58/294.56	* Chain [18]: 1
1140.58/294.56	  with precondition: [V=0,Out=0,V1>=0] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	#### Cost of chains of minus(V1,V,Out):
1140.58/294.56	* Chain [[21],20]: 4*it(21)+10*s(4)+3*s(19)+3
1140.58/294.56	  Such that:aux(7) =< V1+1
1140.58/294.56	aux(5) =< V+Out
1140.58/294.56	it(21) =< Out
1140.58/294.56	s(4) =< aux(5)
1140.58/294.56	s(20) =< it(21)*aux(7)
1140.58/294.56	s(19) =< s(20)
1140.58/294.56	
1140.58/294.56	  with precondition: [V>=1,Out>=1,V1+1>=Out+V] 
1140.58/294.56	
1140.58/294.56	* Chain [22,[21],20]: 14*it(21)+3*s(19)+7
1140.58/294.56	  Such that:aux(7) =< V1+1
1140.58/294.56	aux(8) =< Out
1140.58/294.56	it(21) =< aux(8)
1140.58/294.56	s(20) =< it(21)*aux(7)
1140.58/294.56	s(19) =< s(20)
1140.58/294.56	
1140.58/294.56	  with precondition: [V=0,Out>=2,V1+1>=Out] 
1140.58/294.56	
1140.58/294.56	* Chain [22,20]: 10*s(4)+7
1140.58/294.56	  Such that:aux(5) =< 1
1140.58/294.56	s(4) =< aux(5)
1140.58/294.56	
1140.58/294.56	  with precondition: [V=0,Out=1,V1>=0] 
1140.58/294.56	
1140.58/294.56	* Chain [20]: 10*s(4)+3
1140.58/294.56	  Such that:aux(5) =< V
1140.58/294.56	s(4) =< aux(5)
1140.58/294.56	
1140.58/294.56	  with precondition: [Out=0,V1>=0,V>=0] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	#### Cost of chains of start(V1,V,V5):
1140.58/294.56	* Chain [26]: 10*s(22)+24*s(25)+3*s(27)+14*s(29)+4*s(32)+3*s(35)+1*s(39)+7
1140.58/294.56	  Such that:s(21) =< 1
1140.58/294.56	s(39) =< V1
1140.58/294.56	s(32) =< V1-V+1
1140.58/294.56	aux(11) =< V1+1
1140.58/294.56	aux(12) =< V
1140.58/294.56	s(29) =< aux(12)
1140.58/294.56	s(22) =< s(21)
1140.58/294.56	s(25) =< aux(11)
1140.58/294.56	s(26) =< s(25)*aux(11)
1140.58/294.56	s(27) =< s(26)
1140.58/294.56	s(34) =< s(32)*aux(11)
1140.58/294.56	s(35) =< s(34)
1140.58/294.56	
1140.58/294.56	  with precondition: [V1>=0,V>=0] 
1140.58/294.56	
1140.58/294.56	* Chain [25]: 1
1140.58/294.56	  with precondition: [V1=0,V>=1] 
1140.58/294.56	
1140.58/294.56	* Chain [24]: 10*s(42)+4*s(45)+10*s(46)+3*s(48)+4
1140.58/294.56	  Such that:aux(13) =< V+1
1140.58/294.56	s(45) =< V-V5
1140.58/294.56	s(41) =< V5+1
1140.58/294.56	s(46) =< aux(13)
1140.58/294.56	s(47) =< s(45)*aux(13)
1140.58/294.56	s(48) =< s(47)
1140.58/294.56	s(42) =< s(41)
1140.58/294.56	
1140.58/294.56	  with precondition: [V1=2,V>=0,V5>=0] 
1140.58/294.56	
1140.58/294.56	* Chain [23]: 1*s(49)+1
1140.58/294.56	  Such that:s(49) =< V
1140.58/294.56	
1140.58/294.56	  with precondition: [V1=V,V1>=1] 
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	Closed-form bounds of start(V1,V,V5): 
1140.58/294.56	-------------------------------------
1140.58/294.56	* Chain [26] with precondition: [V1>=0,V>=0] 
1140.58/294.56	    - Upper bound: 25*V1+14*V+41+(3*V1+3)*(V1+1)+(3*V1+3)*nat(V1-V+1)+nat(V1-V+1)*4 
1140.58/294.56	    - Complexity: n^2 
1140.58/294.56	* Chain [25] with precondition: [V1=0,V>=1] 
1140.58/294.56	    - Upper bound: 1 
1140.58/294.56	    - Complexity: constant 
1140.58/294.56	* Chain [24] with precondition: [V1=2,V>=0,V5>=0] 
1140.58/294.56	    - Upper bound: 10*V+14+(3*V+3)*nat(V-V5)+(10*V5+10)+nat(V-V5)*4 
1140.58/294.56	    - Complexity: n^2 
1140.58/294.56	* Chain [23] with precondition: [V1=V,V1>=1] 
1140.58/294.56	    - Upper bound: V+1 
1140.58/294.56	    - Complexity: n 
1140.58/294.56	
1140.58/294.56	### Maximum cost of start(V1,V,V5): max([10*V+13+(3*V+3)*nat(V-V5)+nat(V5+1)*10+nat(V-V5)*4,25*V1+13*V+40+(3*V1+3)*(V1+1)+(3*V1+3)*nat(V1-V+1)+nat(V1-V+1)*4+V])+1 
1140.58/294.56	Asymptotic class: n^2 
1140.58/294.56	* Total analysis performed in 278 ms.
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(12)
1140.58/294.56	BOUNDS(1, n^2)
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1140.58/294.56	Transformed a relative TRS into a decreasing-loop problem.
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(14)
1140.58/294.56	Obligation:
1140.58/294.56	Analyzing the following TRS for decreasing loops:
1140.58/294.56	
1140.58/294.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y)
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y)))
1140.58/294.56	   min(0, v) -> 0
1140.58/294.56	   min(u, 0) -> 0
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v))
1140.58/294.56	   equal(0, 0) -> true
1140.58/294.56	   equal(s(x), 0) -> false
1140.58/294.56	   equal(0, s(y)) -> false
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y)
1140.58/294.56	
1140.58/294.56	S is empty.
1140.58/294.56	Rewrite Strategy: FULL
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(15) DecreasingLoopProof (LOWER BOUND(ID))
1140.58/294.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1140.58/294.56	
1140.58/294.56	The rewrite sequence
1140.58/294.56	
1140.58/294.56	min(s(u), s(v)) ->^+ s(min(u, v))
1140.58/294.56	
1140.58/294.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1140.58/294.56	
1140.58/294.56	The pumping substitution is [u / s(u), v / s(v)].
1140.58/294.56	
1140.58/294.56	The result substitution is [ ].
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(16)
1140.58/294.56	Complex Obligation (BEST)
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(17)
1140.58/294.56	Obligation:
1140.58/294.56	Proved the lower bound n^1 for the following obligation:
1140.58/294.56	
1140.58/294.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y)
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y)))
1140.58/294.56	   min(0, v) -> 0
1140.58/294.56	   min(u, 0) -> 0
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v))
1140.58/294.56	   equal(0, 0) -> true
1140.58/294.56	   equal(s(x), 0) -> false
1140.58/294.56	   equal(0, s(y)) -> false
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y)
1140.58/294.56	
1140.58/294.56	S is empty.
1140.58/294.56	Rewrite Strategy: FULL
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(18) LowerBoundPropagationProof (FINISHED)
1140.58/294.56	Propagated lower bound.
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(19)
1140.58/294.56	BOUNDS(n^1, INF)
1140.58/294.56	
1140.58/294.56	----------------------------------------
1140.58/294.56	
1140.58/294.56	(20)
1140.58/294.56	Obligation:
1140.58/294.56	Analyzing the following TRS for decreasing loops:
1140.58/294.56	
1140.58/294.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
1140.58/294.56	
1140.58/294.56	
1140.58/294.56	The TRS R consists of the following rules:
1140.58/294.56	
1140.58/294.56	   minus(x, y) -> cond(equal(min(x, y), y), x, y)
1140.58/294.56	   cond(true, x, y) -> s(minus(x, s(y)))
1140.58/294.56	   min(0, v) -> 0
1140.58/294.56	   min(u, 0) -> 0
1140.58/294.56	   min(s(u), s(v)) -> s(min(u, v))
1140.58/294.56	   equal(0, 0) -> true
1140.58/294.56	   equal(s(x), 0) -> false
1140.58/294.56	   equal(0, s(y)) -> false
1140.58/294.56	   equal(s(x), s(y)) -> equal(x, y)
1140.58/294.56	
1140.58/294.56	S is empty.
1140.58/294.56	Rewrite Strategy: FULL
1140.58/294.61	EOF