883.24/291.57	WORST_CASE(Omega(n^1), O(n^2))
883.50/291.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
883.50/291.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
883.50/291.59	
883.50/291.59	
883.50/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
883.50/291.59	
883.50/291.59	(0) CpxTRS
883.50/291.59	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
883.50/291.59	(2) CpxWeightedTrs
883.50/291.59	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
883.50/291.59	    (4) CpxTypedWeightedTrs
883.50/291.59	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
883.50/291.59	    (6) CpxTypedWeightedCompleteTrs
883.50/291.59	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
883.50/291.59	    (8) CpxRNTS
883.50/291.59	    (9) CompleteCoflocoProof [FINISHED, 243 ms]
883.50/291.59	    (10) BOUNDS(1, n^2)
883.50/291.59	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
883.50/291.59	(12) TRS for Loop Detection
883.50/291.59	    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
883.50/291.59	    (14) BEST
883.50/291.59	        (15) proven lower bound
883.50/291.59	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
883.50/291.59	            (17) BOUNDS(n^1, INF)
883.50/291.59	        (18) TRS for Loop Detection
883.50/291.59	
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(0)
883.50/291.59	Obligation:
883.50/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
883.50/291.59	
883.50/291.59	
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y)
883.50/291.59	   cond(false, x, y) -> 0
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y)))
883.50/291.59	   gt(0, v) -> false
883.50/291.59	   gt(s(u), 0) -> true
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v)
883.50/291.59	
883.50/291.59	S is empty.
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
883.50/291.59	Transformed relative TRS to weighted TRS
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(2)
883.50/291.59	Obligation:
883.50/291.59	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
883.50/291.59	
883.50/291.59	
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y) [1]
883.50/291.59	   cond(false, x, y) -> 0 [1]
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y))) [1]
883.50/291.59	   gt(0, v) -> false [1]
883.50/291.59	   gt(s(u), 0) -> true [1]
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v) [1]
883.50/291.59	
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
883.50/291.59	Infered types.
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(4)
883.50/291.59	Obligation:
883.50/291.59	Runtime Complexity Weighted TRS with Types.
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y) [1]
883.50/291.59	   cond(false, x, y) -> 0 [1]
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y))) [1]
883.50/291.59	   gt(0, v) -> false [1]
883.50/291.59	   gt(s(u), 0) -> true [1]
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v) [1]
883.50/291.59	
883.50/291.59	The TRS has the following type information:
883.50/291.59	minus :: 0:s -> 0:s -> 0:s
883.50/291.59	cond :: false:true -> 0:s -> 0:s -> 0:s
883.50/291.59	gt :: 0:s -> 0:s -> false:true
883.50/291.59	false :: false:true
883.50/291.59	0 :: 0:s
883.50/291.59	true :: false:true
883.50/291.59	s :: 0:s -> 0:s
883.50/291.59	
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(5) CompletionProof (UPPER BOUND(ID))
883.50/291.59	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
883.50/291.59	none
883.50/291.59	
883.50/291.59	And the following fresh constants: none
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(6)
883.50/291.59	Obligation:
883.50/291.59	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
883.50/291.59	
883.50/291.59	Runtime Complexity Weighted TRS with Types.
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y) [1]
883.50/291.59	   cond(false, x, y) -> 0 [1]
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y))) [1]
883.50/291.59	   gt(0, v) -> false [1]
883.50/291.59	   gt(s(u), 0) -> true [1]
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v) [1]
883.50/291.59	
883.50/291.59	The TRS has the following type information:
883.50/291.59	minus :: 0:s -> 0:s -> 0:s
883.50/291.59	cond :: false:true -> 0:s -> 0:s -> 0:s
883.50/291.59	gt :: 0:s -> 0:s -> false:true
883.50/291.59	false :: false:true
883.50/291.59	0 :: 0:s
883.50/291.59	true :: false:true
883.50/291.59	s :: 0:s -> 0:s
883.50/291.59	
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
883.50/291.59	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
883.50/291.59	The constant constructors are abstracted as follows:
883.50/291.59	
883.50/291.59	   false => 0
883.50/291.59	   0 => 0
883.50/291.59	   true => 1
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(8)
883.50/291.59	Obligation:
883.50/291.59	Complexity RNTS consisting of the following rules:
883.50/291.59	
883.50/291.59	   cond(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
883.50/291.59	   cond(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
883.50/291.59	   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
883.50/291.59	   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
883.50/291.59	   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
883.50/291.59	   minus(z, z') -{ 1 }-> cond(gt(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
883.50/291.59	
883.50/291.59	Only complete derivations are relevant for the runtime complexity.
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(9) CompleteCoflocoProof (FINISHED)
883.50/291.59	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
883.50/291.59	
883.50/291.59	eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
883.50/291.59	eq(start(V1, V, V5),0,[cond(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
883.50/291.59	eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
883.50/291.59	eq(minus(V1, V, Out),1,[gt(V3, V2, Ret0),cond(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
883.50/291.59	eq(cond(V1, V, V5, Out),1,[],[Out = 0,V = V4,V5 = V6,V4 >= 0,V6 >= 0,V1 = 0]).
883.50/291.59	eq(cond(V1, V, V5, Out),1,[minus(V8, 1 + V7, Ret1)],[Out = 1 + Ret1,V = V8,V5 = V7,V1 = 1,V8 >= 0,V7 >= 0]).
883.50/291.59	eq(gt(V1, V, Out),1,[],[Out = 0,V9 >= 0,V = V9,V1 = 0]).
883.50/291.59	eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V10,V = 0,V10 >= 0]).
883.50/291.59	eq(gt(V1, V, Out),1,[gt(V12, V11, Ret2)],[Out = Ret2,V11 >= 0,V = 1 + V11,V1 = 1 + V12,V12 >= 0]).
883.50/291.59	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
883.50/291.59	input_output_vars(cond(V1,V,V5,Out),[V1,V,V5],[Out]).
883.50/291.59	input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
883.50/291.59	
883.50/291.59	
883.50/291.59	CoFloCo proof output:
883.50/291.59	Preprocessing Cost Relations
883.50/291.59	=====================================
883.50/291.59	
883.50/291.59	#### Computed strongly connected components 
883.50/291.59	0. recursive  : [gt/3]
883.50/291.59	1. recursive  : [cond/4,minus/3]
883.50/291.59	2. non_recursive  : [start/3]
883.50/291.59	
883.50/291.59	#### Obtained direct recursion through partial evaluation 
883.50/291.59	0. SCC is partially evaluated into gt/3
883.50/291.59	1. SCC is partially evaluated into minus/3
883.50/291.59	2. SCC is partially evaluated into start/3
883.50/291.59	
883.50/291.59	Control-Flow Refinement of Cost Relations
883.50/291.59	=====================================
883.50/291.59	
883.50/291.59	### Specialization of cost equations gt/3 
883.50/291.59	* CE 9 is refined into CE [10] 
883.50/291.59	* CE 8 is refined into CE [11] 
883.50/291.59	* CE 7 is refined into CE [12] 
883.50/291.59	
883.50/291.59	
883.50/291.59	### Cost equations --> "Loop" of gt/3 
883.50/291.59	* CEs [11] --> Loop 8 
883.50/291.59	* CEs [12] --> Loop 9 
883.50/291.59	* CEs [10] --> Loop 10 
883.50/291.59	
883.50/291.59	### Ranking functions of CR gt(V1,V,Out) 
883.50/291.59	* RF of phase [10]: [V,V1]
883.50/291.59	
883.50/291.59	#### Partial ranking functions of CR gt(V1,V,Out) 
883.50/291.59	* Partial RF of phase [10]:
883.50/291.59	  - RF of loop [10:1]:
883.50/291.59	    V
883.50/291.59	    V1
883.50/291.59	
883.50/291.59	
883.50/291.59	### Specialization of cost equations minus/3 
883.50/291.59	* CE 6 is refined into CE [13,14] 
883.50/291.59	* CE 5 is refined into CE [15,16] 
883.50/291.59	
883.50/291.59	
883.50/291.59	### Cost equations --> "Loop" of minus/3 
883.50/291.59	* CEs [16] --> Loop 11 
883.50/291.59	* CEs [15] --> Loop 12 
883.50/291.59	* CEs [14] --> Loop 13 
883.50/291.59	* CEs [13] --> Loop 14 
883.50/291.59	
883.50/291.59	### Ranking functions of CR minus(V1,V,Out) 
883.50/291.59	* RF of phase [11]: [V1-V]
883.50/291.59	
883.50/291.59	#### Partial ranking functions of CR minus(V1,V,Out) 
883.50/291.59	* Partial RF of phase [11]:
883.50/291.59	  - RF of loop [11:1]:
883.50/291.59	    V1-V
883.50/291.59	
883.50/291.59	
883.50/291.59	### Specialization of cost equations start/3 
883.50/291.59	* CE 1 is refined into CE [17,18,19] 
883.50/291.59	* CE 2 is refined into CE [20] 
883.50/291.59	* CE 3 is refined into CE [21,22,23,24,25] 
883.50/291.59	* CE 4 is refined into CE [26,27,28,29] 
883.50/291.59	
883.50/291.59	
883.50/291.59	### Cost equations --> "Loop" of start/3 
883.50/291.59	* CEs [25,29] --> Loop 15 
883.50/291.59	* CEs [19] --> Loop 16 
883.50/291.59	* CEs [18,24,28] --> Loop 17 
883.50/291.59	* CEs [17,22,23,27] --> Loop 18 
883.50/291.59	* CEs [20,21,26] --> Loop 19 
883.50/291.59	
883.50/291.59	### Ranking functions of CR start(V1,V,V5) 
883.50/291.59	
883.50/291.59	#### Partial ranking functions of CR start(V1,V,V5) 
883.50/291.59	
883.50/291.59	
883.50/291.59	Computing Bounds
883.50/291.59	=====================================
883.50/291.59	
883.50/291.59	#### Cost of chains of gt(V1,V,Out):
883.50/291.59	* Chain [[10],9]: 1*it(10)+1
883.50/291.59	  Such that:it(10) =< V1
883.50/291.59	
883.50/291.59	  with precondition: [Out=0,V1>=1,V>=V1] 
883.50/291.59	
883.50/291.59	* Chain [[10],8]: 1*it(10)+1
883.50/291.59	  Such that:it(10) =< V
883.50/291.59	
883.50/291.59	  with precondition: [Out=1,V>=1,V1>=V+1] 
883.50/291.59	
883.50/291.59	* Chain [9]: 1
883.50/291.59	  with precondition: [V1=0,Out=0,V>=0] 
883.50/291.59	
883.50/291.59	* Chain [8]: 1
883.50/291.59	  with precondition: [V=0,Out=1,V1>=1] 
883.50/291.59	
883.50/291.59	
883.50/291.59	#### Cost of chains of minus(V1,V,Out):
883.50/291.59	* Chain [[11],13]: 3*it(11)+1*s(1)+1*s(4)+3
883.50/291.59	  Such that:it(11) =< Out
883.50/291.59	aux(2) =< V+Out
883.50/291.59	s(1) =< aux(2)
883.50/291.59	s(4) =< it(11)*aux(2)
883.50/291.59	
883.50/291.59	  with precondition: [V1=Out+V,V>=1,V1>=V+1] 
883.50/291.59	
883.50/291.59	* Chain [14]: 3
883.50/291.59	  with precondition: [V1=0,Out=0,V>=0] 
883.50/291.59	
883.50/291.59	* Chain [13]: 1*s(1)+3
883.50/291.59	  Such that:s(1) =< V1
883.50/291.59	
883.50/291.59	  with precondition: [Out=0,V1>=1,V>=V1] 
883.50/291.59	
883.50/291.59	* Chain [12,[11],13]: 4*it(11)+1*s(4)+6
883.50/291.59	  Such that:aux(3) =< Out
883.50/291.59	it(11) =< aux(3)
883.50/291.59	s(4) =< it(11)*aux(3)
883.50/291.59	
883.50/291.59	  with precondition: [V=0,V1=Out,V1>=2] 
883.50/291.59	
883.50/291.59	* Chain [12,13]: 1*s(1)+6
883.50/291.59	  Such that:s(1) =< 1
883.50/291.59	
883.50/291.59	  with precondition: [V1=1,V=0,Out=1] 
883.50/291.59	
883.50/291.59	
883.50/291.59	#### Cost of chains of start(V1,V,V5):
883.50/291.59	* Chain [19]: 3
883.50/291.59	  with precondition: [V1=0,V>=0] 
883.50/291.59	
883.50/291.59	* Chain [18]: 1*s(5)+4*s(7)+1*s(8)+6
883.50/291.59	  Such that:s(5) =< 1
883.50/291.59	s(6) =< V1
883.50/291.59	s(7) =< s(6)
883.50/291.59	s(8) =< s(7)*s(6)
883.50/291.59	
883.50/291.59	  with precondition: [V=0,V1>=1] 
883.50/291.59	
883.50/291.59	* Chain [17]: 1*s(9)+2*s(10)+4
883.50/291.59	  Such that:s(9) =< V
883.50/291.59	aux(4) =< V1
883.50/291.59	s(10) =< aux(4)
883.50/291.59	
883.50/291.59	  with precondition: [V1>=1,V>=V1] 
883.50/291.59	
883.50/291.59	* Chain [16]: 3*s(12)+1*s(14)+1*s(15)+4
883.50/291.59	  Such that:s(13) =< V
883.50/291.59	s(12) =< V-V5
883.50/291.59	s(14) =< s(13)
883.50/291.59	s(15) =< s(12)*s(13)
883.50/291.59	
883.50/291.59	  with precondition: [V1=1,V5>=0,V>=V5+2] 
883.50/291.59	
883.50/291.59	* Chain [15]: 3*s(16)+1*s(18)+1*s(19)+1*s(20)+3
883.50/291.59	  Such that:s(17) =< V1
883.50/291.59	s(16) =< V1-V
883.50/291.59	s(20) =< V
883.50/291.59	s(18) =< s(17)
883.50/291.59	s(19) =< s(16)*s(17)
883.50/291.59	
883.50/291.59	  with precondition: [V>=1,V1>=V+1] 
883.50/291.59	
883.50/291.59	
883.50/291.59	Closed-form bounds of start(V1,V,V5): 
883.50/291.59	-------------------------------------
883.50/291.59	* Chain [19] with precondition: [V1=0,V>=0] 
883.50/291.59	    - Upper bound: 3 
883.50/291.59	    - Complexity: constant 
883.50/291.59	* Chain [18] with precondition: [V=0,V1>=1] 
883.50/291.59	    - Upper bound: 4*V1+7+V1*V1 
883.50/291.59	    - Complexity: n^2 
883.50/291.59	* Chain [17] with precondition: [V1>=1,V>=V1] 
883.50/291.59	    - Upper bound: 2*V1+V+4 
883.50/291.59	    - Complexity: n 
883.50/291.59	* Chain [16] with precondition: [V1=1,V5>=0,V>=V5+2] 
883.50/291.59	    - Upper bound: 3*V-3*V5+(V+4+(V-V5)*V) 
883.50/291.59	    - Complexity: n^2 
883.50/291.59	* Chain [15] with precondition: [V>=1,V1>=V+1] 
883.50/291.59	    - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1+V) 
883.50/291.59	    - Complexity: n^2 
883.50/291.59	
883.50/291.59	### Maximum cost of start(V1,V,V5): max([max([V1+1+max([V,2*V1+3+V1*V1]),nat(V1-V)*V1+V+nat(V1-V)*3])+V1,V+1+nat(V-V5)*V+nat(V-V5)*3])+3 
883.50/291.59	Asymptotic class: n^2 
883.50/291.59	* Total analysis performed in 170 ms.
883.50/291.59	
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(10)
883.50/291.59	BOUNDS(1, n^2)
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
883.50/291.59	Transformed a relative TRS into a decreasing-loop problem.
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(12)
883.50/291.59	Obligation:
883.50/291.59	Analyzing the following TRS for decreasing loops:
883.50/291.59	
883.50/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
883.50/291.59	
883.50/291.59	
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y)
883.50/291.59	   cond(false, x, y) -> 0
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y)))
883.50/291.59	   gt(0, v) -> false
883.50/291.59	   gt(s(u), 0) -> true
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v)
883.50/291.59	
883.50/291.59	S is empty.
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(13) DecreasingLoopProof (LOWER BOUND(ID))
883.50/291.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
883.50/291.59	
883.50/291.59	The rewrite sequence
883.50/291.59	
883.50/291.59	gt(s(u), s(v)) ->^+ gt(u, v)
883.50/291.59	
883.50/291.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
883.50/291.59	
883.50/291.59	The pumping substitution is [u / s(u), v / s(v)].
883.50/291.59	
883.50/291.59	The result substitution is [ ].
883.50/291.59	
883.50/291.59	
883.50/291.59	
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(14)
883.50/291.59	Complex Obligation (BEST)
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(15)
883.50/291.59	Obligation:
883.50/291.59	Proved the lower bound n^1 for the following obligation:
883.50/291.59	
883.50/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
883.50/291.59	
883.50/291.59	
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y)
883.50/291.59	   cond(false, x, y) -> 0
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y)))
883.50/291.59	   gt(0, v) -> false
883.50/291.59	   gt(s(u), 0) -> true
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v)
883.50/291.59	
883.50/291.59	S is empty.
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(16) LowerBoundPropagationProof (FINISHED)
883.50/291.59	Propagated lower bound.
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(17)
883.50/291.59	BOUNDS(n^1, INF)
883.50/291.59	
883.50/291.59	----------------------------------------
883.50/291.59	
883.50/291.59	(18)
883.50/291.59	Obligation:
883.50/291.59	Analyzing the following TRS for decreasing loops:
883.50/291.59	
883.50/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
883.50/291.59	
883.50/291.59	
883.50/291.59	The TRS R consists of the following rules:
883.50/291.59	
883.50/291.59	   minus(x, y) -> cond(gt(x, y), x, y)
883.50/291.59	   cond(false, x, y) -> 0
883.50/291.59	   cond(true, x, y) -> s(minus(x, s(y)))
883.50/291.59	   gt(0, v) -> false
883.50/291.59	   gt(s(u), 0) -> true
883.50/291.59	   gt(s(u), s(v)) -> gt(u, v)
883.50/291.59	
883.50/291.59	S is empty.
883.50/291.59	Rewrite Strategy: INNERMOST
883.50/291.64	EOF