22.50/6.82	WORST_CASE(Omega(n^1), O(n^1))
22.50/6.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.50/6.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.50/6.83	
22.50/6.83	
22.50/6.83	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.50/6.83	
22.50/6.83	(0) CpxTRS
22.50/6.83	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
22.50/6.83	(2) CpxTRS
22.50/6.83	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
22.50/6.83	    (4) CpxWeightedTrs
22.50/6.83	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.50/6.83	    (6) CpxTypedWeightedTrs
22.50/6.83	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
22.50/6.83	    (8) CpxTypedWeightedCompleteTrs
22.50/6.83	    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms]
22.50/6.83	    (10) CpxRNTS
22.50/6.83	    (11) CompleteCoflocoProof [FINISHED, 240 ms]
22.50/6.83	    (12) BOUNDS(1, n^1)
22.50/6.83	(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.50/6.83	(14) CpxTRS
22.50/6.83	    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.50/6.83	    (16) typed CpxTrs
22.50/6.83	    (17) OrderProof [LOWER BOUND(ID), 0 ms]
22.50/6.83	    (18) typed CpxTrs
22.50/6.83	    (19) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
22.50/6.83	    (20) BEST
22.50/6.83	        (21) proven lower bound
22.50/6.83	            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
22.50/6.83	            (23) BOUNDS(n^1, INF)
22.50/6.83	        (24) typed CpxTrs
22.50/6.83	
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(0)
22.50/6.83	Obligation:
22.50/6.83	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.50/6.83	
22.50/6.83	
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0) -> x
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y)
22.50/6.83	   quot(0, s(y)) -> 0
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.83	
22.50/6.83	S is empty.
22.50/6.83	Rewrite Strategy: FULL
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
22.50/6.83	Converted rc-obligation to irc-obligation.
22.50/6.83	
22.50/6.83	The duplicating contexts are:
22.50/6.83	quot(s(x), s([]))
22.50/6.83	
22.50/6.83	
22.50/6.83	The defined contexts are:
22.50/6.83	quot([], s(x1))
22.50/6.83	minus([], x1)
22.50/6.83	
22.50/6.83	
22.50/6.83	[] just represents basic- or constructor-terms in the following defined contexts:
22.50/6.83	quot([], s(x1))
22.50/6.83	
22.50/6.83	
22.50/6.83	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(2)
22.50/6.83	Obligation:
22.50/6.83	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.50/6.83	
22.50/6.83	
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0) -> x
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y)
22.50/6.83	   quot(0, s(y)) -> 0
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.83	
22.50/6.83	S is empty.
22.50/6.83	Rewrite Strategy: INNERMOST
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
22.50/6.83	Transformed relative TRS to weighted TRS
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(4)
22.50/6.83	Obligation:
22.50/6.83	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.50/6.83	
22.50/6.83	
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0) -> x [1]
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y) [1]
22.50/6.83	   quot(0, s(y)) -> 0 [1]
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
22.50/6.83	
22.50/6.83	Rewrite Strategy: INNERMOST
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.50/6.83	Infered types.
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(6)
22.50/6.83	Obligation:
22.50/6.83	Runtime Complexity Weighted TRS with Types.
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0) -> x [1]
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y) [1]
22.50/6.83	   quot(0, s(y)) -> 0 [1]
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
22.50/6.83	
22.50/6.83	The TRS has the following type information:
22.50/6.83	minus :: 0:s -> 0:s -> 0:s
22.50/6.83	0 :: 0:s
22.50/6.83	s :: 0:s -> 0:s
22.50/6.83	quot :: 0:s -> 0:s -> 0:s
22.50/6.83	
22.50/6.83	Rewrite Strategy: INNERMOST
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(7) CompletionProof (UPPER BOUND(ID))
22.50/6.83	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
22.50/6.83	
22.50/6.83	   minus(v0, v1) -> null_minus [0]
22.50/6.83	   quot(v0, v1) -> null_quot [0]
22.50/6.83	
22.50/6.83	And the following fresh constants:    null_minus, null_quot
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(8)
22.50/6.83	Obligation:
22.50/6.83	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
22.50/6.83	
22.50/6.83	Runtime Complexity Weighted TRS with Types.
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0) -> x [1]
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y) [1]
22.50/6.83	   quot(0, s(y)) -> 0 [1]
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
22.50/6.83	   minus(v0, v1) -> null_minus [0]
22.50/6.83	   quot(v0, v1) -> null_quot [0]
22.50/6.83	
22.50/6.83	The TRS has the following type information:
22.50/6.83	minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
22.50/6.83	0 :: 0:s:null_minus:null_quot
22.50/6.83	s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
22.50/6.83	quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
22.50/6.83	null_minus :: 0:s:null_minus:null_quot
22.50/6.83	null_quot :: 0:s:null_minus:null_quot
22.50/6.83	
22.50/6.83	Rewrite Strategy: INNERMOST
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
22.50/6.83	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
22.50/6.83	The constant constructors are abstracted as follows:
22.50/6.83	
22.50/6.83	   0 => 0
22.50/6.83	   null_minus => 0
22.50/6.83	   null_quot => 0
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(10)
22.50/6.83	Obligation:
22.50/6.83	Complexity RNTS consisting of the following rules:
22.50/6.83	
22.50/6.83	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
22.50/6.83	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
22.50/6.83	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.50/6.83	   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
22.50/6.83	   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
22.50/6.83	   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
22.50/6.83	
22.50/6.83	Only complete derivations are relevant for the runtime complexity.
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(11) CompleteCoflocoProof (FINISHED)
22.50/6.83	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
22.50/6.83	
22.50/6.83	eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
22.50/6.83	eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
22.50/6.83	eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
22.50/6.83	eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]).
22.50/6.83	eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V5,V5 >= 0,V1 = 0]).
22.50/6.83	eq(quot(V1, V, Out),1,[minus(V7, V6, Ret10),quot(Ret10, 1 + V6, Ret1)],[Out = 1 + Ret1,V = 1 + V6,V7 >= 0,V6 >= 0,V1 = 1 + V7]).
22.50/6.83	eq(minus(V1, V, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
22.50/6.83	eq(quot(V1, V, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
22.50/6.83	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
22.50/6.83	input_output_vars(quot(V1,V,Out),[V1,V],[Out]).
22.50/6.83	
22.50/6.83	
22.50/6.83	CoFloCo proof output:
22.50/6.83	Preprocessing Cost Relations
22.50/6.83	=====================================
22.50/6.83	
22.50/6.83	#### Computed strongly connected components 
22.50/6.83	0. recursive  : [minus/3]
22.50/6.83	1. recursive  : [quot/3]
22.50/6.83	2. non_recursive  : [start/2]
22.50/6.83	
22.50/6.83	#### Obtained direct recursion through partial evaluation 
22.50/6.83	0. SCC is partially evaluated into minus/3
22.50/6.83	1. SCC is partially evaluated into quot/3
22.50/6.83	2. SCC is partially evaluated into start/2
22.50/6.83	
22.50/6.83	Control-Flow Refinement of Cost Relations
22.50/6.83	=====================================
22.50/6.83	
22.50/6.83	### Specialization of cost equations minus/3 
22.50/6.83	* CE 5 is refined into CE [9] 
22.50/6.83	* CE 3 is refined into CE [10] 
22.50/6.83	* CE 4 is refined into CE [11] 
22.50/6.83	
22.50/6.83	
22.50/6.83	### Cost equations --> "Loop" of minus/3 
22.50/6.83	* CEs [11] --> Loop 7 
22.50/6.83	* CEs [9] --> Loop 8 
22.50/6.83	* CEs [10] --> Loop 9 
22.50/6.83	
22.50/6.83	### Ranking functions of CR minus(V1,V,Out) 
22.50/6.83	* RF of phase [7]: [V,V1]
22.50/6.83	
22.50/6.83	#### Partial ranking functions of CR minus(V1,V,Out) 
22.50/6.83	* Partial RF of phase [7]:
22.50/6.83	  - RF of loop [7:1]:
22.50/6.83	    V
22.50/6.83	    V1
22.50/6.83	
22.50/6.83	
22.50/6.83	### Specialization of cost equations quot/3 
22.50/6.83	* CE 6 is refined into CE [12] 
22.50/6.83	* CE 8 is refined into CE [13] 
22.50/6.83	* CE 7 is refined into CE [14,15,16] 
22.50/6.83	
22.50/6.83	
22.50/6.83	### Cost equations --> "Loop" of quot/3 
22.50/6.83	* CEs [16] --> Loop 10 
22.50/6.83	* CEs [15] --> Loop 11 
22.50/6.83	* CEs [14] --> Loop 12 
22.50/6.83	* CEs [12,13] --> Loop 13 
22.50/6.83	
22.50/6.83	### Ranking functions of CR quot(V1,V,Out) 
22.50/6.83	* RF of phase [10]: [V1-1,V1-V+1]
22.50/6.83	* RF of phase [12]: [V1]
22.50/6.83	
22.50/6.83	#### Partial ranking functions of CR quot(V1,V,Out) 
22.50/6.83	* Partial RF of phase [10]:
22.50/6.83	  - RF of loop [10:1]:
22.50/6.83	    V1-1
22.50/6.83	    V1-V+1
22.50/6.83	* Partial RF of phase [12]:
22.50/6.83	  - RF of loop [12:1]:
22.50/6.83	    V1
22.50/6.83	
22.50/6.83	
22.50/6.83	### Specialization of cost equations start/2 
22.50/6.83	* CE 1 is refined into CE [17,18,19] 
22.50/6.83	* CE 2 is refined into CE [20,21,22,23,24] 
22.50/6.83	
22.50/6.83	
22.50/6.83	### Cost equations --> "Loop" of start/2 
22.50/6.83	* CEs [20] --> Loop 14 
22.50/6.83	* CEs [17,18,19,21,22,23,24] --> Loop 15 
22.50/6.83	
22.50/6.83	### Ranking functions of CR start(V1,V) 
22.50/6.83	
22.50/6.83	#### Partial ranking functions of CR start(V1,V) 
22.50/6.83	
22.50/6.83	
22.50/6.83	Computing Bounds
22.50/6.83	=====================================
22.50/6.83	
22.50/6.83	#### Cost of chains of minus(V1,V,Out):
22.50/6.83	* Chain [[7],9]: 1*it(7)+1
22.50/6.83	  Such that:it(7) =< V
22.50/6.83	
22.50/6.83	  with precondition: [V1=Out+V,V>=1,V1>=V] 
22.50/6.83	
22.50/6.83	* Chain [[7],8]: 1*it(7)+0
22.50/6.83	  Such that:it(7) =< V
22.50/6.83	
22.50/6.83	  with precondition: [Out=0,V1>=1,V>=1] 
22.50/6.83	
22.50/6.83	* Chain [9]: 1
22.50/6.83	  with precondition: [V=0,V1=Out,V1>=0] 
22.50/6.83	
22.50/6.83	* Chain [8]: 0
22.50/6.83	  with precondition: [Out=0,V1>=0,V>=0] 
22.50/6.83	
22.50/6.83	
22.50/6.83	#### Cost of chains of quot(V1,V,Out):
22.50/6.83	* Chain [[12],13]: 2*it(12)+1
22.50/6.83	  Such that:it(12) =< Out
22.50/6.83	
22.50/6.83	  with precondition: [V=1,Out>=1,V1>=Out] 
22.50/6.83	
22.50/6.83	* Chain [[12],11,13]: 2*it(12)+1*s(2)+2
22.50/6.83	  Such that:s(2) =< 1
22.50/6.83	it(12) =< Out
22.50/6.83	
22.50/6.83	  with precondition: [V=1,Out>=2,V1>=Out] 
22.50/6.83	
22.50/6.83	* Chain [[10],13]: 2*it(10)+1*s(5)+1
22.50/6.83	  Such that:it(10) =< V1-V+1
22.50/6.83	aux(3) =< V1
22.50/6.83	it(10) =< aux(3)
22.50/6.83	s(5) =< aux(3)
22.50/6.83	
22.50/6.83	  with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 
22.50/6.83	
22.50/6.83	* Chain [[10],11,13]: 2*it(10)+1*s(2)+1*s(5)+2
22.50/6.83	  Such that:it(10) =< V1-V+1
22.50/6.83	s(2) =< V
22.50/6.83	aux(4) =< V1
22.50/6.83	it(10) =< aux(4)
22.50/6.83	s(5) =< aux(4)
22.50/6.83	
22.50/6.83	  with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] 
22.50/6.83	
22.50/6.83	* Chain [13]: 1
22.50/6.83	  with precondition: [Out=0,V1>=0,V>=0] 
22.50/6.83	
22.50/6.83	* Chain [11,13]: 1*s(2)+2
22.50/6.83	  Such that:s(2) =< V
22.50/6.83	
22.50/6.83	  with precondition: [Out=1,V1>=1,V>=1] 
22.50/6.83	
22.50/6.83	
22.50/6.83	#### Cost of chains of start(V1,V):
22.50/6.83	* Chain [15]: 4*s(9)+4*s(12)+2*s(14)+2
22.50/6.83	  Such that:aux(6) =< V1
22.50/6.83	aux(7) =< V1-V+1
22.50/6.83	aux(8) =< V
22.50/6.83	s(12) =< aux(7)
22.50/6.83	s(9) =< aux(8)
22.50/6.83	s(12) =< aux(6)
22.50/6.83	s(14) =< aux(6)
22.50/6.83	
22.50/6.83	  with precondition: [V1>=0,V>=0] 
22.50/6.83	
22.50/6.83	* Chain [14]: 1*s(19)+4*s(21)+2
22.50/6.83	  Such that:s(19) =< 1
22.50/6.83	s(20) =< V1
22.50/6.83	s(21) =< s(20)
22.50/6.83	
22.50/6.83	  with precondition: [V=1,V1>=1] 
22.50/6.83	
22.50/6.83	
22.50/6.83	Closed-form bounds of start(V1,V): 
22.50/6.83	-------------------------------------
22.50/6.83	* Chain [15] with precondition: [V1>=0,V>=0] 
22.50/6.83	    - Upper bound: 2*V1+4*V+2+nat(V1-V+1)*4 
22.50/6.83	    - Complexity: n 
22.50/6.83	* Chain [14] with precondition: [V=1,V1>=1] 
22.50/6.83	    - Upper bound: 4*V1+3 
22.50/6.83	    - Complexity: n 
22.50/6.83	
22.50/6.83	### Maximum cost of start(V1,V): 2*V1+2+max([2*V1+1,nat(V1-V+1)*4+4*V]) 
22.50/6.83	Asymptotic class: n 
22.50/6.83	* Total analysis performed in 165 ms.
22.50/6.83	
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(12)
22.50/6.83	BOUNDS(1, n^1)
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(13) RenamingProof (BOTH BOUNDS(ID, ID))
22.50/6.83	Renamed function symbols to avoid clashes with predefined symbol.
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(14)
22.50/6.83	Obligation:
22.50/6.83	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
22.50/6.83	
22.50/6.83	
22.50/6.83	The TRS R consists of the following rules:
22.50/6.83	
22.50/6.83	   minus(x, 0') -> x
22.50/6.83	   minus(s(x), s(y)) -> minus(x, y)
22.50/6.83	   quot(0', s(y)) -> 0'
22.50/6.83	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.83	
22.50/6.83	S is empty.
22.50/6.83	Rewrite Strategy: FULL
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.50/6.83	Infered types.
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(16)
22.50/6.83	Obligation:
22.50/6.83	TRS:
22.50/6.83	Rules:
22.50/6.83	minus(x, 0') -> x
22.50/6.83	minus(s(x), s(y)) -> minus(x, y)
22.50/6.83	quot(0', s(y)) -> 0'
22.50/6.83	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.83	
22.50/6.83	Types:
22.50/6.83	minus :: 0':s -> 0':s -> 0':s
22.50/6.83	0' :: 0':s
22.50/6.83	s :: 0':s -> 0':s
22.50/6.83	quot :: 0':s -> 0':s -> 0':s
22.50/6.83	hole_0':s1_0 :: 0':s
22.50/6.83	gen_0':s2_0 :: Nat -> 0':s
22.50/6.83	
22.50/6.83	----------------------------------------
22.50/6.83	
22.50/6.83	(17) OrderProof (LOWER BOUND(ID))
22.50/6.83	Heuristically decided to analyse the following defined symbols:
22.50/6.83	minus, quot
22.50/6.84	
22.50/6.84	They will be analysed ascendingly in the following order:
22.50/6.84	minus < quot
22.50/6.84	
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(18)
22.50/6.84	Obligation:
22.50/6.84	TRS:
22.50/6.84	Rules:
22.50/6.84	minus(x, 0') -> x
22.50/6.84	minus(s(x), s(y)) -> minus(x, y)
22.50/6.84	quot(0', s(y)) -> 0'
22.50/6.84	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.84	
22.50/6.84	Types:
22.50/6.84	minus :: 0':s -> 0':s -> 0':s
22.50/6.84	0' :: 0':s
22.50/6.84	s :: 0':s -> 0':s
22.50/6.84	quot :: 0':s -> 0':s -> 0':s
22.50/6.84	hole_0':s1_0 :: 0':s
22.50/6.84	gen_0':s2_0 :: Nat -> 0':s
22.50/6.84	
22.50/6.84	
22.50/6.84	Generator Equations:
22.50/6.84	gen_0':s2_0(0) <=> 0'
22.50/6.84	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
22.50/6.84	
22.50/6.84	
22.50/6.84	The following defined symbols remain to be analysed:
22.50/6.84	minus, quot
22.50/6.84	
22.50/6.84	They will be analysed ascendingly in the following order:
22.50/6.84	minus < quot
22.50/6.84	
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(19) RewriteLemmaProof (LOWER BOUND(ID))
22.50/6.84	Proved the following rewrite lemma:
22.50/6.84	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
22.50/6.84	
22.50/6.84	Induction Base:
22.50/6.84	minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
22.50/6.84	gen_0':s2_0(0)
22.50/6.84	
22.50/6.84	Induction Step:
22.50/6.84	minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
22.50/6.84	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
22.50/6.84	gen_0':s2_0(0)
22.50/6.84	
22.50/6.84	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(20)
22.50/6.84	Complex Obligation (BEST)
22.50/6.84	
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(21)
22.50/6.84	Obligation:
22.50/6.84	Proved the lower bound n^1 for the following obligation:
22.50/6.84	
22.50/6.84	TRS:
22.50/6.84	Rules:
22.50/6.84	minus(x, 0') -> x
22.50/6.84	minus(s(x), s(y)) -> minus(x, y)
22.50/6.84	quot(0', s(y)) -> 0'
22.50/6.84	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.84	
22.50/6.84	Types:
22.50/6.84	minus :: 0':s -> 0':s -> 0':s
22.50/6.84	0' :: 0':s
22.50/6.84	s :: 0':s -> 0':s
22.50/6.84	quot :: 0':s -> 0':s -> 0':s
22.50/6.84	hole_0':s1_0 :: 0':s
22.50/6.84	gen_0':s2_0 :: Nat -> 0':s
22.50/6.84	
22.50/6.84	
22.50/6.84	Generator Equations:
22.50/6.84	gen_0':s2_0(0) <=> 0'
22.50/6.84	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
22.50/6.84	
22.50/6.84	
22.50/6.84	The following defined symbols remain to be analysed:
22.50/6.84	minus, quot
22.50/6.84	
22.50/6.84	They will be analysed ascendingly in the following order:
22.50/6.84	minus < quot
22.50/6.84	
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(22) LowerBoundPropagationProof (FINISHED)
22.50/6.84	Propagated lower bound.
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(23)
22.50/6.84	BOUNDS(n^1, INF)
22.50/6.84	
22.50/6.84	----------------------------------------
22.50/6.84	
22.50/6.84	(24)
22.50/6.84	Obligation:
22.50/6.84	TRS:
22.50/6.84	Rules:
22.50/6.84	minus(x, 0') -> x
22.50/6.84	minus(s(x), s(y)) -> minus(x, y)
22.50/6.84	quot(0', s(y)) -> 0'
22.50/6.84	quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
22.50/6.84	
22.50/6.84	Types:
22.50/6.84	minus :: 0':s -> 0':s -> 0':s
22.50/6.84	0' :: 0':s
22.50/6.84	s :: 0':s -> 0':s
22.50/6.84	quot :: 0':s -> 0':s -> 0':s
22.50/6.84	hole_0':s1_0 :: 0':s
22.50/6.84	gen_0':s2_0 :: Nat -> 0':s
22.50/6.84	
22.50/6.84	
22.50/6.84	Lemmas:
22.50/6.84	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
22.50/6.84	
22.50/6.84	
22.50/6.84	Generator Equations:
22.50/6.84	gen_0':s2_0(0) <=> 0'
22.50/6.84	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
22.50/6.84	
22.50/6.84	
22.50/6.84	The following defined symbols remain to be analysed:
22.50/6.84	quot
22.50/6.87	EOF