459.77/291.47	WORST_CASE(Omega(n^1), O(n^2))
459.81/291.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
459.81/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
459.81/291.48	
459.81/291.48	
459.81/291.48	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
459.81/291.48	
459.81/291.48	(0) CpxTRS
459.81/291.48	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
459.81/291.48	(2) CpxWeightedTrs
459.81/291.48	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
459.81/291.48	    (4) CpxTypedWeightedTrs
459.81/291.48	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
459.81/291.48	    (6) CpxTypedWeightedCompleteTrs
459.81/291.48	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
459.81/291.48	    (8) CpxRNTS
459.81/291.48	    (9) CompleteCoflocoProof [FINISHED, 289 ms]
459.81/291.48	    (10) BOUNDS(1, n^2)
459.81/291.48	(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
459.81/291.48	(12) CpxTRS
459.81/291.48	    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
459.81/291.48	    (14) typed CpxTrs
459.81/291.48	    (15) OrderProof [LOWER BOUND(ID), 0 ms]
459.81/291.48	    (16) typed CpxTrs
459.81/291.48	    (17) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
459.81/291.48	    (18) BEST
459.81/291.48	        (19) proven lower bound
459.81/291.48	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
459.81/291.48	            (21) BOUNDS(n^1, INF)
459.81/291.48	        (22) typed CpxTrs
459.81/291.48	
459.81/291.48	
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(0)
459.81/291.48	Obligation:
459.81/291.48	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
459.81/291.48	
459.81/291.48	
459.81/291.48	The TRS R consists of the following rules:
459.81/291.48	
459.81/291.48	   cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.48	   gr(0, x) -> false
459.81/291.48	   gr(s(x), 0) -> true
459.81/291.48	   gr(s(x), s(y)) -> gr(x, y)
459.81/291.48	   p(0) -> 0
459.81/291.48	   p(s(x)) -> x
459.81/291.48	
459.81/291.48	S is empty.
459.81/291.48	Rewrite Strategy: INNERMOST
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
459.81/291.48	Transformed relative TRS to weighted TRS
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(2)
459.81/291.48	Obligation:
459.81/291.48	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
459.81/291.48	
459.81/291.48	
459.81/291.48	The TRS R consists of the following rules:
459.81/291.48	
459.81/291.48	   cond(true, x, y) -> cond(gr(x, y), p(x), y) [1]
459.81/291.48	   gr(0, x) -> false [1]
459.81/291.48	   gr(s(x), 0) -> true [1]
459.81/291.48	   gr(s(x), s(y)) -> gr(x, y) [1]
459.81/291.48	   p(0) -> 0 [1]
459.81/291.48	   p(s(x)) -> x [1]
459.81/291.48	
459.81/291.48	Rewrite Strategy: INNERMOST
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
459.81/291.48	Infered types.
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(4)
459.81/291.48	Obligation:
459.81/291.48	Runtime Complexity Weighted TRS with Types.
459.81/291.48	The TRS R consists of the following rules:
459.81/291.48	
459.81/291.48	   cond(true, x, y) -> cond(gr(x, y), p(x), y) [1]
459.81/291.48	   gr(0, x) -> false [1]
459.81/291.48	   gr(s(x), 0) -> true [1]
459.81/291.48	   gr(s(x), s(y)) -> gr(x, y) [1]
459.81/291.48	   p(0) -> 0 [1]
459.81/291.48	   p(s(x)) -> x [1]
459.81/291.48	
459.81/291.48	The TRS has the following type information:
459.81/291.48	cond :: true:false -> 0:s -> 0:s -> cond
459.81/291.48	true :: true:false
459.81/291.48	gr :: 0:s -> 0:s -> true:false
459.81/291.48	p :: 0:s -> 0:s
459.81/291.48	0 :: 0:s
459.81/291.48	false :: true:false
459.81/291.48	s :: 0:s -> 0:s
459.81/291.48	
459.81/291.48	Rewrite Strategy: INNERMOST
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(5) CompletionProof (UPPER BOUND(ID))
459.81/291.48	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
459.81/291.48	
459.81/291.48	   cond(v0, v1, v2) -> null_cond [0]
459.81/291.48	
459.81/291.48	And the following fresh constants:    null_cond
459.81/291.48	
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(6)
459.81/291.48	Obligation:
459.81/291.48	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
459.81/291.48	
459.81/291.48	Runtime Complexity Weighted TRS with Types.
459.81/291.48	The TRS R consists of the following rules:
459.81/291.48	
459.81/291.48	   cond(true, x, y) -> cond(gr(x, y), p(x), y) [1]
459.81/291.48	   gr(0, x) -> false [1]
459.81/291.48	   gr(s(x), 0) -> true [1]
459.81/291.48	   gr(s(x), s(y)) -> gr(x, y) [1]
459.81/291.48	   p(0) -> 0 [1]
459.81/291.48	   p(s(x)) -> x [1]
459.81/291.48	   cond(v0, v1, v2) -> null_cond [0]
459.81/291.48	
459.81/291.48	The TRS has the following type information:
459.81/291.48	cond :: true:false -> 0:s -> 0:s -> null_cond
459.81/291.48	true :: true:false
459.81/291.48	gr :: 0:s -> 0:s -> true:false
459.81/291.48	p :: 0:s -> 0:s
459.81/291.48	0 :: 0:s
459.81/291.48	false :: true:false
459.81/291.48	s :: 0:s -> 0:s
459.81/291.48	null_cond :: null_cond
459.81/291.48	
459.81/291.48	Rewrite Strategy: INNERMOST
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
459.81/291.48	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
459.81/291.48	The constant constructors are abstracted as follows:
459.81/291.48	
459.81/291.48	   true => 1
459.81/291.48	   0 => 0
459.81/291.48	   false => 0
459.81/291.48	   null_cond => 0
459.81/291.48	
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(8)
459.81/291.48	Obligation:
459.81/291.48	Complexity RNTS consisting of the following rules:
459.81/291.48	
459.81/291.48	   cond(z, z', z'') -{ 1 }-> cond(gr(x, y), p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
459.81/291.48	   cond(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
459.81/291.48	   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
459.81/291.48	   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
459.81/291.48	   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
459.81/291.48	   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
459.81/291.48	   p(z) -{ 1 }-> 0 :|: z = 0
459.81/291.48	
459.81/291.48	Only complete derivations are relevant for the runtime complexity.
459.81/291.48	
459.81/291.48	----------------------------------------
459.81/291.48	
459.81/291.48	(9) CompleteCoflocoProof (FINISHED)
459.81/291.48	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
459.81/291.48	
459.81/291.48	eq(start(V1, V, V2),0,[cond(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
459.81/291.48	eq(start(V1, V, V2),0,[gr(V1, V, Out)],[V1 >= 0,V >= 0]).
459.81/291.48	eq(start(V1, V, V2),0,[p(V1, Out)],[V1 >= 0]).
459.81/291.48	eq(cond(V1, V, V2, Out),1,[gr(V4, V3, Ret0),p(V4, Ret1),cond(Ret0, Ret1, V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
459.81/291.48	eq(gr(V1, V, Out),1,[],[Out = 0,V = V5,V5 >= 0,V1 = 0]).
459.81/291.49	eq(gr(V1, V, Out),1,[],[Out = 1,V6 >= 0,V1 = 1 + V6,V = 0]).
459.81/291.49	eq(gr(V1, V, Out),1,[gr(V7, V8, Ret2)],[Out = Ret2,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
459.81/291.49	eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
459.81/291.49	eq(p(V1, Out),1,[],[Out = V9,V9 >= 0,V1 = 1 + V9]).
459.81/291.49	eq(cond(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]).
459.81/291.49	input_output_vars(cond(V1,V,V2,Out),[V1,V,V2],[Out]).
459.81/291.49	input_output_vars(gr(V1,V,Out),[V1,V],[Out]).
459.81/291.49	input_output_vars(p(V1,Out),[V1],[Out]).
459.81/291.49	
459.81/291.49	
459.81/291.49	CoFloCo proof output:
459.81/291.49	Preprocessing Cost Relations
459.81/291.49	=====================================
459.81/291.49	
459.81/291.49	#### Computed strongly connected components 
459.81/291.49	0. recursive  : [gr/3]
459.81/291.49	1. non_recursive  : [p/2]
459.81/291.49	2. recursive  : [cond/4]
459.81/291.49	3. non_recursive  : [start/3]
459.81/291.49	
459.81/291.49	#### Obtained direct recursion through partial evaluation 
459.81/291.49	0. SCC is partially evaluated into gr/3
459.81/291.49	1. SCC is partially evaluated into p/2
459.81/291.49	2. SCC is partially evaluated into cond/4
459.81/291.49	3. SCC is partially evaluated into start/3
459.81/291.49	
459.81/291.49	Control-Flow Refinement of Cost Relations
459.81/291.49	=====================================
459.81/291.49	
459.81/291.49	### Specialization of cost equations gr/3 
459.81/291.49	* CE 8 is refined into CE [11] 
459.81/291.49	* CE 7 is refined into CE [12] 
459.81/291.49	* CE 6 is refined into CE [13] 
459.81/291.49	
459.81/291.49	
459.81/291.49	### Cost equations --> "Loop" of gr/3 
459.81/291.49	* CEs [12] --> Loop 9 
459.81/291.49	* CEs [13] --> Loop 10 
459.81/291.49	* CEs [11] --> Loop 11 
459.81/291.49	
459.81/291.49	### Ranking functions of CR gr(V1,V,Out) 
459.81/291.49	* RF of phase [11]: [V,V1]
459.81/291.49	
459.81/291.49	#### Partial ranking functions of CR gr(V1,V,Out) 
459.81/291.49	* Partial RF of phase [11]:
459.81/291.49	  - RF of loop [11:1]:
459.81/291.49	    V
459.81/291.49	    V1
459.81/291.49	
459.81/291.49	
459.81/291.49	### Specialization of cost equations p/2 
459.81/291.49	* CE 10 is refined into CE [14] 
459.81/291.49	* CE 9 is refined into CE [15] 
459.81/291.49	
459.81/291.49	
459.81/291.49	### Cost equations --> "Loop" of p/2 
459.81/291.49	* CEs [14] --> Loop 12 
459.81/291.49	* CEs [15] --> Loop 13 
459.81/291.49	
459.81/291.49	### Ranking functions of CR p(V1,Out) 
459.81/291.49	
459.81/291.49	#### Partial ranking functions of CR p(V1,Out) 
459.81/291.49	
459.81/291.49	
459.81/291.49	### Specialization of cost equations cond/4 
459.81/291.49	* CE 5 is refined into CE [16] 
459.81/291.49	* CE 4 is refined into CE [17,18,19,20] 
459.81/291.49	
459.81/291.49	
459.81/291.49	### Cost equations --> "Loop" of cond/4 
459.81/291.49	* CEs [20] --> Loop 14 
459.81/291.49	* CEs [19] --> Loop 15 
459.81/291.49	* CEs [18] --> Loop 16 
459.81/291.49	* CEs [17] --> Loop 17 
459.81/291.49	* CEs [16] --> Loop 18 
459.81/291.49	
459.81/291.49	### Ranking functions of CR cond(V1,V,V2,Out) 
459.81/291.49	* RF of phase [14]: [V-1,V-V2]
459.81/291.49	* RF of phase [16]: [V]
459.81/291.49	
459.81/291.49	#### Partial ranking functions of CR cond(V1,V,V2,Out) 
459.81/291.49	* Partial RF of phase [14]:
459.81/291.49	  - RF of loop [14:1]:
459.81/291.49	    V-1
459.81/291.49	    V-V2
459.81/291.49	* Partial RF of phase [16]:
459.81/291.49	  - RF of loop [16:1]:
459.81/291.49	    V
459.81/291.49	
459.81/291.49	
459.81/291.49	### Specialization of cost equations start/3 
459.81/291.49	* CE 1 is refined into CE [21,22,23,24] 
459.81/291.49	* CE 2 is refined into CE [25,26,27,28] 
459.81/291.49	* CE 3 is refined into CE [29,30] 
459.81/291.49	
459.81/291.49	
459.81/291.49	### Cost equations --> "Loop" of start/3 
459.81/291.49	* CEs [26] --> Loop 19 
459.81/291.49	* CEs [24] --> Loop 20 
459.81/291.49	* CEs [23,27,28,30] --> Loop 21 
459.81/291.49	* CEs [21,22] --> Loop 22 
459.81/291.49	* CEs [25,29] --> Loop 23 
459.81/291.49	
459.81/291.49	### Ranking functions of CR start(V1,V,V2) 
459.81/291.49	
459.81/291.49	#### Partial ranking functions of CR start(V1,V,V2) 
459.81/291.49	
459.81/291.49	
459.81/291.49	Computing Bounds
459.81/291.49	=====================================
459.81/291.49	
459.81/291.49	#### Cost of chains of gr(V1,V,Out):
459.81/291.49	* Chain [[11],10]: 1*it(11)+1
459.81/291.49	  Such that:it(11) =< V1
459.81/291.49	
459.81/291.49	  with precondition: [Out=0,V1>=1,V>=V1] 
459.81/291.49	
459.81/291.49	* Chain [[11],9]: 1*it(11)+1
459.81/291.49	  Such that:it(11) =< V
459.81/291.49	
459.81/291.49	  with precondition: [Out=1,V>=1,V1>=V+1] 
459.81/291.49	
459.81/291.49	* Chain [10]: 1
459.81/291.49	  with precondition: [V1=0,Out=0,V>=0] 
459.81/291.49	
459.81/291.49	* Chain [9]: 1
459.81/291.49	  with precondition: [V=0,Out=1,V1>=1] 
459.81/291.49	
459.81/291.49	
459.81/291.49	#### Cost of chains of p(V1,Out):
459.81/291.49	* Chain [13]: 1
459.81/291.49	  with precondition: [V1=0,Out=0] 
459.81/291.49	
459.81/291.49	* Chain [12]: 1
459.81/291.49	  with precondition: [V1=Out+1,V1>=1] 
459.81/291.49	
459.81/291.49	
459.81/291.49	#### Cost of chains of cond(V1,V,V2,Out):
459.81/291.49	* Chain [[16],18]: 3*it(16)+0
459.81/291.49	  Such that:it(16) =< V
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,V2=0,Out=0,V>=1] 
459.81/291.49	
459.81/291.49	* Chain [[16],17,18]: 3*it(16)+3
459.81/291.49	  Such that:it(16) =< V
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,V2=0,Out=0,V>=1] 
459.81/291.49	
459.81/291.49	* Chain [[14],18]: 3*it(14)+1*s(3)+0
459.81/291.49	  Such that:it(14) =< V-V2
459.81/291.49	aux(1) =< V2
459.81/291.49	s(3) =< it(14)*aux(1)
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
459.81/291.49	
459.81/291.49	* Chain [[14],15,18]: 3*it(14)+1*s(3)+1*s(4)+3
459.81/291.49	  Such that:it(14) =< V-V2
459.81/291.49	aux(2) =< V2
459.81/291.49	s(4) =< aux(2)
459.81/291.49	s(3) =< it(14)*aux(2)
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 
459.81/291.49	
459.81/291.49	* Chain [18]: 0
459.81/291.49	  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 
459.81/291.49	
459.81/291.49	* Chain [17,18]: 3
459.81/291.49	  with precondition: [V1=1,V=0,Out=0,V2>=0] 
459.81/291.49	
459.81/291.49	* Chain [15,18]: 1*s(4)+3
459.81/291.49	  Such that:s(4) =< V
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,Out=0,V>=1,V2>=V] 
459.81/291.49	
459.81/291.49	
459.81/291.49	#### Cost of chains of start(V1,V,V2):
459.81/291.49	* Chain [23]: 1
459.81/291.49	  with precondition: [V1=0] 
459.81/291.49	
459.81/291.49	* Chain [22]: 6*s(15)+3
459.81/291.49	  Such that:s(14) =< V
459.81/291.49	s(15) =< s(14)
459.81/291.49	
459.81/291.49	  with precondition: [V1>=0,V>=0,V2>=0] 
459.81/291.49	
459.81/291.49	* Chain [21]: 2*s(16)+1*s(17)+3
459.81/291.49	  Such that:s(17) =< V1
459.81/291.49	aux(6) =< V
459.81/291.49	s(16) =< aux(6)
459.81/291.49	
459.81/291.49	  with precondition: [V1>=1] 
459.81/291.49	
459.81/291.49	* Chain [20]: 6*s(21)+1*s(22)+2*s(23)+3
459.81/291.49	  Such that:s(19) =< V-V2
459.81/291.49	s(20) =< V2
459.81/291.49	s(21) =< s(19)
459.81/291.49	s(22) =< s(20)
459.81/291.49	s(23) =< s(21)*s(20)
459.81/291.49	
459.81/291.49	  with precondition: [V1=1,V2>=1,V>=V2+1] 
459.81/291.49	
459.81/291.49	* Chain [19]: 1
459.81/291.49	  with precondition: [V=0,V1>=1] 
459.81/291.49	
459.81/291.49	
459.81/291.49	Closed-form bounds of start(V1,V,V2): 
459.81/291.49	-------------------------------------
459.81/291.49	* Chain [23] with precondition: [V1=0] 
459.81/291.49	    - Upper bound: 1 
459.81/291.49	    - Complexity: constant 
459.81/291.49	* Chain [22] with precondition: [V1>=0,V>=0,V2>=0] 
459.81/291.49	    - Upper bound: 6*V+3 
459.81/291.49	    - Complexity: n 
459.81/291.49	* Chain [21] with precondition: [V1>=1] 
459.81/291.49	    - Upper bound: V1+3+nat(V)*2 
459.81/291.49	    - Complexity: n 
459.81/291.49	* Chain [20] with precondition: [V1=1,V2>=1,V>=V2+1] 
459.81/291.49	    - Upper bound: 6*V-6*V2+(V2+3+(V-V2)*(2*V2)) 
459.81/291.49	    - Complexity: n^2 
459.81/291.49	* Chain [19] with precondition: [V=0,V1>=1] 
459.81/291.49	    - Upper bound: 1 
459.81/291.49	    - Complexity: constant 
459.81/291.49	
459.81/291.49	### Maximum cost of start(V1,V,V2): max([nat(V)*2+2+max([V1,nat(V)*4]),nat(V2)+2+nat(V2)*2*nat(V-V2)+nat(V-V2)*6])+1 
459.81/291.49	Asymptotic class: n^2 
459.81/291.49	* Total analysis performed in 207 ms.
459.81/291.49	
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(10)
459.81/291.49	BOUNDS(1, n^2)
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(11) RenamingProof (BOTH BOUNDS(ID, ID))
459.81/291.49	Renamed function symbols to avoid clashes with predefined symbol.
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(12)
459.81/291.49	Obligation:
459.81/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
459.81/291.49	
459.81/291.49	
459.81/291.49	The TRS R consists of the following rules:
459.81/291.49	
459.81/291.49	   cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.49	   gr(0', x) -> false
459.81/291.49	   gr(s(x), 0') -> true
459.81/291.49	   gr(s(x), s(y)) -> gr(x, y)
459.81/291.49	   p(0') -> 0'
459.81/291.49	   p(s(x)) -> x
459.81/291.49	
459.81/291.49	S is empty.
459.81/291.49	Rewrite Strategy: INNERMOST
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
459.81/291.49	Infered types.
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(14)
459.81/291.49	Obligation:
459.81/291.49	Innermost TRS:
459.81/291.49	Rules:
459.81/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.49	gr(0', x) -> false
459.81/291.49	gr(s(x), 0') -> true
459.81/291.49	gr(s(x), s(y)) -> gr(x, y)
459.81/291.49	p(0') -> 0'
459.81/291.49	p(s(x)) -> x
459.81/291.49	
459.81/291.49	Types:
459.81/291.49	cond :: true:false -> 0':s -> 0':s -> cond
459.81/291.49	true :: true:false
459.81/291.49	gr :: 0':s -> 0':s -> true:false
459.81/291.49	p :: 0':s -> 0':s
459.81/291.49	0' :: 0':s
459.81/291.49	false :: true:false
459.81/291.49	s :: 0':s -> 0':s
459.81/291.49	hole_cond1_0 :: cond
459.81/291.49	hole_true:false2_0 :: true:false
459.81/291.49	hole_0':s3_0 :: 0':s
459.81/291.49	gen_0':s4_0 :: Nat -> 0':s
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(15) OrderProof (LOWER BOUND(ID))
459.81/291.49	Heuristically decided to analyse the following defined symbols:
459.81/291.49	cond, gr
459.81/291.49	
459.81/291.49	They will be analysed ascendingly in the following order:
459.81/291.49	gr < cond
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(16)
459.81/291.49	Obligation:
459.81/291.49	Innermost TRS:
459.81/291.49	Rules:
459.81/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.49	gr(0', x) -> false
459.81/291.49	gr(s(x), 0') -> true
459.81/291.49	gr(s(x), s(y)) -> gr(x, y)
459.81/291.49	p(0') -> 0'
459.81/291.49	p(s(x)) -> x
459.81/291.49	
459.81/291.49	Types:
459.81/291.49	cond :: true:false -> 0':s -> 0':s -> cond
459.81/291.49	true :: true:false
459.81/291.49	gr :: 0':s -> 0':s -> true:false
459.81/291.49	p :: 0':s -> 0':s
459.81/291.49	0' :: 0':s
459.81/291.49	false :: true:false
459.81/291.49	s :: 0':s -> 0':s
459.81/291.49	hole_cond1_0 :: cond
459.81/291.49	hole_true:false2_0 :: true:false
459.81/291.49	hole_0':s3_0 :: 0':s
459.81/291.49	gen_0':s4_0 :: Nat -> 0':s
459.81/291.49	
459.81/291.49	
459.81/291.49	Generator Equations:
459.81/291.49	gen_0':s4_0(0) <=> 0'
459.81/291.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
459.81/291.49	
459.81/291.49	
459.81/291.49	The following defined symbols remain to be analysed:
459.81/291.49	gr, cond
459.81/291.49	
459.81/291.49	They will be analysed ascendingly in the following order:
459.81/291.49	gr < cond
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(17) RewriteLemmaProof (LOWER BOUND(ID))
459.81/291.49	Proved the following rewrite lemma:
459.81/291.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
459.81/291.49	
459.81/291.49	Induction Base:
459.81/291.49	gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
459.81/291.49	false
459.81/291.49	
459.81/291.49	Induction Step:
459.81/291.49	gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
459.81/291.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
459.81/291.49	false
459.81/291.49	
459.81/291.49	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(18)
459.81/291.49	Complex Obligation (BEST)
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(19)
459.81/291.49	Obligation:
459.81/291.49	Proved the lower bound n^1 for the following obligation:
459.81/291.49	
459.81/291.49	Innermost TRS:
459.81/291.49	Rules:
459.81/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.49	gr(0', x) -> false
459.81/291.49	gr(s(x), 0') -> true
459.81/291.49	gr(s(x), s(y)) -> gr(x, y)
459.81/291.49	p(0') -> 0'
459.81/291.49	p(s(x)) -> x
459.81/291.49	
459.81/291.49	Types:
459.81/291.49	cond :: true:false -> 0':s -> 0':s -> cond
459.81/291.49	true :: true:false
459.81/291.49	gr :: 0':s -> 0':s -> true:false
459.81/291.49	p :: 0':s -> 0':s
459.81/291.49	0' :: 0':s
459.81/291.49	false :: true:false
459.81/291.49	s :: 0':s -> 0':s
459.81/291.49	hole_cond1_0 :: cond
459.81/291.49	hole_true:false2_0 :: true:false
459.81/291.49	hole_0':s3_0 :: 0':s
459.81/291.49	gen_0':s4_0 :: Nat -> 0':s
459.81/291.49	
459.81/291.49	
459.81/291.49	Generator Equations:
459.81/291.49	gen_0':s4_0(0) <=> 0'
459.81/291.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
459.81/291.49	
459.81/291.49	
459.81/291.49	The following defined symbols remain to be analysed:
459.81/291.49	gr, cond
459.81/291.49	
459.81/291.49	They will be analysed ascendingly in the following order:
459.81/291.49	gr < cond
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(20) LowerBoundPropagationProof (FINISHED)
459.81/291.49	Propagated lower bound.
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(21)
459.81/291.49	BOUNDS(n^1, INF)
459.81/291.49	
459.81/291.49	----------------------------------------
459.81/291.49	
459.81/291.49	(22)
459.81/291.49	Obligation:
459.81/291.49	Innermost TRS:
459.81/291.49	Rules:
459.81/291.49	cond(true, x, y) -> cond(gr(x, y), p(x), y)
459.81/291.49	gr(0', x) -> false
459.81/291.49	gr(s(x), 0') -> true
459.81/291.49	gr(s(x), s(y)) -> gr(x, y)
459.81/291.49	p(0') -> 0'
459.81/291.49	p(s(x)) -> x
459.81/291.49	
459.81/291.49	Types:
459.81/291.49	cond :: true:false -> 0':s -> 0':s -> cond
459.81/291.49	true :: true:false
459.81/291.49	gr :: 0':s -> 0':s -> true:false
459.81/291.49	p :: 0':s -> 0':s
459.81/291.49	0' :: 0':s
459.81/291.49	false :: true:false
459.81/291.49	s :: 0':s -> 0':s
459.81/291.49	hole_cond1_0 :: cond
459.81/291.49	hole_true:false2_0 :: true:false
459.81/291.49	hole_0':s3_0 :: 0':s
459.81/291.49	gen_0':s4_0 :: Nat -> 0':s
459.81/291.49	
459.81/291.49	
459.81/291.49	Lemmas:
459.81/291.49	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
459.81/291.49	
459.81/291.49	
459.81/291.49	Generator Equations:
459.81/291.49	gen_0':s4_0(0) <=> 0'
459.81/291.49	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
459.81/291.49	
459.81/291.49	
459.81/291.49	The following defined symbols remain to be analysed:
459.81/291.49	cond
459.81/291.53	EOF