23.71/9.02	WORST_CASE(Omega(n^1), O(n^1))
23.71/9.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.71/9.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.71/9.03	
23.71/9.03	
23.71/9.03	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.71/9.03	
23.71/9.03	(0) CpxTRS
23.71/9.03	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	(2) CpxTRS
23.71/9.03	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	    (4) CpxWeightedTrs
23.71/9.03	    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	    (6) CpxWeightedTrs
23.71/9.03	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	    (8) CpxTypedWeightedTrs
23.71/9.03	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
23.71/9.03	    (10) CpxTypedWeightedCompleteTrs
23.71/9.03	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
23.71/9.03	    (12) CpxRNTS
23.71/9.03	    (13) CompleteCoflocoProof [FINISHED, 360 ms]
23.71/9.03	    (14) BOUNDS(1, n^1)
23.71/9.03	(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	(16) CpxTRS
23.71/9.03	    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
23.71/9.03	    (18) typed CpxTrs
23.71/9.03	    (19) OrderProof [LOWER BOUND(ID), 0 ms]
23.71/9.03	    (20) typed CpxTrs
23.71/9.03	    (21) RewriteLemmaProof [LOWER BOUND(ID), 329 ms]
23.71/9.03	    (22) BEST
23.71/9.03	        (23) proven lower bound
23.71/9.03	            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
23.71/9.03	            (25) BOUNDS(n^1, INF)
23.71/9.03	        (26) typed CpxTrs
23.71/9.03	            (27) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
23.71/9.03	            (28) typed CpxTrs
23.71/9.03	
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(0)
23.71/9.03	Obligation:
23.71/9.03	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
23.71/9.03	
23.71/9.03	
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   -(x, 0) -> x
23.71/9.03	   -(0, s(y)) -> 0
23.71/9.03	   -(s(x), s(y)) -> -(x, y)
23.71/9.03	   lt(x, 0) -> false
23.71/9.03	   lt(0, s(y)) -> true
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	   if(true, x, y) -> x
23.71/9.03	   if(false, x, y) -> y
23.71/9.03	   div(x, 0) -> 0
23.71/9.03	   div(0, y) -> 0
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	S is empty.
23.71/9.03	Rewrite Strategy: FULL
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Converted rc-obligation to irc-obligation.
23.71/9.03	
23.71/9.03	The duplicating contexts are:
23.71/9.03	div(s([]), s(y))
23.71/9.03	div(s(x), s([]))
23.71/9.03	
23.71/9.03	
23.71/9.03	The defined contexts are:
23.71/9.03	if([], 0, s(x1))
23.71/9.03	if(x0, 0, s([]))
23.71/9.03	div([], s(x1))
23.71/9.03	lt([], x1)
23.71/9.03	-([], x1)
23.71/9.03	
23.71/9.03	
23.71/9.03	[] just represents basic- or constructor-terms in the following defined contexts:
23.71/9.03	if([], 0, s(x1))
23.71/9.03	div([], s(x1))
23.71/9.03	
23.71/9.03	
23.71/9.03	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(2)
23.71/9.03	Obligation:
23.71/9.03	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
23.71/9.03	
23.71/9.03	
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   -(x, 0) -> x
23.71/9.03	   -(0, s(y)) -> 0
23.71/9.03	   -(s(x), s(y)) -> -(x, y)
23.71/9.03	   lt(x, 0) -> false
23.71/9.03	   lt(0, s(y)) -> true
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	   if(true, x, y) -> x
23.71/9.03	   if(false, x, y) -> y
23.71/9.03	   div(x, 0) -> 0
23.71/9.03	   div(0, y) -> 0
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	S is empty.
23.71/9.03	Rewrite Strategy: INNERMOST
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Transformed relative TRS to weighted TRS
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(4)
23.71/9.03	Obligation:
23.71/9.03	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
23.71/9.03	
23.71/9.03	
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   -(x, 0) -> x [1]
23.71/9.03	   -(0, s(y)) -> 0 [1]
23.71/9.03	   -(s(x), s(y)) -> -(x, y) [1]
23.71/9.03	   lt(x, 0) -> false [1]
23.71/9.03	   lt(0, s(y)) -> true [1]
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y) [1]
23.71/9.03	   if(true, x, y) -> x [1]
23.71/9.03	   if(false, x, y) -> y [1]
23.71/9.03	   div(x, 0) -> 0 [1]
23.71/9.03	   div(0, y) -> 0 [1]
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y)))) [1]
23.71/9.03	
23.71/9.03	Rewrite Strategy: INNERMOST
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Renamed defined symbols to avoid conflicts with arithmetic symbols:
23.71/9.03	
23.71/9.03	- => minus
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(6)
23.71/9.03	Obligation:
23.71/9.03	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
23.71/9.03	
23.71/9.03	
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   minus(x, 0) -> x [1]
23.71/9.03	   minus(0, s(y)) -> 0 [1]
23.71/9.03	   minus(s(x), s(y)) -> minus(x, y) [1]
23.71/9.03	   lt(x, 0) -> false [1]
23.71/9.03	   lt(0, s(y)) -> true [1]
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y) [1]
23.71/9.03	   if(true, x, y) -> x [1]
23.71/9.03	   if(false, x, y) -> y [1]
23.71/9.03	   div(x, 0) -> 0 [1]
23.71/9.03	   div(0, y) -> 0 [1]
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]
23.71/9.03	
23.71/9.03	Rewrite Strategy: INNERMOST
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Infered types.
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(8)
23.71/9.03	Obligation:
23.71/9.03	Runtime Complexity Weighted TRS with Types.
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   minus(x, 0) -> x [1]
23.71/9.03	   minus(0, s(y)) -> 0 [1]
23.71/9.03	   minus(s(x), s(y)) -> minus(x, y) [1]
23.71/9.03	   lt(x, 0) -> false [1]
23.71/9.03	   lt(0, s(y)) -> true [1]
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y) [1]
23.71/9.03	   if(true, x, y) -> x [1]
23.71/9.03	   if(false, x, y) -> y [1]
23.71/9.03	   div(x, 0) -> 0 [1]
23.71/9.03	   div(0, y) -> 0 [1]
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]
23.71/9.03	
23.71/9.03	The TRS has the following type information:
23.71/9.03	minus :: 0:s -> 0:s -> 0:s
23.71/9.03	0 :: 0:s
23.71/9.03	s :: 0:s -> 0:s
23.71/9.03	lt :: 0:s -> 0:s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0:s -> 0:s -> 0:s
23.71/9.03	div :: 0:s -> 0:s -> 0:s
23.71/9.03	
23.71/9.03	Rewrite Strategy: INNERMOST
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(9) CompletionProof (UPPER BOUND(ID))
23.71/9.03	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
23.71/9.03	none
23.71/9.03	
23.71/9.03	And the following fresh constants: none
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(10)
23.71/9.03	Obligation:
23.71/9.03	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
23.71/9.03	
23.71/9.03	Runtime Complexity Weighted TRS with Types.
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   minus(x, 0) -> x [1]
23.71/9.03	   minus(0, s(y)) -> 0 [1]
23.71/9.03	   minus(s(x), s(y)) -> minus(x, y) [1]
23.71/9.03	   lt(x, 0) -> false [1]
23.71/9.03	   lt(0, s(y)) -> true [1]
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y) [1]
23.71/9.03	   if(true, x, y) -> x [1]
23.71/9.03	   if(false, x, y) -> y [1]
23.71/9.03	   div(x, 0) -> 0 [1]
23.71/9.03	   div(0, y) -> 0 [1]
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]
23.71/9.03	
23.71/9.03	The TRS has the following type information:
23.71/9.03	minus :: 0:s -> 0:s -> 0:s
23.71/9.03	0 :: 0:s
23.71/9.03	s :: 0:s -> 0:s
23.71/9.03	lt :: 0:s -> 0:s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0:s -> 0:s -> 0:s
23.71/9.03	div :: 0:s -> 0:s -> 0:s
23.71/9.03	
23.71/9.03	Rewrite Strategy: INNERMOST
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
23.71/9.03	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
23.71/9.03	The constant constructors are abstracted as follows:
23.71/9.03	
23.71/9.03	   0 => 0
23.71/9.03	   false => 0
23.71/9.03	   true => 1
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(12)
23.71/9.03	Obligation:
23.71/9.03	Complexity RNTS consisting of the following rules:
23.71/9.03	
23.71/9.03	   div(z, z') -{ 1 }-> if(lt(x, y), 0, 1 + div(minus(x, y), 1 + y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
23.71/9.03	   div(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
23.71/9.03	   div(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
23.71/9.03	   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
23.71/9.03	   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
23.71/9.03	   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
23.71/9.03	   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
23.71/9.03	   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
23.71/9.03	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
23.71/9.03	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
23.71/9.03	   minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
23.71/9.03	
23.71/9.03	Only complete derivations are relevant for the runtime complexity.
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(13) CompleteCoflocoProof (FINISHED)
23.71/9.03	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
23.71/9.03	
23.71/9.03	eq(start(V1, V, V12),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
23.71/9.03	eq(start(V1, V, V12),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
23.71/9.03	eq(start(V1, V, V12),0,[if(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
23.71/9.03	eq(start(V1, V, V12),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
23.71/9.03	eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
23.71/9.03	eq(minus(V1, V, Out),1,[],[Out = 0,V = 1 + V3,V3 >= 0,V1 = 0]).
23.71/9.03	eq(minus(V1, V, Out),1,[minus(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
23.71/9.03	eq(lt(V1, V, Out),1,[],[Out = 0,V6 >= 0,V1 = V6,V = 0]).
23.71/9.03	eq(lt(V1, V, Out),1,[],[Out = 1,V = 1 + V7,V7 >= 0,V1 = 0]).
23.71/9.03	eq(lt(V1, V, Out),1,[lt(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
23.71/9.03	eq(if(V1, V, V12, Out),1,[],[Out = V11,V = V11,V12 = V10,V1 = 1,V11 >= 0,V10 >= 0]).
23.71/9.03	eq(if(V1, V, V12, Out),1,[],[Out = V13,V = V14,V12 = V13,V14 >= 0,V13 >= 0,V1 = 0]).
23.71/9.03	eq(div(V1, V, Out),1,[],[Out = 0,V15 >= 0,V1 = V15,V = 0]).
23.71/9.03	eq(div(V1, V, Out),1,[],[Out = 0,V16 >= 0,V1 = 0,V = V16]).
23.71/9.03	eq(div(V1, V, Out),1,[lt(V17, V18, Ret0),minus(V17, V18, Ret210),div(Ret210, 1 + V18, Ret21),if(Ret0, 0, 1 + Ret21, Ret2)],[Out = Ret2,V = 1 + V18,V17 >= 0,V18 >= 0,V1 = 1 + V17]).
23.71/9.03	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
23.71/9.03	input_output_vars(lt(V1,V,Out),[V1,V],[Out]).
23.71/9.03	input_output_vars(if(V1,V,V12,Out),[V1,V,V12],[Out]).
23.71/9.03	input_output_vars(div(V1,V,Out),[V1,V],[Out]).
23.71/9.03	
23.71/9.03	
23.71/9.03	CoFloCo proof output:
23.71/9.03	Preprocessing Cost Relations
23.71/9.03	=====================================
23.71/9.03	
23.71/9.03	#### Computed strongly connected components 
23.71/9.03	0. non_recursive  : [if/4]
23.71/9.03	1. recursive  : [lt/3]
23.71/9.03	2. recursive  : [minus/3]
23.71/9.03	3. recursive [non_tail] : [(div)/3]
23.71/9.03	4. non_recursive  : [start/3]
23.71/9.03	
23.71/9.03	#### Obtained direct recursion through partial evaluation 
23.71/9.03	0. SCC is partially evaluated into if/4
23.71/9.03	1. SCC is partially evaluated into lt/3
23.71/9.03	2. SCC is partially evaluated into minus/3
23.71/9.03	3. SCC is partially evaluated into (div)/3
23.71/9.03	4. SCC is partially evaluated into start/3
23.71/9.03	
23.71/9.03	Control-Flow Refinement of Cost Relations
23.71/9.03	=====================================
23.71/9.03	
23.71/9.03	### Specialization of cost equations if/4 
23.71/9.03	* CE 11 is refined into CE [16] 
23.71/9.03	* CE 12 is refined into CE [17] 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Cost equations --> "Loop" of if/4 
23.71/9.03	* CEs [16] --> Loop 13 
23.71/9.03	* CEs [17] --> Loop 14 
23.71/9.03	
23.71/9.03	### Ranking functions of CR if(V1,V,V12,Out) 
23.71/9.03	
23.71/9.03	#### Partial ranking functions of CR if(V1,V,V12,Out) 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Specialization of cost equations lt/3 
23.71/9.03	* CE 10 is refined into CE [18] 
23.71/9.03	* CE 8 is refined into CE [19] 
23.71/9.03	* CE 9 is refined into CE [20] 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Cost equations --> "Loop" of lt/3 
23.71/9.03	* CEs [19] --> Loop 15 
23.71/9.03	* CEs [20] --> Loop 16 
23.71/9.03	* CEs [18] --> Loop 17 
23.71/9.03	
23.71/9.03	### Ranking functions of CR lt(V1,V,Out) 
23.71/9.03	* RF of phase [17]: [V,V1]
23.71/9.03	
23.71/9.03	#### Partial ranking functions of CR lt(V1,V,Out) 
23.71/9.03	* Partial RF of phase [17]:
23.71/9.03	  - RF of loop [17:1]:
23.71/9.03	    V
23.71/9.03	    V1
23.71/9.03	
23.71/9.03	
23.71/9.03	### Specialization of cost equations minus/3 
23.71/9.03	* CE 7 is refined into CE [21] 
23.71/9.03	* CE 5 is refined into CE [22] 
23.71/9.03	* CE 6 is refined into CE [23] 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Cost equations --> "Loop" of minus/3 
23.71/9.03	* CEs [22] --> Loop 18 
23.71/9.03	* CEs [23] --> Loop 19 
23.71/9.03	* CEs [21] --> Loop 20 
23.71/9.03	
23.71/9.03	### Ranking functions of CR minus(V1,V,Out) 
23.71/9.03	* RF of phase [20]: [V,V1]
23.71/9.03	
23.71/9.03	#### Partial ranking functions of CR minus(V1,V,Out) 
23.71/9.03	* Partial RF of phase [20]:
23.71/9.03	  - RF of loop [20:1]:
23.71/9.03	    V
23.71/9.03	    V1
23.71/9.03	
23.71/9.03	
23.71/9.03	### Specialization of cost equations (div)/3 
23.71/9.03	* CE 15 is refined into CE [24,25,26,27] 
23.71/9.03	* CE 13 is refined into CE [28] 
23.71/9.03	* CE 14 is refined into CE [29] 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Cost equations --> "Loop" of (div)/3 
23.71/9.03	* CEs [28] --> Loop 21 
23.71/9.03	* CEs [29] --> Loop 22 
23.71/9.03	* CEs [26] --> Loop 23 
23.71/9.03	* CEs [27] --> Loop 24 
23.71/9.03	* CEs [25] --> Loop 25 
23.71/9.03	* CEs [24] --> Loop 26 
23.71/9.03	
23.71/9.03	### Ranking functions of CR div(V1,V,Out) 
23.71/9.03	* RF of phase [23]: [V1-1,V1-V+1]
23.71/9.03	* RF of phase [25]: [V1]
23.71/9.03	
23.71/9.03	#### Partial ranking functions of CR div(V1,V,Out) 
23.71/9.03	* Partial RF of phase [23]:
23.71/9.03	  - RF of loop [23:1]:
23.71/9.03	    V1-1
23.71/9.03	    V1-V+1
23.71/9.03	* Partial RF of phase [25]:
23.71/9.03	  - RF of loop [25:1]:
23.71/9.03	    V1
23.71/9.03	
23.71/9.03	
23.71/9.03	### Specialization of cost equations start/3 
23.71/9.03	* CE 1 is refined into CE [30,31,32,33] 
23.71/9.03	* CE 2 is refined into CE [34,35,36,37] 
23.71/9.03	* CE 3 is refined into CE [38,39] 
23.71/9.03	* CE 4 is refined into CE [40,41,42,43,44,45] 
23.71/9.03	
23.71/9.03	
23.71/9.03	### Cost equations --> "Loop" of start/3 
23.71/9.03	* CEs [33,36,43,45] --> Loop 27 
23.71/9.03	* CEs [31,35,42] --> Loop 28 
23.71/9.03	* CEs [32,37,41,44] --> Loop 29 
23.71/9.03	* CEs [39] --> Loop 30 
23.71/9.03	* CEs [30,34,38,40] --> Loop 31 
23.71/9.03	
23.71/9.03	### Ranking functions of CR start(V1,V,V12) 
23.71/9.03	
23.71/9.03	#### Partial ranking functions of CR start(V1,V,V12) 
23.71/9.03	
23.71/9.03	
23.71/9.03	Computing Bounds
23.71/9.03	=====================================
23.71/9.03	
23.71/9.03	#### Cost of chains of if(V1,V,V12,Out):
23.71/9.03	* Chain [14]: 1
23.71/9.03	  with precondition: [V1=0,V12=Out,V>=0,V12>=0] 
23.71/9.03	
23.71/9.03	* Chain [13]: 1
23.71/9.03	  with precondition: [V1=1,V=Out,V>=0,V12>=0] 
23.71/9.03	
23.71/9.03	
23.71/9.03	#### Cost of chains of lt(V1,V,Out):
23.71/9.03	* Chain [[17],16]: 1*it(17)+1
23.71/9.03	  Such that:it(17) =< V1
23.71/9.03	
23.71/9.03	  with precondition: [Out=1,V1>=1,V>=V1+1] 
23.71/9.03	
23.71/9.03	* Chain [[17],15]: 1*it(17)+1
23.71/9.03	  Such that:it(17) =< V
23.71/9.03	
23.71/9.03	  with precondition: [Out=0,V>=1,V1>=V] 
23.71/9.03	
23.71/9.03	* Chain [16]: 1
23.71/9.03	  with precondition: [V1=0,Out=1,V>=1] 
23.71/9.03	
23.71/9.03	* Chain [15]: 1
23.71/9.03	  with precondition: [V=0,Out=0,V1>=0] 
23.71/9.03	
23.71/9.03	
23.71/9.03	#### Cost of chains of minus(V1,V,Out):
23.71/9.03	* Chain [[20],19]: 1*it(20)+1
23.71/9.03	  Such that:it(20) =< V1
23.71/9.03	
23.71/9.03	  with precondition: [Out=0,V1>=1,V>=V1+1] 
23.71/9.03	
23.71/9.03	* Chain [[20],18]: 1*it(20)+1
23.71/9.03	  Such that:it(20) =< V
23.71/9.03	
23.71/9.03	  with precondition: [V1=Out+V,V>=1,V1>=V] 
23.71/9.03	
23.71/9.03	* Chain [19]: 1
23.71/9.03	  with precondition: [V1=0,Out=0,V>=1] 
23.71/9.03	
23.71/9.03	* Chain [18]: 1
23.71/9.03	  with precondition: [V=0,V1=Out,V1>=0] 
23.71/9.03	
23.71/9.03	
23.71/9.03	#### Cost of chains of div(V1,V,Out):
23.71/9.03	* Chain [[25],22]: 4*it(25)+1
23.71/9.03	  Such that:it(25) =< Out
23.71/9.03	
23.71/9.03	  with precondition: [V=1,V1=Out,V1>=1] 
23.71/9.03	
23.71/9.03	* Chain [[23],26,22]: 4*it(23)+2*s(5)+5
23.71/9.03	  Such that:it(23) =< V1-V+1
23.71/9.03	aux(4) =< V1
23.71/9.03	it(23) =< aux(4)
23.71/9.03	s(5) =< aux(4)
23.71/9.03	
23.71/9.03	  with precondition: [V>=2,Out>=1,V1+1>=2*Out+V] 
23.71/9.03	
23.71/9.03	* Chain [[23],24,22]: 4*it(23)+4*s(5)+5
23.71/9.03	  Such that:it(23) =< V1-V+1
23.71/9.03	aux(6) =< V1
23.71/9.03	s(5) =< aux(6)
23.71/9.03	it(23) =< aux(6)
23.71/9.03	
23.71/9.03	  with precondition: [V>=2,Out>=1,V1>=2*Out+V] 
23.71/9.03	
23.71/9.03	* Chain [[23],22]: 4*it(23)+2*s(5)+1
23.71/9.03	  Such that:it(23) =< V1-V+1
23.71/9.03	aux(7) =< V1
23.71/9.03	it(23) =< aux(7)
23.71/9.03	s(5) =< aux(7)
23.71/9.03	
23.71/9.03	  with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] 
23.71/9.03	
23.71/9.03	* Chain [26,22]: 5
23.71/9.03	  with precondition: [V1=1,Out=0,V>=2] 
23.71/9.03	
23.71/9.03	* Chain [24,22]: 2*s(7)+5
23.71/9.03	  Such that:aux(5) =< V1
23.71/9.03	s(7) =< aux(5)
23.71/9.03	
23.71/9.03	  with precondition: [Out=0,V1>=2,V>=V1+1] 
23.71/9.03	
23.71/9.03	* Chain [22]: 1
23.71/9.03	  with precondition: [V1=0,Out=0,V>=0] 
23.71/9.03	
23.71/9.03	* Chain [21]: 1
23.71/9.03	  with precondition: [V=0,Out=0,V1>=0] 
23.71/9.03	
23.71/9.03	
23.71/9.03	#### Cost of chains of start(V1,V,V12):
23.71/9.03	* Chain [31]: 1
23.71/9.03	  with precondition: [V1=0,V>=0] 
23.71/9.03	
23.71/9.03	* Chain [30]: 1
23.71/9.03	  with precondition: [V1=1,V>=0,V12>=0] 
23.71/9.03	
23.71/9.03	* Chain [29]: 4*s(18)+5
23.71/9.03	  Such that:aux(10) =< V1
23.71/9.03	s(18) =< aux(10)
23.71/9.03	
23.71/9.03	  with precondition: [V1>=1,V>=V1+1] 
23.71/9.03	
23.71/9.03	* Chain [28]: 1
23.71/9.03	  with precondition: [V=0,V1>=0] 
23.71/9.03	
23.71/9.03	* Chain [27]: 2*s(22)+12*s(24)+12*s(27)+5
23.71/9.03	  Such that:s(26) =< V1-V+1
23.71/9.03	aux(11) =< V1
23.71/9.03	aux(12) =< V
23.71/9.03	s(24) =< aux(11)
23.71/9.03	s(22) =< aux(12)
23.71/9.03	s(27) =< s(26)
23.71/9.03	s(27) =< aux(11)
23.71/9.03	
23.71/9.03	  with precondition: [V>=1,V1>=V] 
23.71/9.03	
23.71/9.03	
23.71/9.03	Closed-form bounds of start(V1,V,V12): 
23.71/9.03	-------------------------------------
23.71/9.03	* Chain [31] with precondition: [V1=0,V>=0] 
23.71/9.03	    - Upper bound: 1 
23.71/9.03	    - Complexity: constant 
23.71/9.03	* Chain [30] with precondition: [V1=1,V>=0,V12>=0] 
23.71/9.03	    - Upper bound: 1 
23.71/9.03	    - Complexity: constant 
23.71/9.03	* Chain [29] with precondition: [V1>=1,V>=V1+1] 
23.71/9.03	    - Upper bound: 4*V1+5 
23.71/9.03	    - Complexity: n 
23.71/9.03	* Chain [28] with precondition: [V=0,V1>=0] 
23.71/9.03	    - Upper bound: 1 
23.71/9.03	    - Complexity: constant 
23.71/9.03	* Chain [27] with precondition: [V>=1,V1>=V] 
23.71/9.03	    - Upper bound: 24*V1-10*V+17 
23.71/9.03	    - Complexity: n 
23.71/9.03	
23.71/9.03	### Maximum cost of start(V1,V,V12): 8*V1+2*V+nat(V1-V+1)*12+(4*V1+4)+1 
23.71/9.03	Asymptotic class: n 
23.71/9.03	* Total analysis performed in 284 ms.
23.71/9.03	
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(14)
23.71/9.03	BOUNDS(1, n^1)
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(15) RenamingProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Renamed function symbols to avoid clashes with predefined symbol.
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(16)
23.71/9.03	Obligation:
23.71/9.03	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
23.71/9.03	
23.71/9.03	
23.71/9.03	The TRS R consists of the following rules:
23.71/9.03	
23.71/9.03	   -(x, 0') -> x
23.71/9.03	   -(0', s(y)) -> 0'
23.71/9.03	   -(s(x), s(y)) -> -(x, y)
23.71/9.03	   lt(x, 0') -> false
23.71/9.03	   lt(0', s(y)) -> true
23.71/9.03	   lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	   if(true, x, y) -> x
23.71/9.03	   if(false, x, y) -> y
23.71/9.03	   div(x, 0') -> 0'
23.71/9.03	   div(0', y) -> 0'
23.71/9.03	   div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	S is empty.
23.71/9.03	Rewrite Strategy: FULL
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
23.71/9.03	Infered types.
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(18)
23.71/9.03	Obligation:
23.71/9.03	TRS:
23.71/9.03	Rules:
23.71/9.03	-(x, 0') -> x
23.71/9.03	-(0', s(y)) -> 0'
23.71/9.03	-(s(x), s(y)) -> -(x, y)
23.71/9.03	lt(x, 0') -> false
23.71/9.03	lt(0', s(y)) -> true
23.71/9.03	lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	if(true, x, y) -> x
23.71/9.03	if(false, x, y) -> y
23.71/9.03	div(x, 0') -> 0'
23.71/9.03	div(0', y) -> 0'
23.71/9.03	div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	Types:
23.71/9.03	- :: 0':s -> 0':s -> 0':s
23.71/9.03	0' :: 0':s
23.71/9.03	s :: 0':s -> 0':s
23.71/9.03	lt :: 0':s -> 0':s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0':s -> 0':s -> 0':s
23.71/9.03	div :: 0':s -> 0':s -> 0':s
23.71/9.03	hole_0':s1_0 :: 0':s
23.71/9.03	hole_false:true2_0 :: false:true
23.71/9.03	gen_0':s3_0 :: Nat -> 0':s
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(19) OrderProof (LOWER BOUND(ID))
23.71/9.03	Heuristically decided to analyse the following defined symbols:
23.71/9.03	-, lt, div
23.71/9.03	
23.71/9.03	They will be analysed ascendingly in the following order:
23.71/9.03	- < div
23.71/9.03	lt < div
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(20)
23.71/9.03	Obligation:
23.71/9.03	TRS:
23.71/9.03	Rules:
23.71/9.03	-(x, 0') -> x
23.71/9.03	-(0', s(y)) -> 0'
23.71/9.03	-(s(x), s(y)) -> -(x, y)
23.71/9.03	lt(x, 0') -> false
23.71/9.03	lt(0', s(y)) -> true
23.71/9.03	lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	if(true, x, y) -> x
23.71/9.03	if(false, x, y) -> y
23.71/9.03	div(x, 0') -> 0'
23.71/9.03	div(0', y) -> 0'
23.71/9.03	div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	Types:
23.71/9.03	- :: 0':s -> 0':s -> 0':s
23.71/9.03	0' :: 0':s
23.71/9.03	s :: 0':s -> 0':s
23.71/9.03	lt :: 0':s -> 0':s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0':s -> 0':s -> 0':s
23.71/9.03	div :: 0':s -> 0':s -> 0':s
23.71/9.03	hole_0':s1_0 :: 0':s
23.71/9.03	hole_false:true2_0 :: false:true
23.71/9.03	gen_0':s3_0 :: Nat -> 0':s
23.71/9.03	
23.71/9.03	
23.71/9.03	Generator Equations:
23.71/9.03	gen_0':s3_0(0) <=> 0'
23.71/9.03	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
23.71/9.03	
23.71/9.03	
23.71/9.03	The following defined symbols remain to be analysed:
23.71/9.03	-, lt, div
23.71/9.03	
23.71/9.03	They will be analysed ascendingly in the following order:
23.71/9.03	- < div
23.71/9.03	lt < div
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(21) RewriteLemmaProof (LOWER BOUND(ID))
23.71/9.03	Proved the following rewrite lemma:
23.71/9.03	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
23.71/9.03	
23.71/9.03	Induction Base:
23.71/9.03	-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
23.71/9.03	gen_0':s3_0(0)
23.71/9.03	
23.71/9.03	Induction Step:
23.71/9.03	-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
23.71/9.03	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
23.71/9.03	gen_0':s3_0(0)
23.71/9.03	
23.71/9.03	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(22)
23.71/9.03	Complex Obligation (BEST)
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(23)
23.71/9.03	Obligation:
23.71/9.03	Proved the lower bound n^1 for the following obligation:
23.71/9.03	
23.71/9.03	TRS:
23.71/9.03	Rules:
23.71/9.03	-(x, 0') -> x
23.71/9.03	-(0', s(y)) -> 0'
23.71/9.03	-(s(x), s(y)) -> -(x, y)
23.71/9.03	lt(x, 0') -> false
23.71/9.03	lt(0', s(y)) -> true
23.71/9.03	lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	if(true, x, y) -> x
23.71/9.03	if(false, x, y) -> y
23.71/9.03	div(x, 0') -> 0'
23.71/9.03	div(0', y) -> 0'
23.71/9.03	div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	Types:
23.71/9.03	- :: 0':s -> 0':s -> 0':s
23.71/9.03	0' :: 0':s
23.71/9.03	s :: 0':s -> 0':s
23.71/9.03	lt :: 0':s -> 0':s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0':s -> 0':s -> 0':s
23.71/9.03	div :: 0':s -> 0':s -> 0':s
23.71/9.03	hole_0':s1_0 :: 0':s
23.71/9.03	hole_false:true2_0 :: false:true
23.71/9.03	gen_0':s3_0 :: Nat -> 0':s
23.71/9.03	
23.71/9.03	
23.71/9.03	Generator Equations:
23.71/9.03	gen_0':s3_0(0) <=> 0'
23.71/9.03	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
23.71/9.03	
23.71/9.03	
23.71/9.03	The following defined symbols remain to be analysed:
23.71/9.03	-, lt, div
23.71/9.03	
23.71/9.03	They will be analysed ascendingly in the following order:
23.71/9.03	- < div
23.71/9.03	lt < div
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(24) LowerBoundPropagationProof (FINISHED)
23.71/9.03	Propagated lower bound.
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(25)
23.71/9.03	BOUNDS(n^1, INF)
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(26)
23.71/9.03	Obligation:
23.71/9.03	TRS:
23.71/9.03	Rules:
23.71/9.03	-(x, 0') -> x
23.71/9.03	-(0', s(y)) -> 0'
23.71/9.03	-(s(x), s(y)) -> -(x, y)
23.71/9.03	lt(x, 0') -> false
23.71/9.03	lt(0', s(y)) -> true
23.71/9.03	lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	if(true, x, y) -> x
23.71/9.03	if(false, x, y) -> y
23.71/9.03	div(x, 0') -> 0'
23.71/9.03	div(0', y) -> 0'
23.71/9.03	div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	Types:
23.71/9.03	- :: 0':s -> 0':s -> 0':s
23.71/9.03	0' :: 0':s
23.71/9.03	s :: 0':s -> 0':s
23.71/9.03	lt :: 0':s -> 0':s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0':s -> 0':s -> 0':s
23.71/9.03	div :: 0':s -> 0':s -> 0':s
23.71/9.03	hole_0':s1_0 :: 0':s
23.71/9.03	hole_false:true2_0 :: false:true
23.71/9.03	gen_0':s3_0 :: Nat -> 0':s
23.71/9.03	
23.71/9.03	
23.71/9.03	Lemmas:
23.71/9.03	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
23.71/9.03	
23.71/9.03	
23.71/9.03	Generator Equations:
23.71/9.03	gen_0':s3_0(0) <=> 0'
23.71/9.03	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
23.71/9.03	
23.71/9.03	
23.71/9.03	The following defined symbols remain to be analysed:
23.71/9.03	lt, div
23.71/9.03	
23.71/9.03	They will be analysed ascendingly in the following order:
23.71/9.03	lt < div
23.71/9.03	
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(27) RewriteLemmaProof (LOWER BOUND(ID))
23.71/9.03	Proved the following rewrite lemma:
23.71/9.03	lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) -> false, rt in Omega(1 + n368_0)
23.71/9.03	
23.71/9.03	Induction Base:
23.71/9.03	lt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
23.71/9.03	false
23.71/9.03	
23.71/9.03	Induction Step:
23.71/9.03	lt(gen_0':s3_0(+(n368_0, 1)), gen_0':s3_0(+(n368_0, 1))) ->_R^Omega(1)
23.71/9.03	lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) ->_IH
23.71/9.03	false
23.71/9.03	
23.71/9.03	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
23.71/9.03	----------------------------------------
23.71/9.03	
23.71/9.03	(28)
23.71/9.03	Obligation:
23.71/9.03	TRS:
23.71/9.03	Rules:
23.71/9.03	-(x, 0') -> x
23.71/9.03	-(0', s(y)) -> 0'
23.71/9.03	-(s(x), s(y)) -> -(x, y)
23.71/9.03	lt(x, 0') -> false
23.71/9.03	lt(0', s(y)) -> true
23.71/9.03	lt(s(x), s(y)) -> lt(x, y)
23.71/9.03	if(true, x, y) -> x
23.71/9.03	if(false, x, y) -> y
23.71/9.03	div(x, 0') -> 0'
23.71/9.03	div(0', y) -> 0'
23.71/9.03	div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))
23.71/9.03	
23.71/9.03	Types:
23.71/9.03	- :: 0':s -> 0':s -> 0':s
23.71/9.03	0' :: 0':s
23.71/9.03	s :: 0':s -> 0':s
23.71/9.03	lt :: 0':s -> 0':s -> false:true
23.71/9.03	false :: false:true
23.71/9.03	true :: false:true
23.71/9.03	if :: false:true -> 0':s -> 0':s -> 0':s
23.71/9.03	div :: 0':s -> 0':s -> 0':s
23.71/9.03	hole_0':s1_0 :: 0':s
23.71/9.03	hole_false:true2_0 :: false:true
23.71/9.03	gen_0':s3_0 :: Nat -> 0':s
23.71/9.03	
23.71/9.03	
23.71/9.03	Lemmas:
23.71/9.03	-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
23.71/9.03	lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) -> false, rt in Omega(1 + n368_0)
23.71/9.03	
23.71/9.03	
23.71/9.03	Generator Equations:
23.71/9.03	gen_0':s3_0(0) <=> 0'
23.71/9.03	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
23.71/9.03	
23.71/9.03	
23.71/9.03	The following defined symbols remain to be analysed:
23.71/9.03	div
23.97/9.42	EOF