18.98/5.88	WORST_CASE(Omega(n^1), O(n^1))
18.98/5.89	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
18.98/5.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.98/5.89	
18.98/5.89	
18.98/5.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.98/5.89	
18.98/5.89	(0) CpxTRS
18.98/5.89	(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
18.98/5.89	(2) CpxWeightedTrs
18.98/5.89	    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
18.98/5.89	    (4) CpxTypedWeightedTrs
18.98/5.89	    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
18.98/5.89	    (6) CpxTypedWeightedCompleteTrs
18.98/5.89	    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
18.98/5.89	    (8) CpxRNTS
18.98/5.89	    (9) CompleteCoflocoProof [FINISHED, 491 ms]
18.98/5.89	    (10) BOUNDS(1, n^1)
18.98/5.89	(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
18.98/5.89	(12) CpxTRS
18.98/5.89	    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
18.98/5.89	    (14) typed CpxTrs
18.98/5.89	    (15) OrderProof [LOWER BOUND(ID), 0 ms]
18.98/5.89	    (16) typed CpxTrs
18.98/5.89	    (17) RewriteLemmaProof [LOWER BOUND(ID), 274 ms]
18.98/5.89	    (18) BEST
18.98/5.89	        (19) proven lower bound
18.98/5.89	            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
18.98/5.89	            (21) BOUNDS(n^1, INF)
18.98/5.89	        (22) typed CpxTrs
18.98/5.89	            (23) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
18.98/5.89	            (24) typed CpxTrs
18.98/5.89	
18.98/5.89	
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(0)
18.98/5.89	Obligation:
18.98/5.89	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
18.98/5.89	
18.98/5.89	
18.98/5.89	The TRS R consists of the following rules:
18.98/5.89	
18.98/5.89	   le(0, y) -> true
18.98/5.89	   le(s(x), 0) -> false
18.98/5.89	   le(s(x), s(y)) -> le(x, y)
18.98/5.89	   minus(x, 0) -> x
18.98/5.89	   minus(s(x), s(y)) -> minus(x, y)
18.98/5.89	   mod(0, y) -> 0
18.98/5.89	   mod(s(x), 0) -> 0
18.98/5.89	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.89	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.89	   if_mod(false, s(x), s(y)) -> s(x)
18.98/5.89	
18.98/5.89	S is empty.
18.98/5.89	Rewrite Strategy: INNERMOST
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
18.98/5.89	Transformed relative TRS to weighted TRS
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(2)
18.98/5.89	Obligation:
18.98/5.89	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
18.98/5.89	
18.98/5.89	
18.98/5.89	The TRS R consists of the following rules:
18.98/5.89	
18.98/5.89	   le(0, y) -> true [1]
18.98/5.89	   le(s(x), 0) -> false [1]
18.98/5.89	   le(s(x), s(y)) -> le(x, y) [1]
18.98/5.89	   minus(x, 0) -> x [1]
18.98/5.89	   minus(s(x), s(y)) -> minus(x, y) [1]
18.98/5.89	   mod(0, y) -> 0 [1]
18.98/5.89	   mod(s(x), 0) -> 0 [1]
18.98/5.89	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
18.98/5.89	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
18.98/5.89	   if_mod(false, s(x), s(y)) -> s(x) [1]
18.98/5.89	
18.98/5.89	Rewrite Strategy: INNERMOST
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
18.98/5.89	Infered types.
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(4)
18.98/5.89	Obligation:
18.98/5.89	Runtime Complexity Weighted TRS with Types.
18.98/5.89	The TRS R consists of the following rules:
18.98/5.89	
18.98/5.89	   le(0, y) -> true [1]
18.98/5.89	   le(s(x), 0) -> false [1]
18.98/5.89	   le(s(x), s(y)) -> le(x, y) [1]
18.98/5.89	   minus(x, 0) -> x [1]
18.98/5.89	   minus(s(x), s(y)) -> minus(x, y) [1]
18.98/5.89	   mod(0, y) -> 0 [1]
18.98/5.89	   mod(s(x), 0) -> 0 [1]
18.98/5.89	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
18.98/5.89	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
18.98/5.89	   if_mod(false, s(x), s(y)) -> s(x) [1]
18.98/5.89	
18.98/5.89	The TRS has the following type information:
18.98/5.89	le :: 0:s -> 0:s -> true:false
18.98/5.89	0 :: 0:s
18.98/5.89	true :: true:false
18.98/5.89	s :: 0:s -> 0:s
18.98/5.89	false :: true:false
18.98/5.89	minus :: 0:s -> 0:s -> 0:s
18.98/5.89	mod :: 0:s -> 0:s -> 0:s
18.98/5.89	if_mod :: true:false -> 0:s -> 0:s -> 0:s
18.98/5.89	
18.98/5.89	Rewrite Strategy: INNERMOST
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(5) CompletionProof (UPPER BOUND(ID))
18.98/5.89	The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
18.98/5.89	
18.98/5.89	   minus(v0, v1) -> null_minus [0]
18.98/5.89	   if_mod(v0, v1, v2) -> null_if_mod [0]
18.98/5.89	   le(v0, v1) -> null_le [0]
18.98/5.89	   mod(v0, v1) -> null_mod [0]
18.98/5.89	
18.98/5.89	And the following fresh constants:    null_minus, null_if_mod, null_le, null_mod
18.98/5.89	
18.98/5.89	----------------------------------------
18.98/5.89	
18.98/5.89	(6)
18.98/5.89	Obligation:
18.98/5.89	Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:
18.98/5.89	
18.98/5.89	Runtime Complexity Weighted TRS with Types.
18.98/5.89	The TRS R consists of the following rules:
18.98/5.89	
18.98/5.89	   le(0, y) -> true [1]
18.98/5.89	   le(s(x), 0) -> false [1]
18.98/5.89	   le(s(x), s(y)) -> le(x, y) [1]
18.98/5.89	   minus(x, 0) -> x [1]
18.98/5.89	   minus(s(x), s(y)) -> minus(x, y) [1]
18.98/5.90	   mod(0, y) -> 0 [1]
18.98/5.90	   mod(s(x), 0) -> 0 [1]
18.98/5.90	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
18.98/5.90	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
18.98/5.90	   if_mod(false, s(x), s(y)) -> s(x) [1]
18.98/5.90	   minus(v0, v1) -> null_minus [0]
18.98/5.90	   if_mod(v0, v1, v2) -> null_if_mod [0]
18.98/5.90	   le(v0, v1) -> null_le [0]
18.98/5.90	   mod(v0, v1) -> null_mod [0]
18.98/5.90	
18.98/5.90	The TRS has the following type information:
18.98/5.90	le :: 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod -> true:false:null_le
18.98/5.90	0 :: 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	true :: true:false:null_le
18.98/5.90	s :: 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	false :: true:false:null_le
18.98/5.90	minus :: 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	mod :: 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	if_mod :: true:false:null_le -> 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod -> 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	null_minus :: 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	null_if_mod :: 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	null_le :: true:false:null_le
18.98/5.90	null_mod :: 0:s:null_minus:null_if_mod:null_mod
18.98/5.90	
18.98/5.90	Rewrite Strategy: INNERMOST
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
18.98/5.90	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
18.98/5.90	The constant constructors are abstracted as follows:
18.98/5.90	
18.98/5.90	   0 => 0
18.98/5.90	   true => 2
18.98/5.90	   false => 1
18.98/5.90	   null_minus => 0
18.98/5.90	   null_if_mod => 0
18.98/5.90	   null_le => 0
18.98/5.90	   null_mod => 0
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(8)
18.98/5.90	Obligation:
18.98/5.90	Complexity RNTS consisting of the following rules:
18.98/5.90	
18.98/5.90	   if_mod(z, z', z'') -{ 1 }-> mod(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
18.98/5.90	   if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
18.98/5.90	   if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
18.98/5.90	   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
18.98/5.90	   le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
18.98/5.90	   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
18.98/5.90	   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
18.98/5.90	   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
18.98/5.90	   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
18.98/5.90	   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
18.98/5.90	   mod(z, z') -{ 1 }-> if_mod(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
18.98/5.90	   mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
18.98/5.90	   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
18.98/5.90	   mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
18.98/5.90	
18.98/5.90	Only complete derivations are relevant for the runtime complexity.
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(9) CompleteCoflocoProof (FINISHED)
18.98/5.90	Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:
18.98/5.90	
18.98/5.90	eq(start(V1, V, V14),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
18.98/5.90	eq(start(V1, V, V14),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
18.98/5.90	eq(start(V1, V, V14),0,[mod(V1, V, Out)],[V1 >= 0,V >= 0]).
18.98/5.90	eq(start(V1, V, V14),0,[fun(V1, V, V14, Out)],[V1 >= 0,V >= 0,V14 >= 0]).
18.98/5.90	eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]).
18.98/5.90	eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]).
18.98/5.90	eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
18.98/5.90	eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
18.98/5.90	eq(minus(V1, V, Out),1,[minus(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
18.98/5.90	eq(mod(V1, V, Out),1,[],[Out = 0,V9 >= 0,V1 = 0,V = V9]).
18.98/5.90	eq(mod(V1, V, Out),1,[],[Out = 0,V10 >= 0,V1 = 1 + V10,V = 0]).
18.98/5.90	eq(mod(V1, V, Out),1,[le(V11, V12, Ret0),fun(Ret0, 1 + V12, 1 + V11, Ret2)],[Out = Ret2,V = 1 + V11,V12 >= 0,V11 >= 0,V1 = 1 + V12]).
18.98/5.90	eq(fun(V1, V, V14, Out),1,[minus(V15, V13, Ret01),mod(Ret01, 1 + V13, Ret3)],[Out = Ret3,V1 = 2,V = 1 + V15,V15 >= 0,V13 >= 0,V14 = 1 + V13]).
18.98/5.90	eq(fun(V1, V, V14, Out),1,[],[Out = 1 + V16,V = 1 + V16,V1 = 1,V16 >= 0,V17 >= 0,V14 = 1 + V17]).
18.98/5.90	eq(minus(V1, V, Out),0,[],[Out = 0,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
18.98/5.90	eq(fun(V1, V, V14, Out),0,[],[Out = 0,V21 >= 0,V14 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]).
18.98/5.90	eq(le(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
18.98/5.90	eq(mod(V1, V, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V1 = V25,V = V26]).
18.98/5.90	input_output_vars(le(V1,V,Out),[V1,V],[Out]).
18.98/5.90	input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
18.98/5.90	input_output_vars(mod(V1,V,Out),[V1,V],[Out]).
18.98/5.90	input_output_vars(fun(V1,V,V14,Out),[V1,V,V14],[Out]).
18.98/5.90	
18.98/5.90	
18.98/5.90	CoFloCo proof output:
18.98/5.90	Preprocessing Cost Relations
18.98/5.90	=====================================
18.98/5.90	
18.98/5.90	#### Computed strongly connected components 
18.98/5.90	0. recursive  : [minus/3]
18.98/5.90	1. recursive  : [le/3]
18.98/5.90	2. recursive  : [fun/4,(mod)/3]
18.98/5.90	3. non_recursive  : [start/3]
18.98/5.90	
18.98/5.90	#### Obtained direct recursion through partial evaluation 
18.98/5.90	0. SCC is partially evaluated into minus/3
18.98/5.90	1. SCC is partially evaluated into le/3
18.98/5.90	2. SCC is partially evaluated into (mod)/3
18.98/5.90	3. SCC is partially evaluated into start/3
18.98/5.90	
18.98/5.90	Control-Flow Refinement of Cost Relations
18.98/5.90	=====================================
18.98/5.90	
18.98/5.90	### Specialization of cost equations minus/3 
18.98/5.90	* CE 9 is refined into CE [20] 
18.98/5.90	* CE 7 is refined into CE [21] 
18.98/5.90	* CE 8 is refined into CE [22] 
18.98/5.90	
18.98/5.90	
18.98/5.90	### Cost equations --> "Loop" of minus/3 
18.98/5.90	* CEs [22] --> Loop 14 
18.98/5.90	* CEs [20] --> Loop 15 
18.98/5.90	* CEs [21] --> Loop 16 
18.98/5.90	
18.98/5.90	### Ranking functions of CR minus(V1,V,Out) 
18.98/5.90	* RF of phase [14]: [V,V1]
18.98/5.90	
18.98/5.90	#### Partial ranking functions of CR minus(V1,V,Out) 
18.98/5.90	* Partial RF of phase [14]:
18.98/5.90	  - RF of loop [14:1]:
18.98/5.90	    V
18.98/5.90	    V1
18.98/5.90	
18.98/5.90	
18.98/5.90	### Specialization of cost equations le/3 
18.98/5.90	* CE 19 is refined into CE [23] 
18.98/5.90	* CE 17 is refined into CE [24] 
18.98/5.90	* CE 16 is refined into CE [25] 
18.98/5.90	* CE 18 is refined into CE [26] 
18.98/5.90	
18.98/5.90	
18.98/5.90	### Cost equations --> "Loop" of le/3 
18.98/5.90	* CEs [26] --> Loop 17 
18.98/5.90	* CEs [23] --> Loop 18 
18.98/5.90	* CEs [24] --> Loop 19 
18.98/5.90	* CEs [25] --> Loop 20 
18.98/5.90	
18.98/5.90	### Ranking functions of CR le(V1,V,Out) 
18.98/5.90	* RF of phase [17]: [V,V1]
18.98/5.90	
18.98/5.90	#### Partial ranking functions of CR le(V1,V,Out) 
18.98/5.90	* Partial RF of phase [17]:
18.98/5.90	  - RF of loop [17:1]:
18.98/5.90	    V
18.98/5.90	    V1
18.98/5.90	
18.98/5.90	
18.98/5.90	### Specialization of cost equations (mod)/3 
18.98/5.90	* CE 11 is refined into CE [27,28] 
18.98/5.90	* CE 14 is refined into CE [29] 
18.98/5.90	* CE 10 is refined into CE [30,31,32,33,34] 
18.98/5.90	* CE 13 is refined into CE [35] 
18.98/5.90	* CE 15 is refined into CE [36] 
18.98/5.90	* CE 12 is refined into CE [37,38,39,40] 
18.98/5.90	
18.98/5.90	
18.98/5.90	### Cost equations --> "Loop" of (mod)/3 
18.98/5.90	* CEs [40] --> Loop 21 
18.98/5.90	* CEs [39] --> Loop 22 
18.98/5.90	* CEs [37] --> Loop 23 
18.98/5.90	* CEs [38] --> Loop 24 
18.98/5.90	* CEs [28] --> Loop 25 
18.98/5.90	* CEs [30] --> Loop 26 
18.98/5.90	* CEs [29] --> Loop 27 
18.98/5.90	* CEs [27] --> Loop 28 
18.98/5.90	* CEs [31] --> Loop 29 
18.98/5.90	* CEs [32,33,34,35,36] --> Loop 30 
18.98/5.90	
18.98/5.90	### Ranking functions of CR mod(V1,V,Out) 
18.98/5.90	* RF of phase [21]: [V1-1,V1-V+1]
18.98/5.90	* RF of phase [23]: [V1]
18.98/5.90	
18.98/5.90	#### Partial ranking functions of CR mod(V1,V,Out) 
18.98/5.90	* Partial RF of phase [21]:
18.98/5.90	  - RF of loop [21:1]:
18.98/5.90	    V1-1
18.98/5.90	    V1-V+1
18.98/5.90	* Partial RF of phase [23]:
18.98/5.90	  - RF of loop [23:1]:
18.98/5.90	    V1
18.98/5.90	
18.98/5.90	
18.98/5.90	### Specialization of cost equations start/3 
18.98/5.90	* CE 3 is refined into CE [41,42,43,44,45,46,47,48] 
18.98/5.90	* CE 1 is refined into CE [49] 
18.98/5.90	* CE 2 is refined into CE [50] 
18.98/5.90	* CE 4 is refined into CE [51,52,53,54,55] 
18.98/5.90	* CE 5 is refined into CE [56,57,58] 
18.98/5.90	* CE 6 is refined into CE [59,60,61,62,63,64,65] 
18.98/5.90	
18.98/5.90	
18.98/5.90	### Cost equations --> "Loop" of start/3 
18.98/5.90	* CEs [62] --> Loop 31 
18.98/5.90	* CEs [52,56,61] --> Loop 32 
18.98/5.90	* CEs [45] --> Loop 33 
18.98/5.90	* CEs [41,42,43,44,46,47,48] --> Loop 34 
18.98/5.90	* CEs [60] --> Loop 35 
18.98/5.90	* CEs [50] --> Loop 36 
18.98/5.90	* CEs [49,51,53,54,55,57,58,59,63,64,65] --> Loop 37 
18.98/5.90	
18.98/5.90	### Ranking functions of CR start(V1,V,V14) 
18.98/5.90	
18.98/5.90	#### Partial ranking functions of CR start(V1,V,V14) 
18.98/5.90	
18.98/5.90	
18.98/5.90	Computing Bounds
18.98/5.90	=====================================
18.98/5.90	
18.98/5.90	#### Cost of chains of minus(V1,V,Out):
18.98/5.90	* Chain [[14],16]: 1*it(14)+1
18.98/5.90	  Such that:it(14) =< V
18.98/5.90	
18.98/5.90	  with precondition: [V1=Out+V,V>=1,V1>=V] 
18.98/5.90	
18.98/5.90	* Chain [[14],15]: 1*it(14)+0
18.98/5.90	  Such that:it(14) =< V
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V1>=1,V>=1] 
18.98/5.90	
18.98/5.90	* Chain [16]: 1
18.98/5.90	  with precondition: [V=0,V1=Out,V1>=0] 
18.98/5.90	
18.98/5.90	* Chain [15]: 0
18.98/5.90	  with precondition: [Out=0,V1>=0,V>=0] 
18.98/5.90	
18.98/5.90	
18.98/5.90	#### Cost of chains of le(V1,V,Out):
18.98/5.90	* Chain [[17],20]: 1*it(17)+1
18.98/5.90	  Such that:it(17) =< V1
18.98/5.90	
18.98/5.90	  with precondition: [Out=2,V1>=1,V>=V1] 
18.98/5.90	
18.98/5.90	* Chain [[17],19]: 1*it(17)+1
18.98/5.90	  Such that:it(17) =< V
18.98/5.90	
18.98/5.90	  with precondition: [Out=1,V>=1,V1>=V+1] 
18.98/5.90	
18.98/5.90	* Chain [[17],18]: 1*it(17)+0
18.98/5.90	  Such that:it(17) =< V
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V1>=1,V>=1] 
18.98/5.90	
18.98/5.90	* Chain [20]: 1
18.98/5.90	  with precondition: [V1=0,Out=2,V>=0] 
18.98/5.90	
18.98/5.90	* Chain [19]: 1
18.98/5.90	  with precondition: [V=0,Out=1,V1>=1] 
18.98/5.90	
18.98/5.90	* Chain [18]: 0
18.98/5.90	  with precondition: [Out=0,V1>=0,V>=0] 
18.98/5.90	
18.98/5.90	
18.98/5.90	#### Cost of chains of mod(V1,V,Out):
18.98/5.90	* Chain [[23],30]: 6*it(23)+1*s(5)+2
18.98/5.90	  Such that:s(5) =< 1
18.98/5.90	aux(2) =< V1
18.98/5.90	it(23) =< aux(2)
18.98/5.90	
18.98/5.90	  with precondition: [V=1,Out=0,V1>=1] 
18.98/5.90	
18.98/5.90	* Chain [[23],26]: 4*it(23)+2
18.98/5.90	  Such that:it(23) =< V1
18.98/5.90	
18.98/5.90	  with precondition: [V=1,Out=0,V1>=2] 
18.98/5.90	
18.98/5.90	* Chain [[23],24,30]: 4*it(23)+1*s(5)+5
18.98/5.90	  Such that:s(5) =< 1
18.98/5.90	it(23) =< V1
18.98/5.90	
18.98/5.90	  with precondition: [V=1,Out=0,V1>=2] 
18.98/5.90	
18.98/5.90	* Chain [[21],30]: 8*it(21)+1*s(5)+2
18.98/5.90	  Such that:s(5) =< V
18.98/5.90	aux(6) =< V1
18.98/5.90	it(21) =< aux(6)
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V>=2,V1>=V] 
18.98/5.90	
18.98/5.90	* Chain [[21],29]: 4*it(21)+2*s(11)+2
18.98/5.90	  Such that:it(21) =< V1-V+1
18.98/5.90	aux(7) =< V1
18.98/5.90	it(21) =< aux(7)
18.98/5.90	s(11) =< aux(7)
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V>=2,V1>=V+1] 
18.98/5.90	
18.98/5.90	* Chain [[21],28]: 4*it(21)+2*s(11)+3
18.98/5.90	  Such that:it(21) =< V1-V+1
18.98/5.90	aux(8) =< V1
18.98/5.90	it(21) =< aux(8)
18.98/5.90	s(11) =< aux(8)
18.98/5.90	
18.98/5.90	  with precondition: [Out=1,V>=2,V1>=V+1] 
18.98/5.90	
18.98/5.90	* Chain [[21],25]: 4*it(21)+2*s(11)+1*s(13)+3
18.98/5.90	  Such that:aux(4) =< V1
18.98/5.90	it(21) =< V1-V+1
18.98/5.90	aux(5) =< V1-Out
18.98/5.90	s(13) =< Out
18.98/5.90	it(21) =< aux(4)
18.98/5.90	s(12) =< aux(4)
18.98/5.90	it(21) =< aux(5)
18.98/5.90	s(12) =< aux(5)
18.98/5.90	s(11) =< s(12)
18.98/5.90	
18.98/5.90	  with precondition: [Out>=2,V>=Out+1,V1>=Out+V] 
18.98/5.90	
18.98/5.90	* Chain [[21],22,30]: 4*it(21)+3*s(5)+2*s(11)+5
18.98/5.90	  Such that:aux(4) =< V1
18.98/5.90	aux(10) =< V
18.98/5.90	aux(11) =< V1-V
18.98/5.90	it(21) =< aux(11)
18.98/5.90	s(5) =< aux(10)
18.98/5.90	it(21) =< aux(4)
18.98/5.90	s(12) =< aux(4)
18.98/5.90	s(12) =< aux(11)
18.98/5.90	s(11) =< s(12)
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V>=2,V1>=2*V] 
18.98/5.90	
18.98/5.90	* Chain [30]: 2*s(3)+1*s(5)+2
18.98/5.90	  Such that:s(5) =< V
18.98/5.90	aux(1) =< V1
18.98/5.90	s(3) =< aux(1)
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V1>=0,V>=0] 
18.98/5.90	
18.98/5.90	* Chain [29]: 2
18.98/5.90	  with precondition: [V1=1,Out=0,V>=2] 
18.98/5.90	
18.98/5.90	* Chain [28]: 3
18.98/5.90	  with precondition: [V1=1,Out=1,V>=2] 
18.98/5.90	
18.98/5.90	* Chain [27]: 1
18.98/5.90	  with precondition: [V=0,Out=0,V1>=1] 
18.98/5.90	
18.98/5.90	* Chain [26]: 2
18.98/5.90	  with precondition: [V=1,Out=0,V1>=1] 
18.98/5.90	
18.98/5.90	* Chain [25]: 1*s(13)+3
18.98/5.90	  Such that:s(13) =< V1
18.98/5.90	
18.98/5.90	  with precondition: [V1=Out,V1>=2,V>=V1+1] 
18.98/5.90	
18.98/5.90	* Chain [24,30]: 1*s(5)+5
18.98/5.90	  Such that:s(5) =< 1
18.98/5.90	
18.98/5.90	  with precondition: [V=1,Out=0,V1>=1] 
18.98/5.90	
18.98/5.90	* Chain [22,30]: 3*s(5)+5
18.98/5.90	  Such that:aux(10) =< V
18.98/5.90	s(5) =< aux(10)
18.98/5.90	
18.98/5.90	  with precondition: [Out=0,V>=2,V1>=V] 
18.98/5.90	
18.98/5.90	
18.98/5.90	#### Cost of chains of start(V1,V,V14):
18.98/5.90	* Chain [37]: 12*s(41)+19*s(43)+12*s(47)+4*s(51)+2*s(53)+5
18.98/5.90	  Such that:s(46) =< V1-V
18.98/5.90	aux(17) =< V1
18.98/5.90	aux(18) =< V1-V+1
18.98/5.90	aux(19) =< V
18.98/5.90	s(43) =< aux(17)
18.98/5.90	s(47) =< aux(18)
18.98/5.90	s(41) =< aux(19)
18.98/5.90	s(51) =< s(46)
18.98/5.90	s(51) =< aux(17)
18.98/5.90	s(52) =< aux(17)
18.98/5.90	s(52) =< s(46)
18.98/5.90	s(53) =< s(52)
18.98/5.90	s(47) =< aux(17)
18.98/5.90	
18.98/5.90	  with precondition: [V1>=0,V>=0] 
18.98/5.90	
18.98/5.90	* Chain [36]: 1
18.98/5.90	  with precondition: [V1=1,V>=1,V14>=1] 
18.98/5.90	
18.98/5.90	* Chain [35]: 3
18.98/5.90	  with precondition: [V1=1,V>=2] 
18.98/5.90	
18.98/5.90	* Chain [34]: 36*s(66)+11*s(69)+21*s(78)+12*s(90)+4*s(94)+2*s(96)+18*s(97)+7
18.98/5.90	  Such that:s(89) =< V-2*V14
18.98/5.90	aux(24) =< 1
18.98/5.90	aux(25) =< V
18.98/5.90	aux(26) =< V-2*V14+1
18.98/5.90	aux(27) =< V-V14
18.98/5.90	aux(28) =< V14
18.98/5.90	s(90) =< aux(26)
18.98/5.90	s(97) =< aux(27)
18.98/5.90	s(78) =< aux(28)
18.98/5.90	s(66) =< aux(25)
18.98/5.90	s(69) =< aux(24)
18.98/5.90	s(94) =< s(89)
18.98/5.90	s(94) =< aux(27)
18.98/5.90	s(95) =< aux(27)
18.98/5.90	s(95) =< s(89)
18.98/5.90	s(96) =< s(95)
18.98/5.90	s(90) =< aux(27)
18.98/5.90	
18.98/5.90	  with precondition: [V1=2,V>=1,V14>=1] 
18.98/5.90	
18.98/5.90	* Chain [33]: 1*s(111)+5
18.98/5.90	  Such that:s(111) =< V14
18.98/5.90	
18.98/5.90	  with precondition: [V1=2,V=V14+1,V>=3] 
18.98/5.90	
18.98/5.90	* Chain [32]: 1
18.98/5.90	  with precondition: [V=0,V1>=0] 
18.98/5.90	
18.98/5.90	* Chain [31]: 3*s(114)+14*s(115)+5
18.98/5.90	  Such that:s(112) =< 1
18.98/5.90	s(113) =< V1
18.98/5.90	s(114) =< s(112)
18.98/5.90	s(115) =< s(113)
18.98/5.90	
18.98/5.90	  with precondition: [V=1,V1>=1] 
18.98/5.90	
18.98/5.90	
18.98/5.90	Closed-form bounds of start(V1,V,V14): 
18.98/5.90	-------------------------------------
18.98/5.90	* Chain [37] with precondition: [V1>=0,V>=0] 
18.98/5.90	    - Upper bound: 21*V1+12*V+5+nat(V1-V+1)*12+nat(V1-V)*4 
18.98/5.90	    - Complexity: n 
18.98/5.90	* Chain [36] with precondition: [V1=1,V>=1,V14>=1] 
18.98/5.90	    - Upper bound: 1 
18.98/5.90	    - Complexity: constant 
18.98/5.90	* Chain [35] with precondition: [V1=1,V>=2] 
18.98/5.90	    - Upper bound: 3 
18.98/5.90	    - Complexity: constant 
18.98/5.90	* Chain [34] with precondition: [V1=2,V>=1,V14>=1] 
18.98/5.90	    - Upper bound: 36*V+21*V14+18+nat(V-2*V14+1)*12+nat(V-V14)*20+nat(V-2*V14)*4 
18.98/5.90	    - Complexity: n 
18.98/5.90	* Chain [33] with precondition: [V1=2,V=V14+1,V>=3] 
18.98/5.90	    - Upper bound: V14+5 
18.98/5.90	    - Complexity: n 
18.98/5.90	* Chain [32] with precondition: [V=0,V1>=0] 
18.98/5.90	    - Upper bound: 1 
18.98/5.90	    - Complexity: constant 
18.98/5.90	* Chain [31] with precondition: [V=1,V1>=1] 
18.98/5.90	    - Upper bound: 14*V1+8 
18.98/5.90	    - Complexity: n 
18.98/5.90	
18.98/5.90	### Maximum cost of start(V1,V,V14): max([14*V1+2+max([3,7*V1+12*V+nat(V1-V+1)*12+nat(V1-V)*4]),36*V+13+nat(V14)*20+nat(V-2*V14+1)*12+nat(V-V14)*20+nat(V-2*V14)*4+(nat(V14)+2)])+3 
18.98/5.90	Asymptotic class: n 
18.98/5.90	* Total analysis performed in 408 ms.
18.98/5.90	
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(10)
18.98/5.90	BOUNDS(1, n^1)
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(11) RenamingProof (BOTH BOUNDS(ID, ID))
18.98/5.90	Renamed function symbols to avoid clashes with predefined symbol.
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(12)
18.98/5.90	Obligation:
18.98/5.90	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
18.98/5.90	
18.98/5.90	
18.98/5.90	The TRS R consists of the following rules:
18.98/5.90	
18.98/5.90	   le(0', y) -> true
18.98/5.90	   le(s(x), 0') -> false
18.98/5.90	   le(s(x), s(y)) -> le(x, y)
18.98/5.90	   minus(x, 0') -> x
18.98/5.90	   minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	   mod(0', y) -> 0'
18.98/5.90	   mod(s(x), 0') -> 0'
18.98/5.90	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	   if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	S is empty.
18.98/5.90	Rewrite Strategy: INNERMOST
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
18.98/5.90	Infered types.
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(14)
18.98/5.90	Obligation:
18.98/5.90	Innermost TRS:
18.98/5.90	Rules:
18.98/5.90	le(0', y) -> true
18.98/5.90	le(s(x), 0') -> false
18.98/5.90	le(s(x), s(y)) -> le(x, y)
18.98/5.90	minus(x, 0') -> x
18.98/5.90	minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	mod(0', y) -> 0'
18.98/5.90	mod(s(x), 0') -> 0'
18.98/5.90	mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	Types:
18.98/5.90	le :: 0':s -> 0':s -> true:false
18.98/5.90	0' :: 0':s
18.98/5.90	true :: true:false
18.98/5.90	s :: 0':s -> 0':s
18.98/5.90	false :: true:false
18.98/5.90	minus :: 0':s -> 0':s -> 0':s
18.98/5.90	mod :: 0':s -> 0':s -> 0':s
18.98/5.90	if_mod :: true:false -> 0':s -> 0':s -> 0':s
18.98/5.90	hole_true:false1_0 :: true:false
18.98/5.90	hole_0':s2_0 :: 0':s
18.98/5.90	gen_0':s3_0 :: Nat -> 0':s
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(15) OrderProof (LOWER BOUND(ID))
18.98/5.90	Heuristically decided to analyse the following defined symbols:
18.98/5.90	le, minus, mod
18.98/5.90	
18.98/5.90	They will be analysed ascendingly in the following order:
18.98/5.90	le < mod
18.98/5.90	minus < mod
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(16)
18.98/5.90	Obligation:
18.98/5.90	Innermost TRS:
18.98/5.90	Rules:
18.98/5.90	le(0', y) -> true
18.98/5.90	le(s(x), 0') -> false
18.98/5.90	le(s(x), s(y)) -> le(x, y)
18.98/5.90	minus(x, 0') -> x
18.98/5.90	minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	mod(0', y) -> 0'
18.98/5.90	mod(s(x), 0') -> 0'
18.98/5.90	mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	Types:
18.98/5.90	le :: 0':s -> 0':s -> true:false
18.98/5.90	0' :: 0':s
18.98/5.90	true :: true:false
18.98/5.90	s :: 0':s -> 0':s
18.98/5.90	false :: true:false
18.98/5.90	minus :: 0':s -> 0':s -> 0':s
18.98/5.90	mod :: 0':s -> 0':s -> 0':s
18.98/5.90	if_mod :: true:false -> 0':s -> 0':s -> 0':s
18.98/5.90	hole_true:false1_0 :: true:false
18.98/5.90	hole_0':s2_0 :: 0':s
18.98/5.90	gen_0':s3_0 :: Nat -> 0':s
18.98/5.90	
18.98/5.90	
18.98/5.90	Generator Equations:
18.98/5.90	gen_0':s3_0(0) <=> 0'
18.98/5.90	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
18.98/5.90	
18.98/5.90	
18.98/5.90	The following defined symbols remain to be analysed:
18.98/5.90	le, minus, mod
18.98/5.90	
18.98/5.90	They will be analysed ascendingly in the following order:
18.98/5.90	le < mod
18.98/5.90	minus < mod
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(17) RewriteLemmaProof (LOWER BOUND(ID))
18.98/5.90	Proved the following rewrite lemma:
18.98/5.90	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
18.98/5.90	
18.98/5.90	Induction Base:
18.98/5.90	le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
18.98/5.90	true
18.98/5.90	
18.98/5.90	Induction Step:
18.98/5.90	le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
18.98/5.90	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
18.98/5.90	true
18.98/5.90	
18.98/5.90	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(18)
18.98/5.90	Complex Obligation (BEST)
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(19)
18.98/5.90	Obligation:
18.98/5.90	Proved the lower bound n^1 for the following obligation:
18.98/5.90	
18.98/5.90	Innermost TRS:
18.98/5.90	Rules:
18.98/5.90	le(0', y) -> true
18.98/5.90	le(s(x), 0') -> false
18.98/5.90	le(s(x), s(y)) -> le(x, y)
18.98/5.90	minus(x, 0') -> x
18.98/5.90	minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	mod(0', y) -> 0'
18.98/5.90	mod(s(x), 0') -> 0'
18.98/5.90	mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	Types:
18.98/5.90	le :: 0':s -> 0':s -> true:false
18.98/5.90	0' :: 0':s
18.98/5.90	true :: true:false
18.98/5.90	s :: 0':s -> 0':s
18.98/5.90	false :: true:false
18.98/5.90	minus :: 0':s -> 0':s -> 0':s
18.98/5.90	mod :: 0':s -> 0':s -> 0':s
18.98/5.90	if_mod :: true:false -> 0':s -> 0':s -> 0':s
18.98/5.90	hole_true:false1_0 :: true:false
18.98/5.90	hole_0':s2_0 :: 0':s
18.98/5.90	gen_0':s3_0 :: Nat -> 0':s
18.98/5.90	
18.98/5.90	
18.98/5.90	Generator Equations:
18.98/5.90	gen_0':s3_0(0) <=> 0'
18.98/5.90	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
18.98/5.90	
18.98/5.90	
18.98/5.90	The following defined symbols remain to be analysed:
18.98/5.90	le, minus, mod
18.98/5.90	
18.98/5.90	They will be analysed ascendingly in the following order:
18.98/5.90	le < mod
18.98/5.90	minus < mod
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(20) LowerBoundPropagationProof (FINISHED)
18.98/5.90	Propagated lower bound.
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(21)
18.98/5.90	BOUNDS(n^1, INF)
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(22)
18.98/5.90	Obligation:
18.98/5.90	Innermost TRS:
18.98/5.90	Rules:
18.98/5.90	le(0', y) -> true
18.98/5.90	le(s(x), 0') -> false
18.98/5.90	le(s(x), s(y)) -> le(x, y)
18.98/5.90	minus(x, 0') -> x
18.98/5.90	minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	mod(0', y) -> 0'
18.98/5.90	mod(s(x), 0') -> 0'
18.98/5.90	mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	Types:
18.98/5.90	le :: 0':s -> 0':s -> true:false
18.98/5.90	0' :: 0':s
18.98/5.90	true :: true:false
18.98/5.90	s :: 0':s -> 0':s
18.98/5.90	false :: true:false
18.98/5.90	minus :: 0':s -> 0':s -> 0':s
18.98/5.90	mod :: 0':s -> 0':s -> 0':s
18.98/5.90	if_mod :: true:false -> 0':s -> 0':s -> 0':s
18.98/5.90	hole_true:false1_0 :: true:false
18.98/5.90	hole_0':s2_0 :: 0':s
18.98/5.90	gen_0':s3_0 :: Nat -> 0':s
18.98/5.90	
18.98/5.90	
18.98/5.90	Lemmas:
18.98/5.90	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
18.98/5.90	
18.98/5.90	
18.98/5.90	Generator Equations:
18.98/5.90	gen_0':s3_0(0) <=> 0'
18.98/5.90	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
18.98/5.90	
18.98/5.90	
18.98/5.90	The following defined symbols remain to be analysed:
18.98/5.90	minus, mod
18.98/5.90	
18.98/5.90	They will be analysed ascendingly in the following order:
18.98/5.90	minus < mod
18.98/5.90	
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(23) RewriteLemmaProof (LOWER BOUND(ID))
18.98/5.90	Proved the following rewrite lemma:
18.98/5.90	minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) -> gen_0':s3_0(0), rt in Omega(1 + n294_0)
18.98/5.90	
18.98/5.90	Induction Base:
18.98/5.90	minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
18.98/5.90	gen_0':s3_0(0)
18.98/5.90	
18.98/5.90	Induction Step:
18.98/5.90	minus(gen_0':s3_0(+(n294_0, 1)), gen_0':s3_0(+(n294_0, 1))) ->_R^Omega(1)
18.98/5.90	minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) ->_IH
18.98/5.90	gen_0':s3_0(0)
18.98/5.90	
18.98/5.90	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
18.98/5.90	----------------------------------------
18.98/5.90	
18.98/5.90	(24)
18.98/5.90	Obligation:
18.98/5.90	Innermost TRS:
18.98/5.90	Rules:
18.98/5.90	le(0', y) -> true
18.98/5.90	le(s(x), 0') -> false
18.98/5.90	le(s(x), s(y)) -> le(x, y)
18.98/5.90	minus(x, 0') -> x
18.98/5.90	minus(s(x), s(y)) -> minus(x, y)
18.98/5.90	mod(0', y) -> 0'
18.98/5.90	mod(s(x), 0') -> 0'
18.98/5.90	mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
18.98/5.90	if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
18.98/5.90	if_mod(false, s(x), s(y)) -> s(x)
18.98/5.90	
18.98/5.90	Types:
18.98/5.90	le :: 0':s -> 0':s -> true:false
18.98/5.90	0' :: 0':s
18.98/5.90	true :: true:false
18.98/5.90	s :: 0':s -> 0':s
18.98/5.90	false :: true:false
18.98/5.90	minus :: 0':s -> 0':s -> 0':s
18.98/5.90	mod :: 0':s -> 0':s -> 0':s
18.98/5.90	if_mod :: true:false -> 0':s -> 0':s -> 0':s
18.98/5.90	hole_true:false1_0 :: true:false
18.98/5.90	hole_0':s2_0 :: 0':s
18.98/5.90	gen_0':s3_0 :: Nat -> 0':s
18.98/5.90	
18.98/5.90	
18.98/5.90	Lemmas:
18.98/5.90	le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
18.98/5.90	minus(gen_0':s3_0(n294_0), gen_0':s3_0(n294_0)) -> gen_0':s3_0(0), rt in Omega(1 + n294_0)
18.98/5.90	
18.98/5.90	
18.98/5.90	Generator Equations:
18.98/5.90	gen_0':s3_0(0) <=> 0'
18.98/5.90	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
18.98/5.90	
18.98/5.90	
18.98/5.90	The following defined symbols remain to be analysed:
18.98/5.90	mod
18.98/5.92	EOF