22.49/8.20	WORST_CASE(Omega(n^1), O(n^1))
22.72/8.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.72/8.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.72/8.21	
22.72/8.21	
22.72/8.21	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.72/8.21	
22.72/8.21	(0) CpxTRS
22.72/8.21	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	(2) CpxTRS
22.72/8.21	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (4) CpxWeightedTrs
22.72/8.21	    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (6) CpxWeightedTrs
22.72/8.21	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (8) CpxTypedWeightedTrs
22.72/8.21	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
22.72/8.21	    (10) CpxTypedWeightedCompleteTrs
22.72/8.21	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (12) CpxTypedWeightedCompleteTrs
22.72/8.21	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
22.72/8.21	    (14) CpxRNTS
22.72/8.21	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (16) CpxRNTS
22.72/8.21	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (18) CpxRNTS
22.72/8.21	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.72/8.21	    (20) CpxRNTS
22.72/8.21	    (21) IntTrsBoundProof [UPPER BOUND(ID), 344 ms]
22.72/8.21	    (22) CpxRNTS
22.72/8.21	    (23) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
22.72/8.21	    (24) CpxRNTS
22.72/8.21	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
22.72/8.21	    (26) CpxRNTS
22.72/8.21	    (27) IntTrsBoundProof [UPPER BOUND(ID), 1197 ms]
22.72/8.21	    (28) CpxRNTS
22.72/8.21	    (29) IntTrsBoundProof [UPPER BOUND(ID), 587 ms]
22.72/8.21	    (30) CpxRNTS
22.72/8.21	    (31) FinalProof [FINISHED, 0 ms]
22.72/8.21	    (32) BOUNDS(1, n^1)
22.72/8.21	(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	(34) CpxTRS
22.72/8.21	    (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
22.72/8.21	    (36) typed CpxTrs
22.72/8.21	    (37) OrderProof [LOWER BOUND(ID), 0 ms]
22.72/8.21	    (38) typed CpxTrs
22.72/8.21	    (39) RewriteLemmaProof [LOWER BOUND(ID), 409 ms]
22.72/8.21	    (40) proven lower bound
22.72/8.21	    (41) LowerBoundPropagationProof [FINISHED, 0 ms]
22.72/8.21	    (42) BOUNDS(n^1, INF)
22.72/8.21	
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(0)
22.72/8.21	Obligation:
22.72/8.21	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
22.72/8.21	
22.72/8.21	
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
22.72/8.21	   f(true, x, y, z) -> del(.(y, z))
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	   =(nil, nil) -> true
22.72/8.21	   =(.(x, y), nil) -> false
22.72/8.21	   =(nil, .(y, z)) -> false
22.72/8.21	   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))
22.72/8.21	
22.72/8.21	S is empty.
22.72/8.21	Rewrite Strategy: FULL
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Converted rc-obligation to irc-obligation.
22.72/8.21	
22.72/8.21	The duplicating contexts are:
22.72/8.21	del(.([], .(y, z)))
22.72/8.21	del(.(x, .([], z)))
22.72/8.21	
22.72/8.21	
22.72/8.21	The defined contexts are:
22.72/8.21	f([], x1, x2, x3)
22.72/8.21	
22.72/8.21	
22.72/8.21	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(2)
22.72/8.21	Obligation:
22.72/8.21	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
22.72/8.21	
22.72/8.21	
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
22.72/8.21	   f(true, x, y, z) -> del(.(y, z))
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	   =(nil, nil) -> true
22.72/8.21	   =(.(x, y), nil) -> false
22.72/8.21	   =(nil, .(y, z)) -> false
22.72/8.21	   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))
22.72/8.21	
22.72/8.21	S is empty.
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Transformed relative TRS to weighted TRS
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(4)
22.72/8.21	Obligation:
22.72/8.21	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.72/8.21	
22.72/8.21	
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(=(x, y), x, y, z) [1]
22.72/8.21	   f(true, x, y, z) -> del(.(y, z)) [1]
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
22.72/8.21	   =(nil, nil) -> true [1]
22.72/8.21	   =(.(x, y), nil) -> false [1]
22.72/8.21	   =(nil, .(y, z)) -> false [1]
22.72/8.21	   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v)) [1]
22.72/8.21	
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Renamed defined symbols to avoid conflicts with arithmetic symbols:
22.72/8.21	
22.72/8.21	= => eq
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(6)
22.72/8.21	Obligation:
22.72/8.21	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
22.72/8.21	
22.72/8.21	
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
22.72/8.21	   f(true, x, y, z) -> del(.(y, z)) [1]
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
22.72/8.21	   eq(nil, nil) -> true [1]
22.72/8.21	   eq(.(x, y), nil) -> false [1]
22.72/8.21	   eq(nil, .(y, z)) -> false [1]
22.72/8.21	   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
22.72/8.21	
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Infered types.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(8)
22.72/8.21	Obligation:
22.72/8.21	Runtime Complexity Weighted TRS with Types.
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
22.72/8.21	   f(true, x, y, z) -> del(.(y, z)) [1]
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
22.72/8.21	   eq(nil, nil) -> true [1]
22.72/8.21	   eq(.(x, y), nil) -> false [1]
22.72/8.21	   eq(nil, .(y, z)) -> false [1]
22.72/8.21	   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
22.72/8.21	
22.72/8.21	The TRS has the following type information:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	eq :: .:nil:u:v -> .:nil:u:v -> true:false:and
22.72/8.21	true :: true:false:and
22.72/8.21	false :: true:false:and
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and -> true:false:and -> true:false:and
22.72/8.21	
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(9) CompletionProof (UPPER BOUND(ID))
22.72/8.21	The transformation into a RNTS is sound, since:
22.72/8.21	
22.72/8.21	(a) The obligation is a constructor system where every type has a constant constructor,
22.72/8.21	
22.72/8.21	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
22.72/8.21	
22.72/8.21	   del_1
22.72/8.21	   f_4
22.72/8.21	
22.72/8.21	(c) The following functions are completely defined:
22.72/8.21	
22.72/8.21	   eq_2
22.72/8.21	
22.72/8.21	Due to the following rules being added:
22.72/8.21	
22.72/8.21	   eq(v0, v1) -> null_eq [0]
22.72/8.21	
22.72/8.21	And the following fresh constants:    null_eq
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(10)
22.72/8.21	Obligation:
22.72/8.21	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.72/8.21	
22.72/8.21	Runtime Complexity Weighted TRS with Types.
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
22.72/8.21	   f(true, x, y, z) -> del(.(y, z)) [1]
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
22.72/8.21	   eq(nil, nil) -> true [1]
22.72/8.21	   eq(.(x, y), nil) -> false [1]
22.72/8.21	   eq(nil, .(y, z)) -> false [1]
22.72/8.21	   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
22.72/8.21	   eq(v0, v1) -> null_eq [0]
22.72/8.21	
22.72/8.21	The TRS has the following type information:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
22.72/8.21	true :: true:false:and:null_eq
22.72/8.21	false :: true:false:and:null_eq
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
22.72/8.21	null_eq :: true:false:and:null_eq
22.72/8.21	
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(12)
22.72/8.21	Obligation:
22.72/8.21	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
22.72/8.21	
22.72/8.21	Runtime Complexity Weighted TRS with Types.
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(nil, .(nil, z))) -> f(true, nil, nil, z) [2]
22.72/8.21	   del(.(.(x', y'), .(nil, z))) -> f(false, .(x', y'), nil, z) [2]
22.72/8.21	   del(.(nil, .(.(y'', z'), z))) -> f(false, nil, .(y'', z'), z) [2]
22.72/8.21	   del(.(.(x'', y1), .(.(u, v), z))) -> f(and(eq(x'', u), eq(y1, v)), .(x'', y1), .(u, v), z) [2]
22.72/8.21	   del(.(x, .(y, z))) -> f(null_eq, x, y, z) [1]
22.72/8.21	   f(true, x, y, z) -> del(.(y, z)) [1]
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
22.72/8.21	   eq(nil, nil) -> true [1]
22.72/8.21	   eq(.(x, y), nil) -> false [1]
22.72/8.21	   eq(nil, .(y, z)) -> false [1]
22.72/8.21	   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
22.72/8.21	   eq(v0, v1) -> null_eq [0]
22.72/8.21	
22.72/8.21	The TRS has the following type information:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
22.72/8.21	true :: true:false:and:null_eq
22.72/8.21	false :: true:false:and:null_eq
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
22.72/8.21	null_eq :: true:false:and:null_eq
22.72/8.21	
22.72/8.21	Rewrite Strategy: INNERMOST
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
22.72/8.21	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
22.72/8.21	The constant constructors are abstracted as follows:
22.72/8.21	
22.72/8.21	   true => 1
22.72/8.21	   false => 0
22.72/8.21	   nil => 0
22.72/8.21	   u => 1
22.72/8.21	   v => 2
22.72/8.21	   null_eq => 0
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(14)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z) :|: z >= 0, z'' = 1 + 0 + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + y + z) :|: z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = 1, z1 = x
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + x + del(1 + y + z) :|: z'' = 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z1 = x
22.72/8.21	
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(16)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Found the following analysis order by SCC decomposition:
22.72/8.21	
22.72/8.21	   { eq }
22.72/8.21	   { del, f }
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(18)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {eq}, {del,f}
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(19) ResultPropagationProof (UPPER BOUND(ID))
22.72/8.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(20)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {eq}, {del,f}
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(21) IntTrsBoundProof (UPPER BOUND(ID))
22.72/8.21	
22.72/8.21	Computed SIZE bound using CoFloCo for: eq
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(1) with polynomial bound: 1
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(22)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {eq}, {del,f}
22.72/8.21	Previous analysis results are:
22.72/8.21	  eq: runtime: ?, size: O(1) [1]
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(23) IntTrsBoundProof (UPPER BOUND(ID))
22.72/8.21	
22.72/8.21	Computed RUNTIME bound using CoFloCo for: eq
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(1) with polynomial bound: 3
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(24)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {del,f}
22.72/8.21	Previous analysis results are:
22.72/8.21	  eq: runtime: O(1) [3], size: O(1) [1]
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(25) ResultPropagationProof (UPPER BOUND(ID))
22.72/8.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(26)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {del,f}
22.72/8.21	Previous analysis results are:
22.72/8.21	  eq: runtime: O(1) [3], size: O(1) [1]
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(27) IntTrsBoundProof (UPPER BOUND(ID))
22.72/8.21	
22.72/8.21	Computed SIZE bound using CoFloCo for: del
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(1) with polynomial bound: 0
22.72/8.21	
22.72/8.21	Computed SIZE bound using CoFloCo for: f
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(n^1) with polynomial bound: 1 + z1
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(28)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: {del,f}
22.72/8.21	Previous analysis results are:
22.72/8.21	  eq: runtime: O(1) [3], size: O(1) [1]
22.72/8.21	  del: runtime: ?, size: O(1) [0]
22.72/8.21	  f: runtime: ?, size: O(n^1) [1 + z1]
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(29) IntTrsBoundProof (UPPER BOUND(ID))
22.72/8.21	
22.72/8.21	Computed RUNTIME bound using KoAT for: del
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(n^1) with polynomial bound: 34*z''
22.72/8.21	
22.72/8.21	Computed RUNTIME bound using CoFloCo for: f
22.72/8.21	after applying outer abstraction to obtain an ITS,
22.72/8.21	resulting in: O(n^1) with polynomial bound: 35 + 34*z2 + 34*z3
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(30)
22.72/8.21	Obligation:
22.72/8.21	Complexity RNTS consisting of the following rules:
22.72/8.21	
22.72/8.21	   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
22.72/8.21	   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
22.72/8.21	   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
22.72/8.21	   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
22.72/8.21	   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
22.72/8.21	   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
22.72/8.21	   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
22.72/8.21	   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
22.72/8.21	   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
22.72/8.21	   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0
22.72/8.21	
22.72/8.21	Function symbols to be analyzed: 
22.72/8.21	Previous analysis results are:
22.72/8.21	  eq: runtime: O(1) [3], size: O(1) [1]
22.72/8.21	  del: runtime: O(n^1) [34*z''], size: O(1) [0]
22.72/8.21	  f: runtime: O(n^1) [35 + 34*z2 + 34*z3], size: O(n^1) [1 + z1]
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(31) FinalProof (FINISHED)
22.72/8.21	Computed overall runtime complexity
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(32)
22.72/8.21	BOUNDS(1, n^1)
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(33) RenamingProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Renamed function symbols to avoid clashes with predefined symbol.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(34)
22.72/8.21	Obligation:
22.72/8.21	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
22.72/8.21	
22.72/8.21	
22.72/8.21	The TRS R consists of the following rules:
22.72/8.21	
22.72/8.21	   del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
22.72/8.21	   f(true, x, y, z) -> del(.(y, z))
22.72/8.21	   f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	   ='(nil, nil) -> true
22.72/8.21	   ='(.(x, y), nil) -> false
22.72/8.21	   ='(nil, .(y, z)) -> false
22.72/8.21	   ='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))
22.72/8.21	
22.72/8.21	S is empty.
22.72/8.21	Rewrite Strategy: FULL
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(35) TypeInferenceProof (BOTH BOUNDS(ID, ID))
22.72/8.21	Infered types.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(36)
22.72/8.21	Obligation:
22.72/8.21	TRS:
22.72/8.21	Rules:
22.72/8.21	del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
22.72/8.21	f(true, x, y, z) -> del(.(y, z))
22.72/8.21	f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	='(nil, nil) -> true
22.72/8.21	='(.(x, y), nil) -> false
22.72/8.21	='(nil, .(y, z)) -> false
22.72/8.21	='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))
22.72/8.21	
22.72/8.21	Types:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
22.72/8.21	true :: true:false:and
22.72/8.21	false :: true:false:and
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and -> true:false:and -> true:false:and
22.72/8.21	hole_.:nil:u:v1_0 :: .:nil:u:v
22.72/8.21	hole_true:false:and2_0 :: true:false:and
22.72/8.21	gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
22.72/8.21	gen_true:false:and4_0 :: Nat -> true:false:and
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(37) OrderProof (LOWER BOUND(ID))
22.72/8.21	Heuristically decided to analyse the following defined symbols:
22.72/8.21	del, ='
22.72/8.21	
22.72/8.21	They will be analysed ascendingly in the following order:
22.72/8.21	=' < del
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(38)
22.72/8.21	Obligation:
22.72/8.21	TRS:
22.72/8.21	Rules:
22.72/8.21	del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
22.72/8.21	f(true, x, y, z) -> del(.(y, z))
22.72/8.21	f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	='(nil, nil) -> true
22.72/8.21	='(.(x, y), nil) -> false
22.72/8.21	='(nil, .(y, z)) -> false
22.72/8.21	='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))
22.72/8.21	
22.72/8.21	Types:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
22.72/8.21	true :: true:false:and
22.72/8.21	false :: true:false:and
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and -> true:false:and -> true:false:and
22.72/8.21	hole_.:nil:u:v1_0 :: .:nil:u:v
22.72/8.21	hole_true:false:and2_0 :: true:false:and
22.72/8.21	gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
22.72/8.21	gen_true:false:and4_0 :: Nat -> true:false:and
22.72/8.21	
22.72/8.21	
22.72/8.21	Generator Equations:
22.72/8.21	gen_.:nil:u:v3_0(0) <=> nil
22.72/8.21	gen_.:nil:u:v3_0(+(x, 1)) <=> .(nil, gen_.:nil:u:v3_0(x))
22.72/8.21	gen_true:false:and4_0(0) <=> false
22.72/8.21	gen_true:false:and4_0(+(x, 1)) <=> and(false, gen_true:false:and4_0(x))
22.72/8.21	
22.72/8.21	
22.72/8.21	The following defined symbols remain to be analysed:
22.72/8.21	=', del
22.72/8.21	
22.72/8.21	They will be analysed ascendingly in the following order:
22.72/8.21	=' < del
22.72/8.21	
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(39) RewriteLemmaProof (LOWER BOUND(ID))
22.72/8.21	Proved the following rewrite lemma:
22.72/8.21	del(gen_.:nil:u:v3_0(+(2, n56_0))) -> *5_0, rt in Omega(n56_0)
22.72/8.21	
22.72/8.21	Induction Base:
22.72/8.21	del(gen_.:nil:u:v3_0(+(2, 0)))
22.72/8.21	
22.72/8.21	Induction Step:
22.72/8.21	del(gen_.:nil:u:v3_0(+(2, +(n56_0, 1)))) ->_R^Omega(1)
22.72/8.21	f(='(nil, nil), nil, nil, gen_.:nil:u:v3_0(+(1, n56_0))) ->_R^Omega(1)
22.72/8.21	f(true, nil, nil, gen_.:nil:u:v3_0(+(1, n56_0))) ->_R^Omega(1)
22.72/8.21	del(.(nil, gen_.:nil:u:v3_0(+(1, n56_0)))) ->_IH
22.72/8.21	*5_0
22.72/8.21	
22.72/8.21	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(40)
22.72/8.21	Obligation:
22.72/8.21	Proved the lower bound n^1 for the following obligation:
22.72/8.21	
22.72/8.21	TRS:
22.72/8.21	Rules:
22.72/8.21	del(.(x, .(y, z))) -> f(='(x, y), x, y, z)
22.72/8.21	f(true, x, y, z) -> del(.(y, z))
22.72/8.21	f(false, x, y, z) -> .(x, del(.(y, z)))
22.72/8.21	='(nil, nil) -> true
22.72/8.21	='(.(x, y), nil) -> false
22.72/8.21	='(nil, .(y, z)) -> false
22.72/8.21	='(.(x, y), .(u, v)) -> and(='(x, u), ='(y, v))
22.72/8.21	
22.72/8.21	Types:
22.72/8.21	del :: .:nil:u:v -> .:nil:u:v
22.72/8.21	. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
22.72/8.21	=' :: .:nil:u:v -> .:nil:u:v -> true:false:and
22.72/8.21	true :: true:false:and
22.72/8.21	false :: true:false:and
22.72/8.21	nil :: .:nil:u:v
22.72/8.21	u :: .:nil:u:v
22.72/8.21	v :: .:nil:u:v
22.72/8.21	and :: true:false:and -> true:false:and -> true:false:and
22.72/8.21	hole_.:nil:u:v1_0 :: .:nil:u:v
22.72/8.21	hole_true:false:and2_0 :: true:false:and
22.72/8.21	gen_.:nil:u:v3_0 :: Nat -> .:nil:u:v
22.72/8.21	gen_true:false:and4_0 :: Nat -> true:false:and
22.72/8.21	
22.72/8.21	
22.72/8.21	Generator Equations:
22.72/8.21	gen_.:nil:u:v3_0(0) <=> nil
22.72/8.21	gen_.:nil:u:v3_0(+(x, 1)) <=> .(nil, gen_.:nil:u:v3_0(x))
22.72/8.21	gen_true:false:and4_0(0) <=> false
22.72/8.21	gen_true:false:and4_0(+(x, 1)) <=> and(false, gen_true:false:and4_0(x))
22.72/8.21	
22.72/8.21	
22.72/8.21	The following defined symbols remain to be analysed:
22.72/8.21	del
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(41) LowerBoundPropagationProof (FINISHED)
22.72/8.21	Propagated lower bound.
22.72/8.21	----------------------------------------
22.72/8.21	
22.72/8.21	(42)
22.72/8.21	BOUNDS(n^1, INF)
22.72/8.24	EOF