20.42/7.87	WORST_CASE(Omega(n^2), O(n^2))
20.42/7.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.42/7.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.42/7.88	
20.42/7.88	
20.42/7.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).
20.42/7.88	
20.42/7.88	(0) CpxTRS
20.42/7.88	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	(2) CpxTRS
20.42/7.88	    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
20.42/7.88	    (4) CdtProblem
20.42/7.88	    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	    (6) CdtProblem
20.42/7.88	    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	    (8) CdtProblem
20.42/7.88	    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
20.42/7.88	    (10) CdtProblem
20.42/7.88	    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 39 ms]
20.42/7.88	    (12) CdtProblem
20.42/7.88	    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	    (14) BOUNDS(1, 1)
20.42/7.88	(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	(16) CpxTRS
20.42/7.88	    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.42/7.88	    (18) typed CpxTrs
20.42/7.88	    (19) OrderProof [LOWER BOUND(ID), 0 ms]
20.42/7.88	    (20) typed CpxTrs
20.42/7.88	    (21) RewriteLemmaProof [LOWER BOUND(ID), 292 ms]
20.42/7.88	    (22) BEST
20.42/7.88	        (23) proven lower bound
20.42/7.88	            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
20.42/7.88	            (25) BOUNDS(n^1, INF)
20.42/7.88	        (26) typed CpxTrs
20.42/7.88	            (27) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
20.42/7.88	            (28) proven lower bound
20.42/7.88	            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
20.42/7.88	            (30) BOUNDS(n^2, INF)
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(0)
20.42/7.88	Obligation:
20.42/7.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).
20.42/7.88	
20.42/7.88	
20.42/7.88	The TRS R consists of the following rules:
20.42/7.88	
20.42/7.88	   and(tt, X) -> activate(X)
20.42/7.88	   plus(N, 0) -> N
20.42/7.88	   plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	   x(N, 0) -> 0
20.42/7.88	   x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	   activate(X) -> X
20.42/7.88	
20.42/7.88	S is empty.
20.42/7.88	Rewrite Strategy: FULL
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.42/7.88	Converted rc-obligation to irc-obligation.
20.42/7.88	
20.42/7.88	The duplicating contexts are:
20.42/7.88	x([], s(M))
20.42/7.88	
20.42/7.88	
20.42/7.88	The defined contexts are:
20.42/7.88	plus([], x1)
20.42/7.88	
20.42/7.88	
20.42/7.88	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(2)
20.42/7.88	Obligation:
20.42/7.88	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
20.42/7.88	
20.42/7.88	
20.42/7.88	The TRS R consists of the following rules:
20.42/7.88	
20.42/7.88	   and(tt, X) -> activate(X)
20.42/7.88	   plus(N, 0) -> N
20.42/7.88	   plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	   x(N, 0) -> 0
20.42/7.88	   x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	   activate(X) -> X
20.42/7.88	
20.42/7.88	S is empty.
20.42/7.88	Rewrite Strategy: INNERMOST
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(3) CpxTrsToCdtProof (UPPER BOUND(ID))
20.42/7.88	Converted Cpx (relative) TRS to CDT
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(4)
20.42/7.88	Obligation:
20.42/7.88	Complexity Dependency Tuples Problem
20.42/7.88	
20.42/7.88	Rules:
20.42/7.88	   and(tt, z0) -> activate(z0)
20.42/7.88	   plus(z0, 0) -> z0
20.42/7.88	   plus(z0, s(z1)) -> s(plus(z0, z1))
20.42/7.88	   x(z0, 0) -> 0
20.42/7.88	   x(z0, s(z1)) -> plus(x(z0, z1), z0)
20.42/7.88	   activate(z0) -> z0
20.42/7.88	Tuples:
20.42/7.88	   AND(tt, z0) -> c(ACTIVATE(z0))
20.42/7.88	   PLUS(z0, 0) -> c1
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, 0) -> c3
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	   ACTIVATE(z0) -> c5
20.42/7.88	S tuples:
20.42/7.88	   AND(tt, z0) -> c(ACTIVATE(z0))
20.42/7.88	   PLUS(z0, 0) -> c1
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, 0) -> c3
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	   ACTIVATE(z0) -> c5
20.42/7.88	K tuples:none
20.42/7.88	Defined Rule Symbols:   and_2, plus_2, x_2, activate_1
20.42/7.88	
20.42/7.88	Defined Pair Symbols:   AND_2, PLUS_2, X_2, ACTIVATE_1
20.42/7.88	
20.42/7.88	Compound Symbols:   c_1, c1, c2_1, c3, c4_2, c5
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
20.42/7.88	Removed 4 trailing nodes:
20.42/7.88	   X(z0, 0) -> c3
20.42/7.88	   ACTIVATE(z0) -> c5
20.42/7.88	   PLUS(z0, 0) -> c1
20.42/7.88	   AND(tt, z0) -> c(ACTIVATE(z0))
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(6)
20.42/7.88	Obligation:
20.42/7.88	Complexity Dependency Tuples Problem
20.42/7.88	
20.42/7.88	Rules:
20.42/7.88	   and(tt, z0) -> activate(z0)
20.42/7.88	   plus(z0, 0) -> z0
20.42/7.88	   plus(z0, s(z1)) -> s(plus(z0, z1))
20.42/7.88	   x(z0, 0) -> 0
20.42/7.88	   x(z0, s(z1)) -> plus(x(z0, z1), z0)
20.42/7.88	   activate(z0) -> z0
20.42/7.88	Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	S tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	K tuples:none
20.42/7.88	Defined Rule Symbols:   and_2, plus_2, x_2, activate_1
20.42/7.88	
20.42/7.88	Defined Pair Symbols:   PLUS_2, X_2
20.42/7.88	
20.42/7.88	Compound Symbols:   c2_1, c4_2
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
20.42/7.88	The following rules are not usable and were removed:
20.42/7.88	   and(tt, z0) -> activate(z0)
20.42/7.88	   activate(z0) -> z0
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(8)
20.42/7.88	Obligation:
20.42/7.88	Complexity Dependency Tuples Problem
20.42/7.88	
20.42/7.88	Rules:
20.42/7.88	   x(z0, 0) -> 0
20.42/7.88	   x(z0, s(z1)) -> plus(x(z0, z1), z0)
20.42/7.88	   plus(z0, 0) -> z0
20.42/7.88	   plus(z0, s(z1)) -> s(plus(z0, z1))
20.42/7.88	Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	S tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	K tuples:none
20.42/7.88	Defined Rule Symbols:   x_2, plus_2
20.42/7.88	
20.42/7.88	Defined Pair Symbols:   PLUS_2, X_2
20.42/7.88	
20.42/7.88	Compound Symbols:   c2_1, c4_2
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
20.42/7.88	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	We considered the (Usable) Rules:none
20.42/7.88	And the Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	The order we found is given by the following interpretation:
20.42/7.88	
20.42/7.88	Polynomial interpretation :
20.42/7.88	
20.42/7.88	   POL(0) = [1]
20.42/7.88	   POL(PLUS(x_1, x_2)) = 0
20.42/7.88	   POL(X(x_1, x_2)) = x_2
20.42/7.88	   POL(c2(x_1)) = x_1
20.42/7.88	   POL(c4(x_1, x_2)) = x_1 + x_2
20.42/7.88	   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
20.42/7.88	   POL(s(x_1)) = [1] + x_1
20.42/7.88	   POL(x(x_1, x_2)) = [1] + x_1 + x_2
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(10)
20.42/7.88	Obligation:
20.42/7.88	Complexity Dependency Tuples Problem
20.42/7.88	
20.42/7.88	Rules:
20.42/7.88	   x(z0, 0) -> 0
20.42/7.88	   x(z0, s(z1)) -> plus(x(z0, z1), z0)
20.42/7.88	   plus(z0, 0) -> z0
20.42/7.88	   plus(z0, s(z1)) -> s(plus(z0, z1))
20.42/7.88	Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	S tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	K tuples:
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	Defined Rule Symbols:   x_2, plus_2
20.42/7.88	
20.42/7.88	Defined Pair Symbols:   PLUS_2, X_2
20.42/7.88	
20.42/7.88	Compound Symbols:   c2_1, c4_2
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
20.42/7.88	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	We considered the (Usable) Rules:none
20.42/7.88	And the Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	The order we found is given by the following interpretation:
20.42/7.88	
20.42/7.88	Polynomial interpretation :
20.42/7.88	
20.42/7.88	   POL(0) = 0
20.42/7.88	   POL(PLUS(x_1, x_2)) = [2] + [2]x_2
20.42/7.88	   POL(X(x_1, x_2)) = x_2^2 + [2]x_1*x_2
20.42/7.88	   POL(c2(x_1)) = x_1
20.42/7.88	   POL(c4(x_1, x_2)) = x_1 + x_2
20.42/7.88	   POL(plus(x_1, x_2)) = [2] + x_2^2
20.42/7.88	   POL(s(x_1)) = [2] + x_1
20.42/7.88	   POL(x(x_1, x_2)) = x_1^2
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(12)
20.42/7.88	Obligation:
20.42/7.88	Complexity Dependency Tuples Problem
20.42/7.88	
20.42/7.88	Rules:
20.42/7.88	   x(z0, 0) -> 0
20.42/7.88	   x(z0, s(z1)) -> plus(x(z0, z1), z0)
20.42/7.88	   plus(z0, 0) -> z0
20.42/7.88	   plus(z0, s(z1)) -> s(plus(z0, z1))
20.42/7.88	Tuples:
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	S tuples:none
20.42/7.88	K tuples:
20.42/7.88	   X(z0, s(z1)) -> c4(PLUS(x(z0, z1), z0), X(z0, z1))
20.42/7.88	   PLUS(z0, s(z1)) -> c2(PLUS(z0, z1))
20.42/7.88	Defined Rule Symbols:   x_2, plus_2
20.42/7.88	
20.42/7.88	Defined Pair Symbols:   PLUS_2, X_2
20.42/7.88	
20.42/7.88	Compound Symbols:   c2_1, c4_2
20.42/7.88	
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
20.42/7.88	The set S is empty
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(14)
20.42/7.88	BOUNDS(1, 1)
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(15) RenamingProof (BOTH BOUNDS(ID, ID))
20.42/7.88	Renamed function symbols to avoid clashes with predefined symbol.
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(16)
20.42/7.88	Obligation:
20.42/7.88	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
20.42/7.88	
20.42/7.88	
20.42/7.88	The TRS R consists of the following rules:
20.42/7.88	
20.42/7.88	   and(tt, X) -> activate(X)
20.42/7.88	   plus(N, 0') -> N
20.42/7.88	   plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	   x(N, 0') -> 0'
20.42/7.88	   x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	   activate(X) -> X
20.42/7.88	
20.42/7.88	S is empty.
20.42/7.88	Rewrite Strategy: FULL
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.42/7.88	Infered types.
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(18)
20.42/7.88	Obligation:
20.42/7.88	TRS:
20.42/7.88	Rules:
20.42/7.88	and(tt, X) -> activate(X)
20.42/7.88	plus(N, 0') -> N
20.42/7.88	plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	x(N, 0') -> 0'
20.42/7.88	x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	activate(X) -> X
20.42/7.88	
20.42/7.88	Types:
20.42/7.88	and :: tt -> and:activate -> and:activate
20.42/7.88	tt :: tt
20.42/7.88	activate :: and:activate -> and:activate
20.42/7.88	plus :: 0':s -> 0':s -> 0':s
20.42/7.88	0' :: 0':s
20.42/7.88	s :: 0':s -> 0':s
20.42/7.88	x :: 0':s -> 0':s -> 0':s
20.42/7.88	hole_and:activate1_0 :: and:activate
20.42/7.88	hole_tt2_0 :: tt
20.42/7.88	hole_0':s3_0 :: 0':s
20.42/7.88	gen_0':s4_0 :: Nat -> 0':s
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(19) OrderProof (LOWER BOUND(ID))
20.42/7.88	Heuristically decided to analyse the following defined symbols:
20.42/7.88	plus, x
20.42/7.88	
20.42/7.88	They will be analysed ascendingly in the following order:
20.42/7.88	plus < x
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(20)
20.42/7.88	Obligation:
20.42/7.88	TRS:
20.42/7.88	Rules:
20.42/7.88	and(tt, X) -> activate(X)
20.42/7.88	plus(N, 0') -> N
20.42/7.88	plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	x(N, 0') -> 0'
20.42/7.88	x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	activate(X) -> X
20.42/7.88	
20.42/7.88	Types:
20.42/7.88	and :: tt -> and:activate -> and:activate
20.42/7.88	tt :: tt
20.42/7.88	activate :: and:activate -> and:activate
20.42/7.88	plus :: 0':s -> 0':s -> 0':s
20.42/7.88	0' :: 0':s
20.42/7.88	s :: 0':s -> 0':s
20.42/7.88	x :: 0':s -> 0':s -> 0':s
20.42/7.88	hole_and:activate1_0 :: and:activate
20.42/7.88	hole_tt2_0 :: tt
20.42/7.88	hole_0':s3_0 :: 0':s
20.42/7.88	gen_0':s4_0 :: Nat -> 0':s
20.42/7.88	
20.42/7.88	
20.42/7.88	Generator Equations:
20.42/7.88	gen_0':s4_0(0) <=> 0'
20.42/7.88	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
20.42/7.88	
20.42/7.88	
20.42/7.88	The following defined symbols remain to be analysed:
20.42/7.88	plus, x
20.42/7.88	
20.42/7.88	They will be analysed ascendingly in the following order:
20.42/7.88	plus < x
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(21) RewriteLemmaProof (LOWER BOUND(ID))
20.42/7.88	Proved the following rewrite lemma:
20.42/7.88	plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)
20.42/7.88	
20.42/7.88	Induction Base:
20.42/7.88	plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
20.42/7.88	gen_0':s4_0(a)
20.42/7.88	
20.42/7.88	Induction Step:
20.42/7.88	plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
20.42/7.88	s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) ->_IH
20.42/7.88	s(gen_0':s4_0(+(a, c7_0)))
20.42/7.88	
20.42/7.88	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(22)
20.42/7.88	Complex Obligation (BEST)
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(23)
20.42/7.88	Obligation:
20.42/7.88	Proved the lower bound n^1 for the following obligation:
20.42/7.88	
20.42/7.88	TRS:
20.42/7.88	Rules:
20.42/7.88	and(tt, X) -> activate(X)
20.42/7.88	plus(N, 0') -> N
20.42/7.88	plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	x(N, 0') -> 0'
20.42/7.88	x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	activate(X) -> X
20.42/7.88	
20.42/7.88	Types:
20.42/7.88	and :: tt -> and:activate -> and:activate
20.42/7.88	tt :: tt
20.42/7.88	activate :: and:activate -> and:activate
20.42/7.88	plus :: 0':s -> 0':s -> 0':s
20.42/7.88	0' :: 0':s
20.42/7.88	s :: 0':s -> 0':s
20.42/7.88	x :: 0':s -> 0':s -> 0':s
20.42/7.88	hole_and:activate1_0 :: and:activate
20.42/7.88	hole_tt2_0 :: tt
20.42/7.88	hole_0':s3_0 :: 0':s
20.42/7.88	gen_0':s4_0 :: Nat -> 0':s
20.42/7.88	
20.42/7.88	
20.42/7.88	Generator Equations:
20.42/7.88	gen_0':s4_0(0) <=> 0'
20.42/7.88	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
20.42/7.88	
20.42/7.88	
20.42/7.88	The following defined symbols remain to be analysed:
20.42/7.88	plus, x
20.42/7.88	
20.42/7.88	They will be analysed ascendingly in the following order:
20.42/7.88	plus < x
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(24) LowerBoundPropagationProof (FINISHED)
20.42/7.88	Propagated lower bound.
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(25)
20.42/7.88	BOUNDS(n^1, INF)
20.42/7.88	
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(26)
20.42/7.88	Obligation:
20.42/7.88	TRS:
20.42/7.88	Rules:
20.42/7.88	and(tt, X) -> activate(X)
20.42/7.88	plus(N, 0') -> N
20.42/7.88	plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	x(N, 0') -> 0'
20.42/7.88	x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	activate(X) -> X
20.42/7.88	
20.42/7.88	Types:
20.42/7.88	and :: tt -> and:activate -> and:activate
20.42/7.88	tt :: tt
20.42/7.88	activate :: and:activate -> and:activate
20.42/7.88	plus :: 0':s -> 0':s -> 0':s
20.42/7.88	0' :: 0':s
20.42/7.88	s :: 0':s -> 0':s
20.42/7.88	x :: 0':s -> 0':s -> 0':s
20.42/7.88	hole_and:activate1_0 :: and:activate
20.42/7.88	hole_tt2_0 :: tt
20.42/7.88	hole_0':s3_0 :: 0':s
20.42/7.88	gen_0':s4_0 :: Nat -> 0':s
20.42/7.88	
20.42/7.88	
20.42/7.88	Lemmas:
20.42/7.88	plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)
20.42/7.88	
20.42/7.88	
20.42/7.88	Generator Equations:
20.42/7.88	gen_0':s4_0(0) <=> 0'
20.42/7.88	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
20.42/7.88	
20.42/7.88	
20.42/7.88	The following defined symbols remain to be analysed:
20.42/7.88	x
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(27) RewriteLemmaProof (LOWER BOUND(ID))
20.42/7.88	Proved the following rewrite lemma:
20.42/7.88	x(gen_0':s4_0(a), gen_0':s4_0(n441_0)) -> gen_0':s4_0(*(n441_0, a)), rt in Omega(1 + a*n441_0 + n441_0)
20.42/7.88	
20.42/7.88	Induction Base:
20.42/7.88	x(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
20.42/7.88	0'
20.42/7.88	
20.42/7.88	Induction Step:
20.42/7.88	x(gen_0':s4_0(a), gen_0':s4_0(+(n441_0, 1))) ->_R^Omega(1)
20.42/7.88	plus(x(gen_0':s4_0(a), gen_0':s4_0(n441_0)), gen_0':s4_0(a)) ->_IH
20.42/7.88	plus(gen_0':s4_0(*(c442_0, a)), gen_0':s4_0(a)) ->_L^Omega(1 + a)
20.42/7.88	gen_0':s4_0(+(a, *(n441_0, a)))
20.42/7.88	
20.42/7.88	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
20.42/7.88	----------------------------------------
20.42/7.88	
20.42/7.88	(28)
20.42/7.88	Obligation:
20.42/7.88	Proved the lower bound n^2 for the following obligation:
20.42/7.88	
20.42/7.88	TRS:
20.42/7.88	Rules:
20.42/7.88	and(tt, X) -> activate(X)
20.42/7.88	plus(N, 0') -> N
20.42/7.88	plus(N, s(M)) -> s(plus(N, M))
20.42/7.88	x(N, 0') -> 0'
20.42/7.88	x(N, s(M)) -> plus(x(N, M), N)
20.42/7.88	activate(X) -> X
20.42/7.88	
20.42/7.88	Types:
20.42/7.88	and :: tt -> and:activate -> and:activate
20.42/7.88	tt :: tt
20.42/7.88	activate :: and:activate -> and:activate
20.42/7.88	plus :: 0':s -> 0':s -> 0':s
20.42/7.88	0' :: 0':s
20.42/7.88	s :: 0':s -> 0':s
20.42/7.88	x :: 0':s -> 0':s -> 0':s
20.42/7.88	hole_and:activate1_0 :: and:activate
20.42/7.88	hole_tt2_0 :: tt
20.42/7.88	hole_0':s3_0 :: 0':s
20.42/7.88	gen_0':s4_0 :: Nat -> 0':s
20.42/7.88	
20.42/7.88	
20.42/7.88	Lemmas:
20.42/7.88	plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) -> gen_0':s4_0(+(n6_0, a)), rt in Omega(1 + n6_0)
20.42/7.88	
20.42/7.88	
20.42/7.88	Generator Equations:
20.42/7.88	gen_0':s4_0(0) <=> 0'
20.42/7.89	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
20.42/7.89	
20.42/7.89	
20.42/7.89	The following defined symbols remain to be analysed:
20.42/7.89	x
20.42/7.89	----------------------------------------
20.42/7.89	
20.42/7.89	(29) LowerBoundPropagationProof (FINISHED)
20.42/7.89	Propagated lower bound.
20.42/7.89	----------------------------------------
20.42/7.89	
20.42/7.89	(30)
20.42/7.89	BOUNDS(n^2, INF)
20.56/7.92	EOF