30.72/10.13	WORST_CASE(Omega(n^2), O(n^2))
30.72/10.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.72/10.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.72/10.14	
30.72/10.14	
30.72/10.14	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).
30.72/10.14	
30.72/10.14	(0) CpxTRS
30.72/10.14	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
30.72/10.14	(2) CpxTRS
30.72/10.14	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (4) CpxTRS
30.72/10.14	    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 4 ms]
30.72/10.14	    (6) CdtProblem
30.72/10.14	    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (8) CdtProblem
30.72/10.14	    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (10) CdtProblem
30.72/10.14	    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 72 ms]
30.72/10.14	    (12) CdtProblem
30.72/10.14	    (13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (14) CdtProblem
30.72/10.14	    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 45 ms]
30.72/10.14	    (16) CdtProblem
30.72/10.14	    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (18) BOUNDS(1, 1)
30.72/10.14	(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	(20) CpxTRS
30.72/10.14	    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
30.72/10.14	    (22) typed CpxTrs
30.72/10.14	    (23) OrderProof [LOWER BOUND(ID), 0 ms]
30.72/10.14	    (24) typed CpxTrs
30.72/10.14	    (25) RewriteLemmaProof [LOWER BOUND(ID), 297 ms]
30.72/10.14	    (26) BEST
30.72/10.14	        (27) proven lower bound
30.72/10.14	            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
30.72/10.14	            (29) BOUNDS(n^1, INF)
30.72/10.14	        (30) typed CpxTrs
30.72/10.14	            (31) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
30.72/10.14	            (32) BEST
30.72/10.14	                (33) proven lower bound
30.72/10.14	                    (34) LowerBoundPropagationProof [FINISHED, 0 ms]
30.72/10.14	                    (35) BOUNDS(n^2, INF)
30.72/10.14	                (36) typed CpxTrs
30.72/10.14	                    (37) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
30.72/10.14	                    (38) typed CpxTrs
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(0)
30.72/10.14	Obligation:
30.72/10.14	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).
30.72/10.14	
30.72/10.14	
30.72/10.14	The TRS R consists of the following rules:
30.72/10.14	
30.72/10.14	   plus(x, 0) -> x
30.72/10.14	   plus(0, y) -> y
30.72/10.14	   plus(s(x), y) -> s(plus(x, y))
30.72/10.14	   times(0, y) -> 0
30.72/10.14	   times(s(0), y) -> y
30.72/10.14	   times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	   div(0, y) -> 0
30.72/10.14	   div(x, y) -> quot(x, y, y)
30.72/10.14	   quot(0, s(y), z) -> 0
30.72/10.14	   quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	   quot(x, 0, s(z)) -> s(div(x, s(z)))
30.72/10.14	   div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	S is empty.
30.72/10.14	Rewrite Strategy: FULL
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
30.72/10.14	The following defined symbols can occur below the 1th argument of plus: plus, times
30.72/10.14	
30.72/10.14	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(2)
30.72/10.14	Obligation:
30.72/10.14	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
30.72/10.14	
30.72/10.14	
30.72/10.14	The TRS R consists of the following rules:
30.72/10.14	
30.72/10.14	   plus(x, 0) -> x
30.72/10.14	   plus(0, y) -> y
30.72/10.14	   plus(s(x), y) -> s(plus(x, y))
30.72/10.14	   times(0, y) -> 0
30.72/10.14	   times(s(0), y) -> y
30.72/10.14	   times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	   div(0, y) -> 0
30.72/10.14	   div(x, y) -> quot(x, y, y)
30.72/10.14	   quot(0, s(y), z) -> 0
30.72/10.14	   quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	   quot(x, 0, s(z)) -> s(div(x, s(z)))
30.72/10.14	
30.72/10.14	S is empty.
30.72/10.14	Rewrite Strategy: FULL
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
30.72/10.14	Converted rc-obligation to irc-obligation.
30.72/10.14	
30.72/10.14	The duplicating contexts are:
30.72/10.14	times(s(x), [])
30.72/10.14	div(x, [])
30.72/10.14	
30.72/10.14	
30.72/10.14	The defined contexts are:
30.72/10.14	plus(x0, [])
30.72/10.14	
30.72/10.14	
30.72/10.14	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(4)
30.72/10.14	Obligation:
30.72/10.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
30.72/10.14	
30.72/10.14	
30.72/10.14	The TRS R consists of the following rules:
30.72/10.14	
30.72/10.14	   plus(x, 0) -> x
30.72/10.14	   plus(0, y) -> y
30.72/10.14	   plus(s(x), y) -> s(plus(x, y))
30.72/10.14	   times(0, y) -> 0
30.72/10.14	   times(s(0), y) -> y
30.72/10.14	   times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	   div(0, y) -> 0
30.72/10.14	   div(x, y) -> quot(x, y, y)
30.72/10.14	   quot(0, s(y), z) -> 0
30.72/10.14	   quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	   quot(x, 0, s(z)) -> s(div(x, s(z)))
30.72/10.14	
30.72/10.14	S is empty.
30.72/10.14	Rewrite Strategy: INNERMOST
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(5) CpxTrsToCdtProof (UPPER BOUND(ID))
30.72/10.14	Converted Cpx (relative) TRS to CDT
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(6)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   div(0, z0) -> 0
30.72/10.14	   div(z0, z1) -> quot(z0, z1, z1)
30.72/10.14	   quot(0, s(z0), z1) -> 0
30.72/10.14	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
30.72/10.14	   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(z0, 0) -> c
30.72/10.14	   PLUS(0, z0) -> c1
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(0, z0) -> c3
30.72/10.14	   TIMES(s(0), z0) -> c4
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(0, z0) -> c6
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(0, s(z0), z1) -> c8
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:
30.72/10.14	   PLUS(z0, 0) -> c
30.72/10.14	   PLUS(0, z0) -> c1
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(0, z0) -> c3
30.72/10.14	   TIMES(s(0), z0) -> c4
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(0, z0) -> c6
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(0, s(z0), z1) -> c8
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	K tuples:none
30.72/10.14	Defined Rule Symbols:   plus_2, times_2, div_2, quot_3
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6, c7_1, c8, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
30.72/10.14	Removed 6 trailing nodes:
30.72/10.14	   PLUS(0, z0) -> c1
30.72/10.14	   QUOT(0, s(z0), z1) -> c8
30.72/10.14	   DIV(0, z0) -> c6
30.72/10.14	   TIMES(s(0), z0) -> c4
30.72/10.14	   TIMES(0, z0) -> c3
30.72/10.14	   PLUS(z0, 0) -> c
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(8)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   div(0, z0) -> 0
30.72/10.14	   div(z0, z1) -> quot(z0, z1, z1)
30.72/10.14	   quot(0, s(z0), z1) -> 0
30.72/10.14	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
30.72/10.14	   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	K tuples:none
30.72/10.14	Defined Rule Symbols:   plus_2, times_2, div_2, quot_3
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c2_1, c5_2, c7_1, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
30.72/10.14	The following rules are not usable and were removed:
30.72/10.14	   div(0, z0) -> 0
30.72/10.14	   div(z0, z1) -> quot(z0, z1, z1)
30.72/10.14	   quot(0, s(z0), z1) -> 0
30.72/10.14	   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
30.72/10.14	   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(10)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	K tuples:none
30.72/10.14	Defined Rule Symbols:   times_2, plus_2
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c2_1, c5_2, c7_1, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
30.72/10.14	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	We considered the (Usable) Rules:none
30.72/10.14	And the Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	The order we found is given by the following interpretation:
30.72/10.14	
30.72/10.14	Polynomial interpretation :
30.72/10.14	
30.72/10.14	   POL(0) = [1]
30.72/10.14	   POL(DIV(x_1, x_2)) = x_1 + x_2
30.72/10.14	   POL(PLUS(x_1, x_2)) = 0
30.72/10.14	   POL(QUOT(x_1, x_2, x_3)) = x_1 + x_3
30.72/10.14	   POL(TIMES(x_1, x_2)) = x_1
30.72/10.14	   POL(c10(x_1)) = x_1
30.72/10.14	   POL(c2(x_1)) = x_1
30.72/10.14	   POL(c5(x_1, x_2)) = x_1 + x_2
30.72/10.14	   POL(c7(x_1)) = x_1
30.72/10.14	   POL(c9(x_1)) = x_1
30.72/10.14	   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
30.72/10.14	   POL(s(x_1)) = [1] + x_1
30.72/10.14	   POL(times(x_1, x_2)) = [1] + x_1 + x_2
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(12)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	K tuples:
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	Defined Rule Symbols:   times_2, plus_2
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c2_1, c5_2, c7_1, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
30.72/10.14	The following tuples could be moved from S to K by knowledge propagation:
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(14)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	K tuples:
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	Defined Rule Symbols:   times_2, plus_2
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c2_1, c5_2, c7_1, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
30.72/10.14	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	We considered the (Usable) Rules:none
30.72/10.14	And the Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	The order we found is given by the following interpretation:
30.72/10.14	
30.72/10.14	Polynomial interpretation :
30.72/10.14	
30.72/10.14	   POL(0) = 0
30.72/10.14	   POL(DIV(x_1, x_2)) = 0
30.72/10.14	   POL(PLUS(x_1, x_2)) = [2] + x_1
30.72/10.14	   POL(QUOT(x_1, x_2, x_3)) = 0
30.72/10.14	   POL(TIMES(x_1, x_2)) = x_1*x_2 + x_1^2
30.72/10.14	   POL(c10(x_1)) = x_1
30.72/10.14	   POL(c2(x_1)) = x_1
30.72/10.14	   POL(c5(x_1, x_2)) = x_1 + x_2
30.72/10.14	   POL(c7(x_1)) = x_1
30.72/10.14	   POL(c9(x_1)) = x_1
30.72/10.14	   POL(plus(x_1, x_2)) = [2]x_1 + x_2
30.72/10.14	   POL(s(x_1)) = [2] + x_1
30.72/10.14	   POL(times(x_1, x_2)) = x_1*x_2
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(16)
30.72/10.14	Obligation:
30.72/10.14	Complexity Dependency Tuples Problem
30.72/10.14	
30.72/10.14	Rules:
30.72/10.14	   times(0, z0) -> 0
30.72/10.14	   times(s(0), z0) -> z0
30.72/10.14	   times(s(z0), z1) -> plus(z1, times(z0, z1))
30.72/10.14	   plus(z0, 0) -> z0
30.72/10.14	   plus(0, z0) -> z0
30.72/10.14	   plus(s(z0), z1) -> s(plus(z0, z1))
30.72/10.14	Tuples:
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	S tuples:none
30.72/10.14	K tuples:
30.72/10.14	   TIMES(s(z0), z1) -> c5(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
30.72/10.14	   QUOT(s(z0), s(z1), z2) -> c9(QUOT(z0, z1, z2))
30.72/10.14	   QUOT(z0, 0, s(z1)) -> c10(DIV(z0, s(z1)))
30.72/10.14	   DIV(z0, z1) -> c7(QUOT(z0, z1, z1))
30.72/10.14	   PLUS(s(z0), z1) -> c2(PLUS(z0, z1))
30.72/10.14	Defined Rule Symbols:   times_2, plus_2
30.72/10.14	
30.72/10.14	Defined Pair Symbols:   PLUS_2, TIMES_2, DIV_2, QUOT_3
30.72/10.14	
30.72/10.14	Compound Symbols:   c2_1, c5_2, c7_1, c9_1, c10_1
30.72/10.14	
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(17) SIsEmptyProof (BOTH BOUNDS(ID, ID))
30.72/10.14	The set S is empty
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(18)
30.72/10.14	BOUNDS(1, 1)
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(19) RenamingProof (BOTH BOUNDS(ID, ID))
30.72/10.14	Renamed function symbols to avoid clashes with predefined symbol.
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(20)
30.72/10.14	Obligation:
30.72/10.14	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
30.72/10.14	
30.72/10.14	
30.72/10.14	The TRS R consists of the following rules:
30.72/10.14	
30.72/10.14	   plus(x, 0') -> x
30.72/10.14	   plus(0', y) -> y
30.72/10.14	   plus(s(x), y) -> s(plus(x, y))
30.72/10.14	   times(0', y) -> 0'
30.72/10.14	   times(s(0'), y) -> y
30.72/10.14	   times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	   div(0', y) -> 0'
30.72/10.14	   div(x, y) -> quot(x, y, y)
30.72/10.14	   quot(0', s(y), z) -> 0'
30.72/10.14	   quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	   quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	   div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	S is empty.
30.72/10.14	Rewrite Strategy: FULL
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(21) TypeInferenceProof (BOTH BOUNDS(ID, ID))
30.72/10.14	Infered types.
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(22)
30.72/10.14	Obligation:
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(23) OrderProof (LOWER BOUND(ID))
30.72/10.14	Heuristically decided to analyse the following defined symbols:
30.72/10.14	plus, times, div, quot
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	plus < times
30.72/10.14	times < div
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(24)
30.72/10.14	Obligation:
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	plus, times, div, quot
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	plus < times
30.72/10.14	times < div
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(25) RewriteLemmaProof (LOWER BOUND(ID))
30.72/10.14	Proved the following rewrite lemma:
30.72/10.14	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
30.72/10.14	
30.72/10.14	Induction Base:
30.72/10.14	plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
30.72/10.14	gen_0':s2_0(b)
30.72/10.14	
30.72/10.14	Induction Step:
30.72/10.14	plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
30.72/10.14	s(plus(gen_0':s2_0(n4_0), gen_0':s2_0(b))) ->_IH
30.72/10.14	s(gen_0':s2_0(+(b, c5_0)))
30.72/10.14	
30.72/10.14	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(26)
30.72/10.14	Complex Obligation (BEST)
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(27)
30.72/10.14	Obligation:
30.72/10.14	Proved the lower bound n^1 for the following obligation:
30.72/10.14	
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	plus, times, div, quot
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	plus < times
30.72/10.14	times < div
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(28) LowerBoundPropagationProof (FINISHED)
30.72/10.14	Propagated lower bound.
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(29)
30.72/10.14	BOUNDS(n^1, INF)
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(30)
30.72/10.14	Obligation:
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Lemmas:
30.72/10.14	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	times, div, quot
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	times < div
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(31) RewriteLemmaProof (LOWER BOUND(ID))
30.72/10.14	Proved the following rewrite lemma:
30.72/10.14	times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)
30.72/10.14	
30.72/10.14	Induction Base:
30.72/10.14	times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
30.72/10.14	0'
30.72/10.14	
30.72/10.14	Induction Step:
30.72/10.14	times(gen_0':s2_0(+(n577_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
30.72/10.14	plus(gen_0':s2_0(b), times(gen_0':s2_0(n577_0), gen_0':s2_0(b))) ->_IH
30.72/10.14	plus(gen_0':s2_0(b), gen_0':s2_0(*(c578_0, b))) ->_L^Omega(1 + b)
30.72/10.14	gen_0':s2_0(+(b, *(n577_0, b)))
30.72/10.14	
30.72/10.14	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(32)
30.72/10.14	Complex Obligation (BEST)
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(33)
30.72/10.14	Obligation:
30.72/10.14	Proved the lower bound n^2 for the following obligation:
30.72/10.14	
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Lemmas:
30.72/10.14	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	times, div, quot
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	times < div
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(34) LowerBoundPropagationProof (FINISHED)
30.72/10.14	Propagated lower bound.
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(35)
30.72/10.14	BOUNDS(n^2, INF)
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(36)
30.72/10.14	Obligation:
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Lemmas:
30.72/10.14	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
30.72/10.14	times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	quot, div
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	div = quot
30.72/10.14	
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(37) RewriteLemmaProof (LOWER BOUND(ID))
30.72/10.14	Proved the following rewrite lemma:
30.72/10.14	quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)
30.72/10.14	
30.72/10.14	Induction Base:
30.72/10.14	quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(c)) ->_R^Omega(1)
30.72/10.14	0'
30.72/10.14	
30.72/10.14	Induction Step:
30.72/10.14	quot(gen_0':s2_0(+(n1370_0, 1)), gen_0':s2_0(+(1, +(n1370_0, 1))), gen_0':s2_0(c)) ->_R^Omega(1)
30.72/10.14	quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) ->_IH
30.72/10.14	gen_0':s2_0(0)
30.72/10.14	
30.72/10.14	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
30.72/10.14	----------------------------------------
30.72/10.14	
30.72/10.14	(38)
30.72/10.14	Obligation:
30.72/10.14	TRS:
30.72/10.14	Rules:
30.72/10.14	plus(x, 0') -> x
30.72/10.14	plus(0', y) -> y
30.72/10.14	plus(s(x), y) -> s(plus(x, y))
30.72/10.14	times(0', y) -> 0'
30.72/10.14	times(s(0'), y) -> y
30.72/10.14	times(s(x), y) -> plus(y, times(x, y))
30.72/10.14	div(0', y) -> 0'
30.72/10.14	div(x, y) -> quot(x, y, y)
30.72/10.14	quot(0', s(y), z) -> 0'
30.72/10.14	quot(s(x), s(y), z) -> quot(x, y, z)
30.72/10.14	quot(x, 0', s(z)) -> s(div(x, s(z)))
30.72/10.14	div(div(x, y), z) -> div(x, times(y, z))
30.72/10.14	
30.72/10.14	Types:
30.72/10.14	plus :: 0':s -> 0':s -> 0':s
30.72/10.14	0' :: 0':s
30.72/10.14	s :: 0':s -> 0':s
30.72/10.14	times :: 0':s -> 0':s -> 0':s
30.72/10.14	div :: 0':s -> 0':s -> 0':s
30.72/10.14	quot :: 0':s -> 0':s -> 0':s -> 0':s
30.72/10.14	hole_0':s1_0 :: 0':s
30.72/10.14	gen_0':s2_0 :: Nat -> 0':s
30.72/10.14	
30.72/10.14	
30.72/10.14	Lemmas:
30.72/10.14	plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
30.72/10.14	times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)
30.72/10.14	quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)
30.72/10.14	
30.72/10.14	
30.72/10.14	Generator Equations:
30.72/10.14	gen_0':s2_0(0) <=> 0'
30.72/10.14	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
30.72/10.14	
30.72/10.14	
30.72/10.14	The following defined symbols remain to be analysed:
30.72/10.14	div
30.72/10.14	
30.72/10.14	They will be analysed ascendingly in the following order:
30.72/10.14	div = quot
30.90/10.18	EOF