378.69/291.52	WORST_CASE(Omega(n^1), O(n^2))
378.69/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
378.69/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
378.69/291.53	
378.69/291.53	
378.69/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
378.69/291.53	
378.69/291.53	(0) CpxTRS
378.69/291.53	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
378.69/291.53	(2) CpxTRS
378.69/291.53	    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (4) CpxTRS
378.69/291.53	    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
378.69/291.53	    (6) CdtProblem
378.69/291.53	    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (8) CdtProblem
378.69/291.53	    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (10) CdtProblem
378.69/291.53	    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 102 ms]
378.69/291.53	    (12) CdtProblem
378.69/291.53	    (13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (14) CdtProblem
378.69/291.53	    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 124 ms]
378.69/291.53	    (16) CdtProblem
378.69/291.53	    (17) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (18) CdtProblem
378.69/291.53	    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 84 ms]
378.69/291.53	    (20) CdtProblem
378.69/291.53	    (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (22) BOUNDS(1, 1)
378.69/291.53	(23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	(24) CpxTRS
378.69/291.53	    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
378.69/291.53	    (26) typed CpxTrs
378.69/291.53	    (27) OrderProof [LOWER BOUND(ID), 0 ms]
378.69/291.53	    (28) typed CpxTrs
378.69/291.53	    (29) RewriteLemmaProof [LOWER BOUND(ID), 314 ms]
378.69/291.53	    (30) BEST
378.69/291.53	        (31) proven lower bound
378.69/291.53	            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
378.69/291.53	            (33) BOUNDS(n^1, INF)
378.69/291.53	        (34) typed CpxTrs
378.69/291.53	            (35) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
378.69/291.53	            (36) typed CpxTrs
378.69/291.53	
378.69/291.53	
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(0)
378.69/291.53	Obligation:
378.69/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).
378.69/291.53	
378.69/291.53	
378.69/291.53	The TRS R consists of the following rules:
378.69/291.53	
378.69/291.53	   if(true, x, y) -> x
378.69/291.53	   if(false, x, y) -> y
378.69/291.53	   eq(0, 0) -> true
378.69/291.53	   eq(0, s(x)) -> false
378.69/291.53	   eq(s(x), 0) -> false
378.69/291.53	   eq(s(x), s(y)) -> eq(x, y)
378.69/291.53	   app(nil, l) -> l
378.69/291.53	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.53	   app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.53	   mem(x, nil) -> false
378.69/291.53	   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.53	   ifmem(true, x, l) -> true
378.69/291.53	   ifmem(false, x, l) -> mem(x, l)
378.69/291.53	   inter(x, nil) -> nil
378.69/291.53	   inter(nil, x) -> nil
378.69/291.53	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.53	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.53	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.53	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.53	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.53	   ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.53	
378.69/291.53	S is empty.
378.69/291.53	Rewrite Strategy: FULL
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
378.69/291.53	The following defined symbols can occur below the 0th argument of ifinter: ifmem, eq, mem
378.69/291.53	The following defined symbols can occur below the 0th argument of ifmem: eq
378.69/291.53	
378.69/291.53	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
378.69/291.53	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.53	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.53	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.53	
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(2)
378.69/291.53	Obligation:
378.69/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
378.69/291.53	
378.69/291.53	
378.69/291.53	The TRS R consists of the following rules:
378.69/291.53	
378.69/291.53	   if(true, x, y) -> x
378.69/291.53	   if(false, x, y) -> y
378.69/291.53	   eq(0, 0) -> true
378.69/291.53	   eq(0, s(x)) -> false
378.69/291.53	   eq(s(x), 0) -> false
378.69/291.53	   eq(s(x), s(y)) -> eq(x, y)
378.69/291.53	   app(nil, l) -> l
378.69/291.53	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.53	   mem(x, nil) -> false
378.69/291.53	   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.53	   ifmem(true, x, l) -> true
378.69/291.53	   ifmem(false, x, l) -> mem(x, l)
378.69/291.53	   inter(x, nil) -> nil
378.69/291.53	   inter(nil, x) -> nil
378.69/291.53	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.53	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.53	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.53	   ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.53	
378.69/291.53	S is empty.
378.69/291.53	Rewrite Strategy: FULL
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
378.69/291.53	Converted rc-obligation to irc-obligation.
378.69/291.53	
378.69/291.53	The duplicating contexts are:
378.69/291.53	mem([], cons(y, l))
378.69/291.53	inter(cons([], l1), l2)
378.69/291.53	inter(cons(x, l1), [])
378.69/291.53	inter(l1, cons([], l2))
378.69/291.53	inter([], cons(x, l2))
378.69/291.53	
378.69/291.53	
378.69/291.53	The defined contexts are:
378.69/291.53	ifinter([], x1, x2, x3)
378.69/291.53	ifmem([], x1, x2)
378.69/291.53	
378.69/291.53	
378.69/291.53	[] just represents basic- or constructor-terms in the following defined contexts:
378.69/291.53	ifmem([], x1, x2)
378.69/291.53	
378.69/291.53	
378.69/291.53	As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(4)
378.69/291.53	Obligation:
378.69/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).
378.69/291.53	
378.69/291.53	
378.69/291.53	The TRS R consists of the following rules:
378.69/291.53	
378.69/291.53	   if(true, x, y) -> x
378.69/291.53	   if(false, x, y) -> y
378.69/291.53	   eq(0, 0) -> true
378.69/291.53	   eq(0, s(x)) -> false
378.69/291.53	   eq(s(x), 0) -> false
378.69/291.53	   eq(s(x), s(y)) -> eq(x, y)
378.69/291.53	   app(nil, l) -> l
378.69/291.53	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.53	   mem(x, nil) -> false
378.69/291.53	   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.53	   ifmem(true, x, l) -> true
378.69/291.53	   ifmem(false, x, l) -> mem(x, l)
378.69/291.53	   inter(x, nil) -> nil
378.69/291.53	   inter(nil, x) -> nil
378.69/291.53	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.53	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.53	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.53	   ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.53	
378.69/291.53	S is empty.
378.69/291.53	Rewrite Strategy: INNERMOST
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(5) CpxTrsToCdtProof (UPPER BOUND(ID))
378.69/291.53	Converted Cpx (relative) TRS to CDT
378.69/291.53	----------------------------------------
378.69/291.53	
378.69/291.53	(6)
378.69/291.53	Obligation:
378.69/291.53	Complexity Dependency Tuples Problem
378.69/291.53	
378.69/291.53	Rules:
378.69/291.53	   if(true, z0, z1) -> z0
378.69/291.53	   if(false, z0, z1) -> z1
378.69/291.53	   eq(0, 0) -> true
378.69/291.53	   eq(0, s(z0)) -> false
378.69/291.53	   eq(s(z0), 0) -> false
378.69/291.53	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.53	   app(nil, z0) -> z0
378.69/291.53	   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
378.69/291.53	   mem(z0, nil) -> false
378.69/291.53	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.53	   ifmem(true, z0, z1) -> true
378.69/291.53	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.53	   inter(z0, nil) -> nil
378.69/291.53	   inter(nil, z0) -> nil
378.69/291.53	   inter(cons(z0, z1), z2) -> ifinter(mem(z0, z2), z0, z1, z2)
378.69/291.53	   inter(z0, cons(z1, z2)) -> ifinter(mem(z1, z0), z1, z2, z0)
378.69/291.54	   ifinter(true, z0, z1, z2) -> cons(z0, inter(z1, z2))
378.69/291.54	   ifinter(false, z0, z1, z2) -> inter(z1, z2)
378.69/291.54	Tuples:
378.69/291.54	   IF(true, z0, z1) -> c
378.69/291.54	   IF(false, z0, z1) -> c1
378.69/291.54	   EQ(0, 0) -> c2
378.69/291.54	   EQ(0, s(z0)) -> c3
378.69/291.54	   EQ(s(z0), 0) -> c4
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(nil, z0) -> c6
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, nil) -> c8
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(true, z0, z1) -> c10
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(z0, nil) -> c12
378.69/291.54	   INTER(nil, z0) -> c13
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   IF(true, z0, z1) -> c
378.69/291.54	   IF(false, z0, z1) -> c1
378.69/291.54	   EQ(0, 0) -> c2
378.69/291.54	   EQ(0, s(z0)) -> c3
378.69/291.54	   EQ(s(z0), 0) -> c4
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(nil, z0) -> c6
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, nil) -> c8
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(true, z0, z1) -> c10
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(z0, nil) -> c12
378.69/291.54	   INTER(nil, z0) -> c13
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	K tuples:none
378.69/291.54	Defined Rule Symbols:   if_3, eq_2, app_2, mem_2, ifmem_3, inter_2, ifinter_4
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   IF_3, EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c, c1, c2, c3, c4, c5_1, c6, c7_1, c8, c9_2, c10, c11_1, c12, c13, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
378.69/291.54	Removed 10 trailing nodes:
378.69/291.54	   INTER(z0, nil) -> c12
378.69/291.54	   IF(true, z0, z1) -> c
378.69/291.54	   EQ(0, s(z0)) -> c3
378.69/291.54	   EQ(0, 0) -> c2
378.69/291.54	   IFMEM(true, z0, z1) -> c10
378.69/291.54	   APP(nil, z0) -> c6
378.69/291.54	   EQ(s(z0), 0) -> c4
378.69/291.54	   IF(false, z0, z1) -> c1
378.69/291.54	   INTER(nil, z0) -> c13
378.69/291.54	   MEM(z0, nil) -> c8
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(8)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   if(true, z0, z1) -> z0
378.69/291.54	   if(false, z0, z1) -> z1
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   app(nil, z0) -> z0
378.69/291.54	   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	   inter(z0, nil) -> nil
378.69/291.54	   inter(nil, z0) -> nil
378.69/291.54	   inter(cons(z0, z1), z2) -> ifinter(mem(z0, z2), z0, z1, z2)
378.69/291.54	   inter(z0, cons(z1, z2)) -> ifinter(mem(z1, z0), z1, z2, z0)
378.69/291.54	   ifinter(true, z0, z1, z2) -> cons(z0, inter(z1, z2))
378.69/291.54	   ifinter(false, z0, z1, z2) -> inter(z1, z2)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	K tuples:none
378.69/291.54	Defined Rule Symbols:   if_3, eq_2, app_2, mem_2, ifmem_3, inter_2, ifinter_4
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
378.69/291.54	The following rules are not usable and were removed:
378.69/291.54	   if(true, z0, z1) -> z0
378.69/291.54	   if(false, z0, z1) -> z1
378.69/291.54	   app(nil, z0) -> z0
378.69/291.54	   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
378.69/291.54	   inter(z0, nil) -> nil
378.69/291.54	   inter(nil, z0) -> nil
378.69/291.54	   inter(cons(z0, z1), z2) -> ifinter(mem(z0, z2), z0, z1, z2)
378.69/291.54	   inter(z0, cons(z1, z2)) -> ifinter(mem(z1, z0), z1, z2, z0)
378.69/291.54	   ifinter(true, z0, z1, z2) -> cons(z0, inter(z1, z2))
378.69/291.54	   ifinter(false, z0, z1, z2) -> inter(z1, z2)
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(10)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	K tuples:none
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
378.69/291.54	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	We considered the (Usable) Rules:
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	And the Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	The order we found is given by the following interpretation:
378.69/291.54	
378.69/291.54	Polynomial interpretation :
378.69/291.54	
378.69/291.54	   POL(0) = [1]
378.69/291.54	   POL(APP(x_1, x_2)) = x_1
378.69/291.54	   POL(EQ(x_1, x_2)) = 0
378.69/291.54	   POL(IFINTER(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_3 + x_4
378.69/291.54	   POL(IFMEM(x_1, x_2, x_3)) = 0
378.69/291.54	   POL(INTER(x_1, x_2)) = [1] + x_1 + x_2
378.69/291.54	   POL(MEM(x_1, x_2)) = 0
378.69/291.54	   POL(c11(x_1)) = x_1
378.69/291.54	   POL(c14(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c15(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c16(x_1)) = x_1
378.69/291.54	   POL(c17(x_1)) = x_1
378.69/291.54	   POL(c5(x_1)) = x_1
378.69/291.54	   POL(c7(x_1)) = x_1
378.69/291.54	   POL(c9(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
378.69/291.54	   POL(eq(x_1, x_2)) = [1] + x_1 + x_2
378.69/291.54	   POL(false) = [1]
378.69/291.54	   POL(ifmem(x_1, x_2, x_3)) = [1] + x_2
378.69/291.54	   POL(mem(x_1, x_2)) = [1] + x_1
378.69/291.54	   POL(nil) = [1]
378.69/291.54	   POL(s(x_1)) = [1] + x_1
378.69/291.54	   POL(true) = [1]
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(12)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	K tuples:
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
378.69/291.54	The following tuples could be moved from S to K by knowledge propagation:
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(14)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	K tuples:
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
378.69/291.54	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	We considered the (Usable) Rules:none
378.69/291.54	And the Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	The order we found is given by the following interpretation:
378.69/291.54	
378.69/291.54	Polynomial interpretation :
378.69/291.54	
378.69/291.54	   POL(0) = 0
378.69/291.54	   POL(APP(x_1, x_2)) = [2]x_1 + [2]x_1*x_2 + x_1^2
378.69/291.54	   POL(EQ(x_1, x_2)) = 0
378.69/291.54	   POL(IFINTER(x_1, x_2, x_3, x_4)) = x_4 + [2]x_3*x_4
378.69/291.54	   POL(IFMEM(x_1, x_2, x_3)) = x_3
378.69/291.54	   POL(INTER(x_1, x_2)) = [2]x_1*x_2
378.69/291.54	   POL(MEM(x_1, x_2)) = x_2
378.69/291.54	   POL(c11(x_1)) = x_1
378.69/291.54	   POL(c14(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c15(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c16(x_1)) = x_1
378.69/291.54	   POL(c17(x_1)) = x_1
378.69/291.54	   POL(c5(x_1)) = x_1
378.69/291.54	   POL(c7(x_1)) = x_1
378.69/291.54	   POL(c9(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(cons(x_1, x_2)) = [2] + x_2
378.69/291.54	   POL(eq(x_1, x_2)) = [2]
378.69/291.54	   POL(false) = 0
378.69/291.54	   POL(ifmem(x_1, x_2, x_3)) = [2]x_1 + x_3 + [2]x_3^2
378.69/291.54	   POL(mem(x_1, x_2)) = x_2 + [2]x_2^2
378.69/291.54	   POL(nil) = 0
378.69/291.54	   POL(s(x_1)) = 0
378.69/291.54	   POL(true) = 0
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(16)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	K tuples:
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(17) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
378.69/291.54	The following tuples could be moved from S to K by knowledge propagation:
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(18)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	K tuples:
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
378.69/291.54	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	We considered the (Usable) Rules:none
378.69/291.54	And the Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	The order we found is given by the following interpretation:
378.69/291.54	
378.69/291.54	Polynomial interpretation :
378.69/291.54	
378.69/291.54	   POL(0) = 0
378.69/291.54	   POL(APP(x_1, x_2)) = [2]x_1 + [2]x_1*x_2 + [2]x_1^2
378.69/291.54	   POL(EQ(x_1, x_2)) = x_1*x_2
378.69/291.54	   POL(IFINTER(x_1, x_2, x_3, x_4)) = [2]x_4 + [2]x_3*x_4
378.69/291.54	   POL(IFMEM(x_1, x_2, x_3)) = [2]x_2*x_3
378.69/291.54	   POL(INTER(x_1, x_2)) = [2]x_2 + [2]x_1*x_2
378.69/291.54	   POL(MEM(x_1, x_2)) = [2]x_1*x_2
378.69/291.54	   POL(c11(x_1)) = x_1
378.69/291.54	   POL(c14(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c15(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(c16(x_1)) = x_1
378.69/291.54	   POL(c17(x_1)) = x_1
378.69/291.54	   POL(c5(x_1)) = x_1
378.69/291.54	   POL(c7(x_1)) = x_1
378.69/291.54	   POL(c9(x_1, x_2)) = x_1 + x_2
378.69/291.54	   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
378.69/291.54	   POL(eq(x_1, x_2)) = [1]
378.69/291.54	   POL(false) = [2]
378.69/291.54	   POL(ifmem(x_1, x_2, x_3)) = [2]x_1 + [2]x_3^2
378.69/291.54	   POL(mem(x_1, x_2)) = [2] + [2]x_2^2
378.69/291.54	   POL(nil) = 0
378.69/291.54	   POL(s(x_1)) = [1] + x_1
378.69/291.54	   POL(true) = 0
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(20)
378.69/291.54	Obligation:
378.69/291.54	Complexity Dependency Tuples Problem
378.69/291.54	
378.69/291.54	Rules:
378.69/291.54	   eq(0, 0) -> true
378.69/291.54	   eq(0, s(z0)) -> false
378.69/291.54	   eq(s(z0), 0) -> false
378.69/291.54	   eq(s(z0), s(z1)) -> eq(z0, z1)
378.69/291.54	   mem(z0, nil) -> false
378.69/291.54	   mem(z0, cons(z1, z2)) -> ifmem(eq(z0, z1), z0, z2)
378.69/291.54	   ifmem(true, z0, z1) -> true
378.69/291.54	   ifmem(false, z0, z1) -> mem(z0, z1)
378.69/291.54	Tuples:
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	S tuples:none
378.69/291.54	K tuples:
378.69/291.54	   APP(cons(z0, z1), z2) -> c7(APP(z1, z2))
378.69/291.54	   IFINTER(true, z0, z1, z2) -> c16(INTER(z1, z2))
378.69/291.54	   IFINTER(false, z0, z1, z2) -> c17(INTER(z1, z2))
378.69/291.54	   INTER(cons(z0, z1), z2) -> c14(IFINTER(mem(z0, z2), z0, z1, z2), MEM(z0, z2))
378.69/291.54	   INTER(z0, cons(z1, z2)) -> c15(IFINTER(mem(z1, z0), z1, z2, z0), MEM(z1, z0))
378.69/291.54	   MEM(z0, cons(z1, z2)) -> c9(IFMEM(eq(z0, z1), z0, z2), EQ(z0, z1))
378.69/291.54	   IFMEM(false, z0, z1) -> c11(MEM(z0, z1))
378.69/291.54	   EQ(s(z0), s(z1)) -> c5(EQ(z0, z1))
378.69/291.54	Defined Rule Symbols:   eq_2, mem_2, ifmem_3
378.69/291.54	
378.69/291.54	Defined Pair Symbols:   EQ_2, APP_2, MEM_2, IFMEM_3, INTER_2, IFINTER_4
378.69/291.54	
378.69/291.54	Compound Symbols:   c5_1, c7_1, c9_2, c11_1, c14_2, c15_2, c16_1, c17_1
378.69/291.54	
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(21) SIsEmptyProof (BOTH BOUNDS(ID, ID))
378.69/291.54	The set S is empty
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(22)
378.69/291.54	BOUNDS(1, 1)
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(23) RenamingProof (BOTH BOUNDS(ID, ID))
378.69/291.54	Renamed function symbols to avoid clashes with predefined symbol.
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(24)
378.69/291.54	Obligation:
378.69/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
378.69/291.54	
378.69/291.54	
378.69/291.54	The TRS R consists of the following rules:
378.69/291.54	
378.69/291.54	   if(true, x, y) -> x
378.69/291.54	   if(false, x, y) -> y
378.69/291.54	   eq(0', 0') -> true
378.69/291.54	   eq(0', s(x)) -> false
378.69/291.54	   eq(s(x), 0') -> false
378.69/291.54	   eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	   app(nil, l) -> l
378.69/291.54	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	   app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	   mem(x, nil) -> false
378.69/291.54	   mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	   ifmem(true, x, l) -> true
378.69/291.54	   ifmem(false, x, l) -> mem(x, l)
378.69/291.54	   inter(x, nil) -> nil
378.69/291.54	   inter(nil, x) -> nil
378.69/291.54	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	   ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	S is empty.
378.69/291.54	Rewrite Strategy: FULL
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(25) TypeInferenceProof (BOTH BOUNDS(ID, ID))
378.69/291.54	Infered types.
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(26)
378.69/291.54	Obligation:
378.69/291.54	TRS:
378.69/291.54	Rules:
378.69/291.54	if(true, x, y) -> x
378.69/291.54	if(false, x, y) -> y
378.69/291.54	eq(0', 0') -> true
378.69/291.54	eq(0', s(x)) -> false
378.69/291.54	eq(s(x), 0') -> false
378.69/291.54	eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	app(nil, l) -> l
378.69/291.54	app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	mem(x, nil) -> false
378.69/291.54	mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	ifmem(true, x, l) -> true
378.69/291.54	ifmem(false, x, l) -> mem(x, l)
378.69/291.54	inter(x, nil) -> nil
378.69/291.54	inter(nil, x) -> nil
378.69/291.54	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	Types:
378.69/291.54	if :: true:false -> if -> if -> if
378.69/291.54	true :: true:false
378.69/291.54	false :: true:false
378.69/291.54	eq :: 0':s -> 0':s -> true:false
378.69/291.54	0' :: 0':s
378.69/291.54	s :: 0':s -> 0':s
378.69/291.54	app :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	nil :: nil:cons
378.69/291.54	cons :: 0':s -> nil:cons -> nil:cons
378.69/291.54	mem :: 0':s -> nil:cons -> true:false
378.69/291.54	ifmem :: true:false -> 0':s -> nil:cons -> true:false
378.69/291.54	inter :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
378.69/291.54	hole_if1_0 :: if
378.69/291.54	hole_true:false2_0 :: true:false
378.69/291.54	hole_0':s3_0 :: 0':s
378.69/291.54	hole_nil:cons4_0 :: nil:cons
378.69/291.54	gen_0':s5_0 :: Nat -> 0':s
378.69/291.54	gen_nil:cons6_0 :: Nat -> nil:cons
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(27) OrderProof (LOWER BOUND(ID))
378.69/291.54	Heuristically decided to analyse the following defined symbols:
378.69/291.54	eq, app, mem, inter
378.69/291.54	
378.69/291.54	They will be analysed ascendingly in the following order:
378.69/291.54	eq < mem
378.69/291.54	app < inter
378.69/291.54	mem < inter
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(28)
378.69/291.54	Obligation:
378.69/291.54	TRS:
378.69/291.54	Rules:
378.69/291.54	if(true, x, y) -> x
378.69/291.54	if(false, x, y) -> y
378.69/291.54	eq(0', 0') -> true
378.69/291.54	eq(0', s(x)) -> false
378.69/291.54	eq(s(x), 0') -> false
378.69/291.54	eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	app(nil, l) -> l
378.69/291.54	app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	mem(x, nil) -> false
378.69/291.54	mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	ifmem(true, x, l) -> true
378.69/291.54	ifmem(false, x, l) -> mem(x, l)
378.69/291.54	inter(x, nil) -> nil
378.69/291.54	inter(nil, x) -> nil
378.69/291.54	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	Types:
378.69/291.54	if :: true:false -> if -> if -> if
378.69/291.54	true :: true:false
378.69/291.54	false :: true:false
378.69/291.54	eq :: 0':s -> 0':s -> true:false
378.69/291.54	0' :: 0':s
378.69/291.54	s :: 0':s -> 0':s
378.69/291.54	app :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	nil :: nil:cons
378.69/291.54	cons :: 0':s -> nil:cons -> nil:cons
378.69/291.54	mem :: 0':s -> nil:cons -> true:false
378.69/291.54	ifmem :: true:false -> 0':s -> nil:cons -> true:false
378.69/291.54	inter :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
378.69/291.54	hole_if1_0 :: if
378.69/291.54	hole_true:false2_0 :: true:false
378.69/291.54	hole_0':s3_0 :: 0':s
378.69/291.54	hole_nil:cons4_0 :: nil:cons
378.69/291.54	gen_0':s5_0 :: Nat -> 0':s
378.69/291.54	gen_nil:cons6_0 :: Nat -> nil:cons
378.69/291.54	
378.69/291.54	
378.69/291.54	Generator Equations:
378.69/291.54	gen_0':s5_0(0) <=> 0'
378.69/291.54	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
378.69/291.54	gen_nil:cons6_0(0) <=> nil
378.69/291.54	gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))
378.69/291.54	
378.69/291.54	
378.69/291.54	The following defined symbols remain to be analysed:
378.69/291.54	eq, app, mem, inter
378.69/291.54	
378.69/291.54	They will be analysed ascendingly in the following order:
378.69/291.54	eq < mem
378.69/291.54	app < inter
378.69/291.54	mem < inter
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(29) RewriteLemmaProof (LOWER BOUND(ID))
378.69/291.54	Proved the following rewrite lemma:
378.69/291.54	eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
378.69/291.54	
378.69/291.54	Induction Base:
378.69/291.54	eq(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
378.69/291.54	true
378.69/291.54	
378.69/291.54	Induction Step:
378.69/291.54	eq(gen_0':s5_0(+(n8_0, 1)), gen_0':s5_0(+(n8_0, 1))) ->_R^Omega(1)
378.69/291.54	eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) ->_IH
378.69/291.54	true
378.69/291.54	
378.69/291.54	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(30)
378.69/291.54	Complex Obligation (BEST)
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(31)
378.69/291.54	Obligation:
378.69/291.54	Proved the lower bound n^1 for the following obligation:
378.69/291.54	
378.69/291.54	TRS:
378.69/291.54	Rules:
378.69/291.54	if(true, x, y) -> x
378.69/291.54	if(false, x, y) -> y
378.69/291.54	eq(0', 0') -> true
378.69/291.54	eq(0', s(x)) -> false
378.69/291.54	eq(s(x), 0') -> false
378.69/291.54	eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	app(nil, l) -> l
378.69/291.54	app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	mem(x, nil) -> false
378.69/291.54	mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	ifmem(true, x, l) -> true
378.69/291.54	ifmem(false, x, l) -> mem(x, l)
378.69/291.54	inter(x, nil) -> nil
378.69/291.54	inter(nil, x) -> nil
378.69/291.54	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	Types:
378.69/291.54	if :: true:false -> if -> if -> if
378.69/291.54	true :: true:false
378.69/291.54	false :: true:false
378.69/291.54	eq :: 0':s -> 0':s -> true:false
378.69/291.54	0' :: 0':s
378.69/291.54	s :: 0':s -> 0':s
378.69/291.54	app :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	nil :: nil:cons
378.69/291.54	cons :: 0':s -> nil:cons -> nil:cons
378.69/291.54	mem :: 0':s -> nil:cons -> true:false
378.69/291.54	ifmem :: true:false -> 0':s -> nil:cons -> true:false
378.69/291.54	inter :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
378.69/291.54	hole_if1_0 :: if
378.69/291.54	hole_true:false2_0 :: true:false
378.69/291.54	hole_0':s3_0 :: 0':s
378.69/291.54	hole_nil:cons4_0 :: nil:cons
378.69/291.54	gen_0':s5_0 :: Nat -> 0':s
378.69/291.54	gen_nil:cons6_0 :: Nat -> nil:cons
378.69/291.54	
378.69/291.54	
378.69/291.54	Generator Equations:
378.69/291.54	gen_0':s5_0(0) <=> 0'
378.69/291.54	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
378.69/291.54	gen_nil:cons6_0(0) <=> nil
378.69/291.54	gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))
378.69/291.54	
378.69/291.54	
378.69/291.54	The following defined symbols remain to be analysed:
378.69/291.54	eq, app, mem, inter
378.69/291.54	
378.69/291.54	They will be analysed ascendingly in the following order:
378.69/291.54	eq < mem
378.69/291.54	app < inter
378.69/291.54	mem < inter
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(32) LowerBoundPropagationProof (FINISHED)
378.69/291.54	Propagated lower bound.
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(33)
378.69/291.54	BOUNDS(n^1, INF)
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(34)
378.69/291.54	Obligation:
378.69/291.54	TRS:
378.69/291.54	Rules:
378.69/291.54	if(true, x, y) -> x
378.69/291.54	if(false, x, y) -> y
378.69/291.54	eq(0', 0') -> true
378.69/291.54	eq(0', s(x)) -> false
378.69/291.54	eq(s(x), 0') -> false
378.69/291.54	eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	app(nil, l) -> l
378.69/291.54	app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	mem(x, nil) -> false
378.69/291.54	mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	ifmem(true, x, l) -> true
378.69/291.54	ifmem(false, x, l) -> mem(x, l)
378.69/291.54	inter(x, nil) -> nil
378.69/291.54	inter(nil, x) -> nil
378.69/291.54	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	Types:
378.69/291.54	if :: true:false -> if -> if -> if
378.69/291.54	true :: true:false
378.69/291.54	false :: true:false
378.69/291.54	eq :: 0':s -> 0':s -> true:false
378.69/291.54	0' :: 0':s
378.69/291.54	s :: 0':s -> 0':s
378.69/291.54	app :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	nil :: nil:cons
378.69/291.54	cons :: 0':s -> nil:cons -> nil:cons
378.69/291.54	mem :: 0':s -> nil:cons -> true:false
378.69/291.54	ifmem :: true:false -> 0':s -> nil:cons -> true:false
378.69/291.54	inter :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
378.69/291.54	hole_if1_0 :: if
378.69/291.54	hole_true:false2_0 :: true:false
378.69/291.54	hole_0':s3_0 :: 0':s
378.69/291.54	hole_nil:cons4_0 :: nil:cons
378.69/291.54	gen_0':s5_0 :: Nat -> 0':s
378.69/291.54	gen_nil:cons6_0 :: Nat -> nil:cons
378.69/291.54	
378.69/291.54	
378.69/291.54	Lemmas:
378.69/291.54	eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
378.69/291.54	
378.69/291.54	
378.69/291.54	Generator Equations:
378.69/291.54	gen_0':s5_0(0) <=> 0'
378.69/291.54	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
378.69/291.54	gen_nil:cons6_0(0) <=> nil
378.69/291.54	gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))
378.69/291.54	
378.69/291.54	
378.69/291.54	The following defined symbols remain to be analysed:
378.69/291.54	app, mem, inter
378.69/291.54	
378.69/291.54	They will be analysed ascendingly in the following order:
378.69/291.54	app < inter
378.69/291.54	mem < inter
378.69/291.54	
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(35) RewriteLemmaProof (LOWER BOUND(ID))
378.69/291.54	Proved the following rewrite lemma:
378.69/291.54	app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b)) -> gen_nil:cons6_0(+(n537_0, b)), rt in Omega(1 + n537_0)
378.69/291.54	
378.69/291.54	Induction Base:
378.69/291.54	app(gen_nil:cons6_0(0), gen_nil:cons6_0(b)) ->_R^Omega(1)
378.69/291.54	gen_nil:cons6_0(b)
378.69/291.54	
378.69/291.54	Induction Step:
378.69/291.54	app(gen_nil:cons6_0(+(n537_0, 1)), gen_nil:cons6_0(b)) ->_R^Omega(1)
378.69/291.54	cons(0', app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b))) ->_IH
378.69/291.54	cons(0', gen_nil:cons6_0(+(b, c538_0)))
378.69/291.54	
378.69/291.54	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
378.69/291.54	----------------------------------------
378.69/291.54	
378.69/291.54	(36)
378.69/291.54	Obligation:
378.69/291.54	TRS:
378.69/291.54	Rules:
378.69/291.54	if(true, x, y) -> x
378.69/291.54	if(false, x, y) -> y
378.69/291.54	eq(0', 0') -> true
378.69/291.54	eq(0', s(x)) -> false
378.69/291.54	eq(s(x), 0') -> false
378.69/291.54	eq(s(x), s(y)) -> eq(x, y)
378.69/291.54	app(nil, l) -> l
378.69/291.54	app(cons(x, l1), l2) -> cons(x, app(l1, l2))
378.69/291.54	app(app(l1, l2), l3) -> app(l1, app(l2, l3))
378.69/291.54	mem(x, nil) -> false
378.69/291.54	mem(x, cons(y, l)) -> ifmem(eq(x, y), x, l)
378.69/291.54	ifmem(true, x, l) -> true
378.69/291.54	ifmem(false, x, l) -> mem(x, l)
378.69/291.54	inter(x, nil) -> nil
378.69/291.54	inter(nil, x) -> nil
378.69/291.54	inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
378.69/291.54	inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
378.69/291.54	inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
378.69/291.54	inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
378.69/291.54	ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
378.69/291.54	ifinter(false, x, l1, l2) -> inter(l1, l2)
378.69/291.54	
378.69/291.54	Types:
378.69/291.54	if :: true:false -> if -> if -> if
378.69/291.54	true :: true:false
378.69/291.54	false :: true:false
378.69/291.54	eq :: 0':s -> 0':s -> true:false
378.69/291.54	0' :: 0':s
378.69/291.54	s :: 0':s -> 0':s
378.69/291.54	app :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	nil :: nil:cons
378.69/291.54	cons :: 0':s -> nil:cons -> nil:cons
378.69/291.54	mem :: 0':s -> nil:cons -> true:false
378.69/291.54	ifmem :: true:false -> 0':s -> nil:cons -> true:false
378.69/291.54	inter :: nil:cons -> nil:cons -> nil:cons
378.69/291.54	ifinter :: true:false -> 0':s -> nil:cons -> nil:cons -> nil:cons
378.69/291.54	hole_if1_0 :: if
378.69/291.54	hole_true:false2_0 :: true:false
378.69/291.54	hole_0':s3_0 :: 0':s
378.69/291.54	hole_nil:cons4_0 :: nil:cons
378.69/291.54	gen_0':s5_0 :: Nat -> 0':s
378.69/291.54	gen_nil:cons6_0 :: Nat -> nil:cons
378.69/291.54	
378.69/291.54	
378.69/291.54	Lemmas:
378.69/291.54	eq(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) -> true, rt in Omega(1 + n8_0)
378.69/291.54	app(gen_nil:cons6_0(n537_0), gen_nil:cons6_0(b)) -> gen_nil:cons6_0(+(n537_0, b)), rt in Omega(1 + n537_0)
378.69/291.54	
378.69/291.54	
378.69/291.54	Generator Equations:
378.69/291.54	gen_0':s5_0(0) <=> 0'
378.69/291.54	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
378.69/291.54	gen_nil:cons6_0(0) <=> nil
378.69/291.54	gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))
378.69/291.54	
378.69/291.54	
378.69/291.54	The following defined symbols remain to be analysed:
378.69/291.54	mem, inter
378.69/291.54	
378.69/291.54	They will be analysed ascendingly in the following order:
378.69/291.54	mem < inter
378.81/291.57	EOF