419.69/291.55	WORST_CASE(Omega(n^1), O(n^2))
419.69/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
419.69/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
419.69/291.57	
419.69/291.57	
419.69/291.57	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.57	
419.69/291.57	(0) CpxRelTRS
419.69/291.57	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 361 ms]
419.69/291.57	(2) CpxRelTRS
419.69/291.57	(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	(4) CpxWeightedTrs
419.69/291.57	    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	    (6) CpxWeightedTrs
419.69/291.57	    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	    (8) CpxTypedWeightedTrs
419.69/291.57	    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (10) CpxTypedWeightedCompleteTrs
419.69/291.57	    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	    (12) CpxTypedWeightedCompleteTrs
419.69/291.57	    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (14) CpxRNTS
419.69/291.57	    (15) InliningProof [UPPER BOUND(ID), 1026 ms]
419.69/291.57	    (16) CpxRNTS
419.69/291.57	    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	    (18) CpxRNTS
419.69/291.57	    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
419.69/291.57	    (20) CpxRNTS
419.69/291.57	    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (22) CpxRNTS
419.69/291.57	    (23) IntTrsBoundProof [UPPER BOUND(ID), 276 ms]
419.69/291.57	    (24) CpxRNTS
419.69/291.57	    (25) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
419.69/291.57	    (26) CpxRNTS
419.69/291.57	    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (28) CpxRNTS
419.69/291.57	    (29) IntTrsBoundProof [UPPER BOUND(ID), 211 ms]
419.69/291.57	    (30) CpxRNTS
419.69/291.57	    (31) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
419.69/291.57	    (32) CpxRNTS
419.69/291.57	    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (34) CpxRNTS
419.69/291.57	    (35) IntTrsBoundProof [UPPER BOUND(ID), 2478 ms]
419.69/291.57	    (36) CpxRNTS
419.69/291.57	    (37) IntTrsBoundProof [UPPER BOUND(ID), 416 ms]
419.69/291.57	    (38) CpxRNTS
419.69/291.57	    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (40) CpxRNTS
419.69/291.57	    (41) IntTrsBoundProof [UPPER BOUND(ID), 805 ms]
419.69/291.57	    (42) CpxRNTS
419.69/291.57	    (43) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
419.69/291.57	    (44) CpxRNTS
419.69/291.57	    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (46) CpxRNTS
419.69/291.57	    (47) IntTrsBoundProof [UPPER BOUND(ID), 229 ms]
419.69/291.57	    (48) CpxRNTS
419.69/291.57	    (49) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
419.69/291.57	    (50) CpxRNTS
419.69/291.57	    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (52) CpxRNTS
419.69/291.57	    (53) IntTrsBoundProof [UPPER BOUND(ID), 1091 ms]
419.69/291.57	    (54) CpxRNTS
419.69/291.57	    (55) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
419.69/291.57	    (56) CpxRNTS
419.69/291.57	    (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (58) CpxRNTS
419.69/291.57	    (59) IntTrsBoundProof [UPPER BOUND(ID), 827 ms]
419.69/291.57	    (60) CpxRNTS
419.69/291.57	    (61) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
419.69/291.57	    (62) CpxRNTS
419.69/291.57	    (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (64) CpxRNTS
419.69/291.57	    (65) IntTrsBoundProof [UPPER BOUND(ID), 4820 ms]
419.69/291.57	    (66) CpxRNTS
419.69/291.57	    (67) IntTrsBoundProof [UPPER BOUND(ID), 446 ms]
419.69/291.57	    (68) CpxRNTS
419.69/291.57	    (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (70) CpxRNTS
419.69/291.57	    (71) IntTrsBoundProof [UPPER BOUND(ID), 5122 ms]
419.69/291.57	    (72) CpxRNTS
419.69/291.57	    (73) IntTrsBoundProof [UPPER BOUND(ID), 636 ms]
419.69/291.57	    (74) CpxRNTS
419.69/291.57	    (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
419.69/291.57	    (76) CpxRNTS
419.69/291.57	    (77) IntTrsBoundProof [UPPER BOUND(ID), 1756 ms]
419.69/291.57	    (78) CpxRNTS
419.69/291.57	    (79) IntTrsBoundProof [UPPER BOUND(ID), 216 ms]
419.69/291.57	    (80) CpxRNTS
419.69/291.57	    (81) FinalProof [FINISHED, 0 ms]
419.69/291.57	    (82) BOUNDS(1, n^2)
419.69/291.57	(83) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 2 ms]
419.69/291.57	(84) TRS for Loop Detection
419.69/291.57	    (85) DecreasingLoopProof [LOWER BOUND(ID), 42 ms]
419.69/291.57	    (86) BEST
419.69/291.57	        (87) proven lower bound
419.69/291.57	            (88) LowerBoundPropagationProof [FINISHED, 0 ms]
419.69/291.57	            (89) BOUNDS(n^1, INF)
419.69/291.57	        (90) TRS for Loop Detection
419.69/291.57	
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(0)
419.69/291.57	Obligation:
419.69/291.57	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.57	
419.69/291.57	
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   *(@x, @y) -> #mult(@x, @y)
419.69/291.57	   +(@x, @y) -> #add(@x, @y)
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m)
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs)
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc))
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n)
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs)
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil))
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2)
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
419.69/291.57	   matrixMult#1(nil, @m2) -> nil
419.69/291.57	
419.69/291.57	The (relative) TRS S consists of the following rules:
419.69/291.57	
419.69/291.57	   #add(#0, @y) -> @y
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y)
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y)
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
419.69/291.57	   #mult(#0, #0) -> #0
419.69/291.57	   #mult(#0, #neg(@y)) -> #0
419.69/291.57	   #mult(#0, #pos(@y)) -> #0
419.69/291.57	   #mult(#neg(@x), #0) -> #0
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
419.69/291.57	   #mult(#pos(@x), #0) -> #0
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
419.69/291.57	   #natmult(#0, @y) -> #0
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
419.69/291.57	   #pred(#0) -> #neg(#s(#0))
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
419.69/291.57	   #pred(#pos(#s(#0))) -> #0
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
419.69/291.57	   #succ(#0) -> #pos(#s(#0))
419.69/291.57	   #succ(#neg(#s(#0))) -> #0
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
419.69/291.57	proved termination of relative rules
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(2)
419.69/291.57	Obligation:
419.69/291.57	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.57	
419.69/291.57	
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   *(@x, @y) -> #mult(@x, @y)
419.69/291.57	   +(@x, @y) -> #add(@x, @y)
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m)
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs)
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc))
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n)
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs)
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil))
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2)
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
419.69/291.57	   matrixMult#1(nil, @m2) -> nil
419.69/291.57	
419.69/291.57	The (relative) TRS S consists of the following rules:
419.69/291.57	
419.69/291.57	   #add(#0, @y) -> @y
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y)
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y)
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
419.69/291.57	   #mult(#0, #0) -> #0
419.69/291.57	   #mult(#0, #neg(@y)) -> #0
419.69/291.57	   #mult(#0, #pos(@y)) -> #0
419.69/291.57	   #mult(#neg(@x), #0) -> #0
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
419.69/291.57	   #mult(#pos(@x), #0) -> #0
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
419.69/291.57	   #natmult(#0, @y) -> #0
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
419.69/291.57	   #pred(#0) -> #neg(#s(#0))
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
419.69/291.57	   #pred(#pos(#s(#0))) -> #0
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
419.69/291.57	   #succ(#0) -> #pos(#s(#0))
419.69/291.57	   #succ(#neg(#s(#0))) -> #0
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
419.69/291.57	Transformed relative TRS to weighted TRS
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(4)
419.69/291.57	Obligation:
419.69/291.57	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
419.69/291.57	
419.69/291.57	
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   *(@x, @y) -> #mult(@x, @y) [1]
419.69/291.57	   +(@x, @y) -> #add(@x, @y) [1]
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) [1]
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) [1]
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc [1]
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) [1]
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil [1]
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) [1]
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) [1]
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil [1]
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) [1]
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil)) [1]
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) [1]
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2)) [1]
419.69/291.57	   matrixMult#1(nil, @m2) -> nil [1]
419.69/291.57	   #add(#0, @y) -> @y [0]
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y) [0]
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y) [0]
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #mult(#0, #0) -> #0 [0]
419.69/291.57	   #mult(#0, #neg(@y)) -> #0 [0]
419.69/291.57	   #mult(#0, #pos(@y)) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #natmult(#0, @y) -> #0 [0]
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) [0]
419.69/291.57	   #pred(#0) -> #neg(#s(#0)) [0]
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0]
419.69/291.57	   #pred(#pos(#s(#0))) -> #0 [0]
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0]
419.69/291.57	   #succ(#0) -> #pos(#s(#0)) [0]
419.69/291.57	   #succ(#neg(#s(#0))) -> #0 [0]
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0]
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0]
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
419.69/291.57	Renamed defined symbols to avoid conflicts with arithmetic symbols:
419.69/291.57	
419.69/291.57	* => times
419.69/291.57	+ => plus
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(6)
419.69/291.57	Obligation:
419.69/291.57	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
419.69/291.57	
419.69/291.57	
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   times(@x, @y) -> #mult(@x, @y) [1]
419.69/291.57	   plus(@x, @y) -> #add(@x, @y) [1]
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) [1]
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) [1]
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc [1]
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) [1]
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil [1]
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) [1]
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) [1]
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil [1]
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(plus(times(@x, @n), @y), lineMult(@n, @xs, @ys)) [1]
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(times(@x, @n), lineMult(@n, @xs, nil)) [1]
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) [1]
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2)) [1]
419.69/291.57	   matrixMult#1(nil, @m2) -> nil [1]
419.69/291.57	   #add(#0, @y) -> @y [0]
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y) [0]
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y) [0]
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #mult(#0, #0) -> #0 [0]
419.69/291.57	   #mult(#0, #neg(@y)) -> #0 [0]
419.69/291.57	   #mult(#0, #pos(@y)) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #natmult(#0, @y) -> #0 [0]
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) [0]
419.69/291.57	   #pred(#0) -> #neg(#s(#0)) [0]
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0]
419.69/291.57	   #pred(#pos(#s(#0))) -> #0 [0]
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0]
419.69/291.57	   #succ(#0) -> #pos(#s(#0)) [0]
419.69/291.57	   #succ(#neg(#s(#0))) -> #0 [0]
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0]
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0]
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
419.69/291.57	Infered types.
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(8)
419.69/291.57	Obligation:
419.69/291.57	Runtime Complexity Weighted TRS with Types.
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   times(@x, @y) -> #mult(@x, @y) [1]
419.69/291.57	   plus(@x, @y) -> #add(@x, @y) [1]
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) [1]
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) [1]
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc [1]
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) [1]
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil [1]
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) [1]
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) [1]
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil [1]
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(plus(times(@x, @n), @y), lineMult(@n, @xs, @ys)) [1]
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(times(@x, @n), lineMult(@n, @xs, nil)) [1]
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) [1]
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2)) [1]
419.69/291.57	   matrixMult#1(nil, @m2) -> nil [1]
419.69/291.57	   #add(#0, @y) -> @y [0]
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y) [0]
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y) [0]
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #mult(#0, #0) -> #0 [0]
419.69/291.57	   #mult(#0, #neg(@y)) -> #0 [0]
419.69/291.57	   #mult(#0, #pos(@y)) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #natmult(#0, @y) -> #0 [0]
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) [0]
419.69/291.57	   #pred(#0) -> #neg(#s(#0)) [0]
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0]
419.69/291.57	   #pred(#pos(#s(#0))) -> #0 [0]
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0]
419.69/291.57	   #succ(#0) -> #pos(#s(#0)) [0]
419.69/291.57	   #succ(#neg(#s(#0))) -> #0 [0]
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0]
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0]
419.69/291.57	
419.69/291.57	The TRS has the following type information:
419.69/291.57	times :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#mult :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	plus :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#add :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	computeLine :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	computeLine#1 :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	:: :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	computeLine#2 :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	nil :: :::nil:#0:#s:#neg:#pos
419.69/291.57	lineMult :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	lineMult#1 :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	lineMult#2 :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	matrixMult :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	matrixMult#1 :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#0 :: :::nil:#0:#s:#neg:#pos
419.69/291.57	#neg :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#s :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#pred :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#pos :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#succ :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	#natmult :: :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos -> :::nil:#0:#s:#neg:#pos
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(9) CompletionProof (UPPER BOUND(ID))
419.69/291.57	The transformation into a RNTS is sound, since:
419.69/291.57	
419.69/291.57	(a) The obligation is a constructor system where every type has a constant constructor,
419.69/291.57	
419.69/291.57	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
419.69/291.57	
419.69/291.57	   computeLine_3
419.69/291.57	   computeLine#1_3
419.69/291.57	   computeLine#2_4
419.69/291.57	   matrixMult_2
419.69/291.57	   matrixMult#1_2
419.69/291.57	
419.69/291.57	(c) The following functions are completely defined:
419.69/291.57	
419.69/291.57	   times_2
419.69/291.57	   lineMult_3
419.69/291.57	   lineMult#1_3
419.69/291.57	   lineMult#2_4
419.69/291.57	   plus_2
419.69/291.57	   #add_2
419.69/291.57	   #mult_2
419.69/291.57	   #natmult_2
419.69/291.57	   #pred_1
419.69/291.57	   #succ_1
419.69/291.57	
419.69/291.57	Due to the following rules being added:
419.69/291.57	
419.69/291.57	   #add(v0, v1) -> null_#add [0]
419.69/291.57	   #mult(v0, v1) -> null_#mult [0]
419.69/291.57	   #natmult(v0, v1) -> null_#natmult [0]
419.69/291.57	   #pred(v0) -> null_#pred [0]
419.69/291.57	   #succ(v0) -> null_#succ [0]
419.69/291.57	   lineMult#1(v0, v1, v2) -> null_lineMult#1 [0]
419.69/291.57	   lineMult#2(v0, v1, v2, v3) -> null_lineMult#2 [0]
419.69/291.57	
419.69/291.57	And the following fresh constants:    null_#add, null_#mult, null_#natmult, null_#pred, null_#succ, null_lineMult#1, null_lineMult#2
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(10)
419.69/291.57	Obligation:
419.69/291.57	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
419.69/291.57	
419.69/291.57	Runtime Complexity Weighted TRS with Types.
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   times(@x, @y) -> #mult(@x, @y) [1]
419.69/291.57	   plus(@x, @y) -> #add(@x, @y) [1]
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) [1]
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) [1]
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc [1]
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) [1]
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil [1]
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) [1]
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) [1]
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil [1]
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(plus(times(@x, @n), @y), lineMult(@n, @xs, @ys)) [1]
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(times(@x, @n), lineMult(@n, @xs, nil)) [1]
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) [1]
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2)) [1]
419.69/291.57	   matrixMult#1(nil, @m2) -> nil [1]
419.69/291.57	   #add(#0, @y) -> @y [0]
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y) [0]
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y) [0]
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0]
419.69/291.57	   #mult(#0, #0) -> #0 [0]
419.69/291.57	   #mult(#0, #neg(@y)) -> #0 [0]
419.69/291.57	   #mult(#0, #pos(@y)) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #natmult(#0, @y) -> #0 [0]
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) [0]
419.69/291.57	   #pred(#0) -> #neg(#s(#0)) [0]
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0]
419.69/291.57	   #pred(#pos(#s(#0))) -> #0 [0]
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0]
419.69/291.57	   #succ(#0) -> #pos(#s(#0)) [0]
419.69/291.57	   #succ(#neg(#s(#0))) -> #0 [0]
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0]
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0]
419.69/291.57	   #add(v0, v1) -> null_#add [0]
419.69/291.57	   #mult(v0, v1) -> null_#mult [0]
419.69/291.57	   #natmult(v0, v1) -> null_#natmult [0]
419.69/291.57	   #pred(v0) -> null_#pred [0]
419.69/291.57	   #succ(v0) -> null_#succ [0]
419.69/291.57	   lineMult#1(v0, v1, v2) -> null_lineMult#1 [0]
419.69/291.57	   lineMult#2(v0, v1, v2, v3) -> null_lineMult#2 [0]
419.69/291.57	
419.69/291.57	The TRS has the following type information:
419.69/291.57	times :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#mult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	plus :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#add :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	:: :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	nil :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	matrixMult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	matrixMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#0 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#neg :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#s :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#pred :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#pos :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#succ :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#natmult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#add :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#mult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#natmult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#pred :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#succ :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_lineMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_lineMult#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(11) NarrowingProof (BOTH BOUNDS(ID, ID))
419.69/291.57	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(12)
419.69/291.57	Obligation:
419.69/291.57	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
419.69/291.57	
419.69/291.57	Runtime Complexity Weighted TRS with Types.
419.69/291.57	The TRS R consists of the following rules:
419.69/291.57	
419.69/291.57	   times(@x, @y) -> #mult(@x, @y) [1]
419.69/291.57	   plus(@x, @y) -> #add(@x, @y) [1]
419.69/291.57	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) [1]
419.69/291.57	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) [1]
419.69/291.57	   computeLine#1(nil, @acc, @m) -> @acc [1]
419.69/291.57	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult#1(@l, @acc, @x)) [2]
419.69/291.57	   computeLine#2(nil, @acc, @x, @xs) -> nil [1]
419.69/291.57	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) [1]
419.69/291.57	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) [1]
419.69/291.57	   lineMult#1(nil, @l2, @n) -> nil [1]
419.69/291.57	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(plus(#mult(@x, @n), @y), lineMult(@n, @xs, @ys)) [2]
419.69/291.57	   lineMult#2(nil, @n, @x, @xs) -> ::(times(@x, @n), lineMult(@n, @xs, nil)) [1]
419.69/291.57	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) [1]
419.69/291.57	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2)) [1]
419.69/291.57	   matrixMult#1(nil, @m2) -> nil [1]
419.69/291.57	   #add(#0, @y) -> @y [0]
419.69/291.57	   #add(#neg(#s(#0)), @y) -> #pred(@y) [0]
419.69/291.57	   #add(#neg(#s(#s(#0))), @y) -> #pred(#succ(@y)) [0]
419.69/291.57	   #add(#neg(#s(#s(#s(@x')))), @y) -> #pred(#succ(#add(#pos(#s(@x')), @y))) [0]
419.69/291.57	   #add(#neg(#s(#s(@x))), @y) -> #pred(null_#add) [0]
419.69/291.57	   #add(#pos(#s(#0)), @y) -> #succ(@y) [0]
419.69/291.57	   #add(#pos(#s(#s(#0))), @y) -> #succ(#succ(@y)) [0]
419.69/291.57	   #add(#pos(#s(#s(#s(@x'')))), @y) -> #succ(#succ(#add(#pos(#s(@x'')), @y))) [0]
419.69/291.57	   #add(#pos(#s(#s(@x))), @y) -> #succ(null_#add) [0]
419.69/291.57	   #mult(#0, #0) -> #0 [0]
419.69/291.57	   #mult(#0, #neg(@y)) -> #0 [0]
419.69/291.57	   #mult(#0, #pos(@y)) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #0) -> #0 [0]
419.69/291.57	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) [0]
419.69/291.57	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) [0]
419.69/291.57	   #natmult(#0, @y) -> #0 [0]
419.69/291.57	   #natmult(#s(#0), @y) -> #add(#pos(@y), #0) [0]
419.69/291.57	   #natmult(#s(#s(@x1)), @y) -> #add(#pos(@y), #add(#pos(@y), #natmult(@x1, @y))) [0]
419.69/291.57	   #natmult(#s(@x), @y) -> #add(#pos(@y), null_#natmult) [0]
419.69/291.57	   #pred(#0) -> #neg(#s(#0)) [0]
419.69/291.57	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0]
419.69/291.57	   #pred(#pos(#s(#0))) -> #0 [0]
419.69/291.57	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0]
419.69/291.57	   #succ(#0) -> #pos(#s(#0)) [0]
419.69/291.57	   #succ(#neg(#s(#0))) -> #0 [0]
419.69/291.57	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0]
419.69/291.57	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0]
419.69/291.57	   #add(v0, v1) -> null_#add [0]
419.69/291.57	   #mult(v0, v1) -> null_#mult [0]
419.69/291.57	   #natmult(v0, v1) -> null_#natmult [0]
419.69/291.57	   #pred(v0) -> null_#pred [0]
419.69/291.57	   #succ(v0) -> null_#succ [0]
419.69/291.57	   lineMult#1(v0, v1, v2) -> null_lineMult#1 [0]
419.69/291.57	   lineMult#2(v0, v1, v2, v3) -> null_lineMult#2 [0]
419.69/291.57	
419.69/291.57	The TRS has the following type information:
419.69/291.57	times :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#mult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	plus :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#add :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	:: :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	computeLine#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	nil :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	lineMult#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	matrixMult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	matrixMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#0 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#neg :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#s :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#pred :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#pos :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#succ :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	#natmult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2 -> :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#add :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#mult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#natmult :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#pred :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_#succ :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_lineMult#1 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	null_lineMult#2 :: :::nil:#0:#s:#neg:#pos:null_#add:null_#mult:null_#natmult:null_#pred:null_#succ:null_lineMult#1:null_lineMult#2
419.69/291.57	
419.69/291.57	Rewrite Strategy: INNERMOST
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
419.69/291.57	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
419.69/291.57	The constant constructors are abstracted as follows:
419.69/291.57	
419.69/291.57	   nil => 1
419.69/291.57	   #0 => 0
419.69/291.57	   null_#add => 0
419.69/291.57	   null_#mult => 0
419.69/291.57	   null_#natmult => 0
419.69/291.57	   null_#pred => 0
419.69/291.57	   null_#succ => 0
419.69/291.57	   null_lineMult#1 => 0
419.69/291.57	   null_lineMult#2 => 0
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(14)
419.69/291.57	Obligation:
419.69/291.57	Complexity RNTS consisting of the following rules:
419.69/291.57	
419.69/291.57	   #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x''), @y))) :|: z = 1 + (1 + (1 + (1 + @x''))), z' = @y, @x'' >= 0, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x'), @y))) :|: z = 1 + (1 + (1 + (1 + @x'))), z' = @y, @y >= 0, @x' >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 1 + @y, @y >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: @x >= 0, z = 1 + @x, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #mult(z, z') -{ 0 }-> 1 + #natmult(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: z' = @y, z = 0, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, 0) :|: z = 1 + 0, z' = @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, 0) :|: @x >= 0, z = 1 + @x, z' = @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, #add(1 + @y, #natmult(@x1, @y))) :|: @x1 >= 0, z' = @y, z = 1 + (1 + @x1), @y >= 0
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(@line, @acc, @m) :|: z'' = @acc, @m >= 0, @acc >= 0, @line >= 0, z' = @m, z = @line
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> @acc :|: z'' = @m, @acc >= 0, @m >= 0, z = 1, z' = @acc
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(@m, @acc, @x, @xs) :|: z'' = @m, @acc >= 0, @m >= 0, @x >= 0, z = 1 + @x + @xs, z' = @acc, @xs >= 0
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(@xs, @ls, lineMult#1(@l, @acc, @x)) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, @acc >= 0, @x >= 0, z1 = @xs, z' = @acc, @xs >= 0, z'' = @x
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: @acc >= 0, @x >= 0, z = 1, z1 = @xs, z' = @acc, @xs >= 0, z'' = @x
419.69/291.57	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(@l1, @l2, @n) :|: @l1 >= 0, z'' = @l2, @n >= 0, @l2 >= 0, z = @n, z' = @l1
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(@l2, @n, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, z'' = @n, @l2 >= 0, @n >= 0, @xs >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, z'' = @n, z = 1, @l2 >= 0, @n >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(@x, @n) + lineMult(@n, @xs, 1) :|: z' = @n, @x >= 0, z = 1, @n >= 0, z1 = @xs, @xs >= 0, z'' = @x
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(@x, @n), @y) + lineMult(@n, @xs, @ys) :|: z = 1 + @y + @ys, z' = @n, @x >= 0, @n >= 0, z1 = @xs, @xs >= 0, z'' = @x, @y >= 0, @ys >= 0
419.69/291.57	   matrixMult(z, z') -{ 1 }-> matrixMult#1(@m1, @m2) :|: @m1 >= 0, @m2 >= 0, z = @m1, z' = @m2
419.69/291.57	   matrixMult#1(z, z') -{ 1 }-> 1 :|: @m2 >= 0, z = 1, z' = @m2
419.69/291.57	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, @m2, 1) + matrixMult(@ls, @m2) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, @m2 >= 0, z' = @m2
419.69/291.57	   plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
419.69/291.57	   times(z, z') -{ 1 }-> #mult(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
419.69/291.57	
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(15) InliningProof (UPPER BOUND(ID))
419.69/291.57	Inlined the following terminating rules on right-hand sides where appropriate:
419.69/291.57	
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(16)
419.69/291.57	Obligation:
419.69/291.57	Complexity RNTS consisting of the following rules:
419.69/291.57	
419.69/291.57	   #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, v0 >= 0, @y = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, v0 >= 0, 0 = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, v0' >= 0, 0 = v0'
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), v0 >= 0, 1 + (1 + @x) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), 1 + (1 + @x) = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), v0 >= 0, 1 + (1 + (1 + @x)) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + @x) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x))
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + (1 + @x'))
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + (1 + @x'))
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + @x)
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + @x')
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + @x')
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x''), @y))) :|: z = 1 + (1 + (1 + (1 + @x''))), z' = @y, @x'' >= 0, @y >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x))
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + @x))) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x)
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x'), @y))) :|: z = 1 + (1 + (1 + (1 + @x'))), z' = @y, @y >= 0, @x' >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 1 + @y, @y >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: @x >= 0, z = 1 + @x, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #mult(z, z') -{ 0 }-> 1 + #natmult(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: z' = @y, z = 0, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, 0) :|: z = 1 + 0, z' = @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, 0) :|: @x >= 0, z = 1 + @x, z' = @y, @y >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + @y, #add(1 + @y, #natmult(@x1, @y))) :|: @x1 >= 0, z' = @y, z = 1 + (1 + @x1), @y >= 0
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x))
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x)
419.69/291.57	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(@line, @acc, @m) :|: z'' = @acc, @m >= 0, @acc >= 0, @line >= 0, z' = @m, z = @line
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> @acc :|: z'' = @m, @acc >= 0, @m >= 0, z = 1, z' = @acc
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(@m, @acc, @x, @xs) :|: z'' = @m, @acc >= 0, @m >= 0, @x >= 0, z = 1 + @x + @xs, z' = @acc, @xs >= 0
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(@xs, @ls, lineMult#1(@l, @acc, @x)) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, @acc >= 0, @x >= 0, z1 = @xs, z' = @acc, @xs >= 0, z'' = @x
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: @acc >= 0, @x >= 0, z = 1, z1 = @xs, z' = @acc, @xs >= 0, z'' = @x
419.69/291.57	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(@l1, @l2, @n) :|: @l1 >= 0, z'' = @l2, @n >= 0, @l2 >= 0, z = @n, z' = @l1
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(@l2, @n, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, z'' = @n, @l2 >= 0, @n >= 0, @xs >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, z'' = @n, z = 1, @l2 >= 0, @n >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(@x, @n) + lineMult(@n, @xs, 1) :|: z' = @n, @x >= 0, z = 1, @n >= 0, z1 = @xs, @xs >= 0, z'' = @x
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(@x, @n), @y) + lineMult(@n, @xs, @ys) :|: z = 1 + @y + @ys, z' = @n, @x >= 0, @n >= 0, z1 = @xs, @xs >= 0, z'' = @x, @y >= 0, @ys >= 0
419.69/291.57	   matrixMult(z, z') -{ 1 }-> matrixMult#1(@m1, @m2) :|: @m1 >= 0, @m2 >= 0, z = @m1, z' = @m2
419.69/291.57	   matrixMult#1(z, z') -{ 1 }-> 1 :|: @m2 >= 0, z = 1, z' = @m2
419.69/291.57	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, @m2, 1) + matrixMult(@ls, @m2) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, @m2 >= 0, z' = @m2
419.69/291.57	   plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
419.69/291.57	   times(z, z') -{ 1 }-> #mult(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
419.69/291.57	
419.69/291.57	
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(17) SimplificationProof (BOTH BOUNDS(ID, ID))
419.69/291.57	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
419.69/291.57	----------------------------------------
419.69/291.57	
419.69/291.57	(18)
419.69/291.57	Obligation:
419.69/291.57	Complexity RNTS consisting of the following rules:
419.69/291.57	
419.69/291.57	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.57	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.57	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.57	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.57	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.57	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.57	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.57	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.57	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.57	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.57	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.57	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.57	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.57	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.57	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
419.69/291.58	Found the following analysis order by SCC decomposition:
419.69/291.58	
419.69/291.58	   { #pred }
419.69/291.58	   { #succ }
419.69/291.58	   { #add }
419.69/291.58	   { #natmult }
419.69/291.58	   { plus }
419.69/291.58	   { #mult }
419.69/291.58	   { times }
419.69/291.58	   { lineMult, lineMult#2, lineMult#1 }
419.69/291.58	   { computeLine, computeLine#1, computeLine#2 }
419.69/291.58	   { matrixMult#1, matrixMult }
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(20)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#pred}, {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(21) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(22)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#pred}, {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(23) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed SIZE bound using CoFloCo for: #pred
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(n^1) with polynomial bound: 2 + z
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(24)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#pred}, {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: ?, size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(25) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed RUNTIME bound using CoFloCo for: #pred
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(1) with polynomial bound: 0
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(26)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(27) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(28)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(29) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed SIZE bound using CoFloCo for: #succ
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(n^1) with polynomial bound: 2 + z
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(30)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#succ}, {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #succ: runtime: ?, size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(31) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed RUNTIME bound using CoFloCo for: #succ
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(1) with polynomial bound: 0
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(32)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(33) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(34)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(35) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed SIZE bound using KoAT for: #add
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(n^1) with polynomial bound: 2*z + z'
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(36)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#add}, {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #add: runtime: ?, size: O(n^1) [2*z + z']
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(37) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.58	
419.69/291.58	Computed RUNTIME bound using CoFloCo for: #add
419.69/291.58	after applying outer abstraction to obtain an ITS,
419.69/291.58	resulting in: O(1) with polynomial bound: 0
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(38)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z' >= 0, z - 4 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', 0) :|: z - 1 >= 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.58	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.58	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.58	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.58	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.58	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.58	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.58	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.58	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.58	   plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0
419.69/291.58	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.58	
419.69/291.58	Function symbols to be analyzed: {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.58	Previous analysis results are:
419.69/291.58	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.58	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.58	
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(39) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.58	----------------------------------------
419.69/291.58	
419.69/291.58	(40)
419.69/291.58	Obligation:
419.69/291.58	Complexity RNTS consisting of the following rules:
419.69/291.58	
419.69/291.58	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.58	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.58	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.58	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.58	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(41) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.59	
419.69/291.59	Computed SIZE bound using KoAT for: #natmult
419.69/291.59	after applying outer abstraction to obtain an ITS,
419.69/291.59	resulting in: O(n^2) with polynomial bound: 4 + 4*z + 4*z*z' + 4*z'
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(42)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {#natmult}, {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	  #natmult: runtime: ?, size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(43) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.59	
419.69/291.59	Computed RUNTIME bound using CoFloCo for: #natmult
419.69/291.59	after applying outer abstraction to obtain an ITS,
419.69/291.59	resulting in: O(1) with polynomial bound: 0
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(44)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 1 + #natmult(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> #add(1 + z', #add(1 + z', #natmult(z - 2, z'))) :|: z - 2 >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(45) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(46)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(47) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.59	
419.69/291.59	Computed SIZE bound using CoFloCo for: plus
419.69/291.59	after applying outer abstraction to obtain an ITS,
419.69/291.59	resulting in: O(n^1) with polynomial bound: 2*z + z'
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(48)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {plus}, {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.59	  plus: runtime: ?, size: O(n^1) [2*z + z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(49) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.59	
419.69/291.59	Computed RUNTIME bound using CoFloCo for: plus
419.69/291.59	after applying outer abstraction to obtain an ITS,
419.69/291.59	resulting in: O(1) with polynomial bound: 1
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(50)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.59	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.59	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.59	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.59	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.59	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.59	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.59	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.59	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.59	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.59	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.59	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.59	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.59	
419.69/291.59	Function symbols to be analyzed: {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.59	Previous analysis results are:
419.69/291.59	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.59	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.59	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.59	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.59	
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(51) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.59	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.59	----------------------------------------
419.69/291.59	
419.69/291.59	(52)
419.69/291.59	Obligation:
419.69/291.59	Complexity RNTS consisting of the following rules:
419.69/291.59	
419.69/291.59	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.59	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.59	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(53) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed SIZE bound using KoAT for: #mult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^2) with polynomial bound: 1 + 4*z*z'
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(54)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {#mult}, {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: ?, size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(55) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: #mult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(1) with polynomial bound: 0
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(56)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + plus(#mult(z'', z'), @y) + lineMult(z', z1, @ys) :|: z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> #mult(z, z') :|: z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(57) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.60	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(58)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(59) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed SIZE bound using KoAT for: times
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^2) with polynomial bound: 1 + 4*z*z'
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(60)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {times}, {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: ?, size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(61) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: times
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(1) with polynomial bound: 1
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(62)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 1 }-> 1 + times(z'', z') + lineMult(z', z1, 1) :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(63) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.60	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(64)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + s19 + lineMult(z', z1, 1) :|: s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(65) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: lineMult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: lineMult#2
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: lineMult#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(66)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + s19 + lineMult(z', z1, 1) :|: s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {lineMult,lineMult#2,lineMult#1}, {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: ?, size: INF
419.69/291.60	  lineMult#2: runtime: ?, size: INF
419.69/291.60	  lineMult#1: runtime: ?, size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(67) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: lineMult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 7 + 5*z'
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: lineMult#2
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 10 + 5*z1
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: lineMult#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 6 + 5*z
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(68)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 2 }-> computeLine(z1, @ls, lineMult#1(@l, z', z'')) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 1 }-> lineMult#1(z', z'', z) :|: z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> lineMult#2(z', z'', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 3 }-> 1 + s18 + lineMult(z', z1, @ys) :|: s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 2 }-> 1 + s19 + lineMult(z', z1, 1) :|: s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(69) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.60	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(70)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 8 + 5*@l }-> computeLine(z1, @ls, s20) :|: s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(71) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: computeLine
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: computeLine#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: computeLine#2
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(72)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 8 + 5*@l }-> computeLine(z1, @ls, s20) :|: s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {computeLine,computeLine#1,computeLine#2}, {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	  computeLine: runtime: ?, size: INF
419.69/291.60	  computeLine#1: runtime: ?, size: INF
419.69/291.60	  computeLine#2: runtime: ?, size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(73) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: computeLine
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 13 + 2*z + 10*z'
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: computeLine#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 10 + 2*z + 10*z''
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using CoFloCo for: computeLine#2
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^1) with polynomial bound: 11 + 10*z + 2*z1
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(74)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 1 }-> computeLine#1(z, z'', z') :|: z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> computeLine#2(z'', z', @x, @xs) :|: z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 8 + 5*@l }-> computeLine(z1, @ls, s20) :|: s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 + computeLine(@l, z', 1) + matrixMult(@ls, z') :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	  computeLine: runtime: O(n^1) [13 + 2*z + 10*z'], size: INF
419.69/291.60	  computeLine#1: runtime: O(n^1) [10 + 2*z + 10*z''], size: INF
419.69/291.60	  computeLine#2: runtime: O(n^1) [11 + 10*z + 2*z1], size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(75) ResultPropagationProof (UPPER BOUND(ID))
419.69/291.60	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(76)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 11 + 2*z + 10*z' }-> s25 :|: s25 >= 0, s25 <= inf3, z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 12 + 2*@xs + 10*z'' }-> s26 :|: s26 >= 0, s26 <= inf4, z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 21 + 5*@l + 10*@ls + 2*z1 }-> s27 :|: s27 >= 0, s27 <= inf5, s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 14 + 2*@l + 10*z' }-> 1 + s28 + matrixMult(@ls, z') :|: s28 >= 0, s28 <= inf6, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	  computeLine: runtime: O(n^1) [13 + 2*z + 10*z'], size: INF
419.69/291.60	  computeLine#1: runtime: O(n^1) [10 + 2*z + 10*z''], size: INF
419.69/291.60	  computeLine#2: runtime: O(n^1) [11 + 10*z + 2*z1], size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(77) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: matrixMult#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	Computed SIZE bound using CoFloCo for: matrixMult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: INF with polynomial bound: ?
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(78)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 11 + 2*z + 10*z' }-> s25 :|: s25 >= 0, s25 <= inf3, z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 12 + 2*@xs + 10*z'' }-> s26 :|: s26 >= 0, s26 <= inf4, z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 21 + 5*@l + 10*@ls + 2*z1 }-> s27 :|: s27 >= 0, s27 <= inf5, s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 14 + 2*@l + 10*z' }-> 1 + s28 + matrixMult(@ls, z') :|: s28 >= 0, s28 <= inf6, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: {matrixMult#1,matrixMult}
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	  computeLine: runtime: O(n^1) [13 + 2*z + 10*z'], size: INF
419.69/291.60	  computeLine#1: runtime: O(n^1) [10 + 2*z + 10*z''], size: INF
419.69/291.60	  computeLine#2: runtime: O(n^1) [11 + 10*z + 2*z1], size: INF
419.69/291.60	  matrixMult#1: runtime: ?, size: INF
419.69/291.60	  matrixMult: runtime: ?, size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(79) IntTrsBoundProof (UPPER BOUND(ID))
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using KoAT for: matrixMult#1
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^2) with polynomial bound: 1 + 30*z + 20*z*z' + 4*z^2
419.69/291.60	
419.69/291.60	Computed RUNTIME bound using KoAT for: matrixMult
419.69/291.60	after applying outer abstraction to obtain an ITS,
419.69/291.60	resulting in: O(n^2) with polynomial bound: 2 + 30*z + 20*z*z' + 4*z^2
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(80)
419.69/291.60	Obligation:
419.69/291.60	Complexity RNTS consisting of the following rules:
419.69/291.60	
419.69/291.60	   #add(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> s11 :|: s9 >= 0, s9 <= 2 * (1 + (1 + (z - 4))) + z', s10 >= 0, s10 <= s9 + 2, s11 >= 0, s11 <= s10 + 2, z' >= 0, z - 4 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> s8 :|: s6 >= 0, s6 <= 2 * (1 + (1 + (z - 4))) + z', s7 >= 0, s7 <= s6 + 2, s8 >= 0, s8 <= s7 + 2, z - 4 >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0'
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0
419.69/291.60	   #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x'))
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x)
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x')
419.69/291.60	   #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z = 0, z' - 1 >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #mult(z, z') -{ 0 }-> 1 + s12 :|: s12 >= 0, s12 <= 4 * (z' - 1) + 4 * ((z' - 1) * (z - 1)) + 4 * (z - 1) + 4, z - 1 >= 0, z' - 1 >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s15 :|: s13 >= 0, s13 <= 4 * z' + 4 * (z' * (z - 2)) + 4 * (z - 2) + 4, s14 >= 0, s14 <= 2 * (1 + z') + s13, s15 >= 0, s15 <= 2 * (1 + z') + s14, z - 2 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 2 * (1 + z') + 0, z = 1 + 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 2 * (1 + z') + 0, z - 1 >= 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z = 0, z' >= 0
419.69/291.60	   #natmult(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #pred(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0)
419.69/291.60	   #succ(z) -{ 0 }-> 0 :|: z >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0
419.69/291.60	   #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0
419.69/291.60	   computeLine(z, z', z'') -{ 11 + 2*z + 10*z' }-> s25 :|: s25 >= 0, s25 <= inf3, z' >= 0, z'' >= 0, z >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 12 + 2*@xs + 10*z'' }-> s26 :|: s26 >= 0, s26 <= inf4, z' >= 0, z'' >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0
419.69/291.60	   computeLine#1(z, z', z'') -{ 1 }-> z' :|: z' >= 0, z'' >= 0, z = 1
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 21 + 5*@l + 10*@ls + 2*z1 }-> s27 :|: s27 >= 0, s27 <= inf5, s20 >= 0, s20 <= inf, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   computeLine#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z'' >= 0, z = 1, z1 >= 0
419.69/291.60	   lineMult(z, z', z'') -{ 7 + 5*z' }-> s21 :|: s21 >= 0, s21 <= inf', z' >= 0, z >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 11 + 5*@xs }-> s22 :|: s22 >= 0, s22 <= inf'', @x >= 0, z = 1 + @x + @xs, z' >= 0, z'' >= 0, @xs >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 1 }-> 1 :|: z = 1, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 10 + 5*z1 }-> 1 + s18 + s23 :|: s23 >= 0, s23 <= inf1, s17 >= 0, s17 <= 4 * (z' * z'') + 1, s18 >= 0, s18 <= 2 * s17 + @y, z = 1 + @y + @ys, z'' >= 0, z' >= 0, z1 >= 0, @y >= 0, @ys >= 0
419.69/291.60	   lineMult#2(z, z', z'', z1) -{ 9 + 5*z1 }-> 1 + s19 + s24 :|: s24 >= 0, s24 <= inf2, s19 >= 0, s19 <= 4 * (z' * z'') + 1, z'' >= 0, z = 1, z' >= 0, z1 >= 0
419.69/291.60	   matrixMult(z, z') -{ 1 }-> matrixMult#1(z, z') :|: z >= 0, z' >= 0
419.69/291.60	   matrixMult#1(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 1
419.69/291.60	   matrixMult#1(z, z') -{ 14 + 2*@l + 10*z' }-> 1 + s28 + matrixMult(@ls, z') :|: s28 >= 0, s28 <= inf6, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' >= 0
419.69/291.60	   plus(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2 * z + z', z >= 0, z' >= 0
419.69/291.60	   times(z, z') -{ 1 }-> s16 :|: s16 >= 0, s16 <= 4 * (z' * z) + 1, z >= 0, z' >= 0
419.69/291.60	
419.69/291.60	Function symbols to be analyzed: 
419.69/291.60	Previous analysis results are:
419.69/291.60	  #pred: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #succ: runtime: O(1) [0], size: O(n^1) [2 + z]
419.69/291.60	  #add: runtime: O(1) [0], size: O(n^1) [2*z + z']
419.69/291.60	  #natmult: runtime: O(1) [0], size: O(n^2) [4 + 4*z + 4*z*z' + 4*z']
419.69/291.60	  plus: runtime: O(1) [1], size: O(n^1) [2*z + z']
419.69/291.60	  #mult: runtime: O(1) [0], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  times: runtime: O(1) [1], size: O(n^2) [1 + 4*z*z']
419.69/291.60	  lineMult: runtime: O(n^1) [7 + 5*z'], size: INF
419.69/291.60	  lineMult#2: runtime: O(n^1) [10 + 5*z1], size: INF
419.69/291.60	  lineMult#1: runtime: O(n^1) [6 + 5*z], size: INF
419.69/291.60	  computeLine: runtime: O(n^1) [13 + 2*z + 10*z'], size: INF
419.69/291.60	  computeLine#1: runtime: O(n^1) [10 + 2*z + 10*z''], size: INF
419.69/291.60	  computeLine#2: runtime: O(n^1) [11 + 10*z + 2*z1], size: INF
419.69/291.60	  matrixMult#1: runtime: O(n^2) [1 + 30*z + 20*z*z' + 4*z^2], size: INF
419.69/291.60	  matrixMult: runtime: O(n^2) [2 + 30*z + 20*z*z' + 4*z^2], size: INF
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(81) FinalProof (FINISHED)
419.69/291.60	Computed overall runtime complexity
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(82)
419.69/291.60	BOUNDS(1, n^2)
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(83) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
419.69/291.60	Transformed a relative TRS into a decreasing-loop problem.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(84)
419.69/291.60	Obligation:
419.69/291.60	Analyzing the following TRS for decreasing loops:
419.69/291.60	
419.69/291.60	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.60	
419.69/291.60	
419.69/291.60	The TRS R consists of the following rules:
419.69/291.60	
419.69/291.60	   *(@x, @y) -> #mult(@x, @y)
419.69/291.60	   +(@x, @y) -> #add(@x, @y)
419.69/291.60	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m)
419.69/291.60	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs)
419.69/291.60	   computeLine#1(nil, @acc, @m) -> @acc
419.69/291.60	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc))
419.69/291.60	   computeLine#2(nil, @acc, @x, @xs) -> nil
419.69/291.60	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n)
419.69/291.60	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs)
419.69/291.60	   lineMult#1(nil, @l2, @n) -> nil
419.69/291.60	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
419.69/291.60	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil))
419.69/291.60	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2)
419.69/291.60	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
419.69/291.60	   matrixMult#1(nil, @m2) -> nil
419.69/291.60	
419.69/291.60	The (relative) TRS S consists of the following rules:
419.69/291.60	
419.69/291.60	   #add(#0, @y) -> @y
419.69/291.60	   #add(#neg(#s(#0)), @y) -> #pred(@y)
419.69/291.60	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
419.69/291.60	   #add(#pos(#s(#0)), @y) -> #succ(@y)
419.69/291.60	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
419.69/291.60	   #mult(#0, #0) -> #0
419.69/291.60	   #mult(#0, #neg(@y)) -> #0
419.69/291.60	   #mult(#0, #pos(@y)) -> #0
419.69/291.60	   #mult(#neg(@x), #0) -> #0
419.69/291.60	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #0) -> #0
419.69/291.60	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #natmult(#0, @y) -> #0
419.69/291.60	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
419.69/291.60	   #pred(#0) -> #neg(#s(#0))
419.69/291.60	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
419.69/291.60	   #pred(#pos(#s(#0))) -> #0
419.69/291.60	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
419.69/291.60	   #succ(#0) -> #pos(#s(#0))
419.69/291.60	   #succ(#neg(#s(#0))) -> #0
419.69/291.60	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
419.69/291.60	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
419.69/291.60	
419.69/291.60	Rewrite Strategy: INNERMOST
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(85) DecreasingLoopProof (LOWER BOUND(ID))
419.69/291.60	The following loop(s) give(s) rise to the lower bound Omega(n^1):
419.69/291.60	
419.69/291.60	The rewrite sequence
419.69/291.60	
419.69/291.60	matrixMult#1(::(@l, @ls), @m2) ->^+ ::(computeLine(@l, @m2, nil), matrixMult#1(@ls, @m2))
419.69/291.60	
419.69/291.60	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
419.69/291.60	
419.69/291.60	The pumping substitution is [@ls / ::(@l, @ls)].
419.69/291.60	
419.69/291.60	The result substitution is [ ].
419.69/291.60	
419.69/291.60	
419.69/291.60	
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(86)
419.69/291.60	Complex Obligation (BEST)
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(87)
419.69/291.60	Obligation:
419.69/291.60	Proved the lower bound n^1 for the following obligation:
419.69/291.60	
419.69/291.60	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.60	
419.69/291.60	
419.69/291.60	The TRS R consists of the following rules:
419.69/291.60	
419.69/291.60	   *(@x, @y) -> #mult(@x, @y)
419.69/291.60	   +(@x, @y) -> #add(@x, @y)
419.69/291.60	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m)
419.69/291.60	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs)
419.69/291.60	   computeLine#1(nil, @acc, @m) -> @acc
419.69/291.60	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc))
419.69/291.60	   computeLine#2(nil, @acc, @x, @xs) -> nil
419.69/291.60	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n)
419.69/291.60	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs)
419.69/291.60	   lineMult#1(nil, @l2, @n) -> nil
419.69/291.60	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
419.69/291.60	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil))
419.69/291.60	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2)
419.69/291.60	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
419.69/291.60	   matrixMult#1(nil, @m2) -> nil
419.69/291.60	
419.69/291.60	The (relative) TRS S consists of the following rules:
419.69/291.60	
419.69/291.60	   #add(#0, @y) -> @y
419.69/291.60	   #add(#neg(#s(#0)), @y) -> #pred(@y)
419.69/291.60	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
419.69/291.60	   #add(#pos(#s(#0)), @y) -> #succ(@y)
419.69/291.60	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
419.69/291.60	   #mult(#0, #0) -> #0
419.69/291.60	   #mult(#0, #neg(@y)) -> #0
419.69/291.60	   #mult(#0, #pos(@y)) -> #0
419.69/291.60	   #mult(#neg(@x), #0) -> #0
419.69/291.60	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #0) -> #0
419.69/291.60	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #natmult(#0, @y) -> #0
419.69/291.60	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
419.69/291.60	   #pred(#0) -> #neg(#s(#0))
419.69/291.60	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
419.69/291.60	   #pred(#pos(#s(#0))) -> #0
419.69/291.60	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
419.69/291.60	   #succ(#0) -> #pos(#s(#0))
419.69/291.60	   #succ(#neg(#s(#0))) -> #0
419.69/291.60	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
419.69/291.60	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
419.69/291.60	
419.69/291.60	Rewrite Strategy: INNERMOST
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(88) LowerBoundPropagationProof (FINISHED)
419.69/291.60	Propagated lower bound.
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(89)
419.69/291.60	BOUNDS(n^1, INF)
419.69/291.60	
419.69/291.60	----------------------------------------
419.69/291.60	
419.69/291.60	(90)
419.69/291.60	Obligation:
419.69/291.60	Analyzing the following TRS for decreasing loops:
419.69/291.60	
419.69/291.60	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
419.69/291.60	
419.69/291.60	
419.69/291.60	The TRS R consists of the following rules:
419.69/291.60	
419.69/291.60	   *(@x, @y) -> #mult(@x, @y)
419.69/291.60	   +(@x, @y) -> #add(@x, @y)
419.69/291.60	   computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m)
419.69/291.60	   computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs)
419.69/291.60	   computeLine#1(nil, @acc, @m) -> @acc
419.69/291.60	   computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc))
419.69/291.60	   computeLine#2(nil, @acc, @x, @xs) -> nil
419.69/291.60	   lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n)
419.69/291.60	   lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs)
419.69/291.60	   lineMult#1(nil, @l2, @n) -> nil
419.69/291.60	   lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
419.69/291.60	   lineMult#2(nil, @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil))
419.69/291.60	   matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2)
419.69/291.60	   matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
419.69/291.60	   matrixMult#1(nil, @m2) -> nil
419.69/291.60	
419.69/291.60	The (relative) TRS S consists of the following rules:
419.69/291.60	
419.69/291.60	   #add(#0, @y) -> @y
419.69/291.60	   #add(#neg(#s(#0)), @y) -> #pred(@y)
419.69/291.60	   #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
419.69/291.60	   #add(#pos(#s(#0)), @y) -> #succ(@y)
419.69/291.60	   #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
419.69/291.60	   #mult(#0, #0) -> #0
419.69/291.60	   #mult(#0, #neg(@y)) -> #0
419.69/291.60	   #mult(#0, #pos(@y)) -> #0
419.69/291.60	   #mult(#neg(@x), #0) -> #0
419.69/291.60	   #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #0) -> #0
419.69/291.60	   #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
419.69/291.60	   #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
419.69/291.60	   #natmult(#0, @y) -> #0
419.69/291.60	   #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
419.69/291.60	   #pred(#0) -> #neg(#s(#0))
419.69/291.60	   #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
419.69/291.60	   #pred(#pos(#s(#0))) -> #0
419.69/291.60	   #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
419.69/291.60	   #succ(#0) -> #pos(#s(#0))
419.69/291.60	   #succ(#neg(#s(#0))) -> #0
419.69/291.60	   #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
419.69/291.60	   #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
419.69/291.60	
419.69/291.60	Rewrite Strategy: INNERMOST
419.93/291.66	EOF