LTS Termination Proof

by AProVE

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
l5 l5 l5: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l22 l22 l22: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l39 l39 l39: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l1 l1 l1: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l13 l13 l13: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l31 l31 l31: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l18 l18 l18: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l17 l17 l17: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l35 l35 l35: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l21 l21 l21: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l14 l14 l14: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l9 l9 l9: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l6 l6 l6: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l8 l8 l8: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l34 l34 l34: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l27 l27 l27: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l0 l0 l0: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l12 l12 l12: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l19 l19 l19: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l33 l33 l33: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l26 l26 l26: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l40 l40 l40: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l7 l7 l7: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l24 l24 l24: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l37 l37 l37: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l11 l11 l11: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l3 l3 l3: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l41 l41 l41: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l20 l20 l20: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l32 l32 l32: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l28 l28 l28: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l42 l42 l42: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l2 l2 l2: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l23 l23 l23: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l4 l4 l4: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l38 l38 l38: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l10 l10 l10: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l36 l36 l36: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l29 l29 l29: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l15 l15 l15: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l16 l16 l16: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
l30 l30 l30: x1 = x1x2 = x2x3 = x3x4 = x4x5 = x5x6 = x6x7 = x7x8 = x8x9 = x9x10 = x10x11 = x11x12 = x12x13 = x13x14 = x14x15 = x15x16 = x16x17 = x17x18 = x18x19 = x19x20 = x20x21 = x21x22 = x22x23 = x23x24 = x24x25 = x25x26 = x26x27 = x27
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

2.1 SCC Subproblem 1/3

Here we consider the SCC { l39, l38, l37, l40 }.

2.1.1 Transition Removal

We remove transition 50 using the following ranking functions, which are bounded by 0.

l37: x6 + x8
l38: x6 + x8
l39: x6 + x8
l40: x6 + x8

2.1.2 Transition Removal

We remove transition 52 using the following ranking functions, which are bounded by 0.

l37: −1 − x5 + x8
l38: x5 + x8
l39: −1 − x5 + x8
l40: x5 + x8

2.1.3 Transition Removal

We remove transitions 49, 56, 53 using the following ranking functions, which are bounded by 0.

l37: 2
l38: 1
l39: 3
l40: 0

2.1.4 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.2 SCC Subproblem 2/3

Here we consider the SCC { l35, l33 }.

2.2.1 Transition Removal

We remove transition 47 using the following ranking functions, which are bounded by 0.

l33: x5 + x8
l35: x5 + x8

2.2.2 Transition Removal

We remove transition 59 using the following ranking functions, which are bounded by 0.

l35: 0
l33: −1

2.2.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.3 SCC Subproblem 3/3

Here we consider the SCC { l5, l22, l1, l13, l31, l18, l17, l21, l14, l9, l6, l8, l34, l27, l0, l19, l12, l26, l7, l24, l11, l3, l20, l32, l28, l2, l23, l4, l10, l36, l29, l15, l16, l30 }.

2.3.1 Transition Removal

We remove transitions 29, 39 using the following ranking functions, which are bounded by 0.

l0: −3 − 3⋅x6 + 3⋅x8
l1: −3 − 3⋅x6 + 3⋅x8
l2: −3 − 3⋅x6 + 3⋅x8
l6: −3 − 3⋅x6 + 3⋅x8
l10: −3 − 3⋅x6 + 3⋅x8
l5: −3 − 3⋅x6 + 3⋅x8
l7: −3 − 3⋅x6 + 3⋅x8
l11: −3 − 3⋅x6 + 3⋅x8
l13: −3 − 3⋅x6 + 3⋅x8
l20: −3 − 3⋅x6 + 3⋅x8
l15: −3 − 3⋅x6 + 3⋅x8
l18: −3 − 3⋅x6 + 3⋅x8
l14: −3 − 3⋅x6 + 3⋅x8
l19: −3 − 3⋅x6 + 3⋅x8
l17: −3⋅x6 + 3⋅x8
l16: −3⋅x6 + 3⋅x8
l9: −3⋅x6 + 3⋅x8
l8: −3⋅x6 + 3⋅x8
l22: −3⋅x6 + 3⋅x8
l24: −3⋅x6 + 3⋅x8
l29: −3⋅x6 + 3⋅x8
l26: −3⋅x6 + 3⋅x8
l30: −3⋅x6 + 3⋅x8
l34: −3⋅x6 + 3⋅x8
l36: −3⋅x6 + 3⋅x8
l21: −3⋅x6 + 3⋅x8
l4: −3⋅x6 + 3⋅x8
l3: −3⋅x6 + 3⋅x8
l12: −3 − 3⋅x6 + 3⋅x8
l32: −3 − 3⋅x6 + 3⋅x8
l31: −3 − 3⋅x6 + 3⋅x8
l28: −3 − 3⋅x6 + 3⋅x8
l27: −3 − 3⋅x6 + 3⋅x8
l23: −3 − 3⋅x6 + 3⋅x8

2.3.2 Transition Removal

We remove transitions 31, 55, 42 using the following ranking functions, which are bounded by 0.

l0: −1 − x5 + x8
l1: −1 − x5 + x8
l2: −1 − x5 + x8
l6: −1 − x5 + x8
l10: −1 − x5 + x8
l5: −1 − x5 + x8
l7: −1 − x5 + x8
l11: −1 − x5 + x8
l13: −1 − x5 + x8
l20: −1 − x5 + x8
l15: −1 − x5 + x8
l18: −1 − x5 + x8
l14: −1 − x5 + x8
l19: −1 − x5 + x8
l16: −1 − x5 + x8
l17: −1 − x5 + x8
l9: x5 + x8
l8: x5 + x8
l22: x5 + x8
l24: x5 + x8
l29: x5 + x8
l26: x5 + x8
l30: x5 + x8
l34: x5 + x8
l36: x5 + x8
l21: x5 + x8
l4: −1 − x5 + x8
l3: −1 − x5 + x8
l12: −1 − x5 + x8
l32: −1 − x5 + x8
l31: −1 − x5 + x8
l28: −1 − x5 + x8
l27: −1 − x5 + x8
l23: −1 − x5 + x8

2.3.3 Transition Removal

We remove transitions 25, 33, 44 using the following ranking functions, which are bounded by 0.

l0: 98 − 2⋅x4
l1: 98 − 2⋅x4
l2: 98 − 2⋅x4
l6: 98 − 2⋅x4
l10: 98 − 2⋅x4
l5: 98 − 2⋅x4
l7: 98 − 2⋅x4
l11: 98 − 2⋅x4
l13: 98 − 2⋅x4
l20: 99 − 2⋅x4
l15: 98 − 2⋅x4
l18: 98 − 2⋅x4
l14: 99 − 2⋅x4
l19: 99 − 2⋅x4
l16: 98 − 2⋅x4
l17: 98 − 2⋅x4
l8: 98 − 2⋅x4
l9: 98 − 2⋅x4
l22: 99 − 2⋅x4
l24: 99 − 2⋅x4
l29: 99 − 2⋅x4
l26: 99 − 2⋅x4
l30: 100 − 2⋅x4
l34: 100 − 2⋅x4
l36: 98 − 2⋅x4
l21: 98 − 2⋅x4
l4: −1 − 2⋅x4 + 100⋅x6 − 100⋅x8
l3: −1 − 2⋅x4 + 100⋅x6 − 100⋅x8
l12: 98 − 2⋅x4
l32: 98 − 2⋅x4
l31: 98 − 2⋅x4
l28: 98 − 2⋅x4
l27: 98 − 2⋅x4
l23: 98 − 2⋅x4

2.3.4 Transition Removal

We remove transitions 61, 58 using the following ranking functions, which are bounded by 0.

l0: 1 − x7 + x8
l1: 1 − x7 + x8
l2: 2 − x7 + x8
l6: 2 − x7 + x8
l10: 2 − x7 + x8
l5: 2 − x7 + x8
l7: 2 − x7 + x8
l11: 2 − x7 + x8
l13: 2 − x7 + x8
l15: 2 − x7 + x8
l18: 2 − x7 + x8
l14: 2 − x7 + x8
l19: 2 − x7 + x8
l20: 2 − x7 + x8
l16: −3 − x7 + x8
l17: −3 − x7 + x8
l8: −1 − 2⋅x6x7 + 3⋅x8
l9: −1 − 2⋅x6x7 + 3⋅x8
l22: −5 + x4 − 2⋅x6x7 + 3⋅x8
l24: −5 + x4 − 2⋅x6x7 + 3⋅x8
l29: −5 + x4 + 2⋅x5 − 2⋅x6x7 + x8
l26: −5 + x4 + 2⋅x5 − 2⋅x6x7 + x8
l34: −1 − x5 − 2⋅x6x7 + 4⋅x8
l30: −1 − x5 − 2⋅x6x7 + 4⋅x8
l36: −1 − x5 − 2⋅x6x7 + 4⋅x8
l21: −1 − x5 − 2⋅x6x7 + 4⋅x8
l4: −6 + x4 + 2⋅x5 + x6x7 − 2⋅x8
l3: −6 + x4 + 2⋅x5 + x6x7 − 2⋅x8
l12: x7 + x8
l32: x7 + x8
l31: x7 + x8
l28: x7 + x8
l27: x7 + x8
l23: x7 + x8

2.3.5 Transition Removal

We remove transitions 1, 7, 11, 5, 4, 6, 10, 9, 13, 14, 16, 20, 19, 15, 23, 22, 26, 21, 24, 12 using the following ranking functions, which are bounded by 0.

l0: 0
l1: −4
l2: 0
l6: 1
l10: 4
l5: 2
l7: 3
l11: 5
l13: 6
l15: 7
l18: 8
l14: 8
l19: 9
l20: 10
l16: −4
l17: −4
l8: −1 − 3⋅x6 + 3⋅x8
l9: −1 − 3⋅x6 + 3⋅x8
l22: −5 + x4 − 3⋅x6 + 3⋅x8
l24: −5 + x4 − 3⋅x6 + 3⋅x8
l29: −5 + x4 + 2⋅x5 − 3⋅x6 + x8
l26: −5 + x4 + 2⋅x5 − 3⋅x6 + x8
l34: −1 − x5 − 3⋅x6 + 4⋅x8
l30: −1 − x5 − 3⋅x6 + 4⋅x8
l36: −1 − x5 − 3⋅x6 + 4⋅x8
l21: −1 − x5 − 3⋅x6 + 4⋅x8
l4: −5 + x4 + 2⋅x5x6x8
l3: −5 + x4 + 2⋅x5x6x8
l12: −4
l32: −4
l31: −4
l28: −4
l27: −4
l23: −4

2.3.6 Transition Removal

We remove transitions 41, 54, 30 using the following ranking functions, which are bounded by 0.

l2: −1 + 4⋅x1 + 5⋅x4 + 6⋅x5 + 7⋅x6x7 + 8⋅x8 + 9⋅x9 + 10⋅x10 + 11⋅x11 + 12⋅x12 + 13⋅x13 + 14⋅x14 + 15⋅x15 + 16⋅x16 + 17⋅x17 + 18⋅x18 + 19⋅x19 + 20⋅x20 + 21⋅x21 + 22⋅x22 + 23⋅x23 + 24⋅x24 + 25⋅x25 + 26⋅x26 + 27⋅x27
l0: −1 + 4⋅x1 + 5⋅x4 + 6⋅x5 + 7⋅x6x7 + 8⋅x8 + 9⋅x9 + 10⋅x10 + 11⋅x11 + 12⋅x12 + 13⋅x13 + 14⋅x14 + 15⋅x15 + 16⋅x16 + 17⋅x17 + 18⋅x18 + 19⋅x19 + 20⋅x20 + 21⋅x21 + 22⋅x22 + 23⋅x23 + 24⋅x24 + 25⋅x25 + 26⋅x26 + 27⋅x27
l16: −1 + x5 + 2⋅x6 − 3⋅x8
l17: −1 + x5 + 2⋅x6 − 3⋅x8
l8: x5x8
l9: x5x8
l22: x5x8
l24: x5x8
l29: 1 + x5x8
l26: 1 + x5x8
l34: −2 + x5x8
l30: −2 + x5x8
l36: −1 + x5x8
l21: −1 + x5x8
l4: −1 + x5 + 3⋅x6 − 4⋅x8
l3: −1 + x5 + 3⋅x6 − 4⋅x8
l12: 1 + x5 + 2⋅x6 − 3⋅x8
l32: 1 + x5 + 2⋅x6 − 3⋅x8
l31: 1 + x5 + 2⋅x6 − 3⋅x8
l28: 1 + x5 + 2⋅x6 − 3⋅x8
l27: 1 + x5 + 2⋅x6 − 3⋅x8
l23: 1 + x5 + 2⋅x6 − 3⋅x8
l1: 1 + x5 + 2⋅x6 − 3⋅x8

2.3.7 Transition Removal

We remove transitions 65, 62, 48, 38, 3, 64 using the following ranking functions, which are bounded by 0.

l2: −1 + 3⋅x1 + 4⋅x4 + 5⋅x5 + 6⋅x6x7 + 7⋅x8 + 8⋅x9 + 9⋅x10 + 10⋅x11 + 11⋅x12 + 12⋅x13 + 13⋅x14 + 14⋅x15 + 15⋅x16 + 16⋅x17 + 17⋅x18 + 18⋅x19 + 19⋅x20 + 20⋅x21 + 21⋅x22 + 22⋅x23 + 23⋅x24 + 24⋅x25 + 25⋅x26 + 26⋅x27
l0: −2 + 3⋅x1 + 4⋅x4 + 5⋅x5 + 6⋅x6x7 + 7⋅x8 + 8⋅x9 + 9⋅x10 + 10⋅x11 + 11⋅x12 + 12⋅x13 + 13⋅x14 + 14⋅x15 + 15⋅x16 + 16⋅x17 + 17⋅x18 + 18⋅x19 + 19⋅x20 + 20⋅x21 + 21⋅x22 + 22⋅x23 + 23⋅x24 + 24⋅x25 + 25⋅x26 + 26⋅x27
l16: −3 + 2⋅x6 − 3⋅x7 + x8
l17: −3 + 2⋅x6 − 3⋅x7 + x8
l8: −3 + 2⋅x6 − 3⋅x7 + x8
l9: −3 + 2⋅x6 − 3⋅x7 + x8
l22: −7 + x4 + 2⋅x6 − 3⋅x7 + x8
l24: −7 + x4 + 2⋅x6 − 3⋅x7 + x8
l26: 0
l29: −1
l34: 1
l30: 0
l21: 1
l36: 0
l4: 1
l3: 2
l12: −1 + 2⋅x6 − 3⋅x7 + x8
l32: −1 + 2⋅x6 − 3⋅x7 + x8
l31: −1 + 2⋅x6 − 3⋅x7 + x8
l28: −2 + 2⋅x6 − 2⋅x7
l27: −2 + 2⋅x6 − 2⋅x7
l23: −2 + 2⋅x6 − 2⋅x7
l1: −2 + 2⋅x6 − 2⋅x7

2.3.8 Transition Removal

We remove transitions 18, 8, 32, 36, 35, 28, 17, 57, 45, 60, 40, 63, 34 using the following ranking functions, which are bounded by 0.

l2: −1 + 10⋅x1 + 11⋅x4 + 12⋅x5 + 13⋅x6x7 + 14⋅x8 + 15⋅x9 + 16⋅x10 + 17⋅x11 + 18⋅x12 + 19⋅x13 + 20⋅x14 + 21⋅x15 + 22⋅x16 + 23⋅x17 + 24⋅x18 + 25⋅x19 + 26⋅x20 + 27⋅x21 + 28⋅x22 + 29⋅x23 + 30⋅x24 + 31⋅x25 + 32⋅x26 + 33⋅x27
l0: −2 + 10⋅x1 + 11⋅x4 + 12⋅x5 + 13⋅x6x7 + 14⋅x8 + 15⋅x9 + 16⋅x10 + 17⋅x11 + 18⋅x12 + 19⋅x13 + 20⋅x14 + 21⋅x15 + 22⋅x16 + 23⋅x17 + 24⋅x18 + 25⋅x19 + 26⋅x20 + 27⋅x21 + 28⋅x22 + 29⋅x23 + 30⋅x24 + 31⋅x25 + 32⋅x26 + 33⋅x27
l16: 2
l17: 1
l8: 0
l9: −1
l22: 1
l24: 2
l12: 3
l32: 4
l31: 5
l28: 6
l27: 7
l23: 8
l1: 9

2.3.9 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

l2: 2⋅x5 − 2⋅x7 + 1
l0: 2⋅x5 − 2⋅x7

2.3.10 Transition Removal

We remove transition 27 using the following ranking functions, which are bounded by 0.

l2: 0
l0: −1

2.3.11 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

Tool configuration

AProVE