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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
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 = x27x28 = x28x29 = x29x30 = x30x31 = x31x32 = x32x33 = x33x34 = x34x35 = x35x36 = x36x37 = x37x38 = x38x39 = x39x40 = x40x41 = x41x42 = x42x43 = x43x44 = x44
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

2.1 SCC Subproblem 1/1

Here we consider the SCC { l10, l1, l3, l0, l2, l9 }.

2.1.1 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by 0.

l0: −2 + x10 + x11 − 5⋅x13x14 + 2⋅x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l2: −2 + x10 + x11 − 5⋅x13x14 + 2⋅x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l10: −2 + x10 + x11 − 5⋅x13x14 + 2⋅x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l9: x10 + x14 + 2⋅x21x24 + 2⋅x26 − 2⋅x27 + x28x31
l1: −2 + x10 + x14 + 2⋅x21x24 + x28x31
l3: −2 + x10 + x14 + 2⋅x21x24 + x28x31

2.1.2 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 0.

l0: −1 − 3⋅x11x13x14 + x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l2: −1 − 3⋅x11x13x14 + x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l10: −1 − 3⋅x11x13x14 + x21x27 + x28 + x31 + x32 + x33 + x37 + x40x43 + x44
l9: x21x24 + 2⋅x26 − 2⋅x27
l1: −1 + x21x24
l3: −1 + x21x24

2.1.3 Transition Removal

We remove transitions 1, 15, 13, 4, 3, 2 using the following ranking functions, which are bounded by 0.

l0: 2
l2: 1
l10: 3
l9: −4⋅x26 + 4⋅x27
l3: 0
l1: −1

2.1.4 Trivial Cooperation Program

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

Tool configuration

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