YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 138 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 8 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 15 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Rules:
l0(__const_42HAT0, i1HAT0) -> l1(__const_42HATpost, i1HATpost) :|: __const_42HAT0 = __const_42HATpost && i1HATpost = 1 + i1HAT0
l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l3(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6
l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14 && x12 <= x13
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 1 + x17 <= x16
l1(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22
l6(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 0
l7(x28, x29) -> l6(x30, x31) :|: x29 = x31 && x28 = x30
Start term: l7(__const_42HAT0, i1HAT0)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
l0(__const_42HAT0, i1HAT0) -> l1(__const_42HATpost, i1HATpost) :|: __const_42HAT0 = __const_42HATpost && i1HATpost = 1 + i1HAT0
l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l3(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6
l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14 && x12 <= x13
l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 1 + x17 <= x16
l1(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22
l6(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 0
l7(x28, x29) -> l6(x30, x31) :|: x29 = x31 && x28 = x30
Start term: l7(__const_42HAT0, i1HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_42HAT0, i1HAT0) -> l1(__const_42HATpost, i1HATpost) :|: __const_42HAT0 = __const_42HATpost && i1HATpost = 1 + i1HAT0
(2) l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l3(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6
(4) l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(5) l4(x12, x13) -> l5(x14, x15) :|: x13 = x15 && x12 = x14 && x12 <= x13
(6) l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 1 + x17 <= x16
(7) l1(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22
(8) l6(x24, x25) -> l1(x26, x27) :|: x24 = x26 && x27 = 0
(9) l7(x28, x29) -> l6(x30, x31) :|: x29 = x31 && x28 = x30

Arcs:
(1) -> (7)
(2) -> (1)
(3) -> (2)
(4) -> (1)
(6) -> (3), (4)
(7) -> (5), (6)
(8) -> (7)
(9) -> (8)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(__const_42HAT0, i1HAT0) -> l1(__const_42HATpost, i1HATpost) :|: __const_42HAT0 = __const_42HATpost && i1HATpost = 1 + i1HAT0
(2) l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(3) l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(4) l3(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6
(5) l4(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 1 + x17 <= x16
(6) l1(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22

Arcs:
(1) -> (6)
(2) -> (1)
(3) -> (1)
(4) -> (3)
(5) -> (2), (4)
(6) -> (5)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(__const_42HATpost:0, x11:0) -> l4(__const_42HATpost:0, 1 + x11:0) :|: __const_42HATpost:0 >= 1 + x11:0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
l4(__const_42HATpost:0, x11:0) -> l4(__const_42HATpost:0, c) :|: c = 1 + x11:0 && __const_42HATpost:0 >= 1 + x11:0

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1)] = x - x1

The following rules are decreasing:
l4(__const_42HATpost:0, x11:0) -> l4(__const_42HATpost:0, c) :|: c = 1 + x11:0 && __const_42HATpost:0 >= 1 + x11:0
The following rules are bounded:
l4(__const_42HATpost:0, x11:0) -> l4(__const_42HATpost:0, c) :|: c = 1 + x11:0 && __const_42HATpost:0 >= 1 + x11:0

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(10)
YES