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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 234 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 17 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


----------------------------------------

(0)
Obligation:
Rules:
l0(iEXCL14HAT0, iEXCL22HAT0, resultEXCL12HAT0, temp0EXCL15HAT0) -> l1(iEXCL14HATpost, iEXCL22HATpost, resultEXCL12HATpost, temp0EXCL15HATpost) :|: iEXCL14HAT1 = 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 1 + iEXCL14HAT1 <= 10 && iEXCL14HAT2 = 1 + iEXCL14HAT1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 + iEXCL14HAT2 <= 10 && iEXCL14HATpost = 1 + iEXCL14HAT2 && 2 <= iEXCL14HATpost && iEXCL14HATpost <= 2 && iEXCL22HAT0 = iEXCL22HATpost && resultEXCL12HAT0 = resultEXCL12HATpost && temp0EXCL15HAT0 = temp0EXCL15HATpost
l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x3 && 10 <= x
l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && 1 + x9 <= 10 && 1 + x9 <= x12 && x12 <= 1 + x9 && x12 = 1 + x8 && 1 + x8 <= 10
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
Start term: l4(iEXCL14HAT0, iEXCL22HAT0, resultEXCL12HAT0, temp0EXCL15HAT0)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
l0(iEXCL14HAT0, iEXCL22HAT0, resultEXCL12HAT0, temp0EXCL15HAT0) -> l1(iEXCL14HATpost, iEXCL22HATpost, resultEXCL12HATpost, temp0EXCL15HATpost) :|: iEXCL14HAT1 = 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 1 + iEXCL14HAT1 <= 10 && iEXCL14HAT2 = 1 + iEXCL14HAT1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 + iEXCL14HAT2 <= 10 && iEXCL14HATpost = 1 + iEXCL14HAT2 && 2 <= iEXCL14HATpost && iEXCL14HATpost <= 2 && iEXCL22HAT0 = iEXCL22HATpost && resultEXCL12HAT0 = resultEXCL12HATpost && temp0EXCL15HAT0 = temp0EXCL15HATpost
l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x3 && 10 <= x
l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && 1 + x9 <= 10 && 1 + x9 <= x12 && x12 <= 1 + x9 && x12 = 1 + x8 && 1 + x8 <= 10
l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28
Start term: l4(iEXCL14HAT0, iEXCL22HAT0, resultEXCL12HAT0, temp0EXCL15HAT0)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iEXCL14HAT0, iEXCL22HAT0, resultEXCL12HAT0, temp0EXCL15HAT0) -> l1(iEXCL14HATpost, iEXCL22HATpost, resultEXCL12HATpost, temp0EXCL15HATpost) :|: iEXCL14HAT1 = 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 0 <= iEXCL14HAT1 && iEXCL14HAT1 <= 0 && 1 + iEXCL14HAT1 <= 10 && iEXCL14HAT2 = 1 + iEXCL14HAT1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 <= iEXCL14HAT2 && iEXCL14HAT2 <= 1 && 1 + iEXCL14HAT2 <= 10 && iEXCL14HATpost = 1 + iEXCL14HAT2 && 2 <= iEXCL14HATpost && iEXCL14HATpost <= 2 && iEXCL22HAT0 = iEXCL22HATpost && resultEXCL12HAT0 = resultEXCL12HATpost && temp0EXCL15HAT0 = temp0EXCL15HATpost
(2) l1(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x1 = x5 && x = x4 && x6 = x3 && 10 <= x
(3) l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && 1 + x9 <= 10 && 1 + x9 <= x12 && x12 <= 1 + x9 && x12 = 1 + x8 && 1 + x8 <= 10
(4) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20
(5) l4(x24, x25, x26, x27) -> l0(x28, x29, x30, x31) :|: x27 = x31 && x26 = x30 && x25 = x29 && x24 = x28

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

This digraph is fully evaluated!
----------------------------------------

(4)
Obligation:

Termination digraph:
Nodes:
(1) l1(x8, x9, x10, x11) -> l3(x12, x13, x14, x15) :|: x11 = x15 && x10 = x14 && x9 = x13 && 1 + x9 <= 10 && 1 + x9 <= x12 && x12 <= 1 + x9 && x12 = 1 + x8 && 1 + x8 <= 10
(2) l3(x16, x17, x18, x19) -> l1(x20, x21, x22, x23) :|: x19 = x23 && x18 = x22 && x17 = x21 && x16 = x20

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

----------------------------------------

(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(6)
Obligation:
Rules:
l1(x8:0, x13:0, x10:0, x11:0) -> l1(1 + x8:0, x13:0, x10:0, x11:0) :|: x13:0 < 10 && 1 + x8:0 = 1 + x13:0 && x8:0 < 10

----------------------------------------

(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l1(x1, x2, x3, x4) -> l1(x1, x2)

----------------------------------------

(8)
Obligation:
Rules:
l1(x8:0, x13:0) -> l1(1 + x8:0, x13:0) :|: x13:0 < 10 && 1 + x8:0 = 1 + x13:0 && x8:0 < 10

----------------------------------------

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l1(x8:0, x13:0) -> l1(c, x13:0) :|: c = 1 + x8:0 && (x13:0 < 10 && 1 + x8:0 = 1 + x13:0 && x8:0 < 10)

----------------------------------------

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = 9 - x

The following rules are decreasing:
l1(x8:0, x13:0) -> l1(c, x13:0) :|: c = 1 + x8:0 && (x13:0 < 10 && 1 + x8:0 = 1 + x13:0 && x8:0 < 10)
The following rules are bounded:
l1(x8:0, x13:0) -> l1(c, x13:0) :|: c = 1 + x8:0 && (x13:0 < 10 && 1 + x8:0 = 1 + x13:0 && x8:0 < 10)

----------------------------------------

(12)
YES