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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 134 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 14 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Rules:
l0(Result_4HAT0, i_5HAT0) -> l1(Result_4HATpost, i_5HATpost) :|: 0 <= i_5HAT0 && i_5HAT1 = 1 + i_5HAT0 && i_5HATpost = -2 + i_5HAT1 && Result_4HAT0 = Result_4HATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x6 = x6 && 1 + x5 <= 0
l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(Result_4HAT0, i_5HAT0)

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

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

(2)
Obligation:
Rules:
l0(Result_4HAT0, i_5HAT0) -> l1(Result_4HATpost, i_5HATpost) :|: 0 <= i_5HAT0 && i_5HAT1 = 1 + i_5HAT0 && i_5HATpost = -2 + i_5HAT1 && Result_4HAT0 = Result_4HATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l0(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x6 = x6 && 1 + x5 <= 0
l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14
Start term: l4(Result_4HAT0, i_5HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, i_5HAT0) -> l1(Result_4HATpost, i_5HATpost) :|: 0 <= i_5HAT0 && i_5HAT1 = 1 + i_5HAT0 && i_5HATpost = -2 + i_5HAT1 && Result_4HAT0 = Result_4HATpost
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l0(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x6 = x6 && 1 + x5 <= 0
(4) l3(x8, x9) -> l0(x10, x11) :|: x9 = x11 && x8 = x10
(5) l4(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, i_5HAT0) -> l1(Result_4HATpost, i_5HATpost) :|: 0 <= i_5HAT0 && i_5HAT1 = 1 + i_5HAT0 && i_5HATpost = -2 + i_5HAT1 && Result_4HAT0 = Result_4HATpost
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l0(Result_4HAT0:0, i_5HAT0:0) -> l0(Result_4HAT0:0, -2 + (1 + i_5HAT0:0)) :|: i_5HAT0:0 > -1

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

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

   l0(x1, x2) -> l0(x2)

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

(8)
Obligation:
Rules:
l0(i_5HAT0:0) -> l0(-2 + (1 + i_5HAT0:0)) :|: i_5HAT0:0 > -1

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l0(i_5HAT0:0) -> l0(c) :|: c = -2 + (1 + i_5HAT0:0) && i_5HAT0:0 > -1

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x)] = x

The following rules are decreasing:
l0(i_5HAT0:0) -> l0(c) :|: c = -2 + (1 + i_5HAT0:0) && i_5HAT0:0 > -1
The following rules are bounded:
l0(i_5HAT0:0) -> l0(c) :|: c = -2 + (1 + i_5HAT0:0) && i_5HAT0:0 > -1

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

(12)
YES