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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 137 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 15 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
Rules:
l0(__const_10HAT0, iHAT0) -> l1(__const_10HATpost, iHATpost) :|: iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost
l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2 && x <= x1
l2(x4, x5) -> l3(x6, x7) :|: x4 = x6 && x7 = 1 + x5 && 1 + x5 <= x4
l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10
l1(x12, x13) -> l4(x14, x15) :|: x13 = x15 && x12 = x14
l5(x16, x17) -> l3(x18, x19) :|: x16 = x18 && x19 = 0
l6(x20, x21) -> l5(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l6(__const_10HAT0, iHAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_10HAT0, iHAT0) -> l1(__const_10HATpost, iHATpost) :|: iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost
l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2 && x <= x1
l2(x4, x5) -> l3(x6, x7) :|: x4 = x6 && x7 = 1 + x5 && 1 + x5 <= x4
l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10
l1(x12, x13) -> l4(x14, x15) :|: x13 = x15 && x12 = x14
l5(x16, x17) -> l3(x18, x19) :|: x16 = x18 && x19 = 0
l6(x20, x21) -> l5(x22, x23) :|: x21 = x23 && x20 = x22
Start term: l6(__const_10HAT0, iHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, iHAT0) -> l1(__const_10HATpost, iHATpost) :|: iHAT0 = iHATpost && __const_10HAT0 = __const_10HATpost
(2) l2(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2 && x <= x1
(3) l2(x4, x5) -> l3(x6, x7) :|: x4 = x6 && x7 = 1 + x5 && 1 + x5 <= x4
(4) l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10
(5) l1(x12, x13) -> l4(x14, x15) :|: x13 = x15 && x12 = x14
(6) l5(x16, x17) -> l3(x18, x19) :|: x16 = x18 && x19 = 0
(7) l6(x20, x21) -> l5(x22, x23) :|: x21 = x23 && x20 = x22

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x4, x5) -> l3(x6, x7) :|: x4 = x6 && x7 = 1 + x5 && 1 + x5 <= x4
(2) l3(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l2(x10:0, x5:0) -> l2(x10:0, 1 + x5:0) :|: x10:0 >= 1 + x5:0

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

(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(8)
Obligation:
Rules:
l2(x10:0, x5:0) -> l2(x10:0, c) :|: c = 1 + x5:0 && x10:0 >= 1 + x5:0

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

(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = x - x1

The following rules are decreasing:
l2(x10:0, x5:0) -> l2(x10:0, c) :|: c = 1 + x5:0 && x10:0 >= 1 + x5:0
The following rules are bounded:
l2(x10:0, x5:0) -> l2(x10:0, c) :|: c = 1 + x5:0 && x10:0 >= 1 + x5:0

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

(10)
YES