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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 501 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 134 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 10 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (20) IRSwT
        (21) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (22) IRSwT
        (23) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (24) IRSwT
        (25) TempFilterProof [SOUND, 7 ms]
        (26) IntTRS
        (27) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (28) YES
    (29) IRSwT
        (30) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (31) IRSwT
        (32) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (33) IRSwT
        (34) TempFilterProof [SOUND, 74 ms]
        (35) IntTRS
        (36) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
        (37) YES
    (38) IRSwT
        (39) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (40) IRSwT
        (41) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (42) IRSwT
        (43) TempFilterProof [SOUND, 7 ms]
        (44) IntTRS
        (45) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) IRSwT
        (48) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (49) IRSwT
        (50) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (51) IRSwT
        (52) FilterProof [EQUIVALENT, 1 ms]
        (53) IntTRS
        (54) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (55) IntTRS
        (56) RankingReductionPairProof [EQUIVALENT, 3 ms]
        (57) YES


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

(0)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4) -> f162_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && arg2 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f162_0_mk_LE(x, x1, x2, x3) -> f162_0_mk_LE(x4, x5, x6, x7) :|: x3 = x7 && x1 = x6 && x1 - 1 = x5 && 0 <= x4 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x4 <= x
f162_0_mk_LE(x8, x9, x10, x11) -> f276_0_mk_LE(x12, x13, x14, x15) :|: x11 = x15 && x11 = x14 && x11 - 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x12 <= x8 && -1 <= x11 - 1 && x10 <= 0
f276_0_mk_LE(x16, x17, x18, x19) -> f276_0_mk_LE(x20, x21, x22, x23) :|: x19 = x23 && x17 = x22 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x18 - 1 && x20 <= x16
f276_0_mk_LE(x24, x25, x26, x27) -> f401_0_mk_LE(x28, x29, x30, x31) :|: x27 = x29 && x27 - 1 = x28 && 0 <= x24 - 1 && -1 <= x27 - 1 && x26 <= 0
f401_0_mk_LE(x32, x33, x34, x35) -> f576_0_test_LT(x36, x37, x38, x39) :|: 2 = x36 && x33 <= 0
f401_0_mk_LE(x40, x41, x42, x43) -> f401_0_mk_LE(x44, x45, x46, x47) :|: x40 = x45 && x40 - 1 = x44 && 0 <= x41 - 1
f576_0_test_LT(x48, x49, x50, x51) -> f470_0_length_NONNULL(x52, x53, x54, x55) :|: x48 <= 2 && -1 <= x52 - 1 && -1 <= x48 - 1
f576_0_test_LT(x56, x57, x58, x59) -> f576_0_test_LT(x60, x61, x62, x63) :|: x56 - 1 = x60 && x56 <= 2 && -1 <= x56 - 1
f470_0_length_NONNULL(x64, x65, x66, x67) -> f470_0_length_NONNULL(x68, x69, x70, x71) :|: -1 <= x68 - 1 && 0 <= x64 - 1 && x68 + 1 <= x64
__init(x72, x73, x74, x75) -> f1_0_main_ConstantStackPush(x76, x77, x78, x79) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

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

(2)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4) -> f162_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && arg2 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f162_0_mk_LE(x, x1, x2, x3) -> f162_0_mk_LE(x4, x5, x6, x7) :|: x3 = x7 && x1 = x6 && x1 - 1 = x5 && 0 <= x4 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x4 <= x
f162_0_mk_LE(x8, x9, x10, x11) -> f276_0_mk_LE(x12, x13, x14, x15) :|: x11 = x15 && x11 = x14 && x11 - 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x12 <= x8 && -1 <= x11 - 1 && x10 <= 0
f276_0_mk_LE(x16, x17, x18, x19) -> f276_0_mk_LE(x20, x21, x22, x23) :|: x19 = x23 && x17 = x22 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x18 - 1 && x20 <= x16
f276_0_mk_LE(x24, x25, x26, x27) -> f401_0_mk_LE(x28, x29, x30, x31) :|: x27 = x29 && x27 - 1 = x28 && 0 <= x24 - 1 && -1 <= x27 - 1 && x26 <= 0
f401_0_mk_LE(x32, x33, x34, x35) -> f576_0_test_LT(x36, x37, x38, x39) :|: 2 = x36 && x33 <= 0
f401_0_mk_LE(x40, x41, x42, x43) -> f401_0_mk_LE(x44, x45, x46, x47) :|: x40 = x45 && x40 - 1 = x44 && 0 <= x41 - 1
f576_0_test_LT(x48, x49, x50, x51) -> f470_0_length_NONNULL(x52, x53, x54, x55) :|: x48 <= 2 && -1 <= x52 - 1 && -1 <= x48 - 1
f576_0_test_LT(x56, x57, x58, x59) -> f576_0_test_LT(x60, x61, x62, x63) :|: x56 - 1 = x60 && x56 <= 2 && -1 <= x56 - 1
f470_0_length_NONNULL(x64, x65, x66, x67) -> f470_0_length_NONNULL(x68, x69, x70, x71) :|: -1 <= x68 - 1 && 0 <= x64 - 1 && x68 + 1 <= x64
__init(x72, x73, x74, x75) -> f1_0_main_ConstantStackPush(x76, x77, x78, x79) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4) -> f162_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && arg2 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
(2) f162_0_mk_LE(x, x1, x2, x3) -> f162_0_mk_LE(x4, x5, x6, x7) :|: x3 = x7 && x1 = x6 && x1 - 1 = x5 && 0 <= x4 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x4 <= x
(3) f162_0_mk_LE(x8, x9, x10, x11) -> f276_0_mk_LE(x12, x13, x14, x15) :|: x11 = x15 && x11 = x14 && x11 - 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x12 <= x8 && -1 <= x11 - 1 && x10 <= 0
(4) f276_0_mk_LE(x16, x17, x18, x19) -> f276_0_mk_LE(x20, x21, x22, x23) :|: x19 = x23 && x17 = x22 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x18 - 1 && x20 <= x16
(5) f276_0_mk_LE(x24, x25, x26, x27) -> f401_0_mk_LE(x28, x29, x30, x31) :|: x27 = x29 && x27 - 1 = x28 && 0 <= x24 - 1 && -1 <= x27 - 1 && x26 <= 0
(6) f401_0_mk_LE(x32, x33, x34, x35) -> f576_0_test_LT(x36, x37, x38, x39) :|: 2 = x36 && x33 <= 0
(7) f401_0_mk_LE(x40, x41, x42, x43) -> f401_0_mk_LE(x44, x45, x46, x47) :|: x40 = x45 && x40 - 1 = x44 && 0 <= x41 - 1
(8) f576_0_test_LT(x48, x49, x50, x51) -> f470_0_length_NONNULL(x52, x53, x54, x55) :|: x48 <= 2 && -1 <= x52 - 1 && -1 <= x48 - 1
(9) f576_0_test_LT(x56, x57, x58, x59) -> f576_0_test_LT(x60, x61, x62, x63) :|: x56 - 1 = x60 && x56 <= 2 && -1 <= x56 - 1
(10) f470_0_length_NONNULL(x64, x65, x66, x67) -> f470_0_length_NONNULL(x68, x69, x70, x71) :|: -1 <= x68 - 1 && 0 <= x64 - 1 && x68 + 1 <= x64
(11) __init(x72, x73, x74, x75) -> f1_0_main_ConstantStackPush(x76, x77, x78, x79) :|: 0 <= 0

Arcs:
(1) -> (2), (3)
(2) -> (2), (3)
(3) -> (4), (5)
(4) -> (4), (5)
(5) -> (6), (7)
(6) -> (8), (9)
(7) -> (6), (7)
(8) -> (10)
(9) -> (8), (9)
(10) -> (10)
(11) -> (1)

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f162_0_mk_LE(x, x1, x2, x3) -> f162_0_mk_LE(x4, x5, x6, x7) :|: x3 = x7 && x1 = x6 && x1 - 1 = x5 && 0 <= x4 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x4 <= x

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f162_0_mk_LE(x:0, x1:0, x2:0, x3:0) -> f162_0_mk_LE(x4:0, x1:0 - 1, x1:0, x3:0) :|: x2:0 > 0 && x:0 >= x4:0 && x4:0 > 0 && x:0 > 0

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

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

   f162_0_mk_LE(x1, x2, x3, x4) -> f162_0_mk_LE(x1, x2, x3)

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

(9)
Obligation:
Rules:
f162_0_mk_LE(x:0, x1:0, x2:0) -> f162_0_mk_LE(x4:0, x1:0 - 1, x1:0) :|: x2:0 > 0 && x:0 >= x4:0 && x4:0 > 0 && x:0 > 0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f162_0_mk_LE(INTEGER, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f162_0_mk_LE(x:0, x1:0, x2:0) -> f162_0_mk_LE(x4:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x4:0 && x4:0 > 0 && x:0 > 0)

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f162_0_mk_LE(x, x1, x2)] = x1^2 + 2*x2

The following rules are decreasing:
f162_0_mk_LE(x:0, x1:0, x2:0) -> f162_0_mk_LE(x4:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x4:0 && x4:0 > 0 && x:0 > 0)
The following rules are bounded:
f162_0_mk_LE(x:0, x1:0, x2:0) -> f162_0_mk_LE(x4:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x4:0 && x4:0 > 0 && x:0 > 0)

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f276_0_mk_LE(x16, x17, x18, x19) -> f276_0_mk_LE(x20, x21, x22, x23) :|: x19 = x23 && x17 = x22 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x18 - 1 && x20 <= x16

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
f276_0_mk_LE(x16:0, x17:0, x18:0, x19:0) -> f276_0_mk_LE(x20:0, x17:0 - 1, x17:0, x19:0) :|: x18:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0

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

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

   f276_0_mk_LE(x1, x2, x3, x4) -> f276_0_mk_LE(x1, x2, x3)

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

(18)
Obligation:
Rules:
f276_0_mk_LE(x16:0, x17:0, x18:0) -> f276_0_mk_LE(x20:0, x17:0 - 1, x17:0) :|: x18:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0

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

(19) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(20)
Obligation:
Rules:
f276_0_mk_LE(x, x1, x2) -> f276_0_mk_LE(x7, x1 + -2, x1 + -1) :|: TRUE && x2 >= 1 && x3 + -1 * x <= 0 && x3 >= 1 && x >= 1 && x1 >= 1 && x7 + -1 * x3 <= 0 && x7 >= 1

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

(21) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f276_0_mk_LE(x, x1, x2) -> f276_0_mk_LE(x7, x1 + -2, x1 + -1) :|: TRUE && x2 >= 1 && x3 + -1 * x <= 0 && x3 >= 1 && x >= 1 && x1 >= 1 && x7 + -1 * x3 <= 0 && x7 >= 1

Arcs:
(1) -> (1)

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

(22)
Obligation:

Termination digraph:
Nodes:
(1) f276_0_mk_LE(x, x1, x2) -> f276_0_mk_LE(x7, x1 + -2, x1 + -1) :|: TRUE && x2 >= 1 && x3 + -1 * x <= 0 && x3 >= 1 && x >= 1 && x1 >= 1 && x7 + -1 * x3 <= 0 && x7 >= 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(24)
Obligation:
Rules:
f276_0_mk_LE(x:0, x1:0, x2:0) -> f276_0_mk_LE(x7:0, x1:0 - 2, x1:0 - 1) :|: x7:0 + -1 * x3:0 <= 0 && x7:0 > 0 && x1:0 > 0 && x:0 > 0 && x3:0 > 0 && x2:0 > 0 && x3:0 + -1 * x:0 <= 0

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

(25) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f276_0_mk_LE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(26)
Obligation:
Rules:
f276_0_mk_LE(x:0, x1:0, x2:0) -> f276_0_mk_LE(x7:0, c, c1) :|: c1 = x1:0 - 1 && c = x1:0 - 2 && (x7:0 + -1 * x3:0 <= 0 && x7:0 > 0 && x1:0 > 0 && x:0 > 0 && x3:0 > 0 && x2:0 > 0 && x3:0 + -1 * x:0 <= 0)

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

(27) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f276_0_mk_LE ] = 1/2*f276_0_mk_LE_2

The following rules are decreasing:
f276_0_mk_LE(x:0, x1:0, x2:0) -> f276_0_mk_LE(x7:0, c, c1) :|: c1 = x1:0 - 1 && c = x1:0 - 2 && (x7:0 + -1 * x3:0 <= 0 && x7:0 > 0 && x1:0 > 0 && x:0 > 0 && x3:0 > 0 && x2:0 > 0 && x3:0 + -1 * x:0 <= 0)

The following rules are bounded:
f276_0_mk_LE(x:0, x1:0, x2:0) -> f276_0_mk_LE(x7:0, c, c1) :|: c1 = x1:0 - 1 && c = x1:0 - 2 && (x7:0 + -1 * x3:0 <= 0 && x7:0 > 0 && x1:0 > 0 && x:0 > 0 && x3:0 > 0 && x2:0 > 0 && x3:0 + -1 * x:0 <= 0)


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

(28)
YES

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

(29)
Obligation:

Termination digraph:
Nodes:
(1) f401_0_mk_LE(x40, x41, x42, x43) -> f401_0_mk_LE(x44, x45, x46, x47) :|: x40 = x45 && x40 - 1 = x44 && 0 <= x41 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(31)
Obligation:
Rules:
f401_0_mk_LE(x40:0, x41:0, x42:0, x43:0) -> f401_0_mk_LE(x40:0 - 1, x40:0, x46:0, x47:0) :|: x41:0 > 0

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

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

   f401_0_mk_LE(x1, x2, x3, x4) -> f401_0_mk_LE(x1, x2)

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

(33)
Obligation:
Rules:
f401_0_mk_LE(x40:0, x41:0) -> f401_0_mk_LE(x40:0 - 1, x40:0) :|: x41:0 > 0

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

(34) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f401_0_mk_LE(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(35)
Obligation:
Rules:
f401_0_mk_LE(x40:0, x41:0) -> f401_0_mk_LE(c, x40:0) :|: c = x40:0 - 1 && x41:0 > 0

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

(36) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f401_0_mk_LE(x, x1)] = x^2 + 2*x1

The following rules are decreasing:
f401_0_mk_LE(x40:0, x41:0) -> f401_0_mk_LE(c, x40:0) :|: c = x40:0 - 1 && x41:0 > 0
The following rules are bounded:
f401_0_mk_LE(x40:0, x41:0) -> f401_0_mk_LE(c, x40:0) :|: c = x40:0 - 1 && x41:0 > 0

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

(37)
YES

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

(38)
Obligation:

Termination digraph:
Nodes:
(1) f576_0_test_LT(x56, x57, x58, x59) -> f576_0_test_LT(x60, x61, x62, x63) :|: x56 - 1 = x60 && x56 <= 2 && -1 <= x56 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(40)
Obligation:
Rules:
f576_0_test_LT(x56:0, x57:0, x58:0, x59:0) -> f576_0_test_LT(x56:0 - 1, x61:0, x62:0, x63:0) :|: x56:0 > -1 && x56:0 < 3

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

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

   f576_0_test_LT(x1, x2, x3, x4) -> f576_0_test_LT(x1)

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

(42)
Obligation:
Rules:
f576_0_test_LT(x56:0) -> f576_0_test_LT(x56:0 - 1) :|: x56:0 > -1 && x56:0 < 3

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

(43) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f576_0_test_LT(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(44)
Obligation:
Rules:
f576_0_test_LT(x56:0) -> f576_0_test_LT(c) :|: c = x56:0 - 1 && (x56:0 > -1 && x56:0 < 3)

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

(45) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f576_0_test_LT ] = f576_0_test_LT_1

The following rules are decreasing:
f576_0_test_LT(x56:0) -> f576_0_test_LT(c) :|: c = x56:0 - 1 && (x56:0 > -1 && x56:0 < 3)

The following rules are bounded:
f576_0_test_LT(x56:0) -> f576_0_test_LT(c) :|: c = x56:0 - 1 && (x56:0 > -1 && x56:0 < 3)


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

(46)
YES

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

(47)
Obligation:

Termination digraph:
Nodes:
(1) f470_0_length_NONNULL(x64, x65, x66, x67) -> f470_0_length_NONNULL(x68, x69, x70, x71) :|: -1 <= x68 - 1 && 0 <= x64 - 1 && x68 + 1 <= x64

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(49)
Obligation:
Rules:
f470_0_length_NONNULL(x64:0, x65:0, x66:0, x67:0) -> f470_0_length_NONNULL(x68:0, x69:0, x70:0, x71:0) :|: x68:0 > -1 && x64:0 > 0 && x68:0 + 1 <= x64:0

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

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

   f470_0_length_NONNULL(x1, x2, x3, x4) -> f470_0_length_NONNULL(x1)

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

(51)
Obligation:
Rules:
f470_0_length_NONNULL(x64:0) -> f470_0_length_NONNULL(x68:0) :|: x68:0 > -1 && x64:0 > 0 && x68:0 + 1 <= x64:0

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

(52) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f470_0_length_NONNULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(53)
Obligation:
Rules:
f470_0_length_NONNULL(x64:0) -> f470_0_length_NONNULL(x68:0) :|: x68:0 > -1 && x64:0 > 0 && x68:0 + 1 <= x64:0

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

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

(55)
Obligation:
Rules:
f470_0_length_NONNULL(x64:0:0) -> f470_0_length_NONNULL(x68:0:0) :|: x68:0:0 > -1 && x64:0:0 > 0 && x68:0:0 + 1 <= x64:0:0

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

(56) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f470_0_length_NONNULL ] = f470_0_length_NONNULL_1

The following rules are decreasing:
f470_0_length_NONNULL(x64:0:0) -> f470_0_length_NONNULL(x68:0:0) :|: x68:0:0 > -1 && x64:0:0 > 0 && x68:0:0 + 1 <= x64:0:0

The following rules are bounded:
f470_0_length_NONNULL(x64:0:0) -> f470_0_length_NONNULL(x68:0:0) :|: x68:0:0 > -1 && x64:0:0 > 0 && x68:0:0 + 1 <= x64:0:0


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

(57)
YES