WORST_CASE(Omega(n^1),O(n^2))
proof of input_j76vOTytrg.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 224 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 529 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 289 ms]
    (26) CpxRNTS
    (27) FinalProof [FINISHED, 0 ms]
    (28) BOUNDS(1, n^2)
(29) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CdtProblem
    (31) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CpxRelTRS
    (33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CpxRelTRS
    (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) typed CpxTrs
    (37) OrderProof [LOWER BOUND(ID), 3 ms]
    (38) typed CpxTrs
    (39) RewriteLemmaProof [LOWER BOUND(ID), 305 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs
            (47) RewriteLemmaProof [LOWER BOUND(ID), 233 ms]
            (48) typed CpxTrs
            (49) RewriteLemmaProof [LOWER BOUND(ID), 200 ms]
            (50) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1]

The TRS has the following type information:
quot :: 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   quot_3

(c) The following functions are completely defined:

   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) [1]

The TRS has the following type information:
quot :: 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   quot(0, s(y), s(z)) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   quot(x, 0, s(0)) -> s(quot(x, s(0), s(0))) [2]
   quot(x, 0, s(s(x'))) -> s(quot(x, s(plus(x', s(0))), s(s(x')))) [2]

The TRS has the following type information:
quot :: 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(x, 1 + plus(x', 1 + 0), 1 + (1 + x')) :|: z'' = 0, z' = x, x >= 0, x' >= 0, z1 = 1 + (1 + x')
   quot(z', z'', z1) -{ 2 }-> 1 + quot(x, 1 + 0, 1 + 0) :|: z'' = 0, z' = x, z1 = 1 + 0, x >= 0


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

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0


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

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { plus }
   { quot }

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {quot}

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

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {quot}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z' + z''

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {quot}
Previous analysis results are:
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z'

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: ?, size: O(n^1) [2 + 2*z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 5*z' + z'*z1 + z1

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 + z1 }-> 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= z1 - 2 + (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
   quot(z', z'', z1) -{ 2 }-> 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [5 + 5*z' + z'*z1 + z1], size: O(n^1) [2 + 2*z']

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

(27) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(28)
BOUNDS(1, n^2)

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

(29) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   quot(0, s(z0), s(z1)) -> 0
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0, s(z1)) -> s(quot(z0, plus(z1, s(0)), s(z1)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   QUOT(0, s(z0), s(z1)) -> c
   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, plus(z1, s(0)), s(z1)), PLUS(z1, s(0)))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
S tuples:
   QUOT(0, s(z0), s(z1)) -> c
   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, plus(z1, s(0)), s(z1)), PLUS(z1, s(0)))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   quot_3, plus_2

Defined Pair Symbols:   QUOT_3, PLUS_2

Compound Symbols:   c, c1_1, c2_2, c3, c4_1


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

(31) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(32)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   QUOT(0, s(z0), s(z1)) -> c
   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c2(QUOT(z0, plus(z1, s(0)), s(z1)), PLUS(z1, s(0)))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))

The (relative) TRS S consists of the following rules:

   quot(0, s(z0), s(z1)) -> 0
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0, s(z1)) -> s(quot(z0, plus(z1, s(0)), s(z1)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))

Rewrite Strategy: INNERMOST
----------------------------------------

(33) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(34)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   QUOT(0', s(z0), s(z1)) -> c
   QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
   QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
   PLUS(0', z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))

The (relative) TRS S consists of the following rules:

   quot(0', s(z0), s(z1)) -> 0'
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))

Rewrite Strategy: INNERMOST
----------------------------------------

(35) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(36)
Obligation:
Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4

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

(37) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
QUOT, plus, PLUS, quot

They will be analysed ascendingly in the following order:
plus < QUOT
PLUS < QUOT
plus < quot

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

(38)
Obligation:
Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
plus, QUOT, PLUS, quot

They will be analysed ascendingly in the following order:
plus < QUOT
PLUS < QUOT
plus < quot

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s5_5(n8_5), gen_0':s5_5(b)) -> gen_0':s5_5(+(n8_5, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s5_5(0), gen_0':s5_5(b)) ->_R^Omega(0)
gen_0':s5_5(b)

Induction Step:
plus(gen_0':s5_5(+(n8_5, 1)), gen_0':s5_5(b)) ->_R^Omega(0)
s(plus(gen_0':s5_5(n8_5), gen_0':s5_5(b))) ->_IH
s(gen_0':s5_5(+(b, c9_5)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(40)
Obligation:
Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
plus(gen_0':s5_5(n8_5), gen_0':s5_5(b)) -> gen_0':s5_5(+(n8_5, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
PLUS, QUOT, quot

They will be analysed ascendingly in the following order:
PLUS < QUOT

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s5_5(n859_5), gen_0':s5_5(b)) -> gen_c3:c46_5(n859_5), rt in Omega(1 + n859_5)

Induction Base:
PLUS(gen_0':s5_5(0), gen_0':s5_5(b)) ->_R^Omega(1)
c3

Induction Step:
PLUS(gen_0':s5_5(+(n859_5, 1)), gen_0':s5_5(b)) ->_R^Omega(1)
c4(PLUS(gen_0':s5_5(n859_5), gen_0':s5_5(b))) ->_IH
c4(gen_c3:c46_5(c860_5))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(42)
Complex Obligation (BEST)

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

(43)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
plus(gen_0':s5_5(n8_5), gen_0':s5_5(b)) -> gen_0':s5_5(+(n8_5, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
PLUS, QUOT, quot

They will be analysed ascendingly in the following order:
PLUS < QUOT

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

(44) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(45)
BOUNDS(n^1, INF)

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

(46)
Obligation:
Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
plus(gen_0':s5_5(n8_5), gen_0':s5_5(b)) -> gen_0':s5_5(+(n8_5, b)), rt in Omega(0)
PLUS(gen_0':s5_5(n859_5), gen_0':s5_5(b)) -> gen_c3:c46_5(n859_5), rt in Omega(1 + n859_5)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
QUOT, quot
----------------------------------------

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
QUOT(gen_0':s5_5(n1321_5), gen_0':s5_5(+(1, n1321_5)), gen_0':s5_5(1)) -> gen_c:c1:c24_5(n1321_5), rt in Omega(1 + n1321_5)

Induction Base:
QUOT(gen_0':s5_5(0), gen_0':s5_5(+(1, 0)), gen_0':s5_5(1)) ->_R^Omega(1)
c

Induction Step:
QUOT(gen_0':s5_5(+(n1321_5, 1)), gen_0':s5_5(+(1, +(n1321_5, 1))), gen_0':s5_5(1)) ->_R^Omega(1)
c1(QUOT(gen_0':s5_5(n1321_5), gen_0':s5_5(+(1, n1321_5)), gen_0':s5_5(1))) ->_IH
c1(gen_c:c1:c24_5(c1322_5))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(48)
Obligation:
Innermost TRS:
Rules:
QUOT(0', s(z0), s(z1)) -> c
QUOT(s(z0), s(z1), z2) -> c1(QUOT(z0, z1, z2))
QUOT(z0, 0', s(z1)) -> c2(QUOT(z0, plus(z1, s(0')), s(z1)), PLUS(z1, s(0')))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
quot(0', s(z0), s(z1)) -> 0'
quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
quot(z0, 0', s(z1)) -> s(quot(z0, plus(z1, s(0')), s(z1)))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))

Types:
QUOT :: 0':s -> 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c3:c4 -> c:c1:c2
plus :: 0':s -> 0':s -> 0':s
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
plus(gen_0':s5_5(n8_5), gen_0':s5_5(b)) -> gen_0':s5_5(+(n8_5, b)), rt in Omega(0)
PLUS(gen_0':s5_5(n859_5), gen_0':s5_5(b)) -> gen_c3:c46_5(n859_5), rt in Omega(1 + n859_5)
QUOT(gen_0':s5_5(n1321_5), gen_0':s5_5(+(1, n1321_5)), gen_0':s5_5(1)) -> gen_c:c1:c24_5(n1321_5), rt in Omega(1 + n1321_5)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
quot
----------------------------------------

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s5_5(n3408_5), gen_0':s5_5(+(1, n3408_5)), gen_0':s5_5(1)) -> gen_0':s5_5(0), rt in Omega(0)

Induction Base:
quot(gen_0':s5_5(0), gen_0':s5_5(+(1, 0)), gen_0':s5_5(1)) ->_R^Omega(0)
0'

Induction Step:
quot(gen_0':s5_5(+(n3408_5, 1)), gen_0':s5_5(+(1, +(n3408_5, 1))), gen_0':s5_5(1)) ->_R^Omega(0)
quot(gen_0':s5_5(n3408_5), gen_0':s5_5(+(1, n3408_5)), gen_0':s5_5(1)) ->_IH
gen_0':s5_5(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(50)
BOUNDS(1, INF)