WORST_CASE(Omega(n^1),O(n^1))
proof of input_mUTgU207FC.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxWeightedTrs
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedTrs
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) CompleteCoflocoProof [FINISHED, 326 ms]
    (22) BOUNDS(1, n^1)
(23) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxRelTRS
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) typed CpxTrs
    (31) OrderProof [LOWER BOUND(ID), 7 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 360 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 93 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 71 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
            (44) typed CpxTrs


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

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


The TRS R consists of the following rules:

   leq(0, y) -> true
   leq(s(x), 0) -> false
   leq(s(x), s(y)) -> leq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, IF_3, -'_2, MOD_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8, c9_2, c10_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   -'(z0, 0) -> c5
   IF(false, z0, z1) -> c4
   LEQ(s(z0), 0) -> c1
   LEQ(0, z0) -> c
   IF(true, z0, z1) -> c3
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_2, c10_3


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_1, c10_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2

Defined Pair Symbols:   LEQ_2, -'_2, MOD_2

Compound Symbols:   c2_1, c6_1, c9_1, c10_2


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))

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

   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0

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

(11) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0)) [1]
   MOD(s(z0), s(z1)) -> c10(MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) [1]
   -(s(z0), s(z1)) -> -(z0, z1) [0]
   -(z0, 0) -> z0 [0]

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

(13) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0)) [1]
   MOD(s(z0), s(z1)) -> c10(MOD(minus(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0)) [1]
   MOD(s(z0), s(z1)) -> c10(MOD(minus(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]

The TRS has the following type information:
LEQ :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
-' :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
MOD :: s:0 -> s:0 -> c9:c10
c9 :: c2 -> c9:c10
c10 :: c9:c10 -> c6 -> c9:c10
minus :: s:0 -> s:0 -> s:0
0 :: s:0

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

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   LEQ(v0, v1) -> null_LEQ [0]
   -'(v0, v1) -> null_-' [0]
   MOD(v0, v1) -> null_MOD [0]

And the following fresh constants:    null_minus, null_LEQ, null_-', null_MOD

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

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   MOD(s(z0), s(z1)) -> c9(LEQ(z1, z0)) [1]
   MOD(s(z0), s(z1)) -> c10(MOD(minus(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1))) [1]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   minus(z0, 0) -> z0 [0]
   minus(v0, v1) -> null_minus [0]
   LEQ(v0, v1) -> null_LEQ [0]
   -'(v0, v1) -> null_-' [0]
   MOD(v0, v1) -> null_MOD [0]

The TRS has the following type information:
LEQ :: s:0:null_minus -> s:0:null_minus -> c2:null_LEQ
s :: s:0:null_minus -> s:0:null_minus
c2 :: c2:null_LEQ -> c2:null_LEQ
-' :: s:0:null_minus -> s:0:null_minus -> c6:null_-'
c6 :: c6:null_-' -> c6:null_-'
MOD :: s:0:null_minus -> s:0:null_minus -> c9:c10:null_MOD
c9 :: c2:null_LEQ -> c9:c10:null_MOD
c10 :: c9:c10:null_MOD -> c6:null_-' -> c9:c10:null_MOD
minus :: s:0:null_minus -> s:0:null_minus -> s:0:null_minus
0 :: s:0:null_minus
null_minus :: s:0:null_minus
null_LEQ :: c2:null_LEQ
null_-' :: c6:null_-'
null_MOD :: c9:c10:null_MOD

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

(19) 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
   null_minus => 0
   null_LEQ => 0
   null_-' => 0
   null_MOD => 0

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

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

   -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   LEQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LEQ(z, z') -{ 1 }-> 1 + LEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MOD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MOD(z, z') -{ 1 }-> 1 + LEQ(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MOD(z, z') -{ 1 }-> 1 + MOD(minus(1 + z0, 1 + z1), 1 + z1) + -'(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(21) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, Out),1,[fun1(V5, V4, Ret11)],[Out = 1 + Ret11,V4 >= 0,V1 = 1 + V5,V5 >= 0,V = 1 + V4]).
eq(fun2(V1, V, Out),1,[fun(V6, V7, Ret12)],[Out = 1 + Ret12,V6 >= 0,V1 = 1 + V7,V7 >= 0,V = 1 + V6]).
eq(fun2(V1, V, Out),1,[minus(1 + V8, 1 + V9, Ret010),fun2(Ret010, 1 + V9, Ret01),fun1(1 + V8, 1 + V9, Ret13)],[Out = 1 + Ret01 + Ret13,V9 >= 0,V1 = 1 + V8,V8 >= 0,V = 1 + V9]).
eq(minus(V1, V, Out),0,[minus(V11, V10, Ret)],[Out = Ret,V10 >= 0,V1 = 1 + V11,V11 >= 0,V = 1 + V10]).
eq(minus(V1, V, Out),0,[],[Out = V12,V1 = V12,V12 >= 0,V = 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V19 >= 0,V20 >= 0,V1 = V19,V = V20]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive  : [fun1/3]
2. recursive  : [minus/3]
3. recursive [non_tail] : [fun2/3]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into fun1/3
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into fun2/3
4. SCC is partially evaluated into start/2

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun/3 
* CE 6 is refined into CE [15] 
* CE 5 is refined into CE [16] 


### Cost equations --> "Loop" of fun/3 
* CEs [16] --> Loop 12 
* CEs [15] --> Loop 13 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations fun1/3 
* CE 8 is refined into CE [17] 
* CE 7 is refined into CE [18] 


### Cost equations --> "Loop" of fun1/3 
* CEs [18] --> Loop 14 
* CEs [17] --> Loop 15 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 14 is refined into CE [19] 
* CE 13 is refined into CE [20] 
* CE 12 is refined into CE [21] 


### Cost equations --> "Loop" of minus/3 
* CEs [21] --> Loop 16 
* CEs [19] --> Loop 17 
* CEs [20] --> Loop 18 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [16]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V
    V1


### Specialization of cost equations fun2/3 
* CE 9 is refined into CE [22,23] 
* CE 11 is refined into CE [24] 
* CE 10 is refined into CE [25,26,27,28] 


### Cost equations --> "Loop" of fun2/3 
* CEs [28] --> Loop 19 
* CEs [27] --> Loop 20 
* CEs [26] --> Loop 21 
* CEs [25] --> Loop 22 
* CEs [23] --> Loop 23 
* CEs [22] --> Loop 24 
* CEs [24] --> Loop 25 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [19,20]: [V1,V1-V+1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [19,20]:
  - RF of loop [19:1,20:1]:
    V1
    V1-V+1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [29,30] 
* CE 2 is refined into CE [31,32] 
* CE 3 is refined into CE [33,34,35,36] 
* CE 4 is refined into CE [37,38,39] 


### Cost equations --> "Loop" of start/2 
* CEs [29,30,31,32,33,34,35,36,37,38,39] --> Loop 26 

### Ranking functions of CR start(V1,V) 

#### Partial ranking functions of CR start(V1,V) 


Computing Bounds
=====================================

#### Cost of chains of fun(V1,V,Out):
* Chain [[12],13]: 1*it(12)+0
  Such that:it(12) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,Out):
* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< Out

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [15]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[16],18]: 0
  with precondition: [V1=Out+V,V>=1,V1>=V] 

* Chain [[16],17]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [18]: 0
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [17]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun2(V1,V,Out):
* Chain [[19,20],25]: 2*it(19)+1*s(3)+0
  Such that:aux(2) =< V1-V+1
aux(5) =< V1
it(19) =< aux(5)
s(3) =< aux(5)
it(19) =< aux(2)

  with precondition: [V>=1,Out>=1,V1>=V,2*V1+1>=Out+V] 

* Chain [[19,20],24]: 2*it(19)+1*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(6) =< V1
it(19) =< aux(6)
s(3) =< aux(6)
it(19) =< aux(2)

  with precondition: [V>=1,Out>=2,V1>=V+1,2*V1>=Out+V] 

* Chain [[19,20],23]: 2*it(19)+1*s(3)+1*s(4)+1
  Such that:aux(2) =< V1-V+1
s(4) =< V
aux(7) =< V1
it(19) =< aux(7)
s(3) =< aux(7)
it(19) =< aux(2)

  with precondition: [V>=2,Out>=3,V1>=V+2,3*V1>=2*Out+V] 

* Chain [[19,20],22,25]: 2*it(19)+1*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(8) =< V1
it(19) =< aux(8)
s(3) =< aux(8)
it(19) =< aux(2)

  with precondition: [V>=1,Out>=2,V1>=V+1,2*V1>=Out+V] 

* Chain [[19,20],21,25]: 2*it(19)+2*s(3)+1
  Such that:aux(2) =< V1-V+1
aux(9) =< V1
s(3) =< aux(9)
it(19) =< aux(9)
it(19) =< aux(2)

  with precondition: [V>=1,Out>=3,V1>=V+1,2*V1+1>=Out+V] 

* Chain [25]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [24]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [23]: 1*s(4)+1
  Such that:s(4) =< V

  with precondition: [Out>=2,V1>=Out,V>=Out] 

* Chain [22,25]: 1
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [21,25]: 1*s(5)+1
  Such that:s(5) =< V1

  with precondition: [Out>=2,V1+1>=Out,V+1>=Out] 


#### Cost of chains of start(V1,V):
* Chain [26]: 4*s(29)+7*s(34)+10*s(35)+1
  Such that:s(33) =< V1-V+1
aux(12) =< V1
aux(13) =< V
s(34) =< aux(12)
s(29) =< aux(13)
s(35) =< aux(12)
s(35) =< s(33)

  with precondition: [V1>=0,V>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [26] with precondition: [V1>=0,V>=0] 
    - Upper bound: 17*V1+4*V+1 
    - Complexity: n 

### Maximum cost of start(V1,V): 17*V1+4*V+1 
Asymptotic class: n 
* Total analysis performed in 287 ms.


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

(22)
BOUNDS(1, n^1)

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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))
Tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
S tuples:
   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   leq_2, if_3, -_2, mod_2

Defined Pair Symbols:   LEQ_2, IF_3, -'_2, MOD_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5, c6_1, c7, c8, c9_2, c10_3


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

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

(26)
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:

   LEQ(0, z0) -> c
   LEQ(s(z0), 0) -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0, z0) -> c7
   MOD(s(z0), 0) -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))

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

   leq(0, z0) -> true
   leq(s(z0), 0) -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0, z0) -> 0
   mod(s(z0), 0) -> 0
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

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

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

(28)
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:

   LEQ(0', z0) -> c
   LEQ(s(z0), 0') -> c1
   LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
   IF(true, z0, z1) -> c3
   IF(false, z0, z1) -> c4
   -'(z0, 0') -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
   MOD(0', z0) -> c7
   MOD(s(z0), 0') -> c8
   MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
   MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))

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

   leq(0', z0) -> true
   leq(s(z0), 0') -> false
   leq(s(z0), s(z1)) -> leq(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   -(z0, 0') -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   mod(0', z0) -> 0'
   mod(s(z0), 0') -> 0'
   mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

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

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

(30)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10

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

(31) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LEQ, -', MOD, leq, mod, -

They will be analysed ascendingly in the following order:
LEQ < MOD
-' < MOD
leq < MOD
mod < MOD
- < MOD
leq < mod
- < mod

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

(32)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
LEQ, -', MOD, leq, mod, -

They will be analysed ascendingly in the following order:
LEQ < MOD
-' < MOD
leq < MOD
mod < MOD
- < MOD
leq < mod
- < mod

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11)) -> gen_c:c1:c27_11(n12_11), rt in Omega(1 + n12_11)

Induction Base:
LEQ(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1)
c

Induction Step:
LEQ(gen_0':s8_11(+(n12_11, 1)), gen_0':s8_11(+(n12_11, 1))) ->_R^Omega(1)
c2(LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11))) ->_IH
c2(gen_c:c1:c27_11(c13_11))

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

(34)
Complex Obligation (BEST)

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

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
LEQ, -', MOD, leq, mod, -

They will be analysed ascendingly in the following order:
LEQ < MOD
-' < MOD
leq < MOD
mod < MOD
- < MOD
leq < mod
- < mod

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Lemmas:
LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11)) -> gen_c:c1:c27_11(n12_11), rt in Omega(1 + n12_11)


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
-', MOD, leq, mod, -

They will be analysed ascendingly in the following order:
-' < MOD
leq < MOD
mod < MOD
- < MOD
leq < mod
- < mod

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-'(gen_0':s8_11(n607_11), gen_0':s8_11(n607_11)) -> gen_c5:c69_11(n607_11), rt in Omega(1 + n607_11)

Induction Base:
-'(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1)
c5

Induction Step:
-'(gen_0':s8_11(+(n607_11, 1)), gen_0':s8_11(+(n607_11, 1))) ->_R^Omega(1)
c6(-'(gen_0':s8_11(n607_11), gen_0':s8_11(n607_11))) ->_IH
c6(gen_c5:c69_11(c608_11))

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

(40)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Lemmas:
LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11)) -> gen_c:c1:c27_11(n12_11), rt in Omega(1 + n12_11)
-'(gen_0':s8_11(n607_11), gen_0':s8_11(n607_11)) -> gen_c5:c69_11(n607_11), rt in Omega(1 + n607_11)


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
leq, MOD, mod, -

They will be analysed ascendingly in the following order:
leq < MOD
mod < MOD
- < MOD
leq < mod
- < mod

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s8_11(n1092_11), gen_0':s8_11(n1092_11)) -> true, rt in Omega(0)

Induction Base:
leq(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0)
true

Induction Step:
leq(gen_0':s8_11(+(n1092_11, 1)), gen_0':s8_11(+(n1092_11, 1))) ->_R^Omega(0)
leq(gen_0':s8_11(n1092_11), gen_0':s8_11(n1092_11)) ->_IH
true

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

(42)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Lemmas:
LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11)) -> gen_c:c1:c27_11(n12_11), rt in Omega(1 + n12_11)
-'(gen_0':s8_11(n607_11), gen_0':s8_11(n607_11)) -> gen_c5:c69_11(n607_11), rt in Omega(1 + n607_11)
leq(gen_0':s8_11(n1092_11), gen_0':s8_11(n1092_11)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
-, MOD, mod

They will be analysed ascendingly in the following order:
mod < MOD
- < MOD
- < mod

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s8_11(n1471_11), gen_0':s8_11(n1471_11)) -> gen_0':s8_11(0), rt in Omega(0)

Induction Base:
-(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0)
gen_0':s8_11(0)

Induction Step:
-(gen_0':s8_11(+(n1471_11, 1)), gen_0':s8_11(+(n1471_11, 1))) ->_R^Omega(0)
-(gen_0':s8_11(n1471_11), gen_0':s8_11(n1471_11)) ->_IH
gen_0':s8_11(0)

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

(44)
Obligation:
Innermost TRS:
Rules:
LEQ(0', z0) -> c
LEQ(s(z0), 0') -> c1
LEQ(s(z0), s(z1)) -> c2(LEQ(z0, z1))
IF(true, z0, z1) -> c3
IF(false, z0, z1) -> c4
-'(z0, 0') -> c5
-'(s(z0), s(z1)) -> c6(-'(z0, z1))
MOD(0', z0) -> c7
MOD(s(z0), 0') -> c8
MOD(s(z0), s(z1)) -> c9(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), LEQ(z1, z0))
MOD(s(z0), s(z1)) -> c10(IF(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0)), MOD(-(s(z0), s(z1)), s(z1)), -'(s(z0), s(z1)))
leq(0', z0) -> true
leq(s(z0), 0') -> false
leq(s(z0), s(z1)) -> leq(z0, z1)
if(true, z0, z1) -> z0
if(false, z0, z1) -> z1
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
mod(0', z0) -> 0'
mod(s(z0), 0') -> 0'
mod(s(z0), s(z1)) -> if(leq(z1, z0), mod(-(s(z0), s(z1)), s(z1)), s(z0))

Types:
LEQ :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
IF :: true:false -> 0':s -> 0':s -> c3:c4
true :: true:false
c3 :: c3:c4
false :: true:false
c4 :: c3:c4
-' :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
MOD :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
c9 :: c3:c4 -> c:c1:c2 -> c7:c8:c9:c10
leq :: 0':s -> 0':s -> true:false
mod :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
c10 :: c3:c4 -> c7:c8:c9:c10 -> c5:c6 -> c7:c8:c9:c10
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_true:false4_11 :: true:false
hole_c5:c65_11 :: c5:c6
hole_c7:c8:c9:c106_11 :: c7:c8:c9:c10
gen_c:c1:c27_11 :: Nat -> c:c1:c2
gen_0':s8_11 :: Nat -> 0':s
gen_c5:c69_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1010_11 :: Nat -> c7:c8:c9:c10


Lemmas:
LEQ(gen_0':s8_11(n12_11), gen_0':s8_11(n12_11)) -> gen_c:c1:c27_11(n12_11), rt in Omega(1 + n12_11)
-'(gen_0':s8_11(n607_11), gen_0':s8_11(n607_11)) -> gen_c5:c69_11(n607_11), rt in Omega(1 + n607_11)
leq(gen_0':s8_11(n1092_11), gen_0':s8_11(n1092_11)) -> true, rt in Omega(0)
-(gen_0':s8_11(n1471_11), gen_0':s8_11(n1471_11)) -> gen_0':s8_11(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c27_11(0) <=> c
gen_c:c1:c27_11(+(x, 1)) <=> c2(gen_c:c1:c27_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c5:c69_11(0) <=> c5
gen_c5:c69_11(+(x, 1)) <=> c6(gen_c5:c69_11(x))
gen_c7:c8:c9:c1010_11(0) <=> c7
gen_c7:c8:c9:c1010_11(+(x, 1)) <=> c10(c3, gen_c7:c8:c9:c1010_11(x), c5)


The following defined symbols remain to be analysed:
mod, MOD

They will be analysed ascendingly in the following order:
mod < MOD