/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


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


MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) CR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 12 ms]
(8) HASKELL
(9) LetRed [EQUIVALENT, 0 ms]
(10) HASKELL
(11) NumRed [SOUND, 12 ms]
(12) HASKELL


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

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
addToFM :: Ord a => FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM fm key elt = addToFM_C (\old new ->new) fm key elt;

addToFM_C :: Ord b => (a  ->  a  ->  a)  ->  FiniteMap b a  ->  b  ->  a  ->  FiniteMap b a;
addToFM_C combiner EmptyFM key elt = unitFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt  | new_key < key = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r
 | new_key > key = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt)
 | otherwise = Branch new_key (combiner elt new_elt) size fm_l fm_r;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap a b  ->  (a,b);
findMax (Branch key elt _ _ EmptyFM) = (key,elt);
findMax (Branch key elt _ _ fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt _ EmptyFM _) = (key,elt);
findMin (Branch key elt _ fm_l _) = findMin fm_l;

mkBalBranch :: Ord a => a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBalBranch key elt fm_L fm_R  | size_l + size_r < 2 = mkBranch 1 key elt fm_L fm_R
 | size_r > sIZE_RATIO * size_l = case fm_R of { 
Branch _ _ _ fm_rl fm_rr | sizeFM fm_rl < 2 * sizeFM fm_rr -> single_L fm_L fm_R
 | otherwise -> double_L fm_L fm_R; 
 } 
 | size_l > sIZE_RATIO * size_r = case fm_L of { 
Branch _ _ _ fm_ll fm_lr | sizeFM fm_lr < 2 * sizeFM fm_ll -> single_R fm_L fm_R
 | otherwise -> double_R fm_L fm_R; 
 } 
 | otherwise = mkBranch 2 key elt fm_L fm_R where { 
double_L fm_l (Branch key_r elt_r _ (Branch key_rl elt_rl _ fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
double_R (Branch key_l elt_l _ fm_ll (Branch key_lr elt_lr _ fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
single_L fm_l (Branch key_r elt_r _ fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
single_R (Branch key_l elt_l _ fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
size_l = sizeFM fm_L;
size_r = sizeFM fm_R;
 };

mkBranch :: Ord a => Int  ->  a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBranch which key elt fm_l fm_r = let { 
result = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where { 
balance_ok = True;
left_ok = case fm_l of { 
EmptyFM-> True; 
Branch left_key _ _ _ _-> let { 
biggest_left_key = fst (findMax fm_l);
} in biggest_left_key < key; 
 } ;
left_size = sizeFM fm_l;
right_ok = case fm_r of { 
EmptyFM-> True; 
Branch right_key _ _ _ _-> let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key; 
 } ;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

mkVBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkVBalBranch key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch key elt fm_l@(Branch key_l elt_l _ fm_ll fm_lr) fm_r@(Branch key_r elt_r _ fm_rl fm_rr)  | sIZE_RATIO * size_l < size_r = mkBalBranch key_r elt_r (mkVBalBranch key elt fm_l fm_rl) fm_rr
 | sIZE_RATIO * size_r < size_l = mkBalBranch key_l elt_l fm_ll (mkVBalBranch key elt fm_lr fm_r)
 | otherwise = mkBranch 13 key elt fm_l fm_r where { 
size_l = sizeFM fm_l;
size_r = sizeFM fm_r;
 };

plusFM :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 _ left right) = mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where { 
gts = splitGT fm1 split_key;
lts = splitLT fm1 split_key;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap a b  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch _ _ size _ _) = size;

splitGT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitGT EmptyFM split_key = emptyFM;
splitGT (Branch key elt _ fm_l fm_r) split_key  | split_key > key = splitGT fm_r split_key
 | split_key < key = mkVBalBranch key elt (splitGT fm_l split_key) fm_r
 | otherwise = fm_r;

splitLT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitLT EmptyFM split_key = emptyFM;
splitLT (Branch key elt _ fm_l fm_r) split_key  | split_key < key = splitLT fm_l split_key
 | split_key > key = mkVBalBranch key elt fm_l (splitLT fm_r split_key)
 | otherwise = fm_l;

unitFM :: b  ->  a  ->  FiniteMap b a;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\oldnew->new"
is transformed to
"addToFM0 old new = new;
"

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

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
addToFM :: Ord a => FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord a => (b  ->  b  ->  b)  ->  FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM_C combiner EmptyFM key elt = unitFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt  | new_key < key = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r
 | new_key > key = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt)
 | otherwise = Branch new_key (combiner elt new_elt) size fm_l fm_r;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap a b  ->  (a,b);
findMax (Branch key elt _ _ EmptyFM) = (key,elt);
findMax (Branch key elt _ _ fm_r) = findMax fm_r;

findMin :: FiniteMap b a  ->  (b,a);
findMin (Branch key elt _ EmptyFM _) = (key,elt);
findMin (Branch key elt _ fm_l _) = findMin fm_l;

mkBalBranch :: Ord a => a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBalBranch key elt fm_L fm_R  | size_l + size_r < 2 = mkBranch 1 key elt fm_L fm_R
 | size_r > sIZE_RATIO * size_l = case fm_R of { 
Branch _ _ _ fm_rl fm_rr | sizeFM fm_rl < 2 * sizeFM fm_rr -> single_L fm_L fm_R
 | otherwise -> double_L fm_L fm_R; 
 } 
 | size_l > sIZE_RATIO * size_r = case fm_L of { 
Branch _ _ _ fm_ll fm_lr | sizeFM fm_lr < 2 * sizeFM fm_ll -> single_R fm_L fm_R
 | otherwise -> double_R fm_L fm_R; 
 } 
 | otherwise = mkBranch 2 key elt fm_L fm_R where { 
double_L fm_l (Branch key_r elt_r _ (Branch key_rl elt_rl _ fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
double_R (Branch key_l elt_l _ fm_ll (Branch key_lr elt_lr _ fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
single_L fm_l (Branch key_r elt_r _ fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
single_R (Branch key_l elt_l _ fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
size_l = sizeFM fm_L;
size_r = sizeFM fm_R;
 };

mkBranch :: Ord a => Int  ->  a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBranch which key elt fm_l fm_r = let { 
result = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where { 
balance_ok = True;
left_ok = case fm_l of { 
EmptyFM-> True; 
Branch left_key _ _ _ _-> let { 
biggest_left_key = fst (findMax fm_l);
} in biggest_left_key < key; 
 } ;
left_size = sizeFM fm_l;
right_ok = case fm_r of { 
EmptyFM-> True; 
Branch right_key _ _ _ _-> let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key; 
 } ;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

mkVBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkVBalBranch key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch key elt fm_l@(Branch key_l elt_l _ fm_ll fm_lr) fm_r@(Branch key_r elt_r _ fm_rl fm_rr)  | sIZE_RATIO * size_l < size_r = mkBalBranch key_r elt_r (mkVBalBranch key elt fm_l fm_rl) fm_rr
 | sIZE_RATIO * size_r < size_l = mkBalBranch key_l elt_l fm_ll (mkVBalBranch key elt fm_lr fm_r)
 | otherwise = mkBranch 13 key elt fm_l fm_r where { 
size_l = sizeFM fm_l;
size_r = sizeFM fm_r;
 };

plusFM :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 _ left right) = mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where { 
gts = splitGT fm1 split_key;
lts = splitLT fm1 split_key;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap a b  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch _ _ size _ _) = size;

splitGT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitGT EmptyFM split_key = emptyFM;
splitGT (Branch key elt _ fm_l fm_r) split_key  | split_key > key = splitGT fm_r split_key
 | split_key < key = mkVBalBranch key elt (splitGT fm_l split_key) fm_r
 | otherwise = fm_r;

splitLT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitLT EmptyFM split_key = emptyFM;
splitLT (Branch key elt _ fm_l fm_r) split_key  | split_key < key = splitLT fm_l split_key
 | split_key > key = mkVBalBranch key elt fm_l (splitLT fm_r split_key)
 | otherwise = fm_l;

unitFM :: b  ->  a  ->  FiniteMap b a;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case fm_r of {
EmptyFM  -> True;
Branch right_key _ _ _ _  -> let {
smallest_right_key  = fst (findMin fm_r);
} in key < smallest_right_key}
"
is transformed to
"right_ok0 fm_r key EmptyFM = True;
right_ok0 fm_r key (Branch right_key _ _ _ _) = let {
smallest_right_key  = fst (findMin fm_r);
} in key < smallest_right_key;
"
The following Case expression
"case fm_l of {
EmptyFM  -> True;
Branch left_key _ _ _ _  -> let {
biggest_left_key  = fst (findMax fm_l);
} in biggest_left_key < key}
"
is transformed to
"left_ok0 fm_l key EmptyFM = True;
left_ok0 fm_l key (Branch left_key _ _ _ _) = let {
biggest_left_key  = fst (findMax fm_l);
} in biggest_left_key < key;
"
The following Case expression
"case fm_R of {
Branch _ _ _ fm_rl fm_rr |sizeFM fm_rl < 2 * sizeFM fm_rrsingle_L fm_L fm_R|otherwisedouble_L fm_L fm_R}
"
is transformed to
"mkBalBranch0 fm_L fm_R (Branch _ _ _ fm_rl fm_rr)|sizeFM fm_rl < 2 * sizeFM fm_rrsingle_L fm_L fm_R|otherwisedouble_L fm_L fm_R;
"
The following Case expression
"case fm_L of {
Branch _ _ _ fm_ll fm_lr |sizeFM fm_lr < 2 * sizeFM fm_llsingle_R fm_L fm_R|otherwisedouble_R fm_L fm_R}
"
is transformed to
"mkBalBranch1 fm_L fm_R (Branch _ _ _ fm_ll fm_lr)|sizeFM fm_lr < 2 * sizeFM fm_llsingle_R fm_L fm_R|otherwisedouble_R fm_L fm_R;
"

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

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
addToFM :: Ord b => FiniteMap b a  ->  b  ->  a  ->  FiniteMap b a;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord a => (b  ->  b  ->  b)  ->  FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM_C combiner EmptyFM key elt = unitFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt  | new_key < key = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r
 | new_key > key = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt)
 | otherwise = Branch new_key (combiner elt new_elt) size fm_l fm_r;

emptyFM :: FiniteMap b a;
emptyFM = EmptyFM;

findMax :: FiniteMap a b  ->  (a,b);
findMax (Branch key elt _ _ EmptyFM) = (key,elt);
findMax (Branch key elt _ _ fm_r) = findMax fm_r;

findMin :: FiniteMap b a  ->  (b,a);
findMin (Branch key elt _ EmptyFM _) = (key,elt);
findMin (Branch key elt _ fm_l _) = findMin fm_l;

mkBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBalBranch key elt fm_L fm_R  | size_l + size_r < 2 = mkBranch 1 key elt fm_L fm_R
 | size_r > sIZE_RATIO * size_l = mkBalBranch0 fm_L fm_R fm_R
 | size_l > sIZE_RATIO * size_r = mkBalBranch1 fm_L fm_R fm_L
 | otherwise = mkBranch 2 key elt fm_L fm_R where { 
double_L fm_l (Branch key_r elt_r _ (Branch key_rl elt_rl _ fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
double_R (Branch key_l elt_l _ fm_ll (Branch key_lr elt_lr _ fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
mkBalBranch0 fm_L fm_R (Branch _ _ _ fm_rl fm_rr)  | sizeFM fm_rl < 2 * sizeFM fm_rr = single_L fm_L fm_R
 | otherwise = double_L fm_L fm_R;
mkBalBranch1 fm_L fm_R (Branch _ _ _ fm_ll fm_lr)  | sizeFM fm_lr < 2 * sizeFM fm_ll = single_R fm_L fm_R
 | otherwise = double_R fm_L fm_R;
single_L fm_l (Branch key_r elt_r _ fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
single_R (Branch key_l elt_l _ fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
size_l = sizeFM fm_L;
size_r = sizeFM fm_R;
 };

mkBranch :: Ord a => Int  ->  a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBranch which key elt fm_l fm_r = let { 
result = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where { 
balance_ok = True;
left_ok = left_ok0 fm_l key fm_l;
left_ok0 fm_l key EmptyFM = True;
left_ok0 fm_l key (Branch left_key _ _ _ _) = let { 
biggest_left_key = fst (findMax fm_l);
} in biggest_left_key < key;
left_size = sizeFM fm_l;
right_ok = right_ok0 fm_r key fm_r;
right_ok0 fm_r key EmptyFM = True;
right_ok0 fm_r key (Branch right_key _ _ _ _) = let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

mkVBalBranch :: Ord a => a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkVBalBranch key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch key elt fm_l@(Branch key_l elt_l _ fm_ll fm_lr) fm_r@(Branch key_r elt_r _ fm_rl fm_rr)  | sIZE_RATIO * size_l < size_r = mkBalBranch key_r elt_r (mkVBalBranch key elt fm_l fm_rl) fm_rr
 | sIZE_RATIO * size_r < size_l = mkBalBranch key_l elt_l fm_ll (mkVBalBranch key elt fm_lr fm_r)
 | otherwise = mkBranch 13 key elt fm_l fm_r where { 
size_l = sizeFM fm_l;
size_r = sizeFM fm_r;
 };

plusFM :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 _ left right) = mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where { 
gts = splitGT fm1 split_key;
lts = splitLT fm1 split_key;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap a b  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch _ _ size _ _) = size;

splitGT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitGT EmptyFM split_key = emptyFM;
splitGT (Branch key elt _ fm_l fm_r) split_key  | split_key > key = splitGT fm_r split_key
 | split_key < key = mkVBalBranch key elt (splitGT fm_l split_key) fm_r
 | otherwise = fm_r;

splitLT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitLT EmptyFM split_key = emptyFM;
splitLT (Branch key elt _ fm_l fm_r) split_key  | split_key < key = splitLT fm_l split_key
 | split_key > key = mkVBalBranch key elt fm_l (splitLT fm_r split_key)
 | otherwise = fm_l;

unitFM :: a  ->  b  ->  FiniteMap a b;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(5) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.

Binding Reductions:
The bind variable of the following binding Pattern
"fm_l@(Branch wv ww wx wy wz)"
is replaced by the following term
"Branch wv ww wx wy wz"
The bind variable of the following binding Pattern
"fm_r@(Branch xv xw xx xy xz)"
is replaced by the following term
"Branch xv xw xx xy xz"

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

(6)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
addToFM :: Ord b => FiniteMap b a  ->  b  ->  a  ->  FiniteMap b a;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord a => (b  ->  b  ->  b)  ->  FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM_C combiner EmptyFM key elt = unitFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt  | new_key < key = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r
 | new_key > key = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt)
 | otherwise = Branch new_key (combiner elt new_elt) size fm_l fm_r;

emptyFM :: FiniteMap b a;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt zy zz EmptyFM) = (key,elt);
findMax (Branch key elt vuu vuv fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt vxu EmptyFM vxv) = (key,elt);
findMin (Branch key elt vxw fm_l vxx) = findMin fm_l;

mkBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBalBranch key elt fm_L fm_R  | size_l + size_r < 2 = mkBranch 1 key elt fm_L fm_R
 | size_r > sIZE_RATIO * size_l = mkBalBranch0 fm_L fm_R fm_R
 | size_l > sIZE_RATIO * size_r = mkBalBranch1 fm_L fm_R fm_L
 | otherwise = mkBranch 2 key elt fm_L fm_R where { 
double_L fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
double_R (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr)  | sizeFM fm_rl < 2 * sizeFM fm_rr = single_L fm_L fm_R
 | otherwise = double_L fm_L fm_R;
mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr)  | sizeFM fm_lr < 2 * sizeFM fm_ll = single_R fm_L fm_R
 | otherwise = double_R fm_L fm_R;
single_L fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
single_R (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
size_l = sizeFM fm_L;
size_r = sizeFM fm_R;
 };

mkBranch :: Ord b => Int  ->  b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBranch which key elt fm_l fm_r = let { 
result = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where { 
balance_ok = True;
left_ok = left_ok0 fm_l key fm_l;
left_ok0 fm_l key EmptyFM = True;
left_ok0 fm_l key (Branch left_key yw yx yy yz) = let { 
biggest_left_key = fst (findMax fm_l);
} in biggest_left_key < key;
left_size = sizeFM fm_l;
right_ok = right_ok0 fm_r key fm_r;
right_ok0 fm_r key EmptyFM = True;
right_ok0 fm_r key (Branch right_key zu zv zw zx) = let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

mkVBalBranch :: Ord a => a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkVBalBranch key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz)  | sIZE_RATIO * size_l < size_r = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz
 | sIZE_RATIO * size_r < size_l = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz))
 | otherwise = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) where { 
size_l = sizeFM (Branch wv ww wx wy wz);
size_r = sizeFM (Branch xv xw xx xy xz);
 };

plusFM :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 vz left right) = mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where { 
gts = splitGT fm1 split_key;
lts = splitLT fm1 split_key;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch vww vwx size vwy vwz) = size;

splitGT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitGT EmptyFM split_key = emptyFM;
splitGT (Branch key elt yu fm_l fm_r) split_key  | split_key > key = splitGT fm_r split_key
 | split_key < key = mkVBalBranch key elt (splitGT fm_l split_key) fm_r
 | otherwise = fm_r;

splitLT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitLT EmptyFM split_key = emptyFM;
splitLT (Branch key elt yv fm_l fm_r) split_key  | split_key < key = splitLT fm_l split_key
 | split_key > key = mkVBalBranch key elt fm_l (splitLT fm_r split_key)
 | otherwise = fm_l;

unitFM :: a  ->  b  ->  FiniteMap a b;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(7) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"addToFM_C combiner EmptyFM key elt = unitFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt|new_key < keymkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r|new_key > keymkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt)|otherwiseBranch new_key (combiner elt new_elt) size fm_l fm_r;
"
is transformed to
"addToFM_C combiner EmptyFM key elt = addToFM_C4 combiner EmptyFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt;
"
"addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r;
addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt (new_key > key);
"
"addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt True = Branch new_key (combiner elt new_elt) size fm_l fm_r;
"
"addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt);
addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt otherwise;
"
"addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt (new_key < key);
"
"addToFM_C4 combiner EmptyFM key elt = unitFM key elt;
addToFM_C4 vyu vyv vyw vyx = addToFM_C3 vyu vyv vyw vyx;
"
The following Function with conditions
"mkVBalBranch key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz)|sIZE_RATIO * size_l < size_rmkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz|sIZE_RATIO * size_r < size_lmkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz))|otherwisemkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) where {
size_l  = sizeFM (Branch wv ww wx wy wz);
;
size_r  = sizeFM (Branch xv xw xx xy xz);
} 
;
"
is transformed to
"mkVBalBranch key elt EmptyFM fm_r = mkVBalBranch5 key elt EmptyFM fm_r;
mkVBalBranch key elt fm_l EmptyFM = mkVBalBranch4 key elt fm_l EmptyFM;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);
"
"mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_l < size_r) where {
mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);
;
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz otherwise;
;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_r < size_l);
;
size_l  = sizeFM (Branch wv ww wx wy wz);
;
size_r  = sizeFM (Branch xv xw xx xy xz);
} 
;
"
"mkVBalBranch4 key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch4 vzv vzw vzx vzy = mkVBalBranch3 vzv vzw vzx vzy;
"
"mkVBalBranch5 key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch5 wuu wuv wuw wux = mkVBalBranch4 wuu wuv wuw wux;
"
The following Function with conditions
"splitGT EmptyFM split_key = emptyFM;
splitGT (Branch key elt yu fm_l fm_r) split_key|split_key > keysplitGT fm_r split_key|split_key < keymkVBalBranch key elt (splitGT fm_l split_key) fm_r|otherwisefm_r;
"
is transformed to
"splitGT EmptyFM split_key = splitGT4 EmptyFM split_key;
splitGT (Branch key elt yu fm_l fm_r) split_key = splitGT3 (Branch key elt yu fm_l fm_r) split_key;
"
"splitGT0 key elt yu fm_l fm_r split_key True = fm_r;
"
"splitGT1 key elt yu fm_l fm_r split_key True = mkVBalBranch key elt (splitGT fm_l split_key) fm_r;
splitGT1 key elt yu fm_l fm_r split_key False = splitGT0 key elt yu fm_l fm_r split_key otherwise;
"
"splitGT2 key elt yu fm_l fm_r split_key True = splitGT fm_r split_key;
splitGT2 key elt yu fm_l fm_r split_key False = splitGT1 key elt yu fm_l fm_r split_key (split_key < key);
"
"splitGT3 (Branch key elt yu fm_l fm_r) split_key = splitGT2 key elt yu fm_l fm_r split_key (split_key > key);
"
"splitGT4 EmptyFM split_key = emptyFM;
splitGT4 wvu wvv = splitGT3 wvu wvv;
"
The following Function with conditions
"splitLT EmptyFM split_key = emptyFM;
splitLT (Branch key elt yv fm_l fm_r) split_key|split_key < keysplitLT fm_l split_key|split_key > keymkVBalBranch key elt fm_l (splitLT fm_r split_key)|otherwisefm_l;
"
is transformed to
"splitLT EmptyFM split_key = splitLT4 EmptyFM split_key;
splitLT (Branch key elt yv fm_l fm_r) split_key = splitLT3 (Branch key elt yv fm_l fm_r) split_key;
"
"splitLT1 key elt yv fm_l fm_r split_key True = mkVBalBranch key elt fm_l (splitLT fm_r split_key);
splitLT1 key elt yv fm_l fm_r split_key False = splitLT0 key elt yv fm_l fm_r split_key otherwise;
"
"splitLT0 key elt yv fm_l fm_r split_key True = fm_l;
"
"splitLT2 key elt yv fm_l fm_r split_key True = splitLT fm_l split_key;
splitLT2 key elt yv fm_l fm_r split_key False = splitLT1 key elt yv fm_l fm_r split_key (split_key > key);
"
"splitLT3 (Branch key elt yv fm_l fm_r) split_key = splitLT2 key elt yv fm_l fm_r split_key (split_key < key);
"
"splitLT4 EmptyFM split_key = emptyFM;
splitLT4 wvy wvz = splitLT3 wvy wvz;
"
The following Function with conditions
"mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr)|sizeFM fm_lr < 2 * sizeFM fm_llsingle_R fm_L fm_R|otherwisedouble_R fm_L fm_R;
"
is transformed to
"mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);
"
"mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = double_R fm_L fm_R;
"
"mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;
"
"mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
"
The following Function with conditions
"mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr)|sizeFM fm_rl < 2 * sizeFM fm_rrsingle_L fm_L fm_R|otherwisedouble_L fm_L fm_R;
"
is transformed to
"mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);
"
"mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = double_L fm_L fm_R;
"
"mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;
"
"mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
"
The following Function with conditions
"mkBalBranch key elt fm_L fm_R|size_l + size_r < 2mkBranch 1 key elt fm_L fm_R|size_r > sIZE_RATIO * size_lmkBalBranch0 fm_L fm_R fm_R|size_l > sIZE_RATIO * size_rmkBalBranch1 fm_L fm_R fm_L|otherwisemkBranch 2 key elt fm_L fm_R where {
double_L fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
;
double_R (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
;
mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr)|sizeFM fm_rl < 2 * sizeFM fm_rrsingle_L fm_L fm_R|otherwisedouble_L fm_L fm_R;
;
mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr)|sizeFM fm_lr < 2 * sizeFM fm_llsingle_R fm_L fm_R|otherwisedouble_R fm_L fm_R;
;
single_L fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
;
single_R (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
;
size_l  = sizeFM fm_L;
;
size_r  = sizeFM fm_R;
} 
;
"
is transformed to
"mkBalBranch key elt fm_L fm_R = mkBalBranch6 key elt fm_L fm_R;
"
"mkBalBranch6 key elt fm_L fm_R = mkBalBranch5 key elt fm_L fm_R (size_l + size_r < 2) where {
double_L fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
;
double_R (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
;
mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);
;
mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = double_L fm_L fm_R;
;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;
;
mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
;
mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);
;
mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = double_R fm_L fm_R;
;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;
;
mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
;
mkBalBranch2 key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;
;
mkBalBranch3 key elt fm_L fm_R True = mkBalBranch1 fm_L fm_R fm_L;
mkBalBranch3 key elt fm_L fm_R False = mkBalBranch2 key elt fm_L fm_R otherwise;
;
mkBalBranch4 key elt fm_L fm_R True = mkBalBranch0 fm_L fm_R fm_R;
mkBalBranch4 key elt fm_L fm_R False = mkBalBranch3 key elt fm_L fm_R (size_l > sIZE_RATIO * size_r);
;
mkBalBranch5 key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch5 key elt fm_L fm_R False = mkBalBranch4 key elt fm_L fm_R (size_r > sIZE_RATIO * size_l);
;
single_L fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
;
single_R (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
;
size_l  = sizeFM fm_L;
;
size_r  = sizeFM fm_R;
} 
;
"

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

(8)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
addToFM :: Ord a => FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord b => (a  ->  a  ->  a)  ->  FiniteMap b a  ->  b  ->  a  ->  FiniteMap b a;
addToFM_C combiner EmptyFM key elt = addToFM_C4 combiner EmptyFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt;

addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt True = Branch new_key (combiner elt new_elt) size fm_l fm_r;

addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt);
addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt otherwise;

addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r;
addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt (new_key > key);

addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt (new_key < key);

addToFM_C4 combiner EmptyFM key elt = unitFM key elt;
addToFM_C4 vyu vyv vyw vyx = addToFM_C3 vyu vyv vyw vyx;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt zy zz EmptyFM) = (key,elt);
findMax (Branch key elt vuu vuv fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt vxu EmptyFM vxv) = (key,elt);
findMin (Branch key elt vxw fm_l vxx) = findMin fm_l;

mkBalBranch :: Ord a => a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBalBranch key elt fm_L fm_R = mkBalBranch6 key elt fm_L fm_R;

mkBalBranch6 key elt fm_L fm_R = mkBalBranch5 key elt fm_L fm_R (size_l + size_r < 2) where { 
double_L fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
double_R (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);
mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = double_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;
mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);
mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = double_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;
mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
mkBalBranch2 key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;
mkBalBranch3 key elt fm_L fm_R True = mkBalBranch1 fm_L fm_R fm_L;
mkBalBranch3 key elt fm_L fm_R False = mkBalBranch2 key elt fm_L fm_R otherwise;
mkBalBranch4 key elt fm_L fm_R True = mkBalBranch0 fm_L fm_R fm_R;
mkBalBranch4 key elt fm_L fm_R False = mkBalBranch3 key elt fm_L fm_R (size_l > sIZE_RATIO * size_r);
mkBalBranch5 key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch5 key elt fm_L fm_R False = mkBalBranch4 key elt fm_L fm_R (size_r > sIZE_RATIO * size_l);
single_L fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
single_R (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
size_l = sizeFM fm_L;
size_r = sizeFM fm_R;
 };

mkBranch :: Ord b => Int  ->  b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBranch which key elt fm_l fm_r = let { 
result = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where { 
balance_ok = True;
left_ok = left_ok0 fm_l key fm_l;
left_ok0 fm_l key EmptyFM = True;
left_ok0 fm_l key (Branch left_key yw yx yy yz) = let { 
biggest_left_key = fst (findMax fm_l);
} in biggest_left_key < key;
left_size = sizeFM fm_l;
right_ok = right_ok0 fm_r key fm_r;
right_ok0 fm_r key EmptyFM = True;
right_ok0 fm_r key (Branch right_key zu zv zw zx) = let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

mkVBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkVBalBranch key elt EmptyFM fm_r = mkVBalBranch5 key elt EmptyFM fm_r;
mkVBalBranch key elt fm_l EmptyFM = mkVBalBranch4 key elt fm_l EmptyFM;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);

mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_l < size_r) where { 
mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz otherwise;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_r < size_l);
size_l = sizeFM (Branch wv ww wx wy wz);
size_r = sizeFM (Branch xv xw xx xy xz);
 };

mkVBalBranch4 key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch4 vzv vzw vzx vzy = mkVBalBranch3 vzv vzw vzx vzy;

mkVBalBranch5 key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch5 wuu wuv wuw wux = mkVBalBranch4 wuu wuv wuw wux;

plusFM :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 vz left right) = mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where { 
gts = splitGT fm1 split_key;
lts = splitLT fm1 split_key;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch vww vwx size vwy vwz) = size;

splitGT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitGT EmptyFM split_key = splitGT4 EmptyFM split_key;
splitGT (Branch key elt yu fm_l fm_r) split_key = splitGT3 (Branch key elt yu fm_l fm_r) split_key;

splitGT0 key elt yu fm_l fm_r split_key True = fm_r;

splitGT1 key elt yu fm_l fm_r split_key True = mkVBalBranch key elt (splitGT fm_l split_key) fm_r;
splitGT1 key elt yu fm_l fm_r split_key False = splitGT0 key elt yu fm_l fm_r split_key otherwise;

splitGT2 key elt yu fm_l fm_r split_key True = splitGT fm_r split_key;
splitGT2 key elt yu fm_l fm_r split_key False = splitGT1 key elt yu fm_l fm_r split_key (split_key < key);

splitGT3 (Branch key elt yu fm_l fm_r) split_key = splitGT2 key elt yu fm_l fm_r split_key (split_key > key);

splitGT4 EmptyFM split_key = emptyFM;
splitGT4 wvu wvv = splitGT3 wvu wvv;

splitLT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitLT EmptyFM split_key = splitLT4 EmptyFM split_key;
splitLT (Branch key elt yv fm_l fm_r) split_key = splitLT3 (Branch key elt yv fm_l fm_r) split_key;

splitLT0 key elt yv fm_l fm_r split_key True = fm_l;

splitLT1 key elt yv fm_l fm_r split_key True = mkVBalBranch key elt fm_l (splitLT fm_r split_key);
splitLT1 key elt yv fm_l fm_r split_key False = splitLT0 key elt yv fm_l fm_r split_key otherwise;

splitLT2 key elt yv fm_l fm_r split_key True = splitLT fm_l split_key;
splitLT2 key elt yv fm_l fm_r split_key False = splitLT1 key elt yv fm_l fm_r split_key (split_key > key);

splitLT3 (Branch key elt yv fm_l fm_r) split_key = splitLT2 key elt yv fm_l fm_r split_key (split_key < key);

splitLT4 EmptyFM split_key = emptyFM;
splitLT4 wvy wvz = splitLT3 wvy wvz;

unitFM :: a  ->  b  ->  FiniteMap a b;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(9) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"mkBalBranch5 key elt fm_L fm_R (size_l + size_r < 2) where {
double_L fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 key elt fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
;
double_R (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 key elt fm_lrr fm_r);
;
mkBalBranch0 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);
;
mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = double_L fm_L fm_R;
;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;
;
mkBalBranch02 fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
;
mkBalBranch1 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);
;
mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = double_R fm_L fm_R;
;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;
;
mkBalBranch12 fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
;
mkBalBranch2 key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;
;
mkBalBranch3 key elt fm_L fm_R True = mkBalBranch1 fm_L fm_R fm_L;
mkBalBranch3 key elt fm_L fm_R False = mkBalBranch2 key elt fm_L fm_R otherwise;
;
mkBalBranch4 key elt fm_L fm_R True = mkBalBranch0 fm_L fm_R fm_R;
mkBalBranch4 key elt fm_L fm_R False = mkBalBranch3 key elt fm_L fm_R (size_l > sIZE_RATIO * size_r);
;
mkBalBranch5 key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch5 key elt fm_L fm_R False = mkBalBranch4 key elt fm_L fm_R (size_r > sIZE_RATIO * size_l);
;
single_L fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 key elt fm_l fm_rl) fm_rr;
;
single_R (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 key elt fm_lr fm_r);
;
size_l  = sizeFM fm_L;
;
size_r  = sizeFM fm_R;
} 
"
are unpacked to the following functions on top level
"mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_r www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_l www wwx wwy wwz);
"
"mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;
"
"mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_l www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_r www wwx wwy wwz);
"
"mkBalBranch6Double_L www wwx wwy wwz fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 www wwx fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
"
"mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R otherwise;
"
"mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Single_L www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;
"
"mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Double_R www wwx wwy wwz fm_L fm_R;
"
"mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);
"
"mkBalBranch6Single_R www wwx wwy wwz (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 www wwx fm_lr fm_r);
"
"mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);
"
"mkBalBranch6Size_r www wwx wwy wwz = sizeFM wwy;
"
"mkBalBranch6Single_L www wwx wwy wwz fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 www wwx fm_l fm_rl) fm_rr;
"
"mkBalBranch6Double_R www wwx wwy wwz (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 www wwx fm_lrr fm_r);
"
"mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
"
"mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Double_L www wwx wwy wwz fm_L fm_R;
"
"mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Single_R www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;
"
"mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
"
"mkBalBranch6Size_l www wwx wwy wwz = sizeFM wwz;
"
The bindings of the following Let/Where expression
"let {
result  = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result where {
balance_ok  = True;
;
left_ok  = left_ok0 fm_l key fm_l;
;
left_ok0 fm_l key EmptyFM = True;
left_ok0 fm_l key (Branch left_key yw yx yy yz) = let {
biggest_left_key  = fst (findMax fm_l);
} in biggest_left_key < key;
;
left_size  = sizeFM fm_l;
;
right_ok  = right_ok0 fm_r key fm_r;
;
right_ok0 fm_r key EmptyFM = True;
right_ok0 fm_r key (Branch right_key zu zv zw zx) = let {
smallest_right_key  = fst (findMin fm_r);
} in key < smallest_right_key;
;
right_size  = sizeFM fm_r;
;
unbox x = x;
} 
"
are unpacked to the following functions on top level
"mkBranchLeft_ok wxu wxv wxw = mkBranchLeft_ok0 wxu wxv wxw wxu wxv wxu;
"
"mkBranchBalance_ok wxu wxv wxw = True;
"
"mkBranchRight_ok wxu wxv wxw = mkBranchRight_ok0 wxu wxv wxw wxw wxv wxw;
"
"mkBranchLeft_ok0 wxu wxv wxw fm_l key EmptyFM = True;
mkBranchLeft_ok0 wxu wxv wxw fm_l key (Branch left_key yw yx yy yz) = mkBranchLeft_ok0Biggest_left_key fm_l < key;
"
"mkBranchRight_size wxu wxv wxw = sizeFM wxw;
"
"mkBranchUnbox wxu wxv wxw x = x;
"
"mkBranchLeft_size wxu wxv wxw = sizeFM wxu;
"
"mkBranchRight_ok0 wxu wxv wxw fm_r key EmptyFM = True;
mkBranchRight_ok0 wxu wxv wxw fm_r key (Branch right_key zu zv zw zx) = key < mkBranchRight_ok0Smallest_right_key fm_r;
"
The bindings of the following Let/Where expression
"let {
result  = Branch key elt (unbox (1 + left_size + right_size)) fm_l fm_r;
} in result"
are unpacked to the following functions on top level
"mkBranchResult wxx wxy wxz wyu = Branch wxx wxy (mkBranchUnbox wxz wxx wyu (1 + mkBranchLeft_size wxz wxx wyu + mkBranchRight_size wxz wxx wyu)) wxz wyu;
"
The bindings of the following Let/Where expression
"mkVBalBranch split_key elt1 (plusFM lts left) (plusFM gts right) where {
gts  = splitGT fm1 split_key;
;
lts  = splitLT fm1 split_key;
} 
"
are unpacked to the following functions on top level
"plusFMLts wyv wyw = splitLT wyv wyw;
"
"plusFMGts wyv wyw = splitGT wyv wyw;
"
The bindings of the following Let/Where expression
"mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_l < size_r) where {
mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);
;
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch0 key elt wv ww wx wy wz xv xw xx xy xz otherwise;
;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch2 key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch1 key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * size_r < size_l);
;
size_l  = sizeFM (Branch wv ww wx wy wz);
;
size_r  = sizeFM (Branch xv xw xx xy xz);
} 
"
are unpacked to the following functions on top level
"mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wyx wyy wyz wzu wzv);
"
"mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);
"
"mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wzw wzx wzy wzz xuu);
"
"mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz otherwise;
"
"mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu < mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu);
"
The bindings of the following Let/Where expression
"let {
biggest_left_key  = fst (findMax fm_l);
} in biggest_left_key < key"
are unpacked to the following functions on top level
"mkBranchLeft_ok0Biggest_left_key xuv = fst (findMax xuv);
"
The bindings of the following Let/Where expression
"let {
smallest_right_key  = fst (findMin fm_r);
} in key < smallest_right_key"
are unpacked to the following functions on top level
"mkBranchRight_ok0Smallest_right_key xuw = fst (findMin xuw);
"

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

(10)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
addToFM :: Ord b => FiniteMap b a  ->  b  ->  a  ->  FiniteMap b a;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord a => (b  ->  b  ->  b)  ->  FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM_C combiner EmptyFM key elt = addToFM_C4 combiner EmptyFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt;

addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt True = Branch new_key (combiner elt new_elt) size fm_l fm_r;

addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt);
addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt otherwise;

addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r;
addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt (new_key > key);

addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt (new_key < key);

addToFM_C4 combiner EmptyFM key elt = unitFM key elt;
addToFM_C4 vyu vyv vyw vyx = addToFM_C3 vyu vyv vyw vyx;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt zy zz EmptyFM) = (key,elt);
findMax (Branch key elt vuu vuv fm_r) = findMax fm_r;

findMin :: FiniteMap b a  ->  (b,a);
findMin (Branch key elt vxu EmptyFM vxv) = (key,elt);
findMin (Branch key elt vxw fm_l vxx) = findMin fm_l;

mkBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBalBranch key elt fm_L fm_R = mkBalBranch6 key elt fm_L fm_R;

mkBalBranch6 key elt fm_L fm_R = mkBalBranch6MkBalBranch5 key elt fm_R fm_L key elt fm_L fm_R (mkBalBranch6Size_l key elt fm_R fm_L + mkBalBranch6Size_r key elt fm_R fm_L < 2);

mkBalBranch6Double_L www wwx wwy wwz fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 www wwx fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);

mkBalBranch6Double_R www wwx wwy wwz (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 www wwx fm_lrr fm_r);

mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);

mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Double_L www wwx wwy wwz fm_L fm_R;

mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Single_L www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;

mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);

mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);

mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Double_R www wwx wwy wwz fm_L fm_R;

mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Single_R www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;

mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);

mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;

mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R otherwise;

mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_l www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_r www wwx wwy wwz);

mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_r www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_l www wwx wwy wwz);

mkBalBranch6Single_L www wwx wwy wwz fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 www wwx fm_l fm_rl) fm_rr;

mkBalBranch6Single_R www wwx wwy wwz (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 www wwx fm_lr fm_r);

mkBalBranch6Size_l www wwx wwy wwz = sizeFM wwz;

mkBalBranch6Size_r www wwx wwy wwz = sizeFM wwy;

mkBranch :: Ord b => Int  ->  b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBranch which key elt fm_l fm_r = mkBranchResult key elt fm_l fm_r;

mkBranchBalance_ok wxu wxv wxw = True;

mkBranchLeft_ok wxu wxv wxw = mkBranchLeft_ok0 wxu wxv wxw wxu wxv wxu;

mkBranchLeft_ok0 wxu wxv wxw fm_l key EmptyFM = True;
mkBranchLeft_ok0 wxu wxv wxw fm_l key (Branch left_key yw yx yy yz) = mkBranchLeft_ok0Biggest_left_key fm_l < key;

mkBranchLeft_ok0Biggest_left_key xuv = fst (findMax xuv);

mkBranchLeft_size wxu wxv wxw = sizeFM wxu;

mkBranchResult wxx wxy wxz wyu = Branch wxx wxy (mkBranchUnbox wxz wxx wyu (1 + mkBranchLeft_size wxz wxx wyu + mkBranchRight_size wxz wxx wyu)) wxz wyu;

mkBranchRight_ok wxu wxv wxw = mkBranchRight_ok0 wxu wxv wxw wxw wxv wxw;

mkBranchRight_ok0 wxu wxv wxw fm_r key EmptyFM = True;
mkBranchRight_ok0 wxu wxv wxw fm_r key (Branch right_key zu zv zw zx) = key < mkBranchRight_ok0Smallest_right_key fm_r;

mkBranchRight_ok0Smallest_right_key xuw = fst (findMin xuw);

mkBranchRight_size wxu wxv wxw = sizeFM wxw;

mkBranchUnbox :: Ord a =>  ->  (FiniteMap a b) ( ->  a ( ->  (FiniteMap a b) (Int  ->  Int)));
mkBranchUnbox wxu wxv wxw x = x;

mkVBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkVBalBranch key elt EmptyFM fm_r = mkVBalBranch5 key elt EmptyFM fm_r;
mkVBalBranch key elt fm_l EmptyFM = mkVBalBranch4 key elt fm_l EmptyFM;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);

mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3MkVBalBranch2 xv xw xx xy xz wv ww wx wy wz key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * mkVBalBranch3Size_l xv xw xx xy xz wv ww wx wy wz < mkVBalBranch3Size_r xv xw xx xy xz wv ww wx wy wz);

mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch 13 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);

mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz otherwise;

mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu < mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu);

mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wzw wzx wzy wzz xuu);

mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wyx wyy wyz wzu wzv);

mkVBalBranch4 key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch4 vzv vzw vzx vzy = mkVBalBranch3 vzv vzw vzx vzy;

mkVBalBranch5 key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch5 wuu wuv wuw wux = mkVBalBranch4 wuu wuv wuw wux;

plusFM :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 vz left right) = mkVBalBranch split_key elt1 (plusFM (plusFMLts fm1 split_key) left) (plusFM (plusFMGts fm1 split_key) right);

plusFMGts wyv wyw = splitGT wyv wyw;

plusFMLts wyv wyw = splitLT wyv wyw;

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch vww vwx size vwy vwz) = size;

splitGT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitGT EmptyFM split_key = splitGT4 EmptyFM split_key;
splitGT (Branch key elt yu fm_l fm_r) split_key = splitGT3 (Branch key elt yu fm_l fm_r) split_key;

splitGT0 key elt yu fm_l fm_r split_key True = fm_r;

splitGT1 key elt yu fm_l fm_r split_key True = mkVBalBranch key elt (splitGT fm_l split_key) fm_r;
splitGT1 key elt yu fm_l fm_r split_key False = splitGT0 key elt yu fm_l fm_r split_key otherwise;

splitGT2 key elt yu fm_l fm_r split_key True = splitGT fm_r split_key;
splitGT2 key elt yu fm_l fm_r split_key False = splitGT1 key elt yu fm_l fm_r split_key (split_key < key);

splitGT3 (Branch key elt yu fm_l fm_r) split_key = splitGT2 key elt yu fm_l fm_r split_key (split_key > key);

splitGT4 EmptyFM split_key = emptyFM;
splitGT4 wvu wvv = splitGT3 wvu wvv;

splitLT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitLT EmptyFM split_key = splitLT4 EmptyFM split_key;
splitLT (Branch key elt yv fm_l fm_r) split_key = splitLT3 (Branch key elt yv fm_l fm_r) split_key;

splitLT0 key elt yv fm_l fm_r split_key True = fm_l;

splitLT1 key elt yv fm_l fm_r split_key True = mkVBalBranch key elt fm_l (splitLT fm_r split_key);
splitLT1 key elt yv fm_l fm_r split_key False = splitLT0 key elt yv fm_l fm_r split_key otherwise;

splitLT2 key elt yv fm_l fm_r split_key True = splitLT fm_l split_key;
splitLT2 key elt yv fm_l fm_r split_key False = splitLT1 key elt yv fm_l fm_r split_key (split_key > key);

splitLT3 (Branch key elt yv fm_l fm_r) split_key = splitLT2 key elt yv fm_l fm_r split_key (split_key < key);

splitLT4 EmptyFM split_key = emptyFM;
splitLT4 wvy wvz = splitLT3 wvy wvz;

unitFM :: a  ->  b  ->  FiniteMap a b;
unitFM key elt = Branch key elt 1 emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(11) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(12)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
addToFM :: Ord a => FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM fm key elt = addToFM_C addToFM0 fm key elt;

addToFM0 old new = new;

addToFM_C :: Ord a => (b  ->  b  ->  b)  ->  FiniteMap a b  ->  a  ->  b  ->  FiniteMap a b;
addToFM_C combiner EmptyFM key elt = addToFM_C4 combiner EmptyFM key elt;
addToFM_C combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt;

addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt True = Branch new_key (combiner elt new_elt) size fm_l fm_r;

addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt fm_l (addToFM_C combiner fm_r new_key new_elt);
addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C0 combiner key elt size fm_l fm_r new_key new_elt otherwise;

addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt True = mkBalBranch key elt (addToFM_C combiner fm_l new_key new_elt) fm_r;
addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt False = addToFM_C1 combiner key elt size fm_l fm_r new_key new_elt (new_key > key);

addToFM_C3 combiner (Branch key elt size fm_l fm_r) new_key new_elt = addToFM_C2 combiner key elt size fm_l fm_r new_key new_elt (new_key < key);

addToFM_C4 combiner EmptyFM key elt = unitFM key elt;
addToFM_C4 vyu vyv vyw vyx = addToFM_C3 vyu vyv vyw vyx;

emptyFM :: FiniteMap b a;
emptyFM = EmptyFM;

findMax :: FiniteMap a b  ->  (a,b);
findMax (Branch key elt zy zz EmptyFM) = (key,elt);
findMax (Branch key elt vuu vuv fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt vxu EmptyFM vxv) = (key,elt);
findMin (Branch key elt vxw fm_l vxx) = findMin fm_l;

mkBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkBalBranch key elt fm_L fm_R = mkBalBranch6 key elt fm_L fm_R;

mkBalBranch6 key elt fm_L fm_R = mkBalBranch6MkBalBranch5 key elt fm_R fm_L key elt fm_L fm_R (mkBalBranch6Size_l key elt fm_R fm_L + mkBalBranch6Size_r key elt fm_R fm_L < Pos (Succ (Succ Zero)));

mkBalBranch6Double_L www wwx wwy wwz fm_l (Branch key_r elt_r vvw (Branch key_rl elt_rl vvx fm_rll fm_rlr) fm_rr) = mkBranch (Pos (Succ (Succ (Succ (Succ (Succ Zero)))))) key_rl elt_rl (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) www wwx fm_l fm_rll) (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))) key_r elt_r fm_rlr fm_rr);

mkBalBranch6Double_R www wwx wwy wwz (Branch key_l elt_l vux fm_ll (Branch key_lr elt_lr vuy fm_lrl fm_lrr)) fm_r = mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) key_lr elt_lr (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))) key_l elt_l fm_ll fm_lrl) (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))) www wwx fm_lrr fm_r);

mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr);

mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Double_L www wwx wwy wwz fm_L fm_R;

mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr True = mkBalBranch6Single_L www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr False = mkBalBranch6MkBalBranch00 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr otherwise;

mkBalBranch6MkBalBranch02 www wwx wwy wwz fm_L fm_R (Branch vvy vvz vwu fm_rl fm_rr) = mkBalBranch6MkBalBranch01 www wwx wwy wwz fm_L fm_R vvy vvz vwu fm_rl fm_rr (sizeFM fm_rl < Pos (Succ (Succ Zero)) * sizeFM fm_rr);

mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr);

mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Double_R www wwx wwy wwz fm_L fm_R;

mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr True = mkBalBranch6Single_R www wwx wwy wwz fm_L fm_R;
mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr False = mkBalBranch6MkBalBranch10 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr otherwise;

mkBalBranch6MkBalBranch12 www wwx wwy wwz fm_L fm_R (Branch vuz vvu vvv fm_ll fm_lr) = mkBalBranch6MkBalBranch11 www wwx wwy wwz fm_L fm_R vuz vvu vvv fm_ll fm_lr (sizeFM fm_lr < Pos (Succ (Succ Zero)) * sizeFM fm_ll);

mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R True = mkBranch (Pos (Succ (Succ Zero))) key elt fm_L fm_R;

mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 www wwx wwy wwz fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 www wwx wwy wwz key elt fm_L fm_R otherwise;

mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 www wwx wwy wwz fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_l www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_r www wwx wwy wwz);

mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R True = mkBranch (Pos (Succ Zero)) key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 www wwx wwy wwz key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 www wwx wwy wwz key elt fm_L fm_R (mkBalBranch6Size_r www wwx wwy wwz > sIZE_RATIO * mkBalBranch6Size_l www wwx wwy wwz);

mkBalBranch6Single_L www wwx wwy wwz fm_l (Branch key_r elt_r vwv fm_rl fm_rr) = mkBranch (Pos (Succ (Succ (Succ Zero)))) key_r elt_r (mkBranch (Pos (Succ (Succ (Succ (Succ Zero))))) www wwx fm_l fm_rl) fm_rr;

mkBalBranch6Single_R www wwx wwy wwz (Branch key_l elt_l vuw fm_ll fm_lr) fm_r = mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))) key_l elt_l fm_ll (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))) www wwx fm_lr fm_r);

mkBalBranch6Size_l www wwx wwy wwz = sizeFM wwz;

mkBalBranch6Size_r www wwx wwy wwz = sizeFM wwy;

mkBranch :: Ord a => Int  ->  a  ->  b  ->  FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
mkBranch which key elt fm_l fm_r = mkBranchResult key elt fm_l fm_r;

mkBranchBalance_ok wxu wxv wxw = True;

mkBranchLeft_ok wxu wxv wxw = mkBranchLeft_ok0 wxu wxv wxw wxu wxv wxu;

mkBranchLeft_ok0 wxu wxv wxw fm_l key EmptyFM = True;
mkBranchLeft_ok0 wxu wxv wxw fm_l key (Branch left_key yw yx yy yz) = mkBranchLeft_ok0Biggest_left_key fm_l < key;

mkBranchLeft_ok0Biggest_left_key xuv = fst (findMax xuv);

mkBranchLeft_size wxu wxv wxw = sizeFM wxu;

mkBranchResult wxx wxy wxz wyu = Branch wxx wxy (mkBranchUnbox wxz wxx wyu (Pos (Succ Zero) + mkBranchLeft_size wxz wxx wyu + mkBranchRight_size wxz wxx wyu)) wxz wyu;

mkBranchRight_ok wxu wxv wxw = mkBranchRight_ok0 wxu wxv wxw wxw wxv wxw;

mkBranchRight_ok0 wxu wxv wxw fm_r key EmptyFM = True;
mkBranchRight_ok0 wxu wxv wxw fm_r key (Branch right_key zu zv zw zx) = key < mkBranchRight_ok0Smallest_right_key fm_r;

mkBranchRight_ok0Smallest_right_key xuw = fst (findMin xuw);

mkBranchRight_size wxu wxv wxw = sizeFM wxw;

mkBranchUnbox :: Ord a =>  ->  (FiniteMap a b) ( ->  a ( ->  (FiniteMap a b) (Int  ->  Int)));
mkBranchUnbox wxu wxv wxw x = x;

mkVBalBranch :: Ord b => b  ->  a  ->  FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
mkVBalBranch key elt EmptyFM fm_r = mkVBalBranch5 key elt EmptyFM fm_r;
mkVBalBranch key elt fm_l EmptyFM = mkVBalBranch4 key elt fm_l EmptyFM;
mkVBalBranch key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);

mkVBalBranch3 key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz) = mkVBalBranch3MkVBalBranch2 xv xw xx xy xz wv ww wx wy wz key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * mkVBalBranch3Size_l xv xw xx xy xz wv ww wx wy wz < mkVBalBranch3Size_r xv xw xx xy xz wv ww wx wy wz);

mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))) key elt (Branch wv ww wx wy wz) (Branch xv xw xx xy xz);

mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch wv ww wy (mkVBalBranch key elt wz (Branch xv xw xx xy xz));
mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch0 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz otherwise;

mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz True = mkBalBranch xv xw (mkVBalBranch key elt (Branch wv ww wx wy wz) xy) xz;
mkVBalBranch3MkVBalBranch2 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz False = mkVBalBranch3MkVBalBranch1 wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu key elt wv ww wx wy wz xv xw xx xy xz (sIZE_RATIO * mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu < mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu);

mkVBalBranch3Size_l wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wzw wzx wzy wzz xuu);

mkVBalBranch3Size_r wyx wyy wyz wzu wzv wzw wzx wzy wzz xuu = sizeFM (Branch wyx wyy wyz wzu wzv);

mkVBalBranch4 key elt fm_l EmptyFM = addToFM fm_l key elt;
mkVBalBranch4 vzv vzw vzx vzy = mkVBalBranch3 vzv vzw vzx vzy;

mkVBalBranch5 key elt EmptyFM fm_r = addToFM fm_r key elt;
mkVBalBranch5 wuu wuv wuw wux = mkVBalBranch4 wuu wuv wuw wux;

plusFM :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
plusFM EmptyFM fm2 = fm2;
plusFM fm1 EmptyFM = fm1;
plusFM fm1 (Branch split_key elt1 vz left right) = mkVBalBranch split_key elt1 (plusFM (plusFMLts fm1 split_key) left) (plusFM (plusFMGts fm1 split_key) right);

plusFMGts wyv wyw = splitGT wyv wyw;

plusFMLts wyv wyw = splitLT wyv wyw;

sIZE_RATIO :: Int;
sIZE_RATIO = Pos (Succ (Succ (Succ (Succ (Succ Zero)))));

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = Pos Zero;
sizeFM (Branch vww vwx size vwy vwz) = size;

splitGT :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
splitGT EmptyFM split_key = splitGT4 EmptyFM split_key;
splitGT (Branch key elt yu fm_l fm_r) split_key = splitGT3 (Branch key elt yu fm_l fm_r) split_key;

splitGT0 key elt yu fm_l fm_r split_key True = fm_r;

splitGT1 key elt yu fm_l fm_r split_key True = mkVBalBranch key elt (splitGT fm_l split_key) fm_r;
splitGT1 key elt yu fm_l fm_r split_key False = splitGT0 key elt yu fm_l fm_r split_key otherwise;

splitGT2 key elt yu fm_l fm_r split_key True = splitGT fm_r split_key;
splitGT2 key elt yu fm_l fm_r split_key False = splitGT1 key elt yu fm_l fm_r split_key (split_key < key);

splitGT3 (Branch key elt yu fm_l fm_r) split_key = splitGT2 key elt yu fm_l fm_r split_key (split_key > key);

splitGT4 EmptyFM split_key = emptyFM;
splitGT4 wvu wvv = splitGT3 wvu wvv;

splitLT :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
splitLT EmptyFM split_key = splitLT4 EmptyFM split_key;
splitLT (Branch key elt yv fm_l fm_r) split_key = splitLT3 (Branch key elt yv fm_l fm_r) split_key;

splitLT0 key elt yv fm_l fm_r split_key True = fm_l;

splitLT1 key elt yv fm_l fm_r split_key True = mkVBalBranch key elt fm_l (splitLT fm_r split_key);
splitLT1 key elt yv fm_l fm_r split_key False = splitLT0 key elt yv fm_l fm_r split_key otherwise;

splitLT2 key elt yv fm_l fm_r split_key True = splitLT fm_l split_key;
splitLT2 key elt yv fm_l fm_r split_key False = splitLT1 key elt yv fm_l fm_r split_key (split_key > key);

splitLT3 (Branch key elt yv fm_l fm_r) split_key = splitLT2 key elt yv fm_l fm_r split_key (split_key < key);

splitLT4 EmptyFM split_key = emptyFM;
splitLT4 wvy wvz = splitLT3 wvy wvz;

unitFM :: b  ->  a  ->  FiniteMap b a;
unitFM key elt = Branch key elt (Pos (Succ Zero)) emptyFM emptyFM;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}