/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 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 a b where { 
}
delFromFM :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
delFromFM EmptyFM del_key = emptyFM;
delFromFM (Branch key elt size fm_l fm_r) del_key  | del_key > key = mkBalBranch key elt fm_l (delFromFM fm_r del_key)
 | del_key < key = mkBalBranch key elt (delFromFM fm_l del_key) fm_r
 | key == del_key = glueBal fm_l fm_r;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMax (Branch key elt _ fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt _ fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMin (Branch key elt _ EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt _ fm_l fm_r) = mkBalBranch key elt (deleteMin 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;

glueBal :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
glueBal EmptyFM fm2 = fm2;
glueBal fm1 EmptyFM = fm1;
glueBal fm1 fm2  | sizeFM fm2 > sizeFM fm1 = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2)
 | otherwise = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2 where { 
mid_elt1 = (\(_,mid_elt1) ->mid_elt1) vv2;
mid_elt2 = (\(_,mid_elt2) ->mid_elt2) vv3;
mid_key1 = (\(mid_key1,_) ->mid_key1) vv2;
mid_key2 = (\(mid_key2,_) ->mid_key2) vv3;
vv2 = findMax fm1;
vv3 = findMin fm2;
 };

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 = 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;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

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

}
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
"\(_,mid_elt2)->mid_elt2"
is transformed to
"mid_elt20 (_,mid_elt2) = mid_elt2;
"
The following Lambda expression
"\(mid_key2,_)->mid_key2"
is transformed to
"mid_key20 (mid_key2,_) = mid_key2;
"
The following Lambda expression
"\(mid_key1,_)->mid_key1"
is transformed to
"mid_key10 (mid_key1,_) = mid_key1;
"
The following Lambda expression
"\(_,mid_elt1)->mid_elt1"
is transformed to
"mid_elt10 (_,mid_elt1) = mid_elt1;
"

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

(2)
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 { 
}
delFromFM :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
delFromFM EmptyFM del_key = emptyFM;
delFromFM (Branch key elt size fm_l fm_r) del_key  | del_key > key = mkBalBranch key elt fm_l (delFromFM fm_r del_key)
 | del_key < key = mkBalBranch key elt (delFromFM fm_l del_key) fm_r
 | key == del_key = glueBal fm_l fm_r;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMax (Branch key elt _ fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt _ fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

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

glueBal :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
glueBal EmptyFM fm2 = fm2;
glueBal fm1 EmptyFM = fm1;
glueBal fm1 fm2  | sizeFM fm2 > sizeFM fm1 = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2)
 | otherwise = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2 where { 
mid_elt1 = mid_elt10 vv2;
mid_elt10 (_,mid_elt1) = mid_elt1;
mid_elt2 = mid_elt20 vv3;
mid_elt20 (_,mid_elt2) = mid_elt2;
mid_key1 = mid_key10 vv2;
mid_key10 (mid_key1,_) = mid_key1;
mid_key2 = mid_key20 vv3;
mid_key20 (mid_key2,_) = mid_key2;
vv2 = findMax fm1;
vv3 = findMin fm2;
 };

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 = 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;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

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

}
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 b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
delFromFM :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
delFromFM EmptyFM del_key = emptyFM;
delFromFM (Branch key elt size fm_l fm_r) del_key  | del_key > key = mkBalBranch key elt fm_l (delFromFM fm_r del_key)
 | del_key < key = mkBalBranch key elt (delFromFM fm_l del_key) fm_r
 | key == del_key = glueBal fm_l fm_r;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMax (Branch key elt _ fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt _ fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMin (Branch key elt _ EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt _ fm_l fm_r) = mkBalBranch key elt (deleteMin fm_l) fm_r;

emptyFM :: FiniteMap b a;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
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;

glueBal :: Ord a => FiniteMap a b  ->  FiniteMap a b  ->  FiniteMap a b;
glueBal EmptyFM fm2 = fm2;
glueBal fm1 EmptyFM = fm1;
glueBal fm1 fm2  | sizeFM fm2 > sizeFM fm1 = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2)
 | otherwise = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2 where { 
mid_elt1 = mid_elt10 vv2;
mid_elt10 (_,mid_elt1) = mid_elt1;
mid_elt2 = mid_elt20 vv3;
mid_elt20 (_,mid_elt2) = mid_elt2;
mid_key1 = mid_key10 vv2;
mid_key10 (mid_key1,_) = mid_key1;
mid_key2 = mid_key20 vv3;
mid_key20 (mid_key2,_) = mid_key2;
vv2 = findMax fm1;
vv3 = findMin fm2;
 };

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 = 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;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

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

}
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.
----------------------------------------

(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 { 
}
delFromFM :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
delFromFM EmptyFM del_key = emptyFM;
delFromFM (Branch key elt size fm_l fm_r) del_key  | del_key > key = mkBalBranch key elt fm_l (delFromFM fm_r del_key)
 | del_key < key = mkBalBranch key elt (delFromFM fm_l del_key) fm_r
 | key == del_key = glueBal fm_l fm_r;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMax (Branch key elt xx fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt xy fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMin (Branch key elt xz EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt yu fm_l fm_r) = mkBalBranch key elt (deleteMin fm_l) fm_r;

emptyFM :: FiniteMap b a;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt vuv vuw EmptyFM) = (key,elt);
findMax (Branch key elt vux vuy fm_r) = findMax fm_r;

findMin :: FiniteMap b a  ->  (b,a);
findMin (Branch key elt yv EmptyFM yw) = (key,elt);
findMin (Branch key elt yx fm_l yy) = findMin fm_l;

glueBal :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
glueBal EmptyFM fm2 = fm2;
glueBal fm1 EmptyFM = fm1;
glueBal fm1 fm2  | sizeFM fm2 > sizeFM fm1 = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2)
 | otherwise = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2 where { 
mid_elt1 = mid_elt10 vv2;
mid_elt10 (ww,mid_elt1) = mid_elt1;
mid_elt2 = mid_elt20 vv3;
mid_elt20 (wv,mid_elt2) = mid_elt2;
mid_key1 = mid_key10 vv2;
mid_key10 (mid_key1,wx) = mid_key1;
mid_key2 = mid_key20 vv3;
mid_key20 (mid_key2,wy) = mid_key2;
vv2 = findMax fm1;
vv3 = findMin fm2;
 };

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 = 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 vvz (Branch key_rl elt_rl vwu 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 vvu fm_ll (Branch key_lr elt_lr vvv 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 vwv vww vwx 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 vvw vvx vvy 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 vwy 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 vuz 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 yz zu zv zw) = 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 zx zy zz vuu) = let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch wz xu size xv xw) = size;

}
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
"glueBal EmptyFM fm2 = fm2;
glueBal fm1 EmptyFM = fm1;
glueBal fm1 fm2|sizeFM fm2 > sizeFM fm1mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2)|otherwisemkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2 where {
mid_elt1  = mid_elt10 vv2;
;
mid_elt10 (ww,mid_elt1) = mid_elt1;
;
mid_elt2  = mid_elt20 vv3;
;
mid_elt20 (wv,mid_elt2) = mid_elt2;
;
mid_key1  = mid_key10 vv2;
;
mid_key10 (mid_key1,wx) = mid_key1;
;
mid_key2  = mid_key20 vv3;
;
mid_key20 (mid_key2,wy) = mid_key2;
;
vv2  = findMax fm1;
;
vv3  = findMin fm2;
} 
;
"
is transformed to
"glueBal EmptyFM fm2 = glueBal4 EmptyFM fm2;
glueBal fm1 EmptyFM = glueBal3 fm1 EmptyFM;
glueBal fm1 fm2 = glueBal2 fm1 fm2;
"
"glueBal2 fm1 fm2 = glueBal1 fm1 fm2 (sizeFM fm2 > sizeFM fm1) where {
glueBal0 fm1 fm2 True = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2;
;
glueBal1 fm1 fm2 True = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2);
glueBal1 fm1 fm2 False = glueBal0 fm1 fm2 otherwise;
;
mid_elt1  = mid_elt10 vv2;
;
mid_elt10 (ww,mid_elt1) = mid_elt1;
;
mid_elt2  = mid_elt20 vv3;
;
mid_elt20 (wv,mid_elt2) = mid_elt2;
;
mid_key1  = mid_key10 vv2;
;
mid_key10 (mid_key1,wx) = mid_key1;
;
mid_key2  = mid_key20 vv3;
;
mid_key20 (mid_key2,wy) = mid_key2;
;
vv2  = findMax fm1;
;
vv3  = findMin fm2;
} 
;
"
"glueBal3 fm1 EmptyFM = fm1;
glueBal3 vxu vxv = glueBal2 vxu vxv;
"
"glueBal4 EmptyFM fm2 = fm2;
glueBal4 vxx vxy = glueBal3 vxx vxy;
"
The following Function with conditions
"delFromFM EmptyFM del_key = emptyFM;
delFromFM (Branch key elt size fm_l fm_r) del_key|del_key > keymkBalBranch key elt fm_l (delFromFM fm_r del_key)|del_key < keymkBalBranch key elt (delFromFM fm_l del_key) fm_r|key == del_keyglueBal fm_l fm_r;
"
is transformed to
"delFromFM EmptyFM del_key = delFromFM4 EmptyFM del_key;
delFromFM (Branch key elt size fm_l fm_r) del_key = delFromFM3 (Branch key elt size fm_l fm_r) del_key;
"
"delFromFM0 key elt size fm_l fm_r del_key True = glueBal fm_l fm_r;
"
"delFromFM1 key elt size fm_l fm_r del_key True = mkBalBranch key elt (delFromFM fm_l del_key) fm_r;
delFromFM1 key elt size fm_l fm_r del_key False = delFromFM0 key elt size fm_l fm_r del_key (key == del_key);
"
"delFromFM2 key elt size fm_l fm_r del_key True = mkBalBranch key elt fm_l (delFromFM fm_r del_key);
delFromFM2 key elt size fm_l fm_r del_key False = delFromFM1 key elt size fm_l fm_r del_key (del_key < key);
"
"delFromFM3 (Branch key elt size fm_l fm_r) del_key = delFromFM2 key elt size fm_l fm_r del_key (del_key > key);
"
"delFromFM4 EmptyFM del_key = emptyFM;
delFromFM4 vyv vyw = delFromFM3 vyv vyw;
"
The following Function with conditions
"mkBalBranch1 fm_L fm_R (Branch vvw vvx vvy 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 vvw vvx vvy fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);
"
"mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;
"
"mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = double_R fm_L fm_R;
"
"mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
"
The following Function with conditions
"mkBalBranch0 fm_L fm_R (Branch vwv vww vwx 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 vwv vww vwx fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);
"
"mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;
"
"mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr True = double_L fm_L fm_R;
"
"mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vwv vww vwx 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 vvz (Branch key_rl elt_rl vwu 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 vvu fm_ll (Branch key_lr elt_lr vvv 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 vwv vww vwx 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 vvw vvx vvy 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 vwy 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 vuz 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 vvz (Branch key_rl elt_rl vwu 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 vvu fm_ll (Branch key_lr elt_lr vvv 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 vwv vww vwx fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);
;
mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr True = double_L fm_L fm_R;
;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;
;
mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
;
mkBalBranch1 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);
;
mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = double_R fm_L fm_R;
;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;
;
mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vvw vvx vvy 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 vwy 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 vuz 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 b a where { 
}
delFromFM :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
delFromFM EmptyFM del_key = delFromFM4 EmptyFM del_key;
delFromFM (Branch key elt size fm_l fm_r) del_key = delFromFM3 (Branch key elt size fm_l fm_r) del_key;

delFromFM0 key elt size fm_l fm_r del_key True = glueBal fm_l fm_r;

delFromFM1 key elt size fm_l fm_r del_key True = mkBalBranch key elt (delFromFM fm_l del_key) fm_r;
delFromFM1 key elt size fm_l fm_r del_key False = delFromFM0 key elt size fm_l fm_r del_key (key == del_key);

delFromFM2 key elt size fm_l fm_r del_key True = mkBalBranch key elt fm_l (delFromFM fm_r del_key);
delFromFM2 key elt size fm_l fm_r del_key False = delFromFM1 key elt size fm_l fm_r del_key (del_key < key);

delFromFM3 (Branch key elt size fm_l fm_r) del_key = delFromFM2 key elt size fm_l fm_r del_key (del_key > key);

delFromFM4 EmptyFM del_key = emptyFM;
delFromFM4 vyv vyw = delFromFM3 vyv vyw;

delListFromFM :: Ord b => FiniteMap b a  ->  [b]  ->  FiniteMap b a;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMax (Branch key elt xx fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt xy fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMin (Branch key elt xz EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt yu fm_l fm_r) = mkBalBranch key elt (deleteMin fm_l) fm_r;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt vuv vuw EmptyFM) = (key,elt);
findMax (Branch key elt vux vuy fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt yv EmptyFM yw) = (key,elt);
findMin (Branch key elt yx fm_l yy) = findMin fm_l;

glueBal :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
glueBal EmptyFM fm2 = glueBal4 EmptyFM fm2;
glueBal fm1 EmptyFM = glueBal3 fm1 EmptyFM;
glueBal fm1 fm2 = glueBal2 fm1 fm2;

glueBal2 fm1 fm2 = glueBal1 fm1 fm2 (sizeFM fm2 > sizeFM fm1) where { 
glueBal0 fm1 fm2 True = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2;
glueBal1 fm1 fm2 True = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2);
glueBal1 fm1 fm2 False = glueBal0 fm1 fm2 otherwise;
mid_elt1 = mid_elt10 vv2;
mid_elt10 (ww,mid_elt1) = mid_elt1;
mid_elt2 = mid_elt20 vv3;
mid_elt20 (wv,mid_elt2) = mid_elt2;
mid_key1 = mid_key10 vv2;
mid_key10 (mid_key1,wx) = mid_key1;
mid_key2 = mid_key20 vv3;
mid_key20 (mid_key2,wy) = mid_key2;
vv2 = findMax fm1;
vv3 = findMin fm2;
 };

glueBal3 fm1 EmptyFM = fm1;
glueBal3 vxu vxv = glueBal2 vxu vxv;

glueBal4 EmptyFM fm2 = fm2;
glueBal4 vxx vxy = glueBal3 vxx vxy;

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 vvz (Branch key_rl elt_rl vwu 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 vvu fm_ll (Branch key_lr elt_lr vvv 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 vwv vww vwx fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);
mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr True = double_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;
mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
mkBalBranch1 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);
mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = double_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;
mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vvw vvx vvy 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 vwy 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 vuz 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 yz zu zv zw) = 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 zx zy zz vuu) = let { 
smallest_right_key = fst (findMin fm_r);
} in key < smallest_right_key;
right_size = sizeFM fm_r;
unbox :: Int  ->  Int;
unbox x = x;
 };

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap a b  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch wz xu size xv xw) = size;

}
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 vvz (Branch key_rl elt_rl vwu 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 vvu fm_ll (Branch key_lr elt_lr vvv 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 vwv vww vwx fm_rl fm_rr) = mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);
;
mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr True = double_L fm_L fm_R;
;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr True = single_L fm_L fm_R;
mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch00 fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;
;
mkBalBranch02 fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch01 fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
;
mkBalBranch1 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);
;
mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = double_R fm_L fm_R;
;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr True = single_R fm_L fm_R;
mkBalBranch11 fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch10 fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;
;
mkBalBranch12 fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch11 fm_L fm_R vvw vvx vvy 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 vwy 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 vuz 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
"mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;
"
"mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);
"
"mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_l vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_r vyz vzu vzv vzw);
"
"mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Double_L vyz vzu vzv vzw fm_L fm_R;
"
"mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);
"
"mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);
"
"mkBalBranch6Size_r vyz vzu vzv vzw = sizeFM vyz;
"
"mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Single_R vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;
"
"mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Single_L vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;
"
"mkBalBranch6Single_R vyz vzu vzv vzw (Branch key_l elt_l vuz fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 vzu vzv fm_lr fm_r);
"
"mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_r vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_l vyz vzu vzv vzw);
"
"mkBalBranch6Single_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vwy fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 vzu vzv fm_l fm_rl) fm_rr;
"
"mkBalBranch6Double_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vvz (Branch key_rl elt_rl vwu fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 vzu vzv fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);
"
"mkBalBranch6Size_l vyz vzu vzv vzw = sizeFM vzw;
"
"mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Double_R vyz vzu vzv vzw fm_L fm_R;
"
"mkBalBranch6Double_R vyz vzu vzv vzw (Branch key_l elt_l vvu fm_ll (Branch key_lr elt_lr vvv fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 vzu vzv fm_lrr fm_r);
"
"mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);
"
"mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R otherwise;
"
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 yz zu zv zw) = 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 zx zy zz vuu) = 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
"mkBranchRight_ok0 vzx vzy vzz fm_r key EmptyFM = True;
mkBranchRight_ok0 vzx vzy vzz fm_r key (Branch right_key zx zy zz vuu) = key < mkBranchRight_ok0Smallest_right_key fm_r;
"
"mkBranchRight_size vzx vzy vzz = sizeFM vzx;
"
"mkBranchLeft_ok vzx vzy vzz = mkBranchLeft_ok0 vzx vzy vzz vzy vzz vzy;
"
"mkBranchLeft_size vzx vzy vzz = sizeFM vzy;
"
"mkBranchRight_ok vzx vzy vzz = mkBranchRight_ok0 vzx vzy vzz vzx vzz vzx;
"
"mkBranchLeft_ok0 vzx vzy vzz fm_l key EmptyFM = True;
mkBranchLeft_ok0 vzx vzy vzz fm_l key (Branch left_key yz zu zv zw) = mkBranchLeft_ok0Biggest_left_key fm_l < key;
"
"mkBranchBalance_ok vzx vzy vzz = True;
"
"mkBranchUnbox vzx vzy vzz x = x;
"
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 wuu wuv wuw wux = Branch wuu wuv (mkBranchUnbox wuw wux wuu (1 + mkBranchLeft_size wuw wux wuu + mkBranchRight_size wuw wux wuu)) wux wuw;
"
The bindings of the following Let/Where expression
"glueBal1 fm1 fm2 (sizeFM fm2 > sizeFM fm1) where {
glueBal0 fm1 fm2 True = mkBalBranch mid_key1 mid_elt1 (deleteMax fm1) fm2;
;
glueBal1 fm1 fm2 True = mkBalBranch mid_key2 mid_elt2 fm1 (deleteMin fm2);
glueBal1 fm1 fm2 False = glueBal0 fm1 fm2 otherwise;
;
mid_elt1  = mid_elt10 vv2;
;
mid_elt10 (ww,mid_elt1) = mid_elt1;
;
mid_elt2  = mid_elt20 vv3;
;
mid_elt20 (wv,mid_elt2) = mid_elt2;
;
mid_key1  = mid_key10 vv2;
;
mid_key10 (mid_key1,wx) = mid_key1;
;
mid_key2  = mid_key20 vv3;
;
mid_key20 (mid_key2,wy) = mid_key2;
;
vv2  = findMax fm1;
;
vv3  = findMin fm2;
} 
"
are unpacked to the following functions on top level
"glueBal2Mid_elt2 wuy wuz = glueBal2Mid_elt20 wuy wuz (glueBal2Vv3 wuy wuz);
"
"glueBal2Mid_elt1 wuy wuz = glueBal2Mid_elt10 wuy wuz (glueBal2Vv2 wuy wuz);
"
"glueBal2Mid_elt20 wuy wuz (wv,mid_elt2) = mid_elt2;
"
"glueBal2GlueBal1 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key2 wuy wuz) (glueBal2Mid_elt2 wuy wuz) fm1 (deleteMin fm2);
glueBal2GlueBal1 wuy wuz fm1 fm2 False = glueBal2GlueBal0 wuy wuz fm1 fm2 otherwise;
"
"glueBal2GlueBal0 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key1 wuy wuz) (glueBal2Mid_elt1 wuy wuz) (deleteMax fm1) fm2;
"
"glueBal2Mid_key10 wuy wuz (mid_key1,wx) = mid_key1;
"
"glueBal2Vv2 wuy wuz = findMax wuy;
"
"glueBal2Vv3 wuy wuz = findMin wuz;
"
"glueBal2Mid_key1 wuy wuz = glueBal2Mid_key10 wuy wuz (glueBal2Vv2 wuy wuz);
"
"glueBal2Mid_key2 wuy wuz = glueBal2Mid_key20 wuy wuz (glueBal2Vv3 wuy wuz);
"
"glueBal2Mid_elt10 wuy wuz (ww,mid_elt1) = mid_elt1;
"
"glueBal2Mid_key20 wuy wuz (mid_key2,wy) = mid_key2;
"
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 wvu = fst (findMin wvu);
"
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 wvv = fst (findMax wvv);
"

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

(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 a b where { 
}
delFromFM :: Ord a => FiniteMap a b  ->  a  ->  FiniteMap a b;
delFromFM EmptyFM del_key = delFromFM4 EmptyFM del_key;
delFromFM (Branch key elt size fm_l fm_r) del_key = delFromFM3 (Branch key elt size fm_l fm_r) del_key;

delFromFM0 key elt size fm_l fm_r del_key True = glueBal fm_l fm_r;

delFromFM1 key elt size fm_l fm_r del_key True = mkBalBranch key elt (delFromFM fm_l del_key) fm_r;
delFromFM1 key elt size fm_l fm_r del_key False = delFromFM0 key elt size fm_l fm_r del_key (key == del_key);

delFromFM2 key elt size fm_l fm_r del_key True = mkBalBranch key elt fm_l (delFromFM fm_r del_key);
delFromFM2 key elt size fm_l fm_r del_key False = delFromFM1 key elt size fm_l fm_r del_key (del_key < key);

delFromFM3 (Branch key elt size fm_l fm_r) del_key = delFromFM2 key elt size fm_l fm_r del_key (del_key > key);

delFromFM4 EmptyFM del_key = emptyFM;
delFromFM4 vyv vyw = delFromFM3 vyv vyw;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMax (Branch key elt xx fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt xy fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord b => FiniteMap b a  ->  FiniteMap b a;
deleteMin (Branch key elt xz EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt yu fm_l fm_r) = mkBalBranch key elt (deleteMin fm_l) fm_r;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt vuv vuw EmptyFM) = (key,elt);
findMax (Branch key elt vux vuy fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt yv EmptyFM yw) = (key,elt);
findMin (Branch key elt yx fm_l yy) = findMin fm_l;

glueBal :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
glueBal EmptyFM fm2 = glueBal4 EmptyFM fm2;
glueBal fm1 EmptyFM = glueBal3 fm1 EmptyFM;
glueBal fm1 fm2 = glueBal2 fm1 fm2;

glueBal2 fm1 fm2 = glueBal2GlueBal1 fm1 fm2 fm1 fm2 (sizeFM fm2 > sizeFM fm1);

glueBal2GlueBal0 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key1 wuy wuz) (glueBal2Mid_elt1 wuy wuz) (deleteMax fm1) fm2;

glueBal2GlueBal1 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key2 wuy wuz) (glueBal2Mid_elt2 wuy wuz) fm1 (deleteMin fm2);
glueBal2GlueBal1 wuy wuz fm1 fm2 False = glueBal2GlueBal0 wuy wuz fm1 fm2 otherwise;

glueBal2Mid_elt1 wuy wuz = glueBal2Mid_elt10 wuy wuz (glueBal2Vv2 wuy wuz);

glueBal2Mid_elt10 wuy wuz (ww,mid_elt1) = mid_elt1;

glueBal2Mid_elt2 wuy wuz = glueBal2Mid_elt20 wuy wuz (glueBal2Vv3 wuy wuz);

glueBal2Mid_elt20 wuy wuz (wv,mid_elt2) = mid_elt2;

glueBal2Mid_key1 wuy wuz = glueBal2Mid_key10 wuy wuz (glueBal2Vv2 wuy wuz);

glueBal2Mid_key10 wuy wuz (mid_key1,wx) = mid_key1;

glueBal2Mid_key2 wuy wuz = glueBal2Mid_key20 wuy wuz (glueBal2Vv3 wuy wuz);

glueBal2Mid_key20 wuy wuz (mid_key2,wy) = mid_key2;

glueBal2Vv2 wuy wuz = findMax wuy;

glueBal2Vv3 wuy wuz = findMin wuz;

glueBal3 fm1 EmptyFM = fm1;
glueBal3 vxu vxv = glueBal2 vxu vxv;

glueBal4 EmptyFM fm2 = fm2;
glueBal4 vxx vxy = glueBal3 vxx vxy;

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

mkBalBranch6Double_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vvz (Branch key_rl elt_rl vwu fm_rll fm_rlr) fm_rr) = mkBranch 5 key_rl elt_rl (mkBranch 6 vzu vzv fm_l fm_rll) (mkBranch 7 key_r elt_r fm_rlr fm_rr);

mkBalBranch6Double_R vyz vzu vzv vzw (Branch key_l elt_l vvu fm_ll (Branch key_lr elt_lr vvv fm_lrl fm_lrr)) fm_r = mkBranch 10 key_lr elt_lr (mkBranch 11 key_l elt_l fm_ll fm_lrl) (mkBranch 12 vzu vzv fm_lrr fm_r);

mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);

mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Double_L vyz vzu vzv vzw fm_L fm_R;

mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Single_L vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;

mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < 2 * sizeFM fm_rr);

mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);

mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Double_R vyz vzu vzv vzw fm_L fm_R;

mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Single_R vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;

mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr (sizeFM fm_lr < 2 * sizeFM fm_ll);

mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch 2 key elt fm_L fm_R;

mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R otherwise;

mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_l vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_r vyz vzu vzv vzw);

mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch 1 key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_r vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_l vyz vzu vzv vzw);

mkBalBranch6Single_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vwy fm_rl fm_rr) = mkBranch 3 key_r elt_r (mkBranch 4 vzu vzv fm_l fm_rl) fm_rr;

mkBalBranch6Single_R vyz vzu vzv vzw (Branch key_l elt_l vuz fm_ll fm_lr) fm_r = mkBranch 8 key_l elt_l fm_ll (mkBranch 9 vzu vzv fm_lr fm_r);

mkBalBranch6Size_l vyz vzu vzv vzw = sizeFM vzw;

mkBalBranch6Size_r vyz vzu vzv vzw = sizeFM vyz;

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_r fm_l;

mkBranchBalance_ok vzx vzy vzz = True;

mkBranchLeft_ok vzx vzy vzz = mkBranchLeft_ok0 vzx vzy vzz vzy vzz vzy;

mkBranchLeft_ok0 vzx vzy vzz fm_l key EmptyFM = True;
mkBranchLeft_ok0 vzx vzy vzz fm_l key (Branch left_key yz zu zv zw) = mkBranchLeft_ok0Biggest_left_key fm_l < key;

mkBranchLeft_ok0Biggest_left_key wvv = fst (findMax wvv);

mkBranchLeft_size vzx vzy vzz = sizeFM vzy;

mkBranchResult wuu wuv wuw wux = Branch wuu wuv (mkBranchUnbox wuw wux wuu (1 + mkBranchLeft_size wuw wux wuu + mkBranchRight_size wuw wux wuu)) wux wuw;

mkBranchRight_ok vzx vzy vzz = mkBranchRight_ok0 vzx vzy vzz vzx vzz vzx;

mkBranchRight_ok0 vzx vzy vzz fm_r key EmptyFM = True;
mkBranchRight_ok0 vzx vzy vzz fm_r key (Branch right_key zx zy zz vuu) = key < mkBranchRight_ok0Smallest_right_key fm_r;

mkBranchRight_ok0Smallest_right_key wvu = fst (findMin wvu);

mkBranchRight_size vzx vzy vzz = sizeFM vzx;

mkBranchUnbox :: Ord a =>  ->  (FiniteMap a b) ( ->  (FiniteMap a b) ( ->  a (Int  ->  Int)));
mkBranchUnbox vzx vzy vzz x = x;

sIZE_RATIO :: Int;
sIZE_RATIO = 5;

sizeFM :: FiniteMap b a  ->  Int;
sizeFM EmptyFM = 0;
sizeFM (Branch wz xu size xv xw) = size;

}
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 { 
}
delFromFM :: Ord b => FiniteMap b a  ->  b  ->  FiniteMap b a;
delFromFM EmptyFM del_key = delFromFM4 EmptyFM del_key;
delFromFM (Branch key elt size fm_l fm_r) del_key = delFromFM3 (Branch key elt size fm_l fm_r) del_key;

delFromFM0 key elt size fm_l fm_r del_key True = glueBal fm_l fm_r;

delFromFM1 key elt size fm_l fm_r del_key True = mkBalBranch key elt (delFromFM fm_l del_key) fm_r;
delFromFM1 key elt size fm_l fm_r del_key False = delFromFM0 key elt size fm_l fm_r del_key (key == del_key);

delFromFM2 key elt size fm_l fm_r del_key True = mkBalBranch key elt fm_l (delFromFM fm_r del_key);
delFromFM2 key elt size fm_l fm_r del_key False = delFromFM1 key elt size fm_l fm_r del_key (del_key < key);

delFromFM3 (Branch key elt size fm_l fm_r) del_key = delFromFM2 key elt size fm_l fm_r del_key (del_key > key);

delFromFM4 EmptyFM del_key = emptyFM;
delFromFM4 vyv vyw = delFromFM3 vyv vyw;

delListFromFM :: Ord a => FiniteMap a b  ->  [a]  ->  FiniteMap a b;
delListFromFM fm keys = foldl delFromFM fm keys;

deleteMax :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMax (Branch key elt xx fm_l EmptyFM) = fm_l;
deleteMax (Branch key elt xy fm_l fm_r) = mkBalBranch key elt fm_l (deleteMax fm_r);

deleteMin :: Ord a => FiniteMap a b  ->  FiniteMap a b;
deleteMin (Branch key elt xz EmptyFM fm_r) = fm_r;
deleteMin (Branch key elt yu fm_l fm_r) = mkBalBranch key elt (deleteMin fm_l) fm_r;

emptyFM :: FiniteMap a b;
emptyFM = EmptyFM;

findMax :: FiniteMap b a  ->  (b,a);
findMax (Branch key elt vuv vuw EmptyFM) = (key,elt);
findMax (Branch key elt vux vuy fm_r) = findMax fm_r;

findMin :: FiniteMap a b  ->  (a,b);
findMin (Branch key elt yv EmptyFM yw) = (key,elt);
findMin (Branch key elt yx fm_l yy) = findMin fm_l;

glueBal :: Ord b => FiniteMap b a  ->  FiniteMap b a  ->  FiniteMap b a;
glueBal EmptyFM fm2 = glueBal4 EmptyFM fm2;
glueBal fm1 EmptyFM = glueBal3 fm1 EmptyFM;
glueBal fm1 fm2 = glueBal2 fm1 fm2;

glueBal2 fm1 fm2 = glueBal2GlueBal1 fm1 fm2 fm1 fm2 (sizeFM fm2 > sizeFM fm1);

glueBal2GlueBal0 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key1 wuy wuz) (glueBal2Mid_elt1 wuy wuz) (deleteMax fm1) fm2;

glueBal2GlueBal1 wuy wuz fm1 fm2 True = mkBalBranch (glueBal2Mid_key2 wuy wuz) (glueBal2Mid_elt2 wuy wuz) fm1 (deleteMin fm2);
glueBal2GlueBal1 wuy wuz fm1 fm2 False = glueBal2GlueBal0 wuy wuz fm1 fm2 otherwise;

glueBal2Mid_elt1 wuy wuz = glueBal2Mid_elt10 wuy wuz (glueBal2Vv2 wuy wuz);

glueBal2Mid_elt10 wuy wuz (ww,mid_elt1) = mid_elt1;

glueBal2Mid_elt2 wuy wuz = glueBal2Mid_elt20 wuy wuz (glueBal2Vv3 wuy wuz);

glueBal2Mid_elt20 wuy wuz (wv,mid_elt2) = mid_elt2;

glueBal2Mid_key1 wuy wuz = glueBal2Mid_key10 wuy wuz (glueBal2Vv2 wuy wuz);

glueBal2Mid_key10 wuy wuz (mid_key1,wx) = mid_key1;

glueBal2Mid_key2 wuy wuz = glueBal2Mid_key20 wuy wuz (glueBal2Vv3 wuy wuz);

glueBal2Mid_key20 wuy wuz (mid_key2,wy) = mid_key2;

glueBal2Vv2 wuy wuz = findMax wuy;

glueBal2Vv3 wuy wuz = findMin wuz;

glueBal3 fm1 EmptyFM = fm1;
glueBal3 vxu vxv = glueBal2 vxu vxv;

glueBal4 EmptyFM fm2 = fm2;
glueBal4 vxx vxy = glueBal3 vxx vxy;

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

mkBalBranch6Double_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vvz (Branch key_rl elt_rl vwu 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))))))) vzu vzv fm_l fm_rll) (mkBranch (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))) key_r elt_r fm_rlr fm_rr);

mkBalBranch6Double_R vyz vzu vzv vzw (Branch key_l elt_l vvu fm_ll (Branch key_lr elt_lr vvv 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))))))))))))) vzu vzv fm_lrr fm_r);

mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr);

mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Double_L vyz vzu vzv vzw fm_L fm_R;

mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr True = mkBalBranch6Single_L vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr False = mkBalBranch6MkBalBranch00 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr otherwise;

mkBalBranch6MkBalBranch02 vyz vzu vzv vzw fm_L fm_R (Branch vwv vww vwx fm_rl fm_rr) = mkBalBranch6MkBalBranch01 vyz vzu vzv vzw fm_L fm_R vwv vww vwx fm_rl fm_rr (sizeFM fm_rl < Pos (Succ (Succ Zero)) * sizeFM fm_rr);

mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr);

mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Double_R vyz vzu vzv vzw fm_L fm_R;

mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr True = mkBalBranch6Single_R vyz vzu vzv vzw fm_L fm_R;
mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr False = mkBalBranch6MkBalBranch10 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr otherwise;

mkBalBranch6MkBalBranch12 vyz vzu vzv vzw fm_L fm_R (Branch vvw vvx vvy fm_ll fm_lr) = mkBalBranch6MkBalBranch11 vyz vzu vzv vzw fm_L fm_R vvw vvx vvy fm_ll fm_lr (sizeFM fm_lr < Pos (Succ (Succ Zero)) * sizeFM fm_ll);

mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch (Pos (Succ (Succ Zero))) key elt fm_L fm_R;

mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch1 vyz vzu vzv vzw fm_L fm_R fm_L;
mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch2 vyz vzu vzv vzw key elt fm_L fm_R otherwise;

mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R True = mkBalBranch6MkBalBranch0 vyz vzu vzv vzw fm_L fm_R fm_R;
mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch3 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_l vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_r vyz vzu vzv vzw);

mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R True = mkBranch (Pos (Succ Zero)) key elt fm_L fm_R;
mkBalBranch6MkBalBranch5 vyz vzu vzv vzw key elt fm_L fm_R False = mkBalBranch6MkBalBranch4 vyz vzu vzv vzw key elt fm_L fm_R (mkBalBranch6Size_r vyz vzu vzv vzw > sIZE_RATIO * mkBalBranch6Size_l vyz vzu vzv vzw);

mkBalBranch6Single_L vyz vzu vzv vzw fm_l (Branch key_r elt_r vwy fm_rl fm_rr) = mkBranch (Pos (Succ (Succ (Succ Zero)))) key_r elt_r (mkBranch (Pos (Succ (Succ (Succ (Succ Zero))))) vzu vzv fm_l fm_rl) fm_rr;

mkBalBranch6Single_R vyz vzu vzv vzw (Branch key_l elt_l vuz 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)))))))))) vzu vzv fm_lr fm_r);

mkBalBranch6Size_l vyz vzu vzv vzw = sizeFM vzw;

mkBalBranch6Size_r vyz vzu vzv vzw = sizeFM vyz;

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_r fm_l;

mkBranchBalance_ok vzx vzy vzz = True;

mkBranchLeft_ok vzx vzy vzz = mkBranchLeft_ok0 vzx vzy vzz vzy vzz vzy;

mkBranchLeft_ok0 vzx vzy vzz fm_l key EmptyFM = True;
mkBranchLeft_ok0 vzx vzy vzz fm_l key (Branch left_key yz zu zv zw) = mkBranchLeft_ok0Biggest_left_key fm_l < key;

mkBranchLeft_ok0Biggest_left_key wvv = fst (findMax wvv);

mkBranchLeft_size vzx vzy vzz = sizeFM vzy;

mkBranchResult wuu wuv wuw wux = Branch wuu wuv (mkBranchUnbox wuw wux wuu (Pos (Succ Zero) + mkBranchLeft_size wuw wux wuu + mkBranchRight_size wuw wux wuu)) wux wuw;

mkBranchRight_ok vzx vzy vzz = mkBranchRight_ok0 vzx vzy vzz vzx vzz vzx;

mkBranchRight_ok0 vzx vzy vzz fm_r key EmptyFM = True;
mkBranchRight_ok0 vzx vzy vzz fm_r key (Branch right_key zx zy zz vuu) = key < mkBranchRight_ok0Smallest_right_key fm_r;

mkBranchRight_ok0Smallest_right_key wvu = fst (findMin wvu);

mkBranchRight_size vzx vzy vzz = sizeFM vzx;

mkBranchUnbox :: Ord a =>  ->  (FiniteMap a b) ( ->  (FiniteMap a b) ( ->  a (Int  ->  Int)));
mkBranchUnbox vzx vzy vzz x = x;

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

sizeFM :: FiniteMap a b  ->  Int;
sizeFM EmptyFM = Pos Zero;
sizeFM (Branch wz xu size xv xw) = size;

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