/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


MAYBE
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 15 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) LetRed [EQUIVALENT, 23 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(_ :% d')->d'"
is transformed to
"d'0 (_ :% d') = d';
"
The following Lambda expression
"\(_ :% d)->d"
is transformed to
"d0 (_ :% d) = d;
"
The following Lambda expression
"\(n' :% _)->n'"
is transformed to
"n'0 (n' :% _) = n';
"
The following Lambda expression
"\xr->xr"
is transformed to
"xr0 xr = xr;
"
The following Lambda expression
"\(n :% _)->n"
is transformed to
"n0 (n :% _) = n;
"
The following Lambda expression
"\(q,_)->q"
is transformed to
"q1 (q,_) = q;
"
The following Lambda expression
"\(_,r)->r"
is transformed to
"r0 (_,r) = r;
"
The following Lambda expression
"\(_,r')->r'"
is transformed to
"r'0 (_,r') = r';
"
The following Lambda expression
"\(_ :% d'')->d''"
is transformed to
"d''0 (_ :% d'') = d'';
"
The following Lambda expression
"\(q',_)->q'"
is transformed to
"q'0 (q',_) = q';
"
The following Lambda expression
"\(n'' :% _)->n''"
is transformed to
"n''0 (n'' :% _) = n'';
"

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(3) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
is transformed to
"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
primDivNatS0 x y False = Zero;
"
The following If expression
"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
is transformed to
"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
primModNatS0 x y False = Succ x;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"absReal x|x >= 0x|otherwise`negate` x;
"
is transformed to
"absReal x = absReal2 x;
"
"absReal0 x True = `negate` x;
"
"absReal1 x True = x;
absReal1 x False = absReal0 x otherwise;
"
"absReal2 x = absReal1 x (x >= 0);
"
The following Function with conditions
"gcd' x 0 = x;
gcd' x y = gcd' y (x `rem` y);
"
is transformed to
"gcd' x zx = gcd'2 x zx;
gcd' x y = gcd'0 x y;
"
"gcd'0 x y = gcd' y (x `rem` y);
"
"gcd'1 True x zx = x;
gcd'1 zy zz vuu = gcd'0 zz vuu;
"
"gcd'2 x zx = gcd'1 (zx == 0) x zx;
gcd'2 vuv vuw = gcd'0 vuv vuw;
"
The following Function with conditions
"gcd 0 0 = error [];
gcd x y = gcd' (abs x) (abs y) where {
gcd' x 0 = x;
gcd' x y = gcd' y (x `rem` y);
} 
;
"
is transformed to
"gcd vux vuy = gcd3 vux vuy;
gcd x y = gcd0 x y;
"
"gcd0 x y = gcd' (abs x) (abs y) where {
gcd' x zx = gcd'2 x zx;
gcd' x y = gcd'0 x y;
;
gcd'0 x y = gcd' y (x `rem` y);
;
gcd'1 True x zx = x;
gcd'1 zy zz vuu = gcd'0 zz vuu;
;
gcd'2 x zx = gcd'1 (zx == 0) x zx;
gcd'2 vuv vuw = gcd'0 vuv vuw;
} 
;
"
"gcd1 True vux vuy = error [];
gcd1 vuz vvu vvv = gcd0 vvu vvv;
"
"gcd2 True vux vuy = gcd1 (vuy == 0) vux vuy;
gcd2 vvw vvx vvy = gcd0 vvx vvy;
"
"gcd3 vux vuy = gcd2 (vux == 0) vux vuy;
gcd3 vvz vwu = gcd0 vvz vwu;
"
The following Function with conditions
"simplest x y|y < xsimplest y x|x == yxr|x > 0simplest' n d n' d'|y < 0`negate` simplest' (`negate` n') d' (`negate` n) d|otherwise0 :% 1 where {
d  = d0 vu34;
;
d'  = d'0 vu35;
;
d'0 (vx :% d') = d';
;
d0 (vy :% d) = d;
;
n  = n0 vu34;
;
n'  = n'0 vu35;
;
n'0 (n' :% vw) = n';
;
n0 (n :% vv) = n;
;
vu34  = toRational x;
;
vu35  = toRational y;
;
xr  = xr0 vu34;
;
xr0 xr = xr;
} 
;
"
is transformed to
"simplest x y = simplest5 x y;
"
"simplest5 x y = simplest4 x y (y < x) where {
d  = d0 vu34;
;
d'  = d'0 vu35;
;
d'0 (vx :% d') = d';
;
d0 (vy :% d) = d;
;
n  = n0 vu34;
;
n'  = n'0 vu35;
;
n'0 (n' :% vw) = n';
;
n0 (n :% vv) = n;
;
simplest0 x y True = 0 :% 1;
;
simplest1 x y True = `negate` simplest' (`negate` n') d' (`negate` n) d;
simplest1 x y False = simplest0 x y otherwise;
;
simplest2 x y True = simplest' n d n' d';
simplest2 x y False = simplest1 x y (y < 0);
;
simplest3 x y True = xr;
simplest3 x y False = simplest2 x y (x > 0);
;
simplest4 x y True = simplest y x;
simplest4 x y False = simplest3 x y (x == y);
;
vu34  = toRational x;
;
vu35  = toRational y;
;
xr  = xr0 vu34;
;
xr0 xr = xr;
} 
;
"
The following Function with conditions
"simplest' n d n' d'|r == 0q :% 1|q /= q'(q + 1) :% 1|otherwise(q * n'' + d'') :% n'' where {
d''  = d''0 vu38;
;
d''0 (wu :% d'') = d'';
;
n''  = n''0 vu38;
;
n''0 (n'' :% wv) = n'';
;
q  = q1 vu36;
;
q'  = q'0 vu37;
;
q'0 (q',wx) = q';
;
q1 (q,vz) = q;
;
r  = r0 vu36;
;
r'  = r'0 vu37;
;
r'0 (ww,r') = r';
;
r0 (wy,r) = r;
;
vu36  = quotRem n d;
;
vu37  = quotRem n' d';
;
vu38  = simplest' d' r' d r;
} 
;
"
is transformed to
"simplest' n d n' d' = simplest'3 n d n' d';
"
"simplest'3 n d n' d' = simplest'2 n d n' d' (r == 0) where {
d''  = d''0 vu38;
;
d''0 (wu :% d'') = d'';
;
n''  = n''0 vu38;
;
n''0 (n'' :% wv) = n'';
;
q  = q1 vu36;
;
q'  = q'0 vu37;
;
q'0 (q',wx) = q';
;
q1 (q,vz) = q;
;
r  = r0 vu36;
;
r'  = r'0 vu37;
;
r'0 (ww,r') = r';
;
r0 (wy,r) = r;
;
simplest'0 n d n' d' True = (q * n'' + d'') :% n'';
;
simplest'1 n d n' d' True = (q + 1) :% 1;
simplest'1 n d n' d' False = simplest'0 n d n' d' otherwise;
;
simplest'2 n d n' d' True = q :% 1;
simplest'2 n d n' d' False = simplest'1 n d n' d' (q /= q');
;
vu36  = quotRem n d;
;
vu37  = quotRem n' d';
;
vu38  = simplest' d' r' d r;
} 
;
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"reduce x y|y == 0error []|otherwisex `quot` d :% (y `quot` d) where {
d  = gcd x y;
} 
;
"
is transformed to
"reduce x y = reduce2 x y;
"
"reduce2 x y = reduce1 x y (y == 0) where {
d  = gcd x y;
;
reduce0 x y True = x `quot` d :% (y `quot` d);
;
reduce1 x y True = error [];
reduce1 x y False = reduce0 x y otherwise;
} 
;
"
The following Function with conditions
"signumReal x|x == 00|x > 01|otherwise-1;
"
is transformed to
"signumReal x = signumReal3 x;
"
"signumReal2 x True = 0;
signumReal2 x False = signumReal1 x (x > 0);
"
"signumReal0 x True = -1;
"
"signumReal1 x True = 1;
signumReal1 x False = signumReal0 x otherwise;
"
"signumReal3 x = signumReal2 x (x == 0);
"

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

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(9) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"gcd' (abs x) (abs y) where {
gcd' x zx = gcd'2 x zx;
gcd' x y = gcd'0 x y;
;
gcd'0 x y = gcd' y (x `rem` y);
;
gcd'1 True x zx = x;
gcd'1 zy zz vuu = gcd'0 zz vuu;
;
gcd'2 x zx = gcd'1 (zx == 0) x zx;
gcd'2 vuv vuw = gcd'0 vuv vuw;
} 
"
are unpacked to the following functions on top level
"gcd0Gcd'2 x zx = gcd0Gcd'1 (zx == 0) x zx;
gcd0Gcd'2 vuv vuw = gcd0Gcd'0 vuv vuw;
"
"gcd0Gcd'1 True x zx = x;
gcd0Gcd'1 zy zz vuu = gcd0Gcd'0 zz vuu;
"
"gcd0Gcd'0 x y = gcd0Gcd' y (x `rem` y);
"
"gcd0Gcd' x zx = gcd0Gcd'2 x zx;
gcd0Gcd' x y = gcd0Gcd'0 x y;
"
The bindings of the following Let/Where expression
"reduce1 x y (y == 0) where {
d  = gcd x y;
;
reduce0 x y True = x `quot` d :% (y `quot` d);
;
reduce1 x y True = error [];
reduce1 x y False = reduce0 x y otherwise;
} 
"
are unpacked to the following functions on top level
"reduce2D vwv vww = gcd vwv vww;
"
"reduce2Reduce1 vwv vww x y True = error [];
reduce2Reduce1 vwv vww x y False = reduce2Reduce0 vwv vww x y otherwise;
"
"reduce2Reduce0 vwv vww x y True = x `quot` reduce2D vwv vww :% (y `quot` reduce2D vwv vww);
"
The bindings of the following Let/Where expression
"simplest (x - eps) (x + eps) where {
simplest x y = simplest5 x y;
;
simplest' n d n' d' = simplest'3 n d n' d';
;
simplest'3 n d n' d' = simplest'2 n d n' d' (r == 0) where {
d''  = d''0 vu38;
;
d''0 (wu :% d'') = d'';
;
n''  = n''0 vu38;
;
n''0 (n'' :% wv) = n'';
;
q  = q1 vu36;
;
q'  = q'0 vu37;
;
q'0 (q',wx) = q';
;
q1 (q,vz) = q;
;
r  = r0 vu36;
;
r'  = r'0 vu37;
;
r'0 (ww,r') = r';
;
r0 (wy,r) = r;
;
simplest'0 n d n' d' True = (q * n'' + d'') :% n'';
;
simplest'1 n d n' d' True = (q + 1) :% 1;
simplest'1 n d n' d' False = simplest'0 n d n' d' otherwise;
;
simplest'2 n d n' d' True = q :% 1;
simplest'2 n d n' d' False = simplest'1 n d n' d' (q /= q');
;
vu36  = quotRem n d;
;
vu37  = quotRem n' d';
;
vu38  = simplest' d' r' d r;
} 
;
;
simplest5 x y = simplest4 x y (y < x) where {
d  = d0 vu34;
;
d'  = d'0 vu35;
;
d'0 (vx :% d') = d';
;
d0 (vy :% d) = d;
;
n  = n0 vu34;
;
n'  = n'0 vu35;
;
n'0 (n' :% vw) = n';
;
n0 (n :% vv) = n;
;
simplest0 x y True = 0 :% 1;
;
simplest1 x y True = `negate` simplest' (`negate` n') d' (`negate` n) d;
simplest1 x y False = simplest0 x y otherwise;
;
simplest2 x y True = simplest' n d n' d';
simplest2 x y False = simplest1 x y (y < 0);
;
simplest3 x y True = xr;
simplest3 x y False = simplest2 x y (x > 0);
;
simplest4 x y True = simplest y x;
simplest4 x y False = simplest3 x y (x == y);
;
vu34  = toRational x;
;
vu35  = toRational y;
;
xr  = xr0 vu34;
;
xr0 xr = xr;
} 
;
} 
"
are unpacked to the following functions on top level
"approxRationalSimplest'3 n d n' d' = approxRationalSimplest'3Simplest'2 d' d n' n n d n' d' (approxRationalSimplest'3R d' d n' n == 0);
"
"approxRationalSimplest' n d n' d' = approxRationalSimplest'3 n d n' d';
"
"approxRationalSimplest5 x y = approxRationalSimplest5Simplest4 y x x y (y < x);
"
"approxRationalSimplest x y = approxRationalSimplest5 x y;
"
The bindings of the following Let/Where expression
"simplest'2 n d n' d' (r == 0) where {
d''  = d''0 vu38;
;
d''0 (wu :% d'') = d'';
;
n''  = n''0 vu38;
;
n''0 (n'' :% wv) = n'';
;
q  = q1 vu36;
;
q'  = q'0 vu37;
;
q'0 (q',wx) = q';
;
q1 (q,vz) = q;
;
r  = r0 vu36;
;
r'  = r'0 vu37;
;
r'0 (ww,r') = r';
;
r0 (wy,r) = r;
;
simplest'0 n d n' d' True = (q * n'' + d'') :% n'';
;
simplest'1 n d n' d' True = (q + 1) :% 1;
simplest'1 n d n' d' False = simplest'0 n d n' d' otherwise;
;
simplest'2 n d n' d' True = q :% 1;
simplest'2 n d n' d' False = simplest'1 n d n' d' (q /= q');
;
vu36  = quotRem n d;
;
vu37  = quotRem n' d';
;
vu38  = simplest' d' r' d r;
} 
"
are unpacked to the following functions on top level
"approxRationalSimplest'3R vwx vwy vwz vxu = approxRationalSimplest'3R0 vwx vwy vwz vxu (approxRationalSimplest'3Vu36 vwx vwy vwz vxu);
"
"approxRationalSimplest'3Vu38 vwx vwy vwz vxu = approxRationalSimplest' vwx (approxRationalSimplest'3R' vwx vwy vwz vxu) vwy (approxRationalSimplest'3R vwx vwy vwz vxu);
"
"approxRationalSimplest'3R' vwx vwy vwz vxu = approxRationalSimplest'3R'0 vwx vwy vwz vxu (approxRationalSimplest'3Vu37 vwx vwy vwz vxu);
"
"approxRationalSimplest'3N''0 vwx vwy vwz vxu (n'' :% wv) = n'';
"
"approxRationalSimplest'3Vu37 vwx vwy vwz vxu = quotRem vwz vwx;
"
"approxRationalSimplest'3R0 vwx vwy vwz vxu (wy,r) = r;
"
"approxRationalSimplest'3D'' vwx vwy vwz vxu = approxRationalSimplest'3D''0 vwx vwy vwz vxu (approxRationalSimplest'3Vu38 vwx vwy vwz vxu);
"
"approxRationalSimplest'3Simplest'2 vwx vwy vwz vxu n d n' d' True = approxRationalSimplest'3Q vwx vwy vwz vxu :% 1;
approxRationalSimplest'3Simplest'2 vwx vwy vwz vxu n d n' d' False = approxRationalSimplest'3Simplest'1 vwx vwy vwz vxu n d n' d' (approxRationalSimplest'3Q vwx vwy vwz vxu /= approxRationalSimplest'3Q' vwx vwy vwz vxu);
"
"approxRationalSimplest'3Q vwx vwy vwz vxu = approxRationalSimplest'3Q1 vwx vwy vwz vxu (approxRationalSimplest'3Vu36 vwx vwy vwz vxu);
"
"approxRationalSimplest'3Simplest'0 vwx vwy vwz vxu n d n' d' True = (approxRationalSimplest'3Q vwx vwy vwz vxu * approxRationalSimplest'3N'' vwx vwy vwz vxu + approxRationalSimplest'3D'' vwx vwy vwz vxu) :% approxRationalSimplest'3N'' vwx vwy vwz vxu;
"
"approxRationalSimplest'3Simplest'1 vwx vwy vwz vxu n d n' d' True = (approxRationalSimplest'3Q vwx vwy vwz vxu + 1) :% 1;
approxRationalSimplest'3Simplest'1 vwx vwy vwz vxu n d n' d' False = approxRationalSimplest'3Simplest'0 vwx vwy vwz vxu n d n' d' otherwise;
"
"approxRationalSimplest'3Q'0 vwx vwy vwz vxu (q',wx) = q';
"
"approxRationalSimplest'3Vu36 vwx vwy vwz vxu = quotRem vxu vwy;
"
"approxRationalSimplest'3R'0 vwx vwy vwz vxu (ww,r') = r';
"
"approxRationalSimplest'3Q1 vwx vwy vwz vxu (q,vz) = q;
"
"approxRationalSimplest'3D''0 vwx vwy vwz vxu (wu :% d'') = d'';
"
"approxRationalSimplest'3N'' vwx vwy vwz vxu = approxRationalSimplest'3N''0 vwx vwy vwz vxu (approxRationalSimplest'3Vu38 vwx vwy vwz vxu);
"
"approxRationalSimplest'3Q' vwx vwy vwz vxu = approxRationalSimplest'3Q'0 vwx vwy vwz vxu (approxRationalSimplest'3Vu37 vwx vwy vwz vxu);
"
The bindings of the following Let/Where expression
"simplest4 x y (y < x) where {
d  = d0 vu34;
;
d'  = d'0 vu35;
;
d'0 (vx :% d') = d';
;
d0 (vy :% d) = d;
;
n  = n0 vu34;
;
n'  = n'0 vu35;
;
n'0 (n' :% vw) = n';
;
n0 (n :% vv) = n;
;
simplest0 x y True = 0 :% 1;
;
simplest1 x y True = `negate` simplest' (`negate` n') d' (`negate` n) d;
simplest1 x y False = simplest0 x y otherwise;
;
simplest2 x y True = simplest' n d n' d';
simplest2 x y False = simplest1 x y (y < 0);
;
simplest3 x y True = xr;
simplest3 x y False = simplest2 x y (x > 0);
;
simplest4 x y True = simplest y x;
simplest4 x y False = simplest3 x y (x == y);
;
vu34  = toRational x;
;
vu35  = toRational y;
;
xr  = xr0 vu34;
;
xr0 xr = xr;
} 
"
are unpacked to the following functions on top level
"approxRationalSimplest5Simplest3 vxv vxw x y True = approxRationalSimplest5Xr vxv vxw;
approxRationalSimplest5Simplest3 vxv vxw x y False = approxRationalSimplest5Simplest2 vxv vxw x y (x > 0);
"
"approxRationalSimplest5N0 vxv vxw (n :% vv) = n;
"
"approxRationalSimplest5Vu35 vxv vxw = toRational vxv;
"
"approxRationalSimplest5Simplest4 vxv vxw x y True = approxRationalSimplest y x;
approxRationalSimplest5Simplest4 vxv vxw x y False = approxRationalSimplest5Simplest3 vxv vxw x y (x == y);
"
"approxRationalSimplest5D0 vxv vxw (vy :% d) = d;
"
"approxRationalSimplest5N vxv vxw = approxRationalSimplest5N0 vxv vxw (approxRationalSimplest5Vu34 vxv vxw);
"
"approxRationalSimplest5Simplest2 vxv vxw x y True = approxRationalSimplest' (approxRationalSimplest5N vxv vxw) (approxRationalSimplest5D vxv vxw) (approxRationalSimplest5N' vxv vxw) (approxRationalSimplest5D' vxv vxw);
approxRationalSimplest5Simplest2 vxv vxw x y False = approxRationalSimplest5Simplest1 vxv vxw x y (y < 0);
"
"approxRationalSimplest5D vxv vxw = approxRationalSimplest5D0 vxv vxw (approxRationalSimplest5Vu34 vxv vxw);
"
"approxRationalSimplest5Simplest0 vxv vxw x y True = 0 :% 1;
"
"approxRationalSimplest5N'0 vxv vxw (n' :% vw) = n';
"
"approxRationalSimplest5Xr vxv vxw = approxRationalSimplest5Xr0 vxv vxw (approxRationalSimplest5Vu34 vxv vxw);
"
"approxRationalSimplest5Simplest1 vxv vxw x y True = `negate` approxRationalSimplest' (`negate` approxRationalSimplest5N' vxv vxw) (approxRationalSimplest5D' vxv vxw) (`negate` approxRationalSimplest5N vxv vxw) (approxRationalSimplest5D vxv vxw);
approxRationalSimplest5Simplest1 vxv vxw x y False = approxRationalSimplest5Simplest0 vxv vxw x y otherwise;
"
"approxRationalSimplest5D' vxv vxw = approxRationalSimplest5D'0 vxv vxw (approxRationalSimplest5Vu35 vxv vxw);
"
"approxRationalSimplest5Vu34 vxv vxw = toRational vxw;
"
"approxRationalSimplest5Xr0 vxv vxw xr = xr;
"
"approxRationalSimplest5N' vxv vxw = approxRationalSimplest5N'0 vxv vxw (approxRationalSimplest5Vu35 vxv vxw);
"
"approxRationalSimplest5D'0 vxv vxw (vx :% d') = d';
"

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

(10)
Obligation:
mainModule Main 
module Main where {
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 Main where {
import qualified Prelude;
}