YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
if(true(),x,y) -> x
if(false(),x,y) -> y
-(s(x),s(y)) -> -(x,y)
-(x,0()) -> x
-(0(),x) -> 0()
<(s(x),s(y)) -> <(x,y)
<(0(),s(x)) -> true()
<(x,0()) -> false()
mod(0(),y) -> 0()
mod(x,s(y)) -> if(<(x,s(y)),x,mod(-(x,s(y)),s(y)))
mod(x,0()) -> x
gcd(x,y) -> gcd(y,mod(x,y))
gcd(x,0()) -> x
gcd(0(),x) -> x

CP(R):
0() -> 0()
0() -> 0()
0() -> if(<(0(),s(___y353)),0(),mod(-(0(),s(___y353)),s(___y353)))
0() -> 0()
if(<(0(),s(___y392)),0(),mod(-(0(),s(___y392)),s(___y392))) -> 0()
0() -> 0()
gcd(0(),mod(___y487,0())) -> ___y487
gcd(___y491,mod(0(),___y491)) -> ___y491
___y520 -> gcd(0(),mod(___y520,0()))
0() -> 0()
___y554 -> gcd(___y554,mod(0(),___y554))
0() -> 0()

critical-pair-closing system C:
if(true(),x,y) -> x
if(false(),x,y) -> y
-(s(x),s(y)) -> -(x,y)
-(x,0()) -> x
-(0(),x) -> 0()
<(s(x),s(y)) -> <(x,y)
<(0(),s(x)) -> true()
<(x,0()) -> false()
mod(0(),y) -> 0()
mod(x,0()) -> x
gcd(x,0()) -> x
gcd(0(),x) -> x

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
if(true(),x,y) -> x
if(false(),x,y) -> y
-(s(x),s(y)) -> -(x,y)
-(x,0()) -> x
-(0(),x) -> 0()
<(s(x),s(y)) -> <(x,y)
<(0(),s(x)) -> true()
<(x,0()) -> false()
mod(0(),y) -> 0()
mod(x,0()) -> x
gcd(x,0()) -> x
gcd(0(),x) -> x

CP(R):
0() -> 0()
0() -> 0()
0() -> 0()
0() -> 0()
0() -> 0()
0() -> 0()

critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.