YES
Rewrite Rules:
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Apply Direct Methods...
Inner CPs:
   [ f(?x,g(?x)) = g(f(?x,?x)) ]
Outer CPs:
   [  ]
unknown Terminating
not Left-Linear, not Right-Linear
unknown Gomi&Oyamaguchi&Ohta
check Non-Confluence...Unknown
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Sort Assignment:
 a : 12=>12
 f : 12*12=>12
 g : 12=>12
 h : 12=>12
maximal types: {12}
Persistent Decomposition failed: Can't judge

Try Layer Preserving Decomposition for...
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
(lp)   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x ]
Rewrite Rules:
   [ f(?x,a(g(?x))) -> g(f(?x,?x)),
     f(?x,g(?x)) -> g(f(?x,?x)),
     a(?x) -> ?x ]
Apply Direct Methods...
Inner CPs:
   [ f(?x,g(?x)) = g(f(?x,?x)) ]
Outer CPs:
   [  ]
Terminating, WCR: CR
(lp)   [ a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Rewrite Rules:
   [ a(?x) -> ?x,
     h(?x) -> h(a(h(?x))) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [  ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
Linear
Development Closed
Direct Methods: CR
Result by Layer Preserving Decomposition: CR

Final result: CR
89.trs: YES (12 msec.)