diff --git a/NEWS b/NEWS
index 85d2bcd79..ed911a584 100644
--- a/NEWS
+++ b/NEWS
@@ -9,6 +9,12 @@ New in spot 1.0a (not released):
     - "P0.init" is parsed as an atomic even without the double quotes,
       but it was always output with double quotes.  This version will
       not quote this atomic proposition anymore.
+  * Minor improvements:
+    - Four new LTL simplifications rules:
+        GF(a|Xb) = GF(a|b)
+        GF(a|Fb) = GF(a|b)
+        FG(a&Xb) = FG(a&b)
+        FG(a&Gb) = FG(a&b)
   * Pruning:
     - lbtt has been removed from the distribution.  A copy of the last
       version we distributed is still available at
diff --git a/doc/tl/tl.tex b/doc/tl/tl.tex
index 0bee4a013..c11702450 100644
--- a/doc/tl/tl.tex
+++ b/doc/tl/tl.tex
@@ -1313,9 +1313,11 @@ instead of $\equiv$, and they can be disabled by setting the
 The following are simplification rules for unary operators (applied
 from left to right, as usual):
 \begin{align*}
-  \X\F\G f & \equiv \F\G f & \F\X f    & \equiv \X\F f       & \G\X f     & \equiv \X\G f\\
-  \X\G\F f & \equiv \G\F f & \F(f\U g) & \equiv \F g         & \G(f \R g) & \equiv \G g       \\
-           &               & \F(f\M g) & \equiv \F (f\AND g) & \G(f \W g) & \equiv \G(f\OR g)
+  \X\F\G f & \equiv \F\G f & \F\X f           & \equiv \X\F f        & \G\X f          & \equiv \X\G f       \\
+  \X\G\F f & \equiv \G\F f & \F(f\U g)        & \equiv \F g          & \G(f \R g)      & \equiv \G g         \\
+           &               & \F(f\M g)        & \equiv \F (f\AND g)  & \G(f \W g)      & \equiv \G(f\OR g)   \\
+           &               & \F\G(f\AND \X g) & \equiv \F\G(f\AND g) & \G\F(f\OR \X g) & \equiv \G\F(f\OR g) \\
+           &               & \F\G(f\AND \G g) & \equiv \F\G(f\AND g) & \G\F(f\OR \F g) & \equiv \G\F(f\OR g) \\
 \end{align*}
 \begin{align*}
   \G(f_1\OR\ldots\OR f_n \OR \G\F(g_1)\OR\ldots\OR \G\F(g_m)) & \equiv \G(f_1\OR\ldots\OR f_n)\OR \G\F(g_1\OR\ldots\OR g_m)
diff --git a/src/ltltest/reduccmp.test b/src/ltltest/reduccmp.test
index 417da8d2f..ef5117699 100755
--- a/src/ltltest/reduccmp.test
+++ b/src/ltltest/reduccmp.test
@@ -1,8 +1,9 @@
 #! /bin/sh
-# Copyright (C) 2009, 2010, 2011, 2012 Laboratoire de Recherche et
-# Developpement de l'Epita (LRDE).
+# -*- coding: utf-8 -*-
+# Copyright (C) 2009, 2010, 2011, 2012, 2013 Laboratoire de Recherche
+# et Developpement de l'Epita (LRDE).
 # Copyright (C) 2004, 2006 Laboratoire d'Informatique de Paris 6 (LIP6),
-# département Systèmes Répartis Coopératifs (SRC), Université Pierre
+# département Systèmes Répartis Coopératifs (SRC), Université Pierre
 # et Marie Curie.
 #
 # This file is part of Spot, a model checking library.
@@ -98,6 +99,13 @@ for x in ../reduccmp ../reductaustr; do
      run 0 $x 'FX(a)' 'XF(a)'
      run 0 $x 'GX(a)' 'XG(a)'
 
+     run 0 $x 'GF(a | Xb)' 'GF(a | b)'
+     run 0 $x 'GF(a | Fb)' 'GF(a | b)'
+     run 0 $x 'GF(Xa | Fb)' 'GF(a | b)'
+     run 0 $x 'FG(a & Xb)' 'FG(a & b)'
+     run 0 $x 'FG(a & Gb)' 'FG(a & b)'
+     run 0 $x 'FG(Xa & Gb)' 'FG(a & b)'
+
      run 0 $x 'X(a) U X(b)' 'X(a U b)'
      run 0 $x 'X(a) R X(b)' 'X(a R b)'
      run 0 $x 'Xa & Xb' 'X(a & b)'
diff --git a/src/ltlvisit/simplify.cc b/src/ltlvisit/simplify.cc
index 1282cf765..6a572a510 100644
--- a/src/ltlvisit/simplify.cc
+++ b/src/ltlvisit/simplify.cc
@@ -1,5 +1,6 @@
-// Copyright (C) 2011, 2012 Laboratoire de Recherche et Developpement
-// de l'Epita (LRDE).
+// -*- coding: utf-8 -*-
+// Copyright (C) 2011, 2012, 2013 Laboratoire de Recherche et
+// Developpement de l'Epita (LRDE).
 //
 // This file is part of Spot, a model checking library.
 //
@@ -1184,6 +1185,47 @@ namespace spot
 		      result_ = recurse_destroy(res);
 		      return;
 		    }
+
+		  // FG(a & Xb) = FG(a & b)
+		  // FG(a & Gb) = FG(a & b)
+		  if (const unop* g = is_G(result_))
+		    if (const multop* m = is_And(g->child()))
+		      if (!m->is_boolean())
+			{
+			  m->clone();
+			  mospliter s(mospliter::Strip_G | mospliter::Strip_X,
+				      m, c_);
+			  if (!s.res_G->empty() || !s.res_X->empty())
+			    {
+			      result_->destroy();
+			      s.res_other->insert(s.res_other->begin(),
+						  s.res_G->begin(),
+						  s.res_G->end());
+			      delete s.res_G;
+			      s.res_other->insert(s.res_other->begin(),
+						  s.res_X->begin(),
+						  s.res_X->end());
+			      delete s.res_X;
+			      const formula* in =
+				multop::instance(multop::And, s.res_other);
+			      result_ =
+				recurse_destroy(unop_unop(unop::F, unop::G,
+							  in));
+			      return;
+			    }
+			  else
+			    {
+			      for (multop::vec::iterator j =
+				     s.res_other->begin();
+				   j != s.res_other->end(); ++j)
+				if (*j)
+				  (*j)->destroy();
+			      delete s.res_other;
+			      delete s.res_G;
+			      delete s.res_X;
+			      // and continue...
+			    }
+			}
 		}
 
 	      // if Fa => a, keep a.
@@ -1288,6 +1330,47 @@ namespace spot
 			  result_ = mo;
 			}
 		    }
+
+		  // GF(a | Xb) = GF(a | b)
+		  // GF(a | Fb) = GF(a | b)
+		  if (const unop* f = is_F(result_))
+		    if (const multop* m = is_Or(f->child()))
+		      if (!m->is_boolean())
+			{
+			  m->clone();
+			  mospliter s(mospliter::Strip_F | mospliter::Strip_X,
+				      m, c_);
+			  if (!s.res_F->empty() || !s.res_X->empty())
+			    {
+			      result_->destroy();
+			      s.res_other->insert(s.res_other->begin(),
+						  s.res_F->begin(),
+						  s.res_F->end());
+			      delete s.res_F;
+			      s.res_other->insert(s.res_other->begin(),
+						  s.res_X->begin(),
+						  s.res_X->end());
+			      delete s.res_X;
+			      const formula* in =
+				multop::instance(multop::Or, s.res_other);
+			      result_ =
+				recurse_destroy(unop_unop(unop::G, unop::F,
+							  in));
+			      return;
+			    }
+			  else
+			    {
+			      for (multop::vec::iterator j =
+				     s.res_other->begin();
+				   j != s.res_other->end(); ++j)
+				if (*j)
+				  (*j)->destroy();
+			      delete s.res_other;
+			      delete s.res_F;
+			      delete s.res_X;
+			      // and continue...
+			    }
+			}
 		}
 	      // if a => Ga, keep a.
 	      if (opt_.containment_checks_stronger