diff --git a/spot/twaalgos/sccfilter.cc b/spot/twaalgos/sccfilter.cc
index 1b5d452d6..65e022453 100644
--- a/spot/twaalgos/sccfilter.cc
+++ b/spot/twaalgos/sccfilter.cc
@@ -436,14 +436,20 @@ namespace spot
              scc_info* given_si)
   {
     twa_graph_ptr res;
+    scc_info* si = given_si;
     if (aut->prop_inherently_weak())
       {
+        // Create scc_info here, because we will need it to decide
+        // very-weakness.
+        if (!si)
+          si = new scc_info(aut, scc_info_options::TRACK_SUCCS
+                            | scc_info_options::TRACK_STATES_IF_FIN_USED);
         if (aut->acc().is_t() || aut->acc().is_co_buchi())
           res =
-            scc_filter_apply<state_filter<weak_filter<false>>>(aut, given_si);
+            scc_filter_apply<state_filter<weak_filter<false>>>(aut, si);
         else
           res =
-            scc_filter_apply<state_filter<weak_filter<true>>>(aut, given_si);
+            scc_filter_apply<state_filter<weak_filter<true>>>(aut, si);
       }
     else if (aut->acc().is_generalized_buchi())
       {
@@ -485,6 +491,10 @@ namespace spot
       {
         res->prop_weak(true);
         res->prop_state_acc(true);
+        if (si->scc_count() == aut->num_states())
+          res->prop_very_weak(true);
+        if (!given_si)
+          delete si;
       }
     return res;
   }
diff --git a/tests/core/dca2.test b/tests/core/dca2.test
index eef1d8168..74a24b2e7 100755
--- a/tests/core/dca2.test
+++ b/tests/core/dca2.test
@@ -67,8 +67,8 @@ while read l_f; do
         autfilt -q --acceptance-is=Streett-like and.hoa
         # Streett | Streett
         autfilt r.hoa --name="($l_f)|!($r_f)" --product-or=l.hoa -S > or.hoa
-        if ! grep -q ' weak' l.hoa; then
-            if ! grep -q ' weak' r.hoa; then
+        if ! grep -q '[ -]weak' l.hoa; then
+            if ! grep -q '[ -]weak' r.hoa; then
                 autfilt -q -v --acceptance-is=Streett-like or.hoa || exit 1
             fi
         fi
diff --git a/tests/core/monitor.test b/tests/core/monitor.test
index fc43c6b2c..26f5c5667 100755
--- a/tests/core/monitor.test
+++ b/tests/core/monitor.test
@@ -98,7 +98,7 @@ AP: 1 "a"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc complete
-properties: deterministic stutter-invariant weak
+properties: deterministic stutter-invariant very-weak
 --BODY--
 State: 0 {0}
 [0] 0
diff --git a/tests/core/parity2.test b/tests/core/parity2.test
index 40daa2cbc..6d40b7414 100755
--- a/tests/core/parity2.test
+++ b/tests/core/parity2.test
@@ -236,7 +236,7 @@ AP: 1 "a"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc stutter-invariant
-properties: weak
+properties: very-weak
 --BODY--
 State: 0
 [t] 0
@@ -270,7 +270,7 @@ AP: 1 "a"
 acc-name: Rabin 1
 Acceptance: 2 Fin(0) & Inf(1)
 properties: trans-labels explicit-labels state-acc stutter-invariant
-properties: weak
+properties: very-weak
 --BODY--
 State: 0
 [t] 0
@@ -304,7 +304,7 @@ AP: 1 "a"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc stutter-invariant
-properties: weak
+properties: very-weak
 --BODY--
 State: 0
 [t] 0
@@ -338,7 +338,7 @@ AP: 1 "a"
 acc-name: Streett 1
 Acceptance: 2 Fin(0) | Inf(1)
 properties: trans-labels explicit-labels state-acc colored
-properties: stutter-invariant weak
+properties: stutter-invariant very-weak
 --BODY--
 State: 0 {0}
 [t] 0
@@ -372,7 +372,7 @@ AP: 1 "a"
 acc-name: parity min odd 3
 Acceptance: 3 Fin(0) & (Inf(1) | Fin(2))
 properties: trans-labels explicit-labels state-acc colored
-properties: stutter-invariant weak
+properties: stutter-invariant very-weak
 --BODY--
 State: 0 {2}
 [t] 0
@@ -406,7 +406,7 @@ AP: 1 "a"
 acc-name: parity max even 3
 Acceptance: 3 Inf(2) | (Fin(1) & Inf(0))
 properties: trans-labels explicit-labels state-acc colored
-properties: stutter-invariant weak
+properties: stutter-invariant very-weak
 --BODY--
 State: 0 {1}
 [t] 0
diff --git a/tests/core/randomize.test b/tests/core/randomize.test
index de3059486..259f03d0b 100755
--- a/tests/core/randomize.test
+++ b/tests/core/randomize.test
@@ -33,7 +33,7 @@ AP: 4 "a" "b" "c" "d"
 acc-name: all
 Acceptance: 0 t
 properties: trans-labels explicit-labels state-acc stutter-invariant
-properties: weak
+properties: very-weak
 --BODY--
 State: 0
 [0] 1
diff --git a/tests/core/readsave.test b/tests/core/readsave.test
index d38e553f2..55c985170 100755
--- a/tests/core/readsave.test
+++ b/tests/core/readsave.test
@@ -779,7 +779,8 @@ Start: 1
 AP: 2 "a" "b"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
-properties: trans-labels explicit-labels state-acc deterministic weak
+properties: trans-labels explicit-labels state-acc deterministic
+properties: very-weak
 --BODY--
 State: 0
 [1] 2
@@ -810,8 +811,8 @@ AP: 2 "a" "b"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc deterministic
-properties: no-univ-branch unambiguous semi-deterministic weak
-properties: inherently-weak
+properties: no-univ-branch unambiguous semi-deterministic very-weak
+properties: weak inherently-weak
 --BODY--
 State: 0
 [1] 2
diff --git a/tests/core/remfin.test b/tests/core/remfin.test
index 821f9eb61..c3835d294 100755
--- a/tests/core/remfin.test
+++ b/tests/core/remfin.test
@@ -972,7 +972,7 @@ AP: 0
 acc-name: all
 Acceptance: 0 t
 properties: trans-labels explicit-labels state-acc deterministic
-properties: stutter-invariant weak
+properties: stutter-invariant very-weak
 --BODY--
 State: 0
 --END--
diff --git a/tests/core/sccsimpl.test b/tests/core/sccsimpl.test
index 7211244b5..c62989b13 100755
--- a/tests/core/sccsimpl.test
+++ b/tests/core/sccsimpl.test
@@ -1,6 +1,6 @@
 #!/bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2011, 2013, 2015 Laboratoire de Recherche et
+# Copyright (C) 2011, 2013, 2015, 2018 Laboratoire de Recherche et
 # Développement de l'Epita
 #
 # This file is part of Spot, a model checking library.
@@ -316,7 +316,7 @@ AP: 1 "p0"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc complete
-properties: deterministic weak
+properties: deterministic very-weak
 --BODY--
 State: 0 {0}
 [t] 0
@@ -331,7 +331,7 @@ AP: 1 "p0"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
 properties: trans-labels explicit-labels state-acc complete
-properties: deterministic weak
+properties: deterministic very-weak
 --BODY--
 State: 0 {0}
 [t] 0
diff --git a/tests/core/wdba2.test b/tests/core/wdba2.test
index 82ed64823..fb20d0a25 100755
--- a/tests/core/wdba2.test
+++ b/tests/core/wdba2.test
@@ -1,6 +1,6 @@
 #!/bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2012, 2014, 2015 Laboratoire de Recherche et
+# Copyright (C) 2012, 2014, 2015, 2018 Laboratoire de Recherche et
 # Développement de l'Epita (LRDE).
 #
 # This file is part of Spot, a model checking library.
@@ -67,7 +67,7 @@ Start: 2
 AP: 1 "p1"
 acc-name: Buchi
 Acceptance: 1 Inf(0)
-properties: implicit-labels state-acc complete deterministic weak
+properties: implicit-labels state-acc complete deterministic very-weak
 --BODY--
 State: 0 {0}
 0 0
diff --git a/tests/python/dualize.py b/tests/python/dualize.py
index abd3d2ddf..b919eaac0 100755
--- a/tests/python/dualize.py
+++ b/tests/python/dualize.py
@@ -98,7 +98,7 @@ AP: 1 "a"
 acc-name: co-Buchi
 Acceptance: 1 Fin(0)
 properties: trans-labels explicit-labels state-acc complete
-properties: deterministic stutter-invariant weak
+properties: deterministic stutter-invariant very-weak
 --BODY--
 State: 0 {0}
 [t] 0
diff --git a/tests/python/remfin.py b/tests/python/remfin.py
index 5acf3b173..d1bb3793e 100644
--- a/tests/python/remfin.py
+++ b/tests/python/remfin.py
@@ -27,7 +27,8 @@ Start: 0
 AP: 1 "a"
 acc-name: co-Buchi
 Acceptance: 1 Fin(0)
-properties: trans-labels explicit-labels state-acc deterministic weak
+properties: trans-labels explicit-labels state-acc deterministic
+properties: very-weak
 --BODY--
 State: 0 {0}
 [0] 1