diff --git a/Graph-notes-copy2.vue b/Graph-notes-copy2.vue
index 340d24d..a21ea07 100644
--- a/Graph-notes-copy2.vue
+++ b/Graph-notes-copy2.vue
@@ -1,14 +1,14 @@
-<!-- Tufts VUE 3.3.0 concept-map (Graph-notes-copy2.vue) 2017-03-19 -->
+<!-- Tufts VUE 3.3.0 concept-map (Graph-notes-copy2.vue) 2017-03-23 -->
 <!-- Tufts VUE: http://vue.tufts.edu/ -->
 <!-- Do Not Remove: VUE mapping @version(1.1) jar:file:/nix/store/z92y35qgs6g3cvvh0i4f14mg5n47zvvi-vue-3.3.0/share/vue/vue.jar!/tufts/vue/resources/lw_mapping_1_1.xml -->
-<!-- Do Not Remove: Saved date Sun Mar 19 16:06:34 CET 2017 by georges on platform Linux 4.4.40 in JVM 1.8.0_122-04 -->
+<!-- Do Not Remove: Saved date Thu Mar 23 01:54:28 CET 2017 by georges on platform Linux 4.4.40 in JVM 1.8.0_122-04 -->
 <!-- Do Not Remove: Saving version @(#)VUE: built October 8 2015 at 1724 by tomadm on Linux 2.6.32-504.23.4.el6.x86_64 i386 JVM 1.7.0_21-b11(bits=32) -->
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-    <resource referenceCreated="1489935994403" size="216026"
+    <resource referenceCreated="1490230468578" size="216025"
         spec="/home/georges/phc/racket-packages/phc-graph/Graph-notes-copy2.vue"
         type="1" xsi:type="URLResource">
         <title>Graph-notes-copy2.vue</title>
@@ -3913,8 +3913,8 @@
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-    <userZoom>1.0</userZoom>
-    <userOrigin x="-1194.022" y="-194.81644"/>
+    <userZoom>0.75</userZoom>
+    <userOrigin x="-1182.522" y="-189.81644"/>
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diff --git a/free-identifier-tree-equal.rkt b/free-identifier-tree-equal.rkt
index d48412d..9650d57 100644
--- a/free-identifier-tree-equal.rkt
+++ b/free-identifier-tree-equal.rkt
@@ -2,7 +2,7 @@
 
 (require racket/struct
          ;; TODO: move delay-pure/private/immutable-struct to a separate package
-         delay-pure/private/immutable-struct) ;; for immutable-struct? below.
+         phc-toolkit/untyped/meta-struct) ;; for immutable-struct? below.
 
 (provide free-id-tree=?
          free-id-tree-hash-code
@@ -31,7 +31,7 @@
                            (cons/c isyntax isyntax)
                            (vectorof isyntax #:immutable #t)
                            (syntax/c isyntax)
-                           (and/c immutable-struct?
+                           (and/c struct-instance-is-immutable?
                                   (λ (v)
                                     (andmap isyntax/c (struct->list v)))))))
 
diff --git a/invariants-phantom.hl.rkt b/invariants-phantom.hl.rkt
index 49199f6..200335d 100644
--- a/invariants-phantom.hl.rkt
+++ b/invariants-phantom.hl.rkt
@@ -15,12 +15,12 @@
 
 @section{Introduction}
 
-The cautious compiler writer will no doubt want to check that the graph used
-to represent the program verifies some structural properties. For example, the
-compiled language might not allow cycles between types. Another desirable
-property is that the @racket[in-method] field of an instruction points back to
-the method containing it. We will use this second property as a running
-example in this section.
+The cautious compiler writer will no doubt want to check that the Abstract
+Syntax Tree or Graph used to represent the program verifies some structural
+properties. For example, the compiled language might not allow cycles between
+types. Another desirable property is that the @racket[in-method] field of the
+node representing an instruction points back to the method containing it. We
+will use this second property as a running example in this section.
 
 @section{Implementation overview : subtyping, variance and phantom types}
 
@@ -341,6 +341,12 @@ having an empty union.
 
 @subsection{Structural (in)equality and (non-)membership invariants}
 
+@subsubsection{Invariants and their relationships}
+
+@itemlist[
+ @item{Paths, }
+ ]
+
 @subsubsection{Comparison operator tokens}
 
 We define some tokens which will be used to identify the operator which
@@ -377,17 +383,20 @@ relates two nodes in the graph.
        (define-type-expander Cycle
          (syntax-parser
            [(_ field:id … {~literal ↙} loop1:id … (target:id) loop2:id …)
-            #'(List* (NonTarget ε)
-                     (NonTarget 'field) …
-                     (Rec R (List* (NonTarget 'loop1) … ;(NonTarget 'loop1) …
-                                   (Target 'target) ;(NonTarget 'target)
-                                   (NonTarget 'loop2) … ;(NonTarget 'loop2) …
-                                   R)))]
+            #'(→ (List* (NonTarget ε)
+                        (NonTarget 'field) …
+                        (Rec R (List* (NonTarget 'loop1) … ;(NonTarget 'loop1) …
+                                      (Target 'target) ;(NonTarget 'target)
+                                      (U (List* (NonTarget 'loop2) … ;(NonTarget 'loop2) …
+                                                R)
+                                         Null)))) Void)]
            [(_ field … target)
-            ;; TODO: something special at the end?
-            #'(List (NonTarget ε) (NonTarget 'field) … (Target 'target))]
+            #'(→ (List (NonTarget ε)
+                       (NonTarget 'field)
+                       …
+                       (Target 'target)) Void)]
            [(_)
-            #'(List (Target ε))]))]
+            #'(→ (List (Target ε)) Void)]))]
 
 @;{@[
    
@@ -431,57 +440,23 @@ Two sorts of paths inside (in)equality constraints:
 
 @subsection{Putting it all together}
 
-@chunk[<check-a-stronger-b>
-       (define-syntax (check-a-stronger-than-b stx)
-         (syntax-case stx ()
-           [(_ stronger weaker)
-            (syntax/top-loc stx
-              (begin (check-ann (ann witness-value stronger)
-                                weaker)
-                     (check-not-tc
-                      (ann (ann witness-value weaker) stronger))))]))
-       
-       (define-syntax (check-a-same-as-b stx)
-         (syntax-case stx ()
-           [(_ a b)
-            (syntax/top-loc stx
-              (begin
-                (check-ann (ann witness-value a) b)
-                (check-ann (ann witness-value b) a)))]))]
-
 @chunk[<*>
        (require (for-syntax phc-toolkit/untyped
                             syntax/parse))
 
-       (provide Invariants ≡)
-       
+       ;; For testing:
+       (provide Invariants
+                ≡
+                inv≡
+                inv≢
+                Or
+                Target
+                NonTarget
+                ε
+                witness-value)
+
        <witness-value>
        <grouping-invariants>
        <cycles>
        <comparison-operators>
-       <≡>
-
-       (module+ test
-         (require phc-toolkit)
-         <check-a-stronger-b>
-
-         (ann witness-value (Invariants)) ;; No invariants
-         (ann witness-value (Invariants (≡ (_ a) (_ a b c))))
-
-         (check-a-stronger-than-b (Invariants (≡ (_ a) (_ a b c)))
-                                  (Invariants))
-
-         (check-a-same-as-b (Invariants (≡ (_ a) (_ a b c)))
-                         (Invariants (≡ (_ a b c) (_ a))))
-
-         (check-a-stronger-than-b (Invariants (≡ (_) (_ b c))
-                                              (≡ (_) (_ b d)))
-                                  (Invariants (≡ (_) (_ b c))))
-         (check-a-stronger-than-b (Invariants (≡ (_) (_ b d))
-                                              (≡ (_) (_ b c)))
-                                  (Invariants (≡ (_) (_ b c))))
-
-         (check-a-stronger-than-b (Invariants (≡ (_)
-                                                 (_ b d a b d ↙ a b (d))))
-                                  (Invariants (≡ (_)
-                                                 (_ b d ↙ a b (d))))))]
+       <≡>]
diff --git a/test/invariant-phantom/simple.rkt b/test/invariant-phantom/simple.rkt
new file mode 100644
index 0000000..f92a491
--- /dev/null
+++ b/test/invariant-phantom/simple.rkt
@@ -0,0 +1,37 @@
+#lang type-expander
+
+(require (lib "phc-graph/invariants-phantom.hl.rkt")
+         "util.rkt"
+         phc-toolkit)
+
+(check-ann witness-value (Invariants)) ;; No invariants
+(check-ann witness-value (Invariants (≡ (_ a) (_ a b c))))
+
+(check-a-stronger-than-b (Invariants (≡ (_ a) (_ a b c)))
+                         (Invariants))
+
+(check-a-same-as-b (Invariants (≡ (_ a) (_ a b c)))
+                   (Invariants (≡ (_ a b c) (_ a))))
+
+(check-a-stronger-than-b (Invariants (≡ (_) (_ b c))
+                                     (≡ (_) (_ b d)))
+                         (Invariants (≡ (_) (_ b c))))
+(check-a-stronger-than-b (Invariants (≡ (_) (_ b d))
+                                     (≡ (_) (_ b c)))
+                         (Invariants (≡ (_) (_ b c))))
+
+;; ∀ .b.d(.a.b.>d)* of length ≥ 5
+;; is stronger than
+;; ∀ .b.d(.a.b.>d)* of length ≥ 8
+;; as the elements of the latter are included in the former, but
+;; the first element (length = 5) is missing in the latter, so the
+;; former constrains more paths.
+(check-a-stronger-than-b (Invariants (≡ (_)
+                                        (_ b d ↙ a b (d))))
+                         (Invariants (≡ (_)
+                                        (_ b d a b d ↙ a b (d)))))
+
+(check-a-stronger-than-b (Invariants (≡ (_)
+                                        (_ a b c ↙ d (e))))
+                         (Invariants (≡ (_)
+                                        (_ a b c d e))))
\ No newline at end of file
diff --git a/test/invariant-phantom/util.rkt b/test/invariant-phantom/util.rkt
new file mode 100644
index 0000000..9f82749
--- /dev/null
+++ b/test/invariant-phantom/util.rkt
@@ -0,0 +1,25 @@
+#lang type-expander
+
+(provide check-a-same-as-b
+         check-a-stronger-than-b)
+
+(require phc-toolkit
+         (lib "phc-graph/invariants-phantom.hl.rkt")
+         (for-syntax phc-toolkit/untyped))
+
+(define-syntax (check-a-stronger-than-b stx)
+  (syntax-case stx ()
+    [(_ stronger weaker)
+     (syntax/top-loc stx
+       (begin (check-ann (ann witness-value stronger)
+                         weaker)
+              (check-not-tc
+               (ann (ann witness-value weaker) stronger))))]))
+       
+(define-syntax (check-a-same-as-b stx)
+  (syntax-case stx ()
+    [(_ a b)
+     (syntax/top-loc stx
+       (begin
+         (check-ann (ann witness-value a) b)
+         (check-ann (ann witness-value b) a)))]))
\ No newline at end of file