diff --git a/Graph-notes-copy2.vue b/Graph-notes-copy2.vue
index e863db8..b7341f9 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) 2016-12-29 -->
+<!-- Tufts VUE 3.3.0 concept-map (Graph-notes-copy2.vue) 2016-12-30 -->
 <!-- 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 Thu Dec 29 22:27:01 CET 2016 by georges on platform Linux 4.4.38 in JVM 1.8.0_122-04 -->
+<!-- Do Not Remove: Saved date Fri Dec 30 22:07:05 CET 2016 by georges on platform Linux 4.4.38 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="1483132025205" size="204773"
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@@ -1046,8 +1046,8 @@
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@@ -2015,10 +2025,9 @@
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+        label="Define a wrapper, which may alter the input value and input types, and the output value and output types"
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diff --git a/info.rkt b/info.rkt
index 7f9c731..311610d 100644
--- a/info.rkt
+++ b/info.rkt
@@ -11,7 +11,8 @@
                "srfi-lite-lib"
                "delay-pure"
                "backport-template-pr1514"
-               "typed-map"))
+               "typed-map"
+               "scribble-lib"))
 (define build-deps '("scribble-lib"
                      "racket-doc"
                      "remember"
diff --git a/invariants-phantom.hl.rkt b/invariants-phantom.hl.rkt
index b9b0a9d..3ecfb9e 100644
--- a/invariants-phantom.hl.rkt
+++ b/invariants-phantom.hl.rkt
@@ -1,15 +1,19 @@
 #lang aful/unhygienic hyper-literate type-expander/lang
 
-@title[#;#:style #;(with-html5 manual-doc-style)
-       #:tag "invariants-phantom"
-       #:tag-prefix "phc-graph/invariants-phantom"]{Tracking checked contracts
+@require[scribble-math
+         scribble-enhanced/doc
+         "notations.rkt"]
+
+@title[#:style (with-html5 manual-doc-style)
+       #:tag "inv-phantom"
+       #:tag-prefix "phc-graph/inv-phantom"]{Tracking checked contracts
  via phantom types}
 
 @(chunks-toc-prefix
   '("(lib phc-graph/scribblings/phc-graph-implementation.scrbl)"
-    "phc-graph/invariants-phantom"))
+    "phc-graph/inv-phantom"))
 
-@section{Overview}
+@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
@@ -18,6 +22,8 @@ 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.
 
+@section{Implementation overview}
+
 It is possible to express with Typed/Racket that a @racket[Method] should
 contain a list of @racket[Instruction]s, and that @racket[Instruction]s should
 point to a @racket[Method]@note{We are not concerned here about the ability to
@@ -86,13 +92,80 @@ to wrap it in a struct type which is only accessible by the graph library:
        (code:comment "provide only the type, not the constructor or accessor")
        (provide Flag-Wrapper)]
 
-@; size → use a dummy function which errors when called
-@; (accepts an opaque token to prevent calling)
+We would like to be able to indicate that a graph node has validated several
+invariants. For that, we need a way to represent the type of a "set" of
+invariant witnesses. We also want some subtyping relationship between the
+sets: a set @${s₁} with more invariant witnesses should be a subtype of a
+subset @${s₂ ⊆ s₁}. We can order the invariant witnesses and use @racket[Rec]
+to build the type of a list of invariant witnesses, where some may be missing:
 
-@; Subtyping: multiple contracts with case→
+@chunk[<invariant-set-as-List+Rec>
+       (define-type At-Least-InvB+InvD
+         (Rec R₁ (U (Pairof Any R₁)
+                    (Pairof 'InvB (Rec R₂ (U (Pairof Any R₂)
+                                             (Pairof 'InvD (Listof Any))))))))]
 
-@; Subtyping: when C1 ⇒ C2,
-@; make the marker for C1 a subtype of the marker for C2
+@chunk[<invariant-set-as-List+Rec-use>
+       (ann '(InvA InvB InvC InvD InvE) At-Least-InvB+InvD)
+       (ann '(InvB InvD) At-Least-InvB+InvD)
+       (code:comment "Rejected, because it lacks 'InvD")
+       (code:comment "(ann '(InvB InvC InvE) At-Least-InvB+InvD)")
+       (code:comment "The elements must be in the right order,")
+       (code:comment "this would be rejected by the typechecker:")
+       (code:comment "(ann '(InvD InvB) At-Least-InvB+InvD)")]
+
+Another solution is to group the witnesses in an untagged union with
+@racket[U], and place it in a contravariant position:
+
+@chunk[<invariant-set-as-contravariant-U>
+       (define-type At-Least-InvB+InvD
+         (→ (U 'InvB 'InvD) Void))]
+
+This solution also has the advantage that the size of the run-time witness is
+constant, and does not depend on the number of checked contracts (unlike the
+representation using a list). In practice the function should never be called.
+It can however simply be implemented in a way which pleases the type checked
+as a function accepting anything and returning void.
+
+In addition to testifying that a graph node was checked against multiple,
+separate contracts, there might be some contracts which check stronger
+properties than others. A way to encode this relationship in the type system
+is to have subtyping relationships between the contract witnesses, so that
+@; TODO: get rid of the mathit
+@${\mathit{P}₁(x) ⇒ \mathit{P}₂(x) ⇒ \mathit{Inv}₁ @texsubtype \mathit{Inv}₂}:
+
+@chunk[<invariant-contract-subtyping>
+       (struct InvWeak ())
+       (struct InvStrong InvWeak ())]
+
+If the witnesses must appear in a contravariant position (when using
+@racket[U] to group them), the relationship must be reversed:
+
+@chunk[<invariant-contract-subtyping>
+       (struct InvStrongContra ())
+       (struct InvWeakContra InvStrongContra ())]
+
+Alternatively, it is possible to use a second contravariant position to
+reverse the subtyping relationship again:
+
+@chunk[<invariant-contract-subtyping>
+       (struct InvWeak ())
+       (struct InvStrong InvWeak ())
+
+       (define InvWeakContra (→ InvWeak Void))
+       (define InvStrongContra (→ InvStrong Void))]
+
+Finally, we note that the invariants should always be represented using a
+particular struct type, instead of using a symbol, so that name clashes are
+not a problem.
+
+@section{Types for some graph contracts}
+
+@chunk[<structural>
+       (≡ [a : Nd] (get a f1 f2))
+       (∈ [a : Nd] (get a f1 f2))
+       (∉ [a : Nd] (get a f1 f2))
+       (∉ [a : Nd] (get a (* f1 f2 f3 f4) (* f5 f6)))]
 
 @chunk[<*>
        (void)]
\ No newline at end of file
diff --git a/notations.rkt b/notations.rkt
new file mode 100644
index 0000000..e82ddf5
--- /dev/null
+++ b/notations.rkt
@@ -0,0 +1,7 @@
+#lang racket
+
+(require scribble/base)
+
+(provide texsubtype)
+
+(define texsubtype "<:")
\ No newline at end of file
diff --git a/test/test-flexible-with.rkt b/test/test-flexible-with.rkt
index 9b428e6..c4ab572 100644
--- a/test/test-flexible-with.rkt
+++ b/test/test-flexible-with.rkt
@@ -22,15 +22,13 @@
                         [struct struct-field …] …)))]))
 
 (gs bt-fields
-    257
+    16
     (a b c)
     [sab a b]
     [sbc b c]
     [sabc a b c])
 
-;(define-type btac (bt-fields a c))
-
-#|
+(define-type btac (bt-fields a c))
 
 (check-equal?:
  (~> (ann (with-c (sab→tree 1 2) 'nine)
@@ -78,5 +76,4 @@
  (call-with-values
   #λ(tree→sbc (without-a (with-c (sab→tree 1 2) 3)))
   list)
- '(2 3))
-|#
\ No newline at end of file
+ '(2 3))
\ No newline at end of file
diff --git a/times.rkt b/times.rkt.txt
similarity index 100%
rename from times.rkt
rename to times.rkt.txt