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   docs:
   - install
   - language
-
+- title: IVy by example
+  docs:
+  - examples/client_server_example
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+---
+layout: page
+title: Invariants
+---
+
+The most basic technique for proving correctness in IVy is to
+construct an inductive invariant. IVy makes this easier by providing
+tools to visualize the failures of inductive proofs and suggest
+possible refinements to the proof.
+
+An *safety invariant* is a formula or a set of formulas that has the
+following properties:
+
+- *initiation*: It is true in all initial states of the program.
+
+- *safety*: If it is true in a program state, then no exported action
+can cause an assertion failure starting in that state.
+
+- *consecution*: If it is true in a state, then after executing any
+exported action, the formula remains true.
+
+In this example, we will use IVy's CTI method. CTI stands for
+"counter-example to induction". If one of the above conditions fails,
+IVy tries to construct a *simple* example of the failure. We can
+attempt to correct the proof by *generalizing* from this
+counter-example.
+
+# An abstract protocol model
+
+The following is a very abstract model of a protocol that establishes
+connections between clients and servers. Each server has a semaphore
+that is used to guarantee that at any time at most one client can be
+connected to the server.
+
+    #lang ivy1.5
+
+    type client
+    type server
+
+    relation link(X:client, Y:server)
+    relation semaphore(X:server)
+
+    init semaphore(W) & ~link(X,Y)
+
+    action connect(x:client,y:server) = {
+      assume semaphore(y);
+      link(x,y) := true;
+      semaphore(y) := false
+    }
+
+    action disconnect(x:client,y:server) = {
+      assume link(x,y);
+      link(x,y) := false;
+      semaphore(y) := true
+    }
+
+    export connect
+    export disconnect
+
+This program declares two types `client` and `server`. The state of
+the protocol model consists of two relations. The relation `link`
+tells us which clients are connected to which servers, while
+`semaphore` tells us which servers have their semaphore "up".
+
+The program exports two actions to the environment: `connect` and
+`disconnect`. The `connect` actions creates a link from client `x` to
+server `y`, putting the server's semaphore down. Notice that `connect`
+assumes the server's semaphore is initially up. The `disconnect`
+action removes a link and puts the semaphore up. The two `export`
+declarations at the end tell us that the environment may call
+`connect` and `disconnect` in arbitrary sequence, though it must obey
+the stated assumptions.
+
+## Safety and invariant conjectures
+
+A program is *safe* if the environment cannot call it in any way that
+causes an assertion to be false. There are various way to use
+assertions to specify desired safety properties of a program. A simple
+one is to add a test action that asserts some property of the program
+state. In the client/server example above, we might specify that no
+two distinct clients can be connected to a single server using the
+following test action:
+
+    action test = {
+      assert ~(X ~= Z & link(X,Y) & link(Z,Y))
+    }
+
+    export test
+
+The assertion is implicitly universally quantified over (distinct)
+clients `X` and `Z` and server `Y`. 
+
+# Discovering a safety invariant
+
+To try to construct a safety invariant for this program, we run IVy in
+its CTI mode. Here is the command line:
+
+    $ ivy ui=cti client_server_example.ivy
+
+This produces the following rather uninformative display:
+
+![IVy screenshot](images/client_server1.png)
+
+Since we didn't specify any conjectured invariant formulas in the
+input file, IVy starts with the empty set of conjectures. This
+trivially satisfies the initiation and consecution conditions, but
+probably won't satisfy the saftey condition. We'll try anyway and see
+what goes wrong. We select the "check inductiveness" operation, like this:
+
+![IVy screenshot](images/client_server2.png)
+
+Here's what IVY says:
+
+![IVy screenshot](images/client_server3.png)
+
+When we click OK, we see the following display:
+
+![IVy screenshot](images/client_server4.png)
+
+On the left-hand side of the display, we see a transition of the
+program from a state labeled `0` to a state labeled `1`. The action
+labeling the transition arrow can tell us seomthing about hwo we get
+from state `0` to state `1` (an in particular, how the assertion in
+our program fails). For the moment, though, let's concentrate on the
+right-hand side. Here, we see a representation of state `0`, the one just
+before the asertion failure. It shows one server (arbitrarily numbered `0`)
+and two clients (numbered `0` and `1`). The checkboxes on the right allow us to
+display further information about the state. For example, let's check the box
+next to `link(X,Y)` under the label `+`. Here's what we get:
+
+![IVy screenshot](images/client_server5.png)
+
+The red arrows show the link relationships between the clients and the
+server (notice that on the right, the relation `link(X,Y)` is in red).
+This is condition that causes our assertion to fail when the `test`
+action is executed. 
+
+This situation is unrealistic. That is, a cluster of two clients and a
+server as shown should never occur. We will conjecture that in fact
+thsi 'bad pattern' never occurs. To do this we select the `Gather` option
+from the `Conjecture` menu. When then see the following:
+
+![IVy screenshot](images/client_server6.png)
+
+IVy has collect three facts about the displayed state, shown under the
+heading 'Constraints'. These facts are a logical representation of the
+bad pattern we observed graphically. Two of them are obvious: the facts
+`link(0,0)` and `link(1,0)` say that both clients are linked to the server.
+Implicit in the grpahic, though, is a third fact: `0:client ~= 1:client`. 
+This says that `0` and `1` name distinct clients. 
+
+Also notice that the nodes and the arcs in the graph have been highlighted
+to indicate that they are all used in the listed facts.
+
+Since think that this particular pattern should never occur, we will generalize
+it to produce a *conjecture* about the program state. Choosing the `Strengthen` option
+from the `Conjecture` menu, we see:
+
+![IVy screenshot](images/client_server7.png)
+
+IVy is suggesting to add this fact to the list of conjectured invariants:
+
+    ForAll CL0:client, CL1:client, SE0:server. ~(link(CL0,SE0) & link(CL1,SE0) & CL0 ~= CL1
+
+This says that there is are no clients `CL0` and `CL1` and server
+`SE0` matching our bad pattern.  In other words, the bad pattern
+occurrs nowhere in our program state, no matter how many clients and
+servers there are. This is a simple generalization from our
+counterexample.
+
+We click OK, adding this formula to our (thus far empty) list of
+conjectures.  Of course this conjecture is a trivial one. It just says
+that the formula in our assertion is always true.
+
+We can now try checking inductiveness again with our new conjecture.
+Here is what IVy says:
+
+![IVy screenshot](images/client_server8.png)
+
+It is telling us that our new conjecture fails the consecution test.
+When we click OK, we see the counter-example to induction:
+
+![IVy screenshot](images/client_server9.png)
+
+IVy has already displayed the `link` relation, since it occurs in the
+conjecture.
+
+
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