some updates

src/starkex/cairo.md

@@ -2,9 +2,9 @@
 
 Verifying a SHARP Cairo proof on the Starkex contracts is done in a number of transactions:
 
-* 3 Verify Merkle transactions ([example](https://etherscan.io/tx/0x5ad19d4524e0d2f2281dd71b8b1030fca7131ce74b821b936f73df2cba9d65e5))
-* 8 Verify FRI transactions ([example](https://etherscan.io/tx/0xccc7446d9e5e14892496ee3956f0e9579c2f56b8e70441623aa283c302130201))
-* A bunch of Register Continuous Memory Page ([example](https://etherscan.io/tx/0x6862ef5e0ce7599124e7c81625130990a102f483dde292d76b8d869b7d280ea7))
+* 3 [Verify Merkle](https://etherscan.io/address/0x5899efea757e0dbd6d114b3375c23d7540f65fa4) transactions to verify Merkle paths ([example](https://etherscan.io/tx/0x5ad19d4524e0d2f2281dd71b8b1030fca7131ce74b821b936f73df2cba9d65e5))
+* 8 [Verify FRI](https://etherscan.io/address/0x3e6118da317f7a433031f03bb71ab870d87dd2dd) transactions to verify the FRI proofs ([example](https://etherscan.io/tx/0xccc7446d9e5e14892496ee3956f0e9579c2f56b8e70441623aa283c302130201))
+* A bunch of [Register Continuous Memory Page](https://etherscan.io/address/0xfd14567eaf9ba941cb8c8a94eec14831ca7fd1b4) transactions ([example](https://etherscan.io/tx/0x6862ef5e0ce7599124e7c81625130990a102f483dde292d76b8d869b7d280ea7))
 * A single Verify Proof and Register ([example](https://etherscan.io/tx/0x720571bcac39e6b973537d7dd2ba253072e6f82c634ce361de73769d026ce4a1))
 
 The reason the verification of a proof is split in multiple transactions is because proofs are too large, and the verification cost too much gas, to fit in a single transaction. 

src/starkex/facts.md

@@ -81,9 +81,9 @@ where `registerGpsFacts` is defined as:
 
 ## Example of checking if a fact is registered
 
-[Starknet]() is one user of SHARP, and as such their smart contract uses the fact registry.
+[Starknet](https://book.starknet.io/) is the main application making use of SHARP, and as such their smart contract uses the fact registry directly.
 
-The main function of Starknet is [`updateState()`](https://github.com/mimoo/starknet-contracts/blob/main/contracts/Starknet.sol#L176) which updates the state based on proofs that have been verified:
+The main function of Starknet is [`updateState()`](https://github.com/mimoo/starknet-contracts/blob/main/contracts/Starknet.sol#L176), which updates the state based on proofs that have been verified:
 
 ```js
     function updateState(
@@ -103,4 +103,31 @@ The main function of Starknet is [`updateState()`](https://github.com/mimoo/star
         );
 
         // TRUNCATED...
+```
+
+Another example we can look at is within a proof verification. As explained in [Verifying a Cairo proof](./cairo.md), a proof verification is split in multiple transactions.
+
+For example, Merkle membership proofs are verified in segregated transactions, and then the fact that they were verified is used within another execution:
+
+```js
+    function verifyMerkle(
+        uint256[] memory merkleView,
+        uint256[] memory initialMerkleQueue,
+        uint256 height,
+        uint256 expectedRoot
+    ) public {
+        // TRUNCATED...
+
+        bytes32 resRoot = verifyMerkle(channelPtr, merkleQueuePtr, bytes32(expectedRoot), nQueries);
+        bytes32 factHash;
+        assembly {
+            // Append the resulted root (should be the return value of verify) to dataToHashPtr.
+            mstore(dataToHashPtr, resRoot)
+            // Reset dataToHashPtr.
+            dataToHashPtr := add(channelPtr, 0x20)
+            factHash := keccak256(dataToHashPtr, add(mul(nQueries, 0x40), 0x20))
+        }
+
+        registerFact(factHash);
+    }
 ```

src/starkex/starkex.md

@@ -1,6 +1,9 @@
 # Starkex
 
-The starkex contract live in https://github.com/starkware-libs/starkex-contracts
+The starkex contracts are deployed at [0x6cB3EE90C50a38A0e4662bB7e7E6e40B91361BF6](https://etherscan.io/address/0x6cB3EE90C50a38A0e4662bB7e7E6e40B91361BF6#code) and sit behind the upgradable smart contract at [0x47312450b3ac8b5b8e247a6bb6d523e7605bdb60](https://etherscan.io/address/0x47312450b3ac8b5b8e247a6bb6d523e7605bdb60#readProxyContract).
+
+Somewhat outdated implementations live on Github at [starkware-libs/starkex-contracts](https://github.com/starkware-libs/starkex-contracts).
+
 They implement the [SHARP (SHARed Prover)](https://starkware.co/resource/joining-forces-sharp/) verifier, which allows us to verify SHARP proofs on Ethereum.
 
 > Note: A number of contracts refer to "GPS" which is the old name for SHARP (general-proving service).