Aard custom (#49)

* Merge changes to docs

* Fix typo

* Correct SUMMARY so it compiles; update .gitignore

* Clean up statements.md

Make syntax and notation consistent with Rust source code.

* Fix statements for Merkle trees and compound types

* First draft of custom statements and small updates to signedpod.md

* Update book/src/merkletree.md

Co-authored-by: Ahmad Afuni <root@ahmadafuni.com>

* merklestatements correct typo

Co-authored-by: Ahmad Afuni <root@ahmadafuni.com>

* add todo for gadget ids

Co-authored-by: Ahmad Afuni <root@ahmadafuni.com>

* Separate out custom statements version 1

* More details on custom statements version 1

* new file custom2

* Partial draft of version 2

* First draft of version 2 spec, it's kind of a mess

* Another version of the custom predicates spec

* Update book/src/custom2.md

Co-authored-by: Eduard S. <eduardsanou@posteo.net>

* Simple example of deduction rule applied in circuit

* Implement Edu's comments on custom predicates

* Backend predicates must be defined in groups

* Add more examples

* Two diff statements using same constant

* Remove deprecated example

---------

Co-authored-by: Ahmad Afuni <root@ahmadafuni.com>
Co-authored-by: Eduard S. <eduardsanou@posteo.net>

book/src/examples.md

@@ -136,3 +136,96 @@ statement is_great_boy(great_boy: PubKey, good_boy_issuers: MerkleTree):
          # good boy 0 != good boy 1
          - neq(friend_pod_0.signer, friend_pod_1.signer)
 ``` 
+
+## Attested GreatBoy
+
+An Attested Great Boy Pod is like a Great Boy Pod, but the names of the signers are revealed.
+
+```
+statement is_great_boy(great_boy: PubKey, friend0: String, friend1: String, good_boy_issuers: MerkleTree):
+   - OR():
+      - AND(friend_pod_0: Pod, friend_pod_1: Pod):
+         # Two good boys consider this user their friend
+         - is_friend(friend_pod_0.signer, great_boy)
+         - is_friend(friend_pod_1.signer, great_boy)
+         # good boy 0 != good boy 1
+         - neq(friend_pod_0.signer, friend_pod_1.signer)
+         # publicize signer names
+         - value_of(friend_pod_0.name, friend0)
+         - value_of(friend_pod_1.name, friend1)
+``` 
+
+To produce a Great Boy Pod, you need two Friend Pods, `friend_pod0` and `friend_pod1`, each of which reveals its `signer`.
+
+## Tracking PodIDs: Posts and comments
+
+The goal of this example is to model a social network, where posts and comments are pods.
+
+A Post is a signature pod with the following fields:
+```
+content: String
+poster: String
+signer: PubKey
+timestamp: Int
+```
+
+A Comment is a signature pod with the following fields:
+```
+content: String
+referenced_post: PodID
+signer: PubKey
+timestamp: Int
+```
+
+A post is popular if it has at least two comments from different signers.
+
+```
+statement is_popular(post: PodID):
+   - AND():
+      - IsEqual(comment1.referenced_post, post)
+      - IsEqual(comment2.referenced_post, post)
+      - NotEqual(comment1.signer, comment2.signer)
+```
+
+## Multiple people over 18
+
+Suppose I want to prove that two different people are over 18, and a third person is under 18, using the custom predicates `over_18` and `under_18`.
+```
+statement over_18(age):
+   - AND():
+      - ValueOf(eighteen, 18)
+      - GEq(age, eighteen)
+```
+
+```
+statement under_18(age):
+   - AND():
+      - ValueOf(eighteen, 18)
+      - Lt(age, eighteen)
+```
+
+With wildcards:
+```
+statement over_18(*1, *2):
+   - AND():
+      - ValueOf(*3, *4, 18)
+      - GEq(*1, *2, *3, *4)
+```
+
+Maybe I have two input pods `gov_id1` and `gov_id2`, and I want to prove that these pods refer to two different people, both of whom are over 18; and a third pods `gov_id3` refers to someone under 18.  So in my public output statements, I want to have:
+```
+IsUnequal(gov_id1.name, gov_id2.name)
+over_18(gov_id1.age)
+over_18(gov_id2.age)
+under_18(gov_id3.age).
+```
+
+I would prove this with the following sequence of deductions:
+| Statement | Reason |
+| --- | --- |
+| ValueOf(local_eighteen, 18) | (new entry) |
+| over_18(gov_id1.age) | over_18, <br> *1 = _SELF, <br> *2 = "local_eighteen", <br> *3 = gov_id1, <br> *4 = "age" |
+| over_18(gov_id2.age) | over_18, <br> *1 = _SELF, <br> *2 = "local_eighteen", <br> *3 = gov_id2, <br> *4 = "age" |
+| under_18(gov_id3.age) | under_18, <br> *1 = _SELF, <br> *2 = "local_eighteen", <br> *3 = gov_id3, <br> *4 = "age" |
+| IsUnequal(gov_id1.name, gov_id2.name) | (is unequal from entries) |
+