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Aard custom (#49)

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* 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

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Co-authored-by: Ahmad Afuni <root@ahmadafuni.com>
Co-authored-by: Eduard S. <eduardsanou@posteo.net>
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# Custom statements and custom operations
Users of the POD system can introduce _custom predicates_ (previously called _custom statements_) to express complex logical relations not available in the built-in predicates. Every custom predicate is defined as the conjunction (AND) or disjunction (OR) of a small number of other statements.
When a custom predicate is introduced in a MainPod, it becomes available for use in that POD and all PODs that inherit[^inherit] from it.
On the frontend, a custom predicate is defined as a collection of conjunctions and disjunctions of statements. The definition can be recursive: the definition of a predicate can involve the predicate itself, or the definitions of several predicates can depend on each other.
At the backend level, every definition of a predicate is either a conjunction or a disjunction of statements. To convert a frontend custom predicate to the backend, the middleware may need to introduce _sub-predicates_.
On the backend, custom predicates are defined in _groups_. A group can contain one or more custom predicates and their associated sub-predicates. Recursive definition is only possible within a group: the definition of a predicate in a group can only depend on previously existing predicates, itself, and other predicates in the same group.
## Custom predicates and their IDs
A custom predicate, like a built-in predicate, is identified by a _name_ on the front end and an _identifier_ on the back end. The identifier is a cryptographic hash of the definition of the group.
[^inherit]: What to call this? One POD "inherits" from another?