ligo/src/proto/alpha/docs/language.md

Michelson: the language of Smart Contracts in Tezos
===================================================

The language is stack based, with high level data types and primitives
and strict static type checking. Its design cherry picks traits from
several language families. Vigilant readers will notice direct
references to Forth, Scheme, ML and Cat.

A Michelson program is a series of instructions that are run in
sequence: each instruction receives as input the stack resulting of
the previous instruction, and rewrites it for the next one. The stack
contains both immediate values and heap allocated structures. All
values are immutable and garbage collected.

A Michelson program receives as input a single element stack
containing an input value and the contents of a storage space. It must
return a single element stack containing an output value and the new
contents of the storage space. Alternatively, a Michelson program can
fail, explicitly using a specific opcode, or because something went
wrong that could not be caught by the type system (e.g. division by
zero, gas exhaustion).

The types of the input, output and storage are fixed and monomorphic,
and the program is typechecked before being introduced into the
system. No smart contract execution can fail because an instruction
has been executed on a stack of unexpected length or contents.

This specification gives the complete instruction set, type system and
semantics of the language. It is meant as a precise reference manual,
not an easy introduction. Even though, some examples are provided at
the end of the document and can be read first or at the same time as
the specification.

Table of contents
-----------------

  * I - Semantics
  * II - Type system
  * III - Core data types
  * IV - Core instructions
  * V - Operations
  * VI - Domain specific data types
  * VII - Domain specific operations
  * VIII - Concrete syntax
  * IX - Examples
  * X - Full grammar
  * XI - Reference implementation

I - Semantics
-------------

This specification gives a detailed formal semantics of the Michelson
language. It explains in a symbolic way the computation performed by
the Michelson interpreter on a given program and initial stack to
produce the corresponding resulting stack. The Michelson interpreter
is a pure function: it only builds a result stack from the elements of
an initial one, without affecting its environment. This semantics is
then naturally given in what is called a big step form: a symbolic
definition of a recursive reference interpreter. This definition takes
the form of a list of rules that cover all the possible inputs of the
interpreter (program and stack), and describe the computation of the
corresponding resulting stacks.

### Rules form and selection

The rules have the main following form.

    > (syntax pattern) / (initial stack pattern) => (result stack pattern)
      iff (conditions)
      where (recursions)

The left hand side of the `=>` sign is used for selecting the
rule. Given a program and an initial stack, one (and only one) rule
can be selected using the following process. First, the toplevel
structure of the program must match the syntax pattern. This is quite
simple since there is only a few non trivial patterns to deal with
instruction sequences, and the rest is made of trivial pattern that
match one specific instruction. Then, the initial stack must match the
initial stack pattern. Finally, some rules add extra conditions over
the values in the stack that follow the `iff` keyword. Sometimes,
several rules may apply in a given context. In this case, the one that
appears first in this specification is to be selected. If no rule
applies, the result is equivalent to the one for the explicit `FAIL`
instruction. This case does not happen on well-typed programs, as
explained in the next section.

The right hand side describes the result of the interpreter if the
rule applies. It consists in a stack pattern, whose part are either
constants, or elements of the context (program and initial stack) that
have been named on the left hand side of the `=>` sign.

### Recursive rules (big step form)

Sometimes, the result of interpreting a program is derived from the
result of interpreting another one (as in conditionals or function
calls). In these cases, the rule contains a clause of the following
form.

    where (intermediate program) / (intermediate stack) => (partial result)

This means that this rules applies in case interpreting the
intermediate state on the left gives the pattern on the right.

The left hand sign of the `=>` sign is constructed from elements of
the initial state or other partial results, and the right hand side
identify parts that can be used to build the result stack of the rule.

If the partial result pattern does not actually match the result of
the interpretation, then the result of the whole rule is equivalent to
the one for the explicit `FAIL` instruction. Again, this case does not
happen on well-typed programs, as explained in the next section.

### Format of patterns

Code patterns are of one of the following syntactical forms.

  * `INSTR` (an uppercase identifier) is a simple instruction
    (e.g. `DROP`);
  * `INSTR (arg) ...` is a compound instruction,
    whose arguments can be code, data or type patterns
    (e.g. `PUSH nat 3`) ;
  * `{ (instr) ; ... }` is a possibly empty sequence of instructions,
    (e.g. `IF { SWAP ; DROP } { DROP }`),
    nested sequences can drop the braces ;
  * `name` is a pattern that matches any program and names a part of
    the matched program that can be used to build the result ;
  * `_` is a pattern that matches any instruction.

Stack patterns are of one of the following syntactical forms.

  * `[FAIL]` is the special failed state ;
  * `[]` is the empty stack ;
  * `(top) : (rest)` is a stack whose top element is matched by the
    data pattern `(top)` on the left, and whose remaining elements are
    matched by the stack pattern `(rest)` on the right
    (e.g. `x : y : rest`) ;
  * `name` is a pattern that matches any stack and names it in order
    to use it to build the result ;
  * `_` is a pattern that matches any stack.

Data patterns are of one of the following syntactical forms.

  * integer/natural number literals,
    (e.g. `3`) ;
  * string literals,
    (e.g. `"contents"`) ;
  * `Tag` (capitalized) is a symbolic constant,
    (e.g. `Unit`, `True`, `False`) ;
  * `(Tag (arg) ...)` tagged constructed data,
    (e.g. `(Pair 3 4)`) ;
  * a code pattern for first class code values ;
  * `name` to name a value in order to use it to build the result ;
  * `_` to match any value.

The domain of instruction names, symbolic constants and data
constructors is fixed by this specification. Michelson does not let
the programmer introduce its own types.

Be aware that the syntax used in the specification may differ a bit
from the concrete syntax, which is presented in Section VIII. In
particular, some instructions are annotated with types that are not
present in the concrete language because they are synthesized by the
typechecker.

### Shortcuts

Sometimes, it is easier to think (and shorter to write) in terms of
program rewriting than in terms of big step semantics. When it is the
case, and when both are equivalents, we write rules of the form:

    p / S   =>   S''
    where   p' / S'   =>   S''

using the following shortcut:

    p / S   =>   p' / S'

The concrete language also has some syntax sugar to group some common
sequences of operations as one. This is described in this
specification using a simple regular expression style recursive
instruction rewriting.

II - Introduction to the type system and notations
--------------------------------------------------

This specification describes a type system for Michelson. To make
things clear, in particular to readers that are not accustomed to
reading formal programming language specifications, it does not give a
typechecking or inference algorithm. It only gives an intentional
definition of what we consider to be well-typed programs. For each
syntactical form, it describes the stacks that are considered
well-typed inputs, and the resulting outputs.

The type system is sound, meaning that if a program can be given a
type, then if run on a well-typed input stack, the interpreter will
never apply an interpretation rule on a stack of unexpected length or
contents. Also, it will never reach a state where it cannot select an
appropriate rule to continue the execution. Well-typed programs do not
block, and do not go wrong.

### Type notations

The specification introduces notations for the types of values, terms
and stacks. Apart from a subset of value types that appear in the
form of type annotations in some places throughout the language, it is
important to understand that this type language only exists in the
specification.

A stack type can be written:

  * `[]` for the empty stack ;
  * `(top) : (rest)` for the stack whose first value has type `(top)`
    and queue has stack type `(rest)`.

Instructions, programs and primitives of the language are also typed,
their types are written:

    (type of stack before) -> (type of stack after)

The types of values in the stack are written:

  * `identifier` for a primitive data-type
    (e.g. `bool`),
  * `identifier (arg)` for a parametric data-type with one parameter type `(arg)`
    (e.g. `list nat`),
  * `identifier (arg) ...` for a parametric data-type with several parameters
    (e.g. `map string int`),
  * `[ (type of stack before) -> (type of stack after) ]` for a code quotation,
    (e.g. `[ int : int : [] -> int : [] ]`),
  * `lambda (arg) (ret)` is a shortcut for `[ (arg) : [] -> (ret) : [] ]`.

### Meta type variables

The typing rules introduce meta type variables. To be clear, this has
nothing to do with polymorphism, which Michelson does not have. These
variables only live at the specification level, and are used to
express the consistency between the parts of the program. For
instance, the typing rule for the `IF` construct introduces meta
variables to express that both branches must have the same type.

Here are the notations for meta type variables:

  * `'a` for a type variable,
  * `'A` for a stack type variable,
  * `_` for an anonymous type or stack type variable.

### Typing rules

The system is syntax directed, which means here that it defines a
single typing rule for each syntax construct. A typing rule restricts
the type of input stacks that are authorized for this syntax
construct, links the output type to the input type, and links both of
them to the subexpressions when needed, using meta type variables.

Typing rules are of the form:

    (syntax pattern)
    :: (type of stack before) -> (type of stack after) [rule-name]
       iff (premises)

Where premises are typing requirements over subprograms or values in
the stack, both of the form `(x) :: (type)`, meaning that value `(x)`
must have type `(type)`.

A program is shown well-typed if one can find an instance of a rule
that applies to the toplevel program expression, with all meta type
variables replaced by non variable type expressions, and of which all
type requirements in the premises can be proven well-typed in the
same manner. For the reader unfamiliar with formal type systems, this
is called building a typing derivation.

Here is an example typing derivation on a small program that computes
`(x+5)*10` for a given input `x`, obtained by instantiating the typing
rules for instructions `PUSH`, `ADD` and for the sequence, as found in
the next sections. When instantiating, we replace the `iff` with `by`.

    { PUSH nat 5 ; ADD ; PUSH nat 10 ; SWAP ; MUL }
    :: [ nat : [] -> nat : [] ]
       by { PUSH nat 5 ; ADD }
          :: [ nat : [] -> nat : [] ]
             by PUSH nat 5
                :: [ nat : [] -> nat : nat : [] ]
                   by 5 :: nat
            and ADD
                :: [ nat : nat : [] -> nat : [] ]
      and { PUSH nat 10 ; SWAP ; MUL }
          :: [ nat : [] -> nat : [] ]
             by PUSH nat 10
                :: [ nat : [] -> nat : nat : [] ]
                   by 10 :: nat
            and { SWAP ; MUL }
                :: [ nat : nat : [] -> nat : [] ]
                   by SWAP
                      :: [ nat : nat : [] -> nat : nat : [] ]
                  and MUL
                      :: [ nat : nat : [] -> nat : [] ]

Producing such a typing derivation can be done in a number of manners,
such as unification or abstract interpretation. In the implementation
of Michelson, this is done by performing a recursive symbolic
evaluation of the program on an abstract stack representing the input
type provided by the programmer, and checking that the resulting
symbolic stack is consistent with the expected result, also provided
by the programmer.

### Side note

As with most type systems, it is incomplete. There are programs that
cannot be given a type in this type system, yet that would not go
wrong if executed. This is a necessary compromise to make the type
system usable. Also, it is important to remember that the
implementation of Michelson does not accept as many programs as the
type system describes as well-typed. This is because the
implementation uses a simple single pass typechecking algorithm, and
does not handle any form of polymorphism.


III - Core data types and notations
-----------------------------------

   * `string`, `nat`, `int`:
     The core primitive constant types.

   * `bool`:
     The type for booleans whose values are `True` and `False`

   * `unit`:
     The type whose only value is `Unit`, to use as a placeholder when
     some result or parameter is non necessary. For instance, when the
     only goal of a contract is to update its storage.

   * `list (t)`:
     A single, immutable, homogeneous linked list, whose elements are
     of type `(t)`, and that we note `Nil` for the empty list or
     `(Cons (head) (tail))`.

   * `pair (l) (r)`:
     A pair of values `a` and `b` of types `(l)` and `(r)`, that we
     write `(Pair a b)`.

   * `option (t)`:
     Optional value of type `(t)` that we note `None` or `(Some v)`.

   * `or (l) (r)`:
     A union of two types: a value holding either a value `a` of type
     `(l)` or a value `b` of type `(r)`, that we write `(Left a)` or
     `(Right b)`.

   * `set (t)`:
     Immutable sets of values of type `(t)` that we note
     `(Set (item) ...)`.

   * `map (k) (t)`:
     Immutable maps from keys of type `(k)` of values of type `(t)`
     that we note `(Map (Item (key) (value)) ...)`.


IV - Core instructions
----------------------

### Control structures

   * `FAIL`:
     Explicitly abort the current program.

        :: _   ->  _

     This special instruction is callable in any context, since it
     does not use its input stack (first rule below), and makes the
     output useless since all subsequent instruction will simply
     ignore their usual semantics to propagate the failure up to the
     main result (second rule below). Its type is thus completely
     generic.

        > FAIL / _   =>  [FAIL]
        > _ / [FAIL]    =>   [FAIL]

   * `{ I ; C }`:
     Sequence.

        :: 'A   ->   'C
           iff   I :: [ 'A -> 'B ]
                 C :: [ 'B -> 'C ]

        > I ; C / SA    =>   SC
          where   I / SA   =>   SB
            and   C / SB   =>   SC

   * `IF bt bf`:
     Conditional branching.

        :: bool : 'A   ->   'B
           iff   bt :: [ 'A -> 'B ]
                 bf :: [ 'A -> 'B ]

        > IF bt bf / True : S    =>    bt / S
        > IF bt bf / False : S   =>    bf / S

   * `LOOP body`:
     A generic loop.

        :: bool : 'A   ->   'A
           iff   body :: [ 'A -> bool : 'A ]

        > LOOP body / True : S    =>    body ; LOOP body / S
        > LOOP body / False : S   =>    S

   * `DIP code`:
     Runs code protecting the top of the stack.

        :: 'b : 'A   ->   'b : 'C
           iff   code :: [ 'A -> 'C ]

        > DIP code / x : S   =>   x : S'
          where    code / S   =>   S'

   * `DII+P code`:
     A syntactic sugar for working deeper in the stack.

        > DII(\rest=I*)P code / S   =>   DIP (DI(\rest)P code) / S

   * `EXEC`:
     Execute a function from the stack.

        :: 'a : lambda 'a 'b : 'C   ->   'b : 'C

        > EXEC / a : f : S   =>   r : S
          where f / a : []   =>   r : []

### Stack operations

   * `DROP`:
     Drop the top element of the stack.

        :: _ : 'A   ->   'A

        > DROP / _ : S   =>   S

   * `DUP`:
     Duplicate the top of the stack.

        :: 'a : 'A   ->   'a : 'a : 'A

        > DUP / x : S   =>   x : x : S

   * `DUU+P`:
     A syntactic sugar for duplicating the `n`th element of the stack.

        > DUU(\rest=U*)P / S   =>   DIP (DU(\rest)P) ; SWAP / S

   * `SWAP`:
     Exchange the top two elements of the stack.

        :: 'a : 'b : 'A   ->   'b : 'a : 'A

        > SWAP / x : y : S   =>   y : x : S

   * `PUSH 'a x`:
     Push a constant value of a given type onto the stack.


        :: 'A   ->   'a : 'A
           iff   x :: 'a

        > PUSH 'a x / S   =>   x : S

   * `UNIT`:
     Push a unit value onto the stack.

        :: 'A   ->   unit : 'A

        > UNIT / S   =>   Unit : S

   * `LAMBDA 'a 'b code`:
     Push a lambda with given parameter and return types onto the stack.

        :: 'A ->  (lambda 'a 'b) : 'A

### Generic comparison

Comparison only works on a class of types that we call comparable.
A `COMPARE` operation is defined in an ad hoc way for each comparable
type, but the result of compare is always an `int`, which can in
turn be checked in a generic manner using the following
combinators. The result of `COMPARE` is `0` if the top two elements
of the stack are equal, negative if the first element in the stack
is less than the second, and positive otherwise.

   * `EQ`:
     Checks that the top of the stack EQuals zero.

        :: int : 'S   ->   bool : 'S

        > EQ ; C / 0 : S     =>   C / True : S
        > EQ ; C / _ : S     =>   C / False : S


   * `NEQ`:
     Checks that the top of the stack does Not EQual zero.

        :: int : 'S   ->   bool : 'S

        > NEQ ; C / 0 : S     =>   C / False : S
        > NEQ ; C / _ : S     =>   C / True : S

   * `LT`:
     Checks that the top of the stack is Less Than zero.

        :: int : 'S   ->   bool : 'S

        > LT ; C / v : S     =>   C / True : S   iff  v < 0
        > LT ; C / _ : S     =>   C / False : S

   * `GT`:
     Checks that the top of the stack is Greater Than zero.

        :: int : 'S   ->   bool : 'S

        > GT ; C / v : S     =>   C / True : S   iff  v > 0
        > GT ; C / _ : S     =>   C / False : S

   * `LE`:
     Checks that the top of the stack is Less Than of Equal to zero.

        :: int : 'S   ->   bool : 'S

        > LE ; C / v : S     =>   C / True : S   iff  v <= 0
        > LE ; C / _ : S     =>   C / False : S

   * `GE`:
     Checks that the top of the stack is Greater Than of Equal to zero.

        :: int : 'S   ->   bool : 'S

        > GE ; C / v : S     =>   C / True : S   iff  v >= 0
        > GE ; C / _ : S     =>   C / False : S

Syntactic sugar exists for merging `COMPARE` and comparison
combinators, and also for branching.

   * `CMP{EQ|NEQ|LT|GT|LE|GE}`

        > CMP(\op) ; C / S   =>   COMPARE ; (\op) ; C / S

   * `IF{EQ|NEQ|LT|GT|LE|GE} bt bf`

        > IF(\op) ; C / S   =>   (\op) ; IF bt bf ; C / S

   * `IFCMP{EQ|NEQ|LT|GT|LE|GE} bt bf`

        > IFCMP(\op) ; C / S   =>   COMPARE ; (\op) ; IF bt bf ; C / S


V - Operations
--------------

### Operations on booleans

   * `OR`

        :: bool : bool : 'S   ->   bool : 'S

        > OR ; C / x : y : S   =>   C / (x | y) : S

   * `AND`

        :: bool : bool : 'S   ->   bool : 'S

        > AND ; C / x : y : S   =>   C / (x & y) : S

   * `XOR`

        :: bool : bool : 'S   ->   bool : 'S

        > XOR ; C / x : y : S   =>   C / (x ^ y) : S

   * `NOT`

        :: bool : 'S   ->   bool : 'S

        > NOT ; C / x : S   =>   C / ~x : S

### Operations on integers and natural numbers

Integers and naturals are arbitrary-precision,
meaning the only size limit is fuel.

   * `NEG`

        :: int : 'S   ->   int : 'S
        :: nat : 'S   ->   int : 'S

        > NEG ; C / x : S   =>   C / -x : S

   * `ABS`

        :: int : 'S   ->   nat : 'S

        > ABS ; C / x : S   =>   C / abs (x) : S

   * `ADD`

        :: int : int : 'S   ->   int : 'S
        :: int : nat : 'S   ->   int : 'S
        :: nat : int : 'S   ->   int : 'S
        :: nat : nat : 'S   ->   nat : 'S

        > ADD ; C / x : y : S   =>   C / (x + y) : S

   * `SUB`

        :: int : int : 'S   ->   int : 'S
        :: int : nat : 'S   ->   int : 'S
        :: nat : int : 'S   ->   int : 'S
        :: nat : nat : 'S   ->   int : 'S

        > SUB ; C / x : y : S   =>   C / (x - y) : S

   * `MUL`

        :: int : int : 'S   ->   int : 'S
        :: int : nat : 'S   ->   int : 'S
        :: nat : int : 'S   ->   int : 'S
        :: nat : nat : 'S   ->   nat : 'S

        > MUL ; C / x : y : S   =>   C / (x * y) : S

   * `EDIV`
     Perform Euclidian division

        :: int : int : 'S   ->   option (pair int nat) : 'S
        :: int : nat : 'S   ->   option (pair int nat) : 'S
        :: nat : int : 'S   ->   option (pair int nat) : 'S
        :: nat : nat : 'S   ->   option (pair nat nat) : 'S

        > EDIV ; C / x : 0 : S   =>   C / None
        > EDIV ; C / x : y : S   =>   C / Some (Pair (x / y) (x % y)) : S

Bitwise logical operators are also available on unsigned integers.

   * `OR`

        :: nat : nat : 'S   ->   nat : 'S

        > OR ; C / x : y : S   =>   C / (x | y) : S

   * `AND`

        :: nat : nat : 'S   ->   nat : 'S

        > AND ; C / x : y : S   =>   C / (x & y) : S

   * `XOR`

        :: nat : nat : 'S   ->   nat : 'S

        > XOR ; C / x : y : S   =>   C / (x ^ y) : S

   * `NOT`
     The return type of `NOT` is an `int` and not a `nat`.
     This is because the sign is also negated.
     The resulting integer is computed using two's complement.
     For instance, the boolean negation of `0` is `-1`.

        :: nat : 'S   ->   int : 'S
        :: int : 'S   ->   int : 'S

        > NOT ; C / x : S   =>   C / ~x : S

   * `LSL`

        :: nat : nat : 'S   ->   nat : 'S

        > LSL ; C / x : s : S   =>   C / (x << s) : S
          iff   s <= 256
        > LSL ; C / x : s : S   =>   [FAIL]

   * `LSR`

        :: nat : nat : 'S   ->   nat : 'S

        > LSR ; C / x : s : S   =>   C / (x >>> s) : S

   * `COMPARE`:
     Integer/natural comparison

        :: int : int : 'S   ->   int : 'S
        :: nat : nat : 'S   ->   int : 'S

        > COMPARE ; C / x : y : S   =>   C / -1 : S  iff x < y
        > COMPARE ; C / x : y : S   =>   C / 0 : S   iff x = y
        > COMPARE ; C / x : y : S   =>   C / 1 : S   iff x > y

### Operations on strings

Strings are mostly used for naming things without having to rely on
external ID databases. So what can be done is basically use string
constants as is, concatenate them and use them as keys.

   * `CONCAT`:
     String concatenation.

        :: string : string : 'S   -> string : 'S

   * `COMPARE`:
     Lexicographic comparison.

        :: string : string : 'S   ->   int : 'S

### Operations on pairs

   * `PAIR`:
     Build a pair from the stack's top two elements.

        :: 'a : 'b : 'S   ->   pair 'a 'b : 'S

        > PAIR ; C / a : b : S   =>   C / (Pair a b) : S

   * `P(A*AI)+R`:
     A syntactic sugar for building nested pairs in bulk.

        > P(\fst=A*)AI(\rest=(A*AI)+)R ; C / S  =>  P(\fst)AIR ; P(\rest)R ; C / S
        > PA(\rest=A*)AIR ; C / S  =>   DIP (P(\rest)AIR) ; C / S

   * `CAR`:
     Access the left part of a pair.

        :: pair 'a _ : 'S   ->   'a : 'S

        > Car ; C / (Pair a _) : S   =>   C / a : S

   * `CDR`:
     Access the right part of a pair.

        :: pair _ 'b : 'S   ->   'b : 'S

        > Car ; C / (Pair _ b) : S   =>   C / b : S

   * `C[AD]+R`:
     A syntactic sugar for accessing fields in nested pairs.

        > CA(\rest=[AD]+)R ; C / S   =>   CAR ; C(\rest)R ; C / S
        > CD(\rest=[AD]+)R ; C / S   =>   CDR ; C(\rest)R ; C / S

### Operations on sets

   * `EMPTY_SET 'elt`:
     Build a new, empty set for elements of a given type.

        :: 'S   ->   set 'elt : 'S

     The `'elt` type must be comparable (the `COMPARE` primitive must
     be defined over it).

   * `MEM`:
     Check for the presence of an element in a set.

        :: 'elt : set 'elt : 'S   ->  bool : 'S

   * `UPDATE`:
     Inserts or removes an element in a set, replacing a previous value.

        :: 'elt : bool : set 'elt : 'S   ->   set 'elt : 'S

   * `REDUCE`:
     Apply a function on a set passing the result of each
     application to the next one and return the last.

        :: lambda (pair 'elt * 'b) 'b : set 'elt : 'b : 'S   ->   'b : 'S

   * `SIZE`:
     Get the cardinality of the set.

        :: set 'elt : 'S -> nat : 'S

### Operations on maps

   * `EMPTY_MAP 'key 'val`:
     Build a new, empty map.

     The `'key` type must be comparable (the `COMPARE` primitive must be
     defined over it).

        :: 'S -> map 'key 'val : 'S

   * `GET`:
     Access an element in a map, returns an optional value to be
     checked with `IF_SOME`.

        :: 'key : map 'key 'val : 'S   ->   option 'val : 'S

   * `MEM`:
     Check for the presence of an element in a map.

        :: 'key : map 'key 'val : 'S   ->  bool : 'S

   * `UPDATE`:
     Assign or remove an element in a map.

        :: 'key : option 'val : map 'key 'val : 'S   ->   map 'key 'val : 'S

   * `MAP`:
     Apply a function on a map and return the map of results under
     the same bindings.

        :: lambda (pair 'key 'val) 'b : map 'key 'val : 'S   ->   map 'key 'b : 'S

   * `REDUCE`:
     Apply a function on a map passing the result of each
     application to the next one and return the last.

        :: lambda (pair (pair 'key 'val) 'b) 'b : map 'key 'val : 'b : 'S   ->   'b : 'S

   * `SIZE`:
     Get the cardinality of the map.

        :: map 'key 'val : 'S -> nat : 'S

### Operations on optional values

   * `SOME`:
     Pack a present optional value.

        :: 'a : 'S   ->   'a? : 'S

        > SOME ; C / v :: S   =>   C / (Some v) :: S

   * `NONE 'a`:
     The absent optional value.

        :: 'S   ->   'a? : 'S

        > NONE ; C / v :: S   =>   C / None :: S

   * `IF_NONE bt bf`:
     Inspect an optional value.

        :: 'a? : 'S   ->   'b : 'S
           iff   bt :: [ 'S -> 'b : 'S]
                 bf :: [ 'a : 'S -> 'b : 'S]

        > IF_NONE ; C / (None) : S   =>    bt ; C / S
        > IF_NONE ; C / (Some a) : S    =>    bf ; C / a : S


### Operations on unions

   * `LEFT 'b`:
     Pack a value in a union (left case).

        :: 'a : 'S   ->   or 'a 'b : 'S

        > LEFT ; C / v :: S   =>   C / (Left v) :: S

   * `RIGHT 'a`:
     Pack a value in a union (right case).

        :: 'b : 'S   ->   or 'a 'b : 'S

        > RIGHT ; C / v :: S   =>   C / (Right v) :: S

   * `IF_LEFT bt bf`:
     Inspect an optional value.

        :: or 'a 'b : 'S   ->   'c : 'S
           iff   bt :: [ 'a : 'S -> 'c : 'S]
                 bf :: [ 'b : 'S -> 'c : 'S]

        > IF_LEFT ; C / (Left a) : S    =>    bt ; C / a : S
        > IF_LEFT ; C / (Right b) : S   =>    bf ; C / b : S

### Operations on lists

   * `CONS`:
     Prepend an element to a list.

        :: 'a : list 'a : 'S   ->   list 'a : 'S

        > CONS ; C / a : l : S   =>   C / (Cons a l) : S

   * `NIL 'a`:
     The empty list.

        :: 'S   ->   list 'a : 'S

        > NIL ; C / S   =>   C / Nil : S

   * `IF_CONS bt bf`:
     Inspect an optional value.

        :: list 'a : 'S   ->   'b : 'S
           iff   bt :: [ 'a : list 'a : 'S -> 'b : 'S]
                 bf :: [ 'S -> 'b : 'S]

        > IF_CONS ; C / (Cons a rest) : S   =>    bt ; C / a : rest : S
        > IF_CONS ; C / Nil : S   =>    bf ; C / S

   * `MAP`:
     Apply a function on a list from left to right and
     return the list of results in the same order.

        :: lambda 'a 'b : list 'a : 'S -> list 'b : 'S

   * `REDUCE`:
     Apply a function on a list from left to right
     passing the result of each application to the next one
     and return the last.

        :: lambda (pair 'a 'b) 'b : list 'a : 'b : 'S -> 'b : 'S


VI - Domain specific data types
-------------------------------

   * `timestamp`:
     Dates in the real world.

   * `tez`:
     A specific type for manipulating tokens.

   * `contract 'param 'result`:
     A contract, with the type of its code.

   * `key`:
     A public cryptography key.

   * `signature`:
     A cryptographic signature.


VII - Domain specific operations
--------------------------------

### Operations on timestamps

Timestamp immediates can be obtained by the `NOW` operation, or
retrieved from script parameters or globals. The only valid operations
are the addition of a (positive) number of seconds and the comparison.

   * `ADD`
     Increment / decrement a timestamp of the given number of seconds.

        :: timestamp : nat : 'S -> timestamp : 'S
        :: nat : timestamp : 'S -> timestamp : 'S

        > ADD ; C / seconds : nat (t) : S   =>   C / (seconds + t) : S
        > ADD ; C / nat (t) : seconds : S   =>   C / (t + seconds) : S

   * `COMPARE`:
     Timestamp comparison.

        :: timestamp : timestamp : 'S   ->   int : 'S

### Operations on Tez

Tez are internally represented by a 64 bit signed integer.
There are restrictions to prevent creating a negative amount of tez.
Operations are limited to prevent overflow and mixing them with
other numerical types by mistake. They are also mandatorily checked
for under/overflows.

   * `ADD`:

        :: tez : tez : 'S   ->   tez : 'S

        > ADD ; C / x : y : S   =>   [FAIL]   on overflow
        > ADD ; C / x : y : S   =>   C / (x + y) : S

   * `SUB`:

        :: tez : tez : 'S   ->   tez : 'S

        > SUB ; C / x : y : S   =>   [FAIL]   iff   x < y
        > SUB ; C / x : y : S   =>   C / (x - y) : S

   * `MUL`

        :: tez : nat : 'S   ->   tez : 'S
        :: nat : tez : 'S   ->   tez : 'S

        > MUL ; C / x : y : S   =>   [FAIL]   on overflow
        > MUL ; C / x : y : S   =>   C / (x * y) : S

   * `EDIV`

        :: tez : nat : 'S   ->   option (pair tez tez) : 'S
        :: tez : tez : 'S   ->   option (pair nat tez) : 'S

        > EDIV ; C / x : 0 : S   =>   C / None
        > EDIV ; C / x : y : S   =>   C / Some (Pair (x / y) (x % y)) : S

   * `COMPARE`:

        :: tez : tez : 'S   ->   int : 'S

### Operations on contracts

   * `MANAGER`:
     Access the manager of a contract.

        :: contract 'p 'r : 'S   ->   key : 'S

   * `CREATE_CONTRACT`:
     Forge a new contract.


        :: key : key? : bool : bool : tez : lambda (pair 'p 'g) (pair 'r 'g) : 'g : 'S
           -> contract 'p 'r : 'S

     As with non code-emitted originations the
     contract code takes as argument the transferred amount plus an
     ad-hoc argument and returns an ad-hoc value. The code also takes
     the global data and returns it to be stored and retrieved on the
     next transaction. These data are initialized by another
     parameter. The calling convention for the code is as follows:
     `(Pair arg globals)) -> (Pair ret globals)`, as
     extrapolatable from the instruction type. The first parameters are
     the manager, optional delegate, then spendable and delegatable
     flags and finally the initial amount taken from the currently
     executed contract. The contract is returned as a first class
     value to be called immediately or stored.

   * `CREATE_ACCOUNT`:
     Forge an account (a contract without code).

        :: key : key? : bool : tez : 'S   ->   contract unit unit : 'S

     Take as argument the manager, optional delegate, the delegatable
     flag and finally the initial amount taken from the currently
     executed contract.

   * `TRANSFER_TOKENS`:
     Forge and evaluate a transaction.

        :: 'p : tez : contract 'p 'r : 'g : []   ->   'r : 'g : []

     The parameter and return value must be consistent with the ones
     expected by the contract, unit for an account. To preserve the
     global consistency of the system, the current contract's storage
     must be updated before passing the control to another script. For
     this, the script must put the partially updated storage on the
     stack ('g is the type of the contract's storage).  If a recursive
     call to the current contract happened, the updated storage is put
     on the stack next to the return value.  Nothing else can remain
     on the stack during a nested call. If some local values have to
     be kept for after the nested call, they have to be stored
     explicitly in a transient part of the storage. A trivial example
     of that is to reserve a boolean in the storage, initialized to
     false, reset to false at the end of each contract execution, and
     set to true during a nested call.  This thus gives an easy way
     for a contract to prevent recursive call (the contract just fails
     if the boolean is true).

   * `BALANCE`:
     Push the current amount of tez of the current contract.

        :: 'S   ->   tez :: 'S

   * `SOURCE 'p 'r`:
     Push the source contract of the current transaction.

        :: 'S   ->   contract 'p 'r :: 'S

   * `SELF`:
     Push the current contract.

        :: 'S   ->   contract 'p 'r :: 'S
           where   contract 'p 'r is the type of the current contract

   * `AMOUNT`:
     Push the amount of the current transaction.

        :: 'S   ->   tez :: 'S

   * `DEFAULT_ACCOUNT`:
     Return a default contract with the given public/private key pair.
     Any funds deposited in this contract can immediately be spent by the
     holder of the private key. This contract cannot execute Michelson code
     and will always exist on the blockchain.

        :: key : 'S   ->   contract unit unit :: 'S

### Special operations

   * `STEPS_TO_QUOTA`:
     Push the remaining steps before the contract execution must terminate.

        :: 'S   ->   nat :: 'S

   * `NOW`:
     Push the timestamp of the block whose validation triggered this
     execution (does not change during the execution of the contract).

        :: 'S   ->   timestamp :: 'S

### Cryptographic primitives

   * `H`:
     Compute a cryptographic hash of the value contents using the
     Sha256 cryptographic algorithm.

        :: 'a : 'S   ->   string : 'S

   * `CHECK_SIGNATURE`:
     Check that a sequence of bytes has been signed with a given key.

        :: key : pair signature string : 'S   ->   bool : 'S

   * `COMPARE`:

        :: key : key : 'S   ->   int : 'S


VIII - Concrete syntax
----------------------

The concrete language is very close to the formal notation of the
specification. Its structure is extremely simple: an expression in the
language can only be one of the three following constructs.

  1. A constant (integer or string).
  2. The application of a primitive to a sequence of expressions.
  3. A sequence of expressions.

### Constants

There are two kinds of constants:

  1. Integers or naturals in decimal (no prefix),
     hexadecimal (0x prefix), octal (0o prefix) or binary (0b prefix).
  2. Strings with usual escapes `\n`, `\t`, `\b`, `\r`, `\\`, `\"`.
     Strings are encoding agnostic sequences of bytes. Non printable
     characters can be escaped by 3 digits decimal codes `\ddd` or
     2 digit hexadecimal codes `\xHH`.

### Primitive applications

In the specification, primitive applications always luckily fit on a
single line. In this case, the concrete syntax is exactly the formal
notation. However, it is sometimes necessary to break lines in a real
program, which can be done as follows.

As in Python or Haskell, the concrete syntax of the language is
indentation sensitive. The elements of a syntactical block, such as
all the elements of a sequence, or all the parameters of a primitive,
must be written with the exact same left margin in the program source
code. This is unlike in C-like languages, where blocks are delimited
with braces and the margin is ignored by the compiled.

The simplest form requires to break the line after the primitive name
and after every argument. Argument must be indented by at least one
more space than the primitive, and all arguments must sit on the exact
same column.

    PRIM
      arg1
      arg2
      ...

If an argument of a primitive application is a primitive application
itself, its arguments must be pushed even further on the right, to
lift any ambiguity, as in the following example.

    PRIM1
      PRIM2
        arg1_prim2
        arg2_prim2
      arg2_prim1

It is possible to put successive arguments on a single line using
a semicolon as a separator:

    PRIM
      arg1; arg2
      arg3; arg4

It is also possible to add arguments on the same line as the primitive
as a lighter way to write simple expressions. An other representation
of the first example is:

    PRIM arg1 arg2 ...

It is possible to mix both notations as in:

    PRIM arg1 arg2
      arg3
      arg4

Or even:

    PRIM arg1 arg2
      arg3; arg4

Both equivalent to:

    PRIM
      arg1
      arg2
      arg3
      arg4

Trailing semicolons are ignored:

    PRIM
      arg1;
      arg2

Calling a primitive with a compound argument on a single line is
allowed by wrapping with parentheses. Another notation for the second
example is:

    PRIM1 (PRIM2 arg1_prim2 arg2_prim2) arg2_prim1

### Sequences

Successive expression can be grouped as a single sequence expression
using braces delimiters and semicolon separators.

    { expr1 ; expr2 ; expr3 ; expr4 }

A sequence block can be split on several lines. In this situation, the
whole block, including the closing brace, must be indented with
respect to the first instruction.

    { expr1 ; expr2
      expr3 ; expr4 }

Blocks can be passed as argument to a primitive.


    PRIM arg1 arg2
      { arg3_expr1 ; arg3_expr2
        arg3_expr3 ; arg3_expr4 }


### Conventions

The concrete syntax follows the same lexical conventions as the
specification: instructions are represented by uppercase identifiers,
type constructors by lowercase identifiers, and constant constructors
are Capitalized.

Lists can be written in a single shot instead of a succession of `Cons`

    (List 1 2 3) = (Cons 1 (Cons 2 (Cons 3 Nil)))

All domain specific constants are strings with specific formats:

  - `tez` amounts are written using the same notation as JSON schemas
    and the command line client: thousands are optionally separated by
    comas, and centiles, if present, must be prefixed by a period.
     - in regexp form: `([0-9]{1,3}(,[0-9]{3})+)|[0-9]+(\.[0.9]{2})?`
     - `"1234567"`      means 123456700 tez centiles
     - `"1,234,567"`    means 123456700 tez centiles
     - `"1234567.89"`   means 123456789 tez centiles
     - `"1,234,567.00"` means 123456789 tez centiles
     - `"1234,567"`     is invalid
     - `"1,234,567."`   is invalid
     - `"1,234,567.0"`  is invalid
  - `timestamp`s are written using `RFC 339` notation.
  - `contract`s are the raw strings returned by JSON RPCs or the command
    line interface and cannot be forged by hand so their format is of
    no interest here.
  - `key`s are `Sha256` hashes of `ed25519` public keys encoded in
    `base48` format with the following custom alphabet:
    `"eXMNE9qvHPQDdcFx5J86rT7VRm2atAypGhgLfbS3CKjnksB4"`.
  - `signature`s are `ed25519` signatures as a series of hex-encoded bytes.

To prevent errors, control flow primitives that take instructions as
parameters require sequences in the concrete syntax.

    IF { instr1_true ; instr2_true ; ... } { instr1_false ; instr2_false ; ... }

    IF
      { instr1_true ; instr2_true ; ... }
      { instr1_false ; instr2_false ; ... }

### Main program structure

The toplevel of a smart contract file must be an undelimited sequence
of four primitive applications (in no particular order) that provide
its `parameter`, `return` and `storage` types, as well as its `code`.

See the next section for a concrete example.

### Comments

A hash sign (`#`) anywhere outside of a string literal will make the
rest of the line (and itself) completely ignored, as in the following
example.

    PUSH nat 1 # pushes 1
    PUSH nat 2 # pushes 2
    ADD         # computes 2 + 1

IX - Examples
-------------

Contracts in the system are stored as a piece of code and a global
data storage. The type of the global data of the storage is fixed for
each contract at origination time. This is ensured statically by
checking on origination that the code preserves the type of the global
data. For this, the code of the contract is checked to be of the
following type lambda (pair 'arg 'global) -> (pair 'ret
'global) where 'global is the type of the original global store given
on origination. The contract also takes a parameter and
returns a value, hence the complete calling convention above.

### Empty contract

Any contract with the same `parameter` and `return` types
may be written with an empty sequence in its `code` section.
The simplest contract is the contract for which the
`parameter`, `storage`, and `return` are all of type `unit`.
This contract is as follows:

    code {};
    storage unit;
    parameter unit;
    return unit;

### Reservoir contract

We want to create a contract that stores tez until a timestamp `T` or a
maximum amount `N` is reached. Whenever `N` is reached before `T`, all tokens
are reversed to an account `B` (and the contract is automatically
deleted). Any call to the contract's code performed after `T` will
otherwise transfer the tokens to another account `A`.

We want to build this contract in a reusable manner, so we do not
hard-code the parameters. Instead, we assume that the global data of
the contract are `(Pair (Pair T N) (Pair A B))`.

Hence, the global data of the contract has the following type

    'g =
      pair
        pair timestamp tez
        pair (contract unit unit) (contract unit unit)

Following the contract calling convention, the code is a lambda of type

    lambda
      pair unit 'g
      pair unit 'g

written as

    lambda
      pair unit
        pair
          pair timestamp tez
          pair (contract unit unit) (contract unit unit)
      pair unit
        pair
          pair timestamp tez
          pair (contract unit unit) (contract unit unit)

The complete source `reservoir.tz` is:

    parameter timestamp ;
    storage
      pair
        (pair timestamp tez) # T N
        (pair (contract unit unit) (contract unit unit)) ; # A B
    return unit ;
    code
      { DUP ; CDAAR ; # T
        NOW ;
        COMPARE ; LE ;
        IF { DUP ; CDADR ; # N
             BALANCE ;
             COMPARE ; LE ;
             IF { CDR ; UNIT ; PAIR }
                { DUP ; CDDDR ; # B
                  BALANCE ; UNIT ;
                  DIIIP { CDR } ;
                  TRANSFER_TOKENS ;
                  PAIR } }
           { DUP ; CDDAR ; # A
             BALANCE ;
             UNIT ;
             DIIIP { CDR } ;
             TRANSFER_TOKENS ;
             PAIR } }

### Reservoir contract (variant with broker and status)

We basically want the same contract as the previous one, but instead
of destroying it, we want to keep it alive, storing a flag `S` so that
we can tell afterwards if the tokens have been transferred to `A` or `B`.
We also want a broker `X` to get some fee `P` in any case.

We thus add variables `P` and `S` and `X` to the global data of the
contract, now `(Pair (S, Pair (T, Pair (Pair P N) (Pair X (Pair A B)))))`.
`P` is the  fee for broker  `A`, `S` is the  state, as a  string `"open"`,
`"timeout"` or `"success"`.

At the beginning of the transaction:

     S is accessible via a CDAR
     T               via a CDDAR
     P               via a CDDDAAR
     N               via a CDDDADR
     X               via a CDDDDAR
     A               via a CDDDDDAR
     B               via a CDDDDDDR

For the contract to stay alive, we test that all least `(Tez "1.00")` is
still available after each transaction. This value is given as an
example and must be updated according to the actual Tezos minimal
value for contract balance.

The complete source `scrutable_reservoir.tz` is:

    parameter timestamp ;
    storage
      pair
        string # S
        pair
          timestamp # T
          pair
            (pair tez tez) ; # P N
            pair
              (contract unit unit) # X
              pair (contract unit unit) (contract unit unit) ; # A B
    return unit ;
    code
      { DUP ; CDAR # S
        PUSH string "open" ;
        COMPARE ; NEQ ;
        IF { FAIL } # on "success", "timeout" or a bad init value
           { DUP ; CDDAR ; # T
             NOW ;
             COMPARE ; LT ;
             IF { # Before timeout
                  # We compute ((1 + P) + N) tez for keeping the contract alive
                  PUSH tez "1.00" ;
                  DIP { DUP ; CDDDAAR } ; ADD ; # P
                  DIP { DUP ; CDDDADR } ; ADD ; # N
                  # We compare to the cumulated amount
                  BALANCE ;
                  COMPARE; LT ;
                  IF { # Not enough cash, we just accept the transaction
                       # and leave the global untouched
                       CDR }
                     { # Enough cash, successful ending
                       # We update the global
                       CDDR ; PUSH string "success" ; PAIR ;
                       # We transfer the fee to the broker
                       DUP ; CDDAAR ; # P
                       DIP { DUP ; CDDDAR } # X
                       UNIT ; TRANSFER_TOKENS ; DROP ;
                       # We transfer the rest to A
                       DUP ; CDDADR ; # N
                       DIP { DUP ; CDDDDAR } # A
                       UNIT ; TRANSFER_TOKENS ; DROP } }
                { # After timeout, we refund
                  # We update the global
                  CDDR ; PUSH string "timeout" ; PAIR ;
                  # We try to transfer the fee to the broker
                  PUSH tez "1.00" ; BALANCE ; SUB ; # available
                  DIP { DUP ; CDDAAR } ; # P
                  COMPARE ; LT ; # available < P
                  IF { PUSH tez "1.00" ; BALANCE ; SUB ; # available
                       DIP { DUP ; CDDDAR } # X
                       UNIT ; TRANSFER_TOKENS ; DROP }
                     { DUP ; CDDAAR ; # P
                       DIP { DUP ; CDDDAR } # X
                       UNIT ; TRANSFER_TOKENS ; DROP }
                  # We transfer the rest to B
                  PUSH tez "1.00" ; BALANCE ; SUB ; # available
                  DIP { DUP ; CDDDDDR } # B
                  UNIT ; TRANSFER_TOKENS ; DROP } }
        # return Unit
        UNIT ; PAIR }

### Forward contract

We want to write a forward contract on dried peas. The contract takes
as global data the tons of peas `Q`, the expected delivery date `T`, the
contract agreement date `Z`, a strike `K`, a collateral `C` per ton of dried
peas, and the accounts of the buyer `B`, the seller `S` and the warehouse `W`.

These parameters as grouped in the global storage as follows:

    Pair
      Pair (Pair Q (Pair T Z))
      Pair
        (Pair K C)
        (Pair (Pair B S) W)

of type

    pair
      pair nat (pair timestamp timestamp)
      pair
        pair tez tez
        pair (pair account account) account


The 24 hours after timestamp `Z` are for the buyer and seller to store
their collateral `(Q * C)`. For this, the contract takes a string as
parameter, matching `"buyer"` or `"seller"` indicating the party for which
the tokens are transferred. At the end of this day, each of them can
send a transaction to send its tokens back. For this, we need to store
who already paid and how much, as a `(pair tez tez)` where the left
component is the buyer and the right one the seller.

After the first day, nothing cam happen until `T`.

During the 24 hours after `T`, the buyer must pay `(Q * K)` to the
contract, minus the amount already sent.

After this day, if the buyer didn't pay enough then any transaction
will send all the tokens to the seller.

Otherwise, the  seller must deliver at  least `Q` tons of  dried peas to
the warehouse, in  the next 24 hours.  When the amount is  equal to or
exceeds `Q`, all the tokens are transferred to the seller and the contract
is destroyed. For  storing the quantity of peas  already delivered, we
add a counter  of type `nat` in the global  storage. For knowing this
quantity, we accept messages from W with a partial amount of delivered
peas as argument.

After this day, any transaction will send all the tokens to the buyer
(not enough peas have been delivered in time).

Hence, the global storage is a pair, with the counters on the left,
and the constant parameters on the right, initially as follows.

    Pair
      Pair 0 (Pair 0_00 0_00)
      Pair
        Pair (Pair Q (Pair T Z))
        Pair
          (Pair K C)
          (Pair (Pair B S) W)

of type

    pair
      pair nat (pair tez tez)
      pair
        pair nat (pair timestamp timestamp)
        pair
          pair tez tez
          pair (pair account account) account

The parameter of the transaction will be either a transfer from the
buyer or the seller or a delivery notification from the warehouse of
type `(or string nat)`.

At the beginning of the transaction:

    Q is accessible via a CDDAAR
    T               via a CDDADAR
    Z               via a CDDADDR
    K               via a CDDDAAR
    C               via a CDDDADR
    B               via a CDDDDAAR
    S               via a CDDDDADR
    W               via a CDDDDDR
    the delivery counter via a CDAAR
    the amount versed by the seller via a CDADDR
    the argument via a CAR

The contract returns a unit value, and we assume that it is created
with the minimum amount, set to `(Tez "1.00")`.

The complete source `forward.tz` is:

    parameter (or string nat) ;
    return unit ;
    storage
      pair
        pair nat (pair tez tez) # counter from_buyer from_seller
        pair
          pair nat (pair timestamp timestamp) # Q T Z
          pair
            pair tez tez # K C
            pair
              pair (contract unit unit) (contract unit unit) # B S
              (contract unit unit); # W
    code
      { DUP ; CDDADDR ; # Z
        PUSH nat 86400 ; SWAP ; ADD ; # one day in second
        NOW ; COMPARE ; LT ;
        IF { # Before Z + 24
             DUP ; CAR ; # we must receive (Left "buyer") or (Left "seller")
             IF_LEFT
               { DUP ; PUSH string "buyer" ; COMPARE ; EQ ;
                 IF { DROP ;
                      DUP ; CDADAR ; # amount already versed by the buyer
                      DIP { AMOUNT } ; ADD ; # transaction
                      #  then we rebuild the globals
                      DIP { DUP ; CDADDR } ; PAIR ; # seller amount
                      PUSH nat 0 ; PAIR ; # delivery counter at 0
                      DIP { CDDR } ; PAIR ; # parameters
                      # and return Unit
                      UNIT ; PAIR }
                    { PUSH string "seller" ; COMPARE ; EQ ;
                      IF { DUP ; CDADDR ; # amount already versed by the seller
                           DIP { AMOUNT } ; ADD ; # transaction
                           #  then we rebuild the globals
                           DIP { DUP ; CDADAR } ; SWAP ; PAIR ; # buyer amount
                           PUSH nat 0 ; PAIR ; # delivery counter at 0
                           DIP { CDDR } ; PAIR ; # parameters
                           # and return Unit
                           UNIT ; PAIR }
                         { FAIL } } } # (Left _)
               { FAIL } } # (Right _)
           { # After Z + 24
             # test if the required amount is reached
             DUP ; CDDAAR ; # Q
             DIP { DUP ; CDDDADR } ; MUL ; # C
             PUSH nat 2 ; MUL ;
             PUSH tez "1.00" ; ADD ;
             BALANCE ; COMPARE ; LT ; # balance < 2 * (Q * C) + 1
             IF { # refund the parties
                  CDR ; DUP ; CADAR ; # amount versed by the buyer
                  DIP { DUP ; CDDDAAR } # B
                  UNIT ; TRANSFER_TOKENS ; DROP
                  DUP ; CADDR ; # amount versed by the seller
                  DIP { DUP ; CDDDADR } # S
                  UNIT ; TRANSFER_TOKENS ; DROP
                  BALANCE ; # bonus to the warehouse to destroy the account
                  DIP { DUP ; CDDDDR } # W
                  UNIT ; TRANSFER_TOKENS ; DROP
                  # return unit, don't change the global
                  # since the contract will be destroyed
                  UNIT ; PAIR }
                { # otherwise continue
                  DUP ; CDDADAR # T
                  NOW ; COMPARE ; LT
                  IF { FAIL } # Between Z + 24 and T
                     { # after T
                       DUP ; CDDADAR # T
                       PUSH nat 86400 ; ADD # one day in second
                       NOW ; COMPARE ; LT
                       IF { # Between T and T + 24
                            # we only accept transactions from the buyer
                            DUP ; CAR ; # we must receive (Left "buyer")
                            IF_LEFT
                              { PUSH string "buyer" ; COMPARE ; EQ ;
                                IF { DUP ; CDADAR ; # amount already versed by the buyer
                                     DIP { AMOUNT } ; ADD ; # transaction
                                     # The amount must not exceed Q * K
                                     DUP ;
                                     DIIP { DUP ; CDDAAR ; # Q
                                            DIP { DUP ; CDDDAAR } ; MUL ; } ; # K
                                     DIP { COMPARE ; GT ; # new amount > Q * K
                                           IF { FAIL } { } } ; # abort or continue
                                     #  then we rebuild the globals
                                     DIP { DUP ; CDADDR } ; PAIR ; # seller amount
                                     PUSH nat 0 ; PAIR ; # delivery counter at 0
                                     DIP { CDDR } ; PAIR ; # parameters
                                     # and return Unit
                                     UNIT ; PAIR }
                                   { FAIL } } # (Left _)
                              { FAIL } } # (Right _)
                          { # After T + 24
                            # test if the required payment is reached
                            DUP ; CDDAAR ; # Q
                            DIP { DUP ; CDDDAAR } ; MUL ; # K
                            DIP { DUP ; CDADAR } ; # amount already versed by the buyer
                            COMPARE ; NEQ ;
                            IF { # not reached, pay the seller and destroy the contract
                                 BALANCE ;
                                 DIP { DUP ; CDDDDADR } # S
                                 DIIP { CDR } ;
                                 UNIT ; TRANSFER_TOKENS ; DROP ;
                                 # and return Unit
                                 UNIT ; PAIR }
                               { # otherwise continue
                                 DUP ; CDDADAR # T
                                 PUSH nat 86400 ; ADD ;
                                 PUSH nat 86400 ; ADD ; # two days in second
                                 NOW ; COMPARE ; LT
                                 IF { # Between T + 24 and T + 48
                                      # We accept only delivery notifications, from W
                                      DUP ; CDDDDDR ; MANAGER ; # W
                                      SOURCE unit unit ; MANAGER ;
                                      COMPARE ; NEQ ;
                                      IF { FAIL } {} # fail if not the warehouse
                                      DUP ; CAR ; # we must receive (Right amount)
                                      IF_LEFT
                                        { FAIL } # (Left _)
                                        { # We increment the counter
                                          DIP { DUP ; CDAAR } ; ADD ;
                                          # And rebuild the globals in advance
                                          DIP { DUP ; CDADR } ; PAIR ;
                                          DIP { CDDR } ; PAIR ;
                                          UNIT ; PAIR ;
                                          # We test if enough have been delivered
                                          DUP ; CDAAR ;
                                          DIP { DUP ; CDDAAR } ;
                                          COMPARE ; LT ; # counter < Q
                                          IF { CDR } # wait for more
                                             { # Transfer all the money to the seller
                                               BALANCE ; # and destroy the contract
                                               DIP { DUP ; CDDDDADR } # S
                                               DIIP { CDR } ;
                                               UNIT ; TRANSFER_TOKENS ; DROP } } ;
                                      UNIT ; PAIR }
                                    { # after T + 48, transfer everything to the buyer
                                      BALANCE ; # and destroy the contract
                                      DIP { DUP ; CDDDDAAR } # B
                                      DIIP { CDR } ;
                                      UNIT ; TRANSFER_TOKENS ; DROP ;
                                      # and return unit
                                      UNIT ; PAIR } } } } } } }

X - Full grammar
----------------

    <data> ::=
      | <int constant>
      | <natural number constant>
      | <string constant>
      | <timestamp string constant>
      | <signature string constant>
      | <key string constant>
      | <tez string constant>
      | <contract string constant>
      | Unit
      | True
      | False
      | Pair <data> <data>
      | Left <data>
      | Right <data>
      | Some <data>
      | None
      | List <data> ...
      | Set <data> ...
      | Map (Item <data> <data>) ...
      | instruction
    <instruction> ::=
      | { <instruction> ... }
      | DROP
      | DUP
      | SWAP
      | PUSH <type> <data>
      | SOME
      | NONE <type>
      | UNIT
      | IF_NONE { <instruction> ... } { <instruction> ... }
      | PAIR
      | CAR
      | CDR
      | LEFT <type>
      | RIGHT <type>
      | IF_LEFT { <instruction> ... } { <instruction> ... }
      | NIL <type>
      | CONS
      | IF_CONS { <instruction> ... } { <instruction> ... }
      | EMPTY_SET <type>
      | EMPTY_MAP <comparable type> <type>
      | MAP
      | REDUCE
      | MEM
      | GET
      | UPDATE
      | IF { <instruction> ... } { <instruction> ... }
      | LOOP { <instruction> ... }
      | LAMBDA <type> <type> { <instruction> ... }
      | EXEC
      | DIP { <instruction> ... }
      | FAIL
      | CONCAT
      | ADD
      | SUB
      | MUL
      | DIV
      | ABS
      | NEG
      | MOD
      | LSL
      | LSR
      | OR
      | AND
      | XOR
      | NOT
      | COMPARE
      | EQ
      | NEQ
      | LT
      | GT
      | LE
      | GE
      | CAST
      | CHECKED_ABS
      | CHECKED_NEG
      | CHECKED_ADD
      | CHECKED_SUB
      | CHECKED_MUL
      | CHECKED_CAST
      | FLOOR
      | CEIL
      | INF
      | NAN
      | ISNAN
      | NANAN
      | MANAGER
      | SELF
      | TRANSFER_TOKENS
      | CREATE_ACCOUNT
      | CREATE_CONTRACT
      | NOW
      | AMOUNT
      | BALANCE
      | CHECK_SIGNATURE
      | H
      | STEPS_TO_QUOTA
      | SOURCE <type> <type>
    <type> ::=
      | int
      | nat
      | unit
      | string
      | tez
      | bool
      | key
      | timestamp
      | signature
      | option <type>
      | list <type>
      | set <comparable type>
      | contract <type> <type>
      | pair <type> <type>
      | or <type> <type>
      | lambda <type> <type>
      | map <comparable type> <type>
    <comparable type> ::=
      | int
      | nat
      | string
      | tez
      | bool
      | key
      | timestamp

XI - Reference implementation
-----------------------------

The language is implemented in OCaml as follows:

  * The lower internal representation is written as a GADT whose type
    parameters encode exactly the typing rules given in this
    specification. In other words, if a program written in this
    representation is accepted by OCaml's typechecker, it is
    mandatorily type-safe. This of course also valid for programs not
    handwritten but generated by OCaml code, so we are sure that any
    manipulated code is type-safe.

    In the end, what remains to be checked is the encoding of the
    typing rules as OCaml types, which boils down to half a line of
    code for each instruction. Everything else is left to the
    venerable and well trusted OCaml.

  * The interpreter is basically the direct transcription of the
    rewriting rules presented above. It takes an instruction, a stack
    and transforms it. OCaml's typechecker ensures that the
    transformation respects the pre and post stack types declared by
    the GADT case for each instruction.

    The only things that remain to we reviewed are value dependent
    choices, such as that we did not swap true and false when
    interpreting the If instruction.

  * The input, untyped internal representation is an OCaml ADT with
    the only 5 grammar constructions: `String`, `Int`, `Seq` and
    `Prim`.  It is the target language for the parser, since not all
    parsable programs are well typed, and thus could simply not be
    constructed using the GADT.

  * The typechecker is a simple function that recognizes the abstract
    grammar described in section X by pattern matching, producing the
    well-typed, corresponding GADT expressions. It is mostly a
    checker, not a full inferer, and thus takes some annotations
    (basically the input and output of the program, of lambdas and of
    uninitialized maps and sets). It works by performing a
    symbolic evaluation of the program, transforming a symbolic
    stack. It only needs one pass over the whole program.

    Here again, OCaml does most of the checking, the structure of the
    function is very simple, what we have to check is that we
    transform a `Prim ("If", ...)` into an `If`, a `Prim ("Dup", ...)`
    into a `Dup`, etc.