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+## Arithmetic as an embedded domain-specific language
+
+Let's parse another classic grammar, the grammar of arithmetic expressions with
+addition, multiplication, integers, and variables.  In the process, we'll learn
+how to:
+
+- Convert identifiers such as `x` into strings within a macro.
+- add the ability to "escape" the macro context from within the macro. This is useful to interpret identifiers with their _original_ meaning (predefined values)
+  instead of their new meaning within a macro (treat as a symbol).
+
+Let's begin with the simplest thing possible. We'll define an AST, and use operators `+` and `*` to denote
+building an arithmetic AST.
+
+
+Here's the AST that we will be parsing:
+
+```lean,ignore
+-- example on parsing arith language via macros
+inductive Arith : Type where
+  | add : Arith → Arith → Arith -- e + f
+  | mul : Arith → Arith → Arith -- e * f
+  | int : Int → Arith -- constant
+  | symbol : String → Arith -- variable
+```
+
+We declare a syntax category to describe the grammar that we will be parsing.
+See that we control the precedence of `+` and `*` by writing `syntax:50` for addition and `syntax:60` for multiplication,
+indicating that multiplication binds tighter than addition (higher the number, tighter the binding).
+This allows us to declare _precedence_ when defining new syntax.
+
+```lean,ignore
+declare_syntax_cat arith
+syntax num : arith -- int for Arith.int
+syntax str : arith -- strings for Arith.symbol
+syntax:60  arith:60 "+" arith:61 : arith -- Arith.add
+syntax:70 arith:70 "*" arith:71 : arith -- Arith.mul
+syntax "(" arith ")" : arith -- bracketed expressions
+```
+
+Further, if we look at `syntax:60  arith:60 "+" arith:61 : arith`, the
+precedence declarations at `arith:60 "+" arith:61` conveys that the left
+argument must have precedence at least `60` or greater, and the right argument
+must have precedence at least`61` or greater.  Note that this forces left
+associativity. To understand this, let's compare two hypothetical parses:
+
+```
+-- syntax:60  arith:60 "+" arith:61 : arith -- Arith.add
+-- a + b + c
+(a:60 + b:61):60 + c
+a + (b:60 + c:61):60
+```
+
+In the parse tree of `a + (b:60 + c:61):60`, we see that the right argument `(b + c)` is given the precedence `60`. However,
+the rule for addition expects the right argument to have a precedence of **at least** 61, as witnessed by the `arith:61` at
+the right-hand-side of `syntax:60 arith:60 "+" arith:61 : arith`. Thus, the rule `syntax:60  arith:60 "+" arith:61 : arith`
+ensures that addition is left associative.
+
+Since addition is declared arguments of precedence `60/61` and multiplication with `70/71`, this causes multiplication to bind
+tighter than addition. Once again, let's compare two hypothetical parses:
+
+```
+-- syntax:60  arith:60 "+" arith:61 : arith -- Arith.add
+-- syntax:70 arith:70 "*" arith:71 : arith -- Arith.mul
+-- a * b + c
+a * (b:60 + c:61):60
+(a:70 * b:71):70 + c
+```
+
+While parsing `a * (b + c)`, `(b + c)` is assigned a precedence `60` by the addition rule. However, multiplication expects
+the right argument to have precedence **at least** 71. Thus, this parse is invalid. In contrast, `(a * b) + c` assigns
+a precedence of `70` to `(a * b)`. This is compatible with addition which expects the left argument to have precedence
+**at least `60` ** (`70` is greater than `60`). Thus, the string `a * b + c` is parsed as `(a * b) + c`.
+For more details, please look at the [Lean manual on syntax extensions](../syntax.md#notations-and-precedence).
+
+
+
+
+To go from strings into `Arith`, We define the macro `Arith|` to
+translate the syntax category `arith` into an `Arith` inductive value that
+lives in `term`:
+
+
+```lean,ignore
+-- auxiliary notation for translating `arith` into `term`
+syntax "[Arith| " arith "]" : term
+```
+
+Our macro rules perform the "obvious" translation:
+
+```lean,ignore
+macro_rules
+  | `([Arith| $s:strLit ]) => `(Arith.symbol $s)
+  | `([Arith| $num:numLit ]) => `(Arith.int $num)
+  | `([Arith| $x:arith + $y:arith ]) => `(Arith.add [Arith| $x] [Arith| $y])
+  | `([Arith| $x:arith * $y:arith ]) => `(Arith.mul [Arith| $x] [Arith| $y])
+  | `([Arith| ($x:arith) ]) => `([Arith| $x ])
+```
+And some examples:
+
+```lean,ignore
+#check [Arith| "x" * "y"] -- Arith.mul (Arith.symbol "x") (Arith.symbol "y") : Arith
+
+#check [Arith| "x" + "y"] -- add
+-- Arith.add (Arith.symbol "x") (Arith.symbol "y") 
+
+#check  [Arith| "x" + 20] -- symbol + int
+-- Arith.add (Arith.symbol "x") (Arith.int 20)
+
+
+#check [Arith| "x" + "y" * "z" ] -- precedence
+Arith.add (Arith.symbol "x") (Arith.mul (Arith.symbol "y") (Arith.symbol "z"))
+-- 
+#check [Arith| "x" * "y" + "z"] -- precedence
+-- Arith.add (Arith.mul (Arith.symbol "x") (Arith.symbol "y")) (Arith.symbol "z")
+
+
+#check [Arith| ("x" + "y") * "z"] -- brackets
+-- Arith.mul (Arith.add (Arith.symbol "x") (Arith.symbol "y")) (Arith.symbol "z")
+```
+
+
+Writing variables as strings, such as `"x"`  gets old; Wouldn't it be so much
+prettier if we could write `x * y`, and have the macro translate this into `Arith.mul (Arith.Symbol "x") (Arith.mul "y")`?
+We can do this, and this will be our first taste of manipulating macro variables --- we'll use `x.getId` instead of directly evaluating `$x`.
+We also write a macro rule for `Arith|` that translates an identifier into
+a string, using `$(Lean.quote (toString x.getId))`.  (TODO: explain what
+`Lean.quote` does):
+
+```lean,ignore
+syntax ident : arith
+
+macro_rules
+  | `([Arith| $x:ident]) => `(Arith.symbol $(Lean.quote (toString x.getId)))
+```
+
+
+Let's test and see that we can now write expressions such as `x * y` directly instead of having to write `"x" * "y"`:
+
+```lean,ignore
+#check [Arith| x ] -- Arith.symbol "x"
+
+#check [Arith| x + y] -- Arith.add (Arith.symbol "x") (Arith.symbol "y")
+```
+
+We now show an unfortunate consequence of the above definitions. Suppose we want to build `(x + y) + z)`.
+Since we already have defined `x_plus_y` as `x + y`, perhaps we should reuse it! Let's try:
+
+```lean,ignore
+#check [Arith| x_plus_y + z] --Arith.add (Arith.symbol "x_plus_y") (Arith.symbol "z")
+```
+
+Whoops, that didn't work! What happened? Lean treats `x_plus_y` _itself_ as an identifier! So we need to add some syntax
+to be able to "escape" the `Arith|` context. Let's use the syntax `<[ $e:term ]>` to mean: evaluate `$e` as a real term,
+not an identifier. The macro looks like follows:
+
+```lean,ignore
+syntax "<[" term "]>" : arith -- escape for embedding terms into `Arith`
+macro_rules
+  | `([Arith| <[ $e:term ]> ]) => e
+
+```
+
+Let's try our previous example:
+
+```lean,ignore
+#check [Arith| <[ x_plus_y ]>] -- x_plus_y
+```
+
+Perfect!
+
+In this tutorial, we expanded on the previous tutorial to parse a more
+realistic grammar with multiple levels of precedence, how to parse identifiers directly
+within a macro, and how to provide an escape from within the macro context.
+
+#### Full code listing
+
+```lean
+-- example on parsing arith language via macros
+inductive Arith: Type
+  | add : Arith → Arith → Arith -- e + f
+  | mul : Arith → Arith → Arith -- e * f
+  | int : Int → Arith -- constant
+  | symbol : String → Arith -- variable
+
+declare_syntax_cat arith
+
+syntax num : arith -- int for Arith.int
+syntax str : arith -- strings for Arith.symbol
+syntax:60  arith:60 "+" arith:61 : arith -- Arith.add
+syntax:70 arith:70 "*" arith:71 : arith -- Arith.mul
+syntax "(" arith ")" : arith -- bracketed expressions
+
+-- auxiliary notation for translating `arith` into `term`
+syntax "[Arith| " arith "]" : term
+
+macro_rules
+  | `([Arith| $s:strLit ]) => `(Arith.symbol $s )
+  | `([Arith| $num:numLit ]) => `(Arith.int $num )
+  | `([Arith| $x:arith + $y:arith ]) => `(Arith.add [Arith| $x] [Arith| $y] )
+  | `([Arith| $x:arith * $y:arith ]) => `(Arith.mul [Arith| $x] [Arith| $y] )
+  | `([Arith| ($x:arith) ]) => `([Arith| $x ])
+
+def foo := 
+#check [Arith| "x" * "y" ] -- mul
+-- Arith.mul (Arith.symbol "x") (Arith.symbol "y")
+
+def bar := 
+#check [Arith| "x" + "y"] -- add
+-- Arith.add (Arith.symbol "x") (Arith.symbol "y")
+
+def baz := 
+#check [Arith| "x" + 20] -- symbol + int
+-- Arith.add (Arith.symbol "x") (Arith.int 20)
+
+def quux_left := 
+#check [Arith| "x" + "y" * "z" ] -- precedence
+-- Arith.add (Arith.symbol "x") (Arith.mul (Arith.symbol "y") (Arith.symbol "z"))
+
+def quux_right := 
+#check [Arith| "x" * "y" + "z"] -- precedence
+-- Arith.add (Arith.mul (Arith.symbol "x") (Arith.symbol "y")) (Arith.symbol "z")
+
+
+def quuz := 
+#check [Arith| ("x" + "y") * "z"] -- brackets
+-- Arith.mul (Arith.add (Arith.symbol "x") (Arith.symbol "y")) (Arith.symbol "z")
+
+syntax ident : arith
+
+macro_rules
+  | `([Arith| $x:ident]) => `(Arith.symbol $(Lean.quote (toString x.getId)))
+
+#check [Arith| x ] -- Arith.symbol "x"
+#check [Arith| x + y] -- Arith.add (Arith.symbol "x") (Arith.symbol "y")
+#check [Arith| x_plus_y + z] -- Arith.add (Arith.symbol "x_plus_y") (Arith.symbol "z")
+
+syntax "<[" term "]>" : arith -- escape for embedding terms into `Arith`
+ 
+macro_rules
+  | `([Arith| <[ $e:term ]> ]) => e
+  
+
+#check [Arith| <[ x_plus_y ]>] -- x_plus_y
+```
+