Documentation

Lean.Parser.Term

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    A docComment parses a "documentation comment" like /-- foo -/. This is not treated like a regular comment (that is, as whitespace); it is parsed and forms part of the syntax tree structure.

    A docComment node contains a /-- atom and then the remainder of the comment, foo -/ in this example. Use TSyntax.getDocString to extract the body text from a doc string syntax node.

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      sepByIndentSemicolon(p) parses a sequence of p optionally followed by ;, similar to manyIndent(p ";"?), except that if two occurrences of p occur on the same line, the ; is mandatory. This is used by tactic parsing, so that

      example := by
        skip
        skip
      

      is legal, but by skip skip is not - it must be written as by skip; skip.

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        sepBy1IndentSemicolon(p) parses a (nonempty) sequence of p optionally followed by ;, similar to many1Indent(p ";"?), except that if two occurrences of p occur on the same line, the ; is mandatory. This is used by tactic parsing, so that

        example := by
          skip
          skip
        

        is legal, but by skip skip is not - it must be written as by skip; skip.

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            The syntax { tacs } is an alternative syntax for · tacs. It runs the tactics in sequence, and fails if the goal is not solved.

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              A sequence of tactics in brackets, or a delimiter-free indented sequence of tactics. Delimiter-free indentation is determined by the first tactic of the sequence.

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                Same as [tacticSeq] but requires delimiter-free tactic sequence to have strict indentation. The strict indentation requirement only apply to nested bys, as top-level bys do not have a position set.

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                      Specify a termination argument for well-founded termination:

                      termination_by a - b
                      

                      indicates that termination of the currently defined recursive function follows because the difference between the the arguments a and b.

                      If the fuction takes further argument after the colon, you can name them as follows:

                      def example (a : Nat) : NatNatNat :=
                      termination_by b c => a - b
                      

                      If omitted, a termination argument will be inferred.

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                        Manually prove that the termination argument (as specified with termination_by or inferred) decreases at each recursive call.

                        By default, the tactic decreasing_tactic is used.

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                          Termination hints are termination_by and decreasing_by, in that order.

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                            Built-in parsers #

                            by tac constructs a term of the expected type by running the tactic(s) tac.

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                                  A type universe. Type ≡ Type 0, Type u ≡ Sort (u + 1).

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                                    A specific universe in Lean's infinite hierarchy of universes.

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                                      The universe of propositions. Prop ≡ Sort 0.

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                                        A placeholder term, to be synthesized by unification.

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                                          Parses a "synthetic hole", that is, ?foo or ?_. This syntax is used to construct named metavariables.

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                                            A temporary placeholder for a missing proof or value.

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                                              A placeholder for an implicit lambda abstraction's variable. The lambda abstraction is scoped to the surrounding parentheses. For example, (· + ·) is equivalent to fun x y => x + y.

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                                                Type ascription notation: (0 : Int) instructs Lean to process 0 as a value of type Int. An empty type ascription (e :) elaborates e without the expected type. This is occasionally useful when Lean's heuristics for filling arguments from the expected type do not yield the right result.

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                                                  Tuple notation; () is short for Unit.unit, (a, b, c) for Prod.mk a (Prod.mk b c), etc.

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                                                    Parentheses, used for grouping expressions (e.g., a * (b + c)). Can also be used for creating simple functions when combined with ·. Here are some examples:

                                                    • (· + 1) is shorthand for fun x => x + 1
                                                    • (· + ·) is shorthand for fun x y => x + y
                                                    • (f · a b) is shorthand for fun x => f x a b
                                                    • (h (· + 1) ·) is shorthand for fun x => h (fun y => y + 1) x
                                                    • also applies to other parentheses-like notations such as (·, 1)
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                                                      The anonymous constructor ⟨e, ...⟩ is equivalent to c e ... if the expected type is an inductive type with a single constructor c. If more terms are given than c has parameters, the remaining arguments are turned into a new anonymous constructor application. For example, ⟨a, b, c⟩ : α × (β × γ) is equivalent to ⟨a, ⟨b, c⟩⟩.

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                                                          A sufficesDecl represents everything that comes after the suffices keyword: an optional x :, then a term ty, then from val or by tac.

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                                                                          Structure instance. { x := e, ... } assigns e to field x, which may be inherited. If e is itself a variable called x, it can be elided: fun y => { x := 1, y }. A structure update of an existing value can be given via with: { point with x := 1 }. The structure type can be specified if not inferable: { x := 1, y := 2 : Point }.

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                                                                              @x disables automatic insertion of implicit parameters of the constant x. @e for any term e also disables the insertion of implicit lambdas at this position.

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                                                                                .(e) marks an "inaccessible pattern", which does not influence evaluation of the pattern match, but may be necessary for type-checking. In contrast to regular patterns, e may be an arbitrary term of the appropriate type.

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                                                                                            Implicit binder. In regular applications without @, it is automatically inserted and solved by unification whenever all explicit parameters before it are specified.

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                                                                                                  Strict-implicit binder. In contrast to { ... } regular implicit binders, a strict-implicit binder is inserted automatically only when at least one subsequent explicit parameter is specified.

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                                                                                                    Instance-implicit binder. In regular applications without @, it is automatically inserted and solved by typeclass inference of the specified class.

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                                                                                                      A bracketedBinder matches any kind of binder group that uses some kind of brackets:

                                                                                                      • An explicit binder like (x y : A)
                                                                                                      • An implicit binder like {x y : A}
                                                                                                      • A strict implicit binder, ⦃y z : A⦄ or its ASCII alternative {{y z : A}}
                                                                                                      • An instance binder [A] or [x : A] (multiple variables are not allowed here)
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                                                                                                              Useful for syntax quotations. Note that generic patterns such as `(matchAltExpr| | ... => $rhs) should also work with other rhsParsers (of arity 1).

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                                                                                                                instance Lean.Parser.Term.instCoeTSyntaxConsSyntaxNodeKindMkStr4Nil :
                                                                                                                Coe (Lean.TSyntax `Lean.Parser.Term.matchAltExpr) (Lean.TSyntax `Lean.Parser.Term.matchAlt)
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                                                                                                                  matchDiscr matches a "match discriminant", either h : tm or tm, used in match as match h1 : e1, e2, h3 : e3 with ....

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                                                                                                                            Pattern matching. match e, ... with | p, ... => f | ... matches each given term e against each pattern p of a match alternative. When all patterns of an alternative match, the match term evaluates to the value of the corresponding right-hand side f with the pattern variables bound to the respective matched values. If used as match h : e, ... with | p, ... => f | ..., h : e = p is available within f.

                                                                                                                            When not constructing a proof, match does not automatically substitute variables matched on in dependent variables' types. Use match (generalizing := true) ... to enforce this.

                                                                                                                            Syntax quotations can also be used in a pattern match. This matches a Syntax value against quotations, pattern variables, or _.

                                                                                                                            Quoted identifiers only match identical identifiers - custom matching such as by the preresolved names only should be done explicitly.

                                                                                                                            Syntax.atoms are ignored during matching by default except when part of a built-in literal. For users introducing new atoms, we recommend wrapping them in dedicated syntax kinds if they should participate in matching. For example, in

                                                                                                                            syntax "c" ("foo" <|> "bar") ...
                                                                                                                            

                                                                                                                            foo and bar are indistinguishable during matching, but in

                                                                                                                            syntax foo := "foo"
                                                                                                                            syntax "c" (foo <|> "bar") ...
                                                                                                                            

                                                                                                                            they are not.

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                                                                                                                              Empty match/ex falso. nomatch e is of arbitrary type α : Sort u if Lean can show that an empty set of patterns is exhaustive given e's type, e.g. because it has no constructors.

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                                                                                                                                                    A literal of type Name.

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                                                                                                                                                      A resolved name literal. Evaluates to the full name of the given constant if existent in the current context, or else fails.

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                                                                                                                                                                  letDecl matches the body of a let declaration let f x1 x2 := e, let pat := e (where pat is an arbitrary term) or let f | pat1 => e1 | pat2 => e2 ... (a pattern matching declaration), except for the let keyword itself. let rec declarations are not handled here.

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                                                                                                                                                                    let is used to declare a local definition. Example:

                                                                                                                                                                    let x := 1
                                                                                                                                                                    let y := x + 1
                                                                                                                                                                    x + y
                                                                                                                                                                    

                                                                                                                                                                    Since functions are first class citizens in Lean, you can use let to declare local functions too.

                                                                                                                                                                    let double := fun x => 2*x
                                                                                                                                                                    double (double 3)
                                                                                                                                                                    

                                                                                                                                                                    For recursive definitions, you should use let rec. You can also perform pattern matching using let. For example, assume p has type Nat × Nat, then you can write

                                                                                                                                                                    let (x, y) := p
                                                                                                                                                                    x + y
                                                                                                                                                                    
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                                                                                                                                                                      let_fun x := v; b is syntax sugar for (fun x => b) v. It is very similar to let x := v; b, but they are not equivalent. In let_fun, the value v has been abstracted away and cannot be accessed in b.

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                                                                                                                                                                        let_delayed x := v; b is similar to let x := v; b, but b is elaborated before v.

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                                                                                                                                                                          let-declaration that is only included in the elaborated term if variable is still there. It is often used when building macros.

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                                                                                                                                                                                  haveDecl matches the body of a have declaration: have := e, have f x1 x2 := e, have pat := e (where pat is an arbitrary term) or have f | pat1 => e1 | pat2 => e2 ... (a pattern matching declaration), except for the have keyword itself.

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                                                                                                                                                                                          attrKind matches ("scoped" <|> "local")?, used before an attribute like @[local simp].

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                                                                                                                                                                                                letRecDecl matches the body of a let-rec declaration: a doc comment, attributes, and then a let declaration without the let keyword, such as /-- foo -/ @[simp] bar := 1.

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                                                                                                                                                                                                  letRecDecls matches letRecDecl,+, a comma-separated list of let-rec declarations (see letRecDecl).

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                                                                                                                                                                                                            binrel% r a b elaborates r a b as a binary relation using the type propogation protocol in Lean.Elab.Extra.

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                                                                                                                                                                                                              binrel_no_prop% r a b is similar to binrel% r a b, but it coerces Prop arguments into Bool.

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                                                                                                                                                                                                                binop% f a b elaborates f a b as a binary operation using the type propogation protocol in Lean.Elab.Extra.

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                                                                                                                                                                                                                  binop_lazy% is similar to binop% f a b, but it wraps b as a function from Unit.

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                                                                                                                                                                                                                    leftact% f a b elaborates f a b as a left action using the type propogation protocol in Lean.Elab.Extra. In particular, it is like a unary operation with a fixed parameter a, where only the right argument b participates in the operator coercion elaborator.

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                                                                                                                                                                                                                      rightact% f a b elaborates f a b as a right action using the type propogation protocol in Lean.Elab.Extra. In particular, it is like a unary operation with a fixed parameter b, where only the left argument a participates in the operator coercion elaborator.

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                                                                                                                                                                                                                        unop% f a elaborates f a as a unary operation using the type propogation protocol in Lean.Elab.Extra.

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                                                                                                                                                                                                                              A macro which evaluates to the name of the currently elaborating declaration.

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                                                                                                                                                                                                                                • with_decl_name% id e elaborates e in a context while changing the effective declaration name to id.
                                                                                                                                                                                                                                • with_decl_name% ?id e does the same, but resolves id as a new definition name (appending the current namespaces).
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                                                                                                                                                                                                                                          clear% x; e elaborates x after clearing the free variable x from the local context. If x cannot be cleared (due to dependencies), it will keep x without failing.

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                                                                                                                                                                                                                                                      Helper parser for marking match-alternatives that should not trigger errors if unused. We use them to implement macro_rules and elab_rules

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                                                                                                                                                                                                                                                              The extended field notation e.f is roughly short for T.f e where T is the type of e. More precisely,

                                                                                                                                                                                                                                                              • if e is of a function type, e.f is translated to Function.f (p := e) where p is the first explicit parameter of function type
                                                                                                                                                                                                                                                              • if e is of a named type T ... and there is a declaration T.f (possibly from export), e.f is translated to T.f (p := e) where p is the first explicit parameter of type T ...
                                                                                                                                                                                                                                                              • otherwise, if e is of a structure type, the above is repeated for every base type of the structure.

                                                                                                                                                                                                                                                              The field index notation e.i, where i is a positive number, is short for accessing the i-th field (1-indexed) of e if it is of a structure type.

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                                                                                                                                                                                                                                                                    x.{u, ...} explicitly specifies the universes u, ... of the constant x.

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                                                                                                                                                                                                                                                                      x@e or x:h@e matches the pattern e and binds its value to the identifier x. If present, the identifier h is bound to a proof of x = e.

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                                                                                                                                                                                                                                                                        e |>.x is a shorthand for (e).x. It is especially useful for avoiding parentheses with repeated applications.

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                                                                                                                                                                                                                                                                          h ▸ e is a macro built on top of Eq.rec and Eq.symm definitions. Given h : a = b and e : p a, the term h ▸ e has type p b. You can also view h ▸ e as a "type casting" operation where you change the type of e by using h. See the Chapter "Quantifiers and Equality" in the manual "Theorem Proving in Lean" for additional information.

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                                                                                                                                                                                                                                                                              instance Lean.Parser.Term.instCoeTSyntaxConsSyntaxNodeKindMkStr4Nil_1 :
                                                                                                                                                                                                                                                                              Coe (Lean.TSyntax `Lean.Parser.Term.bracketedBinderF) (Lean.TSyntax `Lean.Parser.Term.bracketedBinder)
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                                                                                                                                                                                                                                                                              panic! msg formally evaluates to @Inhabited.default α if the expected type α implements Inhabited. At runtime, msg and the file position are printed to stderr unless the C function lean_set_panic_messages(false) has been executed before. If the C function lean_set_exit_on_panic(true) has been executed before, the process is then aborted.

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                                                                                                                                                                                                                                                                                A shorthand for panic! "unreachable code has been reached".

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                                                                                                                                                                                                                                                                                  dbg_trace e; body evaluates to body and prints e (which can be an interpolated string literal) to stderr. It should only be used for debugging.

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                                                                                                                                                                                                                                                                                    assert! cond panics if cond evaluates to false.

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