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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                    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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                                  The term denotes a term that was omitted by the pretty printer. The presence of in pretty printer output is controlled by the pp.deepTerms and pp.proofs options, and these options can be further adjusted using pp.deepTerms.threshold and pp.proofs.threshold.

                                  It is only meant to be used for pretty printing. However, in case it is copied and pasted from the Infoview, logs a warning and elaborates like _.

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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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                                                                                  Explicit binder, like (x y : A) or (x y). Default values can be specified using (x : A := v) syntax, and tactics using (x : A := by tac).

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                                                                                    Implicit binder, like {x y : A} or {x y}. In regular applications, whenever all parameters before it have been specified, then a _ placeholder is automatically inserted for this parameter. Implicit parameters should be able to be determined from the other arguments and the return type by unification.

                                                                                    In @ explicit mode, implicit binders behave like explicit binders.

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                                                                                          Strict-implicit binder, like ⦃x y : A⦄ or ⦃x y⦄. In contrast to { ... } implicit binders, strict-implicit binders do not automatically insert a _ placeholder until at least one subsequent explicit parameter is specified. Do not use strict-implicit binders unless there is a subsequent explicit parameter. Assuming this rule is followed, for fully applied expressions implicit and strict-implicit binders have the same behavior.

                                                                                          Example: If h : ∀ ⦃x : A⦄, x ∈ s → p x and hs : y ∈ s, then h by itself elaborates to itself without inserting _ for the x : A parameter, and h hs has type p y. In contrast, if h' : ∀ {x : A}, x ∈ s → p x, then h by itself elaborates to have type ?m ∈ s → p ?m with ?m a fresh metavariable.

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                                                                                            Instance-implicit binder, like [C] or [inst : C]. In regular applications without @ explicit mode, it is automatically inserted and solved for by typeclass inference for the specified class C. In @ explicit mode, if _ is used for an an instance-implicit parameter, then it is still solved for by typeclass inference; use (_) to inhibit this and have it be solved for by unification instead, like an implicit argument.

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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_lean :
                                                                                                        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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                                                                                                                                                                                haveI behaves like have, but inlines the value instead of producing a let_fun term.

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                                                                                                                                                                                  letI behaves like let, but inlines the value instead of producing a let_fun term.

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

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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. If written as termination_by?, the inferrred termination argument will be suggested.

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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. If written as termination_by?, the inferrred termination argument will be suggested.

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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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                                                                                                                                                                                                      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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                                                                                                                                                                                                                  unsafe t : α is an expression constructor which allows using unsafe declarations inside the body of t : α, by creating an auxiliary definition containing t and using implementedBy to wrap it in a safe interface. It is required that α is nonempty for this to be sound, but even beyond that, an unsafe block should be carefully inspected for memory safety because the compiler is unable to guarantee the safety of the operation.

                                                                                                                                                                                                                  For example, the evalExpr function is unsafe, because the compiler cannot guarantee that when you call evalExpr Foo ``Foo e that the type Foo corresponds to the name Foo, but in a particular use case, we can ensure this, so unsafe (evalExpr Foo ``Foo e) is a correct usage.

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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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                                                                                                                                                                                                                                                                            Syntax kind for syntax nodes representing the field of a projection in the InfoTree. Specifically, the InfoTree node for a projection s.f contains a child InfoTree node with syntax (Syntax.node .none identProjKind #[`f]).

                                                                                                                                                                                                                                                                            This is necessary because projection syntax cannot always be detected purely syntactically (s.f may refer to either the identifier s.f or a projection s.f depending on the available context).

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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.

                                                                                                                                                                                                                                                                                      The macro tries both orientations of h. If the context provides an expected type, it rewrites the expeced type, else it rewrites the type of e`.

                                                                                                                                                                                                                                                                                      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_lean_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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                                                                                                                                                                                                                                                                                                          Implementation of the show_term term elaborator.

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                                                                                                                                                                                                                                                                                                            match_expr support.

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