mathlib documentation

core / init.meta.expr

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structure pos  :
Type

Column and line position in a Lean source file.

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@[protected, instance]
def pos.decidable_eq  :
decidable_eq pos
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@[protected, instance]
meta def pos.has_to_format  :
has_to_format pos
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inductive binder_info  :
Type

Auxiliary annotation for binders (Lambda and Pi). This information is only used for elaboration. The difference between {} and ⦃⦄ is how implicit arguments are treated that are not followed by explicit arguments. {} arguments are applied eagerly, while ⦃⦄ arguments are left partially applied:

def foo {x : ℕ} : ℕ := x
def bar ⦃x : ℕ⦄ : ℕ := x
#check foo -- foo : ℕ
#check bar -- bar : Π ⦃x : ℕ⦄, ℕ
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@[protected, instance]
def binder_info.has_repr  :
has_repr binder_info
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meta constant macro_def  :
Type

Macros are basically "promises" to build an expr by some C++ code, you can't build them in Lean. You can unfold a macro and force it to evaluate. They are used for

  • sorry.
  • Term placeholders (_) in pexprs.
  • Expression annotations. See expr.is_annotation.
  • Meta-recursive calls. Eg: 
    meta def Y : (α → α) → α | f := f (Y f)
    The Y that appears in f (Y f) is a macro.
  • Builtin projections: 
    structure foo := (mynat : ℕ)
#print foo.mynat
-- @[reducible]
-- def foo.mynat : foo → ℕ :=
-- λ (c : foo), [foo.mynat c]
    The thing in square brackets is a macro.
  • Ephemeral structures inside certain specialised C++ implemented tactics.
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meta inductive expr (elaborated : bool := tt) :
Type

An expression. eg (4+5).

The elab flag is indicates whether the expr has been elaborated and doesn't contain any placeholder macros. For example the equality x = x is represented in expr ff as app (app (const `eq _) x) x while in expr tt it is represented as app (app (app (const `eq _) t) x) x (one more argument). The VM replaces instances of this datatype with the C++ implementation.

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@[protected, instance]
meta def expr.inhabited  :
inhabited expr
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meta constant expr.macro_def_name (d : macro_def) :
name

Get the name of the macro definition.

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meta def expr.mk_var (n : ℕ) :
expr
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meta constant expr.is_annotation {elab : bool} :
expr elab → option (name × expr elab)

Expressions can be annotated using an annotation macro during compilation. For example, a have x:X, from p, q expression will be compiled to (λ x:X,q)(p), but nested in an annotation macro with the name "have". These annotations have no real semantic meaning, but are useful for helping Lean's pretty printer.

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meta constant expr.is_string_macro {elab : bool} :
expr elab → option (expr elab)
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meta def expr.erase_annotations {elab : bool} :
expr elab → expr elab

Remove all macro annotations from the given expr.

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@[instance]
meta constant expr.has_decidable_eq  :
decidable_eq expr

Compares expressions, including binder names.

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meta constant expr.alpha_eqv  :
expr → expr → bool

Compares expressions while ignoring binder names.

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@[protected]
meta constant expr.to_string {elab : bool} :
expr elab → string
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@[protected, instance]
meta def expr.has_to_string {elab : bool} :
has_to_string (expr elab)
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@[protected, instance]
meta def expr.has_to_format {elab : bool} :
has_to_format (expr elab)
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@[protected, instance]
meta def expr.has_coe_to_fun {elab : bool} :
has_coe_to_fun (expr elab) (λ (e : expr elab), expr elab → expr elab)

Coercion for letting users write (f a) instead of (expr.app f a)

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meta constant expr.hash  :
expr → ℕ

Each expression created by Lean carries a hash. This is calculated upon creation of the expression. Two structurally equal expressions will have the same hash.

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meta constant expr.lt  :
expr → expr → bool

Compares expressions, ignoring binder names, and sorting by hash.

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meta constant expr.lex_lt  :
expr → expr → bool

Compares expressions, ignoring binder names.

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meta constant expr.fold {α : Type} :
expr → α → (expr → ℕ → α → α) → α

expr.fold e a f: Traverses each subexpression of e. The nat passed to the folder f is the binder depth.

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meta constant expr.replace  :
expr → (expr → ℕ → option expr) → expr

expr.replace e f Traverse over an expr e with a function f which can decide to replace subexpressions or not. For each subexpression s in the expression tree, f s n is called where n is how many binders are present above the given subexpression s. If f s n returns none, the children of s will be traversed. Otherwise if some s' is returned, s' will replace s and this subexpression will not be traversed further.

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meta constant expr.abstract_local  :
expr → name → expr

abstract_local e n replaces each instance of the local constant with unique (not pretty) name n in e with a de-Bruijn variable.

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meta constant expr.abstract_locals  :
expr → list name → expr

Multi version of abstract_local. Note that the given expression will only be traversed once, so this is not the same as list.foldl expr.abstract_local.

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meta def expr.abstract  :
expr → expr → expr

abstract e x Abstracts the expression e over the local constant x.

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meta constant expr.instantiate_univ_params  :
expr → list (name × level) → expr

Expressions depend on levels, and these may depend on universe parameters which have names. instantiate_univ_params e [(n₁,l₁), ...] will traverse e and replace any universe parameters with name nᵢ with the corresponding level lᵢ.

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meta constant expr.instantiate_nth_var  :
ℕ → expr → expr → expr

instantiate_nth_var n a b takes the nth de-Bruijn variable in a and replaces each occurrence with b.

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meta constant expr.instantiate_var  :
expr → expr → expr

instantiate_var a b takes the 0th de-Bruijn variable in a and replaces each occurrence with b.

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meta constant expr.instantiate_vars  :
expr → list expr → expr

instantiate_vars `(#0 #1 #2) [x,y,z] = `(%%x %%y %%z)

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meta constant expr.instantiate_vars_core  :
expr → ℕ → list expr → expr

Same as instantiate_vars except lifts and shifts the vars by the given amount. instantiate_vars_core `(#0 #1 #2 #3) 0 [x,y] = `(x y #0 #1) instantiate_vars_core `(#0 #1 #2 #3) 1 [x,y] = `(#0 x y #1) instantiate_vars_core `(#0 #1 #2 #3) 2 [x,y] = `(#0 #1 x y)

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@[protected]
meta constant expr.subst {elab : bool} :
expr elab → expr elab → expr elab

Perform beta-reduction if the left expression is a lambda, or construct an application otherwise. That is: expr.subst `(λ x, %%Y) Z = Y[x/Z], and expr.subst X Z = X.app Z otherwise

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meta constant expr.get_free_var_range  :
expr → ℕ

get_free_var_range e returns one plus the maximum de-Bruijn value in e. Eg get_free_var_range(#1 #0)yields2`

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meta constant expr.has_var  :
expr → bool

has_var e returns true iff e has free variables.

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meta constant expr.has_var_idx  :
expr → ℕ → bool

has_var_idx e n returns true iff e has a free variable with de-Bruijn index n.

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meta constant expr.has_local  :
expr → bool

has_local e returns true if e contains a local constant.

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meta constant expr.has_meta_var  :
expr → bool

has_meta_var e returns true iff e contains a metavariable.

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meta constant expr.lower_vars  :
expr → ℕ → ℕ → expr

lower_vars e s d lowers the free variables >= s in e by d. Note that this can cause variable clashes. examples:

  • lower_vars `(#2 #1 #0) 1 1 = `(#1 #0 #0)
  • lower_vars `(λ x, #2 #1 #0) 1 1 = `(λ x, #1 #1 #0 )
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meta constant expr.lift_vars  :
expr → ℕ → ℕ → expr

Lifts free variables. lift_vars e s d will lift all free variables with index ≥ s in e by d.

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@[protected]
meta constant expr.pos {elab : bool} :
expr elab → option pos

Get the position of the given expression in the Lean source file, if anywhere.

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meta constant expr.copy_pos_info  :
expr → expr → expr

copy_pos_info src tgt copies position information from src to tgt.

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meta constant expr.is_internal_cnstr  :
expr → option unsigned

Returns some n when the given expression is a constant with the name ..._cnstr.n

is_internal_cnstr : expr → option unsigned
|(const (mk_numeral n (mk_string "_cnstr" _)) _) := some n
|_ := none

[NOTE] This is not used anywhere in core Lean.

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meta constant expr.get_nat_value  :
expr → option ℕ

There is a macro called a "nat_value_macro" holding a natural number which are used during compilation. This function extracts that to a natural number. [NOTE] This is not used anywhere in Lean.

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meta constant expr.collect_univ_params  :
expr → list name

Get a list of all of the universe parameters that the given expression depends on.

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meta constant expr.occurs  :
expr → expr → bool

occurs e t returns tt iff e occurs in t up to α-equivalence. Purely structural: no unification or definitional equality.

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meta constant expr.has_local_in  :
expr → name_set → bool

Returns true if any of the names in the given name_set are present in the given expr.

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meta constant expr.get_weight  :
expr → ℕ

Computes the number of sub-expressions (constant time).

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meta constant expr.get_depth  :
expr → ℕ

Computes the maximum depth of the expression (constant time).

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meta constant expr.mk_delayed_abstraction  :
expr → list name → expr

mk_delayed_abstraction m ls creates a delayed abstraction on the metavariable m with the unique names of the local constants ls. If m is not a metavariable then this is equivalent to abstract_locals.

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meta constant expr.get_delayed_abstraction_locals  :
expr → option (list name)

If the given expression is a delayed abstraction macro, return some ls where ls is a list of unique names of locals that will be abstracted.

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@[class]
meta def reflected {α : Sort u} :
α → Type

(reflected a) is a special opaque container for a closed expr representing a. It can only be obtained via type class inference, which will use the representation of a in the calling context. Local constants in the representation are replaced by nested inference of reflected instances.

The quotation expression `(a) (outside of patterns) is equivalent to reflect a and thus can be used as an explicit way of inferring an instance of reflected a.

Instances
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meta def reflected.to_expr {α : Sort u} {a : α} :
reflected a → expr
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meta def reflected.subst {α : Sort v} {β : α → Sort u} {f : Π (a : α), β a} {a : α} :
reflected f → reflected a → reflected (f a)
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@[protected, instance]
meta constant expr.reflect {elab : bool} (e : expr elab) :
reflected e
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@[protected, instance]
meta constant string.reflect (s : string) :
reflected s
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@[protected, instance]
meta def expr.has_coe {α : Sort u} (a : α) :
has_coe (reflected a) expr
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@[protected]
meta def reflect {α : Sort u} (a : α) [h : reflected a] :
reflected a
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@[protected, instance]
meta def reflected.has_to_format {α : Sort u_1} (a : α) :
has_to_format (reflected a)
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meta def expr.lt_prop (a b : expr) :
Prop
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@[protected, instance]
meta def expr.lt_prop.decidable_rel  :
decidable_rel expr.lt_prop
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@[protected, instance]
meta def expr.has_lt  :
has_lt expr

Compares expressions, ignoring binder names, and sorting by hash.

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meta def expr.mk_true  :
expr
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meta def expr.mk_false  :
expr
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meta constant expr.mk_sorry (type : expr) :
expr

Returns the sorry macro with the given type.

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meta constant expr.is_sorry (e : expr) :
option expr

Checks whether e is sorry, and returns its type.

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meta def expr.instantiate_local (n : name) (s e : expr) :
expr

Replace each instance of the local constant with name n by the expression s in e.

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meta def expr.instantiate_locals (s : list (name × expr)) (e : expr) :
expr
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meta def expr.is_var  :
expr → bool
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meta def expr.app_of_list  :
expr → list expr → expr
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meta def expr.is_app  :
expr → bool
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meta def expr.app_fn  :
expr → expr
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meta def expr.app_arg  :
expr → expr
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meta def expr.get_app_fn {elab : bool} :
expr elab → expr elab
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meta def expr.get_app_num_args  :
expr → ℕ
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meta def expr.get_app_args_aux  :
list expr → expr → list expr
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meta def expr.get_app_args  :
expr → list expr
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meta def expr.mk_app  :
expr → list expr → expr
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meta def expr.mk_binding (ctor : name → binder_info → expr → expr → expr) (e l : expr) :
expr
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meta def expr.bind_pi (e l : expr) :
expr

(bind_pi e l) abstracts and pi-binds the local l in e

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meta def expr.bind_lambda (e l : expr) :
expr

(bind_lambda e l) abstracts and lambda-binds the local l in e

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meta def expr.ith_arg_aux  :
expr → ℕ → expr
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meta def expr.ith_arg (e : expr) (i : ℕ) :
expr
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meta def expr.const_name {elab : bool} :
expr elab → name
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meta def expr.is_constant {elab : bool} :
expr elab → bool
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meta def expr.is_local_constant  :
expr → bool
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meta def expr.local_uniq_name  :
expr → name
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meta def expr.local_pp_name {elab : bool} :
expr elab → name
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meta def expr.local_type {elab : bool} :
expr elab → expr elab
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meta def expr.is_aux_decl  :
expr → bool
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meta def expr.is_constant_of {elab : bool} :
expr elab → name → bool
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meta def expr.is_app_of (e : expr) (n : name) :
bool
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meta def expr.is_napp_of (e : expr) (c : name) (n : ℕ) :
bool

The same as is_app_of but must also have exactly n arguments.

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meta def expr.is_false  :
expr → bool
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meta def expr.is_not  :
expr → option expr
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meta def expr.is_and  :
expr → option (expr × expr)
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meta def expr.is_or  :
expr → option (expr × expr)
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meta def expr.is_iff  :
expr → option (expr × expr)
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meta def expr.is_eq  :
expr → option (expr × expr)
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meta def expr.is_ne  :
expr → option (expr × expr)
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meta def expr.is_bin_arith_app (e : expr) (op : name) :
option (expr × expr)
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meta def expr.is_lt (e : expr) :
option (expr × expr)
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meta def expr.is_gt (e : expr) :
option (expr × expr)
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meta def expr.is_le (e : expr) :
option (expr × expr)
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meta def expr.is_ge (e : expr) :
option (expr × expr)
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meta def expr.is_heq  :
expr → option (expr × expr × expr × expr)
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meta def expr.is_lambda  :
expr → bool
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meta def expr.is_pi  :
expr → bool
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meta def expr.is_arrow  :
expr → bool
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meta def expr.is_let  :
expr → bool
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meta def expr.binding_name  :
expr → name

The name of the bound variable in a pi, lambda or let expression.

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meta def expr.binding_info  :
expr → binder_info

The binder info of a pi or lambda expression.

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meta def expr.binding_domain  :
expr → expr

The domain (type of bound variable) of a pi, lambda or let expression.

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meta def expr.binding_body  :
expr → expr

The body of a pi, lambda or let expression. This definition doesn't instantiate bound variables, and therefore produces a term that is open. See note [open expressions] in mathlib.

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meta def expr.nth_binding_body  :
ℕ → expr → expr

nth_binding_body n e iterates binding_body n times to an iterated pi expression e. This definition doesn't instantiate bound variables, and therefore produces a term that is open. See note [open expressions] in mathlib.

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meta def expr.is_macro  :
expr → bool
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meta def expr.is_numeral  :
expr → bool
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meta def expr.pi_arity  :
expr → ℕ
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meta def expr.lam_arity  :
expr → ℕ
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meta def expr.imp (a b : expr) :
expr
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meta def expr.lambdas  :
list expr → expr → expr

lambdas cs e lambda binds e with each of the local constants in cs.

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meta def expr.pis  :
list expr → expr → expr

Same as expr.lambdas but with pi.

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meta def expr.extract_opt_auto_param  :
expr → expr
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meta def expr.to_raw_fmt {elab : bool} :
expr elab → format
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meta def expr.mfold {α : Type} {m : Type → Type} [monad m] (e : expr) (a : α) (fn : expr → ℕ → α → m α) :
m α

Fold an accumulator a over each subexpression in the expression e. The nat passed to fn is the number of binders above the subexpression.

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meta def expr_map (data : Type) :
Type

An dictionary from data to expressions.

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meta def expr_map.mk (data : Type) :
expr_map data
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meta def mk_expr_map {data : Type} :
expr_map data
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meta def expr_set  :
Type
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meta def mk_expr_set  :
expr_set