mathlib documentation

Reflexivity conversion: given e returns (e, ⊢ e = e)

Turns a conversion tactic into one that always succeeds, where failure is interpreted as a proof by reflexivity.

Transitivity conversion: given two conversions (which take an expression e and returns (e', ⊢ e = e')), produces another conversion that combines them with transitivity, treating failures as reflexivity conversions.

Given an expr α representing a type with numeral structure, of_nat α n creates the α-valued numeral expression corresponding to n.

Given an expr α representing a type with numeral structure, of_int α n creates the α-valued numeral expression corresponding to n. The output is either a numeral or the negation of a numeral.

Convert a list of expressions to an expression denoting the list of those expressions.

Generates an expression of the form ∃(args), inner. args is assumed to be a list of local constants. When possible, p ∧ q is used instead of ∃(_ : p), q.

traverse f e applies the monadic function f to the direct descendants of e.

mfoldl f a e folds the monadic function f over the subterms of the expression e, with initial value a.

kreplace e old new replaces all occurrences of the expression old in e with new. The occurrences of old in e are determined using keyed matching with transparency md; see kabstract for details. If unify is true, we may assign metavariables in e as we match subterms of e against old.

pre.contains_sorry_aux nm checks whether sorry occurs in the value of the declaration nm or (recusively) in any declarations occurring in the value of nm with namespace pre. Auxiliary function for name.contains_sorry.

nm.contains_sorry checks whether sorry occurs in the value of the declaration nm or in any declarations nm._proof_i (or to be more precise: any declaration in namespace nm). See also expr.contains_sorry.

get_state returns the underlying state inside an interaction monad, from within that monad.

set_state sets the underlying state inside an interaction monad, from within that monad.

run_with_state state tac applies tac to the given state state and returns the result, subsequently restoring the original state. If tac fails, then run_with_state does too.

join' [a,b,c] produces the format object abc. It differs from format.join by using format.nil instead of "" for the empty list.

intercalate x [a, b, c] produces the format object a.x.b.x.c, where . represents format.join.

soft_break is similar to line. Whereas in group (x ++ line ++ y ++ line ++ z) the result either fits on one line or in three, x ++ soft_break ++ y ++ soft_break ++ z each line break is decided independently

Format a list as a comma separated list, without any brackets.

format a list by separating elements with soft_break instead of line

add_local_consts_as_local_hyps vars add the given list vars of expr.local_consts to the tactic state. This is harder than it sounds, since the list of local constants which we have been passed can have dependencies between their types.

For example, suppose we have two local constants n : ℕ and h : n = 3. Then we cannot blindly add h as a local hypothesis, since we need the n to which it refers to be the n created as a new local hypothesis, not the old local constant n with the same name. Of course, these dependencies can be nested arbitrarily deep.

If the list of passed local constants have types which depend on one another (which can only happen by hand-crafting the exprs manually), this function will loop forever.

Compute the arity of explicit arguments of type.

Compute the arity of explicit arguments of fn's type.

For e = f x₁ ... xₙ, get_app_fn_args_whnf e returns (f, [x₁, ..., xₙ]). e is normalised as necessary; for example:

get_app_fn_args_whnf `(let f := g x in f y) = (`(g), [`(x), `(y)])

The returned expression is in whnf, but the arguments are generally not.

get_app_fn_whnf e md unfold_ginductive is like expr.get_app_fn e but e is normalised as necessary (with transparency md). unfold_ginductive controls whether constructors of generalised inductive types are unfolded. The returned expression is in whnf.

get_app_fn_const_whnf e md unfold_ginductive expects that e = C x₁ ... xₙ, where C is a constant, after normalisation with transparency md. If so, the name of C is returned. Otherwise the tactic fails. unfold_ginductive controls whether constructors of generalised inductive types are unfolded.

get_app_args_whnf e md unfold_ginductive is like expr.get_app_args e but e is normalised as necessary (with transparency md). unfold_ginductive controls whether constructors of generalised inductive types are unfolded. The returned expressions are not necessarily in whnf.

pis loc_consts f is used to create a pi expression whose body is f. loc_consts should be a list of local constants. The function will abstract these local constants from f and bind them with pi binders.

For example, if a, b are local constants with types Ta, Tb, pis [a, b] `(f a b) will return the expression Π (a : Ta) (b : Tb), f a b.

lambdas loc_consts f is used to create a lambda expression whose body is f. loc_consts should be a list of local constants. The function will abstract these local constants from f and bind them with lambda binders.

For example, if a, b are local constants with types Ta, Tb, lambdas [a, b] `(f a b) will return the expression λ (a : Ta) (b : Tb), f a b.

mk_theorem n ls t e creates a theorem declaration with name n, universe parameters named ls, type t, and body e.

add_theorem_by n ls type tac uses tac to synthesize a term with type type, and adds this to the environment as a theorem with name n and universe parameters ls.

eval_expr' α e attempts to evaluate the expression e in the type α. This is a variant of eval_expr in core. Due to unexplained behavior in the VM, in rare situations the latter will fail but the former will succeed.

mk_fresh_name returns identifiers starting with underscores, which are not legal when emitted by tactic programs. mk_user_fresh_name turns the useful source of random names provided by mk_fresh_name into names which are usable by tactic programs.

The returned name has four components which are all strings.

has_attribute' attr_name decl_name checks whether decl_name exists and has attribute attr_name.

Checks whether the name is a simp lemma

Checks whether the name is an instance.

local_decls returns a dictionary mapping names to their corresponding declarations. Covers all declarations from the current file.

get_decls_from returns a dictionary mapping names to their corresponding declarations. Covers all declarations the files listed in fs, with the current file listed as none.

The path of the file names is expected to be relative to the root of the project (i.e. the location of leanpkg.toml when it is present); e.g. "src/tactic/core.lean"

Possible issue: get_decls_from uses get_cwd, the current working directory, which may not always point at the root of the project. It would work better if it searched for the root directory or, better yet, if Lean exposed its path information.

If {nm}_{n} doesn't exist in the environment, returns that, otherwise tries {nm}_{n+1}

Return a name which doesn't already exist in the environment. If nm doesn't exist, it returns that, otherwise it tries nm_2, nm_3, ...

Returns a pair (e, t), where e ← mk_const d.to_name, and t = d.type but with universe params updated to match the fresh universe metavariables in e.

This should have the same effect as just

do e ← mk_const d.to_name,
   t ← infer_type e,
   return (e, t)

but is hopefully faster.

Replace every universe metavariable in an expression with a universe parameter.

(This is useful when making new declarations.)

mk_local n creates a dummy local variable with name n. The type of this local constant is a constant with name n, so it is very unlikely to be a meaningful expression.

mk_psigma [x,y,z], with [x,y,z] list of local constants of types x : tx, y : ty x and z : tz x y, creates an expression of sigma type: ⟨x,y,z⟩ : Σ' (x : tx) (y : ty x), tz x y.

Update the type of a local constant or metavariable. For local constants and metavariables obtained via, for example, tactic.get_local, the type stored in the expression is not necessarily the same as the type returned by infer_type. This tactic, given a local constant or metavariable, updates the stored type to match the output of infer_type. If the input is not a local constant or metavariable, update_type does nothing.

elim_gen_prod n e _ ns with e an expression of type psigma _, applies cases on e n times and uses ns to name the resulting variables. Returns a triple: list of new variables, remaining term and unused variable names.

elim_gen_sum n e applies cases on e n times. e is assumed to be a local constant whose type is a (nested) sum ⊕. Returns the list of local constants representing the components of e.

Given elab_def, a tactic to solve the current goal, extract_def n trusted elab_def will create an auxiliary definition named n and use it to close the goal. If trusted is false, it will be a meta definition.

Attempts to close the goal with dec_trivial.

Runs a tactic for a result, reverting the state after completion.

Runs a tactic for a result, reverting the state after completion or error.

Repeat a tactic at least once, calling it recursively on all subgoals, until it fails. This tactic fails if the first invocation fails.

iterate_range m n t: Repeat the given tactic at least m times and at most n times or until t fails. Fails if t does not run at least m times.

Given a tactic tac that takes an expression and returns a new expression and a proof of equality, use that tactic to change the type of the hypotheses listed in hs, as well as the goal if tgt = tt.

Returns tt if any types were successfully changed.

revert_after e reverts all local constants after local constant e.

revert_target_deps reverts all local constants on which the target depends (recursively). Returns the number of local constants that have been reverted.

generalize' e n generalizes the target with respect to e. It creates a new local constant with name n of the same type as e and replaces all occurrences of e by n.

generalize' is similar to generalize but also succeeds when e does not occur in the goal, in which case it just calls assert. In contrast to generalize it already introduces the generalized variable.

intron_no_renames n calls intro n times, using the pretty-printing name provided by the binder to name the new local constant. Unlike intron, it does not rename introduced constants if the names shadow existing constants.

get_univ_level t returns the universe level of a type t

local_def_value e returns the value of the expression e, assuming that e has been defined locally using a let expression. Otherwise it fails.

is_local_def e succeeds when e is a local definition (a local constant of the form e : α := t) and otherwise fails.

Returns the local definitions from the context. A local definition is a local constant of the form e : α := t. The local definitions are returned in the order in which they appear in the context.

partition_local_deps vs, with vs a list of local constants, reorders vs in the order they appear in the local context together with the variables that follow them. If local context is [a,b,c,d,e,f], and that we call partition_local_deps [d,b], we get [[d,e,f], [b,c]]. The head of each list is one of the variables given as a parameter.

clear_value [e₀, e₁, e₂, ...] clears the body of the local definitions e₀, e₁, e₂, ... changing them into regular hypotheses. A hypothesis e : α := t is changed to e : α. The order of locals e₀, e₁, e₂ does not matter as a permutation will be chosen so as to preserve type correctness. This tactic is called clearbody in Coq.

context_has_local_def is true iff there is at least one local definition in the context.

context_upto_hyp_has_local_def h is true iff any of the hypotheses in the context up to and including h is a local definition.

If the expression h is a local variable with type x = t or t = x, where x is a local constant, tactic.subst' h substitutes x by t everywhere in the main goal and then clears h. If h is another local variable, then we find a local constant with type h = t or t = h and substitute t for h.

This is like tactic.subst, but fails with a nicer error message if the substituted variable is a local definition. It is trickier to fix this in core, since tactic.is_local_def is in mathlib.

A variant of simplify_bottom_up. Given a tactic post for rewriting subexpressions, simp_bottom_up post e tries to rewrite e starting at the leaf nodes. Returns the resulting expression and a proof of equality.

Creates an instance_cache for the type α.

If n is the name of a type class with one parameter, get c n tries to find an instance of n c.α by checking the cache c. If there is no entry in the cache, it tries to find the instance via type class resolution, and updates the cache.

If e is a pi expression that binds an instance-implicit variable of type n, append_typeclasses e c l searches c for an instance p of type n and returns p :: l.

Creates the application n c.α p l, where p is a type class instance found in the cache c.

c.of_nat n creates the c.α-valued numeral expression corresponding to n.

c.of_int n creates the c.α-valued numeral expression corresponding to n. The output is either a numeral or the negation of a numeral.

A variation on assert where a (possibly incomplete) proof of the assertion is provided as a parameter.

(h,gs) ← local_proof `h p tac creates a local h : p and use tac to (partially) construct a proof for it. gs is the list of remaining goals in the proof of h.

The benefits over assert are:

• unlike with h ← assert `h p, tac , h cannot be used by tac;
• when tac does not complete the proof of h, returning the list of goals allows one to write a tactic using h and with the confidence that a proof will not boil over to goals left over from the proof of h, unlike what would be the case when using tactic.swap.

var_names e returns a list of the unique names of the initial pi bindings in e.

When struct_n is the name of a structure type, subobject_names struct_n returns two lists of names (instances, fields). The names in instances are the projections from struct_n to the structures that it extends (assuming it was defined with old_structure_cmd false). The names in fields are the standard fields of struct_n.

expanded_field_list struct_n produces a list of the names of the fields of the structure named struct_n. These are returned as pairs of names (prefix, name), where the full name of the projection is prefix.name.

struct_n cannot be a synonym for a structure, it must be itself a structure

Return a list of all type classes which can be instantiated for the given expression.

Finds an instance of an implication cond → tgt. Returns a pair of a local constant e of type cond, and an instance of tgt that can mention e. The local constant e is added as an hypothesis to the tactic state, but should not be used, since it has been "proven" by a metavariable.

Create a list of n fresh metavariables.

Returns the only goal, or fails if there isn't just one goal.

iterate_at_most_on_all_goals n t: repeat the given tactic at most n times on all goals, or until it fails. Always succeeds.

iterate_at_most_on_subgoals n t: repeat the tactic t at most n times on the first goal and on all subgoals thus produced, or until it fails. Fails iff t fails on current goal.

This makes sure that the execution of the tactic does not change the tactic state. This can be helpful while using rewrite, apply, or expr munging. Remember to instantiate your metavariables before you're done!

apply_list l, for l : list (tactic expr), tries to apply the lemmas generated by the tactics in l on the first goal, and fail if none succeeds.

Constructs a list of tactic expr given a list of p-expressions, as follows:

• if the p-expression is the name of a theorem, use i_to_expr_for_apply on it
• if the p-expression is a user attribute, add all the theorems with this attribute to the list.

We need to return a list of tactic expr, rather than just expr, because these expressions will be repeatedly applied against goals, and we need to ensure that metavariables don't get stuck.

apply_rules hs n: apply the list of rules hs (given as pexpr) and assumption on the first goal and the resulting subgoals, iteratively, at most n times.

Unlike solve_by_elim, apply_rules does not do any backtracking, and just greedily applies a lemma from the list until it can't.

replace h p elaborates the pexpr p, clears the existing hypothesis named h from the local context, and adds a new hypothesis named h. The type of this hypothesis is the type of p. Fails if there is nothing named h in the local context.

Auxiliary function for iff_mp and iff_mpr. Takes a name, which should be either `iff.mp or `iff.mpr. If the passed expression is an iterated function type eventually producing an iff, returns an expression with the iff converted to either the forwards or backwards implication, as requested.

iff_mp_core e ty assumes that ty is the type of e. If ty has the shape Π ..., A ↔ B, returns an expression whose type is Π ..., A → B.

iff_mpr_core e ty assumes that ty is the type of e. If ty has the shape Π ..., A ↔ B, returns an expression whose type is Π ..., B → A.

Given an expression whose type is (a possibly iterated function producing) an iff, create the expression which is the forward implication.

Given an expression whose type is (a possibly iterated function producing) an iff, create the expression which is the reverse implication.

Attempts to apply e, and if that fails, if e is an iff, try applying both directions separately.

This is a version of apply_any that takes a list of tactic exprs instead of exprs, and evaluates these as thunks before trying to apply them.

We need to do this to avoid metavariables getting stuck during subsequent rounds of apply.

apply_any lemmas tries to apply one of the list lemmas to the current goal.

apply_any lemmas opt allows control over how lemmas are applied. opt has fields:

• use_symmetry: if no lemma applies, call symmetry and try again. (Defaults to tt.)
• use_exfalso: if no lemma applies, call exfalso and try again. (Defaults to tt.)
• apply: use a tactic other than tactic.apply (e.g. tactic.fapply or tactic.eapply).

apply_any lemmas tac calls the tactic tac after a successful application. Defaults to skip. This is used, for example, by solve_by_elim to arrange recursive invocations of apply_any.

Try to apply a hypothesis from the local context to the goal.

change_core e none is equivalent to change e. It tries to change the goal to e and fails if this is not a definitional equality.

change_core e (some h) assumes h is a local constant, and tries to change the type of h to e by reverting h, changing the goal, and reintroducing hypotheses.

change_with_at olde newe hyp replaces occurences of olde with newe at hypothesis hyp, assuming olde and newe are defeq when elaborated.

Returns a list of all metavariables in the current partial proof. This can differ from the list of goals, since the goals can be manually edited.

sorry_if_contains_sorry will solve any goal already containing sorry in its type with sorry, and fail otherwise.

Fail if the target contains a metavariable.

Succeeds only if the current goal is a proposition.

Succeeds only if we can construct an instance showing the current goal is a subsingleton type.

Succeeds only if the current goal is "terminal", in the sense that no other goals depend on it (except possibly through shared metavariables; see independent_goal).

Succeeds only if the current goal is "independent", in the sense that no other goals depend on it, even through shared meta-variables.

triv' tries to close the first goal with the proof trivial : true. Unlike triv, it only unfolds reducible definitions, so it sometimes fails faster.

Apply a tactic as many times as possible, collecting the results in a list. Fail if the tactic does not succeed at least once.

Introduces one or more variables and returns the new local constants. Fails if intro cannot be applied.

Run a tactic "under binders", by running intros before, and revert afterwards.

Run a tactic "under binders", by running intros before, and revert afterwards.

successes invokes each tactic in turn, returning the list of successful results.

Try all the tactics in a list, each time starting at the original tactic_state, returning the list of successful results, and reverting to the original tactic_state.

Try all the tactics in a list, each time starting at the original tactic_state, returning the list of successful results sorted by the value produced by a subsequent execution of the sort_by tactic, and reverting to the original tactic_state.

Just like split, fsplit applies the constructor when the type of the target is an inductive data type with one constructor. However it does not reorder goals or invoke auto_param tactics.

Just like split, fsplit applies the constructor when the type of the target is an inductive data type with one constructor. However it does not reorder goals or invoke auto_param tactics.

Calls injection on each hypothesis, and then, for each hypothesis on which injection succeeds, clears the old hypothesis.

Calls injection on each hypothesis, and then, for each hypothesis on which injection succeeds, clears the old hypothesis.

Calls cases on every local hypothesis, succeeding if it succeeds on at least one hypothesis.

note_anon t v, given a proof v : t, adds h : t to the current context, where the name h is fresh.

note_anon none v will infer the type t from v.

find_local t returns a local constant with type t, or fails if none exists.

dependent_pose_core l: introduce dependent hypotheses, where the proofs depend on the values of the previous local constants. l is a list of local constants and their values.

Instantiates metavariables that appear in the current goal.

Instantiates metavariables in all goals.

Protect the declaration n

emit_command_here str behaves as if the string str were placed as a user command at the current line.

Inner recursion for emit_code_here.

emit_code_here str behaves as if the string str were placed at the current location in source code.

run_parser p is like run_cmd but for the parser monad. It executes parser p at the top level, giving access to operations like emit_code_here.

get_current_namespace returns the current namespace (it could be name.anonymous).

This function deserves a C++ implementation in core lean, and will fail if it is not called from the body of a command (i.e. anywhere else that the lean.parser monad can be invoked).

get_variables returns a list of existing variable names, along with their types and binder info.

get_included_variables returns those variables v returned by get_variables which have been "included" by an include v statement and are not (yet) omited.

From the lean.parser monad, synthesize a tactic_state which includes all of the local variables referenced in es : list pexpr, and those variables which have been includeed in the local context---precisely those variables which would be ambiently accessible if we were in a tactic-mode block where the goals had types es.mmap to_expr, for example.

Returns a new ts : tactic_state with these local variables added, and mappings : list (expr × expr), for which pairs (var, hyp) correspond to an existing variable var and the local hypothesis hyp which was added to the tactic state ts as a result.

Hole command used to fill in a structure's field when specifying an instance.

In the following:

instance : monad id :=
{! !}

invoking the hole command "Instance Stub" ("Generate a skeleton for the structure under construction.") produces:

instance : monad id :=
{ map := _,
  map_const := _,
  pure := _,
  seq := _,
  seq_left := _,
  seq_right := _,
  bind := _ }

Like resolve_name except when the list of goals is empty. In that situation resolve_name fails whereas resolve_name' simply proceeds on a dummy goal

Strips unnecessary prefixes from a name, e.g. if a namespace is open.

Used to format return strings for the hole commands match_stub and eqn_stub.

Hole command used to generate a match expression.

In the following:

meta def foo (e : expr) : tactic unit :=
{! e !}

invoking hole command "Match Stub" ("Generate a list of equations for a match expression") produces:

meta def foo (e : expr) : tactic unit :=
match e with
| (expr.var a) := _
| (expr.sort a) := _
| (expr.const a a_1) := _
| (expr.mvar a a_1 a_2) := _
| (expr.local_const a a_1 a_2 a_3) := _
| (expr.app a a_1) := _
| (expr.lam a a_1 a_2 a_3) := _
| (expr.pi a a_1 a_2 a_3) := _
| (expr.elet a a_1 a_2 a_3) := _
| (expr.macro a a_1) := _
end

Invoking hole command "Equations Stub" ("Generate a list of equations for a recursive definition") in the following:

meta def foo : {! expr → tactic unit !} -- `:=` is omitted

produces:

meta def foo : expr → tactic unit
| (expr.var a) := _
| (expr.sort a) := _
| (expr.const a a_1) := _
| (expr.mvar a a_1 a_2) := _
| (expr.local_const a a_1 a_2 a_3) := _
| (expr.app a a_1) := _
| (expr.lam a a_1 a_2 a_3) := _
| (expr.pi a a_1 a_2 a_3) := _
| (expr.elet a a_1 a_2 a_3) := _
| (expr.macro a a_1) := _

A similar result can be obtained by invoking "Equations Stub" on the following:

meta def foo : expr → tactic unit := -- do not forget to write `:=`!!
{! !}

meta def foo : expr → tactic unit := -- don't forget to erase `:=`!!
| (expr.var a) := _
| (expr.sort a) := _
| (expr.const a a_1) := _
| (expr.mvar a a_1 a_2) := _
| (expr.local_const a a_1 a_2 a_3) := _
| (expr.app a a_1) := _
| (expr.lam a a_1 a_2 a_3) := _
| (expr.pi a a_1 a_2 a_3) := _
| (expr.elet a a_1 a_2 a_3) := _
| (expr.macro a a_1) := _

This command lists the constructors that can be used to satisfy the expected type.

Invoking "List Constructors" ("Show the list of constructors of the expected type") in the following hole:

def foo : ℤ ⊕ ℕ :=
{! !}

produces:

def foo : ℤ ⊕ ℕ :=
{! sum.inl, sum.inr !}

and will display:

sum.inl : ℤ → ℤ ⊕ ℕ

sum.inr : ℕ → ℤ ⊕ ℕ
```

Makes the declaration classical.prop_decidable available to type class inference. This asserts that all propositions are decidable, but does not have computational content.

mk_comp v e checks whether e is a sequence of nested applications f (g (h v)), and if so, returns the expression f ∘ g ∘ h.

Given two expressions e₀ and e₁, return the expression `(%%e₀ ↔ %%e₁).

From a lemma of the shape ∀ x, f (g x) = h x derive an auxiliary lemma of the form f ∘ g = h for reasoning about higher-order functions.

A user attribute that applies to lemmas of the shape ∀ x, f (g x) = h x. It derives an auxiliary lemma of the form f ∘ g = h for reasoning about higher-order functions.

Copies a definition into the tactic.interactive namespace to make it usable in proof scripts. It allows one to write

@[interactive]
meta def my_tactic := ...

instead of

meta def my_tactic := ...

run_cmd add_interactive [``my_tactic]
```

Similar to existsi, use l will use entries in l to instantiate existential obligations at the beginning of a target. Unlike existsi, the pexprs in l are elaborated with respect to the expected type.

example : ∃ x : ℤ, x = x :=
by tactic.use ``(42)

See the doc string for tactic.interactive.use for more information.

clear_aux_decl_aux l clears all expressions in l that represent aux decls from the local context.

clear_aux_decl clears all expressions from the local context that represent aux decls.

apply_at_aux e et [] h ht (with et the type of e and ht the type of h) finds a list of expressions vs and returns (e.mk_args (vs ++ [h]), vs).

apply_at e h applies implication e on hypothesis h and replaces h with the result.

symmetry_hyp h applies symmetry on hypothesis h.

setup_tactic_parser is a user command that opens the namespaces used in writing interactive tactics, and declares the local postfix notation ? for optional and * for many. It does not use the namespace command, so it will typically be used after namespace tactic.interactive.

finally tac finalizer runs tac first, then runs finalizer even if tac fails. finally tac finalizer fails if either tac or finalizer fails.

on_exception handler tac runs tac first, and then runs handler only if tac failed.

decorate_error add_msg tac prepends add_msg to an exception produced by tac

Applies tactic t. If it succeeds, revert the state, and return the value. If it fails, returns the error message.

Applies tactic t. If it succeeds, return the value. If it fails, returns the error message.

This tactic succeeds if t succeeds or fails with message msg such that p msg is tt.

trace_error msg t executes the tactic t. If t fails, traces msg and the failure message of t.

trace_if_enabled `n msg traces the message msg only if tracing is enabled for the name n.

Create new names registered for tracing with declare_trace n. Then use set_option trace.n true/false to enable or disable tracing for n.

trace_state_if_enabled `n msg prints the tactic state, preceded by the optional string msg, only if tracing is enabled for the name n.

This combinator is for testing purposes. It succeeds if t fails with message msg, and fails otherwise.

Construct a Try this: refine ... or Try this: exact ... string which would construct g.

with_local_goals gs tac runs tac on the goals gs and then restores the initial goals and returns the goals tac ended on.

like with_local_goals but discards the resulting goals

Representation of a proof goal that lends itself to comparison. The following goal:

l₀ : T,
l₁ : T
⊢ ∀ v : T, foo

is represented as

(2, ∀ l₀ l₁ v : T, foo)

The number 2 indicates that first the two bound variables of the ∀ are actually local constant. Comparing two such goals with = rather than =ₐ or is_def_eq tells us that proof script should not see the difference between the two.

proof state made of multiple goal meant for comparing the result of running different tactics

create a packaged_goal corresponding to the current goal

goal_of_mvar g, with g a meta variable, creates a packaged_goal corresponding to g interpretted as a proof goal

get_proof_state lists the user visible goal for each goal of the current state and for each goal, abstracts all of the meta variables of the other gaols.

This produces a list of goals in the form of ℕ × expr where the expr encodes the following proof state:

2 goals
l₁ : t₁,
l₂ : t₂,
l₃ : t₃
⊢ tgt₁

⊢ tgt₂

as

[ (3, ∀ (mv : tgt₁) (mv : tgt₂) (l₁ : t₁) (l₂ : t₂) (l₃ : t₃), tgt₁),
  (0, ∀ (mv : tgt₁) (mv : tgt₂), tgt₂) ]

with 2 goals, the first 2 bound variables encode the meta variable of all the goals, the next 3 (in the first goal) and 0 (in the second goal) are the local constants.

This representation allows us to compare goals and proof states while ignoring information like the unique name of local constants and the equality or difference of meta variables that encode the same goal.

Run tac in a disposable proof state and return the state. See proof_state, goal and get_proof_state.

A type alias for tactic format, standing for "pretty print format".

mk lifts fmt : format to the tactic monad (pformat).

an alias for pp.

See format! in init/meta/interactive_base.lean.

The main differences are that pp is called instead of to_fmt and that we can use arguments of type tactic α in the quotations.

Now, consider the following:

e ← to_expr ``(3 + 7),
trace format!"{e}"  -- outputs `has_add.add.{0} nat nat.has_add
                    -- (bit1.{0} nat nat.has_one nat.has_add (has_one.one.{0} nat nat.has_one)) ...`
trace pformat!"{e}" -- outputs `3 + 7`

The difference is significant. And now, the following is expressible:

e ← to_expr ``(3 + 7),
trace pformat!"{e} : {infer_type e}" -- outputs `3 + 7 : ℕ`

See also: trace! and fail!

The combination of pformat and fail.

The combination of pformat and trace.

A hackish way to get the src directory of any project. Requires as argument any declaration name n in that project, and k, the number of characters in the path of the file where n is declared not part of the src directory. Example: For mathlib_dir_locator this is the length of tactic/project_dir.lean, so 23. Note: does not work in the file where n is declared.

A hackish way to get the src directory of mathlib.

Checks whether a declaration with the given name is declared in mathlib. If you want to run this tactic many times, you should use environment.is_prefix_of_file instead, since it is expensive to execute get_mathlib_dir many times.

Runs a tactic by name. If it is a tactic string, return whatever string it returns. If it is a tactic unit, return the name. (This is mostly used in invoking "self-reporting tactics", e.g. by tidy and hint.)

Assumes pi_expr is of the form Π x1 ... xn xn+1..., _. Creates a pexpr of the form Π x1 ... xn, func (arg x1 ... xn). All arguments (implicit and explicit) to arg should be supplied.

Assumes pi_expr is of the form Π x1 ... xn, _. Creates a pexpr of the form Π x1 ... xn, func (arg x1 ... xn). All arguments (implicit and explicit) to arg should be supplied.

If func is a pexpr representing a function that takes an argument a, get_pexpr_arg_arity_with_tgt func tgt returns the arity of a. When tgt is a pi expr, func is elaborated in a context with the domain of tgt.

Examples:

• get_pexpr_arg_arity ``(ring) `(true) returns 0, since ring takes one non-function argument.
• get_pexpr_arg_arity_with_tgt ``(monad) `(true) returns 1, since monad takes one argument of type α → α.
• get_pexpr_arg_arity_with_tgt ``(module R) `(Π (R : Type), comm_ring R → true) returns 0

find_private_decl n none finds a private declaration named n in any of the imported files.

find_private_decl n (some m) finds a private declaration named n in the same file where a declaration named m can be found.

import_private foo from bar finds a private declaration foo in the same file as bar and creates a local notation to refer to it.

import_private foo looks for foo in all imported files.

When possible, make foo non-private rather than using this feature.

The command mk_simp_attribute simp_name "description" creates a simp set with name simp_name. Lemmas tagged with @[simp_name] will be included when simp with simp_name is called. mk_simp_attribute simp_name none will use a default description.

Appending the command with with attr1 attr2 ... will include all declarations tagged with attr1, attr2, ... in the new simp set.

This command is preferred to using run_cmd mk_simp_attr `simp_name since it adds a doc string to the attribute that is defined. If you need to create a simp set in a file where this command is not available, you should use

run_cmd mk_simp_attr `simp_name
run_cmd add_doc_string `simp_attr.simp_name "Description of the simp set here"

Given a user attribute name attr_name, get_user_attribute_name attr_name returns the name of the declaration that defines this attribute. Fails if there is no user attribute with this name. Example: get_user_attribute_name `norm_cast returns `norm_cast.norm_cast_attr

A tactic to set either a basic attribute or a user attribute. If the user attribute has a parameter, the default value will be used. This tactic raises an error if there is no inhabited instance for the parameter type.

find_defeq red m e looks for a key in m that is defeq to e (up to transparency red), and returns the value associated with this key if it exists. Otherwise, it fails.