Documentation

Init.Conv

conv is the syntax category for a "conv tactic", where "conv" is short for conversion. A conv tactic is a program which receives a target, printed as | a, and is tasked with coming up with some term b and a proof of a = b. It is mainly used for doing targeted term transformations, for example rewriting only on the left side of an equality.

Equations
    Instances For

      The * occurrence list means to apply to all occurrences of the pattern.

      Equations
        Instances For

          A list 1 2 4 of occurrences means to apply to the first, second, and fourth occurrence of the pattern.

          Equations
            Instances For

              An occurrence specification, either * or a list of numbers. The default is [1].

              Equations
                Instances For

                  with_annotate_state stx t annotates the lexical range of stx : Syntax with the initial and final state of running tactic t.

                  Equations
                    Instances For

                      skip does nothing.

                      Equations
                        Instances For

                          Traverses into the left subterm of a binary operator.

                          In general, for an n-ary operator, it traverses into the second to last argument. It is a synonym for arg -2.

                          Equations
                            Instances For

                              Traverses into the right subterm of a binary operator.

                              In general, for an n-ary operator, it traverses into the last argument. It is a synonym for arg -1.

                              Equations
                                Instances For

                                  Traverses into the function of a (unary) function application. For example, | f a b turns into | f a. (Use arg 0 to traverse into f.)

                                  Equations
                                    Instances For

                                      Reduces the target to Weak Head Normal Form. This reduces definitions in "head position" until a constructor is exposed. For example, List.map f [a, b, c] weak head normalizes to f a :: List.map f [b, c].

                                      Equations
                                        Instances For

                                          Expands let-declarations and let-variables.

                                          Equations
                                            Instances For

                                              Puts term in normal form, this tactic is meant for debugging purposes only.

                                              Equations
                                                Instances For

                                                  Performs one step of "congruence", which takes a term and produces subgoals for all the function arguments. For example, if the target is f x y then congr produces two subgoals, one for x and one for y.

                                                  Equations
                                                    Instances For
                                                      • arg i traverses into the i'th argument of the target. For example if the target is f a b c d then arg 1 traverses to a and arg 3 traverses to c. The index may be negative; arg -1 traverses into the last argument, arg -2 into the second-to-last argument, and so on.
                                                      • arg @i is the same as arg i but it counts all arguments instead of just the explicit arguments.
                                                      • arg 0 traverses into the function. If the target is f a b c d, arg 0 traverses into f.
                                                      Equations
                                                        Instances For

                                                          ext x traverses into a binder (a fun x => e or ∀ x, e expression) to target e, introducing name x in the process.

                                                          Equations
                                                            Instances For

                                                              change t' replaces the target t with t', assuming t and t' are definitionally equal.

                                                              Equations
                                                                Instances For

                                                                  delta id1 id2 ... unfolds all occurrences of id1, id2, ... in the target. Like the delta tactic, this ignores any definitional equations and uses primitive delta-reduction instead, which may result in leaking implementation details. Users should prefer unfold for unfolding definitions.

                                                                  Equations
                                                                    Instances For

                                                                      Definitions can be either global or local definitions.

                                                                      For non-recursive global definitions, this tactic is identical to delta. For recursive global definitions, it uses the "unfolding lemma" id.eq_def, which is generated for each recursive definition, to unfold according to the recursive definition given by the user. Only one level of unfolding is performed, in contrast to simp only [id], which unfolds definition id recursively.

                                                                      This is the conv version of the unfold tactic.

                                                                      Equations
                                                                        Instances For
                                                                          • pattern pat traverses to the first subterm of the target that matches pat.
                                                                          • pattern (occs := *) pat traverses to every subterm of the target that matches pat which is not contained in another match of pat. It generates one subgoal for each matching subterm.
                                                                          • pattern (occs := 1 2 4) pat matches occurrences 1, 2, 4 of pat and produces three subgoals. Occurrences are numbered left to right from the outside in.

                                                                          Note that skipping an occurrence of pat will traverse inside that subexpression, which means it may find more matches and this can affect the numbering of subsequent pattern matches. For example, if we are searching for f _ in f (f a) = f b:

                                                                          • occs := 1 2 (and occs := *) returns | f (f a) and | f b
                                                                          • occs := 2 returns | f a
                                                                          • occs := 2 3 returns | f a and | f b
                                                                          • occs := 1 3 is an error, because after skipping f b there is no third match.
                                                                          Equations
                                                                            Instances For

                                                                              rw [thm] rewrites the target using thm. See the rw tactic for more information.

                                                                              Equations
                                                                                Instances For

                                                                                  simp [thm] performs simplification using thm and marked @[simp] lemmas. See the simp tactic for more information.

                                                                                  Equations
                                                                                    Instances For

                                                                                      simp? takes the same arguments as simp, but reports an equivalent call to simp only that would be sufficient to close the goal. See the simp? tactic for more information.

                                                                                      Equations
                                                                                        Instances For

                                                                                          dsimp is the definitional simplifier in conv-mode. It differs from simp in that it only applies theorems that hold by reflexivity.

                                                                                          Examples:

                                                                                          example (a : Nat): (0 + 0) = a - a := by
                                                                                            conv =>
                                                                                              lhs
                                                                                              dsimp
                                                                                              rw [← Nat.sub_self a]
                                                                                          
                                                                                          Equations
                                                                                            Instances For

                                                                                              simp? takes the same arguments as simp, but reports an equivalent call to simp only that would be sufficient to close the goal. See the simp? tactic for more information.

                                                                                              Equations
                                                                                                Instances For

                                                                                                  simp_match simplifies match expressions. For example,

                                                                                                  match [a, b] with
                                                                                                  | [] => 0
                                                                                                  | hd :: tl => hd
                                                                                                  

                                                                                                  simplifies to a.

                                                                                                  Equations
                                                                                                    Instances For

                                                                                                      Executes the given tactic block without converting conv goal into a regular goal.

                                                                                                      Equations
                                                                                                        Instances For

                                                                                                          Focuses, converts the conv goal lhs into a regular goal lhs = rhs, and then executes the given tactic block.

                                                                                                          Equations
                                                                                                            Instances For

                                                                                                              Executes the given conv block without converting regular goal into a conv goal.

                                                                                                              Equations
                                                                                                                Instances For

                                                                                                                  { convs } runs the list of convs on the current target, and any subgoals that remain are trivially closed by skip.

                                                                                                                  Equations
                                                                                                                    Instances For

                                                                                                                      (convs) runs the convs in sequence on the current list of targets. This is pure grouping with no added effects.

                                                                                                                      Equations
                                                                                                                        Instances For

                                                                                                                          rfl closes one conv goal "trivially", by using reflexivity (that is, no rewriting).

                                                                                                                          Equations
                                                                                                                            Instances For

                                                                                                                              done succeeds iff there are no goals remaining.

                                                                                                                              Equations
                                                                                                                                Instances For

                                                                                                                                  trace_state prints the current goal state.

                                                                                                                                  Equations
                                                                                                                                    Instances For

                                                                                                                                      all_goals tac runs tac on each goal, concatenating the resulting goals, if any.

                                                                                                                                      Equations
                                                                                                                                        Instances For

                                                                                                                                          any_goals tac applies the tactic tac to every goal, and succeeds if at least one application succeeds.

                                                                                                                                          Equations
                                                                                                                                            Instances For
                                                                                                                                              • case tag => tac focuses on the goal with case name tag and solves it using tac, or else fails.
                                                                                                                                              • case tag x₁ ... xₙ => tac additionally renames the n most recent hypotheses with inaccessible names to the given names.
                                                                                                                                              • case tag₁ | tag₂ => tac is equivalent to (case tag₁ => tac); (case tag₂ => tac).
                                                                                                                                              Equations
                                                                                                                                                Instances For

                                                                                                                                                  case' is similar to the case tag => tac tactic, but does not ensure the goal has been solved after applying tac, nor admits the goal if tac failed. Recall that case closes the goal using sorry when tac fails, and the tactic execution is not interrupted.

                                                                                                                                                  Equations
                                                                                                                                                    Instances For

                                                                                                                                                      next => tac focuses on the next goal and solves it using tac, or else fails. next x₁ ... xₙ => tac additionally renames the n most recent hypotheses with inaccessible names to the given names.

                                                                                                                                                      Equations
                                                                                                                                                        Instances For

                                                                                                                                                          focus tac focuses on the main goal, suppressing all other goals, and runs tac on it. Usually · tac, which enforces that the goal is closed by tac, should be preferred.

                                                                                                                                                          Equations
                                                                                                                                                            Instances For

                                                                                                                                                              conv => cs runs cs in sequence on the target t, resulting in t', which becomes the new target subgoal.

                                                                                                                                                              Equations
                                                                                                                                                                Instances For

                                                                                                                                                                  · conv focuses on the main conv goal and tries to solve it using s.

                                                                                                                                                                  Equations
                                                                                                                                                                    Instances For

                                                                                                                                                                      fail_if_success t fails if the tactic t succeeds.

                                                                                                                                                                      Equations
                                                                                                                                                                        Instances For

                                                                                                                                                                          rw [rules] applies the given list of rewrite rules to the target. See the rw tactic for more information.

                                                                                                                                                                          Equations
                                                                                                                                                                            Instances For

                                                                                                                                                                              erw [rules] is a shorthand for rw (transparency := .default) [rules]. This does rewriting up to unfolding of regular definitions (by comparison to regular rw which only unfolds @[reducible] definitions).

                                                                                                                                                                              Equations
                                                                                                                                                                                Instances For

                                                                                                                                                                                  args traverses into all arguments. Synonym for congr.

                                                                                                                                                                                  Equations
                                                                                                                                                                                    Instances For

                                                                                                                                                                                      left traverses into the left argument. Synonym for lhs.

                                                                                                                                                                                      Equations
                                                                                                                                                                                        Instances For

                                                                                                                                                                                          right traverses into the right argument. Synonym for rhs.

                                                                                                                                                                                          Equations
                                                                                                                                                                                            Instances For

                                                                                                                                                                                              intro traverses into binders. Synonym for ext.

                                                                                                                                                                                              Equations
                                                                                                                                                                                                Instances For

                                                                                                                                                                                                  enter [arg, ...] is a compact way to describe a path to a subterm. It is a shorthand for other conv tactics as follows:

                                                                                                                                                                                                  • enter [i] is equivalent to arg i.
                                                                                                                                                                                                  • enter [@i] is equivalent to arg @i.
                                                                                                                                                                                                  • enter [x] (where x is an identifier) is equivalent to ext x. For example, given the target f (g a (fun x => x b)), enter [1, 2, x, 1] will traverse to the subterm b.
                                                                                                                                                                                                  Equations
                                                                                                                                                                                                    Instances For

                                                                                                                                                                                                      The apply thm conv tactic is the same as apply thm the tactic. There are no restrictions on thm, but strange results may occur if thm cannot be reasonably interpreted as proving one equality from a list of others.

                                                                                                                                                                                                      Equations
                                                                                                                                                                                                        Instances For

                                                                                                                                                                                                          first | conv | ... runs each conv until one succeeds, or else fails.

                                                                                                                                                                                                          Equations
                                                                                                                                                                                                            Instances For

                                                                                                                                                                                                              try tac runs tac and succeeds even if tac failed.

                                                                                                                                                                                                              Equations
                                                                                                                                                                                                                Instances For

                                                                                                                                                                                                                  tac <;> tac' runs tac on the main goal and tac' on each produced goal, concatenating all goals produced by tac'.

                                                                                                                                                                                                                  Equations
                                                                                                                                                                                                                    Instances For

                                                                                                                                                                                                                      repeat convs runs the sequence convs repeatedly until it fails to apply.

                                                                                                                                                                                                                      Equations
                                                                                                                                                                                                                        Instances For

                                                                                                                                                                                                                          Extracts let and let_fun expressions from within the target expression. This is the conv mode version of the extract_lets tactic.

                                                                                                                                                                                                                          • extract_lets extracts all the lets from the target.
                                                                                                                                                                                                                          • extract_lets x y z extracts all the lets from the target and uses x, y, and z for the first names. Using _ for a name leaves it unnamed.

                                                                                                                                                                                                                          Limitation: the extracted local declarations do not persist outside of the conv goal. See also lift_lets, which does not extract lets as local declarations.

                                                                                                                                                                                                                          Equations
                                                                                                                                                                                                                            Instances For

                                                                                                                                                                                                                              Lifts let and let_fun expressions within the target expression as far out as possible. This is the conv mode version of the lift_lets tactic.

                                                                                                                                                                                                                              Equations
                                                                                                                                                                                                                                Instances For

                                                                                                                                                                                                                                  conv => ... allows the user to perform targeted rewriting on a goal or hypothesis, by focusing on particular subexpressions.

                                                                                                                                                                                                                                  See https://lean-lang.org/theorem_proving_in_lean4/conv.html for more details.

                                                                                                                                                                                                                                  Basic forms:

                                                                                                                                                                                                                                  • conv => cs will rewrite the goal with conv tactics cs.
                                                                                                                                                                                                                                  • conv at h => cs will rewrite hypothesis h.
                                                                                                                                                                                                                                  • conv in pat => cs will rewrite the first subexpression matching pat (see pattern).
                                                                                                                                                                                                                                  Equations
                                                                                                                                                                                                                                    Instances For

                                                                                                                                                                                                                                      norm_cast tactic in conv mode.

                                                                                                                                                                                                                                      Equations
                                                                                                                                                                                                                                        Instances For