# 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.

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

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

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

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

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skip does nothing.

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Traverses into the left subterm of a binary operator. (In general, for an n-ary operator, it traverses into the second to last argument.)

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Traverses into the right subterm of a binary operator. (In general, for an n-ary operator, it traverses into the last argument.)

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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].

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Expands let-declarations and let-variables.

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

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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.

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• 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.
• arg @i is the same as arg i but it counts all arguments instead of just the explicit arguments.
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ext x traverses into a binder (a fun x => e or ∀ x, e expression) to target e, introducing name x in the process.

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

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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.

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• unfold foo unfolds all occurrences of foo in the target.
• unfold id1 id2 ... is equivalent to unfold id1; unfold id2; .... Like the unfold tactic, this uses equational lemmas for the chosen definition to rewrite the target. For recursive definitions, only one layer of unfolding is performed.
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• 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.
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rw [thm] rewrites the target using thm. See the rw tactic for more information.

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

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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]

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simp_match simplifies match expressions. For example,

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


simplifies to a.

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

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

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

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

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

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

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done succeeds iff there are no goals remaining.

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trace_state prints the current goal state.

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

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

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• 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).
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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.

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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.

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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.

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

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

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fail_if_success t fails if the tactic t succeeds.

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

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

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args traverses into all arguments. Synonym for congr.

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left traverses into the left argument. Synonym for lhs.

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right traverses into the right argument. Synonym for rhs.

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intro traverses into binders. Synonym for ext.

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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.
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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.

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

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try tac runs tac and succeeds even if tac failed.

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

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

See https://leanprover.github.io/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).
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