\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

# Reasoning with equalities

There are multiple notions of equality in Coq:

- Leibniz equality is the standard way to define equality in Coq and the Calculus of Inductive Constructions, which is in terms of a binary relation, i.e. a binary function that returns a `Prop`. The standard library defines `eq` similar to this:

  > Inductive eq {A : Type} (x : A) : A -> Prop := eq_refl : eq x x.

  The notation `x = y` represents the term `eq x y`. The notation `x = y :> A` gives the type of x and y explicitly.

- Setoid equality defines equality in terms of an equivalence relation. A setoid is a set that is equipped with an equivalence relation (see https://en.wikipedia.org/wiki/Setoid). These are needed to form a quotient set or quotient (see https://en.wikipedia.org/wiki/Equivalence_Class). In Coq, users generally work with setoids rather than constructing quotients, for which there is no specific support.

- Definitional equality is equality based on the conversion rules, which Coq can determine automatically. When two terms are definitionally equal, Coq knows it can replace one with the other, such as with `change` `X with Y`, among many other advantages. "Convertible" is another way of saying that two terms are definitionally equal.

Tactics for dealing with equality of inductive types such as `injection` and `inversion` are described here.

## Tactics for simple equalities

Tactic reflexivity

  For a goal with the form `forall open_binders ,? t = u`, verifies that `t` and `u` are definitionally equal, and if so, solves the goal (by applying `eq_refl`). If not, it fails.

  The tactic may also be applied to goals with the form `forall open_binders ,? R term1 term2` where `R` is a reflexive relation registered with the `Equivalence` or `Reflexive` typeclasses. See `Class` and `Instance`.

  Error The relation ident is not a declared reflexive relation. Maybe you need to require the Coq.Classes.RelationClasses library

Tactic symmetry simple_occurrences?

  Changes a goal that has the form `forall open_binders ,? t = u` into `u = t`. `simple_occurrences` may be used to apply the change in the selected hypotheses and/or the conclusion.

  The tactic may also be applied to goals with the form `forall open_binders ,? R term1 term2` where `R` is a symmetric relation registered with the `Equivalence` or `Symmetric` typeclasses. See `Class` and `Instance`.

  Error The relation ident is not a declared symmetric relation. Maybe you need to require the Coq.Classes.RelationClasses library

Tactic transitivity one_term

  Changes a goal that has the form `forall open_binders ,? t = u` into the two subgoals `t = one_term` and `one_term = u`.

  The tactic may also be applied to goals with the form `forall open_binders ,? R term1 term2` where `R` is a transitive relation registered with the `Equivalence` or `Transitivity` typeclasses. See `Class` and `Instance`.

  Tactic etransitivity

    This tactic behaves like `transitivity`, using a fresh evar instead of a concrete `one_term`.

  Error The relation ident is not a declared transitive relation. Maybe you need to require the Coq.Classes.RelationClasses library

Tactic f_equal

  For a goal with the form `f a1 ... an = g b1 ... bn`, creates subgoals `f = g` and `ai = bi` for the `n` arguments. Subgoals that can be proven by `reflexivity` or `congruence` are solved automatically.

## Rewriting with Leibniz and setoid equality

Tactic rewrite oriented_rewriter+, occurrences? by ltac_expr3?
  oriented_rewriter`::=`->​<-? natural? ?​!? one_term_with_bindings``one_term_with_bindings`::=`>? one_term with bindings?``

  Replaces subterms with other subterms that have been proven to be equal. The type of `one_term` must have the form:

  > `forall open_binders ,? EQ term1 term2`

  where `EQ` is the Leibniz equality `eq` or a registered setoid equality. Note that `eq term1 term2` is typically written with the infix notation `term1 = term2`. You must `Require Setoid` to use the tactic with a setoid equality or with setoid rewriting.

  `rewrite one_term` finds subterms matching `term1` in the goal, and replaces them with `term2` (or the reverse if `<-` is given). Some of the variables `x`i are solved by unification, and some of the types `A1, …, An` may become new subgoals. `rewrite` won't find occurrences inside `forall` that refer to variables bound by the `forall`; use the more advanced `setoid_rewrite` if you want to find such occurrences.

  `oriented_rewriter+,`

    The `oriented_rewriter`s are applied sequentially to the first goal generated by the previous `oriented_rewriter`. If any of them fail, the tactic fails.

  `->​<-?`

    For `->` (the default), `term1` is rewritten into `term2`. For `<-`, `term2` is rewritten into `term1`.

  `natural? ?​!?`

    `natural` is the number of rewrites to perform. If `?` is given, `natural` is the maximum number of rewrites to perform; otherwise `natural` is the exact number of rewrites to perform.

    `?` (without `natural`) performs the rewrite as many times as possible (possibly zero times). This form never fails. `!` (without `natural`) performs the rewrite as many times as possible and at least once. The tactic fails if the requested number of rewrites can't be performed. `natural !` is equivalent to `natural`.

  `occurrences`

    If `occurrences` specifies multiple occurrences, the tactic succeeds if any of them can be rewritten. If not specified, only the first occurrence in the conclusion is replaced.

    Note

    If `at occs_nums` is specified, rewriting is always done with setoid rewriting, even for Leibniz equality, which means that you must `Require Setoid` to use that form. However, note that `rewrite` (even when using setoid rewriting) and `setoid_rewrite` don't behave identically (as is noted above and below).

  `by ltac_expr3`

    If specified, is used to resolve all side conditions generated by the tactic.

  Note

  For each selected hypothesis and/or the conclusion, `rewrite` finds the first matching subterm in depth-first search order. Only subterms identical to that first matched subterm are rewritten. If the `at` clause is specified, only these subterms are considered when counting occurrences. To select a different set of matching subterms, you can specify how some or all of the free variables are bound by using a `with` clause (see `one_term_with_bindings`).

  For instance, if we want to rewrite the right-hand side in the following goal, this will not work:

  Require Import Arith.
    [Loading ML file ring_plugin.cmxs (using legacy method) ... done]
  Lemma example x y : x + y = y + x.
    1 goal x, y : nat ============================ x + y = y + x
  rewrite Nat.add_comm at 2.
    Toplevel input, characters 0-25: > rewrite Nat.add_comm at 2. > ^^^^^^^^^^^^^^^^^^^^^^^^^ Error: Invalid occurrence number: 2.

  One can explicitly specify how some variables are bound to match a different subterm:

  rewrite Nat.add_comm with (m := x).
    1 goal x, y : nat ============================ x + y = x + y

  Note that the more advanced `setoid_rewrite` tactic behaves differently, and thus the number of occurrences available to rewrite may differ between the two tactics.

  Error Tactic failure: Setoid library not loaded.
  Error Cannot find a relation to rewrite.
  Error Tactic generated a subgoal identical to the original goal.
  Error Found no subterm matching term in ident.
  Error Found no subterm matching term in the current goal.

    This happens if `term` does not occur in, respectively, the named hypothesis or the goal.

  Tactic erewrite oriented_rewriter+, occurrences? by ltac_expr3?

    Works like `rewrite`, but turns unresolved bindings, if any, into existential variables instead of failing. It has the same parameters as `rewrite`.

  Flag Keyed Unification

    This flag makes higher-order unification used by `rewrite` rely on a set of keys to drive unification. The subterms, considered as rewriting candidates, must start with the same key as the left- or right-hand side of the lemma given to rewrite, and the arguments are then unified up to full reduction.

Tactic rewrite * ->​<-? one_term in ident? at rewrite_occs? by ltac_expr3?
Tactic rewrite * ->​<-? one_term at rewrite_occs in ident by ltac_expr3?

Tactic replace one_termfrom with one_termto occurrences? by ltac_expr3?
Tactic replace ->​<-? one_termfrom occurrences?

  The first form replaces all free occurrences of `one_termfrom` in the current goal with `one_termto` and generates an equality `one_termto = one_termfrom` as a subgoal. (Note the generated equality is reversed with respect to the order of the two terms in the tactic syntax; see issue #13480.) This equality is automatically solved if it occurs among the hypotheses, or if its symmetric form occurs.

  The second form, with `->` or no arrow, replaces `one_termfrom` with `termto` using the first hypothesis whose type has the form `one_termfrom = termto`. If `<-` is given, the tactic uses the first hypothesis with the reverse form, i.e. `termto = one_termfrom`.

  `occurrences`

    The `type of` and `value of` forms are not supported. Note you must `Require Setoid` to use the `at` clause in `occurrences`.

  `by ltac_expr3`

    Applies the `ltac_expr3` to solve the generated equality.

  Error Terms do not have convertible types.
  Tactic cutrewrite ->​<-? one_term in ident?

    Where `one_term` is an equality.

    Deprecated since version 8.5: Use `replace` instead.

Tactic substitute ->​<-? one_term with bindings?

Tactic subst ident*

  For each `ident`, in order, for which there is a hypothesis in the form `ident = term` or `term = ident`, replaces `ident` with `term` everywhere in the hypotheses and the conclusion and clears `ident` and the hypothesis from the context. If there are multiple hypotheses that match the `ident`, the first one is used. If no `ident` is given, replacement is done for all hypotheses in the appropriate form in top to bottom order.

  If `ident` is a local definition of the form `ident := term`, it is also unfolded and cleared.

  If `ident` is a section variable it must have no indirect occurrences in the goal, i.e. no global declarations implicitly depending on the section variable may be present in the goal.

  Note

  If the hypothesis is itself dependent in the goal, it is replaced by the proof of reflexivity of equality.

  Flag Regular Subst Tactic

    This flag controls the behavior of `subst`. When it is activated (it is by default), `subst` also deals with the following corner cases:

    - A context with ordered hypotheses `ident1 = ident2` and `ident1 = t`, or `t′ = ident1` with `t′` not a variable, and no other hypotheses of the form `ident2 = u` or `u = ident2`; without the flag, a second call to subst would be necessary to replace `ident2` by `t` or `t′` respectively.

    - The presence of a recursive equation which without the flag would be a cause of failure of `subst`.

    - A context with cyclic dependencies as with hypotheses `ident1 = f ident2` and `ident2 = g ident1` which without the flag would be a cause of failure of `subst`.

    Additionally, it prevents a local definition such as `ident := t` from being unfolded which otherwise it would exceptionally unfold in configurations containing hypotheses of the form `ident = u`, or `u′ = ident` with `u′` not a variable. Finally, it preserves the initial order of hypotheses, which without the flag it may break.

  Error Cannot find any non-recursive equality over ident.
  Error Section variable ident occurs implicitly in global declaration qualid present in hypothesis ident.
  Error Section variable ident occurs implicitly in global declaration qualid present in the conclusion.

    Raised when the variable is a section variable with indirect dependencies in the goal. If `ident` is a section variable, it must not have any indirect occurrences in the goal, i.e. no global declarations implicitly depending on the section variable may be present in the goal.

Tactic simple subst

Tactic stepl one_term by ltac_expr?

  For chaining rewriting steps. It assumes a goal in the form `R term1 term2` where `R` is a binary relation and relies on a database of lemmas of the form `forall x y z, R x y -> eq x z -> R z y` where `eq` is typically a setoid equality. The application of `stepl one_term` then replaces the goal by `R one_term term2` and adds a new goal stating `eq one_term term1`.

  If `ltac_expr` is specified, it is applied to the side condition.

  Command Declare Left Step one_term

    Adds `one_term` to the database used by `stepl`.

  This tactic is especially useful for parametric setoids which are not accepted as regular setoids for `rewrite` and `setoid_replace` (see Generalized rewriting).

  Tactic stepr one_term by ltac_expr?

    This behaves like `stepl` but on the right hand side of the binary relation. Lemmas are expected to be in the form `forall x y z, R x y -> eq y z -> R x z`.

  Command Declare Right Step one_term

    Adds `term` to the database used by `stepr`.

## Rewriting with definitional equality

Tactic change one_termfrom at occs_nums? with? one_termto occurrences?

  Replaces terms with other convertible terms. If `one_termfrom` is not specified, then `one_termto` replaces the conclusion and/or the specified hypotheses. If `one_termfrom` is specified, the tactic replaces occurrences of `one_termto` within the conclusion and/or the specified hypotheses.

  `one_termfrom at occs_nums? with?`

    Replaces the occurrences of `one_termfrom` specified by `occs_nums` with `one_termto`, provided that the two `one_term`s are convertible. `one_termfrom` may contain pattern variables such as `?x`, whose value which will substituted for `x` in `one_termto`, such as in `change (f ?x ?y) with (g (x, y))` or `change (fun x => ?f x) with f`.

    The `at … with …` form is deprecated in 8.14; use `with … at …` instead. For `at … with … in H |-`, use `with … in H at … |-`.

  `occurrences`

    If `with` is not specified, `occurrences` must only specify entire hypotheses and/or the goal; it must not include any `at occs_nums` clauses.

  Error Not convertible.
  Error Found an "at" clause without "with" clause
  Tactic now_show one_term

    A synonym for `change one_term`. It can be used to make some proof steps explicit when refactoring a proof script to make it readable.

  See also

  Applying conversion rules

Tactic change_no_check one_termfrom at occs_nums? with? one_termto occurrences?

  For advanced usage. Similar to `change`, but as an optimization, it skips checking that `one_termto` is convertible with the goal or `one_termfrom`.

  Recall that the Coq kernel typechecks proofs again when they are concluded to ensure correctness. Hence, using `change` checks convertibility twice overall, while `change_no_check` can produce ill-typed terms, but checks convertibility only once. Hence, `change_no_check` can be useful to speed up certain proof scripts, especially if one knows by construction that the argument is indeed convertible to the goal.

  In the following example, `change_no_check` replaces `False` with `True`, but `Qed` then rejects the proof, ensuring consistency.

  Example

  Goal False.
    1 goal ============================ False
  change_no_check True.
    1 goal ============================ True
  exact I.
    No more goals.
  Qed.
    Toplevel input, characters 0-4: > Qed. > ^^^^ Error: The term "I" has type "True" while it is expected to have type "False".

  Example

  Goal True -> False.
    1 goal ============================ True -> False
  intro H.
    1 goal H : True ============================ False
  change_no_check False in H.
    1 goal H : False ============================ False
  exact H.
    No more goals.
  Qed.
    Toplevel input, characters 0-4: > Qed. > ^^^^ Error: The term "fun H : True => H" has type "True -> True" while it is expected to have type "True -> False".

## Applying conversion rules

These tactics apply reductions and expansions, replacing convertible subterms with others that are equal by definition in CIC. They implement different specialized uses of the `change` tactic. Other ways to apply these reductions are through the `Eval` command, the `Eval` clause in the `Definition`/`Example` command and the `eval` tactic.

Tactics described in this section include:

- `lazy` and `cbv`, which allow precise selection of which reduction rules to apply

- `simpl` and `cbn`, which are "clever" tactics meant to give the most readable result

- `hnf` and `red`, which apply reduction rules only to the head of the term

- `vm_compute` and `native_compute`, which are performance-oriented.

Conversion tactics, with two exceptions, only change the types and contexts of existential variables and leave the proof term unchanged. (The `vm_compute` and `native_compute` tactics change existential variables in a way similar to other conversions while also adding a single explicit cast to the proof term to tell the kernel which reduction engine to use. See Type cast.) For example:

> Goal 3 + 4 = 7.
>   1 goal ============================ 3 + 4 = 7
> Show Proof.
>   ?Goal
> Show Existentials.
>   Existential 1 = ?Goal : [ |- 3 + 4 = 7]
> cbv.
>   1 goal ============================ 7 = 7
> Show Proof.
>   (?Goal : 3 + 4 = 7)
> Show Existentials.
>   Existential 1 = ?Goal : [ |- 7 = 7]
> Abort.

Tactic lazy reductions? simple_occurrences
Tactic cbv reductions? simple_occurrences
  reductions`::=`reduction+`````|`delta_reductions``reduction`::=`beta`````|`delta delta_reductions?`````|`match`````|`fix`````|`cofix`````|`iota`````|`zeta``delta_reductions`::=`-? [ reference+ ]``

  Normalize the goal as specified by `reductions`. If no reductions are specified by name, all reductions are applied. If any reductions are specified by name, then only the named reductions are applied. The reductions include:

  `beta`

    beta-reduction of functional application

  `delta delta_reductions?`

    delta-reduction: unfolding of transparent constants, see Controlling reduction strategies and the conversion algorithm. The form in `reductions` without the keyword `delta` includes `beta`, `iota` and `zeta` reductions in addition to `delta` using the given `delta_reductions`.

    `-? [ reference+ ]`

      without the `-`, limits delta unfolding to the listed constants. If the `-` is present, unfolding is applied to all constants that are not listed. Notice that the `delta` doesn't apply to variables bound by a let-in construction inside the term itself (use `zeta` to inline these). Opaque constants are never unfolded except by `vm_compute` and `native_compute` (see #4476 and Controlling reduction strategies and the conversion algorithm).

  `iota`

    iota-reduction of pattern matching (`match`) over a constructed term and reduction of `fix` and `cofix` expressions. Shorthand for `match fix cofix`.

  `zeta`

    zeta-reduction: reduction of let-in definitions

  Normalization is done by first evaluating the head of the expression into weak-head normal form, i.e. until the evaluation is blocked by a variable, an opaque constant, an axiom, such as in `x u1 … un`, `match x with … end`, `(fix f x {struct x} := …) x`, a constructed form (a \(\lambda\)-expression, constructor, cofixpoint, inductive type, product type or sort) or a redex for which flags prevent reduction of the redex. Once a weak-head normal form is obtained, subterms are recursively reduced using the same strategy.

  There are two strategies for reduction to weak-head normal form: lazy (the `lazy` tactic), or call-by-value (the `cbv` tactic). The lazy strategy is a call by need strategy, with sharing of reductions: the arguments of a function call are weakly evaluated only when necessary, and if an argument is used several times then it is weakly computed only once. This reduction is efficient for reducing expressions with dead code. For instance, the proofs of a proposition `exists x. P(x)` reduce to a pair of a witness `t` and a proof that `t` satisfies the predicate `P`. Most of the time, `t` may be computed without computing the proof of `P(t)`, thanks to the lazy strategy.

  Flag Kernel Term Sharing

    Turning this flag off disables the sharing of computations in `lazy`, making it a call-by-name reduction. This also affects the reduction procedure used by the kernel when typechecking. By default sharing is activated.

  The call-by-value strategy is the one used in ML languages: the arguments of a function call are systematically weakly evaluated first. The lazy strategy is similar to how Haskell reduces terms. Although the lazy strategy always does fewer reductions than the call-by-value strategy, the latter is generally more efficient for evaluating purely computational expressions (i.e. with little dead code).

  Tactic compute delta_reductions? simple_occurrences

    A variant form of `cbv`.

  `Debug` `"Cbv"` makes `cbv` (and its derivative `compute`) print information about the constants it encounters and the unfolding decisions it makes.

Tactic simpl delta_reductions? reference_occs​pattern_occs? simple_occurrences
  reference_occs`::=`reference at occs_nums?``pattern_occs`::=`one_term at occs_nums?``

  Reduces a term to something still readable instead of fully normalizing it. It performs a sort of strong normalization with two key differences:

  - It unfolds constants only if they lead to an ι-reduction, i.e. reducing a match or unfolding a fixpoint.

  - When reducing a constant unfolding to (co)fixpoints, the tactic uses the name of the constant the (co)fixpoint comes from instead of the (co)fixpoint definition in recursive calls.

  `simple_occurrences`

    Permits selecting whether to reduce the conclusion and/or one or more hypotheses. While the `at` option of `occurrences` is not allowed here, `reference_occs` and `pattern_occs` have a somewhat less flexible `at` option for selecting specific occurrences.

  `simpl` can unfold transparent constants whose name can be reused in recursive calls as well as those designated by `Arguments` `reference … /` commands. For instance, a constant `plus' := plus` may be unfolded and reused in recursive calls, but a constant such as `succ := plus (S O)` is not unfolded unless it was specifically designated in an `Arguments` command such as `Arguments succ /.`.

  `reference_occs​pattern_occs` can limit the application of `simpl` to:

  - applicative subterms whose head is the constant `qualid` or is the constant used in the notation `string` (see `reference`)

  - subterms matching a pattern `one_term`

Tactic cbn reductions? simple_occurrences

  `cbn` was intended to be a more principled, faster and more predictable replacement for `simpl`.

  The main difference between `cbn` and `simpl` is that `cbn` may unfold constants even when they cannot be reused in recursive calls: in the previous example, `succ t` is reduced to `S t`.

  `Debug` `"RAKAM"` makes `cbn` print various debugging information. `RAKAM` is the Refolding Algebraic Krivine Abstract Machine.

Tactic hnf simple_occurrences

  Replaces the current goal with its weak-head normal form according to the βδιζ-reduction rules, i.e. it reduces the head of the goal until it becomes a product or an irreducible term. All inner βι-redexes are also reduced. While `hnf` behaves similarly to `simpl` and `cbn`, unlike them, it does not recurse into subterms. The behavior of `hnf` can be tuned using the `Arguments` command.

  Example: The term `fun n : nat => S n + S n` is not reduced by `hnf`.

  Note

  The δ rule only applies to transparent constants (see Controlling reduction strategies and the conversion algorithm on transparency and opacity).

Tactic red simple_occurrences

  βιζ-reduces the constant at the head of `T` (which may be called the head constant; head means the beginning of the term), if possible, in the selected hypotheses and/or the goal, which must have the form:

  > `forall open_binders,? T`

  (where `T` does not begin with a `forall`) to `c t1 … tn` where `c` is a constant. If `c` is transparent then it replaces `c` with its definition and reduces again until no further reduction is possible.

  Error No head constant to reduce.

Tactic unfold reference_occs+, occurrences?

  Applies delta-reduction to the constants specified by each `reference_occs`. The selected hypotheses and/or goals are then reduced to βιζ-normal form. Use the general reduction tactics if you want to only apply the δ rule, for example `cbv` `delta [ reference ]`.

  `reference_occs`

    If `reference` is a `qualid`, it must be a defined transparent constant or local definition (see Top-level definitions and Controlling reduction strategies and the conversion algorithm).

    If `reference` is a `string scope_key?`, the `string` is the discriminating symbol of a notation (e.g. "+") or an expression defining a notation (e.g. `"_ + _"`) and the notation is an application whose head symbol is an unfoldable constant, then the tactic unfolds it.

  `occurrences`

    If `occurrences` is specified, the specified occurrences will be replaced in the selected hypotheses and/or goal. Otherwise every occurrence of the constants in the goal is replaced. If multiple `reference_occs` are given, any `at` clauses must be in the `reference_occs` rather than in `occurrences`.

  Error Cannot turn inductive​constructor into an evaluable reference.

    Occurs when trying to unfold something that is defined as an inductive type (or constructor) and not as a definition.

    Example

    Goal 0 <= 1.
      1 goal ============================ 0 <= 1
    unfold le.
      Toplevel input, characters 7-9: > unfold le. > ^^ Error: Cannot turn inductive le into an evaluable reference.
  Error ident is opaque.

    Raised if you are trying to unfold a definition that has been marked opaque.

    Example

    Opaque Nat.add.
    Goal 1 + 0 = 1.
      1 goal ============================ 1 + 0 = 1
    unfold Nat.add.
      Toplevel input, characters 0-14: > unfold Nat.add. > ^^^^^^^^^^^^^^ Error: Nat.add is opaque.
    Error Bad occurrence number of qualid.
    Error qualid does not occur.

Tactic fold one_term+ simple_occurrences

  First, this tactic reduces each `one_term` using the `red` tactic. Then, every occurrence of the resulting terms in the selected hypotheses and/or goal will be replaced by its associated `one_term`. This tactic is particularly useful for reversing undesired unfoldings, which may make the goal very hard to read. The undesired unfoldings may be due to the limited capabilities of other reduction tactics. On the other hand, when an unfolded function applied to its argument has been reduced, the `fold` tactic doesn't do anything.

  `fold` `one_term1 one_term2` is equivalent to `fold one_term1; fold one_term2`.

  Example: `fold` doesn't always undo `unfold`

  Goal ~0=0.
    1 goal ============================ 0 <> 0
  unfold not.
    1 goal ============================ 0 = 0 -> False

  This `fold` doesn't undo the preceeding `unfold` (it makes no change):

  fold not.
    1 goal ============================ 0 = 0 -> False

  However, this `pattern` followed by `fold` does:

  pattern (0 = 0).
    1 goal ============================ (fun P : Prop => P -> False) (0 = 0)
  fold not.
    1 goal ============================ 0 <> 0

  Example: Use `fold` to reverse unfolding of `fold_right`

  Require Import Coq.Lists.List.
  Local Open Scope list_scope.
  Goal forall x xs, fold_right and True (x::xs).
    1 goal ============================ forall (x : Prop) (xs : list Prop), fold_right and True (x :: xs)
  red.
    1 goal ============================ forall (x : Prop) (xs : list Prop), x /\ (fix fold_right (l : list Prop) : Prop := match l with | nil => True | b :: t => b /\ fold_right t end) xs
  fold (fold_right and True).
    1 goal ============================ forall (x : Prop) (xs : list Prop), x /\ fold_right and True xs

Tactic pattern pattern_occs+, occurrences?

  Performs beta-expansion (the inverse of beta-reduction) for the selected hypotheses and/or goals. The `one_term`s in `pattern_occs` must be free subterms in the selected items. The expansion is done for each selected item `T` for a set of `one_term`s in the `pattern_occs` by:

  - replacing all selected occurrences of the `one_term`s in `T` with fresh variables

  - abstracting these variables

  - applying the abstracted goal to the `one_term`s

  For instance, if the current goal `T` is expressible as `φ(t1 … tn)` where the notation captures all the instances of the `ti` in φ, then `pattern` `t1, …, tn` generates the equivalent goal `(fun (x1:A1 … (xn:An) => φ(x1 … xn)) t1 … tn`. If `ti` occurs in one of the generated types `Aj` (for `j > i`), occurrences will also be considered and possibly abstracted.

  This tactic can be used, for instance, when the tactic `apply` fails on matching or to better control the behavior of `rewrite`.

### Fast reduction tactics: vm_compute and native_compute

`vm_compute` is a brute-force but efficient tactic that first normalizes the terms before comparing them. It is based on a bytecode representation of terms similar to the bytecode representation used in the ZINC virtual machine [Ler90]. It is especially useful for intensive computation of algebraic values, such as numbers, and for reflection-based tactics.

`native_compute` is based on on converting the Coq code to OCaml.

Note that both these tactics ignore `Opaque` markings (see issue #4776), nor do they apply unfolding strategies such as from `Strategy`.

`native_compute` is typically two to five times faster than `vm_compute` at applying conversion rules when Coq is running native code, but `native_compute` requires considerably more overhead. We recommend using `native_compute` when all of the following are true (otherwise use `vm_compute`):

- the running time in `vm_compute` at least 5-10 seconds

- the size of the input term is small (e.g. hand-generated code rather than automatically-generated code that may have nested destructs on inductives with dozens or hundreds of constructors)

- the output is small (e.g. you're returning a boolean, a natural number or an integer rather than a large abstract syntax tree)

These tactics change existential variables in a way similar to other conversions while also adding a single explicit cast (see Type cast) to the proof term to tell the kernel which reduction engine to use.

Tactic vm_compute reference_occs​pattern_occs? occurrences?

  Evaluates the goal using the optimized call-by-value evaluation bytecode-based virtual machine described in [GregoireL02]. This algorithm is dramatically more efficient than the algorithm used for the `cbv` tactic, but it cannot be fine-tuned. It is especially useful for full evaluation of algebraic objects. This includes the case of reflection-based tactics.

Tactic native_compute reference_occs​pattern_occs? occurrences?

  Evaluates the goal by compilation to OCaml as described in [BDenesGregoire11]. Depending on the configuration, this tactic can either default to `vm_compute`, recompile dependencies or fail due to some missing precompiled dependencies, see the native-compiler option for details.

  Flag NativeCompute Timing

    This flag causes all calls to the native compiler to print timing information for the conversion to native code, compilation, execution, and reification phases of native compilation. Timing is printed in units of seconds of wall-clock time.

  Flag NativeCompute Profiling

    On Linux, if you have the `perf` profiler installed, this flag makes it possible to profile `native_compute` evaluations.

  Option NativeCompute Profile Filename string

    This option specifies the profile output; the default is `native_compute_profile.data`. The actual filename used will contain extra characters to avoid overwriting an existing file; that filename is reported to the user. That means you can individually profile multiple uses of `native_compute` in a script. From the Linux command line, run `perf report` on the profile file to see the results. Consult the `perf` documentation for more details.

### Computing in a term: eval and Eval

Evaluation of a term can be performed with:

Tactic eval red_expr in term
  red_expr`::=`lazy reductions?`````|`cbv reductions?`````|`compute delta_reductions?`````|`vm_compute reference_occs​pattern_occs?`````|`native_compute reference_occs​pattern_occs?`````|`red`````|`hnf`````|`simpl delta_reductions? reference_occs​pattern_occs?`````|`cbn reductions?`````|`unfold reference_occs+,`````|`fold one_term+`````|`pattern pattern_occs+,`````|`ident``

  `eval` is a `value_tactic`. It returns the result of applying the conversion rules specified by `red_expr`. It does not change the proof state.

  The `red_expr` alternatives that begin with a keyword correspond to the tactic with the same name, though in several cases with simpler syntax than the tactic. `ident` is a named reduction expression created with `Declare Reduction`.

  See also

  Section Applying conversion rules.

Command Eval red_expr in term

  Performs the specified reduction on `term` and displays the resulting term with its type. If a proof is open, `term` may reference hypotheses of the selected goal. `Eval` is a `query_command`, so it may be prefixed with a goal selector.

  Command Compute term

    Evaluates `term` using the bytecode-based virtual machine. It is a shortcut for `Eval` `vm_compute in term`. `Compute` is a `query_command`, so it may be prefixed with a goal selector.

Command Declare Reduction ident := red_expr

  Declares a short name for the reduction expression `red_expr`, for instance `lazy beta delta [foo bar]`. This short name can then be used in `Eval ident in` or `eval` constructs. This command accepts the `local` attribute, which indicates that the reduction will be discarded at the end of the file or module. The name is not qualified. In particular declaring the same name in several modules or in several functor applications will be rejected if these declarations are not local. The name `ident` cannot be used directly as an Ltac tactic, but nothing prevents the user from also performing a `Ltac ident := red_expr`.

## Controlling reduction strategies and the conversion algorithm

The commands to fine-tune the reduction strategies and the lazy conversion algorithm are described in this section. Also see Effects of Arguments on unfolding, which supports additional fine-tuning.

Command Opaque reference+

  Marks the specified constants as opaque so tactics won't unfold them with delta-reduction. "Constants" are items defined by commands such as `Definition`, `Let` (with an explicit body), `Fixpoint`, `CoFixpoint` and `Function`.

  This command accepts the `global` attribute. By default, the scope of `Opaque` is limited to the current section or module.

  `Opaque` also affects Coq's conversion algorithm, causing it to delay unfolding the specified constants as much as possible when it has to check that two distinct applied constants are convertible. See Section Conversion rules.

Command Transparent reference+

  The opposite of `Opaque`, it marks the specified constants as transparent so that tactics may unfold them. See `Opaque` above.

  This command accepts the `global` attribute. By default, the scope of `Transparent` is limited to the current section or module.

  Note that constants defined by proofs ending with `Qed` are irreversibly opaque; `Transparent` will not make them transparent. This is consistent with the usual mathematical practice of proof irrelevance: what matters in a mathematical development is the sequence of lemma statements, not their actual proofs. This distinguishes lemmas from the usual defined constants, whose actual values are of course relevant in general.

  Error The reference qualid was not found in the current environment.

    There is no constant named `qualid` in the environment.

See also

Applying conversion rules, `Qed` and `Defined`

Command Strategy strategy_level [ reference+ ]+
  strategy_level`::=`opaque`````|`integer`````|`expand`````|`transparent``

  Generalizes the behavior of the `Opaque` and `Transparent` commands. It is used to fine-tune the strategy for unfolding constants, both at the tactic level and at the kernel level. This command associates a `strategy_level` with the qualified names in the `reference` sequence. Whenever two expressions with two distinct head constants are compared (for example, typechecking `f x` where `f : A -> B` and `x : C` will result in converting `A` and `C`), the one with lower level is expanded first. In case of a tie, the second one (appearing in the cast type) is expanded.

  This command accepts the `local` attribute, which limits its effect to the current section or module, in which case the section and module behavior is the same as `Opaque` and `Transparent` (without `global`).

  Levels can be one of the following (higher to lower):

  > - `opaque` : level of opaque constants. They cannot be expanded by tactics (behaves like +∞, see next item).

  > - `integer` : levels indexed by an integer. Level 0 corresponds to the default behavior, which corresponds to transparent constants. This level can also be referred to as `transparent`. Negative levels correspond to constants to be expanded before normal transparent constants, while positive levels correspond to constants to be expanded after normal transparent constants.

  > - `expand` : level of constants that should be expanded first (behaves like −∞)

  > - `transparent` : Equivalent to level 0

Command Print Strategy reference

  This command prints the strategy currently associated with `reference`. It fails if `reference` is not an unfoldable reference, that is, neither a variable nor a constant.

  Error The reference is not unfoldable.

Command Print Strategies

  Print all the currently non-transparent strategies.

Tactic with_strategy strategy_level_or_var [ reference+ ] ltac_expr3
  strategy_level_or_var`::=`strategy_level`````|`ident``

  Executes `ltac_expr3`, applying the alternate unfolding behavior that the `Strategy` command controls, but only for `ltac_expr3`. This can be useful for guarding calls to reduction in tactic automation to ensure that certain constants are never unfolded by tactics like `simpl` and `cbn` or to ensure that unfolding does not fail.

  Example

  Opaque id.
  Goal id 10 = 10.
    1 goal ============================ id 10 = 10
  Fail unfold id.
    The command has indeed failed with message: id is opaque.
  with_strategy transparent [id] unfold id.
    1 goal ============================ 10 = 10

  Warning

  Use this tactic with care, as effects do not persist past the end of the proof script. Notably, this fine-tuning of the conversion strategy is not in effect during `Qed` nor `Defined`, so this tactic is most useful either in combination with `abstract`, which will check the proof early while the fine-tuning is still in effect, or to guard calls to conversion in tactic automation to ensure that, e.g., `unfold` does not fail just because the user made a constant `Opaque`.

  This can be illustrated with the following example involving the factorial function.

  Fixpoint fact (n : nat) : nat := match n with | 0 => 1 | S n' => n * fact n' end.
    fact is defined fact is recursively defined (guarded on 1st argument)

  Suppose now that, for whatever reason, we want in general to unfold the `id` function very late during conversion:

  Strategy 1000 [id].

  If we try to prove `id (fact n) = fact n` by `reflexivity`, it will now take time proportional to \(n!\), because Coq will keep unfolding `fact` and `*` and `+` before it unfolds `id`, resulting in a full computation of `fact n` (in unary, because we are using `nat`), which takes time \(n!\). We can see this cross the relevant threshold at around \(n = 9\):

  Goal True.
    1 goal ============================ True
  Time assert (id (fact 8) = fact 8) by reflexivity.
    Finished transaction in 0.413 secs (0.31u,0.101s) (successful) 1 goal H : id (fact 8) = fact 8 ============================ True
  Time assert (id (fact 9) = fact 9) by reflexivity.
    Finished transaction in 2.482 secs (2.468u,0.009s) (successful) 1 goal H : id (fact 8) = fact 8 H0 : id (fact 9) = fact 9 ============================ True

  Note that behavior will be the same if you mark `id` as `Opaque` because while most reduction tactics refuse to unfold `Opaque` constants, conversion treats `Opaque` as merely a hint to unfold this constant last.

  We can get around this issue by using `with_strategy`:

  Goal True.
    1 goal ============================ True
  Fail Timeout 1 assert (id (fact 100) = fact 100) by reflexivity.
    The command has indeed failed with message: Timeout!
  Time assert (id (fact 100) = fact 100) by with_strategy -1 [id] reflexivity.
    Finished transaction in 0.002 secs (0.002u,0.s) (successful) 1 goal H : id (fact 100) = fact 100 ============================ True

  However, when we go to close the proof, we will run into trouble, because the reduction strategy changes are local to the tactic passed to `with_strategy`.

  exact I.
    No more goals.
  Timeout 1 Defined.
    Toplevel input, characters 0-18: > Timeout 1 Defined. > ^^^^^^^^^^^^^^^^^^ Error: Timeout!

  We can fix this issue by using `abstract`:

  Goal True.
    1 goal ============================ True
  Time assert (id (fact 100) = fact 100) by with_strategy -1 [id] abstract reflexivity.
    Finished transaction in 0.004 secs (0.004u,0.s) (successful) 1 goal H : id (fact 100) = fact 100 ============================ True
  exact I.
    No more goals.
  Time Defined.
    Finished transaction in 0.002 secs (0.002u,0.s) (successful)

  On small examples this sort of behavior doesn't matter, but because Coq is a super-linear performance domain in so many places, unless great care is taken, tactic automation using `with_strategy` may not be robustly performant when scaling the size of the input.

  Warning

  In much the same way this tactic does not play well with `Qed` and `Defined` without using `abstract` as an intermediary, this tactic does not play well with `coqchk`, even when used with `abstract`, due to the inability of tactics to persist information about conversion hints in the proof term. See #12200 for more details.

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