\[\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}\]

# Solvers for logic and equality

Tactic tauto

  This tactic implements a decision procedure for intuitionistic propositional calculus based on the contraction-free sequent calculi LJT* of Roy Dyckhoff [Dyc92]. Note that `tauto` succeeds on any instance of an intuitionistic tautological proposition. `tauto` unfolds negations and logical equivalence but does not unfold any other definition.

  Example

  The following goal can be proved by `tauto` whereas `auto` would fail:

  Goal forall (x:nat) (P:nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x.
    1 goal ============================ forall (x : nat) (P : nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x
  intros.
    1 goal x : nat P : nat -> Prop H : x = 0 \/ P x H0 : x <> 0 ============================ P x
  tauto.
    No more goals.

  Moreover, if it has nothing else to do, `tauto` performs introductions. Therefore, the use of `intros` in the previous proof is unnecessary. `tauto` can for instance for:

  Example

  Goal forall (A:Prop) (P:nat -> Prop), A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ A -> P x.
    1 goal ============================ forall (A : Prop) (P : nat -> Prop), A \/ (forall x : nat, ~ A -> P x) -> forall x : nat, ~ A -> P x
  tauto.
    No more goals.

  Note

  In contrast, `tauto` cannot solve the following goal `Goal forall (A:Prop) (P:nat -> Prop), A \/ (forall x:nat, ~ A -> P x) ->` `forall x:nat, ~ ~ (A \/ P x).` because `(forall x:nat, ~ A -> P x)` cannot be treated as atomic and an instantiation of `x` is necessary.

  Tactic dtauto

    While `tauto` recognizes inductively defined connectives isomorphic to the standard connectives `and`, `prod`, `or`, `sum`, `False`, `Empty_set`, `unit` and `True`, `dtauto` also recognizes all inductive types with one constructor and no indices, i.e. record-style connectives.

Tactic intuition ltac_expr?

  Uses the search tree built by the decision procedure for `tauto` to generate a set of subgoals equivalent to the original one (but simpler than it) and applies `ltac_expr` to them [Mun94]. If `ltac_expr` is not specified, it defaults to `Tauto.intuition_solver`. The initial value of `intuition_solver` is `auto with *`. If `ltac_expr` fails on some goals then `intuition` fails. In fact, `tauto` is simply `intuition fail`.

  `intuition` recognizes inductively defined connectives isomorphic to the standard connectives `and`, `prod`, `or`, `sum`, `False`, `Empty_set`, `unit` and `True`.

  Example

  For instance, the tactic `intuition auto` applied to the goal:

      (forall (x:nat), P x) /\ B -> (forall (y:nat), P y) /\ P O \/ B /\ P O

  internally replaces it by the equivalent one:

      (forall (x:nat), P x), B |- P O

  and then uses `auto` which completes the proof.

  Tactic dintuition ltac_expr?

    In addition to the inductively defined connectives recognized by `intuition`, `dintuition` also recognizes all inductive types with one constructor and no indices, i.e. record-style connectives.

  Flag Intuition Negation Unfolding

    This flag controls whether `intuition` unfolds inner negations which do not need to be unfolded. It is on by default.

Tactic rtauto

  Solves propositional tautologies similarly to `tauto`, but the proof term is built using a reflection scheme applied to a sequent calculus proof of the goal. The search procedure is also implemented using a different technique.

  Users should be aware that this difference may result in faster proof search but slower proof checking, and `rtauto` might not solve goals that `tauto` would be able to solve (e.g. goals involving universal quantifiers).

  Note that this tactic is only available after a `Require Import Rtauto`.

Tactic firstorder ltac_expr? using qualid+,? with ident+?

  An experimental extension of `tauto` to first-order reasoning. It is not restricted to usual logical connectives but instead can reason about any first-order class inductive definition.

  `ltac_expr`

    Tries to solve the goal with `ltac_expr` when no logical rule applies. If unspecified, the tactic uses the default from the `Firstorder Solver` option.

  `using qualid+,`

    Adds the lemmas `qualid+,` to the proof search environment. If `qualid` refers to an inductive type, its constructors are added to the proof search environment.

  `with ident+`

    Adds lemmas from `auto` hint bases `ident+` to the proof search environment.

  Option Firstorder Solver ltac_expr

    The default tactic used by `firstorder` when no rule applies in `auto with core`. It can be set locally or globally using this option.

  Command Print Firstorder Solver

    Prints the default tactic used by `firstorder` when no rule applies.

  Option Firstorder Depth natural

    This option controls the proof search depth bound.

  Tactic gintuition ltac_expr?

    An extension of `intuition` to first-order reasoning (similar to how `firstorder` extends `tauto`).

Tactic congruence natural? with one_term+?
  `natural`

    Specifies the maximum number of hypotheses stating quantified equalities that may be added to the problem in order to solve it. The default is 1000.

  `with one_term+?`

    Adds `one_term+` to the pool of terms used by `congruence`. This helps in case you have partially applied constructors in your goal.

  Implements the standard Nelson and Oppen congruence closure algorithm, which is a decision procedure for ground equalities with uninterpreted symbols. It also includes constructor theory (see `injection` and `discriminate`). If the goal is a non-quantified equality, congruence tries to prove it with non-quantified equalities in the context. Otherwise it tries to infer a discriminable equality from those in the context. Alternatively, congruence tries to prove that a hypothesis is equal to the goal or to the negation of another hypothesis.

  `congruence` is also able to take advantage of hypotheses stating quantified equalities, but you have to provide a bound for the number of extra equalities generated that way. Please note that one of the sides of the equality must contain all the quantified variables in order for congruence to match against it.

  Increasing the maximum number of hypotheses may solve problems that would have failed with a smaller value. It will make failures slower but it won't make successes found with the smaller value any slower. You may want to use `assert` to add some lemmas as hypotheses so that `congruence` can use them.

  Tactic simple congruence natural? with one_term+?

    Behaves like `congruence`, but does not unfold definitions.

  Example

  Theorem T (A:Type) (f:A -> A) (g: A -> A -> A) a b: a=(f a) -> (g b (f a))=(f (f a)) -> (g a b)=(f (g b a)) -> (g a b)=a.
    1 goal A : Type f : A -> A g : A -> A -> A a, b : A ============================ a = f a -> g b (f a) = f (f a) -> g a b = f (g b a) -> g a b = a
  intros.
    1 goal A : Type f : A -> A g : A -> A -> A a, b : A H : a = f a H0 : g b (f a) = f (f a) H1 : g a b = f (g b a) ============================ g a b = a
  congruence.
    No more goals.
  Qed.
  Theorem inj (A:Type) (f:A -> A * A) (a c d: A) : f = pair a -> Some (f c) = Some (f d) -> c=d.
    1 goal A : Type f : A -> A * A a, c, d : A ============================ f = pair a -> Some (f c) = Some (f d) -> c = d
  intros.
    1 goal A : Type f : A -> A * A a, c, d : A H : f = pair a H0 : Some (f c) = Some (f d) ============================ c = d
  congruence.
    No more goals.
  Qed.
  Error I don’t know how to handle dependent equality.

    The decision procedure managed to find a proof of the goal or of a discriminable equality but this proof could not be built in Coq because of dependently-typed functions.

  Error Goal is solvable by congruence but some arguments are missing. Try congruence with term+, replacing metavariables by arbitrary terms.

    The decision procedure could solve the goal with the provision that additional arguments are supplied for some partially applied constructors. Any term of an appropriate type will allow the tactic to successfully solve the goal. Those additional arguments can be given to congruence by filling in the holes in the terms given in the error message, using the `with` clause.

  Setting `Debug` `"congruence"` makes `congruence` print debug information.

Tactic btauto

  The tactic `btauto` implements a reflexive solver for boolean tautologies. It solves goals of the form `t = u` where `t` and `u` are constructed over the following grammar:

  btauto_term`::=`ident`````|`true`````|`false`````|`orb btauto_term btauto_term`````|`andb btauto_term btauto_term`````|`xorb btauto_term btauto_term`````|`negb btauto_term`````|`if btauto_term then btauto_term else btauto_term``

  Whenever the formula supplied is not a tautology, it also provides a counter-example.

  Internally, it uses a system very similar to the one of the ring tactic.

  Note that this tactic is only available after a `Require Import Btauto`.

  Error Cannot recognize a boolean equality.

    The goal is not of the form `t = u`. Especially note that `btauto` doesn't introduce variables into the context on its own.

Other versions v: 8.16.1

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