diff options
Diffstat (limited to 'doc/sphinx')
| -rw-r--r-- | doc/sphinx/README.rst | 4 | ||||
| -rw-r--r-- | doc/sphinx/README.template.rst | 4 | ||||
| -rw-r--r-- | doc/sphinx/addendum/generalized-rewriting.rst | 72 | ||||
| -rw-r--r-- | doc/sphinx/addendum/implicit-coercions.rst | 14 | ||||
| -rw-r--r-- | doc/sphinx/addendum/micromega.rst | 2 | ||||
| -rw-r--r-- | doc/sphinx/addendum/ring.rst | 14 | ||||
| -rw-r--r-- | doc/sphinx/language/cic.rst | 115 | ||||
| -rw-r--r-- | doc/sphinx/language/coq-library.rst | 38 | ||||
| -rw-r--r-- | doc/sphinx/language/gallina-extensions.rst | 22 | ||||
| -rw-r--r-- | doc/sphinx/language/gallina-specification-language.rst | 106 | ||||
| -rw-r--r-- | doc/sphinx/proof-engine/ltac.rst | 170 | ||||
| -rw-r--r-- | doc/sphinx/proof-engine/tactics.rst | 701 | ||||
| -rw-r--r-- | doc/sphinx/user-extensions/syntax-extensions.rst | 72 |
13 files changed, 781 insertions, 553 deletions
diff --git a/doc/sphinx/README.rst b/doc/sphinx/README.rst index a20b74822c..e4f078c1d6 100644 --- a/doc/sphinx/README.rst +++ b/doc/sphinx/README.rst @@ -416,12 +416,12 @@ Omitting annotations DO .. code:: - .. tacv:: assert @form as @intro_pattern + .. tacv:: assert @form as @simple_intropattern DON'T .. code:: - .. tacv:: assert form as intro_pattern + .. tacv:: assert form as simple_intropattern Using the ``.. coqtop::`` directive for syntax highlighting ----------------------------------------------------------- diff --git a/doc/sphinx/README.template.rst b/doc/sphinx/README.template.rst index 11f0cdc008..81f25bf274 100644 --- a/doc/sphinx/README.template.rst +++ b/doc/sphinx/README.template.rst @@ -172,12 +172,12 @@ Omitting annotations DO .. code:: - .. tacv:: assert @form as @intro_pattern + .. tacv:: assert @form as @simple_intropattern DON'T .. code:: - .. tacv:: assert form as intro_pattern + .. tacv:: assert form as simple_intropattern Using the ``.. coqtop::`` directive for syntax highlighting ----------------------------------------------------------- diff --git a/doc/sphinx/addendum/generalized-rewriting.rst b/doc/sphinx/addendum/generalized-rewriting.rst index e468cc63cd..b606fb4dd2 100644 --- a/doc/sphinx/addendum/generalized-rewriting.rst +++ b/doc/sphinx/addendum/generalized-rewriting.rst @@ -714,47 +714,47 @@ following grammar: .. productionlist:: rewriting s, t, u : `strategy` - : | `lemma` - : | `lemma_right_to_left` - : | `failure` - : | `identity` - : | `reflexivity` - : | `progress` - : | `failure_catch` - : | `composition` - : | `left_biased_choice` - : | `iteration_one_or_more` - : | `iteration_zero_or_more` - : | `one_subterm` - : | `all_subterms` - : | `innermost_first` - : | `outermost_first` - : | `bottom_up` - : | `top_down` - : | `apply_hint` - : | `any_of_the_terms` - : | `apply_reduction` - : | `fold_expression` + : `lemma` + : `lemma_right_to_left` + : `failure` + : `identity` + : `reflexivity` + : `progress` + : `failure_catch` + : `composition` + : `left_biased_choice` + : `iteration_one_or_more` + : `iteration_zero_or_more` + : `one_subterm` + : `all_subterms` + : `innermost_first` + : `outermost_first` + : `bottom_up` + : `top_down` + : `apply_hint` + : `any_of_the_terms` + : `apply_reduction` + : `fold_expression` .. productionlist:: rewriting - strategy : "(" `s` ")" + strategy : ( `s` ) lemma : `c` - lemma_right_to_left : "<-" `c` - failure : `fail` - identity : `id` - reflexivity : `refl` - progress : `progress` `s` - failure_catch : `try` `s` - composition : `s` ";" `u` + lemma_right_to_left : <- `c` + failure : fail + identity : id + reflexivity : refl + progress : progress `s` + failure_catch : try `s` + composition : `s` ; `u` left_biased_choice : choice `s` `t` - iteration_one_or_more : `repeat` `s` - iteration_zero_or_more : `any` `s` + iteration_one_or_more : repeat `s` + iteration_zero_or_more : any `s` one_subterm : subterm `s` all_subterms : subterms `s` - innermost_first : `innermost` `s` - outermost_first : `outermost` `s` - bottom_up : `bottomup` `s` - top_down : `topdown` `s` + innermost_first : innermost `s` + outermost_first : outermost `s` + bottom_up : bottomup `s` + top_down : topdown `s` apply_hint : hints `hintdb` any_of_the_terms : terms (`c`)+ apply_reduction : eval `redexpr` @@ -767,7 +767,7 @@ primitive fixpoint operator: .. productionlist:: rewriting try `s` : choice `s` `id` any `s` : fix `u`. try (`s` ; `u`) - repeat `s` : `s` ; `any` `s` + repeat `s` : `s` ; any `s` bottomup s : fix `bu`. (choice (progress (subterms bu)) s) ; try bu topdown s : fix `td`. (choice s (progress (subterms td))) ; try td innermost s : fix `i`. (choice (subterm i) s) diff --git a/doc/sphinx/addendum/implicit-coercions.rst b/doc/sphinx/addendum/implicit-coercions.rst index 64e2d7c4ab..e5b41be691 100644 --- a/doc/sphinx/addendum/implicit-coercions.rst +++ b/doc/sphinx/addendum/implicit-coercions.rst @@ -41,8 +41,8 @@ Formally, the syntax of a classes is defined as: .. productionlist:: class: `qualid` - : | Sortclass - : | Funclass + : Sortclass + : Funclass Coercions @@ -184,10 +184,10 @@ Figure :ref:`vernacular` as follows: \comindex{Hypothesis \mbox{\rm (and coercions)}} .. productionlist:: - assumption : assumption_keyword assums . - assums : simple_assums - : | (simple_assums) ... (simple_assums) - simple_assums : ident ... ident :[>] term + assumption : `assumption_keyword` `assums` . + assums : `simple_assums` + : (`simple_assums`) ... (`simple_assums`) + simple_assums : `ident` ... `ident` :[>] `term` If the extra ``>`` is present before the type of some assumptions, these assumptions are declared as coercions. @@ -203,7 +203,7 @@ grammar of inductive types from Figure :ref:`vernacular` as follows: .. productionlist:: inductive : Inductive `ind_body` with ... with `ind_body` - : | CoInductive `ind_body` with ... with `ind_body` + : CoInductive `ind_body` with ... with `ind_body` ind_body : `ident` [ `binders` ] : `term` := [[|] `constructor` | ... | `constructor` ] constructor : `ident` [ `binders` ] [:[>] `term` ] diff --git a/doc/sphinx/addendum/micromega.rst b/doc/sphinx/addendum/micromega.rst index fd66de427c..e799677c59 100644 --- a/doc/sphinx/addendum/micromega.rst +++ b/doc/sphinx/addendum/micromega.rst @@ -38,7 +38,7 @@ The tactics solve propositional formulas parameterized by atomic arithmetic expressions interpreted over a domain :math:`D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}`. The syntax of the formulas is the following: - .. productionlist:: `F` + .. productionlist:: F F : A ∣ P ∣ True ∣ False ∣ F ∧ F ∣ F ∨ F ∣ F ↔ F ∣ F → F ∣ ¬ F A : p = p ∣ p > p ∣ p < p ∣ p ≥ p ∣ p ≤ p p : c ∣ x ∣ −p ∣ p − p ∣ p + p ∣ p × p ∣ p ^ n diff --git a/doc/sphinx/addendum/ring.rst b/doc/sphinx/addendum/ring.rst index 99d689132d..8204d93fa7 100644 --- a/doc/sphinx/addendum/ring.rst +++ b/doc/sphinx/addendum/ring.rst @@ -308,13 +308,13 @@ The syntax for adding a new ring is .. productionlist:: coq ring_mod : abstract | decidable `term` | morphism `term` - : | setoid `term` `term` - : | constants [`ltac`] - : | preprocess [`ltac`] - : | postprocess [`ltac`] - : | power_tac `term` [`ltac`] - : | sign `term` - : | div `term` + : setoid `term` `term` + : constants [`ltac`] + : preprocess [`ltac`] + : postprocess [`ltac`] + : power_tac `term` [`ltac`] + : sign `term` + : div `term` abstract declares the ring as abstract. This is the default. diff --git a/doc/sphinx/language/cic.rst b/doc/sphinx/language/cic.rst index cc5d9d6205..91504089a8 100644 --- a/doc/sphinx/language/cic.rst +++ b/doc/sphinx/language/cic.rst @@ -133,8 +133,8 @@ the following rules. #. if :math:`t` and :math:`u` are terms then :math:`(t~u)` is a term (:g:`t u` in |Coq| concrete syntax). The term :math:`(t~u)` reads as “t applied to u”. -#. if :g:`x` is a variable, and :math:`t`, :math:`T` and :math:`u` are - terms then :g:`let x:=t:T in u` is +#. if :math:`x` is a variable, and :math:`t`, :math:`T` and :math:`u` are + terms then :math:`\letin{x}{t:T}{u}` is a term which denotes the term :math:`u` where the variable :math:`x` is locally bound to :math:`t` of type :math:`T`. This stands for the common “let-in” construction of functional programs such as ML or Scheme. @@ -145,7 +145,7 @@ the following rules. **Free variables.** The notion of free variables is defined as usual. In the expressions -:g:`λx:T. U` and :g:`∀ x:T, U` the occurrences of :math:`x` in :math:`U` are bound. +:math:`λx:T.~U` and :math:`∀ x:T,~U` the occurrences of :math:`x` in :math:`U` are bound. .. _Substitution: @@ -383,7 +383,7 @@ following rules. .. note:: We may have :math:`\letin{x}{t:T}{u}` well-typed without having - :math:`((λ x:T.u) t)` well-typed (where :math:`T` is a type of + :math:`((λ x:T.~u)~t)` well-typed (where :math:`T` is a type of :math:`t`). This is because the value :math:`t` associated to :math:`x` may be used in a conversion rule (see Section :ref:`Conversion-rules`). @@ -407,17 +407,17 @@ can decide if two programs are *intentionally* equal (one says We want to be able to identify some terms as we can identify the application of a function to a given argument with its result. For instance the identity function over a given type T can be written -:math:`λx:T. x`. In any global environment :math:`E` and local context +:math:`λx:T.~x`. In any global environment :math:`E` and local context :math:`Γ`, we want to identify any object :math:`a` (of type -:math:`T`) with the application :math:`((λ x:T. x) a)`. We define for +:math:`T`) with the application :math:`((λ x:T.~x)~a)`. We define for this a *reduction* (or a *conversion*) rule we call :math:`β`: .. math:: - E[Γ] ⊢ ((λx:T. t) u)~\triangleright_β~\subst{t}{x}{u} + E[Γ] ⊢ ((λx:T.~t)~u)~\triangleright_β~\subst{t}{x}{u} We say that :math:`\subst{t}{x}{u}` is the *β-contraction* of -:math:`((λx:T. t) u)` and, conversely, that :math:`((λ x:T. t) u)` is the +:math:`((λx:T.~t)~u)` and, conversely, that :math:`((λ x:T.~t)~u)` is the *β-expansion* of :math:`\subst{t}{x}{u}`. According to β-reduction, terms of the *Calculus of Inductive @@ -481,7 +481,7 @@ destroyed, this reduction differs from δ-reduction. It is called \WTEG{u}{U} \WTE{\Gamma::(x:=u:U)}{t}{T} -------------- - E[Γ] ⊢ \letin{x}{u}{t}~\triangleright_ζ~\subst{t}{x}{u} + E[Γ] ⊢ \letin{x}{u:U}{t}~\triangleright_ζ~\subst{t}{x}{u} .. _eta-expansion: @@ -519,7 +519,7 @@ for :math:`x` an arbitrary variable name fresh in :math:`t`. We could not allow .. math:: - λ x:Type(1),(f x) \triangleright_η f + λ x:Type(1),(f~x) \triangleright_η f because the type of the reduced term :math:`∀ x:\Type(2),\Type(1)` would not be convertible to the type of the original term :math:`∀ x:\Type(1),\Type(1).` @@ -544,7 +544,7 @@ exist terms :math:`u_1` and :math:`u_2` such that :math:`E[Γ] ⊢ t_1 \triangle i.e. :math:`u_1` is :math:`λ x:T. u_1'` and :math:`u_2 x` is recursively convertible to :math:`u_1'` , or, symmetrically, :math:`u_2` is :math:`λx:T. u_2'` -and :math:`u_1 x` is recursively convertible to u_2′ . We then write +and :math:`u_1 x` is recursively convertible to :math:`u_2'`. We then write :math:`E[Γ] ⊢ t_1 =_{βδιζη} t_2` . Apart from this we consider two instances of polymorphic and @@ -625,14 +625,14 @@ a *subtyping* relation inductively defined by: universe levels) with constructors .. math:: - [c_1 : ∀Γ_P ,∀ T_{1,1} … T_{1,n_1} , t~v_{1,1} … v_{1,m} ;…; - c_k : ∀Γ_P ,∀ T_{k,1} … T_{k,n_k} ,t~v_{n,1} … v_{n,m} ] + [c_1 : ∀Γ_P ,∀ T_{1,1} … T_{1,n_1} ,~t~v_{1,1} … v_{1,m} ;…; + c_k : ∀Γ_P ,∀ T_{k,1} … T_{k,n_k} ,~t~v_{k,1} … v_{k,m} ] and .. math:: - [c_1 : ∀Γ_P' ,∀ T_{1,1}' … T_{1,n_1}' , t'~v_{1,1}' … v_{1,m}' ;…; - c_k : ∀Γ_P' ,∀ T_{k,1}' … T_{k,n_k}' ,t'~v_{n,1}' … v_{n,m}' ] + [c_1 : ∀Γ_P' ,∀ T_{1,1}' … T_{1,n_1}' ,~t'~v_{1,1}' … v_{1,m}' ;…; + c_k : ∀Γ_P' ,∀ T_{k,1}' … T_{k,n_k}' ,~t'~v_{k,1}' … v_{k,m}' ] respectively then @@ -771,8 +771,8 @@ the sort of the inductive type t (not to be confused with :math:`\Sort` which is \odd&:&\nat → \Prop \end{array}\right]} {\left[\begin{array}{rcl} \evenO &:& \even~0\\ - \evenS &:& \forall n, \odd~n → \even~(\kw{S}~n)\\ - \oddS &:& \forall n, \even~n → \odd~(\kw{S}~n) + \evenS &:& \forall n, \odd~n → \even~(\nS~n)\\ + \oddS &:& \forall n, \even~n → \odd~(\nS~n) \end{array}\right]} which corresponds to the result of the |Coq| declaration: @@ -820,9 +820,9 @@ contains an inductive declaration. \begin{array}{l} E[Γ] ⊢ \even : \nat→\Prop\\ E[Γ] ⊢ \odd : \nat→\Prop\\ - E[Γ] ⊢ \even\_O : \even~O\\ - E[Γ] ⊢ \even\_S : \forall~n:\nat, \odd~n → \even~(S~n)\\ - E[Γ] ⊢ \odd\_S : \forall~n:\nat, \even~n → \odd~(S~n) + E[Γ] ⊢ \evenO : \even~\nO\\ + E[Γ] ⊢ \evenS : \forall~n:\nat,~\odd~n → \even~(\nS~n)\\ + E[Γ] ⊢ \oddS : \forall~n:\nat,~\even~n → \odd~(\nS~n) \end{array} @@ -861,8 +861,8 @@ sort :math:`s`. :math:`A→ Set` and :math:`∀ A:\Prop,A→ \Prop` are arities. -Type constructor -++++++++++++++++ +Type of constructor ++++++++++++++++++++ We say that T is a *type of constructor of I* in one of the following two cases: @@ -943,7 +943,7 @@ condition* for a constant :math:`X` in the following cases: + Type ``A → (nat → nattree A) → nattree A`` of constructor ``node`` satisfies the positivity condition for ``nattree`` because: - - ``nattree`` occurs only strictly positively in ``A`` ... (bullet 3) + - ``nattree`` occurs only strictly positively in ``A`` ... (bullet 1) - ``nattree`` occurs only strictly positively in ``nat → nattree A`` ... (bullet 3 + 2) @@ -1041,6 +1041,12 @@ in :math:`\Type`. enabled it will prevail over automatic template polymorphism and cause an error when using the ``template`` attribute. +.. warn:: Automatically declaring @ident as template polymorphic. + + Warning ``auto-template`` can be used to find which types are + implicitly declared template polymorphic by :flag:`Auto Template + Polymorphism`. + If :math:`A` is an arity of some sort and :math:`s` is a sort, we write :math:`A_{/s}` for the arity obtained from :math:`A` by replacing its sort with :math:`s`. Especially, if :math:`A` is well-typed in some global environment and local @@ -1199,10 +1205,11 @@ a strongly normalizing reduction, we cannot accept any sort of recursion (even terminating). So the basic idea is to restrict ourselves to primitive recursive functions and functionals. -For instance, assuming a parameter :g:`A:Set` exists in the local context, -we want to build a function length of type :g:`list A -> nat` which computes -the length of the list, such that :g:`(length (nil A)) = O` and :g:`(length -(cons A a l)) = (S (length l))`. We want these equalities to be +For instance, assuming a parameter :math:`A:\Set` exists in the local context, +we want to build a function length of type :math:`\List~A → \nat` which computes +the length of the list, such that :math:`(\length~(\Nil~A)) = \nO` and +:math:`(\length~(\cons~A~a~l)) = (\nS~(\length~l))`. +We want these equalities to be recognized implicitly and taken into account in the conversion rule. From the logical point of view, we have built a type family by giving @@ -1216,22 +1223,22 @@ In case the inductive definition is effectively a recursive one, we want to capture the extra property that we have built the smallest fixed point of this recursive equation. This says that we are only manipulating finite objects. This analysis provides induction -principles. For instance, in order to prove :g:`∀ l:list A,(has_length A l -(length l))` it is enough to prove: +principles. For instance, in order to prove +:math:`∀ l:\List~A,~(\kw{has}\_\kw{length}~A~l~(\length~l))` it is enough to prove: -+ :g:`(has_length A (nil A) (length (nil A)))` -+ :g:`∀ a:A, ∀ l:list A, (has_length A l (length l)) →` - :g:`(has_length A (cons A a l) (length (cons A a l)))` ++ :math:`(\kw{has}\_\kw{length}~A~(\Nil~A)~(\length~(\Nil~A)))` ++ :math:`∀ a:A,~∀ l:\List~A,~(\kw{has}\_\kw{length}~A~l~(\length~l)) →` + :math:`(\kw{has}\_\kw{length}~A~(\cons~A~a~l)~(\length~(\cons~A~a~l)))` which given the conversion equalities satisfied by length is the same as proving: -+ :g:`(has_length A (nil A) O)` -+ :g:`∀ a:A, ∀ l:list A, (has_length A l (length l)) →` - :g:`(has_length A (cons A a l) (S (length l)))` ++ :math:`(\kw{has}\_\kw{length}~A~(\Nil~A)~\nO)` ++ :math:`∀ a:A,~∀ l:\List~A,~(\kw{has}\_\kw{length}~A~l~(\length~l)) →` + :math:`(\kw{has}\_\kw{length}~A~(\cons~A~a~l)~(\nS~(\length~l)))` One conceptually simple way to do that, following the basic scheme @@ -1479,19 +1486,19 @@ We write :math:`\{c\}^P` for :math:`\{c:C\}^P` with :math:`C` the type of :math: According to the definition: .. math:: - \{(\kw{nil}~\nat)\}^P ≡ \{(\kw{nil}~\nat) : (\List~\nat)\}^P ≡ (P~(\kw{nil}~\nat)) + \{(\Nil~\nat)\}^P ≡ \{(\Nil~\nat) : (\List~\nat)\}^P ≡ (P~(\Nil~\nat)) .. math:: \begin{array}{rl} - \{(\kw{cons}~\nat)\}^P & ≡\{(\kw{cons}~\nat) : (\nat→\List~\nat→\List~\nat)\}^P \\ - & ≡∀ n:\nat, \{(\kw{cons}~\nat~n) : \List~\nat→\List~\nat)\}^P \\ - & ≡∀ n:\nat, ∀ l:\List~\nat, \{(\kw{cons}~\nat~n~l) : \List~\nat)\}^P \\ - & ≡∀ n:\nat, ∀ l:\List~\nat,(P~(\kw{cons}~\nat~n~l)). + \{(\cons~\nat)\}^P & ≡\{(\cons~\nat) : (\nat→\List~\nat→\List~\nat)\}^P \\ + & ≡∀ n:\nat, \{(\cons~\nat~n) : \List~\nat→\List~\nat)\}^P \\ + & ≡∀ n:\nat, ∀ l:\List~\nat, \{(\cons~\nat~n~l) : \List~\nat)\}^P \\ + & ≡∀ n:\nat, ∀ l:\List~\nat,(P~(\cons~\nat~n~l)). \end{array} - Given some :math:`P` then :math:`\{(\kw{nil}~\nat)\}^P` represents the expected type of :math:`f_1` , - and :math:`\{(\kw{cons}~\nat)\}^P` represents the expected type of :math:`f_2`. + Given some :math:`P` then :math:`\{(\Nil~\nat)\}^P` represents the expected type of :math:`f_1` , + and :math:`\{(\cons~\nat)\}^P` represents the expected type of :math:`f_2`. .. _Typing-rule: @@ -1527,8 +1534,8 @@ definition :math:`\ind{r}{Γ_I}{Γ_C}` with :math:`Γ_C = [c_1 :C_1 ;…;c_n :C_ E[Γ] ⊢ t : (\List ~\nat) \\ E[Γ] ⊢ P : B \\ [(\List ~\nat)|B] \\ - E[Γ] ⊢ f_1 : {(\kw{nil} ~\nat)}^P \\ - E[Γ] ⊢ f_2 : {(\kw{cons} ~\nat)}^P + E[Γ] ⊢ f_1 : \{(\Nil ~\nat)\}^P \\ + E[Γ] ⊢ f_2 : \{(\cons ~\nat)\}^P \end{array} ------------------------------------------------ E[Γ] ⊢ \case(t,P,f_1 |f_2 ) : (P~t) @@ -1565,14 +1572,14 @@ concrete syntax for a recursive set of mutually recursive declarations is (with :math:`Γ_i` contexts): .. math:: - \fix~f_1 (Γ_1 ) :A_1 :=t_1 \with … \with~f_n (Γ_n ) :A_n :=t_n + \fix~f_1 (Γ_1 ) :A_1 :=t_1~\with … \with~f_n (Γ_n ) :A_n :=t_n The terms are obtained by projections from this set of declarations and are written .. math:: - \fix~f_1 (Γ_1 ) :A_1 :=t_1 \with … \with~f_n (Γ_n ) :A_n :=t_n \for~f_i + \fix~f_1 (Γ_1 ) :A_1 :=t_1~\with … \with~f_n (Γ_n ) :A_n :=t_n~\for~f_i In the inference rules, we represent such a term by @@ -1580,7 +1587,7 @@ In the inference rules, we represent such a term by \Fix~f_i\{f_1 :A_1':=t_1' … f_n :A_n':=t_n'\} with :math:`t_i'` (resp. :math:`A_i'`) representing the term :math:`t_i` abstracted (resp. -generalized) with respect to the bindings in the context Γ_i , namely +generalized) with respect to the bindings in the context :math:`Γ_i`, namely :math:`t_i'=λ Γ_i . t_i` and :math:`A_i'=∀ Γ_i , A_i`. @@ -1608,14 +1615,14 @@ instance in the case of natural numbers, a proof of the induction principle of type .. math:: - ∀ P:\nat→\Prop, (P~O)→(∀ n:\nat, (P~n)→(P~(\kw{S}~n)))→ ∀ n:\nat, (P~n) + ∀ P:\nat→\Prop,~(P~\nO)→(∀ n:\nat,~(P~n)→(P~(\nS~n)))→ ∀ n:\nat,~(P~n) can be represented by the term: .. math:: \begin{array}{l} - λ P:\nat→\Prop. λ f:(P~O). λ g:(∀ n:\nat, (P~n)→(P~(S~n))).\\ - \Fix~h\{h:∀ n:\nat, (P~n):=λ n:\nat. \case(n,P,f | λp:\nat. (g~p~(h~p)))\} + λ P:\nat→\Prop.~λ f:(P~\nO).~λ g:(∀ n:\nat,~(P~n)→(P~(\nS~n))).\\ + \Fix~h\{h:∀ n:\nat,~(P~n):=λ n:\nat.~\case(n,P,f | λp:\nat.~(g~p~(h~p)))\} \end{array} Before accepting a fixpoint definition as being correctly typed, we @@ -1702,7 +1709,7 @@ Let :math:`F` be the set of declarations: The reduction for fixpoints is: .. math:: - (\Fix~f_i \{F\} a_1 …a_{k_i}) \triangleright_ι \subst{t_i}{f_k}{\Fix~f_k \{F\}}_{k=1… n} ~a_1 … a_{k_i} + (\Fix~f_i \{F\}~a_1 …a_{k_i}) \triangleright_ι \subst{t_i}{f_k}{\Fix~f_k \{F\}}_{k=1… n} ~a_1 … a_{k_i} when :math:`a_{k_i}` starts with a constructor. This last restriction is needed in order to keep strong normalization and corresponds to the reduction @@ -1761,17 +1768,17 @@ and :math:`\subst{E}{|Γ|}{|Γ|c}` to mean the parallel substitution .. math:: \frac{\WF{E;c:U;E′;c′:=t:T;E″}{Γ}} - {\WF{E;c:U;E′;c′:=λ x:U. \subst{t}{c}{x}:∀x:U,\subst{T}{c}{x};\subst{E″}{c′}{(c′~c)}} + {\WF{E;c:U;E′;c′:=λ x:U.~\subst{t}{c}{x}:∀x:U,~\subst{T}{c}{x};\subst{E″}{c′}{(c′~c)}} {\subst{Γ}{c}{(c~c′)}}} .. math:: \frac{\WF{E;c:U;E′;c′:T;E″}{Γ}} - {\WF{E;c:U;E′;c′:∀ x:U,\subst{T}{c}{x};\subst{E″}{c′}{(c′~c)}}{Γ{c/(c~c′)}}} + {\WF{E;c:U;E′;c′:∀ x:U,~\subst{T}{c}{x};\subst{E″}{c′}{(c′~c)}}{\subst{Γ}{c}{(c~c′)}}} .. math:: \frac{\WF{E;c:U;E′;\ind{p}{Γ_I}{Γ_C};E″}{Γ}} - {\WFTWOLINES{E;c:U;E′;\ind{p+1}{∀ x:U,\subst{Γ_I}{c}{x}}{∀ x:U,\subst{Γ_C}{c}{x}}; + {\WFTWOLINES{E;c:U;E′;\ind{p+1}{∀ x:U,~\subst{Γ_I}{c}{x}}{∀ x:U,~\subst{Γ_C}{c}{x}}; \subst{E″}{|Γ_I ,Γ_C |}{|Γ_I ,Γ_C | c}} {\subst{Γ}{|Γ_I ,Γ_C|}{|Γ_I ,Γ_C | c}}} diff --git a/doc/sphinx/language/coq-library.rst b/doc/sphinx/language/coq-library.rst index 10650af1d1..b82b3b0e80 100644 --- a/doc/sphinx/language/coq-library.rst +++ b/doc/sphinx/language/coq-library.rst @@ -104,18 +104,18 @@ subclass :token:`form` of the syntactic class :token:`term`. The syntax of a nice last column. Or even better, find a proper way to do this! .. productionlist:: - form : True (True) - : | False (False) - : | ~ `form` (not) - : | `form` /\ `form` (and) - : | `form` \/ `form` (or) - : | `form` -> `form` (primitive implication) - : | `form` <-> `form` (iff) - : | forall `ident` : `type`, `form` (primitive for all) - : | exists `ident` [: `specif`], `form` (ex) - : | exists2 `ident` [: `specif`], `form` & `form` (ex2) - : | `term` = `term` (eq) - : | `term` = `term` :> `specif` (eq) + form : True (True) + : False (False) + : ~ `form` (not) + : `form` /\ `form` (and) + : `form` \/ `form` (or) + : `form` -> `form` (primitive implication) + : `form` <-> `form` (iff) + : forall `ident` : `type`, `form` (primitive for all) + : exists `ident` [: `specif`], `form` (ex) + : exists2 `ident` [: `specif`], `form` & `form` (ex2) + : `term` = `term` (eq) + : `term` = `term` :> `specif` (eq) .. note:: @@ -287,13 +287,13 @@ the next section :ref:`specification`): .. productionlist:: specif : `specif` * `specif` (prod) - : | `specif` + `specif` (sum) - : | `specif` + { `specif` } (sumor) - : | { `specif` } + { `specif` } (sumbool) - : | { `ident` : `specif` | `form` } (sig) - : | { `ident` : `specif` | `form` & `form` } (sig2) - : | { `ident` : `specif` & `specif` } (sigT) - : | { `ident` : `specif` & `specif` & `specif` } (sigT2) + : `specif` + `specif` (sum) + : `specif` + { `specif` } (sumor) + : { `specif` } + { `specif` } (sumbool) + : { `ident` : `specif` | `form` } (sig) + : { `ident` : `specif` | `form` & `form` } (sig2) + : { `ident` : `specif` & `specif` } (sigT) + : { `ident` : `specif` & `specif` & `specif` } (sigT2) term : (`term`, `term`) (pair) diff --git a/doc/sphinx/language/gallina-extensions.rst b/doc/sphinx/language/gallina-extensions.rst index 376a6b8eed..d0e44cd212 100644 --- a/doc/sphinx/language/gallina-extensions.rst +++ b/doc/sphinx/language/gallina-extensions.rst @@ -25,7 +25,7 @@ expressions. In this sense, the :cmd:`Record` construction allows defining record_keyword : Record | Inductive | CoInductive record_body : `ident` [ `binders` ] [: `sort` ] := [ `ident` ] { [ `field` ; … ; `field` ] }. field : `ident` [ `binders` ] : `type` [ where `notation` ] - : | `ident` [ `binders` ] [: `type` ] := `term` + : `ident` [ `binders` ] [: `type` ] := `term` .. cmd:: Record @ident @binders {? : @sort} := {? @ident} { {*; @ident @binders : @type } } @@ -165,8 +165,8 @@ available: .. productionlist:: terms projection : `term` `.` ( `qualid` ) - : | `term` `.` ( `qualid` `arg` … `arg` ) - : | `term` `.` ( @`qualid` `term` … `term` ) + : `term` `.` ( `qualid` `arg` … `arg` ) + : `term` `.` ( @`qualid` `term` … `term` ) Syntax of Record projections @@ -818,14 +818,14 @@ together, as well as a means of massive abstraction. .. productionlist:: modules module_type : `qualid` - : | `module_type` with Definition `qualid` := `term` - : | `module_type` with Module `qualid` := `qualid` - : | `qualid` `qualid` … `qualid` - : | !`qualid` `qualid` … `qualid` + : `module_type` with Definition `qualid` := `term` + : `module_type` with Module `qualid` := `qualid` + : `qualid` `qualid` … `qualid` + : !`qualid` `qualid` … `qualid` module_binding : ( [Import|Export] `ident` … `ident` : `module_type` ) module_bindings : `module_binding` … `module_binding` module_expression : `qualid` … `qualid` - : | !`qualid` … `qualid` + : !`qualid` … `qualid` Syntax of modules @@ -1814,10 +1814,10 @@ This syntax extension is given in the following grammar: .. productionlist:: explicit_apps term : @ `qualid` `term` … `term` - : | @ `qualid` - : | `qualid` `argument` … `argument` + : @ `qualid` + : `qualid` `argument` … `argument` argument : `term` - : | (`ident` := `term`) + : (`ident` := `term`) Syntax for explicitly giving implicit arguments diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 8fa631174f..5ecf007eff 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -127,43 +127,43 @@ is described in Chapter :ref:`syntaxextensionsandinterpretationscopes`. .. productionlist:: coq term : forall `binders` , `term` - : | fun `binders` => `term` - : | fix `fix_bodies` - : | cofix `cofix_bodies` - : | let `ident` [`binders`] [: `term`] := `term` in `term` - : | let fix `fix_body` in `term` - : | let cofix `cofix_body` in `term` - : | let ( [`name` , … , `name`] ) [`dep_ret_type`] := `term` in `term` - : | let ' `pattern` [in `term`] := `term` [`return_type`] in `term` - : | if `term` [`dep_ret_type`] then `term` else `term` - : | `term` : `term` - : | `term` <: `term` - : | `term` :> - : | `term` -> `term` - : | `term` `arg` … `arg` - : | @ `qualid` [`term` … `term`] - : | `term` % `ident` - : | match `match_item` , … , `match_item` [`return_type`] with + : fun `binders` => `term` + : fix `fix_bodies` + : cofix `cofix_bodies` + : let `ident` [`binders`] [: `term`] := `term` in `term` + : let fix `fix_body` in `term` + : let cofix `cofix_body` in `term` + : let ( [`name` , … , `name`] ) [`dep_ret_type`] := `term` in `term` + : let ' `pattern` [in `term`] := `term` [`return_type`] in `term` + : if `term` [`dep_ret_type`] then `term` else `term` + : `term` : `term` + : `term` <: `term` + : `term` :> + : `term` -> `term` + : `term` `arg` … `arg` + : @ `qualid` [`term` … `term`] + : `term` % `ident` + : match `match_item` , … , `match_item` [`return_type`] with : [[|] `equation` | … | `equation`] end - : | `qualid` - : | `sort` - : | `num` - : | _ - : | ( `term` ) + : `qualid` + : `sort` + : `num` + : _ + : ( `term` ) arg : `term` - : | ( `ident` := `term` ) + : ( `ident` := `term` ) binders : `binder` … `binder` binder : `name` - : | ( `name` … `name` : `term` ) - : | ( `name` [: `term`] := `term` ) - : | ' `pattern` + : ( `name` … `name` : `term` ) + : ( `name` [: `term`] := `term` ) + : ' `pattern` name : `ident` | _ qualid : `ident` | `qualid` `access_ident` sort : Prop | Set | Type fix_bodies : `fix_body` - : | `fix_body` with `fix_body` with … with `fix_body` for `ident` + : `fix_body` with `fix_body` with … with `fix_body` for `ident` cofix_bodies : `cofix_body` - : | `cofix_body` with `cofix_body` with … with `cofix_body` for `ident` + : `cofix_body` with `cofix_body` with … with `cofix_body` for `ident` fix_body : `ident` `binders` [`annotation`] [: `term`] := `term` cofix_body : `ident` [`binders`] [: `term`] := `term` annotation : { struct `ident` } @@ -173,13 +173,13 @@ is described in Chapter :ref:`syntaxextensionsandinterpretationscopes`. equation : `mult_pattern` | … | `mult_pattern` => `term` mult_pattern : `pattern` , … , `pattern` pattern : `qualid` `pattern` … `pattern` - : | @ `qualid` `pattern` … `pattern` - : | `pattern` as `ident` - : | `pattern` % `ident` - : | `qualid` - : | _ - : | `num` - : | ( `or_pattern` , … , `or_pattern` ) + : @ `qualid` `pattern` … `pattern` + : `pattern` as `ident` + : `pattern` % `ident` + : `qualid` + : _ + : `num` + : ( `or_pattern` , … , `or_pattern` ) or_pattern : `pattern` | … | `pattern` @@ -524,38 +524,38 @@ The Vernacular .. productionlist:: coq decorated-sentence : [ `decoration` … `decoration` ] `sentence` sentence : `assumption` - : | `definition` - : | `inductive` - : | `fixpoint` - : | `assertion` `proof` + : `definition` + : `inductive` + : `fixpoint` + : `assertion` `proof` assumption : `assumption_keyword` `assums`. assumption_keyword : Axiom | Conjecture - : | Parameter | Parameters - : | Variable | Variables - : | Hypothesis | Hypotheses + : Parameter | Parameters + : Variable | Variables + : Hypothesis | Hypotheses assums : `ident` … `ident` : `term` - : | ( `ident` … `ident` : `term` ) … ( `ident` … `ident` : `term` ) + : ( `ident` … `ident` : `term` ) … ( `ident` … `ident` : `term` ) definition : [Local] Definition `ident` [`binders`] [: `term`] := `term` . - : | Let `ident` [`binders`] [: `term`] := `term` . + : Let `ident` [`binders`] [: `term`] := `term` . inductive : Inductive `ind_body` with … with `ind_body` . - : | CoInductive `ind_body` with … with `ind_body` . + : CoInductive `ind_body` with … with `ind_body` . ind_body : `ident` [`binders`] : `term` := : [[|] `ident` [`binders`] [:`term`] | … | `ident` [`binders`] [:`term`]] fixpoint : Fixpoint `fix_body` with … with `fix_body` . - : | CoFixpoint `cofix_body` with … with `cofix_body` . + : CoFixpoint `cofix_body` with … with `cofix_body` . assertion : `assertion_keyword` `ident` [`binders`] : `term` . assertion_keyword : Theorem | Lemma - : | Remark | Fact - : | Corollary | Proposition - : | Definition | Example + : Remark | Fact + : Corollary | Proposition + : Definition | Example proof : Proof . … Qed . - : | Proof . … Defined . - : | Proof . … Admitted . + : Proof . … Defined . + : Proof . … Admitted . decoration : #[ `attributes` ] attributes : [`attribute`, … , `attribute`] attribute : `ident` - :| `ident` = `string` - :| `ident` ( `attributes` ) + : `ident` = `string` + : `ident` ( `attributes` ) .. todo:: This use of … in this grammar is inconsistent What about removing the proof part of this grammar from this chapter diff --git a/doc/sphinx/proof-engine/ltac.rst b/doc/sphinx/proof-engine/ltac.rst index 1071682ead..442077616f 100644 --- a/doc/sphinx/proof-engine/ltac.rst +++ b/doc/sphinx/proof-engine/ltac.rst @@ -41,117 +41,121 @@ mode but it can also be used in toplevel definitions as shown below. .. note:: - - The infix tacticals “… \|\| …”, “… + …”, and “… ; …” are associative. + - The infix tacticals ``… || …`` , ``… + …`` , and ``… ; …`` are associative. - - In :token:`tacarg`, there is an overlap between qualid as a direct tactic - argument and :token:`qualid` as a particular case of term. The resolution is - done by first looking for a reference of the tactic language and if - it fails, for a reference to a term. To force the resolution as a - reference of the tactic language, use the form :g:`ltac:(@qualid)`. To - force the resolution as a reference to a term, use the syntax - :g:`(@qualid)`. + .. example:: - - As shown by the figure, tactical ``\|\|`` binds more than the prefix - tacticals try, repeat, do and abstract which themselves bind more - than the postfix tactical “… ;[ … ]” which binds more than “… ; …”. + If you want that :n:`@tactic__2; @tactic__3` be fully run on the first + subgoal generated by :n:`@tactic__1`, before running on the other + subgoals, then you should not write + :n:`@tactic__1; (@tactic__2; @tactic__3)` but rather + :n:`@tactic__1; [> @tactic__2; @tactic__3 .. ]`. - For instance + - In :token:`tacarg`, there is an overlap between :token:`qualid` as a + direct tactic argument and :token:`qualid` as a particular case of + :token:`term`. The resolution is done by first looking for a reference + of the tactic language and if it fails, for a reference to a term. + To force the resolution as a reference of the tactic language, use the + form :n:`ltac:(@qualid)`. To force the resolution as a reference to a + term, use the syntax :n:`(@qualid)`. - .. coqtop:: in + - As shown by the figure, tactical ``… || …`` binds more than the prefix + tacticals :tacn:`try`, :tacn:`repeat`, :tacn:`do` and :tacn:`abstract` + which themselves bind more than the postfix tactical ``… ;[ … ]`` + which binds at the same level as ``… ; …`` . - try repeat tac1 || tac2; tac3; [tac31 | ... | tac3n]; tac4. + .. example:: - is understood as + :n:`try repeat @tactic__1 || @tactic__2; @tactic__3; [ {+| @tactic } ]; @tactic__4` - .. coqtop:: in + is understood as: - try (repeat (tac1 || tac2)); - ((tac3; [tac31 | ... | tac3n]); tac4). + :n:`((try (repeat (@tactic__1 || @tactic__2)); @tactic__3); [ {+| @tactic } ]); @tactic__4` .. productionlist:: coq expr : `expr` ; `expr` - : | [> `expr` | ... | `expr` ] - : | `expr` ; [ `expr` | ... | `expr` ] - : | `tacexpr3` - tacexpr3 : do (`natural` | `ident`) tacexpr3 - : | progress `tacexpr3` - : | repeat `tacexpr3` - : | try `tacexpr3` - : | once `tacexpr3` - : | exactly_once `tacexpr3` - : | timeout (`natural` | `ident`) `tacexpr3` - : | time [`string`] `tacexpr3` - : | only `selector`: `tacexpr3` - : | `tacexpr2` + : [> `expr` | ... | `expr` ] + : `expr` ; [ `expr` | ... | `expr` ] + : `tacexpr3` + tacexpr3 : do (`natural` | `ident`) `tacexpr3` + : progress `tacexpr3` + : repeat `tacexpr3` + : try `tacexpr3` + : once `tacexpr3` + : exactly_once `tacexpr3` + : timeout (`natural` | `ident`) `tacexpr3` + : time [`string`] `tacexpr3` + : only `selector`: `tacexpr3` + : `tacexpr2` tacexpr2 : `tacexpr1` || `tacexpr3` - : | `tacexpr1` + `tacexpr3` - : | tryif `tacexpr1` then `tacexpr1` else `tacexpr1` - : | `tacexpr1` + : `tacexpr1` + `tacexpr3` + : tryif `tacexpr1` then `tacexpr1` else `tacexpr1` + : `tacexpr1` tacexpr1 : fun `name` ... `name` => `atom` - : | let [rec] `let_clause` with ... with `let_clause` in `atom` - : | match goal with `context_rule` | ... | `context_rule` end - : | match reverse goal with `context_rule` | ... | `context_rule` end - : | match `expr` with `match_rule` | ... | `match_rule` end - : | lazymatch goal with `context_rule` | ... | `context_rule` end - : | lazymatch reverse goal with `context_rule` | ... | `context_rule` end - : | lazymatch `expr` with `match_rule` | ... | `match_rule` end - : | multimatch goal with `context_rule` | ... | `context_rule` end - : | multimatch reverse goal with `context_rule` | ... | `context_rule` end - : | multimatch `expr` with `match_rule` | ... | `match_rule` end - : | abstract `atom` - : | abstract `atom` using `ident` - : | first [ `expr` | ... | `expr` ] - : | solve [ `expr` | ... | `expr` ] - : | idtac [ `message_token` ... `message_token`] - : | fail [`natural`] [`message_token` ... `message_token`] - : | fresh [ `component` … `component` ] - : | context `ident` [`term`] - : | eval `redexpr` in `term` - : | type of `term` - : | constr : `term` - : | uconstr : `term` - : | type_term `term` - : | numgoals - : | guard `test` - : | assert_fails `tacexpr3` - : | assert_succeeds `tacexpr3` - : | `atomic_tactic` - : | `qualid` `tacarg` ... `tacarg` - : | `atom` + : let [rec] `let_clause` with ... with `let_clause` in `atom` + : match goal with `context_rule` | ... | `context_rule` end + : match reverse goal with `context_rule` | ... | `context_rule` end + : match `expr` with `match_rule` | ... | `match_rule` end + : lazymatch goal with `context_rule` | ... | `context_rule` end + : lazymatch reverse goal with `context_rule` | ... | `context_rule` end + : lazymatch `expr` with `match_rule` | ... | `match_rule` end + : multimatch goal with `context_rule` | ... | `context_rule` end + : multimatch reverse goal with `context_rule` | ... | `context_rule` end + : multimatch `expr` with `match_rule` | ... | `match_rule` end + : abstract `atom` + : abstract `atom` using `ident` + : first [ `expr` | ... | `expr` ] + : solve [ `expr` | ... | `expr` ] + : idtac [ `message_token` ... `message_token`] + : fail [`natural`] [`message_token` ... `message_token`] + : fresh [ `component` … `component` ] + : context `ident` [`term`] + : eval `redexpr` in `term` + : type of `term` + : constr : `term` + : uconstr : `term` + : type_term `term` + : numgoals + : guard `test` + : assert_fails `tacexpr3` + : assert_succeeds `tacexpr3` + : `atomic_tactic` + : `qualid` `tacarg` ... `tacarg` + : `atom` atom : `qualid` - : | () - : | `integer` - : | ( `expr` ) + : () + : `integer` + : ( `expr` ) component : `string` | `qualid` message_token : `string` | `ident` | `integer` tacarg : `qualid` - : | () - : | ltac : `atom` - : | `term` + : () + : ltac : `atom` + : `term` let_clause : `ident` [`name` ... `name`] := `expr` context_rule : `context_hyp`, ..., `context_hyp` |- `cpattern` => `expr` - : | `cpattern` => `expr` - : | |- `cpattern` => `expr` - : | _ => `expr` + : `cpattern` => `expr` + : |- `cpattern` => `expr` + : _ => `expr` context_hyp : `name` : `cpattern` - : | `name` := `cpattern` [: `cpattern`] + : `name` := `cpattern` [: `cpattern`] match_rule : `cpattern` => `expr` - : | context [ident] [ `cpattern` ] => `expr` - : | _ => `expr` + : context [`ident`] [ `cpattern` ] => `expr` + : _ => `expr` test : `integer` = `integer` - : | `integer` (< | <= | > | >=) `integer` + : `integer` (< | <= | > | >=) `integer` selector : [`ident`] - : | `integer` - : | (`integer` | `integer` - `integer`), ..., (`integer` | `integer` - `integer`) + : `integer` + : (`integer` | `integer` - `integer`), ..., (`integer` | `integer` - `integer`) toplevel_selector : `selector` - : | all - : | par - : | ! + : all + : par + : ! .. productionlist:: coq top : [Local] Ltac `ltac_def` with ... with `ltac_def` ltac_def : `ident` [`ident` ... `ident`] := `expr` - : | `qualid` [`ident` ... `ident`] ::= `expr` + : `qualid` [`ident` ... `ident`] ::= `expr` .. _ltac-semantics: diff --git a/doc/sphinx/proof-engine/tactics.rst b/doc/sphinx/proof-engine/tactics.rst index 59602581c7..250d9c3a8a 100644 --- a/doc/sphinx/proof-engine/tactics.rst +++ b/doc/sphinx/proof-engine/tactics.rst @@ -33,10 +33,13 @@ extends the folklore notion of tactical) to combine those atomic tactics. This chapter is devoted to atomic tactics. The tactic language will be described in Chapter :ref:`ltac`. +Common elements of tactics +-------------------------- + .. _invocation-of-tactics: Invocation of tactics -------------------------- +~~~~~~~~~~~~~~~~~~~~~ A tactic is applied as an ordinary command. It may be preceded by a goal selector (see Section :ref:`ltac-semantics`). If no selector is @@ -44,9 +47,9 @@ specified, the default selector is used. .. _tactic_invocation_grammar: - .. productionlist:: `sentence` - tactic_invocation : toplevel_selector : tactic. - : |tactic . + .. productionlist:: sentence + tactic_invocation : `toplevel_selector` : `tactic`. + : `tactic`. .. opt:: Default Goal Selector "@toplevel_selector" :name: Default Goal Selector @@ -71,29 +74,31 @@ specified, the default selector is used. Bindings list ~~~~~~~~~~~~~~~~~~~ -Tactics that take a term as argument may also support a bindings list, -so as to instantiate some parameters of the term by name or position. -The general form of a term equipped with a bindings list is ``term with -bindings_list`` where ``bindings_list`` may be of two different forms: +Tactics that take a term as an argument may also support a bindings list +to instantiate some parameters of the term by name or position. +The general form of a term with a bindings list is +:n:`@term with @bindings_list` where :token:`bindings_list` can take two different forms: .. _bindings_list_grammar: - .. productionlist:: `bindings_list` - bindings_list : (ref := `term`) ... (ref := `term`) + .. productionlist:: bindings_list + ref : `ident` + : `num` + bindings_list : (`ref` := `term`) ... (`ref` := `term`) : `term` ... `term` -+ In a bindings list of the form :n:`{* (ref:= term)}`, :n:`ref` is either an ++ In a bindings list of the form :n:`{+ (@ref:= @term)}`, :n:`@ref` is either an :n:`@ident` or a :n:`@num`. The references are determined according to the type of - ``term``. If :n:`ref` is an identifier, this identifier has to be bound in the - type of ``term`` and the binding provides the tactic with an instance for the - parameter of this name. If :n:`ref` is some number ``n``, this number denotes - the ``n``-th non dependent premise of the ``term``, as determined by the type - of ``term``. + :n:`@term`. If :n:`@ref` is an identifier, this identifier has to be bound in the + type of :n:`@term` and the binding provides the tactic with an instance for the + parameter of this name. If :n:`@ref` is a number ``n``, it refers to + the ``n``-th non dependent premise of the :n:`@term`, as determined by the type + of :n:`@term`. .. exn:: No such binder. :undocumented: -+ A bindings list can also be a simple list of terms :n:`{* term}`. ++ A bindings list can also be a simple list of terms :n:`{* @term}`. In that case the references to which these terms correspond are determined by the tactic. In case of :tacn:`induction`, :tacn:`destruct`, :tacn:`elim` and :tacn:`case`, the terms have to @@ -105,6 +110,350 @@ bindings_list`` where ``bindings_list`` may be of two different forms: .. exn:: Not the right number of missing arguments. :undocumented: +.. _intropatterns: + +Intro patterns +~~~~~~~~~~~~~~ + +Intro patterns let you specify the name to assign to variables and hypotheses +introduced by tactics. They also let you split an introduced hypothesis into +multiple hypotheses or subgoals. Common tactics that accept intro patterns +include :tacn:`assert`, :tacn:`intros` and :tacn:`destruct`. + +.. productionlist:: coq + intropattern_list : `intropattern` ... `intropattern` + : `empty` + empty : + intropattern : * + : ** + : `simple_intropattern` + simple_intropattern : `simple_intropattern_closed` [ % `term` ... % `term` ] + simple_intropattern_closed : `naming_intropattern` + : _ + : `or_and_intropattern` + : `equality_intropattern` + naming_intropattern : `ident` + : ? + : ?`ident` + or_and_intropattern : [ `intropattern_list` | ... | `intropattern_list` ] + : ( `simple_intropattern` , ... , `simple_intropattern` ) + : ( `simple_intropattern` & ... & `simple_intropattern` ) + equality_intropattern : -> + : <- + : [= `intropattern_list` ] + or_and_intropattern_loc : `or_and_intropattern` + : `ident` + +Note that the intro pattern syntax varies between tactics. +Most tactics use :n:`@simple_intropattern` in the grammar. +:tacn:`destruct`, :tacn:`edestruct`, :tacn:`induction`, +:tacn:`einduction`, :tacn:`case`, :tacn:`ecase` and the various +:tacn:`inversion` tactics use :n:`@or_and_intropattern_loc`, while +:tacn:`intros` and :tacn:`eintros` use :n:`@intropattern_list`. +The :n:`eqn:` construct in various tactics uses :n:`@naming_intropattern`. + +**Naming patterns** + +Use these elementary patterns to specify a name: + +* :n:`@ident` - use the specified name +* :n:`?` - let Coq choose a name +* :n:`?@ident` - generate a name that begins with :n:`@ident` +* :n:`_` - discard the matched part (unless it is required for another + hypothesis) +* if a disjunction pattern omits a name, such as :g:`[|H2]`, Coq will choose a name + +**Splitting patterns** + +The most common splitting patterns are: + +* split a hypothesis in the form :n:`A /\ B` into two + hypotheses :g:`H1: A` and :g:`H2: B` using the pattern :g:`(H1 & H2)` or + :g:`(H1, H2)` or :g:`[H1 H2]`. + :ref:`Example <intropattern_conj_ex>`. This also works on :n:`A <-> B`, which + is just a notation representing :n:`(A -> B) /\ (B -> A)`. +* split a hypothesis in the form :g:`A \/ B` into two + subgoals using the pattern :g:`[H1|H2]`. The first subgoal will have the hypothesis + :g:`H1: A` and the second subgoal will have the hypothesis :g:`H2: B`. + :ref:`Example <intropattern_disj_ex>` +* split a hypothesis in either of the forms :g:`A /\ B` or :g:`A \/ B` using the pattern :g:`[]`. + +Patterns can be nested: :n:`[[Ha|Hb] H]` can be used to split :n:`(A \/ B) /\ C`. + +Note that there is no equivalent to intro patterns for goals. For a goal :g:`A /\ B`, +use the :tacn:`split` tactic to replace the current goal with subgoals :g:`A` and :g:`B`. +For a goal :g:`A \/ B`, use :tacn:`left` to replace the current goal with :g:`A`, or +:tacn:`right` to replace the current goal with :g:`B`. + +* :n:`( {+, @simple_intropattern}` ) - matches + a product over an inductive type with a + :ref:`single constructor <intropattern_cons_note>`. + If the number of patterns + equals the number of constructor arguments, then it applies the patterns only to + the arguments, and + :n:`( {+, @simple_intropattern} )` is equivalent to :n:`[{+ @simple_intropattern}]`. + If the number of patterns equals the number of constructor arguments plus the number + of :n:`let-ins`, the patterns are applied to the arguments and :n:`let-in` variables. + +* :n:`( {+& @simple_intropattern} )` - matches a right-hand nested term that consists + of one or more nested binary inductive types such as :g:`a1 OP1 a2 OP2 ...` + (where the :g:`OPn` are right-associative). + (If the :g:`OPn` are left-associative, additional parentheses will be needed to make the + term right-hand nested, such as :g:`a1 OP1 (a2 OP2 ...)`.) + The splitting pattern can have more than 2 names, for example :g:`(H1 & H2 & H3)` + matches :g:`A /\ B /\ C`. + The inductive types must have a + :ref:`single constructor with two parameters <intropattern_cons_note>`. + :ref:`Example <intropattern_ampersand_ex>` + +* :n:`[ {+| @intropattern_list} ]` - splits an inductive type that has + :ref:`multiple constructors <intropattern_cons_note>` + such as :n:`A \/ B` + into multiple subgoals. The number of :token:`intropattern_list` must be the same as the number of + constructors for the matched part. +* :n:`[ {+ @intropattern} ]` - splits an inductive type that has a + :ref:`single constructor with multiple parameters <intropattern_cons_note>` + such as :n:`A /\ B` into multiple hypotheses. Use :n:`[H1 [H2 H3]]` to match :g:`A /\ B /\ C`. +* :n:`[]` - splits an inductive type: If the inductive + type has multiple constructors, such as :n:`A \/ B`, + create one subgoal for each constructor. If the inductive type has a single constructor with + multiple parameters, such as :n:`A /\ B`, split it into multiple hypotheses. + +**Equality patterns** + +These patterns can be used when the hypothesis is an equality: + +* :n:`->` - replaces the right-hand side of the hypothesis with the left-hand + side of the hypothesis in the conclusion of the goal; the hypothesis is + cleared; if the left-hand side of the hypothesis is a variable, it is + substituted everywhere in the context and the variable is removed. + :ref:`Example <intropattern_rarrow_ex>` +* :n:`<-` - similar to :n:`->`, but replaces the left-hand side of the hypothesis + with the right-hand side of the hypothesis. +* :n:`[= {*, @intropattern} ]` - If the product is over an equality type, + applies either :tacn:`injection` or :tacn:`discriminate`. + If :tacn:`injection` is applicable, the intropattern + is used on the hypotheses generated by :tacn:`injection`. If the + number of patterns is smaller than the number of hypotheses generated, the + pattern :n:`?` is used to complete the list. + :ref:`Example <intropattern_inj_discr_ex>` + +**Other patterns** + +* :n:`*` - introduces one or more quantified variables from the result + until there are no more quantified variables. + :ref:`Example <intropattern_star_ex>` + +* :n:`**` - introduces one or more quantified variables or hypotheses from the result until there are + no more quantified variables or implications (:g:`->`). :g:`intros **` is equivalent + to :g:`intros`. + :ref:`Example <intropattern_2stars_ex>` + +* :n:`@simple_intropattern_closed {* % @term}` - first applies each of the terms + with the :tacn:`apply ... in` tactic on the hypothesis to be introduced, then it uses + :n:`@simple_intropattern_closed`. + :ref:`Example <intropattern_injection_ex>` + +.. flag:: Bracketing Last Introduction Pattern + + For :n:`intros @intropattern_list`, controls how to handle a + conjunctive pattern that doesn't give enough simple patterns to match + all the arguments in the constructor. If set (the default), |Coq| generates + additional names to match the number of arguments. + Unsetting the option will put the additional hypotheses in the goal instead, behavior that is more + similar to |SSR|'s intro patterns. + + .. deprecated:: 8.10 + +.. _intropattern_cons_note: + +.. note:: + + :n:`A \/ B` and :n:`A /\ B` use infix notation to refer to the inductive + types :n:`or` and :n:`and`. + :n:`or` has multiple constructors (:n:`or_introl` and :n:`or_intror`), + while :n:`and` has a single constructor (:n:`conj`) with multiple parameters + (:n:`A` and :n:`B`). + These are defined in theories/Init/Logic.v. The "where" clauses define the + infix notation for "or" and "and". + + .. coqdoc:: + + Inductive or (A B:Prop) : Prop := + | or_introl : A -> A \/ B + | or_intror : B -> A \/ B + where "A \/ B" := (or A B) : type_scope. + + Inductive and (A B:Prop) : Prop := + conj : A -> B -> A /\ B + where "A /\ B" := (and A B) : type_scope. + +.. note:: + + :n:`intros {+ p}` is not always equivalent to :n:`intros p; ... ; intros p` + if some of the :n:`p` are :g:`_`. In the first form, all erasures are done + at once, while they're done sequentially for each tactic in the second form. + If the second matched term depends on the first matched term and the pattern + for both is :g:`_` (i.e., both will be erased), the first :n:`intros` in the second + form will fail because the second matched term still has the dependency on the first. + +Examples: + +.. _intropattern_conj_ex: + + .. example:: intro pattern for /\\ + + .. coqtop:: reset none + + Goal forall (A: Prop) (B: Prop), (A /\ B) -> True. + + .. coqtop:: out + + intros. + + .. coqtop:: all + + destruct H as (HA & HB). + +.. _intropattern_disj_ex: + + .. example:: intro pattern for \\/ + + .. coqtop:: reset none + + Goal forall (A: Prop) (B: Prop), (A \/ B) -> True. + + .. coqtop:: out + + intros. + + .. coqtop:: all + + destruct H as [HA|HB]. all: swap 1 2. + +.. _intropattern_rarrow_ex: + + .. example:: -> intro pattern + + .. coqtop:: reset none + + Goal forall (x:nat) (y:nat) (z:nat), (x = y) -> (y = z) -> (x = z). + + .. coqtop:: out + + intros * H. + + .. coqtop:: all + + intros ->. + +.. _intropattern_inj_discr_ex: + + .. example:: [=] intro pattern + + The first :n:`intros [=]` uses :tacn:`injection` to strip :n:`(S ...)` from + both sides of the matched equality. The second uses :tacn:`discriminate` on + the contradiction :n:`1 = 2` (internally represented as :n:`(S O) = (S (S O))`) + to complete the goal. + + .. coqtop:: reset none + + Goal forall (n m:nat), (S n) = (S m) -> (S O)=(S (S O)) -> False. + + .. coqtop:: out + + intros *. + + .. coqtop:: all + + intros [= H]. + + .. coqtop:: all + + intros [=]. + +.. _intropattern_ampersand_ex: + + .. example:: (A & B & ...) intro pattern + + .. coqtop:: reset none + + Variables (A : Prop) (B: nat -> Prop) (C: Prop). + + .. coqtop:: out + + Goal A /\ (exists x:nat, B x /\ C) -> True. + + .. coqtop:: all + + intros (a & x & b & c). + +.. _intropattern_star_ex: + + .. example:: * intro pattern + + .. coqtop:: reset out + + Goal forall (A: Prop) (B: Prop), A -> B. + + .. coqtop:: all + + intros *. + +.. _intropattern_2stars_ex: + + .. example:: ** pattern ("intros \**" is equivalent to "intros") + + .. coqtop:: reset out + + Goal forall (A: Prop) (B: Prop), A -> B. + + .. coqtop:: all + + intros **. + + .. example:: compound intro pattern + + .. coqtop:: reset out + + Goal forall A B C:Prop, A \/ B /\ C -> (A -> C) -> C. + + .. coqtop:: all + + intros * [a | (_,c)] f. + all: swap 1 2. + +.. _intropattern_injection_ex: + + .. example:: combined intro pattern using [=] -> and % + + .. coqtop:: reset none + + Require Import Coq.Lists.List. + Section IntroPatterns. + Variables (A : Type) (xs ys : list A). + + .. coqtop:: out + + Example ThreeIntroPatternsCombined : + S (length ys) = 1 -> xs ++ ys = xs. + + .. coqtop:: all + + intros [=->%length_zero_iff_nil]. + + * `intros` would add :g:`H : S (length ys) = 1` + * `intros [=]` would additionally apply :tacn:`injection` to :g:`H` to yield :g:`H0 : length ys = 0` + * `intros [=->%length_zero_iff_nil]` applies the theorem, making H the equality :g:`l=nil`, + which is then applied as for :g:`->`. + + .. coqdoc:: + + Theorem length_zero_iff_nil (l : list A): + length l = 0 <-> l=nil. + + The example is based on `Tej Chajed's coq-tricks <https://github.com/tchajed/coq-tricks/blob/8e6efe4971ed828ac8bdb5512c1f615d7d62691e/src/IntroPatterns.v>`_ + .. _occurrencessets: Occurrence sets and occurrence clauses @@ -113,11 +462,11 @@ Occurrence sets and occurrence clauses An occurrence clause is a modifier to some tactics that obeys the following syntax: - .. productionlist:: `sentence` + .. productionlist:: sentence occurrence_clause : in `goal_occurrences` - goal_occurrences : [`ident` [`at_occurrences`], ... , ident [`at_occurrences`] [|- [* [`at_occurrences`]]]] - :| * |- [* [`at_occurrences`]] - :| * + goal_occurrences : [`ident` [`at_occurrences`], ... , `ident` [`at_occurrences`] [|- [* [`at_occurrences`]]]] + : * |- [* [`at_occurrences`]] + : * at_occurrences : at `occurrences` occurrences : [-] `num` ... `num` @@ -508,10 +857,10 @@ Applying theorems This works as :tacn:`apply ... in` but turns unresolved bindings into existential variables, if any, instead of failing. - .. tacv:: apply {+, @term {? with @bindings_list } } in @ident as @intro_pattern + .. tacv:: apply {+, @term {? with @bindings_list } } in @ident as @simple_intropattern :name: apply ... in ... as - This works as :tacn:`apply ... in` then applies the :token:`intro_pattern` + This works as :tacn:`apply ... in` then applies the :token:`simple_intropattern` to the hypothesis :token:`ident`. .. tacv:: simple apply @term in @ident @@ -525,8 +874,8 @@ Applying theorems Tactic :n:`simple apply @term in @ident` does not either traverse tuples as :n:`apply @term in @ident` does. - .. tacv:: {? simple} apply {+, @term {? with @bindings_list}} in @ident {? as @intro_pattern} - {? simple} eapply {+, @term {? with @bindings_list}} in @ident {? as @intro_pattern} + .. tacv:: {? simple} apply {+, @term {? with @bindings_list}} in @ident {? as @simple_intropattern} + {? simple} eapply {+, @term {? with @bindings_list}} in @ident {? as @simple_intropattern} This summarizes the different syntactic variants of :n:`apply @term in @ident` and :n:`eapply @term in @ident`. @@ -726,149 +1075,17 @@ Managing the local context .. exn:: No such hypothesis: @ident. :undocumented: -.. tacn:: intros @intro_pattern_list +.. tacn:: intros @intropattern_list :name: intros ... - This extension of the tactic :n:`intros` allows to apply tactics on the fly - on the variables or hypotheses which have been introduced. An - *introduction pattern list* :n:`@intro_pattern_list` is a list of - introduction patterns possibly containing the filling introduction - patterns `*` and `**`. An *introduction pattern* is either: - - + a *naming introduction pattern*, i.e. either one of: - - + the pattern :n:`?` - - + the pattern :n:`?ident` - - + an identifier - - + an *action introduction pattern* which itself classifies into: - - + a *disjunctive/conjunctive introduction pattern*, i.e. either one of - - + a disjunction of lists of patterns - :n:`[@intro_pattern_list | ... | @intro_pattern_list]` - - + a conjunction of patterns: :n:`({+, p})` - - + a list of patterns - :n:`({+& p})` - for sequence of right-associative binary constructs - - + an *equality introduction pattern*, i.e. either one of: - - + a pattern for decomposing an equality: :n:`[= {+ p}]` - + the rewriting orientations: :n:`->` or :n:`<-` - - + the on-the-fly application of lemmas: :n:`p{+ %term}` where :n:`p` - itself is not a pattern for on-the-fly application of lemmas (note: - syntax is in experimental stage) - - + the wildcard: :n:`_` - - - Assuming a goal of type :g:`Q → P` (non-dependent product), or of type - :g:`forall x:T, P` (dependent product), the behavior of - :n:`intros p` is defined inductively over the structure of the introduction - pattern :n:`p`: - - Introduction on :n:`?` performs the introduction, and lets Coq choose a fresh - name for the variable; - - Introduction on :n:`?@ident` performs the introduction, and lets Coq choose a - fresh name for the variable based on :n:`@ident`; - - Introduction on :n:`@ident` behaves as described in :tacn:`intro` - - Introduction over a disjunction of list of patterns - :n:`[@intro_pattern_list | ... | @intro_pattern_list ]` expects the product - to be over an inductive type whose number of constructors is `n` (or more - generally over a type of conclusion an inductive type built from `n` - constructors, e.g. :g:`C -> A\/B` with `n=2` since :g:`A\/B` has `2` - constructors): it destructs the introduced hypothesis as :n:`destruct` (see - :tacn:`destruct`) would and applies on each generated subgoal the - corresponding tactic; - - The introduction patterns in :n:`@intro_pattern_list` are expected to consume - no more than the number of arguments of the `i`-th constructor. If it - consumes less, then Coq completes the pattern so that all the arguments of - the constructors of the inductive type are introduced (for instance, the - list of patterns :n:`[ | ] H` applied on goal :g:`forall x:nat, x=0 -> 0=x` - behaves the same as the list of patterns :n:`[ | ? ] H`); - - Introduction over a conjunction of patterns :n:`({+, p})` expects - the goal to be a product over an inductive type :g:`I` with a single - constructor that itself has at least `n` arguments: It performs a case - analysis over the hypothesis, as :n:`destruct` would, and applies the - patterns :n:`{+ p}` to the arguments of the constructor of :g:`I` (observe - that :n:`({+ p})` is an alternative notation for :n:`[{+ p}]`); - - Introduction via :n:`({+& p})` is a shortcut for introduction via - :n:`(p,( ... ,( ..., p ) ... ))`; it expects the hypothesis to be a sequence of - right-associative binary inductive constructors such as :g:`conj` or - :g:`ex_intro`; for instance, a hypothesis with type - :g:`A /\(exists x, B /\ C /\ D)` can be introduced via pattern - :n:`(a & x & b & c & d)`; - - If the product is over an equality type, then a pattern of the form - :n:`[= {+ p}]` applies either :tacn:`injection` or :tacn:`discriminate` - instead of :tacn:`destruct`; if :tacn:`injection` is applicable, the patterns - :n:`{+, p}` are used on the hypotheses generated by :tacn:`injection`; if the - number of patterns is smaller than the number of hypotheses generated, the - pattern :n:`?` is used to complete the list. - - Introduction over ``->`` (respectively over ``<-``) - expects the hypothesis to be an equality and the right-hand-side - (respectively the left-hand-side) is replaced by the left-hand-side - (respectively the right-hand-side) in the conclusion of the goal; - the hypothesis itself is erased; if the term to substitute is a variable, it - is substituted also in the context of goal and the variable is removed too. - - Introduction over a pattern :n:`p{+ %term}` first applies :n:`{+ term}` - on the hypothesis to be introduced (as in :n:`apply {+, term}`) prior to the - application of the introduction pattern :n:`p`; - - Introduction on the wildcard depends on whether the product is dependent or not: - in the non-dependent case, it erases the corresponding hypothesis (i.e. it - behaves as an :tacn:`intro` followed by a :tacn:`clear`) while in the - dependent case, it succeeds and erases the variable only if the wildcard is part - of a more complex list of introduction patterns that also erases the hypotheses - depending on this variable; - - Introduction over :n:`*` introduces all forthcoming quantified variables - appearing in a row; introduction over :n:`**` introduces all forthcoming - quantified variables or hypotheses until the goal is not any more a - quantification or an implication. - - .. example:: - - .. coqtop:: reset all - - Goal forall A B C:Prop, A \/ B /\ C -> (A -> C) -> C. - intros * [a | (_,c)] f. - -.. note:: - - :n:`intros {+ p}` is not equivalent to :n:`intros p; ... ; intros p` - for the following reason: If one of the :n:`p` is a wildcard pattern, it - might succeed in the first case because the further hypotheses it - depends on are eventually erased too while it might fail in the second - case because of dependencies in hypotheses which are not yet - introduced (and a fortiori not yet erased). - -.. note:: - - In :n:`intros @intro_pattern_list`, if the last introduction pattern - is a disjunctive or conjunctive pattern - :n:`[{+| @intro_pattern_list}]`, the completion of :n:`@intro_pattern_list` - so that all the arguments of the i-th constructors of the corresponding - inductive type are introduced can be controlled with the following option: + Introduces one or more variables or hypotheses from the goal by matching the + intro patterns. See the description in :ref:`intropatterns`. - .. flag:: Bracketing Last Introduction Pattern +.. tacn:: eintros @intropattern_list + :name: eintros - Force completion, if needed, when the last introduction pattern is a - disjunctive or conjunctive pattern (on by default). + Works just like :tacn:`intros ...` except that it creates existential variables + for any unresolved variables rather than failing. .. tacn:: clear @ident :name: clear @@ -1057,19 +1274,19 @@ Managing the local context used as a synonym of :tacn:`epose`, i.e. when the :token:`term` does not occur in the goal. -.. tacn:: remember @term as @ident__1 {? eqn:@ident__2 } +.. tacn:: remember @term as @ident__1 {? eqn:@naming_intropattern } :name: remember - This behaves as :n:`set (@ident__1 := @term) in *`, using a logical + This behaves as :n:`set (@ident := @term) in *`, using a logical (Leibniz’s) equality instead of a local definition. - If :n:`@ident__2` is provided, it will be the name of the new equation. + Use :n:`@naming_intropattern` to name or split up the new equation. - .. tacv:: remember @term as @ident__1 {? eqn:@ident__2 } in @goal_occurrences + .. tacv:: remember @term as @ident__1 {? eqn:@naming_intropattern } in @goal_occurrences This is a more general form of :tacn:`remember` that remembers the occurrences of :token:`term` specified by an occurrence set. - .. tacv:: eremember @term as @ident__1 {? eqn:@ident__2 } {? in @goal_occurrences } + .. tacv:: eremember @term as @ident__1 {? eqn:@naming_intropattern } {? in @goal_occurrences } :name: eremember While the different variants of :tacn:`remember` expect that no @@ -1163,16 +1380,16 @@ Controlling the proof flow :name: Proof is not complete. (assert) :undocumented: - .. tacv:: assert @type as @intro_pattern + .. tacv:: assert @type as @simple_intropattern - If :n:`intro_pattern` is a naming introduction pattern (see :tacn:`intro`), + If :n:`simple_intropattern` is an intro pattern (see :ref:`intropatterns`), the hypothesis is named after this introduction pattern (in particular, if - :n:`intro_pattern` is :n:`@ident`, the tactic behaves like - :n:`assert (@ident : @type)`). If :n:`intro_pattern` is an action + :n:`simple_intropattern` is :n:`@ident`, the tactic behaves like + :n:`assert (@ident : @type)`). If :n:`simple_intropattern` is an action introduction pattern, the tactic behaves like :n:`assert @type` followed by the action done by this introduction pattern. - .. tacv:: assert @type as @intro_pattern by @tactic + .. tacv:: assert @type as @simple_intropattern by @tactic This combines the two previous variants of :tacn:`assert`. @@ -1186,7 +1403,7 @@ Controlling the proof flow .. exn:: Variable @ident is already declared. :undocumented: -.. tacv:: eassert @type as @intro_pattern by @tactic +.. tacv:: eassert @type as @simple_intropattern by @tactic :name: eassert While the different variants of :tacn:`assert` expect that no existential @@ -1194,16 +1411,16 @@ Controlling the proof flow This allows not to specify the asserted statement completeley before starting to prove it. -.. tacv:: pose proof @term {? as @intro_pattern} +.. tacv:: pose proof @term {? as @simple_intropattern} :name: pose proof - This tactic behaves like :n:`assert @type {? as @intro_pattern} by exact @term` + This tactic behaves like :n:`assert @type {? as @simple_intropattern} by exact @term` where :token:`type` is the type of :token:`term`. In particular, :n:`pose proof @term as @ident` behaves as :n:`assert (@ident := @term)` - and :n:`pose proof @term as @intro_pattern` is the same as applying the - :token:`intro_pattern` to :token:`term`. + and :n:`pose proof @term as @simple_intropattern` is the same as applying the + :token:`simple_intropattern` to :token:`term`. -.. tacv:: epose proof @term {? as @intro_pattern} +.. tacv:: epose proof @term {? as @simple_intropattern} :name: epose proof While :tacn:`pose proof` expects that no existential variables are generated by @@ -1221,20 +1438,20 @@ Controlling the proof flow This behaves like :n:`enough (@ident : @type)` with the name :token:`ident` of the hypothesis generated by Coq. -.. tacv:: enough @type as @intro_pattern +.. tacv:: enough @type as @simple_intropattern - This behaves like :n:`enough @type` using :token:`intro_pattern` to name or + This behaves like :n:`enough @type` using :token:`simple_intropattern` to name or destruct the new hypothesis. .. tacv:: enough (@ident : @type) by @tactic - enough @type {? as @intro_pattern } by @tactic + enough @type {? as @simple_intropattern } by @tactic This behaves as above but with :token:`tactic` expected to solve the initial goal after the extra assumption :token:`type` is added and possibly destructed. If the - :n:`as @intro_pattern` clause generates more than one subgoal, :token:`tactic` is + :n:`as @simple_intropattern` clause generates more than one subgoal, :token:`tactic` is applied to all of them. -.. tacv:: eenough @type {? as @intro_pattern } {? by @tactic } +.. tacv:: eenough @type {? as @simple_intropattern } {? by @tactic } eenough (@ident : @type) {? by @tactic } :name: eenough; _ @@ -1250,8 +1467,8 @@ Controlling the proof flow subgoals: :g:`U -> T` and :g:`U`. The subgoal :g:`U -> T` comes first in the list of remaining subgoal to prove. -.. tacv:: specialize (@ident {* @term}) {? as @intro_pattern} - specialize @ident with @bindings_list {? as @intro_pattern} +.. tacv:: specialize (@ident {* @term}) {? as @simple_intropattern} + specialize @ident with @bindings_list {? as @simple_intropattern} :name: specialize; _ This tactic works on local hypothesis :n:`@ident`. The @@ -1264,7 +1481,7 @@ Controlling the proof flow uninstantiated arguments are inferred by unification if possible or left quantified in the hypothesis otherwise. With the :n:`as` clause, the local hypothesis :n:`@ident` is left unchanged and instead, the modified hypothesis - is introduced as specified by the :token:`intro_pattern`. The name :n:`@ident` + is introduced as specified by the :token:`simple_intropattern`. The name :n:`@ident` can also refer to a global lemma or hypothesis. In this case, for compatibility reasons, the behavior of :tacn:`specialize` is close to that of :tacn:`generalize`: the instantiated statement becomes an additional premise of @@ -1477,11 +1694,11 @@ analysis on inductive or co-inductive objects (see :ref:`inductive-definitions`) This is a shortcut for :n:`destruct @term; ...; destruct @term`. - .. tacv:: destruct @term as @disj_conj_intro_pattern + .. tacv:: destruct @term as @or_and_intropattern_loc This behaves as :n:`destruct @term` but uses the names - in :token:`disj_conj_intro_pattern` to name the variables introduced in the - context. The :token:`disj_conj_intro_pattern` must have the + in :token:`or_and_intropattern_loc` to name the variables introduced in the + context. The :token:`or_and_intropattern_loc` must have the form :n:`[p11 ... p1n | ... | pm1 ... pmn ]` with ``m`` being the number of constructors of the type of :token:`term`. Each variable introduced by :tacn:`destruct` in the context of the ``i``-th goal @@ -1491,13 +1708,13 @@ analysis on inductive or co-inductive objects (see :ref:`inductive-definitions`) pattern (see :tacn:`intros`). This provides a concise notation for chaining destruction of a hypothesis. - .. tacv:: destruct @term eqn:@naming_intro_pattern + .. tacv:: destruct @term eqn:@naming_intropattern :name: destruct ... eqn: This behaves as :n:`destruct @term` but adds an equation between :token:`term` and the value that it takes in each of the possible cases. The name of the equation is specified - by :token:`naming_intro_pattern` (see :tacn:`intros`), + by :token:`naming_intropattern` (see :tacn:`intros`), in particular ``?`` can be used to let Coq generate a fresh name. .. tacv:: destruct @term with @bindings_list @@ -1525,8 +1742,8 @@ analysis on inductive or co-inductive objects (see :ref:`inductive-definitions`) clause is an occurrence clause whose syntax and behavior is described in :ref:`occurrences sets <occurrencessets>`. - .. tacv:: destruct @term {? with @bindings_list } {? as @disj_conj_intro_pattern } {? eqn:@naming_intro_pattern } {? using @term {? with @bindings_list } } {? in @goal_occurrences } - edestruct @term {? with @bindings_list } {? as @disj_conj_intro_pattern } {? eqn:@naming_intro_pattern } {? using @term {? with @bindings_list } } {? in @goal_occurrences } + .. tacv:: destruct @term {? with @bindings_list } {? as @or_and_intropattern_loc } {? eqn:@naming_intropattern } {? using @term {? with @bindings_list } } {? in @goal_occurrences } + edestruct @term {? with @bindings_list } {? as @or_and_intropattern_loc } {? eqn:@naming_intropattern } {? using @term {? with @bindings_list } } {? in @goal_occurrences } These are the general forms of :tacn:`destruct` and :tacn:`edestruct`. They combine the effects of the ``with``, ``as``, ``eqn:``, ``using``, @@ -1622,11 +1839,11 @@ analysis on inductive or co-inductive objects (see :ref:`inductive-definitions`) Use in this case the variant :tacn:`elim ... with` below. -.. tacv:: induction @term as @disj_conj_intro_pattern +.. tacv:: induction @term as @or_and_intropattern_loc This behaves as :tacn:`induction` but uses the names in - :n:`@disj_conj_intro_pattern` to name the variables introduced in the - context. The :n:`@disj_conj_intro_pattern` must typically be of the form + :n:`@or_and_intropattern_loc` to name the variables introduced in the + context. The :n:`@or_and_intropattern_loc` must typically be of the form :n:`[ p` :sub:`11` :n:`... p` :sub:`1n` :n:`| ... | p`:sub:`m1` :n:`... p`:sub:`mn` :n:`]` with :n:`m` being the number of constructors of the type of :n:`@term`. Each variable introduced by induction in the context of the i-th goal gets its @@ -1686,8 +1903,8 @@ analysis on inductive or co-inductive objects (see :ref:`inductive-definitions`) induction y in x |- *. Show 2. -.. tacv:: induction @term with @bindings_list as @disj_conj_intro_pattern using @term with @bindings_list in @goal_occurrences - einduction @term with @bindings_list as @disj_conj_intro_pattern using @term with @bindings_list in @goal_occurrences +.. tacv:: induction @term with @bindings_list as @or_and_intropattern_loc using @term with @bindings_list in @goal_occurrences + einduction @term with @bindings_list as @or_and_intropattern_loc using @term with @bindings_list in @goal_occurrences These are the most general forms of :tacn:`induction` and :tacn:`einduction`. It combines the effects of the with, as, using, and in clauses. @@ -1898,7 +2115,7 @@ and an explanation of the underlying technique. .. exn:: Not the right number of induction arguments. :undocumented: -.. tacv:: functional induction (@qualid {+ @term}) as @disj_conj_intro_pattern using @term with @bindings_list +.. tacv:: functional induction (@qualid {+ @term}) as @simple_intropattern using @term with @bindings_list Similarly to :tacn:`induction` and :tacn:`elim`, this allows giving explicitly the name of the introduced variables, the induction principle, and @@ -2053,18 +2270,18 @@ and an explanation of the underlying technique. .. exn:: goal does not satisfy the expected preconditions. :undocumented: - .. tacv:: injection @term {? with @bindings_list} as {+ @intro_pattern} - injection @num as {+ intro_pattern} - injection as {+ intro_pattern} - einjection @term {? with @bindings_list} as {+ intro_pattern} - einjection @num as {+ intro_pattern} - einjection as {+ intro_pattern} + .. tacv:: injection @term {? with @bindings_list} as {+ @simple_intropattern} + injection @num as {+ simple_intropattern} + injection as {+ simple_intropattern} + einjection @term {? with @bindings_list} as {+ simple_intropattern} + einjection @num as {+ simple_intropattern} + einjection as {+ simple_intropattern} - These variants apply :n:`intros {+ @intro_pattern}` after the call to + These variants apply :n:`intros {+ @simple_intropattern}` after the call to :tacn:`injection` or :tacn:`einjection` so that all equalities generated are moved in - the context of hypotheses. The number of :n:`@intro_pattern` must not exceed + the context of hypotheses. The number of :n:`@simple_intropattern` must not exceed the number of equalities newly generated. If it is smaller, fresh - names are automatically generated to adjust the list of :n:`@intro_pattern` + names are automatically generated to adjust the list of :n:`@simple_intropattern` to the number of new equalities. The original equality is erased if it corresponds to a hypothesis. @@ -2118,10 +2335,10 @@ and an explanation of the underlying technique. This behaves as :n:`inversion` and then erases :n:`@ident` from the context. -.. tacv:: inversion @ident as @intro_pattern +.. tacv:: inversion @ident as @or_and_intropattern_loc - This generally behaves as inversion but using names in :n:`@intro_pattern` - for naming hypotheses. The :n:`@intro_pattern` must have the form + This generally behaves as inversion but using names in :n:`@or_and_intropattern_loc` + for naming hypotheses. The :n:`@or_and_intropattern_loc` must have the form :n:`[p`:sub:`11` :n:`... p`:sub:`1n` :n:`| ... | p`:sub:`m1` :n:`... p`:sub:`mn` :n:`]` with `m` being the number of constructors of the type of :n:`@ident`. Be careful that the list must be of length `m` even if ``inversion`` discards @@ -2153,12 +2370,12 @@ and an explanation of the underlying technique. Goal forall l:list nat, contains0 (1 :: l) -> contains0 l. intros l H; inversion H as [ | l' p Hl' [Heqp Heql'] ]. -.. tacv:: inversion @num as @intro_pattern +.. tacv:: inversion @num as @or_and_intropattern_loc This allows naming the hypotheses introduced by :n:`inversion @num` in the context. -.. tacv:: inversion_clear @ident as @intro_pattern +.. tacv:: inversion_clear @ident as @or_and_intropattern_loc This allows naming the hypotheses introduced by ``inversion_clear`` in the context. Notice that hypothesis names can be provided as if ``inversion`` @@ -2170,7 +2387,7 @@ and an explanation of the underlying technique. Let :n:`{+ @ident}` be identifiers in the local context. This tactic behaves as generalizing :n:`{+ @ident}`, and then performing ``inversion``. -.. tacv:: inversion @ident as @intro_pattern in {+ @ident} +.. tacv:: inversion @ident as @or_and_intropattern_loc in {+ @ident} This allows naming the hypotheses introduced in the context by :n:`inversion @ident in {+ @ident}`. @@ -2180,7 +2397,7 @@ and an explanation of the underlying technique. Let :n:`{+ @ident}` be identifiers in the local context. This tactic behaves as generalizing :n:`{+ @ident}`, and then performing ``inversion_clear``. -.. tacv:: inversion_clear @ident as @intro_pattern in {+ @ident} +.. tacv:: inversion_clear @ident as @or_and_intropattern_loc in {+ @ident} This allows naming the hypotheses introduced in the context by :n:`inversion_clear @ident in {+ @ident}`. @@ -2192,7 +2409,7 @@ and an explanation of the underlying technique. ``inversion`` and then substitutes :n:`@ident` for the corresponding :n:`@@term` in the goal. -.. tacv:: dependent inversion @ident as @intro_pattern +.. tacv:: dependent inversion @ident as @or_and_intropattern_loc This allows naming the hypotheses introduced in the context by :n:`dependent inversion @ident`. @@ -2202,7 +2419,7 @@ and an explanation of the underlying technique. Like ``dependent inversion``, except that :n:`@ident` is cleared from the local context. -.. tacv:: dependent inversion_clear @ident as @intro_pattern +.. tacv:: dependent inversion_clear @ident as @or_and_intropattern_loc This allows naming the hypotheses introduced in the context by :n:`dependent inversion_clear @ident`. @@ -2216,7 +2433,7 @@ and an explanation of the underlying technique. then :n:`@term` must be of type :g:`I:forall (x:T), I x -> s'` where :g:`s'` is the type of the goal. -.. tacv:: dependent inversion @ident as @intro_pattern with @term +.. tacv:: dependent inversion @ident as @or_and_intropattern_loc with @term This allows naming the hypotheses introduced in the context by :n:`dependent inversion @ident with @term`. @@ -2226,7 +2443,7 @@ and an explanation of the underlying technique. Like :tacn:`dependent inversion ... with ...` with but clears :n:`@ident` from the local context. -.. tacv:: dependent inversion_clear @ident as @intro_pattern with @term +.. tacv:: dependent inversion_clear @ident as @or_and_intropattern_loc with @term This allows naming the hypotheses introduced in the context by :n:`dependent inversion_clear @ident with @term`. @@ -2237,7 +2454,7 @@ and an explanation of the underlying technique. It is a very primitive inversion tactic that derives all the necessary equalities but it does not simplify the constraints as ``inversion`` does. -.. tacv:: simple inversion @ident as @intro_pattern +.. tacv:: simple inversion @ident as @or_and_intropattern_loc This allows naming the hypotheses introduced in the context by ``simple inversion``. @@ -3586,15 +3803,15 @@ The general command to add a hint to some databases :n:`{+ @ident}` is the following. Beware, there is no operator precedence during parsing, one can check with :cmd:`Print HintDb` to verify the current cut expression: - .. productionlist:: `regexp` - e : ident hint or instance identifier - :| _ any hint - :| e\|e′ disjunction - :| e e′ sequence - :| e * Kleene star - :| emp empty - :| eps epsilon - :| ( e ) + .. productionlist:: regexp + e : `ident` hint or instance identifier + : _ any hint + : `e` | `e` disjunction + : `e` `e` sequence + : `e` * Kleene star + : emp empty + : eps epsilon + : ( `e` ) The `emp` regexp does not match any search path while `eps` matches the empty path. During proof search, the path of @@ -4299,15 +4516,15 @@ Automating .. _btauto_grammar: - .. productionlist:: `sentence` - t : x - :∣ true - :∣ false - :∣ orb t1 t2 - :∣ andb t1 t2 - :∣ xorb t1 t2 - :∣ negb t - :∣ if t1 then t2 else t3 + .. productionlist:: sentence + t : `x` + : true + : false + : orb `t` `t` + : andb `t` `t` + : xorb `t` `t` + : negb `t` + : if `t` then `t` else `t` Whenever the formula supplied is not a tautology, it also provides a counter-example. @@ -4343,7 +4560,7 @@ Automating distributivity, constant propagation) and comparing syntactically the results. -.. tacn:: ring_simplify {+ @term} +.. tacn:: ring_simplify {* @term} :name: ring_simplify This tactic applies the normalization procedure described above to @@ -4357,7 +4574,7 @@ the tactic and how to declare new ring structures. All declared field structures can be printed with the ``Print Rings`` command. .. tacn:: field - field_simplify {+ @term} + field_simplify {* @term} field_simplify_eq :name: field; field_simplify; field_simplify_eq diff --git a/doc/sphinx/user-extensions/syntax-extensions.rst b/doc/sphinx/user-extensions/syntax-extensions.rst index 47afa5ba0c..c707da1353 100644 --- a/doc/sphinx/user-extensions/syntax-extensions.rst +++ b/doc/sphinx/user-extensions/syntax-extensions.rst @@ -859,41 +859,41 @@ notations are given below. The optional :production:`scope` is described in .. productionlist:: coq notation : [Local] Notation `string` := `term` [`modifiers`] [: `scope`]. - : | [Local] Infix `string` := `qualid` [`modifiers`] [: `scope`]. - : | [Local] Reserved Notation `string` [`modifiers`] . - : | Inductive `ind_body` [`decl_notation`] with … with `ind_body` [`decl_notation`]. - : | CoInductive `ind_body` [`decl_notation`] with … with `ind_body` [`decl_notation`]. - : | Fixpoint `fix_body` [`decl_notation`] with … with `fix_body` [`decl_notation`]. - : | CoFixpoint `cofix_body` [`decl_notation`] with … with `cofix_body` [`decl_notation`]. - : | [Local] Declare Custom Entry `ident`. + : [Local] Infix `string` := `qualid` [`modifiers`] [: `scope`]. + : [Local] Reserved Notation `string` [`modifiers`] . + : Inductive `ind_body` [`decl_notation`] with … with `ind_body` [`decl_notation`]. + : CoInductive `ind_body` [`decl_notation`] with … with `ind_body` [`decl_notation`]. + : Fixpoint `fix_body` [`decl_notation`] with … with `fix_body` [`decl_notation`]. + : CoFixpoint `cofix_body` [`decl_notation`] with … with `cofix_body` [`decl_notation`]. + : [Local] Declare Custom Entry `ident`. decl_notation : [where `string` := `term` [: `scope`] and … and `string` := `term` [: `scope`]]. modifiers : at level `num` : in custom `ident` : in custom `ident` at level `num` - : | `ident` , … , `ident` at level `num` [`binderinterp`] - : | `ident` , … , `ident` at next level [`binderinterp`] - : | `ident` `explicit_subentry` - : | left associativity - : | right associativity - : | no associativity - : | only parsing - : | only printing - : | format `string` + : `ident` , … , `ident` at level `num` [`binderinterp`] + : `ident` , … , `ident` at next level [`binderinterp`] + : `ident` `explicit_subentry` + : left associativity + : right associativity + : no associativity + : only parsing + : only printing + : format `string` explicit_subentry : ident - : | global - : | bigint - : | [strict] pattern [at level `num`] - : | binder - : | closed binder - : | constr [`binderinterp`] - : | constr at level `num` [`binderinterp`] - : | constr at next level [`binderinterp`] - : | custom [`binderinterp`] - : | custom at level `num` [`binderinterp`] - : | custom at next level [`binderinterp`] + : global + : bigint + : [strict] pattern [at level `num`] + : binder + : closed binder + : constr [`binderinterp`] + : constr at level `num` [`binderinterp`] + : constr at next level [`binderinterp`] + : custom [`binderinterp`] + : custom at level `num` [`binderinterp`] + : custom at next level [`binderinterp`] binderinterp : as ident - : | as pattern - : | as strict pattern + : as pattern + : as strict pattern .. note:: No typing of the denoted expression is performed at definition time. Type checking is done only at the time of use of the notation. @@ -1692,13 +1692,13 @@ Tactic notations allow to customize the syntax of tactics. They have the followi tacn : Tactic Notation [`tactic_level`] [`prod_item` … `prod_item`] := `tactic`. prod_item : `string` | `tactic_argument_type`(`ident`) tactic_level : (at level `num`) - tactic_argument_type : ident | simple_intropattern | reference - : | hyp | hyp_list | ne_hyp_list - : | constr | uconstr | constr_list | ne_constr_list - : | integer | integer_list | ne_integer_list - : | int_or_var | int_or_var_list | ne_int_or_var_list - : | tactic | tactic0 | tactic1 | tactic2 | tactic3 - : | tactic4 | tactic5 + tactic_argument_type : `ident` | `simple_intropattern` | `reference` + : `hyp` | `hyp_list` | `ne_hyp_list` + : `constr` | `uconstr` | `constr_list` | `ne_constr_list` + : `integer` | `integer_list` | `ne_integer_list` + : `int_or_var` | `int_or_var_list` | `ne_int_or_var_list` + : `tactic` | `tactic0` | `tactic1` | `tactic2` | `tactic3` + : `tactic4` | `tactic5` .. cmd:: Tactic Notation {? (at level @level)} {+ @prod_item} := @tactic. |