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-rw-r--r-- | doc/sphinx/language/gallina-specification-language.rst | 56 |
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diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 1a8e9a5dc..2d96adc5f 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -999,41 +999,47 @@ constructors. Infinite objects are introduced by a non-ending (but effective) process of construction, defined in terms of the constructors of the type. -An example of a co-inductive type is the type of infinite sequences of -natural numbers, usually called streams. It can be introduced in -Coq using the ``CoInductive`` command: +.. cmd:: CoInductive @ident @binders {? : @type } := {? | } @ident : @type {* | @ident : @type} -.. coqtop:: all + This command introduces a co-inductive type. + The syntax of the command is the same as the command :cmd:`Inductive`. + No principle of induction is derived from the definition of a co-inductive + type, since such principles only make sense for inductive types. + For co-inductive types, the only elimination principle is case analysis. + +.. example:: + An example of a co-inductive type is the type of infinite sequences of + natural numbers, usually called streams. - CoInductive Stream : Set := - Seq : nat -> Stream -> Stream. + .. coqtop:: in -The syntax of this command is the same as the command :cmd:`Inductive`. Notice -that no principle of induction is derived from the definition of a co-inductive -type, since such principles only make sense for inductive ones. For co-inductive -ones, the only elimination principle is case analysis. For example, the usual -destructors on streams :g:`hd:Stream->nat` and :g:`tl:Str->Str` can be defined -as follows: + CoInductive Stream : Set := Seq : nat -> Stream -> Stream. -.. coqtop:: all + The usual destructors on streams :g:`hd:Stream->nat` and :g:`tl:Str->Str` + can be defined as follows: - Definition hd (x:Stream) := let (a,s) := x in a. - Definition tl (x:Stream) := let (a,s) := x in s. + .. coqtop:: in + + Definition hd (x:Stream) := let (a,s) := x in a. + Definition tl (x:Stream) := let (a,s) := x in s. Definition of co-inductive predicates and blocks of mutually -co-inductive definitions are also allowed. An example of a co-inductive -predicate is the extensional equality on streams: +co-inductive definitions are also allowed. -.. coqtop:: all +.. example:: + An example of a co-inductive predicate is the extensional equality on + streams: + + .. coqtop:: in - CoInductive EqSt : Stream -> Stream -> Prop := - eqst : forall s1 s2:Stream, - hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2. + CoInductive EqSt : Stream -> Stream -> Prop := + eqst : forall s1 s2:Stream, + hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2. -In order to prove the extensionally equality of two streams :g:`s1` and :g:`s2` -we have to construct an infinite proof of equality, that is, an infinite object -of type :g:`(EqSt s1 s2)`. We will see how to introduce infinite objects in -Section :ref:`cofixpoint`. + In order to prove the extensional equality of two streams :g:`s1` and :g:`s2` + we have to construct an infinite proof of equality, that is, an infinite + object of type :g:`(EqSt s1 s2)`. We will see how to introduce infinite + objects in Section :ref:`cofixpoint`. Definition of recursive functions --------------------------------- |