diff options
Diffstat (limited to 'doc/sphinx/language')
| -rw-r--r-- | doc/sphinx/language/cic.rst | 30 | ||||
| -rw-r--r-- | doc/sphinx/language/coq-library.rst | 31 | ||||
| -rw-r--r-- | doc/sphinx/language/gallina-extensions.rst | 19 | ||||
| -rw-r--r-- | doc/sphinx/language/gallina-specification-language.rst | 6 |
4 files changed, 63 insertions, 23 deletions
diff --git a/doc/sphinx/language/cic.rst b/doc/sphinx/language/cic.rst index b01a4ef0f9..6e0c1e1b61 100644 --- a/doc/sphinx/language/cic.rst +++ b/doc/sphinx/language/cic.rst @@ -723,6 +723,7 @@ each :math:`T` in :math:`(t:T)∈Γ_I` can be written as: :math:`∀Γ_P,∀Γ_{ the sort of the inductive type t (not to be confused with :math:`\Sort` which is the set of sorts). .. example:: + The declaration for parameterized lists is: .. math:: @@ -741,6 +742,7 @@ the sort of the inductive type t (not to be confused with :math:`\Sort` which is | cons : A -> list A -> list A. .. example:: + The declaration for a mutual inductive definition of tree and forest is: @@ -763,6 +765,7 @@ the sort of the inductive type t (not to be confused with :math:`\Sort` which is | consf : tree -> forest -> forest. .. example:: + The declaration for a mutual inductive definition of even and odd is: .. math:: @@ -811,6 +814,7 @@ contains an inductive declaration. E[Γ] ⊢ c : C .. example:: + Provided that our environment :math:`E` contains inductive definitions we showed before, these two inference rules above enable us to conclude that: @@ -919,6 +923,7 @@ condition* for a constant :math:`X` in the following cases: .. example:: + For instance, if one considers the following variant of a tree type branching over the natural numbers: @@ -985,6 +990,7 @@ the Type hierarchy. .. example:: + It is well known that the existential quantifier can be encoded as an inductive definition. The following declaration introduces the second- order existential quantifier :math:`∃ X.P(X)`. @@ -1102,6 +1108,7 @@ sorts at each instance of a pattern-matching (see Section :ref:`Destructors`). A an example, let us consider the following definition: .. example:: + .. coqtop:: in Inductive option (A:Type) : Type := @@ -1118,6 +1125,7 @@ if :g:`option` is applied to a type in :math:`\Prop`, then, the result is not se if set in :math:`\Prop`. .. example:: + .. coqtop:: all Check (fun A:Set => option A). @@ -1126,6 +1134,7 @@ if set in :math:`\Prop`. Here is another example. .. example:: + .. coqtop:: in Inductive prod (A B:Type) : Type := pair : A -> B -> prod A B. @@ -1136,6 +1145,7 @@ none in :math:`\Type`, and in :math:`\Type` otherwise. In all cases, the three k eliminations schemes are allowed. .. example:: + .. coqtop:: all Check (fun A:Set => prod A). @@ -1175,7 +1185,7 @@ 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, so such that :g:`(length (nil A)) = O` and :g:`(length +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 recognized implicitly and taken into account in the conversion rule. @@ -1324,6 +1334,7 @@ the extraction mechanism. Assume :math:`A` and :math:`B` are two propositions, a logical disjunction :math:`A ∨ B` is defined inductively by: .. example:: + .. coqtop:: in Inductive or (A B:Prop) : Prop := @@ -1334,6 +1345,7 @@ The following definition which computes a boolean value by case over the proof of :g:`or A B` is not accepted: .. example:: + .. coqtop:: all Fail Definition choice (A B: Prop) (x:or A B) := @@ -1357,6 +1369,7 @@ property which are provably different, contradicting the proof- irrelevance property which is sometimes a useful axiom: .. example:: + .. coqtop:: all Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y. @@ -1364,7 +1377,7 @@ irrelevance property which is sometimes a useful axiom: The elimination of an inductive definition of type :math:`\Prop` on a predicate :math:`P` of type :math:`I→ Type` leads to a paradox when applied to impredicative inductive definition like the second-order existential quantifier -:g:`exProp` defined above, because it give access to the two projections on +:g:`exProp` defined above, because it gives access to the two projections on this type. @@ -1390,6 +1403,7 @@ be used for rewriting not only in logical propositions but also in any type. .. example:: + .. coqtop:: all Print eq_rec. @@ -1421,6 +1435,7 @@ We write :math:`\{c\}^P` for :math:`\{c:C\}^P` with :math:`C` the type of :math: .. example:: + The following term in concrete syntax:: match t as l return P' with @@ -1485,6 +1500,7 @@ definition :math:`\ind{r}{Γ_I}{Γ_C}` with :math:`Γ_C = [c_1 :C_1 ;…;c_n :C_ .. example:: + Below is a typing rule for the term shown in the previous example: .. inference:: list example @@ -1613,7 +1629,7 @@ then the recursive arguments will correspond to :math:`T_i` in which one of the :math:`I_l` occurs. The main rules for being structurally smaller are the following. -Given a variable :math:`y` of type an inductive definition in a declaration +Given a variable :math:`y` of an inductively defined type in a declaration :math:`\ind{r}{Γ_I}{Γ_C}` where :math:`Γ_I` is :math:`[I_1 :A_1 ;…;I_k :A_k]`, and :math:`Γ_C` is :math:`[c_1 :C_1 ;…;c_n :C_n ]`, the terms structurally smaller than :math:`y` are: @@ -1625,7 +1641,7 @@ Given a variable :math:`y` of type an inductive definition in a declaration Each :math:`f_i` corresponds to a type of constructor :math:`C_q ≡ ∀ p_1 :P_1 ,…,∀ p_r :P_r , ∀ y_1 :B_1 , … ∀ y_k :B_k , (I~a_1 … a_k )` and can consequently be written :math:`λ y_1 :B_1' . … λ y_k :B_k'. g_i`. (:math:`B_i'` is - obtained from :math:`B_i` by substituting parameters variables) the variables + obtained from :math:`B_i` by substituting parameters for variables) the variables :math:`y_j` occurring in :math:`g_i` corresponding to recursive arguments :math:`B_i` (the ones in which one of the :math:`I_l` occurs) are structurally smaller than y. @@ -1634,6 +1650,7 @@ The following definitions are correct, we enter them using the :cmd:`Fixpoint` command and show the internal representation. .. example:: + .. coqtop:: all Fixpoint plus (n m:nat) {struct n} : nat := @@ -1801,7 +1818,7 @@ definitions can be found in :cite:`Gimenez95b,Gim98,GimCas05`. .. _The-Calculus-of-Inductive-Construction-with-impredicative-Set: -The Calculus of Inductive Construction with impredicative Set +The Calculus of Inductive Constructions with impredicative Set ----------------------------------------------------------------- |Coq| can be used as a type-checker for the Calculus of Inductive @@ -1810,6 +1827,7 @@ option ``-impredicative-set``. For example, using the ordinary `coqtop` command, the following is rejected, .. example:: + .. coqtop:: all Fail Definition id: Set := forall X:Set,X->X. @@ -1834,7 +1852,7 @@ inductive definitions* like the example of second-order existential quantifier (:g:`exSet`). There should be restrictions on the eliminations which can be -performed on such definitions. The eliminations rules in the +performed on such definitions. The elimination rules in the impredicative system for sort :math:`\Set` become: diff --git a/doc/sphinx/language/coq-library.rst b/doc/sphinx/language/coq-library.rst index afb49413dd..9de30e2190 100644 --- a/doc/sphinx/language/coq-library.rst +++ b/doc/sphinx/language/coq-library.rst @@ -705,21 +705,29 @@ fixpoint equation can be proved. Accessing the Type level ~~~~~~~~~~~~~~~~~~~~~~~~ -The basic library includes the definitions of the counterparts of some data-types and logical -quantifiers at the ``Type``: level: negation, pair, and properties -of ``identity``. This is the module ``Logic_Type.v``. +The standard library includes ``Type`` level definitions of counterparts of some +logic concepts and basic lemmas about them. + +The module ``Datatypes`` defines ``identity``, which is the ``Type`` level counterpart +of equality: + +.. index:: + single: identity (term) + +.. coqtop:: in + + Inductive identity (A:Type) (a:A) : A -> Type := + identity_refl : identity a a. + +Some properties of ``identity`` are proved in the module ``Logic_Type``, which also +provides the definition of ``Type`` level negation: .. index:: single: notT (term) - single: prodT (term) - single: pairT (term) .. coqtop:: in Definition notT (A:Type) := A -> False. - Inductive prodT (A B:Type) : Type := pairT (_:A) (_:B). - -At the end, it defines data-types at the ``Type`` level. Tactics ~~~~~~~ @@ -840,6 +848,7 @@ Notation Interpretation Precedence Associativity .. example:: + .. coqtop:: all reset Require Import ZArith. @@ -879,6 +888,7 @@ Notation Interpretation =============== =================== .. example:: + .. coqtop:: all reset Require Import Reals. @@ -889,7 +899,7 @@ Notation Interpretation Some tactics for real numbers +++++++++++++++++++++++++++++ -In addition to the powerful ``ring``, ``field`` and ``fourier`` +In addition to the powerful ``ring``, ``field`` and ``lra`` tactics (see Chapter :ref:`tactics`), there are also: .. tacn:: discrR @@ -898,6 +908,7 @@ tactics (see Chapter :ref:`tactics`), there are also: Proves that two real integer constants are different. .. example:: + .. coqtop:: all reset Require Import DiscrR. @@ -911,6 +922,7 @@ tactics (see Chapter :ref:`tactics`), there are also: Allows unfolding the ``Rabs`` constant and splits corresponding conjunctions. .. example:: + .. coqtop:: all reset Require Import Reals. @@ -925,6 +937,7 @@ tactics (see Chapter :ref:`tactics`), there are also: corresponding to the condition on each operand of the product. .. example:: + .. coqtop:: all reset Require Import Reals. diff --git a/doc/sphinx/language/gallina-extensions.rst b/doc/sphinx/language/gallina-extensions.rst index 509ac92f81..7dd0a6e383 100644 --- a/doc/sphinx/language/gallina-extensions.rst +++ b/doc/sphinx/language/gallina-extensions.rst @@ -70,7 +70,9 @@ generates a variant type definition with just one constructor: To build an object of type :n:`@ident`, one should provide the constructor :n:`@ident₀` with the appropriate number of terms filling the fields of the record. -.. example:: Let us define the rational :math:`1/2`: +.. example:: + + Let us define the rational :math:`1/2`: .. coqtop:: in @@ -781,7 +783,8 @@ Section :ref:`gallina-definitions`). .. cmd:: Section @ident - This command is used to open a section named `ident`. + This command is used to open a section named :token:`ident`. + Section names do not need to be unique. .. cmd:: End @ident @@ -1079,7 +1082,7 @@ The definition of ``N`` using the module type expression ``SIG`` with Module N : SIG' := M. -If we just want to be sure that the our implementation satisfies a +If we just want to be sure that our implementation satisfies a given module type without restricting the interface, we can use a transparent constraint @@ -1848,15 +1851,15 @@ are named as expected. .. example:: (continued) -.. coqtop:: all + .. coqtop:: all - Arguments p [s t] _ [u] _: rename. + Arguments p [s t] _ [u] _: rename. - Check (p r1 (u:=c)). + Check (p r1 (u:=c)). - Check (p (s:=a) (t:=b) r1 (u:=c) r2). + Check (p (s:=a) (t:=b) r1 (u:=c) r2). - Fail Arguments p [s t] _ [w] _ : assert. + Fail Arguments p [s t] _ [w] _ : assert. .. _displaying-implicit-args: diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 8250b4b3d6..da5cd00d72 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -758,6 +758,7 @@ Simple inductive types the case of annotated inductive types — cf. next section). .. example:: + The set of natural numbers is defined as: .. coqtop:: all @@ -976,6 +977,7 @@ Mutually defined inductive types reason, the parameters must be strictly the same for each inductive types. .. example:: + The typical example of a mutual inductive data type is the one for trees and forests. We assume given two types :g:`A` and :g:`B` as variables. It can be declared the following way. @@ -1048,6 +1050,7 @@ of the type. 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. @@ -1067,6 +1070,7 @@ Definition of co-inductive predicates and blocks of mutually co-inductive definitions are also allowed. .. example:: + An example of a co-inductive predicate is the extensional equality on streams: @@ -1129,6 +1133,7 @@ constructions. .. example:: + One can define the addition function as : .. coqtop:: all @@ -1201,6 +1206,7 @@ constructions. inductive types. .. example:: + The size of trees and forests can be defined the following way: .. coqtop:: all |