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[coq/en] Fix code width
Many lines exceed the 80 character limit that makes the code wrap around on the website. This mainly addresses comments. The comment style has been preserved. For the code lines that exceeded 80 characters, I have not adhered to any style.
This commit is contained in:
Simon Shine
parent
06e05c4ce3
commit
672b3804f9
@ -25,19 +25,20 @@ Inside Proof General `Ctrl+C Ctrl+<Enter>` will evaluate up to your cursor.
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(*** Variables and functions ***)
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(* The Coq proof assistant can be controlled and queried by a command language called
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the vernacular. Vernacular keywords are capitalized and the commands end with a period.
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Variable and function declarations are formed with the Definition vernacular. *)
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(* The Coq proof assistant can be controlled and queried by a command
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language called the vernacular. Vernacular keywords are capitalized and
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the commands end with a period. Variable and function declarations are
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formed with the Definition vernacular. *)
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Definition x := 10.
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(* Coq can sometimes infer the types of arguments, but it is common practice to annotate
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with types. *)
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(* Coq can sometimes infer the types of arguments, but it is common practice
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to annotate with types. *)
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Definition inc_nat (x : nat) : nat := x + 1.
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(* There exists a large number of vernacular commands for querying information.
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These can be very useful. *)
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(* There exists a large number of vernacular commands for querying
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information. These can be very useful. *)
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Compute (1 + 1). (* 2 : nat *) (* Compute a result. *)
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@ -46,48 +47,50 @@ Check tt. (* tt : unit *) (* Check the type of an expressions *)
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About plus. (* Prints information about an object *)
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(* Print information including the definition *)
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Print true. (* Inductive bool : Set := true : Bool | false : Bool *)
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Print true. (* Inductive bool : Set := true : Bool | false : Bool *)
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Search nat. (* Returns a large list of nat related values *)
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Search "_ + _". (* You can also search on patterns *)
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Search (?a -> ?a -> bool). (* Patterns can have named parameters *)
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Search (?a * ?a).
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(* Locate tells you where notation is coming from. Very helpful when you encounter
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new notation. *)
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Locate "+".
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(* Locate tells you where notation is coming from. Very helpful when you
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encounter new notation. *)
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(* Calling a function with insufficient number of arguments
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does not cause an error, it produces a new function. *)
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Locate "+".
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(* Calling a function with insufficient number of arguments does not cause
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an error, it produces a new function. *)
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Definition make_inc x y := x + y. (* make_inc is int -> int -> int *)
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Definition inc_2 := make_inc 2. (* inc_2 is int -> int *)
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Compute inc_2 3. (* Evaluates to 5 *)
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(* Definitions can be chained with "let ... in" construct.
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This is roughly the same to assigning values to multiple
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variables before using them in expressions in imperative
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languages. *)
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(* Definitions can be chained with "let ... in" construct. This is roughly
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the same to assigning values to multiple variables before using them in
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expressions in imperative languages. *)
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Definition add_xy : nat := let x := 10 in
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let y := 20 in
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x + y.
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(* Pattern matching is somewhat similar to switch statement in imperative
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languages, but offers a lot more expressive power. *)
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Definition is_zero (x : nat) :=
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match x with
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| 0 => true
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| _ => false (* The "_" pattern means "anything else". *)
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end.
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(* You can define recursive function definition using the Fixpoint
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vernacular.*)
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(* You can define recursive function definition using the Fixpoint vernacular.*)
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Fixpoint factorial n := match n with
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| 0 => 1
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| (S n') => n * factorial n'
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end.
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(* Function application usually doesn't need parentheses around arguments *)
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Compute factorial 5. (* 120 : nat *)
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@ -104,11 +107,12 @@ end with
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| (S n) => is_even n
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end.
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(* As Coq is a total programming language, it will only accept programs when it can
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understand they terminate. It can be most easily seen when the recursive call is
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on a pattern matched out subpiece of the input, as then the input is always decreasing
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in size. Getting Coq to understand that functions terminate is not always easy. See the
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references at the end of the article for more on this topic. *)
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(* As Coq is a total programming language, it will only accept programs when
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it can understand they terminate. It can be most easily seen when the
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recursive call is on a pattern matched out subpiece of the input, as then
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the input is always decreasing in size. Getting Coq to understand that
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functions terminate is not always easy. See the references at the end of
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the article for more on this topic. *)
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(* Anonymous functions use the following syntax: *)
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@ -119,16 +123,18 @@ Definition my_id2 : forall A : Type, A -> A := fun A x => x.
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Compute my_id nat 3. (* 3 : nat *)
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(* You can ask Coq to infer terms with an underscore *)
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Compute my_id _ 3.
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Compute my_id _ 3.
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(* An implicit argument of a function is an argument which can be inferred from contextual
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knowledge. Parameters enclosed in {} are implicit by default *)
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(* An implicit argument of a function is an argument which can be inferred
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from contextual knowledge. Parameters enclosed in {} are implicit by
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default *)
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Definition my_id3 {A : Type} (x : A) : A := x.
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Compute my_id3 3. (* 3 : nat *)
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(* Sometimes it may be necessary to turn this off. You can make all arguments explicit
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again with @ *)
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(* Sometimes it may be necessary to turn this off. You can make all
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arguments explicit again with @ *)
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Compute @my_id3 nat 3.
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(* Or give arguments by name *)
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@ -168,17 +174,19 @@ let rec factorial n = match n with
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(*** Notation ***)
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(* Coq has a very powerful Notation system that can be used to write expressions in more
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natural forms. *)
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(* Coq has a very powerful Notation system that can be used to write
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expressions in more natural forms. *)
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Compute Nat.add 3 4. (* 7 : nat *)
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Compute 3 + 4. (* 7 : nat *)
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(* Notation is a syntactic transformation applied to the text of the program before being
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evaluated. Notation is organized into notation scopes. Using different notation scopes
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allows for a weak notion of overloading. *)
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(* Notation is a syntactic transformation applied to the text of the program
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before being evaluated. Notation is organized into notation scopes. Using
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different notation scopes allows for a weak notion of overloading. *)
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(* Imports the Zarith module containing definitions related to the integers Z *)
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Require Import ZArith.
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(* Imports the Zarith module holding definitions related to the integers Z *)
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Require Import ZArith.
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(* Notation scopes can be opened *)
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Open Scope Z_scope.
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@ -187,7 +195,7 @@ Open Scope Z_scope.
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Compute 1 + 7. (* 8 : Z *)
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(* Integer equality checking *)
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Compute 1 =? 2. (* false : bool *)
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Compute 1 =? 2. (* false : bool *)
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(* Locate is useful for finding the origin and definition of notations *)
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Locate "_ =? _". (* Z.eqb x y : Z_scope *)
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@ -199,10 +207,10 @@ Compute 1 + 7. (* 8 : nat *)
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(* Scopes can also be opened inline with the shorthand % *)
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Compute (3 * -7)%Z. (* -21%Z : Z *)
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(* Coq declares by default the following interpretation scopes: core_scope, type_scope,
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function_scope, nat_scope, bool_scope, list_scope, int_scope, uint_scope. You may also
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want the numerical scopes Z_scope (integers) and Q_scope (fractions) held in the ZArith
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and QArith module respectively. *)
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(* Coq declares by default the following interpretation scopes: core_scope,
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type_scope, function_scope, nat_scope, bool_scope, list_scope, int_scope,
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uint_scope. You may also want the numerical scopes Z_scope (integers) and
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Q_scope (fractions) held in the ZArith and QArith module respectively. *)
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(* You can print the contents of scopes *)
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Print Scope nat_scope.
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@ -230,17 +238,19 @@ Bound to classes nat Nat.t
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"x * y" := Init.Nat.mul x y
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*)
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(* Coq has exact fractions available as the type Q in the QArith module.
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Floating point numbers and real numbers are also available but are a more advanced
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topic, as proving properties about them is rather tricky. *)
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(* Coq has exact fractions available as the type Q in the QArith module.
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Floating point numbers and real numbers are also available but are a more
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advanced topic, as proving properties about them is rather tricky. *)
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Require Import QArith.
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Open Scope Q_scope.
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Compute 1. (* 1 : Q *)
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Compute 2. (* 2 : nat *) (* only 1 and 0 are interpreted as fractions by Q_scope *)
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(* Only 1 and 0 are interpreted as fractions by Q_scope *)
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Compute 2. (* 2 : nat *)
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Compute (2 # 3). (* The fraction 2/3 *)
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Compute (1 # 3) ?= (2 # 6). (* Eq : comparison *)
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Compute (1 # 3) ?= (2 # 6). (* Eq : comparison *)
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Close Scope Q_scope.
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Compute ( (2 # 3) / (1 # 5) )%Q. (* 10 # 3 : Q *)
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@ -279,40 +289,43 @@ Definition my_fst2 {A B : Type} (x : A * B) : A := let (a,b) := x in
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(*** Lists ***)
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(* Lists are built by using cons and nil or by using notation available in list_scope. *)
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(* Lists are built by using cons and nil or by using notation available in
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list_scope. *)
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Compute cons 1 (cons 2 (cons 3 nil)). (* (1 :: 2 :: 3 :: nil)%list : list nat *)
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Compute (1 :: 2 :: 3 :: nil)%list.
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Compute (1 :: 2 :: 3 :: nil)%list.
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(* There is also list notation available in the ListNotations modules *)
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Require Import List.
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Import ListNotations.
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Import ListNotations.
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Compute [1 ; 2 ; 3]. (* [1; 2; 3] : list nat *)
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(*
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There are a large number of list manipulation functions available, including:
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(* There is a large number of list manipulation functions available,
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including:
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• length
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• head : first element (with default)
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• head : first element (with default)
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• tail : all but first element
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• app : appending
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• rev : reverse
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• nth : accessing n-th element (with default)
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• map : applying a function
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• flat_map : applying a function returning lists
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• flat_map : applying a function returning lists
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• fold_left : iterator (from head to tail)
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• fold_right : iterator (from tail to head)
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• fold_right : iterator (from tail to head)
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*)
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Definition my_list : list nat := [47; 18; 34].
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Compute List.length my_list. (* 3 : nat *)
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(* All functions in coq must be total, so indexing requires a default value *)
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Compute List.nth 1 my_list 0. (* 18 : nat *)
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Compute List.nth 1 my_list 0. (* 18 : nat *)
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Compute List.map (fun x => x * 2) my_list. (* [94; 36; 68] : list nat *)
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Compute List.filter (fun x => Nat.eqb (Nat.modulo x 2) 0) my_list. (* [18; 34] : list nat *)
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Compute (my_list ++ my_list)%list. (* [47; 18; 34; 47; 18; 34] : list nat *)
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Compute List.filter (fun x => Nat.eqb (Nat.modulo x 2) 0) my_list.
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(* [18; 34] : list nat *)
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Compute (my_list ++ my_list)%list. (* [47; 18; 34; 47; 18; 34] : list nat *)
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(*** Strings ***)
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@ -342,16 +355,19 @@ Close Scope string_scope.
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• PArith : Basic positive integer arithmetic
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• NArith : Basic binary natural number arithmetic
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• ZArith : Basic relative integer arithmetic
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• Numbers : Various approaches to natural, integer and cyclic numbers (currently
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axiomatically and on top of 2^31 binary words)
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• Numbers : Various approaches to natural, integer and cyclic numbers
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(currently axiomatically and on top of 2^31 binary words)
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• Bool : Booleans (basic functions and results)
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• Lists : Monomorphic and polymorphic lists (basic functions and results),
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Streams (infinite sequences defined with co-inductive types)
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• Sets : Sets (classical, constructive, finite, infinite, power set, etc.)
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• FSets : Specification and implementations of finite sets and finite maps
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• FSets : Specification and implementations of finite sets and finite maps
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(by lists and by AVL trees)
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• Reals : Axiomatization of real numbers (classical, basic functions, integer part,
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fractional part, limit, derivative, Cauchy series, power series and results,...)
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• Reals : Axiomatization of real numbers (classical, basic functions,
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integer part, fractional part, limit, derivative, Cauchy series,
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power series and results,...)
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• Relations : Relations (definitions and basic results)
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• Sorting : Sorted list (basic definitions and heapsort correctness)
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• Strings : 8-bits characters and strings
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@ -360,18 +376,20 @@ Close Scope string_scope.
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(*** User-defined data types ***)
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(* Because Coq is dependently typed, defining type aliases is no different than defining
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an alias for a value. *)
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(* Because Coq is dependently typed, defining type aliases is no different
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than defining an alias for a value. *)
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Definition my_three : nat := 3.
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Definition my_nat : Type := nat.
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(* More interesting types can be defined using the Inductive vernacular. Simple enumeration
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can be defined like so *)
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(* More interesting types can be defined using the Inductive vernacular.
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Simple enumeration can be defined like so *)
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Inductive ml := OCaml | StandardML | Coq.
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Definition lang := Coq. (* Has type "ml". *)
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(* For more complicated types, you will need to specify the types of the constructors. *)
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(* For more complicated types, you will need to specify the types of the
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constructors. *)
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(* Type constructors don't need to be empty. *)
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Inductive my_number := plus_infinity
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@ -379,23 +397,28 @@ Inductive my_number := plus_infinity
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Compute nat_value 3. (* nat_value 3 : my_number *)
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(* Record syntax is sugar for tuple-like types. It defines named accessor functions for
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the components. Record types are defined with the notation {...} *)
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(* Record syntax is sugar for tuple-like types. It defines named accessor
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functions for the components. Record types are defined with the notation
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{...} *)
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Record Point2d (A : Set) := mkPoint2d { x2 : A ; y2 : A }.
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(* Record values are constructed with the notation {|...|} *)
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Definition mypoint : Point2d nat := {| x2 := 2 ; y2 := 3 |}.
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Compute x2 nat mypoint. (* 2 : nat *)
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Compute mypoint.(x2 nat). (* 2 : nat *)
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Compute mypoint.(x2 nat). (* 2 : nat *)
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(* Types can be parameterized, like in this type for "list of lists of
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anything". 'a can be substituted with any type. *)
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(* Types can be parameterized, like in this type for "list of lists
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of anything". 'a can be substituted with any type. *)
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Definition list_of_lists a := list (list a).
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Definition list_list_nat := list_of_lists nat.
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(* Types can also be recursive. Like in this type analogous to
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built-in list of naturals. *)
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Inductive my_nat_list := EmptyList | NatList : nat -> my_nat_list -> my_nat_list.
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Inductive my_nat_list :=
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EmptyList | NatList : nat -> my_nat_list -> my_nat_list.
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Compute NatList 1 EmptyList. (* NatList 1 EmptyList : my_nat_list *)
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(** Matching type constructors **)
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@ -427,31 +450,38 @@ Compute sum_list [1; 2; 3]. (* Evaluates to 6 *)
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(*** A Taste of Proving ***)
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(* Explaining the proof language is out of scope for this tutorial, but here is a taste to
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whet your appetite. Check the resources below for more. *)
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(* Explaining the proof language is out of scope for this tutorial, but here
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is a taste to whet your appetite. Check the resources below for more. *)
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(* A fascinating feature of dependently type based theorem provers is that the same
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primitive constructs underly the proof language as the programming features.
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For example, we can write and prove the proposition A and B implies A in raw Gallina *)
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(* A fascinating feature of dependently type based theorem provers is that
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the same primitive constructs underly the proof language as the
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programming features. For example, we can write and prove the
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proposition A and B implies A in raw Gallina *)
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Definition my_theorem : forall A B, A /\ B -> A := fun A B ab => match ab with
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| (conj a b) => a
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end.
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Definition my_theorem : forall A B, A /\ B -> A :=
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fun A B ab => match ab with
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| (conj a b) => a
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end.
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(* Or we can prove it using tactics. Tactics are a macro language to help
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build proof terms in a more natural style and automate away some
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drudgery. *)
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(* Or we can prove it using tactics. Tactics are a macro language to help build proof terms
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in a more natural style and automate away some drudgery. *)
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Theorem my_theorem2 : forall A B, A /\ B -> A.
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Proof.
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intros A B ab. destruct ab as [ a b ]. apply a.
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Qed.
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(* We can prove easily prove simple polynomial equalities using the automated tactic ring. *)
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(* We can prove easily prove simple polynomial equalities using the
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automated tactic ring. *)
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Require Import Ring.
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Require Import Arith.
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Theorem simple_poly : forall (x : nat), (x + 1) * (x + 2) = x * x + 3 * x + 2.
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Proof. intros. ring. Qed.
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(* Here we prove the closed form for the sum of all numbers 1 to n using induction *)
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(* Here we prove the closed form for the sum of all numbers 1 to n using
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induction *)
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Fixpoint sumn (n : nat) : nat :=
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match n with
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@ -465,8 +495,10 @@ Proof. intros n. induction n.
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- simpl. ring [IHn]. (* induction step *)
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Qed.
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```
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With this we have only scratched the surface of Coq. It is a massive ecosystem with many interesting and peculiar topics leading all the way up to modern research.
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With this we have only scratched the surface of Coq. It is a massive
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ecosystem with many interesting and peculiar topics leading all the way up
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to modern research.
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## Further reading
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