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Coq J.-F. Monin Polymorphism Lists
Coq J.-F. Monin Polymorphism Lists
The Coq proof assistant : principles and practice J.-F. Monin - - PowerPoint PPT Presentation
The Coq proof assistant : principles and practice J.-F. Monin - - PowerPoint PPT Presentation
Coq J.-F. Monin Polymorphism Lists The Coq proof assistant : principles and practice J.-F. Monin Universit Grenoble Alpes 2016 Lecture 7 Outline Coq J.-F. Monin Polymorphism Lists Polymorphism Outline Coq J.-F. Monin
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Coq J.-F. Monin Polymorphism Lists
Outline
Polymorphism Lists
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Coq J.-F. Monin Polymorphism Lists
Outline
Polymorphism Lists
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Coq J.-F. Monin Polymorphism Lists
Polymorphism
A type can be a parameter of a function
Example: the identity function Definition ide := fun (X: Type) => fun (x: X) => x. Definition ide (X: Type) (x: X) := x.
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Coq J.-F. Monin Polymorphism Lists
Implicit arguments
When using the identity function, the first argument can be automatically inferred from the second
Example
id nat 3 id 3
Local declaration
Definition id {X: Type} (x: X) := x.
Simplified application
id 3
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Coq J.-F. Monin Polymorphism Lists
Implicit arguments
When using the identity function, the first argument can be automatically inferred from the second
Example
id nat 3 id 3
Local declaration
Definition id {X: Type} (x: X) := x.
Simplified application
id 3
Recovering explicit application
@id nat id (X:=nat)
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Coq J.-F. Monin Polymorphism Lists
Implicit arguments
When using the identity function, the first argument can be automatically inferred from the second
Example
id nat 3 id 3
Local declaration
Definition id {X: Type} (x: X) := x.
Simplified application
id 3
Recovering explicit application
@id nat id (X:=nat)
Global declaration
Set Implicit Arguments.
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Coq J.-F. Monin Polymorphism Lists
Outline
Polymorphism Lists
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Coq J.-F. Monin Polymorphism Lists
Lists
Polymorphic inductive definition
Inductive list (X: Set) : Set := | nil : list X | cons : X -> list X -> list X.
On Type
Can be used in more situations (e.g., lists of predicates) Inductive list (X: Type) : Type := | nil : list X | cons : X -> list X -> list X.
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Coq J.-F. Monin Polymorphism Lists
Basic important properties
app : for appending two lists
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Coq J.-F. Monin Polymorphism Lists
Basic important properties
app : for appending two lists nil is neutral on the left and on the right for app
◮ left : by reflexivity ◮ right : by induction
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Coq J.-F. Monin Polymorphism Lists
Basic important properties
app : for appending two lists nil is neutral on the left and on the right for app
◮ left : by reflexivity ◮ right : by induction
app is associative
◮ app (app u v) w = app u (app v w)
just by induction on u
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Coq J.-F. Monin Polymorphism Lists