The Coq proof assistant : From graphical presentation to - PowerPoint PPT Presentation

Coq J.-F. Monin Case analysis Enumerated types General case The Coq proof assistant : From graphical presentation to principles and practice Coq syntax Simple inductive definitions Case analysis Reduction of a case analysis J.-F. Monin Functions Remarks Université Grenoble Alpes 2016 Lecture 2

Outline Coq J.-F. Monin Case analysis Enumerated types Case analysis General case From graphical Enumerated types presentation to Coq syntax General case Simple inductive definitions Case analysis Reduction of a From graphical presentation to Coq syntax case analysis Simple inductive definitions Functions Case analysis Remarks Reduction of a case analysis Functions Remarks

Outline Coq J.-F. Monin Case analysis Enumerated types Case analysis General case From graphical Enumerated types presentation to Coq syntax General case Simple inductive definitions Case analysis Reduction of a From graphical presentation to Coq syntax case analysis Simple inductive definitions Functions Case analysis Remarks Reduction of a case analysis Functions Remarks

Definition by cases on an enumerated type Coq J.-F. Monin Case analysis Enumerated types Question General case From graphical Give a color for each possible value in rgb presentation to Coq syntax Simple inductive definitions Rf Gf Bf Case analysis rgb rgb rgb Reduction of a case analysis Functions Example Remarks Rf maps to Red , Gf maps to Green , Bf maps to Blue ↓ r Red Green Blue rgb color color color case color

Definition by cases on an enumerated type Coq J.-F. Monin Case analysis Enumerated types General case Question From graphical presentation to Give a rgb for each possible value in rgb Coq syntax Simple inductive definitions Case analysis Rf Gf Bf Reduction of a rgb rgb rgb case analysis Functions Remarks Example Rf maps to Bf , Gf maps to Gf , Bf maps to Rf ↓ r Bf Gf Rf rgb rgb rgb rgb case rgb

Building block Coq J.-F. Monin Case analysis Enumerated types General case In this presentation, the order of contructors matters: From graphical Rf , Gf , Bf presentation to Coq syntax The case construct is driven by 2 parameters Simple inductive definitions Case analysis ◮ the type of the value to be analyzed Reduction of a each enumerated type (e.g. rgb ) comes automatically case analysis Functions with its case construct, which should actually be Remarks written, e.g. case rgb ◮ the type of the result ↓ ↓ ↓ ↓ ↓ A r x 1 x 2 x 3 Set rgb A A A case A

Correct version of previous examples Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax ↓ color r R G B Simple inductive definitions Case analysis Set rgb color co co Reduction of a case case analysis color Functions Remarks ↓ rgb r Bf Gf Rf Set rgb rgb rgb rgb case rgb

Definition by cases on a general inductive type Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax Simple inductive definitions Case analysis Reduction of a case analysis To be introduced below... Functions Remarks

Outline Coq J.-F. Monin Case analysis Enumerated types Case analysis General case From graphical Enumerated types presentation to Coq syntax General case Simple inductive definitions Case analysis Reduction of a From graphical presentation to Coq syntax case analysis Simple inductive definitions Functions Case analysis Remarks Reduction of a case analysis Functions Remarks

From graphical presentation to Coq syntax Red Orange Yellow Green color color color color Blue Indigo Violet color color color color And simultaneously Set Inductive color: Set := | Red : color | Orange : color | Yellow : color | Green : color | Blue : color | Indigo : color | Violet : color .

Coq syntax of tuple4 Making a 4-tuple of rgb ↓ ↓ ↓ ↓ x 1 x 2 x 3 x 4 rgb rgb rgb rgb Mk4rgb tuple4 Inductive tuple4 : Set := | Mk4rgb : forall x1: rgb, forall x2: rgb, forall x3: rgb, forall x4: rgb, tuple4

Coq syntax of tuple4, shorthand Coq J.-F. Monin Case analysis Enumerated types General case From graphical Making a 4-tuple of rgb presentation to Coq syntax Simple inductive definitions Case analysis ↓ ↓ ↓ ↓ x 1 x 2 x 3 x 4 Reduction of a case analysis rgb rgb rgb rgb Mk4rgb Functions tuple4 Remarks Inductive tuple4 : Set := | Mk4rgb : forall x1 x2 x3 x4: rgb, tuple4.

More general 4-tuples: several constructors Coq J.-F. Monin ↓ ↓ ↓ ↓ Case analysis x 1 x 2 x 3 x 4 Enumerated types rgb rgb rgb rgb General case Mk4rgb From graphical tuple4 presentation to Coq syntax Simple inductive definitions ↓ ↓ ↓ ↓ Case analysis x 1 x 2 x 3 x 4 Reduction of a color color color color case analysis Mk4co Functions tuple4 Remarks ↓ ↓ ↓ ↓ x 1 x 2 x 3 x 4 tuple4 tuple4 tuple4 tuple4 Mk4t4 tuple4 Inductive tuple4 : Set := | Mk4rgb : forall x1 x2 x3 x4: rgb, tuple4 | Mk4color : forall x1 x2 x3 x4: color, tuple4 | Mk4t4 : forall x1 x2 x3 x4: tuple4, tuple4 .

Pluging trees: the interactive way Coq J.-F. Monin Case analysis A concrete 4-tuple of rgb Enumerated types General case From graphical presentation to Gf Rf Gf Bf Coq syntax rgb rgb rgb rgb Simple inductive definitions Mk4rgb Case analysis tuple4 Reduction of a case analysis Functions Remarks Definition t1: tuple4. apply Mk4rgb. apply Gf. apply Rf. apply Gf. apply Bf. Defined .

Open trees Coq J.-F. Monin Case analysis ↓ ↓ Enumerated types Gf x 2 Rf x 4 General case rgb rgb rgb rgb From graphical Mk4rgb presentation to Coq syntax tuple4 Simple inductive definitions Case analysis In Coq Reduction of a case analysis Functions Section a_tuple_with_variable. Remarks Variable x2: rgb. Variable x4: rgb. Definition t4 etc. End a_tuple_with_variable.

Exercises Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax Simple inductive definitions Case analysis Write trees for examples of 4-tuples of 4-tuples using Reduction of a case analysis tuple4 . Functions Some of them, closed, some of them open Remarks E.g. �� R , Y , B , B � , � B , O , x 4 , R � , � x 7 , x 7 , x 7 , V � , � V , Y , O , R ��

Definition by cases on an enumerated type Coq J.-F. Monin Case analysis Rf maps to Red , Gf maps to Green , Bf maps to Blue Enumerated types General case From graphical presentation to ↓ Coq syntax color r R G B Simple inductive definitions Set rgb color co co Case analysis case Reduction of a color case analysis Functions Remarks Definition color_of_r: color. destruct r. apply Red. apply Green. apply Blue. Defined .

General case Coq J.-F. Monin Case analysis Enumerated types General case ↓ ↓ ↓ ↓ ↓ From graphical A r x 1 x 2 x 3 presentation to Set rgb A A A Coq syntax Simple inductive definitions case Case analysis A Reduction of a case analysis Functions Definition A_of_r: A. Remarks destruct r. apply x1. apply x2. apply x3. Defined .

Direct definition Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax Simple inductive definitions Case analysis Definition color_of_r : color := Reduction of a match r with case analysis | Rf => Red Functions | Gf => Green Remarks | Bf => Blue end.

Case analysis on a tree Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax Simple inductive definitions Case analysis Reduction of a case analysis See interactive session Functions Remarks

Outline Coq J.-F. Monin Case analysis Enumerated types Case analysis General case From graphical Enumerated types presentation to Coq syntax General case Simple inductive definitions Case analysis Reduction of a From graphical presentation to Coq syntax case analysis Simple inductive definitions Functions Case analysis Remarks Reduction of a case analysis Functions Remarks

Case analysis on a constant Coq J.-F. Monin Case analysis Enumerated types General case From graphical presentation to Coq syntax Simple inductive definitions Case analysis Definition color_of_Bf : color := Reduction of a match Bf with case analysis | Rf => Red Functions | Gf => Green Remarks | Bf => Blue end .

Reduction Coq J.-F. Monin Case analysis Enumerated types General case From graphical match Bf with presentation to Coq syntax | Rf => Red Simple inductive definitions Case analysis | Gf => Green Reduction of a | Bf => Blue case analysis Functions end . Remarks Reduces to Blue

Outline Coq J.-F. Monin Case analysis Enumerated types Case analysis General case From graphical Enumerated types presentation to Coq syntax General case Simple inductive definitions Case analysis Reduction of a From graphical presentation to Coq syntax case analysis Simple inductive definitions Functions Case analysis Remarks Reduction of a case analysis Functions Remarks