I need the full code explained and answered please ( PLEASE DO NOT DISTRIBUTE SOLUTIONS PUBLICLY ) Set Default Goal Selector Require Import Coq Strings Ascii Require Import Coq Lists List Import ListNotations Open Scope char scope Require Import Hw 5 Util ( ) ( Use yield def ) Theorem yield eq forall G A w , List In ( A , w ) ( Hw 5 Util grammar rules G ) Hw 5 Util Yield G A w Proof intros G A w H apply Hw 5 Util Yield def exists w split assumption reflexivity Qed ( Use yield def and app assoc to correct the parenthesis ) Theorem yield right forall G w 1 w 2 r w 3 w 4 , Hw 5 Util Yield G w 1 w 2 w 3 w 1 r w 4 w 2 r Hw 5 Util Yield G w 3 w 4 Proof Admitted ( Similar proof than yield right , but you should rewrite with app assoc after using yield def , not before ) Theorem yield left forall G w 1 w 2 l w 3 w 4 , Hw 5 Util Yield G w 1 w 2 w 3 l w 1 w 4 l w 2 Hw 5 Util Yield G w 3 w 4 Proof intros G w 1 w 2 r w 3 w 4 HYield H 1 H 2 rewrite H 1 , H 2 apply yield def in HYield Admitted ( Consider using one of the lemmas you have just proved When your goal is In ( C , ) ( grammar rules g 1 ) , simpl auto should solve it ) Theorem g 1 rule 1 Hw 5 Util Yield g 1 C C Proof simpl auto Qed Admitted ( Consider using one of the lemmas you have just proved When your goal is In ( C , ) ( grammar rules g 1 ) , simpl auto should solve it ) Theorem g 1 rule 2 Hw 5 Util Yield g 1 C C C Proof Admitted ( Consider using one of the lemmas you have just proved When your goal is In ( C , ) ( grammar rules g 1 ) , simpl auto should solve it ) Theorem g 1 rule 3 Hw 5 Util Yield g 1 C Proof Admitted ( The proof should proceed by inversion and then case analysis on string u ) Theorem yield inv start forall G w , Hw 5 Util Yield G Hw 5 Util grammar start G w In ( Hw 5 Util grammar start G , w ) ( Hw 5 Util grammar rules G ) Proof Admitted ( You will want to use yield inv start Recall that that List In simplifies to a series of disjunctions ) Theorem g 1 ex 1 Hw 5 Util Yield g 1 C Proof Admitted ( The idea is to use either yield left , yield right , or yield def ) Theorem g 1 step 1 Hw 5 Util Yield g 1 C C C C Proof Admitted ( The idea is to use either yield left , yield right , or yield def ) Theorem g 1 step 2 Hw 5 Util Yield g 1 C C C Proof Admitted ( The idea is to use either yield left , yield right , or yield def ) Theorem g 1 step 3 Hw 5 Util Yield g 1 C C Proof Admitted ( The idea is to use either yield left , yield right , or yield def ) Theorem g 1 step 4 Hw 5 Util Yield g 1 C Proof Admitted ( Use either derivation nil or derivation cons ) Theorem g 1 der 1 Hw 5 Util Derivation g 1 C Proof Admitted ( Use either derivation nil or derivation cons ) Theorem g 1 der 2 Hw 5 Util Derivation g 1 C C C Proof Admitted ( Use either derivation nil or derivation cons ) Theorem g 1 der 3 Hw 5 Util Derivation g 1 C C C C C Proof Admitted Theorem g 1 der 4 Hw 5 Util Derivation g 1 C C C C C C Proof Admitted Theorem g 1 der 5 Hw 5 Util Derivation g 1 C C C C C C C

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Question: I need the full code explained and answered please - ( * ##################################################### ### PLEASE DO NOT DISTRIBUTE SOLUTIONS PUBLICLY ### ##################################################### * ) Set

I need the full code explained and answered please

- (*

#####################################################

### PLEASE DO NOT DISTRIBUTE SOLUTIONS PUBLICLY ###

#####################################################

*)

Set Default Goal Selector

"! " .

Require Import Coq.Strings.Ascii.

Require Import Coq.Lists.List.

Import ListNotations.

Open Scope char

_

scope.

Require Import Hw

5

Util.

(* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - *)

(* *

Use

`

yield

_

def

` .

*)

Theorem yield

_

eq:

forall G A w

,

List.In

(

,

) (

5

Util.grammar

_

rules G

) - >

5

Util.Yield G

[

]

.

Proof.

intros G A w H

.

apply Hw

5

Util.Yield

_

def.

exists w

.

split;

[

assumption

|

reflexivity

] .

Qed.

(* *

Use

`

yield

_

def

`

and

`

app

_

assoc

`

to correct the parenthesis.

*)

Theorem yield

_

right:

forall G w

1

2

r w

3

4,

5

Util.Yield G w

1

2 - >

3 =

1 + +

- >

4 =

2 + +

- >

5

Util.Yield G w

3

4 .

Proof.

Admitted.

(* *

Similar proof than

`

yield

_

right

`,

but you should rewrite with

`

app

_

assoc

`

after using

`

yield

_

def

`,

not before.

*)

Theorem yield

_

left:

forall G w

1

2

l w

3

4,

5

Util.Yield G w

1

2 - >

3 =

+ +

1 - >

4 =

+ +

2 - >

5

Util.Yield G w

3

4 .

Proof.

intros G w

1

2

r w

3

4

HYield H

1

2 .

rewrite

< -

1, < -

2 .

apply yield

_

def in HYield.

Admitted.

(* *

Consider using one of the lemmas you have just proved.

When your goal is

`

("

", . . .) (

grammar

_

rules g

1) `, `

simpl; auto

`

should solve it

.

*)

Theorem g

1_

rule

_1

5

Util.Yield g

1 ["

"] [" {"

;

"

"

;

"} "] .

Proof.

simpl; auto.

Qed.

Admitted.

(* *

Consider using one of the lemmas you have just proved.

When your goal is

`

("

", . . .) (

grammar

_

rules g

1) `, `

simpl; auto

`

should solve it

.

*)

Theorem g

1_

rule

_2

5

Util.Yield g

1 ["

"] ["

"

;

"

"] .

Proof.

Admitted.

(* *

Consider using one of the lemmas you have just proved.

When your goal is

`

("

", . . .) (

grammar

_

rules g

1) `, `

simpl; auto

`

should solve it

.

*)

Theorem g

1_

rule

_3

5

Util.Yield g

1 ["

"] [] .

Proof.

Admitted.

(* *

The proof should proceed by inversion and then case analysis on

string

`

` .

*)

Theorem yield

_

inv

_

start:

forall G w

,

5

Util.Yield G

[

5

Util.grammar

_

start G

]

- >

(

5

Util.grammar

_

start G

,

) (

5

Util.grammar

_

rules G

) .

Proof.

Admitted.

(* *

You will want to use

`

yield

_

inv

_

start

` .

Recall that that

`

List

.

`

simplifies to a series of disjunctions.

*)

Theorem g

1_

1

~ Hw

5

Util.Yield g

1 ["

"] [" {"] .

Proof.

Admitted.

(* *

The idea is to use either:

`

yield

_

left

`, `

yield

_

right

`,

`

yield

_

def

` .

*)

Theorem g

1_

step

_1

5

Util.Yield g

1 ["

"

;

"

"] [" {"

;

"

"

;

"} "

;

"

"] .

Proof.

Admitted.

(* *

The idea is to use either:

`

yield

_

left

`, `

yield

_

right

`,

`

yield

_

def

` .

*)

Theorem g

1_

step

_2

5

Util.Yield g

1 [" {"

;

"

"

;

"} "

;

"

"] [" {"

;

"} "

;

"

"] .

Proof.

Admitted.

(* *

The idea is to use either:

`

yield

_

left

`, `

yield

_

right

`,

`

yield

_

def

` .

*)

Theorem g

1_

step

_3

5

Util.Yield g

1 [" {"

;

"} "

;

"

"] [" {"

;

"} "

;

" {"

;

"

"

;

"} "] .

Proof.

Admitted.

(* *

The idea is to use either:

`

yield

_

left

`, `

yield

_

right

`,

`

yield

_

def

` .

*)

Theorem g

1_

step

_4

5

Util.Yield g

1 [" {"

;

"} "

;

" {"

;

"

"

;

"} "] [" {"

;

"} "

;

" {"

;

"} "] .

Proof.

Admitted.

(* *

Use either

`

derivation

_

nil

`

`

derivation

_

cons

` .

*)

Theorem g

1_

der

_1

5

Util.Derivation g

1 [["

"]] .

Proof.

Admitted.

(* *

Use either

`

derivation

_

nil

`

`

derivation

_

cons

` .

*)

Theorem g

1_

der

_2

5

Util.Derivation g

1 [["

"

;

"

"]

;

["

"]] .

Proof.

Admitted.

(* *

Use either

`

derivation

_

nil

`

`

derivation

_

cons

` .

*)

Theorem g

1_

der

_3

5

Util.Derivation g

1 [[" {"

;

"

"

;

"} "

;

"

"]

;

["

"

;

"

"]

;

["

"]] .

Proof.

Admitted.

Theorem g

1_

der

_4

5

Util.Derivation g

1 [

[" {"

;

"} "

;

"

"]

;

[" {"

;

"

"

;

"} "

;

"

"]

;

["

"

;

"

"]

;

["

"]

] .

Proof.

Admitted.

Theorem g

1_

der

_5

5

Util.Derivation g

1 [

[" {"

;

"} "

;

" {"

;

"

"

;

"} "]

;

[" {"

;

"} "

;

"

"]

;

[" {"

;

"

"

;

"} "

;

"

"]

;

["

"

;

"

"]

;

["

"] <

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