fixed merge on Lists

courses/tspl/2019/Assignment1.lagda.md

@@ -20,9 +20,13 @@ You don't need to do all of these, but should attempt at least a few.
 Exercises labelled "(practice)" are included for those who want extra practice.
 
 Submit your homework using the "submit" command.
+```bash
+submit tspl cw1 Assignment1.lagda.md
+```
 Please ensure your files execute correctly under Agda!
 
 
+
 ## Good Scholarly Practice.
 
 Please remember the University requirement as

courses/tspl/2019/Assignment2.lagda.md

@@ -20,6 +20,9 @@ You don't need to do all of these, but should attempt at least a few.
 Exercises labelled "(practice)" are included for those who want extra practice.
 
 Submit your homework using the "submit" command.
+```bash
+submit tspl cw2 Assignment2.lagda.md
+```
 Please ensure your files execute correctly under Agda!
 
 

courses/tspl/2019/Assignment3.lagda.md

@@ -0,0 +1,524 @@
+---
+title     : "Assignment3: TSPL Assignment 3"
+layout    : page
+permalink : /TSPL/2019/Assignment3/
+---
+
+```
+module Assignment3 where
+```
+
+## YOUR NAME AND EMAIL GOES HERE
+
+## Introduction
+
+You must do _all_ the exercises labelled "(recommended)".
+
+Exercises labelled "(stretch)" are there to provide an extra challenge.
+You don't need to do all of these, but should attempt at least a few.
+
+Exercises labelled "(practice)" are included for those who want extra practice.
+
+Submit your homework using the "submit" command.
+Please ensure your files execute correctly under Agda!
+
+
+## Good Scholarly Practice.
+
+Please remember the University requirement as
+regards all assessed work. Details about this can be found at:
+
+> [http://web.inf.ed.ac.uk/infweb/admin/policies/academic-misconduct](http://web.inf.ed.ac.uk/infweb/admin/policies/academic-misconduct)
+
+Furthermore, you are required to take reasonable measures to protect
+your assessed work from unauthorised access. For example, if you put
+any such work on a public repository then you must set access
+permissions appropriately (generally permitting access only to
+yourself, or your group in the case of group practicals).
+
+
+## Imports
+
+```
+import Relation.Binary.PropositionalEquality as Eq
+open Eq using (_≡_; refl; cong; sym)
+open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _≡⟨_⟩_; _∎)
+open import Data.Bool.Base using (Bool; true; false; T; _∧_; _∨_; not)
+open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
+open import Data.Nat.Properties using
+  (+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ)
+open import Relation.Nullary using (¬_; Dec; yes; no)
+open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
+open import Data.Empty using (⊥; ⊥-elim)
+open import Function using (_∘_)
+open import Algebra.Structures using (IsMonoid)
+open import Level using (Level)
+open import Relation.Unary using (Decidable)
+open import plfa.part1.Relations using (_<_; z<s; s<s)
+open import plfa.part1.Isomorphism using (_≃_; ≃-sym; ≃-trans; _≲_; extensionality)
+open plfa.part1.Isomorphism.≃-Reasoning
+open import plfa.part1.Lists using (List; []; _∷_; [_]; [_,_]; [_,_,_]; [_,_,_,_];
+  _++_; reverse; map; foldr; sum; All; Any; here; there; _∈_)
+open import plfa.part2.Lambda hiding (ƛ′_⇒_; case′_[zero⇒_|suc_⇒_]; μ′_⇒_; plus′)
+open import plfa.part2.Properties hiding (value?; unstuck; preserves; wttdgs)
+```
+
+
+## Lists
+
+#### Exercise `reverse-++-distrib` (recommended)
+
+Show that the reverse of one list appended to another is the
+reverse of the second appended to the reverse of the first:
+
+    reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
+
+
+#### Exercise `reverse-involutive` (recommended)
+
+A function is an _involution_ if when applied twice it acts
+as the identity function.  Show that reverse is an involution:
+
+    reverse (reverse xs) ≡ xs
+
+
+#### Exercise `map-compose` (practice)
+
+Prove that the map of a composition is equal to the composition of two maps:
+
+    map (g ∘ f) ≡ map g ∘ map f
+
+The last step of the proof requires extensionality.
+
+#### Exercise `map-++-distribute` (practice)
+
+Prove the following relationship between map and append:
+
+   map f (xs ++ ys) ≡ map f xs ++ map f ys
+
+#### Exercise `map-Tree` (practice)
+
+Define a type of trees with leaves of type `A` and internal
+nodes of type `B`:
+```
+data Tree (A B : Set) : Set where
+  leaf : A → Tree A B
+  node : Tree A B → B → Tree A B → Tree A B
+```
+Define a suitable map operator over trees:
+```
+postulate
+  map-Tree : ∀ {A B C D : Set}
+    → (A → C) → (B → D) → Tree A B → Tree C D
+```
+
+
+#### Exercise `product` (recommended)
+
+Use fold to define a function to find the product of a list of numbers.
+For example:
+
+    product [ 1 , 2 , 3 , 4 ] ≡ 24
+
+```
+-- Your code goes here
+```
+
+#### Exercise `foldr-++` (recommended)
+
+Show that fold and append are related as follows:
+```
+postulate
+  foldr-++ : ∀ {A B : Set} (_⊗_ : A → B → B) (e : B) (xs ys : List A) →
+    foldr _⊗_ e (xs ++ ys) ≡ foldr _⊗_ (foldr _⊗_ e ys) xs
+```
+
+
+#### Exercise `map-is-foldr` (practice)
+
+Show that map can be defined using fold:
+```
+postulate
+  map-is-foldr : ∀ {A B : Set} {f : A → B} →
+    map f ≡ foldr (λ x xs → f x ∷ xs) []
+```
+This requires extensionality.
+
+#### Exercise `fold-Tree` (practice)
+
+Define a suitable fold function for the type of trees given earlier:
+```
+postulate
+  fold-Tree : ∀ {A B C : Set}
+    → (A → C) → (C → B → C → C) → Tree A B → C
+```
+
+```
+-- Your code goes here
+```
+
+#### Exercise `map-is-fold-Tree` (practice)
+
+Demonstrate an analogue of `map-is-foldr` for the type of trees.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `sum-downFrom` (stretch)
+
+Define a function that counts down as follows:
+```
+downFrom : ℕ → List ℕ
+downFrom zero     =  []
+downFrom (suc n)  =  n ∷ downFrom n
+```
+For example:
+```
+_ : downFrom 3 ≡ [ 2 , 1 , 0 ]
+_ = refl
+```
+Prove that the sum of the numbers `(n - 1) + ⋯ + 0` is
+equal to `n * (n ∸ 1) / 2`:
+```
+postulate
+  sum-downFrom : ∀ (n : ℕ)
+    → sum (downFrom n) * 2 ≡ n * (n ∸ 1)
+```
+
+
+#### Exercise `foldl` (practice)
+
+Define a function `foldl` which is analogous to `foldr`, but where
+operations associate to the left rather than the right.  For example:
+
+    foldr _⊗_ e [ x , y , z ]  =  x ⊗ (y ⊗ (z ⊗ e))
+    foldl _⊗_ e [ x , y , z ]  =  ((e ⊗ x) ⊗ y) ⊗ z
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `foldr-monoid-foldl` (practice)
+
+Show that if `_⊗_` and `e` form a monoid, then `foldr _⊗_ e` and
+`foldl _⊗_ e` always compute the same result.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `Any-++-⇔` (recommended)
+
+Prove a result similar to `All-++-⇔`, but with `Any` in place of `All`, and a suitable
+replacement for `_×_`.  As a consequence, demonstrate an equivalence relating
+`_∈_` and `_++_`.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `All-++-≃` (stretch)
+
+Show that the equivalence `All-++-⇔` can be extended to an isomorphism.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `¬Any≃All¬` (recommended)
+
+Show that `Any` and `All` satisfy a version of De Morgan's Law:
+
+    (¬_ ∘ Any P) xs ≃ All (¬_ ∘ P) xs
+
+(Can you see why it is important that here `_∘_` is generalised
+to arbitrary levels, as described in the section on
+[universe polymorphism]({{ site.baseurl }}/Equality/#unipoly)?)
+
+Do we also have the following?
+
+    (¬_ ∘ All P) xs ≃ Any (¬_ ∘ P) xs
+
+If so, prove; if not, explain why.
+
+
+#### Exercise `All-∀` (practice)
+
+Show that `All P xs` is isomorphic to `∀ {x} → x ∈ xs → P x`.
+
+```
+-- You code goes here
+```
+
+
+#### Exercise `Any-∃` (practice)
+
+Show that `Any P xs` is isomorphic to `∃[ x ] (x ∈ xs × P x)`.
+
+```
+-- You code goes here
+```
+
+
+#### Exercise `any?` (stretch)
+
+Just as `All` has analogues `all` and `All?` which determine whether a
+predicate holds for every element of a list, so does `Any` have
+analogues `any` and `Any?` which determine whether a predicate holds
+for some element of a list.  Give their definitions.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `filter?` (stretch)
+
+Define the following variant of the traditional `filter` function on lists,
+which given a decidable predicate and a list returns all elements of the
+list satisfying the predicate:
+```
+postulate
+  filter? : ∀ {A : Set} {P : A → Set}
+    → (P? : Decidable P) → List A → ∃[ ys ]( All P ys )
+```
+
+
+
+## Lambda
+
+#### Exercise `mul` (recommended)
+
+Write out the definition of a lambda term that multiplies
+two natural numbers.  Your definition may use `plus` as
+defined earlier.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `mulᶜ` (practice)
+
+Write out the definition of a lambda term that multiplies
+two natural numbers represented as Church numerals. Your
+definition may use `plusᶜ` as defined earlier (or may not
+— there are nice definitions both ways).
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `primed` (stretch) {#primed}
+
+Some people find it annoying to write `` ` "x" `` instead of `x`.
+We can make examples with lambda terms slightly easier to write
+by adding the following definitions:
+```
+ƛ′_⇒_ : Term → Term → Term
+ƛ′ (` x) ⇒ N  =  ƛ x ⇒ N
+ƛ′ _ ⇒ _      =  ⊥-elim impossible
+  where postulate impossible : ⊥
+
+case′_[zero⇒_|suc_⇒_] : Term → Term → Term → Term → Term
+case′ L [zero⇒ M |suc (` x) ⇒ N ]  =  case L [zero⇒ M |suc x ⇒ N ]
+case′ _ [zero⇒ _ |suc _ ⇒ _ ]      =  ⊥-elim impossible
+  where postulate impossible : ⊥
+
+μ′_⇒_ : Term → Term → Term
+μ′ (` x) ⇒ N  =  μ x ⇒ N
+μ′ _ ⇒ _      =  ⊥-elim impossible
+  where postulate impossible : ⊥
+```
+We intend to apply the function only when the first term is a variable, which we
+indicate by postulating a term `impossible` of the empty type `⊥`.  If we use
+C-c C-n to normalise the term
+
+  ƛ′ two ⇒ two
+
+Agda will return an answer warning us that the impossible has occurred:
+
+  ⊥-elim (plfa.part2.Lambda.impossible (`` `suc (`suc `zero)) (`suc (`suc `zero)) ``)
+
+While postulating the impossible is a useful technique, it must be
+used with care, since such postulation could allow us to provide
+evidence of _any_ proposition whatsoever, regardless of its truth.
+
+The definition of `plus` can now be written as follows:
+```
+plus′ : Term
+plus′ = μ′ + ⇒ ƛ′ m ⇒ ƛ′ n ⇒
+          case′ m
+            [zero⇒ n
+            |suc m ⇒ `suc (+ · m · n) ]
+  where
+  +  =  ` "+"
+  m  =  ` "m"
+  n  =  ` "n"
+```
+Write out the definition of multiplication in the same style.
+
+
+#### Exercise `_[_:=_]′` (stretch)
+
+The definition of substitution above has three clauses (`ƛ`, `case`,
+and `μ`) that invoke a `with` clause to deal with bound variables.
+Rewrite the definition to factor the common part of these three
+clauses into a single function, defined by mutual recursion with
+substitution.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `—↠≲—↠′` (practice)
+
+Show that the first notion of reflexive and transitive closure
+above embeds into the second. Why are they not isomorphic?
+
+```
+-- Your code goes here
+```
+
+#### Exercise `plus-example` (practice)
+
+Write out the reduction sequence demonstrating that one plus one is two.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `Context-≃` (practice)
+
+Show that `Context` is isomorphic to `List (Id × Type)`.
+For instance, the isomorphism relates the context
+
+    ∅ , "s" ⦂ `ℕ ⇒ `ℕ , "z" ⦂ `ℕ
+
+to the list
+
+    [ ⟨ "z" , `ℕ ⟩ , ⟨ "s" , `ℕ ⇒ `ℕ ⟩ ]
+
+```
+-- Your code goes here
+```
+
+#### Exercise `mul-type` (recommended)
+
+Using the term `mul` you defined earlier, write out the derivation
+showing that it is well typed.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `mulᶜ-type` (practice)
+
+Using the term `mulᶜ` you defined earlier, write out the derivation
+showing that it is well typed.
+
+```
+-- Your code goes here
+```
+
+
+
+## Properties
+
+#### Exercise `Progress-≃` (practice)
+
+Show that `Progress M` is isomorphic to `Value M ⊎ ∃[ N ](M —→ N)`.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `progress′` (practice)
+
+Write out the proof of `progress′` in full, and compare it to the
+proof of `progress` above.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `value?` (practice)
+
+Combine `progress` and `—→¬V` to write a program that decides
+whether a well-typed term is a value:
+```
+postulate
+  value? : ∀ {A M} → ∅ ⊢ M ⦂ A → Dec (Value M)
+```
+
+#### Exercise `subst′` (stretch)
+
+Rewrite `subst` to work with the modified definition `_[_:=_]′`
+from the exercise in the previous chapter.  As before, this
+should factor dealing with bound variables into a single function,
+defined by mutual recursion with the proof that substitution
+preserves types.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `mul-eval` (recommended)
+
+Using the evaluator, confirm that two times two is four.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise: `progress-preservation` (practice)
+
+Without peeking at their statements above, write down the progress
+and preservation theorems for the simply typed lambda-calculus.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `subject_expansion` (practice)
+
+We say that `M` _reduces_ to `N` if `M —→ N`,
+but we can also describe the same situation by saying
+that `N` _expands_ to `M`.
+The preservation property is sometimes called _subject reduction_.
+Its opposite is _subject expansion_, which holds if
+`M —→ N` and `∅ ⊢ N ⦂ A` imply `∅ ⊢ M ⦂ A`.
+Find two counter-examples to subject expansion, one
+with case expressions and one not involving case expressions.
+
+```
+-- Your code goes here
+```
+
+
+#### Exercise `stuck` (practice)
+
+Give an example of an ill-typed term that does get stuck.
+
+```
+-- Your code goes here
+```
+
+#### Exercise `unstuck` (recommended)
+
+Provide proofs of the three postulates, `unstuck`, `preserves`, and `wttdgs` above.
+
+```
+-- Your code goes here
+```
+

courses/tspl/2019/tspl2019.md

@@ -13,7 +13,7 @@ permalink : /TSPL/2019/
 
 ## Lectures
 
-Lectures take place Monday, Wednesday, and Friday in AT 5.07. (Room provisional.)
+Lectures take place Monday, Wednesday, and Friday in AT 5.07.
 * **10.00--10.50am** Lecture
 * **11.10--12.00noon** Tutorial
 
@@ -47,48 +47,48 @@ Lectures take place Monday, Wednesday, and Friday in AT 5.07. (Room provisional.
   <td>4</td>
   <td><b>7 Oct</b> <a href="{{ site.baseurl }}/Quantifiers/">Quantifiers</a></td>
   <td><b>9 Oct</b> <a href="{{ site.baseurl }}/Decidable/">Decidable</a></td>
-  <td><b>11 Oct</b> (tutorial only)</td>
+  <td><b>11 Oct</b> (tutorial only, 10.00&ndash;10.50)</td>
  </tr>
  <tr>
   <td>5</td>
   <td><b>14 Oct</b> <a href="{{ site.baseurl }}/Lists/">Lists</a></td>
-  <td><b>16 Oct</b> <!-- (tutorial only) --></td>
+  <td><b>16 Oct</b> <a href="{{ site.baseurl }}/Lambda/">Lambda</a></td>
-  <td><b>18 Oct</b> <!-- <a href="{{ site.baseurl }}/Lists/">Lists</a> --></td>
+  <td><b>18 Oct</b> <a href="{{ site.baseurl }}/Properties/">Properties</a></td>
  </tr>
  <tr>
   <td>6</td>
-  <td><b>21 Oct</b> <!-- <a href="{{ site.baseurl }}/Lambda/">Lambda</a> --></td>
+  <td><b>21 Oct</b> <a href="{{ site.baseurl }}/DeBruijn/">DeBruijn</a></td>
-  <td><b>23 Oct</b> <!-- (no class) --></td>
+  <td><b>23 Oct</b> <a href="{{ site.baseurl }}/More/">More</a></td>
-  <td><b>25 Oct</b> <!-- <a href="{{ site.baseurl }}/Properties/">Properties</a> --></td>
+  <td><b>25 Oct</b> <a href="{{ site.baseurl }}/Inference/">Inference</a></td>
  </tr>
  <tr>
   <td>7</td>
-  <td><b>28 Oct</b> <!-- <a href="{{ site.baseurl }}/DeBruijn/">DeBruijn</a> --></td>
+  <td><b>28 Oct</b> <a href="{{ site.baseurl }}/Untyped/">Untyped</a></td>
-  <td><b>30 Oct</b> <!-- <a href="{{ site.baseurl }}/More/">More</a> --></td>
+  <td><b>30 Oct</b> (to be decided) </td>
-  <td><b>1 Nov</b> <!-- <a href="{{ site.baseurl }}/Inference/">Inference</a> --></td>
+  <td><b>1 Nov</b>  (no class) </td>
  </tr>
  <tr>
   <td>8</td>
-  <td><b>4 Nov</b> <!-- (no class) --></td>
+  <td><b>4 Nov</b> (no class) </td>
-  <td><b>6 Nov</b> <!-- (tutorial only) --></td>
+  <td><b>6 Nov</b> (tutorial only) </td>
-  <td><b>8 Nov</b> <!-- <a href="{{ site.baseurl }}/Untyped/">Untyped</a> --></td>
+  <td><b>8 Nov</b> (no class) </td>
  </tr>
  <tr>
   <td>9</td>
-  <td><b>11 Nov</b> <!-- (no class) --></td>
+  <td><b>11 Nov</b> (no class) </td>
-  <td><b>13 Nov</b> <!-- (tutorial only) --></td>
+  <td><b>13 Nov</b> (tutorial only) </td>
-  <td><b>15 Nov</b> <!-- (no class) --></td>
+  <td><b>15 Nov</b> (no class) </td>
  </tr>
  <tr>
   <td>10</td>
-  <td><b>18 Nov</b> <!-- (no class) --></td>
+  <td><b>18 Nov</b> (no class) </td>
-  <td><b>20 Nov</b> <!-- Propositions as Types --></td>
+  <td><b>20 Nov</b> Propositions as Types </td>
-  <td><b>22 Nov</b> <!-- (no class) --></td>
+  <td><b>22 Nov</b> (no class) </td>
  </tr>
  <tr>
   <td>11</td>
-  <td><b>25 Nov</b> <!-- (no class) --></td>
+  <td><b>25 Nov</b> (no class) </td>
-  <td><b>27 Nov</b> <!-- Quantitative (Wen)--></td>
+  <td><b>27 Nov</b> Quantitative (Wen) </td>
   <td><b>29 Nov</b> (mock exam) </td> 
  </tr>
 </table>
@@ -119,7 +119,7 @@ For instructions on how to set up Agda for PLFA see [Getting Started]({{ site.ba
 
 * [Assignment 1]({{ site.baseurl }}/TSPL/2019/Assignment1/) cw1 due 4pm Thursday 3 October (Week 3)
 * [Assignment 2]({{ site.baseurl }}/TSPL/2019/Assignment2/) cw2 due 4pm Thursday 17 October (Week 5)
-* Assignment 3 <!-- [Assignment 3]({{ site.baseurl }}/TSPL/2019/Assignment3/) --> cw3 due 4pm Thursday 31 October (Week 7)
+* [Assignment 3]({{ site.baseurl }}/TSPL/2019/Assignment3/) cw3 due 4pm Thursday 31 October (Week 7)
 * Assignment 4 <!-- [Assignment 4]({{ site.baseurl }}/TSPL/2019/Assignment4/) --> cw4 due 4pm Thursday 14 November (Week 9)
 * Assignment 5 <!-- [Assignment 5]({{ site.baseurl }}/courses/tspl/2010/Mock1.pdf) --> cw5 due 4pm Thursday 21 November (Week 10)
   <!-- <br />

exercise.sh

@@ -1,14 +0,0 @@
-#!/bin/bash
-
-# Script to extract exercises from PLFA chapters, e.g., `src/plfa/part1/Naturals.lagda.md`.
-# Usage:
-#
-#   ./exercise.sh [SOURCE] [TARGET]
-
-SRC="$1"
-shift
-
-DST="$1"
-shift
-
-awk '/^#/{flag=0} /^#### Exercise/{flag=1} flag' "$SRC" > "$DST"

extra/extra/Lists.lagda.md

@@ -0,0 +1,18 @@
+#### Exercise `Any-∃` (practice)
+
+Show that `Any P xs` is isomorphic to `∃[ x ] (x ∈ xs × P x)`.
+
+```
+Any-∃ : ∀ {A : Set} {P : A → Set} {xs : List A} → Any P xs ≃ ∃[ x ] (x ∈ xs × P x)
+Any-∃ {A} {P} {xs} = record
+  { to = to
+  ; from = {!!}
+  ; from∘to = {!!}
+  ; to∘from = {!!} }
+  where
+  to : ∀ {A : Set} {P : A → Set} {xs : List A} → Any P xs → ∃[ x ] (x ∈ xs × P x)
+  to (here {x = x} px) = ⟨ x , ⟨ here refl , px ⟩ ⟩
+  to (there anyp) with to anyp
+  ...                | ⟨ x , ⟨ x∈xs , px ⟩ ⟩ = ⟨ x , ⟨ there x∈xs , px ⟩ ⟩
+
+```

extra/extra/Rec1.agda

@@ -0,0 +1,7 @@
+module Rec1 where
+
+  import Rec2
+
+  y : ℕ
+  y = x
+

extra/extra/Rec2.agda

@@ -0,0 +1,10 @@
+module Rec2 where
+
+  open import Data.Nat
+  open import Rec1
+
+  x : ℕ
+  x = 42
+
+  z : ℕ
+  z = y

make_assignment.sh

@@ -0,0 +1,91 @@
+#!/bin/bash
+
+# This script can be used to automatically generate assignment files from PLFA source files.
+# It takes a course abbreviation, e.g. TSPL, a year, and the number of the assignment.
+# At the moment, it outputs the University of Edinburgh guidelines for good scholarly practice,
+# making it somewhat specific to courses run there, but the header should be easy to edit.
+#
+# Usage:
+#
+#   ./make_assignment.sh [COURSE_NAME] [COURSE_YEAR] [ASSIGNMENT_NUMBER] [PLFA_SOURCE_FILE...]
+
+COURSE="$1"
+shift
+
+YEAR="$1"
+shift
+
+NUM="$1"
+shift
+
+## Make assignment header
+
+cat <<-EOF
+---
+title     : "Assignment$NUM: $COURSE Assignment $NUM"
+layout    : page
+permalink : /$COURSE/$YEAR/Assignment$NUM/
+---
+
+\`\`\`
+module Assignment$NUM where
+\`\`\`
+
+## YOUR NAME AND EMAIL GOES HERE
+
+## Introduction
+
+You must do _all_ the exercises labelled "(recommended)".
+
+Exercises labelled "(stretch)" are there to provide an extra challenge.
+You don't need to do all of these, but should attempt at least a few.
+
+Exercises labelled "(practice)" are included for those who want extra practice.
+
+Submit your homework using the "submit" command.
+Please ensure your files execute correctly under Agda!
+
+
+## Good Scholarly Practice.
+
+Please remember the University requirement as
+regards all assessed work. Details about this can be found at:
+
+> [http://web.inf.ed.ac.uk/infweb/admin/policies/academic-misconduct](http://web.inf.ed.ac.uk/infweb/admin/policies/academic-misconduct)
+
+Furthermore, you are required to take reasonable measures to protect
+your assessed work from unauthorised access. For example, if you put
+any such work on a public repository then you must set access
+permissions appropriately (generally permitting access only to
+yourself, or your group in the case of group practicals).
+
+EOF
+
+## Make import statements
+
+cat <<-EOF
+
+## Imports
+
+\`\`\`
+EOF
+for SRC in "$@"; do
+    AGDA_MODULE=$(eval "echo \"$SRC\" | sed -e \"s|src/||g; s|\\\.lagda\\\.md||g; s|/|\\\.|g;\"")
+    echo "open import $AGDA_MODULE"
+done
+cat <<-EOF
+\`\`\`
+
+EOF
+
+## Extract exercises
+
+for SRC in "$@"; do
+    NAME=$(basename "${SRC%.lagda.md}")
+    cat <<-EOF
+
+## $NAME
+
+EOF
+    awk '/^#/{flag=0} /^#### Exercise/{flag=1} flag' "$SRC"
+done

src/plfa/part1/Equality.lagda.md

@@ -688,6 +688,14 @@ Before the signature used `Set₁` as the type of a term that includes
 `Set`, whereas here the signature uses `Set (lsuc ℓ)` as the type of a
 term that includes `Set ℓ`.
 
+Most other functions in the standard library are also generalised to
+arbitrary levels. For instance, here is the definition of composition.
+```
+_∘_ : ∀ {ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set ℓ₁} {B : Set ℓ₂} {C : Set ℓ₃}
+  → (B → C) → (A → B) → A → C
+(g ∘ f) x  =  g (f x)
+```
+
 Further information on levels can be found in the [Agda Wiki][wiki].
 
 [wiki]: http://wiki.portal.chalmers.se/agda/pmwiki.php?n=ReferenceManual.UniversePolymorphism

src/plfa/part1/Lists.lagda.md

@@ -351,21 +351,15 @@ list, and the sum of the numbers up to `n - 1` is `n * (n - 1) / 2`.
 
 Show that the reverse of one list appended to another is the
 reverse of the second appended to the reverse of the first:
-```
+
-postulate
+    reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
-  reverse-++-distrib : ∀ {A : Set} (xs ys : List A)
-    → reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
-```
 
 #### Exercise `reverse-involutive` (recommended)
 
 A function is an _involution_ if when applied twice it acts
 as the identity function.  Show that reverse is an involution:
-```
+
-postulate
+    reverse (reverse xs) ≡ xs
-  reverse-involutive : ∀ {A : Set} {xs : List A}
-    → reverse (reverse xs) ≡ xs
-```
 
 
 ## Faster reverse
@@ -950,32 +944,41 @@ Show that the equivalence `All-++-⇔` can be extended to an isomorphism.
 -- Your code goes here
 ```
 
-#### Exercise `¬Any≃All¬` (stretch)
+#### Exercise `¬Any≃All¬` (recommended)
-
-First generalise composition to arbitrary levels, using
-[universe polymorphism]({{ site.baseurl }}/Equality/#unipoly):
-```
-_∘′_ : ∀ {ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set ℓ₁} {B : Set ℓ₂} {C : Set ℓ₃}
-  → (B → C) → (A → B) → A → C
-(g ∘′ f) x  =  g (f x)
-```
 
 Show that `Any` and `All` satisfy a version of De Morgan's Law:
-```
+
-postulate
+    (¬_ ∘ Any P) xs ≃ All (¬_ ∘ P) xs
-  ¬Any≃All¬ : ∀ {A : Set} (P : A → Set) (xs : List A)
+
-    → (¬_ ∘′ Any P) xs ≃ All (¬_ ∘′ P) xs
+(Can you see why it is important that here `_∘_` is generalised
-```
+to arbitrary levels, as described in the section on
+[universe polymorphism]({{ site.baseurl }}/Equality/#unipoly)?)
 
 Do we also have the following?
-```
+
-postulate
+    (¬_ ∘ All P) xs ≃ Any (¬_ ∘ P) xs
-  ¬All≃Any¬ : ∀ {A : Set} (P : A → Set) (xs : List A)
+
-    → (¬_ ∘′ All P) xs ≃ Any (¬_ ∘′ P) xs
-```
 If so, prove; if not, explain why.
 
 
+#### Exercise `All-∀` (practice)
+
+Show that `All P xs` is isomorphic to `∀ {x} → x ∈ xs → P x`.
+
+```
+-- You code goes here
+```
+
+
+#### Exercise `Any-∃` (practice)
+
+Show that `Any P xs` is isomorphic to `∃[ x ] (x ∈ xs × P x)`.
+
+```
+-- You code goes here
+```
+
+
 ## Decidability of All
 
 If we consider a predicate as a function that yields a boolean,
@@ -1026,24 +1029,6 @@ for some element of a list.  Give their definitions.
 ```
 
 
-#### Exercise `All-∀` (practice)
-
-Show that `All P xs` is isomorphic to `∀ {x} → x ∈ xs → P x`.
-
-```
--- You code goes here
-```
-
-
-#### Exercise `Any-∃` (practice)
-
-Show that `Any P xs` is isomorphic to `∃[ x ∈ xs ] P x`.
-
-```
--- You code goes here
-```
-
-
 #### Exercise `filter?` (stretch)
 
 Define the following variant of the traditional `filter` function on lists,

src/plfa/part1/Quantifiers.lagda.md

@@ -27,16 +27,23 @@ open import plfa.part1.Isomorphism using (_≃_; extensionality)
 
 ## Universals
 
-We formalise universal quantification using the
+We formalise universal quantification using the dependent function
-dependent function type, which has appeared throughout this book.
+type, which has appeared throughout this book.  For instance, in
+Chapter Induction we showed addition is associative:
 
-Given a variable `x` of type `A` and a proposition `B x` which
+    +-assoc : ∀ (m n p : ℕ) → (m + n) + p ≡ m + (n + p)
-contains `x` as a free variable, the universally quantified
+
-proposition `∀ (x : A) → B x` holds if for every term `M` of type
+which asserts for all natural numbers `m`, `n`, and `p`
-`A` the proposition `B M` holds.  Here `B M` stands for
+that `(m + n) + p ≡ m + (n + p)` holds.  It is a dependent
-the proposition `B x` with each free occurrence of `x` replaced by
+function, which given values for `m`, `n`, and `p` returns
-`M`.  Variable `x` appears free in `B x` but bound in
+evidence for the corresponding equation.
-`∀ (x : A) → B x`.
+
+In general, given a variable `x` of type `A` and a proposition `B x`
+which contains `x` as a free variable, the universally quantified
+proposition `∀ (x : A) → B x` holds if for every term `M` of type `A`
+the proposition `B M` holds.  Here `B M` stands for the proposition
+`B x` with each free occurrence of `x` replaced by `M`.  Variable `x`
+appears free in `B x` but bound in `∀ (x : A) → B x`.
 
 Evidence that `∀ (x : A) → B x` holds is of the form
 
@@ -371,7 +378,7 @@ restated in this way.
 -- Your code goes here
 ```
 
-#### Exercise `∃-|-≤` (practice)
+#### Exercise `∃-+-≤` (practice)
 
 Show that `y ≤ z` holds if and only if there exists a `x` such that
 `x + y ≡ z`.