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Fixed syntax for assigning anchors.

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Wen Kokke 2021-08-25 00:17:03 +01:00
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3
book/epub.mk
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@ -23,7 +23,7 @@ $(SITE_DIR)/plfa.epub: \
$(RAW_DIR)/epub.md $(EPUB_DIR)/epub.css $(RAW_DIR)/epub.xml $(FRANKENFONT) \ $(RAW_DIR)/epub.md $(EPUB_DIR)/epub.css $(RAW_DIR)/epub.xml $(FRANKENFONT) \
$(MD_FILES) $(EPUB_LUA_SCRIPTS) | setup-install-pandoc $(MD_FILES) $(EPUB_LUA_SCRIPTS) | setup-install-pandoc
@echo "Building EPUB" @echo "Building EPUB"
$(PANDOC) \ @$(PANDOC) \
--strip-comments \ --strip-comments \
--css=$(EPUB_DIR)/epub.css \ --css=$(EPUB_DIR)/epub.css \
--epub-embed-font=$(FRANKENFONT) \ --epub-embed-font=$(FRANKENFONT) \
@ -33,6 +33,7 @@ $(SITE_DIR)/plfa.epub: \
--lua-filter=$(EPUB_LUA_DIR)/remove-badges.lua -M badge-url=https://img.shields.io/badge/ \ --lua-filter=$(EPUB_LUA_DIR)/remove-badges.lua -M badge-url=https://img.shields.io/badge/ \
--lua-filter=$(EPUB_LUA_DIR)/epub-clean-html.lua \ --lua-filter=$(EPUB_LUA_DIR)/epub-clean-html.lua \
--lua-filter=$(EPUB_LUA_DIR)/single-file-links.lua \ --lua-filter=$(EPUB_LUA_DIR)/single-file-links.lua \
--lua-filter=$(EPUB_LUA_DIR)/single-file-identifiers.lua \
--standalone \ --standalone \
--toc --toc-depth=2 \ --toc --toc-depth=2 \
--epub-chapter-level=2 \ --epub-chapter-level=2 \

2
posts/2020-06-02-plfa-as-epub.md
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@ -4,6 +4,6 @@ title: "PLFA as EPUB"
It has been just over a year and a half since this feature was first requested in [#112][issue112]… Thanks to hard work by [Michael Reed][mreed20], and a little elbow grease on our part, it is finally here! [An EPUB version of PLFA!][EPUB] It has been just over a year and a half since this feature was first requested in [#112][issue112]… Thanks to hard work by [Michael Reed][mreed20], and a little elbow grease on our part, it is finally here! [An EPUB version of PLFA!][EPUB]
[EPUB]: https://plfa.github.io/out/epub/plfa.epub [EPUB]: https://plfa.github.io/epub/plfa.epub
[issue112]: https://github.com/plfa/plfa.github.io/issues/112 [issue112]: https://github.com/plfa/plfa.github.io/issues/112
[mreed20]: https://github.com/mreed20 [mreed20]: https://github.com/mreed20

4
posts/2021-08-24-plfa-as-pdf-and-epub.md
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@ -21,8 +21,8 @@ As always, bug reports and pull requests are very, very welcome!
[Altariarite]: https://github.com/Altariarite [Altariarite]: https://github.com/Altariarite
[mreed20]: https://github.com/mreed20 [mreed20]: https://github.com/mreed20
[PDF]: https://plfa.github.io/out/epub/plfa.pdf [PDF]: https://plfa.github.io/epub/plfa.pdf
[EPUB]: https://plfa.github.io/out/epub/plfa.epub [EPUB]: https://plfa.github.io/epub/plfa.epub
[issue577]: https://github.com/plfa/plfa.github.io/issues/577 [issue577]: https://github.com/plfa/plfa.github.io/issues/577
[issue578]: https://github.com/plfa/plfa.github.io/issues/578 [issue578]: https://github.com/plfa/plfa.github.io/issues/578

2
src/plfa/part1/Connectives.lagda.md
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@ -536,7 +536,7 @@ Show empty is the right identity of sums up to isomorphism.
-- Your code goes here -- Your code goes here
``` ```
## Implication is function {name=implication} ## Implication is function {#implication}
Given two propositions `A` and `B`, the implication `A → B` holds if Given two propositions `A` and `B`, the implication `A → B` holds if
whenever `A` holds then `B` must also hold. We formalise implication using whenever `A` holds then `B` must also hold. We formalise implication using

2
src/plfa/part1/Decidable.lagda.md
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@ -561,7 +561,7 @@ postulate
-- Your code goes here -- Your code goes here
``` ```
## Proof by reflection {name=proof-by-reflection} ## Proof by reflection {#proof-by-reflection}
Let's revisit our definition of monus from Let's revisit our definition of monus from
Chapter [Naturals](/Naturals/). Chapter [Naturals](/Naturals/).

4
src/plfa/part1/Equality.lagda.md
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@ -132,7 +132,7 @@ Again, a useful exercise is to carry out an interactive development,
checking how Agda's knowledge changes as each of the two arguments is checking how Agda's knowledge changes as each of the two arguments is
instantiated. instantiated.
## Congruence and substitution {name=cong} ## Congruence and substitution {#cong}
Equality satisfies _congruence_. If two terms are equal, Equality satisfies _congruence_. If two terms are equal,
they remain so after the same function is applied to both: they remain so after the same function is applied to both:
@ -635,7 +635,7 @@ Jesper Cockx, Dominique Devries, Andreas Nuyts, and Philip Wadler,
draft, 2017.) draft, 2017.)
## Universe polymorphism {name=unipoly} ## Universe polymorphism {#unipoly}
As we have seen, not every type belongs to `Set`, but instead every As we have seen, not every type belongs to `Set`, but instead every
type belongs somewhere in the hierarchy `Set₀`, `Set₁`, `Set₂`, and so on, type belongs somewhere in the hierarchy `Set₀`, `Set₁`, `Set₂`, and so on,

20
src/plfa/part1/Induction.lagda.md
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@ -71,7 +71,7 @@ distributes over another operator. A careful author will often call
out these properties---or their lack---for instance by pointing out out these properties---or their lack---for instance by pointing out
that a newly introduced operator is associative but not commutative. that a newly introduced operator is associative but not commutative.
#### Exercise `operators` (practice) {name=operators} #### Exercise `operators` (practice) {#operators}
Give another example of a pair of operators that have an identity Give another example of a pair of operators that have an identity
and are associative, commutative, and distribute over one another. and are associative, commutative, and distribute over one another.
@ -599,7 +599,7 @@ the main proposition first, and the equations required to do so
will suggest what lemmas to prove. will suggest what lemmas to prove.
## Our first corollary: rearranging {name=sections} ## Our first corollary: rearranging {#sections}
We can apply associativity to rearrange parentheses however we like. We can apply associativity to rearrange parentheses however we like.
Here is an example: Here is an example:
@ -695,7 +695,7 @@ judgments where the first number is less than _m_.
There is also a completely finite approach to generating the same equations, There is also a completely finite approach to generating the same equations,
which is left as an exercise for the reader. which is left as an exercise for the reader.
#### Exercise `finite-|-assoc` (stretch) {name=finite-plus-assoc} #### Exercise `finite-|-assoc` (stretch) {#finite-plus-assoc}
Write out what is known about associativity of addition on each of the Write out what is known about associativity of addition on each of the
first four days using a finite story of creation, as first four days using a finite story of creation, as
@ -857,7 +857,7 @@ typing `C-c C-r` will fill it in, completing the proof:
+-assoc′ (suc m) n p rewrite +-assoc′ m n p = refl +-assoc′ (suc m) n p rewrite +-assoc′ m n p = refl
#### Exercise `+-swap` (recommended) {name=plus-swap} #### Exercise `+-swap` (recommended) {#plus-swap}
Show Show
@ -872,7 +872,7 @@ is associative and commutative.
``` ```
#### Exercise `*-distrib-+` (recommended) {name=times-distrib-plus} #### Exercise `*-distrib-+` (recommended) {#times-distrib-plus}
Show multiplication distributes over addition, that is, Show multiplication distributes over addition, that is,
@ -885,7 +885,7 @@ for all naturals `m`, `n`, and `p`.
``` ```
#### Exercise `*-assoc` (recommended) {name=times-assoc} #### Exercise `*-assoc` (recommended) {#times-assoc}
Show multiplication is associative, that is, Show multiplication is associative, that is,
@ -898,7 +898,7 @@ for all naturals `m`, `n`, and `p`.
``` ```
#### Exercise `*-comm` (practice) {name=times-comm} #### Exercise `*-comm` (practice) {#times-comm}
Show multiplication is commutative, that is, Show multiplication is commutative, that is,
@ -912,7 +912,7 @@ you will need to formulate and prove suitable lemmas.
``` ```
#### Exercise `0∸n≡0` (practice) {name=zero-monus} #### Exercise `0∸n≡0` (practice) {#zero-monus}
Show Show
@ -925,7 +925,7 @@ for all naturals `n`. Did your proof require induction?
``` ```
#### Exercise `∸-|-assoc` (practice) {name=monus-plus-assoc} #### Exercise `∸-|-assoc` (practice) {#monus-plus-assoc}
Show that monus associates with addition, that is, Show that monus associates with addition, that is,
@ -949,7 +949,7 @@ Show the following three laws
for all `m`, `n`, and `p`. for all `m`, `n`, and `p`.
#### Exercise `Bin-laws` (stretch) {name=Bin-laws} #### Exercise `Bin-laws` (stretch) {#Bin-laws}
Recall that Recall that
Exercise [Bin](/Naturals/#Bin) Exercise [Bin](/Naturals/#Bin)

6
src/plfa/part1/Isomorphism.lagda.md
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@ -83,7 +83,7 @@ g ∘′ f = λ x → g (f x)
``` ```
## Extensionality {name=extensionality} ## Extensionality {#extensionality}
Extensionality asserts that the only way to distinguish functions is Extensionality asserts that the only way to distinguish functions is
by applying them; if two functions applied to the same argument always by applying them; if two functions applied to the same argument always
@ -450,7 +450,7 @@ postulate
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `_⇔_` (practice) {name=iff} #### Exercise `_⇔_` (practice) {#iff}
Define equivalence of propositions (also known as "if and only if") as follows: Define equivalence of propositions (also known as "if and only if") as follows:
``` ```
@ -465,7 +465,7 @@ Show that equivalence is reflexive, symmetric, and transitive.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `Bin-embedding` (stretch) {name=Bin-embedding} #### Exercise `Bin-embedding` (stretch) {#Bin-embedding}
Recall that Exercises Recall that Exercises
[Bin](/Naturals/#Bin) and [Bin](/Naturals/#Bin) and

6
src/plfa/part1/Lists.lagda.md
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@ -458,7 +458,7 @@ _ =
``` ```
Now the time to reverse a list is linear in the length of the list. Now the time to reverse a list is linear in the length of the list.
## Map {name=Map} ## Map {#Map}
Map applies a function to every element of a list to generate a corresponding list. Map applies a function to every element of a list to generate a corresponding list.
Map is an example of a _higher-order function_, one which takes a function as an Map is an example of a _higher-order function_, one which takes a function as an
@ -558,7 +558,7 @@ Define a suitable map operator over trees:
-- Your code goes here -- Your code goes here
``` ```
## Fold {name=Fold} ## Fold {#Fold}
Fold takes an operator and a value, and uses the operator to combine Fold takes an operator and a value, and uses the operator to combine
each of the elements of the list, taking the given value as the result each of the elements of the list, taking the given value as the result
@ -833,7 +833,7 @@ Show that if `_⊗_` and `e` form a monoid, then `foldr _⊗_ e` and
``` ```
## All {name=All} ## All {#All}
We can also define predicates over lists. Two of the most important We can also define predicates over lists. Two of the most important
are `All` and `Any`. are `All` and `Any`.

16
src/plfa/part1/Naturals.lagda.md
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@ -76,7 +76,7 @@ after zero; and `2` is shorthand for `suc (suc zero)`, which is the
same as `suc 1`, the successor of one; and `3` is shorthand for the same as `suc 1`, the successor of one; and `3` is shorthand for the
successor of two; and so on. successor of two; and so on.
#### Exercise `seven` (practice) {name=seven} #### Exercise `seven` (practice) {#seven}
Write out `7` in longhand. Write out `7` in longhand.
@ -287,7 +287,7 @@ Parentheses and semicolons are among the few characters that cannot
appear in names, so we do not need extra spaces in the `using` list. appear in names, so we do not need extra spaces in the `using` list.
## Operations on naturals are recursive functions {name=plus} ## Operations on naturals are recursive functions {#plus}
Now that we have the natural numbers, what can we do with them? Now that we have the natural numbers, what can we do with them?
For instance, can we define arithmetic operations such as For instance, can we define arithmetic operations such as
@ -425,7 +425,7 @@ is not like testimony in a court which must be weighed to determine
whether the witness is trustworthy. Rather, it is ironclad. The whether the witness is trustworthy. Rather, it is ironclad. The
other word for evidence, which we will use interchangeably, is _proof_. other word for evidence, which we will use interchangeably, is _proof_.
#### Exercise `+-example` (practice) {name=plus-example} #### Exercise `+-example` (practice) {#plus-example}
Compute `3 + 4`, writing out your reasoning as a chain of equations, using the equations for `+`. Compute `3 + 4`, writing out your reasoning as a chain of equations, using the equations for `+`.
@ -486,7 +486,7 @@ Here we have omitted the signature declaring `_ : 2 * 3 ≡ 6`, since
it can easily be inferred from the corresponding term. it can easily be inferred from the corresponding term.
#### Exercise `*-example` (practice) {name=times-example} #### Exercise `*-example` (practice) {#times-example}
Compute `3 * 4`, writing out your reasoning as a chain of equations, using the equations for `*`. Compute `3 * 4`, writing out your reasoning as a chain of equations, using the equations for `*`.
(You do not need to step through the evaluation of `+`.) (You do not need to step through the evaluation of `+`.)
@ -496,7 +496,7 @@ Compute `3 * 4`, writing out your reasoning as a chain of equations, using the e
``` ```
#### Exercise `_^_` (recommended) {name=power} #### Exercise `_^_` (recommended) {#power}
Define exponentiation, which is given by the following equations: Define exponentiation, which is given by the following equations:
@ -566,7 +566,7 @@ _ =
∎ ∎
``` ```
#### Exercise `∸-example₁` and `∸-example₂` (recommended) {name=monus-examples} #### Exercise `∸-example₁` and `∸-example₂` (recommended) {#monus-examples}
Compute `5 ∸ 3` and `3 ∸ 5`, writing out your reasoning as a chain of equations. Compute `5 ∸ 3` and `3 ∸ 5`, writing out your reasoning as a chain of equations.
@ -701,7 +701,7 @@ definitions is quite similar. They might be considered two sides of
the same coin. the same coin.
## The story of creation, finitely {name=finite-creation} ## The story of creation, finitely {#finite-creation}
The above story was told in a stratified way. First, we create The above story was told in a stratified way. First, we create
the infinite set of naturals. We take that set as given when the infinite set of naturals. We take that set as given when
@ -872,7 +872,7 @@ Haskell requires time proportional to the sum of the logarithms of
_m_ and _n_. _m_ and _n_.
#### Exercise `Bin` (stretch) {name=Bin} #### Exercise `Bin` (stretch) {#Bin}
A more efficient representation of natural numbers uses a binary A more efficient representation of natural numbers uses a binary
rather than a unary system. We represent a number as a bitstring: rather than a unary system. We represent a number as a bitstring:

2
src/plfa/part1/Quantifiers.lagda.md
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@ -438,7 +438,7 @@ postulate
Does the converse hold? If so, prove; if not, explain why. Does the converse hold? If so, prove; if not, explain why.
#### Exercise `Bin-isomorphism` (stretch) {name=Bin-isomorphism} #### Exercise `Bin-isomorphism` (stretch) {#Bin-isomorphism}
Recall that Exercises Recall that Exercises
[Bin](/Naturals/#Bin), [Bin](/Naturals/#Bin),

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src/plfa/part1/Relations.lagda.md
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@ -241,7 +241,7 @@ lack---for instance by saying that a newly introduced relation is a
partial order but not a total order. partial order but not a total order.
#### Exercise `orderings` (practice) {name=orderings} #### Exercise `orderings` (practice) {#orderings}
Give an example of a preorder that is not a partial order. Give an example of a preorder that is not a partial order.
@ -359,7 +359,7 @@ and `suc n ≤ suc m` and must show `suc m ≡ suc n`. The inductive
hypothesis `≤-antisym m≤n n≤m` establishes that `m ≡ n`, and our goal hypothesis `≤-antisym m≤n n≤m` establishes that `m ≡ n`, and our goal
follows by congruence. follows by congruence.
#### Exercise `≤-antisym-cases` (practice) {name=leq-antisym-cases} #### Exercise `≤-antisym-cases` (practice) {#leq-antisym-cases}
The above proof omits cases where one argument is `z≤n` and one The above proof omits cases where one argument is `z≤n` and one
argument is `s≤s`. Why is it ok to omit them? argument is `s≤s`. Why is it ok to omit them?
@ -559,7 +559,7 @@ Show that multiplication is monotonic with regard to inequality.
``` ```
## Strict inequality {name=strict-inequality} ## Strict inequality {#strict-inequality}
We can define strict inequality similarly to inequality: We can define strict inequality similarly to inequality:
``` ```
@ -597,7 +597,7 @@ and conversely. One can then give an alternative derivation of the
properties of strict inequality, such as transitivity, by properties of strict inequality, such as transitivity, by
exploiting the corresponding properties of inequality. exploiting the corresponding properties of inequality.
#### Exercise `<-trans` (recommended) {name=less-trans} #### Exercise `<-trans` (recommended) {#less-trans}
Show that strict inequality is transitive. Show that strict inequality is transitive.
@ -605,7 +605,7 @@ Show that strict inequality is transitive.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `trichotomy` (practice) {name=trichotomy} #### Exercise `trichotomy` (practice) {#trichotomy}
Show that strict inequality satisfies a weak version of trichotomy, in Show that strict inequality satisfies a weak version of trichotomy, in
the sense that for any `m` and `n` that one of the following holds: the sense that for any `m` and `n` that one of the following holds:
@ -623,7 +623,7 @@ similar to that used for totality.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `+-mono-<` (practice) {name=plus-mono-less} #### Exercise `+-mono-<` (practice) {#plus-mono-less}
Show that addition is monotonic with respect to strict inequality. Show that addition is monotonic with respect to strict inequality.
As with inequality, some additional definitions may be required. As with inequality, some additional definitions may be required.
@ -632,7 +632,7 @@ As with inequality, some additional definitions may be required.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `≤-iff-<` (recommended) {name=leq-iff-less} #### Exercise `≤-iff-<` (recommended) {#leq-iff-less}
Show that `suc m ≤ n` implies `m < n`, and conversely. Show that `suc m ≤ n` implies `m < n`, and conversely.
@ -640,7 +640,7 @@ Show that `suc m ≤ n` implies `m < n`, and conversely.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `<-trans-revisited` (practice) {name=less-trans-revisited} #### Exercise `<-trans-revisited` (practice) {#less-trans-revisited}
Give an alternative proof that strict inequality is transitive, Give an alternative proof that strict inequality is transitive,
using the relation between strict inequality and inequality and using the relation between strict inequality and inequality and
@ -749,7 +749,7 @@ evidence that the first number is odd. If it is because it is the
successor of an even number, then the result is odd because it is the successor of an even number, then the result is odd because it is the
successor of the sum of two even numbers, which is even. successor of the sum of two even numbers, which is even.
#### Exercise `o+o≡e` (stretch) {name=odd-plus-odd} #### Exercise `o+o≡e` (stretch) {#odd-plus-odd}
Show that the sum of two odd numbers is even. Show that the sum of two odd numbers is even.
@ -757,7 +757,7 @@ Show that the sum of two odd numbers is even.
-- Your code goes here -- Your code goes here
``` ```
#### Exercise `Bin-predicates` (stretch) {name=Bin-predicates} #### Exercise `Bin-predicates` (stretch) {#Bin-predicates}
Recall that Recall that
Exercise [Bin](/Naturals/#Bin) Exercise [Bin](/Naturals/#Bin)

6
src/plfa/part2/Inference.lagda.md
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@ -32,7 +32,7 @@ Chapter [Lambda](/Lambda/),
and from it we compute an intrinsically-typed term, in the style of and from it we compute an intrinsically-typed term, in the style of
Chapter [DeBruijn](/DeBruijn/). Chapter [DeBruijn](/DeBruijn/).
## Introduction: Inference rules as algorithms {name=algorithms} ## Introduction: Inference rules as algorithms {#algorithms}
In the calculus we have considered so far, a term may have more than In the calculus we have considered so far, a term may have more than
one type. For example, one type. For example,
@ -470,7 +470,7 @@ the equality test in the application rule in the first
[section](/Inference/#algorithms). [section](/Inference/#algorithms).
#### Exercise `bidirectional-mul` (recommended) {name=bidirectional-mul} #### Exercise `bidirectional-mul` (recommended) {#bidirectional-mul}
Rewrite your definition of multiplication from Rewrite your definition of multiplication from
Chapter [Lambda](/Lambda/), decorated to support inference. Chapter [Lambda](/Lambda/), decorated to support inference.
@ -480,7 +480,7 @@ Chapter [Lambda](/Lambda/), decorated to support inference.
``` ```
#### Exercise `bidirectional-products` (recommended) {name=bidirectional-products} #### Exercise `bidirectional-products` (recommended) {#bidirectional-products}
Extend the bidirectional type rules to include products from Extend the bidirectional type rules to include products from
Chapter [More](/More/). Chapter [More](/More/).

4
src/plfa/part2/Lambda.lagda.md
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@ -213,7 +213,7 @@ definition may use `plusᶜ` as defined earlier (or may not
``` ```
#### Exercise `primed` (stretch) {name=primed} #### Exercise `primed` (stretch) {#primed}
Some people find it annoying to write `` ` "x" `` instead of `x`. Some people find it annoying to write `` ` "x" `` instead of `x`.
We can make examples with lambda terms slightly easier to write We can make examples with lambda terms slightly easier to write
@ -1201,7 +1201,7 @@ the three places where a bound variable is introduced.
The rules are deterministic, in that at most one rule applies to every term. The rules are deterministic, in that at most one rule applies to every term.
### Example type derivations {name=derivation} ### Example type derivations {#derivation}
Type derivations correspond to trees. In informal notation, here Type derivations correspond to trees. In informal notation, here
is a type derivation for the Church numeral two, is a type derivation for the Church numeral two,

4
src/plfa/part2/More.lagda.md
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@ -147,7 +147,7 @@ Here `M †` is the translation of term `M` from a calculus with the
construct to a calculus without the construct. construct to a calculus without the construct.
## Products {name=products} ## Products {#products}
### Syntax ### Syntax
@ -285,7 +285,7 @@ We can also translate back the other way:
(`proj₁ L) ‡ = case× (L ‡) [⟨ x , y ⟩⇒ x ] (`proj₁ L) ‡ = case× (L ‡) [⟨ x , y ⟩⇒ x ]
(`proj₂ L) ‡ = case× (L ‡) [⟨ x , y ⟩⇒ y ] (`proj₂ L) ‡ = case× (L ‡) [⟨ x , y ⟩⇒ y ]
## Sums {name=sums} ## Sums {#sums}
### Syntax ### Syntax

4
src/plfa/part2/Subtyping.lagda.md
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@ -879,7 +879,7 @@ typed (C-suc c) = ⊢suc (typed c)
typed (C-rcd ⊢Ms dks As<:Bs) = ⊢<: (⊢rcd ⊢Ms dks) As<:Bs typed (C-rcd ⊢Ms dks As<:Bs) = ⊢<: (⊢rcd ⊢Ms dks) As<:Bs
``` ```
## Progress <a name="subtyping-progress"></a> ## Progress <a #"subtyping-progress"></a>
The Progress theorem states that a well-typed term may either take a The Progress theorem states that a well-typed term may either take a
reduction step or it is already a value. The proof of Progress is like reduction step or it is already a value. The proof of Progress is like
@ -952,7 +952,7 @@ progress (⊢<: {A = A}{B} ⊢M A<:B) = progress ⊢M
* Case `⊢<:`: we invoke the induction hypothesis on sub-derivation of `Γ ⊢ M ⦂ A`. * Case `⊢<:`: we invoke the induction hypothesis on sub-derivation of `Γ ⊢ M ⦂ A`.
## Preservation <a name="subtyping-preservation"></a> ## Preservation <a #"subtyping-preservation"></a>
In this section we prove that when a well-typed term takes a reduction In this section we prove that when a well-typed term takes a reduction
step, the result is another well-typed term with the same type. step, the result is another well-typed term with the same type.