`apply`

*exactly*the same as some hypothesis- can be used to
**finish**a proof (shorter than`rewrite`

then`reflexivity`

)

It also works with *conditional* hypotheses:

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n, m, o, p : nat
eq1 : n = m
eq2 : forall q r : nat, q = r -> [q; o] = [r; p]
============================
[n; o] = [m; p]
apply eq2.
n = m

It works by working backwards.
It will try to *pattern match* the universally quantified `q r`

. (i.e. universal var)
We match the *conclusion* and generates the *hypothesis* as a *subgoal*.

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Theorem trans_eq : forall (X:Type) (n m o : X), n = m -> m = o -> n = o.
Example trans_eq_example' : forall (a b c d e f : nat),
[a;b] = [c;d] -> [c;d] = [e;f] -> [a;b] = [e;f].
Proof.
intros a b c d e f eq1 eq2.
apply trans_eq. (* Error: Unable to find an instance for the variable m. *)

The *unification algo* won’t happy since:

- it can find instance for
`n = o`

from`[a;b] = [e;f]`

(matching both conclusion) - but what should be
`m`

? It could be anything as long as`n = m`

and`m = o`

holds.

So we need to tell Coq explicitly which value should be picked for `m`

:

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apply trans_eq with (m:=[c;d]). (* <- supplying extra info, [m:=] can be ommited *)

Prof Mtf: As a PL person, you should feel this is a little bit awkward since now function argument name must be remembered. (but it’s just local and should be able to do any alpha-conversion). named argument is more like a record.

In Coq Intensive 2 (2018), someone proposed the below which works:

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Example trans_eq_example'' : forall (a b c d e f : nat),
[a;b] = [c;d] -> [c;d] = [e;f] -> [a;b] = [e;f].
Proof.
intros a b c d e f.
apply trans_eq. (* Coq was able to match three at all at this time...hmm *)
Qed.

`injection`

and `discrinimate`

### Side Note on Terminologys of Function

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relation

function is defined as

a special kind of binary relation. it requires`xRy1 ∧ xRy2 → y1 = y2`

called “functional” or “univalent”, “right-unique”, or “deterministic” and also`∀x ∈ X, ∃y ∈ Y s.t. xRy`

called “left-total”

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x ↦ f(x)
input ↦ output
argument ↦ value
X ↦ Y
domain 域 ↦ co-domain 陪域
what can go into ↦ what possibly come out
A ⊆ X ↦ f(A) = {f(x) | x ∈ A}
↦ image
↦ what actually come out
f⁻¹(B)={x ∈ X|f(x) ∈ B} ↦ B ⊆ Y
preimage ↦
when A = X ↦ Y
↦ range
image of domain

Besides subset, the notation of `image`

and `pre-image`

can be applied to *element* as well.
However, by definition:

- the image of an element
`x`

of domain ↦ always single element of codomain (singleton set) - the preimage of an element
`y`

of codomain ↦ may be empty, or one, or many!`<= 1 ↦ 1`

: injective (left-unique)`>= 1 ↦ 1`

: surjective (right-total)- ` 1 ↦ 1` : bijective

Noted that the definition of “function” doesn’t require “right-total”ity) until we have `surjective`

.

graph = `[(x, f(x))]`

, these points form a “curve”, 真的是图像

### Total vs Partial

For math, we seldon use partial function since we can simply “define a perfect domain for that”.
But in Type Theory, Category Theory, we usually consider the *domain* `X`

and the *domain of definition* `X'`

.

Besides, `f(x)`

can be `undefined`

. (not “left-total”, might not have “right”)

### Conclusion - the road from Relation to Function

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bi-relation
| + right-unique
partial function
| + left-total
(total) function
+ left-unique / \ + right-total
injection surjection
\ /
bijection

### Original notes on Injective, surjective, Bijective

All talk about the propeties of *preimage*!

- Injective:
`<= 1 ↦ 1`

or`0, 1 ↦ 1`

(distinctness) - Surjective:
`>= 1 ↦ 1`

(at least 1 in the domain) - Bijective: ` 1 ↦ 1
`(intersection of Inj and Surj, so only`

1` preimage,*one-to-one correspondence*)

*injectivitiy* and *disjointness*, or `inversion`

.

Recall the definition of `nat`

:

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Inductive nat : Type :=
| O : nat
| S : nat → nat.

Besides there are two forms of `nat`

(for `destruct`

and `induction`

), there are more facts:

- The constructor
`S`

is*injective*(distinct), i.e`S n = S m -> n = m`

. - The constructors
`O`

and`S`

are*disjoint*, i.e.`forall n, O != S n`

.

`injection`

- can be used to prove the
*preimages*are the same. `injection`

leave things in conclusion rather than hypo. with`as`

would be in hypo.

`disjoint`

*principle of explosion*(a logical principle)- asserts a contraditory hypothesis entails anything. (even false things)
*vacously true*

`false = true`

is contraditory because they are distinct constructors.

`inversion`

- the big hammer: inversion of the definition.
- combining
`injection`

and`disjoint`

and even some more`rewrite`

.- IMH, which one to use depends on
*semantics*

- IMH, which one to use depends on

from Coq Intensive (not sure why it’s not the case in book version).

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Theorem S_injective_inv : forall (n m : nat),
S n = S m -> n = m.
Proof.
intros n m H. inversion H. reflexivity. Qed.
Theorem inversion_ex1 : forall (n m : nat),
[n] = [m] -> n = m.
Proof.
intros n m H. inversion H. reflexivity. Qed.

Side question: could Coq derive equality function for inductive type? A: nope. Equality for some inductive types are

undecidable.

### Converse of injectivity

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Theorem f_equal : ∀(A B : Type) (f: A → B) (x y: A),
x = y → f x = f y.
Proof.
intros A B f x y eq.
rewrite eq. reflexivity. Qed.

### Slide Q&A 1

- The tactic fails because tho
`negb`

is injective but`injection`

only workks on constructors.

## Using Tactics in Hypotheses

### Reasoning Backwards and Reasoning Forward (from Coq Intensive 2)

Style of reasoning

- Backwards: start with
*goal*, applying tactics`simpl/destruct/induction`

, generate*subgoals*, until proved.- iteratively reasons about what would imply the goal, until premises or previously proven theorems are reached.

- Forwards: start with
*hypo*, applying tactics, iteratively draws conclusions, until the goal is reached.

Backwards reasoning is dominated stratgy of theroem prover (and execution of prolog). But not natural in informal proof.

True forward reasoning derives fact, but in Coq it’s like hypo deriving hypo, very imperative.

`in`

most tactics also have a variant that performs a similar operation on a statement in the context.

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simpl in H.
simpl in *. (* in all hypo and goal *)
symmetry in H.
apply L in H.

`apply`

ing in hypothesis and in conclusion

`apply`

ing in hypo is very different with `apply`

ing in conclusion.

it’s not we unify the ultimate conclusion and generate premises as new goal, but trying to find a hypothesis to match and left the residual conclusion as new hypothesis.

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Theorem silly3'' : forall (n : nat),
(true = (n =? 5) -> true = ((S (S n)) =? 7)) ->
true = (n =? 5) ->
true = ((S (S n)) =? 7).
Proof.
intros n eq H.
apply eq in H. (* or *) apply eq. (* would be different *)
apply H. Qed.

Also if we add one more premises `true = true ->`

,
the subgoal generated by `apply`

would be in reversed order:

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Theorem silly3'' : forall (n : nat),
(true = true -> true = (n =? 5) -> true = ((S (S n)) =? 7)) ->
true = (n =? 5) ->
true = ((S (S n)) =? 7).
Proof.

Again: “proof engineering”: proof can be done in so many different ways and in different orders.

## Varying the Induction Hypothesis

Sometimes it’s important to control the exact form of the induction hypothesis!!

Considering:

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Theorem double_injective: ∀n m,
double n = double m → n = m.

if we begin with `intros n m. induction n.`

then we get stuck in the inductive case of `n`

, where the induction hypothesis `IHn'`

generated is:

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IHn' : double n' = double m -> n' = m
IHn' : double n' = double (S m') -> n' = S m' (* m = S m' *)

This is not what we want!!

To prove `double_injective`

, we hope `IHn'`

can give us `double n' = double m' -> n' = m'`

(i.e. the `P(n-1)`

case).

The problem is `intros`

implies *for these particular n and m*. (not more

`forall`

but *const*). And when we

`intros n m. induction n`

, we are trying to prove a statement involving *every*n but just a

*single*m…

*How to keep *`m`

generic (universal)?

`m`

generic (universal)?By either `induction n`

before `intros m`

or using `generalize dependent m`

, we can have:

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IHn' : forall m : nat, double n' = double m -> n' = m

where the `m`

here is still universally quantified, so we can instaniate `m`

with `m'`

by `apply`

ing it with `double n' = double m'`

to yield `n' = m'`

or vice versa. (recall conditional statements can be `apply`

ed in 2 ways.)

### Notes on `generalize dependent`

Usually used when the argument order is conflict with instantiate (`intros`

) order.

?

reflection: turing a computational result into a propositional result

## Unfolding Definitions.

tactics like

`simpl`

,`reflexivity`

, and`apply`

will often unfold the definitions of functions automatically. However, this automatic unfolding is somewhatconservative.

`simpl.`

only do unfolding when it can furthur simplify after unfolding. But sometimes you might want to explicitly `unfold`

then do furthur works on that.

## Using `destruct`

on Compound Expressions

destruct the whole arbitrary expression.

`destruct`

by default throw away the whole expression after it, which might leave you into a stuck state.
So explicitly saying `eqn:Name`

would help with that!

## Micro Sermon - Mindless proof-hacking

From Coq Intensive…

- a lot of fun
- …w/o thinking at all
- terrible temptation
- you shouldn’t always resist…

But after 5 mins…you should step back and try to think

A typical coq user

- sitting and does not have their brain engaged all the time…
- at some point…(get stuck)
- oh I have to reengage brain..

what is this really saying…

One way: good old paper and pencil

5 mins is good time!