You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
60 lines
1.5 KiB
Plaintext
60 lines
1.5 KiB
Plaintext
import NNG.Metadata
|
|
import NNG.MyNat.Addition
|
|
|
|
Game "NNG"
|
|
World "Proposition"
|
|
Level 4
|
|
Title "apply"
|
|
|
|
open MyNat
|
|
|
|
Introduction
|
|
"
|
|
Let's do the same level again:
|
|
|
|
$$
|
|
\\begin{CD}
|
|
P @>{h}>> Q @>{i}>> R \\\\
|
|
@. @VV{j}V \\\\
|
|
S @>>{k}> T @>>{l}> U
|
|
\\end{CD}
|
|
$$
|
|
|
|
We are given a proof $p$ of $P$ and our goal is to find a proof of $U$, or
|
|
in other words to find a path through the maze that links $P$ to $U$.
|
|
In level 3 we solved this by using `have`s to move forward, from $P$
|
|
to $Q$ to $T$ to $U$. Using the `apply` tactic we can instead construct
|
|
the path backwards, moving from $U$ to $T$ to $Q$ to $P$.
|
|
"
|
|
|
|
Statement
|
|
"We can solve a maze."
|
|
(P Q R S T U: Prop) (p : P) (h : P → Q) (i : Q → R)
|
|
(j : Q → T) (k : S → T) (l : T → U) : U := by
|
|
Hint "Our goal is to prove $U$. But $l:T\\implies U$ is
|
|
an implication which we are assuming, so it would suffice to prove $T$.
|
|
Tell Lean this by starting the proof below with
|
|
|
|
```
|
|
apply l
|
|
```
|
|
"
|
|
apply l
|
|
Hint "Notice that our assumptions don't change but *the goal changes*
|
|
from `U` to `T`.
|
|
|
|
Keep `apply`ing implications until your goal is `P`, and try not
|
|
to get lost!"
|
|
Branch
|
|
apply k
|
|
Hint "Looks like you got lost. \"Undo\" the last step."
|
|
apply j
|
|
apply h
|
|
Hint "Now solve this goal
|
|
with `exact p`. Note: you will need to learn the difference between
|
|
`exact p` (which works) and `exact P` (which doesn't, because $P$ is
|
|
not a proof of $P$)."
|
|
exact p
|
|
|
|
DisabledTactic «have»
|