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.
77 lines
1.8 KiB
Plaintext
77 lines
1.8 KiB
Plaintext
import NNG.Metadata
|
|
import NNG.MyNat.Addition
|
|
|
|
Game "NNG"
|
|
World "AdvProposition"
|
|
Level 5
|
|
Title "Easter eggs."
|
|
|
|
open MyNat
|
|
|
|
Introduction
|
|
"
|
|
Let's try this again. Try proving it in other ways. (Note that `rcases` is temporarily disabled.)
|
|
|
|
### A trick.
|
|
|
|
Instead of using `rcases` on `h : P ↔ Q` you can just access the proofs of `P → Q` and `Q → P`
|
|
directly with `h.1` and `h.2`. So you can solve this level with
|
|
|
|
```
|
|
intro hpq hqr
|
|
constructor
|
|
intro p
|
|
apply hqr.1
|
|
…
|
|
```
|
|
|
|
### Another trick
|
|
|
|
Instead of using `rcases` on `h : P ↔ Q`, you can just `rw [h]`, and this will change all `P`s to `Q`s
|
|
in the goal. You can use this to create a much shorter proof. Note that
|
|
this is an argument for *not* running the `rcases` tactic on an iff statement;
|
|
you cannot rewrite one-way implications, but you can rewrite two-way implications.
|
|
|
|
|
|
"
|
|
-- TODO
|
|
-- ### Another trick
|
|
-- `cc` works on this sort of goal too.
|
|
|
|
Statement --iff_trans
|
|
"If $P$, $Q$ and $R$ are true/false statements, then `P ↔ Q` and `Q ↔ R` together imply `P ↔ R`.
|
|
"
|
|
(P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) := by
|
|
intro hpq hqr
|
|
Hint "Make a choice and continue either with `constructor` or `rw`.
|
|
|
|
* if you use `constructor`, you will use `{hqr}.1, {hqr}.2, …` later.
|
|
* if you use `rw`, you can replace all `P`s with `Q`s using `rw [{hpq}]`"
|
|
Branch
|
|
rw [hpq]
|
|
Branch
|
|
exact hqr
|
|
rw [hqr]
|
|
Hint "Now `rfl` can close this goal.
|
|
|
|
TODO: Note that the current modification of `rfl` is too weak to prove this. For now, you can
|
|
use `simp` instead (which calls the \"real\" `rfl` internally)."
|
|
simp
|
|
constructor
|
|
intro p
|
|
Hint "Now you can directly `apply {hqr}.1`"
|
|
apply hqr.1
|
|
apply hpq.1
|
|
assumption
|
|
intro r
|
|
apply hpq.2
|
|
apply hqr.2
|
|
assumption
|
|
|
|
DisabledTactic rcases
|
|
|
|
Conclusion
|
|
"
|
|
|
|
"
|