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import NNG.Levels.AdvAddition.Level_1
Game "NNG"
World "AdvAddition"
Level 2
Title "succ_succ_inj"
open MyNat
Introduction
"
In the below theorem, we need to apply `succ_inj` twice. Once to prove
$\\operatorname{succ}(\\operatorname{succ}(a))=\\operatorname{succ}(\\operatorname{succ}(b))
\\implies \\operatorname{succ}(a)=\\operatorname{succ}(b)$, and then again
to prove $\\operatorname{succ}(a)=\\operatorname{succ}(b)\\implies a=b$.
However `succ a = succ b` is
nowhere to be found, it's neither an assumption or a goal when we start
this level. You can make it with `have` or you can use `apply`.
"
Statement
"For all naturals $a$ and $b$, if we assume
$\\operatorname{succ}(\\operatorname{succ}(a))=\\operatorname{succ}(\\operatorname{succ}(b))$,
then we can deduce $a=b$. "
{a b : ℕ
Branch
exact succ_inj (succ_inj h)
apply succ_inj
apply succ_inj
assumption
LemmaTab "Nat"
Conclusion
"
## Sample solutions to this level.
Make sure you understand them all. And remember that `rw` should not be used
with `succ_inj` -- `rw` works only with equalities or `↔` statements,
not implications or functions.
```
example {a b : ℕ
apply succ_inj
apply succ_inj
exact h
example {a b : ℕ
apply succ_inj
exact succ_inj h
example {a b : ℕ
exact succ_inj (succ_inj h)
```
"