78 lines
1.8 KiB
Lean4
78 lines
1.8 KiB
Lean4
import Game.Levels.AdvAddition.Level_9
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import Std.Tactic.RCases
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Game "NNG"
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World "AdvAddition"
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Level 10
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Title "add_left_eq_zero"
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open MyNat
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Introduction
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"
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In Lean, `a ≠ b` is *defined to mean* `(a = b) → False`.
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This means that if you see `a ≠ b` you can *literally treat
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it as saying* `(a = b) → False`. Computer scientists would
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say that these two terms are *definitionally equal*.
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The following lemma, $a+b=0\\implies b=0$, will be useful in inequality world.
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"
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Statement MyNat.add_left_eq_zero
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"If $a$ and $b$ are natural numbers such that
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$$ a + b = 0, $$
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then $b = 0$."
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{a b : ℕ} (h : a + b = 0) : b = 0 := by
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Hint "
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You want to start of by making a distinction `b = 0` or `b = succ d` for some
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`d`. You can do this with
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```
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induction b
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```
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even if you are never going to use the induction hypothesis.
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"
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-- TODO: induction vs rcases
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Branch
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rcases b
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-- TODO: `⊢ zero = 0` :(
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induction b with d
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· rfl
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· Hint "This goal seems impossible! However, our hypothesis `h` is also impossible,
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meaning that we still have a chance!
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First let's see why `h` is impossible. We can
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```
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rw [add_succ] at h
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```
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"
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rw [add_succ] at h
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Hint "
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Because `succ_ne_zero (a + {d})` is a proof that `succ (a + {d}) ≠ 0`,
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it is also a proof of the implication `succ (a + {d}) = 0 → False`.
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Hence `succ_ne_zero (a + {d}) h` is a proof of `False`!
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Unfortunately our goal is not `False`, it's a generic
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false statement.
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Recall however that the `exfalso` command turns any goal into `False`
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(it's logically OK because `False` implies every proposition, true or false).
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You can probably take it from here.
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"
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Branch
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have j := succ_ne_zero (a + d) h
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exfalso
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exact j
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exfalso
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apply succ_ne_zero (a + d)
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exact h
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LemmaTab "Add"
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Conclusion
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"
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"
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