Fix a few minor Lean misformalisations #247
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This was disprovable because the `IsZigZag_def` predicate was missing parentheses. Because of this, the following was in fact true: ```lean have (k : ℕ) [NeZero k] (y : Fin k → ℝ) : k = 1 ↔ IsZigZag y := sorry ``` The issue being that for all sufficiently large `i`, `i + 2 < k` was `False` and thus `i + 2 < k → y (i + 2) < y (i + 1)` was `True` and thus within the second clause of the disjunction in `IsZigZag_def`, we have `y i < y (i + 1)` for all sufficiently large `i : ℕ`. However `y` is a finite sequence and so cannot decrease forever. Thus the second clause was always `False` so that the predicate collapsed to just `k = 1`.
This had contradictory hypotheses and had a trivially provable goal. The hypotheses were contradictory because the `pow` function was universally defined but required to satisfy axioms for each semigroup structure that a type might carry. Thus if we invoke the hypotheses on a type `S` twice, once for one semigroup structure and once for another, we obtain a contradiction unless they happen to agree about `x ^ n` for all `x` and `n`. The goal was trivial because it amounted to a statement of the form `... → Nonempty (Group S)` which is trivial if `S` is non-empty.
eric-wieser
reviewed
Nov 15, 2024
Co-authored-by: Eric Wieser <[email protected]>
GeorgeTsoukalas
approved these changes
Jan 2, 2025
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