feat(data/finset): not_mem theorems #44
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digama0
reviewed
Feb 8, 2018
| variables {s s₁ s₂ : finset α} [decidable_eq α] {a b : α} | ||
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| theorem ne_of_mem_of_not_mem (h_mem : a ∈ s) (h_not_mem : b ∉ s) : a ≠ b := | ||
| ne_of_mem_erase (@@eq.substr _ (erase_insert h_not_mem) h_mem) |
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You know, this can be proved for literally anything that has a has_mem, it has nothing to do with finsets.
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Fair enough. For whatever reason that didn't occur to me before.
theorem ne_of_mem_of_not_mem {A S : Type} [has_mem A S] {a b : A} {s : S} (h_mem : a ∈ s) (h_not_mem : b ∉ s) : a ≠ b :=
λ p : a = b, by subst p; contradictionThere was a problem hiding this comment.
Now that I think about it, does this not already exist somewhere?
theorem ne_of_p_of_not_p {A : Type} {a b : A} (P : A → Prop) (pa : P a) (npb : ¬ P b) : a ≠ b :=
λ h, by subst h; contradiction
theorem ne_of_mem_of_not_mem {A S : Type} [has_mem A S] {a b : A} {s : S} : a ∈ s → b ∉ s → a ≠ b :=
ne_of_p_of_not_p (λ a : A, a ∈ s)There was a problem hiding this comment.
Let me know how you'd like the PR changed. I can add the above two theorems, though they would not naturally go into data/finset.lean.
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Added in 6ef721e |
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These are some theorems I've been using that involve not being a member (
∉) of afinset.