[ refactor ] Restate, and use, the definitions of Monotonic etc. operations
#2580
Conversation
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Thanks - these already are easier to understand. |
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How do I remove #1579 from the |
Not sure how you did it, but it's gone for me 🎉 |
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Now that the abstract definitions are mostly in place, can look at the instantiations in |
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Remark: all of the |
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So this should now be ready, modulo:
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src/Data/Nat/Properties.agda
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| *-monoʳ-≤ : ∀ n → (n *_) Preserves _≤_ ⟶ _≤_ | ||
| *-monoʳ-≤ n m≤o = *-mono-≤ (≤-refl {n}) m≤o | ||
| *-monoʳ-≤ : LeftMonotonic _≤_ _≤_ _*_ | ||
| *-monoʳ-≤ = mono₂⇒monoˡ {≤₁ = _≤_} {≤₂ = _≤_} {≤₃ = _≤_} ≤-refl *-mono-≤ |
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This providing of implicits is annoying. Suggests that some of them should be explicit in the proof?
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Yes, I've been scratching my head over how it might be avoided... without a conclusion yet.
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So if you make all 3 underscores, which ones are yellow? I notice that the previous proof provided the n explicitly, might that work here as well?
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It seems that only {≤₃ = _≤_} is required here...
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Aaargh: there's a difference depending on whether we consider left or right, presumably because of the left-biased implementations. Will need to investigate further. Sigh.
*-monoˡ-≤ : RightMonotonic _≤_ _≤_ _*_
*-monoˡ-≤ = mono₂⇒monoʳ {≤₃ = _≤_} ≤-refl *-mono-≤
*-monoʳ-≤ : LeftMonotonic _≤_ _≤_ _*_
*-monoʳ-≤ = mono₂⇒monoˡ {≤₁ = _≤_} {≤₂ = _≤_} {≤₃ = _≤_} ≤-refl *-mono-≤
src/Data/Nat/Properties.agda
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| +-monoʳ-≤ : ∀ n → (n +_) Preserves _≤_ ⟶ _≤_ | ||
| +-monoʳ-≤ n m≤o = +-mono-≤ (≤-refl {n}) m≤o | ||
| +-monoʳ-≤ : LeftMonotonic _≤_ _≤_ _+_ | ||
| +-monoʳ-≤ = mono₂⇒monoˡ {≤₃ = _≤_} ≤-refl +-mono-≤ |
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But here just 1 suffices:
+-monoˡ-≤ : RightMonotonic _≤_ _≤_ _+_
+-monoˡ-≤ = mono₂⇒monoʳ {≤₃ = _≤_} ≤-refl +-mono-≤
+-monoʳ-≤ : LeftMonotonic _≤_ _≤_ _+_
+-monoʳ-≤ = mono₂⇒monoˡ {≤₃ = _≤_} ≤-refl +-mono-≤
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I'll resolve the merge conflict later, meanwhile tricky questions above needing more thought... ;-)... so have converted back to DRAFT. |
src/Data/Nat/Properties.agda
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| *-monoʳ-≤ : ∀ n → (n *_) Preserves _≤_ ⟶ _≤_ | ||
| *-monoʳ-≤ n m≤o = *-mono-≤ (≤-refl {n}) m≤o | ||
| *-monoʳ-≤ : LeftMonotonic _≤_ _≤_ _*_ | ||
| *-monoʳ-≤ = mono₂⇒monoˡ {≤₁ = _≤_} {≤₂ = _≤_} {≤₃ = _≤_} ≤-refl *-mono-≤ |
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So if you make all 3 underscores, which ones are yellow? I notice that the previous proof provided the n explicitly, might that work here as well?
Aim: towards tackling #1579
TODO:
Data.Nat.*UPDATED: unsolved metas suggest some more revision is necessary :-(Data.Integer.*Data.Rational.*breakingchanges (v3.0: separate PR once the rest is done?)