Folding Cheat Sheet #3 - third in a series
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This document summarizes the universal property of fold, which states that for finite lists the fold function is the unique function that satisfies its defining recursive equations. It provides examples of how common list functions like sum, product, length, concatenation (⧺) can be defined in terms of fold. It also notes that the universal property can be generalized to handle partial and infinite lists. Finally, it states that map, filter and fold form the "triad" or basic building blocks of functional programming.
![𝑔 = 𝑣
𝑔 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 𝑔 𝑥𝑠
𝑠𝑢𝑚 ∷ 𝐼𝑛𝑡 → 𝐼𝑛𝑡
𝑠𝑢𝑚 = 0
𝑠𝑢𝑚 𝑥 ∶ 𝑥𝑠 = 𝑥 + 𝑠𝑢𝑚 𝑥𝑠
𝑠𝑢𝑚 = 𝑓𝑜𝑙𝑑 + 0
⟺
𝑔 = 𝑓𝑜𝑙𝑑 𝑓 𝑣
⟺
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 ∷ 𝐼𝑛𝑡 → 𝐼𝑛𝑡
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 = 1
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑥 ∶ 𝑥𝑠 = 𝑥 × 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑥𝑠
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 = 𝑓𝑜𝑙𝑑 (×) 1
⟺
𝑙𝑒𝑛𝑔𝑡ℎ ∷ [α] → 𝐼𝑛𝑡
𝑙𝑒𝑛𝑔𝑡ℎ [ ] = 0
𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 ∶ 𝑥𝑠 = 1 + 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥𝑠
𝑙𝑒𝑛𝑔𝑡ℎ = 𝑓𝑜𝑙𝑑 (𝜆𝑥. 𝜆𝑛. 1 + 𝑛) 0
⟺
(⧺) ∷ [α] → [α] → [α]
⧺ 𝑦𝑠 = 𝑦𝑠
𝑥 ∶ 𝑥𝑠 ⧺ 𝑦𝑠 = 𝑥 ∶ (𝑥𝑠 ⧺ 𝑦𝑠)
(⧺ 𝑦𝑠) = 𝑓𝑜𝑙𝑑 ∶ 𝑦𝑠
⟺
concat ∷ [ α ] → [α]
concat =
concat 𝑥𝑠 ∶ 𝑥𝑠𝑠 = 𝑥𝑠 ⧺ 𝑐𝑜𝑛𝑐𝑎𝑡 𝑥𝑠𝑠
⟺ concat = 𝑓𝑜𝑙𝑑 (⧺) [ ]](https://image.slidesharecdn.com/folding-cheat-sheet-3-240402173712-07198905/85/Folding-Cheat-Sheet-3-third-in-a-series-3-320.jpg)
![𝑔 = 𝑣
𝑔 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 𝑔 𝑥𝑠
𝑚𝑎𝑝 ∷ 𝛼 → 𝛽 → 𝛼 → 𝛽
𝑚𝑎𝑝 𝑓 =
𝑚𝑎𝑝 𝑓 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 ∶ 𝑚𝑎𝑝 𝑓 𝑥𝑠
𝑚𝑎𝑝 𝑓 = 𝑓𝑜𝑙𝑑𝑟 𝜆𝑥. 𝜆𝑥𝑠. 𝑓 𝑥 ∶ 𝑥𝑠 [ ]
⟺
𝑓𝑖𝑙𝑡𝑒𝑟 ∷ (𝛼 → 𝐵𝑜𝑜𝑙) → 𝛼 → 𝑎
𝑓𝑖𝑙𝑡𝑒𝑟 p =
𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥 ∶ 𝑥𝑠 = 𝐢𝐟 𝑝 𝑥
𝐭𝐡𝐞𝐧 𝑥 ∶ 𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥𝑠
𝐞𝐥𝐬𝐞 𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥𝑠
𝑓𝑖𝑙𝑡𝑒𝑟 p = 𝑓𝑜𝑙𝑑𝑟 (𝜆𝑥. 𝜆𝑥𝑠. 𝐢𝐟 𝑝 𝑥 𝐭𝐡𝐞𝐧 𝑥 ∶ 𝑥𝑠 𝐞𝐥𝐬𝐞 𝑥𝑠) [ ]
⟺
𝑔 = 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑣
⟺
𝑚𝑎𝑝
λ
𝑓𝑜𝑙𝑑
=](https://image.slidesharecdn.com/folding-cheat-sheet-3-240402173712-07198905/85/Folding-Cheat-Sheet-3-third-in-a-series-5-320.jpg)
Folding Cheat Sheet #3 - third in a series
- 1. CHEAT-SHEET Folding #3 ∶ / 𝒂𝟎 ∶ / 𝒂𝟏 ∶ / 𝒂𝟐 ∶ / 𝒂𝟑 𝒇 / 𝒂𝟎 𝒇 / 𝒂𝟏 𝒇 / 𝒂𝟐 𝒇 / 𝒂𝟑 𝒆 @philip_schwarz slides by https://fpilluminated.com/
- 2. The universal property of 𝒇𝒐𝒍𝒅 ... For finite lists, the universal property of 𝒇𝒐𝒍𝒅 can be stated as the following equivalence between two definitions for a function 𝒈 that processes lists: 𝒈 = 𝒗 ⟺ 𝒈 = 𝒇𝒐𝒍𝒅 𝒇 𝒗 𝒈 𝑥 ∶ 𝑥𝑠 = 𝒇 𝑥 𝒈 𝑥𝑠 In the right-to-left direction, substituting 𝒈 = 𝒇𝒐𝒍𝒅 𝒇 𝒗 into the two equations for 𝒈 gives the recursive definition for 𝒇𝒐𝒍𝒅. Conversely, in the left-to-right direction the two equations for g are precisely the assumptions required to show that 𝒈 = 𝒇𝒐𝒍𝒅 𝒇 𝒗 using a simple proof by induction on finite lists… Taken as a whole, the universal property states that for finite lists the function 𝒇𝒐𝒍𝒅 𝒇 𝒗 is not just a solution to its defining equations, but in fact the unique solution…. The universal property of 𝒇𝒐𝒍𝒅 can be generalised to handle partial and infinite lists… Graham Hutton @haskelhutt 𝑓𝑜𝑙𝑑 :: 𝛼 → 𝛽 → 𝛽 → 𝛽 → 𝛼 → 𝛽 𝑓𝑜𝑙𝑑 𝑓 𝑣 = 𝑣 𝑓𝑜𝑙𝑑 𝑓 𝑣 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 𝑓𝑜𝑙𝑑 𝑓 𝑣 𝑥𝑠
- 3. 𝑔 = 𝑣 𝑔 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 𝑔 𝑥𝑠 𝑠𝑢𝑚 ∷ 𝐼𝑛𝑡 → 𝐼𝑛𝑡 𝑠𝑢𝑚 = 0 𝑠𝑢𝑚 𝑥 ∶ 𝑥𝑠 = 𝑥 + 𝑠𝑢𝑚 𝑥𝑠 𝑠𝑢𝑚 = 𝑓𝑜𝑙𝑑 + 0 ⟺ 𝑔 = 𝑓𝑜𝑙𝑑 𝑓 𝑣 ⟺ 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 ∷ 𝐼𝑛𝑡 → 𝐼𝑛𝑡 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 = 1 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑥 ∶ 𝑥𝑠 = 𝑥 × 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑥𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 = 𝑓𝑜𝑙𝑑 (×) 1 ⟺ 𝑙𝑒𝑛𝑔𝑡ℎ ∷ [α] → 𝐼𝑛𝑡 𝑙𝑒𝑛𝑔𝑡ℎ [ ] = 0 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 ∶ 𝑥𝑠 = 1 + 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥𝑠 𝑙𝑒𝑛𝑔𝑡ℎ = 𝑓𝑜𝑙𝑑 (𝜆𝑥. 𝜆𝑛. 1 + 𝑛) 0 ⟺ (⧺) ∷ [α] → [α] → [α] ⧺ 𝑦𝑠 = 𝑦𝑠 𝑥 ∶ 𝑥𝑠 ⧺ 𝑦𝑠 = 𝑥 ∶ (𝑥𝑠 ⧺ 𝑦𝑠) (⧺ 𝑦𝑠) = 𝑓𝑜𝑙𝑑 ∶ 𝑦𝑠 ⟺ concat ∷ [ α ] → [α] concat = concat 𝑥𝑠 ∶ 𝑥𝑠𝑠 = 𝑥𝑠 ⧺ 𝑐𝑜𝑛𝑐𝑎𝑡 𝑥𝑠𝑠 ⟺ concat = 𝑓𝑜𝑙𝑑 (⧺) [ ]
- 4. The Triad of 𝑚𝑎𝑝, 𝑓𝑖𝑙𝑡𝑒𝑟 and 𝑓𝑜𝑙𝑑 𝑚𝑎𝑝 λ The 𝑏𝑟𝑒𝑎𝑑, 𝑏𝑢𝑡𝑡𝑒𝑟, and 𝑗𝑎𝑚 of Functional Programming =
- 5. 𝑔 = 𝑣 𝑔 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 𝑔 𝑥𝑠 𝑚𝑎𝑝 ∷ 𝛼 → 𝛽 → 𝛼 → 𝛽 𝑚𝑎𝑝 𝑓 = 𝑚𝑎𝑝 𝑓 𝑥 ∶ 𝑥𝑠 = 𝑓 𝑥 ∶ 𝑚𝑎𝑝 𝑓 𝑥𝑠 𝑚𝑎𝑝 𝑓 = 𝑓𝑜𝑙𝑑𝑟 𝜆𝑥. 𝜆𝑥𝑠. 𝑓 𝑥 ∶ 𝑥𝑠 [ ] ⟺ 𝑓𝑖𝑙𝑡𝑒𝑟 ∷ (𝛼 → 𝐵𝑜𝑜𝑙) → 𝛼 → 𝑎 𝑓𝑖𝑙𝑡𝑒𝑟 p = 𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥 ∶ 𝑥𝑠 = 𝐢𝐟 𝑝 𝑥 𝐭𝐡𝐞𝐧 𝑥 ∶ 𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥𝑠 𝐞𝐥𝐬𝐞 𝑓𝑖𝑙𝑡𝑒𝑟 p 𝑥𝑠 𝑓𝑖𝑙𝑡𝑒𝑟 p = 𝑓𝑜𝑙𝑑𝑟 (𝜆𝑥. 𝜆𝑥𝑠. 𝐢𝐟 𝑝 𝑥 𝐭𝐡𝐞𝐧 𝑥 ∶ 𝑥𝑠 𝐞𝐥𝐬𝐞 𝑥𝑠) [ ] ⟺ 𝑔 = 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑣 ⟺ 𝑚𝑎𝑝 λ 𝑓𝑜𝑙𝑑 =