Folding Cheat Sheet #2 - second in a series
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Programmatic and Mathematical definitions of right fold and left fold.
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𝑎0 ∶
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𝑎1 ∶
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𝑎2 ∶
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𝑎3
𝑓
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𝑓 𝑎3
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𝑓 𝑎2
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𝑓 𝑎1
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𝑏 𝑎0
𝑓
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𝑎0 𝑓
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𝑎1 𝑓
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𝑎2 𝑓
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𝑎3 𝑏
𝑎0: (𝑎1: 𝑎2: 𝑎3: )
𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3
var 𝑎𝑐𝑐 = 𝑏
foreach(𝑎 in 𝑎𝑠)
𝑎𝑐𝑐 = 𝑓(𝑎𝑐𝑐, 𝑎)
return 𝑎𝑐𝑐
𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑎𝑠
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 𝑎𝑠
𝑎𝑠 = [𝑎0, 𝑎1, 𝑎2, 𝑎3]
𝑓𝑜𝑙𝑑𝑟 ∷ 𝛼 → 𝛽 → 𝛽 → 𝛽 → 𝛼 → 𝛽
𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 = 𝑏
𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑥: 𝑥𝑠 = 𝑓 𝑥 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑥𝑠
𝑓𝑜𝑙𝑑𝑙 ∷ 𝛽 → 𝛼 → 𝛽 → 𝛽 → 𝛼 → 𝛽
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 = 𝑏
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 𝑥: 𝑥𝑠 = 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑏 𝑥 𝑥𝑠
𝑟𝑒𝑝𝑙𝑎𝑐𝑒:
∶ 𝑤𝑖𝑡ℎ 𝑓
𝑤𝑖𝑡ℎ 𝑏
𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎0, 𝑎1, 𝑎2, 𝑎3]
𝑓 𝑎0 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎1, 𝑎2, 𝑎3]
𝑓 𝑎0 (𝑓 𝑎1 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎2, 𝑎3]))
𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎3])))
𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [ ]))))
𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 𝑏)))
𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 𝑏)))
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 [𝑎0, 𝑎1, 𝑎2, 𝑎3]
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑏 𝑎0 [𝑎1, 𝑎2, 𝑎3]
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 [𝑎2, 𝑎3]
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 [𝑎3]
𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3 [ ]
𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3
Programmatic definition of right fold and left fold
𝑓𝑜𝑙𝑑𝑟 associates 𝑓 from the right
𝑓𝑜𝑙𝑑𝑙 associates 𝑓 from the left](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Folding Cheat Sheet #2 - second in a series
- 1. CHEAT-SHEET Folding #2 ∶ / 𝒂𝟎 ∶ / 𝒂𝟏 ∶ / 𝒂𝟐 ∶ / 𝒂𝟑 𝒇 / 𝒂𝟎 𝒇 / 𝒂𝟏 𝒇 / 𝒂𝟐 𝒇 / 𝒂𝟑 𝒆 @philip_schwarz slides by https://fpilluminated.com/
- 2. ∶ / 𝑎0 ∶ / 𝑎1 ∶ / 𝑎2 ∶ / 𝑎3 𝑓 / 𝑓 𝑎3 / 𝑓 𝑎2 / 𝑓 𝑎1 / 𝑏 𝑎0 𝑓 / 𝑎0 𝑓 / 𝑎1 𝑓 / 𝑎2 𝑓 / 𝑎3 𝑏 𝑎0: (𝑎1: 𝑎2: 𝑎3: ) 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3 var 𝑎𝑐𝑐 = 𝑏 foreach(𝑎 in 𝑎𝑠) 𝑎𝑐𝑐 = 𝑓(𝑎𝑐𝑐, 𝑎) return 𝑎𝑐𝑐 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑎𝑠 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 𝑎𝑠 𝑎𝑠 = [𝑎0, 𝑎1, 𝑎2, 𝑎3] 𝑓𝑜𝑙𝑑𝑟 ∷ 𝛼 → 𝛽 → 𝛽 → 𝛽 → 𝛼 → 𝛽 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 = 𝑏 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑥: 𝑥𝑠 = 𝑓 𝑥 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 𝑥𝑠 𝑓𝑜𝑙𝑑𝑙 ∷ 𝛽 → 𝛼 → 𝛽 → 𝛽 → 𝛼 → 𝛽 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 = 𝑏 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 𝑥: 𝑥𝑠 = 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑏 𝑥 𝑥𝑠 𝑟𝑒𝑝𝑙𝑎𝑐𝑒: ∶ 𝑤𝑖𝑡ℎ 𝑓 𝑤𝑖𝑡ℎ 𝑏 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎0, 𝑎1, 𝑎2, 𝑎3] 𝑓 𝑎0 𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎1, 𝑎2, 𝑎3] 𝑓 𝑎0 (𝑓 𝑎1 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎2, 𝑎3])) 𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [𝑎3]))) 𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 (𝑓𝑜𝑙𝑑𝑟 𝑓 𝑏 [ ])))) 𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 𝑏))) 𝑓 𝑎0 (𝑓 𝑎1 (𝑓 𝑎2 (𝑓 𝑎3 𝑏))) 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑏 [𝑎0, 𝑎1, 𝑎2, 𝑎3] 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑏 𝑎0 [𝑎1, 𝑎2, 𝑎3] 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 [𝑎2, 𝑎3] 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 [𝑎3] 𝑓𝑜𝑙𝑑𝑙 𝑓 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3 [ ] 𝑓 𝑓 𝑓 𝑓 𝑏 𝑎0 𝑎1 𝑎2 𝑎3 Programmatic definition of right fold and left fold 𝑓𝑜𝑙𝑑𝑟 associates 𝑓 from the right 𝑓𝑜𝑙𝑑𝑙 associates 𝑓 from the left
- 3. ∶ / 𝑎0 ∶ / 𝑎1 ∶ / 𝑎2 ∶ / 𝑎3 𝑓 / 𝑓 𝑎3 / 𝑓 𝑎2 / 𝑓 𝑎1 / 𝑏 𝑎0 𝑓 / 𝑎0 𝑓 / 𝑎1 𝑓 / 𝑎2 𝑓 / 𝑎3 𝑏 𝑎0: (𝑎1: 𝑎2: 𝑎3: ) 𝑓(𝑓 𝑓 𝑓 𝑏, 𝑎0 , 𝑎1 , 𝑎2 , 𝑎3) 𝑓𝑜𝑙𝑑𝑟 𝑎𝑠 𝑓𝑜𝑙𝑑𝑙 𝑎𝑠 𝑎𝑠 = [𝑎0, 𝑎1, 𝑎2, 𝑎3] 𝑓(𝑎0, 𝑓(𝑎1, 𝑓(𝑎2, 𝑓(𝑎3, 𝑏)))) 𝑓𝑜𝑙𝑑𝑙 = 𝑏; 𝑓𝑜𝑙𝑑𝑙 𝑥𝑠 ⧺ [𝑥] = 𝑓(𝑓𝑜𝑙𝑑𝑙 𝑥𝑠 , 𝑥) 𝑓𝑜𝑙𝑑𝑟 = 𝑏; 𝑓𝑜𝑙𝑑𝑟 𝑥 ⧺ 𝑥𝑠 = 𝑓(𝑥, 𝑓𝑜𝑙𝑑𝑟(𝑥𝑠)) 𝑓𝑜𝑙𝑑𝑙 𝑎0, 𝑎1, 𝑎2, 𝑎3 , 𝑓 𝑓𝑜𝑙𝑑𝑙 𝑎0, 𝑎1, 𝑎2 , 𝑎3 𝑓(𝑓(𝑓𝑜𝑙𝑑𝑙 [𝑎0, 𝑎1] , 𝑎2), 𝑎3) 𝑓 𝑓 𝑓 𝑓𝑜𝑙𝑑𝑙 𝑎0 , 𝑎1 , 𝑎2 , 𝑎3 𝑓 𝑓 𝑓 𝑓 𝑓𝑜𝑙𝑑𝑙 [ ] , 𝑎0 , 𝑎1 , 𝑎2 , 𝑎3 𝑓 𝑓 𝑓 𝑓 𝑏, 𝑎0 , 𝑎1 , 𝑎2 , 𝑎3 𝑓𝑜𝑙𝑑𝑟([𝑎0, 𝑎1, 𝑎2, 𝑎3]) 𝑓(𝑎0, 𝑓𝑜𝑙𝑑𝑟([𝑎1, 𝑎2, 𝑎3])) 𝑓(𝑎0, 𝑓(𝑎1, 𝑓𝑜𝑙𝑑𝑟([𝑎2, 𝑎3]))) 𝑓(𝑎0, 𝑓(𝑎1, 𝑓(𝑎2, 𝑓𝑜𝑙𝑑𝑟([𝑎3])))) 𝑓(𝑎0, 𝑓(𝑎1, 𝑓(𝑎2, 𝑓(𝑎3, 𝑓𝑜𝑙𝑑𝑟([]))))) 𝑓(𝑎0, 𝑓(𝑎1, 𝑓(𝑎2, 𝑓(𝑎3, 𝑏)))) Mathematical definition of right fold and left fold 𝑓𝑜𝑙𝑑𝑟 associates 𝑓 from the right 𝑓𝑜𝑙𝑑𝑙 associates 𝑓 from the left 𝑥 = the Birst element 𝑥s = all but the Birst element 𝑥 = the last element 𝑥𝑠 = all but the last element