1 | include "ASM/Util.ma". |
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2 | include "basics/russell.ma". |
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3 | |
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4 | let rec foldl_strong_internal |
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5 | (A: Type[0]) (P: list A → Type[0]) (l: list A) |
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6 | (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd])) |
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7 | (prefix: list A) (suffix: list A) (acc: P prefix) on suffix: |
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8 | l = prefix @ suffix → P(prefix @ suffix) ≝ |
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9 | match suffix return λl'. l = prefix @ l' → P (prefix @ l') with |
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10 | [ nil ⇒ λprf. ? |
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11 | | cons hd tl ⇒ λprf. ? |
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12 | ]. |
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13 | [ > (append_nil ?) |
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14 | @ acc |
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15 | | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?) |
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16 | [ @ (H prefix hd tl prf acc) |
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17 | | applyS prf |
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18 | ] |
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19 | ] |
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20 | qed. |
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21 | |
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22 | definition foldl_strong ≝ |
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23 | λA: Type[0]. |
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24 | λP: list A → Type[0]. |
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25 | λl: list A. |
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26 | λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
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27 | λacc: P [ ]. |
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28 | foldl_strong_internal A P l H [ ] l acc (refl …). |
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29 | |
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30 | let rec foldr_strong_internal |
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31 | (A:Type[0]) |
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32 | (P: list A → Type[0]) |
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33 | (l: list A) |
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34 | (H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl)) |
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35 | (prefix: list A) (suffix: list A) (acc: P [ ]) on suffix : l = prefix@suffix → P suffix ≝ |
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36 | match suffix return λl'. l = prefix @ l' → P (l') with |
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37 | [ nil ⇒ λprf. acc |
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38 | | cons hd tl ⇒ λprf. H prefix hd tl prf (foldr_strong_internal A P l H (prefix @ [hd]) tl acc ?) ]. |
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39 | applyS prf |
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40 | qed. |
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41 | |
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42 | lemma foldr_strong: |
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43 | ∀A:Type[0]. |
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44 | ∀P: list A → Type[0]. |
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45 | ∀l: list A. |
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46 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl). |
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47 | ∀acc:P [ ]. P l |
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48 | ≝ λA,P,l,H,acc. foldr_strong_internal A P l H [ ] l acc (refl …). |
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49 | |
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50 | lemma pair_destruct: ∀A,B,a1,a2,b1,b2. mk_Prod A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2. |
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51 | #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/ |
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52 | qed. |
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