1 | include "basics/lists/listb.ma". |
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2 | |
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3 | definition bool_to_Prop : bool → Prop ≝ |
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4 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
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5 | |
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6 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
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7 | |
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8 | definition defined : ∀T:Type[0]. option T → bool ≝ |
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9 | λT,x. match x with [ None ⇒ false | Some _ ⇒ true ]. |
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10 | |
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11 | let rec for_each (T:Type[0]) (f: T → bool) (l: list T) on l : bool ≝ |
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12 | match l with |
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13 | [ nil ⇒ true |
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14 | | cons hd tl ⇒ for_each T f tl ∧ f hd ]. |
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15 | |
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16 | lemma for_each_to_memb_to_true: |
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17 | ∀T:DeqSet. ∀f: T → bool. ∀t:T. ∀l: list T. |
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18 | memb T t l → for_each T f l = true → f t = true. |
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19 | #T #f #t #l elim l |
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20 | [ #abs cases abs |
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21 | | #hd #tl #IH normalize in ⊢ (% → ?); inversion (t == hd) #H normalize |
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22 | [ #_ cases (for_each ???) normalize |
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23 | [ <(\P H) // | #abs destruct ] |
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24 | | cases (t ∈ tl) in IH; normalize |
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25 | [2: #_ #abs cases abs |
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26 | | cases (for_each ???) normalize /2/]] |
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27 | qed. |
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28 | |
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29 | let rec sublist (S:DeqSet) (l1:list S) (l2:list S) on l1 : bool ≝ |
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30 | match l1 with |
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31 | [ nil ⇒ true |
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32 | | cons hd tl ⇒ memb S hd l2 ∧ sublist S tl l2 ]. |
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33 | |
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34 | |
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35 | lemma sublist_hd_tl_to_memb: |
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36 | ∀S:DeqSet. ∀hd:S. ∀tl:list S. ∀l2. |
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37 | sublist S (hd::tl) l2 → memb S hd l2. |
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38 | #S #hd #tl #l2 normalize cases (hd ∈ l2) normalize // |
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39 | qed. |
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40 | |
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41 | lemma sublist_hd_tl_to_sublist: |
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42 | ∀S:DeqSet. ∀hd:S. ∀tl:list S. ∀l2. |
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43 | sublist S (hd::tl) l2 → sublist S tl l2. |
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44 | #S #hd #tl #l2 normalize cases (hd ∈ l2) normalize // #H cases H |
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45 | qed. |
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46 | |
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47 | lemma memb_to_sublist_to_memb: |
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48 | ∀T:DeqSet. ∀x:T. ∀l1,l2: list T. |
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49 | memb … x l1 → sublist … l1 l2 → memb … x l2. |
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50 | #T #x #l1 elim l1 |
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51 | [ #l2 #abs cases abs |
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52 | | #hd #tl #IH #l2 whd in match (memb ???); inversion (x==hd) /3/ |
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53 | #H #_ whd in match (sublist ???); <(\P H) cases (memb ???) // ] |
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54 | qed. |
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55 | |
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56 | lemma sublist_insert_snd_pos: |
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57 | ∀S:DeqSet. ∀x,y:S. ∀l1,l2:list S. |
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58 | sublist S l1 (x::l2) → sublist S l1 (x::y::l2). |
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59 | #S #x #y #l1 elim l1 [//] |
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60 | #hd #tl #IH #l2 #H |
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61 | lapply (sublist_hd_tl_to_sublist … H) #K |
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62 | whd in match (sublist ???); >(?:hd∈x::y::l2 = true) /2/ |
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63 | lapply (sublist_hd_tl_to_memb … H) normalize cases (hd==x) normalize // |
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64 | cases (hd==y) normalize // cases (hd∈l2) // #H cases H |
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65 | qed. |
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66 | |
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67 | lemma sublist_tl_hd_tl: |
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68 | ∀S:DeqSet. ∀hd:S. ∀tl:list S. sublist S tl (hd::tl). |
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69 | #S #hd #tl elim tl try % |
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70 | #hd' #tl' #IH whd in match (sublist ???); |
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71 | whd in match (hd' ∈ hd::?::?); cases (hd'==hd) |
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72 | whd in match (hd' ∈ ?::?); >(\b ?) try % normalize |
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73 | lapply (sublist_insert_snd_pos … hd' … IH) |
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74 | cases (sublist ???) // |
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75 | qed. |
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76 | |
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77 | lemma sublist_refl: ∀S:DeqSet. ∀l: list S. sublist S l l. |
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78 | #S #l elim l [//] #hd #tl #IH normalize >(\b ?) // |
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79 | qed. |
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