1 | (* RUSSEL **) |
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2 | |
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3 | include "basics/jmeq.ma". |
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4 | include "basics/types.ma". |
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5 | include "basics/list.ma". |
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6 | |
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7 | notation > "hvbox(a break ≃ b)" |
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8 | non associative with precedence 45 |
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9 | for @{ 'jmeq ? $a ? $b }. |
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10 | |
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11 | notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)" |
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12 | non associative with precedence 45 |
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13 | for @{ 'jmeq $t $a $u $b }. |
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14 | |
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15 | interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y). |
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16 | |
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17 | lemma eq_to_jmeq: |
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18 | ∀A: Type[0]. |
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19 | ∀x, y: A. |
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20 | x = y → x ≃ y. |
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21 | // |
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22 | qed. |
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23 | |
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24 | definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p. |
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25 | definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w]. |
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26 | |
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27 | coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?. |
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28 | coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?. |
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29 | |
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30 | axiom VOID: Type[0]. |
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31 | axiom assert_false: VOID. |
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32 | definition bigbang: ∀A:Type[0].False → VOID → A. |
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33 | #A #abs cases abs |
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34 | qed. |
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35 | |
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36 | coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?. |
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37 | |
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38 | lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p). |
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39 | #A #P #p cases p #w #q @q |
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40 | qed. |
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41 | |
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42 | lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y. |
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43 | #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) % |
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44 | qed. |
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45 | |
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46 | coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?. |
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47 | |
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48 | (* END RUSSELL **) |
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49 | |
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50 | include "ASM/Util.ma". |
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51 | |
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52 | let rec foldl_strong_internal |
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53 | (A: Type[0]) (P: list A → Type[0]) (l: list A) |
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54 | (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd])) |
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55 | (prefix: list A) (suffix: list A) (acc: P prefix) on suffix: |
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56 | l = prefix @ suffix → P(prefix @ suffix) ≝ |
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57 | match suffix return λl'. l = prefix @ l' → P (prefix @ l') with |
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58 | [ nil ⇒ λprf. ? |
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59 | | cons hd tl ⇒ λprf. ? |
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60 | ]. |
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61 | [ > (append_nil ?) |
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62 | @ acc |
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63 | | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?) |
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64 | [ @ (H prefix hd tl prf acc) |
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65 | | applyS prf |
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66 | ] |
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67 | ] |
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68 | qed. |
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69 | |
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70 | definition foldl_strong ≝ |
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71 | λA: Type[0]. |
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72 | λP: list A → Type[0]. |
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73 | λl: list A. |
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74 | λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
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75 | λacc: P [ ]. |
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76 | foldl_strong_internal A P l H [ ] l acc (refl …). |
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77 | |
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78 | let rec foldr_strong_internal |
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79 | (A:Type[0]) |
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80 | (P: list A → Type[0]) |
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81 | (l: list A) |
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82 | (H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl)) |
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83 | (prefix: list A) (suffix: list A) (acc: P [ ]) on suffix : l = prefix@suffix → P suffix ≝ |
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84 | match suffix return λl'. l = prefix @ l' → P (l') with |
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85 | [ nil ⇒ λprf. acc |
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86 | | cons hd tl ⇒ λprf. H prefix hd tl prf (foldr_strong_internal A P l H (prefix @ [hd]) tl acc ?) ]. |
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87 | applyS prf |
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88 | qed. |
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89 | |
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90 | lemma foldr_strong: |
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91 | ∀A:Type[0]. |
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92 | ∀P: list A → Type[0]. |
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93 | ∀l: list A. |
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94 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P tl → P (hd::tl). |
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95 | ∀acc:P [ ]. P l |
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96 | ≝ λA,P,l,H,acc. foldr_strong_internal A P l H [ ] l acc (refl …). |
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97 | |
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98 | lemma pair_destruct: ∀A,B,a1,a2,b1,b2. pair A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2. |
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99 | #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/ |
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100 | qed. |
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