1 | include "Plogic/russell_support.ma". |
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2 | include "extralib.ma". |
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3 | include "Errors.ma". |
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4 | |
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5 | (* IO monad *) |
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6 | |
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7 | ninductive IO (input,output:Type) (T:Type) : Type ≝ |
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8 | | Interact : output → (input → IO input output T) → IO input output T |
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9 | | Value : T → IO input output T |
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10 | | Wrong : IO input output T. |
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11 | |
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12 | nlet rec bindIO (I,O,T,T':Type) (v:IO I O T) (f:T → IO I O T') on v : IO I O T' ≝ |
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13 | match v with |
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14 | [ Interact out k ⇒ (Interact ??? out (λres. bindIO I O T T' (k res) f)) |
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15 | | Value v' ⇒ (f v') |
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16 | | Wrong ⇒ Wrong I O T' |
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17 | ]. |
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18 | |
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19 | nlet rec bindIO2 (I,O,T1,T2,T':Type) (v:IO I O (T1×T2)) (f:T1 → T2 → IO I O T') on v : IO I O T' ≝ |
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20 | match v with |
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21 | [ Interact out k ⇒ (Interact ??? out (λres. bindIO2 ?? T1 T2 T' (k res) f)) |
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22 | | Value v' ⇒ match v' with [ mk_pair v1 v2 ⇒ f v1 v2 ] |
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23 | | Wrong ⇒ Wrong ?? T' |
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24 | ]. |
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25 | |
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26 | ndefinition err_to_io : ∀I,O,T. res T → IO I O T ≝ |
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27 | λI,O,T,v. match v with [ OK v' ⇒ Value I O T v' | Error ⇒ Wrong I O T ]. |
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28 | (*ncoercion err_to_io : ∀I,O,A.∀c:res A.IO I O A ≝ err_to_io on _c:res ? to IO ???.*) |
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29 | ndefinition err_to_io_sig : ∀I,O,T.∀P:T → Prop. res (sigma T P) → IO I O (sigma T P) ≝ |
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30 | λI,O,T,P,v. match v with [ OK v' ⇒ Value I O (sigma T P) v' | Error ⇒ Wrong I O (sigma T P) ]. |
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31 | ncoercion err_to_io_sig : ∀I,O,A.∀P:A → Prop.∀c:res (sigma A P).IO I O (sigma A P) ≝ err_to_io_sig on _c:res (sigma ??) to IO ?? (sigma ??). |
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32 | |
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33 | |
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34 | (* If the original definitions are vague enough, do I need to do this? *) |
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35 | notation "! ident v ← e;: e'" right associative with precedence 40 for @{'bindIO ${e} (λ${ident v}.${e'})}. |
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36 | notation "! 〈ident v1, ident v2〉 ← e;: e'" right associative with precedence 40 for @{'bindIO2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}. |
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37 | interpretation "IO monad bind" 'bindIO e f = (bindIO ???? e f). |
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38 | interpretation "IO monad pair bind" 'bindIO2 e f = (bindIO2 ????? e f). |
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39 | (**) |
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40 | nlet rec P_io (I,O,A:Type) (P:A → Prop) (v:IO I O A) on v : Prop ≝ |
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41 | match v return λ_.Prop with |
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42 | [ Wrong ⇒ True |
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43 | | Value z ⇒ P z |
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44 | | Interact out k ⇒ ∀v'.P_io I O A P (k v') |
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45 | ]. |
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46 | |
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47 | nlet rec P_io' (I,O,A:Type) (P:A → Prop) (v:IO I O A) on v : Prop ≝ |
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48 | match v return λ_.Prop with |
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49 | [ Wrong ⇒ False |
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50 | | Value z ⇒ P z |
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51 | | Interact out k ⇒ ∀v'.P_io' I O A P (k v') |
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52 | ]. |
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53 | |
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54 | ndefinition P_to_P_option_io : ∀I,O,A.∀P:A → Prop.option (IO I O A) → Prop ≝ |
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55 | λI,O,A,P,a.match a with |
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56 | [ None ⇒ False |
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57 | | Some y ⇒ P_io I O A P y |
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58 | ]. |
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59 | |
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60 | nlet rec io_inject_0 (I,O,A:Type) (P:A → Prop) (a:IO I O A) (p:P_io I O A P a) on a : IO I O (sigma A P) ≝ |
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61 | (match a return λa'.a=a' → ? with |
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62 | [ Wrong ⇒ λ_. Wrong I O ? |
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63 | | Value c ⇒ λe2. Value ??? (sig_intro A P c ?) |
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64 | | Interact out k ⇒ λe2. Interact ??? out (λv. io_inject_0 I O A P (k v) ?) |
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65 | ]) (refl ? a). |
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66 | nrewrite > e2 in p; nwhd in ⊢ (% → ?); //; |
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67 | nqed. |
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68 | |
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69 | ndefinition io_inject : ∀I,O,A.∀P:A → Prop.∀a:option (IO I O A).∀p:P_to_P_option_io I O A P a.IO I O (sigma A P) ≝ |
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70 | λI,O,A.λP:A → Prop.λa:option (IO I O A).λp:P_to_P_option_io I O A P a. |
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71 | (match a return λa'.a=a' → IO I O (sigma A P) with |
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72 | [ None ⇒ λe1.? |
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73 | | Some b ⇒ λe1. io_inject_0 I O A P b ? |
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74 | ]) (refl ? a). |
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75 | ##[ nrewrite > e1 in p; nnormalize; *; |
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76 | ##| nrewrite > e1 in p; nnormalize; // |
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77 | ##] nqed. |
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78 | |
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79 | nlet rec io_eject (I,O,A:Type) (P: A → Prop) (a:IO I O (sigma A P)) on a : IO I O A ≝ |
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80 | match a with |
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81 | [ Wrong ⇒ Wrong ??? |
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82 | | Value b ⇒ match b with [ sig_intro w p ⇒ Value ??? w] |
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83 | | Interact out k ⇒ Interact ??? out (λv. io_eject ?? A P (k v)) |
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84 | ]. |
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85 | |
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86 | ncoercion io_inject : |
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87 | ∀I,O,A.∀P:A → Prop.∀a.∀p:P_to_P_option_io I O ? P a.IO I O (sigma A P) ≝ io_inject |
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88 | on a:option (IO ???) to IO ?? (sigma ? ?). |
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89 | ncoercion io_eject : ∀I,O,A.∀P:A → Prop.∀c:IO I O (sigma A P).IO I O A ≝ io_eject |
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90 | on _c:IO ?? (sigma ? ?) to IO ???. |
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91 | |
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92 | ndefinition opt_to_io : ∀I,O,T.option T → IO I O T ≝ |
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93 | λI,O,T,v. match v with [ None ⇒ Wrong I O T | Some v' ⇒ Value I O T v' ]. |
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94 | ncoercion opt_to_io : ∀I,O,T.∀v:option T. IO I O T ≝ opt_to_io on _v:option ? to IO ???. |
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95 | |
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96 | nlemma sig_bindIO_OK: ∀I,O,A,B. ∀P:A → Prop. ∀P':B → Prop. ∀e:IO I O (sigma A P). ∀f:sigma A P → IO I O B. |
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97 | (∀v:A. ∀p:P v. P_io I O ? P' (f (sig_intro A P v p))) → |
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98 | P_io I O ? P' (bindIO I O (sigma A P) B e f). |
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99 | #I O A B P P' e f; nelim e; |
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100 | ##[ #out k IH; #IH'; nwhd; #res; napply IH; //; |
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101 | ##| #v0; nelim v0; #v Hv IH; nwhd; napply IH; |
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102 | ##| //; |
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103 | ##] nqed. |
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104 | |
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105 | nlemma sig_bindIO2_OK: ∀I,O,A,B,C. ∀P:(A×B) → Prop. ∀P':C → Prop. ∀e:IO I O (sigma (A×B) P). ∀f: A → B → IO I O C. |
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106 | (∀vA:A.∀vB:B. ∀p:P 〈vA,vB〉. P_io I O ? P' (f vA vB)) → |
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107 | P_io I O ? P' (bindIO2 I O A B C e f). |
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108 | #I O A B C P P' e f; nelim e; |
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109 | ##[ #out k IH; #IH'; nwhd; #res; napply IH; napply IH'; |
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110 | ##| #v0; nelim v0; #v; nelim v; #vA vB Hv IH; napply IH; //; |
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111 | ##| //; |
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112 | ##] nqed. |
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113 | |
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114 | nlemma opt_bindIO_OK: ∀I,O,A,B. ∀P:B → Prop. ∀e:option A. ∀f: A → IO I O B. |
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115 | (∀v:A. e = Some A v → P_io I O ? P (f v)) → |
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116 | P_io I O ? P (bindIO I O A B e f). |
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117 | #I O A B P e; nelim e; //; #v f H; napply H; //; |
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118 | nqed. |
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119 | |
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120 | nlemma bindIO_OK: ∀I,O,A,B. ∀P:B → Prop. ∀e:IO I O A. ∀f: A → IO I O B. |
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121 | (∀v:A. P_io I O ? P (f v)) → |
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122 | P_io I O ? P (bindIO I O A B e f). |
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123 | #I O A B P e; nelim e; |
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124 | ##[ #out k IH; #f H; nwhd; #res; napply IH; //; |
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125 | ##| #v f H; napply H; |
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126 | ##| //; |
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127 | ##] nqed. |
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128 | |
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129 | (* TODO: is there a way to prove this without extensionality? |
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130 | |
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131 | nlemma bind_assoc_r: ∀A,B,C,e,f,g. |
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132 | bindIO B C (bindIO A B e f) g = bindIO A C e (λx.bindIO B C (f x) g). |
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133 | #A B C e f g; nelim e; |
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134 | ##[ #fn args k IH; nwhd in ⊢ (???%); |
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135 | nnormalize; |
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136 | *) |
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137 | |
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138 | nlemma extract_subset_pair_io: ∀I,O,A,B,C,P. ∀e:{e:A×B | P e}. ∀Q:A→B→IO I O C. ∀R:C→Prop. |
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139 | (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → P_io I O ? R (Q a b)) → |
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140 | P_io I O ? R (match eject ?? e with [ mk_pair a b ⇒ Q a b ]). |
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141 | #I O A B C P e Q R; ncases e; #e'; ncases e'; nnormalize; |
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142 | ##[ *; |
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143 | ##| #e''; ncases e''; #a b Pab H; nnormalize; /2/; |
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144 | ##] nqed. |
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145 | |
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