1 | include "extralib.ma". |
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2 | include "Errors.ma". |
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3 | |
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4 | (* IO monad *) |
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5 | |
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6 | inductive IO (output:Type[0]) (input:output → Type[0]) (T:Type[0]) : Type[0] ≝ |
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7 | | Interact : ∀o:output. (input o → IO output input T) → IO output input T |
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8 | | Value : T → IO output input T |
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9 | | Wrong : IO output input T. |
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10 | |
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11 | let rec bindIO (O:Type[0]) (I:O → Type[0]) (T,T':Type[0]) (v:IO O I T) (f:T → IO O I T') on v : IO O I T' ≝ |
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12 | match v with |
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13 | [ Interact out k ⇒ (Interact ??? out (λres. bindIO O I T T' (k res) f)) |
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14 | | Value v' ⇒ (f v') |
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15 | | Wrong ⇒ Wrong O I T' |
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16 | ]. |
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17 | |
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18 | let rec bindIO2 (O:Type[0]) (I:O → Type[0]) (T1,T2,T':Type[0]) (v:IO O I (T1×T2)) (f:T1 → T2 → IO O I T') on v : IO O I T' ≝ |
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19 | match v with |
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20 | [ Interact out k ⇒ (Interact ??? out (λres. bindIO2 ?? T1 T2 T' (k res) f)) |
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21 | | Value v' ⇒ match v' with [ pair v1 v2 ⇒ f v1 v2 ] |
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22 | | Wrong ⇒ Wrong ?? T' |
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23 | ]. |
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24 | |
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25 | definition err_to_io : ∀O,I,T. res T → IO O I T ≝ |
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26 | λO,I,T,v. match v with [ OK v' ⇒ Value O I T v' | Error ⇒ Wrong O I T ]. |
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27 | coercion err_to_io : ∀O,I,A.∀c:res A.IO O I A ≝ err_to_io on _c:res ? to IO ???. |
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28 | definition err_to_io_sig : ∀O,I,T.∀P:T → Prop. res (Sig T P) → IO O I (Sig T P) ≝ |
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29 | λO,I,T,P,v. match v with [ OK v' ⇒ Value O I (Sig T P) v' | Error ⇒ Wrong O I (Sig T P) ]. |
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30 | (*coercion err_to_io_sig : ∀O,I,A.∀P:A → Prop.∀c:res (Sig A P).IO O I (Sig A P) ≝ err_to_io_sig on _c:res (Sig ??) to IO ?? (Sig ??).*) |
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31 | |
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32 | |
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33 | (* If the original definitions are vague enough, do I need to do this? *) |
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34 | notation > "! ident v ← e; e'" with precedence 40 for @{'bindIO ${e} (λ${ident v}.${e'})}. |
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35 | notation > "! ident v : ty ← e; e'" with precedence 40 for @{'bindIO ${e} (λ${ident v} : ${ty}.${e'})}. |
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36 | notation < "vbox(! \nbsp ident v ← e; break e')" with precedence 40 for @{'bindIO ${e} (λ${ident v}.${e'})}. |
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37 | notation < "vbox(! \nbsp ident v : ty ← e; break e')" with precedence 40 for @{'bindIO ${e} (λ${ident v} : ${ty}.${e'})}. |
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38 | notation > "! 〈ident v1, ident v2〉 ← e; e'" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}. |
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39 | notation > "! 〈ident v1 : ty1, ident v2 : ty2〉 ← e; e'" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}. |
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40 | notation < "vbox(! \nbsp 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}. |
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41 | notation < "vbox(! \nbsp 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bindIO2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}. |
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42 | interpretation "IO monad bind" 'bindIO e f = (bindIO ???? e f). |
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43 | interpretation "IO monad Prod bind" 'bindIO2 e f = (bindIO2 ????? e f). |
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44 | (**) |
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45 | let rec P_io O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝ |
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46 | match v return λ_.Prop with |
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47 | [ Wrong ⇒ True |
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48 | | Value z ⇒ P z |
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49 | | Interact out k ⇒ ∀v'.P_io O I A P (k v') |
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50 | ]. |
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51 | |
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52 | let rec P_io' O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝ |
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53 | match v return λ_.Prop with |
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54 | [ Wrong ⇒ False |
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55 | | Value z ⇒ P z |
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56 | | Interact out k ⇒ ∀v'.P_io' O I A P (k v') |
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57 | ]. |
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58 | |
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59 | definition P_to_P_option_io : ∀O,I,A.∀P:A → Prop.option (IO O I A) → Prop ≝ |
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60 | λO,I,A,P,a.match a with |
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61 | [ None ⇒ False |
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62 | | Some y ⇒ P_io O I A P y |
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63 | ]. |
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64 | |
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65 | let rec io_inject_0 O I (A:Type[0]) (P:A → Prop) (a:IO O I A) (p:P_io O I A P a) on a : IO O I (Sig A P) ≝ |
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66 | (match a return λa'.P_io O I A P a' → ? with |
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67 | [ Wrong ⇒ λ_. Wrong O I ? |
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68 | | Value c ⇒ λp'. Value ??? (dp A P c p') |
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69 | | Interact out k ⇒ λp'. Interact ??? out (λv. io_inject_0 O I A P (k v) (p' v)) |
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70 | ]) p. |
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71 | |
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72 | definition io_inject : ∀O,I,A.∀P:A → Prop.∀a:option (IO O I A).∀p:P_to_P_option_io O I A P a.IO O I (Sig A P) ≝ |
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73 | λO,I,A.λP:A → Prop.λa:option (IO O I A).λp:P_to_P_option_io O I A P a. |
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74 | (match a return λa'.P_to_P_option_io O I A P a' → IO O I (Sig A P) with |
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75 | [ None ⇒ λp'.? |
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76 | | Some b ⇒ λp'. io_inject_0 O I A P b p' |
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77 | ]) p. |
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78 | elim p'; qed. |
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79 | |
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80 | let rec io_eject O I (A:Type[0]) (P: A → Prop) (a:IO O I (Sig A P)) on a : IO O I A ≝ |
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81 | match a with |
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82 | [ Wrong ⇒ Wrong ??? |
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83 | | Value b ⇒ match b with [ dp w p ⇒ Value ??? w] |
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84 | | Interact out k ⇒ Interact ??? out (λv. io_eject ?? A P (k v)) |
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85 | ]. |
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86 | |
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87 | coercion io_inject : |
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88 | ∀O,I,A.∀P:A → Prop.∀a.∀p:P_to_P_option_io O I ? P a.IO O I (Sig A P) ≝ io_inject |
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89 | on a:option (IO ???) to IO ?? (Sig ? ?). |
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90 | coercion io_eject : ∀O,I,A.∀P:A → Prop.∀c:IO O I (Sig A P).IO O I A ≝ io_eject |
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91 | on _c:IO ?? (Sig ? ?) to IO ???. |
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92 | |
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93 | definition opt_to_io : ∀O,I,T.option T → IO O I T ≝ |
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94 | λO,I,T,v. match v with [ None ⇒ Wrong ?? T | Some v' ⇒ Value ??? v' ]. |
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95 | coercion opt_to_io : ∀O,I,T.∀v:option T. IO O I T ≝ opt_to_io on _v:option ? to IO ???. |
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96 | |
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97 | lemma sig_bindIO_OK: ∀O,I,A,B. ∀P:A → Prop. ∀P':B → Prop. ∀e:IO O I (Sig A P). ∀f:Sig A P → IO O I B. |
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98 | (∀v:A. ∀p:P v. P_io O I ? P' (f (dp A P v p))) → |
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99 | P_io O I ? P' (bindIO O I (Sig A P) B e f). |
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100 | #O #I #A #B #P #P' #e #f elim e; |
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101 | [ #out #k #IH #IH' whd; #res @IH //; |
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102 | | #v0 elim v0; #v #Hv #IH whd; @IH |
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103 | | //; |
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104 | ] qed. |
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105 | |
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106 | lemma sig_bindIO2_OK: ∀O,I,A,B,C. ∀P:(A×B) → Prop. ∀P':C → Prop. ∀e:IO O I (Sig (A×B) P). ∀f: A → B → IO O I C. |
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107 | (∀vA:A.∀vB:B. ∀p:P 〈vA,vB〉. P_io O I ? P' (f vA vB)) → |
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108 | P_io O I ? P' (bindIO2 O I A B C e f). |
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109 | #I #O #A #B #C #P #P' #e #f elim e; |
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110 | [ #out #k #IH #IH' whd; #res @IH @IH' |
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111 | | #v0 elim v0; #v elim v; #vA #vB #Hv #IH @IH //; |
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112 | | //; |
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113 | ] qed. |
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114 | |
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115 | lemma opt_bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:option A. ∀f: A → IO O I B. |
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116 | (∀v:A. e = Some A v → P_io O I ? P (f v)) → |
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117 | P_io O I ? P (bindIO O I A B e f). |
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118 | #I #O #A #B #P #e elim e; //; #v #f #H @H //; |
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119 | qed. |
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120 | |
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121 | lemma opt_bindIO2_OK: ∀O,I,A,B,C. ∀P:C → Prop. ∀e:option (A×B). ∀f: A → B → IO O I C. |
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122 | (∀vA:A.∀vB:B. e = Some (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) → |
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123 | P_io O I ? P (bindIO2 O I A B C e f). |
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124 | #I #O #A #B #C #P #e elim e; //; #v cases v; #vA #vB #f #H @H //; |
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125 | qed. |
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126 | |
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127 | lemma res_bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:res A. ∀f: A → IO O I B. |
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128 | (∀v:A. e = OK A v → P_io O I ? P (f v)) → |
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129 | P_io O I ? P (bindIO O I A B e f). |
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130 | #I #O #A #B #P #e elim e; //; #v #f #H @H //; |
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131 | qed. |
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132 | |
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133 | lemma res_bindIO2_OK: ∀O,I,A,B,C. ∀P:C → Prop. ∀e:res (A×B). ∀f: A → B → IO O I C. |
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134 | (∀vA:A.∀vB:B. e = OK (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) → |
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135 | P_io O I ? P (bindIO2 O I A B C e f). |
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136 | #I #O #A #B #C #P #e elim e; //; #v cases v; #vA #vB #f #H @H //; |
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137 | qed. |
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138 | |
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139 | lemma bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:IO O I A. ∀f: A → IO O I B. |
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140 | (∀v:A. P_io O I ? P (f v)) → |
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141 | P_io O I ? P (bindIO O I A B e f). |
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142 | #I #O #A #B #P #e elim e; |
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143 | [ #out #k #IH #f #H whd; #res @IH //; |
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144 | | #v #f #H @H |
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145 | | //; |
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146 | ] qed. |
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147 | |
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148 | lemma bindIO2_OK: ∀O,I,A,B,C. ∀P:C → Prop. ∀e:IO O I (A×B). ∀f: A → B → IO O I C. |
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149 | (∀v1:A.∀v2:B. P_io O I ? P (f v1 v2)) → |
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150 | P_io O I ? P (bindIO2 O I A B C e f). |
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151 | #I #O #A #B #C #P #e elim e; |
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152 | [ #out #k #IH #f #H whd; #res @IH //; |
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153 | | #v cases v; #v1 #v2 #f #H @H |
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154 | | //; |
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155 | ] qed. |
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156 | |
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157 | lemma P_bindIO_OK: ∀O,I,A,B. ∀P':A → Prop. ∀P:B → Prop. ∀e:IO O I A. ∀f: A → IO O I B. |
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158 | P_io … P' e → |
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159 | (∀v:A. P' v → P_io O I ? P (f v)) → |
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160 | P_io O I ? P (bindIO O I A B e f). |
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161 | #I #O #A #B #P' #P #e elim e; |
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162 | [ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2/; |
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163 | | #v #f #He #H @H @He |
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164 | | //; |
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165 | ] qed. |
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166 | |
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167 | lemma P_bindIO2_OK: ∀O,I,A,B,C. ∀P':A×B → Prop. ∀P:C → Prop. ∀e:IO O I (A×B). ∀f: A → B → IO O I C. |
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168 | P_io … P' e → |
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169 | (∀v1:A.∀v2:B. P' 〈v1,v2〉 → P_io O I ? P (f v1 v2)) → |
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170 | P_io O I ? P (bindIO2 O I A B C e f). |
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171 | #I #O #A #B #C #P' #P #e elim e; |
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172 | [ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2/; |
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173 | | #v cases v; #v1 #v2 #f #He #H @H @He |
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174 | | //; |
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175 | ] qed. |
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176 | |
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177 | |
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178 | (* Is there a way to prove this without extensionality? *) |
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179 | |
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180 | lemma bind_assoc_r: ∀O,I,A,B,C,e,f,g. |
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181 | ∀ext:(∀T1,T2:Type[0].∀f,f':T1 → T2.(∀x.f x = f' x) → f = f'). |
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182 | bindIO O I B C (bindIO O I A B e f) g = bindIO O I A C e (λx.bindIO O I B C (f x) g). |
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183 | #O #I #A #B #C #e #f #g #ext elim e; |
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184 | [ #o #k #IH whd in ⊢ (??%%); @eq_f |
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185 | @ext @IH |
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186 | | #v @refl |
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187 | | @refl |
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188 | ] qed. |
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189 | |
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190 | (* |
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191 | lemma extract_subset_pair_io: ∀O,I,A,B,C,P. ∀e:{e:A×B | P e}. ∀Q:A→B→IO O I C. ∀R:C→Prop. |
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192 | (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → P_io O I ? R (Q a b)) → |
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193 | P_io O I ? R (match eject ?? e with [ pair a b ⇒ Q a b ]). |
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194 | #I #O #A #B #C #P #e #Q #R cases e; #e' cases e'; normalize; |
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195 | [ *; |
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196 | | #e'' cases e''; #a #b #Pab #H normalize; /2/; |
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197 | ] qed. |
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198 | *) |
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