1 | include "utilities/extralib.ma". |
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2 | include "common/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 : errmsg → IO output input T. |
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10 | |
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11 | include "utilities/proper.ma". |
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12 | (* a weak form of extensionality *) |
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13 | axiom interact_proper : |
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14 | ∀O,I,T,o.∀f,g : I o → IO O I T.(∀i. f i = g i) → Interact … o f = Interact … o g. |
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15 | |
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16 | 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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17 | match v with |
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18 | [ Interact out k ⇒ (Interact ??? out (λres. bindIO O I T T' (k res) f)) |
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19 | | Value v' ⇒ (f v') |
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20 | | Wrong m ⇒ Wrong O I T' m |
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21 | ]. |
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22 | |
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23 | include "utilities/monad.ma". |
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24 | |
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25 | definition IOMonad ≝ λO,I. |
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26 | MakeMonadProps |
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27 | (* the monad *) |
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28 | (IO O I) |
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29 | (* return *) |
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30 | (λX.Value … X) |
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31 | (* bind *) |
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32 | (bindIO O I) |
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33 | ???. / by / |
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34 | [(* bind_ret *) |
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35 | #X#m elim m normalize // #o#f#Hi @interact_proper // |
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36 | |(* bind_bind *) |
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37 | #X#Y#Z#m#f#g elim m normalize [2,3://] |
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38 | (* Interact *) |
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39 | #o#f #Hi @interact_proper // |
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40 | ] |
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41 | qed. |
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42 | |
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43 | definition bindIO2 ≝ λO,I. m_bind2 (IOMonad O I). |
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44 | |
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45 | include "hints_declaration.ma". |
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46 | |
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47 | unification hint 0 ≔ O, I, T; |
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48 | N ≟ IOMonad O I, M ≟ max_def N, M' ≟ m_def M |
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49 | (*******************************************) ⊢ |
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50 | IO O I T ≡ monad M' T |
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51 | . |
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52 | |
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53 | |
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54 | definition err_to_io : ∀O,I,T. res T → IO O I T ≝ |
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55 | λO,I,T,v. match v with [ OK v' ⇒ Value O I T v' | Error m ⇒ Wrong O I T m ]. |
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56 | |
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57 | 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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58 | |
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59 | 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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60 | λO,I,T,P,v. match v with [ OK v' ⇒ Value O I (Sig T P) v' | Error m ⇒ Wrong O I (Sig T P) m ]. |
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61 | (*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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62 | |
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63 | let rec P_io O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝ |
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64 | match v return λ_.Prop with |
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65 | [ Wrong _ ⇒ True |
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66 | | Value z ⇒ P z |
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67 | | Interact out k ⇒ ∀v'.P_io O I A P (k v') |
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68 | ]. |
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69 | |
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70 | let rec P_io' O I (A:Type[0]) (P:A → Prop) (v:IO O I A) on v : Prop ≝ |
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71 | match v return λ_.Prop with |
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72 | [ Wrong _ ⇒ False |
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73 | | Value z ⇒ P z |
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74 | | Interact out k ⇒ ∀v'.P_io' O I A P (k v') |
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75 | ]. |
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76 | |
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77 | definition P_to_P_option_io : ∀O,I,A.∀P:A → Prop.option (IO O I A) → Prop ≝ |
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78 | λO,I,A,P,a.match a with |
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79 | [ None ⇒ False |
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80 | | Some y ⇒ P_io O I A P y |
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81 | ]. |
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82 | |
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83 | 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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84 | (match a return λa'.P_io O I A P a' → ? with |
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85 | [ Wrong m ⇒ λ_. Wrong O I ? m |
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86 | | Value c ⇒ λp'. Value ??? (mk_Sig A P c p') |
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87 | | Interact out k ⇒ λp'. Interact ??? out (λv. io_inject_0 O I A P (k v) (p' v)) |
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88 | ]) p. |
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89 | |
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90 | 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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91 | 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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92 | (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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93 | [ None ⇒ λp'.? |
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94 | | Some b ⇒ λp'. io_inject_0 O I A P b p' |
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95 | ]) p. |
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96 | elim p'; qed. |
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97 | |
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98 | 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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99 | match a with |
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100 | [ Wrong m ⇒ Wrong ??? m |
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101 | | Value b ⇒ match b with [ mk_Sig w p ⇒ Value ??? w] |
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102 | | Interact out k ⇒ Interact ??? out (λv. io_eject ?? A P (k v)) |
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103 | ]. |
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104 | |
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105 | coercion io_inject : |
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106 | ∀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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107 | on a:option (IO ???) to IO ?? (Sig ? ?). |
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108 | 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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109 | on _c:IO ?? (Sig ? ?) to IO ???. |
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110 | |
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111 | definition opt_to_io : ∀O,I,T.errmsg → option T → IO O I T ≝ |
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112 | λO,I,T,m,v. match v with [ None ⇒ Wrong ?? T m | Some v' ⇒ Value ??? v' ]. |
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113 | |
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114 | 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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115 | (∀v:A. ∀p:P v. P_io O I ? P' (f (mk_Sig A P v p))) → |
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116 | P_io O I ? P' (bindIO O I (Sig A P) B e f). |
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117 | #O #I #A #B #P #P' #e #f elim e; |
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118 | [ #out #k #IH #IH' whd; #res @IH //; |
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119 | | #v0 elim v0; #v #Hv #IH whd; @IH |
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120 | | //; |
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121 | ] qed. |
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122 | |
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123 | 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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124 | (∀vA:A.∀vB:B. ∀p:P 〈vA,vB〉. P_io O I ? P' (f vA vB)) → |
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125 | P_io O I ? P' (bindIO2 O I A B C e f). |
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126 | #I #O #A #B #C #P #P' #e #f elim e; |
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127 | [ #out #k #IH #IH' whd; #res @IH @IH' |
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128 | | #v0 elim v0; #v elim v; #vA #vB #Hv #IH @IH //; |
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129 | | //; |
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130 | ] qed. |
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131 | |
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132 | lemma opt_bindIO_OK: ∀O,I,A,B,m. ∀P:B → Prop. ∀e:option A. ∀f: A → IO O I B. |
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133 | (∀v:A. e = Some A v → P_io O I ? P (f v)) → |
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134 | P_io O I ? P (bindIO O I A B (opt_to_io ??? m e) f). |
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135 | #I #O #A #B #m #P #e elim e; //; #v #f #H @H //; |
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136 | qed. |
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137 | |
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138 | lemma opt_bindIO2_OK: ∀O,I,A,B,C,m. ∀P:C → Prop. ∀e:option (A×B). ∀f: A → B → IO O I C. |
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139 | (∀vA:A.∀vB:B. e = Some (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) → |
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140 | P_io O I ? P (bindIO2 O I A B C (opt_to_io ??? m e) f). |
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141 | #I #O #A #B #C #m #P #e elim e; //; #v cases v; #vA #vB #f #H @H //; |
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142 | qed. |
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143 | |
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144 | lemma res_bindIO_OK: ∀O,I,A,B. ∀P:B → Prop. ∀e:res A. ∀f: A → IO O I B. |
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145 | (∀v:A. e = OK A v → P_io O I ? P (f v)) → |
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146 | P_io O I ? P (bindIO O I A B e f). |
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147 | #I #O #A #B #P #e elim e; //; #v #f #H @H //; |
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148 | qed. |
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149 | |
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150 | 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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151 | (∀vA:A.∀vB:B. e = OK (A×B) 〈vA,vB〉 → P_io O I ? P (f vA vB)) → |
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152 | P_io O I ? P (bindIO2 O I A B C e f). |
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153 | #I #O #A #B #C #P #e elim e; //; #v cases v; #vA #vB #f #H @H //; |
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154 | qed. |
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155 | |
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156 | 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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157 | (∀v:A. P_io O I ? P (f v)) → |
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158 | P_io O I ? P (bindIO O I A B e f). |
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159 | #I #O #A #B #P #e elim e; |
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160 | [ #out #k #IH #f #H whd; #res @IH //; |
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161 | | #v #f #H @H |
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162 | | //; |
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163 | ] qed. |
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164 | |
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165 | 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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166 | (∀v1:A.∀v2:B. P_io O I ? P (f v1 v2)) → |
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167 | P_io O I ? P (bindIO2 O I A B C e f). |
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168 | #I #O #A #B #C #P #e elim e; |
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169 | [ #out #k #IH #f #H whd; #res @IH //; |
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170 | | #v cases v; #v1 #v2 #f #H @H |
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171 | | //; |
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172 | ] qed. |
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173 | |
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174 | 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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175 | P_io … P' e → |
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176 | (∀v:A. P' v → P_io O I ? P (f v)) → |
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177 | P_io O I ? P (bindIO O I A B e f). |
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178 | #I #O #A #B #P' #P #e elim e; |
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179 | [ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2 by /; |
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180 | | #v #f #He #H @H @He |
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181 | | //; |
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182 | ] qed. |
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183 | |
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184 | 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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185 | P_io … P' e → |
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186 | (∀v1:A.∀v2:B. P' 〈v1,v2〉 → P_io O I ? P (f v1 v2)) → |
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187 | P_io O I ? P (bindIO2 O I A B C e f). |
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188 | #I #O #A #B #C #P' #P #e elim e; |
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189 | [ #out #k #IH #f #He #H whd in He ⊢ %; #res @IH /2 by /; |
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190 | | #v cases v; #v1 #v2 #f #He #H @H @He |
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191 | | //; |
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192 | ] qed. |
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193 | |
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194 | |
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195 | (* Is there a way to prove this without extensionality? *) |
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196 | (* |
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197 | lemma bind_assoc_r: ∀O,I,A,B,C,e,f,g. |
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198 | ∀ext:(∀T1,T2:Type[0].∀f,f':T1 → T2.(∀x.f x = f' x) → f = f'). |
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199 | 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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200 | #O #I #A #B #C #e #f #g #ext elim e; |
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201 | [ #o #k #IH whd in ⊢ (??%%); @eq_f |
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202 | @ext @IH |
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203 | | #v @refl |
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204 | | #m @refl |
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205 | ] qed. |
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206 | *) |
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207 | definition bind_assoc_r ≝ λO,I.m_bind_bind (IOMonad O I). |
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208 | |
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209 | (* |
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210 | 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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211 | (∀a,b. eject ?? e = 〈a,b〉 → P 〈a,b〉 → P_io O I ? R (Q a b)) → |
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212 | P_io O I ? R (match eject ?? e with [ pair a b ⇒ Q a b ]). |
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213 | #I #O #A #B #C #P #e #Q #R cases e; #e' cases e'; normalize; |
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214 | [ *; |
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215 | | #e'' cases e''; #a #b #Pab #H normalize; /2 by _/; |
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216 | ] qed. |
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217 | *) |
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