1 | include "utilities/monad.ma". |
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2 | include "basics/types.ma". |
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3 | |
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4 | definition Option ≝ MakeMonadProps |
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5 | (* the monad *) |
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6 | option |
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7 | (* return *) |
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8 | (λX.Some X) |
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9 | (* bind *) |
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10 | (λX,Y,m,f. match m with [ Some x ⇒ f x | None ⇒ None ?]) |
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11 | ????. |
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12 | // #X[|*:#Y[#Z]]*normalize//qed. |
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13 | |
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14 | include "hints_declaration.ma". |
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15 | alias symbol "hint_decl" (instance 1) = "hint_decl_Type1". |
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16 | unification hint 0 ≔ X; |
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17 | N ≟ max_def Option |
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18 | (*---------------*) ⊢ |
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19 | option X ≡ monad N X |
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20 | . |
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21 | |
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22 | definition opt_safe : ∀X.∀m : option X. m ≠ None X → X ≝ |
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23 | λX,m,prf.match m return λx.m = x → X with |
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24 | [ Some t ⇒ λ_.t |
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25 | | None ⇒ λeq_m.? |
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26 | ] (refl …). elim (absurd … eq_m prf) qed. |
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27 | |
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28 | lemma opt_to_opt_safe : ∀A,m,prf. m = Some ? (opt_safe A m prf). |
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29 | #A * |
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30 | [ #ABS elim (absurd ? (refl ??) ABS) |
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31 | | // |
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32 | ] qed. |
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33 | |
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34 | lemma opt_safe_elim : ∀A.∀P : A → Prop.∀m,prf. |
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35 | (∀x.m = Some ? x → P x) → P (opt_safe ? m prf). |
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36 | #A#P* |
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37 | [#prf elim(absurd … (refl ??) prf) |
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38 | |#x normalize #_ #H @H @refl |
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39 | ] qed. |
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40 | |
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41 | lemma opt_safe_proof_irr : ∀A,m,prf1,prf2. |
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42 | opt_safe A m prf1 = opt_safe A m prf2. |
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43 | #A* // qed. |
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44 | |
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45 | definition opt_try_catch : ∀X.option X → (unit → X) → X ≝ |
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46 | λX,m,f.match m with [Some x ⇒ x | None ⇒ f it]. |
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47 | |
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48 | notation > "'Try' m 'catch' opt (ident e opt( : ty)) ⇒ f" with precedence 46 for |
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49 | ${ default |
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50 | @{ ${ default |
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51 | @{'trycatch $m (λ${ident e}:$ty.$f)} |
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52 | @{'trycatch $m (λ${ident e}.$f)} |
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53 | }} |
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54 | @{'trycatch $m (λ_.$f)} |
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55 | }. |
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56 | |
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57 | notation < "hvbox('Try' \nbsp break m \nbsp break 'catch' \nbsp ident e : ty \nbsp ⇒ \nbsp break f)" with precedence 46 for |
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58 | @{'trycatch $m (λ${ident e}:$ty.$f)}. |
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59 | notation < "hvbox('Try' \nbsp break m \nbsp break 'catch' \nbsp ⇒ \nbsp break f)" with precedence 46 for |
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60 | @{'trycatch $m (λ_:$ty.$f)}. |
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61 | |
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62 | interpretation "option try and catch" 'trycatch m f = (opt_try_catch ? m f). |
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63 | |
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64 | definition OptPred ≝ λPdef : Prop.mk_InjMonadPred Option |
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65 | (mk_MonadPred ? |
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66 | (λX,P,x.Try ! y ← x ; return P y catch ⇒ Pdef) |
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67 | ???) |
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68 | (λX,P,m_sig.match m_sig with [ mk_Sig m prf ⇒ match m return λx.Try ! y ← x ; return P y catch ⇒ Pdef → option (Σy.?) with [ Some x ⇒ λprf.Some ? x | None ⇒ λ_.None ? ] prf ]) |
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69 | ?. |
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70 | [4: @prf |
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71 | |*: |
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72 | #X[2:#Y]#P1[1,2:#P2[2:#H]] |
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73 | [1,2,4: * [5: *]] normalize /2/ |
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74 | qed. |
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75 | |
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76 | definition opt_All : ∀X.(X → Prop) → option X → Prop ≝ m_pred … (OptPred True). |
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77 | |
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78 | lemma opt_All_mp : ∀A,P,Q.(∀x. P x → Q x) → ∀o.opt_All A P o → opt_All ? Q o |
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79 | ≝ m_pred_mp …. |
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80 | |
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81 | definition opt_Exists : ∀X.(X → Prop) → option X → Prop ≝ m_pred … (OptPred False). |
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82 | |
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83 | lemma opt_Exists_mp : ∀A,P,Q.(∀x. P x → Q x) → ∀o.opt_Exists A P o → opt_Exists ? Q o |
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84 | ≝ m_pred_mp …. |
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85 | |
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86 | (* moved from ASM/ASM.ma! *) |
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87 | definition partial_injection : ∀A,B.(A → option B) → Prop ≝ |
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88 | λA,B,f.∀x,y,z.f x = Some ? z → f y = Some ? z → x = y. |
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