1 | include "arithmetics/nat.ma". |
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2 | |
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3 | inductive nat_compared : nat → nat → Type[0] ≝ |
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4 | | nat_lt : ∀n,m:nat. nat_compared n (n+S m) |
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5 | | nat_eq : ∀n:nat. nat_compared n n |
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6 | | nat_gt : ∀n,m:nat. nat_compared (m+S n) m. |
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7 | |
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8 | let rec nat_compare (n:nat) (m:nat) : nat_compared n m ≝ |
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9 | match n return λx. nat_compared x m with |
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10 | [ O ⇒ match m return λy. nat_compared O y with [ O ⇒ nat_eq ? | S m' ⇒ nat_lt ?? ] |
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11 | | S n' ⇒ |
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12 | match m return λy. nat_compared (S n') y with |
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13 | [ O ⇒ nat_gt n' O |
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14 | | S m' ⇒ match nat_compare n' m' return λx,y.λ_. nat_compared (S x) (S y) with |
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15 | [ nat_lt x y ⇒ nat_lt ?? |
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16 | | nat_eq x ⇒ nat_eq ? |
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17 | | nat_gt x y ⇒ nat_gt ? (S y) |
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18 | ] |
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19 | ] |
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20 | ]. |
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21 | |
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22 | lemma nat_compare_eq : ∀n. nat_compare n n = nat_eq n. |
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23 | #n elim n |
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24 | [ @refl |
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25 | | #m #IH whd in ⊢ (??%?) > IH @refl |
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26 | ] qed. |
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27 | |
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28 | lemma nat_compare_lt : ∀n,m. nat_compare n (n+S m) = nat_lt n m. |
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29 | #n #m elim n |
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30 | [ // |
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31 | | #n' #IH whd in ⊢ (??%?) > IH @refl |
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32 | ] qed. |
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33 | |
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34 | lemma nat_compare_gt : ∀n,m. nat_compare (m+S n) m = nat_gt n m. |
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35 | #n #m elim m |
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36 | [ // |
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37 | | #m' #IH whd in ⊢ (??%?) > IH @refl |
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38 | ] qed. |
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39 | |
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40 | lemma max_l : ∀m,n,o:nat. o ≤ m → o ≤ max m n. |
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41 | #m #n #o #H whd in ⊢ (??%) @leb_elim #H' |
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42 | [ @(transitive_le ? m ? H H') |
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43 | | @H |
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44 | ] qed. |
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45 | |
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46 | lemma max_r : ∀m,n,o:nat. o ≤ n → o ≤ max m n. |
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47 | #m #n #o #H whd in ⊢ (??%) @leb_elim #H' |
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48 | [ @H |
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49 | | @(transitive_le … H) @(transitive_le … (not_le_to_lt … H')) // |
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50 | ] qed. |
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51 | |
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52 | |
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53 | (* "Fast" proofs: some proofs get reduced during normalization (in particular, |
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54 | some functions which use a proof for rewriting are applied to constants and |
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55 | get reduced during a proof or while matita is searching for a term; |
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56 | they may also be normalized during testing), and so here are some more |
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57 | efficient versions. Perhaps they could be replaced using some kind of proof |
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58 | irrelevance? *) |
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59 | |
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60 | let rec plus_n_Sm_fast (n:nat) on n : ∀m:nat. S (n+m) = n+S m ≝ |
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61 | match n return λn'.∀m.S(n'+m) = n'+S m with |
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62 | [ O ⇒ λm.refl ?? |
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63 | | S n' ⇒ λm. ? |
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64 | ]. normalize @(match plus_n_Sm_fast n' m with [ refl ⇒ ? ]) @refl qed. |
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65 | |
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66 | let rec plus_n_O_faster (n:nat) : n = n + O ≝ |
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67 | match n return λn.n=n+O with |
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68 | [ O ⇒ refl ?? |
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69 | | S n' ⇒ match plus_n_O_faster n' return λx.λ_.S n'=S x with [ refl ⇒ refl ?? ] |
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70 | ]. |
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71 | |
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72 | let rec commutative_plus_faster (n,m:nat) : n+m = m+n ≝ |
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73 | match n return λn.n+m = m+n with |
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74 | [ O ⇒ plus_n_O_faster ? |
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75 | | S n' ⇒ ? |
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76 | ]. @(match plus_n_Sm_fast m n' return λx.λ_. ? = x with [ refl ⇒ ? ]) |
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77 | @(match commutative_plus_faster n' m return λx.λ_.? = S x with [refl ⇒ ?]) @refl qed. |
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78 | |
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