[1928] | 1 | include "ASM/Util.ma". |
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| 2 | include "ASM/Arithmetic.ma". |
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| 3 | |
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| 4 | definition nat_of_bool: bool → nat ≝ |
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| 5 | λb: bool. |
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| 6 | match b with |
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| 7 | [ true ⇒ 1 |
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| 8 | | false ⇒ 0 |
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| 9 | ]. |
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| 10 | |
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| 11 | lemma blah: |
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| 12 | ∀n: nat. |
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| 13 | ∀bv: BitVector n. |
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| 14 | ∀buffer: nat. |
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| 15 | nat_of_bitvector_aux n buffer bv = nat_of_bitvector n bv + (buffer * 2^n). |
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| 16 | #n #bv elim bv |
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| 17 | [1: |
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| 18 | #buffer elim buffer try % |
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| 19 | #buffer' #inductive_hypothesis |
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| 20 | normalize <times_n_1 % |
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| 21 | |2: |
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| 22 | #n' #hd #tl #inductive_hypothesis |
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| 23 | #buffer cases hd normalize |
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| 24 | >inductive_hypothesis |
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| 25 | >inductive_hypothesis |
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| 26 | [1: |
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| 27 | change with ( |
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| 28 | ? + (2 * buffer + 1) * ?) in ⊢ (??%?); |
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| 29 | change with ( |
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| 30 | ? + buffer * (2 * 2^n')) in ⊢ (???%); |
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| 31 | cases daemon |
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| 32 | ] |
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| 33 | ] |
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| 34 | cases daemon |
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| 35 | qed. |
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| 36 | |
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| 37 | lemma nat_of_bitvector_aux_hd_tl: |
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| 38 | ∀n: nat. |
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| 39 | ∀tl: BitVector n. |
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| 40 | ∀hd: bool. |
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| 41 | nat_of_bitvector (S n) (hd:::tl) = |
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| 42 | nat_of_bitvector n tl + (nat_of_bool hd * 2^n). |
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| 43 | #n #tl elim tl |
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| 44 | [1: |
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| 45 | #hd cases hd % |
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| 46 | |2: |
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| 47 | #n' #hd' #tl' #inductive_hypothesis #hd |
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| 48 | cases hd whd in ⊢ (??%?); normalize nodelta |
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| 49 | >inductive_hypothesis cases hd' normalize nodelta |
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| 50 | normalize in match (nat_of_bool ?); |
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| 51 | normalize in match (nat_of_bool ?); |
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| 52 | [4: |
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| 53 | normalize in match (2 * ?); |
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| 54 | <plus_n_O <plus_n_O % |
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| 55 | |3: |
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| 56 | normalize in match (2 * ?); |
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| 57 | normalize in match (0 + 1); |
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| 58 | <plus_n_O |
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| 59 | normalize in match (1 * ?); |
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| 60 | <plus_n_O |
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| 61 | cases daemon |
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| 62 | (* XXX: lemma *) |
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| 63 | |*: |
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| 64 | cases daemon |
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| 65 | ] |
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| 66 | ] |
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| 67 | qed. |
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| 68 | |
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| 69 | lemma succ_nat_of_bitvector_aux_half_add_1: |
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| 70 | ∀n: nat. |
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| 71 | ∀bv: BitVector n. |
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| 72 | ∀buffer: nat. |
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| 73 | ∀power_proof: nat_of_bitvector … bv < 2^n - 1. |
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| 74 | S (nat_of_bitvector_aux n buffer bv) = |
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[1946] | 75 | nat_of_bitvector … (add n (bitvector_of_nat … 1) bv). |
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[1928] | 76 | #n #bv elim bv |
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| 77 | [1: |
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| 78 | #buffer normalize #absurd |
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| 79 | cases (lt_to_not_zero … absurd) |
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| 80 | |2: |
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| 81 | #n' #hd #tl #inductive_hypothesis #buffer |
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| 82 | cases daemon |
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| 83 | ] |
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| 84 | qed. |
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| 85 | |
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| 86 | lemma succ_nat_of_bitvector_half_add_1: |
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| 87 | ∀n: nat. |
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| 88 | ∀bv: BitVector n. |
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| 89 | ∀power_proof: nat_of_bitvector … bv < 2^n - 1. |
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| 90 | S (nat_of_bitvector … bv) = nat_of_bitvector … |
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[1946] | 91 | (add n (bitvector_of_nat … 1) bv). |
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[1928] | 92 | #n #bv elim bv |
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| 93 | [1: |
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| 94 | normalize #absurd |
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| 95 | cases (lt_to_not_zero … absurd) |
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| 96 | |2: |
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| 97 | #n' #hd #tl #inductive_hypothesis |
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| 98 | cases daemon |
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| 99 | ] |
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[1946] | 100 | qed. |
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