[248] | 1 | include "Universes.ma". |
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| 2 | include "Equality.ma". |
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| 3 | include "Connectives.ma". |
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| 4 | include "Nat.ma". |
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| 5 | include "Exponential.ma". |
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| 6 | include "Bool.ma". |
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| 7 | include "BitVector.ma". |
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| 8 | include "List.ma". |
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[246] | 9 | |
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| 10 | ndefinition eight ≝ exponential (S(S(Z))) (S(S(S(Z)))). |
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| 11 | ndefinition sixteen ≝ eight + eight. |
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[248] | 12 | ndefinition one_hundred_and_twenty_eight ≝ sixteen * eight. |
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[246] | 13 | ndefinition two_hundred_and_fifty_six ≝ |
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| 14 | one_hundred_and_twenty_eight + one_hundred_and_twenty_eight. |
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| 15 | |
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[248] | 16 | ndefinition nat_of_bool ≝ |
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| 17 | λb: Bool. |
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| 18 | match b with |
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| 19 | [ False ⇒ Z |
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| 20 | | True ⇒ S Z |
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| 21 | ]. |
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| 22 | |
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[246] | 23 | ndefinition add_n_with_carry: |
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| 24 | ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (List Bool) ≝ |
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| 25 | λn: Nat. |
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| 26 | λb: BitVector n. |
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| 27 | λc: BitVector n. |
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| 28 | λcarry: Bool. |
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| 29 | let b_as_nat ≝ nat_of_bitvector n b in |
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| 30 | let c_as_nat ≝ nat_of_bitvector n c in |
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| 31 | let carry_as_nat ≝ nat_of_bool carry in |
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| 32 | let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in |
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[248] | 33 | let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) + |
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| 34 | (modulus c_as_nat ((S (S Z)) * n)) + |
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| 35 | c_as_nat) ≥ ((S (S Z)) * n) in |
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[246] | 36 | let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) + |
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| 37 | (modulus c_as_nat ((S (S Z))^(n - (S Z)))) + |
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| 38 | c_as_nat) ≥ ((S (S Z))^(n - (S Z)))) in |
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| 39 | let result ≝ modulus result_old ((S (S Z))^n) in |
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| 40 | let cy_flag ≝ (result_old ≥ ((S (S Z))^n)) in |
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| 41 | let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in |
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[248] | 42 | ? (mk_Cartesian (BitVector n) ? (? (bitvector_of_nat n result)) |
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[246] | 43 | (cy_flag :: ac_flag :: ov_flag :: Empty Bool)). |
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| 44 | //. |
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| 45 | nqed. |
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| 46 | |
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| 47 | (* |
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| 48 | ndefinition sub_8_with_carry ≝ |
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| 49 | λb: BitVector eight. |
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| 50 | λc: BitVector eight. |
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| 51 | λcarry: Bool. |
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| 52 | let b_as_nat ≝ nat_of_bitvector eight b in |
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| 53 | let c_as_nat ≝ nat_of_bitvector eight c in |
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| 54 | let carry_as_nat ≝ nat_of_bool carry in |
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| 55 | let result_old_1 ≝ subtraction_underflow b_as_nat c_as_nat in |
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| 56 | match result_old_1 with |
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| 57 | [ Nothing ⇒ |
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| 58 | let ac_flag ≝ True in |
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| 59 | |
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| 60 | | Just result_old_1' ⇒ |
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| 61 | ] |
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| 62 | *) |
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| 63 | |
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| 64 | ndefinition add_8_with_carry ≝ add_n_with_carry eight. |
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| 65 | ndefinition add_16_with_carry ≝ add_n_with_carry sixteen. |
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| 66 | |
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[248] | 67 | (* |
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[246] | 68 | ndefinition increment ≝ |
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| 69 | λn: Nat. |
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| 70 | λb: BitVector n. |
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| 71 | let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in |
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| 72 | let overflow ≝ b_as_nat ≥ (S (S Z))^n in |
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| 73 | match overflow with |
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[248] | 74 | [ False ⇒ bitvector_of_nat n b_as_nat |
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| 75 | | True ⇒ bitvector_of_nat n Z |
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[246] | 76 | ]. |
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| 77 | |
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| 78 | ndefinition decrement ≝ |
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| 79 | λn: Nat. |
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| 80 | λb: BitVector n. |
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| 81 | let b_as_nat ≝ nat_of_bitvector n b in |
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| 82 | match b_as_nat with |
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| 83 | [ Z ⇒ max n |
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| 84 | | S o ⇒ bitvector_of_nat n o |
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| 85 | ]. |
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| 86 | |
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| 87 | alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop". |
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| 88 | |
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| 89 | ndefinition bitvector_of_bool: |
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| 90 | ∀n: Nat. ∀b: Bool. BitVector n ≝ |
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| 91 | λn: Nat. |
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| 92 | λb: Bool. |
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| 93 | ? (pad (n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))). |
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| 94 | //. |
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[248] | 95 | nqed. |
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| 96 | |
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| 97 | *) |
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