[671] | 1 | include "Exponential.ma". |
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| 2 | include "BitVector.ma". |
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| 3 | |
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| 4 | ndefinition nat_of_bool ≝ |
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| 5 | λb: Bool. |
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| 6 | match b with |
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| 7 | [ false ⇒ Z |
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| 8 | | true ⇒ S Z |
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| 9 | ]. |
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| 10 | |
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| 11 | ndefinition add_n_with_carry: |
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| 12 | ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (BitVector three) ≝ |
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| 13 | λn: Nat. |
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| 14 | λb: BitVector n. |
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| 15 | λc: BitVector n. |
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| 16 | λcarry: Bool. |
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| 17 | let b_as_nat ≝ nat_of_bitvector n b in |
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| 18 | let c_as_nat ≝ nat_of_bitvector n c in |
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| 19 | let carry_as_nat ≝ nat_of_bool carry in |
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| 20 | let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in |
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| 21 | let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) + |
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| 22 | (modulus c_as_nat ((S (S Z)) * n)) + |
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| 23 | c_as_nat) ≳ ((S (S Z)) * n) in |
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| 24 | let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) + |
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| 25 | (modulus c_as_nat ((S (S Z))^(n - (S Z)))) + |
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| 26 | c_as_nat) ≳ ((S (S Z))^(n - (S Z)))) in |
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| 27 | let result ≝ modulus result_old ((S (S Z))^n) in |
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| 28 | let cy_flag ≝ (result_old ≳ ((S (S Z))^n)) in |
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| 29 | let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in |
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| 30 | mk_Cartesian ? ? (bitvector_of_nat n result) |
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| 31 | ([[ cy_flag ; ac_flag ; ov_flag ]]). |
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| 32 | |
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| 33 | ndefinition sub_n_with_carry: ∀n: Nat. ∀b,c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (BitVector three) ≝ |
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| 34 | λn: Nat. |
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| 35 | λb: BitVector n. |
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| 36 | λc: BitVector n. |
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| 37 | λcarry: Bool. |
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| 38 | let b_as_nat ≝ nat_of_bitvector n b in |
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| 39 | let c_as_nat ≝ nat_of_bitvector n c in |
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| 40 | let carry_as_nat ≝ nat_of_bool carry in |
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| 41 | let temporary ≝ (b_as_nat mod (two * n)) - (c_as_nat mod (two * n)) in |
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| 42 | let ac_flag ≝ less_than_b (b_as_nat mod (two * n)) ((c_as_nat mod (two * n)) + carry_as_nat) in |
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| 43 | let bit_six ≝ less_than_b (b_as_nat mod (two^(n - one))) ((c_as_nat mod (two^(n - one))) + carry_as_nat) in |
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| 44 | let 〈b',cy_flag〉 ≝ |
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| 45 | if greater_than_or_equal_b b_as_nat (c_as_nat + carry_as_nat) then |
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| 46 | 〈b_as_nat, false〉 |
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| 47 | else |
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| 48 | 〈b_as_nat + (two^n), true〉 |
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| 49 | in |
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| 50 | let ov_flag ≝ exclusive_disjunction cy_flag bit_six in |
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| 51 | 〈bitvector_of_nat n ((b' - c_as_nat) - carry_as_nat), [[ cy_flag; ac_flag; ov_flag ]]〉. |
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| 52 | |
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| 53 | |
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| 54 | ndefinition sub_8_with_carry ≝ sub_n_with_carry eight. |
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| 55 | ndefinition sub_16_with_carry ≝ sub_n_with_carry sixteen. |
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| 56 | |
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| 57 | ndefinition add_8_with_carry ≝ add_n_with_carry eight. |
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| 58 | ndefinition add_16_with_carry ≝ add_n_with_carry sixteen. |
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| 59 | |
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| 60 | ndefinition increment ≝ |
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| 61 | λn: Nat. |
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| 62 | λb: BitVector n. |
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| 63 | let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in |
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| 64 | let overflow ≝ b_as_nat ≳ (S (S Z))^n in |
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| 65 | match overflow with |
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| 66 | [ false ⇒ bitvector_of_nat n b_as_nat |
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| 67 | | true ⇒ zero n |
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| 68 | ]. |
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| 69 | |
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| 70 | ndefinition decrement ≝ |
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| 71 | λn: Nat. |
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| 72 | λb: BitVector n. |
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| 73 | let b_as_nat ≝ nat_of_bitvector n b in |
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| 74 | match b_as_nat with |
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| 75 | [ Z ⇒ max n |
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| 76 | | S o ⇒ bitvector_of_nat n o |
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| 77 | ]. |
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| 78 | |
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| 79 | alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop". |
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| 80 | |
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| 81 | ndefinition bitvector_of_bool: |
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| 82 | ∀n: Nat. ∀b: Bool. BitVector (S n) ≝ |
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| 83 | λn: Nat. |
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| 84 | λb: Bool. |
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| 85 | (pad (S n - (S Z)) (S Z) [[b]])⌈(S n - (S Z)) + S Z ↦ S n⌉. |
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| 86 | /2/. |
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| 87 | nqed. |
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| 88 | |
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| 89 | ndefinition full_add ≝ |
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| 90 | λn: Nat. |
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| 91 | λb, c: BitVector n. |
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| 92 | λd: Bit. |
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| 93 | fold_right2_i ? ? ? ( |
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| 94 | λn. |
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| 95 | λb1, b2: Bool. |
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| 96 | λd: Bit × (BitVector n). |
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| 97 | let 〈c1,r〉 ≝ d in |
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| 98 | 〈inclusive_disjunction (conjunction b1 b2) |
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| 99 | (conjunction c1 (inclusive_disjunction b1 b2)), |
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| 100 | (exclusive_disjunction (exclusive_disjunction b1 b2) c1) ::: r〉) |
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| 101 | 〈d, [[ ]]〉 ? b c. |
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| 102 | |
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| 103 | ndefinition half_add ≝ |
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| 104 | λn: Nat. |
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| 105 | λb, c: BitVector n. |
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| 106 | full_add n b c false. |
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