source: Deliverables/D4.1/Matita/Arithmetic.ma @ 357

Last change on this file since 357 was 357, checked in by sacerdot, 9 years ago
  • stupid bug fixed in BitVectorTrie?
  • dependencies minimized, dead code removed
File size: 4.0 KB
Line 
1include "Exponential.ma".
2include "BitVector.ma".
3
4ndefinition nat_of_bool ≝
5  λb: Bool.
6    match b with
7      [ false ⇒ Z
8      | true ⇒ S Z
9      ].
10   
11ndefinition add_n_with_carry:
12      ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (List Bool) ≝
13  λn: Nat.
14  λb: BitVector n.
15  λc: BitVector n.
16  λcarry: Bool.
17    let b_as_nat ≝ nat_of_bitvector n b in
18    let c_as_nat ≝ nat_of_bitvector n c in
19    let carry_as_nat ≝ nat_of_bool carry in
20    let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in
21    let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) +
22                  (modulus c_as_nat ((S (S Z)) * n)) +
23                  c_as_nat) ≳ ((S (S Z)) * n) in
24    let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) +
25                  (modulus c_as_nat ((S (S Z))^(n - (S Z)))) +
26                  c_as_nat) ≳ ((S (S Z))^(n - (S Z)))) in
27    let result ≝ modulus result_old ((S (S Z))^n) in
28    let cy_flag ≝ (result_old ≳ ((S (S Z))^n)) in
29    let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in
30      ? (mk_Cartesian (BitVector n) ? (? (bitvector_of_nat n result))
31                          (cy_flag :: ac_flag :: ov_flag :: Empty Bool)).
32    #H; nassumption;
33nqed.
34
35ndefinition sub_8_with_carry: ∀b,c: BitVector eight. ∀carry: Bool. Cartesian (BitVector eight) (List Bool) ≝
36  λb: BitVector eight.
37  λc: BitVector eight.
38  λcarry: Bool.
39    let b_as_nat ≝ nat_of_bitvector eight b in
40    let c_as_nat ≝ nat_of_bitvector eight c in
41    let carry_as_nat ≝ nat_of_bool carry in
42    let temporary ≝ b_as_nat mod sixteen - c_as_nat mod sixteen in
43    let ac_flag ≝ negation (conjunction ((b_as_nat mod sixteen) ≲ (c_as_nat mod sixteen)) (temporary ≲ carry_as_nat)) in
44    let bit_six ≝ negation (conjunction ((b_as_nat mod one_hundred_and_twenty_eight) ≲ (c_as_nat mod one_hundred_and_twenty_eight)) (temporary ≲ carry_as_nat)) in
45    let old_result_1 ≝ b_as_nat - c_as_nat in
46    let old_result_2 ≝ old_result_1 - carry_as_nat in
47    let ov_flag ≝ exclusive_disjunction carry bit_six in
48      match conjunction (b_as_nat ≲ c_as_nat) (old_result_1 ≲ carry_as_nat) with
49        [ false ⇒
50           let cy_flag ≝ false in
51            〈 bitvector_of_nat eight old_result_2, [cy_flag ; ac_flag ; ov_flag ] 〉
52        | true ⇒
53           let cy_flag ≝ true in
54           let new_result ≝ b_as_nat + two_hundred_and_fifty_six - c_as_nat - carry_as_nat in
55            〈 bitvector_of_nat eight new_result, [ cy_flag ; ac_flag ; ov_flag ] 〉
56        ].
57         
58ndefinition add_8_with_carry ≝ add_n_with_carry eight.
59ndefinition add_16_with_carry ≝ add_n_with_carry sixteen.
60
61ndefinition increment ≝
62  λn: Nat.
63  λb: BitVector n.
64    let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in
65    let overflow ≝ b_as_nat ≳ (S (S Z))^n in
66      match overflow with
67        [ false ⇒ bitvector_of_nat n b_as_nat
68        | true ⇒ zero n
69        ].
70       
71ndefinition decrement ≝
72  λn: Nat.
73  λb: BitVector n.
74    let b_as_nat ≝ nat_of_bitvector n b in
75      match b_as_nat with
76        [ Z ⇒ max n
77        | S o ⇒ bitvector_of_nat n o
78        ].
79       
80alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop".
81
82ndefinition bitvector_of_bool:
83      ∀n: Nat. ∀b: Bool. BitVector (S n) ≝
84  λn: Nat.
85  λb: Bool.
86    ? (pad (S n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))).
87  nrewrite > plus_minus_inverse_right
88   [ napply (λx.x) | /2/]
89nqed.
90
91ndefinition full_add ≝
92  λn: Nat.
93  λb, c: BitVector n.
94  λd: Bit.
95    fold_right2_i ? ? ? (
96      λn.
97       λb1, b2: Bool.
98        λd: Bit × (BitVector n).
99        let 〈c1,r〉 ≝ d in
100          〈inclusive_disjunction (conjunction b1 b2)
101                                 (conjunction c1 (inclusive_disjunction b1 b2)),
102           (exclusive_disjunction (exclusive_disjunction b1 b2) c1) ::: r〉)
103     〈d, [[ ]]〉 ? b c.
104   
105ndefinition half_add ≝
106  λn: Nat.
107  λb, c: BitVector n.
108    full_add n b c false.
109
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