source: Deliverables/D4.1/Matita/Arithmetic.ma @ 272

Last change on this file since 272 was 272, checked in by mulligan, 10 years ago

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1include "Universes.ma".
2include "Plogic/equality.ma".
3include "Connectives.ma".
4include "Nat.ma".
5include "Exponential.ma".
6include "Bool.ma".
7include "BitVector.ma".
8include "List.ma".
9
10ndefinition one ≝ S Z.
11ndefinition two ≝ (S(S(Z))).
12ndefinition three ≝ two + one.
13ndefinition four ≝ two + two.
14ndefinition five ≝ three + two.
15ndefinition six ≝ three + three.
16ndefinition seven ≝ three + four.
17ndefinition eight ≝ four + four.
18ndefinition nine ≝ five + four.
19ndefinition ten ≝ five + five.
20ndefinition eleven ≝ six + five.
21ndefinition twelve ≝ six + six.
22ndefinition thirteen ≝ seven + six.
23ndefinition fourteen ≝ seven + seven.
24ndefinition fifteen ≝ eight + seven.
25ndefinition sixteen ≝ eight + eight.
26ndefinition seventeen ≝ nine + eight.
27ndefinition eighteen ≝ nine + nine.
28ndefinition nineteen ≝ ten + nine.
29ndefinition one_hundred_and_twenty_eight ≝ sixteen * eight.
30ndefinition two_hundred_and_fifty_six ≝
31  one_hundred_and_twenty_eight + one_hundred_and_twenty_eight.                                         
32   
33ndefinition nat_of_bool ≝
34  λb: Bool.
35    match b with
36      [ false ⇒ Z
37      | true ⇒ S Z
38      ].
39   
40ndefinition add_n_with_carry:
41      ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (List Bool) ≝
42  λn: Nat.
43  λb: BitVector n.
44  λc: BitVector n.
45  λcarry: Bool.
46    let b_as_nat ≝ nat_of_bitvector n b in
47    let c_as_nat ≝ nat_of_bitvector n c in
48    let carry_as_nat ≝ nat_of_bool carry in
49    let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in
50    let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) +
51                  (modulus c_as_nat ((S (S Z)) * n)) +
52                  c_as_nat) ≥ ((S (S Z)) * n) in
53    let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) +
54                  (modulus c_as_nat ((S (S Z))^(n - (S Z)))) +
55                  c_as_nat) ≥ ((S (S Z))^(n - (S Z)))) in
56    let result ≝ modulus result_old ((S (S Z))^n) in
57    let cy_flag ≝ (result_old ≥ ((S (S Z))^n)) in
58    let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in
59      ? (mk_Cartesian (BitVector n) ? (? (bitvector_of_nat n result))
60                          (cy_flag :: ac_flag :: ov_flag :: Empty Bool)).
61    //.
62nqed.
63
64naxiom less_than_b: Nat → Nat → Bool.
65
66ndefinition sub_8_with_carry ≝
67  λb: BitVector eight.
68  λc: BitVector eight.
69  λcarry: Bool.
70    let b_as_nat ≝ nat_of_bitvector eight b in
71    let c_as_nat ≝ nat_of_bitvector eight c in
72    let carry_as_nat ≝ nat_of_bool carry in
73    let result_old_1 ≝ minus (minus b_as_nat c_as_nat) carry_as_nat in
74    let modulus_1 ≝ (modulus b_as_nat sixteen) - (modulus c_as_nat sixteen) in
75      match less_than_b (modulus b_as_nat sixteen) (modulus c_as_nat sixteen) with
76        [ true ⇒
77          let ac_flag ≝ true in
78          let result_old_2 ≝ (minus (modulus b_as_nat one_hundred_and_twenty_eight)
79                                   (modulus c_as_nat one_hundred_and_twenty_eight)) in
80            match less_than_b (modulus b_as_nat one_hundred_and_twenty_eight)
81                              (modulus c_as_nat one_hundred_and_twenty_eight) with
82            [ true ⇒
83              let bit_six ≝ true in
84              let result_carry ≝ mk_Cartesian … result_old_1 false in
85              let ov_flag ≝ exclusive_disjunction cy_flag bit_six in
86                mk_Cartesian … (first … result_carry) (second … result_carry :: ac_flag :: ov_flag :: Empty Bool)
87            | false ⇒ ?
88            ]
89        | false ⇒ ?
90        ].
91   
92   
93ndefinition add_8_with_carry ≝ add_n_with_carry eight.
94ndefinition add_16_with_carry ≝ add_n_with_carry sixteen.
95
96(*
97ndefinition increment ≝
98  λn: Nat.
99  λb: BitVector n.
100    let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in
101    let overflow ≝ b_as_nat ≥ (S (S Z))^n in
102      match overflow with
103        [ False ⇒ bitvector_of_nat n b_as_nat
104        | True ⇒ bitvector_of_nat n Z
105        ].
106       
107ndefinition decrement ≝
108  λn: Nat.
109  λb: BitVector n.
110    let b_as_nat ≝ nat_of_bitvector n b in
111      match b_as_nat with
112        [ Z ⇒ max n
113        | S o ⇒ bitvector_of_nat n o
114        ].
115       
116alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop".
117
118ndefinition bitvector_of_bool:
119      ∀n: Nat. ∀b: Bool. BitVector n ≝
120  λn: Nat.
121  λb: Bool.
122    ? (pad (n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))).
123  //.
124nqed.
125
126*)
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