source: Deliverables/D4.1/Matita/Arithmetic.ma @ 273

Last change on this file since 273 was 273, checked in by mulligan, 10 years ago

Some fault functions were rewritten.

File size: 3.4 KB
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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    #H; nassumption;
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   
74ndefinition add_8_with_carry ≝ add_n_with_carry eight.
75ndefinition add_16_with_carry ≝ add_n_with_carry sixteen.
76
77ndefinition increment ≝
78  λn: Nat.
79  λb: BitVector n.
80    let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in
81    let overflow ≝ b_as_nat ≥ (S (S Z))^n in
82      match overflow with
83        [ false ⇒ bitvector_of_nat n b_as_nat
84        | true ⇒ zero n
85        ].
86       
87ndefinition decrement ≝
88  λn: Nat.
89  λb: BitVector n.
90    let b_as_nat ≝ nat_of_bitvector n b in
91      match b_as_nat with
92        [ Z ⇒ max n
93        | S o ⇒ bitvector_of_nat n o
94        ].
95       
96alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop".
97
98ndefinition bitvector_of_bool:
99      ∀n: Nat. ∀b: Bool. BitVector n ≝
100  λn: Nat.
101  λb: Bool.
102    ? (pad (n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))).
103  nrewrite > (plus_minus_inverse_right n ?);
104  #H;
105  nassumption;
106nqed.
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