source: Deliverables/D4.1/Matita/Arithmetic.ma @ 283

Last change on this file since 283 was 281, 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 ≝ ten * ten.
30ndefinition one_hundred_and_twenty_eight ≝ sixteen * eight.
31ndefinition one_hundred_and_twenty_nine ≝ one_hundred_and_twenty_eight + one.
32ndefinition one_hundred_and_thirty ≝ one_hundred_and_twenty_nine + one.
33ndefinition one_hundred_and_thirty_one ≝ one_hundred_and_thirty + one.
34ndefinition one_hundred_and_thirty_five ≝ one_hundred_and_twenty_eight + seven.
35ndefinition one_hundred_and_thirty_six ≝ one_hundred_and_thirty_five + one.
36ndefinition one_hundred_and_thirty_seven ≝ one_hundred_and_thirty_six + one.
37ndefinition one_hundred_and_thirty_eight ≝ one_hundred_and_twenty_eight + ten.
38ndefinition one_hundred_and_thirty_nine ≝ one_hundred_and_thirty_eight + one.
39ndefinition one_hundred_and_forty ≝ one_hundred_and_thirty_nine + one.
40ndefinition one_hundred_and_forty_one ≝ one_hundred_and_forty + one.
41ndefinition one_hundred_and_forty_four ≝ one_hundred_and_twenty_eight + sixteen.
42ndefinition one_hundred_and_fifty_two ≝ one_hundred_and_forty_four + eight.
43ndefinition one_hundred_and_fifty_three ≝ one_hundred_and_forty_four + nine.
44ndefinition one_hundred_and_sixty ≝ one_hundred_and_forty_four + sixteen.
45ndefinition one_hundred_and_sixty_eight ≝ one_hundred_and_sixty + eight.
46ndefinition one_hundred_and_seventy_six ≝ one_hundred_and_sixty + sixteen.
47ndefinition one_hundred_and_eighty_four ≝ one_hundred_and_seventy_six + eight.
48ndefinition two_hundred ≝ one_hundred + one_hundred.
49ndefinition two_hundred_and_two ≝ two_hundred + two.
50ndefinition two_hundred_and_three ≝ two_hundred_and_two + one.
51ndefinition two_hundred_and_four ≝ two_hundred_and_three + one.
52ndefinition two_hundred_and_five ≝ two_hundred_and_four + one.
53ndefinition two_hundred_and_eight ≝ two_hundred_and_five + three.
54ndefinition two_hundred_and_twenty_four ≝ two_hundred_and_eight + sixteen.
55ndefinition two_hundred_and_forty ≝ two_hundred_and_twenty_four + sixteen.
56ndefinition two_hundred_and_fifty_six ≝
57  one_hundred_and_twenty_eight + one_hundred_and_twenty_eight.                                       
58   
59ndefinition nat_of_bool ≝
60  λb: Bool.
61    match b with
62      [ false ⇒ Z
63      | true ⇒ S Z
64      ].
65   
66ndefinition add_n_with_carry:
67      ∀n: Nat. ∀b, c: BitVector n. ∀carry: Bool. Cartesian (BitVector n) (List Bool) ≝
68  λn: Nat.
69  λb: BitVector n.
70  λc: BitVector n.
71  λcarry: Bool.
72    let b_as_nat ≝ nat_of_bitvector n b in
73    let c_as_nat ≝ nat_of_bitvector n c in
74    let carry_as_nat ≝ nat_of_bool carry in
75    let result_old ≝ b_as_nat + c_as_nat + carry_as_nat in
76    let ac_flag ≝ ((modulus b_as_nat ((S (S Z)) * n)) +
77                  (modulus c_as_nat ((S (S Z)) * n)) +
78                  c_as_nat) ≥ ((S (S Z)) * n) in
79    let bit_xxx ≝ (((modulus b_as_nat ((S (S Z))^(n - (S Z)))) +
80                  (modulus c_as_nat ((S (S Z))^(n - (S Z)))) +
81                  c_as_nat) ≥ ((S (S Z))^(n - (S Z)))) in
82    let result ≝ modulus result_old ((S (S Z))^n) in
83    let cy_flag ≝ (result_old ≥ ((S (S Z))^n)) in
84    let ov_flag ≝ exclusive_disjunction cy_flag bit_xxx in
85      ? (mk_Cartesian (BitVector n) ? (? (bitvector_of_nat n result))
86                          (cy_flag :: ac_flag :: ov_flag :: Empty Bool)).
87    #H; nassumption;
88nqed.
89
90ndefinition sub_8_with_carry: ∀b,c: BitVector eight. ∀carry: Bool. Cartesian (BitVector eight) (List Bool) ≝
91  λb: BitVector eight.
92  λc: BitVector eight.
93  λcarry: Bool.
94    let b_as_nat ≝ nat_of_bitvector eight b in
95    let c_as_nat ≝ nat_of_bitvector eight c in
96    let carry_as_nat ≝ nat_of_bool carry in
97    let temporary ≝ b_as_nat mod sixteen - c_as_nat mod sixteen in
98    let ac_flag ≝ negation (conjunction ((b_as_nat mod sixteen) ≤ (c_as_nat mod sixteen)) (temporary ≤ carry_as_nat)) in
99    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
100    let old_result_1 ≝ b_as_nat - c_as_nat in
101    let old_result_2 ≝ old_result_1 - carry_as_nat in
102    let ov_flag ≝ exclusive_disjunction carry bit_six in
103      match conjunction (b_as_nat ≤ c_as_nat) (old_result_1 ≤ carry_as_nat) with
104        [ false ⇒
105           let cy_flag ≝ false in
106            〈 bitvector_of_nat eight old_result_2, [cy_flag ; ac_flag ; ov_flag ] 〉
107        | true ⇒
108           let cy_flag ≝ true in
109           let new_result ≝ b_as_nat + two_hundred_and_fifty_six - c_as_nat - carry_as_nat in
110            〈 bitvector_of_nat eight new_result, [ cy_flag ; ac_flag ; ov_flag ] 〉
111        ].
112         
113ndefinition add_8_with_carry ≝ add_n_with_carry eight.
114ndefinition add_16_with_carry ≝ add_n_with_carry sixteen.
115
116ndefinition increment ≝
117  λn: Nat.
118  λb: BitVector n.
119    let b_as_nat ≝ (nat_of_bitvector n b) + (S Z) in
120    let overflow ≝ b_as_nat ≥ (S (S Z))^n in
121      match overflow with
122        [ false ⇒ bitvector_of_nat n b_as_nat
123        | true ⇒ zero n
124        ].
125       
126ndefinition decrement ≝
127  λn: Nat.
128  λb: BitVector n.
129    let b_as_nat ≝ nat_of_bitvector n b in
130      match b_as_nat with
131        [ Z ⇒ max n
132        | S o ⇒ bitvector_of_nat n o
133        ].
134       
135alias symbol "greater_than_or_equal" (instance 1) = "Nat greater than or equal prop".
136
137ndefinition bitvector_of_bool:
138      ∀n: Nat. ∀b: Bool. BitVector n ≝
139  λn: Nat.
140  λb: Bool.
141    ? (pad (n - (S Z)) (S Z) (Cons Bool ? b (Empty Bool))).
142  nrewrite > (plus_minus_inverse_right n ?);
143  #H;
144  nassumption;
145nqed.
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