blob: 22c1860136ee028a7da3109f8a6bb6c69fcfe1e2 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
|
theory Nat
imports Identity
begin
axiomatization
Nat :: \<open>o\<close> and
zero :: \<open>o\<close> ("0") and
suc :: \<open>o \<Rightarrow> o\<close> and
NatInd :: \<open>(o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>
where
NatF: "Nat: U i" and
Nat_zero: "0: Nat" and
Nat_suc: "n: Nat \<Longrightarrow> suc n: Nat" and
NatE: "\<lbrakk>
n: Nat;
c\<^sub>0: C 0;
\<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
\<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n: C n" and
Nat_comp_zero: "\<lbrakk>
c\<^sub>0: C 0;
\<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
\<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) 0 \<equiv> c\<^sub>0" and
Nat_comp_suc: "\<lbrakk>
n: Nat;
c\<^sub>0: C 0;
\<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k);
\<And>n. n: Nat \<Longrightarrow> C n: U i
\<rbrakk> \<Longrightarrow>
NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) (suc n) \<equiv>
f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n)"
lemmas
[intros] = NatF Nat_zero Nat_suc and
[elims "?n"] = NatE and
[comps] = Nat_comp_zero Nat_comp_suc
abbreviation "NatRec C \<equiv> NatInd (\<lambda>_. C)"
abbreviation one ("1") where "1 \<equiv> suc 0"
abbreviation two ("2") where "2 \<equiv> suc 1"
abbreviation three ("3") where "3 \<equiv> suc 2"
abbreviation four ("4") where "4 \<equiv> suc 3"
abbreviation five ("5") where "5 \<equiv> suc 4"
abbreviation six ("6") where "6 \<equiv> suc 5"
abbreviation seven ("7") where "7 \<equiv> suc 6"
abbreviation eight ("8") where "8 \<equiv> suc 7"
abbreviation nine ("9") where "9 \<equiv> suc 8"
section \<open>Basic arithmetic\<close>
subsection \<open>Addition\<close>
definition add (infixl "+" 120) where "m + n \<equiv> NatRec Nat m (K suc) n"
lemma add_type [typechk]:
assumes "m: Nat" "n: Nat"
shows "m + n: Nat"
unfolding add_def by typechk
lemma add_zero [comps]:
assumes "m: Nat"
shows "m + 0 \<equiv> m"
unfolding add_def by reduce
lemma add_suc [comps]:
assumes "m: Nat" "n: Nat"
shows "m + suc n \<equiv> suc (m + n)"
unfolding add_def by reduce
Lemma (derive) zero_add:
assumes "n: Nat"
shows "0 + n = n"
apply (elim n)
\<guillemotright> by (reduce; intro)
\<guillemotright> vars _ ih by reduce (eq ih; intro)
done
Lemma (derive) suc_add:
assumes "m: Nat" "n: Nat"
shows "suc m + n = suc (m + n)"
apply (elim n)
\<guillemotright> by reduce refl
\<guillemotright> vars _ ih by reduce (eq ih; intro)
done
Lemma (derive) suc_eq:
assumes "m: Nat" "n: Nat"
shows "p: m = n \<Longrightarrow> suc m = suc n"
by (eq p) intro
Lemma (derive) add_assoc:
assumes "l: Nat" "m: Nat" "n: Nat"
shows "l + (m + n) = l + m+ n"
apply (elim n)
\<guillemotright> by reduce intro
\<guillemotright> vars _ ih by reduce (eq ih; intro)
done
Lemma (derive) add_comm:
assumes "m: Nat" "n: Nat"
shows "m + n = n + m"
apply (elim n)
\<guillemotright> by (reduce; rule zero_add[symmetric])
\<guillemotright> prems vars n ih
proof reduce
have "suc (m + n) = suc (n + m)" by (eq ih) intro
also have ".. = suc n + m" by (transport eq: suc_add) refl
finally show "{}" by this
qed
done
subsection \<open>Multiplication\<close>
definition mul (infixl "*" 121) where "m * n \<equiv> NatRec Nat 0 (K $ add m) n"
lemma mul_type [typechk]:
assumes "m: Nat" "n: Nat"
shows "m * n: Nat"
unfolding mul_def by typechk
lemma mul_zero [comps]:
assumes "n: Nat"
shows "n * 0 \<equiv> 0"
unfolding mul_def by reduce
lemma mul_one [comps]:
assumes "n: Nat"
shows "n * 1 \<equiv> n"
unfolding mul_def by reduce
lemma mul_suc [comps]:
assumes "m: Nat" "n: Nat"
shows "m * suc n \<equiv> m + m * n"
unfolding mul_def by reduce
Lemma (derive) zero_mul:
assumes "n: Nat"
shows "0 * n = 0"
apply (elim n)
\<guillemotright> by reduce refl
\<guillemotright> prems vars n ih
proof reduce
have "0 + 0 * n = 0 + 0 " by (eq ih) refl
also have ".. = 0" by reduce refl
finally show "0 + 0 * n = 0" by this
qed
done
Lemma (derive) suc_mul:
assumes "m: Nat" "n: Nat"
shows "suc m * n = m * n + n"
apply (elim n)
\<guillemotright> by reduce refl
\<guillemotright> prems vars n ih
proof (reduce, transport eq: \<open>ih:_\<close>)
have "suc m + (m * n + n) = suc (m + {})" by (rule suc_add)
also have ".. = suc (m + m * n + n)" by (transport eq: add_assoc) refl
finally show "{}" by this
qed
done
Lemma (derive) mul_dist_add:
assumes "l: Nat" "m: Nat" "n: Nat"
shows "l * (m + n) = l * m + l * n"
apply (elim n)
\<guillemotright> by reduce refl
\<guillemotright> prems prms vars n ih
proof reduce
have "l + l * (m + n) = l + (l * m + l * n)" by (eq ih) refl
also have ".. = l + l * m + l * n" by (rule add_assoc)
also have ".. = l * m + l + l * n" by (transport eq: add_comm) refl
also have ".. = l * m + (l + l * n)" by (transport eq: add_assoc) refl
finally show "{}" by this
qed
done
Lemma (derive) mul_assoc:
assumes "l: Nat" "m: Nat" "n: Nat"
shows "l * (m * n) = l * m * n"
apply (elim n)
\<guillemotright> by reduce refl
\<guillemotright> prems vars n ih
proof reduce
have "l * (m + m * n) = l * m + l * (m * n)" by (rule mul_dist_add)
also have ".. = l * m + l * m * n" by (transport eq: \<open>ih:_\<close>) refl
finally show "{}" by this
qed
done
Lemma (derive) mul_comm:
assumes "m: Nat" "n: Nat"
shows "m * n = n * m"
apply (elim n)
\<guillemotright> by reduce (transport eq: zero_mul, refl)
\<guillemotright> prems vars n ih
proof (reduce, rule pathinv)
have "suc n * m = n * m + m" by (rule suc_mul)
also have ".. = m + m * n"
by (transport eq: \<open>ih:_\<close>, transport eq: add_comm) refl
finally show "{}" by this
qed
done
end
|
