aboutsummaryrefslogtreecommitdiff
path: root/hott/Nat.thy
blob: f45387c4be64abd9e17cb81f1652cbe4e518dd4a (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
215
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 (fn n. C n) c\<^sub>0 (fn 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 (fn n. C n) c\<^sub>0 (fn 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 (fn n. C n) c\<^sub>0 (fn k rec. f k rec) (suc n) \<equiv>
        f n (NatInd (fn n. C n) c\<^sub>0 (fn k rec. f k rec) n)"

lemmas
  [form] = NatF and
  [intr, intro] = Nat_zero Nat_suc and
  [elim "?n"] = NatE and
  [comp] = Nat_comp_zero Nat_comp_suc

abbreviation "NatRec C \<equiv> NatInd (fn _. 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 [type]:
  assumes "m: Nat" "n: Nat"
  shows "m + n: Nat"
  unfolding add_def by typechk

Lemma add_zero [comp]:
  assumes "m: Nat"
  shows "m + 0 \<equiv> m"
  unfolding add_def by reduce

Lemma add_suc [comp]:
  assumes "m: Nat" "n: Nat"
  shows "m + suc n \<equiv> suc (m + n)"
  unfolding add_def by reduce

Lemma (def) zero_add:
  assumes "n: Nat"
  shows "0 + n = n"
  apply (elim n)
    \<^item> by (reduce; intro)
    \<^item> vars _ ih by reduce (eq ih; refl)
  done

Lemma (def) suc_add:
  assumes "m: Nat" "n: Nat"
  shows "suc m + n = suc (m + n)"
  apply (elim n)
    \<^item> by reduce refl
    \<^item> vars _ ih by reduce (eq ih; refl)
  done

Lemma (def) suc_eq:
  assumes "m: Nat" "n: Nat"
  shows "p: m = n \<Longrightarrow> suc m = suc n"
  by (eq p) intro

Lemma (def) add_assoc:
  assumes "l: Nat" "m: Nat" "n: Nat"
  shows "l + (m + n) = l + m+ n"
  apply (elim n)
    \<^item> by reduce intro
    \<^item> vars _ ih by reduce (eq ih; refl)
  done

Lemma (def) add_comm:
  assumes "m: Nat" "n: Nat"
  shows "m + n = n + m"
  apply (elim n)
    \<^item> by (reduce; rule zero_add[symmetric])
    \<^item> vars n ih
      proof reduce
        have "suc (m + n) = suc (n + m)" by (eq ih) refl
        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 [type]:
  assumes "m: Nat" "n: Nat"
  shows "m * n: Nat"
  unfolding mul_def by typechk

Lemma mul_zero [comp]:
  assumes "n: Nat"
  shows "n * 0 \<equiv> 0"
  unfolding mul_def by reduce

Lemma mul_one [comp]:
  assumes "n: Nat"
  shows "n * 1 \<equiv> n"
  unfolding mul_def by reduce

Lemma mul_suc [comp]:
  assumes "m: Nat" "n: Nat"
  shows "m * suc n \<equiv> m + m * n"
  unfolding mul_def by reduce

Lemma (def) zero_mul:
  assumes "n: Nat"
  shows "0 * n = 0"
  apply (elim n)
  \<^item> by reduce refl
  \<^item> vars n ih
    proof reduce
      have "0 + 0 * n = 0 + 0 " by (eq ih) refl
      also have ".. = 0" by reduce refl
      finally show "{}" by this
    qed
  done

Lemma (def) suc_mul:
  assumes "m: Nat" "n: Nat"
  shows "suc m * n = m * n + n"
  apply (elim n)
  \<^item> by reduce refl
  \<^item> 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 (def) mul_dist_add:
  assumes "l: Nat" "m: Nat" "n: Nat"
  shows "l * (m + n) = l * m + l * n"
  apply (elim n)
  \<^item> by reduce refl
  \<^item> 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 (def) mul_assoc:
  assumes "l: Nat" "m: Nat" "n: Nat"
  shows "l * (m * n) = l * m * n"
  apply (elim n)
  \<^item> by reduce refl
  \<^item> 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 (def) mul_comm:
  assumes "m: Nat" "n: Nat"
  shows "m * n = n * m"
  apply (elim n)
  \<^item> by reduce (transport eq: zero_mul, refl)
  \<^item> 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