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
|
(* Title: HoTT/Sum.thy
Author: Josh Chen
Date: Jun 2018
Dependent sum type.
*)
theory Sum
imports HoTT_Base
begin
section \<open>Constants and syntax\<close>
axiomatization
Sum :: "[Term, Typefam] \<Rightarrow> Term" and
pair :: "[Term, Term] \<Rightarrow> Term" ("(1<_,/ _>)") and
indSum :: "[[Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<Sum>)")
syntax
"_SUM" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Sum>_:_./ _)" 20)
"_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3SUM _:_./ _)" 20)
translations
"\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
"SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)"
text "Nondependent pair."
abbreviation Pair :: "[Term, Term] \<Rightarrow> Term" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
section \<open>Type rules\<close>
axiomatization where
Sum_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x): U(i)"
and
Sum_intro: "\<lbrakk>B: A \<longrightarrow> U(i); a: A; b: B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)"
and
Sum_elim: "\<lbrakk>
C: \<Sum>x:A. B(x) \<longrightarrow> U(i);
\<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y) : C(<x,y>);
p : \<Sum>x:A. B(x)
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(p) : C(p)" (* What does writing \<lambda>x y. f(x, y) change? *)
and
Sum_comp: "\<lbrakk>
C: \<Sum>x:A. B(x) \<longrightarrow> U(i);
B: A \<longrightarrow> U(i);
\<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y) : C(<x,y>);
a: A;
b: B(a)
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(<a,b>) \<equiv> f(a)(b)"
text "Admissible inference rules for sum formation:"
axiomatization where
Sum_form_cond1: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
and
Sum_form_cond2: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
lemmas Sum_rules [intro] = Sum_form Sum_intro Sum_elim Sum_comp
lemmas Sum_wellform [wellform] = Sum_form_cond1 Sum_form_cond2
lemmas Sum_comps [comp] = Sum_comp
section \<open>Empty type\<close>
axiomatization
Empty :: Term ("\<zero>") and
indEmpty :: "Term \<Rightarrow> Term" ("(1ind\<^sub>\<zero>)")
where
Empty_form: "\<zero> : U(O)"
and
Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U(i); z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero>(z): C(z)"
lemmas Empty_rules [intro] = Empty_form Empty_elim
end
|
