blob: 9ecab4d8022a42f06906f46c101625b390dd57f1 (
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
|
(* Title: HoTT/Prod.thy
Author: Josh Chen
Dependent product (function) type for the HoTT logic.
*)
theory Prod
imports HoTT_Base
begin
axiomatization
Prod :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60)
syntax
"_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30)
"_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<^bold>\<lambda>_:_./ _)" 30)
"_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30)
"_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3%%_:_./ _)" 30)
\<comment> \<open>The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>.\<close>
translations
"\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)"
"\<^bold>\<lambda>x:A. b" \<rightleftharpoons> "CONST lambda A (\<lambda>x. b)"
"PROD x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)"
"%%x:A. b" \<rightharpoonup> "CONST lambda A (\<lambda>x. b)"
\<comment> \<open>Type rules\<close>
axiomatization where
Prod_form [intro]: "\<And>A B. \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) : U"
and
Prod_intro [intro]: "\<And>A B b. (\<And>x. x : A \<Longrightarrow> b(x) : B(x)) \<Longrightarrow> \<^bold>\<lambda>x:A. b(x) : \<Prod>x:A. B(x)"
and
Prod_elim [elim]: "\<And>A B f a. \<lbrakk>f : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> f`a : B(a)"
and
Prod_comp [simp]: "\<And>A b a. a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
and
Prod_uniq [simp]: "\<And>A f. \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
text "Note that the syntax \<open>\<^bold>\<lambda>\<close> (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation)."
\<comment> \<open>Nondependent functions are a special case.\<close>
abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 40)
where "A \<rightarrow> B \<equiv> \<Prod>_:A. B"
end
|
