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
|
(* Title: HoTT/EqualProps.thy
Author: Josh Chen
Date: Jun 2018
Properties of equality.
*)
theory EqualProps
imports
HoTT_Methods
Equal
Prod
begin
section \<open>Symmetry / Path inverse\<close>
definition inv :: "[Term, Term, Term] \<Rightarrow> Term" ("(1inv[_,/ _,/ _])")
where "inv[A,x,y] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. indEqual[A] (\<lambda>x y _. y =\<^sub>A x) (\<lambda>x. refl(x)) x y p"
lemma inv_type:
assumes "A : U(i)" and "x : A" and "y : A" shows "inv[A,x,y] : x =\<^sub>A y \<rightarrow> y =\<^sub>A x"
unfolding inv_def
by (derive lems: assms)
corollary assumes "x =\<^sub>A y : U(i)" and "p : x =\<^sub>A y" shows "inv[A,x,y]`p : y =\<^sub>A x"
by (derive lems: inv_type assms)
lemma inv_comp: assumes "A : U(i)" and "a : A" shows "inv[A,a,a]`refl(a) \<equiv> refl(a)"
unfolding inv_def by (simplify lems: assms)
section \<open>Transitivity / Path composition\<close>
text "``Raw'' composition function, of type \<open>\<Prod>x,y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close>."
definition rcompose :: "Term \<Rightarrow> Term" ("(1rcompose[_])")
where "rcompose[A] \<equiv> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:(x =\<^sub>A y). indEqual[A]
(\<lambda>x y _. \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)
(\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>p:(x =\<^sub>A z). indEqual[A](\<lambda>x z _. x =\<^sub>A z) (\<lambda>x. refl(x)) x z p)
x y p"
text "``Natural'' composition function, of type \<open>\<Prod>x,y,z:A. x =\<^sub>A y \<rightarrow> y =\<^sub>A z \<rightarrow> x =\<^sub>A z\<close>."
abbreviation compose :: "[Term, Term, Term, Term] \<Rightarrow> Term" ("(1compose[_,/ _,/ _,/ _])")
where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:(x =\<^sub>A y). \<^bold>\<lambda>q:(y =\<^sub>A z). rcompose[A]`x`y`p`z`q"
lemma compose_type:
assumes "A : U(i)" and "x : A" and "y : A" and "z : A"
shows "compose[A,x,y,z] : x =\<^sub>A y \<rightarrow> y =\<^sub>A z \<rightarrow> x =\<^sub>A z"
unfolding rcompose_def
by (derive lems: assms)
lemma compose_comp:
assumes "A : U(i)" and "a : A" shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)"
unfolding rcompose_def
by (simplify lems: assms)
lemmas EqualProps_rules [intro] = inv_type inv_comp compose_type compose_comp
lemmas EqualProps_comps [comp] = inv_comp compose_comp
end
|
