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(* 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] \<Rightarrow> (Term \<Rightarrow> Term)" ("(1inv[_,/ _])")
where "inv[x,y] \<equiv> \<lambda>p. ind\<^sub>= (\<lambda>x. refl(x)) x y p"
lemma inv_type:
assumes "A : U(i)" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "inv[x,y](p): y =\<^sub>A x"
unfolding inv_def
proof
show "\<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> y =\<^sub>A x: U(i)" using assms(1) ..
show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" ..
qed (fact assms)+
lemma inv_comp: assumes "A : U(i)" and "a : A" shows "inv[a,a](refl(a)) \<equiv> refl(a)"
unfolding inv_def
proof
show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" ..
show "\<And>x. x: A \<Longrightarrow> x =\<^sub>A x: U(i)" using assms(1) ..
qed (fact 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"
definition compose :: "[Term, Term, Term] \<Rightarrow> [Term, Term] \<Rightarrow> Term" ("(1compose[ _,/ _,/ _])")
where "compose[x,y,z] \<equiv> \<lambda>p."
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
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