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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 ‹Symmetry / Path inverse›
definition inv :: "[Term, Term, Term] ⇒ Term" ("(1inv[_,/ _,/ _])")
where "inv[A,x,y] ≡ ❙λp: (x =⇩A y). indEqual[A] (λx y _. y =⇩A x) (λx. refl(x)) x y p"
lemma inv_type:
assumes "p : x =⇩A y"
shows "inv[A,x,y]`p : y =⇩A x"
proof
show "inv[A,x,y] : (x =⇩A y) → (y =⇩A x)"
proof (unfold inv_def, standard)
fix p assume asm: "p : x =⇩A y"
show "indEqual[A] (λx y _. y =[A] x) refl x y p : y =⇩A x"
proof standard+
show "x : A" by (wellformed jdgmt: asm)
show "y : A" by (wellformed jdgmt: asm)
qed (assumption | rule | rule asm)+
qed (wellformed jdgmt: assms)
qed (rule assms)
lemma inv_comp:
assumes "a : A"
shows "inv[A,a,a]`refl(a) ≡ refl(a)"
proof -
have "inv[A,a,a]`refl(a) ≡ indEqual[A] (λx y _. y =⇩A x) (λx. refl(x)) a a refl(a)"
proof (unfold inv_def, standard)
show "refl(a) : a =⇩A a" using assms ..
fix p assume asm: "p : a =⇩A a"
show "indEqual[A] (λx y _. y =⇩A x) refl a a p : a =⇩A a"
proof standard+
show "a : A" by (wellformed jdgmt: asm)
then show "a : A" . ― ‹The elimination rule requires that both arguments to ‹indEqual› be shown to have the correct type.›
qed (assumption | rule | rule asm)+
qed
also have "indEqual[A] (λx y _. y =⇩A x) (λx. refl(x)) a a refl(a) ≡ refl(a)"
by (standard | assumption | rule assms)+
finally show "inv[A,a,a]`refl(a) ≡ refl(a)" .
qed
section ‹Transitivity / Path composition›
― ‹"Raw" composition function›
definition compose' :: "Term ⇒ Term" ("(1compose''[_])")
where "compose'[A] ≡
indEqual[A] (λx y _. ∏z:A. ∏q: y =⇩A z. x =⇩A z) (indEqual[A](λx z _. x =⇩A z) (❙λx:A. refl(x)))"
― ‹"Natural" composition function›
abbreviation compose :: "[Term, Term, Term, Term] ⇒ Term" ("(1compose[_,/ _,/ _,/ _])")
where "compose[A,x,y,z] ≡ ❙λp:x =⇩A y. ❙λq:y =⇩A z. compose'[A]`x`y`p`z`q"
(**** GOOD CANDIDATE FOR AUTOMATION ****)
lemma compose_comp:
assumes "a : A"
shows "compose[A,a,a,a]`refl(a)`refl(a) ≡ refl(a)" using assms Equal_intro[OF assms] unfolding compose'_def by simp
text "The above proof is a good candidate for proof automation; in particular we would like the system to be able to automatically find the conditions of the ‹using› clause in the proof.
This would likely involve something like:
1. Recognizing that there is a function application that can be simplified.
2. Noting that the obstruction to applying ‹Prod_comp› is the requirement that ‹refl(a) : a =⇩A a›.
3. Obtaining such a condition, using the known fact ‹a : A› and the introduction rule ‹Equal_intro›."
lemmas Equal_simps [simp] = inv_comp compose_comp
subsubsection ‹Pretty printing›
abbreviation inv_pretty :: "[Term, Term, Term, Term] ⇒ Term" ("(1_⇧-⇧1[_, _, _])" 500)
where "p⇧-⇧1[A,x,y] ≡ inv[A,x,y]`p"
abbreviation compose_pretty :: "[Term, Term, Term, Term, Term, Term] ⇒ Term" ("(1_ ∙[_, _, _, _]/ _)")
where "p ∙[A,x,y,z] q ≡ compose[A,x,y,z]`p`q"
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