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(* Title: HoTT/Sum.thy
Author: Josh Chen
Date: Jun 2018
Dependent sum type.
*)
theory Sum
imports Prod
begin
axiomatization
Sum :: "[Term, Typefam] \<Rightarrow> Term" and
pair :: "[Term, Term] \<Rightarrow> Term" ("(1'(_,/ _'))") and
indSum :: "[Term, Typefam, Typefam, [Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1indSum[_,/ _])")
section \<open>Syntax\<close>
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)"
section \<open>Type rules\<close>
axiomatization where
Sum_form: "\<And>A B. \<lbrakk>A : U; B: A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x : U"
and
Sum_form_cond1: "\<And>A B. (\<Sum>x:A. B x : U) \<Longrightarrow> A : U"
and
Sum_form_cond2: "\<And>A B. (\<Sum>x:A. B x : U) \<Longrightarrow> B: A \<rightarrow> U"
and
Sum_intro: "\<And>A B a b. \<lbrakk>B: A \<rightarrow> U; a : A; b : B a\<rbrakk> \<Longrightarrow> (a,b) : \<Sum>x:A. B x"
and
Sum_elim: "\<And>A B C f p. \<lbrakk>
C: \<Sum>x:A. B x \<rightarrow> U;
\<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> indSum[A,B] C f p : C p"
and
Sum_comp: "\<And>A B C f a b. \<lbrakk>
C: \<Sum>x:A. B x \<rightarrow> U;
\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> f x y : C (x,y);
a : A;
b : B a
\<rbrakk> \<Longrightarrow> indSum[A,B] C f (a,b) \<equiv> f a b"
lemmas Sum_rules [intro] = Sum_form Sum_intro Sum_elim Sum_comp
lemmas Sum_form_conds [elim] = Sum_form_cond1 Sum_form_cond2
\<comment> \<open>Nondependent pair\<close>
abbreviation Pair :: "[Term, Term] \<Rightarrow> Term" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
section \<open>Projection functions\<close>
consts
fst :: "[Term, 'a] \<Rightarrow> Term" ("(1fst[/_,/ _])")
snd :: "[Term, 'a] \<Rightarrow> Term" ("(1snd[/_,/ _])")
overloading
fst_dep \<equiv> fst
fst_nondep \<equiv> fst
begin
definition fst_dep :: "[Term, Typefam] \<Rightarrow> Term" where
"fst_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p"
definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where
"fst_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) p"
end
overloading
snd_dep \<equiv> snd
snd_nondep \<equiv> snd
begin
definition snd_dep :: "[Term, Typefam] \<Rightarrow> Term" where
"snd_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p"
definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where
"snd_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) p"
end
text "Results about projections:"
lemma fst_dep_type:
assumes "p : \<Sum>x:A. B x"
shows "fst[A,B]`p : A"
proof \<comment> \<open>The standard reasoner knows to backchain with the product elimination rule here...\<close>
\<comment> \<open>Also write about this proof: compare the effect of letting the standard reasoner do simplifications, as opposed to using the minus switch and writing everything out explicitly from first principles.\<close>
have *: "\<Sum>x:A. B x : U" using assms ..
show "fst[A,B]: (\<Sum>x:A. B x) \<rightarrow> A"
proof (unfold fst_dep_def, standard) \<comment> \<open>...and with the product introduction rule here...\<close>
show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p : A"
proof \<comment> \<open>...and with sum elimination here.\<close>
show "A : U" using * ..
qed
qed (rule *)
qed (rule assms)
lemma fst_dep_comp: (* Potential for automation *)
assumes "B: A \<rightarrow> U" and "a : A" and "b : B a"
shows "fst[A,B]`(a,b) \<equiv> a"
proof -
\<comment> "Write about this proof: unfolding, how we set up the introduction rules (explain \<open>..\<close>), do a trace of the proof, explain the meaning of keywords, etc."
have *: "A : U" using assms(2) .. (* I keep thinking this should not have to be done explicitly, but rather automated. *)
have "fst[A,B]`(a,b) \<equiv> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)"
proof (unfold fst_dep_def, standard)
show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p : A" using * ..
show "(a,b) : \<Sum>x:A. B x" using assms ..
qed
also have "indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a"
by (rule Sum_comp) (rule *, assumption, (rule assms)+)
finally show "fst[A,B]`(a,b) \<equiv> a" .
qed
lemma snd_dep_type:
assumes "p : \<Sum>x:A. B x"
shows "snd[A,B]`p : B (fst[A,B]`p)"
oops
\<comment> \<open>Do we need this for the following lemma?\<close>
lemma snd_dep_comp:
assumes "B: A \<rightarrow> U" and "a : A" and "b : B a"
shows "snd[A,B]`(a,b) \<equiv> b"
proof -
\<comment> \<open>We use the following two lemmas:\<close>
have lemma1: "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> B (fst[A,B]`p) : U"
proof -
fix p assume "p : \<Sum>x:A. B x"
then have "fst[A,B]`p : A" by (rule fst_dep_type)
then show "B (fst[A,B]`p) : U" by (rule assms(1))
qed
have lemma2: "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> y : B (fst[A,B]`(x, y))"
proof -
fix x y assume "x : A" and "y : B x"
moreover with assms(1) have "fst[A,B]`(x,y) \<equiv> x" by (rule fst_dep_comp)
ultimately show "y : B (fst[A,B]`(x,y))" by simp
qed
\<comment> \<open>And now the proof.\<close>
have "snd[A,B]`(a,b) \<equiv> indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b)"
proof (unfold snd_dep_def, standard)
show "(a,b) : \<Sum>x:A. B x" using assms ..
fix p assume *: "p : \<Sum>x:A. B x"
show "indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p : B (fst[A,B]`p)"
by (standard, elim lemma1, elim lemma2) (assumption, rule *)
qed
also have "indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b) \<equiv> b"
by (standard, elim lemma1, elim lemma2) (assumption, (rule assms)+)
finally show "snd[A,B]`(a,b) \<equiv> b" .
qed
lemma fst_nondep_comp:
assumes "a : A" and "b : B"
shows "fst[A,B]`(a,b) \<equiv> a"
proof -
have "A : U" using assms(1) ..
then show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def by simp
qed
lemma snd_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b"
proof -
assume "a : A" and "b : B"
then have "(a, b) : A \<times> B" ..
then show "snd[A,B]`(a,b) \<equiv> b" unfolding snd_nondep_def by simp
qed
lemmas proj_simps [simp] = fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp
end
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