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(* Title: HoTT/Proj.thy
Author: Josh Chen
Date: Jun 2018
Projection functions \<open>fst\<close> and \<open>snd\<close> for the dependent sum type.
*)
theory Proj
imports
HoTT_Methods
Prod
Sum
begin
consts
fst :: "[Term, 'a] \<Rightarrow> Term" ("(1fst[/_,/ _])")
snd :: "[Term, 'a] \<Rightarrow> Term" ("(1snd[/_,/ _])")
section \<open>Overloaded syntax for dependent and nondependent pairs\<close>
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
section \<open>Properties\<close>
text "Typing judgments and computation rules for the dependent and non-dependent projection functions."
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
text "In proving results about the second dependent projection function we often use the following two lemmas."
lemma snd_dep_welltyped:
assumes "p : \<Sum>x:A. B x"
shows "B (fst[A,B]`p) : U"
proof -
have "\<Sum>x:A. B x : U" using assms ..
then have *: "B: A \<rightarrow> U" ..
have "fst[A,B]`p : A" using assms by (rule fst_dep_type)
then show "B (fst[A,B]`p) : U" by (rule *)
qed
lemma snd_dep_const_type:
assumes "B: A \<rightarrow> U" and "x : A" and "y : B x"
shows "y : B (fst[A,B]`(x,y))"
proof -
have "fst[A,B]`(x,y) \<equiv> x" using assms by (rule fst_dep_comp)
then show "y : B (fst[A,B]`(x,y))" using assms by simp
qed
lemma snd_dep_type:
assumes "p : \<Sum>x:A. B x"
shows "snd[A,B]`p : B (fst[A,B]`p)"
proof
have *: "\<Sum>x:A. B x : U" using assms ..
show "snd[A, B] : \<Prod>p:(\<Sum>x:A. B x). B (fst[A,B]`p)"
proof (unfold snd_dep_def, standard)
show "\<And>p. p : \<Sum>x:A. B x \<Longrightarrow> indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p : B (fst[A,B]`p)"
proof (standard, elim snd_dep_welltyped)
fix x y assume 1: "x : A" and 2: "y : B x"
show "y : B (fst[A,B]`(x,y))"
proof -
have "B: A \<rightarrow> U" using * ..
then show "y : B (fst[A,B]`(x,y))" using 1 2 by (rule snd_dep_const_type)
qed
qed
qed (rule *)
qed (rule assms)
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 -
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)"
proof (standard, elim snd_dep_welltyped)
show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> y : B (fst[A,B]`(x,y))" using assms
by (elim snd_dep_const_type)
qed (rule *)
qed
also have "indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b) \<equiv> b"
proof (standard, elim snd_dep_welltyped)
show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> y : B (fst[A,B]`(x,y))" using assms
by (elim snd_dep_const_type)
qed (rule assms)+
finally show "snd[A,B]`(a,b) \<equiv> b" .
qed
text "For non-dependent projection functions:"
print_statement fst_dep_type
print_statement fst_dep_type[where ?p = p and ?A = A and ?B = "\<lambda>_. B"]
lemma fst_nondep_type: "p : A \<times> B \<Longrightarrow> fst[A,B]`p : A"
by (rule fst_dep_type[where ?B = "\<lambda>_. B"])
lemma fst_nondep_comp:
assumes "a : A" and "b : B"
shows "fst[A,B]`(a,b) \<equiv> a"
proof (unfold fst_nondep_def)
have "A : U" using assms(1) ..
then show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def
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
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