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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"
unfolding fst_dep_def
by (derive lems: assms)
lemma fst_dep_comp:
assumes "B: A \<rightarrow> U" and "a : A" and "b : B a"
shows "fst[A,B]`(a,b) \<equiv> a"
unfolding fst_dep_def
by (simplify lems: assms)
\<comment> \<open> (* Old proof *)
proof -
have "fst[A,B]`(a,b) \<equiv> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)"
by (derive lems: assms unfolds: fst_dep_def)
also have "indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a"
by (derive lems: assms)
finally show "fst[A,B]`(a,b) \<equiv> a" .
qed
\<close>
\<comment> \<open> (* Old lemma *)
text "In proving results about the second dependent projection function we often use the following lemma."
lemma lem:
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
\<close>
lemma snd_dep_type:
assumes "p : \<Sum>x:A. B x"
shows "snd[A,B]`p : B (fst[A,B]`p)"
unfolding fst_dep_def snd_dep_def
by (simplify lems: assms)
\<comment> \<open> (* Old proof *)
proof (derive lems: assms unfolds: snd_dep_def)
show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
fix x y assume asm: "x : A" "y : B x"
have "B: A \<rightarrow> U" by (wellformed jdgmt: assms)
then show "y : B (fst[A, B]`(x,y))" using asm by (rule lem)
qed (assumption | rule assms)+
\<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"
unfolding snd_dep_def fst_dep_def
by (simplify lems: assms)
\<comment> \<open> (* Old proof *)
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 (derive lems: assms unfolds: snd_dep_def)
show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
fix x y assume asm: "x : A" "y : B x"
with assms(1) show "y : B (fst[A, B]`(x,y))" by (rule lem)
qed (assumption | derive lems: assms)+
also have "indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b) \<equiv> b"
proof (simple lems: assms)
show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
fix x y assume "x : A" and "y : B x"
with assms(1) show "y : B (fst[A,B]`(x,y))" by (rule lem)
qed (assumption | rule assms)+
finally show "snd[A,B]`(a,b) \<equiv> b" .
qed
\<close>
text "For non-dependent projection functions:"
lemma fst_nondep_type:
assumes "p : A \<times> B"
shows "fst[A,B]`p : A"
unfolding fst_nondep_def
by (derive lems: assms)
lemma fst_nondep_comp:
assumes "a : A" and "b : B"
shows "fst[A,B]`(a,b) \<equiv> a"
unfolding fst_nondep_def
by (simplify lems: assms)
\<comment> \<open> (* Old proof *)
proof -
have "fst[A,B]`(a,b) \<equiv> indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)"
by (derive lems: assms unfolds: fst_nondep_def)
also have "indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a"
by (derive lems: assms)
finally show "fst[A,B]`(a,b) \<equiv> a" .
qed
\<close>
lemma snd_nondep_type:
assumes "p : A \<times> B"
shows "snd[A,B]`p : B"
unfolding snd_nondep_def
by (derive lems: assms)
\<comment> \<open> (* Old proof *)
proof
show "snd[A,B] : A \<times> B \<rightarrow> B"
proof (derive unfolds: snd_nondep_def)
fix q assume asm: "q : A \<times> B"
show "indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) q : B" by (derive lems: asm)
qed (wellformed jdgmt: assms)+
qed (rule assms)
\<close>
lemma snd_nondep_comp:
assumes "a : A" and "b : B"
shows "snd[A,B]`(a,b) \<equiv> b"
unfolding snd_nondep_def
by (simplify lems: assms)
\<comment> \<open> (* Old proof *)
proof -
have "snd[A,B]`(a,b) \<equiv> indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) (a,b)"
by (derive lems: assms unfolds: snd_nondep_def)
also have "indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) (a,b) \<equiv> b"
by (derive lems: assms)
finally show "snd[A,B]`(a,b) \<equiv> b" .
qed
\<close>
end
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