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(* Title: HoTT/Prod.thy
Author: Josh Chen
Date: Aug 2018
Dependent product (function) type.
*)
theory Prod
imports HoTT_Base
begin
section \<open>Constants and syntax\<close>
axiomatization
Prod :: "[Term, Typefam] \<Rightarrow> Term" and
lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
appl :: "[Term, Term] \<Rightarrow> Term" ("(1_`_)" [61, 60] 60)
\<comment> \<open>Application binds tighter than abstraction.\<close>
syntax
"_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30)
"_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term" ("(1\<^bold>\<lambda>_:_./ _)" 30)
"_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 30)
"_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3%%_:_./ _)" 30)
text "The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>."
translations
"\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)"
"\<^bold>\<lambda>x:A. b" \<rightleftharpoons> "CONST lambda(A, \<lambda>x. b)"
"PROD x:A. B" \<rightharpoonup> "CONST Prod(A, \<lambda>x. B)"
"%%x:A. b" \<rightharpoonup> "CONST lambda(A, \<lambda>x. b)"
text "Nondependent functions are a special case."
abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 40)
where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
section \<open>Type rules\<close>
axiomatization where
Prod_form: "\<And>i A B. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)"
and
Prod_intro: "\<And>i A B b. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b(x): \<Prod>x:A. B(x)"
and
Prod_elim: "\<And>A B f a. \<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)"
and
Prod_comp: "\<And>i A B b a. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
and
Prod_uniq: "\<And>A B f. f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
text "
Note that the syntax \<open>\<^bold>\<lambda>\<close> (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation).
"
text "
In addition to the usual type rules, it is a meta-theorem (*PROVE THIS!*) that whenever \<open>\<Prod>x:A. B x: U(i)\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A: U(i)\<close> and \<open>B: A \<longrightarrow> U(i)\<close>.
That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly.
"
axiomatization where
Prod_form_cond1: "\<And>i A B. (\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
and
Prod_form_cond2: "\<And>i A B. (\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
text "Set up the standard reasoner to use the type rules:"
lemmas Prod_rules = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq
lemmas Prod_form_conds [intro (*elim, wellform*)] = Prod_form_cond1 Prod_form_cond2
lemmas Prod_comps [comp] = Prod_comp Prod_uniq
section \<open>Unit type\<close>
axiomatization
Unit :: Term ("\<one>") and
pt :: Term ("\<star>") and
indUnit :: "[Typefam, Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<one>)")
where
Unit_form: "\<one> : U(O)"
and
Unit_intro: "\<star> : \<one>"
and
Unit_elim: "\<And>i C c a. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C(\<star>); a : \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(C, c, a) : C(a)"
and
Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(C, c, \<star>) \<equiv> c"
lemmas Unit_rules [intro] = Unit_form Unit_intro Unit_elim Unit_comp
lemmas Unit_comps [comp] = Unit_comp
end
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