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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 \<Rightarrow> Term) \<Rightarrow> Term" (binder "\<^bold>\<lambda>" 30) and
appl :: "[Term, Term] \<Rightarrow> Term" (infixl "`" 60)
\<comment> \<open>Application binds tighter than abstraction.\<close>
syntax
"_PROD" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 30)
"_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3PROD _:_./ _)" 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)"
"PROD x:A. B" \<rightharpoonup> "CONST Prod 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: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)"
and
Prod_intro: "\<lbrakk>A: U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x): \<Prod>x:A. B(x)"
and
Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)"
and
Prod_comp: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)"
and
Prod_uniq: "f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f"
and
Prod_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b(x) \<equiv> b'(x); A: U(i)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) \<equiv> \<^bold>\<lambda>x. b'(x)"
text "
The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions; Prod_eq is the actual definitional equality rule.
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 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: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
and
Prod_form_cond2: "(\<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 [intro] = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq Prod_eq
lemmas Prod_wellform [wellform] = Prod_form_cond1 Prod_form_cond2
lemmas Prod_comps [comp] = Prod_comp Prod_uniq Prod_eq
section \<open>Function composition\<close>
definition compose :: "[Term, Term] \<Rightarrow> Term" (infixr "o" 70) where "g o f \<equiv> \<^bold>\<lambda>x. g`(f`x)"
syntax "_COMPOSE" :: "[Term, Term] \<Rightarrow> Term" (infixr "\<circ>" 70)
translations "g \<circ> f" \<rightleftharpoons> "g o f"
section \<open>Unit type\<close>
axiomatization
Unit :: Term ("\<one>") and
pt :: Term ("\<star>") and
indUnit :: "[Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<one>)")
where
Unit_form: "\<one>: U(O)"
and
Unit_intro: "\<star>: \<one>"
and
Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)"
and
Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(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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