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(********
Isabelle/HoTT: Dependent product (dependent function)
Jan 2019
********)
theory Prod
imports HoTT_Base HoTT_Methods
begin
section \<open>Basic type definitions\<close>
axiomatization
Prod :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
lam :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
app :: "[t, t] \<Rightarrow> t" ("(1_ ` _)" [120, 121] 120)
\<comment> \<open>Application should bind tighter than abstraction.\<close>
syntax
"_Prod" :: "[idt, t, t] \<Rightarrow> t" ("(3TT '(_: _')./ _)" 30)
"_Prod'" :: "[idt, t, t] \<Rightarrow> t" ("(3TT _: _./ _)" 30)
"_lam" :: "[idt, t, t] \<Rightarrow> t" ("(3,\\ '(_: _')./ _)" 30)
"_lam'" :: "[idt, t, t] \<Rightarrow> t" ("(3,\\ _: _./ _)" 30)
translations
"TT(x: A). B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
"TT x: A. B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
",\\(x: A). b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
",\\x: A. b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
text \<open>
The syntax translations above bind the variable @{term x} in the expressions @{term B} and @{term b}.
\<close>
text \<open>Non-dependent functions are a special case:\<close>
abbreviation Fun :: "[t, t] \<Rightarrow> t" (infixr "\<rightarrow>" 40)
where "A \<rightarrow> B \<equiv> TT(_: A). B"
axiomatization where
\<comment> \<open>Type rules\<close>
Prod_form: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> TT x: A. B x: U i" and
Prod_intro: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> ,\\x: A. b x: TT x: A. B x" and
Prod_elim: "\<lbrakk>f: TT x: A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and
Prod_cmp: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (,\\x: A. b x)`a \<equiv> b a" and
Prod_uniq: "f: TT x: A. B x \<Longrightarrow> ,\\x: A. f`x \<equiv> f" and
\<comment> \<open>Congruence rules\<close>
Prod_form_eq: "\<lbrakk>A: U i; B: A \<leadsto> U i; C: A \<leadsto> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk>
\<Longrightarrow> TT x: A. B x \<equiv> TT x: A. C x" and
Prod_intro_eq: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> ,\\x: A. b x \<equiv> ,\\x: A. c x"
text \<open>
The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions.
The actual definitional equality rule in the type theory is @{thm Prod_intro_eq}.
Note that this is a separate rule from function extensionality.
\<close>
lemmas Prod_form [form]
lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
lemmas Prod_comp [comp] = Prod_cmp Prod_uniq
lemmas Prod_cong [cong] = Prod_form_eq Prod_intro_eq
section \<open>Function composition\<close>
definition compose :: "[t, t, t] \<Rightarrow> t"
where "compose A g f \<equiv> ,\\x: A. g`(f`x)"
declare compose_def [comp]
syntax "_compose" :: "[t, t] \<Rightarrow> t" (infixr "o" 110)
parse_translation \<open>
let fun compose_tr ctxt tms =
let
val g :: f :: _ = tms |> map (Typing.tm_of_ptm ctxt)
val dom =
case f of
Const ("Prod.lam", _) $ T $ _ => T
| _ => (case Typing.get_typing f (Typing.typing_assms ctxt) of
SOME (Const ("Prod.Prod", _) $ T $ _) => T
| SOME _ => Exn.error "Can't compose with a non-function"
| NONE => Exn.error "Cannot infer domain of composition: please state this explicitly")
in
@{const compose} $ dom $ g $ f
end
in
[("_compose", compose_tr)]
end
\<close>
lemma compose_assoc:
assumes "A: U i" "f: A \<rightarrow> B" "g: B \<rightarrow> C" "h: TT x: C. D x"
shows "compose A (compose B h g) f \<equiv> compose A h (compose A g f)"
by (derive lems: assms cong)
lemma compose_comp:
assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x"
shows "(,\\x: B. c x) o (,\\x: A. b x) \<equiv> ,\\x: A. c (b x)"
by (derive lems: assms cong)
abbreviation id :: "t \<Rightarrow> t" where "id A \<equiv> ,\\x: A. x"
end
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