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-theory HoTT
-  imports Pure
-begin
-
-section \<open>Setup\<close>
-text "For ML files, routines and setup."
-
-section \<open>Basic definitions\<close>
-text "A single meta-level type \<open>Term\<close> suffices to implement the object-level types and terms.
-We do not implement universes, but simply follow the informal notation in the HoTT book."
-
-typedecl Term
-
-section \<open>Judgments\<close>
-
-consts
-  is_a_type :: "Term \<Rightarrow> prop"           ("(_ : U)" [0] 1000)
-  is_of_type :: "[Term, Term] \<Rightarrow> prop"  ("(3_ :/ _)" [0, 0] 1000)
-
-section \<open>Definitional equality\<close>
-text "We take the meta-equality \<open>\<equiv>\<close>, defined by the Pure framework for any of its terms, and use it additionally for definitional/judgmental equality of types and terms in our theory.
-
-Note that the Pure framework already provides axioms and results for various properties of \<open>\<equiv>\<close>, which we make use of and extend where necessary."
-
-theorem equal_types:
-  assumes "A \<equiv> B" and "A : U"
-  shows "B : U" using assms by simp
-
-theorem equal_type_element:
-  assumes "A \<equiv> B" and "x : A"
-  shows "x : B" using assms by simp
-
-lemmas type_equality [intro, simp] = equal_types equal_types[rotated] equal_type_element equal_type_element[rotated]
-
-section \<open>Type families\<close>
-text "Type families are implemented using meta-level lambda expressions \<open>P::Term \<Rightarrow> Term\<close> that further satisfy the following property."
-
-abbreviation is_type_family :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> prop"  ("(3_:/ _ \<rightarrow> U)")
-  where "P: A \<rightarrow> U \<equiv> (\<And>x::Term. x : A \<Longrightarrow> P(x) : U)"
-
-section \<open>Types\<close>
-
-subsection \<open>Dependent function/product\<close>
-
-axiomatization
-  Prod :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
-  lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term"
-syntax
-  "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term"          ("(3\<Prod>_:_./ _)" 30)
-  "_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term"        ("(3\<^bold>\<lambda>_:_./ _)" 30)
-  "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"    ("(3PROD _:_./ _)" 30)
-  "_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3%%_:_./ _)" 30)
-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)"
-  (* The above syntax translations bind the x in the expressions B, b. *)
-
-abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
-  where "A\<rightarrow>B \<equiv> \<Prod>_:A. B"
-
-axiomatization
-  appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
-where
-  Prod_form: "\<And>(A::Term) (B::Term \<Rightarrow> Term). \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) : U"
-and
-  Prod_intro [intro]:
-    "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term). (\<And>x::Term. x : A \<Longrightarrow> b(x) : B(x)) \<Longrightarrow> \<^bold>\<lambda>x:A. b(x) : \<Prod>x:A. B(x)"
-and
-  Prod_elim [elim]:
-    "\<And>(A::Term) (B::Term \<Rightarrow> Term) (f::Term) (a::Term). \<lbrakk>f : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> f`a : B(a)"
-and
-  Prod_comp [simp]: "\<And>(A::Term) (b::Term \<Rightarrow> Term) (a::Term). a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
-and
-  Prod_uniq [simp]: "\<And>A f::Term. \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
-
-lemmas Prod_formation [intro] = Prod_form Prod_form[rotated]
-
-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)."
-
-subsection \<open>Dependent pair/sum\<close>
-
-axiomatization
-  Sum :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term"
-syntax
-  "_SUM" :: "[idt, Term, Term] \<Rightarrow> Term"        ("(3\<Sum>_:_./ _)" 20)
-  "_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3SUM _:_./ _)" 20)
-translations
-  "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
-  "SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)"
-
-abbreviation Pair :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<times>" 50)
-  where "A\<times>B \<equiv> \<Sum>_:A. B"
-
-axiomatization
-  pair :: "[Term, Term] \<Rightarrow> Term"  ("(1'(_,/ _'))") and
-  indSum :: "(Term \<Rightarrow> Term) \<Rightarrow> Term"
-where
-  Sum_form: "\<And>(A::Term) (B::Term \<Rightarrow> Term). \<lbrakk>A : U; B: A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) : U"
-and
-  Sum_intro [intro]:
-    "\<And>(A::Term) (B::Term \<Rightarrow> Term) (a::Term) (b::Term). \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> (a, b) : \<Sum>x:A. B(x)"
-and
-  Sum_elim [elim]:
-    "\<And>(A::Term) (B::Term \<Rightarrow> Term) (C::Term \<Rightarrow> Term) (f::Term) (p::Term).
-      \<lbrakk>C: \<Sum>x:A. B(x) \<rightarrow> U; f : \<Prod>x:A. \<Prod>y:B(x). C((x,y)); p : \<Sum>x:A. B(x)\<rbrakk> \<Longrightarrow> indSum(C)`f`p : C(p)"
-and
-  Sum_comp [simp]: "\<And>(C::Term \<Rightarrow> Term) (f::Term) (a::Term) (b::Term). indSum(C)`f`(a,b) \<equiv> f`a`b"
-
-lemmas Sum_formation [intro] = Sum_form Sum_form[rotated]
-
-text "We choose to formulate the elimination rule by using the object-level function type and function application as much as possible.
-Hence only the type family \<open>C\<close> is left as a meta-level argument to the inductor indSum."
-
-subsubsection \<open>Projections\<close>
-
-consts
-  fst :: "[Term, 'a] \<Rightarrow> Term"  ("(1fst[/_,/ _])")
-  snd :: "[Term, 'a] \<Rightarrow> Term"  ("(1snd[/_,/ _])")
-overloading
-  fst_dep \<equiv> fst
-  snd_dep \<equiv> snd
-  fst_nondep \<equiv> fst
-  snd_nondep \<equiv> snd
-begin
-definition fst_dep :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" where
-  "fst_dep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). x)"
-
-definition snd_dep :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" where
-  "snd_dep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)"
-
-definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where
-  "fst_nondep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x)"
-
-definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where
-  "snd_nondep A B \<equiv> indSum(\<lambda>_. A)`(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. y)"
-end
-
-lemma fst_dep_comp: "\<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> fst[A,B]`(a,b) \<equiv> a" unfolding fst_dep_def by simp
-lemma snd_dep_comp: "\<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" unfolding snd_dep_def by simp
-
-lemma fst_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def by simp
-lemma snd_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" unfolding snd_nondep_def by simp
-
-\<comment> \<open>Simplification rules for projections\<close>
-lemmas fst_snd_simps [simp] = fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp
-
-subsection \<open>Equality type\<close>
-
-axiomatization
-  Equal :: "[Term, Term, Term] \<Rightarrow> Term"
-syntax
-  "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term"        ("(3_ =\<^sub>_/ _)" [101, 101] 100)
-  "_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term"  ("(3_ =[_]/ _)" [101, 101] 100)
-translations
-  "a =\<^sub>A b" \<rightleftharpoons> "CONST Equal A a b"
-  "a =[A] b" \<rightharpoonup> "CONST Equal A a b"
-
-axiomatization
-  refl :: "Term \<Rightarrow> Term"  ("(refl'(_'))") and
-  indEqual :: "[Term, [Term, Term, Term] \<Rightarrow> Term] \<Rightarrow> Term"  ("(indEqual[_])")
-where
-  Equal_form: "\<And>A a b::Term. \<lbrakk>A : U; a : A; b : A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U"
-  (* Should I write a permuted version \<open>\<lbrakk>A : U; b : A; a : A\<rbrakk> \<Longrightarrow> \<dots>\<close>? *)
-and
-  Equal_intro [intro]: "\<And>A x::Term. x : A \<Longrightarrow> refl(x) : x =\<^sub>A x"
-and
-  Equal_elim [elim]:
-    "\<And>(A::Term) (C::[Term, Term, Term] \<Rightarrow> Term) (f::Term) (a::Term) (b::Term) (p::Term).
-      \<lbrakk> \<And>x y::Term. \<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<rightarrow> U;
-        f : \<Prod>x:A. C(x)(x)(refl(x));
-        a : A;
-        b : A;
-        p : a =\<^sub>A b \<rbrakk>
-    \<Longrightarrow> indEqual[A](C)`f`a`b`p : C(a)(b)(p)"
-and
-  Equal_comp [simp]:
-    "\<And>(A::Term) (C::[Term, Term, Term] \<Rightarrow> Term) (f::Term) (a::Term). indEqual[A](C)`f`a`a`refl(a) \<equiv> f`a"
-
-lemmas Equal_formation [intro] = Equal_form Equal_form[rotated 1] Equal_form[rotated 2]
-
-subsubsection \<open>Properties of equality\<close>
-
-text "Symmetry/Path inverse"
-
-definition inv :: "[Term, Term, Term] \<Rightarrow> Term"  ("(1inv[_,/ _,/ _])")
-  where "inv[A,x,y] \<equiv> indEqual[A](\<lambda>x y _. y =\<^sub>A x)`(\<^bold>\<lambda>x:A. refl(x))`x`y"
-
-lemma inv_comp: "\<And>A a::Term. a : A \<Longrightarrow> inv[A,a,a]`refl(a) \<equiv> refl(a)" unfolding inv_def by simp
-
-text "Transitivity/Path composition"
-
-\<comment> \<open>"Raw" composition function\<close>
-abbreviation compose' :: "Term \<Rightarrow> Term"  ("(1compose''[_])")
-  where "compose'[A] \<equiv> indEqual[A](\<lambda>x y _. \<Prod>z:A. \<Prod>q: y =\<^sub>A z. x =\<^sub>A z)`(indEqual[A](\<lambda>x z _. x =\<^sub>A z)`(\<^bold>\<lambda>x:A. refl(x)))"
-
-\<comment> \<open>"Natural" composition function\<close>
-abbreviation compose :: "[Term, Term, Term, Term] \<Rightarrow> Term"  ("(1compose[_,/ _,/ _,/ _])")
-  where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. \<^bold>\<lambda>q:y =\<^sub>A z. compose'[A]`x`y`p`z`q"
-
-(**** GOOD CANDIDATE FOR AUTOMATION ****)
-lemma compose_comp:
-  assumes "a : A"
-  shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" using assms Equal_intro[OF assms] by simp
-
-text "The above proof is a good candidate for proof automation; in particular we would like the system to be able to automatically find the conditions of the \<open>using\<close> clause in the proof.
-This would likely involve something like:
-  1. Recognizing that there is a function application that can be simplified.
-  2. Noting that the obstruction to applying \<open>Prod_comp\<close> is the requirement that \<open>refl(a) : a =\<^sub>A a\<close>.
-  3. Obtaining such a condition, using the known fact \<open>a : A\<close> and the introduction rule \<open>Equal_intro\<close>."
-
-lemmas Equal_simps [simp] = inv_comp compose_comp
-
-subsubsection \<open>Pretty printing\<close>
-
-abbreviation inv_pretty :: "[Term, Term, Term, Term] \<Rightarrow> Term"  ("(1_\<^sup>-\<^sup>1\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_)" 500)
-  where "p\<^sup>-\<^sup>1\<^sub>A\<^sub>,\<^sub>x\<^sub>,\<^sub>y \<equiv> inv[A,x,y]`p"
-
-abbreviation compose_pretty :: "[Term, Term, Term, Term, Term, Term] \<Rightarrow> Term"  ("(1_ \<bullet>\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_/ _)")
-  where "p \<bullet>\<^sub>A\<^sub>,\<^sub>x\<^sub>,\<^sub>y\<^sub>,\<^sub>z q \<equiv> compose[A,x,y,z]`p`q"
-
-end
-
-(*
-subsubsection \<open>Empty type\<close>
-
-axiomatization
-  Null :: Term and
-  ind_Null :: "Term \<Rightarrow> Term \<Rightarrow> Term" ("(ind'_Null'(_,/ _'))")
-where
-  Null_form: "Null : U" and
-  Null_elim: "\<And>C x a. \<lbrakk>x : Null \<Longrightarrow> C(x) : U; a : Null\<rbrakk> \<Longrightarrow> ind_Null(C(x), a) : C(a)"
-
-subsubsection \<open>Natural numbers\<close>
-
-axiomatization
-  Nat :: Term and
-  zero :: Term ("0") and
-  succ :: "Term \<Rightarrow> Term" and (* how to enforce \<open>succ : Nat\<rightarrow>Nat\<close>? *)
-  ind_Nat :: "Term \<Rightarrow> Term \<Rightarrow> Term \<Rightarrow> Term \<Rightarrow> Term"
-where
-  Nat_form: "Nat : U" and
-  Nat_intro1: "0 : Nat" and
-  Nat_intro2: "\<And>n. n : Nat \<Longrightarrow> succ n : Nat"
-  (* computation rules *)
-
-*)
\ No newline at end of file