From 593faab277de53cbe2cb0c2feca5de307d9334ac Mon Sep 17 00:00:00 2001 From: Josh Chen Date: Sat, 9 Jun 2018 00:11:39 +0200 Subject: Reorganize code --- HoTT.thy | 248 --------------------------------------------------------------- 1 file changed, 248 deletions(-) delete mode 100644 HoTT.thy (limited to 'HoTT.thy') diff --git a/HoTT.thy b/HoTT.thy deleted file mode 100644 index cfb29df..0000000 --- a/HoTT.thy +++ /dev/null @@ -1,248 +0,0 @@ -theory HoTT - imports Pure -begin - -section \Setup\ -text "For ML files, routines and setup." - -section \Basic definitions\ -text "A single meta-level type \Term\ 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 \Judgments\ - -consts - is_a_type :: "Term \ prop" ("(_ : U)" [0] 1000) - is_of_type :: "[Term, Term] \ prop" ("(3_ :/ _)" [0, 0] 1000) - -section \Definitional equality\ -text "We take the meta-equality \\\, 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 \\\, which we make use of and extend where necessary." - -theorem equal_types: - assumes "A \ B" and "A : U" - shows "B : U" using assms by simp - -theorem equal_type_element: - assumes "A \ 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 \Type families\ -text "Type families are implemented using meta-level lambda expressions \P::Term \ Term\ that further satisfy the following property." - -abbreviation is_type_family :: "[Term \ Term, Term] \ prop" ("(3_:/ _ \ U)") - where "P: A \ U \ (\x::Term. x : A \ P(x) : U)" - -section \Types\ - -subsection \Dependent function/product\ - -axiomatization - Prod :: "[Term, Term \ Term] \ Term" and - lambda :: "[Term, Term \ Term] \ Term" -syntax - "_PROD" :: "[idt, Term, Term] \ Term" ("(3\_:_./ _)" 30) - "_LAMBDA" :: "[idt, Term, Term] \ Term" ("(3\<^bold>\_:_./ _)" 30) - "_PROD_ASCII" :: "[idt, Term, Term] \ Term" ("(3PROD _:_./ _)" 30) - "_LAMBDA_ASCII" :: "[idt, Term, Term] \ Term" ("(3%%_:_./ _)" 30) -translations - "\x:A. B" \ "CONST Prod A (\x. B)" - "\<^bold>\x:A. b" \ "CONST lambda A (\x. b)" - "PROD x:A. B" \ "CONST Prod A (\x. B)" - "%%x:A. b" \ "CONST lambda A (\x. b)" - (* The above syntax translations bind the x in the expressions B, b. *) - -abbreviation Function :: "[Term, Term] \ Term" (infixr "\" 40) - where "A\B \ \_:A. B" - -axiomatization - appl :: "[Term, Term] \ Term" (infixl "`" 60) -where - Prod_form: "\(A::Term) (B::Term \ Term). \A : U; B : A \ U\ \ \x:A. B(x) : U" -and - Prod_intro [intro]: - "\(A::Term) (B::Term \ Term) (b::Term \ Term). (\x::Term. x : A \ b(x) : B(x)) \ \<^bold>\x:A. b(x) : \x:A. B(x)" -and - Prod_elim [elim]: - "\(A::Term) (B::Term \ Term) (f::Term) (a::Term). \f : \x:A. B(x); a : A\ \ f`a : B(a)" -and - Prod_comp [simp]: "\(A::Term) (b::Term \ Term) (a::Term). a : A \ (\<^bold>\x:A. b(x))`a \ b(a)" -and - Prod_uniq [simp]: "\A f::Term. \<^bold>\x:A. (f`x) \ f" - -lemmas Prod_formation [intro] = Prod_form Prod_form[rotated] - -text "Note that the syntax \\<^bold>\\ (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \
2.5.2 of the Isabelle/Isar Implementation)." - -subsection \Dependent pair/sum\ - -axiomatization - Sum :: "[Term, Term \ Term] \ Term" -syntax - "_SUM" :: "[idt, Term, Term] \ Term" ("(3\_:_./ _)" 20) - "_SUM_ASCII" :: "[idt, Term, Term] \ Term" ("(3SUM _:_./ _)" 20) -translations - "\x:A. B" \ "CONST Sum A (\x. B)" - "SUM x:A. B" \ "CONST Sum A (\x. B)" - -abbreviation Pair :: "[Term, Term] \ Term" (infixr "\" 50) - where "A\B \ \_:A. B" - -axiomatization - pair :: "[Term, Term] \ Term" ("(1'(_,/ _'))") and - indSum :: "(Term \ Term) \ Term" -where - Sum_form: "\(A::Term) (B::Term \ Term). \A : U; B: A \ U\ \ \x:A. B(x) : U" -and - Sum_intro [intro]: - "\(A::Term) (B::Term \ Term) (a::Term) (b::Term). \a : A; b : B(a)\ \ (a, b) : \x:A. B(x)" -and - Sum_elim [elim]: - "\(A::Term) (B::Term \ Term) (C::Term \ Term) (f::Term) (p::Term). - \C: \x:A. B(x) \ U; f : \x:A. \y:B(x). C((x,y)); p : \x:A. B(x)\ \ indSum(C)`f`p : C(p)" -and - Sum_comp [simp]: "\(C::Term \ Term) (f::Term) (a::Term) (b::Term). indSum(C)`f`(a,b) \ 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 \C\ is left as a meta-level argument to the inductor indSum." - -subsubsection \Projections\ - -consts - fst :: "[Term, 'a] \ Term" ("(1fst[/_,/ _])") - snd :: "[Term, 'a] \ Term" ("(1snd[/_,/ _])") -overloading - fst_dep \ fst - snd_dep \ snd - fst_nondep \ fst - snd_nondep \ snd -begin -definition fst_dep :: "[Term, Term \ Term] \ Term" where - "fst_dep A B \ indSum(\_. A)`(\<^bold>\x:A. \<^bold>\y:B(x). x)" - -definition snd_dep :: "[Term, Term \ Term] \ Term" where - "snd_dep A B \ indSum(\_. A)`(\<^bold>\x:A. \<^bold>\y:B(x). y)" - -definition fst_nondep :: "[Term, Term] \ Term" where - "fst_nondep A B \ indSum(\_. A)`(\<^bold>\x:A. \<^bold>\y:B. x)" - -definition snd_nondep :: "[Term, Term] \ Term" where - "snd_nondep A B \ indSum(\_. A)`(\<^bold>\x:A. \<^bold>\y:B. y)" -end - -lemma fst_dep_comp: "\a : A; b : B(a)\ \ fst[A,B]`(a,b) \ a" unfolding fst_dep_def by simp -lemma snd_dep_comp: "\a : A; b : B(a)\ \ snd[A,B]`(a,b) \ b" unfolding snd_dep_def by simp - -lemma fst_nondep_comp: "\a : A; b : B\ \ fst[A,B]`(a,b) \ a" unfolding fst_nondep_def by simp -lemma snd_nondep_comp: "\a : A; b : B\ \ snd[A,B]`(a,b) \ b" unfolding snd_nondep_def by simp - -\ \Simplification rules for projections\ -lemmas fst_snd_simps [simp] = fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp - -subsection \Equality type\ - -axiomatization - Equal :: "[Term, Term, Term] \ Term" -syntax - "_EQUAL" :: "[Term, Term, Term] \ Term" ("(3_ =\<^sub>_/ _)" [101, 101] 100) - "_EQUAL_ASCII" :: "[Term, Term, Term] \ Term" ("(3_ =[_]/ _)" [101, 101] 100) -translations - "a =\<^sub>A b" \ "CONST Equal A a b" - "a =[A] b" \ "CONST Equal A a b" - -axiomatization - refl :: "Term \ Term" ("(refl'(_'))") and - indEqual :: "[Term, [Term, Term, Term] \ Term] \ Term" ("(indEqual[_])") -where - Equal_form: "\A a b::Term. \A : U; a : A; b : A\ \ a =\<^sub>A b : U" - (* Should I write a permuted version \\A : U; b : A; a : A\ \ \\? *) -and - Equal_intro [intro]: "\A x::Term. x : A \ refl(x) : x =\<^sub>A x" -and - Equal_elim [elim]: - "\(A::Term) (C::[Term, Term, Term] \ Term) (f::Term) (a::Term) (b::Term) (p::Term). - \ \x y::Term. \x : A; y : A\ \ C(x)(y): x =\<^sub>A y \ U; - f : \x:A. C(x)(x)(refl(x)); - a : A; - b : A; - p : a =\<^sub>A b \ - \ indEqual[A](C)`f`a`b`p : C(a)(b)(p)" -and - Equal_comp [simp]: - "\(A::Term) (C::[Term, Term, Term] \ Term) (f::Term) (a::Term). indEqual[A](C)`f`a`a`refl(a) \ f`a" - -lemmas Equal_formation [intro] = Equal_form Equal_form[rotated 1] Equal_form[rotated 2] - -subsubsection \Properties of equality\ - -text "Symmetry/Path inverse" - -definition inv :: "[Term, Term, Term] \ Term" ("(1inv[_,/ _,/ _])") - where "inv[A,x,y] \ indEqual[A](\x y _. y =\<^sub>A x)`(\<^bold>\x:A. refl(x))`x`y" - -lemma inv_comp: "\A a::Term. a : A \ inv[A,a,a]`refl(a) \ refl(a)" unfolding inv_def by simp - -text "Transitivity/Path composition" - -\ \"Raw" composition function\ -abbreviation compose' :: "Term \ Term" ("(1compose''[_])") - where "compose'[A] \ indEqual[A](\x y _. \z:A. \q: y =\<^sub>A z. x =\<^sub>A z)`(indEqual[A](\x z _. x =\<^sub>A z)`(\<^bold>\x:A. refl(x)))" - -\ \"Natural" composition function\ -abbreviation compose :: "[Term, Term, Term, Term] \ Term" ("(1compose[_,/ _,/ _,/ _])") - where "compose[A,x,y,z] \ \<^bold>\p:x =\<^sub>A y. \<^bold>\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) \ 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 \using\ 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 \Prod_comp\ is the requirement that \refl(a) : a =\<^sub>A a\. - 3. Obtaining such a condition, using the known fact \a : A\ and the introduction rule \Equal_intro\." - -lemmas Equal_simps [simp] = inv_comp compose_comp - -subsubsection \Pretty printing\ - -abbreviation inv_pretty :: "[Term, Term, Term, Term] \ Term" ("(1_\<^sup>-\<^sup>1\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_)" 500) - where "p\<^sup>-\<^sup>1\<^sub>A\<^sub>,\<^sub>x\<^sub>,\<^sub>y \ inv[A,x,y]`p" - -abbreviation compose_pretty :: "[Term, Term, Term, Term, Term, Term] \ Term" ("(1_ \\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_\<^sub>,\<^sub>_/ _)") - where "p \\<^sub>A\<^sub>,\<^sub>x\<^sub>,\<^sub>y\<^sub>,\<^sub>z q \ compose[A,x,y,z]`p`q" - -end - -(* -subsubsection \Empty type\ - -axiomatization - Null :: Term and - ind_Null :: "Term \ Term \ Term" ("(ind'_Null'(_,/ _'))") -where - Null_form: "Null : U" and - Null_elim: "\C x a. \x : Null \ C(x) : U; a : Null\ \ ind_Null(C(x), a) : C(a)" - -subsubsection \Natural numbers\ - -axiomatization - Nat :: Term and - zero :: Term ("0") and - succ :: "Term \ Term" and (* how to enforce \succ : Nat\Nat\? *) - ind_Nat :: "Term \ Term \ Term \ Term \ Term" -where - Nat_form: "Nat : U" and - Nat_intro1: "0 : Nat" and - Nat_intro2: "\n. n : Nat \ succ n : Nat" - (* computation rules *) - -*) \ No newline at end of file -- cgit v1.2.3