diff options
| -rw-r--r-- | HoTT.thy | 75 | ||||
| -rw-r--r-- | HoTT_Test.thy | 58 | ||||
| -rw-r--r-- | HoTT_Theorems.thy | 68 |
3 files changed, 108 insertions, 93 deletions
@@ -1,63 +1,66 @@ theory HoTT -imports Pure + imports Pure begin section \<open>Setup\<close> -text \<open> -For ML files, routines and setup. -\<close> + +text "For ML files, routines and setup." section \<open>Basic definitions\<close> -text \<open> -A single meta-level type \<open>Term\<close> suffices to implement the object-level types and terms. -For now we do not implement universes, but simply follow the informal notation in the HoTT book. -\<close> (* Actually unsure if a single meta-type suffices... *) + +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 -subsection \<open>Judgements\<close> +subsection \<open>Judgments\<close> + consts is_a_type :: "Term \<Rightarrow> prop" ("(_ : U)" [0] 1000) - is_of_type :: "Term \<Rightarrow> Term \<Rightarrow> prop" ("(3_ :/ _)" [0, 0] 1000) + is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(3_ :/ _)" [0, 0] 1000) subsection \<open>Basic axioms\<close> + subsubsection \<open>Definitional equality\<close> -text\<open> -We take the meta-equality \<equiv>, 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 the various properties of \<equiv>, -which we make use of and extend where necessary. -\<close> +text "We take the meta-equality \<equiv>, 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 \<equiv>, which we make use of and extend where necessary." subsubsection \<open>Type-related properties\<close> axiomatization where - term_substitution: "\<And>(A::Term) (x::Term) y::Term. x \<equiv> y \<Longrightarrow> A(x) \<equiv> A(y)" and - equal_types: "\<And>(A::Term) (B::Term) x::Term. \<lbrakk>A \<equiv> B; x : A\<rbrakk> \<Longrightarrow> x : B" and - inhabited_implies_type: "\<And>(x::Term) A::Term. x : A \<Longrightarrow> A : U" + equal_types: "\<And>(A::Term) (B::Term) (x::Term). \<lbrakk>A \<equiv> B; x : A\<rbrakk> \<Longrightarrow> x : B" and + inhabited_implies_type: "\<And>(x::Term) (A::Term). x : A \<Longrightarrow> A : U" subsection \<open>Basic types\<close> -subsubsection \<open>Nondependent function type\<close> -(* -Write this for now to test stuff, later I should make -this an instance of the dependent function. -Same for the nondependent product below. +subsubsection \<open>Dependent product type\<close> -Note that the syntax \<^bold>\<lambda> (bold lambda) clashes with the proof term syntax! -(See \<section>2.5.2, The Isabelle/Isar Implementation.) -*) +consts + Prod :: "[Term, (Term \<Rightarrow> Term)] \<Rightarrow> Term" +syntax + "_Prod" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Prod>_:_./ _)" 10) +translations + "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)" +(* The above syntax translation binds the x in B *) axiomatization - Function :: "Term \<Rightarrow> Term \<Rightarrow> Term" (infixr "\<rightarrow>" 10) and - lambda :: "Term \<Rightarrow> Term \<Rightarrow> Term \<Rightarrow> Term" ("(3\<^bold>\<lambda>_:_./ _)" [1000, 0, 0] 10) and - appl :: "Term \<Rightarrow> Term \<Rightarrow> Term" ("(3_`/_)" [10, 10] 10) + lambda :: "(Term \<Rightarrow> Term) \<Rightarrow> Term" (binder "\<^bold>\<lambda>" 10) and + appl :: "[Term, Term] \<Rightarrow> Term" ("(3_`/_)" [10, 10] 60) where - Function_form: "\<And>(A::Term) B::Term. \<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> A\<rightarrow>B : U" and - Function_intro: "\<And>(A::Term) (B::Term) b::Term. (\<And>x. x : A \<Longrightarrow> b(x) : B) \<Longrightarrow> \<^bold>\<lambda>x:A. b(x) : A\<rightarrow>B" and - Function_elim: "\<And>A B f a. \<lbrakk>f : A\<rightarrow>B; a : A\<rbrakk> \<Longrightarrow> f`a : B" - (* beta and eta reductions should go here *) + Prod_form: "\<And>(A::Term) (B::Term \<Rightarrow> Term). \<lbrakk>A : U; \<And>(x::Term). x : A \<Longrightarrow> B(x) : U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) : U" and + Prod_intro: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (b::Term \<Rightarrow> Term). \<lbrakk>A : U; \<And>(x::Term). x : A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and + Prod_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: "\<And>(b::Term \<Rightarrow> Term) (a::Term). (\<^bold>\<lambda>x. b(x))`a \<equiv> b(a)" and + Prod_uniq: "\<And>(A::Term) (B::Term \<Rightarrow> Term) (f::Term). \<lbrakk>f : \<Prod>x:A. B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. (f`x) \<equiv> f" + +text "Observe that the metatype \<open>Term \<Rightarrow> Term\<close> is used to represent type families, for example \<open>Prod\<close> takes a type \<open>A\<close> and a type family \<open>B\<close> and constructs a dependent function type from them." + +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)." + +abbreviation Function :: "[Term, Term] \<Rightarrow> Term" (infixr "\<rightarrow>" 30) +where "A\<rightarrow>B \<equiv> \<Prod>_:A. B" subsubsection \<open>Nondependent product type\<close> @@ -72,8 +75,10 @@ where Product_comp: "\<And>A B C g a b. \<lbrakk>C : U; g : A\<rightarrow>B\<rightarrow>C; a : A; b : B\<rbrakk> \<Longrightarrow> rec_Product(A,B,C,g)`(a,b) \<equiv> (g`a)`b" \<comment> \<open>Projection onto first component\<close> +(* definition proj1 :: "Term \<Rightarrow> Term \<Rightarrow> Term" ("(proj1\<langle>_,_\<rangle>)") where - "proj1\<langle>A,B\<rangle> \<equiv> rec_Product(A, B, A, \<^bold>\<lambda>x. \<^bold>\<lambda>y. x)" + "\<And>A B x y. proj1\<langle>A,B\<rangle> \<equiv> rec_Product(A, B, A, \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. (\<lambda>x. x))" +*) subsubsection \<open>Empty type\<close> diff --git a/HoTT_Test.thy b/HoTT_Test.thy deleted file mode 100644 index b37ac76..0000000 --- a/HoTT_Test.thy +++ /dev/null @@ -1,58 +0,0 @@ -theory HoTT_Test -imports HoTT -begin - -text \<open>Check trivial stuff:\<close> - -lemma "Null : U" - by (rule Null_form) - -lemma "\<lbrakk>A \<equiv> B; x : B\<rbrakk> \<Longrightarrow> x : A" -proof - - assume "A \<equiv> B" - then have "B \<equiv> A" by (rule Pure.symmetric) - then have "x : B \<Longrightarrow> x :A" by (rule equal_types) - oops -(* qed---figure out how to discharge assumptions *) - -text \<open> -Do functions behave like we expect them to? -(Or, is my implementation at least somewhat correct?... -\<close> - -\<comment> {* NOTE!!! The issues with substitution here. -We need the HoTT term \<open>b\<close> and the type family \<open>B\<close> to indeed be a Pure \<lambda>-term! -This seems to mean that I have to implement the HoTT types in more than one Pure type. -See CTT.thy for examples. -*} - -lemma "A \<equiv> B \<Longrightarrow> (\<^bold>\<lambda>x:A. x) : B\<rightarrow>A" -proof - - have "A \<equiv> B \<Longrightarrow> B \<equiv> A" by (rule Pure.symmetric) - then have "\<And>x::Term. A \<equiv> B \<Longrightarrow> x : B \<Longrightarrow> x : A" by (rule equal_types) - thus "A \<equiv> B \<Longrightarrow> (\<^bold>\<lambda>x:A. x) : B\<rightarrow>A" by (rule Function_intro) -qed - -lemma "\<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A" -proof - - have "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. x : B \<rightarrow> A" by (rule Function_intro) - thus "\<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A" by (rule Function_intro) -qed - -text \<open>Here's a dumb proof that 2 is a natural number:\<close> - -lemma "succ(succ 0) : Nat" -proof - - have "0 : Nat" by (rule Nat_intro1) - from this have "(succ 0) : Nat" by (rule Nat_intro2) - thus "succ(succ 0) : Nat" by (rule Nat_intro2) -qed - -text \<open> -We can of course iterate the above for as many applications of \<open>succ\<close> as we like. -The next thing to do is to implement some kind of induction tactic to automate such proofs. - -When we get more stuff working, I'd like to aim for formalizing the encode-decode method. -\<close> - -end
\ No newline at end of file diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy new file mode 100644 index 0000000..bea3dfe --- /dev/null +++ b/HoTT_Theorems.thy @@ -0,0 +1,68 @@ +theory HoTT_Theorems + imports HoTT +begin + +text "A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified." + +section \<open>Foundational stuff\<close> + +theorem "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> B : U" by simp + +section \<open>Functions\<close> + +lemma "A : U \<Longrightarrow> \<^bold>\<lambda>x. x : A\<rightarrow>A" + by (rule Prod_intro) + +text "Note that there is no provision for declaring the type of bound variables outside of the scope of a lambda expression. +Hence a statement like \<open>x : A\<close> is not needed (nor possible!) in the premises of the following lemma." + +lemma "\<lbrakk>A : U; A \<equiv> B\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. x : B\<rightarrow>A" +proof - + assume + 0: "A : U" and + 1: "A \<equiv> B" + from 0 have 2: "\<^bold>\<lambda>x. x : A\<rightarrow>A" by (rule Prod_intro) + from 1 have 3: "A\<rightarrow>A \<equiv> B\<rightarrow>A" by simp + from 3 and 2 show "\<^bold>\<lambda>x. x : B\<rightarrow>A" by (rule equal_types) + qed + +lemma "\<lbrakk>A : U; B : U; x : A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. x : B\<rightarrow>A" +proof - +assume + 1: "A : U" and + 2: "B : U" and + 3: "x : A" +then show "\<^bold>\<lambda>y. x : B\<rightarrow>A" +proof - +from 3 have "\<^bold>\<lambda>y. x : B\<rightarrow>A" by (rule Prod_intro) +qed +qed + +lemma "\<lbrakk>A : U; B : U\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. \<^bold>\<lambda>y. x : A\<rightarrow>B\<rightarrow>A" +proof - + fix x + assume + "A : U" and + "B : U" and + "x : A" + then have "\<^bold>\<lambda>y. x : B\<rightarrow>A" by (rule Prod_intro) + +qed + +section \<open>Nats\<close> + +text "Here's a dumb proof that 2 is a natural number." + +lemma "succ(succ 0) : Nat" +proof - + have "0 : Nat" by (rule Nat_intro1) + from this have "(succ 0) : Nat" by (rule Nat_intro2) + thus "succ(succ 0) : Nat" by (rule Nat_intro2) +qed + +text "We can of course iterate the above for as many applications of \<open>succ\<close> as we like. +The next thing to do is to implement induction to automate such proofs. + +When we get more stuff working, I'd like to aim for formalizing the encode-decode method to be able to prove the only naturals are 0 and those obtained from it by \<open>succ\<close>." + +end
\ No newline at end of file |