diff options
| -rw-r--r-- | HoTT_Base.thy | 40 | ||||
| -rw-r--r-- | HoTT_Theorems.thy | 3 | ||||
| -rw-r--r-- | Sum.thy | 41 |
3 files changed, 44 insertions, 40 deletions
diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 9b422c4..7794601 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -2,7 +2,7 @@ Author: Josh Chen Date: Jun 2018 -Basic setup and definitions of a homotopy type theory object logic. +Basic setup and definitions of a homotopy type theory object logic without universes. *) theory HoTT_Base @@ -18,16 +18,23 @@ text "Set up type checking routines, proof methods etc." section \<open>Metalogical definitions\<close> text "A single meta-type \<open>Term\<close> suffices to implement the object-logic types and terms. -Our implementation does not have universes, and we simply use \<open>a : U\<close> as a convenient shorthand meaning ``\<open>a\<close> is a type''." +We do not implement universes, and simply use \<open>a : U\<close> as a convenient shorthand to mean ``\<open>a\<close> is a type''." typedecl Term section \<open>Judgments\<close> +text "We formalize the judgments \<open>a : A\<close> and \<open>A : U\<close> separately, in contrast to the HoTT book where the latter is considered an instance of the former. + +For judgmental equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it." + consts -is_a_type :: "Term \<Rightarrow> prop" ("(1_ :/ U)" [0] 1000) -is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ :/ _)" [0, 0] 1000) + is_a_type :: "Term \<Rightarrow> prop" ("(1_ :/ U)" [0] 1000) + is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ :/ _)" [0, 0] 1000) + +axiomatization where + inhabited_implies_type [intro]: "\<And>a A. a : A \<Longrightarrow> A : U" section \<open>Type families\<close> @@ -36,31 +43,10 @@ text "A (one-variable) type family is a meta lambda term \<open>P :: Term \<Righ type_synonym Typefam = "Term \<Rightarrow> Term" -abbreviation is_type_family :: "[Typefam, Term] \<Rightarrow> prop" ("(3_:/ _ \<rightarrow> U)") +abbreviation (input) is_type_family :: "[Typefam, Term] \<Rightarrow> prop" ("(3_:/ _ \<rightarrow> U)") where "P: A \<rightarrow> U \<equiv> (\<And>x. x : A \<Longrightarrow> P x : U)" -text "There is an obvious generalization to multivariate type families, but implementing such an abbreviation involves writing ML and is for the moment not really crucial." - - -section \<open>Definitional equality\<close> - -text "The Pure equality \<open>\<equiv>\<close> is used for definitional aka judgmental equality of types and terms." - -\<comment> \<open>Do these ever need to be used? - -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 +text "There is an obvious generalization to multivariate type families, but implementing such an abbreviation involves writing ML code, and is for the moment not really crucial." -lemmas type_equality = - equal_types - equal_types[rotated] - equal_type_element - equal_type_element[rotated] -\<close> end
\ No newline at end of file diff --git a/HoTT_Theorems.thy b/HoTT_Theorems.thy index 2c2a31d..ab5374d 100644 --- a/HoTT_Theorems.thy +++ b/HoTT_Theorems.thy @@ -34,6 +34,9 @@ proof fix a assume "a : A" then show "\<^bold>\<lambda>y:B. a : B \<rightarrow> A" .. + + ML_val \<open>@{context} |> Variable.names_of\<close> + qed @@ -6,7 +6,7 @@ Dependent sum type. *) theory Sum - imports HoTT_Base Prod + imports Prod begin @@ -68,32 +68,47 @@ overloading snd_nondep \<equiv> snd begin definition snd_dep :: "[Term, Typefam] \<Rightarrow> Term" where - "snd_dep A B \<equiv> indSum[A,B] (\<lambda>p. B(fst[A,B]`p)) (\<lambda>x y. y)" + "snd_dep A B \<equiv> indSum[A,B] (\<lambda>p. B fst[A,B]`p) (\<lambda>x y. y)" definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where - "snd_nondep A B \<equiv> indSum[A, \<lambda>_. B] (\<lambda>p. B((fst A B)`p)) (\<lambda>x y. y)" + "snd_nondep A B \<equiv> indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y)" end -text "Simplification rules:" +text "Properties of projections:" + +lemma fst_dep_type: + assumes "p : \<Sum>x:A. B x" + shows "fst[A,B]`p : A" +proof - + have "\<Sum>x:A. B x : U" using assms .. + then have "A : U" by (rule Sum_intro) + unfolding fst_dep_def using assms by (rule Sum_comp) + lemma fst_dep_comp: - assumes "a : A" and "b : B(a)" + assumes "a : A" and "b : B a" shows "fst[A,B]`(a,b) \<equiv> a" proof - - show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_dep_def using assms by simp + have "A : U" using assms(1) .. + then have "\<lambda>_. A: \<Sum>x:A. B x \<rightarrow> U" . + moreover have "\<And>x y. x : A \<Longrightarrow> (\<lambda>x y. x) x y : A" . + ultimately show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_dep_def using assms by (rule Sum_comp) qed -lemma snd_dep_comp: "\<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> snd[A,B]`(a,b) \<equiv> b" +lemma snd_dep_comp: + assumes "a : A" and "b : B a" + shows "snd[A,B]`(a,b) \<equiv> b" proof - - assume "a : A" and "b : B(a)" - then have "(a, b) : \<Sum>x:A. B(x)" .. - then show "snd[A,B]`(a,b) \<equiv> b" unfolding snd_dep_def by simp + have "\<lambda>p. B fst[A,B]`p: \<Sum>x:A. B x \<rightarrow> U" + + ultimately show "snd[A,B]`(a,b) \<equiv> b" unfolding snd_dep_def by (rule Sum_comp) qed -lemma fst_nondep_comp: "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> fst[A,B]`(a,b) \<equiv> a" +lemma fst_nondep_comp: + assumes "a : A" and "b : B" + shows "fst[A,B]`(a,b) \<equiv> a" proof - - assume "a : A" and "b : B" - then have "(a, b) : A \<times> B" .. + have "A : U" using assms(1) .. then show "fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def by simp qed |