diff options
| -rw-r--r-- | HoTT_Test.thy (renamed from HoTT_Theorems.thy) | 59 | ||||
| -rw-r--r-- | ex/HoTT book/Ch1.thy (renamed from ex/HoTT Book/Ch1.thy) | 7 |
2 files changed, 42 insertions, 24 deletions
diff --git a/HoTT_Theorems.thy b/HoTT_Test.thy index 79614d3..4c87e78 100644 --- a/HoTT_Theorems.thy +++ b/HoTT_Test.thy @@ -1,54 +1,65 @@ -theory HoTT_Theorems +(* Title: HoTT/HoTT_Test.thy + Author: Josh Chen + Date: Aug 2018 + +This is an old "test suite" from early implementations of the theory, updated for the most recent version. +It is not always guaranteed to be up to date. +*) + +theory HoTT_Test imports HoTT begin -text "A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified. -Things that *should* be automated: - \<bullet> Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>. - \<bullet> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair?" +text " + A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified. + + Things that *should* be automated: + - Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>. + - Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair? +" -\<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close> declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=5]] + \<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close> -section \<open>\<Prod> type\<close> + +section \<open>\<Pi>-type\<close> subsection \<open>Typing functions\<close> -text "Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following." +text " + Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following. +" -proposition assumes "A : U" shows "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" using assms .. +proposition assumes "A : U(i)" shows "\<^bold>\<lambda>x. x: A \<rightarrow> A" using assms .. proposition - assumes "A : U" and "A \<equiv> B" - shows "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" + assumes "A : U(i)" and "A \<equiv> B" + shows "\<^bold>\<lambda>x. x : B \<rightarrow> A" proof - have "A\<rightarrow>A \<equiv> B\<rightarrow>A" using assms by simp - moreover have "\<^bold>\<lambda>x:A. x : A\<rightarrow>A" using assms(1) .. - ultimately show "\<^bold>\<lambda>x:A. x : B\<rightarrow>A" by simp + moreover have "\<^bold>\<lambda>x. x : A \<rightarrow> A" using assms(1) .. + ultimately show "\<^bold>\<lambda>x. x : B \<rightarrow> A" by simp qed proposition - assumes "A : U" and "B : U" - shows "\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B. x : A\<rightarrow>B\<rightarrow>A" -proof - fix x - assume "x : A" - with assms(2) show "\<^bold>\<lambda>y:B. x : B\<rightarrow>A" .. -qed (rule assms) + assumes "A : U(i)" and "B : U(i)" + shows "\<^bold>\<lambda>x y. x : A \<rightarrow> B \<rightarrow> A" +by (simple lems: assms) subsection \<open>Function application\<close> -proposition assumes "a : A" shows "(\<^bold>\<lambda>x:A. x)`a \<equiv> a" using assms by simp +proposition assumes "a : A" shows "(\<^bold>\<lambda>x. x)`a \<equiv> a" by (simple lems: assms) text "Currying:" lemma - assumes "a : A" - shows "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)`a \<equiv> \<^bold>\<lambda>y:B(a). y" + assumes "a : A" and "\<And>x. x: A \<Longrightarrow> B(x): U(i)" + shows "(\<^bold>\<lambda>x y. y)`a \<equiv> \<^bold>\<lambda>y. y" proof - show "\<And>x. a : A \<Longrightarrow> x : A \<Longrightarrow> \<^bold>\<lambda>y:B x. y : B x \<rightarrow> B x" + show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. y : B(x) \<rightarrow> B(x)" by (simple lems: assms) +qed fact lemma "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:B(x). y)`a`b \<equiv> b" by simp diff --git a/ex/HoTT Book/Ch1.thy b/ex/HoTT book/Ch1.thy index 84a5cf4..65de875 100644 --- a/ex/HoTT Book/Ch1.thy +++ b/ex/HoTT book/Ch1.thy @@ -1,3 +1,10 @@ +(* Title: HoTT/ex/HoTT book/Ch1.thy + Author: Josh Chen + Date: Aug 2018 + +A formalization of some content of Chapter 1 of the Homotopy Type Theory book. +*) + theory Ch1 imports "../../HoTT" begin |