diff options
Diffstat (limited to '')
| -rw-r--r-- | tests/Test.thy | 110 |
1 files changed, 0 insertions, 110 deletions
diff --git a/tests/Test.thy b/tests/Test.thy deleted file mode 100644 index 6f9f996..0000000 --- a/tests/Test.thy +++ /dev/null @@ -1,110 +0,0 @@ -(* -Title: tests/Test.thy -Author: Joshua Chen -Date: 2018 - -This is an old test suite from early implementations. -It is not always guaranteed to be up to date or to reflect most recent versions of the theory. -*) - -theory Test -imports "../HoTT" - -begin - - -text \<open> -A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified. - -Things that *should* be automated: -\<^item> Checking that @{term A} is a well-formed type, when writing things like @{prop "x: A"} and @{prop "A: U i"}. -\<^item> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair? -\<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> - -subsection \<open>Typing functions\<close> - -text \<open>Declaring @{thm Prod_intro} with the @{attribute intro} attribute enables @{method rule} to prove the following.\<close> - -proposition assumes "A : U(i)" shows "\<^bold>\<lambda>x. x: A \<rightarrow> A" -by (routine add: assms) - -proposition - 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. x : A \<rightarrow> A" by (routine add: assms) - ultimately show "\<^bold>\<lambda>x. x : B \<rightarrow> A" by simp -qed - -proposition - assumes "A : U(i)" and "B : U(i)" - shows "\<^bold>\<lambda>x y. x : A \<rightarrow> B \<rightarrow> A" -by (routine add: assms) - -subsection \<open>Function application\<close> - -proposition assumes "a : A" shows "(\<^bold>\<lambda>x. x)`a \<equiv> a" -by (derive lems: assms) - -lemma - 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" -by (derive lems: assms) - -lemma "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. y)`a`b \<equiv> b" -by derive - -lemma "\<lbrakk>A: U(i); a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. f x y)`a \<equiv> \<^bold>\<lambda>y. f a y" -proof derive -oops \<comment> \<open>Missing some premises.\<close> - -lemma "\<lbrakk>a : A; b : B(a); c : C(a)(b)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. \<^bold>\<lambda>y. \<^bold>\<lambda>z. f x y z)`a`b`c \<equiv> f a b c" -proof derive -oops - - -subsection \<open>Currying functions\<close> - -proposition curried_function_formation: - assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "\<And>x. C(x): B(x) \<longrightarrow> U(i)" - shows "\<Prod>x:A. \<Prod>y:B(x). C x y : U(i)" -by (routine add: assms) - -proposition higher_order_currying_formation: - assumes - "A: U(i)" and "B: A \<longrightarrow> U(i)" and - "\<And>x y. y: B(x) \<Longrightarrow> C(x)(y): U(i)" and - "\<And>x y z. z : C(x)(y) \<Longrightarrow> D(x)(y)(z): U(i)" - shows "\<Prod>x:A. \<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z): U(i)" -by (routine add: assms) - -lemma curried_type_judgment: - assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "\<And>x y. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y" - shows "\<^bold>\<lambda>x y. f x y : \<Prod>x:A. \<Prod>y:B(x). C x y" -by (routine add: assms) - - -text \<open> -Polymorphic identity function is now trivial due to lambda expression polymorphism. -It was more involved in previous monomorphic incarnations. -\<close> - -lemma "x: A \<Longrightarrow> id`x \<equiv> x" -by derive - - -section \<open>Natural numbers\<close> - -text \<open>Automatic proof methods recognize natural numbers.\<close> - -proposition "succ(succ(succ 0)): \<nat>" by routine - - -end |