diff options
Diffstat (limited to '')
| -rw-r--r-- | Proj.thy | 30 |
1 files changed, 22 insertions, 8 deletions
@@ -25,10 +25,10 @@ overloading fst_nondep \<equiv> fst begin definition fst_dep :: "[Term, Typefam] \<Rightarrow> Term" where - "fst_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) p" + "fst_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). ind\<^sub>\<Sum>[A,B] (\<lambda>_. A) (\<lambda>x y. x) p" definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where - "fst_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) p" + "fst_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. ind\<^sub>\<Sum>[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) p" end overloading @@ -36,10 +36,10 @@ overloading snd_nondep \<equiv> snd begin definition snd_dep :: "[Term, Typefam] \<Rightarrow> Term" where - "snd_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). indSum[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p" + "snd_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). ind\<^sub>\<Sum>[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p" definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where - "snd_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) p" + "snd_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. ind\<^sub>\<Sum>[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) p" end @@ -48,15 +48,29 @@ section \<open>Properties\<close> text "Typing judgments and computation rules for the dependent and non-dependent projection functions." lemma fst_dep_type: assumes "\<Sum>x:A. B x : U(i)" and "p : \<Sum>x:A. B x" shows "fst[A,B]`p : A" - unfolding fst_dep_def - by (derive lems: assms) +unfolding fst_dep_def +proof + show "lambda (Sum A B) (ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x)) : (\<Sum>x:A. B x) \<rightarrow> A" + proof + show "Sum A B : U(i)" by (rule assms) + show "\<And>p. p : Sum A B \<Longrightarrow> ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) p : A" + proof + show "A : U(i)" using assms(1) .. + qed + qed +qed (rule assms) lemma fst_dep_comp: assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "a : A" and "b : B a" shows "fst[A,B]`(a,b) \<equiv> a" - unfolding fst_dep_def - by (simplify lems: assms) +unfolding fst_dep_def +proof (subst comp) + show "\<And>x. x : Sum A B \<Longrightarrow> ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) x : A" by (standard, rule assms(1), assumption+) + show "(a,b) : Sum A B" using assms(2-4) .. + show "ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) (a, b) \<equiv> a" + proof + oops \<comment> \<open> (* Old proof *) proof - |