diff options
Diffstat (limited to '')
-rw-r--r-- | Proj.thy | 26 |
1 files changed, 18 insertions, 8 deletions
@@ -47,12 +47,14 @@ section \<open>Properties\<close> text "Typing judgments and computation rules for the dependent and non-dependent projection functions." -lemma fst_dep_type: assumes "p : \<Sum>x:A. B x" shows "fst[A,B]`p : A" +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) -lemma fst_dep_comp: assumes "B: A \<rightarrow> U" and "a : A" and "b : B a" shows "fst[A,B]`(a,b) \<equiv> a" +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) @@ -82,7 +84,9 @@ qed \<close> -lemma snd_dep_type: assumes "p : \<Sum>x:A. B x" shows "snd[A,B]`p : B (fst[A,B]`p)" +lemma snd_dep_type: + assumes "\<Sum>x:A. B x : U(i)" and "p : \<Sum>x:A. B x" + shows "snd[A,B]`p : B (fst[A,B]`p)" unfolding fst_dep_def snd_dep_def by (simplify lems: assms) @@ -97,7 +101,9 @@ qed (assumption | rule assms)+ \<close> -lemma snd_dep_comp: assumes "B: A \<rightarrow> U" and "a : A" and "b : B a" shows "snd[A,B]`(a,b) \<equiv> b" +lemma snd_dep_comp: + assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "a : A" and "b : B a" + shows "snd[A,B]`(a,b) \<equiv> b" unfolding snd_dep_def fst_dep_def by (simplify lems: assms) @@ -126,12 +132,14 @@ qed text "Nondependent projections:" -lemma fst_nondep_type: assumes "p : A \<times> B" shows "fst[A,B]`p : A" +lemma fst_nondep_type: assumes "A \<times> B : U(i)" and "p : A \<times> B" shows "fst[A,B]`p : A" unfolding fst_nondep_def by (derive lems: assms) -lemma fst_nondep_comp: assumes "a : A" and "b : B" shows "fst[A,B]`(a,b) \<equiv> a" +lemma fst_nondep_comp: + assumes "A : U(i)" and "B : U(i)" and "a : A" and "b : B" + shows "fst[A,B]`(a,b) \<equiv> a" unfolding fst_nondep_def by (simplify lems: assms) @@ -148,7 +156,7 @@ qed \<close> -lemma snd_nondep_type: assumes "p : A \<times> B" shows "snd[A,B]`p : B" +lemma snd_nondep_type: assumes "A \<times> B : U(i)" and "p : A \<times> B" shows "snd[A,B]`p : B" unfolding snd_nondep_def by (derive lems: assms) @@ -163,7 +171,9 @@ qed (rule assms) \<close> -lemma snd_nondep_comp: assumes "a : A" and "b : B" shows "snd[A,B]`(a,b) \<equiv> b" +lemma snd_nondep_comp: + assumes "A : U(i)" and "B : U(i)" and "a : A" and "b : B" + shows "snd[A,B]`(a,b) \<equiv> b" unfolding snd_nondep_def by (simplify lems: assms) |