diff options
Diffstat (limited to 'Proj.thy')
| -rw-r--r-- | Proj.thy | 19 |
1 files changed, 13 insertions, 6 deletions
@@ -13,21 +13,23 @@ theory Proj begin -abbreviation fst :: "Term \<Rightarrow> Term" where "fst \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. x)(p)" -abbreviation snd :: "Term \<Rightarrow> Term" where "snd \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. y)(p)" +axiomatization fst :: "Term \<Rightarrow> Term" where fst_def: "fst \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. x)(p)" +axiomatization snd :: "Term \<Rightarrow> Term" where snd_def: "snd \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. y)(p)" text "Typing judgments and computation rules for the dependent and non-dependent projection functions." lemma fst_type: assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A" +unfolding fst_def proof - show "A: U(i)" using assms(1) by (rule Sum_forms) + show "A: U(i)" using assms(1) by (rule Sum_wellform) qed (fact assms | assumption)+ lemma fst_comp: assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "fst(<a,b>) \<equiv> a" +unfolding fst_def proof show "\<And>x. x: A \<Longrightarrow> x: A" . qed (rule assms)+ @@ -35,25 +37,27 @@ qed (rule assms)+ lemma snd_type: assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "snd(p): B(fst p)" +unfolding snd_def proof show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" proof - have "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> fst(p): A" using assms(1) by (rule fst_type) - with assms(1) show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (rule Sum_forms) + with assms(1) show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (rule Sum_wellform) qed fix x y assume asm: "x: A" "y: B(x)" show "y: B(fst <x,y>)" proof (subst fst_comp) - show "A: U(i)" using assms(1) by (rule Sum_forms) - show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" using assms(1) by (rule Sum_forms) + show "A: U(i)" using assms(1) by (rule Sum_wellform) + show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" using assms(1) by (rule Sum_wellform) qed (rule asm)+ qed (fact assms) lemma snd_comp: assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "snd(<a,b>) \<equiv> b" +unfolding snd_def proof show "\<And>x y. y: B(x) \<Longrightarrow> y: B(x)" . show "a: A" by (fact assms) @@ -63,4 +67,7 @@ proof qed +lemmas Proj_comps [intro] = fst_comp snd_comp + + end
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