diff options
Diffstat (limited to 'Proj.thy')
| -rw-r--r-- | Proj.thy | 45 |
1 files changed, 43 insertions, 2 deletions
@@ -18,8 +18,49 @@ abbreviation snd :: "Term \<Rightarrow> Term" where "snd \<equiv> \<lambda>p. in text "Typing judgments and computation rules for the dependent and non-dependent projection functions." -lemma assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A" - by (derive lems: assms + +lemma fst_type: + assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A" +proof + show "A: U(i)" using assms(1) by (rule Sum_forms) +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" +proof + show "\<And>x. x: A \<Longrightarrow> x: A" . +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)" +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) + 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) + 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" +proof + show "\<And>x y. y: B(x) \<Longrightarrow> y: B(x)" . + show "a: A" by (fact assms) + show "b: B(a)" by (fact assms) + show *: "B(a): U(i)" using assms(3) by (rule assms(2)) + show "B(a): U(i)" by (fact *) +qed end
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