equal
deleted
inserted
replaced
1969 then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert) |
1969 then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert) |
1970 qed simp |
1970 qed simp |
1971 |
1971 |
1972 interpretation gcd_lcm_complete_lattice_nat: |
1972 interpretation gcd_lcm_complete_lattice_nat: |
1973 complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" |
1973 complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" |
1974 where "Inf.INFIMUM Gcd A f = Gcd (f ` A :: nat set)" |
1974 rewrites "Inf.INFIMUM Gcd A f = Gcd (f ` A :: nat set)" |
1975 and "Sup.SUPREMUM Lcm A f = Lcm (f ` A)" |
1975 and "Sup.SUPREMUM Lcm A f = Lcm (f ` A)" |
1976 proof - |
1976 proof - |
1977 show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)" |
1977 show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)" |
1978 by standard (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite) |
1978 by standard (auto simp add: Gcd_nat_def Lcm_nat_empty Lcm_nat_infinite) |
1979 then interpret gcd_lcm_complete_lattice_nat: |
1979 then interpret gcd_lcm_complete_lattice_nat: |