src/HOL/GCD.thy
changeset 58623 2db1df2c8467
parent 57514 bdc2c6b40bf2
child 58770 ae5e9b4f8daf
equal deleted inserted replaced
58622:aa99568f56de 58623:2db1df2c8467
     6 lemmas are proved uniformly for the natural numbers and integers.
     6 lemmas are proved uniformly for the natural numbers and integers.
     7 
     7 
     8 This file combines and revises a number of prior developments.
     8 This file combines and revises a number of prior developments.
     9 
     9 
    10 The original theories "GCD" and "Primes" were by Christophe Tabacznyj
    10 The original theories "GCD" and "Primes" were by Christophe Tabacznyj
    11 and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
    11 and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
    12 gcd, lcm, and prime for the natural numbers.
    12 gcd, lcm, and prime for the natural numbers.
    13 
    13 
    14 The original theory "IntPrimes" was by Thomas M. Rasmussen, and
    14 The original theory "IntPrimes" was by Thomas M. Rasmussen, and
    15 extended gcd, lcm, primes to the integers. Amine Chaieb provided
    15 extended gcd, lcm, primes to the integers. Amine Chaieb provided
    16 another extension of the notions to the integers, and added a number
    16 another extension of the notions to the integers, and added a number
   362 text {*
   362 text {*
   363   \medskip Multiplication laws
   363   \medskip Multiplication laws
   364 *}
   364 *}
   365 
   365 
   366 lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
   366 lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
   367     -- {* \cite[page 27]{davenport92} *}
   367     -- {* @{cite \<open>page 27\<close> davenport92} *}
   368   apply (induct m n rule: gcd_nat_induct)
   368   apply (induct m n rule: gcd_nat_induct)
   369   apply simp
   369   apply simp
   370   apply (case_tac "k = 0")
   370   apply (case_tac "k = 0")
   371   apply (simp_all add: gcd_non_0_nat)
   371   apply (simp_all add: gcd_non_0_nat)
   372 done
   372 done