src/HOL/ex/Sqrt_Script.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 36778 739a9379e29b
child 57514 bdc2c6b40bf2
permissions -rw-r--r--
introduce order topology
     1 (*  Title:      HOL/ex/Sqrt_Script.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   2001  University of Cambridge
     4 *)
     5 
     6 header {* Square roots of primes are irrational (script version) *}
     7 
     8 theory Sqrt_Script
     9 imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
    10 begin
    11 
    12 text {*
    13   \medskip Contrast this linear Isabelle/Isar script with Markus
    14   Wenzel's more mathematical version.
    15 *}
    16 
    17 subsection {* Preliminaries *}
    18 
    19 lemma prime_nonzero:  "prime (p::nat) \<Longrightarrow> p \<noteq> 0"
    20   by (force simp add: prime_nat_def)
    21 
    22 lemma prime_dvd_other_side:
    23     "(n::nat) * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
    24   apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult_nat)
    25   apply auto
    26   done
    27 
    28 lemma reduction: "prime (p::nat) \<Longrightarrow>
    29     0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j"
    30   apply (rule ccontr)
    31   apply (simp add: linorder_not_less)
    32   apply (erule disjE)
    33    apply (frule mult_le_mono, assumption)
    34    apply auto
    35   apply (force simp add: prime_nat_def)
    36   done
    37 
    38 lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
    39   by (simp add: mult_ac)
    40 
    41 lemma prime_not_square:
    42     "prime (p::nat) \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
    43   apply (induct m rule: nat_less_induct)
    44   apply clarify
    45   apply (frule prime_dvd_other_side, assumption)
    46   apply (erule dvdE)
    47   apply (simp add: nat_mult_eq_cancel_disj prime_nonzero)
    48   apply (blast dest: rearrange reduction)
    49   done
    50 
    51 
    52 subsection {* Main theorem *}
    53 
    54 text {*
    55   The square root of any prime number (including @{text 2}) is
    56   irrational.
    57 *}
    58 
    59 theorem prime_sqrt_irrational:
    60     "prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
    61   apply (rule notI)
    62   apply (erule Rats_abs_nat_div_natE)
    63   apply (simp del: real_of_nat_mult
    64               add: abs_if divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
    65   done
    66 
    67 lemmas two_sqrt_irrational =
    68   prime_sqrt_irrational [OF two_is_prime_nat]
    69 
    70 end