src/HOL/ex/Commutative_RingEx.thy
author haftmann
Wed Mar 12 19:38:14 2008 +0100 (2008-03-12)
changeset 26265 4b63b9e9b10d
parent 17508 c84af7f39a6b
permissions -rw-r--r--
separated Random.thy from Quickcheck.thy
     1 (*  ID:         $Id$
     2     Author:     Bernhard Haeupler
     3 *)
     4 
     5 header {* Some examples demonstrating the comm-ring method *}
     6 
     7 theory Commutative_RingEx
     8 imports Commutative_Ring
     9 begin
    10 
    11 lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
    12 by comm_ring
    13 
    14 lemma "((x::int) + y)^2  = x^2 + y^2 + 2*x*y"
    15 by comm_ring
    16 
    17 lemma "((x::int) + y)^3  = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
    18 by comm_ring
    19 
    20 lemma "((x::int) - y)^3  = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
    21 by comm_ring
    22 
    23 lemma "((x::int) - y)^2  = x^2 + y^2 - 2*x*y"
    24 by comm_ring
    25 
    26 lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
    27 by comm_ring
    28 
    29 lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
    30 by comm_ring
    31 
    32 lemma "(a::int)*b + a*c = a*(b+c)"
    33 by comm_ring
    34 
    35 lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
    36 by comm_ring
    37 
    38 lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
    39 by comm_ring
    40 
    41 lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
    42 by comm_ring
    43 
    44 lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
    45 by comm_ring
    46 
    47 lemma "(a::int)^10 - b^10 = (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9 )"
    48 by comm_ring
    49 
    50 end