src/HOL/ex/BinEx.thy

lucas - fixed bug with name capture variables bound outside redex could (previously)conflict with scheme variables that occur in the conditions of an equation, and which were renamed to avoid conflict with another instantiation. This has now been fixed.

(*  Title:      HOL/ex/BinEx.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge
*)

header {* Binary arithmetic examples *}

theory BinEx = Main:

subsection {* Regression Testing for Cancellation Simprocs *}

(*taken from HOL/Integ/int_arith1.ML *)


lemma "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)"
apply simp  oops

lemma "2*u = (u::int)"
apply simp  oops

lemma "(i + j + 12 + (k::int)) - 15 = y"
apply simp  oops

lemma "(i + j + 12 + (k::int)) - 5 = y"
apply simp  oops

lemma "y - b < (b::int)"
apply simp  oops

lemma "y - (3*b + c) < (b::int) - 2*c"
apply simp  oops

lemma "(2*x - (u*v) + y) - v*3*u = (w::int)"
apply simp  oops

lemma "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)"
apply simp  oops

lemma "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)"
apply simp  oops

lemma "u*v - (x*u*v + (u*v)*4 + y) = (w::int)"
apply simp  oops

lemma "(i + j + 12 + (k::int)) = u + 15 + y"
apply simp  oops

lemma "(i + j*2 + 12 + (k::int)) = j + 5 + y"
apply simp  oops

lemma "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)"
apply simp  oops

lemma "a + -(b+c) + b = (d::int)"
apply simp  oops

lemma "a + -(b+c) - b = (d::int)"
apply simp  oops

(*negative numerals*)
lemma "(i + j + -2 + (k::int)) - (u + 5 + y) = zz"
apply simp  oops

lemma "(i + j + -3 + (k::int)) < u + 5 + y"
apply simp  oops

lemma "(i + j + 3 + (k::int)) < u + -6 + y"
apply simp  oops

lemma "(i + j + -12 + (k::int)) - 15 = y"
apply simp  oops

lemma "(i + j + 12 + (k::int)) - -15 = y"
apply simp  oops

lemma "(i + j + -12 + (k::int)) - -15 = y"
apply simp  oops

lemma "- (2*i) + 3  + (2*i + 4) = (0::int)"
apply simp  oops



subsection {* Arithmetic Method Tests *}


lemma "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"
by arith

lemma "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)"
by arith

lemma "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d"
by arith

lemma "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c"
by arith

lemma "!!a::int. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j"
by arith

lemma "!!a::int. [| a+b < i+j; a<b; i<j |] ==> a+a - - -1 < j+j - 3"
by arith

lemma "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k"
by arith

lemma "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |]
      ==> a <= l"
by arith

lemma "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |]
      ==> a+a+a+a <= l+l+l+l"
by arith

lemma "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |]
      ==> a+a+a+a+a <= l+l+l+l+i"
by arith

lemma "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |]
      ==> a+a+a+a+a+a <= l+l+l+l+i+l"
by arith

lemma "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |]
      ==> 6*a <= 5*l+i"
by arith



subsection {* The Integers *}

text {* Addition *}

lemma "(13::int) + 19 = 32"
  by simp

lemma "(1234::int) + 5678 = 6912"
  by simp

lemma "(1359::int) + -2468 = -1109"
  by simp

lemma "(93746::int) + -46375 = 47371"
  by simp


text {* \medskip Negation *}

lemma "- (65745::int) = -65745"
  by simp

lemma "- (-54321::int) = 54321"
  by simp


text {* \medskip Multiplication *}

lemma "(13::int) * 19 = 247"
  by simp

lemma "(-84::int) * 51 = -4284"
  by simp

lemma "(255::int) * 255 = 65025"
  by simp

lemma "(1359::int) * -2468 = -3354012"
  by simp

lemma "(89::int) * 10 \<noteq> 889"
  by simp

lemma "(13::int) < 18 - 4"
  by simp

lemma "(-345::int) < -242 + -100"
  by simp

lemma "(13557456::int) < 18678654"
  by simp

lemma "(999999::int) \<le> (1000001 + 1) - 2"
  by simp

lemma "(1234567::int) \<le> 1234567"
  by simp


text {* \medskip Quotient and Remainder *}

lemma "(10::int) div 3 = 3"
  by simp

lemma "(10::int) mod 3 = 1"
  by simp

text {* A negative divisor *}

lemma "(10::int) div -3 = -4"
  by simp

lemma "(10::int) mod -3 = -2"
  by simp

text {*
  A negative dividend\footnote{The definition agrees with mathematical
  convention but not with the hardware of most computers}
*}

lemma "(-10::int) div 3 = -4"
  by simp

lemma "(-10::int) mod 3 = 2"
  by simp

text {* A negative dividend \emph{and} divisor *}

lemma "(-10::int) div -3 = 3"
  by simp

lemma "(-10::int) mod -3 = -1"
  by simp

text {* A few bigger examples *}

lemma "(8452::int) mod 3 = 1"
  by simp

lemma "(59485::int) div 434 = 137"
  by simp

lemma "(1000006::int) mod 10 = 6"
  by simp


text {* \medskip Division by shifting *}

lemma "10000000 div 2 = (5000000::int)"
  by simp

lemma "10000001 mod 2 = (1::int)"
  by simp

lemma "10000055 div 32 = (312501::int)"
  by simp

lemma "10000055 mod 32 = (23::int)"
  by simp

lemma "100094 div 144 = (695::int)"
  by simp

lemma "100094 mod 144 = (14::int)"
  by simp


text {* \medskip Powers *}

lemma "2 ^ 10 = (1024::int)"
  by simp

lemma "-3 ^ 7 = (-2187::int)"
  by simp

lemma "13 ^ 7 = (62748517::int)"
  by simp

lemma "3 ^ 15 = (14348907::int)"
  by simp

lemma "-5 ^ 11 = (-48828125::int)"
  by simp


subsection {* The Natural Numbers *}

text {* Successor *}

lemma "Suc 99999 = 100000"
  by (simp add: Suc_nat_number_of)
    -- {* not a default rewrite since sometimes we want to have @{text "Suc #nnn"} *}


text {* \medskip Addition *}

lemma "(13::nat) + 19 = 32"
  by simp

lemma "(1234::nat) + 5678 = 6912"
  by simp

lemma "(973646::nat) + 6475 = 980121"
  by simp


text {* \medskip Subtraction *}

lemma "(32::nat) - 14 = 18"
  by simp

lemma "(14::nat) - 15 = 0"
  by simp

lemma "(14::nat) - 1576644 = 0"
  by simp

lemma "(48273776::nat) - 3873737 = 44400039"
  by simp


text {* \medskip Multiplication *}

lemma "(12::nat) * 11 = 132"
  by simp

lemma "(647::nat) * 3643 = 2357021"
  by simp


text {* \medskip Quotient and Remainder *}

lemma "(10::nat) div 3 = 3"
  by simp

lemma "(10::nat) mod 3 = 1"
  by simp

lemma "(10000::nat) div 9 = 1111"
  by simp

lemma "(10000::nat) mod 9 = 1"
  by simp

lemma "(10000::nat) div 16 = 625"
  by simp

lemma "(10000::nat) mod 16 = 0"
  by simp


text {* \medskip Powers *}

lemma "2 ^ 12 = (4096::nat)"
  by simp

lemma "3 ^ 10 = (59049::nat)"
  by simp

lemma "12 ^ 7 = (35831808::nat)"
  by simp

lemma "3 ^ 14 = (4782969::nat)"
  by simp

lemma "5 ^ 11 = (48828125::nat)"
  by simp


text {* \medskip Testing the cancellation of complementary terms *}

lemma "y + (x + -x) = (0::int) + y"
  by simp

lemma "y + (-x + (- y + x)) = (0::int)"
  by simp

lemma "-x + (y + (- y + x)) = (0::int)"
  by simp

lemma "x + (x + (- x + (- x + (- y + - z)))) = (0::int) - y - z"
  by simp

lemma "x + x - x - x - y - z = (0::int) - y - z"
  by simp

lemma "x + y + z - (x + z) = y - (0::int)"
  by simp

lemma "x + (y + (y + (y + (-x + -x)))) = (0::int) + y - x + y + y"
  by simp

lemma "x + (y + (y + (y + (-y + -x)))) = y + (0::int) + y"
  by simp

lemma "x + y - x + z - x - y - z + x < (1::int)"
  by simp


text{*The proofs about arithmetic yielding normal forms have been deleted:
 they are irrelevant with the new treatment of numerals.*}

end