diff -r ee0b835ca8fa -r 26384af93359 src/HOL/Real/ex/BinEx.ML
--- a/src/HOL/Real/ex/BinEx.ML	Thu Sep 23 14:39:39 1999 +0200
+++ b/src/HOL/Real/ex/BinEx.ML	Thu Sep 23 18:39:05 1999 +0200
@@ -67,3 +67,291 @@
 Goal "(#1234567::real) <= #1234567";  
 by (Simp_tac 1); 
 qed "";
+
+(** Tests **)
+Goal "(x + y = x) = (y = (#0::real))";
+by(arith_tac 1);
+
+Goal "(x + y = y) = (x = (#0::real))";
+by(arith_tac 1);
+
+Goal "(x + y = (#0::real)) = (x = -y)";
+by(arith_tac 1);
+
+Goal "(x + y = (#0::real)) = (y = -x)";
+by(arith_tac 1);
+
+Goal "((x + y) < (x + z)) = (y < (z::real))";
+by(arith_tac 1);
+
+Goal "((x + z) < (y + z)) = (x < (y::real))";
+by(arith_tac 1);
+
+Goal "(~ x < y) = (y <= (x::real))";
+by(arith_tac 1);
+
+Goal "~(x < y & y < (x::real))";
+by(arith_tac 1);
+
+Goal "(x::real) < y ==> ~ y < x";
+by(arith_tac 1);
+
+Goal "((x::real) ~= y) = (x < y | y < x)";
+by(arith_tac 1);
+
+Goal "(~ x <= y) = (y < (x::real))";
+by(arith_tac 1);
+
+Goal "x <= y | y <= (x::real)";
+by(arith_tac 1);
+
+Goal "x <= y | y < (x::real)";
+by(arith_tac 1);
+
+Goal "x < y | y <= (x::real)";
+by(arith_tac 1);
+
+Goal "x <= (x::real)";
+by(arith_tac 1);
+
+Goal "((x::real) <= y) = (x < y | x = y)";
+by(arith_tac 1);
+
+Goal "((x::real) <= y & y <= x) = (x = y)";
+by(arith_tac 1);
+
+Goal "~(x < y & y <= (x::real))";
+by(arith_tac 1);
+
+Goal "~(x <= y & y < (x::real))";
+by(arith_tac 1);
+
+Goal "(-x < (#0::real)) = (#0 < x)";
+by(arith_tac 1);
+
+Goal "((#0::real) < -x) = (x < #0)";
+by(arith_tac 1);
+
+Goal "(-x <= (#0::real)) = (#0 <= x)";
+by(arith_tac 1);
+
+Goal "((#0::real) <= -x) = (x <= #0)";
+by(arith_tac 1);
+
+Goal "(x::real) = y | x < y | y < x";
+by(arith_tac 1);
+
+Goal "(x::real) = #0 | #0 < x | #0 < -x";
+by(arith_tac 1);
+
+Goal "(#0::real) <= x | #0 <= -x";
+by(arith_tac 1);
+
+Goal "((x::real) + y <= x + z) = (y <= z)";
+by(arith_tac 1);
+
+Goal "((x::real) + z <= y + z) = (x <= y)";
+by(arith_tac 1);
+
+Goal "(w::real) < x & y < z ==> w + y < x + z";
+by(arith_tac 1);
+
+Goal "(w::real) <= x & y <= z ==> w + y <= x + z";
+by(arith_tac 1);
+
+Goal "(#0::real) <= x & #0 <= y ==> #0 <= x + y";
+by(arith_tac 1);
+
+Goal "(#0::real) < x & #0 < y ==> #0 < x + y";
+by(arith_tac 1);
+
+Goal "(-x < y) = (#0 < x + (y::real))";
+by(arith_tac 1);
+
+Goal "(x < -y) = (x + y < (#0::real))";
+by(arith_tac 1);
+
+Goal "(y < x + -z) = (y + z < (x::real))";
+by(arith_tac 1);
+
+Goal "(x + -y < z) = (x < z + (y::real))";
+by(arith_tac 1);
+
+Goal "x <= y ==> x < y + (#1::real)";
+by(arith_tac 1);
+
+Goal "(x - y) + y = (x::real)";
+by(arith_tac 1);
+
+Goal "y + (x - y) = (x::real)";
+by(arith_tac 1);
+
+Goal "x - x = (#0::real)";
+by(arith_tac 1);
+
+Goal "(x - y = #0) = (x = (y::real))";
+by(arith_tac 1);
+
+Goal "((#0::real) <= x + x) = (#0 <= x)";
+by(arith_tac 1);
+
+Goal "(-x <= x) = ((#0::real) <= x)";
+by(arith_tac 1);
+
+Goal "(x <= -x) = (x <= (#0::real))";
+by(arith_tac 1);
+
+Goal "(-x = (#0::real)) = (x = #0)";
+by(arith_tac 1);
+
+Goal "-(x - y) = y - (x::real)";
+by(arith_tac 1);
+
+Goal "((#0::real) < x - y) = (y < x)";
+by(arith_tac 1);
+
+Goal "((#0::real) <= x - y) = (y <= x)";
+by(arith_tac 1);
+
+Goal "(x + y) - x = (y::real)";
+by(arith_tac 1);
+
+Goal "(-x = y) = (x = (-y::real))";
+by(arith_tac 1);
+
+Goal "x < (y::real) ==> ~(x = y)";
+by(arith_tac 1);
+
+Goal "(x <= x + y) = ((#0::real) <= y)";
+by(arith_tac 1);
+
+Goal "(y <= x + y) = ((#0::real) <= x)";
+by(arith_tac 1);
+
+Goal "(x < x + y) = ((#0::real) < y)";
+by(arith_tac 1);
+
+Goal "(y < x + y) = ((#0::real) < x)";
+by(arith_tac 1);
+
+Goal "(x - y) - x = (-y::real)";
+by(arith_tac 1);
+
+Goal "(x + y < z) = (x < z - (y::real))";
+by(arith_tac 1);
+
+Goal "(x - y < z) = (x < z + (y::real))";
+by(arith_tac 1);
+
+Goal "(x < y - z) = (x + z < (y::real))";
+by(arith_tac 1);
+
+Goal "(x <= y - z) = (x + z <= (y::real))";
+by(arith_tac 1);
+
+Goal "(x - y <= z) = (x <= z + (y::real))";
+by(arith_tac 1);
+
+Goal "(-x < -y) = (y < (x::real))";
+by(arith_tac 1);
+
+Goal "(-x <= -y) = (y <= (x::real))";
+by(arith_tac 1);
+
+Goal "(a + b) - (c + d) = (a - c) + (b - (d::real))";
+by(arith_tac 1);
+
+Goal "(#0::real) - x = -x";
+by(arith_tac 1);
+
+Goal "x - (#0::real) = x";
+by(arith_tac 1);
+
+Goal "w <= x & y < z ==> w + y < x + (z::real)";
+by(arith_tac 1);
+
+Goal "w < x & y <= z ==> w + y < x + (z::real)";
+by(arith_tac 1);
+
+Goal "(#0::real) <= x & #0 < y ==> #0 < x + (y::real)";
+by(arith_tac 1);
+
+Goal "(#0::real) < x & #0 <= y ==> #0 < x + y";
+by(arith_tac 1);
+
+Goal "-x - y = -(x + (y::real))";
+by(arith_tac 1);
+
+Goal "x - (-y) = x + (y::real)";
+by(arith_tac 1);
+
+Goal "-x - -y = y - (x::real)";
+by(arith_tac 1);
+
+Goal "(a - b) + (b - c) = a - (c::real)";
+by(arith_tac 1);
+
+Goal "(x = y - z) = (x + z = (y::real))";
+by(arith_tac 1);
+
+Goal "(x - y = z) = (x = z + (y::real))";
+by(arith_tac 1);
+
+Goal "x - (x - y) = (y::real)";
+by(arith_tac 1);
+
+Goal "x - (x + y) = -(y::real)";
+by(arith_tac 1);
+
+Goal "x = y ==> x <= (y::real)";
+by(arith_tac 1);
+
+Goal "(#0::real) < x ==> ~(x = #0)";
+by(arith_tac 1);
+
+Goal "(x + y) * (x - y) = (x * x) - (y * y)";
+
+Goal "(-x = -y) = (x = (y::real))";
+by(arith_tac 1);
+
+Goal "(-x < -y) = (y < (x::real))";
+by(arith_tac 1);
+
+Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. [| a < b; c < d |] ==> a-d <= b+(-c)";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] \
+\     ==> a+a <= j+j";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. [| a+b < i+j; a<b; i<j |] \
+\     ==> a+a < j+j";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
+by (arith_tac 1);
+
+Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
+\     ==> a <= l";
+by (fast_arith_tac 1);
+
+Goal "!!a::real. [| 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 (fast_arith_tac 1);
+
+(* Too slow. Needs "combine_coefficients" simproc
+Goal "!!a::real. [| 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 (fast_arith_tac 1);
+
+Goal "!!a::real. [| 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 (fast_arith_tac 1);
+*)
+