--- a/src/HOL/Real/ex/BinEx.thy Thu Sep 27 18:43:40 2001 +0200
+++ b/src/HOL/Real/ex/BinEx.thy Thu Sep 27 18:44:12 2001 +0200
@@ -1,2 +1,352 @@
+(* Title: HOL/Real/ex/BinEx.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1999 University of Cambridge
+*)
-BinEx = Real
+header {* Binary arithmetic examples *}
+
+theory BinEx = Real:
+
+text {*
+ Examples of performing binary arithmetic by simplification This time
+ we use the reals, though the representation is just of integers.
+*}
+
+text {* \medskip Addition *}
+
+lemma "(#1359::real) + #-2468 = #-1109"
+ by simp
+
+lemma "(#93746::real) + #-46375 = #47371"
+ by simp
+
+
+text {* \medskip Negation *}
+
+lemma "- (#65745::real) = #-65745"
+ by simp
+
+lemma "- (#-54321::real) = #54321"
+ by simp
+
+
+text {* \medskip Multiplication *}
+
+lemma "(#-84::real) * #51 = #-4284"
+ by simp
+
+lemma "(#255::real) * #255 = #65025"
+ by simp
+
+lemma "(#1359::real) * #-2468 = #-3354012"
+ by simp
+
+
+text {* \medskip Inequalities *}
+
+lemma "(#89::real) * #10 \<noteq> #889"
+ by simp
+
+lemma "(#13::real) < #18 - #4"
+ by simp
+
+lemma "(#-345::real) < #-242 + #-100"
+ by simp
+
+lemma "(#13557456::real) < #18678654"
+ by simp
+
+lemma "(#999999::real) \<le> (#1000001 + #1)-#2"
+ by simp
+
+lemma "(#1234567::real) \<le> #1234567"
+ by simp
+
+
+text {* \medskip Tests *}
+
+lemma "(x + y = x) = (y = (#0::real))"
+ by arith
+
+lemma "(x + y = y) = (x = (#0::real))"
+ by arith
+
+lemma "(x + y = (#0::real)) = (x = -y)"
+ by arith
+
+lemma "(x + y = (#0::real)) = (y = -x)"
+ by arith
+
+lemma "((x + y) < (x + z)) = (y < (z::real))"
+ by arith
+
+lemma "((x + z) < (y + z)) = (x < (y::real))"
+ by arith
+
+lemma "(\<not> x < y) = (y \<le> (x::real))"
+ by arith
+
+lemma "\<not> (x < y \<and> y < (x::real))"
+ by arith
+
+lemma "(x::real) < y ==> \<not> y < x"
+ by arith
+
+lemma "((x::real) \<noteq> y) = (x < y \<or> y < x)"
+ by arith
+
+lemma "(\<not> x \<le> y) = (y < (x::real))"
+ by arith
+
+lemma "x \<le> y \<or> y \<le> (x::real)"
+ by arith
+
+lemma "x \<le> y \<or> y < (x::real)"
+ by arith
+
+lemma "x < y \<or> y \<le> (x::real)"
+ by arith
+
+lemma "x \<le> (x::real)"
+ by arith
+
+lemma "((x::real) \<le> y) = (x < y \<or> x = y)"
+ by arith
+
+lemma "((x::real) \<le> y \<and> y \<le> x) = (x = y)"
+ by arith
+
+lemma "\<not>(x < y \<and> y \<le> (x::real))"
+ by arith
+
+lemma "\<not>(x \<le> y \<and> y < (x::real))"
+ by arith
+
+lemma "(-x < (#0::real)) = (#0 < x)"
+ by arith
+
+lemma "((#0::real) < -x) = (x < #0)"
+ by arith
+
+lemma "(-x \<le> (#0::real)) = (#0 \<le> x)"
+ by arith
+
+lemma "((#0::real) \<le> -x) = (x \<le> #0)"
+ by arith
+
+lemma "(x::real) = y \<or> x < y \<or> y < x"
+ by arith
+
+lemma "(x::real) = #0 \<or> #0 < x \<or> #0 < -x"
+ by arith
+
+lemma "(#0::real) \<le> x \<or> #0 \<le> -x"
+ by arith
+
+lemma "((x::real) + y \<le> x + z) = (y \<le> z)"
+ by arith
+
+lemma "((x::real) + z \<le> y + z) = (x \<le> y)"
+ by arith
+
+lemma "(w::real) < x \<and> y < z ==> w + y < x + z"
+ by arith
+
+lemma "(w::real) \<le> x \<and> y \<le> z ==> w + y \<le> x + z"
+ by arith
+
+lemma "(#0::real) \<le> x \<and> #0 \<le> y ==> #0 \<le> x + y"
+ by arith
+
+lemma "(#0::real) < x \<and> #0 < y ==> #0 < x + y"
+ by arith
+
+lemma "(-x < y) = (#0 < x + (y::real))"
+ by arith
+
+lemma "(x < -y) = (x + y < (#0::real))"
+ by arith
+
+lemma "(y < x + -z) = (y + z < (x::real))"
+ by arith
+
+lemma "(x + -y < z) = (x < z + (y::real))"
+ by arith
+
+lemma "x \<le> y ==> x < y + (#1::real)"
+ by arith
+
+lemma "(x - y) + y = (x::real)"
+ by arith
+
+lemma "y + (x - y) = (x::real)"
+ by arith
+
+lemma "x - x = (#0::real)"
+ by arith
+
+lemma "(x - y = #0) = (x = (y::real))"
+ by arith
+
+lemma "((#0::real) \<le> x + x) = (#0 \<le> x)"
+ by arith
+
+lemma "(-x \<le> x) = ((#0::real) \<le> x)"
+ by arith
+
+lemma "(x \<le> -x) = (x \<le> (#0::real))"
+ by arith
+
+lemma "(-x = (#0::real)) = (x = #0)"
+ by arith
+
+lemma "-(x - y) = y - (x::real)"
+ by arith
+
+lemma "((#0::real) < x - y) = (y < x)"
+ by arith
+
+lemma "((#0::real) \<le> x - y) = (y \<le> x)"
+ by arith
+
+lemma "(x + y) - x = (y::real)"
+ by arith
+
+lemma "(-x = y) = (x = (-y::real))"
+ by arith
+
+lemma "x < (y::real) ==> \<not>(x = y)"
+ by arith
+
+lemma "(x \<le> x + y) = ((#0::real) \<le> y)"
+ by arith
+
+lemma "(y \<le> x + y) = ((#0::real) \<le> x)"
+ by arith
+
+lemma "(x < x + y) = ((#0::real) < y)"
+ by arith
+
+lemma "(y < x + y) = ((#0::real) < x)"
+ by arith
+
+lemma "(x - y) - x = (-y::real)"
+ by arith
+
+lemma "(x + y < z) = (x < z - (y::real))"
+ by arith
+
+lemma "(x - y < z) = (x < z + (y::real))"
+ by arith
+
+lemma "(x < y - z) = (x + z < (y::real))"
+ by arith
+
+lemma "(x \<le> y - z) = (x + z \<le> (y::real))"
+ by arith
+
+lemma "(x - y \<le> z) = (x \<le> z + (y::real))"
+ by arith
+
+lemma "(-x < -y) = (y < (x::real))"
+ by arith
+
+lemma "(-x \<le> -y) = (y \<le> (x::real))"
+ by arith
+
+lemma "(a + b) - (c + d) = (a - c) + (b - (d::real))"
+ by arith
+
+lemma "(#0::real) - x = -x"
+ by arith
+
+lemma "x - (#0::real) = x"
+ by arith
+
+lemma "w \<le> x \<and> y < z ==> w + y < x + (z::real)"
+ by arith
+
+lemma "w < x \<and> y \<le> z ==> w + y < x + (z::real)"
+ by arith
+
+lemma "(#0::real) \<le> x \<and> #0 < y ==> #0 < x + (y::real)"
+ by arith
+
+lemma "(#0::real) < x \<and> #0 \<le> y ==> #0 < x + y"
+ by arith
+
+lemma "-x - y = -(x + (y::real))"
+ by arith
+
+lemma "x - (-y) = x + (y::real)"
+ by arith
+
+lemma "-x - -y = y - (x::real)"
+ by arith
+
+lemma "(a - b) + (b - c) = a - (c::real)"
+ by arith
+
+lemma "(x = y - z) = (x + z = (y::real))"
+ by arith
+
+lemma "(x - y = z) = (x = z + (y::real))"
+ by arith
+
+lemma "x - (x - y) = (y::real)"
+ by arith
+
+lemma "x - (x + y) = -(y::real)"
+ by arith
+
+lemma "x = y ==> x \<le> (y::real)"
+ by arith
+
+lemma "(#0::real) < x ==> \<not>(x = #0)"
+ by arith
+
+lemma "(x + y) * (x - y) = (x * x) - (y * y)"
+ oops
+
+lemma "(-x = -y) = (x = (y::real))"
+ by arith
+
+lemma "(-x < -y) = (y < (x::real))"
+ by arith
+
+lemma "!!a::real. a \<le> b ==> c \<le> d ==> x + y < z ==> a + c \<le> b + d"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a < b ==> c < d ==> a - d \<le> b + (-c)"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a \<le> b ==> b + b \<le> c ==> a + a \<le> c"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b \<le> i + j ==> a \<le> b ==> i \<le> j ==> a + a \<le> j + j"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b < i + j ==> a < b ==> i < j ==> a + a < j + j"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b + c \<le> i + j + k \<and> a \<le> b \<and> b \<le> c \<and> i \<le> j \<and> j \<le> k --> a + a + a \<le> k + k + k"
+ by arith
+
+lemma "!!a::real. a + b + c + d \<le> i + j + k + l ==> a \<le> b ==> b \<le> c
+ ==> c \<le> d ==> i \<le> j ==> j \<le> k ==> k \<le> l ==> a \<le> l"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b + c + d \<le> i + j + k + l ==> a \<le> b ==> b \<le> c
+ ==> c \<le> d ==> i \<le> j ==> j \<le> k ==> k \<le> l ==> a + a + a + a \<le> l + l + l + l"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b + c + d \<le> i + j + k + l ==> a \<le> b ==> b \<le> c
+ ==> c \<le> d ==> i \<le> j ==> j \<le> k ==> k \<le> l ==> a + a + a + a + a \<le> l + l + l + l + i"
+ by (tactic "fast_arith_tac 1")
+
+lemma "!!a::real. a + b + c + d \<le> i + j + k + l ==> a \<le> b ==> b \<le> c
+ ==> c \<le> d ==> i \<le> j ==> j \<le> k ==> k \<le> l ==> a + a + a + a + a + a \<le> l + l + l + l + i + l"
+ by (tactic "fast_arith_tac 1")
+
+end