--- a/src/HOL/Real/ex/BinEx.thy Tue May 06 12:29:49 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,370 +0,0 @@
-(* Title: HOL/Real/ex/BinEx.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1999 University of Cambridge
-*)
-
-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 Powers *}
-
-lemma "2 ^ 15 = (32768::real)"
- by simp
-
-lemma "-3 ^ 7 = (-2187::real)"
- by simp
-
-lemma "13 ^ 7 = (62748517::real)"
- by simp
-
-lemma "3 ^ 15 = (14348907::real)"
- by simp
-
-lemma "-5 ^ 11 = (-48828125::real)"
- 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