(* Title: HOL/ex/BinEx.thy ID: $Id: BinEx.thy,v 1.21 2006/10/01 16:29:24 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1998 University of Cambridge *) header {* Binary arithmetic examples *} theory BinEx imports Main begin subsection {* Regression Testing for Cancellation Simprocs *} 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 ≠ 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) ≤ (1000001 + 1) - 2" by simp lemma "(1234567::int) ≤ 1234567" by simp text{*No integer overflow!*} lemma "1234567 * (1234567::int) < 1234567*1234567*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 and with ML, 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 end
lemma
[| a ≤ b; c ≤ d; x + y < z |] ==> a + c ≤ b + d
lemma
[| a < b; c < d |] ==> a - d + 2 ≤ b + - c
lemma
[| a < b; c < d |] ==> a + c + 1 < b + d
lemma
[| a ≤ b; b + b ≤ c |] ==> a + a ≤ c
lemma
[| a + b ≤ i + j; a ≤ b; i ≤ j |] ==> a + a ≤ j + j
lemma
[| a + b < i + j; a < b; i < j |] ==> a + a - - -1 < j + j - 3
lemma
a + b + c ≤ i + j + k ∧ a ≤ b ∧ b ≤ c ∧ i ≤ j ∧ j ≤ k --> a + a + a ≤ k + k + k
lemma
[| a + b + c + d ≤ i + j + k + l; a ≤ b; b ≤ c; c ≤ d; i ≤ j; j ≤ k; k ≤ l |]
==> a ≤ l
lemma
[| 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
lemma
[| 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
lemma
[| 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
lemma
[| 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
lemma
13 + 19 = 32
lemma
1234 + 5678 = 6912
lemma
1359 + -2468 = -1109
lemma
93746 + -46375 = 47371
lemma
- 65745 = -65745
lemma
- -54321 = 54321
lemma
13 * 19 = 247
lemma
-84 * 51 = -4284
lemma
255 * 255 = 65025
lemma
1359 * -2468 = -3354012
lemma
89 * 10 ≠ 889
lemma
13 < 18 - 4
lemma
-345 < -242 + -100
lemma
13557456 < 18678654
lemma
999999 ≤ 1000001 + 1 - 2
lemma
1234567 ≤ 1234567
lemma
1234567 * 1234567 < 1234567 * 1234567 * 1234567
lemma
10 div 3 = 3
lemma
10 mod 3 = 1
lemma
10 div -3 = -4
lemma
10 mod -3 = -2
lemma
-10 div 3 = -4
lemma
-10 mod 3 = 2
lemma
-10 div -3 = 3
lemma
-10 mod -3 = -1
lemma
8452 mod 3 = 1
lemma
59485 div 434 = 137
lemma
1000006 mod 10 = 6
lemma
10000000 div 2 = 5000000
lemma
10000001 mod 2 = 1
lemma
10000055 div 32 = 312501
lemma
10000055 mod 32 = 23
lemma
100094 div 144 = 695
lemma
100094 mod 144 = 14
lemma
2 ^ 10 = 1024
lemma
-3 ^ 7 = -2187
lemma
13 ^ 7 = 62748517
lemma
3 ^ 15 = 14348907
lemma
-5 ^ 11 = -48828125
lemma
Suc 99999 = 100000
lemma
13 + 19 = 32
lemma
1234 + 5678 = 6912
lemma
973646 + 6475 = 980121
lemma
32 - 14 = 18
lemma
14 - 15 = 0
lemma
14 - 1576644 = 0
lemma
48273776 - 3873737 = 44400039
lemma
12 * 11 = 132
lemma
647 * 3643 = 2357021
lemma
10 div 3 = 3
lemma
10 mod 3 = 1
lemma
10000 div 9 = 1111
lemma
10000 mod 9 = 1
lemma
10000 div 16 = 625
lemma
10000 mod 16 = 0
lemma
2 ^ 12 = 4096
lemma
3 ^ 10 = 59049
lemma
12 ^ 7 = 35831808
lemma
3 ^ 14 = 4782969
lemma
5 ^ 11 = 48828125
lemma
y + (x + - x) = 0 + y
lemma
y + (- x + (- y + x)) = 0
lemma
- x + (y + (- y + x)) = 0
lemma
x + (x + (- x + (- x + (- y + - z)))) = 0 - y - z
lemma
x + x - x - x - y - z = 0 - y - z
lemma
x + y + z - (x + z) = y - 0
lemma
x + (y + (y + (y + (- x + - x)))) = 0 + y - x + y + y
lemma
x + (y + (y + (y + (- y + - x)))) = y + 0 + y
lemma
x + y - x + z - x - y - z + x < 1