src/HOL/ex/Arith_Examples.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 47108 2a1953f0d20d child 58889 5b7a9633cfa8 permissions -rw-r--r--
introduce order topology
1 (*  Title:  HOL/ex/Arith_Examples.thy
2     Author: Tjark Weber
3 *)
5 header {* Arithmetic *}
7 theory Arith_Examples
8 imports Main
9 begin
11 text {*
12   The @{text arith} method is used frequently throughout the Isabelle
13   distribution.  This file merely contains some additional tests and special
14   corner cases.  Some rather technical remarks:
16   @{ML Lin_Arith.simple_tac} is a very basic version of the tactic.  It performs no
17   meta-to-object-logic conversion, and only some splitting of operators.
18   @{ML Lin_Arith.tac} performs meta-to-object-logic conversion, full
19   splitting of operators, and NNF normalization of the goal.  The @{text arith}
20   method combines them both, and tries other methods (e.g.~@{text presburger})
21   as well.  This is the one that you should use in your proofs!
23   An @{text arith}-based simproc is available as well (see @{ML
24   Lin_Arith.simproc}), which---for performance
25   reasons---however does even less splitting than @{ML Lin_Arith.simple_tac}
26   at the moment (namely inequalities only).  (On the other hand, it
27   does take apart conjunctions, which @{ML Lin_Arith.simple_tac} currently
28   does not do.)
29 *}
32 subsection {* Splitting of Operators: @{term max}, @{term min}, @{term abs},
33            @{term minus}, @{term nat}, @{term Divides.mod},
34            @{term Divides.div} *}
36 lemma "(i::nat) <= max i j"
37   by linarith
39 lemma "(i::int) <= max i j"
40   by linarith
42 lemma "min i j <= (i::nat)"
43   by linarith
45 lemma "min i j <= (i::int)"
46   by linarith
48 lemma "min (i::nat) j <= max i j"
49   by linarith
51 lemma "min (i::int) j <= max i j"
52   by linarith
54 lemma "min (i::nat) j + max i j = i + j"
55   by linarith
57 lemma "min (i::int) j + max i j = i + j"
58   by linarith
60 lemma "(i::nat) < j ==> min i j < max i j"
61   by linarith
63 lemma "(i::int) < j ==> min i j < max i j"
64   by linarith
66 lemma "(0::int) <= abs i"
67   by linarith
69 lemma "(i::int) <= abs i"
70   by linarith
72 lemma "abs (abs (i::int)) = abs i"
73   by linarith
75 text {* Also testing subgoals with bound variables. *}
77 lemma "!!x. (x::nat) <= y ==> x - y = 0"
78   by linarith
80 lemma "!!x. (x::nat) - y = 0 ==> x <= y"
81   by linarith
83 lemma "!!x. ((x::nat) <= y) = (x - y = 0)"
84   by linarith
86 lemma "[| (x::nat) < y; d < 1 |] ==> x - y = d"
87   by linarith
89 lemma "[| (x::nat) < y; d < 1 |] ==> x - y - x = d - x"
90   by linarith
92 lemma "(x::int) < y ==> x - y < 0"
93   by linarith
95 lemma "nat (i + j) <= nat i + nat j"
96   by linarith
98 lemma "i < j ==> nat (i - j) = 0"
99   by linarith
101 lemma "(i::nat) mod 0 = i"
102   (* rule split_mod is only declared by default for numerals *)
103   using split_mod [of _ _ "0", arith_split]
104   by linarith
106 lemma "(i::nat) mod 1 = 0"
107   (* rule split_mod is only declared by default for numerals *)
108   using split_mod [of _ _ "1", arith_split]
109   by linarith
111 lemma "(i::nat) mod 42 <= 41"
112   by linarith
114 lemma "(i::int) mod 0 = i"
115   (* rule split_zmod is only declared by default for numerals *)
116   using split_zmod [of _ _ "0", arith_split]
117   by linarith
119 lemma "(i::int) mod 1 = 0"
120   (* rule split_zmod is only declared by default for numerals *)
121   using split_zmod [of _ _ "1", arith_split]
122   by linarith
124 lemma "(i::int) mod 42 <= 41"
125   by linarith
127 lemma "-(i::int) * 1 = 0 ==> i = 0"
128   by linarith
130 lemma "[| (0::int) < abs i; abs i * 1 < abs i * j |] ==> 1 < abs i * j"
131   by linarith
134 subsection {* Meta-Logic *}
136 lemma "x < Suc y == x <= y"
137   by linarith
139 lemma "((x::nat) == z ==> x ~= y) ==> x ~= y | z ~= y"
140   by linarith
143 subsection {* Various Other Examples *}
145 lemma "(x < Suc y) = (x <= y)"
146   by linarith
148 lemma "[| (x::nat) < y; y < z |] ==> x < z"
149   by linarith
151 lemma "(x::nat) < y & y < z ==> x < z"
152   by linarith
154 text {* This example involves no arithmetic at all, but is solved by
155   preprocessing (i.e. NNF normalization) alone. *}
157 lemma "(P::bool) = Q ==> Q = P"
158   by linarith
160 lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> min (x::nat) y = 0"
161   by linarith
163 lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> max (x::nat) y = x + y"
164   by linarith
166 lemma "[| (x::nat) ~= y; a + 2 = b; a < y; y < b; a < x; x < b |] ==> False"
167   by linarith
169 lemma "[| (x::nat) > y; y > z; z > x |] ==> False"
170   by linarith
172 lemma "(x::nat) - 5 > y ==> y < x"
173   by linarith
175 lemma "(x::nat) ~= 0 ==> 0 < x"
176   by linarith
178 lemma "[| (x::nat) ~= y; x <= y |] ==> x < y"
179   by linarith
181 lemma "[| (x::nat) < y; P (x - y) |] ==> P 0"
182   by linarith
184 lemma "(x - y) - (x::nat) = (x - x) - y"
185   by linarith
187 lemma "[| (a::nat) < b; c < d |] ==> (a - b) = (c - d)"
188   by linarith
190 lemma "((a::nat) - (b - (c - (d - e)))) = (a - (b - (c - (d - e))))"
191   by linarith
193 lemma "(n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
194   (n = n' & n' < m) | (n = m & m < n') |
195   (n' < m & m < n) | (n' < m & m = n) |
196   (n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) |
197   (m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) |
198   (m = n & n < n') | (m = n' & n' < n) |
199   (n' = m & m = (n::nat))"
200 (* FIXME: this should work in principle, but is extremely slow because     *)
201 (*        preprocessing negates the goal and tries to compute its negation *)
202 (*        normal form, which creates lots of separate cases for this       *)
203 (*        disjunction of conjunctions                                      *)
204 (* by (tactic {* Lin_Arith.tac 1 *}) *)
205 oops
207 lemma "2 * (x::nat) ~= 1"
208 (* FIXME: this is beyond the scope of the decision procedure at the moment, *)
209 (*        because its negation is satisfiable in the rationals?             *)
210 (* by (tactic {* Lin_Arith.simple_tac 1 *}) *)
211 oops
213 text {* Constants. *}
215 lemma "(0::nat) < 1"
216   by linarith
218 lemma "(0::int) < 1"
219   by linarith
221 lemma "(47::nat) + 11 < 8 * 15"
222   by linarith
224 lemma "(47::int) + 11 < 8 * 15"
225   by linarith
227 text {* Splitting of inequalities of different type. *}
229 lemma "[| (a::nat) ~= b; (i::int) ~= j; a < 2; b < 2 |] ==>
230   a + b <= nat (max (abs i) (abs j))"
231   by linarith
233 text {* Again, but different order. *}
235 lemma "[| (i::int) ~= j; (a::nat) ~= b; a < 2; b < 2 |] ==>
236   a + b <= nat (max (abs i) (abs j))"
237   by linarith
239 end