src/HOL/ex/Arith_Examples.thy
author haftmann
Fri Oct 10 19:55:32 2014 +0200 (2014-10-10)
changeset 58646 cd63a4b12a33
parent 47108 2a1953f0d20d
child 58889 5b7a9633cfa8
permissions -rw-r--r--
specialized specification: avoid trivial instances
     1 (*  Title:  HOL/ex/Arith_Examples.thy
     2     Author: Tjark Weber
     3 *)
     4 
     5 header {* Arithmetic *}
     6 
     7 theory Arith_Examples
     8 imports Main
     9 begin
    10 
    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:
    15 
    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!
    22 
    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 *}
    30 
    31 
    32 subsection {* Splitting of Operators: @{term max}, @{term min}, @{term abs},
    33            @{term minus}, @{term nat}, @{term Divides.mod},
    34            @{term Divides.div} *}
    35 
    36 lemma "(i::nat) <= max i j"
    37   by linarith
    38 
    39 lemma "(i::int) <= max i j"
    40   by linarith
    41 
    42 lemma "min i j <= (i::nat)"
    43   by linarith
    44 
    45 lemma "min i j <= (i::int)"
    46   by linarith
    47 
    48 lemma "min (i::nat) j <= max i j"
    49   by linarith
    50 
    51 lemma "min (i::int) j <= max i j"
    52   by linarith
    53 
    54 lemma "min (i::nat) j + max i j = i + j"
    55   by linarith
    56 
    57 lemma "min (i::int) j + max i j = i + j"
    58   by linarith
    59 
    60 lemma "(i::nat) < j ==> min i j < max i j"
    61   by linarith
    62 
    63 lemma "(i::int) < j ==> min i j < max i j"
    64   by linarith
    65 
    66 lemma "(0::int) <= abs i"
    67   by linarith
    68 
    69 lemma "(i::int) <= abs i"
    70   by linarith
    71 
    72 lemma "abs (abs (i::int)) = abs i"
    73   by linarith
    74 
    75 text {* Also testing subgoals with bound variables. *}
    76 
    77 lemma "!!x. (x::nat) <= y ==> x - y = 0"
    78   by linarith
    79 
    80 lemma "!!x. (x::nat) - y = 0 ==> x <= y"
    81   by linarith
    82 
    83 lemma "!!x. ((x::nat) <= y) = (x - y = 0)"
    84   by linarith
    85 
    86 lemma "[| (x::nat) < y; d < 1 |] ==> x - y = d"
    87   by linarith
    88 
    89 lemma "[| (x::nat) < y; d < 1 |] ==> x - y - x = d - x"
    90   by linarith
    91 
    92 lemma "(x::int) < y ==> x - y < 0"
    93   by linarith
    94 
    95 lemma "nat (i + j) <= nat i + nat j"
    96   by linarith
    97 
    98 lemma "i < j ==> nat (i - j) = 0"
    99   by linarith
   100 
   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
   105 
   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
   110 
   111 lemma "(i::nat) mod 42 <= 41"
   112   by linarith
   113 
   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
   118 
   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
   123 
   124 lemma "(i::int) mod 42 <= 41"
   125   by linarith
   126 
   127 lemma "-(i::int) * 1 = 0 ==> i = 0"
   128   by linarith
   129 
   130 lemma "[| (0::int) < abs i; abs i * 1 < abs i * j |] ==> 1 < abs i * j"
   131   by linarith
   132 
   133 
   134 subsection {* Meta-Logic *}
   135 
   136 lemma "x < Suc y == x <= y"
   137   by linarith
   138 
   139 lemma "((x::nat) == z ==> x ~= y) ==> x ~= y | z ~= y"
   140   by linarith
   141 
   142 
   143 subsection {* Various Other Examples *}
   144 
   145 lemma "(x < Suc y) = (x <= y)"
   146   by linarith
   147 
   148 lemma "[| (x::nat) < y; y < z |] ==> x < z"
   149   by linarith
   150 
   151 lemma "(x::nat) < y & y < z ==> x < z"
   152   by linarith
   153 
   154 text {* This example involves no arithmetic at all, but is solved by
   155   preprocessing (i.e. NNF normalization) alone. *}
   156 
   157 lemma "(P::bool) = Q ==> Q = P"
   158   by linarith
   159 
   160 lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> min (x::nat) y = 0"
   161   by linarith
   162 
   163 lemma "[| P = (x = 0); (~P) = (y = 0) |] ==> max (x::nat) y = x + y"
   164   by linarith
   165 
   166 lemma "[| (x::nat) ~= y; a + 2 = b; a < y; y < b; a < x; x < b |] ==> False"
   167   by linarith
   168 
   169 lemma "[| (x::nat) > y; y > z; z > x |] ==> False"
   170   by linarith
   171 
   172 lemma "(x::nat) - 5 > y ==> y < x"
   173   by linarith
   174 
   175 lemma "(x::nat) ~= 0 ==> 0 < x"
   176   by linarith
   177 
   178 lemma "[| (x::nat) ~= y; x <= y |] ==> x < y"
   179   by linarith
   180 
   181 lemma "[| (x::nat) < y; P (x - y) |] ==> P 0"
   182   by linarith
   183 
   184 lemma "(x - y) - (x::nat) = (x - x) - y"
   185   by linarith
   186 
   187 lemma "[| (a::nat) < b; c < d |] ==> (a - b) = (c - d)"
   188   by linarith
   189 
   190 lemma "((a::nat) - (b - (c - (d - e)))) = (a - (b - (c - (d - e))))"
   191   by linarith
   192 
   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
   206 
   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
   212 
   213 text {* Constants. *}
   214 
   215 lemma "(0::nat) < 1"
   216   by linarith
   217 
   218 lemma "(0::int) < 1"
   219   by linarith
   220 
   221 lemma "(47::nat) + 11 < 8 * 15"
   222   by linarith
   223 
   224 lemma "(47::int) + 11 < 8 * 15"
   225   by linarith
   226 
   227 text {* Splitting of inequalities of different type. *}
   228 
   229 lemma "[| (a::nat) ~= b; (i::int) ~= j; a < 2; b < 2 |] ==>
   230   a + b <= nat (max (abs i) (abs j))"
   231   by linarith
   232 
   233 text {* Again, but different order. *}
   234 
   235 lemma "[| (i::int) ~= j; (a::nat) ~= b; a < 2; b < 2 |] ==>
   236   a + b <= nat (max (abs i) (abs j))"
   237   by linarith
   238 
   239 end