src/HOL/Library/SetsAndFunctions.thy
 author chaieb Mon Jun 11 11:06:04 2007 +0200 (2007-06-11) changeset 23315 df3a7e9ebadb parent 21404 eb85850d3eb7 child 23477 f4b83f03cac9 permissions -rwxr-xr-x
tuned Proof
```     1 (*  Title:      HOL/Library/SetsAndFunctions.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Jeremy Avigad and Kevin Donnelly
```
```     4 *)
```
```     5
```
```     6 header {* Operations on sets and functions *}
```
```     7
```
```     8 theory SetsAndFunctions
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 text {*
```
```    13 This library lifts operations like addition and muliplication to sets and
```
```    14 functions of appropriate types. It was designed to support asymptotic
```
```    15 calculations. See the comments at the top of theory @{text BigO}.
```
```    16 *}
```
```    17
```
```    18 subsection {* Basic definitions *}
```
```    19
```
```    20 instance set :: (plus) plus ..
```
```    21 instance "fun" :: (type, plus) plus ..
```
```    22
```
```    23 defs (overloaded)
```
```    24   func_plus: "f + g == (%x. f x + g x)"
```
```    25   set_plus: "A + B == {c. EX a:A. EX b:B. c = a + b}"
```
```    26
```
```    27 instance set :: (times) times ..
```
```    28 instance "fun" :: (type, times) times ..
```
```    29
```
```    30 defs (overloaded)
```
```    31   func_times: "f * g == (%x. f x * g x)"
```
```    32   set_times:"A * B == {c. EX a:A. EX b:B. c = a * b}"
```
```    33
```
```    34 instance "fun" :: (type, minus) minus ..
```
```    35
```
```    36 defs (overloaded)
```
```    37   func_minus: "- f == (%x. - f x)"
```
```    38   func_diff: "f - g == %x. f x - g x"
```
```    39
```
```    40 instance "fun" :: (type, zero) zero ..
```
```    41 instance set :: (zero) zero ..
```
```    42
```
```    43 defs (overloaded)
```
```    44   func_zero: "0::(('a::type) => ('b::zero)) == %x. 0"
```
```    45   set_zero: "0::('a::zero)set == {0}"
```
```    46
```
```    47 instance "fun" :: (type, one) one ..
```
```    48 instance set :: (one) one ..
```
```    49
```
```    50 defs (overloaded)
```
```    51   func_one: "1::(('a::type) => ('b::one)) == %x. 1"
```
```    52   set_one: "1::('a::one)set == {1}"
```
```    53
```
```    54 definition
```
```    55   elt_set_plus :: "'a::plus => 'a set => 'a set"  (infixl "+o" 70) where
```
```    56   "a +o B = {c. EX b:B. c = a + b}"
```
```    57
```
```    58 definition
```
```    59   elt_set_times :: "'a::times => 'a set => 'a set"  (infixl "*o" 80) where
```
```    60   "a *o B = {c. EX b:B. c = a * b}"
```
```    61
```
```    62 abbreviation (input)
```
```    63   elt_set_eq :: "'a => 'a set => bool"  (infix "=o" 50) where
```
```    64   "x =o A == x : A"
```
```    65
```
```    66 instance "fun" :: (type,semigroup_add)semigroup_add
```
```    67   by default (auto simp add: func_plus add_assoc)
```
```    68
```
```    69 instance "fun" :: (type,comm_monoid_add)comm_monoid_add
```
```    70   by default (auto simp add: func_zero func_plus add_ac)
```
```    71
```
```    72 instance "fun" :: (type,ab_group_add)ab_group_add
```
```    73   apply default
```
```    74    apply (simp add: func_minus func_plus func_zero)
```
```    75   apply (simp add: func_minus func_plus func_diff diff_minus)
```
```    76   done
```
```    77
```
```    78 instance "fun" :: (type,semigroup_mult)semigroup_mult
```
```    79   apply default
```
```    80   apply (auto simp add: func_times mult_assoc)
```
```    81   done
```
```    82
```
```    83 instance "fun" :: (type,comm_monoid_mult)comm_monoid_mult
```
```    84   apply default
```
```    85    apply (auto simp add: func_one func_times mult_ac)
```
```    86   done
```
```    87
```
```    88 instance "fun" :: (type,comm_ring_1)comm_ring_1
```
```    89   apply default
```
```    90    apply (auto simp add: func_plus func_times func_minus func_diff ext
```
```    91      func_one func_zero ring_eq_simps)
```
```    92   apply (drule fun_cong)
```
```    93   apply simp
```
```    94   done
```
```    95
```
```    96 instance set :: (semigroup_add)semigroup_add
```
```    97   apply default
```
```    98   apply (unfold set_plus)
```
```    99   apply (force simp add: add_assoc)
```
```   100   done
```
```   101
```
```   102 instance set :: (semigroup_mult)semigroup_mult
```
```   103   apply default
```
```   104   apply (unfold set_times)
```
```   105   apply (force simp add: mult_assoc)
```
```   106   done
```
```   107
```
```   108 instance set :: (comm_monoid_add)comm_monoid_add
```
```   109   apply default
```
```   110    apply (unfold set_plus)
```
```   111    apply (force simp add: add_ac)
```
```   112   apply (unfold set_zero)
```
```   113   apply force
```
```   114   done
```
```   115
```
```   116 instance set :: (comm_monoid_mult)comm_monoid_mult
```
```   117   apply default
```
```   118    apply (unfold set_times)
```
```   119    apply (force simp add: mult_ac)
```
```   120   apply (unfold set_one)
```
```   121   apply force
```
```   122   done
```
```   123
```
```   124
```
```   125 subsection {* Basic properties *}
```
```   126
```
```   127 lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"
```
```   128   by (auto simp add: set_plus)
```
```   129
```
```   130 lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
```
```   131   by (auto simp add: elt_set_plus_def)
```
```   132
```
```   133 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) +
```
```   134     (b +o D) = (a + b) +o (C + D)"
```
```   135   apply (auto simp add: elt_set_plus_def set_plus add_ac)
```
```   136    apply (rule_tac x = "ba + bb" in exI)
```
```   137   apply (auto simp add: add_ac)
```
```   138   apply (rule_tac x = "aa + a" in exI)
```
```   139   apply (auto simp add: add_ac)
```
```   140   done
```
```   141
```
```   142 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
```
```   143     (a + b) +o C"
```
```   144   by (auto simp add: elt_set_plus_def add_assoc)
```
```   145
```
```   146 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =
```
```   147     a +o (B + C)"
```
```   148   apply (auto simp add: elt_set_plus_def set_plus)
```
```   149    apply (blast intro: add_ac)
```
```   150   apply (rule_tac x = "a + aa" in exI)
```
```   151   apply (rule conjI)
```
```   152    apply (rule_tac x = "aa" in bexI)
```
```   153     apply auto
```
```   154   apply (rule_tac x = "ba" in bexI)
```
```   155    apply (auto simp add: add_ac)
```
```   156   done
```
```   157
```
```   158 theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
```
```   159     a +o (C + D)"
```
```   160   apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus add_ac)
```
```   161    apply (rule_tac x = "aa + ba" in exI)
```
```   162    apply (auto simp add: add_ac)
```
```   163   done
```
```   164
```
```   165 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
```
```   166   set_plus_rearrange3 set_plus_rearrange4
```
```   167
```
```   168 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
```
```   169   by (auto simp add: elt_set_plus_def)
```
```   170
```
```   171 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
```
```   172     C + E <= D + F"
```
```   173   by (auto simp add: set_plus)
```
```   174
```
```   175 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
```
```   176   by (auto simp add: elt_set_plus_def set_plus)
```
```   177
```
```   178 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
```
```   179     a +o D <= D + C"
```
```   180   by (auto simp add: elt_set_plus_def set_plus add_ac)
```
```   181
```
```   182 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"
```
```   183   apply (subgoal_tac "a +o B <= a +o D")
```
```   184    apply (erule order_trans)
```
```   185    apply (erule set_plus_mono3)
```
```   186   apply (erule set_plus_mono)
```
```   187   done
```
```   188
```
```   189 lemma set_plus_mono_b: "C <= D ==> x : a +o C
```
```   190     ==> x : a +o D"
```
```   191   apply (frule set_plus_mono)
```
```   192   apply auto
```
```   193   done
```
```   194
```
```   195 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==>
```
```   196     x : D + F"
```
```   197   apply (frule set_plus_mono2)
```
```   198    prefer 2
```
```   199    apply force
```
```   200   apply assumption
```
```   201   done
```
```   202
```
```   203 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"
```
```   204   apply (frule set_plus_mono3)
```
```   205   apply auto
```
```   206   done
```
```   207
```
```   208 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
```
```   209     x : a +o D ==> x : D + C"
```
```   210   apply (frule set_plus_mono4)
```
```   211   apply auto
```
```   212   done
```
```   213
```
```   214 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
```
```   215   by (auto simp add: elt_set_plus_def)
```
```   216
```
```   217 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
```
```   218   apply (auto intro!: subsetI simp add: set_plus)
```
```   219   apply (rule_tac x = 0 in bexI)
```
```   220    apply (rule_tac x = x in bexI)
```
```   221     apply (auto simp add: add_ac)
```
```   222   done
```
```   223
```
```   224 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
```
```   225   by (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   226
```
```   227 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
```
```   228   apply (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   229   apply (subgoal_tac "a = (a + - b) + b")
```
```   230    apply (rule bexI, assumption, assumption)
```
```   231   apply (auto simp add: add_ac)
```
```   232   done
```
```   233
```
```   234 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
```
```   235   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
```
```   236     assumption)
```
```   237
```
```   238 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"
```
```   239   by (auto simp add: set_times)
```
```   240
```
```   241 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
```
```   242   by (auto simp add: elt_set_times_def)
```
```   243
```
```   244 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) *
```
```   245     (b *o D) = (a * b) *o (C * D)"
```
```   246   apply (auto simp add: elt_set_times_def set_times)
```
```   247    apply (rule_tac x = "ba * bb" in exI)
```
```   248    apply (auto simp add: mult_ac)
```
```   249   apply (rule_tac x = "aa * a" in exI)
```
```   250   apply (auto simp add: mult_ac)
```
```   251   done
```
```   252
```
```   253 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
```
```   254     (a * b) *o C"
```
```   255   by (auto simp add: elt_set_times_def mult_assoc)
```
```   256
```
```   257 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =
```
```   258     a *o (B * C)"
```
```   259   apply (auto simp add: elt_set_times_def set_times)
```
```   260    apply (blast intro: mult_ac)
```
```   261   apply (rule_tac x = "a * aa" in exI)
```
```   262   apply (rule conjI)
```
```   263    apply (rule_tac x = "aa" in bexI)
```
```   264     apply auto
```
```   265   apply (rule_tac x = "ba" in bexI)
```
```   266    apply (auto simp add: mult_ac)
```
```   267   done
```
```   268
```
```   269 theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
```
```   270     a *o (C * D)"
```
```   271   apply (auto intro!: subsetI simp add: elt_set_times_def set_times
```
```   272     mult_ac)
```
```   273    apply (rule_tac x = "aa * ba" in exI)
```
```   274    apply (auto simp add: mult_ac)
```
```   275   done
```
```   276
```
```   277 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
```
```   278   set_times_rearrange3 set_times_rearrange4
```
```   279
```
```   280 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
```
```   281   by (auto simp add: elt_set_times_def)
```
```   282
```
```   283 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
```
```   284     C * E <= D * F"
```
```   285   by (auto simp add: set_times)
```
```   286
```
```   287 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
```
```   288   by (auto simp add: elt_set_times_def set_times)
```
```   289
```
```   290 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
```
```   291     a *o D <= D * C"
```
```   292   by (auto simp add: elt_set_times_def set_times mult_ac)
```
```   293
```
```   294 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"
```
```   295   apply (subgoal_tac "a *o B <= a *o D")
```
```   296    apply (erule order_trans)
```
```   297    apply (erule set_times_mono3)
```
```   298   apply (erule set_times_mono)
```
```   299   done
```
```   300
```
```   301 lemma set_times_mono_b: "C <= D ==> x : a *o C
```
```   302     ==> x : a *o D"
```
```   303   apply (frule set_times_mono)
```
```   304   apply auto
```
```   305   done
```
```   306
```
```   307 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==>
```
```   308     x : D * F"
```
```   309   apply (frule set_times_mono2)
```
```   310    prefer 2
```
```   311    apply force
```
```   312   apply assumption
```
```   313   done
```
```   314
```
```   315 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"
```
```   316   apply (frule set_times_mono3)
```
```   317   apply auto
```
```   318   done
```
```   319
```
```   320 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
```
```   321     x : a *o D ==> x : D * C"
```
```   322   apply (frule set_times_mono4)
```
```   323   apply auto
```
```   324   done
```
```   325
```
```   326 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
```
```   327   by (auto simp add: elt_set_times_def)
```
```   328
```
```   329 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
```
```   330     (a * b) +o (a *o C)"
```
```   331   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distrib)
```
```   332
```
```   333 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
```
```   334     (a *o B) + (a *o C)"
```
```   335   apply (auto simp add: set_plus elt_set_times_def ring_distrib)
```
```   336    apply blast
```
```   337   apply (rule_tac x = "b + bb" in exI)
```
```   338   apply (auto simp add: ring_distrib)
```
```   339   done
```
```   340
```
```   341 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
```
```   342     a *o D + C * D"
```
```   343   apply (auto intro!: subsetI simp add:
```
```   344     elt_set_plus_def elt_set_times_def set_times
```
```   345     set_plus ring_distrib)
```
```   346   apply auto
```
```   347   done
```
```   348
```
```   349 theorems set_times_plus_distribs =
```
```   350   set_times_plus_distrib
```
```   351   set_times_plus_distrib2
```
```   352
```
```   353 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
```
```   354     - a : C"
```
```   355   by (auto simp add: elt_set_times_def)
```
```   356
```
```   357 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
```
```   358     - a : (- 1) *o C"
```
```   359   by (auto simp add: elt_set_times_def)
```
```   360
```
```   361 end
```