src/HOL/ex/Set_Algebras.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 41582 c34415351b6d child 44890 22f665a2e91c permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/ex/Set_Algebras.thy
```
```     2     Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
```
```     3 *)
```
```     4
```
```     5 header {* Algebraic operations on sets *}
```
```     6
```
```     7 theory Set_Algebras
```
```     8 imports Main Interpretation_with_Defs
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12   This library lifts operations like addition and muliplication to
```
```    13   sets.  It was designed to support asymptotic calculations. See the
```
```    14   comments at the top of theory @{text BigO}.
```
```    15 *}
```
```    16
```
```    17 definition set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<oplus>" 65) where
```
```    18   "A \<oplus> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
```
```    19
```
```    20 definition set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<otimes>" 70) where
```
```    21   "A \<otimes> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
```
```    22
```
```    23 definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
```
```    24   "a +o B = {c. \<exists>b\<in>B. c = a + b}"
```
```    25
```
```    26 definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
```
```    27   "a *o B = {c. \<exists>b\<in>B. c = a * b}"
```
```    28
```
```    29 abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
```
```    30   "x =o A \<equiv> x \<in> A"
```
```    31
```
```    32 interpretation set_add!: semigroup "set_plus :: 'a::semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
```
```    33 qed (force simp add: set_plus_def add.assoc)
```
```    34
```
```    35 interpretation set_add!: abel_semigroup "set_plus :: 'a::ab_semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
```
```    36 qed (force simp add: set_plus_def add.commute)
```
```    37
```
```    38 interpretation set_add!: monoid "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
```
```    39 qed (simp_all add: set_plus_def)
```
```    40
```
```    41 interpretation set_add!: comm_monoid "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
```
```    42 qed (simp add: set_plus_def)
```
```    43
```
```    44 interpretation set_add!: monoid_add "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
```
```    45   defines listsum_set is set_add.listsum
```
```    46 proof
```
```    47 qed (simp_all add: set_add.assoc)
```
```    48
```
```    49 interpretation set_add!: comm_monoid_add "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
```
```    50   defines setsum_set is set_add.setsum
```
```    51   where "monoid_add.listsum set_plus {0::'a} = listsum_set"
```
```    52 proof -
```
```    53   show "class.comm_monoid_add (set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {0}" proof
```
```    54   qed (simp_all add: set_add.commute)
```
```    55   then interpret set_add!: comm_monoid_add "set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" .
```
```    56   show "monoid_add.listsum set_plus {0::'a} = listsum_set"
```
```    57     by (simp only: listsum_set_def)
```
```    58 qed
```
```    59
```
```    60 interpretation set_mult!: semigroup "set_times :: 'a::semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
```
```    61 qed (force simp add: set_times_def mult.assoc)
```
```    62
```
```    63 interpretation set_mult!: abel_semigroup "set_times :: 'a::ab_semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
```
```    64 qed (force simp add: set_times_def mult.commute)
```
```    65
```
```    66 interpretation set_mult!: monoid "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
```
```    67 qed (simp_all add: set_times_def)
```
```    68
```
```    69 interpretation set_mult!: comm_monoid "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
```
```    70 qed (simp add: set_times_def)
```
```    71
```
```    72 interpretation set_mult!: monoid_mult "{1}" "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
```
```    73   defines power_set is set_mult.power
```
```    74 proof
```
```    75 qed (simp_all add: set_mult.assoc)
```
```    76
```
```    77 interpretation set_mult!: comm_monoid_mult "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
```
```    78   defines setprod_set is set_mult.setprod
```
```    79   where "power.power {1} set_times = power_set"
```
```    80 proof -
```
```    81   show "class.comm_monoid_mult (set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {1}" proof
```
```    82   qed (simp_all add: set_mult.commute)
```
```    83   then interpret set_mult!: comm_monoid_mult "set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" .
```
```    84   show "power.power {1} set_times = power_set"
```
```    85     by (simp add: power_set_def)
```
```    86 qed
```
```    87
```
```    88 lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
```
```    89   by (auto simp add: set_plus_def)
```
```    90
```
```    91 lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
```
```    92   by (auto simp add: elt_set_plus_def)
```
```    93
```
```    94 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus>
```
```    95     (b +o D) = (a + b) +o (C \<oplus> D)"
```
```    96   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
```
```    97    apply (rule_tac x = "ba + bb" in exI)
```
```    98   apply (auto simp add: add_ac)
```
```    99   apply (rule_tac x = "aa + a" in exI)
```
```   100   apply (auto simp add: add_ac)
```
```   101   done
```
```   102
```
```   103 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
```
```   104     (a + b) +o C"
```
```   105   by (auto simp add: elt_set_plus_def add_assoc)
```
```   106
```
```   107 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C =
```
```   108     a +o (B \<oplus> C)"
```
```   109   apply (auto simp add: elt_set_plus_def set_plus_def)
```
```   110    apply (blast intro: add_ac)
```
```   111   apply (rule_tac x = "a + aa" in exI)
```
```   112   apply (rule conjI)
```
```   113    apply (rule_tac x = "aa" in bexI)
```
```   114     apply auto
```
```   115   apply (rule_tac x = "ba" in bexI)
```
```   116    apply (auto simp add: add_ac)
```
```   117   done
```
```   118
```
```   119 theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
```
```   120     a +o (C \<oplus> D)"
```
```   121   apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus_def add_ac)
```
```   122    apply (rule_tac x = "aa + ba" in exI)
```
```   123    apply (auto simp add: add_ac)
```
```   124   done
```
```   125
```
```   126 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
```
```   127   set_plus_rearrange3 set_plus_rearrange4
```
```   128
```
```   129 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
```
```   130   by (auto simp add: elt_set_plus_def)
```
```   131
```
```   132 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
```
```   133     C \<oplus> E <= D \<oplus> F"
```
```   134   by (auto simp add: set_plus_def)
```
```   135
```
```   136 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
```
```   137   by (auto simp add: elt_set_plus_def set_plus_def)
```
```   138
```
```   139 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
```
```   140     a +o D <= D \<oplus> C"
```
```   141   by (auto simp add: elt_set_plus_def set_plus_def add_ac)
```
```   142
```
```   143 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
```
```   144   apply (subgoal_tac "a +o B <= a +o D")
```
```   145    apply (erule order_trans)
```
```   146    apply (erule set_plus_mono3)
```
```   147   apply (erule set_plus_mono)
```
```   148   done
```
```   149
```
```   150 lemma set_plus_mono_b: "C <= D ==> x : a +o C
```
```   151     ==> x : a +o D"
```
```   152   apply (frule set_plus_mono)
```
```   153   apply auto
```
```   154   done
```
```   155
```
```   156 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
```
```   157     x : D \<oplus> F"
```
```   158   apply (frule set_plus_mono2)
```
```   159    prefer 2
```
```   160    apply force
```
```   161   apply assumption
```
```   162   done
```
```   163
```
```   164 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D"
```
```   165   apply (frule set_plus_mono3)
```
```   166   apply auto
```
```   167   done
```
```   168
```
```   169 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
```
```   170     x : a +o D ==> x : D \<oplus> C"
```
```   171   apply (frule set_plus_mono4)
```
```   172   apply auto
```
```   173   done
```
```   174
```
```   175 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
```
```   176   by (auto simp add: elt_set_plus_def)
```
```   177
```
```   178 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
```
```   179   apply (auto intro!: subsetI simp add: set_plus_def)
```
```   180   apply (rule_tac x = 0 in bexI)
```
```   181    apply (rule_tac x = x in bexI)
```
```   182     apply (auto simp add: add_ac)
```
```   183   done
```
```   184
```
```   185 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
```
```   186   by (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   187
```
```   188 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
```
```   189   apply (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   190   apply (subgoal_tac "a = (a + - b) + b")
```
```   191    apply (rule bexI, assumption, assumption)
```
```   192   apply (auto simp add: add_ac)
```
```   193   done
```
```   194
```
```   195 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
```
```   196   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
```
```   197     assumption)
```
```   198
```
```   199 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
```
```   200   by (auto simp add: set_times_def)
```
```   201
```
```   202 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
```
```   203   by (auto simp add: elt_set_times_def)
```
```   204
```
```   205 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
```
```   206     (b *o D) = (a * b) *o (C \<otimes> D)"
```
```   207   apply (auto simp add: elt_set_times_def set_times_def)
```
```   208    apply (rule_tac x = "ba * bb" in exI)
```
```   209    apply (auto simp add: mult_ac)
```
```   210   apply (rule_tac x = "aa * a" in exI)
```
```   211   apply (auto simp add: mult_ac)
```
```   212   done
```
```   213
```
```   214 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
```
```   215     (a * b) *o C"
```
```   216   by (auto simp add: elt_set_times_def mult_assoc)
```
```   217
```
```   218 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
```
```   219     a *o (B \<otimes> C)"
```
```   220   apply (auto simp add: elt_set_times_def set_times_def)
```
```   221    apply (blast intro: mult_ac)
```
```   222   apply (rule_tac x = "a * aa" in exI)
```
```   223   apply (rule conjI)
```
```   224    apply (rule_tac x = "aa" in bexI)
```
```   225     apply auto
```
```   226   apply (rule_tac x = "ba" in bexI)
```
```   227    apply (auto simp add: mult_ac)
```
```   228   done
```
```   229
```
```   230 theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
```
```   231     a *o (C \<otimes> D)"
```
```   232   apply (auto intro!: subsetI simp add: elt_set_times_def set_times_def
```
```   233     mult_ac)
```
```   234    apply (rule_tac x = "aa * ba" in exI)
```
```   235    apply (auto simp add: mult_ac)
```
```   236   done
```
```   237
```
```   238 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
```
```   239   set_times_rearrange3 set_times_rearrange4
```
```   240
```
```   241 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
```
```   242   by (auto simp add: elt_set_times_def)
```
```   243
```
```   244 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
```
```   245     C \<otimes> E <= D \<otimes> F"
```
```   246   by (auto simp add: set_times_def)
```
```   247
```
```   248 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
```
```   249   by (auto simp add: elt_set_times_def set_times_def)
```
```   250
```
```   251 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
```
```   252     a *o D <= D \<otimes> C"
```
```   253   by (auto simp add: elt_set_times_def set_times_def mult_ac)
```
```   254
```
```   255 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
```
```   256   apply (subgoal_tac "a *o B <= a *o D")
```
```   257    apply (erule order_trans)
```
```   258    apply (erule set_times_mono3)
```
```   259   apply (erule set_times_mono)
```
```   260   done
```
```   261
```
```   262 lemma set_times_mono_b: "C <= D ==> x : a *o C
```
```   263     ==> x : a *o D"
```
```   264   apply (frule set_times_mono)
```
```   265   apply auto
```
```   266   done
```
```   267
```
```   268 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
```
```   269     x : D \<otimes> F"
```
```   270   apply (frule set_times_mono2)
```
```   271    prefer 2
```
```   272    apply force
```
```   273   apply assumption
```
```   274   done
```
```   275
```
```   276 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
```
```   277   apply (frule set_times_mono3)
```
```   278   apply auto
```
```   279   done
```
```   280
```
```   281 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
```
```   282     x : a *o D ==> x : D \<otimes> C"
```
```   283   apply (frule set_times_mono4)
```
```   284   apply auto
```
```   285   done
```
```   286
```
```   287 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
```
```   288   by (auto simp add: elt_set_times_def)
```
```   289
```
```   290 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
```
```   291     (a * b) +o (a *o C)"
```
```   292   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
```
```   293
```
```   294 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
```
```   295     (a *o B) \<oplus> (a *o C)"
```
```   296   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
```
```   297    apply blast
```
```   298   apply (rule_tac x = "b + bb" in exI)
```
```   299   apply (auto simp add: ring_distribs)
```
```   300   done
```
```   301
```
```   302 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
```
```   303     a *o D \<oplus> C \<otimes> D"
```
```   304   apply (auto intro!: subsetI simp add:
```
```   305     elt_set_plus_def elt_set_times_def set_times_def
```
```   306     set_plus_def ring_distribs)
```
```   307   apply auto
```
```   308   done
```
```   309
```
```   310 theorems set_times_plus_distribs =
```
```   311   set_times_plus_distrib
```
```   312   set_times_plus_distrib2
```
```   313
```
```   314 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
```
```   315     - a : C"
```
```   316   by (auto simp add: elt_set_times_def)
```
```   317
```
```   318 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
```
```   319     - a : (- 1) *o C"
```
```   320   by (auto simp add: elt_set_times_def)
```
```   321
```
```   322 lemma set_plus_image:
```
```   323   fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
```
```   324   unfolding set_plus_def by (fastsimp simp: image_iff)
```
```   325
```
```   326 lemma set_setsum_alt:
```
```   327   assumes fin: "finite I"
```
```   328   shows "setsum_set S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
```
```   329     (is "_ = ?setsum I")
```
```   330 using fin proof induct
```
```   331   case (insert x F)
```
```   332   have "setsum_set S (insert x F) = S x \<oplus> ?setsum F"
```
```   333     using insert.hyps by auto
```
```   334   also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
```
```   335     unfolding set_plus_def
```
```   336   proof safe
```
```   337     fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
```
```   338     then show "\<exists>s'. y + setsum s F = s' x + setsum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)"
```
```   339       using insert.hyps
```
```   340       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
```
```   341   qed auto
```
```   342   finally show ?case
```
```   343     using insert.hyps by auto
```
```   344 qed auto
```
```   345
```
```   346 lemma setsum_set_cond_linear:
```
```   347   fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
```
```   348   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A \<oplus> B)" "P {0}"
```
```   349     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
```
```   350   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
```
```   351   shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
```
```   352 proof cases
```
```   353   assume "finite I" from this all show ?thesis
```
```   354   proof induct
```
```   355     case (insert x F)
```
```   356     from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum_set S F)"
```
```   357       by induct auto
```
```   358     with insert show ?case
```
```   359       by (simp, subst f) auto
```
```   360   qed (auto intro!: f)
```
```   361 qed (auto intro!: f)
```
```   362
```
```   363 lemma setsum_set_linear:
```
```   364   fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
```
```   365   assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
```
```   366   shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
```
```   367   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
```
```   368
```
```   369 end
```