1.1 --- a/src/HOL/IsaMakefile	Sat Jan 15 18:49:42 2011 +0100
     1.2 +++ b/src/HOL/IsaMakefile	Sat Jan 15 20:05:29 2011 +0100
     1.3 @@ -1034,19 +1034,20 @@
     1.4    ex/HarmonicSeries.thy ex/Hebrew.thy ex/Hex_Bin_Examples.thy		\
     1.5    ex/Higher_Order_Logic.thy ex/Iff_Oracle.thy ex/Induction_Schema.thy	\
     1.6    ex/InductiveInvariant.thy ex/InductiveInvariant_examples.thy		\
     1.7 -  ex/Intuitionistic.thy ex/Lagrange.thy \
     1.8 -  ex/List_to_Set_Comprehension_Examples.thy ex/LocaleTest2.thy ex/MT.thy	\
     1.9 -  ex/MergeSort.thy ex/Meson_Test.thy ex/MonoidGroup.thy			\
    1.10 +  ex/Interpretation_with_Defs.thy ex/Intuitionistic.thy ex/Lagrange.thy	\
    1.11 +  ex/List_to_Set_Comprehension_Examples.thy ex/LocaleTest2.thy		\
    1.12 +  ex/MT.thy ex/MergeSort.thy ex/Meson_Test.thy ex/MonoidGroup.thy	\
    1.13    ex/Multiquote.thy ex/NatSum.thy ex/Normalization_by_Evaluation.thy	\
    1.14    ex/Numeral.thy ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy		\
    1.15    ex/Quickcheck_Examples.thy ex/Quickcheck_Lattice_Examples.thy		\
    1.16    ex/Quicksort.thy ex/ROOT.ML ex/Recdefs.thy ex/Records.thy		\
    1.17    ex/ReflectionEx.thy ex/Refute_Examples.thy ex/SAT_Examples.thy	\
    1.18 -  ex/SVC_Oracle.thy ex/Serbian.thy ex/Sqrt.thy ex/Sqrt_Script.thy	\
    1.19 -  ex/Sudoku.thy ex/Tarski.thy ex/Termination.thy ex/Transfer_Ex.thy	\
    1.20 -  ex/Tree23.thy ex/Unification.thy ex/While_Combinator_Example.thy	\
    1.21 -  ex/document/root.bib ex/document/root.tex ex/set.thy ex/svc_funcs.ML	\
    1.22 -  ex/svc_test.thy
    1.23 +  ex/SVC_Oracle.thy ex/Serbian.thy ex/Set_Algebras.thy ex/Sqrt.thy	\
    1.24 +  ex/Sqrt_Script.thy ex/Sudoku.thy ex/Tarski.thy ex/Termination.thy	\
    1.25 +  ex/Transfer_Ex.thy ex/Tree23.thy ex/Unification.thy			\
    1.26 +  ex/While_Combinator_Example.thy ex/document/root.bib			\
    1.27 +  ex/document/root.tex ex/set.thy ex/svc_funcs.ML ex/svc_test.thy	\
    1.28 +  ../Tools/interpretation_with_defs.ML
    1.29  	@$(ISABELLE_TOOL) usedir $(OUT)/HOL ex
    1.30  
    1.31  

     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     2.2 +++ b/src/HOL/ex/Interpretation_with_Defs.thy	Sat Jan 15 20:05:29 2011 +0100
     2.3 @@ -0,0 +1,12 @@
     2.4 +(*  Title:      HOL/ex/Interpretation_with_Defs.thy
     2.5 +    Author:     Florian Haftmann, TU Muenchen
     2.6 +*)
     2.7 +
     2.8 +header {* Interpretation accompanied with mixin definitions.  EXPERIMENTAL. *}
     2.9 +
    2.10 +theory Interpretation_with_Defs
    2.11 +imports Pure
    2.12 +uses "~~/src/Tools/interpretation_with_defs.ML"
    2.13 +begin
    2.14 +
    2.15 +end

     3.1 --- a/src/HOL/ex/ROOT.ML	Sat Jan 15 18:49:42 2011 +0100
     3.2 +++ b/src/HOL/ex/ROOT.ML	Sat Jan 15 20:05:29 2011 +0100
     3.3 @@ -72,7 +72,8 @@
     3.4    "Dedekind_Real",
     3.5    "Quicksort",
     3.6    "Birthday_Paradoxon",
     3.7 -  "List_to_Set_Comprehension_Examples"
     3.8 +  "List_to_Set_Comprehension_Examples",
     3.9 +  "Set_Algebras"
    3.10  ];
    3.11  
    3.12  use_thy "SVC_Oracle";

     4.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     4.2 +++ b/src/HOL/ex/Set_Algebras.thy	Sat Jan 15 20:05:29 2011 +0100
     4.3 @@ -0,0 +1,369 @@
     4.4 +(*  Title:      HOL/ex/Set_Algebras.thy
     4.5 +    Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
     4.6 +*)
     4.7 +
     4.8 +header {* Algebraic operations on sets *}
     4.9 +
    4.10 +theory Set_Algebras
    4.11 +imports Main Interpretation_with_Defs
    4.12 +begin
    4.13 +
    4.14 +text {*
    4.15 +  This library lifts operations like addition and muliplication to
    4.16 +  sets.  It was designed to support asymptotic calculations. See the
    4.17 +  comments at the top of theory @{text BigO}.
    4.18 +*}
    4.19 +
    4.20 +definition set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<oplus>" 65) where
    4.21 +  "A \<oplus> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
    4.22 +
    4.23 +definition set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<otimes>" 70) where
    4.24 +  "A \<otimes> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
    4.25 +
    4.26 +definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
    4.27 +  "a +o B = {c. \<exists>b\<in>B. c = a + b}"
    4.28 +
    4.29 +definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
    4.30 +  "a *o B = {c. \<exists>b\<in>B. c = a * b}"
    4.31 +
    4.32 +abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
    4.33 +  "x =o A \<equiv> x \<in> A"
    4.34 +
    4.35 +interpretation set_add!: semigroup "set_plus :: 'a::semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
    4.36 +qed (force simp add: set_plus_def add.assoc)
    4.37 +
    4.38 +interpretation set_add!: abel_semigroup "set_plus :: 'a::ab_semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
    4.39 +qed (force simp add: set_plus_def add.commute)
    4.40 +
    4.41 +interpretation set_add!: monoid "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
    4.42 +qed (simp_all add: set_plus_def)
    4.43 +
    4.44 +interpretation set_add!: comm_monoid "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
    4.45 +qed (simp add: set_plus_def)
    4.46 +
    4.47 +interpretation set_add!: monoid_add "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
    4.48 +  defines listsum_set is set_add.listsum
    4.49 +proof
    4.50 +qed (simp_all add: set_add.assoc)
    4.51 +
    4.52 +interpretation set_add!: comm_monoid_add "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
    4.53 +  defines setsum_set is set_add.setsum
    4.54 +  where "monoid_add.listsum set_plus {0::'a} = listsum_set"
    4.55 +proof -
    4.56 +  show "class.comm_monoid_add (set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {0}" proof
    4.57 +  qed (simp_all add: set_add.commute)
    4.58 +  then interpret set_add!: comm_monoid_add "set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" .
    4.59 +  show "monoid_add.listsum set_plus {0::'a} = listsum_set"
    4.60 +    by (simp only: listsum_set_def)
    4.61 +qed
    4.62 +
    4.63 +interpretation set_mult!: semigroup "set_times :: 'a::semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
    4.64 +qed (force simp add: set_times_def mult.assoc)
    4.65 +
    4.66 +interpretation set_mult!: abel_semigroup "set_times :: 'a::ab_semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
    4.67 +qed (force simp add: set_times_def mult.commute)
    4.68 +
    4.69 +interpretation set_mult!: monoid "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
    4.70 +qed (simp_all add: set_times_def)
    4.71 +
    4.72 +interpretation set_mult!: comm_monoid "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
    4.73 +qed (simp add: set_times_def)
    4.74 +
    4.75 +interpretation set_mult!: monoid_mult "{1}" "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
    4.76 +  defines power_set is set_mult.power
    4.77 +proof
    4.78 +qed (simp_all add: set_mult.assoc)
    4.79 +
    4.80 +interpretation set_mult!: comm_monoid_mult "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
    4.81 +  defines setprod_set is set_mult.setprod
    4.82 +  where "power.power {1} set_times = power_set"
    4.83 +proof -
    4.84 +  show "class.comm_monoid_mult (set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {1}" proof
    4.85 +  qed (simp_all add: set_mult.commute)
    4.86 +  then interpret set_mult!: comm_monoid_mult "set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" .
    4.87 +  show "power.power {1} set_times = power_set"
    4.88 +    by (simp add: power_set_def)
    4.89 +qed
    4.90 +
    4.91 +lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
    4.92 +  by (auto simp add: set_plus_def)
    4.93 +
    4.94 +lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
    4.95 +  by (auto simp add: elt_set_plus_def)
    4.96 +
    4.97 +lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus>
    4.98 +    (b +o D) = (a + b) +o (C \<oplus> D)"
    4.99 +  apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
   4.100 +   apply (rule_tac x = "ba + bb" in exI)
   4.101 +  apply (auto simp add: add_ac)
   4.102 +  apply (rule_tac x = "aa + a" in exI)
   4.103 +  apply (auto simp add: add_ac)
   4.104 +  done
   4.105 +
   4.106 +lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
   4.107 +    (a + b) +o C"
   4.108 +  by (auto simp add: elt_set_plus_def add_assoc)
   4.109 +
   4.110 +lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C =
   4.111 +    a +o (B \<oplus> C)"
   4.112 +  apply (auto simp add: elt_set_plus_def set_plus_def)
   4.113 +   apply (blast intro: add_ac)
   4.114 +  apply (rule_tac x = "a + aa" in exI)
   4.115 +  apply (rule conjI)
   4.116 +   apply (rule_tac x = "aa" in bexI)
   4.117 +    apply auto
   4.118 +  apply (rule_tac x = "ba" in bexI)
   4.119 +   apply (auto simp add: add_ac)
   4.120 +  done
   4.121 +
   4.122 +theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
   4.123 +    a +o (C \<oplus> D)"
   4.124 +  apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus_def add_ac)
   4.125 +   apply (rule_tac x = "aa + ba" in exI)
   4.126 +   apply (auto simp add: add_ac)
   4.127 +  done
   4.128 +
   4.129 +theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
   4.130 +  set_plus_rearrange3 set_plus_rearrange4
   4.131 +
   4.132 +lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
   4.133 +  by (auto simp add: elt_set_plus_def)
   4.134 +
   4.135 +lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
   4.136 +    C \<oplus> E <= D \<oplus> F"
   4.137 +  by (auto simp add: set_plus_def)
   4.138 +
   4.139 +lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
   4.140 +  by (auto simp add: elt_set_plus_def set_plus_def)
   4.141 +
   4.142 +lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
   4.143 +    a +o D <= D \<oplus> C"
   4.144 +  by (auto simp add: elt_set_plus_def set_plus_def add_ac)
   4.145 +
   4.146 +lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
   4.147 +  apply (subgoal_tac "a +o B <= a +o D")
   4.148 +   apply (erule order_trans)
   4.149 +   apply (erule set_plus_mono3)
   4.150 +  apply (erule set_plus_mono)
   4.151 +  done
   4.152 +
   4.153 +lemma set_plus_mono_b: "C <= D ==> x : a +o C
   4.154 +    ==> x : a +o D"
   4.155 +  apply (frule set_plus_mono)
   4.156 +  apply auto
   4.157 +  done
   4.158 +
   4.159 +lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
   4.160 +    x : D \<oplus> F"
   4.161 +  apply (frule set_plus_mono2)
   4.162 +   prefer 2
   4.163 +   apply force
   4.164 +  apply assumption
   4.165 +  done
   4.166 +
   4.167 +lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D"
   4.168 +  apply (frule set_plus_mono3)
   4.169 +  apply auto
   4.170 +  done
   4.171 +
   4.172 +lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
   4.173 +    x : a +o D ==> x : D \<oplus> C"
   4.174 +  apply (frule set_plus_mono4)
   4.175 +  apply auto
   4.176 +  done
   4.177 +
   4.178 +lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
   4.179 +  by (auto simp add: elt_set_plus_def)
   4.180 +
   4.181 +lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
   4.182 +  apply (auto intro!: subsetI simp add: set_plus_def)
   4.183 +  apply (rule_tac x = 0 in bexI)
   4.184 +   apply (rule_tac x = x in bexI)
   4.185 +    apply (auto simp add: add_ac)
   4.186 +  done
   4.187 +
   4.188 +lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
   4.189 +  by (auto simp add: elt_set_plus_def add_ac diff_minus)
   4.190 +
   4.191 +lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
   4.192 +  apply (auto simp add: elt_set_plus_def add_ac diff_minus)
   4.193 +  apply (subgoal_tac "a = (a + - b) + b")
   4.194 +   apply (rule bexI, assumption, assumption)
   4.195 +  apply (auto simp add: add_ac)
   4.196 +  done
   4.197 +
   4.198 +lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
   4.199 +  by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
   4.200 +    assumption)
   4.201 +
   4.202 +lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
   4.203 +  by (auto simp add: set_times_def)
   4.204 +
   4.205 +lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
   4.206 +  by (auto simp add: elt_set_times_def)
   4.207 +
   4.208 +lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
   4.209 +    (b *o D) = (a * b) *o (C \<otimes> D)"
   4.210 +  apply (auto simp add: elt_set_times_def set_times_def)
   4.211 +   apply (rule_tac x = "ba * bb" in exI)
   4.212 +   apply (auto simp add: mult_ac)
   4.213 +  apply (rule_tac x = "aa * a" in exI)
   4.214 +  apply (auto simp add: mult_ac)
   4.215 +  done
   4.216 +
   4.217 +lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
   4.218 +    (a * b) *o C"
   4.219 +  by (auto simp add: elt_set_times_def mult_assoc)
   4.220 +
   4.221 +lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
   4.222 +    a *o (B \<otimes> C)"
   4.223 +  apply (auto simp add: elt_set_times_def set_times_def)
   4.224 +   apply (blast intro: mult_ac)
   4.225 +  apply (rule_tac x = "a * aa" in exI)
   4.226 +  apply (rule conjI)
   4.227 +   apply (rule_tac x = "aa" in bexI)
   4.228 +    apply auto
   4.229 +  apply (rule_tac x = "ba" in bexI)
   4.230 +   apply (auto simp add: mult_ac)
   4.231 +  done
   4.232 +
   4.233 +theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
   4.234 +    a *o (C \<otimes> D)"
   4.235 +  apply (auto intro!: subsetI simp add: elt_set_times_def set_times_def
   4.236 +    mult_ac)
   4.237 +   apply (rule_tac x = "aa * ba" in exI)
   4.238 +   apply (auto simp add: mult_ac)
   4.239 +  done
   4.240 +
   4.241 +theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
   4.242 +  set_times_rearrange3 set_times_rearrange4
   4.243 +
   4.244 +lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
   4.245 +  by (auto simp add: elt_set_times_def)
   4.246 +
   4.247 +lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
   4.248 +    C \<otimes> E <= D \<otimes> F"
   4.249 +  by (auto simp add: set_times_def)
   4.250 +
   4.251 +lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
   4.252 +  by (auto simp add: elt_set_times_def set_times_def)
   4.253 +
   4.254 +lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
   4.255 +    a *o D <= D \<otimes> C"
   4.256 +  by (auto simp add: elt_set_times_def set_times_def mult_ac)
   4.257 +
   4.258 +lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
   4.259 +  apply (subgoal_tac "a *o B <= a *o D")
   4.260 +   apply (erule order_trans)
   4.261 +   apply (erule set_times_mono3)
   4.262 +  apply (erule set_times_mono)
   4.263 +  done
   4.264 +
   4.265 +lemma set_times_mono_b: "C <= D ==> x : a *o C
   4.266 +    ==> x : a *o D"
   4.267 +  apply (frule set_times_mono)
   4.268 +  apply auto
   4.269 +  done
   4.270 +
   4.271 +lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
   4.272 +    x : D \<otimes> F"
   4.273 +  apply (frule set_times_mono2)
   4.274 +   prefer 2
   4.275 +   apply force
   4.276 +  apply assumption
   4.277 +  done
   4.278 +
   4.279 +lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
   4.280 +  apply (frule set_times_mono3)
   4.281 +  apply auto
   4.282 +  done
   4.283 +
   4.284 +lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
   4.285 +    x : a *o D ==> x : D \<otimes> C"
   4.286 +  apply (frule set_times_mono4)
   4.287 +  apply auto
   4.288 +  done
   4.289 +
   4.290 +lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
   4.291 +  by (auto simp add: elt_set_times_def)
   4.292 +
   4.293 +lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
   4.294 +    (a * b) +o (a *o C)"
   4.295 +  by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
   4.296 +
   4.297 +lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
   4.298 +    (a *o B) \<oplus> (a *o C)"
   4.299 +  apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
   4.300 +   apply blast
   4.301 +  apply (rule_tac x = "b + bb" in exI)
   4.302 +  apply (auto simp add: ring_distribs)
   4.303 +  done
   4.304 +
   4.305 +lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
   4.306 +    a *o D \<oplus> C \<otimes> D"
   4.307 +  apply (auto intro!: subsetI simp add:
   4.308 +    elt_set_plus_def elt_set_times_def set_times_def
   4.309 +    set_plus_def ring_distribs)
   4.310 +  apply auto
   4.311 +  done
   4.312 +
   4.313 +theorems set_times_plus_distribs =
   4.314 +  set_times_plus_distrib
   4.315 +  set_times_plus_distrib2
   4.316 +
   4.317 +lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
   4.318 +    - a : C"
   4.319 +  by (auto simp add: elt_set_times_def)
   4.320 +
   4.321 +lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
   4.322 +    - a : (- 1) *o C"
   4.323 +  by (auto simp add: elt_set_times_def)
   4.324 +
   4.325 +lemma set_plus_image:
   4.326 +  fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
   4.327 +  unfolding set_plus_def by (fastsimp simp: image_iff)
   4.328 +
   4.329 +lemma set_setsum_alt:
   4.330 +  assumes fin: "finite I"
   4.331 +  shows "setsum_set S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
   4.332 +    (is "_ = ?setsum I")
   4.333 +using fin proof induct
   4.334 +  case (insert x F)
   4.335 +  have "setsum_set S (insert x F) = S x \<oplus> ?setsum F"
   4.336 +    using insert.hyps by auto
   4.337 +  also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
   4.338 +    unfolding set_plus_def
   4.339 +  proof safe
   4.340 +    fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
   4.341 +    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)"
   4.342 +      using insert.hyps
   4.343 +      by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
   4.344 +  qed auto
   4.345 +  finally show ?case
   4.346 +    using insert.hyps by auto
   4.347 +qed auto
   4.348 +
   4.349 +lemma setsum_set_cond_linear:
   4.350 +  fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
   4.351 +  assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A \<oplus> B)" "P {0}"
   4.352 +    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
   4.353 +  assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
   4.354 +  shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
   4.355 +proof cases
   4.356 +  assume "finite I" from this all show ?thesis
   4.357 +  proof induct
   4.358 +    case (insert x F)
   4.359 +    from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum_set S F)"
   4.360 +      by induct auto
   4.361 +    with insert show ?case
   4.362 +      by (simp, subst f) auto
   4.363 +  qed (auto intro!: f)
   4.364 +qed (auto intro!: f)
   4.365 +
   4.366 +lemma setsum_set_linear:
   4.367 +  fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
   4.368 +  assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
   4.369 +  shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
   4.370 +  using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
   4.371 +
   4.372 +end

     5.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     5.2 +++ b/src/Tools/interpretation_with_defs.ML	Sat Jan 15 20:05:29 2011 +0100
     5.3 @@ -0,0 +1,96 @@
     5.4 +(*  Title:      Tools/interpretation_with_defs.ML
     5.5 +    Author:     Florian Haftmann, TU Muenchen
     5.6 +
     5.7 +Interpretation accompanied with mixin definitions.  EXPERIMENTAL.
     5.8 +*)
     5.9 +
    5.10 +signature INTERPRETATION_WITH_DEFS =
    5.11 +sig
    5.12 +  val interpretation: Expression.expression_i ->
    5.13 +    (Attrib.binding * ((binding * mixfix) * term)) list -> (Attrib.binding * term) list ->
    5.14 +    theory -> Proof.state
    5.15 +  val interpretation_cmd: Expression.expression ->
    5.16 +    (Attrib.binding * ((binding * mixfix) * string)) list -> (Attrib.binding * string) list ->
    5.17 +    theory -> Proof.state
    5.18 +end;
    5.19 +
    5.20 +structure Interpretation_With_Defs : INTERPRETATION_WITH_DEFS =
    5.21 +struct
    5.22 +
    5.23 +fun note_eqns_register deps witss def_eqns attrss eqns export export' context =
    5.24 +  let
    5.25 +    fun meta_rewrite context =
    5.26 +      map (Local_Defs.meta_rewrite_rule (Context.proof_of context) #> Drule.abs_def) o
    5.27 +        maps snd;
    5.28 +  in
    5.29 +    context
    5.30 +    |> Element.generic_note_thmss Thm.lemmaK
    5.31 +      (attrss ~~ map (fn eqn => [([Morphism.thm (export' $> export) eqn], [])]) eqns)
    5.32 +    |-> (fn facts => `(fn context => meta_rewrite context facts))
    5.33 +    |-> (fn eqns => fold (fn ((dep, morph), wits) =>
    5.34 +      fn context =>
    5.35 +        Locale.add_registration (dep, morph $> Element.satisfy_morphism
    5.36 +            (map (Element.morph_witness export') wits))
    5.37 +          (Element.eq_morphism (Context.theory_of context) (def_eqns @ eqns) |>
    5.38 +            Option.map (rpair true))
    5.39 +          export context) (deps ~~ witss))
    5.40 +  end;
    5.41 +
    5.42 +local
    5.43 +
    5.44 +fun gen_interpretation prep_expr prep_decl parse_term parse_prop prep_attr
    5.45 +    expression raw_defs raw_eqns theory =
    5.46 +  let
    5.47 +    val (_, (_, defs_ctxt)) =
    5.48 +      prep_decl expression I [] (ProofContext.init_global theory);
    5.49 +
    5.50 +    val rhss = map (parse_term defs_ctxt o snd o snd) raw_defs
    5.51 +      |> Syntax.check_terms defs_ctxt;
    5.52 +    val defs = map2 (fn (binding_thm, (binding_syn, _)) => fn rhs =>
    5.53 +      (binding_syn, (binding_thm, rhs))) raw_defs rhss;
    5.54 +
    5.55 +    val (def_eqns, theory') = theory
    5.56 +      |> Named_Target.theory_init
    5.57 +      |> fold_map (Local_Theory.define) defs
    5.58 +      |>> map (Thm.symmetric o snd o snd)
    5.59 +      |> Local_Theory.exit_result_global (map o Morphism.thm);
    5.60 +
    5.61 +    val ((propss, deps, export), expr_ctxt) = theory'
    5.62 +      |> ProofContext.init_global
    5.63 +      |> prep_expr expression;
    5.64 +
    5.65 +    val eqns = map (parse_prop expr_ctxt o snd) raw_eqns
    5.66 +      |> Syntax.check_terms expr_ctxt;
    5.67 +    val attrss = map ((apsnd o map) (prep_attr theory) o fst) raw_eqns;
    5.68 +    val goal_ctxt = fold Variable.auto_fixes eqns expr_ctxt;
    5.69 +    val export' = Variable.export_morphism goal_ctxt expr_ctxt;
    5.70 +
    5.71 +    fun after_qed witss eqns =
    5.72 +      (ProofContext.background_theory o Context.theory_map)
    5.73 +        (note_eqns_register deps witss def_eqns attrss eqns export export');
    5.74 +
    5.75 +  in Element.witness_proof_eqs after_qed propss eqns goal_ctxt end;
    5.76 +
    5.77 +in
    5.78 +
    5.79 +fun interpretation x = gen_interpretation Expression.cert_goal_expression
    5.80 +  Expression.cert_declaration (K I) (K I) (K I) x;
    5.81 +fun interpretation_cmd x = gen_interpretation Expression.read_goal_expression
    5.82 +  Expression.read_declaration Syntax.parse_term Syntax.parse_prop Attrib.intern_src x;
    5.83 +
    5.84 +end;
    5.85 +
    5.86 +val definesK = "defines";
    5.87 +val _ = Keyword.keyword definesK;
    5.88 +
    5.89 +val _ =
    5.90 +  Outer_Syntax.command "interpretation"
    5.91 +    "prove interpretation of locale expression in theory" Keyword.thy_goal
    5.92 +    (Parse.!!! (Parse_Spec.locale_expression true) --
    5.93 +      Scan.optional (Parse.$$$ definesK |-- Parse.and_list1 (Parse_Spec.opt_thm_name ":"
    5.94 +        -- ((Parse.binding -- Parse.opt_mixfix') --| Parse.$$$ "is" -- Parse.term))) [] --
    5.95 +      Scan.optional (Parse.where_ |-- Parse.and_list1 (Parse_Spec.opt_thm_name ":" -- Parse.prop)) []
    5.96 +      >> (fn ((expr, defs), equations) => Toplevel.print o
    5.97 +          Toplevel.theory_to_proof (interpretation_cmd expr defs equations)));
    5.98 +
    5.99 +end;