--- a/src/HOL/IsaMakefile Sat Apr 14 15:08:59 2012 +0100
+++ b/src/HOL/IsaMakefile Sat Apr 14 19:29:31 2012 +0200
@@ -1027,7 +1027,7 @@
ex/Quicksort.thy ex/ROOT.ML \
ex/Records.thy ex/ReflectionEx.thy ex/Refute_Examples.thy \
ex/SAT_Examples.thy ex/Serbian.thy ex/Set_Theory.thy \
- ex/Set_Algebras.thy ex/Simproc_Tests.thy ex/SVC_Oracle.thy \
+ ex/Simproc_Tests.thy ex/SVC_Oracle.thy \
ex/sledgehammer_tactics.ML ex/Seq.thy ex/Sqrt.thy ex/Sqrt_Script.thy \
ex/Sudoku.thy ex/Tarski.thy ex/Termination.thy ex/Transfer_Ex.thy \
ex/Tree23.thy ex/Unification.thy ex/While_Combinator_Example.thy \
--- a/src/HOL/ex/ROOT.ML Sat Apr 14 15:08:59 2012 +0100
+++ b/src/HOL/ex/ROOT.ML Sat Apr 14 19:29:31 2012 +0200
@@ -67,7 +67,6 @@
"Quicksort",
"Birthday_Paradox",
"List_to_Set_Comprehension_Examples",
- "Set_Algebras",
"Seq",
"Simproc_Tests",
"Executable_Relation"
--- a/src/HOL/ex/Set_Algebras.thy Sat Apr 14 15:08:59 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,371 +0,0 @@
-(* Title: HOL/ex/Set_Algebras.thy
- Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
-*)
-
-header {* Algebraic operations on sets *}
-
-theory Set_Algebras
-imports Main Interpretation_with_Defs
-begin
-
-text {*
- This library lifts operations like addition and muliplication to
- sets. It was designed to support asymptotic calculations. See the
- comments at the top of theory @{text BigO}.
-*}
-
-definition set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<oplus>" 65) where
- "A \<oplus> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
-
-definition set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<otimes>" 70) where
- "A \<otimes> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
-
-definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70) where
- "a +o B = {c. \<exists>b\<in>B. c = a + b}"
-
-definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80) where
- "a *o B = {c. \<exists>b\<in>B. c = a * b}"
-
-abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50) where
- "x =o A \<equiv> x \<in> A"
-
-interpretation set_add!: semigroup "set_plus :: 'a::semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set"
- by default (force simp add: set_plus_def add.assoc)
-
-interpretation set_add!: abel_semigroup "set_plus :: 'a::ab_semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set"
- by default (force simp add: set_plus_def add.commute)
-
-interpretation set_add!: monoid "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
- by default (simp_all add: set_plus_def)
-
-interpretation set_add!: comm_monoid "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
- by default (simp add: set_plus_def)
-
-interpretation set_add!: monoid_add "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
- defines listsum_set is set_add.listsum
- by default (simp_all add: set_add.assoc)
-
-interpretation
- set_add!: comm_monoid_add "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
- defines setsum_set is set_add.setsum
- where "monoid_add.listsum set_plus {0::'a} = listsum_set"
-proof -
- show "class.comm_monoid_add (set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {0}"
- by default (simp_all add: set_add.commute)
- then interpret set_add!: comm_monoid_add "set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" .
- show "monoid_add.listsum set_plus {0::'a} = listsum_set"
- by (simp only: listsum_set_def)
-qed
-
-interpretation set_mult!: semigroup "set_times :: 'a::semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
- by default (force simp add: set_times_def mult.assoc)
-
-interpretation
- set_mult!: abel_semigroup "set_times :: 'a::ab_semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
- by default (force simp add: set_times_def mult.commute)
-
-interpretation set_mult!: monoid "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
- by default (simp_all add: set_times_def)
-
-interpretation
- set_mult!: comm_monoid "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
- by default (simp add: set_times_def)
-
-interpretation set_mult!: monoid_mult "{1}" "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
- defines power_set is set_mult.power
- by default (simp_all add: set_mult.assoc)
-
-interpretation
- set_mult!: comm_monoid_mult "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
- defines setprod_set is set_mult.setprod
- where "power.power {1} set_times = power_set"
-proof -
- show "class.comm_monoid_mult (set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {1}"
- by default (simp_all add: set_mult.commute)
- then interpret set_mult!: comm_monoid_mult "set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" .
- show "power.power {1} set_times = power_set"
- by (simp add: power_set_def)
-qed
-
-lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
- by (auto simp add: set_plus_def)
-
-lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
- by (auto simp add: elt_set_plus_def)
-
-lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus> (b +o D) = (a + b) +o (C \<oplus> D)"
- apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
- apply (rule_tac x = "ba + bb" in exI)
- apply (auto simp add: add_ac)
- apply (rule_tac x = "aa + a" in exI)
- apply (auto simp add: add_ac)
- done
-
-lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
- by (auto simp add: elt_set_plus_def add_assoc)
-
-lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C = a +o (B \<oplus> C)"
- apply (auto simp add: elt_set_plus_def set_plus_def)
- apply (blast intro: add_ac)
- apply (rule_tac x = "a + aa" in exI)
- apply (rule conjI)
- apply (rule_tac x = "aa" in bexI)
- apply auto
- apply (rule_tac x = "ba" in bexI)
- apply (auto simp add: add_ac)
- done
-
-theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) = a +o (C \<oplus> D)"
- apply (auto intro!: simp add: elt_set_plus_def set_plus_def add_ac)
- apply (rule_tac x = "aa + ba" in exI)
- apply (auto simp add: add_ac)
- done
-
-theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
- set_plus_rearrange3 set_plus_rearrange4
-
-lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
- by (auto simp add: elt_set_plus_def)
-
-lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
- C \<oplus> E <= D \<oplus> F"
- by (auto simp add: set_plus_def)
-
-lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
- by (auto simp add: elt_set_plus_def set_plus_def)
-
-lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
- a +o D <= D \<oplus> C"
- by (auto simp add: elt_set_plus_def set_plus_def add_ac)
-
-lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
- apply (subgoal_tac "a +o B <= a +o D")
- apply (erule order_trans)
- apply (erule set_plus_mono3)
- apply (erule set_plus_mono)
- done
-
-lemma set_plus_mono_b: "C <= D ==> x : a +o C
- ==> x : a +o D"
- apply (frule set_plus_mono)
- apply auto
- done
-
-lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
- x : D \<oplus> F"
- apply (frule set_plus_mono2)
- prefer 2
- apply force
- apply assumption
- done
-
-lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D"
- apply (frule set_plus_mono3)
- apply auto
- done
-
-lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
- x : a +o D ==> x : D \<oplus> C"
- apply (frule set_plus_mono4)
- apply auto
- done
-
-lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
- by (auto simp add: elt_set_plus_def)
-
-lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
- apply (auto intro!: simp add: set_plus_def)
- apply (rule_tac x = 0 in bexI)
- apply (rule_tac x = x in bexI)
- apply (auto simp add: add_ac)
- done
-
-lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
- by (auto simp add: elt_set_plus_def add_ac diff_minus)
-
-lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
- apply (auto simp add: elt_set_plus_def add_ac diff_minus)
- apply (subgoal_tac "a = (a + - b) + b")
- apply (rule bexI, assumption, assumption)
- apply (auto simp add: add_ac)
- done
-
-lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
- by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
- assumption)
-
-lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
- by (auto simp add: set_times_def)
-
-lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
- by (auto simp add: elt_set_times_def)
-
-lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
- (b *o D) = (a * b) *o (C \<otimes> D)"
- apply (auto simp add: elt_set_times_def set_times_def)
- apply (rule_tac x = "ba * bb" in exI)
- apply (auto simp add: mult_ac)
- apply (rule_tac x = "aa * a" in exI)
- apply (auto simp add: mult_ac)
- done
-
-lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
- (a * b) *o C"
- by (auto simp add: elt_set_times_def mult_assoc)
-
-lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
- a *o (B \<otimes> C)"
- apply (auto simp add: elt_set_times_def set_times_def)
- apply (blast intro: mult_ac)
- apply (rule_tac x = "a * aa" in exI)
- apply (rule conjI)
- apply (rule_tac x = "aa" in bexI)
- apply auto
- apply (rule_tac x = "ba" in bexI)
- apply (auto simp add: mult_ac)
- done
-
-theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
- a *o (C \<otimes> D)"
- apply (auto intro!: simp add: elt_set_times_def set_times_def
- mult_ac)
- apply (rule_tac x = "aa * ba" in exI)
- apply (auto simp add: mult_ac)
- done
-
-theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
- set_times_rearrange3 set_times_rearrange4
-
-lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
- by (auto simp add: elt_set_times_def)
-
-lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
- C \<otimes> E <= D \<otimes> F"
- by (auto simp add: set_times_def)
-
-lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
- by (auto simp add: elt_set_times_def set_times_def)
-
-lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
- a *o D <= D \<otimes> C"
- by (auto simp add: elt_set_times_def set_times_def mult_ac)
-
-lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
- apply (subgoal_tac "a *o B <= a *o D")
- apply (erule order_trans)
- apply (erule set_times_mono3)
- apply (erule set_times_mono)
- done
-
-lemma set_times_mono_b: "C <= D ==> x : a *o C
- ==> x : a *o D"
- apply (frule set_times_mono)
- apply auto
- done
-
-lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
- x : D \<otimes> F"
- apply (frule set_times_mono2)
- prefer 2
- apply force
- apply assumption
- done
-
-lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
- apply (frule set_times_mono3)
- apply auto
- done
-
-lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
- x : a *o D ==> x : D \<otimes> C"
- apply (frule set_times_mono4)
- apply auto
- done
-
-lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
- by (auto simp add: elt_set_times_def)
-
-lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
- (a * b) +o (a *o C)"
- by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
-
-lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
- (a *o B) \<oplus> (a *o C)"
- apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
- apply blast
- apply (rule_tac x = "b + bb" in exI)
- apply (auto simp add: ring_distribs)
- done
-
-lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
- a *o D \<oplus> C \<otimes> D"
- apply (auto simp add:
- elt_set_plus_def elt_set_times_def set_times_def
- set_plus_def ring_distribs)
- apply auto
- done
-
-theorems set_times_plus_distribs =
- set_times_plus_distrib
- set_times_plus_distrib2
-
-lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
- - a : C"
- by (auto simp add: elt_set_times_def)
-
-lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
- - a : (- 1) *o C"
- by (auto simp add: elt_set_times_def)
-
-lemma set_plus_image:
- fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
- unfolding set_plus_def by (fastforce simp: image_iff)
-
-lemma set_setsum_alt:
- assumes fin: "finite I"
- shows "setsum_set S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
- (is "_ = ?setsum I")
-using fin
-proof induct
- case (insert x F)
- have "setsum_set S (insert x F) = S x \<oplus> ?setsum F"
- using insert.hyps by auto
- also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
- proof -
- {
- fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
- then have "\<exists>s'. y + setsum s F = s' x + setsum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)"
- using insert.hyps
- by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
- }
- then show ?thesis
- unfolding set_plus_def by auto
- qed
- finally show ?case
- using insert.hyps by auto
-qed auto
-
-lemma setsum_set_cond_linear:
- fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
- assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A \<oplus> B)" "P {0}"
- and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
- assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
- shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
-proof cases
- assume "finite I" from this all show ?thesis
- proof induct
- case (insert x F)
- from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum_set S F)"
- by induct auto
- with insert show ?case
- by (simp, subst f) auto
- qed (auto intro!: f)
-qed (auto intro!: f)
-
-lemma setsum_set_linear:
- fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
- assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
- shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
- using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
-
-end