Added Presburger theory.
(* Title: Summation Operator for Abelian Groups
ID: $Id$
Author: Clemens Ballarin, started 19 November 2002
Copyright: TU Muenchen
*)
header {* Summation Operator *}
theory Summation = Group:
(* Instantiation of LC from Finite_Set.thy is not possible,
because here we have explicit typing rules like x : carrier G.
We introduce an explicit argument for the domain D *)
consts
foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
inductive "foldSetD D f e"
intros
emptyI [intro]: "e : D ==> ({}, e) : foldSetD D f e"
insertI [intro]: "[| x ~: A; f x y : D; (A, y) : foldSetD D f e |] ==>
(insert x A, f x y) : foldSetD D f e"
inductive_cases empty_foldSetDE [elim!]: "({}, x) : foldSetD D f e"
constdefs
foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
"foldD D f e A == THE x. (A, x) : foldSetD D f e"
lemma foldSetD_closed:
"[| (A, z) : foldSetD D f e ; e : D; !!x y. [| x : A; y : D |] ==> f x y : D
|] ==> z : D";
by (erule foldSetD.elims) auto
lemma Diff1_foldSetD:
"[| (A - {x}, y) : foldSetD D f e; x : A; f x y : D |] ==>
(A, f x y) : foldSetD D f e"
apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
apply auto
done
lemma foldSetD_imp_finite [simp]: "(A, x) : foldSetD D f e ==> finite A"
by (induct set: foldSetD) auto
lemma finite_imp_foldSetD:
"[| finite A; e : D; !!x y. [| x : A; y : D |] ==> f x y : D |] ==>
EX x. (A, x) : foldSetD D f e"
proof (induct set: Finites)
case empty then show ?case by auto
next
case (insert F x)
then obtain y where y: "(F, y) : foldSetD D f e" by auto
with insert have "y : D" by (auto dest: foldSetD_closed)
with y and insert have "(insert x F, f x y) : foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed
subsection {* Left-commutative operations *}
locale LCD =
fixes B :: "'b set"
and D :: "'a set"
and f :: "'b => 'a => 'a" (infixl "\<cdot>" 70)
assumes left_commute: "[| x : B; y : B; z : D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
and f_closed [simp, intro!]: "!!x y. [| x : B; y : D |] ==> f x y : D"
lemma (in LCD) foldSetD_closed [dest]:
"(A, z) : foldSetD D f e ==> z : D";
by (erule foldSetD.elims) auto
lemma (in LCD) Diff1_foldSetD:
"[| (A - {x}, y) : foldSetD D f e; x : A; A <= B |] ==>
(A, f x y) : foldSetD D f e"
apply (subgoal_tac "x : B")
prefer 2 apply fast
apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
apply auto
done
lemma (in LCD) foldSetD_imp_finite [simp]:
"(A, x) : foldSetD D f e ==> finite A"
by (induct set: foldSetD) auto
lemma (in LCD) finite_imp_foldSetD:
"[| finite A; A <= B; e : D |] ==> EX x. (A, x) : foldSetD D f e"
proof (induct set: Finites)
case empty then show ?case by auto
next
case (insert F x)
then obtain y where y: "(F, y) : foldSetD D f e" by auto
with insert have "y : D" by auto
with y and insert have "(insert x F, f x y) : foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed
lemma (in LCD) foldSetD_determ_aux:
"e : D ==> ALL A x. A <= B & card A < n --> (A, x) : foldSetD D f e -->
(ALL y. (A, y) : foldSetD D f e --> y = x)"
apply (induct n)
apply (auto simp add: less_Suc_eq)
apply (erule foldSetD.cases)
apply blast
apply (erule foldSetD.cases)
apply blast
apply clarify
txt {* force simplification of @{text "card A < card (insert ...)"}. *}
apply (erule rev_mp)
apply (simp add: less_Suc_eq_le)
apply (rule impI)
apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
apply (subgoal_tac "Aa = Ab")
prefer 2 apply (blast elim!: equalityE)
apply blast
txt {* case @{prop "xa \<notin> xb"}. *}
apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab")
prefer 2 apply (blast elim!: equalityE)
apply clarify
apply (subgoal_tac "Aa = insert xb Ab - {xa}")
prefer 2 apply blast
apply (subgoal_tac "card Aa <= card Ab")
prefer 2
apply (rule Suc_le_mono [THEN subst])
apply (simp add: card_Suc_Diff1)
apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
apply (blast intro: foldSetD_imp_finite finite_Diff)
(* new subgoal from finite_imp_foldSetD *)
apply best (* blast doesn't seem to solve this *)
apply assumption
apply (frule (1) Diff1_foldSetD)
(* new subgoal from Diff1_foldSetD *)
apply best
(*
apply (best del: foldSetD_closed elim: foldSetD_closed)
apply (rule f_closed) apply assumption apply (rule foldSetD_closed)
prefer 3 apply assumption apply (rule e_closed)
apply (rule f_closed) apply force apply assumption
*)
apply (subgoal_tac "ya = f xb x")
prefer 2
(* new subgoal to make IH applicable *)
apply (subgoal_tac "Aa <= B")
prefer 2 apply best
apply (blast del: equalityCE)
apply (subgoal_tac "(Ab - {xa}, x) : foldSetD D f e")
prefer 2 apply simp
apply (subgoal_tac "yb = f xa x")
prefer 2
(* apply (drule_tac x = xa in Diff1_foldSetD)
apply assumption
apply (rule f_closed) apply best apply (rule foldSetD_closed)
prefer 3 apply assumption apply (rule e_closed)
apply (rule f_closed) apply best apply assumption
*)
apply (blast del: equalityCE dest: Diff1_foldSetD)
apply (simp (no_asm_simp))
apply (rule left_commute)
apply assumption apply best apply best
done
lemma (in LCD) foldSetD_determ:
"[| (A, x) : foldSetD D f e; (A, y) : foldSetD D f e; e : D; A <= B |]
==> y = x"
by (blast intro: foldSetD_determ_aux [rule_format])
lemma (in LCD) foldD_equality:
"[| (A, y) : foldSetD D f e; e : D; A <= B |] ==> foldD D f e A = y"
by (unfold foldD_def) (blast intro: foldSetD_determ)
lemma foldD_empty [simp]:
"e : D ==> foldD D f e {} = e"
by (unfold foldD_def) blast
lemma (in LCD) foldD_insert_aux:
"[| x ~: A; x : B; e : D; A <= B |] ==>
((insert x A, v) : foldSetD D f e) =
(EX y. (A, y) : foldSetD D f e & v = f x y)"
apply auto
apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
apply (fastsimp dest: foldSetD_imp_finite)
(* new subgoal by finite_imp_foldSetD *)
apply assumption
apply assumption
apply (blast intro: foldSetD_determ)
done
lemma (in LCD) foldD_insert:
"[| finite A; x ~: A; x : B; e : D; A <= B |] ==>
foldD D f e (insert x A) = f x (foldD D f e A)"
apply (unfold foldD_def)
apply (simp add: foldD_insert_aux)
apply (rule the_equality)
apply (auto intro: finite_imp_foldSetD
cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
done
lemma (in LCD) foldD_closed [simp]:
"[| finite A; e : D; A <= B |] ==> foldD D f e A : D"
proof (induct set: Finites)
case empty then show ?case by (simp add: foldD_empty)
next
case insert then show ?case by (simp add: foldD_insert)
qed
lemma (in LCD) foldD_commute:
"[| finite A; x : B; e : D; A <= B |] ==>
f x (foldD D f e A) = foldD D f (f x e) A"
apply (induct set: Finites)
apply simp
apply (auto simp add: left_commute foldD_insert)
done
lemma Int_mono2:
"[| A <= C; B <= C |] ==> A Int B <= C"
by blast
lemma (in LCD) foldD_nest_Un_Int:
"[| finite A; finite C; e : D; A <= B; C <= B |] ==>
foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
apply (induct set: Finites)
apply simp
apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
Int_mono2 Un_subset_iff)
done
lemma (in LCD) foldD_nest_Un_disjoint:
"[| finite A; finite B; A Int B = {}; e : D; A <= B; C <= B |]
==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
by (simp add: foldD_nest_Un_Int)
-- {* Delete rules to do with @{text foldSetD} relation. *}
declare foldSetD_imp_finite [simp del]
empty_foldSetDE [rule del]
foldSetD.intros [rule del]
declare (in LCD)
foldSetD_closed [rule del]
subsection {* Commutative monoids *}
text {*
We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
instead of @{text "'b => 'a => 'a"}.
*}
locale ACeD =
fixes D :: "'a set"
and f :: "'a => 'a => 'a" (infixl "\<cdot>" 70)
and e :: 'a
assumes ident [simp]: "x : D ==> x \<cdot> e = x"
and commute: "[| x : D; y : D |] ==> x \<cdot> y = y \<cdot> x"
and assoc: "[| x : D; y : D; z : D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
and e_closed [simp]: "e : D"
and f_closed [simp]: "[| x : D; y : D |] ==> x \<cdot> y : D"
lemma (in ACeD) left_commute:
"[| x : D; y : D; z : D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
proof -
assume D: "x : D" "y : D" "z : D"
then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
finally show ?thesis .
qed
lemmas (in ACeD) AC = assoc commute left_commute
lemma (in ACeD) left_ident [simp]: "x : D ==> e \<cdot> x = x"
proof -
assume D: "x : D"
have "x \<cdot> e = x" by (rule ident)
with D show ?thesis by (simp add: commute)
qed
lemma (in ACeD) foldD_Un_Int:
"[| finite A; finite B; A <= D; B <= D |] ==>
foldD D f e A \<cdot> foldD D f e B =
foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
apply (induct set: Finites)
apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
(* left_commute is required to show premise of LCD.intro *)
apply (simp add: AC insert_absorb Int_insert_left
LCD.foldD_insert [OF LCD.intro [of D]]
LCD.foldD_closed [OF LCD.intro [of D]]
Int_mono2 Un_subset_iff)
done
lemma (in ACeD) foldD_Un_disjoint:
"[| finite A; finite B; A Int B = {}; A <= D; B <= D |] ==>
foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
by (simp add: foldD_Un_Int
left_commute LCD.foldD_closed [OF LCD.intro [of D]] Un_subset_iff)
subsection {* Abelian groups with summation operator *}
lemma (in abelian_group) sum_lcomm:
"[| x : carrier G; y : carrier G; z : carrier G |] ==>
x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
proof -
assume "x : carrier G" "y : carrier G" "z : carrier G"
then have "x \<oplus> (y \<oplus> z) = (x \<oplus> y) \<oplus> z" by (simp add: sum_assoc)
also from prems have "... = (y \<oplus> x) \<oplus> z" by (simp add: sum_commute)
also from prems have "... = y \<oplus> (x \<oplus> z)" by (simp add: sum_assoc)
finally show ?thesis .
qed
lemmas (in abelian_group) AC = sum_assoc sum_commute sum_lcomm
record ('a, 'b) group_with_sum = "'a group" +
setSum :: "['b => 'a, 'b set] => 'a"
(* TODO: nice syntax for the summation operator inside the locale
like \<Oplus>\<index> i\<in>A. f i, probably needs hand-coded translation *)
locale agroup_with_sum = abelian_group +
assumes setSum_def:
"setSum G f A = (if finite A then foldD (carrier G) (op \<oplus> o f) \<zero> A else \<zero>)"
ML_setup {*
Context.>> (fn thy => (simpset_ref_of thy :=
simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
lemma (in agroup_with_sum) setSum_empty [simp]:
"setSum G f {} = \<zero>"
by (simp add: setSum_def)
ML_setup {*
Context.>> (fn thy => (simpset_ref_of thy :=
simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
lemma insert_conj:
"[| a = b; a : B |] ==> a : insert b B"
by blast
declare funcsetI [intro]
funcset_mem [dest]
lemma (in agroup_with_sum) setSum_insert [simp]:
"[| finite F; a \<notin> F; f : F -> carrier G; f a : carrier G |] ==>
setSum G f (insert a F) = f a \<oplus> setSum G f F"
apply (rule trans)
apply (simp add: setSum_def)
apply (rule trans)
apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
apply simp
apply (rule sum_lcomm)
apply fast apply fast apply assumption
apply (fastsimp intro: sum_closed)
apply simp+ apply fast
apply (auto simp add: setSum_def)
done
lemma (in agroup_with_sum) setSum_0:
"setSum G (%i. \<zero>) A = \<zero>"
(* apply (case_tac "finite A")
prefer 2 apply (simp add: setSum_def) *)
proof (cases "finite A")
case True then show ?thesis
proof (induct set: Finites)
case empty show ?case by simp
next
case (insert A a)
have "(%i. \<zero>) : A -> carrier G" by auto
with insert show ?case by simp
qed
next
case False then show ?thesis by (simp add: setSum_def)
qed
(*
lemma setSum_eq_0_iff [simp]:
"finite F ==> (setSum f F = 0) = (ALL a:F. f a = (0::nat))"
by (induct set: Finites) auto
lemma setSum_SucD: "setSum f A = Suc n ==> EX a:A. 0 < f a"
apply (case_tac "finite A")
prefer 2 apply (simp add: setSum_def)
apply (erule rev_mp)
apply (erule finite_induct)
apply auto
done
lemma card_eq_setSum: "finite A ==> card A = setSum (\<lambda>x. 1) A"
*) -- {* Could allow many @{text "card"} proofs to be simplified. *}
(*
by (induct set: Finites) auto
*)
lemma (in agroup_with_sum) setSum_closed:
"[| finite A; f : A -> carrier G |] ==> setSum G f A : carrier G"
proof (induct set: Finites)
case empty show ?case by simp
next
case (insert A a)
then have a: "f a : carrier G" by fast
from insert have A: "f : A -> carrier G" by fast
from insert A a show ?case by simp
qed
(*
lemma (in agroup_with_sum) setSum_closed:
"[| finite A; f``A <= carrier G |] ==> setSum G f A : carrier G"
lemma (in agroup_with_sum) setSum_closed:
"[| finite A; !!i. i : A ==> f i : carrier G |] ==>
setSum G f A : carrier G"
*)
lemma funcset_Int_left [simp, intro]:
"[| f : A -> C; f : B -> C |] ==> f : A Int B -> C"
by fast
lemma funcset_Int_right:
"(f : A -> B Int C) = (f : A -> B & f : A -> C)"
by fast
lemma funcset_Un_right:
"[| f : A -> B; f : A -> C |] ==> f : A -> B Un C"
by fast
lemma funcset_Un_left [iff]:
"(f : A Un B -> C) = (f : A -> C & f : B -> C)"
by fast
lemma (in agroup_with_sum) setSum_Un_Int:
"[| finite A; finite B; g : A -> carrier G; g : B -> carrier G |] ==>
setSum G g (A Un B) \<oplus> setSum G g (A Int B) = setSum G g A \<oplus> setSum G g B"
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
proof (induct set: Finites)
case empty then show ?case by (simp add: setSum_closed)
next
case (insert A a)
then have a: "g a : carrier G" by fast
from insert have A: "g : A -> carrier G" by fast
from insert A a show ?case
by (simp add: AC Int_insert_left insert_absorb setSum_closed
Int_mono2 Un_subset_iff)
qed
lemma (in agroup_with_sum) setSum_Un_disjoint:
"[| finite A; finite B; A Int B = {};
g : A -> carrier G; g : B -> carrier G |]
==> setSum G g (A Un B) = setSum G g A \<oplus> setSum G g B"
apply (subst setSum_Un_Int [symmetric])
apply (auto simp add: setSum_closed)
done
(*
lemma setSum_UN_disjoint:
fixes f :: "'a => 'b::plus_ac0"
shows
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
setSum f (UNION I A) = setSum (\<lambda>i. setSum f (A i)) I"
apply (induct set: Finites)
apply simp
apply atomize
apply (subgoal_tac "ALL i:F. x \<noteq> i")
prefer 2 apply blast
apply (subgoal_tac "A x Int UNION F A = {}")
prefer 2 apply blast
apply (simp add: setSum_Un_disjoint)
done
*)
lemma (in agroup_with_sum) setSum_addf:
"[| finite A; f : A -> carrier G; g : A -> carrier G |] ==>
setSum G (%x. f x \<oplus> g x) A = (setSum G f A \<oplus> setSum G g A)"
proof (induct set: Finites)
case empty show ?case by simp
next
case (insert A a) then
have fA: "f : A -> carrier G" by fast
from insert have fa: "f a : carrier G" by fast
from insert have gA: "g : A -> carrier G" by fast
from insert have ga: "g a : carrier G" by fast
from insert have fga: "(%x. f x \<oplus> g x) a : carrier G" by (simp add: Pi_def)
from insert have fgA: "(%x. f x \<oplus> g x) : A -> carrier G"
by (simp add: Pi_def)
show ?case (* check if all simps are really necessary *)
by (simp add: insert fA fa gA ga fgA fga AC setSum_closed Int_insert_left insert_absorb Int_mono2 Un_subset_iff)
qed
(*
lemma setSum_Un: "finite A ==> finite B ==>
(setSum f (A Un B) :: nat) = setSum f A + setSum f B - setSum f (A Int B)"
-- {* For the natural numbers, we have subtraction. *}
apply (subst setSum_Un_Int [symmetric])
apply auto
done
lemma setSum_diff1: "(setSum f (A - {a}) :: nat) =
(if a:A then setSum f A - f a else setSum f A)"
apply (case_tac "finite A")
prefer 2 apply (simp add: setSum_def)
apply (erule finite_induct)
apply (auto simp add: insert_Diff_if)
apply (drule_tac a = a in mk_disjoint_insert)
apply auto
done
*)
lemma (in agroup_with_sum) setSum_cong:
"[| A = B; g : B -> carrier G;
!!i. i : B ==> f i = g i |] ==> setSum G f A = setSum G g B"
proof -
assume prems: "A = B" "g : B -> carrier G"
"!!i. i : B ==> f i = g i"
show ?thesis
proof (cases "finite B")
case True
then have "!!A. [| A = B; g : B -> carrier G;
!!i. i : B ==> f i = g i |] ==> setSum G f A = setSum G g B"
proof induct
case empty thus ?case by simp
next
case (insert B x)
then have "setSum G f A = setSum G f (insert x B)" by simp
also from insert have "... = f x \<oplus> setSum G f B"
proof (intro setSum_insert)
show "finite B" .
next
show "x ~: B" .
next
assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
"g \<in> insert x B \<rightarrow> carrier G"
thus "f : B -> carrier G" by fastsimp
next
assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
"g \<in> insert x B \<rightarrow> carrier G"
thus "f x \<in> carrier G" by fastsimp
qed
also from insert have "... = g x \<oplus> setSum G g B" by fastsimp
also from insert have "... = setSum G g (insert x B)"
by (intro setSum_insert [THEN sym]) auto
finally show ?case .
qed
with prems show ?thesis by simp
next
case False with prems show ?thesis by (simp add: setSum_def)
qed
qed
lemma (in agroup_with_sum) setSum_cong1 [cong]:
"[| A = B; !!i. i : B ==> f i = g i;
g : B -> carrier G = True |] ==> setSum G f A = setSum G g B"
by (rule setSum_cong) fast+
text {*Usually, if this rule causes a failed congruence proof error,
the reason is that the premise @{text "g : B -> carrier G"} cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful. *}
declare funcsetI [rule del]
funcset_mem [rule del]
(*** Examples --- Summation over the integer interval {..n} ***)
(* New locale where index set is restricted to nat *)
locale agroup_with_natsum = agroup_with_sum +
assumes "False ==> setSum G f (A::nat set) = setSum G f A"
lemma (in agroup_with_natsum) natSum_0 [simp]:
"f : {0::nat} -> carrier G ==> setSum G f {..0} = f 0"
by (simp add: Pi_def)
lemma (in agroup_with_natsum) natsum_Suc [simp]:
"f : {..Suc n} -> carrier G ==>
setSum G f {..Suc n} = (f (Suc n) \<oplus> setSum G f {..n})"
by (simp add: Pi_def atMost_Suc)
lemma (in agroup_with_natsum) natsum_Suc2:
"f : {..Suc n} -> carrier G ==>
setSum G f {..Suc n} = (setSum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
proof (induct n)
case 0 thus ?case by (simp add: Pi_def)
next
case Suc thus ?case by (simp add: sum_assoc Pi_def setSum_closed)
qed
lemma (in agroup_with_natsum) natsum_zero [simp]:
"setSum G (%i. \<zero>) {..n::nat} = \<zero>"
by (induct n) (simp_all add: Pi_def)
lemma (in agroup_with_natsum) natsum_add [simp]:
"[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
setSum G (%i. f i \<oplus> g i) {..n::nat} = setSum G f {..n} \<oplus> setSum G g {..n}"
by (induct n) (simp_all add: AC Pi_def setSum_closed)
end