Theory FoldSet (Isabelle2005: October 2005)

# Theory FoldSet

Up to index of Isabelle/ZF/Induct

theory FoldSet
imports Main
begin

    (*  Title:      ZF/Induct/FoldSet.thy
        ID:         $Id: FoldSet.thy,v 1.7 2005/06/17 14:15:11 haftmann Exp $
        Author:     Sidi O Ehmety, Cambridge University Computer Laboratory

    A "fold" functional for finite sets.  For n non-negative we have
    fold f e {x1,...,xn} = f x1 (... (f xn e)) where f is at
    least left-commutative.
    *)

    theory FoldSet imports Main begin

    consts fold_set :: "[i, i, [i,i]=>i, i] => i"

    inductive
      domains "fold_set(A, B, f,e)" <= "Fin(A)*B"
      intros
        emptyI: "e∈B ==> <0, e>∈fold_set(A, B, f,e)"
        consI:  "[| x∈A; x ∉C;  <C,y> : fold_set(A, B,f,e); f(x,y):B |]
                    ==>  <cons(x,C), f(x,y)>∈fold_set(A, B, f, e)"
      type_intros Fin.intros

    constdefs

      fold :: "[i, [i,i]=>i, i, i] => i"  ("fold[_]'(_,_,_')")
       "fold[B](f,e, A) == THE x. <A, x>∈fold_set(A, B, f,e)"

       setsum :: "[i=>i, i] => i"
       "setsum(g, C) == if Finite(C) then
                         fold[int](%x y. g(x) $+ y, #0, C) else #0"

    (** foldSet **)

    inductive_cases empty_fold_setE: "<0, x> : fold_set(A, B, f,e)"
    inductive_cases cons_fold_setE: "<cons(x,C), y> : fold_set(A, B, f,e)"

    (* add-hoc lemmas *)

    lemma cons_lemma1: "[| x∉C; x∉B |] ==> cons(x,B)=cons(x,C) <-> B = C"
    by (auto elim: equalityE)

    lemma cons_lemma2: "[| cons(x, B)=cons(y, C); x≠y; x∉B; y∉C |]
        ==>  B - {y} = C-{x} & x∈C & y∈B"
    apply (auto elim: equalityE)
    done

    (* fold_set monotonicity *)
    lemma fold_set_mono_lemma:
         "<C, x> : fold_set(A, B, f, e)
          ==> ALL D. A<=D --> <C, x> : fold_set(D, B, f, e)"
    apply (erule fold_set.induct)
    apply (auto intro: fold_set.intros)
    done

    lemma fold_set_mono: " C<=A ==> fold_set(C, B, f, e) <= fold_set(A, B, f, e)"
    apply clarify
    apply (frule fold_set.dom_subset [THEN subsetD], clarify)
    apply (auto dest: fold_set_mono_lemma)
    done

    lemma fold_set_lemma:
         "<C, x>∈fold_set(A, B, f, e) ==> <C, x>∈fold_set(C, B, f, e) & C<=A"
    apply (erule fold_set.induct)
    apply (auto intro!: fold_set.intros intro: fold_set_mono [THEN subsetD])
    done

    (* Proving that fold_set is deterministic *)
    lemma Diff1_fold_set:
         "[| <C-{x},y> : fold_set(A, B, f,e);  x∈C; x∈A; f(x, y):B |]
          ==> <C, f(x, y)> : fold_set(A, B, f, e)"
    apply (frule fold_set.dom_subset [THEN subsetD])
    apply (erule cons_Diff [THEN subst], rule fold_set.intros, auto)
    done

    locale fold_typing =
     fixes A and B and e and f
     assumes ftype [intro,simp]:  "[|x ∈ A; y ∈ B|] ==> f(x,y) ∈ B"
         and etype [intro,simp]:  "e ∈ B"
         and fcomm:  "[|x ∈ A; y ∈ A; z ∈ B|] ==> f(x, f(y, z))=f(y, f(x, z))"

    lemma (in fold_typing) Fin_imp_fold_set:
         "C∈Fin(A) ==> (EX x. <C, x> : fold_set(A, B, f,e))"
    apply (erule Fin_induct)
    apply (auto dest: fold_set.dom_subset [THEN subsetD]
                intro: fold_set.intros etype ftype)
    done

    lemma Diff_sing_imp:
         "[|C - {b} = D - {a}; a ≠ b; b ∈ C|] ==> C = cons(b,D) - {a}"
    by (blast elim: equalityE)

    lemma (in fold_typing) fold_set_determ_lemma [rule_format]:
    "n∈nat
     ==> ALL C. |C|<n -->
       (ALL x. <C, x> : fold_set(A, B, f,e)-->
               (ALL y. <C, y> : fold_set(A, B, f,e) --> y=x))"
    apply (erule nat_induct)
     apply (auto simp add: le_iff)
    apply (erule fold_set.cases)
     apply (force elim!: empty_fold_setE)
    apply (erule fold_set.cases)
     apply (force elim!: empty_fold_setE, clarify)
    (*force simplification of "|C| < |cons(...)|"*)
    apply (frule_tac a = Ca in fold_set.dom_subset [THEN subsetD, THEN SigmaD1])
    apply (frule_tac a = Cb in fold_set.dom_subset [THEN subsetD, THEN SigmaD1])
    apply (simp add: Fin_into_Finite [THEN Finite_imp_cardinal_cons])
    apply (case_tac "x=xb", auto)
    apply (simp add: cons_lemma1, blast)
    txt{*case @{term "x≠xb"}*}
    apply (drule cons_lemma2, safe)
    apply (frule Diff_sing_imp, assumption+)
    txt{** LEVEL 17*}
    apply (subgoal_tac "|Ca| le |Cb|")
     prefer 2
     apply (rule succ_le_imp_le)
     apply (simp add: Fin_into_Finite Finite_imp_succ_cardinal_Diff
                      Fin_into_Finite [THEN Finite_imp_cardinal_cons])
    apply (rule_tac C1 = "Ca-{xb}" in Fin_imp_fold_set [THEN exE])
     apply (blast intro: Diff_subset [THEN Fin_subset])
    txt{** LEVEL 24 **}
    apply (frule Diff1_fold_set, blast, blast)
    apply (blast dest!: ftype fold_set.dom_subset [THEN subsetD])
    apply (subgoal_tac "ya = f(xb,xa) ")
     prefer 2 apply (blast del: equalityCE)
    apply (subgoal_tac "<Cb-{x}, xa> : fold_set(A,B,f,e)")
     prefer 2 apply simp
    apply (subgoal_tac "yb = f (x, xa) ")
     apply (drule_tac [2] C = Cb in Diff1_fold_set, simp_all)
      apply (blast intro: fcomm dest!: fold_set.dom_subset [THEN subsetD])
     apply (blast intro: ftype dest!: fold_set.dom_subset [THEN subsetD], blast)
    done

    lemma (in fold_typing) fold_set_determ:
         "[| <C, x>∈fold_set(A, B, f, e);
             <C, y>∈fold_set(A, B, f, e)|] ==> y=x"
    apply (frule fold_set.dom_subset [THEN subsetD], clarify)
    apply (drule Fin_into_Finite)
    apply (unfold Finite_def, clarify)
    apply (rule_tac n = "succ (n)" in fold_set_determ_lemma)
    apply (auto intro: eqpoll_imp_lepoll [THEN lepoll_cardinal_le])
    done

    (** The fold function **)

    lemma (in fold_typing) fold_equality:
         "<C,y> : fold_set(A,B,f,e) ==> fold[B](f,e,C) = y"
    apply (unfold fold_def)
    apply (frule fold_set.dom_subset [THEN subsetD], clarify)
    apply (rule the_equality)
     apply (rule_tac [2] A=C in fold_typing.fold_set_determ)
    apply (force dest: fold_set_lemma)
    apply (auto dest: fold_set_lemma)
    apply (simp add: fold_typing_def, auto)
    apply (auto dest: fold_set_lemma intro: ftype etype fcomm)
    done

    lemma fold_0 [simp]: "e : B ==> fold[B](f,e,0) = e"
    apply (unfold fold_def)
    apply (blast elim!: empty_fold_setE intro: fold_set.intros)
    done

    text{*This result is the right-to-left direction of the subsequent result*}
    lemma (in fold_typing) fold_set_imp_cons:
         "[| <C, y> : fold_set(C, B, f, e); C : Fin(A); c : A; c∉C |]
          ==> <cons(c, C), f(c,y)> : fold_set(cons(c, C), B, f, e)"
    apply (frule FinD [THEN fold_set_mono, THEN subsetD])
     apply assumption
    apply (frule fold_set.dom_subset [of A, THEN subsetD])
    apply (blast intro!: fold_set.consI intro: fold_set_mono [THEN subsetD])
    done

    lemma (in fold_typing) fold_cons_lemma [rule_format]:
    "[| C : Fin(A); c : A; c∉C |]
         ==> <cons(c, C), v> : fold_set(cons(c, C), B, f, e) <->
             (EX y. <C, y> : fold_set(C, B, f, e) & v = f(c, y))"
    apply auto
     prefer 2 apply (blast intro: fold_set_imp_cons)
     apply (frule_tac Fin.consI [of c, THEN FinD, THEN fold_set_mono, THEN subsetD], assumption+)
    apply (frule_tac fold_set.dom_subset [of A, THEN subsetD])
    apply (drule FinD)
    apply (rule_tac A1 = "cons(c,C)" and f1=f and B1=B and C1=C and e1=e in fold_typing.Fin_imp_fold_set [THEN exE])
    apply (blast intro: fold_typing.intro ftype etype fcomm)
    apply (blast intro: Fin_subset [of _ "cons(c,C)"] Finite_into_Fin
                 dest: Fin_into_Finite)
    apply (rule_tac x = x in exI)
    apply (auto intro: fold_set.intros)
    apply (drule_tac fold_set_lemma [of C], blast)
    apply (blast intro!: fold_set.consI
                 intro: fold_set_determ fold_set_mono [THEN subsetD]
                 dest: fold_set.dom_subset [THEN subsetD])
    done

    lemma (in fold_typing) fold_cons:
         "[| C∈Fin(A); c∈A; c∉C|]
          ==> fold[B](f, e, cons(c, C)) = f(c, fold[B](f, e, C))"
    apply (unfold fold_def)
    apply (simp add: fold_cons_lemma)
    apply (rule the_equality, auto)
     apply (subgoal_tac [2] "⟨C, y⟩ ∈ fold_set(A, B, f, e)")
      apply (drule Fin_imp_fold_set)
    apply (auto dest: fold_set_lemma  simp add: fold_def [symmetric] fold_equality)
    apply (blast intro: fold_set_mono [THEN subsetD] dest!: FinD)
    done

    lemma (in fold_typing) fold_type [simp,TC]:
         "C∈Fin(A) ==> fold[B](f,e,C):B"
    apply (erule Fin_induct)
    apply (simp_all add: fold_cons ftype etype)
    done

    lemma (in fold_typing) fold_commute [rule_format]:
         "[| C∈Fin(A); c∈A |]
          ==> (∀y∈B. f(c, fold[B](f, y, C)) = fold[B](f, f(c, y), C))"
    apply (erule Fin_induct)
    apply (simp_all add: fold_typing.fold_cons [of A B _ f]
                         fold_typing.fold_type [of A B _ f]
                         fold_typing_def fcomm)
    done

    lemma (in fold_typing) fold_nest_Un_Int:
         "[| C∈Fin(A); D∈Fin(A) |]
          ==> fold[B](f, fold[B](f, e, D), C) =
              fold[B](f, fold[B](f, e, (C Int D)), C Un D)"
    apply (erule Fin_induct, auto)
    apply (simp add: Un_cons Int_cons_left fold_type fold_commute
                     fold_typing.fold_cons [of A _ _ f]
                     fold_typing_def fcomm cons_absorb)
    done

    lemma (in fold_typing) fold_nest_Un_disjoint:
         "[| C∈Fin(A); D∈Fin(A); C Int D = 0 |]
          ==> fold[B](f,e,C Un D) =  fold[B](f, fold[B](f,e,D), C)"
    by (simp add: fold_nest_Un_Int)

    lemma Finite_cons_lemma: "Finite(C) ==> C∈Fin(cons(c, C))"
    apply (drule Finite_into_Fin)
    apply (blast intro: Fin_mono [THEN subsetD])
    done

    subsection{*The Operator @{term setsum}*}

    lemma setsum_0 [simp]: "setsum(g, 0) = #0"
    by (simp add: setsum_def)

    lemma setsum_cons [simp]:
         "Finite(C) ==>
          setsum(g, cons(c,C)) =
            (if c : C then setsum(g,C) else g(c) $+ setsum(g,C))"
    apply (auto simp add: setsum_def Finite_cons cons_absorb)
    apply (rule_tac A = "cons (c, C)" in fold_typing.fold_cons)
    apply (auto intro: fold_typing.intro Finite_cons_lemma)
    done

    lemma setsum_K0: "setsum((%i. #0), C) = #0"
    apply (case_tac "Finite (C) ")
     prefer 2 apply (simp add: setsum_def)
    apply (erule Finite_induct, auto)
    done

    (*The reversed orientation looks more natural, but LOOPS as a simprule!*)
    lemma setsum_Un_Int:
         "[| Finite(C); Finite(D) |]
          ==> setsum(g, C Un D) $+ setsum(g, C Int D)
            = setsum(g, C) $+ setsum(g, D)"
    apply (erule Finite_induct)
    apply (simp_all add: Int_cons_right cons_absorb Un_cons Int_commute Finite_Un
                         Int_lower1 [THEN subset_Finite])
    done

    lemma setsum_type [simp,TC]: "setsum(g, C):int"
    apply (case_tac "Finite (C) ")
     prefer 2 apply (simp add: setsum_def)
    apply (erule Finite_induct, auto)
    done

    lemma setsum_Un_disjoint:
         "[| Finite(C); Finite(D); C Int D = 0 |]
          ==> setsum(g, C Un D) = setsum(g, C) $+ setsum(g,D)"
    apply (subst setsum_Un_Int [symmetric])
    apply (subgoal_tac [3] "Finite (C Un D) ")
    apply (auto intro: Finite_Un)
    done

    lemma Finite_RepFun [rule_format (no_asm)]:
         "Finite(I) ==> (∀i∈I. Finite(C(i))) --> Finite(RepFun(I, C))"
    apply (erule Finite_induct, auto)
    done

    lemma setsum_UN_disjoint [rule_format (no_asm)]:
         "Finite(I)
          ==> (∀i∈I. Finite(C(i))) -->
              (∀i∈I. ∀j∈I. i≠j --> C(i) Int C(j) = 0) -->
              setsum(f, \<Union>i∈I. C(i)) = setsum (%i. setsum(f, C(i)), I)"
    apply (erule Finite_induct, auto)
    apply (subgoal_tac "∀i∈B. x ≠ i")
     prefer 2 apply blast
    apply (subgoal_tac "C (x) Int (\<Union>i∈B. C (i)) = 0")
     prefer 2 apply blast
    apply (subgoal_tac "Finite (\<Union>i∈B. C (i)) & Finite (C (x)) & Finite (B) ")
    apply (simp (no_asm_simp) add: setsum_Un_disjoint)
    apply (auto intro: Finite_Union Finite_RepFun)
    done

    lemma setsum_addf: "setsum(%x. f(x) $+ g(x),C) = setsum(f, C) $+ setsum(g, C)"
    apply (case_tac "Finite (C) ")
     prefer 2 apply (simp add: setsum_def)
    apply (erule Finite_induct, auto)
    done

    lemma fold_set_cong:
         "[| A=A'; B=B'; e=e'; (∀x∈A'. ∀y∈B'. f(x,y) = f'(x,y)) |]
          ==> fold_set(A,B,f,e) = fold_set(A',B',f',e')"
    apply (simp add: fold_set_def)
    apply (intro refl iff_refl lfp_cong Collect_cong disj_cong ex_cong, auto)
    done

    lemma fold_cong:
    "[| B=B'; A=A'; e=e';
        !!x y. [|x∈A'; y∈B'|] ==> f(x,y) = f'(x,y) |] ==>
       fold[B](f,e,A) = fold[B'](f', e', A')"
    apply (simp add: fold_def)
    apply (subst fold_set_cong)
    apply (rule_tac [5] refl, simp_all)
    done

    lemma setsum_cong:
     "[| A=B; !!x. x∈B ==> f(x) = g(x) |] ==>
         setsum(f, A) = setsum(g, B)"
    by (simp add: setsum_def cong add: fold_cong)

    lemma setsum_Un:
         "[| Finite(A); Finite(B) |]
          ==> setsum(f, A Un B) =
              setsum(f, A) $+ setsum(f, B) $- setsum(f, A Int B)"
    apply (subst setsum_Un_Int [symmetric], auto)
    done

    lemma setsum_zneg_or_0 [rule_format (no_asm)]:
         "Finite(A) ==> (∀x∈A. g(x) $<= #0) --> setsum(g, A) $<= #0"
    apply (erule Finite_induct)
    apply (auto intro: zneg_or_0_add_zneg_or_0_imp_zneg_or_0)
    done

    lemma setsum_succD_lemma [rule_format]:
         "Finite(A)
          ==> ∀n∈nat. setsum(f,A) = $# succ(n) --> (∃a∈A. #0 $< f(a))"
    apply (erule Finite_induct)
    apply (auto simp del: int_of_0 int_of_succ simp add: not_zless_iff_zle int_of_0 [symmetric])
    apply (subgoal_tac "setsum (f, B) $<= #0")
    apply simp_all
    prefer 2 apply (blast intro: setsum_zneg_or_0)
    apply (subgoal_tac "$# 1 $<= f (x) $+ setsum (f, B) ")
    apply (drule zdiff_zle_iff [THEN iffD2])
    apply (subgoal_tac "$# 1 $<= $# 1 $- setsum (f,B) ")
    apply (drule_tac x = "$# 1" in zle_trans)
    apply (rule_tac [2] j = "#1" in zless_zle_trans, auto)
    done

    lemma setsum_succD:
         "[| setsum(f, A) = $# succ(n); n∈nat |]==> ∃a∈A. #0 $< f(a)"
    apply (case_tac "Finite (A) ")
    apply (blast intro: setsum_succD_lemma)
    apply (unfold setsum_def)
    apply (auto simp del: int_of_0 int_of_succ simp add: int_succ_int_1 [symmetric] int_of_0 [symmetric])
    done

    lemma g_zpos_imp_setsum_zpos [rule_format]:
         "Finite(A) ==> (∀x∈A. #0 $<= g(x)) --> #0 $<= setsum(g, A)"
    apply (erule Finite_induct)
    apply (simp (no_asm))
    apply (auto intro: zpos_add_zpos_imp_zpos)
    done

    lemma g_zpos_imp_setsum_zpos2 [rule_format]:
         "[| Finite(A); ∀x. #0 $<= g(x) |] ==> #0 $<= setsum(g, A)"
    apply (erule Finite_induct)
    apply (auto intro: zpos_add_zpos_imp_zpos)
    done

    lemma g_zspos_imp_setsum_zspos [rule_format]:
         "Finite(A)
          ==> (∀x∈A. #0 $< g(x)) --> A ≠ 0 --> (#0 $< setsum(g, A))"
    apply (erule Finite_induct)
    apply (auto intro: zspos_add_zspos_imp_zspos)
    done

    lemma setsum_Diff [rule_format]:
         "Finite(A) ==> ∀a. M(a) = #0 --> setsum(M, A) = setsum(M, A-{a})"
    apply (erule Finite_induct)
    apply (simp_all add: Diff_cons_eq Finite_Diff)
    done

    ML
    {*
    val fold_set_mono = thm "fold_set_mono";
    val Diff1_fold_set = thm "Diff1_fold_set";
    val Diff_sing_imp = thm "Diff_sing_imp";
    val fold_0 = thm "fold_0";
    val setsum_0 = thm "setsum_0";
    val setsum_cons = thm "setsum_cons";
    val setsum_K0 = thm "setsum_K0";
    val setsum_Un_Int = thm "setsum_Un_Int";
    val setsum_type = thm "setsum_type";
    val setsum_Un_disjoint = thm "setsum_Un_disjoint";
    val Finite_RepFun = thm "Finite_RepFun";
    val setsum_UN_disjoint = thm "setsum_UN_disjoint";
    val setsum_addf = thm "setsum_addf";
    val fold_set_cong = thm "fold_set_cong";
    val fold_cong = thm "fold_cong";
    val setsum_cong = thm "setsum_cong";
    val setsum_Un = thm "setsum_Un";
    val setsum_zneg_or_0 = thm "setsum_zneg_or_0";
    val setsum_succD = thm "setsum_succD";
    val g_zpos_imp_setsum_zpos = thm "g_zpos_imp_setsum_zpos";
    val g_zpos_imp_setsum_zpos2 = thm "g_zpos_imp_setsum_zpos2";
    val g_zspos_imp_setsum_zspos = thm "g_zspos_imp_setsum_zspos";
    val setsum_Diff = thm "setsum_Diff";
    *}

    end

lemmas empty_fold_setE:

      [| ⟨0, x⟩ ∈ fold_set(A, B, f, e); [| e ∈ B; x = e |] ==> Q |] ==> Q

lemmas cons_fold_setE:

      [| ⟨cons(x, C), y⟩ ∈ fold_set(A, B, f, e);
         !!C x y.
            [| x ∈ A; x ∉ C; ⟨C, y⟩ ∈ fold_set(A, B, %x y. f(x, y), e); f(x, y) ∈ B;
               cons(x, C) = cons(x, C); y = f(x, y) |]
            ==> Q |]
      ==> Q

lemma cons_lemma1:

      [| x ∉ C; x ∉ B |] ==> cons(x, B) = cons(x, C) <-> B = C

lemma cons_lemma2:

      [| cons(x, B) = cons(y, C); x ≠ y; x ∉ B; y ∉ C |]
      ==> B - {y} = C - {x} ∧ x ∈ C ∧ y ∈ B

lemma fold_set_mono_lemma:

      ⟨C, x⟩ ∈ fold_set(A, B, f, e) ==> ∀D. A ⊆ D --> ⟨C, x⟩ ∈ fold_set(D, B, f, e)

lemma fold_set_mono:

      C ⊆ A ==> fold_set(C, B, f, e) ⊆ fold_set(A, B, f, e)

lemma fold_set_lemma:

      ⟨C, x⟩ ∈ fold_set(A, B, f, e) ==> ⟨C, x⟩ ∈ fold_set(C, B, f, e) ∧ C ⊆ A

lemma Diff1_fold_set:

      [| ⟨C - {x}, y⟩ ∈ fold_set(A, B, f, e); x ∈ C; x ∈ A; f(x, y) ∈ B |]
      ==> ⟨C, f(x, y)⟩ ∈ fold_set(A, B, f, e)

lemma Fin_imp_fold_set:

      [| fold_typing(A, B, e, f); C ∈ Fin(A) |] ==> ∃x. ⟨C, x⟩ ∈ fold_set(A, B, f, e)

lemma Diff_sing_imp:

      [| C - {b} = D - {a}; a ≠ b; b ∈ C |] ==> C = cons(b, D) - {a}

lemma fold_set_determ_lemma:

      [| fold_typing(A, B, e, f); n ∈ nat |]
      ==> ∀C. |C| < n -->
              (∀x. ⟨C, x⟩ ∈ fold_set(A, B, f, e) -->
                   (∀y. ⟨C, y⟩ ∈ fold_set(A, B, f, e) --> y = x))

lemma fold_set_determ:

      [| fold_typing(A, B, e, f); ⟨C, x⟩ ∈ fold_set(A, B, f, e);
         ⟨C, y⟩ ∈ fold_set(A, B, f, e) |]
      ==> y = x

lemma fold_equality:

      [| fold_typing(A, B, e, f); ⟨C, y⟩ ∈ fold_set(A, B, f, e) |]
      ==> fold[B](f,e,C) = y

lemma fold_0:

      e ∈ B ==> fold[B](f,e,0) = e

lemma fold_set_imp_cons:

      [| fold_typing(A, B, e, f); ⟨C, y⟩ ∈ fold_set(C, B, f, e); C ∈ Fin(A); c ∈ A;
         c ∉ C |]
      ==> ⟨cons(c, C), f(c, y)⟩ ∈ fold_set(cons(c, C), B, f, e)

lemma fold_cons_lemma:

      [| fold_typing(A, B, e, f); C ∈ Fin(A); c ∈ A; c ∉ C |]
      ==> ⟨cons(c, C), v⟩ ∈ fold_set(cons(c, C), B, f, e) <->
          (∃y. ⟨C, y⟩ ∈ fold_set(C, B, f, e) ∧ v = f(c, y))

lemma fold_cons:

      [| fold_typing(A, B, e, f); C ∈ Fin(A); c ∈ A; c ∉ C |]
      ==> fold[B](f,e,cons(c, C)) = f(c, fold[B](f,e,C))

lemma fold_type:

      [| fold_typing(A, B, e, f); C ∈ Fin(A) |] ==> fold[B](f,e,C) ∈ B

lemma fold_commute:

      [| fold_typing(A, B, e, f); C ∈ Fin(A); c ∈ A |]
      ==> ∀y∈B. f(c, fold[B](f,y,C)) = fold[B](f,f(c, y),C)

lemma fold_nest_Un_Int:

      [| fold_typing(A, B, e, f); C ∈ Fin(A); D ∈ Fin(A) |]
      ==> fold[B](f,fold[B](f,e,D),C) = fold[B](f,fold[B](f,e,C ∩ D),C ∪ D)

lemma fold_nest_Un_disjoint:

      [| fold_typing(A, B, e, f); C ∈ Fin(A); D ∈ Fin(A); C ∩ D = 0 |]
      ==> fold[B](f,e,C ∪ D) = fold[B](f,fold[B](f,e,D),C)

lemma Finite_cons_lemma:

      Finite(C) ==> C ∈ Fin(cons(c, C))

### The Operator @{term setsum}

lemma setsum_0:

      setsum(g, 0) = #0

lemma setsum_cons:

      Finite(C)
      ==> setsum(g, cons(c, C)) =
          (if c ∈ C then setsum(g, C) else g(c) $+ setsum(g, C))

lemma setsum_K0:

      setsum(%i. #0, C) = #0

lemma setsum_Un_Int:

      [| Finite(C); Finite(D) |]
      ==> setsum(g, C ∪ D) $+ setsum(g, C ∩ D) = setsum(g, C) $+ setsum(g, D)

lemma setsum_type:

      setsum(g, C) ∈ int

lemma setsum_Un_disjoint:

      [| Finite(C); Finite(D); C ∩ D = 0 |]
      ==> setsum(g, C ∪ D) = setsum(g, C) $+ setsum(g, D)

lemma Finite_RepFun:

      [| Finite(I); ∀i∈I. Finite(C(i)) |] ==> Finite(RepFun(I, C))

lemma setsum_UN_disjoint:

      [| Finite(I); ∀i∈I. Finite(C(i)); ∀i∈I. ∀j∈I. i ≠ j --> C(i) ∩ C(j) = 0 |]
      ==> setsum(f, \<Union>RepFun(I, C)) = setsum(%i. setsum(f, C(i)), I)

lemma setsum_addf:

      setsum(%x. f(x) $+ g(x), C) = setsum(f, C) $+ setsum(g, C)

lemma fold_set_cong:

      [| A = A'; B = B'; e = e'; ∀x∈A'. ∀y∈B'. f(x, y) = f'(x, y) |]
      ==> fold_set(A, B, f, e) = fold_set(A', B', f', e')

lemma fold_cong:

      [| B = B'; A = A'; e = e'; !!x y. [| x ∈ A'; y ∈ B' |] ==> f(x, y) = f'(x, y) |]
      ==> fold[B](f,e,A) = fold[B'](f',e',A')

lemma setsum_cong:

      [| A = B; !!x. x ∈ B ==> f(x) = g(x) |] ==> setsum(f, A) = setsum(g, B)

lemma setsum_Un:

      [| Finite(A); Finite(B) |]
      ==> setsum(f, A ∪ B) = setsum(f, A) $+ setsum(f, B) $- setsum(f, A ∩ B)

lemma setsum_zneg_or_0:

      [| Finite(A); ∀x∈A. g(x) $≤ #0 |] ==> setsum(g, A) $≤ #0

lemma setsum_succD_lemma:

      [| Finite(A); n ∈ nat; setsum(f, A) = $# succ(n) |] ==> ∃a∈A. #0 $< f(a)

lemma setsum_succD:

      [| setsum(f, A) = $# succ(n); n ∈ nat |] ==> ∃a∈A. #0 $< f(a)

lemma g_zpos_imp_setsum_zpos:

      [| Finite(A); !!x. x ∈ A ==> #0 $≤ g(x) |] ==> #0 $≤ setsum(g, A)

lemma g_zpos_imp_setsum_zpos2:

      [| Finite(A); !!x. #0 $≤ g(x) |] ==> #0 $≤ setsum(g, A)

lemma g_zspos_imp_setsum_zspos:

      [| Finite(A); !!x. x ∈ A ==> #0 $< g(x); A ≠ 0 |] ==> #0 $< setsum(g, A)

lemma setsum_Diff:

      [| Finite(A); M(a) = #0 |] ==> setsum(M, A) = setsum(M, A - {a})