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); [| eB; x = e |] ==> Q |] ==> Q

lemmas cons_fold_setE:

  [| ⟨cons(x, C), y⟩ ∈ fold_set(A, B, f, e);
     !!C x y.
        [| xA; xC; ⟨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:

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

lemma cons_lemma2:

  [| cons(x, B) = cons(y, C); xy; xB; yC |]
  ==> B - {y} = C - {x} ∧ xCyB

lemma fold_set_mono_lemma:

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

lemma fold_set_mono:

  CA ==> 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) ∧ CA

lemma Diff1_fold_set:

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

  eB ==> 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); cA;
     cC |]
  ==> ⟨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); cA; cC |]
  ==> ⟨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); cA; cC |]
  ==> 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); cA |]
  ==> ∀yB. 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,CD),CD)

lemma fold_nest_Un_disjoint:

  [| fold_typing(A, B, e, f); C ∈ Fin(A); D ∈ Fin(A); CD = 0 |]
  ==> fold[B](f,e,CD) = 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 cC 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, CD) $+ setsum(g, CD) = setsum(g, C) $+ setsum(g, D)

lemma setsum_type:

  setsum(g, C) ∈ int

lemma setsum_Un_disjoint:

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

lemma Finite_RepFun:

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

lemma setsum_UN_disjoint:

  [| Finite(I); ∀iI. Finite(C(i)); ∀iI. ∀jI. ij --> 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'; ∀xA'. ∀yB'. 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. [| xA'; yB' |] ==> f(x, y) = f'(x, y) |]
  ==> fold[B](f,e,A) = fold[B'](f',e',A')

lemma setsum_cong:

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

lemma setsum_Un:

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

lemma setsum_zneg_or_0:

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

lemma setsum_succD_lemma:

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

lemma setsum_succD:

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

lemma g_zpos_imp_setsum_zpos:

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

lemma setsum_Diff:

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