Theory FoldSet

(*  Title:      ZF/Induct/FoldSet.thy
    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 ZF 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
  
definition
  fold :: "[i, [i,i]⇒i, i, i] ⇒ i"  (‹fold[_]'(_,_,_')›)  where
   "fold[B](f,e, A) ≡ THE x. ⟨A, x⟩∈fold_set(A, B, f,e)"

definition
   setsum :: "[i⇒i, i] ⇒ i"  where
   "setsum(g, C) ≡ if Finite(C) then
                     fold[int](λx 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)  
      ⟹ ∀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) ⟹ (∃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
 ⟹ ∀C. |C|<n ⟶  
   (∀x. ⟨C, x⟩ ∈ fold_set(A, B, f,e)⟶ 
           (∀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| ≤ |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"
  unfolding 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"
  unfolding 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) ⟷   
         (∃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))"
  unfolding 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 ∩ D)), C ∪ 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 ∩ D = 0⟧  
      ⟹ fold[B](f,e,C ∪ 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((λi. #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 ∪ D) $+ setsum(g, C ∩ 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 ∩ D = 0⟧  
      ⟹ setsum(g, C ∪ D) = setsum(g, C) $+ setsum(g,D)"
apply (subst setsum_Un_Int [symmetric])
apply (subgoal_tac [3] "Finite (C ∪ 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) ∩ C(j) = 0) ⟶  
          setsum(f, ⋃i∈I. C(i)) = setsum (λi. 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) ∩ (⋃i∈B. C (i)) = 0")
 prefer 2 apply blast
apply (subgoal_tac "Finite (⋃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(λx. 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 ∪ B) =  
          setsum(f, A) $+ setsum(f, B) $- setsum(f, A ∩ 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)
  unfolding 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

end