Theory OrderArith

(*  Title:      ZF/OrderArith.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

section‹Combining Orderings: Foundations of Ordinal Arithmetic›

theory OrderArith imports Order Sum Ordinal begin

  (*disjoint sum of two relations; underlies ordinal addition*)
  radd    :: "[i,i,i,i]i"  where
                {z: (A+B) * (A+B).
                    (x y. z = Inl(x), Inr(y))   |
                    (x' x. z = Inl(x'), Inl(x)  x',x:r)   |
                    (y' y. z = Inr(y'), Inr(y)  y',y:s)}"

  (*lexicographic product of two relations; underlies ordinal multiplication*)
  rmult   :: "[i,i,i,i]i"  where
                {z: (A*B) * (A*B).
                    x' y' x y. z = x',y', x,y 
                       (x',x: r | (x'=x  y',y: s))}"

  (*inverse image of a relation*)
  rvimage :: "[i,i,i]i"  where
    "rvimage(A,f,r)  {z  A*A. x y. z = x,y  f`x,f`y: r}"

  measure :: "[i, ii]  i"  where
    "measure(A,f)  {x,y: A*A. f(x) < f(y)}"

subsection‹Addition of Relations -- Disjoint Sum›

subsubsection‹Rewrite rules.  Can be used to obtain introduction rules›

lemma radd_Inl_Inr_iff [iff]:
    "Inl(a), Inr(b)  radd(A,r,B,s)    a  A  b  B"
by (unfold radd_def, blast)

lemma radd_Inl_iff [iff]:
    "Inl(a'), Inl(a)  radd(A,r,B,s)    a':A  a  A  a',a:r"
by (unfold radd_def, blast)

lemma radd_Inr_iff [iff]:
    "Inr(b'), Inr(b)  radd(A,r,B,s)   b':B  b  B  b',b:s"
by (unfold radd_def, blast)

lemma radd_Inr_Inl_iff [simp]:
    "Inr(b), Inl(a)  radd(A,r,B,s)  False"
by (unfold radd_def, blast)

declare radd_Inr_Inl_iff [THEN iffD1, dest!]

subsubsection‹Elimination Rule›

lemma raddE:
    "p',p  radd(A,r,B,s);
        x y. p'=Inl(x); x  A; p=Inr(y); y  B  Q;
        x' x. p'=Inl(x'); p=Inl(x); x',x: r; x':A; x  A  Q;
        y' y. p'=Inr(y'); p=Inr(y); y',y: s; y':B; y  B  Q
by (unfold radd_def, blast)

subsubsection‹Type checking›

lemma radd_type: "radd(A,r,B,s)  (A+B) * (A+B)"
  unfolding radd_def
apply (rule Collect_subset)

lemmas field_radd = radd_type [THEN field_rel_subset]


lemma linear_radd:
    "linear(A,r);  linear(B,s)  linear(A+B,radd(A,r,B,s))"
by (unfold linear_def, blast)


lemma wf_on_radd: "wf[A](r);  wf[B](s)  wf[A+B](radd(A,r,B,s))"
apply (rule wf_onI2)
apply (subgoal_tac "xA. Inl (x)  Ba")
 ― ‹Proving the lemma, which is needed twice!›
 prefer 2
 apply (erule_tac V = "y  A + B" in thin_rl)
 apply (rule_tac ballI)
 apply (erule_tac r = r and a = x in wf_on_induct, assumption)
 apply blast
txt‹Returning to main part of proof›
apply safe
apply blast
apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast)

lemma wf_radd: "wf(r);  wf(s)  wf(radd(field(r),r,field(s),s))"
apply (simp add: wf_iff_wf_on_field)
apply (rule wf_on_subset_A [OF _ field_radd])
apply (blast intro: wf_on_radd)

lemma well_ord_radd:
     "well_ord(A,r);  well_ord(B,s)  well_ord(A+B, radd(A,r,B,s))"
apply (rule well_ordI)
apply (simp add: well_ord_def wf_on_radd)
apply (simp add: well_ord_def tot_ord_def linear_radd)

subsubsection‹An termord_iso congruence law›

lemma sum_bij:
     "f  bij(A,C);  g  bij(B,D)
       (λzA+B. case(λx. Inl(f`x), λy. Inr(g`y), z))  bij(A+B, C+D)"
apply (rule_tac d = "case (λx. Inl (converse(f)`x), λy. Inr(converse(g)`y))"
       in lam_bijective)
apply (typecheck add: bij_is_inj inj_is_fun)
apply (auto simp add: left_inverse_bij right_inverse_bij)

lemma sum_ord_iso_cong:
    "f  ord_iso(A,r,A',r');  g  ord_iso(B,s,B',s') 
            (λzA+B. case(λx. Inl(f`x), λy. Inr(g`y), z))
             ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
  unfolding ord_iso_def
apply (safe intro!: sum_bij)
(*Do the beta-reductions now*)
apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])

(*Could we prove an ord_iso result?  Perhaps
     ord_iso(A+B, radd(A,r,B,s), A ∪ B, r ∪ s) *)
lemma sum_disjoint_bij: "A  B = 0 
            (λzA+B. case(λx. x, λy. y, z))  bij(A+B, A  B)"
apply (rule_tac d = "λz. if z  A then Inl (z) else Inr (z) " in lam_bijective)
apply auto


lemma sum_assoc_bij:
     "(λz(A+B)+C. case(case(Inl, λy. Inr(Inl(y))), λy. Inr(Inr(y)), z))
       bij((A+B)+C, A+(B+C))"
apply (rule_tac d = "case (λx. Inl (Inl (x)), case (λx. Inl (Inr (x)), Inr))"
       in lam_bijective)
apply auto

lemma sum_assoc_ord_iso:
     "(λz(A+B)+C. case(case(Inl, λy. Inr(Inl(y))), λy. Inr(Inr(y)), z))
       ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
                A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
by (rule sum_assoc_bij [THEN ord_isoI], auto)

subsection‹Multiplication of Relations -- Lexicographic Product›

subsubsection‹Rewrite rule.  Can be used to obtain introduction rules›

lemma  rmult_iff [iff]:
    "a',b', a,b  rmult(A,r,B,s) 
            (a',a: r   a':A  a  A  b': B  b  B) |
            (b',b: s   a'=a  a  A  b': B  b  B)"

by (unfold rmult_def, blast)

lemma rmultE:
    "a',b', a,b  rmult(A,r,B,s);
        a',a: r;  a':A;  a  A;  b':B;  b  B  Q;
        b',b: s;  a  A;  a'=a;  b':B;  b  B  Q
by blast

subsubsection‹Type checking›

lemma rmult_type: "rmult(A,r,B,s)  (A*B) * (A*B)"
by (unfold rmult_def, rule Collect_subset)

lemmas field_rmult = rmult_type [THEN field_rel_subset]


lemma linear_rmult:
    "linear(A,r);  linear(B,s)  linear(A*B,rmult(A,r,B,s))"
by (simp add: linear_def, blast)


lemma wf_on_rmult: "wf[A](r);  wf[B](s)  wf[A*B](rmult(A,r,B,s))"
apply (rule wf_onI2)
apply (erule SigmaE)
apply (erule ssubst)
apply (subgoal_tac "bB. x,b: Ba", blast)
apply (erule_tac a = x in wf_on_induct, assumption)
apply (rule ballI)
apply (erule_tac a = b in wf_on_induct, assumption)
apply (best elim!: rmultE bspec [THEN mp])

lemma wf_rmult: "wf(r);  wf(s)  wf(rmult(field(r),r,field(s),s))"
apply (simp add: wf_iff_wf_on_field)
apply (rule wf_on_subset_A [OF _ field_rmult])
apply (blast intro: wf_on_rmult)

lemma well_ord_rmult:
     "well_ord(A,r);  well_ord(B,s)  well_ord(A*B, rmult(A,r,B,s))"
apply (rule well_ordI)
apply (simp add: well_ord_def wf_on_rmult)
apply (simp add: well_ord_def tot_ord_def linear_rmult)

subsubsection‹An termord_iso congruence law›

lemma prod_bij:
     "f  bij(A,C);  g  bij(B,D)
       (lam x,y:A*B. f`x, g`y)  bij(A*B, C*D)"
apply (rule_tac d = "λx,y. converse (f) `x, converse (g) `y"
       in lam_bijective)
apply (typecheck add: bij_is_inj inj_is_fun)
apply (auto simp add: left_inverse_bij right_inverse_bij)

lemma prod_ord_iso_cong:
    "f  ord_iso(A,r,A',r');  g  ord_iso(B,s,B',s')
      (lam x,y:A*B. f`x, g`y)
          ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
  unfolding ord_iso_def
apply (safe intro!: prod_bij)
apply (simp_all add: bij_is_fun [THEN apply_type])
apply (blast intro: bij_is_inj [THEN inj_apply_equality])

lemma singleton_prod_bij: "(λzA. x,z)  bij(A, {x}*A)"
by (rule_tac d = snd in lam_bijective, auto)

lemma singleton_prod_ord_iso:
          (λzA. x,z)  ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
apply (rule singleton_prod_bij [THEN ord_isoI])
apply (simp (no_asm_simp))
apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])

(*Here we build a complicated function term, then simplify it using
  case_cong, id_conv, comp_lam, case_case.*)
lemma prod_sum_singleton_bij:
       (λxC*B + D. case(λx. x, λy.a,y, x))
        bij(C*B + D, C*B  {a}*D)"
apply (rule subst_elem)
apply (rule id_bij [THEN sum_bij, THEN comp_bij])
apply (rule singleton_prod_bij)
apply (rule sum_disjoint_bij, blast)
apply (simp (no_asm_simp) cong add: case_cong)
apply (rule comp_lam [THEN trans, symmetric])
apply (fast elim!: case_type)
apply (simp (no_asm_simp) add: case_case)

lemma prod_sum_singleton_ord_iso:
 "a  A;  well_ord(A,r) 
    (λxpred(A,a,r)*B + pred(B,b,s). case(λx. x, λy.a,y, x))
     ord_iso(pred(A,a,r)*B + pred(B,b,s),
                  radd(A*B, rmult(A,r,B,s), B, s),
              pred(A,a,r)*B  {a}*pred(B,b,s), rmult(A,r,B,s))"
apply (rule prod_sum_singleton_bij [THEN ord_isoI])
apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)

subsubsection‹Distributive law›

lemma sum_prod_distrib_bij:
     "(lam x,z:(A+B)*C. case(λy. Inl(y,z), λy. Inr(y,z), x))
       bij((A+B)*C, (A*C)+(B*C))"
by (rule_tac d = "case (λx,y.Inl (x),y, λx,y.Inr (x),y) "
    in lam_bijective, auto)

lemma sum_prod_distrib_ord_iso:
 "(lam x,z:(A+B)*C. case(λy. Inl(y,z), λy. Inr(y,z), x))
   ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
            (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)


lemma prod_assoc_bij:
     "(lam x,y, z:(A*B)*C. x,y,z)  bij((A*B)*C, A*(B*C))"
by (rule_tac d = "λx, y,z. x,y, z" in lam_bijective, auto)

lemma prod_assoc_ord_iso:
 "(lam x,y, z:(A*B)*C. x,y,z)
   ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
            A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
by (rule prod_assoc_bij [THEN ord_isoI], auto)

subsection‹Inverse Image of a Relation›

subsubsection‹Rewrite rule›

lemma rvimage_iff: "a,b  rvimage(A,f,r)    f`a,f`b: r  a  A  b  A"
by (unfold rvimage_def, blast)

subsubsection‹Type checking›

lemma rvimage_type: "rvimage(A,f,r)  A*A"
by (unfold rvimage_def, rule Collect_subset)

lemmas field_rvimage = rvimage_type [THEN field_rel_subset]

lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
by (unfold rvimage_def, blast)

subsubsection‹Partial Ordering Properties›

lemma irrefl_rvimage:
    "f  inj(A,B);  irrefl(B,r)  irrefl(A, rvimage(A,f,r))"
  unfolding irrefl_def rvimage_def
apply (blast intro: inj_is_fun [THEN apply_type])

lemma trans_on_rvimage:
    "f  inj(A,B);  trans[B](r)  trans[A](rvimage(A,f,r))"
  unfolding trans_on_def rvimage_def
apply (blast intro: inj_is_fun [THEN apply_type])

lemma part_ord_rvimage:
    "f  inj(A,B);  part_ord(B,r)  part_ord(A, rvimage(A,f,r))"
  unfolding part_ord_def
apply (blast intro!: irrefl_rvimage trans_on_rvimage)


lemma linear_rvimage:
    "f  inj(A,B);  linear(B,r)  linear(A,rvimage(A,f,r))"
apply (simp add: inj_def linear_def rvimage_iff)
apply (blast intro: apply_funtype)

lemma tot_ord_rvimage:
    "f  inj(A,B);  tot_ord(B,r)  tot_ord(A, rvimage(A,f,r))"
  unfolding tot_ord_def
apply (blast intro!: part_ord_rvimage linear_rvimage)


lemma wf_rvimage [intro!]: "wf(r)  wf(rvimage(A,f,r))"
apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
apply clarify
apply (subgoal_tac "w. w  {w: {f`x. x  Q}. x. x  Q  (f`x = w) }")
 apply (erule allE)
 apply (erule impE)
 apply assumption
 apply blast
apply blast

text‹But note that the combination of wf_imp_wf_on› and
 wf_rvimage› gives propwf(r)  wf[C](rvimage(A,f,r))
lemma wf_on_rvimage: "f  AB;  wf[B](r)  wf[A](rvimage(A,f,r))"
apply (rule wf_onI2)
apply (subgoal_tac "zA. f`z=f`y  z  Ba")
 apply blast
apply (erule_tac a = "f`y" in wf_on_induct)
 apply (blast intro!: apply_funtype)
apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])

(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
lemma well_ord_rvimage:
     "f  inj(A,B);  well_ord(B,r)  well_ord(A, rvimage(A,f,r))"
apply (rule well_ordI)
  unfolding well_ord_def tot_ord_def
apply (blast intro!: wf_on_rvimage inj_is_fun)
apply (blast intro!: linear_rvimage)

lemma ord_iso_rvimage:
    "f  bij(A,B)  f  ord_iso(A, rvimage(A,f,s), B, s)"
  unfolding ord_iso_def
apply (simp add: rvimage_iff)

lemma ord_iso_rvimage_eq:
    "f  ord_iso(A,r, B,s)  rvimage(A,f,s) = r  A*A"
by (unfold ord_iso_def rvimage_def, blast)

subsection‹Every well-founded relation is a subset of some inverse image of
      an ordinal›

lemma wf_rvimage_Ord: "Ord(i)  wf(rvimage(A, f, Memrel(i)))"
by (blast intro: wf_rvimage wf_Memrel)

  wfrank :: "[i,i]i"  where
    "wfrank(r,a)  wfrec(r, a, λx f. y  r-``{x}. succ(f`y))"

  wftype :: "ii"  where
    "wftype(r)  y  range(r). succ(wfrank(r,y))"

lemma wfrank: "wf(r)  wfrank(r,a) = (y  r-``{a}. succ(wfrank(r,y)))"
by (subst wfrank_def [THEN def_wfrec], simp_all)

lemma Ord_wfrank: "wf(r)  Ord(wfrank(r,a))"
apply (rule_tac a=a in wf_induct, assumption)
apply (subst wfrank, assumption)
apply (rule Ord_succ [THEN Ord_UN], blast)

lemma wfrank_lt: "wf(r); a,b  r  wfrank(r,a) < wfrank(r,b)"
apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
apply (rule UN_I [THEN ltI])
apply (simp add: Ord_wfrank vimage_iff)+

lemma Ord_wftype: "wf(r)  Ord(wftype(r))"
by (simp add: wftype_def Ord_wfrank)

lemma wftypeI: "wf(r);  x  field(r)  wfrank(r,x)  wftype(r)"
apply (simp add: wftype_def)
apply (blast intro: wfrank_lt [THEN ltD])

lemma wf_imp_subset_rvimage:
     "wf(r); r  A*A  i f. Ord(i)  r  rvimage(A, f, Memrel(i))"
apply (rule_tac x="wftype(r)" in exI)
apply (rule_tac x="λxA. wfrank(r,x)" in exI)
apply (simp add: Ord_wftype, clarify)
apply (frule subsetD, assumption, clarify)
apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
apply (blast intro: wftypeI)

theorem wf_iff_subset_rvimage:
  "relation(r)  wf(r)  (i f A. Ord(i)  r  rvimage(A, f, Memrel(i)))"
by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
          intro: wf_rvimage_Ord [THEN wf_subset])

subsection‹Other Results›

lemma wf_times: "A  B = 0  wf(A*B)"
by (simp add: wf_def, blast)

text‹Could also be used to prove wf_radd›
lemma wf_Un:
     "range(r)  domain(s) = 0; wf(r);  wf(s)  wf(r  s)"
apply (simp add: wf_def, clarify)
apply (rule equalityI)
 prefer 2 apply blast
apply clarify
apply (drule_tac x=Z in spec)
apply (drule_tac x="Z  domain(s)" in spec)
apply simp
apply (blast intro: elim: equalityE)

subsubsection‹The Empty Relation›

lemma wf0: "wf(0)"
by (simp add: wf_def, blast)

lemma linear0: "linear(0,0)"
by (simp add: linear_def)

lemma well_ord0: "well_ord(0,0)"
by (blast intro: wf_imp_wf_on well_ordI wf0 linear0)

subsubsection‹The "measure" relation is useful with wfrec›

lemma measure_eq_rvimage_Memrel:
     "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
apply (rule equalityI, auto)
apply (auto intro: Ord_in_Ord simp add: lt_def)

lemma wf_measure [iff]: "wf(measure(A,f))"
by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)

lemma measure_iff [iff]: "x,y  measure(A,f)  x  A  y  A  f(x)<f(y)"
by (simp (no_asm) add: measure_def)

lemma linear_measure:
 assumes Ordf: "x. x  A  Ord(f(x))"
     and inj:  "x y. x  A; y  A; f(x) = f(y)  x=y"
 shows "linear(A, measure(A,f))"
apply (auto simp add: linear_def)
apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt)
    apply (simp_all add: Ordf)
apply (blast intro: inj)

lemma wf_on_measure: "wf[B](measure(A,f))"
by (rule wf_imp_wf_on [OF wf_measure])

lemma well_ord_measure:
 assumes Ordf: "x. x  A  Ord(f(x))"
     and inj:  "x y. x  A; y  A; f(x) = f(y)  x=y"
 shows "well_ord(A, measure(A,f))"
apply (rule well_ordI)
apply (rule wf_on_measure)
apply (blast intro: linear_measure Ordf inj)

lemma measure_type: "measure(A,f)  A*A"
by (auto simp add: measure_def)

subsubsection‹Well-foundedness of Unions›

lemma wf_on_Union:
 assumes wfA: "wf[A](r)"
     and wfB: "a. aA  wf[B(a)](s)"
     and ok: "a u v. u,v  s; v  B(a); a  A
                        (a'A. a',a  r  u  B(a')) | u  B(a)"
 shows "wf[aA. B(a)](s)"
apply (rule wf_onI2)
apply (erule UN_E)
apply (subgoal_tac "z  B(a). z  Ba", blast)
apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
apply (rule ballI)
apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
apply (rename_tac u)
apply (drule_tac x=u in bspec, blast)
apply (erule mp, clarify)
apply (frule ok, assumption+, blast)

subsubsection‹Bijections involving Powersets›

lemma Pow_sum_bij:
    "(λZ  Pow(A+B). {x  A. Inl(x)  Z}, {y  B. Inr(y)  Z})
      bij(Pow(A+B), Pow(A)*Pow(B))"
apply (rule_tac d = "λX,Y. {Inl (x). x  X}  {Inr (y). y  Y}"
       in lam_bijective)
apply force+

text‹As a special case, we have termbij(Pow(A*B), A  Pow(B))
lemma Pow_Sigma_bij:
    "(λr  Pow(Sigma(A,B)). λx  A. r``{x})
      bij(Pow(Sigma(A,B)), x  A. Pow(B(x)))"
apply (rule_tac d = "λf. x  A. y  f`x. {x,y}" in lam_bijective)
apply (blast intro: lam_type)
apply (blast dest: apply_type, simp_all)
apply fast (*strange, but blast can't do it*)
apply (rule fun_extension, auto)
by blast