(*  Title:      ZF/OrderType.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1994  University of Cambridge

 *)

 header{*Order Types and Ordinal Arithmetic*}

 theory OrderType imports OrderArith OrdQuant Nat_ZF begin

 text{*The order type of a well-ordering is the least ordinal isomorphic to it.

 Ordinal arithmetic is traditionally defined in terms of order types, as it is

 here.  But a definition by transfinite recursion would be much simpler!*}

 definition  

   ordermap  :: "[i,i]=>i"  where

    "ordermap(A,r) == lam x:A. wfrec[A](r, x, %x f. f `` pred(A,x,r))"

 definition  

   ordertype :: "[i,i]=>i"  where

    "ordertype(A,r) == ordermap(A,r)``A"

 definition  

   (*alternative definition of ordinal numbers*)

   Ord_alt   :: "i => o"  where

    "Ord_alt(X) == well_ord(X, Memrel(X)) & (ALL u:X. u=pred(X, u, Memrel(X)))"

 definition  

   (*coercion to ordinal: if not, just 0*)

   ordify    :: "i=>i"  where

     "ordify(x) == if Ord(x) then x else 0"

 definition  

   (*ordinal multiplication*)

   omult      :: "[i,i]=>i"           (infixl "**" 70)  where

    "i ** j == ordertype(j*i, rmult(j,Memrel(j),i,Memrel(i)))"

 definition  

   (*ordinal addition*)

   raw_oadd   :: "[i,i]=>i"  where

     "raw_oadd(i,j) == ordertype(i+j, radd(i,Memrel(i),j,Memrel(j)))"

 definition  

   oadd      :: "[i,i]=>i"           (infixl "++" 65)  where

     "i ++ j == raw_oadd(ordify(i),ordify(j))"

 definition  

   (*ordinal subtraction*)

   odiff      :: "[i,i]=>i"           (infixl "--" 65)  where

     "i -- j == ordertype(i-j, Memrel(i))"

 notation (xsymbols)

   omult  (infixl "\<times>\<times>" 70)

 notation (HTML output)

   omult  (infixl "\<times>\<times>" 70)

 subsection{*Proofs needing the combination of Ordinal.thy and Order.thy*}

 lemma le_well_ord_Memrel: "j le i ==> well_ord(j, Memrel(i))"

 apply (rule well_ordI)

 apply (rule wf_Memrel [THEN wf_imp_wf_on])

 apply (simp add: ltD lt_Ord linear_def

                  ltI [THEN lt_trans2 [of _ j i]])

 apply (intro ballI Ord_linear)

 apply (blast intro: Ord_in_Ord lt_Ord)+

 done

 (*"Ord(i) ==> well_ord(i, Memrel(i))"*)

 lemmas well_ord_Memrel = le_refl [THEN le_well_ord_Memrel]

 (*Kunen's Theorem 7.3 (i), page 16;  see also Ordinal/Ord_in_Ord

   The smaller ordinal is an initial segment of the larger *)

 lemma lt_pred_Memrel: 

     "j<i ==> pred(i, j, Memrel(i)) = j"

 apply (unfold pred_def lt_def)

 apply (simp (no_asm_simp))

 apply (blast intro: Ord_trans)

 done

 lemma pred_Memrel: 

       "x:A ==> pred(A, x, Memrel(A)) = A Int x"

 by (unfold pred_def Memrel_def, blast)

 lemma Ord_iso_implies_eq_lemma:

      "[| j<i;  f: ord_iso(i,Memrel(i),j,Memrel(j)) |] ==> R"

 apply (frule lt_pred_Memrel)

 apply (erule ltE)

 apply (rule well_ord_Memrel [THEN well_ord_iso_predE, of i f j], auto) 

 apply (unfold ord_iso_def)

 (*Combining the two simplifications causes looping*)

 apply (simp (no_asm_simp))

 apply (blast intro: bij_is_fun [THEN apply_type] Ord_trans)

 done

 (*Kunen's Theorem 7.3 (ii), page 16.  Isomorphic ordinals are equal*)

 lemma Ord_iso_implies_eq:

      "[| Ord(i);  Ord(j);  f:  ord_iso(i,Memrel(i),j,Memrel(j)) |]     

       ==> i=j"

 apply (rule_tac i = i and j = j in Ord_linear_lt)

 apply (blast intro: ord_iso_sym Ord_iso_implies_eq_lemma)+

 done

 subsection{*Ordermap and ordertype*}

 lemma ordermap_type: 

     "ordermap(A,r) : A -> ordertype(A,r)"

 apply (unfold ordermap_def ordertype_def)

 apply (rule lam_type)

 apply (rule lamI [THEN imageI], assumption+)

 done

 subsubsection{*Unfolding of ordermap *}

 (*Useful for cardinality reasoning; see CardinalArith.ML*)

 lemma ordermap_eq_image: 

     "[| wf[A](r);  x:A |]

      ==> ordermap(A,r) ` x = ordermap(A,r) `` pred(A,x,r)"

 apply (unfold ordermap_def pred_def)

 apply (simp (no_asm_simp))

 apply (erule wfrec_on [THEN trans], assumption)

 apply (simp (no_asm_simp) add: subset_iff image_lam vimage_singleton_iff)

 done

 (*Useful for rewriting PROVIDED pred is not unfolded until later!*)

 lemma ordermap_pred_unfold:

      "[| wf[A](r);  x:A |]

       ==> ordermap(A,r) ` x = {ordermap(A,r)`y . y : pred(A,x,r)}"

 by (simp add: ordermap_eq_image pred_subset ordermap_type [THEN image_fun])

 (*pred-unfolded version.  NOT suitable for rewriting -- loops!*)

 lemmas ordermap_unfold = ordermap_pred_unfold [simplified pred_def] 

 (*The theorem above is 

 [| wf[A](r); x : A |]

 ==> ordermap(A,r) ` x = {ordermap(A,r) ` y . y: {y: A . <y,x> : r}}

 NOTE: the definition of ordermap used here delivers ordinals only if r is

 transitive.  If r is the predecessor relation on the naturals then

 ordermap(nat,predr) ` n equals {n-1} and not n.  A more complicated definition,

 like

   ordermap(A,r) ` x = Union{succ(ordermap(A,r) ` y) . y: {y: A . <y,x> : r}},

 might eliminate the need for r to be transitive.

 *)

 subsubsection{*Showing that ordermap, ordertype yield ordinals *}

 lemma Ord_ordermap: 

     "[| well_ord(A,r);  x:A |] ==> Ord(ordermap(A,r) ` x)"

 apply (unfold well_ord_def tot_ord_def part_ord_def, safe)

 apply (rule_tac a=x in wf_on_induct, assumption+)

 apply (simp (no_asm_simp) add: ordermap_pred_unfold)

 apply (rule OrdI [OF _ Ord_is_Transset])

 apply (unfold pred_def Transset_def)

 apply (blast intro: trans_onD

              dest!: ordermap_unfold [THEN equalityD1])+ 

 done

 lemma Ord_ordertype: 

     "well_ord(A,r) ==> Ord(ordertype(A,r))"

 apply (unfold ordertype_def)

 apply (subst image_fun [OF ordermap_type subset_refl])

 apply (rule OrdI [OF _ Ord_is_Transset])

 prefer 2 apply (blast intro: Ord_ordermap)

 apply (unfold Transset_def well_ord_def)

 apply (blast intro: trans_onD

              dest!: ordermap_unfold [THEN equalityD1])

 done

 subsubsection{*ordermap preserves the orderings in both directions *}

 lemma ordermap_mono:

      "[| <w,x>: r;  wf[A](r);  w: A; x: A |]

       ==> ordermap(A,r)`w : ordermap(A,r)`x"

 apply (erule_tac x1 = x in ordermap_unfold [THEN ssubst], assumption, blast)

 done

 (*linearity of r is crucial here*)

 lemma converse_ordermap_mono: 

     "[| ordermap(A,r)`w : ordermap(A,r)`x;  well_ord(A,r); w: A; x: A |]

      ==> <w,x>: r"

 apply (unfold well_ord_def tot_ord_def, safe)

 apply (erule_tac x=w and y=x in linearE, assumption+) 

 apply (blast elim!: mem_not_refl [THEN notE])

 apply (blast dest: ordermap_mono intro: mem_asym) 

 done

 lemmas ordermap_surj = 

     ordermap_type [THEN surj_image, unfolded ordertype_def [symmetric]]

 lemma ordermap_bij: 

     "well_ord(A,r) ==> ordermap(A,r) : bij(A, ordertype(A,r))"

 apply (unfold well_ord_def tot_ord_def bij_def inj_def)

 apply (force intro!: ordermap_type ordermap_surj 

              elim: linearE dest: ordermap_mono 

              simp add: mem_not_refl)

 done

 subsubsection{*Isomorphisms involving ordertype *}

 lemma ordertype_ord_iso: 

  "well_ord(A,r)

   ==> ordermap(A,r) : ord_iso(A,r, ordertype(A,r), Memrel(ordertype(A,r)))"

 apply (unfold ord_iso_def)

 apply (safe elim!: well_ord_is_wf 

             intro!: ordermap_type [THEN apply_type] ordermap_mono ordermap_bij)

 apply (blast dest!: converse_ordermap_mono)

 done

 lemma ordertype_eq:

      "[| f: ord_iso(A,r,B,s);  well_ord(B,s) |]

       ==> ordertype(A,r) = ordertype(B,s)"

 apply (frule well_ord_ord_iso, assumption)

 apply (rule Ord_iso_implies_eq, (erule Ord_ordertype)+)

 apply (blast intro: ord_iso_trans ord_iso_sym ordertype_ord_iso)

 done

 lemma ordertype_eq_imp_ord_iso:

      "[| ordertype(A,r) = ordertype(B,s); well_ord(A,r);  well_ord(B,s) |] 

       ==> EX f. f: ord_iso(A,r,B,s)"

 apply (rule exI)

 apply (rule ordertype_ord_iso [THEN ord_iso_trans], assumption)

 apply (erule ssubst)

 apply (erule ordertype_ord_iso [THEN ord_iso_sym])

 done

 subsubsection{*Basic equalities for ordertype *}

 (*Ordertype of Memrel*)

 lemma le_ordertype_Memrel: "j le i ==> ordertype(j,Memrel(i)) = j"

 apply (rule Ord_iso_implies_eq [symmetric])

 apply (erule ltE, assumption)

 apply (blast intro: le_well_ord_Memrel Ord_ordertype)

 apply (rule ord_iso_trans)

 apply (erule_tac [2] le_well_ord_Memrel [THEN ordertype_ord_iso])

 apply (rule id_bij [THEN ord_isoI])

 apply (simp (no_asm_simp))

 apply (fast elim: ltE Ord_in_Ord Ord_trans)

 done

 (*"Ord(i) ==> ordertype(i, Memrel(i)) = i"*)

 lemmas ordertype_Memrel = le_refl [THEN le_ordertype_Memrel]

 lemma ordertype_0 [simp]: "ordertype(0,r) = 0"

 apply (rule id_bij [THEN ord_isoI, THEN ordertype_eq, THEN trans])

 apply (erule emptyE)

 apply (rule well_ord_0)

 apply (rule Ord_0 [THEN ordertype_Memrel])

 done

 (*Ordertype of rvimage:  [| f: bij(A,B);  well_ord(B,s) |] ==>

                          ordertype(A, rvimage(A,f,s)) = ordertype(B,s) *)

 lemmas bij_ordertype_vimage = ord_iso_rvimage [THEN ordertype_eq]

 subsubsection{*A fundamental unfolding law for ordertype. *}

 (*Ordermap returns the same result if applied to an initial segment*)

 lemma ordermap_pred_eq_ordermap:

      "[| well_ord(A,r);  y:A;  z: pred(A,y,r) |]

       ==> ordermap(pred(A,y,r), r) ` z = ordermap(A, r) ` z"

 apply (frule wf_on_subset_A [OF well_ord_is_wf pred_subset])

 apply (rule_tac a=z in wf_on_induct, assumption+)

 apply (safe elim!: predE)

 apply (simp (no_asm_simp) add: ordermap_pred_unfold well_ord_is_wf pred_iff)

 (*combining these two simplifications LOOPS! *)

 apply (simp (no_asm_simp) add: pred_pred_eq)

 apply (simp add: pred_def)

 apply (rule RepFun_cong [OF _ refl])

 apply (drule well_ord_is_trans_on)

 apply (fast elim!: trans_onD)

 done

 lemma ordertype_unfold: 

     "ordertype(A,r) = {ordermap(A,r)`y . y : A}"

 apply (unfold ordertype_def)

 apply (rule image_fun [OF ordermap_type subset_refl])

 done

 text{*Theorems by Krzysztof Grabczewski; proofs simplified by lcp *}

 lemma ordertype_pred_subset: "[| well_ord(A,r);  x:A |] ==>              

           ordertype(pred(A,x,r),r) <= ordertype(A,r)"

 apply (simp add: ordertype_unfold well_ord_subset [OF _ pred_subset])

 apply (fast intro: ordermap_pred_eq_ordermap elim: predE)

 done

 lemma ordertype_pred_lt:

      "[| well_ord(A,r);  x:A |]

       ==> ordertype(pred(A,x,r),r) < ordertype(A,r)"

 apply (rule ordertype_pred_subset [THEN subset_imp_le, THEN leE])

 apply (simp_all add: Ord_ordertype well_ord_subset [OF _ pred_subset])

 apply (erule sym [THEN ordertype_eq_imp_ord_iso, THEN exE])

 apply (erule_tac [3] well_ord_iso_predE)

 apply (simp_all add: well_ord_subset [OF _ pred_subset])

 done

 (*May rewrite with this -- provided no rules are supplied for proving that

         well_ord(pred(A,x,r), r) *)

 lemma ordertype_pred_unfold:

      "well_ord(A,r)

       ==> ordertype(A,r) = {ordertype(pred(A,x,r),r). x:A}"

 apply (rule equalityI)

 apply (safe intro!: ordertype_pred_lt [THEN ltD])

 apply (auto simp add: ordertype_def well_ord_is_wf [THEN ordermap_eq_image]

                       ordermap_type [THEN image_fun]

                       ordermap_pred_eq_ordermap pred_subset)

 done

 subsection{*Alternative definition of ordinal*}

 (*proof by Krzysztof Grabczewski*)

 lemma Ord_is_Ord_alt: "Ord(i) ==> Ord_alt(i)"

 apply (unfold Ord_alt_def)

 apply (rule conjI)

 apply (erule well_ord_Memrel)

 apply (unfold Ord_def Transset_def pred_def Memrel_def, blast) 

 done

 (*proof by lcp*)

 lemma Ord_alt_is_Ord: 

     "Ord_alt(i) ==> Ord(i)"

 apply (unfold Ord_alt_def Ord_def Transset_def well_ord_def 

                      tot_ord_def part_ord_def trans_on_def)

 apply (simp add: pred_Memrel)

 apply (blast elim!: equalityE)

 done

 subsection{*Ordinal Addition*}

 subsubsection{*Order Type calculations for radd *}

 text{*Addition with 0 *}

 lemma bij_sum_0: "(lam z:A+0. case(%x. x, %y. y, z)) : bij(A+0, A)"

 apply (rule_tac d = Inl in lam_bijective, safe)

 apply (simp_all (no_asm_simp))

 done

 lemma ordertype_sum_0_eq:

      "well_ord(A,r) ==> ordertype(A+0, radd(A,r,0,s)) = ordertype(A,r)"

 apply (rule bij_sum_0 [THEN ord_isoI, THEN ordertype_eq])

 prefer 2 apply assumption

 apply force

 done

 lemma bij_0_sum: "(lam z:0+A. case(%x. x, %y. y, z)) : bij(0+A, A)"

 apply (rule_tac d = Inr in lam_bijective, safe)

 apply (simp_all (no_asm_simp))

 done

 lemma ordertype_0_sum_eq:

      "well_ord(A,r) ==> ordertype(0+A, radd(0,s,A,r)) = ordertype(A,r)"

 apply (rule bij_0_sum [THEN ord_isoI, THEN ordertype_eq])

 prefer 2 apply assumption

 apply force

 done

 text{*Initial segments of radd.  Statements by Grabczewski *}

 (*In fact, pred(A+B, Inl(a), radd(A,r,B,s)) = pred(A,a,r)+0 *)

 lemma pred_Inl_bij: 

  "a:A ==> (lam x:pred(A,a,r). Inl(x))     

           : bij(pred(A,a,r), pred(A+B, Inl(a), radd(A,r,B,s)))"

 apply (unfold pred_def)

 apply (rule_tac d = "case (%x. x, %y. y) " in lam_bijective)

 apply auto

 done

 lemma ordertype_pred_Inl_eq:

      "[| a:A;  well_ord(A,r) |]

       ==> ordertype(pred(A+B, Inl(a), radd(A,r,B,s)), radd(A,r,B,s)) =  

           ordertype(pred(A,a,r), r)"

 apply (rule pred_Inl_bij [THEN ord_isoI, THEN ord_iso_sym, THEN ordertype_eq])

 apply (simp_all add: well_ord_subset [OF _ pred_subset])

 apply (simp add: pred_def)

 done

 lemma pred_Inr_bij: 

  "b:B ==>   

          id(A+pred(B,b,s))       

          : bij(A+pred(B,b,s), pred(A+B, Inr(b), radd(A,r,B,s)))"

 apply (unfold pred_def id_def)

 apply (rule_tac d = "%z. z" in lam_bijective, auto) 

 done

 lemma ordertype_pred_Inr_eq:

      "[| b:B;  well_ord(A,r);  well_ord(B,s) |]

       ==> ordertype(pred(A+B, Inr(b), radd(A,r,B,s)), radd(A,r,B,s)) =  

           ordertype(A+pred(B,b,s), radd(A,r,pred(B,b,s),s))"

 apply (rule pred_Inr_bij [THEN ord_isoI, THEN ord_iso_sym, THEN ordertype_eq])

 prefer 2 apply (force simp add: pred_def id_def, assumption)

 apply (blast intro: well_ord_radd well_ord_subset [OF _ pred_subset])

 done

 subsubsection{*ordify: trivial coercion to an ordinal *}

 lemma Ord_ordify [iff, TC]: "Ord(ordify(x))"

 by (simp add: ordify_def)

 (*Collapsing*)

 lemma ordify_idem [simp]: "ordify(ordify(x)) = ordify(x)"

 by (simp add: ordify_def)

 subsubsection{*Basic laws for ordinal addition *}

 lemma Ord_raw_oadd: "[|Ord(i); Ord(j)|] ==> Ord(raw_oadd(i,j))"

 by (simp add: raw_oadd_def ordify_def Ord_ordertype well_ord_radd

               well_ord_Memrel)

 lemma Ord_oadd [iff,TC]: "Ord(i++j)"

 by (simp add: oadd_def Ord_raw_oadd)

 text{*Ordinal addition with zero *}

 lemma raw_oadd_0: "Ord(i) ==> raw_oadd(i,0) = i"

 by (simp add: raw_oadd_def ordify_def ordertype_sum_0_eq

               ordertype_Memrel well_ord_Memrel)

 lemma oadd_0 [simp]: "Ord(i) ==> i++0 = i"

 apply (simp (no_asm_simp) add: oadd_def raw_oadd_0 ordify_def)

 done

 lemma raw_oadd_0_left: "Ord(i) ==> raw_oadd(0,i) = i"

 by (simp add: raw_oadd_def ordify_def ordertype_0_sum_eq ordertype_Memrel

               well_ord_Memrel)

 lemma oadd_0_left [simp]: "Ord(i) ==> 0++i = i"

 by (simp add: oadd_def raw_oadd_0_left ordify_def)

 lemma oadd_eq_if_raw_oadd:

      "i++j = (if Ord(i) then (if Ord(j) then raw_oadd(i,j) else i)  

               else (if Ord(j) then j else 0))"

 by (simp add: oadd_def ordify_def raw_oadd_0_left raw_oadd_0)

 lemma raw_oadd_eq_oadd: "[|Ord(i); Ord(j)|] ==> raw_oadd(i,j) = i++j"

 by (simp add: oadd_def ordify_def)

 (*** Further properties of ordinal addition.  Statements by Grabczewski,

     proofs by lcp. ***)

 (*Surely also provable by transfinite induction on j?*)

 lemma lt_oadd1: "k<i ==> k < i++j"

 apply (simp add: oadd_def ordify_def lt_Ord2 raw_oadd_0, clarify)

 apply (simp add: raw_oadd_def)

 apply (rule ltE, assumption)

 apply (rule ltI)

 apply (force simp add: ordertype_pred_unfold well_ord_radd well_ord_Memrel

           ordertype_pred_Inl_eq lt_pred_Memrel leI [THEN le_ordertype_Memrel])

 apply (blast intro: Ord_ordertype well_ord_radd well_ord_Memrel)

 done

 (*Thus also we obtain the rule  i++j = k ==> i le k *)

 lemma oadd_le_self: "Ord(i) ==> i le i++j"

 apply (rule all_lt_imp_le)

 apply (auto simp add: Ord_oadd lt_oadd1) 

 done

 text{*Various other results *}

 lemma id_ord_iso_Memrel: "A<=B ==> id(A) : ord_iso(A, Memrel(A), A, Memrel(B))"

 apply (rule id_bij [THEN ord_isoI])

 apply (simp (no_asm_simp))

 apply blast

 done

 lemma subset_ord_iso_Memrel:

      "[| f: ord_iso(A,Memrel(B),C,r); A<=B |] ==> f: ord_iso(A,Memrel(A),C,r)"

 apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN fun_is_rel]) 

 apply (frule ord_iso_trans [OF id_ord_iso_Memrel], assumption) 

 apply (simp add: right_comp_id) 

 done

 lemma restrict_ord_iso:

      "[| f \<in> ord_iso(i, Memrel(i), Order.pred(A,a,r), r);  a \<in> A; j < i; 

        trans[A](r) |]

       ==> restrict(f,j) \<in> ord_iso(j, Memrel(j), Order.pred(A,f`j,r), r)"

 apply (frule ltD) 

 apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) 

 apply (frule ord_iso_restrict_pred, assumption) 

 apply (simp add: pred_iff trans_pred_pred_eq lt_pred_Memrel)

 apply (blast intro!: subset_ord_iso_Memrel le_imp_subset [OF leI]) 

 done

 lemma restrict_ord_iso2:

      "[| f \<in> ord_iso(Order.pred(A,a,r), r, i, Memrel(i));  a \<in> A; 

        j < i; trans[A](r) |]

       ==> converse(restrict(converse(f), j)) 

           \<in> ord_iso(Order.pred(A, converse(f)`j, r), r, j, Memrel(j))"

 by (blast intro: restrict_ord_iso ord_iso_sym ltI)

 lemma ordertype_sum_Memrel:

      "[| well_ord(A,r);  k<j |]

       ==> ordertype(A+k, radd(A, r, k, Memrel(j))) =  

           ordertype(A+k, radd(A, r, k, Memrel(k)))"

 apply (erule ltE)

 apply (rule ord_iso_refl [THEN sum_ord_iso_cong, THEN ordertype_eq])

 apply (erule OrdmemD [THEN id_ord_iso_Memrel, THEN ord_iso_sym])

 apply (simp_all add: well_ord_radd well_ord_Memrel)

 done

 lemma oadd_lt_mono2: "k<j ==> i++k < i++j"

 apply (simp add: oadd_def ordify_def raw_oadd_0_left lt_Ord lt_Ord2, clarify)

 apply (simp add: raw_oadd_def)

 apply (rule ltE, assumption)

 apply (rule ordertype_pred_unfold [THEN equalityD2, THEN subsetD, THEN ltI])

 apply (simp_all add: Ord_ordertype well_ord_radd well_ord_Memrel)

 apply (rule bexI)

 apply (erule_tac [2] InrI)

 apply (simp add: ordertype_pred_Inr_eq well_ord_Memrel lt_pred_Memrel

                  leI [THEN le_ordertype_Memrel] ordertype_sum_Memrel)

 done

 lemma oadd_lt_cancel2: "[| i++j < i++k;  Ord(j) |] ==> j<k"

 apply (simp (asm_lr) add: oadd_eq_if_raw_oadd split add: split_if_asm)

  prefer 2

  apply (frule_tac i = i and j = j in oadd_le_self)

  apply (simp (asm_lr) add: oadd_def ordify_def lt_Ord not_lt_iff_le [THEN iff_sym])

 apply (rule Ord_linear_lt, auto) 

 apply (simp_all add: raw_oadd_eq_oadd)

 apply (blast dest: oadd_lt_mono2 elim: lt_irrefl lt_asym)+

 done

 lemma oadd_lt_iff2: "Ord(j) ==> i++j < i++k <-> j<k"

 by (blast intro!: oadd_lt_mono2 dest!: oadd_lt_cancel2)

 lemma oadd_inject: "[| i++j = i++k;  Ord(j); Ord(k) |] ==> j=k"

 apply (simp add: oadd_eq_if_raw_oadd split add: split_if_asm)

 apply (simp add: raw_oadd_eq_oadd)

 apply (rule Ord_linear_lt, auto) 

 apply (force dest: oadd_lt_mono2 [of concl: i] simp add: lt_not_refl)+

 done

 lemma lt_oadd_disj: "k < i++j ==> k<i | (EX l:j. k = i++l )"

 apply (simp add: Ord_in_Ord' [of _ j] oadd_eq_if_raw_oadd

             split add: split_if_asm)

  prefer 2

  apply (simp add: Ord_in_Ord' [of _ j] lt_def)

 apply (simp add: ordertype_pred_unfold well_ord_radd well_ord_Memrel raw_oadd_def)

 apply (erule ltD [THEN RepFunE])

 apply (force simp add: ordertype_pred_Inl_eq well_ord_Memrel ltI 

                        lt_pred_Memrel le_ordertype_Memrel leI

                        ordertype_pred_Inr_eq ordertype_sum_Memrel)

 done

 subsubsection{*Ordinal addition with successor -- via associativity! *}

 lemma oadd_assoc: "(i++j)++k = i++(j++k)"

 apply (simp add: oadd_eq_if_raw_oadd Ord_raw_oadd raw_oadd_0 raw_oadd_0_left, clarify)

 apply (simp add: raw_oadd_def)

 apply (rule ordertype_eq [THEN trans])

 apply (rule sum_ord_iso_cong [OF ordertype_ord_iso [THEN ord_iso_sym] 

                                  ord_iso_refl])

 apply (simp_all add: Ord_ordertype well_ord_radd well_ord_Memrel)

 apply (rule sum_assoc_ord_iso [THEN ordertype_eq, THEN trans])

 apply (rule_tac [2] ordertype_eq)

 apply (rule_tac [2] sum_ord_iso_cong [OF ord_iso_refl ordertype_ord_iso])

 apply (blast intro: Ord_ordertype well_ord_radd well_ord_Memrel)+

 done

 lemma oadd_unfold: "[| Ord(i);  Ord(j) |] ==> i++j = i Un (\<Union>k\<in>j. {i++k})"

 apply (rule subsetI [THEN equalityI])

 apply (erule ltI [THEN lt_oadd_disj, THEN disjE])

 apply (blast intro: Ord_oadd) 

 apply (blast elim!: ltE, blast) 

 apply (force intro: lt_oadd1 oadd_lt_mono2 simp add: Ord_mem_iff_lt)

 done

 lemma oadd_1: "Ord(i) ==> i++1 = succ(i)"

 apply (simp (no_asm_simp) add: oadd_unfold Ord_1 oadd_0)

 apply blast

 done

 lemma oadd_succ [simp]: "Ord(j) ==> i++succ(j) = succ(i++j)"

 apply (simp add: oadd_eq_if_raw_oadd, clarify)

 apply (simp add: raw_oadd_eq_oadd)

 apply (simp add: oadd_1 [of j, symmetric] oadd_1 [of "i++j", symmetric]

                  oadd_assoc)

 done

 text{*Ordinal addition with limit ordinals *}

 lemma oadd_UN:

      "[| !!x. x:A ==> Ord(j(x));  a:A |]

       ==> i ++ (\<Union>x\<in>A. j(x)) = (\<Union>x\<in>A. i++j(x))"

 by (blast intro: ltI Ord_UN Ord_oadd lt_oadd1 [THEN ltD] 

                  oadd_lt_mono2 [THEN ltD] 

           elim!: ltE dest!: ltI [THEN lt_oadd_disj])

 lemma oadd_Limit: "Limit(j) ==> i++j = (\<Union>k\<in>j. i++k)"

 apply (frule Limit_has_0 [THEN ltD])

 apply (simp add: Limit_is_Ord [THEN Ord_in_Ord] oadd_UN [symmetric] 

                  Union_eq_UN [symmetric] Limit_Union_eq)

 done

 lemma oadd_eq_0_iff: "[| Ord(i); Ord(j) |] ==> (i ++ j) = 0 <-> i=0 & j=0"

 apply (erule trans_induct3 [of j])

 apply (simp_all add: oadd_Limit)

 apply (simp add: Union_empty_iff Limit_def lt_def, blast)

 done

 lemma oadd_eq_lt_iff: "[| Ord(i); Ord(j) |] ==> 0 < (i ++ j) <-> 0<i | 0<j"

 by (simp add: Ord_0_lt_iff [symmetric] oadd_eq_0_iff)

 lemma oadd_LimitI: "[| Ord(i); Limit(j) |] ==> Limit(i ++ j)"

 apply (simp add: oadd_Limit)

 apply (frule Limit_has_1 [THEN ltD])

 apply (rule increasing_LimitI)

  apply (rule Ord_0_lt)

   apply (blast intro: Ord_in_Ord [OF Limit_is_Ord])

  apply (force simp add: Union_empty_iff oadd_eq_0_iff

                         Limit_is_Ord [of j, THEN Ord_in_Ord], auto)

 apply (rule_tac x="succ(y)" in bexI)

  apply (simp add: ltI Limit_is_Ord [of j, THEN Ord_in_Ord])

 apply (simp add: Limit_def lt_def) 

 done

 text{*Order/monotonicity properties of ordinal addition *}

 lemma oadd_le_self2: "Ord(i) ==> i le j++i"

 apply (erule_tac i = i in trans_induct3)

 apply (simp (no_asm_simp) add: Ord_0_le)

 apply (simp (no_asm_simp) add: oadd_succ succ_leI)

 apply (simp (no_asm_simp) add: oadd_Limit)

 apply (rule le_trans)

 apply (rule_tac [2] le_implies_UN_le_UN)

 apply (erule_tac [2] bspec)

  prefer 2 apply assumption

 apply (simp add: Union_eq_UN [symmetric] Limit_Union_eq le_refl Limit_is_Ord)

 done

 lemma oadd_le_mono1: "k le j ==> k++i le j++i"

 apply (frule lt_Ord)

 apply (frule le_Ord2)

 apply (simp add: oadd_eq_if_raw_oadd, clarify)

 apply (simp add: raw_oadd_eq_oadd)

 apply (erule_tac i = i in trans_induct3)

 apply (simp (no_asm_simp))

 apply (simp (no_asm_simp) add: oadd_succ succ_le_iff)

 apply (simp (no_asm_simp) add: oadd_Limit)

 apply (rule le_implies_UN_le_UN, blast)

 done

 lemma oadd_lt_mono: "[| i' le i;  j'<j |] ==> i'++j' < i++j"

 by (blast intro: lt_trans1 oadd_le_mono1 oadd_lt_mono2 Ord_succD elim: ltE)

 lemma oadd_le_mono: "[| i' le i;  j' le j |] ==> i'++j' le i++j"

 by (simp del: oadd_succ add: oadd_succ [symmetric] le_Ord2 oadd_lt_mono)

 lemma oadd_le_iff2: "[| Ord(j); Ord(k) |] ==> i++j le i++k <-> j le k"

 by (simp del: oadd_succ add: oadd_lt_iff2 oadd_succ [symmetric] Ord_succ)

 lemma oadd_lt_self: "[| Ord(i);  0<j |] ==> i < i++j"

 apply (rule lt_trans2) 

 apply (erule le_refl) 

 apply (simp only: lt_Ord2  oadd_1 [of i, symmetric]) 

 apply (blast intro: succ_leI oadd_le_mono)

 done

 text{*Every ordinal is exceeded by some limit ordinal.*}

 lemma Ord_imp_greater_Limit: "Ord(i) ==> \<exists>k. i<k & Limit(k)"

 apply (rule_tac x="i ++ nat" in exI) 

 apply (blast intro: oadd_LimitI  oadd_lt_self  Limit_nat [THEN Limit_has_0])

 done

 lemma Ord2_imp_greater_Limit: "[|Ord(i); Ord(j)|] ==> \<exists>k. i<k & j<k & Limit(k)"

 apply (insert Ord_Un [of i j, THEN Ord_imp_greater_Limit]) 

 apply (simp add: Un_least_lt_iff) 

 done

 subsection{*Ordinal Subtraction*}

 text{*The difference is @{term "ordertype(j-i, Memrel(j))"}.

     It's probably simpler to define the difference recursively!*}

 lemma bij_sum_Diff:

      "A<=B ==> (lam y:B. if(y:A, Inl(y), Inr(y))) : bij(B, A+(B-A))"

 apply (rule_tac d = "case (%x. x, %y. y) " in lam_bijective)

 apply (blast intro!: if_type)

 apply (fast intro!: case_type)

 apply (erule_tac [2] sumE)

 apply (simp_all (no_asm_simp))

 done

 lemma ordertype_sum_Diff:

      "i le j ==>   

             ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j))) =        

             ordertype(j, Memrel(j))"

 apply (safe dest!: le_subset_iff [THEN iffD1])

 apply (rule bij_sum_Diff [THEN ord_isoI, THEN ord_iso_sym, THEN ordertype_eq])

 apply (erule_tac [3] well_ord_Memrel, assumption)

 apply (simp (no_asm_simp))

 apply (frule_tac j = y in Ord_in_Ord, assumption)

 apply (frule_tac j = x in Ord_in_Ord, assumption)

 apply (simp (no_asm_simp) add: Ord_mem_iff_lt lt_Ord not_lt_iff_le)

 apply (blast intro: lt_trans2 lt_trans)

 done

 lemma Ord_odiff [simp,TC]: 

     "[| Ord(i);  Ord(j) |] ==> Ord(i--j)"

 apply (unfold odiff_def)

 apply (blast intro: Ord_ordertype Diff_subset well_ord_subset well_ord_Memrel)

 done

 lemma raw_oadd_ordertype_Diff: 

    "i le j   

     ==> raw_oadd(i,j--i) = ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j)))"

 apply (simp add: raw_oadd_def odiff_def)

 apply (safe dest!: le_subset_iff [THEN iffD1])

 apply (rule sum_ord_iso_cong [THEN ordertype_eq])

 apply (erule id_ord_iso_Memrel)

 apply (rule ordertype_ord_iso [THEN ord_iso_sym])

 apply (blast intro: well_ord_radd Diff_subset well_ord_subset well_ord_Memrel)+

 done

 lemma oadd_odiff_inverse: "i le j ==> i ++ (j--i) = j"

 by (simp add: lt_Ord le_Ord2 oadd_def ordify_def raw_oadd_ordertype_Diff

               ordertype_sum_Diff ordertype_Memrel lt_Ord2 [THEN Ord_succD])

 (*By oadd_inject, the difference between i and j is unique.  Note that we get

   i++j = k  ==>  j = k--i.  *)

 lemma odiff_oadd_inverse: "[| Ord(i); Ord(j) |] ==> (i++j) -- i = j"

 apply (rule oadd_inject)

 apply (blast intro: oadd_odiff_inverse oadd_le_self)

 apply (blast intro: Ord_ordertype Ord_oadd Ord_odiff)+

 done

 lemma odiff_lt_mono2: "[| i<j;  k le i |] ==> i--k < j--k"

 apply (rule_tac i = k in oadd_lt_cancel2)

 apply (simp add: oadd_odiff_inverse)

 apply (subst oadd_odiff_inverse)

 apply (blast intro: le_trans leI, assumption)

 apply (simp (no_asm_simp) add: lt_Ord le_Ord2)

 done

 subsection{*Ordinal Multiplication*}

 lemma Ord_omult [simp,TC]: 

     "[| Ord(i);  Ord(j) |] ==> Ord(i**j)"

 apply (unfold omult_def)

 apply (blast intro: Ord_ordertype well_ord_rmult well_ord_Memrel)

 done

 subsubsection{*A useful unfolding law *}

 lemma pred_Pair_eq: 

  "[| a:A;  b:B |] ==> pred(A*B, <a,b>, rmult(A,r,B,s)) =      

                       pred(A,a,r)*B Un ({a} * pred(B,b,s))"

 apply (unfold pred_def, blast)

 done

 lemma ordertype_pred_Pair_eq:

      "[| a:A;  b:B;  well_ord(A,r);  well_ord(B,s) |] ==>            

          ordertype(pred(A*B, <a,b>, rmult(A,r,B,s)), rmult(A,r,B,s)) =  

          ordertype(pred(A,a,r)*B + pred(B,b,s),                         

                   radd(A*B, rmult(A,r,B,s), B, s))"

 apply (simp (no_asm_simp) add: pred_Pair_eq)

 apply (rule ordertype_eq [symmetric])

 apply (rule prod_sum_singleton_ord_iso)

 apply (simp_all add: pred_subset well_ord_rmult [THEN well_ord_subset])

 apply (blast intro: pred_subset well_ord_rmult [THEN well_ord_subset] 

              elim!: predE)

 done

 lemma ordertype_pred_Pair_lemma: 

     "[| i'<i;  j'<j |]

      ==> ordertype(pred(i*j, <i',j'>, rmult(i,Memrel(i),j,Memrel(j))),  

                    rmult(i,Memrel(i),j,Memrel(j))) =                    

          raw_oadd (j**i', j')"

 apply (unfold raw_oadd_def omult_def)

 apply (simp add: ordertype_pred_Pair_eq lt_pred_Memrel ltD lt_Ord2 

                  well_ord_Memrel)

 apply (rule trans)

  apply (rule_tac [2] ordertype_ord_iso 

                       [THEN sum_ord_iso_cong, THEN ordertype_eq])

   apply (rule_tac [3] ord_iso_refl)

 apply (rule id_bij [THEN ord_isoI, THEN ordertype_eq])

 apply (elim SigmaE sumE ltE ssubst)

 apply (simp_all add: well_ord_rmult well_ord_radd well_ord_Memrel

                      Ord_ordertype lt_Ord lt_Ord2) 

 apply (blast intro: Ord_trans)+

 done

 lemma lt_omult: 

  "[| Ord(i);  Ord(j);  k<j**i |]

   ==> EX j' i'. k = j**i' ++ j' & j'<j & i'<i"

 apply (unfold omult_def)

 apply (simp add: ordertype_pred_unfold well_ord_rmult well_ord_Memrel)

 apply (safe elim!: ltE)

 apply (simp add: ordertype_pred_Pair_lemma ltI raw_oadd_eq_oadd 

             omult_def [symmetric] Ord_in_Ord' [of _ i] Ord_in_Ord' [of _ j])

 apply (blast intro: ltI)

 done

 lemma omult_oadd_lt: 

      "[| j'<j;  i'<i |] ==> j**i' ++ j'  <  j**i"

 apply (unfold omult_def)

 apply (rule ltI)

  prefer 2

  apply (simp add: Ord_ordertype well_ord_rmult well_ord_Memrel lt_Ord2)

 apply (simp add: ordertype_pred_unfold well_ord_rmult well_ord_Memrel lt_Ord2)

 apply (rule bexI [of _ i']) 

 apply (rule bexI [of _ j']) 

 apply (simp add: ordertype_pred_Pair_lemma ltI omult_def [symmetric])

 apply (simp add: lt_Ord lt_Ord2 raw_oadd_eq_oadd)

 apply (simp_all add: lt_def) 

 done

 lemma omult_unfold:

      "[| Ord(i);  Ord(j) |] ==> j**i = (\<Union>j'\<in>j. \<Union>i'\<in>i. {j**i' ++ j'})"

 apply (rule subsetI [THEN equalityI])

 apply (rule lt_omult [THEN exE])

 apply (erule_tac [3] ltI)

 apply (simp_all add: Ord_omult) 

 apply (blast elim!: ltE)

 apply (blast intro: omult_oadd_lt [THEN ltD] ltI)

 done

 subsubsection{*Basic laws for ordinal multiplication *}

 text{*Ordinal multiplication by zero *}

 lemma omult_0 [simp]: "i**0 = 0"

 apply (unfold omult_def)

 apply (simp (no_asm_simp))

 done

 lemma omult_0_left [simp]: "0**i = 0"

 apply (unfold omult_def)

 apply (simp (no_asm_simp))

 done

 text{*Ordinal multiplication by 1 *}

 lemma omult_1 [simp]: "Ord(i) ==> i**1 = i"

 apply (unfold omult_def)

 apply (rule_tac s1="Memrel(i)" 

        in ord_isoI [THEN ordertype_eq, THEN trans])

 apply (rule_tac c = snd and d = "%z.<0,z>"  in lam_bijective)

 apply (auto elim!: snd_type well_ord_Memrel ordertype_Memrel)

 done

 lemma omult_1_left [simp]: "Ord(i) ==> 1**i = i"

 apply (unfold omult_def)

 apply (rule_tac s1="Memrel(i)" 

        in ord_isoI [THEN ordertype_eq, THEN trans])

 apply (rule_tac c = fst and d = "%z.<z,0>" in lam_bijective)

 apply (auto elim!: fst_type well_ord_Memrel ordertype_Memrel)

 done

 text{*Distributive law for ordinal multiplication and addition *}

 lemma oadd_omult_distrib:

      "[| Ord(i);  Ord(j);  Ord(k) |] ==> i**(j++k) = (i**j)++(i**k)"

 apply (simp add: oadd_eq_if_raw_oadd)

 apply (simp add: omult_def raw_oadd_def)

 apply (rule ordertype_eq [THEN trans])

 apply (rule prod_ord_iso_cong [OF ordertype_ord_iso [THEN ord_iso_sym] 

                                   ord_iso_refl])

 apply (simp_all add: well_ord_rmult well_ord_radd well_ord_Memrel 

                      Ord_ordertype)

 apply (rule sum_prod_distrib_ord_iso [THEN ordertype_eq, THEN trans])

 apply (rule_tac [2] ordertype_eq)

 apply (rule_tac [2] sum_ord_iso_cong [OF ordertype_ord_iso ordertype_ord_iso])

 apply (simp_all add: well_ord_rmult well_ord_radd well_ord_Memrel 

                      Ord_ordertype)

 done

 lemma omult_succ: "[| Ord(i);  Ord(j) |] ==> i**succ(j) = (i**j)++i"

 by (simp del: oadd_succ add: oadd_1 [of j, symmetric] oadd_omult_distrib)

 text{*Associative law *}

 lemma omult_assoc: 

     "[| Ord(i);  Ord(j);  Ord(k) |] ==> (i**j)**k = i**(j**k)"

 apply (unfold omult_def)

 apply (rule ordertype_eq [THEN trans])

 apply (rule prod_ord_iso_cong [OF ord_iso_refl 

                                   ordertype_ord_iso [THEN ord_iso_sym]])

 apply (blast intro: well_ord_rmult well_ord_Memrel)+

 apply (rule prod_assoc_ord_iso 

              [THEN ord_iso_sym, THEN ordertype_eq, THEN trans])

 apply (rule_tac [2] ordertype_eq)

 apply (rule_tac [2] prod_ord_iso_cong [OF ordertype_ord_iso ord_iso_refl])

 apply (blast intro: well_ord_rmult well_ord_Memrel Ord_ordertype)+

 done

 text{*Ordinal multiplication with limit ordinals *}

 lemma omult_UN: 

      "[| Ord(i);  !!x. x:A ==> Ord(j(x)) |]

       ==> i ** (\<Union>x\<in>A. j(x)) = (\<Union>x\<in>A. i**j(x))"

 by (simp (no_asm_simp) add: Ord_UN omult_unfold, blast)

 lemma omult_Limit: "[| Ord(i);  Limit(j) |] ==> i**j = (\<Union>k\<in>j. i**k)"

 by (simp add: Limit_is_Ord [THEN Ord_in_Ord] omult_UN [symmetric] 

               Union_eq_UN [symmetric] Limit_Union_eq)

 subsubsection{*Ordering/monotonicity properties of ordinal multiplication *}

 (*As a special case we have "[| 0<i;  0<j |] ==> 0 < i**j" *)

 lemma lt_omult1: "[| k<i;  0<j |] ==> k < i**j"

 apply (safe elim!: ltE intro!: ltI Ord_omult)

 apply (force simp add: omult_unfold)

 done

 lemma omult_le_self: "[| Ord(i);  0<j |] ==> i le i**j"

 by (blast intro: all_lt_imp_le Ord_omult lt_omult1 lt_Ord2)

 lemma omult_le_mono1: "[| k le j;  Ord(i) |] ==> k**i le j**i"

 apply (frule lt_Ord)

 apply (frule le_Ord2)

 apply (erule trans_induct3)

 apply (simp (no_asm_simp) add: le_refl Ord_0)

 apply (simp (no_asm_simp) add: omult_succ oadd_le_mono)

 apply (simp (no_asm_simp) add: omult_Limit)

 apply (rule le_implies_UN_le_UN, blast)

 done

 lemma omult_lt_mono2: "[| k<j;  0<i |] ==> i**k < i**j"

 apply (rule ltI)

 apply (simp (no_asm_simp) add: omult_unfold lt_Ord2)

 apply (safe elim!: ltE intro!: Ord_omult)

 apply (force simp add: Ord_omult)

 done

 lemma omult_le_mono2: "[| k le j;  Ord(i) |] ==> i**k le i**j"

 apply (rule subset_imp_le)

 apply (safe elim!: ltE dest!: Ord_succD intro!: Ord_omult)

 apply (simp add: omult_unfold)

 apply (blast intro: Ord_trans) 

 done

 lemma omult_le_mono: "[| i' le i;  j' le j |] ==> i'**j' le i**j"

 by (blast intro: le_trans omult_le_mono1 omult_le_mono2 Ord_succD elim: ltE)

 lemma omult_lt_mono: "[| i' le i;  j'<j;  0<i |] ==> i'**j' < i**j"

 by (blast intro: lt_trans1 omult_le_mono1 omult_lt_mono2 Ord_succD elim: ltE)

 lemma omult_le_self2: "[| Ord(i);  0<j |] ==> i le j**i"

 apply (frule lt_Ord2)

 apply (erule_tac i = i in trans_induct3)

 apply (simp (no_asm_simp))

 apply (simp (no_asm_simp) add: omult_succ)

 apply (erule lt_trans1)

 apply (rule_tac b = "j**x" in oadd_0 [THEN subst], rule_tac [2] oadd_lt_mono2)

 apply (blast intro: Ord_omult, assumption)

 apply (simp (no_asm_simp) add: omult_Limit)

 apply (rule le_trans)

 apply (rule_tac [2] le_implies_UN_le_UN)

 prefer 2 apply blast

 apply (simp (no_asm_simp) add: Union_eq_UN [symmetric] Limit_Union_eq Limit_is_Ord)

 done

 text{*Further properties of ordinal multiplication *}

 lemma omult_inject: "[| i**j = i**k;  0<i;  Ord(j);  Ord(k) |] ==> j=k"

 apply (rule Ord_linear_lt)

 prefer 4 apply assumption

 apply auto 

 apply (force dest: omult_lt_mono2 simp add: lt_not_refl)+

 done

 subsection{*The Relation @{term Lt}*}

 lemma wf_Lt: "wf(Lt)"

 apply (rule wf_subset) 

 apply (rule wf_Memrel) 

 apply (auto simp add: Lt_def Memrel_def lt_def) 

 done

 lemma irrefl_Lt: "irrefl(A,Lt)"

 by (auto simp add: Lt_def irrefl_def)

 lemma trans_Lt: "trans[A](Lt)"

 apply (simp add: Lt_def trans_on_def) 

 apply (blast intro: lt_trans) 

 done

 lemma part_ord_Lt: "part_ord(A,Lt)"

 by (simp add: part_ord_def irrefl_Lt trans_Lt)

 lemma linear_Lt: "linear(nat,Lt)"

 apply (auto dest!: not_lt_imp_le simp add: Lt_def linear_def le_iff) 

 apply (drule lt_asym, auto) 

 done

 lemma tot_ord_Lt: "tot_ord(nat,Lt)"

 by (simp add: tot_ord_def linear_Lt part_ord_Lt)

 lemma well_ord_Lt: "well_ord(nat,Lt)"

 by (simp add: well_ord_def wf_Lt wf_imp_wf_on tot_ord_Lt)

 end