Theory Order

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

Results from the book "Set Theory: an Introduction to Independence Proofs"
        by Kenneth Kunen.  Chapter 1, section 6.
Additional definitions and lemmas for reflexive orders.

section‹Partial and Total Orderings: Basic Definitions and Properties›

theory Order imports WF Perm begin

text ‹We adopt the following convention: ord› is used for
  strict orders and order› is used for their reflexive

  part_ord :: "[i,i]o"                (*Strict partial ordering*)  where
   "part_ord(A,r)  irrefl(A,r)  trans[A](r)"

  linear   :: "[i,i]o"                (*Strict total ordering*)  where
   "linear(A,r)  (xA. yA. x,y:r | x=y | y,x:r)"

  tot_ord  :: "[i,i]o"                (*Strict total ordering*)  where
   "tot_ord(A,r)  part_ord(A,r)  linear(A,r)"

  "preorder_on(A, r)  refl(A, r)  trans[A](r)"

definition                              (*Partial ordering*)
  "partial_order_on(A, r)  preorder_on(A, r)  antisym(r)"

  "Preorder(r)  preorder_on(field(r), r)"

  "Partial_order(r)  partial_order_on(field(r), r)"

  well_ord :: "[i,i]o"                (*Well-ordering*)  where
   "well_ord(A,r)  tot_ord(A,r)  wf[A](r)"

  mono_map :: "[i,i,i,i]i"            (*Order-preserving maps*)  where
              {f  A->B. xA. yA. x,y:r  <f`x,f`y>:s}"

  ord_iso  :: "[i,i,i,i]i"  ((_, _ / _, _) 51)  (*Order isomorphisms*)  where
   "A,r  B,s 
              {f  bij(A,B). xA. yA. x,y:r  <f`x,f`y>:s}"

  pred     :: "[i,i,i]i"              (*Set of predecessors*)  where
   "pred(A,x,r)  {y  A. y,x:r}"

  ord_iso_map :: "[i,i,i,i]i"         (*Construction for linearity theorem*)  where
     xA. yB. f  ord_iso(pred(A,x,r), r, pred(B,y,s), s). {x,y}"

  first :: "[i, i, i]  o"  where
    "first(u, X, R)  u  X  (vX. vu  u,v  R)"

subsection‹Immediate Consequences of the Definitions›

lemma part_ord_Imp_asym:
    "part_ord(A,r)  asym(r  A*A)"
by (unfold part_ord_def irrefl_def trans_on_def asym_def, blast)

lemma linearE:
    "linear(A,r);  x  A;  y  A;
        x,y:r  P;  x=y  P;  y,x:r  P
by (simp add: linear_def, blast)

(** General properties of well_ord **)

lemma well_ordI:
    "wf[A](r); linear(A,r)  well_ord(A,r)"
apply (simp add: irrefl_def part_ord_def tot_ord_def
                 trans_on_def well_ord_def wf_on_not_refl)
apply (fast elim: linearE wf_on_asym wf_on_chain3)

lemma well_ord_is_wf:
    "well_ord(A,r)  wf[A](r)"
by (unfold well_ord_def, safe)

lemma well_ord_is_trans_on:
    "well_ord(A,r)  trans[A](r)"
by (unfold well_ord_def tot_ord_def part_ord_def, safe)

lemma well_ord_is_linear: "well_ord(A,r)  linear(A,r)"
by (unfold well_ord_def tot_ord_def, blast)

(** Derived rules for pred(A,x,r) **)

lemma pred_iff: "y  pred(A,x,r)  y,x:r  y  A"
by (unfold pred_def, blast)

lemmas predI = conjI [THEN pred_iff [THEN iffD2]]

lemma predE: "y  pred(A,x,r);  y  A; y,x:r  P  P"
by (simp add: pred_def)

lemma pred_subset_under: "pred(A,x,r)  r -`` {x}"
by (simp add: pred_def, blast)

lemma pred_subset: "pred(A,x,r)  A"
by (simp add: pred_def, blast)

lemma pred_pred_eq:
    "pred(pred(A,x,r), y, r) = pred(A,x,r)  pred(A,y,r)"
by (simp add: pred_def, blast)

lemma trans_pred_pred_eq:
    "trans[A](r);  y,x:r;  x  A;  y  A
      pred(pred(A,x,r), y, r) = pred(A,y,r)"
by (unfold trans_on_def pred_def, blast)

subsection‹Restricting an Ordering's Domain›

(** The ordering's properties hold over all subsets of its domain
    [including initial segments of the form pred(A,x,r) **)

(*Note: a relation s such that s<=r need not be a partial ordering*)
lemma part_ord_subset:
    "part_ord(A,r);  B<=A  part_ord(B,r)"
by (unfold part_ord_def irrefl_def trans_on_def, blast)

lemma linear_subset:
    "linear(A,r);  B<=A  linear(B,r)"
by (unfold linear_def, blast)

lemma tot_ord_subset:
    "tot_ord(A,r);  B<=A  tot_ord(B,r)"
  unfolding tot_ord_def
apply (fast elim!: part_ord_subset linear_subset)

lemma well_ord_subset:
    "well_ord(A,r);  B<=A  well_ord(B,r)"
  unfolding well_ord_def
apply (fast elim!: tot_ord_subset wf_on_subset_A)

(** Relations restricted to a smaller domain, by Krzysztof Grabczewski **)

lemma irrefl_Int_iff: "irrefl(A,r  A*A)  irrefl(A,r)"
by (unfold irrefl_def, blast)

lemma trans_on_Int_iff: "trans[A](r  A*A)  trans[A](r)"
by (unfold trans_on_def, blast)

lemma part_ord_Int_iff: "part_ord(A,r  A*A)  part_ord(A,r)"
  unfolding part_ord_def
apply (simp add: irrefl_Int_iff trans_on_Int_iff)

lemma linear_Int_iff: "linear(A,r  A*A)  linear(A,r)"
by (unfold linear_def, blast)

lemma tot_ord_Int_iff: "tot_ord(A,r  A*A)  tot_ord(A,r)"
  unfolding tot_ord_def
apply (simp add: part_ord_Int_iff linear_Int_iff)

lemma wf_on_Int_iff: "wf[A](r  A*A)  wf[A](r)"
apply (unfold wf_on_def wf_def, fast) (*10 times faster than blast!*)

lemma well_ord_Int_iff: "well_ord(A,r  A*A)  well_ord(A,r)"
  unfolding well_ord_def
apply (simp add: tot_ord_Int_iff wf_on_Int_iff)

subsection‹Empty and Unit Domains›

(*The empty relation is well-founded*)
lemma wf_on_any_0: "wf[A](0)"
by (simp add: wf_on_def wf_def, fast)

subsubsection‹Relations over the Empty Set›

lemma irrefl_0: "irrefl(0,r)"
by (unfold irrefl_def, blast)

lemma trans_on_0: "trans[0](r)"
by (unfold trans_on_def, blast)

lemma part_ord_0: "part_ord(0,r)"
  unfolding part_ord_def
apply (simp add: irrefl_0 trans_on_0)

lemma linear_0: "linear(0,r)"
by (unfold linear_def, blast)

lemma tot_ord_0: "tot_ord(0,r)"
  unfolding tot_ord_def
apply (simp add: part_ord_0 linear_0)

lemma wf_on_0: "wf[0](r)"
by (unfold wf_on_def wf_def, blast)

lemma well_ord_0: "well_ord(0,r)"
  unfolding well_ord_def
apply (simp add: tot_ord_0 wf_on_0)

subsubsection‹The Empty Relation Well-Orders the Unit Set›

text‹by Grabczewski›

lemma tot_ord_unit: "tot_ord({a},0)"
by (simp add: irrefl_def trans_on_def part_ord_def linear_def tot_ord_def)

lemma well_ord_unit: "well_ord({a},0)"
  unfolding well_ord_def
apply (simp add: tot_ord_unit wf_on_any_0)


text‹Suppes calls them "similarities"›

(** Order-preserving (monotone) maps **)

lemma mono_map_is_fun: "f  mono_map(A,r,B,s)  f  A->B"
by (simp add: mono_map_def)

lemma mono_map_is_inj:
    "linear(A,r);  wf[B](s);  f  mono_map(A,r,B,s)  f  inj(A,B)"
apply (unfold mono_map_def inj_def, clarify)
apply (erule_tac x=w and y=x in linearE, assumption+)
apply (force intro: apply_type dest: wf_on_not_refl)+

lemma ord_isoI:
    "f  bij(A, B);
        x y. x  A; y  A  x, y  r  <f`x, f`y>  s
      f  ord_iso(A,r,B,s)"
by (simp add: ord_iso_def)

lemma ord_iso_is_mono_map:
    "f  ord_iso(A,r,B,s)  f  mono_map(A,r,B,s)"
apply (simp add: ord_iso_def mono_map_def)
apply (blast dest!: bij_is_fun)

lemma ord_iso_is_bij:
    "f  ord_iso(A,r,B,s)  f  bij(A,B)"
by (simp add: ord_iso_def)

(*Needed?  But ord_iso_converse is!*)
lemma ord_iso_apply:
    "f  ord_iso(A,r,B,s);  x,y: r;  x  A;  y  A  <f`x, f`y>  s"
by (simp add: ord_iso_def)

lemma ord_iso_converse:
    "f  ord_iso(A,r,B,s);  x,y: s;  x  B;  y  B
      <converse(f) ` x, converse(f) ` y>  r"
apply (simp add: ord_iso_def, clarify)
apply (erule bspec [THEN bspec, THEN iffD2])
apply (erule asm_rl bij_converse_bij [THEN bij_is_fun, THEN apply_type])+
apply (auto simp add: right_inverse_bij)

(** Symmetry and Transitivity Rules **)

(*Reflexivity of similarity*)
lemma ord_iso_refl: "id(A): ord_iso(A,r,A,r)"
by (rule id_bij [THEN ord_isoI], simp)

(*Symmetry of similarity*)
lemma ord_iso_sym: "f  ord_iso(A,r,B,s)  converse(f): ord_iso(B,s,A,r)"
apply (simp add: ord_iso_def)
apply (auto simp add: right_inverse_bij bij_converse_bij
                      bij_is_fun [THEN apply_funtype])

(*Transitivity of similarity*)
lemma mono_map_trans:
    "g  mono_map(A,r,B,s);  f  mono_map(B,s,C,t)
      (f O g): mono_map(A,r,C,t)"
  unfolding mono_map_def
apply (auto simp add: comp_fun)

(*Transitivity of similarity: the order-isomorphism relation*)
lemma ord_iso_trans:
    "g  ord_iso(A,r,B,s);  f  ord_iso(B,s,C,t)
      (f O g): ord_iso(A,r,C,t)"
apply (unfold ord_iso_def, clarify)
apply (frule bij_is_fun [of f])
apply (frule bij_is_fun [of g])
apply (auto simp add: comp_bij)

(** Two monotone maps can make an order-isomorphism **)

lemma mono_ord_isoI:
    "f  mono_map(A,r,B,s);  g  mono_map(B,s,A,r);
        f O g = id(B);  g O f = id(A)  f  ord_iso(A,r,B,s)"
apply (simp add: ord_iso_def mono_map_def, safe)
apply (intro fg_imp_bijective, auto)
apply (subgoal_tac "<g` (f`x), g` (f`y) >  r")
apply (simp add: comp_eq_id_iff [THEN iffD1])
apply (blast intro: apply_funtype)

lemma well_ord_mono_ord_isoI:
     "well_ord(A,r);  well_ord(B,s);
         f  mono_map(A,r,B,s);  converse(f): mono_map(B,s,A,r)
       f  ord_iso(A,r,B,s)"
apply (intro mono_ord_isoI, auto)
apply (frule mono_map_is_fun [THEN fun_is_rel])
apply (erule converse_converse [THEN subst], rule left_comp_inverse)
apply (blast intro: left_comp_inverse mono_map_is_inj well_ord_is_linear

(** Order-isomorphisms preserve the ordering's properties **)

lemma part_ord_ord_iso:
    "part_ord(B,s);  f  ord_iso(A,r,B,s)  part_ord(A,r)"
apply (simp add: part_ord_def irrefl_def trans_on_def ord_iso_def)
apply (fast intro: bij_is_fun [THEN apply_type])

lemma linear_ord_iso:
    "linear(B,s);  f  ord_iso(A,r,B,s)  linear(A,r)"
apply (simp add: linear_def ord_iso_def, safe)
apply (drule_tac x1 = "f`x" and x = "f`y" in bspec [THEN bspec])
apply (safe elim!: bij_is_fun [THEN apply_type])
apply (drule_tac t = "(`) (converse (f))" in subst_context)
apply (simp add: left_inverse_bij)

lemma wf_on_ord_iso:
    "wf[B](s);  f  ord_iso(A,r,B,s)  wf[A](r)"
apply (simp add: wf_on_def wf_def ord_iso_def, safe)
apply (drule_tac x = "{f`z. z  Z  A}" in spec)
apply (safe intro!: equalityI)
apply (blast dest!: equalityD1 intro: bij_is_fun [THEN apply_type])+

lemma well_ord_ord_iso:
    "well_ord(B,s);  f  ord_iso(A,r,B,s)  well_ord(A,r)"
  unfolding well_ord_def tot_ord_def
apply (fast elim!: part_ord_ord_iso linear_ord_iso wf_on_ord_iso)

subsection‹Main results of Kunen, Chapter 1 section 6›

(*Inductive argument for Kunen's Lemma 6.1, etc.
  Simple proof from Halmos, page 72*)
lemma well_ord_iso_subset_lemma:
     "well_ord(A,r);  f  ord_iso(A,r, A',r);  A'<= A;  y  A
       ¬ <f`y, y>: r"
apply (simp add: well_ord_def ord_iso_def)
apply (elim conjE CollectE)
apply (rule_tac a=y in wf_on_induct, assumption+)
apply (blast dest: bij_is_fun [THEN apply_type])

(*Kunen's Lemma 6.1 ∈ there's no order-isomorphism to an initial segment
                     of a well-ordering*)
lemma well_ord_iso_predE:
     "well_ord(A,r);  f  ord_iso(A, r, pred(A,x,r), r);  x  A  P"
apply (insert well_ord_iso_subset_lemma [of A r f "pred(A,x,r)" x])
apply (simp add: pred_subset)
(*Now we know  f`x < x *)
apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption)
(*Now we also know @{term"f`x ∈ pred(A,x,r)"}: contradiction! *)
apply (simp add: well_ord_def pred_def)

(*Simple consequence of Lemma 6.1*)
lemma well_ord_iso_pred_eq:
     "well_ord(A,r);  f  ord_iso(pred(A,a,r), r, pred(A,c,r), r);
         a  A;  c  A  a=c"
apply (frule well_ord_is_trans_on)
apply (frule well_ord_is_linear)
apply (erule_tac x=a and y=c in linearE, assumption+)
apply (drule ord_iso_sym)
(*two symmetric cases*)
apply (auto elim!: well_ord_subset [OF _ pred_subset, THEN well_ord_iso_predE]
            intro!: predI
            simp add: trans_pred_pred_eq)

(*Does not assume r is a wellordering!*)
lemma ord_iso_image_pred:
     "f  ord_iso(A,r,B,s);  a  A  f `` pred(A,a,r) = pred(B, f`a, s)"
  unfolding ord_iso_def pred_def
apply (erule CollectE)
apply (simp (no_asm_simp) add: image_fun [OF bij_is_fun Collect_subset])
apply (rule equalityI)
apply (safe elim!: bij_is_fun [THEN apply_type])
apply (rule RepFun_eqI)
apply (blast intro!: right_inverse_bij [symmetric])
apply (auto simp add: right_inverse_bij  bij_is_fun [THEN apply_funtype])

lemma ord_iso_restrict_image:
     "f  ord_iso(A,r,B,s);  C<=A
       restrict(f,C)  ord_iso(C, r, f``C, s)"
apply (simp add: ord_iso_def)
apply (blast intro: bij_is_inj restrict_bij)

(*But in use, A and B may themselves be initial segments.  Then use
  trans_pred_pred_eq to simplify the pred(pred...) terms.  See just below.*)
lemma ord_iso_restrict_pred:
   "f  ord_iso(A,r,B,s);   a  A
     restrict(f, pred(A,a,r))  ord_iso(pred(A,a,r), r, pred(B, f`a, s), s)"
apply (simp add: ord_iso_image_pred [symmetric])
apply (blast intro: ord_iso_restrict_image elim: predE)

(*Tricky; a lot of forward proof!*)
lemma well_ord_iso_preserving:
     "well_ord(A,r);  well_ord(B,s);  a,c: r;
         f  ord_iso(pred(A,a,r), r, pred(B,b,s), s);
         g  ord_iso(pred(A,c,r), r, pred(B,d,s), s);
         a  A;  c  A;  b  B;  d  B  b,d: s"
apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], (erule asm_rl predI predE)+)
apply (subgoal_tac "b = g`a")
apply (simp (no_asm_simp))
apply (rule well_ord_iso_pred_eq, auto)
apply (frule ord_iso_restrict_pred, (erule asm_rl predI)+)
apply (simp add: well_ord_is_trans_on trans_pred_pred_eq)
apply (erule ord_iso_sym [THEN ord_iso_trans], assumption)

(*See Halmos, page 72*)
lemma well_ord_iso_unique_lemma:
         f  ord_iso(A,r, B,s);  g  ord_iso(A,r, B,s);  y  A
       ¬ <g`y, f`y>  s"
apply (frule well_ord_iso_subset_lemma)
apply (rule_tac f = "converse (f) " and g = g in ord_iso_trans)
apply auto
apply (blast intro: ord_iso_sym)
apply (frule ord_iso_is_bij [of f])
apply (frule ord_iso_is_bij [of g])
apply (frule ord_iso_converse)
apply (blast intro!: bij_converse_bij
             intro: bij_is_fun apply_funtype)+
apply (erule notE)
apply (simp add: left_inverse_bij bij_is_fun comp_fun_apply [of _ A B])

(*Kunen's Lemma 6.2: Order-isomorphisms between well-orderings are unique*)
lemma well_ord_iso_unique: "well_ord(A,r);
         f  ord_iso(A,r, B,s);  g  ord_iso(A,r, B,s)  f = g"
apply (rule fun_extension)
apply (erule ord_iso_is_bij [THEN bij_is_fun])+
apply (subgoal_tac "f`x  B  g`x  B  linear(B,s)")
 apply (simp add: linear_def)
 apply (blast dest: well_ord_iso_unique_lemma)
apply (blast intro: ord_iso_is_bij bij_is_fun apply_funtype
                    well_ord_is_linear well_ord_ord_iso ord_iso_sym)

subsection‹Towards Kunen's Theorem 6.3: Linearity of the Similarity Relation›

lemma ord_iso_map_subset: "ord_iso_map(A,r,B,s)  A*B"
by (unfold ord_iso_map_def, blast)

lemma domain_ord_iso_map: "domain(ord_iso_map(A,r,B,s))  A"
by (unfold ord_iso_map_def, blast)

lemma range_ord_iso_map: "range(ord_iso_map(A,r,B,s))  B"
by (unfold ord_iso_map_def, blast)

lemma converse_ord_iso_map:
    "converse(ord_iso_map(A,r,B,s)) = ord_iso_map(B,s,A,r)"
  unfolding ord_iso_map_def
apply (blast intro: ord_iso_sym)

lemma function_ord_iso_map:
    "well_ord(B,s)  function(ord_iso_map(A,r,B,s))"
  unfolding ord_iso_map_def function_def
apply (blast intro: well_ord_iso_pred_eq ord_iso_sym ord_iso_trans)

lemma ord_iso_map_fun: "well_ord(B,s)  ord_iso_map(A,r,B,s)
            domain(ord_iso_map(A,r,B,s)) -> range(ord_iso_map(A,r,B,s))"
by (simp add: Pi_iff function_ord_iso_map
                 ord_iso_map_subset [THEN domain_times_range])

lemma ord_iso_map_mono_map:
    "well_ord(A,r);  well_ord(B,s)
            mono_map(domain(ord_iso_map(A,r,B,s)), r,
                      range(ord_iso_map(A,r,B,s)), s)"
  unfolding mono_map_def
apply (simp (no_asm_simp) add: ord_iso_map_fun)
apply safe
apply (subgoal_tac "x  A  ya:A  y  B  yb:B")
 apply (simp add: apply_equality [OF _  ord_iso_map_fun])
   unfolding ord_iso_map_def
 apply (blast intro: well_ord_iso_preserving, blast)

lemma ord_iso_map_ord_iso:
    "well_ord(A,r);  well_ord(B,s)  ord_iso_map(A,r,B,s)
            ord_iso(domain(ord_iso_map(A,r,B,s)), r,
                      range(ord_iso_map(A,r,B,s)), s)"
apply (rule well_ord_mono_ord_isoI)
   prefer 4
   apply (rule converse_ord_iso_map [THEN subst])
   apply (simp add: ord_iso_map_mono_map
                    ord_iso_map_subset [THEN converse_converse])
apply (blast intro!: domain_ord_iso_map range_ord_iso_map
             intro: well_ord_subset ord_iso_map_mono_map)+

(*One way of saying that domain(ord_iso_map(A,r,B,s)) is downwards-closed*)
lemma domain_ord_iso_map_subset:
     "well_ord(A,r);  well_ord(B,s);
         a  A;  a  domain(ord_iso_map(A,r,B,s))
        domain(ord_iso_map(A,r,B,s))  pred(A, a, r)"
  unfolding ord_iso_map_def
apply (safe intro!: predI)
(*Case analysis on  xa vs a in r *)
apply (simp (no_asm_simp))
apply (frule_tac A = A in well_ord_is_linear)
apply (rename_tac b y f)
apply (erule_tac x=b and y=a in linearE, assumption+)
(*Trivial case: b=a*)
apply clarify
apply blast
(*Harder case: ⟨a, xa⟩: r*)
apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type],
       (erule asm_rl predI predE)+)
apply (frule ord_iso_restrict_pred)
 apply (simp add: pred_iff)
apply (simp split: split_if_asm
          add: well_ord_is_trans_on trans_pred_pred_eq domain_UN domain_Union, blast)

(*For the 4-way case analysis in the main result*)
lemma domain_ord_iso_map_cases:
     "well_ord(A,r);  well_ord(B,s)
       domain(ord_iso_map(A,r,B,s)) = A |
          (xA. domain(ord_iso_map(A,r,B,s)) = pred(A,x,r))"
apply (frule well_ord_is_wf)
  unfolding wf_on_def wf_def
apply (drule_tac x = "A-domain (ord_iso_map (A,r,B,s))" in spec)
apply safe
(*The first case: the domain equals A*)
apply (rule domain_ord_iso_map [THEN equalityI])
apply (erule Diff_eq_0_iff [THEN iffD1])
(*The other case: the domain equals an initial segment*)
apply (blast del: domainI subsetI
             elim!: predE
             intro!: domain_ord_iso_map_subset
             intro: subsetI)+

(*As above, by duality*)
lemma range_ord_iso_map_cases:
    "well_ord(A,r);  well_ord(B,s)
      range(ord_iso_map(A,r,B,s)) = B |
         (yB. range(ord_iso_map(A,r,B,s)) = pred(B,y,s))"
apply (rule converse_ord_iso_map [THEN subst])
apply (simp add: domain_ord_iso_map_cases)

text‹Kunen's Theorem 6.3: Fundamental Theorem for Well-Ordered Sets›
theorem well_ord_trichotomy:
   "well_ord(A,r);  well_ord(B,s)
     ord_iso_map(A,r,B,s)  ord_iso(A, r, B, s) |
        (xA. ord_iso_map(A,r,B,s)  ord_iso(pred(A,x,r), r, B, s)) |
        (yB. ord_iso_map(A,r,B,s)  ord_iso(A, r, pred(B,y,s), s))"
apply (frule_tac B = B in domain_ord_iso_map_cases, assumption)
apply (frule_tac B = B in range_ord_iso_map_cases, assumption)
apply (drule ord_iso_map_ord_iso, assumption)
apply (elim disjE bexE)
   apply (simp_all add: bexI)
apply (rule wf_on_not_refl [THEN notE])
  apply (erule well_ord_is_wf)
 apply assumption
apply (subgoal_tac "x,y: ord_iso_map (A,r,B,s) ")
 apply (drule rangeI)
 apply (simp add: pred_def)
apply (unfold ord_iso_map_def, blast)

subsection‹Miscellaneous Results by Krzysztof Grabczewski›

(** Properties of converse(r) **)

lemma irrefl_converse: "irrefl(A,r)  irrefl(A,converse(r))"
by (unfold irrefl_def, blast)

lemma trans_on_converse: "trans[A](r)  trans[A](converse(r))"
by (unfold trans_on_def, blast)

lemma part_ord_converse: "part_ord(A,r)  part_ord(A,converse(r))"
  unfolding part_ord_def
apply (blast intro!: irrefl_converse trans_on_converse)

lemma linear_converse: "linear(A,r)  linear(A,converse(r))"
by (unfold linear_def, blast)

lemma tot_ord_converse: "tot_ord(A,r)  tot_ord(A,converse(r))"
  unfolding tot_ord_def
apply (blast intro!: part_ord_converse linear_converse)

(** By Krzysztof Grabczewski.
    Lemmas involving the first element of a well ordered set **)

lemma first_is_elem: "first(b,B,r)  b  B"
by (unfold first_def, blast)

lemma well_ord_imp_ex1_first:
        "well_ord(A,r); B<=A; B0  (∃!b. first(b,B,r))"
  unfolding well_ord_def wf_on_def wf_def first_def
apply (elim conjE allE disjE, blast)
apply (erule bexE)
apply (rule_tac a = x in ex1I, auto)
apply (unfold tot_ord_def linear_def, blast)

lemma the_first_in:
     "well_ord(A,r); B<=A; B0  (THE b. first(b,B,r))  B"
apply (drule well_ord_imp_ex1_first, assumption+)
apply (rule first_is_elem)
apply (erule theI)

subsection ‹Lemmas for the Reflexive Orders›

lemma subset_vimage_vimage_iff:
  "Preorder(r); A  field(r); B  field(r) 
  r -`` A  r -`` B  (aA. bB. a, b  r)"
  apply (auto simp: subset_def preorder_on_def refl_def vimage_def image_def)
   apply blast
  unfolding trans_on_def
  apply (erule_tac P = "(λx. yfield(r).
          zfield(r). x, y  r  y, z  r  x, z  r)" for r in rev_ballE)
    (* instance obtained from proof term generated by best *)
   apply best
  apply blast

lemma subset_vimage1_vimage1_iff:
  "Preorder(r); a  field(r); b  field(r) 
  r -`` {a}  r -`` {b}  a, b  r"
  by (simp add: subset_vimage_vimage_iff)

lemma Refl_antisym_eq_Image1_Image1_iff:
  "refl(field(r), r); antisym(r); a  field(r); b  field(r) 
  r `` {a} = r `` {b}  a = b"
  apply rule
   apply (frule equality_iffD)
   apply (drule equality_iffD)
   apply (simp add: antisym_def refl_def)
   apply best
  apply (simp add: antisym_def refl_def)

lemma Partial_order_eq_Image1_Image1_iff:
  "Partial_order(r); a  field(r); b  field(r) 
  r `` {a} = r `` {b}  a = b"
  by (simp add: partial_order_on_def preorder_on_def

lemma Refl_antisym_eq_vimage1_vimage1_iff:
  "refl(field(r), r); antisym(r); a  field(r); b  field(r) 
  r -`` {a} = r -`` {b}  a = b"
  apply rule
   apply (frule equality_iffD)
   apply (drule equality_iffD)
   apply (simp add: antisym_def refl_def)
   apply best
  apply (simp add: antisym_def refl_def)

lemma Partial_order_eq_vimage1_vimage1_iff:
  "Partial_order(r); a  field(r); b  field(r) 
  r -`` {a} = r -`` {b}  a = b"
  by (simp add: partial_order_on_def preorder_on_def