src/ZF/Order.ML

nat_arith_tac -> arith_tac

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

Orders in Zermelo-Fraenkel Set Theory 

Results from the book "Set Theory: an Introduction to Independence Proofs"
        by Ken Kunen.  Chapter 1, section 6.
*)

open Order;

(** Basic properties of the definitions **)

(*needed?*)
Goalw [part_ord_def, irrefl_def, trans_on_def, asym_def]
    "part_ord(A,r) ==> asym(r Int A*A)";
by (Blast_tac 1);
qed "part_ord_Imp_asym";

val major::premx::premy::prems = Goalw [linear_def]
    "[| linear(A,r);  x:A;  y:A;  \
\       <x,y>:r ==> P;  x=y ==> P;  <y,x>:r ==> P |] ==> P";
by (cut_facts_tac [major,premx,premy] 1);
by (REPEAT_FIRST (eresolve_tac [ballE,disjE]));
by (EVERY1 (map etac prems));
by (ALLGOALS contr_tac);
qed "linearE";

(*Does the case analysis, deleting linear(A,r) and proving trivial subgoals*)
val linear_case_tac =
    SELECT_GOAL (EVERY [etac linearE 1, assume_tac 1, assume_tac 1,
                        REPEAT_SOME (assume_tac ORELSE' contr_tac)]);

(** General properties of well_ord **)

Goalw [irrefl_def, part_ord_def, tot_ord_def, 
                 trans_on_def, well_ord_def]
    "[| wf[A](r); linear(A,r) |] ==> well_ord(A,r)";
by (asm_simp_tac (simpset() addsimps [wf_on_not_refl]) 1);
by (fast_tac (claset() addEs [linearE, wf_on_asym, wf_on_chain3]) 1);
qed "well_ordI";

Goalw [well_ord_def]
    "well_ord(A,r) ==> wf[A](r)";
by Safe_tac;
qed "well_ord_is_wf";

Goalw [well_ord_def, tot_ord_def, part_ord_def]
    "well_ord(A,r) ==> trans[A](r)";
by Safe_tac;
qed "well_ord_is_trans_on";

Goalw [well_ord_def, tot_ord_def]
  "well_ord(A,r) ==> linear(A,r)";
by (Blast_tac 1);
qed "well_ord_is_linear";


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

Goalw [pred_def] "y : pred(A,x,r) <-> <y,x>:r & y:A";
by (Blast_tac 1);
qed "pred_iff";

bind_thm ("predI", conjI RS (pred_iff RS iffD2));

val [major,minor] = Goalw [pred_def]
    "[| y: pred(A,x,r);  [| y:A; <y,x>:r |] ==> P |] ==> P";
by (rtac (major RS CollectE) 1);
by (REPEAT (ares_tac [minor] 1));
qed "predE";

Goalw [pred_def] "pred(A,x,r) <= r -`` {x}";
by (Blast_tac 1);
qed "pred_subset_under";

Goalw [pred_def] "pred(A,x,r) <= A";
by (Blast_tac 1);
qed "pred_subset";

Goalw [pred_def]
    "pred(pred(A,x,r), y, r) = pred(A,x,r) Int pred(A,y,r)";
by (Blast_tac 1);
qed "pred_pred_eq";

Goalw [trans_on_def, pred_def]
    "[| trans[A](r);  <y,x>:r;  x:A;  y:A \
\         |] ==> pred(pred(A,x,r), y, r) = pred(A,y,r)";
by (Blast_tac 1);
qed "trans_pred_pred_eq";


(** 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*)
Goalw [part_ord_def, irrefl_def, trans_on_def]
    "[| part_ord(A,r);  B<=A |] ==> part_ord(B,r)";
by (Blast_tac 1);
qed "part_ord_subset";

Goalw [linear_def]
    "[| linear(A,r);  B<=A |] ==> linear(B,r)";
by (Blast_tac 1);
qed "linear_subset";

Goalw [tot_ord_def]
    "[| tot_ord(A,r);  B<=A |] ==> tot_ord(B,r)";
by (fast_tac (claset() addSEs [part_ord_subset, linear_subset]) 1);
qed "tot_ord_subset";

Goalw [well_ord_def]
    "[| well_ord(A,r);  B<=A |] ==> well_ord(B,r)";
by (fast_tac (claset() addSEs [tot_ord_subset, wf_on_subset_A]) 1);
qed "well_ord_subset";


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

Goalw [irrefl_def] "irrefl(A,r Int A*A) <-> irrefl(A,r)";
by (Blast_tac 1);
qed "irrefl_Int_iff";

Goalw [trans_on_def] "trans[A](r Int A*A) <-> trans[A](r)";
by (Blast_tac 1);
qed "trans_on_Int_iff";

Goalw [part_ord_def] "part_ord(A,r Int A*A) <-> part_ord(A,r)";
by (simp_tac (simpset() addsimps [irrefl_Int_iff, trans_on_Int_iff]) 1);
qed "part_ord_Int_iff";

Goalw [linear_def] "linear(A,r Int A*A) <-> linear(A,r)";
by (Blast_tac 1);
qed "linear_Int_iff";

Goalw [tot_ord_def] "tot_ord(A,r Int A*A) <-> tot_ord(A,r)";
by (simp_tac (simpset() addsimps [part_ord_Int_iff, linear_Int_iff]) 1);
qed "tot_ord_Int_iff";

Goalw [wf_on_def, wf_def] "wf[A](r Int A*A) <-> wf[A](r)";
by (Fast_tac 1);   (*10 times faster than Blast_tac!*) 
qed "wf_on_Int_iff";

Goalw [well_ord_def] "well_ord(A,r Int A*A) <-> well_ord(A,r)";
by (simp_tac (simpset() addsimps [tot_ord_Int_iff, wf_on_Int_iff]) 1);
qed "well_ord_Int_iff";


(** Relations over the Empty Set **)

Goalw [irrefl_def] "irrefl(0,r)";
by (Blast_tac 1);
qed "irrefl_0";

Goalw [trans_on_def] "trans[0](r)";
by (Blast_tac 1);
qed "trans_on_0";

Goalw [part_ord_def] "part_ord(0,r)";
by (simp_tac (simpset() addsimps [irrefl_0, trans_on_0]) 1);
qed "part_ord_0";

Goalw [linear_def] "linear(0,r)";
by (Blast_tac 1);
qed "linear_0";

Goalw [tot_ord_def] "tot_ord(0,r)";
by (simp_tac (simpset() addsimps [part_ord_0, linear_0]) 1);
qed "tot_ord_0";

Goalw [wf_on_def, wf_def] "wf[0](r)";
by (Blast_tac 1);
qed "wf_on_0";

Goalw [well_ord_def] "well_ord(0,r)";
by (simp_tac (simpset() addsimps [tot_ord_0, wf_on_0]) 1);
qed "well_ord_0";


(** The unit set is well-ordered by the empty relation (Grabczewski) **)

Goalw [irrefl_def, trans_on_def, part_ord_def, linear_def, tot_ord_def]
        "tot_ord({a},0)";
by (Simp_tac 1);
qed "tot_ord_unit";

Goalw [wf_on_def, wf_def] "wf[{a}](0)";
by (Fast_tac 1);
qed "wf_on_unit";

Goalw [well_ord_def] "well_ord({a},0)";
by (simp_tac (simpset() addsimps [tot_ord_unit, wf_on_unit]) 1);
qed "well_ord_unit";


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

Goalw [mono_map_def] 
    "f: mono_map(A,r,B,s) ==> f: A->B";
by (etac CollectD1 1);
qed "mono_map_is_fun";

Goalw [mono_map_def, inj_def] 
    "[| linear(A,r);  wf[B](s);  f: mono_map(A,r,B,s) |] ==> f: inj(A,B)";
by (Clarify_tac 1);
by (linear_case_tac 1);
by (REPEAT 
    (EVERY [eresolve_tac [wf_on_not_refl RS notE] 1,
            etac ssubst 2,
            Fast_tac 2,
            REPEAT (ares_tac [apply_type] 1)]));
qed "mono_map_is_inj";


(** Order-isomorphisms -- or similarities, as Suppes calls them **)

val prems = Goalw [ord_iso_def]
    "[| 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 (blast_tac (claset() addSIs prems) 1);
qed "ord_isoI";

Goalw [ord_iso_def, mono_map_def]
    "f: ord_iso(A,r,B,s) ==> f: mono_map(A,r,B,s)";
by (blast_tac (claset() addSDs [bij_is_fun]) 1);
qed "ord_iso_is_mono_map";

Goalw [ord_iso_def] 
    "f: ord_iso(A,r,B,s) ==> f: bij(A,B)";
by (etac CollectD1 1);
qed "ord_iso_is_bij";

(*Needed?  But ord_iso_converse is!*)
Goalw [ord_iso_def] 
    "[| f: ord_iso(A,r,B,s);  <x,y>: r;  x:A;  y:A |] ==> \
\         <f`x, f`y> : s";
by (Blast_tac 1);
qed "ord_iso_apply";

Goalw [ord_iso_def] 
    "[| f: ord_iso(A,r,B,s);  <x,y>: s;  x:B;  y:B |] ==> \
\         <converse(f) ` x, converse(f) ` y> : r";
by (etac CollectE 1);
by (etac (bspec RS bspec RS iffD2) 1);
by (REPEAT (eresolve_tac [asm_rl, 
                          bij_converse_bij RS bij_is_fun RS apply_type] 1));
by (asm_simp_tac (simpset() addsimps [right_inverse_bij]) 1);
qed "ord_iso_converse";


(*Rewriting with bijections and converse (function inverse)*)
val bij_inverse_ss = 
    simpset() setSolver (type_auto_tac [ord_iso_is_bij, bij_is_fun, apply_type, 
                                       bij_converse_bij, comp_fun, comp_bij])
          addsimps [right_inverse_bij, left_inverse_bij];

(** Symmetry and Transitivity Rules **)

(*Reflexivity of similarity*)
Goal "id(A): ord_iso(A,r,A,r)";
by (resolve_tac [id_bij RS ord_isoI] 1);
by (Asm_simp_tac 1);
qed "ord_iso_refl";

(*Symmetry of similarity*)
Goalw [ord_iso_def] 
    "f: ord_iso(A,r,B,s) ==> converse(f): ord_iso(B,s,A,r)";
by (fast_tac (claset() addss bij_inverse_ss) 1);
qed "ord_iso_sym";

(*Transitivity of similarity*)
Goalw [mono_map_def] 
    "[| g: mono_map(A,r,B,s);  f: mono_map(B,s,C,t) |] ==> \
\         (f O g): mono_map(A,r,C,t)";
by (fast_tac (claset() addss bij_inverse_ss) 1);
qed "mono_map_trans";

(*Transitivity of similarity: the order-isomorphism relation*)
Goalw [ord_iso_def] 
    "[| g: ord_iso(A,r,B,s);  f: ord_iso(B,s,C,t) |] ==> \
\         (f O g): ord_iso(A,r,C,t)";
by (fast_tac (claset() addss bij_inverse_ss) 1);
qed "ord_iso_trans";

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

Goalw [ord_iso_def, mono_map_def]
    "[| 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)";
by Safe_tac;
by (REPEAT_FIRST (ares_tac [fg_imp_bijective]));
by (Blast_tac 1);
by (subgoal_tac "<g`(f`x), g`(f`y)> : r" 1);
by (blast_tac (claset() addIs [apply_funtype]) 2);
by (asm_full_simp_tac (simpset() addsimps [comp_eq_id_iff RS iffD1]) 1);
qed "mono_ord_isoI";

Goal "[| 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)";
by (REPEAT (ares_tac [mono_ord_isoI] 1));
by (forward_tac [mono_map_is_fun RS fun_is_rel] 1);
by (etac (converse_converse RS subst) 1 THEN rtac left_comp_inverse 1);
by (DEPTH_SOLVE (ares_tac [mono_map_is_inj, left_comp_inverse] 1
          ORELSE eresolve_tac [well_ord_is_linear, well_ord_is_wf] 1));
qed "well_ord_mono_ord_isoI";


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

Goalw [part_ord_def, irrefl_def, trans_on_def, ord_iso_def]
    "[| part_ord(B,s);  f: ord_iso(A,r,B,s) |] ==> part_ord(A,r)";
by (Asm_simp_tac 1);
by (fast_tac (claset() addIs [bij_is_fun RS apply_type]) 1);
qed "part_ord_ord_iso";

Goalw [linear_def, ord_iso_def]
    "[| linear(B,s);  f: ord_iso(A,r,B,s) |] ==> linear(A,r)";
by (Asm_simp_tac 1);
by Safe_tac;
by (dres_inst_tac [("x1", "f`x"), ("x", "f`xa")] (bspec RS bspec) 1);
by (safe_tac (claset() addSEs [bij_is_fun RS apply_type]));
by (dres_inst_tac [("t", "op `(converse(f))")] subst_context 1);
by (asm_full_simp_tac (simpset() addsimps [left_inverse_bij]) 1);
qed "linear_ord_iso";

Goalw [wf_on_def, wf_def, ord_iso_def]
    "[| wf[B](s);  f: ord_iso(A,r,B,s) |] ==> wf[A](r)";
(*reversed &-congruence rule handles context of membership in A*)
by (asm_full_simp_tac (simpset() addcongs [conj_cong2]) 1);
by Safe_tac;
by (dres_inst_tac [("x", "{f`z. z:Z Int A}")] spec 1);
by (safe_tac (claset() addSIs [equalityI]));
by (ALLGOALS (blast_tac
	      (claset() addSDs [equalityD1] addIs [bij_is_fun RS apply_type])));
qed "wf_on_ord_iso";

Goalw [well_ord_def, tot_ord_def]
    "[| well_ord(B,s);  f: ord_iso(A,r,B,s) |] ==> well_ord(A,r)";
by (fast_tac
    (claset() addSEs [part_ord_ord_iso, linear_ord_iso, wf_on_ord_iso]) 1);
qed "well_ord_ord_iso";


(*** Main results of Kunen, Chapter 1 section 6 ***)

(*Inductive argument for Kunen's Lemma 6.1, etc. 
  Simple proof from Halmos, page 72*)
Goalw [well_ord_def, ord_iso_def]
  "[| well_ord(A,r);  f: ord_iso(A,r, A',r);  A'<= A;  y: A |] \
\       ==> ~ <f`y, y>: r";
by (REPEAT (eresolve_tac [conjE, CollectE] 1));
by (wf_on_ind_tac "y" [] 1);
by (dres_inst_tac [("a","y1")] (bij_is_fun RS apply_type) 1);
by (assume_tac 1);
by (Blast_tac 1);
qed "well_ord_iso_subset_lemma";

(*Kunen's Lemma 6.1: there's no order-isomorphism to an initial segment
                     of a well-ordering*)
Goal "[| well_ord(A,r);  f : ord_iso(A, r, pred(A,x,r), r);  x:A |] ==> P";
by (metacut_tac well_ord_iso_subset_lemma 1);
by (REPEAT_FIRST (ares_tac [pred_subset]));
(*Now we know  f`x < x *)
by (EVERY1 [dtac (ord_iso_is_bij RS bij_is_fun RS apply_type),
             assume_tac]);
(*Now we also know f`x : pred(A,x,r);  contradiction! *)
by (asm_full_simp_tac (simpset() addsimps [well_ord_def, pred_def]) 1);
qed "well_ord_iso_predE";

(*Simple consequence of Lemma 6.1*)
Goal "[| well_ord(A,r);  f : ord_iso(pred(A,a,r), r, pred(A,c,r), r);  \
\         a:A;  c:A |] ==> a=c";
by (forward_tac [well_ord_is_trans_on] 1);
by (forward_tac [well_ord_is_linear] 1);
by (linear_case_tac 1);
by (dtac ord_iso_sym 1);
by (REPEAT   (*because there are two symmetric cases*)
    (EVERY [eresolve_tac [pred_subset RSN (2, well_ord_subset) RS
                          well_ord_iso_predE] 1,
            blast_tac (claset() addSIs [predI]) 2,
            asm_simp_tac (simpset() addsimps [trans_pred_pred_eq]) 1]));
qed "well_ord_iso_pred_eq";

(*Does not assume r is a wellordering!*)
Goalw [ord_iso_def, pred_def]
 "[| f : ord_iso(A,r,B,s);   a:A |] ==>    \
\      f `` pred(A,a,r) = pred(B, f`a, s)";
by (etac CollectE 1);
by (asm_simp_tac 
    (simpset() addsimps [[bij_is_fun, Collect_subset] MRS image_fun]) 1);
by (rtac equalityI 1);
by (safe_tac (claset() addSEs [bij_is_fun RS apply_type]));
by (rtac RepFun_eqI 1);
by (blast_tac (claset() addSIs [right_inverse_bij RS sym]) 1);
by (asm_simp_tac bij_inverse_ss 1);
qed "ord_iso_image_pred";

(*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.*)
Goal "[| 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)";
by (asm_simp_tac (simpset() addsimps [ord_iso_image_pred RS sym]) 1);
by (rewtac ord_iso_def);
by (etac CollectE 1);
by (rtac CollectI 1);
by (asm_full_simp_tac (simpset() addsimps [pred_def]) 2);
by (eresolve_tac [[bij_is_inj, pred_subset] MRS restrict_bij] 1);
qed "ord_iso_restrict_pred";

(*Tricky; a lot of forward proof!*)
Goal "[| 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";
by (forward_tac [ord_iso_is_bij RS bij_is_fun RS apply_type] 1  THEN
    REPEAT1 (eresolve_tac [asm_rl, predI, predE] 1));
by (subgoal_tac "b = g`a" 1);
by (Asm_simp_tac 1);
by (rtac well_ord_iso_pred_eq 1);
by (REPEAT_SOME assume_tac);
by (forward_tac [ord_iso_restrict_pred] 1  THEN
    REPEAT1 (eresolve_tac [asm_rl, predI] 1));
by (asm_full_simp_tac
    (simpset() addsimps [well_ord_is_trans_on, trans_pred_pred_eq]) 1);
by (eresolve_tac [ord_iso_sym RS ord_iso_trans] 1);
by (assume_tac 1);
qed "well_ord_iso_preserving";

val  bij_apply_cs = claset() addSIs [bij_converse_bij]
                            addIs  [ord_iso_is_bij, bij_is_fun, apply_funtype];

(*See Halmos, page 72*)
Goal "[| well_ord(A,r);  \
\            f: ord_iso(A,r, B,s);  g: ord_iso(A,r, B,s);  y: A |] \
\         ==> ~ <g`y, f`y> : s";
by (forward_tac [well_ord_iso_subset_lemma] 1);
by (res_inst_tac [("f","converse(f)"), ("g","g")] ord_iso_trans 1);
by (REPEAT_FIRST (ares_tac [subset_refl, ord_iso_sym]));
by Safe_tac;
by (forward_tac [ord_iso_converse] 1);
by (EVERY (map (blast_tac bij_apply_cs) [1,1,1]));
by (asm_full_simp_tac bij_inverse_ss 1);
qed "well_ord_iso_unique_lemma";

(*Kunen's Lemma 6.2: Order-isomorphisms between well-orderings are unique*)
Goal "[| well_ord(A,r);  \
\            f: ord_iso(A,r, B,s);  g: ord_iso(A,r, B,s) |] ==> f = g";
by (rtac fun_extension 1);
by (REPEAT (etac (ord_iso_is_bij RS bij_is_fun) 1));
by (subgoals_tac ["f`x : B", "g`x : B", "linear(B,s)"] 1);
by (REPEAT (blast_tac bij_apply_cs 3));
by (dtac well_ord_ord_iso 2 THEN etac ord_iso_sym 2);
by (asm_full_simp_tac (simpset() addsimps [tot_ord_def, well_ord_def]) 2);
by (linear_case_tac 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, well_ord_iso_unique_lemma RS notE] 1));
qed "well_ord_iso_unique";


(** Towards Kunen's Theorem 6.3: linearity of the similarity relation **)

Goalw [ord_iso_map_def] "ord_iso_map(A,r,B,s) <= A*B";
by (Blast_tac 1);
qed "ord_iso_map_subset";

Goalw [ord_iso_map_def] "domain(ord_iso_map(A,r,B,s)) <= A";
by (Blast_tac 1);
qed "domain_ord_iso_map";

Goalw [ord_iso_map_def] "range(ord_iso_map(A,r,B,s)) <= B";
by (Blast_tac 1);
qed "range_ord_iso_map";

Goalw [ord_iso_map_def]
    "converse(ord_iso_map(A,r,B,s)) = ord_iso_map(B,s,A,r)";
by (blast_tac (claset() addIs [ord_iso_sym]) 1);
qed "converse_ord_iso_map";

Goalw [ord_iso_map_def, function_def]
    "well_ord(B,s) ==> function(ord_iso_map(A,r,B,s))";
by (blast_tac (claset() addIs [well_ord_iso_pred_eq, 
			      ord_iso_sym, ord_iso_trans]) 1);
qed "function_ord_iso_map";

Goal "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 (asm_simp_tac 
    (simpset() addsimps [Pi_iff, function_ord_iso_map,
                     ord_iso_map_subset RS domain_times_range]) 1);
qed "ord_iso_map_fun";

Goalw [mono_map_def]
    "[| well_ord(A,r);  well_ord(B,s) |] ==> ord_iso_map(A,r,B,s)  \
\          : mono_map(domain(ord_iso_map(A,r,B,s)), r,  \
\                     range(ord_iso_map(A,r,B,s)), s)";
by (asm_simp_tac (simpset() addsimps [ord_iso_map_fun]) 1);
by Safe_tac;
by (subgoals_tac ["x:A", "xa:A", "y:B", "ya:B"] 1);
by (REPEAT 
    (blast_tac (claset() addSEs [ord_iso_map_subset RS subsetD RS SigmaE]) 2));
by (asm_simp_tac 
    (simpset() addsimps [ord_iso_map_fun RSN (2,apply_equality)]) 1);
by (rewtac ord_iso_map_def);
by Safe_tac;
by (rtac well_ord_iso_preserving 1 THEN REPEAT_FIRST assume_tac);
qed "ord_iso_map_mono_map";

Goalw [mono_map_def]
    "[| 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)";
by (rtac well_ord_mono_ord_isoI 1);
by (resolve_tac [converse_ord_iso_map RS subst] 4);
by (asm_simp_tac 
    (simpset() addsimps [ord_iso_map_subset RS converse_converse]) 4);
by (REPEAT (ares_tac [ord_iso_map_mono_map] 3));
by (ALLGOALS (etac well_ord_subset));
by (ALLGOALS (resolve_tac [domain_ord_iso_map, range_ord_iso_map]));
qed "ord_iso_map_ord_iso";

(*One way of saying that domain(ord_iso_map(A,r,B,s)) is downwards-closed*)
Goalw [ord_iso_map_def]
  "[| 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)";
by (safe_tac (claset() addSIs [predI]));
(*Case analysis on  xaa vs a in r *)
by (forw_inst_tac [("A","A")] well_ord_is_linear 1);
by (linear_case_tac 1);
(*Trivial case: a=xa*)
by (Blast_tac 2);
(*Harder case: <a, xa>: r*)
by (forward_tac [ord_iso_is_bij RS bij_is_fun RS apply_type] 1  THEN
    REPEAT1 (eresolve_tac [asm_rl, predI, predE] 1));
by (forward_tac [ord_iso_restrict_pred] 1  THEN
    REPEAT1 (eresolve_tac [asm_rl, predI] 1));
by (asm_full_simp_tac
    (simpset() addsimps [well_ord_is_trans_on, trans_pred_pred_eq]) 1);
by (Blast_tac 1);
qed "domain_ord_iso_map_subset";

(*For the 4-way case analysis in the main result*)
Goal "[| well_ord(A,r);  well_ord(B,s) |] ==> \
\       domain(ord_iso_map(A,r,B,s)) = A |      \
\       (EX x:A. domain(ord_iso_map(A,r,B,s)) = pred(A,x,r))";
by (forward_tac [well_ord_is_wf] 1);
by (rewrite_goals_tac [wf_on_def, wf_def]);
by (dres_inst_tac [("x", "A-domain(ord_iso_map(A,r,B,s))")] spec 1);
by Safe_tac;
(*The first case: the domain equals A*)
by (rtac (domain_ord_iso_map RS equalityI) 1);
by (etac (Diff_eq_0_iff RS iffD1) 1);
(*The other case: the domain equals an initial segment*)
by (swap_res_tac [bexI] 1);
by (assume_tac 2);
by (rtac equalityI 1);
(*not (claset()) below; that would use rules like domainE!*)
by (blast_tac (claset() addSEs [predE]) 2);
by (REPEAT (ares_tac [domain_ord_iso_map_subset] 1));
qed "domain_ord_iso_map_cases";

(*As above, by duality*)
Goal "[| well_ord(A,r);  well_ord(B,s) |] ==> \
\       range(ord_iso_map(A,r,B,s)) = B |       \
\       (EX y:B. range(ord_iso_map(A,r,B,s))= pred(B,y,s))";
by (resolve_tac [converse_ord_iso_map RS subst] 1);
by (asm_simp_tac
    (simpset() addsimps [range_converse, domain_ord_iso_map_cases]) 1);
qed "range_ord_iso_map_cases";

(*Kunen's Theorem 6.3: Fundamental Theorem for Well-Ordered Sets*)
Goal "[| well_ord(A,r);  well_ord(B,s) |] ==>         \
\       ord_iso_map(A,r,B,s) : ord_iso(A, r, B, s) |    \
\       (EX x:A. ord_iso_map(A,r,B,s) : ord_iso(pred(A,x,r), r, B, s)) | \
\       (EX y:B. ord_iso_map(A,r,B,s) : ord_iso(A, r, pred(B,y,s), s))";
by (forw_inst_tac [("B","B")] domain_ord_iso_map_cases 1);
by (forw_inst_tac [("B","B")] range_ord_iso_map_cases 2);
by (REPEAT_FIRST (eresolve_tac [asm_rl, disjE, bexE]));
by (ALLGOALS (dtac ord_iso_map_ord_iso THEN' assume_tac THEN' 
              asm_full_simp_tac (simpset() addsimps [bexI])));
by (resolve_tac [wf_on_not_refl RS notE] 1);
by (etac well_ord_is_wf 1);
by (assume_tac 1);
by (subgoal_tac "<x,y>: ord_iso_map(A,r,B,s)" 1);
by (dtac rangeI 1);
by (asm_full_simp_tac (simpset() addsimps [pred_def]) 1);
by (rewtac ord_iso_map_def);
by (Blast_tac 1);
qed "well_ord_trichotomy";


(*** Properties of converse(r), by Krzysztof Grabczewski ***)

Goalw [irrefl_def] 
            "irrefl(A,r) ==> irrefl(A,converse(r))";
by (Blast_tac 1);
qed "irrefl_converse";

Goalw [trans_on_def] 
    "trans[A](r) ==> trans[A](converse(r))";
by (Blast_tac 1);
qed "trans_on_converse";

Goalw [part_ord_def] 
    "part_ord(A,r) ==> part_ord(A,converse(r))";
by (blast_tac (claset() addSIs [irrefl_converse, trans_on_converse]) 1);
qed "part_ord_converse";

Goalw [linear_def] 
    "linear(A,r) ==> linear(A,converse(r))";
by (Blast_tac 1);
qed "linear_converse";

Goalw [tot_ord_def] 
    "tot_ord(A,r) ==> tot_ord(A,converse(r))";
by (blast_tac (claset() addSIs [part_ord_converse, linear_converse]) 1);
qed "tot_ord_converse";


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

Goalw [first_def] "first(b,B,r) ==> b:B";
by (Blast_tac 1);
qed "first_is_elem";

Goalw [well_ord_def, wf_on_def, wf_def,     first_def] 
        "[| well_ord(A,r); B<=A; B~=0 |] ==> (EX! b. first(b,B,r))";
by (REPEAT (eresolve_tac [conjE,allE,disjE] 1));
by (contr_tac 1);
by (etac bexE 1);
by (res_inst_tac [("a","x")] ex1I 1);
by (Blast_tac 2);
by (rewrite_goals_tac [tot_ord_def, linear_def]);
by (Blast.depth_tac (claset()) 7 1);
qed "well_ord_imp_ex1_first";

Goal "[| well_ord(A,r); B<=A; B~=0 |] ==> (THE b. first(b,B,r)) : B";
by (dtac well_ord_imp_ex1_first 1 THEN REPEAT (assume_tac 1));
by (rtac first_is_elem 1);
by (etac theI 1);
qed "the_first_in";