(* Title: ZF/Constructible/Rank.thy ID: $Id: Rank.thy,v 1.6 2006/11/17 01:20:03 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) header {*Absoluteness for Order Types, Rank Functions and Well-Founded Relations*} theory Rank imports WF_absolute begin subsection {*Order Types: A Direct Construction by Replacement*} locale M_ordertype = M_basic + assumes well_ord_iso_separation: "[| M(A); M(f); M(r) |] ==> separation (M, λx. x∈A --> (∃y[M]. (∃p[M]. fun_apply(M,f,x,y) & pair(M,y,x,p) & p ∈ r)))" and obase_separation: --{*part of the order type formalization*} "[| M(A); M(r) |] ==> separation(M, λa. ∃x[M]. ∃g[M]. ∃mx[M]. ∃par[M]. ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))" and obase_equals_separation: "[| M(A); M(r) |] ==> separation (M, λx. x∈A --> ~(∃y[M]. ∃g[M]. ordinal(M,y) & (∃my[M]. ∃pxr[M]. membership(M,y,my) & pred_set(M,A,x,r,pxr) & order_isomorphism(M,pxr,r,y,my,g))))" and omap_replacement: "[| M(A); M(r) |] ==> strong_replacement(M, λa z. ∃x[M]. ∃g[M]. ∃mx[M]. ∃par[M]. ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) & pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))" text{*Inductive argument for Kunen's Lemma I 6.1, etc. Simple proof from Halmos, page 72*} lemma (in M_ordertype) wellordered_iso_subset_lemma: "[| wellordered(M,A,r); f ∈ ord_iso(A,r, A',r); A'<= A; y ∈ A; M(A); M(f); M(r) |] ==> ~ <f`y, y> ∈ r" apply (unfold wellordered_def ord_iso_def) apply (elim conjE CollectE) apply (erule wellfounded_on_induct, assumption+) apply (insert well_ord_iso_separation [of A f r]) apply (simp, clarify) apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast) done text{*Kunen's Lemma I 6.1, page 14: there's no order-isomorphism to an initial segment of a well-ordering*} lemma (in M_ordertype) wellordered_iso_predD: "[| wellordered(M,A,r); f ∈ ord_iso(A, r, Order.pred(A,x,r), r); M(A); M(f); M(r) |] ==> x ∉ A" apply (rule notI) apply (frule wellordered_iso_subset_lemma, assumption) apply (auto elim: predE) (*Now we know ~ (f`x < x) *) apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) (*Now we also know f`x ∈ pred(A,x,r); contradiction! *) apply (simp add: Order.pred_def) done lemma (in M_ordertype) wellordered_iso_pred_eq_lemma: "[| f ∈ 〈Order.pred(A,y,r), r〉 ≅ 〈Order.pred(A,x,r), r〉; wellordered(M,A,r); x∈A; y∈A; M(A); M(f); M(r) |] ==> <x,y> ∉ r" apply (frule wellordered_is_trans_on, assumption) apply (rule notI) apply (drule_tac x2=y and x=x and r2=r in wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD]) apply (simp add: trans_pred_pred_eq) apply (blast intro: predI dest: transM)+ done text{*Simple consequence of Lemma 6.1*} lemma (in M_ordertype) wellordered_iso_pred_eq: "[| wellordered(M,A,r); f ∈ ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r); M(A); M(f); M(r); a∈A; c∈A |] ==> a=c" apply (frule wellordered_is_trans_on, assumption) apply (frule wellordered_is_linear, assumption) apply (erule_tac x=a and y=c in linearE, auto) apply (drule ord_iso_sym) (*two symmetric cases*) apply (blast dest: wellordered_iso_pred_eq_lemma)+ done text{*Following Kunen's Theorem I 7.6, page 17. Note that this material is not required elsewhere.*} text{*Can't use @{text well_ord_iso_preserving} because it needs the strong premise @{term "well_ord(A,r)"}*} lemma (in M_ordertype) ord_iso_pred_imp_lt: "[| f ∈ ord_iso(Order.pred(A,x,r), r, i, Memrel(i)); g ∈ ord_iso(Order.pred(A,y,r), r, j, Memrel(j)); wellordered(M,A,r); x ∈ A; y ∈ A; M(A); M(r); M(f); M(g); M(j); Ord(i); Ord(j); 〈x,y〉 ∈ r |] ==> i < j" apply (frule wellordered_is_trans_on, assumption) apply (frule_tac y=y in transM, assumption) apply (rule_tac i=i and j=j in Ord_linear_lt, auto) txt{*case @{term "i=j"} yields a contradiction*} apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in wellordered_iso_predD [THEN notE]) apply (blast intro: wellordered_subset [OF _ pred_subset]) apply (simp add: trans_pred_pred_eq) apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) apply (simp_all add: pred_iff pred_closed converse_closed comp_closed) txt{*case @{term "j<i"} also yields a contradiction*} apply (frule restrict_ord_iso2, assumption+) apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun]) apply (frule apply_type, blast intro: ltD) --{*thus @{term "converse(f)`j ∈ Order.pred(A,x,r)"}*} apply (simp add: pred_iff) apply (subgoal_tac "∃h[M]. h ∈ ord_iso(Order.pred(A,y,r), r, Order.pred(A, converse(f)`j, r), r)") apply (clarify, frule wellordered_iso_pred_eq, assumption+) apply (blast dest: wellordered_asym) apply (intro rexI) apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+ done lemma ord_iso_converse1: "[| f: ord_iso(A,r,B,s); <b, f`a>: s; a:A; b:B |] ==> <converse(f) ` b, a> ∈ r" apply (frule ord_iso_converse, assumption+) apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype]) apply (simp add: left_inverse_bij [OF ord_iso_is_bij]) done definition obase :: "[i=>o,i,i] => i" where --{*the domain of @{text om}, eventually shown to equal @{text A}*} "obase(M,A,r) == {a∈A. ∃x[M]. ∃g[M]. Ord(x) & g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}" definition omap :: "[i=>o,i,i,i] => o" where --{*the function that maps wosets to order types*} "omap(M,A,r,f) == ∀z[M]. z ∈ f <-> (∃a∈A. ∃x[M]. ∃g[M]. z = <a,x> & Ord(x) & g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" definition otype :: "[i=>o,i,i,i] => o" where --{*the order types themselves*} "otype(M,A,r,i) == ∃f[M]. omap(M,A,r,f) & is_range(M,f,i)" text{*Can also be proved with the premise @{term "M(z)"} instead of @{term "M(f)"}, but that version is less useful. This lemma is also more useful than the definition, @{text omap_def}.*} lemma (in M_ordertype) omap_iff: "[| omap(M,A,r,f); M(A); M(f) |] ==> z ∈ f <-> (∃a∈A. ∃x[M]. ∃g[M]. z = <a,x> & Ord(x) & g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" apply (simp add: omap_def Memrel_closed pred_closed) apply (rule iffI) apply (drule_tac [2] x=z in rspec) apply (drule_tac x=z in rspec) apply (blast dest: transM)+ done lemma (in M_ordertype) omap_unique: "[| omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') |] ==> f' = f" apply (rule equality_iffI) apply (simp add: omap_iff) done lemma (in M_ordertype) omap_yields_Ord: "[| omap(M,A,r,f); 〈a,x〉 ∈ f; M(a); M(x) |] ==> Ord(x)" by (simp add: omap_def) lemma (in M_ordertype) otype_iff: "[| otype(M,A,r,i); M(A); M(r); M(i) |] ==> x ∈ i <-> (M(x) & Ord(x) & (∃a∈A. ∃g[M]. g ∈ ord_iso(Order.pred(A,a,r),r,x,Memrel(x))))" apply (auto simp add: omap_iff otype_def) apply (blast intro: transM) apply (rule rangeI) apply (frule transM, assumption) apply (simp add: omap_iff, blast) done lemma (in M_ordertype) otype_eq_range: "[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> i = range(f)" apply (auto simp add: otype_def omap_iff) apply (blast dest: omap_unique) done lemma (in M_ordertype) Ord_otype: "[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)" apply (rule OrdI) prefer 2 apply (simp add: Ord_def otype_def omap_def) apply clarify apply (frule pair_components_in_M, assumption) apply blast apply (auto simp add: Transset_def otype_iff) apply (blast intro: transM) apply (blast intro: Ord_in_Ord) apply (rename_tac y a g) apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun, THEN apply_funtype], assumption) apply (rule_tac x="converse(g)`y" in bexI) apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption) apply (safe elim!: predE) apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM) done lemma (in M_ordertype) domain_omap: "[| omap(M,A,r,f); M(A); M(r); M(B); M(f) |] ==> domain(f) = obase(M,A,r)" apply (simp add: domain_closed obase_def) apply (rule equality_iffI) apply (simp add: domain_iff omap_iff, blast) done lemma (in M_ordertype) omap_subset: "[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(B); M(i) |] ==> f ⊆ obase(M,A,r) * i" apply clarify apply (simp add: omap_iff obase_def) apply (force simp add: otype_iff) done lemma (in M_ordertype) omap_funtype: "[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> f ∈ obase(M,A,r) -> i" apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff) apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) done lemma (in M_ordertype) wellordered_omap_bij: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> f ∈ bij(obase(M,A,r),i)" apply (insert omap_funtype [of A r f i]) apply (auto simp add: bij_def inj_def) prefer 2 apply (blast intro: fun_is_surj dest: otype_eq_range) apply (frule_tac a=w in apply_Pair, assumption) apply (frule_tac a=x in apply_Pair, assumption) apply (simp add: omap_iff) apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans) done text{*This is not the final result: we must show @{term "oB(A,r) = A"}*} lemma (in M_ordertype) omap_ord_iso: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> f ∈ ord_iso(obase(M,A,r),r,i,Memrel(i))" apply (rule ord_isoI) apply (erule wellordered_omap_bij, assumption+) apply (insert omap_funtype [of A r f i], simp) apply (frule_tac a=x in apply_Pair, assumption) apply (frule_tac a=y in apply_Pair, assumption) apply (auto simp add: omap_iff) txt{*direction 1: assuming @{term "〈x,y〉 ∈ r"}*} apply (blast intro: ltD ord_iso_pred_imp_lt) txt{*direction 2: proving @{term "〈x,y〉 ∈ r"} using linearity of @{term r}*} apply (rename_tac x y g ga) apply (frule wellordered_is_linear, assumption, erule_tac x=x and y=y in linearE, assumption+) txt{*the case @{term "x=y"} leads to immediate contradiction*} apply (blast elim: mem_irrefl) txt{*the case @{term "〈y,x〉 ∈ r"}: handle like the opposite direction*} apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym) done lemma (in M_ordertype) Ord_omap_image_pred: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i); b ∈ A |] ==> Ord(f `` Order.pred(A,b,r))" apply (frule wellordered_is_trans_on, assumption) apply (rule OrdI) prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast) txt{*Hard part is to show that the image is a transitive set.*} apply (simp add: Transset_def, clarify) apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f i]]) apply (rename_tac c j, clarify) apply (frule omap_funtype [of A r f, THEN apply_funtype], assumption+) apply (subgoal_tac "j ∈ i") prefer 2 apply (blast intro: Ord_trans Ord_otype) apply (subgoal_tac "converse(f) ` j ∈ obase(M,A,r)") prefer 2 apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij, THEN bij_is_fun, THEN apply_funtype]) apply (rule_tac x="converse(f) ` j" in bexI) apply (simp add: right_inverse_bij [OF wellordered_omap_bij]) apply (intro predI conjI) apply (erule_tac b=c in trans_onD) apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f i]]) apply (auto simp add: obase_def) done lemma (in M_ordertype) restrict_omap_ord_iso: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); D ⊆ obase(M,A,r); M(A); M(r); M(f); M(i) |] ==> restrict(f,D) ∈ (〈D,r〉 ≅ 〈f``D, Memrel(f``D)〉)" apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f i]], assumption+) apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel]) apply (blast dest: subsetD [OF omap_subset]) apply (drule ord_iso_sym, simp) done lemma (in M_ordertype) obase_equals: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> obase(M,A,r) = A" apply (rule equalityI, force simp add: obase_def, clarify) apply (unfold obase_def, simp) apply (frule wellordered_is_wellfounded_on, assumption) apply (erule wellfounded_on_induct, assumption+) apply (frule obase_equals_separation [of A r], assumption) apply (simp, clarify) apply (rename_tac b) apply (subgoal_tac "Order.pred(A,b,r) <= obase(M,A,r)") apply (blast intro!: restrict_omap_ord_iso Ord_omap_image_pred) apply (force simp add: pred_iff obase_def) done text{*Main result: @{term om} gives the order-isomorphism @{term "〈A,r〉 ≅ 〈i, Memrel(i)〉"} *} theorem (in M_ordertype) omap_ord_iso_otype: "[| wellordered(M,A,r); omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> f ∈ ord_iso(A, r, i, Memrel(i))" apply (frule omap_ord_iso, assumption+) apply (simp add: obase_equals) done lemma (in M_ordertype) obase_exists: "[| M(A); M(r) |] ==> M(obase(M,A,r))" apply (simp add: obase_def) apply (insert obase_separation [of A r]) apply (simp add: separation_def) done lemma (in M_ordertype) omap_exists: "[| M(A); M(r) |] ==> ∃z[M]. omap(M,A,r,z)" apply (simp add: omap_def) apply (insert omap_replacement [of A r]) apply (simp add: strong_replacement_def) apply (drule_tac x="obase(M,A,r)" in rspec) apply (simp add: obase_exists) apply (simp add: Memrel_closed pred_closed obase_def) apply (erule impE) apply (clarsimp simp add: univalent_def) apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify) apply (rule_tac x=Y in rexI) apply (simp add: Memrel_closed pred_closed obase_def, blast, assumption) done declare rall_simps [simp] rex_simps [simp] lemma (in M_ordertype) otype_exists: "[| wellordered(M,A,r); M(A); M(r) |] ==> ∃i[M]. otype(M,A,r,i)" apply (insert omap_exists [of A r]) apply (simp add: otype_def, safe) apply (rule_tac x="range(x)" in rexI) apply blast+ done lemma (in M_ordertype) ordertype_exists: "[| wellordered(M,A,r); M(A); M(r) |] ==> ∃f[M]. (∃i[M]. Ord(i) & f ∈ ord_iso(A, r, i, Memrel(i)))" apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) apply (rename_tac i) apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) apply (rule Ord_otype) apply (force simp add: otype_def range_closed) apply (simp_all add: wellordered_is_trans_on) done lemma (in M_ordertype) relativized_imp_well_ord: "[| wellordered(M,A,r); M(A); M(r) |] ==> well_ord(A,r)" apply (insert ordertype_exists [of A r], simp) apply (blast intro: well_ord_ord_iso well_ord_Memrel) done subsection {*Kunen's theorem 5.4, page 127*} text{*(a) The notion of Wellordering is absolute*} theorem (in M_ordertype) well_ord_abs [simp]: "[| M(A); M(r) |] ==> wellordered(M,A,r) <-> well_ord(A,r)" by (blast intro: well_ord_imp_relativized relativized_imp_well_ord) text{*(b) Order types are absolute*} theorem (in M_ordertype) "[| wellordered(M,A,r); f ∈ ord_iso(A, r, i, Memrel(i)); M(A); M(r); M(f); M(i); Ord(i) |] ==> i = ordertype(A,r)" by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso Ord_iso_implies_eq ord_iso_sym ord_iso_trans) subsection{*Ordinal Arithmetic: Two Examples of Recursion*} text{*Note: the remainder of this theory is not needed elsewhere.*} subsubsection{*Ordinal Addition*} (*FIXME: update to use new techniques!!*) (*This expresses ordinal addition in the language of ZF. It also provides an abbreviation that can be used in the instance of strong replacement below. Here j is used to define the relation, namely Memrel(succ(j)), while x determines the domain of f.*) definition is_oadd_fun :: "[i=>o,i,i,i,i] => o" where "is_oadd_fun(M,i,j,x,f) == (∀sj msj. M(sj) --> M(msj) --> successor(M,j,sj) --> membership(M,sj,msj) --> M_is_recfun(M, %x g y. ∃gx[M]. image(M,g,x,gx) & union(M,i,gx,y), msj, x, f))" definition is_oadd :: "[i=>o,i,i,i] => o" where "is_oadd(M,i,j,k) == (~ ordinal(M,i) & ~ ordinal(M,j) & k=0) | (~ ordinal(M,i) & ordinal(M,j) & k=j) | (ordinal(M,i) & ~ ordinal(M,j) & k=i) | (ordinal(M,i) & ordinal(M,j) & (∃f fj sj. M(f) & M(fj) & M(sj) & successor(M,j,sj) & is_oadd_fun(M,i,sj,sj,f) & fun_apply(M,f,j,fj) & fj = k))" definition (*NEEDS RELATIVIZATION*) omult_eqns :: "[i,i,i,i] => o" where "omult_eqns(i,x,g,z) == Ord(x) & (x=0 --> z=0) & (∀j. x = succ(j) --> z = g`j ++ i) & (Limit(x) --> z = \<Union>(g``x))" definition is_omult_fun :: "[i=>o,i,i,i] => o" where "is_omult_fun(M,i,j,f) == (∃df. M(df) & is_function(M,f) & is_domain(M,f,df) & subset(M, j, df)) & (∀x∈j. omult_eqns(i,x,f,f`x))" definition is_omult :: "[i=>o,i,i,i] => o" where "is_omult(M,i,j,k) == ∃f fj sj. M(f) & M(fj) & M(sj) & successor(M,j,sj) & is_omult_fun(M,i,sj,f) & fun_apply(M,f,j,fj) & fj = k" locale M_ord_arith = M_ordertype + assumes oadd_strong_replacement: "[| M(i); M(j) |] ==> strong_replacement(M, λx z. ∃y[M]. pair(M,x,y,z) & (∃f[M]. ∃fx[M]. is_oadd_fun(M,i,j,x,f) & image(M,f,x,fx) & y = i Un fx))" and omult_strong_replacement': "[| M(i); M(j) |] ==> strong_replacement(M, λx z. ∃y[M]. z = <x,y> & (∃g[M]. is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) & y = (THE z. omult_eqns(i, x, g, z))))" text{*@{text is_oadd_fun}: Relating the pure "language of set theory" to Isabelle/ZF*} lemma (in M_ord_arith) is_oadd_fun_iff: "[| a≤j; M(i); M(j); M(a); M(f) |] ==> is_oadd_fun(M,i,j,a,f) <-> f ∈ a -> range(f) & (∀x. M(x) --> x < a --> f`x = i Un f``x)" apply (frule lt_Ord) apply (simp add: is_oadd_fun_def Memrel_closed Un_closed relation2_def is_recfun_abs [of "%x g. i Un g``x"] image_closed is_recfun_iff_equation Ball_def lt_trans [OF ltI, of _ a] lt_Memrel) apply (simp add: lt_def) apply (blast dest: transM) done lemma (in M_ord_arith) oadd_strong_replacement': "[| M(i); M(j) |] ==> strong_replacement(M, λx z. ∃y[M]. z = <x,y> & (∃g[M]. is_recfun(Memrel(succ(j)),x,%x g. i Un g``x,g) & y = i Un g``x))" apply (insert oadd_strong_replacement [of i j]) apply (simp add: is_oadd_fun_def relation2_def is_recfun_abs [of "%x g. i Un g``x"]) done lemma (in M_ord_arith) exists_oadd: "[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_recfun(Memrel(succ(j)), j, %x g. i Un g``x, f)" apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel]) apply (simp_all add: Memrel_type oadd_strong_replacement') done lemma (in M_ord_arith) exists_oadd_fun: "[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_oadd_fun(M,i,succ(j),succ(j),f)" apply (rule exists_oadd [THEN rexE]) apply (erule Ord_succ, assumption, simp) apply (rename_tac f) apply (frule is_recfun_type) apply (rule_tac x=f in rexI) apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def is_oadd_fun_iff Ord_trans [OF _ succI1], assumption) done lemma (in M_ord_arith) is_oadd_fun_apply: "[| x < j; M(i); M(j); M(f); is_oadd_fun(M,i,j,j,f) |] ==> f`x = i Un (\<Union>k∈x. {f ` k})" apply (simp add: is_oadd_fun_iff lt_Ord2, clarify) apply (frule lt_closed, simp) apply (frule leI [THEN le_imp_subset]) apply (simp add: image_fun, blast) done lemma (in M_ord_arith) is_oadd_fun_iff_oadd [rule_format]: "[| is_oadd_fun(M,i,J,J,f); M(i); M(J); M(f); Ord(i); Ord(j) |] ==> j<J --> f`j = i++j" apply (erule_tac i=j in trans_induct, clarify) apply (subgoal_tac "∀k∈x. k<J") apply (simp (no_asm_simp) add: is_oadd_def oadd_unfold is_oadd_fun_apply) apply (blast intro: lt_trans ltI lt_Ord) done lemma (in M_ord_arith) Ord_oadd_abs: "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_oadd(M,i,j,k) <-> k = i++j" apply (simp add: is_oadd_def is_oadd_fun_iff_oadd) apply (frule exists_oadd_fun [of j i], blast+) done lemma (in M_ord_arith) oadd_abs: "[| M(i); M(j); M(k) |] ==> is_oadd(M,i,j,k) <-> k = i++j" apply (case_tac "Ord(i) & Ord(j)") apply (simp add: Ord_oadd_abs) apply (auto simp add: is_oadd_def oadd_eq_if_raw_oadd) done lemma (in M_ord_arith) oadd_closed [intro,simp]: "[| M(i); M(j) |] ==> M(i++j)" apply (simp add: oadd_eq_if_raw_oadd, clarify) apply (simp add: raw_oadd_eq_oadd) apply (frule exists_oadd_fun [of j i], auto) apply (simp add: apply_closed is_oadd_fun_iff_oadd [symmetric]) done subsubsection{*Ordinal Multiplication*} lemma omult_eqns_unique: "[| omult_eqns(i,x,g,z); omult_eqns(i,x,g,z') |] ==> z=z'"; apply (simp add: omult_eqns_def, clarify) apply (erule Ord_cases, simp_all) done lemma omult_eqns_0: "omult_eqns(i,0,g,z) <-> z=0" by (simp add: omult_eqns_def) lemma the_omult_eqns_0: "(THE z. omult_eqns(i,0,g,z)) = 0" by (simp add: omult_eqns_0) lemma omult_eqns_succ: "omult_eqns(i,succ(j),g,z) <-> Ord(j) & z = g`j ++ i" by (simp add: omult_eqns_def) lemma the_omult_eqns_succ: "Ord(j) ==> (THE z. omult_eqns(i,succ(j),g,z)) = g`j ++ i" by (simp add: omult_eqns_succ) lemma omult_eqns_Limit: "Limit(x) ==> omult_eqns(i,x,g,z) <-> z = \<Union>(g``x)" apply (simp add: omult_eqns_def) apply (blast intro: Limit_is_Ord) done lemma the_omult_eqns_Limit: "Limit(x) ==> (THE z. omult_eqns(i,x,g,z)) = \<Union>(g``x)" by (simp add: omult_eqns_Limit) lemma omult_eqns_Not: "~ Ord(x) ==> ~ omult_eqns(i,x,g,z)" by (simp add: omult_eqns_def) lemma (in M_ord_arith) the_omult_eqns_closed: "[| M(i); M(x); M(g); function(g) |] ==> M(THE z. omult_eqns(i, x, g, z))" apply (case_tac "Ord(x)") prefer 2 apply (simp add: omult_eqns_Not) --{*trivial, non-Ord case*} apply (erule Ord_cases) apply (simp add: omult_eqns_0) apply (simp add: omult_eqns_succ apply_closed oadd_closed) apply (simp add: omult_eqns_Limit) done lemma (in M_ord_arith) exists_omult: "[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_recfun(Memrel(succ(j)), j, %x g. THE z. omult_eqns(i,x,g,z), f)" apply (rule wf_exists_is_recfun [OF wf_Memrel trans_Memrel]) apply (simp_all add: Memrel_type omult_strong_replacement') apply (blast intro: the_omult_eqns_closed) done lemma (in M_ord_arith) exists_omult_fun: "[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_omult_fun(M,i,succ(j),f)" apply (rule exists_omult [THEN rexE]) apply (erule Ord_succ, assumption, simp) apply (rename_tac f) apply (frule is_recfun_type) apply (rule_tac x=f in rexI) apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def is_omult_fun_def Ord_trans [OF _ succI1]) apply (force dest: Ord_in_Ord' simp add: omult_eqns_def the_omult_eqns_0 the_omult_eqns_succ the_omult_eqns_Limit, assumption) done lemma (in M_ord_arith) is_omult_fun_apply_0: "[| 0 < j; is_omult_fun(M,i,j,f) |] ==> f`0 = 0" by (simp add: is_omult_fun_def omult_eqns_def lt_def ball_conj_distrib) lemma (in M_ord_arith) is_omult_fun_apply_succ: "[| succ(x) < j; is_omult_fun(M,i,j,f) |] ==> f`succ(x) = f`x ++ i" by (simp add: is_omult_fun_def omult_eqns_def lt_def, blast) lemma (in M_ord_arith) is_omult_fun_apply_Limit: "[| x < j; Limit(x); M(j); M(f); is_omult_fun(M,i,j,f) |] ==> f ` x = (\<Union>y∈x. f`y)" apply (simp add: is_omult_fun_def omult_eqns_def domain_closed lt_def, clarify) apply (drule subset_trans [OF OrdmemD], assumption+) apply (simp add: ball_conj_distrib omult_Limit image_function) done lemma (in M_ord_arith) is_omult_fun_eq_omult: "[| is_omult_fun(M,i,J,f); M(J); M(f); Ord(i); Ord(j) |] ==> j<J --> f`j = i**j" apply (erule_tac i=j in trans_induct3) apply (safe del: impCE) apply (simp add: is_omult_fun_apply_0) apply (subgoal_tac "x<J") apply (simp add: is_omult_fun_apply_succ omult_succ) apply (blast intro: lt_trans) apply (subgoal_tac "∀k∈x. k<J") apply (simp add: is_omult_fun_apply_Limit omult_Limit) apply (blast intro: lt_trans ltI lt_Ord) done lemma (in M_ord_arith) omult_abs: "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_omult(M,i,j,k) <-> k = i**j" apply (simp add: is_omult_def is_omult_fun_eq_omult) apply (frule exists_omult_fun [of j i], blast+) done subsection {*Absoluteness of Well-Founded Relations*} text{*Relativized to @{term M}: Every well-founded relation is a subset of some inverse image of an ordinal. Key step is the construction (in @{term M}) of a rank function.*} locale M_wfrank = M_trancl + assumes wfrank_separation: "M(r) ==> separation (M, λx. ∀rplus[M]. tran_closure(M,r,rplus) --> ~ (∃f[M]. M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f)))" and wfrank_strong_replacement: "M(r) ==> strong_replacement(M, λx z. ∀rplus[M]. tran_closure(M,r,rplus) --> (∃y[M]. ∃f[M]. pair(M,x,y,z) & M_is_recfun(M, %x f y. is_range(M,f,y), rplus, x, f) & is_range(M,f,y)))" and Ord_wfrank_separation: "M(r) ==> separation (M, λx. ∀rplus[M]. tran_closure(M,r,rplus) --> ~ (∀f[M]. ∀rangef[M]. is_range(M,f,rangef) --> M_is_recfun(M, λx f y. is_range(M,f,y), rplus, x, f) --> ordinal(M,rangef)))" text{*Proving that the relativized instances of Separation or Replacement agree with the "real" ones.*} lemma (in M_wfrank) wfrank_separation': "M(r) ==> separation (M, λx. ~ (∃f[M]. is_recfun(r^+, x, %x f. range(f), f)))" apply (insert wfrank_separation [of r]) apply (simp add: relation2_def is_recfun_abs [of "%x. range"]) done lemma (in M_wfrank) wfrank_strong_replacement': "M(r) ==> strong_replacement(M, λx z. ∃y[M]. ∃f[M]. pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) & y = range(f))" apply (insert wfrank_strong_replacement [of r]) apply (simp add: relation2_def is_recfun_abs [of "%x. range"]) done lemma (in M_wfrank) Ord_wfrank_separation': "M(r) ==> separation (M, λx. ~ (∀f[M]. is_recfun(r^+, x, λx. range, f) --> Ord(range(f))))" apply (insert Ord_wfrank_separation [of r]) apply (simp add: relation2_def is_recfun_abs [of "%x. range"]) done text{*This function, defined using replacement, is a rank function for well-founded relations within the class M.*} definition wellfoundedrank :: "[i=>o,i,i] => i" where "wellfoundedrank(M,r,A) == {p. x∈A, ∃y[M]. ∃f[M]. p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) & y = range(f)}" lemma (in M_wfrank) exists_wfrank: "[| wellfounded(M,r); M(a); M(r) |] ==> ∃f[M]. is_recfun(r^+, a, %x f. range(f), f)" apply (rule wellfounded_exists_is_recfun) apply (blast intro: wellfounded_trancl) apply (rule trans_trancl) apply (erule wfrank_separation') apply (erule wfrank_strong_replacement') apply (simp_all add: trancl_subset_times) done lemma (in M_wfrank) M_wellfoundedrank: "[| wellfounded(M,r); M(r); M(A) |] ==> M(wellfoundedrank(M,r,A))" apply (insert wfrank_strong_replacement' [of r]) apply (simp add: wellfoundedrank_def) apply (rule strong_replacement_closed) apply assumption+ apply (rule univalent_is_recfun) apply (blast intro: wellfounded_trancl) apply (rule trans_trancl) apply (simp add: trancl_subset_times) apply (blast dest: transM) done lemma (in M_wfrank) Ord_wfrank_range [rule_format]: "[| wellfounded(M,r); a∈A; M(r); M(A) |] ==> ∀f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))" apply (drule wellfounded_trancl, assumption) apply (rule wellfounded_induct, assumption, erule (1) transM) apply simp apply (blast intro: Ord_wfrank_separation', clarify) txt{*The reasoning in both cases is that we get @{term y} such that @{term "〈y, x〉 ∈ r^+"}. We find that @{term "f`y = restrict(f, r^+ -`` {y})"}. *} apply (rule OrdI [OF _ Ord_is_Transset]) txt{*An ordinal is a transitive set...*} apply (simp add: Transset_def) apply clarify apply (frule apply_recfun2, assumption) apply (force simp add: restrict_iff) txt{*...of ordinals. This second case requires the induction hyp.*} apply clarify apply (rename_tac i y) apply (frule apply_recfun2, assumption) apply (frule is_recfun_imp_in_r, assumption) apply (frule is_recfun_restrict) (*simp_all won't work*) apply (simp add: trans_trancl trancl_subset_times)+ apply (drule spec [THEN mp], assumption) apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))") apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec) apply assumption apply (simp add: function_apply_equality [OF _ is_recfun_imp_function]) apply (blast dest: pair_components_in_M) done lemma (in M_wfrank) Ord_range_wellfoundedrank: "[| wellfounded(M,r); r ⊆ A*A; M(r); M(A) |] ==> Ord (range(wellfoundedrank(M,r,A)))" apply (frule wellfounded_trancl, assumption) apply (frule trancl_subset_times) apply (simp add: wellfoundedrank_def) apply (rule OrdI [OF _ Ord_is_Transset]) prefer 2 txt{*by our previous result the range consists of ordinals.*} apply (blast intro: Ord_wfrank_range) txt{*We still must show that the range is a transitive set.*} apply (simp add: Transset_def, clarify, simp) apply (rename_tac x i f u) apply (frule is_recfun_imp_in_r, assumption) apply (subgoal_tac "M(u) & M(i) & M(x)") prefer 2 apply (blast dest: transM, clarify) apply (rule_tac a=u in rangeI) apply (rule_tac x=u in ReplaceI) apply simp apply (rule_tac x="restrict(f, r^+ -`` {u})" in rexI) apply (blast intro: is_recfun_restrict trans_trancl dest: apply_recfun2) apply simp apply blast txt{*Unicity requirement of Replacement*} apply clarify apply (frule apply_recfun2, assumption) apply (simp add: trans_trancl is_recfun_cut) done lemma (in M_wfrank) function_wellfoundedrank: "[| wellfounded(M,r); M(r); M(A)|] ==> function(wellfoundedrank(M,r,A))" apply (simp add: wellfoundedrank_def function_def, clarify) txt{*Uniqueness: repeated below!*} apply (drule is_recfun_functional, assumption) apply (blast intro: wellfounded_trancl) apply (simp_all add: trancl_subset_times trans_trancl) done lemma (in M_wfrank) domain_wellfoundedrank: "[| wellfounded(M,r); M(r); M(A)|] ==> domain(wellfoundedrank(M,r,A)) = A" apply (simp add: wellfoundedrank_def function_def) apply (rule equalityI, auto) apply (frule transM, assumption) apply (frule_tac a=x in exists_wfrank, assumption+, clarify) apply (rule_tac b="range(f)" in domainI) apply (rule_tac x=x in ReplaceI) apply simp apply (rule_tac x=f in rexI, blast, simp_all) txt{*Uniqueness (for Replacement): repeated above!*} apply clarify apply (drule is_recfun_functional, assumption) apply (blast intro: wellfounded_trancl) apply (simp_all add: trancl_subset_times trans_trancl) done lemma (in M_wfrank) wellfoundedrank_type: "[| wellfounded(M,r); M(r); M(A)|] ==> wellfoundedrank(M,r,A) ∈ A -> range(wellfoundedrank(M,r,A))" apply (frule function_wellfoundedrank [of r A], assumption+) apply (frule function_imp_Pi) apply (simp add: wellfoundedrank_def relation_def) apply blast apply (simp add: domain_wellfoundedrank) done lemma (in M_wfrank) Ord_wellfoundedrank: "[| wellfounded(M,r); a ∈ A; r ⊆ A*A; M(r); M(A) |] ==> Ord(wellfoundedrank(M,r,A) ` a)" by (blast intro: apply_funtype [OF wellfoundedrank_type] Ord_in_Ord [OF Ord_range_wellfoundedrank]) lemma (in M_wfrank) wellfoundedrank_eq: "[| is_recfun(r^+, a, %x. range, f); wellfounded(M,r); a ∈ A; M(f); M(r); M(A)|] ==> wellfoundedrank(M,r,A) ` a = range(f)" apply (rule apply_equality) prefer 2 apply (blast intro: wellfoundedrank_type) apply (simp add: wellfoundedrank_def) apply (rule ReplaceI) apply (rule_tac x="range(f)" in rexI) apply blast apply simp_all txt{*Unicity requirement of Replacement*} apply clarify apply (drule is_recfun_functional, assumption) apply (blast intro: wellfounded_trancl) apply (simp_all add: trancl_subset_times trans_trancl) done lemma (in M_wfrank) wellfoundedrank_lt: "[| <a,b> ∈ r; wellfounded(M,r); r ⊆ A*A; M(r); M(A)|] ==> wellfoundedrank(M,r,A) ` a < wellfoundedrank(M,r,A) ` b" apply (frule wellfounded_trancl, assumption) apply (subgoal_tac "a∈A & b∈A") prefer 2 apply blast apply (simp add: lt_def Ord_wellfoundedrank, clarify) apply (frule exists_wfrank [of concl: _ b], erule (1) transM, assumption) apply clarify apply (rename_tac fb) apply (frule is_recfun_restrict [of concl: "r^+" a]) apply (rule trans_trancl, assumption) apply (simp_all add: r_into_trancl trancl_subset_times) txt{*Still the same goal, but with new @{text is_recfun} assumptions.*} apply (simp add: wellfoundedrank_eq) apply (frule_tac a=a in wellfoundedrank_eq, assumption+) apply (simp_all add: transM [of a]) txt{*We have used equations for wellfoundedrank and now must use some for @{text is_recfun}. *} apply (rule_tac a=a in rangeI) apply (simp add: is_recfun_type [THEN apply_iff] vimage_singleton_iff r_into_trancl apply_recfun r_into_trancl) done lemma (in M_wfrank) wellfounded_imp_subset_rvimage: "[|wellfounded(M,r); r ⊆ A*A; M(r); M(A)|] ==> ∃i f. Ord(i) & r <= rvimage(A, f, Memrel(i))" apply (rule_tac x="range(wellfoundedrank(M,r,A))" in exI) apply (rule_tac x="wellfoundedrank(M,r,A)" in exI) apply (simp add: Ord_range_wellfoundedrank, clarify) apply (frule subsetD, assumption, clarify) apply (simp add: rvimage_iff wellfoundedrank_lt [THEN ltD]) apply (blast intro: apply_rangeI wellfoundedrank_type) done lemma (in M_wfrank) wellfounded_imp_wf: "[|wellfounded(M,r); relation(r); M(r)|] ==> wf(r)" by (blast dest!: relation_field_times_field wellfounded_imp_subset_rvimage intro: wf_rvimage_Ord [THEN wf_subset]) lemma (in M_wfrank) wellfounded_on_imp_wf_on: "[|wellfounded_on(M,A,r); relation(r); M(r); M(A)|] ==> wf[A](r)" apply (simp add: wellfounded_on_iff_wellfounded wf_on_def) apply (rule wellfounded_imp_wf) apply (simp_all add: relation_def) done theorem (in M_wfrank) wf_abs: "[|relation(r); M(r)|] ==> wellfounded(M,r) <-> wf(r)" by (blast intro: wellfounded_imp_wf wf_imp_relativized) theorem (in M_wfrank) wf_on_abs: "[|relation(r); M(r); M(A)|] ==> wellfounded_on(M,A,r) <-> wf[A](r)" by (blast intro: wellfounded_on_imp_wf_on wf_on_imp_relativized) end
lemma wellordered_iso_subset_lemma:
[| wellordered(M, A, r); f ∈ ord_iso(A, r, A', r); A' ⊆ A; y ∈ A; M(A); M(f);
M(r) |]
==> 〈f ` y, y〉 ∉ r
lemma wellordered_iso_predD:
[| wellordered(M, A, r); f ∈ ord_iso(A, r, Order.pred(A, x, r), r); M(A); M(f);
M(r) |]
==> x ∉ A
lemma wellordered_iso_pred_eq_lemma:
[| f ∈ ord_iso(Order.pred(A, y, r), r, Order.pred(A, x, r), r);
wellordered(M, A, r); x ∈ A; y ∈ A; M(A); M(f); M(r) |]
==> 〈x, y〉 ∉ r
lemma wellordered_iso_pred_eq:
[| wellordered(M, A, r);
f ∈ ord_iso(Order.pred(A, a, r), r, Order.pred(A, c, r), r); M(A); M(f);
M(r); a ∈ A; c ∈ A |]
==> a = c
lemma ord_iso_pred_imp_lt:
[| f ∈ ord_iso(Order.pred(A, x, r), r, i, Memrel(i));
g ∈ ord_iso(Order.pred(A, y, r), r, j, Memrel(j)); wellordered(M, A, r);
x ∈ A; y ∈ A; M(A); M(r); M(f); M(g); M(j); Ord(i); Ord(j); 〈x, y〉 ∈ r |]
==> i < j
lemma ord_iso_converse1:
[| f ∈ ord_iso(A, r, B, s); 〈b, f ` a〉 ∈ s; a ∈ A; b ∈ B |]
==> 〈converse(f) ` b, a〉 ∈ r
lemma omap_iff:
[| omap(M, A, r, f); M(A); M(f) |]
==> z ∈ f <->
(∃a∈A. ∃x[M]. ∃g[M]. z = 〈a, x〉 ∧
Ord(x) ∧
g ∈ ord_iso(Order.pred(A, a, r), r, x, Memrel(x)))
lemma omap_unique:
[| omap(M, A, r, f); omap(M, A, r, f'); M(A); M(r); M(f); M(f') |] ==> f' = f
lemma omap_yields_Ord:
[| omap(M, A, r, f); 〈a, x〉 ∈ f; M(a); M(x) |] ==> Ord(x)
lemma otype_iff:
[| otype(M, A, r, i); M(A); M(r); M(i) |]
==> x ∈ i <->
M(x) ∧
Ord(x) ∧ (∃a∈A. ∃g[M]. g ∈ ord_iso(Order.pred(A, a, r), r, x, Memrel(x)))
lemma otype_eq_range:
[| omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f); M(i) |]
==> i = range(f)
lemma Ord_otype:
[| otype(M, A, r, i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)
lemma domain_omap:
[| omap(M, A, r, f); M(A); M(r); M(B); M(f) |] ==> domain(f) = obase(M, A, r)
lemma omap_subset:
[| omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f); M(B); M(i) |]
==> f ⊆ obase(M, A, r) × i
lemma omap_funtype:
[| omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f); M(i) |]
==> f ∈ obase(M, A, r) -> i
lemma wellordered_omap_bij:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f);
M(i) |]
==> f ∈ bij(obase(M, A, r), i)
lemma omap_ord_iso:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f);
M(i) |]
==> f ∈ ord_iso(obase(M, A, r), r, i, Memrel(i))
lemma Ord_omap_image_pred:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f);
M(i); b ∈ A |]
==> Ord(f `` Order.pred(A, b, r))
lemma restrict_omap_ord_iso:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i);
D ⊆ obase(M, A, r); M(A); M(r); M(f); M(i) |]
==> restrict(f, D) ∈ ord_iso(D, r, f `` D, Memrel(f `` D))
lemma obase_equals:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f);
M(i) |]
==> obase(M, A, r) = A
theorem omap_ord_iso_otype:
[| wellordered(M, A, r); omap(M, A, r, f); otype(M, A, r, i); M(A); M(r); M(f);
M(i) |]
==> f ∈ ord_iso(A, r, i, Memrel(i))
lemma obase_exists:
[| M(A); M(r) |] ==> M(obase(M, A, r))
lemma omap_exists:
[| M(A); M(r) |] ==> ∃z[M]. omap(M, A, r, z)
lemma otype_exists:
[| wellordered(M, A, r); M(A); M(r) |] ==> ∃i[M]. otype(M, A, r, i)
lemma ordertype_exists:
[| wellordered(M, A, r); M(A); M(r) |]
==> ∃f[M]. ∃i[M]. Ord(i) ∧ f ∈ ord_iso(A, r, i, Memrel(i))
lemma relativized_imp_well_ord:
[| wellordered(M, A, r); M(A); M(r) |] ==> well_ord(A, r)
theorem well_ord_abs:
[| M(A); M(r) |] ==> wellordered(M, A, r) <-> well_ord(A, r)
theorem
[| wellordered(M, A, r); f ∈ ord_iso(A, r, i, Memrel(i)); M(A); M(r); M(f);
M(i); Ord(i) |]
==> i = ordertype(A, r)
lemma is_oadd_fun_iff:
[| a ≤ j; M(i); M(j); M(a); M(f) |]
==> is_oadd_fun(M, i, j, a, f) <->
f ∈ a -> range(f) ∧ (∀x. M(x) --> x < a --> f ` x = i ∪ f `` x)
lemma oadd_strong_replacement':
[| M(i); M(j) |]
==> strong_replacement
(M, λx z. ∃y[M]. z = 〈x, y〉 ∧
(∃g[M]. is_recfun
(Memrel(succ(j)), x, λx g. i ∪ g `` x, g) ∧
y = i ∪ g `` x))
lemma exists_oadd:
[| Ord(j); M(i); M(j) |]
==> ∃f[M]. is_recfun(Memrel(succ(j)), j, λx g. i ∪ g `` x, f)
lemma exists_oadd_fun:
[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_oadd_fun(M, i, succ(j), succ(j), f)
lemma is_oadd_fun_apply:
[| x < j; M(i); M(j); M(f); is_oadd_fun(M, i, j, j, f) |]
==> f ` x = i ∪ (\<Union>k∈x. {f ` k})
lemma is_oadd_fun_iff_oadd:
[| is_oadd_fun(M, i, J, J, f); M(i); M(J); M(f); Ord(i); Ord(j); j < J |]
==> f ` j = i ++ j
lemma Ord_oadd_abs:
[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_oadd(M, i, j, k) <-> k = i ++ j
lemma oadd_abs:
[| M(i); M(j); M(k) |] ==> is_oadd(M, i, j, k) <-> k = i ++ j
lemma oadd_closed:
[| M(i); M(j) |] ==> M(i ++ j)
lemma omult_eqns_unique:
[| omult_eqns(i, x, g, z); omult_eqns(i, x, g, z') |] ==> z = z'
lemma omult_eqns_0:
omult_eqns(i, 0, g, z) <-> z = 0
lemma the_omult_eqns_0:
(THE z. omult_eqns(i, 0, g, z)) = 0
lemma omult_eqns_succ:
omult_eqns(i, succ(j), g, z) <-> Ord(j) ∧ z = g ` j ++ i
lemma the_omult_eqns_succ:
Ord(j) ==> (THE z. omult_eqns(i, succ(j), g, z)) = g ` j ++ i
lemma omult_eqns_Limit:
Limit(x) ==> omult_eqns(i, x, g, z) <-> z = \<Union>g `` x
lemma the_omult_eqns_Limit:
Limit(x) ==> (THE z. omult_eqns(i, x, g, z)) = \<Union>g `` x
lemma omult_eqns_Not:
¬ Ord(x) ==> ¬ omult_eqns(i, x, g, z)
lemma the_omult_eqns_closed:
[| M(i); M(x); M(g); function(g) |] ==> M(THE z. omult_eqns(i, x, g, z))
lemma exists_omult:
[| Ord(j); M(i); M(j) |]
==> ∃f[M]. is_recfun(Memrel(succ(j)), j, λx g. THE z. omult_eqns(i, x, g, z), f)
lemma exists_omult_fun:
[| Ord(j); M(i); M(j) |] ==> ∃f[M]. is_omult_fun(M, i, succ(j), f)
lemma is_omult_fun_apply_0:
[| 0 < j; is_omult_fun(M, i, j, f) |] ==> f ` 0 = 0
lemma is_omult_fun_apply_succ:
[| succ(x) < j; is_omult_fun(M, i, j, f) |] ==> f ` succ(x) = f ` x ++ i
lemma is_omult_fun_apply_Limit:
[| x < j; Limit(x); M(j); M(f); is_omult_fun(M, i, j, f) |]
==> f ` x = (\<Union>y∈x. f ` y)
lemma is_omult_fun_eq_omult:
[| is_omult_fun(M, i, J, f); M(J); M(f); Ord(i); Ord(j) |]
==> j < J --> f ` j = i ×× j
lemma omult_abs:
[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_omult(M, i, j, k) <-> k = i ×× j
lemma wfrank_separation':
M(r) ==> separation(M, λx. ¬ (∃f[M]. is_recfun(r^+, x, λx f. range(f), f)))
lemma wfrank_strong_replacement':
M(r)
==> strong_replacement
(M, λx z. ∃y[M]. ∃f[M]. pair(M, x, y, z) ∧
is_recfun(r^+, x, λx f. range(f), f) ∧
y = range(f))
lemma Ord_wfrank_separation':
M(r)
==> separation
(M, λx. ¬ (∀f[M]. is_recfun(r^+, x, λx. range, f) --> Ord(range(f))))
lemma exists_wfrank:
[| wellfounded(M, r); M(a); M(r) |]
==> ∃f[M]. is_recfun(r^+, a, λx f. range(f), f)
lemma M_wellfoundedrank:
[| wellfounded(M, r); M(r); M(A) |] ==> M(wellfoundedrank(M, r, A))
lemma Ord_wfrank_range:
[| wellfounded(M, r); a ∈ A; M(r); M(A); M(f);
is_recfun(r^+, a, λx. range, f) |]
==> Ord(range(f))
lemma Ord_range_wellfoundedrank:
[| wellfounded(M, r); r ⊆ A × A; M(r); M(A) |]
==> Ord(range(wellfoundedrank(M, r, A)))
lemma function_wellfoundedrank:
[| wellfounded(M, r); M(r); M(A) |] ==> function(wellfoundedrank(M, r, A))
lemma domain_wellfoundedrank:
[| wellfounded(M, r); M(r); M(A) |] ==> domain(wellfoundedrank(M, r, A)) = A
lemma wellfoundedrank_type:
[| wellfounded(M, r); M(r); M(A) |]
==> wellfoundedrank(M, r, A) ∈ A -> range(wellfoundedrank(M, r, A))
lemma Ord_wellfoundedrank:
[| wellfounded(M, r); a ∈ A; r ⊆ A × A; M(r); M(A) |]
==> Ord(wellfoundedrank(M, r, A) ` a)
lemma wellfoundedrank_eq:
[| is_recfun(r^+, a, λx. range, f); wellfounded(M, r); a ∈ A; M(f); M(r);
M(A) |]
==> wellfoundedrank(M, r, A) ` a = range(f)
lemma wellfoundedrank_lt:
[| 〈a, b〉 ∈ r; wellfounded(M, r); r ⊆ A × A; M(r); M(A) |]
==> wellfoundedrank(M, r, A) ` a < wellfoundedrank(M, r, A) ` b
lemma wellfounded_imp_subset_rvimage:
[| wellfounded(M, r); r ⊆ A × A; M(r); M(A) |]
==> ∃i f. Ord(i) ∧ r ⊆ rvimage(A, f, Memrel(i))
lemma wellfounded_imp_wf:
[| wellfounded(M, r); relation(r); M(r) |] ==> wf(r)
lemma wellfounded_on_imp_wf_on:
[| wellfounded_on(M, A, r); relation(r); M(r); M(A) |] ==> wf[A](r)
theorem wf_abs:
[| relation(r); M(r) |] ==> wellfounded(M, r) <-> wf(r)
theorem wf_on_abs:
[| relation(r); M(r); M(A) |] ==> wellfounded_on(M, A, r) <-> wf[A](r)