(* Title: ZF/OrderArith.thy ID: $Id: OrderArith.thy,v 1.19 2007/10/07 19:19:32 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) header{*Combining Orderings: Foundations of Ordinal Arithmetic*} theory OrderArith imports Order Sum Ordinal begin definition (*disjoint sum of two relations; underlies ordinal addition*) radd :: "[i,i,i,i]=>i" where "radd(A,r,B,s) == {z: (A+B) * (A+B). (EX x y. z = <Inl(x), Inr(y)>) | (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) | (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}" definition (*lexicographic product of two relations; underlies ordinal multiplication*) rmult :: "[i,i,i,i]=>i" where "rmult(A,r,B,s) == {z: (A*B) * (A*B). EX x' y' x y. z = <<x',y'>, <x,y>> & (<x',x>: r | (x'=x & <y',y>: s))}" definition (*inverse image of a relation*) rvimage :: "[i,i,i]=>i" where "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}" definition measure :: "[i, i=>i] => i" where "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}" subsection{*Addition of Relations -- Disjoint Sum*} subsubsection{*Rewrite rules. Can be used to obtain introduction rules*} lemma radd_Inl_Inr_iff [iff]: "<Inl(a), Inr(b)> : radd(A,r,B,s) <-> a:A & b:B" by (unfold radd_def, blast) lemma radd_Inl_iff [iff]: "<Inl(a'), Inl(a)> : radd(A,r,B,s) <-> a':A & a:A & <a',a>:r" by (unfold radd_def, blast) lemma radd_Inr_iff [iff]: "<Inr(b'), Inr(b)> : radd(A,r,B,s) <-> b':B & b:B & <b',b>:s" by (unfold radd_def, blast) lemma radd_Inr_Inl_iff [simp]: "<Inr(b), Inl(a)> : radd(A,r,B,s) <-> False" by (unfold radd_def, blast) declare radd_Inr_Inl_iff [THEN iffD1, dest!] subsubsection{*Elimination Rule*} lemma raddE: "[| <p',p> : radd(A,r,B,s); !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q; !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q; !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q |] ==> Q" by (unfold radd_def, blast) subsubsection{*Type checking*} lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)" apply (unfold radd_def) apply (rule Collect_subset) done lemmas field_radd = radd_type [THEN field_rel_subset] subsubsection{*Linearity*} lemma linear_radd: "[| linear(A,r); linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))" by (unfold linear_def, blast) subsubsection{*Well-foundedness*} lemma wf_on_radd: "[| wf[A](r); wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))" apply (rule wf_onI2) apply (subgoal_tac "ALL x:A. Inl (x) : Ba") --{*Proving the lemma, which is needed twice!*} prefer 2 apply (erule_tac V = "y : A + B" in thin_rl) apply (rule_tac ballI) apply (erule_tac r = r and a = x in wf_on_induct, assumption) apply blast txt{*Returning to main part of proof*} apply safe apply blast apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast) done lemma wf_radd: "[| wf(r); wf(s) |] ==> wf(radd(field(r),r,field(s),s))" apply (simp add: wf_iff_wf_on_field) apply (rule wf_on_subset_A [OF _ field_radd]) apply (blast intro: wf_on_radd) done lemma well_ord_radd: "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))" apply (rule well_ordI) apply (simp add: well_ord_def wf_on_radd) apply (simp add: well_ord_def tot_ord_def linear_radd) done subsubsection{*An @{term ord_iso} congruence law*} lemma sum_bij: "[| f: bij(A,C); g: bij(B,D) |] ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)" apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))" in lam_bijective) apply (typecheck add: bij_is_inj inj_is_fun) apply (auto simp add: left_inverse_bij right_inverse_bij) done lemma sum_ord_iso_cong: "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))" apply (unfold ord_iso_def) apply (safe intro!: sum_bij) (*Do the beta-reductions now*) apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type]) done (*Could we prove an ord_iso result? Perhaps ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *) lemma sum_disjoint_bij: "A Int B = 0 ==> (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)" apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective) apply auto done subsubsection{*Associativity*} lemma sum_assoc_bij: "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) : bij((A+B)+C, A+(B+C))" apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" in lam_bijective) apply auto done lemma sum_assoc_ord_iso: "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t), A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))" by (rule sum_assoc_bij [THEN ord_isoI], auto) subsection{*Multiplication of Relations -- Lexicographic Product*} subsubsection{*Rewrite rule. Can be used to obtain introduction rules*} lemma rmult_iff [iff]: "<<a',b'>, <a,b>> : rmult(A,r,B,s) <-> (<a',a>: r & a':A & a:A & b': B & b: B) | (<b',b>: s & a'=a & a:A & b': B & b: B)" by (unfold rmult_def, blast) lemma rmultE: "[| <<a',b'>, <a,b>> : rmult(A,r,B,s); [| <a',a>: r; a':A; a:A; b':B; b:B |] ==> Q; [| <b',b>: s; a:A; a'=a; b':B; b:B |] ==> Q |] ==> Q" by blast subsubsection{*Type checking*} lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)" by (unfold rmult_def, rule Collect_subset) lemmas field_rmult = rmult_type [THEN field_rel_subset] subsubsection{*Linearity*} lemma linear_rmult: "[| linear(A,r); linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))" by (simp add: linear_def, blast) subsubsection{*Well-foundedness*} lemma wf_on_rmult: "[| wf[A](r); wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))" apply (rule wf_onI2) apply (erule SigmaE) apply (erule ssubst) apply (subgoal_tac "ALL b:B. <x,b>: Ba", blast) apply (erule_tac a = x in wf_on_induct, assumption) apply (rule ballI) apply (erule_tac a = b in wf_on_induct, assumption) apply (best elim!: rmultE bspec [THEN mp]) done lemma wf_rmult: "[| wf(r); wf(s) |] ==> wf(rmult(field(r),r,field(s),s))" apply (simp add: wf_iff_wf_on_field) apply (rule wf_on_subset_A [OF _ field_rmult]) apply (blast intro: wf_on_rmult) done lemma well_ord_rmult: "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))" apply (rule well_ordI) apply (simp add: well_ord_def wf_on_rmult) apply (simp add: well_ord_def tot_ord_def linear_rmult) done subsubsection{*An @{term ord_iso} congruence law*} lemma prod_bij: "[| f: bij(A,C); g: bij(B,D) |] ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)" apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>" in lam_bijective) apply (typecheck add: bij_is_inj inj_is_fun) apply (auto simp add: left_inverse_bij right_inverse_bij) done lemma prod_ord_iso_cong: "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==> (lam <x,y>:A*B. <f`x, g`y>) : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))" apply (unfold ord_iso_def) apply (safe intro!: prod_bij) apply (simp_all add: bij_is_fun [THEN apply_type]) apply (blast intro: bij_is_inj [THEN inj_apply_equality]) done lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)" by (rule_tac d = snd in lam_bijective, auto) (*Used??*) lemma singleton_prod_ord_iso: "well_ord({x},xr) ==> (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))" apply (rule singleton_prod_bij [THEN ord_isoI]) apply (simp (no_asm_simp)) apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl]) done (*Here we build a complicated function term, then simplify it using case_cong, id_conv, comp_lam, case_case.*) lemma prod_sum_singleton_bij: "a~:C ==> (lam x:C*B + D. case(%x. x, %y.<a,y>, x)) : bij(C*B + D, C*B Un {a}*D)" apply (rule subst_elem) apply (rule id_bij [THEN sum_bij, THEN comp_bij]) apply (rule singleton_prod_bij) apply (rule sum_disjoint_bij, blast) apply (simp (no_asm_simp) cong add: case_cong) apply (rule comp_lam [THEN trans, symmetric]) apply (fast elim!: case_type) apply (simp (no_asm_simp) add: case_case) done lemma prod_sum_singleton_ord_iso: "[| a:A; well_ord(A,r) |] ==> (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x)) : ord_iso(pred(A,a,r)*B + pred(B,b,s), radd(A*B, rmult(A,r,B,s), B, s), pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))" apply (rule prod_sum_singleton_bij [THEN ord_isoI]) apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl]) apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE) done subsubsection{*Distributive law*} lemma sum_prod_distrib_bij: "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x)) : bij((A+B)*C, (A*C)+(B*C))" by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) " in lam_bijective, auto) lemma sum_prod_distrib_ord_iso: "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x)) : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t), (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))" by (rule sum_prod_distrib_bij [THEN ord_isoI], auto) subsubsection{*Associativity*} lemma prod_assoc_bij: "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))" by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto) lemma prod_assoc_ord_iso: "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t), A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))" by (rule prod_assoc_bij [THEN ord_isoI], auto) subsection{*Inverse Image of a Relation*} subsubsection{*Rewrite rule*} lemma rvimage_iff: "<a,b> : rvimage(A,f,r) <-> <f`a,f`b>: r & a:A & b:A" by (unfold rvimage_def, blast) subsubsection{*Type checking*} lemma rvimage_type: "rvimage(A,f,r) <= A*A" by (unfold rvimage_def, rule Collect_subset) lemmas field_rvimage = rvimage_type [THEN field_rel_subset] lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))" by (unfold rvimage_def, blast) subsubsection{*Partial Ordering Properties*} lemma irrefl_rvimage: "[| f: inj(A,B); irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))" apply (unfold irrefl_def rvimage_def) apply (blast intro: inj_is_fun [THEN apply_type]) done lemma trans_on_rvimage: "[| f: inj(A,B); trans[B](r) |] ==> trans[A](rvimage(A,f,r))" apply (unfold trans_on_def rvimage_def) apply (blast intro: inj_is_fun [THEN apply_type]) done lemma part_ord_rvimage: "[| f: inj(A,B); part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))" apply (unfold part_ord_def) apply (blast intro!: irrefl_rvimage trans_on_rvimage) done subsubsection{*Linearity*} lemma linear_rvimage: "[| f: inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))" apply (simp add: inj_def linear_def rvimage_iff) apply (blast intro: apply_funtype) done lemma tot_ord_rvimage: "[| f: inj(A,B); tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))" apply (unfold tot_ord_def) apply (blast intro!: part_ord_rvimage linear_rvimage) done subsubsection{*Well-foundedness*} lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))" apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal) apply clarify apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }") apply (erule allE) apply (erule impE) apply assumption apply blast apply blast done text{*But note that the combination of @{text wf_imp_wf_on} and @{text wf_rvimage} gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}*} lemma wf_on_rvimage: "[| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))" apply (rule wf_onI2) apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba") apply blast apply (erule_tac a = "f`y" in wf_on_induct) apply (blast intro!: apply_funtype) apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1]) done (*Note that we need only wf[A](...) and linear(A,...) to get the result!*) lemma well_ord_rvimage: "[| f: inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))" apply (rule well_ordI) apply (unfold well_ord_def tot_ord_def) apply (blast intro!: wf_on_rvimage inj_is_fun) apply (blast intro!: linear_rvimage) done lemma ord_iso_rvimage: "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)" apply (unfold ord_iso_def) apply (simp add: rvimage_iff) done lemma ord_iso_rvimage_eq: "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A" by (unfold ord_iso_def rvimage_def, blast) subsection{*Every well-founded relation is a subset of some inverse image of an ordinal*} lemma wf_rvimage_Ord: "Ord(i) ==> wf(rvimage(A, f, Memrel(i)))" by (blast intro: wf_rvimage wf_Memrel) definition wfrank :: "[i,i]=>i" where "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y ∈ r-``{x}. succ(f`y))" definition wftype :: "i=>i" where "wftype(r) == \<Union>y ∈ range(r). succ(wfrank(r,y))" lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y ∈ r-``{a}. succ(wfrank(r,y)))" by (subst wfrank_def [THEN def_wfrec], simp_all) lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))" apply (rule_tac a=a in wf_induct, assumption) apply (subst wfrank, assumption) apply (rule Ord_succ [THEN Ord_UN], blast) done lemma wfrank_lt: "[|wf(r); <a,b> ∈ r|] ==> wfrank(r,a) < wfrank(r,b)" apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption) apply (rule UN_I [THEN ltI]) apply (simp add: Ord_wfrank vimage_iff)+ done lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))" by (simp add: wftype_def Ord_wfrank) lemma wftypeI: "[|wf(r); x ∈ field(r)|] ==> wfrank(r,x) ∈ wftype(r)" apply (simp add: wftype_def) apply (blast intro: wfrank_lt [THEN ltD]) done lemma wf_imp_subset_rvimage: "[|wf(r); r ⊆ A*A|] ==> ∃i f. Ord(i) & r <= rvimage(A, f, Memrel(i))" apply (rule_tac x="wftype(r)" in exI) apply (rule_tac x="λx∈A. wfrank(r,x)" in exI) apply (simp add: Ord_wftype, clarify) apply (frule subsetD, assumption, clarify) apply (simp add: rvimage_iff wfrank_lt [THEN ltD]) apply (blast intro: wftypeI) done theorem wf_iff_subset_rvimage: "relation(r) ==> wf(r) <-> (∃i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))" by (blast dest!: relation_field_times_field wf_imp_subset_rvimage intro: wf_rvimage_Ord [THEN wf_subset]) subsection{*Other Results*} lemma wf_times: "A Int B = 0 ==> wf(A*B)" by (simp add: wf_def, blast) text{*Could also be used to prove @{text wf_radd}*} lemma wf_Un: "[| range(r) Int domain(s) = 0; wf(r); wf(s) |] ==> wf(r Un s)" apply (simp add: wf_def, clarify) apply (rule equalityI) prefer 2 apply blast apply clarify apply (drule_tac x=Z in spec) apply (drule_tac x="Z Int domain(s)" in spec) apply simp apply (blast intro: elim: equalityE) done subsubsection{*The Empty Relation*} lemma wf0: "wf(0)" by (simp add: wf_def, blast) lemma linear0: "linear(0,0)" by (simp add: linear_def) lemma well_ord0: "well_ord(0,0)" by (blast intro: wf_imp_wf_on well_ordI wf0 linear0) subsubsection{*The "measure" relation is useful with wfrec*} lemma measure_eq_rvimage_Memrel: "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))" apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff) apply (rule equalityI, auto) apply (auto intro: Ord_in_Ord simp add: lt_def) done lemma wf_measure [iff]: "wf(measure(A,f))" by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage) lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)" by (simp (no_asm) add: measure_def) lemma linear_measure: assumes Ordf: "!!x. x ∈ A ==> Ord(f(x))" and inj: "!!x y. [|x ∈ A; y ∈ A; f(x) = f(y) |] ==> x=y" shows "linear(A, measure(A,f))" apply (auto simp add: linear_def) apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt) apply (simp_all add: Ordf) apply (blast intro: inj) done lemma wf_on_measure: "wf[B](measure(A,f))" by (rule wf_imp_wf_on [OF wf_measure]) lemma well_ord_measure: assumes Ordf: "!!x. x ∈ A ==> Ord(f(x))" and inj: "!!x y. [|x ∈ A; y ∈ A; f(x) = f(y) |] ==> x=y" shows "well_ord(A, measure(A,f))" apply (rule well_ordI) apply (rule wf_on_measure) apply (blast intro: linear_measure Ordf inj) done lemma measure_type: "measure(A,f) <= A*A" by (auto simp add: measure_def) subsubsection{*Well-foundedness of Unions*} lemma wf_on_Union: assumes wfA: "wf[A](r)" and wfB: "!!a. a∈A ==> wf[B(a)](s)" and ok: "!!a u v. [|<u,v> ∈ s; v ∈ B(a); a ∈ A|] ==> (∃a'∈A. <a',a> ∈ r & u ∈ B(a')) | u ∈ B(a)" shows "wf[\<Union>a∈A. B(a)](s)" apply (rule wf_onI2) apply (erule UN_E) apply (subgoal_tac "∀z ∈ B(a). z ∈ Ba", blast) apply (rule_tac a = a in wf_on_induct [OF wfA], assumption) apply (rule ballI) apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption) apply (rename_tac u) apply (drule_tac x=u in bspec, blast) apply (erule mp, clarify) apply (frule ok, assumption+, blast) done subsubsection{*Bijections involving Powersets*} lemma Pow_sum_bij: "(λZ ∈ Pow(A+B). <{x ∈ A. Inl(x) ∈ Z}, {y ∈ B. Inr(y) ∈ Z}>) ∈ bij(Pow(A+B), Pow(A)*Pow(B))" apply (rule_tac d = "%<X,Y>. {Inl (x). x ∈ X} Un {Inr (y). y ∈ Y}" in lam_bijective) apply force+ done text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *} lemma Pow_Sigma_bij: "(λr ∈ Pow(Sigma(A,B)). λx ∈ A. r``{x}) ∈ bij(Pow(Sigma(A,B)), Π x ∈ A. Pow(B(x)))" apply (rule_tac d = "%f. \<Union>x ∈ A. \<Union>y ∈ f`x. {<x,y>}" in lam_bijective) apply (blast intro: lam_type) apply (blast dest: apply_type, simp_all) apply fast (*strange, but blast can't do it*) apply (rule fun_extension, auto) by blast end
lemma radd_Inl_Inr_iff:
〈Inl(a), Inr(b)〉 ∈ radd(A, r, B, s) <-> a ∈ A ∧ b ∈ B
lemma radd_Inl_iff:
〈Inl(a'), Inl(a)〉 ∈ radd(A, r, B, s) <-> a' ∈ A ∧ a ∈ A ∧ 〈a', a〉 ∈ r
lemma radd_Inr_iff:
〈Inr(b'), Inr(b)〉 ∈ radd(A, r, B, s) <-> b' ∈ B ∧ b ∈ B ∧ 〈b', b〉 ∈ s
lemma radd_Inr_Inl_iff:
〈Inr(b), Inl(a)〉 ∈ radd(A, r, B, s) <-> False
lemma raddE:
[| 〈p', p〉 ∈ radd(A, r, B, s);
!!x y. [| p' = Inl(x); x ∈ A; p = Inr(y); y ∈ B |] ==> Q;
!!x' x. [| p' = Inl(x'); p = Inl(x); 〈x', x〉 ∈ r; x' ∈ A; x ∈ A |] ==> Q;
!!y' y. [| p' = Inr(y'); p = Inr(y); 〈y', y〉 ∈ s; y' ∈ B; y ∈ B |] ==> Q |]
==> Q
lemma radd_type:
radd(A, r, B, s) ⊆ (A + B) × (A + B)
lemma field_radd:
field(radd(A1, r1, B1, s1)) ⊆ A1 + B1
lemma linear_radd:
[| linear(A, r); linear(B, s) |] ==> linear(A + B, radd(A, r, B, s))
lemma wf_on_radd:
[| wf[A](r); wf[B](s) |] ==> wf[A + B](radd(A, r, B, s))
lemma wf_radd:
[| wf(r); wf(s) |] ==> wf(radd(field(r), r, field(s), s))
lemma well_ord_radd:
[| well_ord(A, r); well_ord(B, s) |] ==> well_ord(A + B, radd(A, r, B, s))
lemma sum_bij:
[| f ∈ bij(A, C); g ∈ bij(B, D) |]
==> (λz∈A + B. case(λx. Inl(f ` x), λy. Inr(g ` y), z)) ∈ bij(A + B, C + D)
lemma sum_ord_iso_cong:
[| f ∈ ord_iso(A, r, A', r'); g ∈ ord_iso(B, s, B', s') |]
==> (λz∈A + B. case(λx. Inl(f ` x), λy. Inr(g ` y), z)) ∈
ord_iso(A + B, radd(A, r, B, s), A' + B', radd(A', r', B', s'))
lemma sum_disjoint_bij:
A ∩ B = 0 ==> (λz∈A + B. case(λx. x, λy. y, z)) ∈ bij(A + B, A ∪ B)
lemma sum_assoc_bij:
(λz∈(A + B) + C. case(case(Inl, λy. Inr(Inl(y))), λy. Inr(Inr(y)), z)) ∈
bij((A + B) + C, A + B + C)
lemma sum_assoc_ord_iso:
(λz∈(A + B) + C. case(case(Inl, λy. Inr(Inl(y))), λy. Inr(Inr(y)), z)) ∈
ord_iso
((A + B) + C, radd(A + B, radd(A, r, B, s), C, t), A + B + C,
radd(A, r, B + C, radd(B, s, C, t)))
lemma rmult_iff:
〈〈a', b'〉, a, b〉 ∈ rmult(A, r, B, s) <->
〈a', a〉 ∈ r ∧ a' ∈ A ∧ a ∈ A ∧ b' ∈ B ∧ b ∈ B ∨
〈b', b〉 ∈ s ∧ a' = a ∧ a ∈ A ∧ b' ∈ B ∧ b ∈ B
lemma rmultE:
[| 〈〈a', b'〉, a, b〉 ∈ rmult(A, r, B, s);
[| 〈a', a〉 ∈ r; a' ∈ A; a ∈ A; b' ∈ B; b ∈ B |] ==> Q;
[| 〈b', b〉 ∈ s; a ∈ A; a' = a; b' ∈ B; b ∈ B |] ==> Q |]
==> Q
lemma rmult_type:
rmult(A, r, B, s) ⊆ (A × B) × A × B
lemma field_rmult:
field(rmult(A1, r1, B1, s1)) ⊆ A1 × B1
lemma linear_rmult:
[| linear(A, r); linear(B, s) |] ==> linear(A × B, rmult(A, r, B, s))
lemma wf_on_rmult:
[| wf[A](r); wf[B](s) |] ==> wf[A × B](rmult(A, r, B, s))
lemma wf_rmult:
[| wf(r); wf(s) |] ==> wf(rmult(field(r), r, field(s), s))
lemma well_ord_rmult:
[| well_ord(A, r); well_ord(B, s) |] ==> well_ord(A × B, rmult(A, r, B, s))
lemma prod_bij:
[| f ∈ bij(A, C); g ∈ bij(B, D) |]
==> (λ〈x,y〉∈A × B. 〈f ` x, g ` y〉) ∈ bij(A × B, C × D)
lemma prod_ord_iso_cong:
[| f ∈ ord_iso(A, r, A', r'); g ∈ ord_iso(B, s, B', s') |]
==> (λ〈x,y〉∈A × B. 〈f ` x, g ` y〉) ∈
ord_iso(A × B, rmult(A, r, B, s), A' × B', rmult(A', r', B', s'))
lemma singleton_prod_bij:
(λz∈A. 〈x, z〉) ∈ bij(A, {x} × A)
lemma singleton_prod_ord_iso:
well_ord({x}, xr)
==> (λz∈A. 〈x, z〉) ∈ ord_iso(A, r, {x} × A, rmult({x}, xr, A, r))
lemma prod_sum_singleton_bij:
a ∉ C
==> (λx∈C × B + D. case(λx. x, λy. 〈a, y〉, x)) ∈ bij(C × B + D, C × B ∪ {a} × D)
lemma prod_sum_singleton_ord_iso:
[| a ∈ A; well_ord(A, r) |]
==> (λx∈pred(A, a, r) × B + pred(B, b, s). case(λx. x, λy. 〈a, y〉, x)) ∈
ord_iso
(pred(A, a, r) × B + pred(B, b, s), radd(A × B, rmult(A, r, B, s), B, s),
pred(A, a, r) × B ∪ {a} × pred(B, b, s), rmult(A, r, B, s))
lemma sum_prod_distrib_bij:
(λ〈x,z〉∈(A + B) × C. case(λy. Inl(〈y, z〉), λy. Inr(〈y, z〉), x)) ∈
bij((A + B) × C, A × C + B × C)
lemma sum_prod_distrib_ord_iso:
(λ〈x,z〉∈(A + B) × C. case(λy. Inl(〈y, z〉), λy. Inr(〈y, z〉), x)) ∈
ord_iso
((A + B) × C, rmult(A + B, radd(A, r, B, s), C, t), A × C + B × C,
radd(A × C, rmult(A, r, C, t), B × C, rmult(B, s, C, t)))
lemma prod_assoc_bij:
(λ〈〈x,y〉,z〉∈(A × B) × C. 〈x, y, z〉) ∈ bij((A × B) × C, A × B × C)
lemma prod_assoc_ord_iso:
(λ〈〈x,y〉,z〉∈(A × B) × C. 〈x, y, z〉) ∈
ord_iso
((A × B) × C, rmult(A × B, rmult(A, r, B, s), C, t), A × B × C,
rmult(A, r, B × C, rmult(B, s, C, t)))
lemma rvimage_iff:
〈a, b〉 ∈ rvimage(A, f, r) <-> 〈f ` a, f ` b〉 ∈ r ∧ a ∈ A ∧ b ∈ A
lemma rvimage_type:
rvimage(A, f, r) ⊆ A × A
lemma field_rvimage:
field(rvimage(A, f1, r1)) ⊆ A
lemma rvimage_converse:
rvimage(A, f, converse(r)) = converse(rvimage(A, f, r))
lemma irrefl_rvimage:
[| f ∈ inj(A, B); irrefl(B, r) |] ==> irrefl(A, rvimage(A, f, r))
lemma trans_on_rvimage:
[| f ∈ inj(A, B); trans[B](r) |] ==> trans[A](rvimage(A, f, r))
lemma part_ord_rvimage:
[| f ∈ inj(A, B); part_ord(B, r) |] ==> part_ord(A, rvimage(A, f, r))
lemma linear_rvimage:
[| f ∈ inj(A, B); linear(B, r) |] ==> linear(A, rvimage(A, f, r))
lemma tot_ord_rvimage:
[| f ∈ inj(A, B); tot_ord(B, r) |] ==> tot_ord(A, rvimage(A, f, r))
lemma wf_rvimage:
wf(r) ==> wf(rvimage(A, f, r))
lemma wf_on_rvimage:
[| f ∈ A -> B; wf[B](r) |] ==> wf[A](rvimage(A, f, r))
lemma well_ord_rvimage:
[| f ∈ inj(A, B); well_ord(B, r) |] ==> well_ord(A, rvimage(A, f, r))
lemma ord_iso_rvimage:
f ∈ bij(A, B) ==> f ∈ ord_iso(A, rvimage(A, f, s), B, s)
lemma ord_iso_rvimage_eq:
f ∈ ord_iso(A, r, B, s) ==> rvimage(A, f, s) = r ∩ A × A
lemma wf_rvimage_Ord:
Ord(i) ==> wf(rvimage(A, f, Memrel(i)))
lemma wfrank:
wf(r) ==> wfrank(r, a) = (\<Union>y∈r -`` {a}. succ(wfrank(r, y)))
lemma Ord_wfrank:
wf(r) ==> Ord(wfrank(r, a))
lemma wfrank_lt:
[| wf(r); 〈a, b〉 ∈ r |] ==> wfrank(r, a) < wfrank(r, b)
lemma Ord_wftype:
wf(r) ==> Ord(wftype(r))
lemma wftypeI:
[| wf(r); x ∈ field(r) |] ==> wfrank(r, x) ∈ wftype(r)
lemma wf_imp_subset_rvimage:
[| wf(r); r ⊆ A × A |] ==> ∃i f. Ord(i) ∧ r ⊆ rvimage(A, f, Memrel(i))
theorem wf_iff_subset_rvimage:
relation(r) ==> wf(r) <-> (∃i f A. Ord(i) ∧ r ⊆ rvimage(A, f, Memrel(i)))
lemma wf_times:
A ∩ B = 0 ==> wf(A × B)
lemma wf_Un:
[| range(r) ∩ domain(s) = 0; wf(r); wf(s) |] ==> wf(r ∪ s)
lemma wf0:
wf(0)
lemma linear0:
linear(0, 0)
lemma well_ord0:
well_ord(0, 0)
lemma measure_eq_rvimage_Memrel:
measure(A, f) = rvimage(A, Lambda(A, f), Memrel(Collect(RepFun(A, f), Ord)))
lemma wf_measure:
wf(measure(A, f))
lemma measure_iff:
〈x, y〉 ∈ measure(A, f) <-> x ∈ A ∧ y ∈ A ∧ f(x) < f(y)
lemma linear_measure:
[| !!x. x ∈ A ==> Ord(f(x)); !!x y. [| x ∈ A; y ∈ A; f(x) = f(y) |] ==> x = y |]
==> linear(A, measure(A, f))
lemma wf_on_measure:
wf[B](measure(A, f))
lemma well_ord_measure:
[| !!x. x ∈ A ==> Ord(f(x)); !!x y. [| x ∈ A; y ∈ A; f(x) = f(y) |] ==> x = y |]
==> well_ord(A, measure(A, f))
lemma measure_type:
measure(A, f) ⊆ A × A
lemma wf_on_Union:
[| wf[A](r); !!a. a ∈ A ==> wf[B(a)](s);
!!a u v.
[| 〈u, v〉 ∈ s; v ∈ B(a); a ∈ A |]
==> (∃a'∈A. 〈a', a〉 ∈ r ∧ u ∈ B(a')) ∨ u ∈ B(a) |]
==> wf[\<Union>a∈A. B(a)](s)
lemma Pow_sum_bij:
(λZ∈Pow(A + B). 〈{x ∈ A . Inl(x) ∈ Z}, {y ∈ B . Inr(y) ∈ Z}〉) ∈
bij(Pow(A + B), Pow(A) × Pow(B))
lemma Pow_Sigma_bij:
(λr∈Pow(Sigma(A, B)). λx∈A. r `` {x}) ∈ bij(Pow(Sigma(A, B)), Πx∈A. Pow(B(x)))