src/ZF/OrderArith.thy
 author paulson Mon Jul 21 13:02:07 2003 +0200 (2003-07-21 ago) changeset 14120 3a73850c6c7d parent 13823 d49ffd9f9662 child 14171 0cab06e3bbd0 permissions -rw-r--r--
Tidied some examples
 clasohm@1478 ` 1` ```(* Title: ZF/OrderArith.thy ``` lcp@437 ` 2` ``` ID: \$Id\$ ``` clasohm@1478 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` lcp@437 ` 4` ``` Copyright 1994 University of Cambridge ``` lcp@437 ` 5` lcp@437 ` 6` ```*) ``` lcp@437 ` 7` paulson@13356 ` 8` ```header{*Combining Orderings: Foundations of Ordinal Arithmetic*} ``` paulson@13356 ` 9` paulson@13140 ` 10` ```theory OrderArith = Order + Sum + Ordinal: ``` paulson@13140 ` 11` ```constdefs ``` lcp@437 ` 12` lcp@437 ` 13` ``` (*disjoint sum of two relations; underlies ordinal addition*) ``` paulson@13140 ` 14` ``` radd :: "[i,i,i,i]=>i" ``` paulson@13140 ` 15` ``` "radd(A,r,B,s) == ``` clasohm@1155 ` 16` ``` {z: (A+B) * (A+B). ``` clasohm@1478 ` 17` ``` (EX x y. z = ) | ``` clasohm@1478 ` 18` ``` (EX x' x. z = & :r) | ``` clasohm@1155 ` 19` ``` (EX y' y. z = & :s)}" ``` lcp@437 ` 20` lcp@437 ` 21` ``` (*lexicographic product of two relations; underlies ordinal multiplication*) ``` paulson@13140 ` 22` ``` rmult :: "[i,i,i,i]=>i" ``` paulson@13140 ` 23` ``` "rmult(A,r,B,s) == ``` clasohm@1155 ` 24` ``` {z: (A*B) * (A*B). ``` clasohm@1478 ` 25` ``` EX x' y' x y. z = <, > & ``` clasohm@1155 ` 26` ``` (: r | (x'=x & : s))}" ``` lcp@437 ` 27` lcp@437 ` 28` ``` (*inverse image of a relation*) ``` paulson@13140 ` 29` ``` rvimage :: "[i,i,i]=>i" ``` paulson@13140 ` 30` ``` "rvimage(A,f,r) == {z: A*A. EX x y. z = & : r}" ``` paulson@13140 ` 31` paulson@13140 ` 32` ``` measure :: "[i, i\i] \ i" ``` paulson@13140 ` 33` ``` "measure(A,f) == {: A*A. f(x) < f(y)}" ``` paulson@13140 ` 34` paulson@13140 ` 35` paulson@13356 ` 36` ```subsection{*Addition of Relations -- Disjoint Sum*} ``` paulson@13140 ` 37` paulson@13512 ` 38` ```subsubsection{*Rewrite rules. Can be used to obtain introduction rules*} ``` paulson@13140 ` 39` paulson@13140 ` 40` ```lemma radd_Inl_Inr_iff [iff]: ``` paulson@13140 ` 41` ``` " : radd(A,r,B,s) <-> a:A & b:B" ``` paulson@13356 ` 42` ```by (unfold radd_def, blast) ``` paulson@13140 ` 43` paulson@13140 ` 44` ```lemma radd_Inl_iff [iff]: ``` paulson@13140 ` 45` ``` " : radd(A,r,B,s) <-> a':A & a:A & :r" ``` paulson@13356 ` 46` ```by (unfold radd_def, blast) ``` paulson@13140 ` 47` paulson@13140 ` 48` ```lemma radd_Inr_iff [iff]: ``` paulson@13140 ` 49` ``` " : radd(A,r,B,s) <-> b':B & b:B & :s" ``` paulson@13356 ` 50` ```by (unfold radd_def, blast) ``` paulson@13140 ` 51` paulson@13823 ` 52` ```lemma radd_Inr_Inl_iff [simp]: ``` paulson@13823 ` 53` ``` " : radd(A,r,B,s) <-> False" ``` paulson@13356 ` 54` ```by (unfold radd_def, blast) ``` paulson@13140 ` 55` paulson@13823 ` 56` ```declare radd_Inr_Inl_iff [THEN iffD1, dest!] ``` paulson@13823 ` 57` paulson@13512 ` 58` ```subsubsection{*Elimination Rule*} ``` paulson@13140 ` 59` paulson@13140 ` 60` ```lemma raddE: ``` paulson@13140 ` 61` ``` "[| : radd(A,r,B,s); ``` paulson@13140 ` 62` ``` !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q; ``` paulson@13140 ` 63` ``` !!x' x. [| p'=Inl(x'); p=Inl(x); : r; x':A; x:A |] ==> Q; ``` paulson@13140 ` 64` ``` !!y' y. [| p'=Inr(y'); p=Inr(y); : s; y':B; y:B |] ==> Q ``` paulson@13140 ` 65` ``` |] ==> Q" ``` paulson@13356 ` 66` ```by (unfold radd_def, blast) ``` paulson@13140 ` 67` paulson@13512 ` 68` ```subsubsection{*Type checking*} ``` paulson@13140 ` 69` paulson@13140 ` 70` ```lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)" ``` paulson@13140 ` 71` ```apply (unfold radd_def) ``` paulson@13140 ` 72` ```apply (rule Collect_subset) ``` paulson@13140 ` 73` ```done ``` paulson@13140 ` 74` paulson@13140 ` 75` ```lemmas field_radd = radd_type [THEN field_rel_subset] ``` paulson@13140 ` 76` paulson@13512 ` 77` ```subsubsection{*Linearity*} ``` paulson@13140 ` 78` paulson@13140 ` 79` ```lemma linear_radd: ``` paulson@13140 ` 80` ``` "[| linear(A,r); linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))" ``` paulson@13356 ` 81` ```by (unfold linear_def, blast) ``` paulson@13140 ` 82` paulson@13140 ` 83` paulson@13512 ` 84` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 85` paulson@13140 ` 86` ```lemma wf_on_radd: "[| wf[A](r); wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))" ``` paulson@13140 ` 87` ```apply (rule wf_onI2) ``` paulson@13140 ` 88` ```apply (subgoal_tac "ALL x:A. Inl (x) : Ba") ``` paulson@13512 ` 89` ``` --{*Proving the lemma, which is needed twice!*} ``` paulson@13140 ` 90` ``` prefer 2 ``` paulson@13140 ` 91` ``` apply (erule_tac V = "y : A + B" in thin_rl) ``` paulson@13140 ` 92` ``` apply (rule_tac ballI) ``` paulson@13784 ` 93` ``` apply (erule_tac r = r and a = x in wf_on_induct, assumption) ``` paulson@13269 ` 94` ``` apply blast ``` paulson@13512 ` 95` ```txt{*Returning to main part of proof*} ``` paulson@13140 ` 96` ```apply safe ``` paulson@13140 ` 97` ```apply blast ``` paulson@13784 ` 98` ```apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast) ``` paulson@13140 ` 99` ```done ``` paulson@13140 ` 100` paulson@13140 ` 101` ```lemma wf_radd: "[| wf(r); wf(s) |] ==> wf(radd(field(r),r,field(s),s))" ``` paulson@13140 ` 102` ```apply (simp add: wf_iff_wf_on_field) ``` paulson@13140 ` 103` ```apply (rule wf_on_subset_A [OF _ field_radd]) ``` paulson@13140 ` 104` ```apply (blast intro: wf_on_radd) ``` paulson@13140 ` 105` ```done ``` paulson@13140 ` 106` paulson@13140 ` 107` ```lemma well_ord_radd: ``` paulson@13140 ` 108` ``` "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))" ``` paulson@13140 ` 109` ```apply (rule well_ordI) ``` paulson@13140 ` 110` ```apply (simp add: well_ord_def wf_on_radd) ``` paulson@13140 ` 111` ```apply (simp add: well_ord_def tot_ord_def linear_radd) ``` paulson@13140 ` 112` ```done ``` paulson@13140 ` 113` paulson@13512 ` 114` ```subsubsection{*An @{term ord_iso} congruence law*} ``` lcp@437 ` 115` paulson@13140 ` 116` ```lemma sum_bij: ``` paulson@13140 ` 117` ``` "[| f: bij(A,C); g: bij(B,D) |] ``` paulson@13140 ` 118` ``` ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)" ``` paulson@13356 ` 119` ```apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))" ``` paulson@13356 ` 120` ``` in lam_bijective) ``` paulson@13140 ` 121` ```apply (typecheck add: bij_is_inj inj_is_fun) ``` paulson@13140 ` 122` ```apply (auto simp add: left_inverse_bij right_inverse_bij) ``` paulson@13140 ` 123` ```done ``` paulson@13140 ` 124` paulson@13140 ` 125` ```lemma sum_ord_iso_cong: ``` paulson@13140 ` 126` ``` "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==> ``` paulson@13140 ` 127` ``` (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) ``` paulson@13140 ` 128` ``` : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))" ``` paulson@13140 ` 129` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 130` ```apply (safe intro!: sum_bij) ``` paulson@13140 ` 131` ```(*Do the beta-reductions now*) ``` paulson@13140 ` 132` ```apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type]) ``` paulson@13140 ` 133` ```done ``` paulson@13140 ` 134` paulson@13140 ` 135` ```(*Could we prove an ord_iso result? Perhaps ``` paulson@13140 ` 136` ``` ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *) ``` paulson@13140 ` 137` ```lemma sum_disjoint_bij: "A Int B = 0 ==> ``` paulson@13140 ` 138` ``` (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)" ``` paulson@13140 ` 139` ```apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective) ``` paulson@13140 ` 140` ```apply auto ``` paulson@13140 ` 141` ```done ``` paulson@13140 ` 142` paulson@13512 ` 143` ```subsubsection{*Associativity*} ``` paulson@13140 ` 144` paulson@13140 ` 145` ```lemma sum_assoc_bij: ``` paulson@13140 ` 146` ``` "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) ``` paulson@13140 ` 147` ``` : bij((A+B)+C, A+(B+C))" ``` paulson@13140 ` 148` ```apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" ``` paulson@13140 ` 149` ``` in lam_bijective) ``` paulson@13140 ` 150` ```apply auto ``` paulson@13140 ` 151` ```done ``` paulson@13140 ` 152` paulson@13140 ` 153` ```lemma sum_assoc_ord_iso: ``` paulson@13140 ` 154` ``` "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) ``` paulson@13140 ` 155` ``` : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t), ``` paulson@13140 ` 156` ``` A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))" ``` paulson@13356 ` 157` ```by (rule sum_assoc_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 158` paulson@13140 ` 159` paulson@13356 ` 160` ```subsection{*Multiplication of Relations -- Lexicographic Product*} ``` paulson@13140 ` 161` paulson@13512 ` 162` ```subsubsection{*Rewrite rule. Can be used to obtain introduction rules*} ``` paulson@13140 ` 163` paulson@13140 ` 164` ```lemma rmult_iff [iff]: ``` paulson@13140 ` 165` ``` "<, > : rmult(A,r,B,s) <-> ``` paulson@13140 ` 166` ``` (: r & a':A & a:A & b': B & b: B) | ``` paulson@13140 ` 167` ``` (: s & a'=a & a:A & b': B & b: B)" ``` paulson@13140 ` 168` paulson@13356 ` 169` ```by (unfold rmult_def, blast) ``` paulson@13140 ` 170` paulson@13140 ` 171` ```lemma rmultE: ``` paulson@13140 ` 172` ``` "[| <, > : rmult(A,r,B,s); ``` paulson@13140 ` 173` ``` [| : r; a':A; a:A; b':B; b:B |] ==> Q; ``` paulson@13140 ` 174` ``` [| : s; a:A; a'=a; b':B; b:B |] ==> Q ``` paulson@13140 ` 175` ``` |] ==> Q" ``` paulson@13356 ` 176` ```by blast ``` paulson@13140 ` 177` paulson@13512 ` 178` ```subsubsection{*Type checking*} ``` paulson@13140 ` 179` paulson@13140 ` 180` ```lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)" ``` paulson@13356 ` 181` ```by (unfold rmult_def, rule Collect_subset) ``` paulson@13140 ` 182` paulson@13140 ` 183` ```lemmas field_rmult = rmult_type [THEN field_rel_subset] ``` paulson@13140 ` 184` paulson@13512 ` 185` ```subsubsection{*Linearity*} ``` paulson@13140 ` 186` paulson@13140 ` 187` ```lemma linear_rmult: ``` paulson@13140 ` 188` ``` "[| linear(A,r); linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))" ``` paulson@13356 ` 189` ```by (simp add: linear_def, blast) ``` paulson@13140 ` 190` paulson@13512 ` 191` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 192` paulson@13140 ` 193` ```lemma wf_on_rmult: "[| wf[A](r); wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))" ``` paulson@13140 ` 194` ```apply (rule wf_onI2) ``` paulson@13140 ` 195` ```apply (erule SigmaE) ``` paulson@13140 ` 196` ```apply (erule ssubst) ``` paulson@13269 ` 197` ```apply (subgoal_tac "ALL b:B. : Ba", blast) ``` paulson@13784 ` 198` ```apply (erule_tac a = x in wf_on_induct, assumption) ``` paulson@13140 ` 199` ```apply (rule ballI) ``` paulson@13784 ` 200` ```apply (erule_tac a = b in wf_on_induct, assumption) ``` paulson@13140 ` 201` ```apply (best elim!: rmultE bspec [THEN mp]) ``` paulson@13140 ` 202` ```done ``` paulson@13140 ` 203` paulson@13140 ` 204` paulson@13140 ` 205` ```lemma wf_rmult: "[| wf(r); wf(s) |] ==> wf(rmult(field(r),r,field(s),s))" ``` paulson@13140 ` 206` ```apply (simp add: wf_iff_wf_on_field) ``` paulson@13140 ` 207` ```apply (rule wf_on_subset_A [OF _ field_rmult]) ``` paulson@13140 ` 208` ```apply (blast intro: wf_on_rmult) ``` paulson@13140 ` 209` ```done ``` paulson@13140 ` 210` paulson@13140 ` 211` ```lemma well_ord_rmult: ``` paulson@13140 ` 212` ``` "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))" ``` paulson@13140 ` 213` ```apply (rule well_ordI) ``` paulson@13140 ` 214` ```apply (simp add: well_ord_def wf_on_rmult) ``` paulson@13140 ` 215` ```apply (simp add: well_ord_def tot_ord_def linear_rmult) ``` paulson@13140 ` 216` ```done ``` paulson@9883 ` 217` paulson@9883 ` 218` paulson@13512 ` 219` ```subsubsection{*An @{term ord_iso} congruence law*} ``` paulson@13140 ` 220` paulson@13140 ` 221` ```lemma prod_bij: ``` paulson@13140 ` 222` ``` "[| f: bij(A,C); g: bij(B,D) |] ``` paulson@13140 ` 223` ``` ==> (lam :A*B. ) : bij(A*B, C*D)" ``` paulson@13140 ` 224` ```apply (rule_tac d = "%. " ``` paulson@13140 ` 225` ``` in lam_bijective) ``` paulson@13140 ` 226` ```apply (typecheck add: bij_is_inj inj_is_fun) ``` paulson@13140 ` 227` ```apply (auto simp add: left_inverse_bij right_inverse_bij) ``` paulson@13140 ` 228` ```done ``` paulson@13140 ` 229` paulson@13140 ` 230` ```lemma prod_ord_iso_cong: ``` paulson@13140 ` 231` ``` "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ``` paulson@13140 ` 232` ``` ==> (lam :A*B. ) ``` paulson@13140 ` 233` ``` : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))" ``` paulson@13140 ` 234` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 235` ```apply (safe intro!: prod_bij) ``` paulson@13140 ` 236` ```apply (simp_all add: bij_is_fun [THEN apply_type]) ``` paulson@13140 ` 237` ```apply (blast intro: bij_is_inj [THEN inj_apply_equality]) ``` paulson@13140 ` 238` ```done ``` paulson@13140 ` 239` paulson@13140 ` 240` ```lemma singleton_prod_bij: "(lam z:A. ) : bij(A, {x}*A)" ``` paulson@13784 ` 241` ```by (rule_tac d = snd in lam_bijective, auto) ``` paulson@13140 ` 242` paulson@13140 ` 243` ```(*Used??*) ``` paulson@13140 ` 244` ```lemma singleton_prod_ord_iso: ``` paulson@13140 ` 245` ``` "well_ord({x},xr) ==> ``` paulson@13140 ` 246` ``` (lam z:A. ) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))" ``` paulson@13140 ` 247` ```apply (rule singleton_prod_bij [THEN ord_isoI]) ``` paulson@13140 ` 248` ```apply (simp (no_asm_simp)) ``` paulson@13140 ` 249` ```apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl]) ``` paulson@13140 ` 250` ```done ``` paulson@13140 ` 251` paulson@13140 ` 252` ```(*Here we build a complicated function term, then simplify it using ``` paulson@13140 ` 253` ``` case_cong, id_conv, comp_lam, case_case.*) ``` paulson@13140 ` 254` ```lemma prod_sum_singleton_bij: ``` paulson@13140 ` 255` ``` "a~:C ==> ``` paulson@13140 ` 256` ``` (lam x:C*B + D. case(%x. x, %y., x)) ``` paulson@13140 ` 257` ``` : bij(C*B + D, C*B Un {a}*D)" ``` paulson@13140 ` 258` ```apply (rule subst_elem) ``` paulson@13140 ` 259` ```apply (rule id_bij [THEN sum_bij, THEN comp_bij]) ``` paulson@13140 ` 260` ```apply (rule singleton_prod_bij) ``` paulson@13269 ` 261` ```apply (rule sum_disjoint_bij, blast) ``` paulson@13140 ` 262` ```apply (simp (no_asm_simp) cong add: case_cong) ``` paulson@13140 ` 263` ```apply (rule comp_lam [THEN trans, symmetric]) ``` paulson@13140 ` 264` ```apply (fast elim!: case_type) ``` paulson@13140 ` 265` ```apply (simp (no_asm_simp) add: case_case) ``` paulson@13140 ` 266` ```done ``` paulson@13140 ` 267` paulson@13140 ` 268` ```lemma prod_sum_singleton_ord_iso: ``` paulson@13140 ` 269` ``` "[| a:A; well_ord(A,r) |] ==> ``` paulson@13140 ` 270` ``` (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y., x)) ``` paulson@13140 ` 271` ``` : ord_iso(pred(A,a,r)*B + pred(B,b,s), ``` paulson@13140 ` 272` ``` radd(A*B, rmult(A,r,B,s), B, s), ``` paulson@13140 ` 273` ``` pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))" ``` paulson@13140 ` 274` ```apply (rule prod_sum_singleton_bij [THEN ord_isoI]) ``` paulson@13140 ` 275` ```apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl]) ``` paulson@13140 ` 276` ```apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE) ``` paulson@13140 ` 277` ```done ``` paulson@13140 ` 278` paulson@13512 ` 279` ```subsubsection{*Distributive law*} ``` paulson@13140 ` 280` paulson@13140 ` 281` ```lemma sum_prod_distrib_bij: ``` paulson@13140 ` 282` ``` "(lam :(A+B)*C. case(%y. Inl(), %y. Inr(), x)) ``` paulson@13140 ` 283` ``` : bij((A+B)*C, (A*C)+(B*C))" ``` paulson@13356 ` 284` ```by (rule_tac d = "case (%., %.) " ``` paulson@13356 ` 285` ``` in lam_bijective, auto) ``` paulson@13140 ` 286` paulson@13140 ` 287` ```lemma sum_prod_distrib_ord_iso: ``` paulson@13140 ` 288` ``` "(lam :(A+B)*C. case(%y. Inl(), %y. Inr(), x)) ``` paulson@13140 ` 289` ``` : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t), ``` paulson@13140 ` 290` ``` (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))" ``` paulson@13356 ` 291` ```by (rule sum_prod_distrib_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 292` paulson@13512 ` 293` ```subsubsection{*Associativity*} ``` paulson@13140 ` 294` paulson@13140 ` 295` ```lemma prod_assoc_bij: ``` paulson@13140 ` 296` ``` "(lam <, z>:(A*B)*C. >) : bij((A*B)*C, A*(B*C))" ``` paulson@13356 ` 297` ```by (rule_tac d = "%>. <, z>" in lam_bijective, auto) ``` paulson@13140 ` 298` paulson@13140 ` 299` ```lemma prod_assoc_ord_iso: ``` paulson@13140 ` 300` ``` "(lam <, z>:(A*B)*C. >) ``` paulson@13140 ` 301` ``` : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t), ``` paulson@13140 ` 302` ``` A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))" ``` paulson@13356 ` 303` ```by (rule prod_assoc_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 304` paulson@13356 ` 305` ```subsection{*Inverse Image of a Relation*} ``` paulson@13140 ` 306` paulson@13512 ` 307` ```subsubsection{*Rewrite rule*} ``` paulson@13140 ` 308` paulson@13140 ` 309` ```lemma rvimage_iff: " : rvimage(A,f,r) <-> : r & a:A & b:A" ``` paulson@13269 ` 310` ```by (unfold rvimage_def, blast) ``` paulson@13140 ` 311` paulson@13512 ` 312` ```subsubsection{*Type checking*} ``` paulson@13140 ` 313` paulson@13140 ` 314` ```lemma rvimage_type: "rvimage(A,f,r) <= A*A" ``` paulson@13784 ` 315` ```by (unfold rvimage_def, rule Collect_subset) ``` paulson@13140 ` 316` paulson@13140 ` 317` ```lemmas field_rvimage = rvimage_type [THEN field_rel_subset] ``` paulson@13140 ` 318` paulson@13140 ` 319` ```lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))" ``` paulson@13269 ` 320` ```by (unfold rvimage_def, blast) ``` paulson@13140 ` 321` paulson@13140 ` 322` paulson@13512 ` 323` ```subsubsection{*Partial Ordering Properties*} ``` paulson@13140 ` 324` paulson@13140 ` 325` ```lemma irrefl_rvimage: ``` paulson@13140 ` 326` ``` "[| f: inj(A,B); irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))" ``` paulson@13140 ` 327` ```apply (unfold irrefl_def rvimage_def) ``` paulson@13140 ` 328` ```apply (blast intro: inj_is_fun [THEN apply_type]) ``` paulson@13140 ` 329` ```done ``` paulson@13140 ` 330` paulson@13140 ` 331` ```lemma trans_on_rvimage: ``` paulson@13140 ` 332` ``` "[| f: inj(A,B); trans[B](r) |] ==> trans[A](rvimage(A,f,r))" ``` paulson@13140 ` 333` ```apply (unfold trans_on_def rvimage_def) ``` paulson@13140 ` 334` ```apply (blast intro: inj_is_fun [THEN apply_type]) ``` paulson@13140 ` 335` ```done ``` paulson@13140 ` 336` paulson@13140 ` 337` ```lemma part_ord_rvimage: ``` paulson@13140 ` 338` ``` "[| f: inj(A,B); part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 339` ```apply (unfold part_ord_def) ``` paulson@13140 ` 340` ```apply (blast intro!: irrefl_rvimage trans_on_rvimage) ``` paulson@13140 ` 341` ```done ``` paulson@13140 ` 342` paulson@13512 ` 343` ```subsubsection{*Linearity*} ``` paulson@13140 ` 344` paulson@13140 ` 345` ```lemma linear_rvimage: ``` paulson@13140 ` 346` ``` "[| f: inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))" ``` paulson@13140 ` 347` ```apply (simp add: inj_def linear_def rvimage_iff) ``` paulson@13269 ` 348` ```apply (blast intro: apply_funtype) ``` paulson@13140 ` 349` ```done ``` paulson@13140 ` 350` paulson@13140 ` 351` ```lemma tot_ord_rvimage: ``` paulson@13140 ` 352` ``` "[| f: inj(A,B); tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 353` ```apply (unfold tot_ord_def) ``` paulson@13140 ` 354` ```apply (blast intro!: part_ord_rvimage linear_rvimage) ``` paulson@13140 ` 355` ```done ``` paulson@13140 ` 356` paulson@13140 ` 357` paulson@13512 ` 358` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 359` paulson@13140 ` 360` ```lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))" ``` paulson@13140 ` 361` ```apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal) ``` paulson@13140 ` 362` ```apply clarify ``` paulson@13140 ` 363` ```apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }") ``` paulson@13140 ` 364` ``` apply (erule allE) ``` paulson@13140 ` 365` ``` apply (erule impE) ``` paulson@13269 ` 366` ``` apply assumption ``` paulson@13140 ` 367` ``` apply blast ``` paulson@13269 ` 368` ```apply blast ``` paulson@13140 ` 369` ```done ``` paulson@13140 ` 370` paulson@13544 ` 371` ```text{*But note that the combination of @{text wf_imp_wf_on} and ``` paulson@13544 ` 372` ``` @{text wf_rvimage} gives @{term "wf(r) ==> wf[C](rvimage(A,f,r))"}*} ``` paulson@13140 ` 373` ```lemma wf_on_rvimage: "[| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))" ``` paulson@13140 ` 374` ```apply (rule wf_onI2) ``` paulson@13140 ` 375` ```apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba") ``` paulson@13140 ` 376` ``` apply blast ``` paulson@13140 ` 377` ```apply (erule_tac a = "f`y" in wf_on_induct) ``` paulson@13140 ` 378` ``` apply (blast intro!: apply_funtype) ``` paulson@13140 ` 379` ```apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1]) ``` paulson@13140 ` 380` ```done ``` paulson@13140 ` 381` paulson@13140 ` 382` ```(*Note that we need only wf[A](...) and linear(A,...) to get the result!*) ``` paulson@13140 ` 383` ```lemma well_ord_rvimage: ``` paulson@13140 ` 384` ``` "[| f: inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 385` ```apply (rule well_ordI) ``` paulson@13140 ` 386` ```apply (unfold well_ord_def tot_ord_def) ``` paulson@13140 ` 387` ```apply (blast intro!: wf_on_rvimage inj_is_fun) ``` paulson@13140 ` 388` ```apply (blast intro!: linear_rvimage) ``` paulson@13140 ` 389` ```done ``` paulson@13140 ` 390` paulson@13140 ` 391` ```lemma ord_iso_rvimage: ``` paulson@13140 ` 392` ``` "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)" ``` paulson@13140 ` 393` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 394` ```apply (simp add: rvimage_iff) ``` paulson@13140 ` 395` ```done ``` paulson@13140 ` 396` paulson@13140 ` 397` ```lemma ord_iso_rvimage_eq: ``` paulson@13140 ` 398` ``` "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A" ``` paulson@13356 ` 399` ```by (unfold ord_iso_def rvimage_def, blast) ``` paulson@13140 ` 400` paulson@13140 ` 401` paulson@13634 ` 402` ```subsection{*Every well-founded relation is a subset of some inverse image of ``` paulson@13634 ` 403` ``` an ordinal*} ``` paulson@13634 ` 404` paulson@13634 ` 405` ```lemma wf_rvimage_Ord: "Ord(i) \ wf(rvimage(A, f, Memrel(i)))" ``` paulson@13634 ` 406` ```by (blast intro: wf_rvimage wf_Memrel) ``` paulson@13634 ` 407` paulson@13634 ` 408` paulson@13634 ` 409` ```constdefs ``` paulson@13634 ` 410` ``` wfrank :: "[i,i]=>i" ``` paulson@13634 ` 411` ``` "wfrank(r,a) == wfrec(r, a, %x f. \y \ r-``{x}. succ(f`y))" ``` paulson@13634 ` 412` paulson@13634 ` 413` ```constdefs ``` paulson@13634 ` 414` ``` wftype :: "i=>i" ``` paulson@13634 ` 415` ``` "wftype(r) == \y \ range(r). succ(wfrank(r,y))" ``` paulson@13634 ` 416` paulson@13634 ` 417` ```lemma wfrank: "wf(r) ==> wfrank(r,a) = (\y \ r-``{a}. succ(wfrank(r,y)))" ``` paulson@13634 ` 418` ```by (subst wfrank_def [THEN def_wfrec], simp_all) ``` paulson@13634 ` 419` paulson@13634 ` 420` ```lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))" ``` paulson@13634 ` 421` ```apply (rule_tac a=a in wf_induct, assumption) ``` paulson@13634 ` 422` ```apply (subst wfrank, assumption) ``` paulson@13634 ` 423` ```apply (rule Ord_succ [THEN Ord_UN], blast) ``` paulson@13634 ` 424` ```done ``` paulson@13634 ` 425` paulson@13634 ` 426` ```lemma wfrank_lt: "[|wf(r); \ r|] ==> wfrank(r,a) < wfrank(r,b)" ``` paulson@13634 ` 427` ```apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption) ``` paulson@13634 ` 428` ```apply (rule UN_I [THEN ltI]) ``` paulson@13634 ` 429` ```apply (simp add: Ord_wfrank vimage_iff)+ ``` paulson@13634 ` 430` ```done ``` paulson@13634 ` 431` paulson@13634 ` 432` ```lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))" ``` paulson@13634 ` 433` ```by (simp add: wftype_def Ord_wfrank) ``` paulson@13634 ` 434` paulson@13634 ` 435` ```lemma wftypeI: "\wf(r); x \ field(r)\ \ wfrank(r,x) \ wftype(r)" ``` paulson@13634 ` 436` ```apply (simp add: wftype_def) ``` paulson@13634 ` 437` ```apply (blast intro: wfrank_lt [THEN ltD]) ``` paulson@13634 ` 438` ```done ``` paulson@13634 ` 439` paulson@13634 ` 440` paulson@13634 ` 441` ```lemma wf_imp_subset_rvimage: ``` paulson@13634 ` 442` ``` "[|wf(r); r \ A*A|] ==> \i f. Ord(i) & r <= rvimage(A, f, Memrel(i))" ``` paulson@13634 ` 443` ```apply (rule_tac x="wftype(r)" in exI) ``` paulson@13634 ` 444` ```apply (rule_tac x="\x\A. wfrank(r,x)" in exI) ``` paulson@13634 ` 445` ```apply (simp add: Ord_wftype, clarify) ``` paulson@13634 ` 446` ```apply (frule subsetD, assumption, clarify) ``` paulson@13634 ` 447` ```apply (simp add: rvimage_iff wfrank_lt [THEN ltD]) ``` paulson@13634 ` 448` ```apply (blast intro: wftypeI) ``` paulson@13634 ` 449` ```done ``` paulson@13634 ` 450` paulson@13634 ` 451` ```theorem wf_iff_subset_rvimage: ``` paulson@13634 ` 452` ``` "relation(r) ==> wf(r) <-> (\i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))" ``` paulson@13634 ` 453` ```by (blast dest!: relation_field_times_field wf_imp_subset_rvimage ``` paulson@13634 ` 454` ``` intro: wf_rvimage_Ord [THEN wf_subset]) ``` paulson@13634 ` 455` paulson@13634 ` 456` paulson@13544 ` 457` ```subsection{*Other Results*} ``` paulson@13544 ` 458` paulson@13544 ` 459` ```lemma wf_times: "A Int B = 0 ==> wf(A*B)" ``` paulson@13544 ` 460` ```by (simp add: wf_def, blast) ``` paulson@13544 ` 461` paulson@13544 ` 462` ```text{*Could also be used to prove @{text wf_radd}*} ``` paulson@13544 ` 463` ```lemma wf_Un: ``` paulson@13544 ` 464` ``` "[| range(r) Int domain(s) = 0; wf(r); wf(s) |] ==> wf(r Un s)" ``` paulson@13544 ` 465` ```apply (simp add: wf_def, clarify) ``` paulson@13544 ` 466` ```apply (rule equalityI) ``` paulson@13544 ` 467` ``` prefer 2 apply blast ``` paulson@13544 ` 468` ```apply clarify ``` paulson@13544 ` 469` ```apply (drule_tac x=Z in spec) ``` paulson@13544 ` 470` ```apply (drule_tac x="Z Int domain(s)" in spec) ``` paulson@13544 ` 471` ```apply simp ``` paulson@13544 ` 472` ```apply (blast intro: elim: equalityE) ``` paulson@13544 ` 473` ```done ``` paulson@13544 ` 474` paulson@13544 ` 475` ```subsubsection{*The Empty Relation*} ``` paulson@13544 ` 476` paulson@13544 ` 477` ```lemma wf0: "wf(0)" ``` paulson@13544 ` 478` ```by (simp add: wf_def, blast) ``` paulson@13544 ` 479` paulson@13544 ` 480` ```lemma linear0: "linear(0,0)" ``` paulson@13544 ` 481` ```by (simp add: linear_def) ``` paulson@13544 ` 482` paulson@13544 ` 483` ```lemma well_ord0: "well_ord(0,0)" ``` paulson@13544 ` 484` ```by (blast intro: wf_imp_wf_on well_ordI wf0 linear0) ``` paulson@13512 ` 485` paulson@13512 ` 486` ```subsubsection{*The "measure" relation is useful with wfrec*} ``` paulson@13140 ` 487` paulson@13140 ` 488` ```lemma measure_eq_rvimage_Memrel: ``` paulson@13140 ` 489` ``` "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))" ``` paulson@13140 ` 490` ```apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff) ``` paulson@13269 ` 491` ```apply (rule equalityI, auto) ``` paulson@13140 ` 492` ```apply (auto intro: Ord_in_Ord simp add: lt_def) ``` paulson@13140 ` 493` ```done ``` paulson@13140 ` 494` paulson@13140 ` 495` ```lemma wf_measure [iff]: "wf(measure(A,f))" ``` paulson@13356 ` 496` ```by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage) ``` paulson@13140 ` 497` paulson@13140 ` 498` ```lemma measure_iff [iff]: " : measure(A,f) <-> x:A & y:A & f(x) A ==> Ord(f(x))" ``` paulson@13544 ` 503` ``` and inj: "!!x y. [|x \ A; y \ A; f(x) = f(y) |] ==> x=y" ``` paulson@13544 ` 504` ``` shows "linear(A, measure(A,f))" ``` paulson@13544 ` 505` ```apply (auto simp add: linear_def) ``` paulson@13544 ` 506` ```apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt) ``` paulson@13544 ` 507` ``` apply (simp_all add: Ordf) ``` paulson@13544 ` 508` ```apply (blast intro: inj) ``` paulson@13544 ` 509` ```done ``` paulson@13544 ` 510` paulson@13544 ` 511` ```lemma wf_on_measure: "wf[B](measure(A,f))" ``` paulson@13544 ` 512` ```by (rule wf_imp_wf_on [OF wf_measure]) ``` paulson@13544 ` 513` paulson@13544 ` 514` ```lemma well_ord_measure: ``` paulson@13544 ` 515` ``` assumes Ordf: "!!x. x \ A ==> Ord(f(x))" ``` paulson@13544 ` 516` ``` and inj: "!!x y. [|x \ A; y \ A; f(x) = f(y) |] ==> x=y" ``` paulson@13544 ` 517` ``` shows "well_ord(A, measure(A,f))" ``` paulson@13544 ` 518` ```apply (rule well_ordI) ``` paulson@13544 ` 519` ```apply (rule wf_on_measure) ``` paulson@13544 ` 520` ```apply (blast intro: linear_measure Ordf inj) ``` paulson@13544 ` 521` ```done ``` paulson@13544 ` 522` paulson@13544 ` 523` ```lemma measure_type: "measure(A,f) <= A*A" ``` paulson@13544 ` 524` ```by (auto simp add: measure_def) ``` paulson@13544 ` 525` paulson@13512 ` 526` ```subsubsection{*Well-foundedness of Unions*} ``` paulson@13512 ` 527` paulson@13512 ` 528` ```lemma wf_on_Union: ``` paulson@13512 ` 529` ``` assumes wfA: "wf[A](r)" ``` paulson@13512 ` 530` ``` and wfB: "!!a. a\A ==> wf[B(a)](s)" ``` paulson@13512 ` 531` ``` and ok: "!!a u v. [| \ s; v \ B(a); a \ A|] ``` paulson@13512 ` 532` ``` ==> (\a'\A. \ r & u \ B(a')) | u \ B(a)" ``` paulson@13512 ` 533` ``` shows "wf[\a\A. B(a)](s)" ``` paulson@13512 ` 534` ```apply (rule wf_onI2) ``` paulson@13512 ` 535` ```apply (erule UN_E) ``` paulson@13512 ` 536` ```apply (subgoal_tac "\z \ B(a). z \ Ba", blast) ``` paulson@13512 ` 537` ```apply (rule_tac a = a in wf_on_induct [OF wfA], assumption) ``` paulson@13512 ` 538` ```apply (rule ballI) ``` paulson@13512 ` 539` ```apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption) ``` paulson@13512 ` 540` ```apply (rename_tac u) ``` paulson@13512 ` 541` ```apply (drule_tac x=u in bspec, blast) ``` paulson@13512 ` 542` ```apply (erule mp, clarify) ``` paulson@13784 ` 543` ```apply (frule ok, assumption+, blast) ``` paulson@13512 ` 544` ```done ``` paulson@13512 ` 545` paulson@14120 ` 546` ```subsubsection{*Bijections involving Powersets*} ``` paulson@14120 ` 547` paulson@14120 ` 548` ```lemma Pow_sum_bij: ``` paulson@14120 ` 549` ``` "(\Z \ Pow(A+B). <{x \ A. Inl(x) \ Z}, {y \ B. Inr(y) \ Z}>) ``` paulson@14120 ` 550` ``` \ bij(Pow(A+B), Pow(A)*Pow(B))" ``` paulson@14120 ` 551` ```apply (rule_tac d = "%. {Inl (x). x \ X} Un {Inr (y). y \ Y}" ``` paulson@14120 ` 552` ``` in lam_bijective) ``` paulson@14120 ` 553` ```apply force+ ``` paulson@14120 ` 554` ```done ``` paulson@14120 ` 555` paulson@14120 ` 556` ```text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *} ``` paulson@14120 ` 557` ```lemma Pow_Sigma_bij: ``` paulson@14120 ` 558` ``` "(\r \ Pow(Sigma(A,B)). \x \ A. r``{x}) ``` paulson@14120 ` 559` ``` \ bij(Pow(Sigma(A,B)), \x \ A. Pow(B(x)))" ``` paulson@14120 ` 560` ```apply (rule_tac d = "%f. \x \ A. \y \ f`x. {}" in lam_bijective) ``` paulson@14120 ` 561` ```apply (blast intro: lam_type) ``` paulson@14120 ` 562` ```apply (blast dest: apply_type, simp_all) ``` paulson@14120 ` 563` ```apply fast (*strange, but blast can't do it*) ``` paulson@14120 ` 564` ```apply (rule fun_extension, auto) ``` paulson@14120 ` 565` ```by blast ``` paulson@14120 ` 566` paulson@13512 ` 567` paulson@13140 ` 568` ```ML {* ``` paulson@13140 ` 569` ```val measure_def = thm "measure_def"; ``` paulson@13140 ` 570` ```val radd_Inl_Inr_iff = thm "radd_Inl_Inr_iff"; ``` paulson@13140 ` 571` ```val radd_Inl_iff = thm "radd_Inl_iff"; ``` paulson@13140 ` 572` ```val radd_Inr_iff = thm "radd_Inr_iff"; ``` paulson@13140 ` 573` ```val radd_Inr_Inl_iff = thm "radd_Inr_Inl_iff"; ``` paulson@13140 ` 574` ```val raddE = thm "raddE"; ``` paulson@13140 ` 575` ```val radd_type = thm "radd_type"; ``` paulson@13140 ` 576` ```val field_radd = thm "field_radd"; ``` paulson@13140 ` 577` ```val linear_radd = thm "linear_radd"; ``` paulson@13140 ` 578` ```val wf_on_radd = thm "wf_on_radd"; ``` paulson@13140 ` 579` ```val wf_radd = thm "wf_radd"; ``` paulson@13140 ` 580` ```val well_ord_radd = thm "well_ord_radd"; ``` paulson@13140 ` 581` ```val sum_bij = thm "sum_bij"; ``` paulson@13140 ` 582` ```val sum_ord_iso_cong = thm "sum_ord_iso_cong"; ``` paulson@13140 ` 583` ```val sum_disjoint_bij = thm "sum_disjoint_bij"; ``` paulson@13140 ` 584` ```val sum_assoc_bij = thm "sum_assoc_bij"; ``` paulson@13140 ` 585` ```val sum_assoc_ord_iso = thm "sum_assoc_ord_iso"; ``` paulson@13140 ` 586` ```val rmult_iff = thm "rmult_iff"; ``` paulson@13140 ` 587` ```val rmultE = thm "rmultE"; ``` paulson@13140 ` 588` ```val rmult_type = thm "rmult_type"; ``` paulson@13140 ` 589` ```val field_rmult = thm "field_rmult"; ``` paulson@13140 ` 590` ```val linear_rmult = thm "linear_rmult"; ``` paulson@13140 ` 591` ```val wf_on_rmult = thm "wf_on_rmult"; ``` paulson@13140 ` 592` ```val wf_rmult = thm "wf_rmult"; ``` paulson@13140 ` 593` ```val well_ord_rmult = thm "well_ord_rmult"; ``` paulson@13140 ` 594` ```val prod_bij = thm "prod_bij"; ``` paulson@13140 ` 595` ```val prod_ord_iso_cong = thm "prod_ord_iso_cong"; ``` paulson@13140 ` 596` ```val singleton_prod_bij = thm "singleton_prod_bij"; ``` paulson@13140 ` 597` ```val singleton_prod_ord_iso = thm "singleton_prod_ord_iso"; ``` paulson@13140 ` 598` ```val prod_sum_singleton_bij = thm "prod_sum_singleton_bij"; ``` paulson@13140 ` 599` ```val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso"; ``` paulson@13140 ` 600` ```val sum_prod_distrib_bij = thm "sum_prod_distrib_bij"; ``` paulson@13140 ` 601` ```val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso"; ``` paulson@13140 ` 602` ```val prod_assoc_bij = thm "prod_assoc_bij"; ``` paulson@13140 ` 603` ```val prod_assoc_ord_iso = thm "prod_assoc_ord_iso"; ``` paulson@13140 ` 604` ```val rvimage_iff = thm "rvimage_iff"; ``` paulson@13140 ` 605` ```val rvimage_type = thm "rvimage_type"; ``` paulson@13140 ` 606` ```val field_rvimage = thm "field_rvimage"; ``` paulson@13140 ` 607` ```val rvimage_converse = thm "rvimage_converse"; ``` paulson@13140 ` 608` ```val irrefl_rvimage = thm "irrefl_rvimage"; ``` paulson@13140 ` 609` ```val trans_on_rvimage = thm "trans_on_rvimage"; ``` paulson@13140 ` 610` ```val part_ord_rvimage = thm "part_ord_rvimage"; ``` paulson@13140 ` 611` ```val linear_rvimage = thm "linear_rvimage"; ``` paulson@13140 ` 612` ```val tot_ord_rvimage = thm "tot_ord_rvimage"; ``` paulson@13140 ` 613` ```val wf_rvimage = thm "wf_rvimage"; ``` paulson@13140 ` 614` ```val wf_on_rvimage = thm "wf_on_rvimage"; ``` paulson@13140 ` 615` ```val well_ord_rvimage = thm "well_ord_rvimage"; ``` paulson@13140 ` 616` ```val ord_iso_rvimage = thm "ord_iso_rvimage"; ``` paulson@13140 ` 617` ```val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq"; ``` paulson@13140 ` 618` ```val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel"; ``` paulson@13140 ` 619` ```val wf_measure = thm "wf_measure"; ``` paulson@13140 ` 620` ```val measure_iff = thm "measure_iff"; ``` paulson@13140 ` 621` ```*} ``` paulson@13140 ` 622` lcp@437 ` 623` ```end ```