src/ZF/OrderArith.thy
 author paulson Wed Aug 28 13:08:50 2002 +0200 (2002-08-28) changeset 13544 895994073bdf parent 13512 80edb859fd24 child 13634 99a593b49b04 permissions -rw-r--r--
various new lemmas for Constructible
 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@13140 ` 52` ```lemma radd_Inr_Inl_iff [iff]: ``` paulson@13140 ` 53` ``` " : radd(A,r,B,s) <-> False" ``` paulson@13356 ` 54` ```by (unfold radd_def, blast) ``` paulson@13140 ` 55` paulson@13512 ` 56` ```subsubsection{*Elimination Rule*} ``` paulson@13140 ` 57` paulson@13140 ` 58` ```lemma raddE: ``` paulson@13140 ` 59` ``` "[| : radd(A,r,B,s); ``` paulson@13140 ` 60` ``` !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q; ``` paulson@13140 ` 61` ``` !!x' x. [| p'=Inl(x'); p=Inl(x); : r; x':A; x:A |] ==> Q; ``` paulson@13140 ` 62` ``` !!y' y. [| p'=Inr(y'); p=Inr(y); : s; y':B; y:B |] ==> Q ``` paulson@13140 ` 63` ``` |] ==> Q" ``` paulson@13356 ` 64` ```by (unfold radd_def, blast) ``` paulson@13140 ` 65` paulson@13512 ` 66` ```subsubsection{*Type checking*} ``` paulson@13140 ` 67` paulson@13140 ` 68` ```lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)" ``` paulson@13140 ` 69` ```apply (unfold radd_def) ``` paulson@13140 ` 70` ```apply (rule Collect_subset) ``` paulson@13140 ` 71` ```done ``` paulson@13140 ` 72` paulson@13140 ` 73` ```lemmas field_radd = radd_type [THEN field_rel_subset] ``` paulson@13140 ` 74` paulson@13512 ` 75` ```subsubsection{*Linearity*} ``` paulson@13140 ` 76` paulson@13140 ` 77` ```lemma linear_radd: ``` paulson@13140 ` 78` ``` "[| linear(A,r); linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))" ``` paulson@13356 ` 79` ```by (unfold linear_def, blast) ``` paulson@13140 ` 80` paulson@13140 ` 81` paulson@13512 ` 82` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 83` paulson@13140 ` 84` ```lemma wf_on_radd: "[| wf[A](r); wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))" ``` paulson@13140 ` 85` ```apply (rule wf_onI2) ``` paulson@13140 ` 86` ```apply (subgoal_tac "ALL x:A. Inl (x) : Ba") ``` paulson@13512 ` 87` ``` --{*Proving the lemma, which is needed twice!*} ``` paulson@13140 ` 88` ``` prefer 2 ``` paulson@13140 ` 89` ``` apply (erule_tac V = "y : A + B" in thin_rl) ``` paulson@13140 ` 90` ``` apply (rule_tac ballI) ``` paulson@13140 ` 91` ``` apply (erule_tac r = "r" and a = "x" in wf_on_induct, assumption) ``` paulson@13269 ` 92` ``` apply blast ``` paulson@13512 ` 93` ```txt{*Returning to main part of proof*} ``` paulson@13140 ` 94` ```apply safe ``` paulson@13140 ` 95` ```apply blast ``` paulson@13269 ` 96` ```apply (erule_tac r = "s" and a = "ya" in wf_on_induct, assumption, blast) ``` paulson@13140 ` 97` ```done ``` paulson@13140 ` 98` paulson@13140 ` 99` ```lemma wf_radd: "[| wf(r); wf(s) |] ==> wf(radd(field(r),r,field(s),s))" ``` paulson@13140 ` 100` ```apply (simp add: wf_iff_wf_on_field) ``` paulson@13140 ` 101` ```apply (rule wf_on_subset_A [OF _ field_radd]) ``` paulson@13140 ` 102` ```apply (blast intro: wf_on_radd) ``` paulson@13140 ` 103` ```done ``` paulson@13140 ` 104` paulson@13140 ` 105` ```lemma well_ord_radd: ``` paulson@13140 ` 106` ``` "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))" ``` paulson@13140 ` 107` ```apply (rule well_ordI) ``` paulson@13140 ` 108` ```apply (simp add: well_ord_def wf_on_radd) ``` paulson@13140 ` 109` ```apply (simp add: well_ord_def tot_ord_def linear_radd) ``` paulson@13140 ` 110` ```done ``` paulson@13140 ` 111` paulson@13512 ` 112` ```subsubsection{*An @{term ord_iso} congruence law*} ``` lcp@437 ` 113` paulson@13140 ` 114` ```lemma sum_bij: ``` paulson@13140 ` 115` ``` "[| f: bij(A,C); g: bij(B,D) |] ``` paulson@13140 ` 116` ``` ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)" ``` paulson@13356 ` 117` ```apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))" ``` paulson@13356 ` 118` ``` in lam_bijective) ``` paulson@13140 ` 119` ```apply (typecheck add: bij_is_inj inj_is_fun) ``` paulson@13140 ` 120` ```apply (auto simp add: left_inverse_bij right_inverse_bij) ``` paulson@13140 ` 121` ```done ``` paulson@13140 ` 122` paulson@13140 ` 123` ```lemma sum_ord_iso_cong: ``` paulson@13140 ` 124` ``` "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==> ``` paulson@13140 ` 125` ``` (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) ``` paulson@13140 ` 126` ``` : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))" ``` paulson@13140 ` 127` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 128` ```apply (safe intro!: sum_bij) ``` paulson@13140 ` 129` ```(*Do the beta-reductions now*) ``` paulson@13140 ` 130` ```apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type]) ``` paulson@13140 ` 131` ```done ``` paulson@13140 ` 132` paulson@13140 ` 133` ```(*Could we prove an ord_iso result? Perhaps ``` paulson@13140 ` 134` ``` ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *) ``` paulson@13140 ` 135` ```lemma sum_disjoint_bij: "A Int B = 0 ==> ``` paulson@13140 ` 136` ``` (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)" ``` paulson@13140 ` 137` ```apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective) ``` paulson@13140 ` 138` ```apply auto ``` paulson@13140 ` 139` ```done ``` paulson@13140 ` 140` paulson@13512 ` 141` ```subsubsection{*Associativity*} ``` paulson@13140 ` 142` paulson@13140 ` 143` ```lemma sum_assoc_bij: ``` paulson@13140 ` 144` ``` "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) ``` paulson@13140 ` 145` ``` : bij((A+B)+C, A+(B+C))" ``` paulson@13140 ` 146` ```apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" ``` paulson@13140 ` 147` ``` in lam_bijective) ``` paulson@13140 ` 148` ```apply auto ``` paulson@13140 ` 149` ```done ``` paulson@13140 ` 150` paulson@13140 ` 151` ```lemma sum_assoc_ord_iso: ``` paulson@13140 ` 152` ``` "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z)) ``` paulson@13140 ` 153` ``` : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t), ``` paulson@13140 ` 154` ``` A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))" ``` paulson@13356 ` 155` ```by (rule sum_assoc_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 156` paulson@13140 ` 157` paulson@13356 ` 158` ```subsection{*Multiplication of Relations -- Lexicographic Product*} ``` paulson@13140 ` 159` paulson@13512 ` 160` ```subsubsection{*Rewrite rule. Can be used to obtain introduction rules*} ``` paulson@13140 ` 161` paulson@13140 ` 162` ```lemma rmult_iff [iff]: ``` paulson@13140 ` 163` ``` "<, > : rmult(A,r,B,s) <-> ``` paulson@13140 ` 164` ``` (: r & a':A & a:A & b': B & b: B) | ``` paulson@13140 ` 165` ``` (: s & a'=a & a:A & b': B & b: B)" ``` paulson@13140 ` 166` paulson@13356 ` 167` ```by (unfold rmult_def, blast) ``` paulson@13140 ` 168` paulson@13140 ` 169` ```lemma rmultE: ``` paulson@13140 ` 170` ``` "[| <, > : rmult(A,r,B,s); ``` paulson@13140 ` 171` ``` [| : r; a':A; a:A; b':B; b:B |] ==> Q; ``` paulson@13140 ` 172` ``` [| : s; a:A; a'=a; b':B; b:B |] ==> Q ``` paulson@13140 ` 173` ``` |] ==> Q" ``` paulson@13356 ` 174` ```by blast ``` paulson@13140 ` 175` paulson@13512 ` 176` ```subsubsection{*Type checking*} ``` paulson@13140 ` 177` paulson@13140 ` 178` ```lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)" ``` paulson@13356 ` 179` ```by (unfold rmult_def, rule Collect_subset) ``` paulson@13140 ` 180` paulson@13140 ` 181` ```lemmas field_rmult = rmult_type [THEN field_rel_subset] ``` paulson@13140 ` 182` paulson@13512 ` 183` ```subsubsection{*Linearity*} ``` paulson@13140 ` 184` paulson@13140 ` 185` ```lemma linear_rmult: ``` paulson@13140 ` 186` ``` "[| linear(A,r); linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))" ``` paulson@13356 ` 187` ```by (simp add: linear_def, blast) ``` paulson@13140 ` 188` paulson@13512 ` 189` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 190` paulson@13140 ` 191` ```lemma wf_on_rmult: "[| wf[A](r); wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))" ``` paulson@13140 ` 192` ```apply (rule wf_onI2) ``` paulson@13140 ` 193` ```apply (erule SigmaE) ``` paulson@13140 ` 194` ```apply (erule ssubst) ``` paulson@13269 ` 195` ```apply (subgoal_tac "ALL b:B. : Ba", blast) ``` paulson@13269 ` 196` ```apply (erule_tac a = "x" in wf_on_induct, assumption) ``` paulson@13140 ` 197` ```apply (rule ballI) ``` paulson@13269 ` 198` ```apply (erule_tac a = "b" in wf_on_induct, assumption) ``` paulson@13140 ` 199` ```apply (best elim!: rmultE bspec [THEN mp]) ``` paulson@13140 ` 200` ```done ``` paulson@13140 ` 201` paulson@13140 ` 202` paulson@13140 ` 203` ```lemma wf_rmult: "[| wf(r); wf(s) |] ==> wf(rmult(field(r),r,field(s),s))" ``` paulson@13140 ` 204` ```apply (simp add: wf_iff_wf_on_field) ``` paulson@13140 ` 205` ```apply (rule wf_on_subset_A [OF _ field_rmult]) ``` paulson@13140 ` 206` ```apply (blast intro: wf_on_rmult) ``` paulson@13140 ` 207` ```done ``` paulson@13140 ` 208` paulson@13140 ` 209` ```lemma well_ord_rmult: ``` paulson@13140 ` 210` ``` "[| well_ord(A,r); well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))" ``` paulson@13140 ` 211` ```apply (rule well_ordI) ``` paulson@13140 ` 212` ```apply (simp add: well_ord_def wf_on_rmult) ``` paulson@13140 ` 213` ```apply (simp add: well_ord_def tot_ord_def linear_rmult) ``` paulson@13140 ` 214` ```done ``` paulson@9883 ` 215` paulson@9883 ` 216` paulson@13512 ` 217` ```subsubsection{*An @{term ord_iso} congruence law*} ``` paulson@13140 ` 218` paulson@13140 ` 219` ```lemma prod_bij: ``` paulson@13140 ` 220` ``` "[| f: bij(A,C); g: bij(B,D) |] ``` paulson@13140 ` 221` ``` ==> (lam :A*B. ) : bij(A*B, C*D)" ``` paulson@13140 ` 222` ```apply (rule_tac d = "%. " ``` paulson@13140 ` 223` ``` in lam_bijective) ``` paulson@13140 ` 224` ```apply (typecheck add: bij_is_inj inj_is_fun) ``` paulson@13140 ` 225` ```apply (auto simp add: left_inverse_bij right_inverse_bij) ``` paulson@13140 ` 226` ```done ``` paulson@13140 ` 227` paulson@13140 ` 228` ```lemma prod_ord_iso_cong: ``` paulson@13140 ` 229` ``` "[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ``` paulson@13140 ` 230` ``` ==> (lam :A*B. ) ``` paulson@13140 ` 231` ``` : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))" ``` paulson@13140 ` 232` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 233` ```apply (safe intro!: prod_bij) ``` paulson@13140 ` 234` ```apply (simp_all add: bij_is_fun [THEN apply_type]) ``` paulson@13140 ` 235` ```apply (blast intro: bij_is_inj [THEN inj_apply_equality]) ``` paulson@13140 ` 236` ```done ``` paulson@13140 ` 237` paulson@13140 ` 238` ```lemma singleton_prod_bij: "(lam z:A. ) : bij(A, {x}*A)" ``` paulson@13269 ` 239` ```by (rule_tac d = "snd" in lam_bijective, auto) ``` paulson@13140 ` 240` paulson@13140 ` 241` ```(*Used??*) ``` paulson@13140 ` 242` ```lemma singleton_prod_ord_iso: ``` paulson@13140 ` 243` ``` "well_ord({x},xr) ==> ``` paulson@13140 ` 244` ``` (lam z:A. ) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))" ``` paulson@13140 ` 245` ```apply (rule singleton_prod_bij [THEN ord_isoI]) ``` paulson@13140 ` 246` ```apply (simp (no_asm_simp)) ``` paulson@13140 ` 247` ```apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl]) ``` paulson@13140 ` 248` ```done ``` paulson@13140 ` 249` paulson@13140 ` 250` ```(*Here we build a complicated function term, then simplify it using ``` paulson@13140 ` 251` ``` case_cong, id_conv, comp_lam, case_case.*) ``` paulson@13140 ` 252` ```lemma prod_sum_singleton_bij: ``` paulson@13140 ` 253` ``` "a~:C ==> ``` paulson@13140 ` 254` ``` (lam x:C*B + D. case(%x. x, %y., x)) ``` paulson@13140 ` 255` ``` : bij(C*B + D, C*B Un {a}*D)" ``` paulson@13140 ` 256` ```apply (rule subst_elem) ``` paulson@13140 ` 257` ```apply (rule id_bij [THEN sum_bij, THEN comp_bij]) ``` paulson@13140 ` 258` ```apply (rule singleton_prod_bij) ``` paulson@13269 ` 259` ```apply (rule sum_disjoint_bij, blast) ``` paulson@13140 ` 260` ```apply (simp (no_asm_simp) cong add: case_cong) ``` paulson@13140 ` 261` ```apply (rule comp_lam [THEN trans, symmetric]) ``` paulson@13140 ` 262` ```apply (fast elim!: case_type) ``` paulson@13140 ` 263` ```apply (simp (no_asm_simp) add: case_case) ``` paulson@13140 ` 264` ```done ``` paulson@13140 ` 265` paulson@13140 ` 266` ```lemma prod_sum_singleton_ord_iso: ``` paulson@13140 ` 267` ``` "[| a:A; well_ord(A,r) |] ==> ``` paulson@13140 ` 268` ``` (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y., x)) ``` paulson@13140 ` 269` ``` : ord_iso(pred(A,a,r)*B + pred(B,b,s), ``` paulson@13140 ` 270` ``` radd(A*B, rmult(A,r,B,s), B, s), ``` paulson@13140 ` 271` ``` pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))" ``` paulson@13140 ` 272` ```apply (rule prod_sum_singleton_bij [THEN ord_isoI]) ``` paulson@13140 ` 273` ```apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl]) ``` paulson@13140 ` 274` ```apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE) ``` paulson@13140 ` 275` ```done ``` paulson@13140 ` 276` paulson@13512 ` 277` ```subsubsection{*Distributive law*} ``` paulson@13140 ` 278` paulson@13140 ` 279` ```lemma sum_prod_distrib_bij: ``` paulson@13140 ` 280` ``` "(lam :(A+B)*C. case(%y. Inl(), %y. Inr(), x)) ``` paulson@13140 ` 281` ``` : bij((A+B)*C, (A*C)+(B*C))" ``` paulson@13356 ` 282` ```by (rule_tac d = "case (%., %.) " ``` paulson@13356 ` 283` ``` in lam_bijective, auto) ``` paulson@13140 ` 284` paulson@13140 ` 285` ```lemma sum_prod_distrib_ord_iso: ``` paulson@13140 ` 286` ``` "(lam :(A+B)*C. case(%y. Inl(), %y. Inr(), x)) ``` paulson@13140 ` 287` ``` : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t), ``` paulson@13140 ` 288` ``` (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))" ``` paulson@13356 ` 289` ```by (rule sum_prod_distrib_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 290` paulson@13512 ` 291` ```subsubsection{*Associativity*} ``` paulson@13140 ` 292` paulson@13140 ` 293` ```lemma prod_assoc_bij: ``` paulson@13140 ` 294` ``` "(lam <, z>:(A*B)*C. >) : bij((A*B)*C, A*(B*C))" ``` paulson@13356 ` 295` ```by (rule_tac d = "%>. <, z>" in lam_bijective, auto) ``` paulson@13140 ` 296` paulson@13140 ` 297` ```lemma prod_assoc_ord_iso: ``` paulson@13140 ` 298` ``` "(lam <, z>:(A*B)*C. >) ``` paulson@13140 ` 299` ``` : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t), ``` paulson@13140 ` 300` ``` A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))" ``` paulson@13356 ` 301` ```by (rule prod_assoc_bij [THEN ord_isoI], auto) ``` paulson@13140 ` 302` paulson@13356 ` 303` ```subsection{*Inverse Image of a Relation*} ``` paulson@13140 ` 304` paulson@13512 ` 305` ```subsubsection{*Rewrite rule*} ``` paulson@13140 ` 306` paulson@13140 ` 307` ```lemma rvimage_iff: " : rvimage(A,f,r) <-> : r & a:A & b:A" ``` paulson@13269 ` 308` ```by (unfold rvimage_def, blast) ``` paulson@13140 ` 309` paulson@13512 ` 310` ```subsubsection{*Type checking*} ``` paulson@13140 ` 311` paulson@13140 ` 312` ```lemma rvimage_type: "rvimage(A,f,r) <= A*A" ``` paulson@13356 ` 313` ```apply (unfold rvimage_def, rule Collect_subset) ``` paulson@13140 ` 314` ```done ``` paulson@13140 ` 315` paulson@13140 ` 316` ```lemmas field_rvimage = rvimage_type [THEN field_rel_subset] ``` paulson@13140 ` 317` paulson@13140 ` 318` ```lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))" ``` paulson@13269 ` 319` ```by (unfold rvimage_def, blast) ``` paulson@13140 ` 320` paulson@13140 ` 321` paulson@13512 ` 322` ```subsubsection{*Partial Ordering Properties*} ``` paulson@13140 ` 323` paulson@13140 ` 324` ```lemma irrefl_rvimage: ``` paulson@13140 ` 325` ``` "[| f: inj(A,B); irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))" ``` paulson@13140 ` 326` ```apply (unfold irrefl_def rvimage_def) ``` paulson@13140 ` 327` ```apply (blast intro: inj_is_fun [THEN apply_type]) ``` paulson@13140 ` 328` ```done ``` paulson@13140 ` 329` paulson@13140 ` 330` ```lemma trans_on_rvimage: ``` paulson@13140 ` 331` ``` "[| f: inj(A,B); trans[B](r) |] ==> trans[A](rvimage(A,f,r))" ``` paulson@13140 ` 332` ```apply (unfold trans_on_def rvimage_def) ``` paulson@13140 ` 333` ```apply (blast intro: inj_is_fun [THEN apply_type]) ``` paulson@13140 ` 334` ```done ``` paulson@13140 ` 335` paulson@13140 ` 336` ```lemma part_ord_rvimage: ``` paulson@13140 ` 337` ``` "[| f: inj(A,B); part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 338` ```apply (unfold part_ord_def) ``` paulson@13140 ` 339` ```apply (blast intro!: irrefl_rvimage trans_on_rvimage) ``` paulson@13140 ` 340` ```done ``` paulson@13140 ` 341` paulson@13512 ` 342` ```subsubsection{*Linearity*} ``` paulson@13140 ` 343` paulson@13140 ` 344` ```lemma linear_rvimage: ``` paulson@13140 ` 345` ``` "[| f: inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))" ``` paulson@13140 ` 346` ```apply (simp add: inj_def linear_def rvimage_iff) ``` paulson@13269 ` 347` ```apply (blast intro: apply_funtype) ``` paulson@13140 ` 348` ```done ``` paulson@13140 ` 349` paulson@13140 ` 350` ```lemma tot_ord_rvimage: ``` paulson@13140 ` 351` ``` "[| f: inj(A,B); tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 352` ```apply (unfold tot_ord_def) ``` paulson@13140 ` 353` ```apply (blast intro!: part_ord_rvimage linear_rvimage) ``` paulson@13140 ` 354` ```done ``` paulson@13140 ` 355` paulson@13140 ` 356` paulson@13512 ` 357` ```subsubsection{*Well-foundedness*} ``` paulson@13140 ` 358` paulson@13140 ` 359` ```lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))" ``` paulson@13140 ` 360` ```apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal) ``` paulson@13140 ` 361` ```apply clarify ``` paulson@13140 ` 362` ```apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }") ``` paulson@13140 ` 363` ``` apply (erule allE) ``` paulson@13140 ` 364` ``` apply (erule impE) ``` paulson@13269 ` 365` ``` apply assumption ``` paulson@13140 ` 366` ``` apply blast ``` paulson@13269 ` 367` ```apply blast ``` paulson@13140 ` 368` ```done ``` paulson@13140 ` 369` paulson@13544 ` 370` ```text{*But note that the combination of @{text wf_imp_wf_on} and ``` paulson@13544 ` 371` ``` @{text wf_rvimage} gives @{term "wf(r) ==> wf[C](rvimage(A,f,r))"}*} ``` paulson@13140 ` 372` ```lemma wf_on_rvimage: "[| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))" ``` paulson@13140 ` 373` ```apply (rule wf_onI2) ``` paulson@13140 ` 374` ```apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba") ``` paulson@13140 ` 375` ``` apply blast ``` paulson@13140 ` 376` ```apply (erule_tac a = "f`y" in wf_on_induct) ``` paulson@13140 ` 377` ``` apply (blast intro!: apply_funtype) ``` paulson@13140 ` 378` ```apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1]) ``` paulson@13140 ` 379` ```done ``` paulson@13140 ` 380` paulson@13140 ` 381` ```(*Note that we need only wf[A](...) and linear(A,...) to get the result!*) ``` paulson@13140 ` 382` ```lemma well_ord_rvimage: ``` paulson@13140 ` 383` ``` "[| f: inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))" ``` paulson@13140 ` 384` ```apply (rule well_ordI) ``` paulson@13140 ` 385` ```apply (unfold well_ord_def tot_ord_def) ``` paulson@13140 ` 386` ```apply (blast intro!: wf_on_rvimage inj_is_fun) ``` paulson@13140 ` 387` ```apply (blast intro!: linear_rvimage) ``` paulson@13140 ` 388` ```done ``` paulson@13140 ` 389` paulson@13140 ` 390` ```lemma ord_iso_rvimage: ``` paulson@13140 ` 391` ``` "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)" ``` paulson@13140 ` 392` ```apply (unfold ord_iso_def) ``` paulson@13140 ` 393` ```apply (simp add: rvimage_iff) ``` paulson@13140 ` 394` ```done ``` paulson@13140 ` 395` paulson@13140 ` 396` ```lemma ord_iso_rvimage_eq: ``` paulson@13140 ` 397` ``` "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A" ``` paulson@13356 ` 398` ```by (unfold ord_iso_def rvimage_def, blast) ``` paulson@13140 ` 399` paulson@13140 ` 400` paulson@13544 ` 401` ```subsection{*Other Results*} ``` paulson@13544 ` 402` paulson@13544 ` 403` ```lemma wf_times: "A Int B = 0 ==> wf(A*B)" ``` paulson@13544 ` 404` ```by (simp add: wf_def, blast) ``` paulson@13544 ` 405` paulson@13544 ` 406` ```text{*Could also be used to prove @{text wf_radd}*} ``` paulson@13544 ` 407` ```lemma wf_Un: ``` paulson@13544 ` 408` ``` "[| range(r) Int domain(s) = 0; wf(r); wf(s) |] ==> wf(r Un s)" ``` paulson@13544 ` 409` ```apply (simp add: wf_def, clarify) ``` paulson@13544 ` 410` ```apply (rule equalityI) ``` paulson@13544 ` 411` ``` prefer 2 apply blast ``` paulson@13544 ` 412` ```apply clarify ``` paulson@13544 ` 413` ```apply (drule_tac x=Z in spec) ``` paulson@13544 ` 414` ```apply (drule_tac x="Z Int domain(s)" in spec) ``` paulson@13544 ` 415` ```apply simp ``` paulson@13544 ` 416` ```apply (blast intro: elim: equalityE) ``` paulson@13544 ` 417` ```done ``` paulson@13544 ` 418` paulson@13544 ` 419` ```subsubsection{*The Empty Relation*} ``` paulson@13544 ` 420` paulson@13544 ` 421` ```lemma wf0: "wf(0)" ``` paulson@13544 ` 422` ```by (simp add: wf_def, blast) ``` paulson@13544 ` 423` paulson@13544 ` 424` ```lemma linear0: "linear(0,0)" ``` paulson@13544 ` 425` ```by (simp add: linear_def) ``` paulson@13544 ` 426` paulson@13544 ` 427` ```lemma well_ord0: "well_ord(0,0)" ``` paulson@13544 ` 428` ```by (blast intro: wf_imp_wf_on well_ordI wf0 linear0) ``` paulson@13512 ` 429` paulson@13512 ` 430` ```subsubsection{*The "measure" relation is useful with wfrec*} ``` paulson@13140 ` 431` paulson@13140 ` 432` ```lemma measure_eq_rvimage_Memrel: ``` paulson@13140 ` 433` ``` "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))" ``` paulson@13140 ` 434` ```apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff) ``` paulson@13269 ` 435` ```apply (rule equalityI, auto) ``` paulson@13140 ` 436` ```apply (auto intro: Ord_in_Ord simp add: lt_def) ``` paulson@13140 ` 437` ```done ``` paulson@13140 ` 438` paulson@13140 ` 439` ```lemma wf_measure [iff]: "wf(measure(A,f))" ``` paulson@13356 ` 440` ```by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage) ``` paulson@13140 ` 441` paulson@13140 ` 442` ```lemma measure_iff [iff]: " : measure(A,f) <-> x:A & y:A & f(x) A ==> Ord(f(x))" ``` paulson@13544 ` 447` ``` and inj: "!!x y. [|x \ A; y \ A; f(x) = f(y) |] ==> x=y" ``` paulson@13544 ` 448` ``` shows "linear(A, measure(A,f))" ``` paulson@13544 ` 449` ```apply (auto simp add: linear_def) ``` paulson@13544 ` 450` ```apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt) ``` paulson@13544 ` 451` ``` apply (simp_all add: Ordf) ``` paulson@13544 ` 452` ```apply (blast intro: inj) ``` paulson@13544 ` 453` ```done ``` paulson@13544 ` 454` paulson@13544 ` 455` ```lemma wf_on_measure: "wf[B](measure(A,f))" ``` paulson@13544 ` 456` ```by (rule wf_imp_wf_on [OF wf_measure]) ``` paulson@13544 ` 457` paulson@13544 ` 458` ```lemma well_ord_measure: ``` paulson@13544 ` 459` ``` assumes Ordf: "!!x. x \ A ==> Ord(f(x))" ``` paulson@13544 ` 460` ``` and inj: "!!x y. [|x \ A; y \ A; f(x) = f(y) |] ==> x=y" ``` paulson@13544 ` 461` ``` shows "well_ord(A, measure(A,f))" ``` paulson@13544 ` 462` ```apply (rule well_ordI) ``` paulson@13544 ` 463` ```apply (rule wf_on_measure) ``` paulson@13544 ` 464` ```apply (blast intro: linear_measure Ordf inj) ``` paulson@13544 ` 465` ```done ``` paulson@13544 ` 466` paulson@13544 ` 467` ```lemma measure_type: "measure(A,f) <= A*A" ``` paulson@13544 ` 468` ```by (auto simp add: measure_def) ``` paulson@13544 ` 469` paulson@13512 ` 470` ```subsubsection{*Well-foundedness of Unions*} ``` paulson@13512 ` 471` paulson@13512 ` 472` ```lemma wf_on_Union: ``` paulson@13512 ` 473` ``` assumes wfA: "wf[A](r)" ``` paulson@13512 ` 474` ``` and wfB: "!!a. a\A ==> wf[B(a)](s)" ``` paulson@13512 ` 475` ``` and ok: "!!a u v. [| \ s; v \ B(a); a \ A|] ``` paulson@13512 ` 476` ``` ==> (\a'\A. \ r & u \ B(a')) | u \ B(a)" ``` paulson@13512 ` 477` ``` shows "wf[\a\A. B(a)](s)" ``` paulson@13512 ` 478` ```apply (rule wf_onI2) ``` paulson@13512 ` 479` ```apply (erule UN_E) ``` paulson@13512 ` 480` ```apply (subgoal_tac "\z \ B(a). z \ Ba", blast) ``` paulson@13512 ` 481` ```apply (rule_tac a = a in wf_on_induct [OF wfA], assumption) ``` paulson@13512 ` 482` ```apply (rule ballI) ``` paulson@13512 ` 483` ```apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption) ``` paulson@13512 ` 484` ```apply (rename_tac u) ``` paulson@13512 ` 485` ```apply (drule_tac x=u in bspec, blast) ``` paulson@13512 ` 486` ```apply (erule mp, clarify) ``` paulson@13512 ` 487` ```apply (frule ok, assumption+); ``` paulson@13512 ` 488` ```apply blast ``` paulson@13512 ` 489` ```done ``` paulson@13512 ` 490` paulson@13512 ` 491` paulson@13140 ` 492` ```ML {* ``` paulson@13140 ` 493` ```val measure_def = thm "measure_def"; ``` paulson@13140 ` 494` ```val radd_Inl_Inr_iff = thm "radd_Inl_Inr_iff"; ``` paulson@13140 ` 495` ```val radd_Inl_iff = thm "radd_Inl_iff"; ``` paulson@13140 ` 496` ```val radd_Inr_iff = thm "radd_Inr_iff"; ``` paulson@13140 ` 497` ```val radd_Inr_Inl_iff = thm "radd_Inr_Inl_iff"; ``` paulson@13140 ` 498` ```val raddE = thm "raddE"; ``` paulson@13140 ` 499` ```val radd_type = thm "radd_type"; ``` paulson@13140 ` 500` ```val field_radd = thm "field_radd"; ``` paulson@13140 ` 501` ```val linear_radd = thm "linear_radd"; ``` paulson@13140 ` 502` ```val wf_on_radd = thm "wf_on_radd"; ``` paulson@13140 ` 503` ```val wf_radd = thm "wf_radd"; ``` paulson@13140 ` 504` ```val well_ord_radd = thm "well_ord_radd"; ``` paulson@13140 ` 505` ```val sum_bij = thm "sum_bij"; ``` paulson@13140 ` 506` ```val sum_ord_iso_cong = thm "sum_ord_iso_cong"; ``` paulson@13140 ` 507` ```val sum_disjoint_bij = thm "sum_disjoint_bij"; ``` paulson@13140 ` 508` ```val sum_assoc_bij = thm "sum_assoc_bij"; ``` paulson@13140 ` 509` ```val sum_assoc_ord_iso = thm "sum_assoc_ord_iso"; ``` paulson@13140 ` 510` ```val rmult_iff = thm "rmult_iff"; ``` paulson@13140 ` 511` ```val rmultE = thm "rmultE"; ``` paulson@13140 ` 512` ```val rmult_type = thm "rmult_type"; ``` paulson@13140 ` 513` ```val field_rmult = thm "field_rmult"; ``` paulson@13140 ` 514` ```val linear_rmult = thm "linear_rmult"; ``` paulson@13140 ` 515` ```val wf_on_rmult = thm "wf_on_rmult"; ``` paulson@13140 ` 516` ```val wf_rmult = thm "wf_rmult"; ``` paulson@13140 ` 517` ```val well_ord_rmult = thm "well_ord_rmult"; ``` paulson@13140 ` 518` ```val prod_bij = thm "prod_bij"; ``` paulson@13140 ` 519` ```val prod_ord_iso_cong = thm "prod_ord_iso_cong"; ``` paulson@13140 ` 520` ```val singleton_prod_bij = thm "singleton_prod_bij"; ``` paulson@13140 ` 521` ```val singleton_prod_ord_iso = thm "singleton_prod_ord_iso"; ``` paulson@13140 ` 522` ```val prod_sum_singleton_bij = thm "prod_sum_singleton_bij"; ``` paulson@13140 ` 523` ```val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso"; ``` paulson@13140 ` 524` ```val sum_prod_distrib_bij = thm "sum_prod_distrib_bij"; ``` paulson@13140 ` 525` ```val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso"; ``` paulson@13140 ` 526` ```val prod_assoc_bij = thm "prod_assoc_bij"; ``` paulson@13140 ` 527` ```val prod_assoc_ord_iso = thm "prod_assoc_ord_iso"; ``` paulson@13140 ` 528` ```val rvimage_iff = thm "rvimage_iff"; ``` paulson@13140 ` 529` ```val rvimage_type = thm "rvimage_type"; ``` paulson@13140 ` 530` ```val field_rvimage = thm "field_rvimage"; ``` paulson@13140 ` 531` ```val rvimage_converse = thm "rvimage_converse"; ``` paulson@13140 ` 532` ```val irrefl_rvimage = thm "irrefl_rvimage"; ``` paulson@13140 ` 533` ```val trans_on_rvimage = thm "trans_on_rvimage"; ``` paulson@13140 ` 534` ```val part_ord_rvimage = thm "part_ord_rvimage"; ``` paulson@13140 ` 535` ```val linear_rvimage = thm "linear_rvimage"; ``` paulson@13140 ` 536` ```val tot_ord_rvimage = thm "tot_ord_rvimage"; ``` paulson@13140 ` 537` ```val wf_rvimage = thm "wf_rvimage"; ``` paulson@13140 ` 538` ```val wf_on_rvimage = thm "wf_on_rvimage"; ``` paulson@13140 ` 539` ```val well_ord_rvimage = thm "well_ord_rvimage"; ``` paulson@13140 ` 540` ```val ord_iso_rvimage = thm "ord_iso_rvimage"; ``` paulson@13140 ` 541` ```val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq"; ``` paulson@13140 ` 542` ```val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel"; ``` paulson@13140 ` 543` ```val wf_measure = thm "wf_measure"; ``` paulson@13140 ` 544` ```val measure_iff = thm "measure_iff"; ``` paulson@13140 ` 545` ```*} ``` paulson@13140 ` 546` lcp@437 ` 547` ```end ```