diff -r 6e7244ae43b8 -r 745547bdc1c7 thys/UF_Rec.thy --- a/thys/UF_Rec.thy Thu May 02 13:19:50 2013 +0100 +++ b/thys/UF_Rec.thy Thu May 09 18:16:36 2013 +0100 @@ -2,11 +2,11 @@ imports Recs Turing_Hoare begin -section {* Universal Function *} -text{* coding of the configuration *} + +section {* Universal Function in HOL *} text {* @{text "Entry sr i"} returns the @{text "i"}-th entry of a list of natural @@ -15,33 +15,16 @@ fun Entry where "Entry sr i = Lo sr (Pi (Suc i))" -definition - "rec_entry = CN rec_lo [Id 2 0, CN rec_pi [CN S [Id 2 1]]]" - -lemma entry_lemma [simp]: - "rec_eval rec_entry [sr, i] = Entry sr i" -by(simp add: rec_entry_def) - - fun Listsum2 :: "nat list \ nat \ nat" where "Listsum2 xs 0 = 0" | "Listsum2 xs (Suc n) = Listsum2 xs n + xs ! n" -fun rec_listsum2 :: "nat \ nat \ recf" - where - "rec_listsum2 vl 0 = CN Z [Id vl 0]" -| "rec_listsum2 vl (Suc n) = CN rec_add [rec_listsum2 vl n, Id vl n]" - -lemma listsum2_lemma [simp]: - "length xs = vl \ rec_eval (rec_listsum2 vl n) xs = Listsum2 xs n" -by (induct n) (simp_all) - text {* @{text "Strt"} corresponds to the @{text "strt"} function on page 90 of the - B book, but this definition generalises the original one to deal with multiple - input arguments. - *} + B book, but this definition generalises the original one in order to deal + with multiple input arguments. *} + fun Strt' :: "nat list \ nat \ nat" where "Strt' xs 0 = 0" @@ -52,55 +35,11 @@ where "Strt xs = (let ys = map Suc xs in Strt' ys (length ys))" -fun rec_strt' :: "nat \ nat \ recf" - where - "rec_strt' xs 0 = Z" -| "rec_strt' xs (Suc n) = - (let dbound = CN rec_add [rec_listsum2 xs n, constn n] in - let t1 = CN rec_power [constn 2, dbound] in - let t2 = CN rec_power [constn 2, CN rec_add [Id xs n, dbound]] in - CN rec_add [rec_strt' xs n, CN rec_minus [t2, t1]])" - -fun rec_map :: "recf \ nat \ recf list" - where - "rec_map rf vl = map (\i. CN rf [Id vl i]) [0.. recf" - where - "rec_strt xs = CN (rec_strt' xs xs) (rec_map S xs)" - -lemma strt'_lemma [simp]: - "length xs = vl \ rec_eval (rec_strt' vl n) xs = Strt' xs n" -by (induct n) (simp_all add: Let_def) - - -lemma map_suc: - "map (\x. Suc (xs ! x)) [0.. (\x. xs ! x) = (\x. Suc (xs ! x))" by (simp add: comp_def) - then have "map (\x. Suc (xs ! x)) [0.. (\x. xs ! x)) [0..x. xs ! x) [0..x. Suc (xs ! x)) [0.. rec_eval (rec_strt vl) xs = Strt xs" -by (simp add: comp_def map_suc[symmetric]) - - text {* The @{text "Scan"} function on page 90 of B book. *} fun Scan :: "nat \ nat" where "Scan r = r mod 2" -definition - "rec_scan = CN rec_mod [Id 1 0, constn 2]" - -lemma scan_lemma [simp]: - "rec_eval rec_scan [r] = r mod 2" -by(simp add: rec_scan_def) - text {* The @{text Newleft} and @{text Newright} functions on page 91 of B book. *} fun Newleft :: "nat \ nat \ nat \ nat" @@ -118,119 +57,52 @@ else if a = 3 then r div 2 else r)" -definition - "rec_newleft = - (let cond1 = CN rec_disj [CN rec_eq [Id 3 2, Z], CN rec_eq [Id 3 2, constn 1]] in - let cond2 = CN rec_eq [Id 3 2, constn 2] in - let cond3 = CN rec_eq [Id 3 2, constn 3] in - let case3 = CN rec_add [CN rec_mult [constn 2, Id 3 0], - CN rec_mod [Id 3 1, constn 2]] in - CN rec_if [cond1, Id 3 0, - CN rec_if [cond2, CN rec_quo [Id 3 0, constn 2], - CN rec_if [cond3, case3, Id 3 0]]])" - -definition - "rec_newright = - (let condn = \n. CN rec_eq [Id 3 2, constn n] in - let case0 = CN rec_minus [Id 3 1, CN rec_scan [Id 3 1]] in - let case1 = CN rec_minus [CN rec_add [Id 3 1, constn 1], CN rec_scan [Id 3 1]] in - let case2 = CN rec_add [CN rec_mult [constn 2, Id 3 1], - CN rec_mod [Id 3 0, constn 2]] in - let case3 = CN rec_quo [Id 2 1, constn 2] in - CN rec_if [condn 0, case0, - CN rec_if [condn 1, case1, - CN rec_if [condn 2, case2, - CN rec_if [condn 3, case3, Id 3 1]]]])" - -lemma newleft_lemma [simp]: - "rec_eval rec_newleft [p, r, a] = Newleft p r a" -by (simp add: rec_newleft_def Let_def) - -lemma newright_lemma [simp]: - "rec_eval rec_newright [p, r, a] = Newright p r a" -by (simp add: rec_newright_def Let_def) - text {* The @{text "Actn"} function given on page 92 of B book, which is used to fetch Turing Machine intructions. In @{text "Actn m q r"}, @{text "m"} is the Goedel coding of a Turing Machine, @{text "q"} is the current state of - Turing Machine, @{text "r"} is the right number of Turing Machine tape. - *} + Turing Machine, @{text "r"} is the right number of Turing Machine tape. *} + fun Actn :: "nat \ nat \ nat \ nat" where "Actn m q r = (if q \ 0 then Entry m (4 * (q - 1) + 2 * Scan r) else 4)" -definition - "rec_actn = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in - let add2 = CN rec_mult [constn 2, CN rec_scan [Id 3 2]] in - let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]] - in CN rec_if [Id 3 1, entry, constn 4])" - -lemma actn_lemma [simp]: - "rec_eval rec_actn [m, q, r] = Actn m q r" -by (simp add: rec_actn_def) - fun Newstat :: "nat \ nat \ nat \ nat" where "Newstat m q r = (if q \ 0 then Entry m (4 * (q - 1) + 2 * Scan r + 1) else 0)" -definition rec_newstat :: "recf" - where - "rec_newstat = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in - let add2 = CN S [CN rec_mult [constn 2, CN rec_scan [Id 3 2]]] in - let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]] - in CN rec_if [Id 3 1, entry, Z])" - -lemma newstat_lemma [simp]: - "rec_eval rec_newstat [m, q, r] = Newstat m q r" -by (simp add: rec_newstat_def) - - fun Trpl :: "nat \ nat \ nat \ nat" where "Trpl p q r = (Pi 0) ^ p * (Pi 1) ^ q * (Pi 2) ^ r" -definition - "rec_trpl = CN rec_mult [CN rec_mult - [CN rec_power [constn (Pi 0), Id 3 0], - CN rec_power [constn (Pi 1), Id 3 1]], - CN rec_power [constn (Pi 2), Id 3 2]]" - -lemma trpl_lemma [simp]: - "rec_eval rec_trpl [p, q, r] = Trpl p q r" -by (simp add: rec_trpl_def) - - - fun Left where "Left c = Lo c (Pi 0)" -definition - "rec_left = CN rec_lo [Id 1 0, constn (Pi 0)]" - -lemma left_lemma [simp]: - "rec_eval rec_left [c] = Left c" -by(simp add: rec_left_def) - fun Right where "Right c = Lo c (Pi 2)" -definition - "rec_right = CN rec_lo [Id 1 0, constn (Pi 2)]" - -lemma right_lemma [simp]: - "rec_eval rec_right [c] = Right c" -by(simp add: rec_right_def) - fun Stat where "Stat c = Lo c (Pi 1)" -definition - "rec_stat = CN rec_lo [Id 1 0, constn (Pi 1)]" +lemma mod_dvd_simp: + "(x mod y = (0::nat)) = (y dvd x)" +by(auto simp add: dvd_def) -lemma stat_lemma [simp]: - "rec_eval rec_stat [c] = Stat c" -by(simp add: rec_stat_def) +lemma Trpl_Left [simp]: + "Left (Trpl p q r) = p" +apply(simp) +apply(subst Lo_def) +apply(subst dvd_eq_mod_eq_0[symmetric]) +apply(simp) +apply(auto) +thm Lo_def +thm mod_dvd_simp +apply(simp add: left.simps trpl.simps lo.simps + loR.simps mod_dvd_simp, auto simp: conf_decode1) +apply(case_tac "Pi 0 ^ l * Pi (Suc 0) ^ st * Pi (Suc (Suc 0)) ^ r", + auto) +apply(erule_tac x = l in allE, auto) + fun Inpt :: "nat \ nat list \ nat" where @@ -242,43 +114,20 @@ (Newstat m (Stat c) (Right c)) (Newright (Left c) (Right c) (Actn m (Stat c) (Right c)))" -definition - "rec_newconf = (let act = CN rec_actn [Id 2 0, CN rec_stat [Id 2 1], CN rec_right [Id 2 1]] in - let left = CN rec_left [Id 2 1] in - let right = CN rec_right [Id 2 1] in - let stat = CN rec_stat [Id 2 1] in - let one = CN rec_newleft [left, right, act] in - let two = CN rec_newstat [Id 2 0, stat, right] in - let three = CN rec_newright [left, right, act] - in CN rec_trpl [one, two, three])" - -lemma newconf_lemma [simp]: - "rec_eval rec_newconf [m, c] = Newconf m c" -by (simp add: rec_newconf_def Let_def) - text {* @{text "Conf k m r"} computes the TM configuration after @{text "k"} steps of execution of TM coded as @{text "m"} starting from the initial configuration where the left - number equals @{text "0"}, right number equals @{text "r"}. - *} + number equals @{text "0"}, right number equals @{text "r"}. *} + fun Conf :: "nat \ nat \ nat \ nat" where "Conf 0 m r = Trpl 0 1 r" | "Conf (Suc k) m r = Newconf m (Conf k m r)" -definition - "rec_conf = PR (CN rec_trpl [constn 0, constn 1, Id 2 1]) - (CN rec_newconf [Id 4 2 , Id 4 1])" - -lemma conf_lemma [simp]: - "rec_eval rec_conf [k, m, r] = Conf k m r" -by(induct k) (simp_all add: rec_conf_def) - - text {* @{text "Nstd c"} returns true if the configuration coded - by @{text "c"} is not a stardard final configuration. - *} + by @{text "c"} is not a stardard final configuration. *} + fun Nstd :: "nat \ bool" where "Nstd c = (Stat c \ 0 \ @@ -286,70 +135,39 @@ Right c \ 2 ^ (Lg (Suc (Right c)) 2) - 1 \ Right c = 0)" -definition - "rec_nstd = (let disj1 = CN rec_noteq [rec_stat, constn 0] in - let disj2 = CN rec_noteq [rec_left, constn 0] in - let rhs = CN rec_pred [CN rec_power [constn 2, CN rec_lg [CN S [rec_right], constn 2]]] in - let disj3 = CN rec_noteq [rec_right, rhs] in - let disj4 = CN rec_eq [rec_right, constn 0] in - CN rec_disj [CN rec_disj [CN rec_disj [disj1, disj2], disj3], disj4])" - -lemma nstd_lemma [simp]: - "rec_eval rec_nstd [c] = (if Nstd c then 1 else 0)" -by(simp add: rec_nstd_def) - text{* @{text "Nostop t m r"} means that afer @{text "t"} steps of - execution, the TM coded by @{text "m"} is not at a stardard - final configuration. - *} + execution the TM coded by @{text "m"} is not at a stardard + final configuration. *} + fun Nostop :: "nat \ nat \ nat \ bool" where "Nostop t m r = Nstd (Conf t m r)" -definition - "rec_nostop = CN rec_nstd [rec_conf]" - -lemma nostop_lemma [simp]: - "rec_eval rec_nostop [t, m, r] = (if Nostop t m r then 1 else 0)" -by (simp add: rec_nostop_def) - - fun Value where "Value x = (Lg (Suc x) 2) - 1" -definition - "rec_value = CN rec_pred [CN rec_lg [S, constn 2]]" - -lemma value_lemma [simp]: - "rec_eval rec_value [x] = Value x" -by (simp add: rec_value_def) - text{* @{text "rec_halt"} is the recursive function calculating the steps a TM needs to execute before to reach a stardard final configuration. This recursive function is the only one using @{text "Mn"} combinator. So it is the only non-primitive recursive function - needs to be used in the construction of the universal function @{text "rec_uf"}. - *} + needs to be used in the construction of the universal function @{text "rec_uf"}. *} -definition - "rec_halt = MN rec_nostop" +fun Halt :: "nat \ nat \ nat" + where + "Halt m r = (LEAST t. \ Nostop t m r)" + +fun UF :: "nat \ nat \ nat" + where + "UF c m = Value (Right (Conf (Halt c m) c m))" -definition - "rec_uf = CN rec_value [CN rec_right [CN rec_conf [rec_halt, Id 2 0, Id 2 1]]]" +section {* Coding of Turing Machines *} text {* - The correctness of @{text "rec_uf"}, nonhalt case. - *} - -subsection {* Coding function of TMs *} - -text {* - The purpose of this section is to get the coding function of Turing Machine, - which is going to be named @{text "code"}. - *} + The purpose of this section is to construct the coding function of Turing + Machine, which is going to be named @{text "code"}. *} fun bl2nat :: "cell list \ nat \ nat" where @@ -361,9 +179,29 @@ where "bl2wc xs = bl2nat xs 0" -fun trpl_code :: "config \ nat" +lemma bl2nat_double [simp]: + "bl2nat xs (Suc n) = 2 * bl2nat xs n" +apply(induct xs arbitrary: n) +apply(auto) +apply(case_tac a) +apply(auto) +done + +lemma bl2nat_simps1 [simp]: + shows "bl2nat (Bk \ y) n = 0" +by (induct y) (auto) + +lemma bl2nat_simps2 [simp]: + shows "bl2nat (Oc \ y) 0 = 2 ^ y - 1" +apply(induct y) +apply(auto) +apply(case_tac "(2::nat)^ y") +apply(auto) +done + +fun Trpl_code :: "config \ nat" where - "trpl_code (st, l, r) = Trpl (bl2wc l) st (bl2wc r)" + "Trpl_code (st, l, r) = Trpl (bl2wc l) st (bl2wc r)" fun action_map :: "action \ nat" where @@ -405,13 +243,259 @@ where "Code tp = Goedel_code (modify_tprog tp)" + +section {* Universal Function as Recursive Functions *} + +definition + "rec_entry = CN rec_lo [Id 2 0, CN rec_pi [CN S [Id 2 1]]]" + +fun rec_listsum2 :: "nat \ nat \ recf" + where + "rec_listsum2 vl 0 = CN Z [Id vl 0]" +| "rec_listsum2 vl (Suc n) = CN rec_add [rec_listsum2 vl n, Id vl n]" + +fun rec_strt' :: "nat \ nat \ recf" + where + "rec_strt' xs 0 = Z" +| "rec_strt' xs (Suc n) = + (let dbound = CN rec_add [rec_listsum2 xs n, constn n] in + let t1 = CN rec_power [constn 2, dbound] in + let t2 = CN rec_power [constn 2, CN rec_add [Id xs n, dbound]] in + CN rec_add [rec_strt' xs n, CN rec_minus [t2, t1]])" + +fun rec_map :: "recf \ nat \ recf list" + where + "rec_map rf vl = map (\i. CN rf [Id vl i]) [0.. recf" + where + "rec_strt xs = CN (rec_strt' xs xs) (rec_map S xs)" + +definition + "rec_scan = CN rec_mod [Id 1 0, constn 2]" + +definition + "rec_newleft = + (let cond1 = CN rec_disj [CN rec_eq [Id 3 2, Z], CN rec_eq [Id 3 2, constn 1]] in + let cond2 = CN rec_eq [Id 3 2, constn 2] in + let cond3 = CN rec_eq [Id 3 2, constn 3] in + let case3 = CN rec_add [CN rec_mult [constn 2, Id 3 0], + CN rec_mod [Id 3 1, constn 2]] in + CN rec_if [cond1, Id 3 0, + CN rec_if [cond2, CN rec_quo [Id 3 0, constn 2], + CN rec_if [cond3, case3, Id 3 0]]])" + +definition + "rec_newright = + (let condn = \n. CN rec_eq [Id 3 2, constn n] in + let case0 = CN rec_minus [Id 3 1, CN rec_scan [Id 3 1]] in + let case1 = CN rec_minus [CN rec_add [Id 3 1, constn 1], CN rec_scan [Id 3 1]] in + let case2 = CN rec_add [CN rec_mult [constn 2, Id 3 1], + CN rec_mod [Id 3 0, constn 2]] in + let case3 = CN rec_quo [Id 2 1, constn 2] in + CN rec_if [condn 0, case0, + CN rec_if [condn 1, case1, + CN rec_if [condn 2, case2, + CN rec_if [condn 3, case3, Id 3 1]]]])" + +definition + "rec_actn = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in + let add2 = CN rec_mult [constn 2, CN rec_scan [Id 3 2]] in + let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]] + in CN rec_if [Id 3 1, entry, constn 4])" + +definition rec_newstat :: "recf" + where + "rec_newstat = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in + let add2 = CN S [CN rec_mult [constn 2, CN rec_scan [Id 3 2]]] in + let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]] + in CN rec_if [Id 3 1, entry, Z])" + +definition + "rec_trpl = CN rec_mult [CN rec_mult + [CN rec_power [constn (Pi 0), Id 3 0], + CN rec_power [constn (Pi 1), Id 3 1]], + CN rec_power [constn (Pi 2), Id 3 2]]" + +definition + "rec_left = CN rec_lo [Id 1 0, constn (Pi 0)]" + +definition + "rec_right = CN rec_lo [Id 1 0, constn (Pi 2)]" + +definition + "rec_stat = CN rec_lo [Id 1 0, constn (Pi 1)]" + +definition + "rec_newconf = (let act = CN rec_actn [Id 2 0, CN rec_stat [Id 2 1], CN rec_right [Id 2 1]] in + let left = CN rec_left [Id 2 1] in + let right = CN rec_right [Id 2 1] in + let stat = CN rec_stat [Id 2 1] in + let one = CN rec_newleft [left, right, act] in + let two = CN rec_newstat [Id 2 0, stat, right] in + let three = CN rec_newright [left, right, act] + in CN rec_trpl [one, two, three])" + +definition + "rec_conf = PR (CN rec_trpl [constn 0, constn 1, Id 2 1]) + (CN rec_newconf [Id 4 2 , Id 4 1])" + +definition + "rec_nstd = (let disj1 = CN rec_noteq [rec_stat, constn 0] in + let disj2 = CN rec_noteq [rec_left, constn 0] in + let rhs = CN rec_pred [CN rec_power [constn 2, CN rec_lg [CN S [rec_right], constn 2]]] in + let disj3 = CN rec_noteq [rec_right, rhs] in + let disj4 = CN rec_eq [rec_right, constn 0] in + CN rec_disj [CN rec_disj [CN rec_disj [disj1, disj2], disj3], disj4])" + +definition + "rec_nostop = CN rec_nstd [rec_conf]" + +definition + "rec_value = CN rec_pred [CN rec_lg [S, constn 2]]" + +definition + "rec_halt = MN rec_nostop" + +definition + "rec_uf = CN rec_value [CN rec_right [CN rec_conf [rec_halt, Id 2 0, Id 2 1]]]" + + + +section {* Correctness Proofs for Recursive Functions *} + +lemma entry_lemma [simp]: + "rec_eval rec_entry [sr, i] = Entry sr i" +by(simp add: rec_entry_def) + +lemma listsum2_lemma [simp]: + "length xs = vl \ rec_eval (rec_listsum2 vl n) xs = Listsum2 xs n" +by (induct n) (simp_all) + +lemma strt'_lemma [simp]: + "length xs = vl \ rec_eval (rec_strt' vl n) xs = Strt' xs n" +by (induct n) (simp_all add: Let_def) + +lemma map_suc: + "map (\x. Suc (xs ! x)) [0.. (\x. xs ! x) = (\x. Suc (xs ! x))" by (simp add: comp_def) + then have "map (\x. Suc (xs ! x)) [0.. (\x. xs ! x)) [0..x. xs ! x) [0..x. Suc (xs ! x)) [0.. rec_eval (rec_strt vl) xs = Strt xs" +by (simp add: comp_def map_suc[symmetric]) + +lemma scan_lemma [simp]: + "rec_eval rec_scan [r] = r mod 2" +by(simp add: rec_scan_def) + +lemma newleft_lemma [simp]: + "rec_eval rec_newleft [p, r, a] = Newleft p r a" +by (simp add: rec_newleft_def Let_def) + +lemma newright_lemma [simp]: + "rec_eval rec_newright [p, r, a] = Newright p r a" +by (simp add: rec_newright_def Let_def) + +lemma actn_lemma [simp]: + "rec_eval rec_actn [m, q, r] = Actn m q r" +by (simp add: rec_actn_def) + +lemma newstat_lemma [simp]: + "rec_eval rec_newstat [m, q, r] = Newstat m q r" +by (simp add: rec_newstat_def) + +lemma trpl_lemma [simp]: + "rec_eval rec_trpl [p, q, r] = Trpl p q r" +by (simp add: rec_trpl_def) + +lemma left_lemma [simp]: + "rec_eval rec_left [c] = Left c" +by(simp add: rec_left_def) + +lemma right_lemma [simp]: + "rec_eval rec_right [c] = Right c" +by(simp add: rec_right_def) + +lemma stat_lemma [simp]: + "rec_eval rec_stat [c] = Stat c" +by(simp add: rec_stat_def) + +lemma newconf_lemma [simp]: + "rec_eval rec_newconf [m, c] = Newconf m c" +by (simp add: rec_newconf_def Let_def) + +lemma conf_lemma [simp]: + "rec_eval rec_conf [k, m, r] = Conf k m r" +by(induct k) (simp_all add: rec_conf_def) + +lemma nstd_lemma [simp]: + "rec_eval rec_nstd [c] = (if Nstd c then 1 else 0)" +by(simp add: rec_nstd_def) + +lemma nostop_lemma [simp]: + "rec_eval rec_nostop [t, m, r] = (if Nostop t m r then 1 else 0)" +by (simp add: rec_nostop_def) + +lemma value_lemma [simp]: + "rec_eval rec_value [x] = Value x" +by (simp add: rec_value_def) + +lemma halt_lemma [simp]: + "rec_eval rec_halt [m, r] = Halt m r" +by (simp add: rec_halt_def) + +lemma uf_lemma [simp]: + "rec_eval rec_uf [m, r] = UF m r" +by (simp add: rec_uf_def) + + subsection {* Relating interperter functions to the execution of TMs *} +lemma rec_step: + assumes "(\ (s, l, r). s \ length tp div 2) c" + shows "Trpl_code (step0 c tp) = Newconf (Code tp) (Trpl_code c)" +apply(cases c) +apply(simp only: Trpl_code.simps) +apply(simp only: Let_def step.simps) +apply(case_tac "fetch tp (a - 0) (read ca)") +apply(simp only: prod.cases) +apply(case_tac "update aa (b, ca)") +apply(simp only: prod.cases) +apply(simp only: Trpl_code.simps) +apply(simp only: Newconf.simps) +apply(simp only: Left.simps) +oops + +lemma rec_steps: + assumes "tm_wf0 tp" + shows "Trpl_code (steps0 (1, Bk \ l, ) tp stp) = Conf stp (Code tp) (bl2wc ())" +apply(induct stp) +apply(simp) +apply(simp) +oops + lemma F_correct: - assumes tp: "steps0 (1, Bk \ l, ) tp stp = (0, Bk \ m, Oc \ rs @ Bk \ n)" + assumes tm: "steps0 (1, Bk \ l, ) tp stp = (0, Bk \ m, Oc \ rs @ Bk \ n)" and wf: "tm_wf0 tp" "0 < rs" shows "rec_eval rec_uf [Code tp, bl2wc ()] = (rs - Suc 0)" +proof - + from least_steps[OF tm] + obtain stp_least where + before: "\stp' < stp_least. \ is_final (steps0 (1, Bk \ l, ) tp stp')" and + after: "\stp' \ stp_least. is_final (steps0 (1, Bk \ l, ) tp stp')" by blast + have "Halt (Code tp) (bl2wc ()) = stp_least" sorry + show ?thesis + apply(simp only: uf_lemma) + apply(simp only: UF.simps) + apply(simp only: Halt.simps) + oops end