theory UF_Rec
imports Recs Turing_Hoare
begin
section {* Universal Function in HOL *}
text {*
@{text "Entry sr i"} returns the @{text "i"}-th entry of a list of natural
numbers encoded by number @{text "sr"} using Godel's coding.
*}
fun Entry where
"Entry sr i = Lo sr (Pi (Suc i))"
fun Listsum2 :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
where
"Listsum2 xs 0 = 0"
| "Listsum2 xs (Suc n) = Listsum2 xs n + xs ! n"
text {*
@{text "Strt"} corresponds to the @{text "strt"} function on page 90 of the
B book, but this definition generalises the original one in order to deal
with multiple input arguments. *}
fun Strt' :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
where
"Strt' xs 0 = 0"
| "Strt' xs (Suc n) = (let dbound = Listsum2 xs n + n
in Strt' xs n + (2 ^ (xs ! n + dbound) - 2 ^ dbound))"
fun Strt :: "nat list \<Rightarrow> nat"
where
"Strt xs = (let ys = map Suc xs in Strt' ys (length ys))"
text {* The @{text "Scan"} function on page 90 of B book. *}
fun Scan :: "nat \<Rightarrow> nat"
where
"Scan r = r mod 2"
text {* The @{text Newleft} and @{text Newright} functions on page 91 of B book. *}
fun Newleft :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Newleft p r a = (if a = 0 \<or> a = 1 then p
else if a = 2 then p div 2
else if a = 3 then 2 * p + r mod 2
else p)"
fun Newright :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Newright p r a = (if a = 0 then r - Scan r
else if a = 1 then r + 1 - Scan r
else if a = 2 then 2 * r + p mod 2
else if a = 3 then r div 2
else r)"
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. *}
fun Actn :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Actn m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r) else 4)"
fun Newstat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Newstat m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r + 1) else 0)"
fun Trpl :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Trpl p q r = (Pi 0) ^ p * (Pi 1) ^ q * (Pi 2) ^ r"
fun Left where
"Left c = Lo c (Pi 0)"
fun Right where
"Right c = Lo c (Pi 2)"
fun Stat where
"Stat c = Lo c (Pi 1)"
lemma mod_dvd_simp:
"(x mod y = (0::nat)) = (y dvd x)"
by(auto simp add: dvd_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 \<Rightarrow> nat list \<Rightarrow> nat"
where
"Inpt m xs = Trpl 0 1 (Strt xs)"
fun Newconf :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Newconf m c = Trpl (Newleft (Left c) (Right c) (Actn m (Stat c) (Right c)))
(Newstat m (Stat c) (Right c))
(Newright (Left c) (Right c) (Actn m (Stat c) (Right c)))"
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"}. *}
fun Conf :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Conf 0 m r = Trpl 0 1 r"
| "Conf (Suc k) m r = Newconf m (Conf k m r)"
text {*
@{text "Nstd c"} returns true if the configuration coded
by @{text "c"} is not a stardard final configuration. *}
fun Nstd :: "nat \<Rightarrow> bool"
where
"Nstd c = (Stat c \<noteq> 0 \<or>
Left c \<noteq> 0 \<or>
Right c \<noteq> 2 ^ (Lg (Suc (Right c)) 2) - 1 \<or>
Right c = 0)"
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. *}
fun Nostop :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
where
"Nostop t m r = Nstd (Conf t m r)"
fun Value where
"Value x = (Lg (Suc x) 2) - 1"
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"}. *}
fun Halt :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"Halt m r = (LEAST t. \<not> Nostop t m r)"
fun UF :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"UF c m = Value (Right (Conf (Halt c m) c m))"
section {* Coding of Turing Machines *}
text {*
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 \<Rightarrow> nat \<Rightarrow> nat"
where
"bl2nat [] n = 0"
| "bl2nat (Bk # bl) n = bl2nat bl (Suc n)"
| "bl2nat (Oc # bl) n = 2 ^ n + bl2nat bl (Suc n)"
fun bl2wc :: "cell list \<Rightarrow> nat"
where
"bl2wc xs = bl2nat xs 0"
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 \<up> y) n = 0"
by (induct y) (auto)
lemma bl2nat_simps2 [simp]:
shows "bl2nat (Oc \<up> y) 0 = 2 ^ y - 1"
apply(induct y)
apply(auto)
apply(case_tac "(2::nat)^ y")
apply(auto)
done
fun Trpl_code :: "config \<Rightarrow> nat"
where
"Trpl_code (st, l, r) = Trpl (bl2wc l) st (bl2wc r)"
fun action_map :: "action \<Rightarrow> nat"
where
"action_map W0 = 0"
| "action_map W1 = 1"
| "action_map L = 2"
| "action_map R = 3"
| "action_map Nop = 4"
fun action_map_iff :: "nat \<Rightarrow> action"
where
"action_map_iff (0::nat) = W0"
| "action_map_iff (Suc 0) = W1"
| "action_map_iff (Suc (Suc 0)) = L"
| "action_map_iff (Suc (Suc (Suc 0))) = R"
| "action_map_iff n = Nop"
fun block_map :: "cell \<Rightarrow> nat"
where
"block_map Bk = 0"
| "block_map Oc = 1"
fun Goedel_code' :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
where
"Goedel_code' [] n = 1"
| "Goedel_code' (x # xs) n = (Pi n) ^ x * Goedel_code' xs (Suc n) "
fun Goedel_code :: "nat list \<Rightarrow> nat"
where
"Goedel_code xs = 2 ^ (length xs) * (Goedel_code' xs 1)"
fun modify_tprog :: "instr list \<Rightarrow> nat list"
where
"modify_tprog [] = []"
| "modify_tprog ((a, s) # nl) = action_map a # s # modify_tprog nl"
text {* @{text "Code tp"} gives the Goedel coding of TM program @{text "tp"}. *}
fun Code :: "instr list \<Rightarrow> nat"
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 \<Rightarrow> nat \<Rightarrow> 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 \<Rightarrow> nat \<Rightarrow> 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 \<Rightarrow> nat \<Rightarrow> recf list"
where
"rec_map rf vl = map (\<lambda>i. CN rf [Id vl i]) [0..<vl]"
fun rec_strt :: "nat \<Rightarrow> 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 = \<lambda>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 \<Longrightarrow> rec_eval (rec_listsum2 vl n) xs = Listsum2 xs n"
by (induct n) (simp_all)
lemma strt'_lemma [simp]:
"length xs = vl \<Longrightarrow> rec_eval (rec_strt' vl n) xs = Strt' xs n"
by (induct n) (simp_all add: Let_def)
lemma map_suc:
"map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs"
proof -
have "Suc \<circ> (\<lambda>x. xs ! x) = (\<lambda>x. Suc (xs ! x))" by (simp add: comp_def)
then have "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map (Suc \<circ> (\<lambda>x. xs ! x)) [0..<length xs]" by simp
also have "... = map Suc (map (\<lambda>x. xs ! x) [0..<length xs])" by simp
also have "... = map Suc xs" by (simp add: map_nth)
finally show "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs" .
qed
lemma strt_lemma [simp]:
"length xs = vl \<Longrightarrow> 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 "(\<lambda> (s, l, r). s \<le> 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 \<up> l, <lm>) tp stp) = Conf stp (Code tp) (bl2wc (<lm>))"
apply(induct stp)
apply(simp)
apply(simp)
oops
lemma F_correct:
assumes tm: "steps0 (1, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
and wf: "tm_wf0 tp" "0 < rs"
shows "rec_eval rec_uf [Code tp, bl2wc (<lm>)] = (rs - Suc 0)"
proof -
from least_steps[OF tm]
obtain stp_least where
before: "\<forall>stp' < stp_least. \<not> is_final (steps0 (1, Bk \<up> l, <lm>) tp stp')" and
after: "\<forall>stp' \<ge> stp_least. is_final (steps0 (1, Bk \<up> l, <lm>) tp stp')" by blast
have "Halt (Code tp) (bl2wc (<lm>)) = stp_least" sorry
show ?thesis
apply(simp only: uf_lemma)
apply(simp only: UF.simps)
apply(simp only: Halt.simps)
oops
end