theory UF_Rec

imports Recs Turing2

begin

section {* Coding of Turing Machines and tapes*}

text {*

  The purpose of this section is to construct the coding function of Turing 

  Machine, which is going to be named @{text "code"}. *}

text {* Encoding of actions as numbers *}

fun action_num :: "action \<Rightarrow> nat"

  where

  "action_num W0 = 0"

| "action_num W1 = 1"

| "action_num L  = 2"

| "action_num R  = 3"

| "action_num Nop = 4"

fun cell_num :: "cell \<Rightarrow> nat" where

  "cell_num Bk = 0"

| "cell_num Oc = 1"

fun code_tp :: "cell list \<Rightarrow> nat list"

  where

  "code_tp [] = []"

| "code_tp (c # tp) = (cell_num c) # code_tp tp"

fun Code_tp where

  "Code_tp tp = lenc (code_tp tp)"

fun Code_conf where

  "Code_conf (s, l, r) = (s, Code_tp l, Code_tp r)"

fun code_instr :: "instr \<Rightarrow> nat" where

  "code_instr i = penc (action_num (fst i)) (snd i)"

fun Code_instr :: "instr \<times> instr \<Rightarrow> nat" where

  "Code_instr i = penc (code_instr (fst i)) (code_instr (snd i))"

fun code_tprog :: "tprog \<Rightarrow> nat list"

  where

  "code_tprog [] =  []"

| "code_tprog (i # tm) = Code_instr i # code_tprog tm"

lemma code_tprog_length [simp]:

  "length (code_tprog tp) = length tp"

by (induct tp) (simp_all)

lemma code_tprog_nth [simp]:

  "n < length tp \<Longrightarrow> (code_tprog tp) ! n = Code_instr (tp ! n)"

by (induct tp arbitrary: n) (simp_all add: nth_Cons')

fun Code_tprog :: "tprog \<Rightarrow> nat"

  where 

  "Code_tprog tm = lenc (code_tprog tm)"

section {* Universal Function in HOL *}

fun Write where

text {* 

  The @{text Newleft} and @{text Newright} functions on page 91 of B book. 

  They calculate the new left and right tape (@{text p} and @{text r}) according 

  to an action @{text a}.

*}

fun Newleft :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"

  where

fun Newright :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"

  where

                      else if a = 1 then Write 1 r

                      else if a = 3 then pdec2 r

                      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 code of the Turing Machine, @{text "q"} is the current state of 

  Turing Machine, and @{text "r"} is the scanned cell of is the right tape. 

*}

fun actn :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

  "actn n 0 = pdec1 (pdec1 n)"

| "actn n _ = pdec1 (pdec2 n)"

fun Actn :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"

  where

  where

fun Conf :: "nat \<times> (nat \<times> nat) \<Rightarrow> nat"

  where

  "Conf (q, (l, r)) = lenc [q, l, r]"

fun Left where

fun Right where

  where

text {*

fun Steps :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"

  where

  "Steps cf p 0  = cf"

text {*

  where

text{* 

fun Nostop :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"

  where

text{*

*}

fun Halt :: "nat \<Rightarrow> nat \<Rightarrow> nat"

  where

fun UF :: "nat \<Rightarrow> nat \<Rightarrow> nat"

  where

section {* The UF can simulate Turing machines *}

lemma Update_left_simulate:

  shows "Newleft (Code_tp l) (Code_tp r) (action_num a) = Code_tp (fst (update a (l, r)))"

apply(induct a)

apply(simp_all)

apply(case_tac l)

apply(simp_all)

apply(case_tac r)

apply(simp_all)

done

lemma Update_right_simulate:

  shows "Newright (Code_tp l) (Code_tp r) (action_num a) = Code_tp (snd (update a (l, r)))"

apply(induct a)

apply(simp_all)

apply(case_tac r)

apply(simp_all)

apply(case_tac r)

apply(simp_all)

apply(case_tac l)

apply(simp_all)

apply(case_tac r)

apply(simp_all)

done

lemma Fetch_state_simulate:

apply(induct tp st c rule: fetch.induct)

apply(simp_all add: list_encode_inverse split: cell.split)

done

lemma Fetch_action_simulate:

apply(induct tp st c rule: fetch.induct)

apply(simp_all add: list_encode_inverse split: cell.split)

done

apply(case_tac tp)

apply(simp_all)

done

lemma misc:

  "2 < (3::nat)"

  "1 < (3::nat)"

  "0 < (3::nat)" 

  "length [x] = 1"

  "length [x, y] = 2"

  "length [x, y , z] = 3"

  "[x, y, z] ! 0 = x"

  "[x, y, z] ! 1 = y"

  "[x, y, z] ! 2 = z"

apply(simp_all)

done

apply(subst step.simps) 

apply(simp only: Let_def)

apply(simp only: Conf.simps Code_conf.simps Right.simps Left.simps)

apply(simp only: list_encode_inverse)

apply(simp only: misc if_True Code_tp.simps)

apply(simp only: prod_case_beta) 

apply(subst Fetch_state_simulate[OF assms, symmetric])

apply(simp only: list_encode_inverse)

apply(simp only: misc if_True)

apply(simp only: Fetch_action_simulate[OF assms])

apply(simp only: Update_left_simulate[simplified Code_tp.simps])

apply(simp only: Update_right_simulate[simplified Code_tp.simps])

apply(case_tac "update (fst (fetch tp st (read r))) (l, r)")

apply(simp only: Code_conf.simps)

apply(simp only: Conf.simps)

apply(simp)

done

  shows "Steps (Conf (Code_conf cf)) (Code_tprog tp) n = Conf (Code_conf (steps cf tp n))"

apply(induct n arbitrary: cf) 

apply(simp)

apply(simp only: Steps.simps steps.simps)

apply(case_tac cf)

apply(simp only: )

apply(rule assms)

apply(drule_tac x="step (a, b, c) tp" in meta_spec)

apply(simp)

done

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_penc [CN rec_penc [Id 3 0, Id 3 1], Id 3 2]"

definition

  "rec_left = rec_pdec1"

definition 

  "rec_right = CN rec_pdec2 [rec_pdec1]"

definition 

  "rec_stat = CN rec_pdec2 [rec_pdec2]"

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"

apply(simp)

apply (simp add: rec_trpl_def)

lemma left_lemma [simp]:

  "rec_eval rec_left [c] = Left c" 

by(simp add: rec_left_def)