theory UF_Rec
imports Recs
begin
section {* Universal Function *}
text {* @{text "NextPrime x"} returns the first prime number after @{text "x"}. *}
fun NextPrime ::"nat \<Rightarrow> nat"
where
"NextPrime x = (LEAST y. y \<le> Suc (fact x) \<and> x < y \<and> prime y)"
definition rec_nextprime :: "recf"
where
"rec_nextprime = (let conj1 = CN rec_le [Id 2 0, CN S [CN rec_fact [Id 2 1]]] in
let conj2 = CN rec_less [Id 2 1, Id 2 0] in
let conj3 = CN rec_prime [Id 2 0] in
let conjs = CN rec_conj [CN rec_conj [conj2, conj1], conj3]
in MN (CN rec_not [conjs]))"
lemma nextprime_lemma [simp]:
"rec_eval rec_nextprime [x] = NextPrime x"
by (simp add: rec_nextprime_def Let_def)
fun Pi :: "nat \<Rightarrow> nat"
where
"Pi 0 = 2" |
"Pi (Suc x) = NextPrime (Pi x)"
definition
"rec_pi = PR (constn 2) (CN rec_nextprime [Id 2 1])"
lemma pi_lemma [simp]:
"rec_eval rec_pi [x] = Pi x"
by (induct x) (simp_all add: rec_pi_def)
text{* coding of the configuration *}
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))"
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 \<Rightarrow> nat \<Rightarrow> nat"
where
"Listsum2 xs 0 = 0"
| "Listsum2 xs (Suc n) = Listsum2 xs n + xs ! n"
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]"
lemma listsum2_lemma [simp]:
"length xs = vl \<Longrightarrow> 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.
*}
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))"
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)"
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])
text {* The @{text "Scan"} function on page 90 of B book. *}
fun Scan :: "nat \<Rightarrow> nat"
where
"Scan r = r mod 2"
definition
"rec_scan = CN rec_rem [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 \<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)"
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_rem [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_rem [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.
*}
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)"
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 \<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)"
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 \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 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 stat_lemma [simp]:
"rec_eval rec_stat [c] = Stat c"
by(simp add: rec_stat_def)
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)))"
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"}.
*}
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)"
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.
*}
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)"
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.
*}
fun Nostop :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 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"}.
*}
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]]]"
end
(*
fun mtest where
"mtest R 0 = if R 0 then 0 else 1"
| "mtest R (Suc n) = (if R n then mtest R n else (Suc n))"
lemma
"rec_eval (rec_maxr2 f) [x, y1, y2] = XXX"
apply(simp only: rec_maxr2_def)
apply(case_tac x)
apply(clarify)
apply(subst rec_eval.simps)
apply(simp only: constn_lemma)
defer
apply(clarify)
apply(subst rec_eval.simps)
apply(simp only: rec_maxr2_def[symmetric])
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
definition
"rec_minr2 f = rec_sigma2 (rec_accum2 (CN rec_not [f]))"
definition
"rec_maxr2 f = rec_sigma2 (CN rec_sign [CN (rec_sigma2 f) [S]])"
definition Minr :: "(nat \<Rightarrow> nat list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> nat"
where "Minr R x ys = Min ({z | z. z \<le> x \<and> R z ys} \<union> {Suc x})"
definition Maxr :: "(nat \<Rightarrow> nat list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> nat"
where "Maxr R x ys = Max ({z | z. z \<le> x \<and> R z ys} \<union> {0})"
lemma rec_minr2_le_Suc:
"rec_eval (rec_minr2 f) [x, y1, y2] \<le> Suc x"
apply(simp add: rec_minr2_def)
apply(auto intro!: setsum_one_le setprod_one_le)
done
lemma rec_minr2_eq_Suc:
assumes "\<forall>z \<le> x. rec_eval f [z, y1, y2] = 0"
shows "rec_eval (rec_minr2 f) [x, y1, y2] = Suc x"
using assms by (simp add: rec_minr2_def)
lemma rec_minr2_le:
assumes h1: "y \<le> x"
and h2: "0 < rec_eval f [y, y1, y2]"
shows "rec_eval (rec_minr2 f) [x, y1, y2] \<le> y"
apply(simp add: rec_minr2_def)
apply(subst setsum_cut_off_le[OF h1])
using h2 apply(auto)
apply(auto intro!: setsum_one_less setprod_one_le)
done
(* ??? *)
lemma setsum_eq_one_le2:
fixes n::nat
assumes "\<forall>i \<le> n. f i \<le> 1"
shows "((\<Sum>i \<le> n. f i) \<ge> Suc n) \<Longrightarrow> (\<forall>i \<le> n. f i = 1)"
using assms
apply(induct n)
apply(simp add: One_nat_def)
apply(simp)
apply(auto simp add: One_nat_def)
apply(drule_tac x="Suc n" in spec)
apply(auto)
by (metis le_Suc_eq)
lemma rec_min2_not:
assumes "rec_eval (rec_minr2 f) [x, y1, y2] = Suc x"
shows "\<forall>z \<le> x. rec_eval f [z, y1, y2] = 0"
using assms
apply(simp add: rec_minr2_def)
apply(subgoal_tac "\<forall>i \<le> x. (\<Prod>z\<le>i. if rec_eval f [z, y1, y2] = 0 then 1 else 0) = (1::nat)")
apply(simp)
apply (metis atMost_iff le_refl zero_neq_one)
apply(rule setsum_eq_one_le2)
apply(auto)
apply(rule setprod_one_le)
apply(auto)
done
lemma disjCI2:
assumes "~P ==> Q" shows "P|Q"
apply (rule classical)
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
done
lemma minr_lemma [simp]:
shows "rec_eval (rec_minr2 f) [x, y1, y2] = (LEAST z. (z \<le> x \<and> 0 < rec_eval f [z, y1, y2]) \<or> z = Suc x)"
apply(induct x)
apply(rule Least_equality[symmetric])
apply(simp add: rec_minr2_def)
apply(erule disjE)
apply(rule rec_minr2_le)
apply(auto)[2]
apply(simp)
apply(rule rec_minr2_le_Suc)
apply(simp)
apply(rule rec_minr2_le)
apply(rule rec_minr2_le_Suc)
apply(rule disjCI)
apply(simp add: rec_minr2_def)
apply(ru le setsum_one_less)
apply(clarify)
apply(rule setprod_one_le)
apply(auto)[1]
apply(simp)
apply(rule setsum_one_le)
apply(clarify)
apply(rule setprod_one_le)
apply(auto)[1]
thm disj_CE
apply(auto)
lemma minr_lemma:
shows "rec_eval (rec_minr2 f) [x, y1, y2] = Minr (\<lambda>x xs. (0 < rec_eval f (x #xs))) x [y1, y2]"
apply(simp only: Minr_def)
apply(rule sym)
apply(rule Min_eqI)
apply(auto)[1]
apply(simp)
apply(erule disjE)
apply(simp)
apply(rule rec_minr2_le_Suc)
apply(rule rec_minr2_le)
apply(auto)[2]
apply(simp)
apply(induct x)
apply(simp add: rec_minr2_def)
apply(
apply(rule disjCI)
apply(rule rec_minr2_eq_Suc)
apply(simp)
apply(auto)
apply(rule setsum_one_le)
apply(auto)[1]
apply(rule setprod_one_le)
apply(auto)[1]
apply(subst setsum_cut_off_le)
apply(auto)[2]
apply(rule setsum_one_less)
apply(auto)[1]
apply(rule setprod_one_le)
apply(auto)[1]
apply(simp)
thm setsum_eq_one_le
apply(subgoal_tac "(\<forall>z\<le>x. (\<Prod>z\<le>z. if rec_eval f [z, y1, y2] = (0::nat) then 1 else 0) > (0::nat)) \<or>
(\<not> (\<forall>z\<le>x. (\<Prod>z\<le>z. if rec_eval f [z, y1, y2] = (0::nat) then 1 else 0) > (0::nat)))")
prefer 2
apply(auto)[1]
apply(erule disjE)
apply(rule disjI1)
apply(rule setsum_eq_one_le)
apply(simp)
apply(rule disjI2)
apply(simp)
apply(clarify)
apply(subgoal_tac "\<exists>l. l = (LEAST z. 0 < rec_eval f [z, y1, y2])")
defer
apply metis
apply(erule exE)
apply(subgoal_tac "l \<le> x")
defer
apply (metis not_leE not_less_Least order_trans)
apply(subst setsum_least_eq)
apply(rotate_tac 4)
apply(assumption)
apply(auto)[1]
apply(subgoal_tac "a < (LEAST z. 0 < rec_eval f [z, y1, y2])")
prefer 2
apply(auto)[1]
apply(rotate_tac 5)
apply(drule not_less_Least)
apply(auto)[1]
apply(auto)
apply(rule_tac x="(LEAST z. 0 < rec_eval f [z, y1, y2])" in exI)
apply(simp)
apply (metis LeastI_ex)
apply(subst setsum_least_eq)
apply(rotate_tac 3)
apply(assumption)
apply(simp)
apply(auto)[1]
apply(subgoal_tac "a < (LEAST z. 0 < rec_eval f [z, y1, y2])")
prefer 2
apply(auto)[1]
apply (metis neq0_conv not_less_Least)
apply(auto)[1]
apply (metis (mono_tags) LeastI_ex)
by (metis LeastI_ex)
lemma minr_lemma:
shows "rec_eval (rec_minr2 f) [x, y1, y2] = Minr (\<lambda>x xs. (0 < rec_eval f (x #xs))) x [y1, y2]"
apply(simp only: Minr_def rec_minr2_def)
apply(simp)
apply(rule sym)
apply(rule Min_eqI)
apply(auto)[1]
apply(simp)
apply(erule disjE)
apply(simp)
apply(rule setsum_one_le)
apply(auto)[1]
apply(rule setprod_one_le)
apply(auto)[1]
apply(subst setsum_cut_off_le)
apply(auto)[2]
apply(rule setsum_one_less)
apply(auto)[1]
apply(rule setprod_one_le)
apply(auto)[1]
apply(simp)
thm setsum_eq_one_le
apply(subgoal_tac "(\<forall>z\<le>x. (\<Prod>z\<le>z. if rec_eval f [z, y1, y2] = (0::nat) then 1 else 0) > (0::nat)) \<or>
(\<not> (\<forall>z\<le>x. (\<Prod>z\<le>z. if rec_eval f [z, y1, y2] = (0::nat) then 1 else 0) > (0::nat)))")
prefer 2
apply(auto)[1]
apply(erule disjE)
apply(rule disjI1)
apply(rule setsum_eq_one_le)
apply(simp)
apply(rule disjI2)
apply(simp)
apply(clarify)
apply(subgoal_tac "\<exists>l. l = (LEAST z. 0 < rec_eval f [z, y1, y2])")
defer
apply metis
apply(erule exE)
apply(subgoal_tac "l \<le> x")
defer
apply (metis not_leE not_less_Least order_trans)
apply(subst setsum_least_eq)
apply(rotate_tac 4)
apply(assumption)
apply(auto)[1]
apply(subgoal_tac "a < (LEAST z. 0 < rec_eval f [z, y1, y2])")
prefer 2
apply(auto)[1]
apply(rotate_tac 5)
apply(drule not_less_Least)
apply(auto)[1]
apply(auto)
apply(rule_tac x="(LEAST z. 0 < rec_eval f [z, y1, y2])" in exI)
apply(simp)
apply (metis LeastI_ex)
apply(subst setsum_least_eq)
apply(rotate_tac 3)
apply(assumption)
apply(simp)
apply(auto)[1]
apply(subgoal_tac "a < (LEAST z. 0 < rec_eval f [z, y1, y2])")
prefer 2
apply(auto)[1]
apply (metis neq0_conv not_less_Least)
apply(auto)[1]
apply (metis (mono_tags) LeastI_ex)
by (metis LeastI_ex)
lemma disjCI2:
assumes "~P ==> Q" shows "P|Q"
apply (rule classical)
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
done
lemma minr_lemma [simp]:
shows "rec_eval (rec_minr2 f) [x, y1, y2] = (LEAST z. (z \<le> x \<and> 0 < rec_eval f [z, y1, y2]) \<or> z = Suc x)"
(*apply(simp add: rec_minr2_def)*)
apply(rule Least_equality[symmetric])
prefer 2
apply(erule disjE)
apply(rule rec_minr2_le)
apply(auto)[2]
apply(clarify)
apply(rule rec_minr2_le_Suc)
apply(rule disjCI)
apply(simp add: rec_minr2_def)
apply(ru le setsum_one_less)
apply(clarify)
apply(rule setprod_one_le)
apply(auto)[1]
apply(simp)
apply(rule setsum_one_le)
apply(clarify)
apply(rule setprod_one_le)
apply(auto)[1]
thm disj_CE
apply(auto)
apply(auto)
prefer 2
apply(rule le_tran
apply(rule le_trans)
apply(simp)
defer
apply(auto)
apply(subst setsum_cut_off_le)
lemma
"Minr R x ys = (LEAST z. (R z ys \<or> z = Suc x))"
apply(simp add: Minr_def)
apply(rule Min_eqI)
apply(auto)[1]
apply(simp)
apply(rule Least_le)
apply(auto)[1]
apply(rule LeastI2_wellorder)
apply(auto)
done
definition (in ord)
Great :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREAT " 10) where
"Great P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le> x))"
(*
lemma Great_equality:
assumes "P x"
and "\<And>y. P y \<Longrightarrow> y \<le> x"
shows "Great P = x"
unfolding Great_def
apply(rule the_equality)
apply(blast intro: assms)
*)
lemma
"Maxr R x ys = (GREAT z. (R z ys \<or> z = 0))"
apply(simp add: Maxr_def)
apply(rule Max_eqI)
apply(auto)[1]
apply(simp)
thm Least_le Greatest_le
oops
lemma
"Maxr R x ys = x - Minr (\<lambda>z. R (x - z)) x ys"
apply(simp add: Maxr_def Minr_def)
apply(rule Max_eqI)
apply(auto)[1]
apply(simp)
apply(erule disjE)
apply(simp)
apply(auto)[1]
defer
apply(simp)
apply(auto)[1]
thm Min_insert
defer
oops
definition quo :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"quo x y = (if y = 0 then 0 else x div y)"
definition
"rec_quo = CN rec_mult [CN rec_sign [Id 2 1],
CN (rec_minr2 (CN rec_less [Id 3 1, CN rec_mult [CN S [Id 3 0], Id 3 2]]))
[Id 2 0, Id 2 0, Id 2 1]]"
lemma div_lemma [simp]:
"rec_eval rec_quo [x, y] = quo x y"
apply(simp add: rec_quo_def quo_def)
apply(rule impI)
apply(rule Min_eqI)
apply(auto)[1]
apply(simp)
apply(erule disjE)
apply(simp)
apply(auto)[1]
apply (metis div_le_dividend le_SucI)
defer
apply(simp)
apply(auto)[1]
apply (metis mult_Suc_right nat_mult_commute split_div_lemma)
apply(auto)[1]
apply (smt div_le_dividend fst_divmod_nat)
apply(simp add: quo_def)
apply(auto)[1]
apply(auto)
fun Minr :: "(nat list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where "Minr R x y = (let setx = {z | z. z \<le> x \<and> R [z, y]} in
if (setx = {}) then (Suc x) else (Min setx))"
definition
"rec_minr f = rec_sigma1 (rec_accum1 (CN rec_not [f]))"
lemma minr_lemma [simp]:
shows "rec_eval (rec_minr f) [x, y] = Minr (\<lambda>xs. (0 < rec_eval f xs)) x y"
apply(simp only: rec_minr_def)
apply(simp only: sigma1_lemma)
apply(simp only: accum1_lemma)
apply(subst rec_eval.simps)
apply(simp only: map.simps nth)
apply(simp only: Minr.simps Let_def)
apply(auto simp del: not_lemma)
apply(simp)
apply(simp only: not_lemma sign_lemma)
apply(rule sym)
apply(rule Min_eqI)
apply(auto)[1]
apply(simp)
apply(subst setsum_cut_off_le[where m="ya"])
apply(simp)
apply(simp)
apply(metis Icc_subset_Ici_iff atLeast_def in_mono le_refl mem_Collect_eq)
apply(rule setsum_one_less)
apply(default)
apply(rule impI)
apply(rule setprod_one_le)
apply(auto split: if_splits)[1]
apply(simp)
apply(rule conjI)
apply(subst setsum_cut_off_le[where m="xa"])
apply(simp)
apply(simp)
apply (metis Icc_subset_Ici_iff atLeast_def in_mono le_refl mem_Collect_eq)
apply(rule le_trans)
apply(rule setsum_one_less)
apply(default)
apply(rule impI)
apply(rule setprod_one_le)
apply(auto split: if_splits)[1]
apply(simp)
apply(subgoal_tac "\<exists>l. l = (LEAST z. 0 < rec_eval f [z, y])")
defer
apply metis
apply(erule exE)
apply(subgoal_tac "l \<le> x")
defer
apply (metis not_leE not_less_Least order_trans)
apply(subst setsum_least_eq)
apply(rotate_tac 3)
apply(assumption)
prefer 3
apply (metis LeastI_ex)
apply(auto)[1]
apply(subgoal_tac "a < (LEAST z. 0 < rec_eval f [z, y])")
prefer 2
apply(auto)[1]
apply(rotate_tac 5)
apply(drule not_less_Least)
apply(auto)[1]
apply(auto)
by (metis (mono_tags) LeastI_ex)
fun Minr2 :: "(nat list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
where "Minr2 R x y = (let setx = {z | z. z \<le> x \<and> R [z, y]} in
Min (setx \<union> {Suc x}))"
lemma Minr2_Minr:
"Minr2 R x y = Minr R x y"
apply(auto)
apply (metis (lifting, no_types) Min_singleton empty_Collect_eq)
apply(rule min_absorb2)
apply(subst Min_le_iff)
apply(auto)
done
*)