diff -r 32c5e8d1f6ff -r 4524c5edcafb thys/Recs.thy
--- a/thys/Recs.thy	Tue May 21 13:50:15 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,828 +0,0 @@
-theory Recs
-imports Main Fact 
-        "~~/src/HOL/Number_Theory/Primes" 
-        "~~/src/HOL/Library/Nat_Bijection"
-        "~~/src/HOL/Library/Discrete"
-begin
-
-declare One_nat_def[simp del]
-
-(*
-  some definitions from 
-
-    A Course in Formal Languages, Automata and Groups
-    I M Chiswell 
-
-  and
-
-    Lecture on undecidability
-    Michael M. Wolf 
-*)
-
-lemma if_zero_one [simp]:
-  "(if P then 1 else 0) = (0::nat) \<longleftrightarrow> \<not> P"
-  "(0::nat) < (if P then 1 else 0) = P"
-  "(if P then 0 else 1) = (if \<not>P then 1 else (0::nat))"
-by (simp_all)
-
-lemma nth:
-  "(x # xs) ! 0 = x"
-  "(x # y # xs) ! 1 = y"
-  "(x # y # z # xs) ! 2 = z"
-  "(x # y # z # u # xs) ! 3 = u"
-by (simp_all)
-
-
-section {* Some auxiliary lemmas about @{text "\<Sum>"} and @{text "\<Prod>"} *}
-
-lemma setprod_atMost_Suc[simp]: 
-  "(\<Prod>i \<le> Suc n. f i) = (\<Prod>i \<le> n. f i) * f(Suc n)"
-by(simp add:atMost_Suc mult_ac)
-
-lemma setprod_lessThan_Suc[simp]: 
-  "(\<Prod>i < Suc n. f i) = (\<Prod>i < n. f i) * f n"
-by (simp add:lessThan_Suc mult_ac)
-
-lemma setsum_add_nat_ivl2: "n \<le> p  \<Longrightarrow>
-  setsum f {..<n} + setsum f {n..p} = setsum f {..p::nat}"
-apply(subst setsum_Un_disjoint[symmetric])
-apply(auto simp add: ivl_disj_un_one)
-done
-
-lemma setsum_eq_zero [simp]:
-  fixes f::"nat \<Rightarrow> nat"
-  shows "(\<Sum>i < n. f i) = 0 \<longleftrightarrow> (\<forall>i < n. f i = 0)" 
-        "(\<Sum>i \<le> n. f i) = 0 \<longleftrightarrow> (\<forall>i \<le> n. f i = 0)" 
-by (auto)
-
-lemma setprod_eq_zero [simp]:
-  fixes f::"nat \<Rightarrow> nat"
-  shows "(\<Prod>i < n. f i) = 0 \<longleftrightarrow> (\<exists>i < n. f i = 0)" 
-        "(\<Prod>i \<le> n. f i) = 0 \<longleftrightarrow> (\<exists>i \<le> n. f i = 0)" 
-by (auto)
-
-lemma setsum_one_less:
-  fixes n::nat
-  assumes "\<forall>i < n. f i \<le> 1" 
-  shows "(\<Sum>i < n. f i) \<le> n"  
-using assms
-by (induct n) (auto)
-
-lemma setsum_one_le:
-  fixes n::nat
-  assumes "\<forall>i \<le> n. f i \<le> 1" 
-  shows "(\<Sum>i \<le> n. f i) \<le> Suc n"  
-using assms
-by (induct n) (auto)
-
-lemma setsum_eq_one_le:
-  fixes n::nat
-  assumes "\<forall>i \<le> n. f i = 1" 
-  shows "(\<Sum>i \<le> n. f i) = Suc n"  
-using assms
-by (induct n) (auto)
-
-lemma setsum_least_eq:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes h0: "p \<le> n"
-  assumes h1: "\<forall>i \<in> {..<p}. f i = 1"
-  assumes h2: "\<forall>i \<in> {p..n}. f i = 0"
-  shows "(\<Sum>i \<le> n. f i) = p"  
-proof -
-  have eq_p: "(\<Sum>i \<in> {..<p}. f i) = p" 
-    using h1 by (induct p) (simp_all)
-  have eq_zero: "(\<Sum>i \<in> {p..n}. f i) = 0" 
-    using h2 by auto
-  have "(\<Sum>i \<le> n. f i) = (\<Sum>i \<in> {..<p}. f i) + (\<Sum>i \<in> {p..n}. f i)"
-    using h0 by (simp add: setsum_add_nat_ivl2) 
-  also have "... = (\<Sum>i \<in> {..<p}. f i)" using eq_zero by simp
-  finally show "(\<Sum>i \<le> n. f i) = p" using eq_p by simp
-qed
-
-lemma nat_mult_le_one:
-  fixes m n::nat
-  assumes "m \<le> 1" "n \<le> 1"
-  shows "m * n \<le> 1"
-using assms by (induct n) (auto)
-
-lemma setprod_one_le:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes "\<forall>i \<le> n. f i \<le> 1" 
-  shows "(\<Prod>i \<le> n. f i) \<le> 1" 
-using assms 
-by (induct n) (auto intro: nat_mult_le_one)
-
-lemma setprod_greater_zero:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes "\<forall>i \<le> n. f i \<ge> 0" 
-  shows "(\<Prod>i \<le> n. f i) \<ge> 0" 
-using assms by (induct n) (auto)
-
-lemma setprod_eq_one:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes "\<forall>i \<le> n. f i = Suc 0" 
-  shows "(\<Prod>i \<le> n. f i) = Suc 0" 
-using assms by (induct n) (auto)
-
-lemma setsum_cut_off_less:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes h1: "m \<le> n"
-  and     h2: "\<forall>i \<in> {m..<n}. f i = 0"
-  shows "(\<Sum>i < n. f i) = (\<Sum>i < m. f i)"
-proof -
-  have eq_zero: "(\<Sum>i \<in> {m..<n}. f i) = 0" 
-    using h2 by auto
-  have "(\<Sum>i < n. f i) = (\<Sum>i \<in> {..<m}. f i) + (\<Sum>i \<in> {m..<n}. f i)"
-    using h1 by (metis atLeast0LessThan le0 setsum_add_nat_ivl) 
-  also have "... = (\<Sum>i \<in> {..<m}. f i)" using eq_zero by simp
-  finally show "(\<Sum>i < n. f i) = (\<Sum>i < m. f i)" by simp
-qed
-
-lemma setsum_cut_off_le:
-  fixes f::"nat \<Rightarrow> nat"
-  assumes h1: "m \<le> n"
-  and     h2: "\<forall>i \<in> {m..n}. f i = 0"
-  shows "(\<Sum>i \<le> n. f i) = (\<Sum>i < m. f i)"
-proof -
-  have eq_zero: "(\<Sum>i \<in> {m..n}. f i) = 0" 
-    using h2 by auto
-  have "(\<Sum>i \<le> n. f i) = (\<Sum>i \<in> {..<m}. f i) + (\<Sum>i \<in> {m..n}. f i)"
-    using h1 by (simp add: setsum_add_nat_ivl2)
-  also have "... = (\<Sum>i \<in> {..<m}. f i)" using eq_zero by simp
-  finally show "(\<Sum>i \<le> n. f i) = (\<Sum>i < m. f i)" by simp
-qed
-
-lemma setprod_one [simp]:
-  fixes n::nat
-  shows "(\<Prod>i < n. Suc 0) = Suc 0"
-        "(\<Prod>i \<le> n. Suc 0) = Suc 0"
-by (induct n) (simp_all)
-
-
-
-section {* Recursive Functions *}
-
-datatype recf =  Z
-              |  S
-              |  Id nat nat
-              |  Cn nat recf "recf list"
-              |  Pr nat recf recf
-              |  Mn nat recf 
-
-fun arity :: "recf \<Rightarrow> nat"
-  where
-  "arity Z = 1" 
-| "arity S = 1"
-| "arity (Id m n) = m"
-| "arity (Cn n f gs) = n"
-| "arity (Pr n f g) = Suc n"
-| "arity (Mn n f) = n"
-
-abbreviation
-  "CN f gs \<equiv> Cn (arity (hd gs)) f gs"
-
-abbreviation
-  "PR f g \<equiv> Pr (arity f) f g"
-
-abbreviation
-  "MN f \<equiv> Mn (arity f - 1) f"
-
-fun rec_eval :: "recf \<Rightarrow> nat list \<Rightarrow> nat"
-  where
-  "rec_eval Z xs = 0" 
-| "rec_eval S xs = Suc (xs ! 0)" 
-| "rec_eval (Id m n) xs = xs ! n" 
-| "rec_eval (Cn n f gs) xs = rec_eval f (map (\<lambda>x. rec_eval x xs) gs)" 
-| "rec_eval (Pr n f g) (0 # xs) = rec_eval f xs"
-| "rec_eval (Pr n f g) (Suc x # xs) = 
-     rec_eval g (x # (rec_eval (Pr n f g) (x # xs)) # xs)"
-| "rec_eval (Mn n f) xs = (LEAST x. rec_eval f (x # xs) = 0)"
-
-inductive 
-  terminates :: "recf \<Rightarrow> nat list \<Rightarrow> bool"
-where
-  termi_z: "terminates Z [n]"
-| termi_s: "terminates S [n]"
-| termi_id: "\<lbrakk>n < m; length xs = m\<rbrakk> \<Longrightarrow> terminates (Id m n) xs"
-| termi_cn: "\<lbrakk>terminates f (map (\<lambda>g. rec_eval g xs) gs); 
-              \<forall>g \<in> set gs. terminates g xs; length xs = n\<rbrakk> \<Longrightarrow> terminates (Cn n f gs) xs"
-| termi_pr: "\<lbrakk>\<forall> y < x. terminates g (y # (rec_eval (Pr n f g) (y # xs) # xs));
-              terminates f xs;
-              length xs = n\<rbrakk> 
-              \<Longrightarrow> terminates (Pr n f g) (xs @ [x])"
-| termi_mn: "\<lbrakk>length xs = n; terminates f (r # xs); 
-              rec_eval f (r # xs) = 0;
-              \<forall> i < r. terminates f (i # xs) \<and> rec_eval f (i # xs) > 0\<rbrakk> \<Longrightarrow> terminates (Mn n f) xs"
-
-
-section {* Recursive Function Definitions *}
-
-text {*
-  @{text "constn n"} is the recursive function which computes 
-  natural number @{text "n"}.
-*}
-fun constn :: "nat \<Rightarrow> recf"
-  where
-  "constn 0 = Z"  |
-  "constn (Suc n) = CN S [constn n]"
-
-definition
-  "rec_swap f = CN f [Id 2 1, Id 2 0]"
-
-definition
-  "rec_add = PR (Id 1 0) (CN S [Id 3 1])"
-
-definition 
-  "rec_mult = PR Z (CN rec_add [Id 3 1, Id 3 2])"
-
-definition 
-  "rec_power = rec_swap (PR (constn 1) (CN rec_mult [Id 3 1, Id 3 2]))"
-
-definition
-  "rec_fact = PR (constn 1) (CN rec_mult [CN S [Id 3 0], Id 3 1])"
-
-definition 
-  "rec_pred = CN (PR Z (Id 3 0)) [Id 1 0, Id 1 0]"
-
-definition 
-  "rec_minus = rec_swap (PR (Id 1 0) (CN rec_pred [Id 3 1]))"
-
-
-text {* 
-  The @{text "sign"} function returns 1 when the input argument 
-  is greater than @{text "0"}. *}
-
-definition 
-  "rec_sign = CN rec_minus [constn 1, CN rec_minus [constn 1, Id 1 0]]"
-
-definition 
-  "rec_not = CN rec_minus [constn 1, Id 1 0]"
-
-text {*
-  @{text "rec_eq"} compares two arguments: returns @{text "1"}
-  if they are equal; @{text "0"} otherwise. *}
-definition 
-  "rec_eq = CN rec_minus [constn 1, CN rec_add [rec_minus, rec_swap rec_minus]]"
-
-definition 
-  "rec_noteq = CN rec_not [rec_eq]"
-
-definition 
-  "rec_conj = CN rec_sign [rec_mult]"
-
-definition 
-  "rec_disj = CN rec_sign [rec_add]"
-
-definition 
-  "rec_imp = CN rec_disj [CN rec_not [Id 2 0], Id 2 1]"
-
-text {* @{term "rec_ifz [z, x, y]"} returns x if z is zero,
-  y otherwise;  @{term "rec_if [z, x, y]"} returns x if z is *not*
-  zero, y otherwise *}
-
-definition 
-  "rec_ifz = PR (Id 2 0) (Id 4 3)"
-
-definition 
-  "rec_if = CN rec_ifz [CN rec_not [Id 3 0], Id 3 1, Id 3 2]"
-
-text {*
-  @{text "rec_less"} compares two arguments and returns @{text "1"} if
-  the first is less than the second; otherwise returns @{text "0"}. *}
-
-definition 
-  "rec_less = CN rec_sign [rec_swap rec_minus]"
-
-definition 
-  "rec_le = CN rec_disj [rec_less, rec_eq]"
-
-text {* Sigma and Accum for function with one and two arguments *}
-
-definition 
-  "rec_sigma1 f = PR (CN f [Z, Id 1 0]) (CN rec_add [Id 3 1, CN f [S, Id 3 2]])"
-
-definition 
-  "rec_sigma2 f = PR (CN f [Z, Id 2 0, Id 2 1]) (CN rec_add [Id 4 1, CN f [S, Id 4 2, Id 4 3]])"
-
-definition 
-  "rec_accum1 f = PR (CN f [Z, Id 1 0]) (CN rec_mult [Id 3 1, CN f [S, Id 3 2]])"
-
-definition 
-  "rec_accum2 f = PR (CN f [Z, Id 2 0, Id 2 1]) (CN rec_mult [Id 4 1, CN f [S, Id 4 2, Id 4 3]])"
-
-text {* Bounded quantifiers for one and two arguments *}
-
-definition
-  "rec_all1 f = CN rec_sign [rec_accum1 f]"
-
-definition
-  "rec_all2 f = CN rec_sign [rec_accum2 f]"
-
-definition
-  "rec_ex1 f = CN rec_sign [rec_sigma1 f]"
-
-definition
-  "rec_ex2 f = CN rec_sign [rec_sigma2 f]"
-
-text {* Dvd, Quotient, Modulo *}
-
-definition 
-  "rec_dvd = 
-     rec_swap (CN (rec_ex2 (CN rec_eq [Id 3 2, CN rec_mult [Id 3 1, Id 3 0]])) [Id 2 0, Id 2 1, Id 2 0])"  
-
-definition
-  "rec_quo = (let lhs = CN S [Id 3 0] in
-              let rhs = CN rec_mult [Id 3 2, CN S [Id 3 1]] in
-              let cond = CN rec_eq [lhs, rhs] in
-              let if_stmt = CN rec_if [cond, CN S [Id 3 1], Id 3 1]
-              in PR Z if_stmt)"
-
-definition
-  "rec_mod = CN rec_minus [Id 2 0, CN rec_mult [Id 2 1, rec_quo]]"
-
-
-section {* Prime Numbers *}
-
-definition 
-  "rec_prime = 
-    (let conj1 = CN rec_less [constn 1, Id 1 0] in
-     let disj  = CN rec_disj [CN rec_eq [Id 2 0, constn 1], rec_eq] in
-     let imp   = CN rec_imp [rec_dvd, disj] in
-     let conj2 = CN (rec_all1 imp) [Id 1 0, Id 1 0] in
-     CN rec_conj [conj1, conj2])"
-
-
-section {* Correctness of Recursive Functions *}
-
-lemma constn_lemma [simp]: 
-  "rec_eval (constn n) xs = n"
-by (induct n) (simp_all)
-
-lemma swap_lemma [simp]:
-  "rec_eval (rec_swap f) [x, y] = rec_eval f [y, x]"
-by (simp add: rec_swap_def)
-
-lemma add_lemma [simp]: 
-  "rec_eval rec_add [x, y] =  x + y"
-by (induct x) (simp_all add: rec_add_def)
-
-lemma mult_lemma [simp]: 
-  "rec_eval rec_mult [x, y] = x * y"
-by (induct x) (simp_all add: rec_mult_def)
-
-lemma power_lemma [simp]: 
-  "rec_eval rec_power [x, y] = x ^ y"
-by (induct y) (simp_all add: rec_power_def)
-
-lemma fact_lemma [simp]: 
-  "rec_eval rec_fact [x] = fact x"
-by (induct x) (simp_all add: rec_fact_def)
-
-lemma pred_lemma [simp]: 
-  "rec_eval rec_pred [x] =  x - 1"
-by (induct x) (simp_all add: rec_pred_def)
-
-lemma minus_lemma [simp]: 
-  "rec_eval rec_minus [x, y] = x - y"
-by (induct y) (simp_all add: rec_minus_def)
-
-lemma sign_lemma [simp]: 
-  "rec_eval rec_sign [x] = (if x = 0 then 0 else 1)"
-by (simp add: rec_sign_def)
-
-lemma not_lemma [simp]: 
-  "rec_eval rec_not [x] = (if x = 0 then 1 else 0)"
-by (simp add: rec_not_def)
-
-lemma eq_lemma [simp]: 
-  "rec_eval rec_eq [x, y] = (if x = y then 1 else 0)"
-by (simp add: rec_eq_def)
-
-lemma noteq_lemma [simp]: 
-  "rec_eval rec_noteq [x, y] = (if x \<noteq> y then 1 else 0)"
-by (simp add: rec_noteq_def)
-
-lemma conj_lemma [simp]: 
-  "rec_eval rec_conj [x, y] = (if x = 0 \<or> y = 0 then 0 else 1)"
-by (simp add: rec_conj_def)
-
-lemma disj_lemma [simp]: 
-  "rec_eval rec_disj [x, y] = (if x = 0 \<and> y = 0 then 0 else 1)"
-by (simp add: rec_disj_def)
-
-lemma imp_lemma [simp]: 
-  "rec_eval rec_imp [x, y] = (if 0 < x \<and> y = 0 then 0 else 1)"
-by (simp add: rec_imp_def)
-
-lemma less_lemma [simp]: 
-  "rec_eval rec_less [x, y] = (if x < y then 1 else 0)"
-by (simp add: rec_less_def)
-
-lemma le_lemma [simp]: 
-  "rec_eval rec_le [x, y] = (if (x \<le> y) then 1 else 0)"
-by(simp add: rec_le_def)
-
-lemma ifz_lemma [simp]:
-  "rec_eval rec_ifz [z, x, y] = (if z = 0 then x else y)" 
-by (case_tac z) (simp_all add: rec_ifz_def)
-
-lemma if_lemma [simp]:
-  "rec_eval rec_if [z, x, y] = (if 0 < z then x else y)" 
-by (simp add: rec_if_def)
-
-lemma sigma1_lemma [simp]: 
-  shows "rec_eval (rec_sigma1 f) [x, y] = (\<Sum> z \<le> x. (rec_eval f)  [z, y])"
-by (induct x) (simp_all add: rec_sigma1_def)
-
-lemma sigma2_lemma [simp]: 
-  shows "rec_eval (rec_sigma2 f) [x, y1, y2] = (\<Sum> z \<le> x. (rec_eval f)  [z, y1, y2])"
-by (induct x) (simp_all add: rec_sigma2_def)
-
-lemma accum1_lemma [simp]: 
-  shows "rec_eval (rec_accum1 f) [x, y] = (\<Prod> z \<le> x. (rec_eval f)  [z, y])"
-by (induct x) (simp_all add: rec_accum1_def)
-
-lemma accum2_lemma [simp]: 
-  shows "rec_eval (rec_accum2 f) [x, y1, y2] = (\<Prod> z \<le> x. (rec_eval f)  [z, y1, y2])"
-by (induct x) (simp_all add: rec_accum2_def)
-
-lemma ex1_lemma [simp]:
- "rec_eval (rec_ex1 f) [x, y] = (if (\<exists>z \<le> x. 0 < rec_eval f [z, y]) then 1 else 0)"
-by (simp add: rec_ex1_def)
-
-lemma ex2_lemma [simp]:
- "rec_eval (rec_ex2 f) [x, y1, y2] = (if (\<exists>z \<le> x. 0 < rec_eval f [z, y1, y2]) then 1 else 0)"
-by (simp add: rec_ex2_def)
-
-lemma all1_lemma [simp]:
- "rec_eval (rec_all1 f) [x, y] = (if (\<forall>z \<le> x. 0 < rec_eval f [z, y]) then 1 else 0)"
-by (simp add: rec_all1_def)
-
-lemma all2_lemma [simp]:
- "rec_eval (rec_all2 f) [x, y1, y2] = (if (\<forall>z \<le> x. 0 < rec_eval f [z, y1, y2]) then 1 else 0)"
-by (simp add: rec_all2_def)
-
-
-lemma dvd_alt_def:
-  fixes x y k:: nat
-  shows "(x dvd y) = (\<exists> k \<le> y. y = x * k)"
-apply(auto simp add: dvd_def)
-apply(case_tac x)
-apply(auto)
-done
-
-lemma dvd_lemma [simp]:
-  "rec_eval rec_dvd [x, y] = (if x dvd y then 1 else 0)"
-unfolding dvd_alt_def
-by (auto simp add: rec_dvd_def)
-
-fun Quo where
-  "Quo x 0 = 0"
-| "Quo x (Suc y) = (if (Suc y = x * (Suc (Quo x y))) then Suc (Quo x y) else Quo x y)"
-
-lemma Quo0:
-  shows "Quo 0 y = 0"
-apply(induct y)
-apply(auto)
-done
-
-lemma Quo1:
-  "x * (Quo x y) \<le> y"
-by (induct y) (simp_all)
-
-lemma Quo2: 
-  "b * (Quo b a) + a mod b = a"
-by (induct a) (auto simp add: mod_Suc)
-
-lemma Quo3:
-  "n * (Quo n m) = m - m mod n"
-using Quo2[of n m] by (auto)
-
-lemma Quo4:
-  assumes h: "0 < x"
-  shows "y < x + x * Quo x y"
-proof -
-  have "x - (y mod x) > 0" using mod_less_divisor assms by auto
-  then have "y < y + (x - (y mod x))" by simp
-  then have "y < x + (y - (y mod x))" by simp
-  then show "y < x + x * (Quo x y)" by (simp add: Quo3) 
-qed
-
-lemma Quo_div: 
-  shows "Quo x y = y div x"
-apply(case_tac "x = 0")
-apply(simp add: Quo0)
-apply(subst split_div_lemma[symmetric])
-apply(auto intro: Quo1 Quo4)
-done
-
-lemma Quo_rec_quo:
-  shows "rec_eval rec_quo [y, x] = Quo x y"
-by (induct y) (simp_all add: rec_quo_def)
-
-lemma quo_lemma [simp]:
-  shows "rec_eval rec_quo [y, x] = y div x"
-by (simp add: Quo_div Quo_rec_quo)
-
-lemma rem_lemma [simp]:
-  shows "rec_eval rec_mod [y, x] = y mod x"
-by (simp add: rec_mod_def mod_div_equality' nat_mult_commute)
-
-
-section {* Prime Numbers *}
-
-lemma prime_alt_def:
-  fixes p::nat
-  shows "prime p = (1 < p \<and> (\<forall>m \<le> p. m dvd p \<longrightarrow> m = 1 \<or> m = p))"
-apply(auto simp add: prime_nat_def dvd_def)
-apply(drule_tac x="k" in spec)
-apply(auto)
-done
-
-lemma prime_lemma [simp]: 
-  "rec_eval rec_prime [x] = (if prime x then 1 else 0)"
-by (auto simp add: rec_prime_def Let_def prime_alt_def)
-
-section {* Bounded Maximisation *}
-
-fun BMax_rec where
-  "BMax_rec R 0 = 0"
-| "BMax_rec R (Suc n) = (if R (Suc n) then (Suc n) else BMax_rec R n)"
-
-definition BMax_set :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat"
-  where "BMax_set R x = Max ({z. z \<le> x \<and> R z} \<union> {0})"
-
-lemma BMax_rec_eq1:
- "BMax_rec R x = (GREATEST z. (R z \<and> z \<le> x) \<or> z = 0)"
-apply(induct x)
-apply(auto intro: Greatest_equality Greatest_equality[symmetric])
-apply(simp add: le_Suc_eq)
-by metis
-
-lemma BMax_rec_eq2:
-  "BMax_rec R x = Max ({z. z \<le> x \<and> R z} \<union> {0})"
-apply(induct x)
-apply(auto intro: Max_eqI Max_eqI[symmetric])
-apply(simp add: le_Suc_eq)
-by metis
-
-lemma BMax_rec_eq3:
-  "BMax_rec R x = Max (Set.filter (\<lambda>z. R z) {..x} \<union> {0})"
-by (simp add: BMax_rec_eq2 Set.filter_def)
-
-definition 
-  "rec_max1 f = PR (constn 0) (CN rec_ifz [CN f [CN S [Id 3 0], Id 3 2], CN S [Id 4 0], Id 4 1])"
- 
-lemma max1_lemma [simp]:
-  "rec_eval (rec_max1 f) [x, y] = BMax_rec (\<lambda>u. rec_eval f [u, y] = 0) x"
-by (induct x) (simp_all add: rec_max1_def)
-
-definition 
-  "rec_max2 f = PR (constn 0) (CN rec_ifz [CN f [CN S [Id 4 0], Id 4 2, Id 4 3], CN S [Id 4 0], Id 4 1])"
- 
-lemma max2_lemma [simp]:
-  "rec_eval (rec_max2 f) [x, y1, y2] = BMax_rec (\<lambda>u. rec_eval f [u, y1, y2] = 0) x"
-by (induct x) (simp_all add: rec_max2_def)
-
-section {* Encodings using Cantor's pairing function *}
-
-text {*
-  We use Cantor's pairing function from Nat_Bijection.
-  However, we need to prove that the formulation of the
-  decoding function there is recursive. For this we first 
-  prove that we can extract the maximal triangle number 
-  using @{term prod_decode}.
-*}
-
-abbreviation Max_triangle_aux where
-  "Max_triangle_aux k z \<equiv> fst (prod_decode_aux k z) + snd (prod_decode_aux k z)"
-
-abbreviation Max_triangle where
-  "Max_triangle z \<equiv> Max_triangle_aux 0 z"
-
-abbreviation
-  "pdec1 z \<equiv> fst (prod_decode z)"
-
-abbreviation
-  "pdec2 z \<equiv> snd (prod_decode z)"
-
-abbreviation 
-  "penc m n \<equiv> prod_encode (m, n)"
-
-lemma fst_prod_decode: 
-  "pdec1 z = z - triangle (Max_triangle z)"
-by (subst (3) prod_decode_inverse[symmetric]) 
-   (simp add: prod_encode_def prod_decode_def split: prod.split)
-
-lemma snd_prod_decode: 
-  "pdec2 z = Max_triangle z - pdec1 z"
-by (simp only: prod_decode_def)
-
-lemma le_triangle:
-  "m \<le> triangle (n + m)"
-by (induct_tac m) (simp_all)
-
-lemma Max_triangle_triangle_le:
-  "triangle (Max_triangle z) \<le> z"
-by (subst (9) prod_decode_inverse[symmetric])
-   (simp add: prod_decode_def prod_encode_def split: prod.split)
-
-lemma Max_triangle_le: 
-  "Max_triangle z \<le> z"
-proof -
-  have "Max_triangle z \<le> triangle (Max_triangle z)"
-    using le_triangle[of _ 0, simplified] by simp
-  also have "... \<le> z" by (rule Max_triangle_triangle_le)
-  finally show "Max_triangle z \<le> z" .
-qed
-
-lemma w_aux: 
-  "Max_triangle (triangle k + m) = Max_triangle_aux k m"
-by (simp add: prod_decode_def[symmetric] prod_decode_triangle_add)
-
-lemma y_aux: "y \<le> Max_triangle_aux y k"
-apply(induct k arbitrary: y rule: nat_less_induct)
-apply(subst (1 2) prod_decode_aux.simps)
-apply(simp)
-apply(rule impI)
-apply(drule_tac x="n - Suc y" in spec)
-apply(drule mp)
-apply(auto)[1]
-apply(drule_tac x="Suc y" in spec)
-apply(erule Suc_leD)
-done
-
-lemma Max_triangle_greatest: 
-  "Max_triangle z = (GREATEST k. (triangle k \<le> z \<and> k \<le> z) \<or> k = 0)"
-apply(rule Greatest_equality[symmetric])
-apply(rule disjI1)
-apply(rule conjI)
-apply(rule Max_triangle_triangle_le)
-apply(rule Max_triangle_le)
-apply(erule disjE)
-apply(erule conjE)
-apply(subst (asm) (1) le_iff_add)
-apply(erule exE)
-apply(clarify)
-apply(simp only: w_aux)
-apply(rule y_aux)
-apply(simp)
-done
-
-definition 
-  "rec_triangle = CN rec_quo [CN rec_mult [Id 1 0, S], constn 2]"
-
-lemma triangle_lemma [simp]:
-  "rec_eval rec_triangle [x] = triangle x"
-by (simp add: rec_triangle_def triangle_def)
- 
-definition
-  "rec_max_triangle = 
-       (let cond = CN rec_not [CN rec_le [CN rec_triangle [Id 2 0], Id 2 1]] in
-        CN (rec_max1 cond) [Id 1 0, Id 1 0])"
-
-lemma max_triangle_lemma [simp]:
-  "rec_eval rec_max_triangle [x] = Max_triangle x"
-by (simp add: Max_triangle_greatest rec_max_triangle_def Let_def BMax_rec_eq1) 
-
-definition
-  "rec_penc = CN rec_add [CN rec_triangle [CN rec_add [Id 2 0, Id 2 1]], Id 2 0]"
-
-definition 
-  "rec_pdec1 = CN rec_minus [Id 1 0, CN rec_triangle [CN rec_max_triangle [Id 1 0]]]" 
-
-definition 
-  "rec_pdec2 = CN rec_minus [CN rec_max_triangle [Id 1 0], CN rec_pdec1 [Id 1 0]]" 
-
-lemma pdec1_lemma [simp]:
-  "rec_eval rec_pdec1 [z] = pdec1 z"
-by (simp add: rec_pdec1_def fst_prod_decode)
-
-lemma pdec2_lemma [simp]:
-  "rec_eval rec_pdec2 [z] = pdec2 z"
-by (simp add: rec_pdec2_def snd_prod_decode)
-
-lemma penc_lemma [simp]:
-  "rec_eval rec_penc [m, n] = penc m n"
-by (simp add: rec_penc_def prod_encode_def)
-
-fun 
-  lenc :: "nat list \<Rightarrow> nat" 
-where
-  "lenc [] = 0"
-| "lenc (x # xs) = penc (Suc x) (lenc xs)"
-
-fun
-  ldec :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "ldec z 0 = (pdec1 z) - 1"
-| "ldec z (Suc n) = ldec (pdec2 z) n"
-
-lemma pdec_zero_simps [simp]:
-  "pdec1 0 = 0" 
-  "pdec2 0 = 0"
-by (simp_all add: prod_decode_def prod_decode_aux.simps)
-
-lemma w:
-  "ldec 0 n = 0"
-by (induct n) (simp_all add: prod_decode_def prod_decode_aux.simps)
-
-lemma list_encode_inverse: 
-  "ldec (lenc xs) n = (if n < length xs then xs ! n else 0)"
-apply(induct xs arbitrary: n rule: lenc.induct) 
-apply(simp_all add: w)
-apply(case_tac n)
-apply(simp_all)
-done
-
-fun within :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
-  "within z 0 = (0 < z)"
-| "within z (Suc n) = within (pdec2 z) n"
-    
-
-section {* Discrete Logarithms *}
-
-definition
-  "rec_lg = 
-     (let calc = CN rec_not [CN rec_le [CN rec_power [Id 3 2, Id 3 0], Id 3 1]] in
-      let max  = CN (rec_max2 calc) [Id 2 0, Id 2 0, Id 2 1] in
-      let cond = CN rec_conj [CN rec_less [constn 1, Id 2 0], CN rec_less [constn 1, Id 2 1]] 
-      in CN rec_ifz [cond, Z, max])"
-
-definition
-  "Lg x y = (if 1 < x \<and> 1 < y then BMax_rec (\<lambda>u. y ^ u \<le> x) x else 0)"
-
-lemma lg_lemma [simp]:
-  "rec_eval rec_lg [x, y] = Lg x y"
-by (simp add: rec_lg_def Lg_def Let_def)
-
-definition
-  "Lo x y = (if 1 < x \<and> 1 < y then BMax_rec (\<lambda>u. x mod (y ^ u) = 0) x else 0)"
-
-definition
-  "rec_lo = 
-     (let calc = CN rec_noteq [CN rec_mod [Id 3 1, CN rec_power [Id 3 2, Id 3 0]], Z] in
-      let max  = CN (rec_max2 calc) [Id 2 0, Id 2 0, Id 2 1] in
-      let cond = CN rec_conj [CN rec_less [constn 1, Id 2 0], CN rec_less [constn 1, Id 2 1]] 
-      in CN rec_ifz [cond, Z, max])"
-
-lemma lo_lemma [simp]:
-  "rec_eval rec_lo [x, y] = Lo x y"
-by (simp add: rec_lo_def Lo_def Let_def)
-
-section {* NextPrime number function *}
-
-text {* 
-  @{text "NextPrime x"} returns the first prime number after @{text "x"};
-  @{text "Pi i"} returns the i-th prime number. *}
-
-definition 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 NextPrime_def)
-
-lemma NextPrime_simps [simp]:
-  shows "NextPrime 2 = 3"
-  and   "NextPrime 3 = 5"
-apply(simp_all add: NextPrime_def)
-apply(rule Least_equality)
-apply(auto)
-apply(eval)
-apply(rule Least_equality)
-apply(auto)
-apply(eval)
-apply(case_tac "y = 4")
-apply(auto)
-done
-
-fun Pi :: "nat \<Rightarrow> nat"
-  where
-  "Pi 0 = 2" |
-  "Pi (Suc x) = NextPrime (Pi x)"
-
-lemma Pi_simps [simp]:
-  shows "Pi 1 = 3"
-  and   "Pi 2 = 5"
-using NextPrime_simps
-by(simp_all add: numeral_eq_Suc One_nat_def)
-
-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)
-
-end
-