theory Recs

  imports Main

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>

  sum f {..<n} + sum f {n..p} = sum f {..p::nat}"

  apply(subst sum.union_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 sum_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"

text {* Abbreviations for calculating the arity of the constructors *}

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"

text {* the evaluation function and termination relation *}

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) (x # xs)"

| 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 {* Arithmetic Functions *}

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_aux = PR (constn 1) (CN rec_mult [CN S [Id 3 0], Id 3 1])"

definition

  "rec_fact = CN rec_fact_aux [Id 1 0, Id 1 0]"

definition 

  "rec_predi = CN (PR Z (Id 3 0)) [Id 1 0, Id 1 0]"

definition 

  "rec_minus = rec_swap (PR (Id 1 0) (CN rec_predi [Id 3 1]))"

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_aux_lemma [simp]: 

  "rec_eval rec_fact_aux [x, y] = fact x"

by (induct x) (simp_all add: rec_fact_aux_def)

lemma fact_lemma [simp]: 

  "rec_eval rec_fact [x] = fact x"

by (simp add: rec_fact_def)

lemma pred_lemma [simp]: 

  "rec_eval rec_predi [x] =  x - 1"

by (induct x) (simp_all add: rec_predi_def)

lemma minus_lemma [simp]: 

  "rec_eval rec_minus [x, y] = x - y"

by (induct y) (simp_all add: rec_minus_def)

section {* Logical functions *}

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 [CN (constn 1) [Id 2 0], 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]"

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 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)

section {* Less and Le Relations *}

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]"

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)

section {* Summation and Product Functions *}

definition 

  "rec_sigma1 f = PR (CN f [CN Z [Id 1 0], Id 1 0]) 

                     (CN rec_add [Id 3 1, CN f [CN S [Id 3 0], Id 3 2]])"

definition 

  "rec_sigma2 f = PR (CN f [CN Z [Id 2 0], Id 2 0, Id 2 1]) 

                     (CN rec_add [Id 4 1, CN f [CN S [Id 4 0], Id 4 2, Id 4 3]])"

definition 

  "rec_accum1 f = PR (CN f [CN Z [Id 1 0], Id 1 0]) 

                     (CN rec_mult [Id 3 1, CN f [CN S [Id 3 0], Id 3 2]])"

definition 

  "rec_accum2 f = PR (CN f [CN Z [Id 2 0], Id 2 0, Id 2 1]) 

                     (CN rec_mult [Id 4 1, CN f [CN S [Id 4 0], Id 4 2, Id 4 3]])"

definition 

  "rec_accum3 f = PR (CN f [CN Z [Id 3 0], Id 3 0, Id 3 1, Id 3 2]) 

                     (CN rec_mult [Id 5 1, CN f [CN S [Id 5 0], Id 5 2, Id 5 3, Id 5 4]])"

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 accum3_lemma [simp]: 

  shows "rec_eval (rec_accum3 f) [x, y1, y2, y3] = (\<Prod> z \<le> x. (rec_eval f)  [z, y1, y2, y3])"

by (induct x) (simp_all add: rec_accum3_def)

section {* Bounded Quantifiers *}

definition

  "rec_all1 f = CN rec_sign [rec_accum1 f]"

definition

  "rec_all2 f = CN rec_sign [rec_accum2 f]"

definition

  "rec_all3 f = CN rec_sign [rec_accum3 f]"

definition

  "rec_all1_less f = (let cond1 = CN rec_eq [Id 3 0, Id 3 1] in

                      let cond2 = CN f [Id 3 0, Id 3 2] 

                      in CN (rec_all2 (CN rec_disj [cond1, cond2])) [Id 2 0, Id 2 0, Id 2 1])"

definition

  "rec_all2_less f = (let cond1 = CN rec_eq [Id 4 0, Id 4 1] in 

                      let cond2 = CN f [Id 4 0, Id 4 2, Id 4 3] in 

                      CN (rec_all3 (CN rec_disj [cond1, cond2])) [Id 3 0, Id 3 0, Id 3 1, Id 3 2])"

definition

  "rec_ex1 f = CN rec_sign [rec_sigma1 f]"

definition

  "rec_ex2 f = CN rec_sign [rec_sigma2 f]"

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 all3_lemma [simp]:

 "rec_eval (rec_all3 f) [x, y1, y2, y3] = (if (\<forall>z \<le> x. 0 < rec_eval f [z, y1, y2, y3]) then 1 else 0)"

by (simp add: rec_all3_def)

lemma all1_less_lemma [simp]:

  "rec_eval (rec_all1_less f) [x, y] = (if (\<forall>z < x. 0 < rec_eval f [z, y]) then 1 else 0)"

apply(auto simp add: Let_def rec_all1_less_def)

apply (metis nat_less_le)+

done

lemma all2_less_lemma [simp]:

  "rec_eval (rec_all2_less f) [x, y1, y2] = (if (\<forall>z < x. 0 < rec_eval f [z, y1, y2]) then 1 else 0)"

apply(auto simp add: Let_def rec_all2_less_def)

apply(metis nat_less_le)+

done

section {* Quotients *}

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)"