Theory Primrec

theory Primrec
imports Main
(*  Title:      HOL/ex/Primrec.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge

Ackermann's Function and the
Primitive Recursive Functions.
*)

header {* Primitive Recursive Functions *}

theory Primrec imports Main begin

text {*
  Proof adopted from

  Nora Szasz, A Machine Checked Proof that Ackermann's Function is not
  Primitive Recursive, In: Huet \& Plotkin, eds., Logical Environments
  (CUP, 1993), 317-338.

  See also E. Mendelson, Introduction to Mathematical Logic.  (Van
  Nostrand, 1964), page 250, exercise 11.
  \medskip
*}


subsection{* Ackermann's Function *}

fun ack :: "nat => nat => nat" where
"ack 0 n =  Suc n" |
"ack (Suc m) 0 = ack m 1" |
"ack (Suc m) (Suc n) = ack m (ack (Suc m) n)"


text {* PROPERTY A 4 *}

lemma less_ack2 [iff]: "j < ack i j"
by (induct i j rule: ack.induct) simp_all


text {* PROPERTY A 5-, the single-step lemma *}

lemma ack_less_ack_Suc2 [iff]: "ack i j < ack i (Suc j)"
by (induct i j rule: ack.induct) simp_all


text {* PROPERTY A 5, monotonicity for @{text "<"} *}

lemma ack_less_mono2: "j < k ==> ack i j < ack i k"
using lift_Suc_mono_less[where f = "ack i"]
by (metis ack_less_ack_Suc2)


text {* PROPERTY A 5', monotonicity for @{text ≤} *}

lemma ack_le_mono2: "j ≤ k ==> ack i j ≤ ack i k"
apply (simp add: order_le_less)
apply (blast intro: ack_less_mono2)
done


text {* PROPERTY A 6 *}

lemma ack2_le_ack1 [iff]: "ack i (Suc j) ≤ ack (Suc i) j"
proof (induct j)
  case 0 show ?case by simp
next
  case (Suc j) show ?case 
    by (auto intro!: ack_le_mono2)
      (metis Suc Suc_leI Suc_lessI less_ack2 linorder_not_less)
qed


text {* PROPERTY A 7-, the single-step lemma *}

lemma ack_less_ack_Suc1 [iff]: "ack i j < ack (Suc i) j"
by (blast intro: ack_less_mono2 less_le_trans)


text {* PROPERTY A 4'? Extra lemma needed for @{term CONSTANT} case, constant functions *}

lemma less_ack1 [iff]: "i < ack i j"
apply (induct i)
 apply simp_all
apply (blast intro: Suc_leI le_less_trans)
done


text {* PROPERTY A 8 *}

lemma ack_1 [simp]: "ack (Suc 0) j = j + 2"
by (induct j) simp_all


text {* PROPERTY A 9.  The unary @{text 1} and @{text 2} in @{term
  ack} is essential for the rewriting. *}

lemma ack_2 [simp]: "ack (Suc (Suc 0)) j = 2 * j + 3"
by (induct j) simp_all


text {* PROPERTY A 7, monotonicity for @{text "<"} [not clear why
  @{thm [source] ack_1} is now needed first!] *}

lemma ack_less_mono1_aux: "ack i k < ack (Suc (i +i')) k"
proof (induct i k rule: ack.induct)
  case (1 n) show ?case
    by (simp, metis ack_less_ack_Suc1 less_ack2 less_trans_Suc) 
next
  case (2 m) thus ?case by simp
next
  case (3 m n) thus ?case
    by (simp, blast intro: less_trans ack_less_mono2)
qed

lemma ack_less_mono1: "i < j ==> ack i k < ack j k"
apply (drule less_imp_Suc_add)
apply (blast intro!: ack_less_mono1_aux)
done


text {* PROPERTY A 7', monotonicity for @{text "≤"} *}

lemma ack_le_mono1: "i ≤ j ==> ack i k ≤ ack j k"
apply (simp add: order_le_less)
apply (blast intro: ack_less_mono1)
done


text {* PROPERTY A 10 *}

lemma ack_nest_bound: "ack i1 (ack i2 j) < ack (2 + (i1 + i2)) j"
apply (simp add: numerals)
apply (rule ack2_le_ack1 [THEN [2] less_le_trans])
apply simp
apply (rule le_add1 [THEN ack_le_mono1, THEN le_less_trans])
apply (rule ack_less_mono1 [THEN ack_less_mono2])
apply (simp add: le_imp_less_Suc le_add2)
done


text {* PROPERTY A 11 *}

lemma ack_add_bound: "ack i1 j + ack i2 j < ack (4 + (i1 + i2)) j"
apply (rule less_trans [of _ "ack (Suc (Suc 0)) (ack (i1 + i2) j)"])
 prefer 2
 apply (rule ack_nest_bound [THEN less_le_trans])
 apply (simp add: Suc3_eq_add_3)
apply simp
apply (cut_tac i = i1 and m1 = i2 and k = j in le_add1 [THEN ack_le_mono1])
apply (cut_tac i = "i2" and m1 = i1 and k = j in le_add2 [THEN ack_le_mono1])
apply auto
done


text {* PROPERTY A 12.  Article uses existential quantifier but the ALF proof
  used @{text "k + 4"}.  Quantified version must be nested @{text
  "∃k'. ∀i j. ..."} *}

lemma ack_add_bound2: "i < ack k j ==> i + j < ack (4 + k) j"
apply (rule less_trans [of _ "ack k j + ack 0 j"])
 apply (blast intro: add_less_mono) 
apply (rule ack_add_bound [THEN less_le_trans])
apply simp
done


subsection{*Primitive Recursive Functions*}

primrec hd0 :: "nat list => nat" where
"hd0 [] = 0" |
"hd0 (m # ms) = m"


text {* Inductive definition of the set of primitive recursive functions of type @{typ "nat list => nat"}. *}

definition SC :: "nat list => nat" where
"SC l = Suc (hd0 l)"

definition CONSTANT :: "nat => nat list => nat" where
"CONSTANT k l = k"

definition PROJ :: "nat => nat list => nat" where
"PROJ i l = hd0 (drop i l)"

definition
COMP :: "(nat list => nat) => (nat list => nat) list => nat list => nat"
where "COMP g fs l = g (map (λf. f l) fs)"

definition PREC :: "(nat list => nat) => (nat list => nat) => nat list => nat"
where
  "PREC f g l =
    (case l of
      [] => 0
    | x # l' => rec_nat (f l') (λy r. g (r # y # l')) x)"
  -- {* Note that @{term g} is applied first to @{term "PREC f g y"} and then to @{term y}! *}

inductive PRIMREC :: "(nat list => nat) => bool" where
SC: "PRIMREC SC" |
CONSTANT: "PRIMREC (CONSTANT k)" |
PROJ: "PRIMREC (PROJ i)" |
COMP: "PRIMREC g ==> ∀f ∈ set fs. PRIMREC f ==> PRIMREC (COMP g fs)" |
PREC: "PRIMREC f ==> PRIMREC g ==> PRIMREC (PREC f g)"


text {* Useful special cases of evaluation *}

lemma SC [simp]: "SC (x # l) = Suc x"
by (simp add: SC_def)

lemma CONSTANT [simp]: "CONSTANT k l = k"
by (simp add: CONSTANT_def)

lemma PROJ_0 [simp]: "PROJ 0 (x # l) = x"
by (simp add: PROJ_def)

lemma COMP_1 [simp]: "COMP g [f] l = g [f l]"
by (simp add: COMP_def)

lemma PREC_0 [simp]: "PREC f g (0 # l) = f l"
by (simp add: PREC_def)

lemma PREC_Suc [simp]: "PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)"
by (simp add: PREC_def)


text {* MAIN RESULT *}

lemma SC_case: "SC l < ack 1 (listsum l)"
apply (unfold SC_def)
apply (induct l)
apply (simp_all add: le_add1 le_imp_less_Suc)
done

lemma CONSTANT_case: "CONSTANT k l < ack k (listsum l)"
by simp

lemma PROJ_case: "PROJ i l < ack 0 (listsum l)"
apply (simp add: PROJ_def)
apply (induct l arbitrary:i)
 apply (auto simp add: drop_Cons split: nat.split)
apply (blast intro: less_le_trans le_add2)
done


text {* @{term COMP} case *}

lemma COMP_map_aux: "∀f ∈ set fs. PRIMREC f ∧ (∃kf. ∀l. f l < ack kf (listsum l))
  ==> ∃k. ∀l. listsum (map (λf. f l) fs) < ack k (listsum l)"
apply (induct fs)
 apply (rule_tac x = 0 in exI)
 apply simp
apply simp
apply (blast intro: add_less_mono ack_add_bound less_trans)
done

lemma COMP_case:
  "∀l. g l < ack kg (listsum l) ==>
  ∀f ∈ set fs. PRIMREC f ∧ (∃kf. ∀l. f l < ack kf (listsum l))
  ==> ∃k. ∀l. COMP g fs  l < ack k (listsum l)"
apply (unfold COMP_def)
apply (drule COMP_map_aux)
apply (meson ack_less_mono2 ack_nest_bound less_trans)
done


text {* @{term PREC} case *}

lemma PREC_case_aux:
  "∀l. f l + listsum l < ack kf (listsum l) ==>
    ∀l. g l + listsum l < ack kg (listsum l) ==>
    PREC f g l + listsum l < ack (Suc (kf + kg)) (listsum l)"
apply (unfold PREC_def)
apply (case_tac l)
 apply simp_all
 apply (blast intro: less_trans)
apply (erule ssubst) -- {* get rid of the needless assumption *}
apply (induct_tac a)
 apply simp_all
 txt {* base case *}
 apply (blast intro: le_add1 [THEN le_imp_less_Suc, THEN ack_less_mono1] less_trans)
txt {* induction step *}
apply (rule Suc_leI [THEN le_less_trans])
 apply (rule le_refl [THEN add_le_mono, THEN le_less_trans])
  prefer 2
  apply (erule spec)
 apply (simp add: le_add2)
txt {* final part of the simplification *}
apply simp
apply (rule le_add2 [THEN ack_le_mono1, THEN le_less_trans])
apply (erule ack_less_mono2)
done

lemma PREC_case:
  "∀l. f l < ack kf (listsum l) ==>
    ∀l. g l < ack kg (listsum l) ==>
    ∃k. ∀l. PREC f g l < ack k (listsum l)"
by (metis le_less_trans [OF le_add1 PREC_case_aux] ack_add_bound2)

lemma ack_bounds_PRIMREC: "PRIMREC f ==> ∃k. ∀l. f l < ack k (listsum l)"
apply (erule PRIMREC.induct)
    apply (blast intro: SC_case CONSTANT_case PROJ_case COMP_case PREC_case)+
done

theorem ack_not_PRIMREC:
  "¬ PRIMREC (λl. case l of [] => 0 | x # l' => ack x x)"
apply (rule notI)
apply (erule ack_bounds_PRIMREC [THEN exE])
apply (rule less_irrefl [THEN notE])
apply (drule_tac x = "[x]" in spec)
apply simp
done

end