# Theory Primrec

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

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

section ‹Primitive Recursive Functions›

theory Primrec imports Main begin

text ‹

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

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 ‹<››

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 ‹≤››

lemma ack_le_mono2: "j ≤ k ==> ack i j ≤ ack i k"
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 ‹1› and ‹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 ‹<› [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 (blast intro!: ack_less_mono1_aux)
done

text ‹PROPERTY A 7', monotonicity for ‹≤››

lemma ack_le_mono1: "i ≤ j ==> ack i k ≤ ack j k"
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
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])
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
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 ‹k + 4›.  Quantified version must be nested ‹∃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 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"

lemma CONSTANT [simp]: "CONSTANT k l = k"

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

lemma COMP_1 [simp]: "COMP g [f] l = g [f l]"

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

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

text ‹MAIN RESULT›

lemma SC_case: "SC l < ack 1 (sum_list l)"
apply (unfold SC_def)
apply (induct l)
done

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

lemma PROJ_case: "PROJ i l < ack 0 (sum_list l)"
apply (induct l arbitrary:i)
apply (auto simp add: drop_Cons split: nat.split)
done

text ‹@{term COMP} case›

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

lemma COMP_case:
"∀l. g l < ack kg (sum_list l) ==>
∀f ∈ set fs. PRIMREC f ∧ (∃kf. ∀l. f l < ack kf (sum_list l))
==> ∃k. ∀l. COMP g fs  l < ack k (sum_list 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 + sum_list l < ack kf (sum_list l) ==>
∀l. g l + sum_list l < ack kg (sum_list l) ==>
PREC f g l + sum_list l < ack (Suc (kf + kg)) (sum_list 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)
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 (sum_list l) ==>
∀l. g l < ack kg (sum_list l) ==>
∃k. ∀l. PREC f g l < ack k (sum_list l)"

lemma ack_bounds_PRIMREC: "PRIMREC f ==> ∃k. ∀l. f l < ack k (sum_list 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
```