(* Title: HOL/ex/Primrec.thy ID: $Id: Primrec.thy,v 1.14 2005/07/06 18:00:56 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1997 University of Cambridge Primitive Recursive Functions. Demonstrates recursive definitions, the TFL package. *) 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 *} consts ack :: "nat * nat => nat" recdef ack "less_than <*lex*> less_than" "ack (0, n) = Suc n" "ack (Suc m, 0) = ack (m, 1)" "ack (Suc m, Suc n) = ack (m, ack (Suc m, n))" consts list_add :: "nat list => nat" primrec "list_add [] = 0" "list_add (m # ms) = m + list_add ms" consts zeroHd :: "nat list => nat" primrec "zeroHd [] = 0" "zeroHd (m # ms) = m" text {* The set of primitive recursive functions of type @{typ "nat list => nat"}. *} constdefs SC :: "nat list => nat" "SC l == Suc (zeroHd l)" CONST :: "nat => nat list => nat" "CONST k l == k" PROJ :: "nat => nat list => nat" "PROJ i l == zeroHd (drop i l)" COMP :: "(nat list => nat) => (nat list => nat) list => nat list => nat" "COMP g fs l == g (map (λf. f l) fs)" PREC :: "(nat list => nat) => (nat list => nat) => nat list => nat" "PREC f g l == case l of [] => 0 | x # l' => nat_rec (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}! *} consts PRIMREC :: "(nat list => nat) set" inductive PRIMREC intros SC: "SC ∈ PRIMREC" CONST: "CONST k ∈ PRIMREC" PROJ: "PROJ i ∈ PRIMREC" COMP: "g ∈ PRIMREC ==> fs ∈ lists PRIMREC ==> COMP g fs ∈ PRIMREC" PREC: "f ∈ PRIMREC ==> g ∈ PRIMREC ==> PREC f g ∈ PRIMREC" text {* Useful special cases of evaluation *} lemma SC [simp]: "SC (x # l) = Suc x" apply (simp add: SC_def) done lemma CONST [simp]: "CONST k l = k" apply (simp add: CONST_def) done lemma PROJ_0 [simp]: "PROJ 0 (x # l) = x" apply (simp add: PROJ_def) done lemma COMP_1 [simp]: "COMP g [f] l = g [f l]" apply (simp add: COMP_def) done lemma PREC_0 [simp]: "PREC f g (0 # l) = f l" apply (simp add: PREC_def) done lemma PREC_Suc [simp]: "PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)" apply (simp add: PREC_def) done text {* PROPERTY A 4 *} lemma less_ack2 [iff]: "j < ack (i, j)" apply (induct i j rule: ack.induct) apply simp_all done text {* PROPERTY A 5-, the single-step lemma *} lemma ack_less_ack_Suc2 [iff]: "ack(i, j) < ack (i, Suc j)" apply (induct i j rule: ack.induct) apply simp_all done text {* PROPERTY A 5, monotonicity for @{text "<"} *} lemma ack_less_mono2: "j < k ==> ack (i, j) < ack (i, k)" apply (induct i k rule: ack.induct) apply simp_all apply (blast elim!: less_SucE intro: less_trans) done 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)" apply (induct j) apply simp_all apply (blast intro: ack_le_mono2 less_ack2 [THEN Suc_leI] le_trans) done text {* PROPERTY A 7-, the single-step lemma *} lemma ack_less_ack_Suc1 [iff]: "ack (i, j) < ack (Suc i, j)" apply (blast intro: ack_less_mono2 less_le_trans) done text {* PROPERTY A 4'? Extra lemma needed for @{term CONST} 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" apply (induct j) apply simp_all done 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" apply (induct j) apply simp_all done 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)" apply (induct i k rule: ack.induct) apply simp_all prefer 2 apply (blast intro: less_trans ack_less_mono2) apply (induct_tac i' n rule: ack.induct) apply simp_all apply (blast intro: Suc_leI [THEN le_less_trans] ack_less_mono2) done 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_tac j = "ack (Suc (Suc 0), ack (i1 + i2, j))" in less_trans) 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_tac j = "ack (k, j) + ack (0, j)" in less_trans) prefer 2 apply (rule ack_add_bound [THEN less_le_trans]) apply simp apply (rule add_less_mono less_ack2 | assumption)+ done text {* Inductive definition of the @{term PR} functions *} text {* MAIN RESULT *} lemma SC_case: "SC l < ack (1, list_add l)" apply (unfold SC_def) apply (induct l) apply (simp_all add: le_add1 le_imp_less_Suc) done lemma CONST_case: "CONST k l < ack (k, list_add l)" apply simp done lemma PROJ_case [rule_format]: "∀i. PROJ i l < ack (0, list_add l)" apply (simp add: PROJ_def) apply (induct l) apply simp_all apply (rule allI) apply (case_tac i) apply (simp (no_asm_simp) add: le_add1 le_imp_less_Suc) apply (simp (no_asm_simp)) apply (blast intro: less_le_trans intro!: le_add2) done text {* @{term COMP} case *} lemma COMP_map_aux: "fs ∈ lists (PRIMREC ∩ {f. ∃kf. ∀l. f l < ack (kf, list_add l)}) ==> ∃k. ∀l. list_add (map (λf. f l) fs) < ack (k, list_add l)" apply (erule lists.induct) apply (rule_tac x = 0 in exI) apply simp apply safe apply simp apply (rule exI) apply (blast intro: add_less_mono ack_add_bound less_trans) done lemma COMP_case: "∀l. g l < ack (kg, list_add l) ==> fs ∈ lists(PRIMREC Int {f. ∃kf. ∀l. f l < ack(kf, list_add l)}) ==> ∃k. ∀l. COMP g fs l < ack(k, list_add l)" apply (unfold COMP_def) apply (frule Int_lower1 [THEN lists_mono, THEN subsetD]) --{*Now, if meson tolerated map, we could finish with @{text "(drule COMP_map_aux, meson ack_less_mono2 ack_nest_bound less_trans)"} *} apply (erule COMP_map_aux [THEN exE]) apply (rule exI) apply (rule allI) apply (drule spec)+ apply (erule less_trans) apply (blast intro: ack_less_mono2 ack_nest_bound less_trans) done text {* @{term PREC} case *} lemma PREC_case_aux: "∀l. f l + list_add l < ack (kf, list_add l) ==> ∀l. g l + list_add l < ack (kg, list_add l) ==> PREC f g l + list_add l < ack (Suc (kf + kg), list_add 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, list_add l) ==> ∀l. g l < ack (kg, list_add l) ==> ∃k. ∀l. PREC f g l < ack (k, list_add l)" apply (rule exI) apply (rule allI) apply (rule le_less_trans [OF le_add1 PREC_case_aux]) apply (blast intro: ack_add_bound2)+ done lemma ack_bounds_PRIMREC: "f ∈ PRIMREC ==> ∃k. ∀l. f l < ack (k, list_add l)" apply (erule PRIMREC.induct) apply (blast intro: SC_case CONST_case PROJ_case COMP_case PREC_case)+ done lemma ack_not_PRIMREC: "(λl. case l of [] => 0 | x # l' => ack (x, x)) ∉ PRIMREC" apply (rule notI) apply (erule ack_bounds_PRIMREC [THEN exE]) apply (rule less_irrefl) apply (drule_tac x = "[x]" in spec) apply simp done end
lemma SC:
SC (x # l) = Suc x
lemma CONST:
CONST k l = k
lemma PROJ_0:
PROJ 0 (x # l) = x
lemma COMP_1:
COMP g [f] l = g [f l]
lemma PREC_0:
PREC f g (0 # l) = f l
lemma PREC_Suc:
PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)
lemma less_ack2:
j < ack (i, j)
lemma ack_less_ack_Suc2:
ack (i, j) < ack (i, Suc j)
lemma ack_less_mono2:
j < k ==> ack (i, j) < ack (i, k)
lemma ack_le_mono2:
j ≤ k ==> ack (i, j) ≤ ack (i, k)
lemma ack2_le_ack1:
ack (i, Suc j) ≤ ack (Suc i, j)
lemma ack_less_ack_Suc1:
ack (i, j) < ack (Suc i, j)
lemma less_ack1:
i < ack (i, j)
lemma ack_1:
ack (Suc 0, j) = j + 2
lemma ack_2:
ack (Suc (Suc 0), j) = 2 * j + 3
lemma ack_less_mono1_aux:
ack (i, k) < ack (Suc (i + i'), k)
lemma ack_less_mono1:
i < j ==> ack (i, k) < ack (j, k)
lemma ack_le_mono1:
i ≤ j ==> ack (i, k) ≤ ack (j, k)
lemma ack_nest_bound:
ack (i1.0, ack (i2.0, j)) < ack (2 + (i1.0 + i2.0), j)
lemma ack_add_bound:
ack (i1.0, j) + ack (i2.0, j) < ack (4 + (i1.0 + i2.0), j)
lemma ack_add_bound2:
i < ack (k, j) ==> i + j < ack (4 + k, j)
lemma SC_case:
SC l < ack (1, list_add l)
lemma CONST_case:
CONST k l < ack (k, list_add l)
lemma PROJ_case:
PROJ i l < ack (0, list_add l)
lemma COMP_map_aux:
fs ∈ lists (PRIMREC ∩ {f. ∃kf. ∀l. f l < ack (kf, list_add l)}) ==> ∃k. ∀l. list_add (map (%f. f l) fs) < ack (k, list_add l)
lemma COMP_case:
[| ∀l. g l < ack (kg, list_add l); fs ∈ lists (PRIMREC ∩ {f. ∃kf. ∀l. f l < ack (kf, list_add l)}) |] ==> ∃k. ∀l. COMP g fs l < ack (k, list_add l)
lemma PREC_case_aux:
[| ∀l. f l + list_add l < ack (kf, list_add l); ∀l. g l + list_add l < ack (kg, list_add l) |] ==> PREC f g l + list_add l < ack (Suc (kf + kg), list_add l)
lemma PREC_case:
[| ∀l. f l < ack (kf, list_add l); ∀l. g l < ack (kg, list_add l) |] ==> ∃k. ∀l. PREC f g l < ack (k, list_add l)
lemma ack_bounds_PRIMREC:
f ∈ PRIMREC ==> ∃k. ∀l. f l < ack (k, list_add l)
lemma ack_not_PRIMREC:
list_case 0 (%x l'. ack (x, x)) ∉ PRIMREC