(* Title: HOL/ex/Primrec
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
Primitive Recursive Functions
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.
*)
(** Useful special cases of evaluation ***)
goalw thy [SC_def] "SC (x#l) = Suc x";
by (Asm_simp_tac 1);
qed "SC";
goalw thy [CONST_def] "CONST k l = k";
by (Asm_simp_tac 1);
qed "CONST";
goalw thy [PROJ_def] "PROJ(0) (x#l) = x";
by (Asm_simp_tac 1);
qed "PROJ_0";
goalw thy [COMP_def] "COMP g [f] l = g [f l]";
by (Asm_simp_tac 1);
qed "COMP_1";
goalw thy [PREC_def] "PREC f g (0#l) = f l";
by (Asm_simp_tac 1);
qed "PREC_0";
goalw thy [PREC_def] "PREC f g (Suc x # l) = g (PREC f g (x#l) # x # l)";
by (Asm_simp_tac 1);
qed "PREC_Suc";
Addsimps [SC, CONST, PROJ_0, COMP_1, PREC_0, PREC_Suc];
Addsimps ack.rules;
(*PROPERTY A 4*)
goal thy "j < ack(i,j)";
by (res_inst_tac [("u","i"),("v","j")] ack.induct 1);
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS trans_tac);
qed "less_ack2";
AddIffs [less_ack2];
(*PROPERTY A 5-, the single-step lemma*)
goal thy "ack(i,j) < ack(i, Suc(j))";
by (res_inst_tac [("u","i"),("v","j")] ack.induct 1);
by (ALLGOALS Asm_simp_tac);
qed "ack_less_ack_Suc2";
AddIffs [ack_less_ack_Suc2];
(*PROPERTY A 5, monotonicity for < *)
goal thy "j<k --> ack(i,j) < ack(i,k)";
by (res_inst_tac [("u","i"),("v","k")] ack.induct 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
qed_spec_mp "ack_less_mono2";
(*PROPERTY A 5', monotonicity for<=*)
goal thy "!!i j k. j<=k ==> ack(i,j)<=ack(i,k)";
by (full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
by (blast_tac (claset() addIs [ack_less_mono2]) 1);
qed "ack_le_mono2";
(*PROPERTY A 6*)
goal thy "ack(i, Suc(j)) <= ack(Suc(i), j)";
by (induct_tac "j" 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (claset() addIs [ack_le_mono2, less_ack2 RS Suc_leI,
le_trans]) 1);
qed "ack2_le_ack1";
AddIffs [ack2_le_ack1];
(*PROPERTY A 7-, the single-step lemma*)
goal thy "ack(i,j) < ack(Suc(i),j)";
by (blast_tac (claset() addIs [ack_less_mono2, less_le_trans]) 1);
qed "ack_less_ack_Suc1";
AddIffs [ack_less_ack_Suc1];
(*PROPERTY A 4'? Extra lemma needed for CONST case, constant functions*)
goal thy "i < ack(i,j)";
by (induct_tac "i" 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (claset() addIs [Suc_leI, le_less_trans]) 1);
qed "less_ack1";
AddIffs [less_ack1];
(*PROPERTY A 8*)
goal thy "ack(1,j) = Suc(Suc(j))";
by (induct_tac "j" 1);
by (ALLGOALS Asm_simp_tac);
qed "ack_1";
Addsimps [ack_1];
(*PROPERTY A 9*)
goal thy "ack(Suc(1),j) = Suc(Suc(Suc(j+j)))";
by (induct_tac "j" 1);
by (ALLGOALS Asm_simp_tac);
qed "ack_2";
Addsimps [ack_2];
(*PROPERTY A 7, monotonicity for < [not clear why ack_1 is now needed first!]*)
goal thy "ack(i,k) < ack(Suc(i+i'),k)";
by (res_inst_tac [("u","i"),("v","k")] ack.induct 1);
by (ALLGOALS Asm_full_simp_tac);
by (blast_tac (claset() addIs [less_trans, ack_less_mono2]) 2);
by (res_inst_tac [("u","i'"),("v","n")] ack.induct 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (claset() addIs [less_trans, ack_less_mono2]) 1);
by (blast_tac (claset() addIs [Suc_leI RS le_less_trans, ack_less_mono2]) 1);
val lemma = result();
goal thy "!!i j k. i<j ==> ack(i,k) < ack(j,k)";
by (etac less_natE 1);
by (blast_tac (claset() addSIs [lemma]) 1);
qed "ack_less_mono1";
(*PROPERTY A 7', monotonicity for<=*)
goal thy "!!i j k. i<=j ==> ack(i,k)<=ack(j,k)";
by (full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
by (blast_tac (claset() addIs [ack_less_mono1]) 1);
qed "ack_le_mono1";
(*PROPERTY A 10*)
goal thy "ack(i1, ack(i2,j)) < ack(Suc(Suc(i1+i2)), j)";
by (rtac (ack2_le_ack1 RSN (2,less_le_trans)) 1);
by (Asm_simp_tac 1);
by (rtac (le_add1 RS ack_le_mono1 RS le_less_trans) 1);
by (rtac (ack_less_mono1 RS ack_less_mono2) 1);
by (simp_tac (simpset() addsimps [le_imp_less_Suc, le_add2]) 1);
qed "ack_nest_bound";
(*PROPERTY A 11*)
goal thy "ack(i1,j) + ack(i2,j) < ack(Suc(Suc(Suc(Suc(i1+i2)))), j)";
by (res_inst_tac [("j", "ack(Suc(1), ack(i1 + i2, j))")] less_trans 1);
by (Asm_simp_tac 1);
by (rtac (ack_nest_bound RS less_le_trans) 2);
by (Asm_simp_tac 2);
by (blast_tac (claset() addSIs [le_add1, le_add2]
addIs [le_imp_less_Suc, ack_le_mono1, le_SucI,
add_le_mono]) 1);
qed "ack_add_bound";
(*PROPERTY A 12. Article uses existential quantifier but the ALF proof
used k+4. Quantified version must be nested EX k'. ALL i,j... *)
goal thy "!!i j k. i < ack(k,j) ==> i+j < ack(Suc(Suc(Suc(Suc(k)))), j)";
by (res_inst_tac [("j", "ack(k,j) + ack(0,j)")] less_trans 1);
by (rtac (ack_add_bound RS less_le_trans) 2);
by (Asm_simp_tac 2);
by (REPEAT (ares_tac ([add_less_mono, less_ack2]) 1));
qed "ack_add_bound2";
(*** Inductive definition of the PR functions ***)
(*** MAIN RESULT ***)
goalw thy [SC_def] "SC l < ack(1, list_add l)";
by (induct_tac "l" 1);
by (ALLGOALS (simp_tac (simpset() addsimps [le_add1, le_imp_less_Suc])));
qed "SC_case";
goal thy "CONST k l < ack(k, list_add l)";
by (Simp_tac 1);
qed "CONST_case";
goalw thy [PROJ_def] "ALL i. PROJ i l < ack(0, list_add l)";
by (Simp_tac 1);
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
by (rtac allI 1);
by (exhaust_tac "i" 1);
by (asm_simp_tac (simpset() addsimps [le_add1, le_imp_less_Suc]) 1);
by (Asm_simp_tac 1);
by (blast_tac (claset() addIs [less_le_trans]
addSIs [le_add2]) 1);
qed_spec_mp "PROJ_case";
(** COMP case **)
goal thy
"!!fs. fs : lists(PRIMREC Int {f. EX kf. ALL l. f l < ack(kf, list_add l)}) \
\ ==> EX k. ALL l. list_add (map(%f. f l) fs) < ack(k, list_add l)";
by (etac lists.induct 1);
by (res_inst_tac [("x","0")] exI 1 THEN Asm_simp_tac 1);
by (safe_tac (claset()));
by (Asm_simp_tac 1);
by (blast_tac (claset() addIs [add_less_mono, ack_add_bound, less_trans]) 1);
qed "COMP_map_lemma";
goalw thy [COMP_def]
"!!g. [| ALL l. g l < ack(kg, list_add l); \
\ fs: lists(PRIMREC Int {f. EX kf. ALL l. f l < ack(kf, list_add l)}) \
\ |] ==> EX k. ALL l. COMP g fs l < ack(k, list_add l)";
by (forward_tac [impOfSubs (Int_lower1 RS lists_mono)] 1);
by (etac (COMP_map_lemma RS exE) 1);
by (rtac exI 1);
by (rtac allI 1);
by (REPEAT (dtac spec 1));
by (etac less_trans 1);
by (blast_tac (claset() addIs [ack_less_mono2, ack_nest_bound, less_trans]) 1);
qed "COMP_case";
(** PREC case **)
goalw thy [PREC_def]
"!!f g. [| ALL l. f l + list_add l < ack(kf, list_add l); \
\ ALL 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)";
by (exhaust_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (claset() addIs [less_trans]) 1);
by (etac ssubst 1); (*get rid of the needless assumption*)
by (induct_tac "a" 1);
by (ALLGOALS Asm_simp_tac);
(*base case*)
by (blast_tac (claset() addIs [le_add1 RS le_imp_less_Suc RS ack_less_mono1,
less_trans]) 1);
(*induction step*)
by (rtac (Suc_leI RS le_less_trans) 1);
by (rtac (le_refl RS add_le_mono RS le_less_trans) 1);
by (etac spec 2);
by (asm_simp_tac (simpset() addsimps [le_add2]) 1);
(*final part of the simplification*)
by (Asm_simp_tac 1);
by (rtac (le_add2 RS ack_le_mono1 RS le_less_trans) 1);
by (etac ack_less_mono2 1);
qed "PREC_case_lemma";
goal thy
"!!f g. [| ALL l. f l < ack(kf, list_add l); \
\ ALL l. g l < ack(kg, list_add l) \
\ |] ==> EX k. ALL l. PREC f g l< ack(k, list_add l)";
by (rtac exI 1);
by (rtac allI 1);
by (rtac ([le_add1, PREC_case_lemma] MRS le_less_trans) 1);
by (REPEAT (blast_tac (claset() addIs [ack_add_bound2]) 1));
qed "PREC_case";
goal thy "!!f. f:PRIMREC ==> EX k. ALL l. f l < ack(k, list_add l)";
by (etac PRIMREC.induct 1);
by (ALLGOALS
(blast_tac (claset() addIs [SC_case, CONST_case, PROJ_case, COMP_case,
PREC_case])));
qed "ack_bounds_PRIMREC";
goal thy "(%l. case l of [] => 0 | x#l' => ack(x,x)) ~: PRIMREC";
by (rtac notI 1);
by (etac (ack_bounds_PRIMREC RS exE) 1);
by (rtac less_irrefl 1);
by (dres_inst_tac [("x", "[x]")] spec 1);
by (Asm_full_simp_tac 1);
qed "ack_not_PRIMREC";