(* Title: ZF/ex/Primrec
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 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.
*)
open Primrec;
val pr_typechecks =
nat_typechecks @ list.intrs @
[lam_type, list_case_type, drop_type, map_type, apply_type, rec_type];
(** Useful special cases of evaluation ***)
val pr_ss = arith_ss
addsimps list.case_eqns
addsimps [list_rec_Nil, list_rec_Cons,
drop_0, drop_Nil, drop_succ_Cons,
map_Nil, map_Cons]
setsolver (type_auto_tac pr_typechecks);
goalw Primrec.thy [SC_def]
"!!x l. [| x:nat; l: list(nat) |] ==> SC ` (Cons(x,l)) = succ(x)";
by (asm_simp_tac pr_ss 1);
qed "SC";
goalw Primrec.thy [CONST_def]
"!!l. [| l: list(nat) |] ==> CONST(k) ` l = k";
by (asm_simp_tac pr_ss 1);
qed "CONST";
goalw Primrec.thy [PROJ_def]
"!!l. [| x: nat; l: list(nat) |] ==> PROJ(0) ` (Cons(x,l)) = x";
by (asm_simp_tac pr_ss 1);
qed "PROJ_0";
goalw Primrec.thy [COMP_def]
"!!l. [| l: list(nat) |] ==> COMP(g,[f]) ` l = g` [f`l]";
by (asm_simp_tac pr_ss 1);
qed "COMP_1";
goalw Primrec.thy [PREC_def]
"!!l. l: list(nat) ==> PREC(f,g) ` (Cons(0,l)) = f`l";
by (asm_simp_tac pr_ss 1);
qed "PREC_0";
goalw Primrec.thy [PREC_def]
"!!l. [| x:nat; l: list(nat) |] ==> \
\ PREC(f,g) ` (Cons(succ(x),l)) = \
\ g ` Cons(PREC(f,g)`(Cons(x,l)), Cons(x,l))";
by (asm_simp_tac pr_ss 1);
qed "PREC_succ";
(*** Inductive definition of the PR functions ***)
(* c: primrec ==> c: list(nat) -> nat *)
val primrec_into_fun = primrec.dom_subset RS subsetD;
val pr_ss = pr_ss
setsolver (type_auto_tac ([primrec_into_fun] @
pr_typechecks @ primrec.intrs));
goalw Primrec.thy [ACK_def] "!!i. i:nat ==> ACK(i): primrec";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac pr_ss));
qed "ACK_in_primrec";
val ack_typechecks =
[ACK_in_primrec, primrec_into_fun RS apply_type,
add_type, list_add_type, nat_into_Ord] @
nat_typechecks @ list.intrs @ primrec.intrs;
(*strict typechecking for the Ackermann proof; instantiates no vars*)
fun tc_tac rls =
REPEAT
(SOMEGOAL (test_assume_tac ORELSE' match_tac (rls @ ack_typechecks)));
goal Primrec.thy "!!i j. [| i:nat; j:nat |] ==> ack(i,j): nat";
by (tc_tac []);
qed "ack_type";
(** Ackermann's function cases **)
(*PROPERTY A 1*)
goalw Primrec.thy [ACK_def] "!!j. j:nat ==> ack(0,j) = succ(j)";
by (asm_simp_tac (pr_ss addsimps [SC]) 1);
qed "ack_0";
(*PROPERTY A 2*)
goalw Primrec.thy [ACK_def] "ack(succ(i), 0) = ack(i,1)";
by (asm_simp_tac (pr_ss addsimps [CONST,PREC_0]) 1);
qed "ack_succ_0";
(*PROPERTY A 3*)
(*Could be proved in Primrec0, like the previous two cases, but using
primrec_into_fun makes type-checking easier!*)
goalw Primrec.thy [ACK_def]
"!!i j. [| i:nat; j:nat |] ==> \
\ ack(succ(i), succ(j)) = ack(i, ack(succ(i), j))";
by (asm_simp_tac (pr_ss addsimps [CONST,PREC_succ,COMP_1,PROJ_0]) 1);
qed "ack_succ_succ";
val ack_ss =
pr_ss addsimps [ack_0, ack_succ_0, ack_succ_succ,
ack_type, nat_into_Ord];
(*PROPERTY A 4*)
goal Primrec.thy "!!i. i:nat ==> ALL j:nat. j < ack(i,j)";
by (etac nat_induct 1);
by (asm_simp_tac ack_ss 1);
by (rtac ballI 1);
by (eres_inst_tac [("n","j")] nat_induct 1);
by (DO_GOAL [rtac (nat_0I RS nat_0_le RS lt_trans),
asm_simp_tac ack_ss] 1);
by (DO_GOAL [etac (succ_leI RS lt_trans1),
asm_simp_tac ack_ss] 1);
qed "lt_ack2_lemma";
bind_thm ("lt_ack2", (lt_ack2_lemma RS bspec));
(*PROPERTY A 5-, the single-step lemma*)
goal Primrec.thy "!!i j. [| i:nat; j:nat |] ==> ack(i,j) < ack(i, succ(j))";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac (ack_ss addsimps [lt_ack2])));
qed "ack_lt_ack_succ2";
(*PROPERTY A 5, monotonicity for < *)
goal Primrec.thy "!!i j k. [| j<k; i:nat; k:nat |] ==> ack(i,j) < ack(i,k)";
by (forward_tac [lt_nat_in_nat] 1 THEN assume_tac 1);
by (etac succ_lt_induct 1);
by (assume_tac 1);
by (rtac lt_trans 2);
by (REPEAT (ares_tac ([ack_lt_ack_succ2, ack_type] @ pr_typechecks) 1));
qed "ack_lt_mono2";
(*PROPERTY A 5', monotonicity for le *)
goal Primrec.thy
"!!i j k. [| j le k; i: nat; k:nat |] ==> ack(i,j) le ack(i,k)";
by (res_inst_tac [("f", "%j.ack(i,j)")] Ord_lt_mono_imp_le_mono 1);
by (REPEAT (ares_tac [ack_lt_mono2, ack_type RS nat_into_Ord] 1));
qed "ack_le_mono2";
(*PROPERTY A 6*)
goal Primrec.thy
"!!i j. [| i:nat; j:nat |] ==> ack(i, succ(j)) le ack(succ(i), j)";
by (nat_ind_tac "j" [] 1);
by (ALLGOALS (asm_simp_tac ack_ss));
by (rtac ack_le_mono2 1);
by (rtac (lt_ack2 RS succ_leI RS le_trans) 1);
by (REPEAT (ares_tac (ack_typechecks) 1));
qed "ack2_le_ack1";
(*PROPERTY A 7-, the single-step lemma*)
goal Primrec.thy "!!i j. [| i:nat; j:nat |] ==> ack(i,j) < ack(succ(i),j)";
by (rtac (ack_lt_mono2 RS lt_trans2) 1);
by (rtac ack2_le_ack1 4);
by (REPEAT (ares_tac ([nat_le_refl, ack_type] @ pr_typechecks) 1));
qed "ack_lt_ack_succ1";
(*PROPERTY A 7, monotonicity for < *)
goal Primrec.thy "!!i j k. [| i<j; j:nat; k:nat |] ==> ack(i,k) < ack(j,k)";
by (forward_tac [lt_nat_in_nat] 1 THEN assume_tac 1);
by (etac succ_lt_induct 1);
by (assume_tac 1);
by (rtac lt_trans 2);
by (REPEAT (ares_tac ([ack_lt_ack_succ1, ack_type] @ pr_typechecks) 1));
qed "ack_lt_mono1";
(*PROPERTY A 7', monotonicity for le *)
goal Primrec.thy
"!!i j k. [| i le j; j:nat; k:nat |] ==> ack(i,k) le ack(j,k)";
by (res_inst_tac [("f", "%j.ack(j,k)")] Ord_lt_mono_imp_le_mono 1);
by (REPEAT (ares_tac [ack_lt_mono1, ack_type RS nat_into_Ord] 1));
qed "ack_le_mono1";
(*PROPERTY A 8*)
goal Primrec.thy "!!j. j:nat ==> ack(1,j) = succ(succ(j))";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac ack_ss));
qed "ack_1";
(*PROPERTY A 9*)
goal Primrec.thy "!!j. j:nat ==> ack(succ(1),j) = succ(succ(succ(j#+j)))";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac (ack_ss addsimps [ack_1, add_succ_right])));
qed "ack_2";
(*PROPERTY A 10*)
goal Primrec.thy
"!!i1 i2 j. [| i1:nat; i2:nat; j:nat |] ==> \
\ ack(i1, ack(i2,j)) < ack(succ(succ(i1#+i2)), j)";
by (rtac (ack2_le_ack1 RSN (2,lt_trans2)) 1);
by (asm_simp_tac ack_ss 1);
by (rtac (add_le_self RS ack_le_mono1 RS lt_trans1) 1);
by (rtac (add_le_self2 RS ack_lt_mono1 RS ack_lt_mono2) 5);
by (tc_tac []);
qed "ack_nest_bound";
(*PROPERTY A 11*)
goal Primrec.thy
"!!i1 i2 j. [| i1:nat; i2:nat; j:nat |] ==> \
\ ack(i1,j) #+ ack(i2,j) < ack(succ(succ(succ(succ(i1#+i2)))), j)";
by (res_inst_tac [("j", "ack(succ(1), ack(i1 #+ i2, j))")] lt_trans 1);
by (asm_simp_tac (ack_ss addsimps [ack_2]) 1);
by (rtac (ack_nest_bound RS lt_trans2) 2);
by (asm_simp_tac ack_ss 5);
by (rtac (add_le_mono RS leI RS leI) 1);
by (REPEAT (ares_tac ([add_le_self, add_le_self2, ack_le_mono1] @
ack_typechecks) 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 Primrec.thy
"!!i j k. [| i < ack(k,j); j:nat; k:nat |] ==> \
\ i#+j < ack(succ(succ(succ(succ(k)))), j)";
by (res_inst_tac [("j", "ack(k,j) #+ ack(0,j)")] lt_trans 1);
by (rtac (ack_add_bound RS lt_trans2) 2);
by (asm_simp_tac (ack_ss addsimps [add_0_right]) 5);
by (REPEAT (ares_tac ([add_lt_mono, lt_ack2] @ ack_typechecks) 1));
qed "ack_add_bound2";
(*** MAIN RESULT ***)
val ack2_ss =
ack_ss addsimps [list_add_Nil, list_add_Cons, list_add_type, nat_into_Ord];
goalw Primrec.thy [SC_def]
"!!l. l: list(nat) ==> SC ` l < ack(1, list_add(l))";
by (etac list.elim 1);
by (asm_simp_tac (ack2_ss addsimps [succ_iff]) 1);
by (asm_simp_tac (ack2_ss addsimps [ack_1, add_le_self]) 1);
qed "SC_case";
(*PROPERTY A 4'? Extra lemma needed for CONST case, constant functions*)
goal Primrec.thy "!!j. [| i:nat; j:nat |] ==> i < ack(i,j)";
by (etac nat_induct 1);
by (asm_simp_tac (ack_ss addsimps [nat_0_le]) 1);
by (etac ([succ_leI, ack_lt_ack_succ1] MRS lt_trans1) 1);
by (tc_tac []);
qed "lt_ack1";
goalw Primrec.thy [CONST_def]
"!!l. [| l: list(nat); k: nat |] ==> CONST(k) ` l < ack(k, list_add(l))";
by (asm_simp_tac (ack2_ss addsimps [lt_ack1]) 1);
qed "CONST_case";
goalw Primrec.thy [PROJ_def]
"!!l. l: list(nat) ==> ALL i:nat. PROJ(i) ` l < ack(0, list_add(l))";
by (asm_simp_tac ack2_ss 1);
by (etac list.induct 1);
by (asm_simp_tac (ack2_ss addsimps [nat_0_le]) 1);
by (asm_simp_tac ack2_ss 1);
by (rtac ballI 1);
by (eres_inst_tac [("n","x")] natE 1);
by (asm_simp_tac (ack2_ss addsimps [add_le_self]) 1);
by (asm_simp_tac ack2_ss 1);
by (etac (bspec RS lt_trans2) 1);
by (rtac (add_le_self2 RS succ_leI) 2);
by (tc_tac []);
qed "PROJ_case_lemma";
val PROJ_case = PROJ_case_lemma RS bspec;
(** COMP case **)
goal Primrec.thy
"!!fs. fs : list({f: primrec . \
\ EX kf:nat. ALL l:list(nat). \
\ f`l < ack(kf, list_add(l))}) \
\ ==> EX k:nat. ALL l: list(nat). \
\ list_add(map(%f. f ` l, fs)) < ack(k, list_add(l))";
by (etac list.induct 1);
by (DO_GOAL [res_inst_tac [("x","0")] bexI,
asm_simp_tac (ack2_ss addsimps [lt_ack1, nat_0_le]),
resolve_tac nat_typechecks] 1);
by (safe_tac ZF_cs);
by (asm_simp_tac ack2_ss 1);
by (rtac (ballI RS bexI) 1);
by (rtac (add_lt_mono RS lt_trans) 1);
by (REPEAT (FIRSTGOAL (etac bspec)));
by (rtac ack_add_bound 5);
by (tc_tac []);
qed "COMP_map_lemma";
goalw Primrec.thy [COMP_def]
"!!g. [| g: primrec; kg: nat; \
\ ALL l:list(nat). g`l < ack(kg, list_add(l)); \
\ fs : list({f: primrec . \
\ EX kf:nat. ALL l:list(nat). \
\ f`l < ack(kf, list_add(l))}) \
\ |] ==> EX k:nat. ALL l: list(nat). COMP(g,fs)`l < ack(k, list_add(l))";
by (asm_simp_tac ZF_ss 1);
by (forward_tac [list_CollectD] 1);
by (etac (COMP_map_lemma RS bexE) 1);
by (rtac (ballI RS bexI) 1);
by (etac (bspec RS lt_trans) 1);
by (rtac lt_trans 2);
by (rtac ack_nest_bound 3);
by (etac (bspec RS ack_lt_mono2) 2);
by (tc_tac [map_type]);
qed "COMP_case";
(** PREC case **)
goalw Primrec.thy [PREC_def]
"!!f g. [| ALL l:list(nat). f`l #+ list_add(l) < ack(kf, list_add(l)); \
\ ALL l:list(nat). g`l #+ list_add(l) < ack(kg, list_add(l)); \
\ f: primrec; kf: nat; \
\ g: primrec; kg: nat; \
\ l: list(nat) \
\ |] ==> PREC(f,g)`l #+ list_add(l) < ack(succ(kf#+kg), list_add(l))";
by (etac list.elim 1);
by (asm_simp_tac (ack2_ss addsimps [[nat_le_refl, lt_ack2] MRS lt_trans]) 1);
by (asm_simp_tac ack2_ss 1);
by (etac ssubst 1); (*get rid of the needless assumption*)
by (eres_inst_tac [("n","a")] nat_induct 1);
(*base case*)
by (DO_GOAL [asm_simp_tac ack2_ss, rtac lt_trans, etac bspec,
assume_tac, rtac (add_le_self RS ack_lt_mono1),
REPEAT o ares_tac (ack_typechecks)] 1);
(*ind step*)
by (asm_simp_tac (ack2_ss addsimps [add_succ_right]) 1);
by (rtac (succ_leI RS lt_trans1) 1);
by (res_inst_tac [("j", "g ` ?ll #+ ?mm")] lt_trans1 1);
by (etac bspec 2);
by (rtac (nat_le_refl RS add_le_mono) 1);
by (tc_tac []);
by (asm_simp_tac (ack2_ss addsimps [add_le_self2]) 1);
(*final part of the simplification*)
by (asm_simp_tac ack2_ss 1);
by (rtac (add_le_self2 RS ack_le_mono1 RS lt_trans1) 1);
by (etac ack_lt_mono2 5);
by (tc_tac []);
qed "PREC_case_lemma";
goal Primrec.thy
"!!f g. [| f: primrec; kf: nat; \
\ g: primrec; kg: nat; \
\ ALL l:list(nat). f`l < ack(kf, list_add(l)); \
\ ALL l:list(nat). g`l < ack(kg, list_add(l)) \
\ |] ==> EX k:nat. ALL l: list(nat). \
\ PREC(f,g)`l< ack(k, list_add(l))";
by (rtac (ballI RS bexI) 1);
by (rtac ([add_le_self, PREC_case_lemma] MRS lt_trans1) 1);
by (REPEAT
(SOMEGOAL
(FIRST' [test_assume_tac,
match_tac (ack_typechecks),
rtac (ack_add_bound2 RS ballI) THEN' etac bspec])));
qed "PREC_case";
goal Primrec.thy
"!!f. f:primrec ==> EX k:nat. ALL l:list(nat). f`l < ack(k, list_add(l))";
by (etac primrec.induct 1);
by (safe_tac ZF_cs);
by (DEPTH_SOLVE
(ares_tac ([SC_case, CONST_case, PROJ_case, COMP_case, PREC_case,
bexI, ballI] @ nat_typechecks) 1));
qed "ack_bounds_primrec";
goal Primrec.thy
"~ (lam l:list(nat). list_case(0, %x xs. ack(x,x), l)) : primrec";
by (rtac notI 1);
by (etac (ack_bounds_primrec RS bexE) 1);
by (rtac lt_irrefl 1);
by (dres_inst_tac [("x", "[x]")] bspec 1);
by (asm_simp_tac ack2_ss 1);
by (asm_full_simp_tac (ack2_ss addsimps [add_0_right]) 1);
qed "ack_not_primrec";