--- a/src/HOL/ex/Primrec.thy Thu Feb 01 20:48:58 2001 +0100
+++ b/src/HOL/ex/Primrec.thy Thu Feb 01 20:51:13 2001 +0100
@@ -1,72 +1,348 @@
-(* Title: HOL/ex/Primrec
+(* Title: HOL/ex/Primrec.thy
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.
-
-Demonstrates recursive definitions, the TFL package
+Primitive Recursive Functions. Demonstrates recursive definitions,
+the TFL package.
*)
-Primrec = Main +
+header {* Primitive Recursive Functions *}
+
+theory Primrec = Main:
+
+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)"
-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))"
+ 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 (\<lambda>f. 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') (\<lambda>y 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 \<in> PRIMREC"
+ CONST: "CONST k \<in> PRIMREC"
+ PROJ: "PROJ i \<in> PRIMREC"
+ COMP: "g \<in> PRIMREC ==> fs \<in> lists PRIMREC ==> COMP g fs \<in> PRIMREC"
+ PREC: "f \<in> PRIMREC ==> g \<in> PRIMREC ==> PREC f g \<in> 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
-consts list_add :: nat list => nat
-primrec
- "list_add [] = 0"
- "list_add (m#ms) = m + list_add ms"
+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 \<le>} *}
+
+lemma ack_le_mono2: "j \<le> k ==> ack (i, j) \<le> ack (i, k)"
+ apply (simp add: order_le_less)
+ apply (blast intro: ack_less_mono2)
+ done
-consts zeroHd :: nat list => nat
-primrec
- "zeroHd [] = 0"
- "zeroHd (m#ms) = m"
+
+text {* PROPERTY A 6 *}
+
+lemma ack2_le_ack1 [iff]: "ack (i, Suc j) \<le> 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 (1, j) = j + #2"
+ apply (induct j)
+ apply simp_all
+ done
+
+
+text {* PROPERTY A 9. The unary @{term 1} and @{term 2} in @{term
+ ack} is essential for the rewriting. *}
+
+lemma ack_2 [simp]: "ack (2, j) = #2 * j + #3"
+ apply (induct j)
+ apply simp_all
+ done
-(** The set of primitive recursive functions of type nat list => nat **)
-consts
- PRIMREC :: (nat list => nat) set
- SC :: nat list => nat
- CONST :: [nat, nat list] => nat
- PROJ :: [nat, nat list] => nat
- COMP :: [nat list => nat, (nat list => nat)list, nat list] => nat
- PREC :: [nat list => nat, nat list => nat, nat list] => nat
+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 "\<le>"} *}
+
+lemma ack_le_mono1: "i \<le> j ==> ack (i, k) \<le> 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
+
-defs
+text {* PROPERTY A 11 *}
- SC_def "SC l == Suc (zeroHd l)"
+lemma ack_add_bound: "ack (i1, j) + ack (i2, j) < ack (#4 + (i1 + i2), j)"
+ apply (rule_tac j = "ack (2, 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
+ "\<exists>k'. \<forall>i j. ..."} *}
- CONST_def "CONST k l == k"
+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 *}
- PROJ_def "PROJ i l == zeroHd (drop i l)"
+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
- COMP_def "COMP g fs l == g (map (%f. f l) fs)"
+lemma PROJ_case [rule_format]: "\<forall>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 *}
- (*Note that g is applied first to PREC f g y and then to y!*)
- PREC_def "PREC f g l == case l of
- [] => 0
- | x#l' => nat_rec (f l') (%y r. g (r#y#l')) x"
+lemma COMP_map_aux: "fs \<in> lists (PRIMREC \<inter> {f. \<exists>kf. \<forall>l. f l < ack (kf, list_add l)})
+ ==> \<exists>k. \<forall>l. list_add (map (\<lambda>f. 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:
+ "\<forall>l. g l < ack (kg, list_add l) ==>
+ fs \<in> lists(PRIMREC Int {f. \<exists>kf. \<forall>l. f l < ack(kf, list_add l)})
+ ==> \<exists>k. \<forall>l. COMP g fs l < ack(k, list_add l)"
+ apply (unfold COMP_def)
+ apply (frule Int_lower1 [THEN lists_mono, THEN subsetD])
+ 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 *}
-
-inductive PRIMREC
- intrs
- 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"
- monos lists_mono
+lemma PREC_case_aux:
+ "\<forall>l. f l + list_add l < ack (kf, list_add l) ==>
+ \<forall>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:
+ "\<forall>l. f l < ack (kf, list_add l) ==>
+ \<forall>l. g l < ack (kg, list_add l) ==>
+ \<exists>k. \<forall>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 \<in> PRIMREC ==> \<exists>k. \<forall>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: "(\<lambda>l. case l of [] => 0 | x # l' => ack (x, x)) \<notin> 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