# HG changeset patch
# User nipkow
# Date 1216295433 -7200
# Node ID 1a3507f86b394badf60ae78c5a4074fa0a50129b
# Parent  3a45b555001aa0cb25cd1383a937b225f54459e3
beautified proofs

diff -r 3a45b555001a -r 1a3507f86b39 src/HOL/ex/Primrec.thy
--- a/src/HOL/ex/Primrec.thy	Thu Jul 17 13:50:17 2008 +0200
+++ b/src/HOL/ex/Primrec.thy	Thu Jul 17 13:50:33 2008 +0200
@@ -3,8 +3,8 @@
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1997  University of Cambridge
 
-Primitive Recursive Functions.  Demonstrates recursive definitions,
-the TFL package.
+Ackermann's Function and the
+Primitive Recursive Functions.
 *)
 
 header {* Primitive Recursive Functions *}
@@ -23,121 +23,45 @@
   \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"}. *}
-
-definition
-  SC :: "nat list => nat" where
-  "SC l = Suc (zeroHd l)"
-
-definition
-  CONSTANT :: "nat => nat list => nat" where
-  "CONSTANT k l = k"
-
-definition
-  PROJ :: "nat => nat list => nat" where
-  "PROJ i l = zeroHd (drop i l)"
-
-definition
-  COMP :: "(nat list => nat) => (nat list => nat) list => nat list => nat" where
-  "COMP g fs l = g (map (\<lambda>f. f l) fs)"
+subsection{* Ackermann's Function *}
 
-definition
-  PREC :: "(nat list => nat) => (nat list => nat) => nat list => nat" where
-  "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}! *}
-
-inductive PRIMREC :: "(nat list => nat) => bool"
-  where
-    SC: "PRIMREC SC"
-  | CONSTANT: "PRIMREC (CONSTANT k)"
-  | PROJ: "PRIMREC (PROJ i)"
-  | COMP: "PRIMREC g ==> \<forall>f \<in> 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"
-  apply (simp add: SC_def)
-  done
-
-lemma CONSTANT [simp]: "CONSTANT k l = k"
-  apply (simp add: CONSTANT_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
+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)"
-  apply (induct i j rule: ack.induct)
-    apply simp_all
-  done
+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)"
-  apply (induct i j rule: ack.induct)
-    apply simp_all
-  done
+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 @{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
+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 @{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
+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
 
 
 text {* PROPERTY A 6 *}
 
-lemma ack2_le_ack1 [iff]: "ack (i, Suc j) \<le> ack (Suc i, j)"
+lemma ack2_le_ack1 [iff]: "ack i (Suc j) \<le> ack (Suc i) j"
 proof (induct j)
   case 0 show ?case by simp
 next
@@ -149,195 +73,248 @@
 
 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
+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
+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
+lemma ack_1 [simp]: "ack (Suc 0) j = j + 2"
+by (induct j) simp_all
 
 
 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
+lemma ack_2 [simp]: "ack (Suc (Suc 0)) j = 2 * j + 3"
+by (induct j) simp_all
 
 
 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_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
+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
+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 *}
+ML{*ResAtp.set_prover "vampire"*}
 
-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
+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 less_trans [of _ "ack (Suc (Suc 0), ack (i1 + i2, j))" _])
-   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
+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 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. ..."} *}
 
-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 (blast intro: add_less_mono less_ack2) 
-   apply (rule ack_add_bound [THEN less_le_trans])
-   apply simp
-  done
+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 (blast intro: add_less_mono less_ack2) 
+apply (rule ack_add_bound [THEN less_le_trans])
+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"}. *}
 
-text {* Inductive definition of the @{term PR} functions *}
+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 (\<lambda>f. 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' => 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}! *}
+
+inductive PRIMREC :: "(nat list => nat) => bool" where
+SC: "PRIMREC SC" |
+CONSTANT: "PRIMREC (CONSTANT k)" |
+PROJ: "PRIMREC (PROJ i)" |
+COMP: "PRIMREC g ==> \<forall>f \<in> 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"
+by (simp add: SC_def)
+
+lemma CONSTANT [simp]: "CONSTANT k l = k"
+by (simp add: CONSTANT_def)
+
+lemma PROJ_0 [simp]: "PROJ 0 (x # l) = x"
+by (simp add: PROJ_def)
+
+lemma COMP_1 [simp]: "COMP g [f] l = g [f l]"
+by (simp add: COMP_def)
+
+lemma PREC_0 [simp]: "PREC f g (0 # l) = f l"
+by (simp add: PREC_def)
+
+lemma PREC_Suc [simp]: "PREC f g (Suc x # l) = g (PREC f g (x # l) # x # l)"
+by (simp add: PREC_def)
+
 
 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 SC_case: "SC l < ack 1 (listsum l)"
+apply (unfold SC_def)
+apply (induct l)
+apply (simp_all add: le_add1 le_imp_less_Suc)
+done
 
-lemma CONSTANT_case: "CONSTANT k l < ack (k, list_add l)"
-  by simp
+lemma CONSTANT_case: "CONSTANT k l < ack k (listsum l)"
+by simp
 
-lemma PROJ_case [rule_format]: "\<forall>i. PROJ i l < ack (0, list_add l)"
-  apply (simp add: PROJ_def)
-  apply (induct l)
-   apply (auto simp add: drop_Cons split: nat.split) 
-  apply (blast intro: less_le_trans le_add2)
-  done
+lemma PROJ_case: "PROJ i l < ack 0 (listsum l)"
+apply (simp add: PROJ_def)
+apply (induct l arbitrary:i)
+ apply (auto simp add: drop_Cons split: nat.split)
+apply (blast intro: less_le_trans le_add2)
+done
 
 
 text {* @{term COMP} case *}
 
-lemma COMP_map_aux: "\<forall>f \<in> set fs. PRIMREC f \<and> (\<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 (induct fs)
-  apply (rule_tac x = 0 in exI) 
-   apply simp
-  apply simp
-  apply (blast intro: add_less_mono ack_add_bound less_trans)
-  done
+lemma COMP_map_aux: "\<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (listsum l))
+  ==> \<exists>k. \<forall>l. listsum (map (\<lambda>f. f l) fs) < ack k (listsum l)"
+apply (induct fs)
+ apply (rule_tac x = 0 in exI)
+ apply simp
+apply simp
+apply (blast intro: add_less_mono ack_add_bound less_trans)
+done
 
 lemma COMP_case:
-  "\<forall>l. g l < ack (kg, list_add l) ==>
-  \<forall>f \<in> set fs. PRIMREC f \<and> (\<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)
-    --{*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
+  "\<forall>l. g l < ack kg (listsum l) ==>
+  \<forall>f \<in> set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (listsum l))
+  ==> \<exists>k. \<forall>l. COMP g fs  l < ack k (listsum l)"
+apply (unfold COMP_def)
+  --{*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:
-  "\<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
+  "\<forall>l. f l + listsum l < ack kf (listsum l) ==>
+    \<forall>l. g l + listsum l < ack kg (listsum l) ==>
+    PREC f g l + listsum l < ack (Suc (kf + kg)) (listsum 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)"
-  by (metis le_less_trans [OF le_add1 PREC_case_aux] ack_add_bound2) 
+  "\<forall>l. f l < ack kf (listsum l) ==>
+    \<forall>l. g l < ack kg (listsum l) ==>
+    \<exists>k. \<forall>l. PREC f g l < ack k (listsum l)"
+by (metis le_less_trans [OF le_add1 PREC_case_aux] ack_add_bound2)
 
-lemma ack_bounds_PRIMREC: "PRIMREC f ==> \<exists>k. \<forall>l. f l < ack (k, list_add l)"
-  apply (erule PRIMREC.induct)
-      apply (blast intro: SC_case CONSTANT_case PROJ_case COMP_case PREC_case)+
-  done
+lemma ack_bounds_PRIMREC: "PRIMREC f ==> \<exists>k. \<forall>l. f l < ack k (listsum l)"
+apply (erule PRIMREC.induct)
+    apply (blast intro: SC_case CONSTANT_case PROJ_case COMP_case PREC_case)+
+done
 
-lemma ack_not_PRIMREC: "\<not> PRIMREC (\<lambda>l. 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
+theorem ack_not_PRIMREC:
+  "\<not> PRIMREC (\<lambda>l. 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