src/ZF/Arith.thy
author wenzelm
Sat Nov 04 19:17:19 2017 +0100 (21 months ago)
changeset 67006 b1278ed3cd46
parent 61798 27f3c10b0b50
child 69587 53982d5ec0bb
permissions -rw-r--r--
prefer main entry points of HOL;
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(*  Title:      ZF/Arith.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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(*"Difference" is subtraction of natural numbers.
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  There are no negative numbers; we have
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     m #- n = 0  iff  m<=n   and     m #- n = succ(k) iff m>n.
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  Also, rec(m, 0, %z w.z) is pred(m).
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*)
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section\<open>Arithmetic Operators and Their Definitions\<close>
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theory Arith imports Univ begin
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text\<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close>
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definition
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  pred   :: "i=>i"    (*inverse of succ*)  where
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    "pred(y) == nat_case(0, %x. x, y)"
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definition
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  natify :: "i=>i"    (*coerces non-nats to nats*)  where
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    "natify == Vrecursor(%f a. if a = succ(pred(a)) then succ(f`pred(a))
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                                                    else 0)"
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consts
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  raw_add  :: "[i,i]=>i"
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  raw_diff  :: "[i,i]=>i"
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  raw_mult  :: "[i,i]=>i"
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primrec
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  "raw_add (0, n) = n"
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  "raw_add (succ(m), n) = succ(raw_add(m, n))"
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primrec
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  raw_diff_0:     "raw_diff(m, 0) = m"
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  raw_diff_succ:  "raw_diff(m, succ(n)) =
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                     nat_case(0, %x. x, raw_diff(m, n))"
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primrec
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  "raw_mult(0, n) = 0"
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  "raw_mult(succ(m), n) = raw_add (n, raw_mult(m, n))"
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definition
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  add :: "[i,i]=>i"                    (infixl "#+" 65)  where
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    "m #+ n == raw_add (natify(m), natify(n))"
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definition
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  diff :: "[i,i]=>i"                    (infixl "#-" 65)  where
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    "m #- n == raw_diff (natify(m), natify(n))"
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definition
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  mult :: "[i,i]=>i"                    (infixl "#*" 70)  where
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    "m #* n == raw_mult (natify(m), natify(n))"
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definition
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  raw_div  :: "[i,i]=>i"  where
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    "raw_div (m, n) ==
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       transrec(m, %j f. if j<n | n=0 then 0 else succ(f`(j#-n)))"
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definition
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  raw_mod  :: "[i,i]=>i"  where
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    "raw_mod (m, n) ==
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       transrec(m, %j f. if j<n | n=0 then j else f`(j#-n))"
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definition
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  div  :: "[i,i]=>i"                    (infixl "div" 70)  where
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    "m div n == raw_div (natify(m), natify(n))"
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definition
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  mod  :: "[i,i]=>i"                    (infixl "mod" 70)  where
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    "m mod n == raw_mod (natify(m), natify(n))"
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declare rec_type [simp]
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        nat_0_le [simp]
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lemma zero_lt_lemma: "[| 0<k; k \<in> nat |] ==> \<exists>j\<in>nat. k = succ(j)"
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apply (erule rev_mp)
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apply (induct_tac "k", auto)
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done
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(* @{term"[| 0 < k; k \<in> nat; !!j. [| j \<in> nat; k = succ(j) |] ==> Q |] ==> Q"} *)
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lemmas zero_lt_natE = zero_lt_lemma [THEN bexE]
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subsection\<open>\<open>natify\<close>, the Coercion to @{term nat}\<close>
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lemma pred_succ_eq [simp]: "pred(succ(y)) = y"
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by (unfold pred_def, auto)
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lemma natify_succ: "natify(succ(x)) = succ(natify(x))"
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by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)
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lemma natify_0 [simp]: "natify(0) = 0"
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by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)
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lemma natify_non_succ: "\<forall>z. x \<noteq> succ(z) ==> natify(x) = 0"
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by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)
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lemma natify_in_nat [iff,TC]: "natify(x) \<in> nat"
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apply (rule_tac a=x in eps_induct)
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apply (case_tac "\<exists>z. x = succ(z)")
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apply (auto simp add: natify_succ natify_non_succ)
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done
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lemma natify_ident [simp]: "n \<in> nat ==> natify(n) = n"
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apply (induct_tac "n")
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apply (auto simp add: natify_succ)
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done
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lemma natify_eqE: "[|natify(x) = y;  x \<in> nat|] ==> x=y"
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by auto
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(*** Collapsing rules: to remove natify from arithmetic expressions ***)
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lemma natify_idem [simp]: "natify(natify(x)) = natify(x)"
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by simp
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(** Addition **)
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lemma add_natify1 [simp]: "natify(m) #+ n = m #+ n"
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by (simp add: add_def)
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lemma add_natify2 [simp]: "m #+ natify(n) = m #+ n"
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by (simp add: add_def)
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(** Multiplication **)
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lemma mult_natify1 [simp]: "natify(m) #* n = m #* n"
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by (simp add: mult_def)
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lemma mult_natify2 [simp]: "m #* natify(n) = m #* n"
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by (simp add: mult_def)
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(** Difference **)
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lemma diff_natify1 [simp]: "natify(m) #- n = m #- n"
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by (simp add: diff_def)
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lemma diff_natify2 [simp]: "m #- natify(n) = m #- n"
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by (simp add: diff_def)
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(** Remainder **)
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lemma mod_natify1 [simp]: "natify(m) mod n = m mod n"
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by (simp add: mod_def)
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lemma mod_natify2 [simp]: "m mod natify(n) = m mod n"
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by (simp add: mod_def)
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(** Quotient **)
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lemma div_natify1 [simp]: "natify(m) div n = m div n"
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by (simp add: div_def)
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lemma div_natify2 [simp]: "m div natify(n) = m div n"
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by (simp add: div_def)
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subsection\<open>Typing rules\<close>
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(** Addition **)
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lemma raw_add_type: "[| m\<in>nat;  n\<in>nat |] ==> raw_add (m, n) \<in> nat"
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by (induct_tac "m", auto)
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lemma add_type [iff,TC]: "m #+ n \<in> nat"
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by (simp add: add_def raw_add_type)
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(** Multiplication **)
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lemma raw_mult_type: "[| m\<in>nat;  n\<in>nat |] ==> raw_mult (m, n) \<in> nat"
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apply (induct_tac "m")
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apply (simp_all add: raw_add_type)
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done
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lemma mult_type [iff,TC]: "m #* n \<in> nat"
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by (simp add: mult_def raw_mult_type)
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(** Difference **)
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lemma raw_diff_type: "[| m\<in>nat;  n\<in>nat |] ==> raw_diff (m, n) \<in> nat"
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by (induct_tac "n", auto)
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lemma diff_type [iff,TC]: "m #- n \<in> nat"
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by (simp add: diff_def raw_diff_type)
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lemma diff_0_eq_0 [simp]: "0 #- n = 0"
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apply (unfold diff_def)
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apply (rule natify_in_nat [THEN nat_induct], auto)
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done
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(*Must simplify BEFORE the induction: else we get a critical pair*)
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lemma diff_succ_succ [simp]: "succ(m) #- succ(n) = m #- n"
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apply (simp add: natify_succ diff_def)
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apply (rule_tac x1 = n in natify_in_nat [THEN nat_induct], auto)
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done
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(*This defining property is no longer wanted*)
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declare raw_diff_succ [simp del]
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(*Natify has weakened this law, compared with the older approach*)
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lemma diff_0 [simp]: "m #- 0 = natify(m)"
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by (simp add: diff_def)
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lemma diff_le_self: "m\<in>nat ==> (m #- n) \<le> m"
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apply (subgoal_tac " (m #- natify (n)) \<le> m")
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apply (rule_tac [2] m = m and n = "natify (n) " in diff_induct)
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apply (erule_tac [6] leE)
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apply (simp_all add: le_iff)
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done
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subsection\<open>Addition\<close>
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(*Natify has weakened this law, compared with the older approach*)
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lemma add_0_natify [simp]: "0 #+ m = natify(m)"
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by (simp add: add_def)
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lemma add_succ [simp]: "succ(m) #+ n = succ(m #+ n)"
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by (simp add: natify_succ add_def)
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lemma add_0: "m \<in> nat ==> 0 #+ m = m"
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by simp
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(*Associative law for addition*)
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lemma add_assoc: "(m #+ n) #+ k = m #+ (n #+ k)"
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apply (subgoal_tac "(natify(m) #+ natify(n)) #+ natify(k) =
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                    natify(m) #+ (natify(n) #+ natify(k))")
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apply (rule_tac [2] n = "natify(m)" in nat_induct)
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apply auto
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done
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(*The following two lemmas are used for add_commute and sometimes
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  elsewhere, since they are safe for rewriting.*)
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lemma add_0_right_natify [simp]: "m #+ 0 = natify(m)"
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apply (subgoal_tac "natify(m) #+ 0 = natify(m)")
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apply (rule_tac [2] n = "natify(m)" in nat_induct)
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apply auto
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done
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lemma add_succ_right [simp]: "m #+ succ(n) = succ(m #+ n)"
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apply (unfold add_def)
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apply (rule_tac n = "natify(m) " in nat_induct)
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apply (auto simp add: natify_succ)
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done
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lemma add_0_right: "m \<in> nat ==> m #+ 0 = m"
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by auto
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(*Commutative law for addition*)
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lemma add_commute: "m #+ n = n #+ m"
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apply (subgoal_tac "natify(m) #+ natify(n) = natify(n) #+ natify(m) ")
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apply (rule_tac [2] n = "natify(m) " in nat_induct)
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apply auto
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done
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(*for a/c rewriting*)
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lemma add_left_commute: "m#+(n#+k)=n#+(m#+k)"
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apply (rule add_commute [THEN trans])
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apply (rule add_assoc [THEN trans])
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apply (rule add_commute [THEN subst_context])
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done
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(*Addition is an AC-operator*)
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lemmas add_ac = add_assoc add_commute add_left_commute
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(*Cancellation law on the left*)
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lemma raw_add_left_cancel:
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     "[| raw_add(k, m) = raw_add(k, n);  k\<in>nat |] ==> m=n"
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apply (erule rev_mp)
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apply (induct_tac "k", auto)
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done
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lemma add_left_cancel_natify: "k #+ m = k #+ n ==> natify(m) = natify(n)"
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apply (unfold add_def)
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apply (drule raw_add_left_cancel, auto)
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done
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lemma add_left_cancel:
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     "[| i = j;  i #+ m = j #+ n;  m\<in>nat;  n\<in>nat |] ==> m = n"
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by (force dest!: add_left_cancel_natify)
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(*Thanks to Sten Agerholm*)
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lemma add_le_elim1_natify: "k#+m \<le> k#+n ==> natify(m) \<le> natify(n)"
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apply (rule_tac P = "natify(k) #+m \<le> natify(k) #+n" in rev_mp)
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apply (rule_tac [2] n = "natify(k) " in nat_induct)
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apply auto
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done
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lemma add_le_elim1: "[| k#+m \<le> k#+n; m \<in> nat; n \<in> nat |] ==> m \<le> n"
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by (drule add_le_elim1_natify, auto)
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lemma add_lt_elim1_natify: "k#+m < k#+n ==> natify(m) < natify(n)"
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apply (rule_tac P = "natify(k) #+m < natify(k) #+n" in rev_mp)
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apply (rule_tac [2] n = "natify(k) " in nat_induct)
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apply auto
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done
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lemma add_lt_elim1: "[| k#+m < k#+n; m \<in> nat; n \<in> nat |] ==> m < n"
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by (drule add_lt_elim1_natify, auto)
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lemma zero_less_add: "[| n \<in> nat; m \<in> nat |] ==> 0 < m #+ n \<longleftrightarrow> (0<m | 0<n)"
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by (induct_tac "n", auto)
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subsection\<open>Monotonicity of Addition\<close>
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(*strict, in 1st argument; proof is by rule induction on 'less than'.
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  Still need j\<in>nat, for consider j = omega.  Then we can have i<omega,
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  which is the same as i\<in>nat, but natify(j)=0, so the conclusion fails.*)
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lemma add_lt_mono1: "[| i<j; j\<in>nat |] ==> i#+k < j#+k"
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apply (frule lt_nat_in_nat, assumption)
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apply (erule succ_lt_induct)
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apply (simp_all add: leI)
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done
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text\<open>strict, in second argument\<close>
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lemma add_lt_mono2: "[| i<j; j\<in>nat |] ==> k#+i < k#+j"
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by (simp add: add_commute [of k] add_lt_mono1)
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text\<open>A [clumsy] way of lifting < monotonicity to \<open>\<le>\<close> monotonicity\<close>
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   328
lemma Ord_lt_mono_imp_le_mono:
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  assumes lt_mono: "!!i j. [| i<j; j:k |] ==> f(i) < f(j)"
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   330
      and ford:    "!!i. i:k ==> Ord(f(i))"
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   331
      and leij:    "i \<le> j"
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   332
      and jink:    "j:k"
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  shows "f(i) \<le> f(j)"
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   334
apply (insert leij jink)
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   335
apply (blast intro!: leCI lt_mono ford elim!: leE)
paulson@13163
   336
done
paulson@13163
   337
wenzelm@61798
   338
text\<open>\<open>\<le>\<close> monotonicity, 1st argument\<close>
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   339
lemma add_le_mono1: "[| i \<le> j; j\<in>nat |] ==> i#+k \<le> j#+k"
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   340
apply (rule_tac f = "%j. j#+k" in Ord_lt_mono_imp_le_mono, typecheck)
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   341
apply (blast intro: add_lt_mono1 add_type [THEN nat_into_Ord])+
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   342
done
paulson@13163
   343
wenzelm@61798
   344
text\<open>\<open>\<le>\<close> monotonicity, both arguments\<close>
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   345
lemma add_le_mono: "[| i \<le> j; k \<le> l; j\<in>nat; l\<in>nat |] ==> i#+k \<le> j#+l"
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   346
apply (rule add_le_mono1 [THEN le_trans], assumption+)
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   347
apply (subst add_commute, subst add_commute, rule add_le_mono1, assumption+)
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   348
done
paulson@13163
   349
wenzelm@60770
   350
text\<open>Combinations of less-than and less-than-or-equals\<close>
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   351
paulson@14060
   352
lemma add_lt_le_mono: "[| i<j; k\<le>l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
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   353
apply (rule add_lt_mono1 [THEN lt_trans2], assumption+)
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   354
apply (subst add_commute, subst add_commute, rule add_le_mono1, assumption+)
paulson@14060
   355
done
paulson@14060
   356
paulson@14060
   357
lemma add_le_lt_mono: "[| i\<le>j; k<l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
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   358
by (subst add_commute, subst add_commute, erule add_lt_le_mono, assumption+)
paulson@14060
   359
wenzelm@60770
   360
text\<open>Less-than: in other words, strict in both arguments\<close>
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   361
lemma add_lt_mono: "[| i<j; k<l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
paulson@46820
   362
apply (rule add_lt_le_mono)
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   363
apply (auto intro: leI)
paulson@14060
   364
done
paulson@14060
   365
paulson@13163
   366
(** Subtraction is the inverse of addition. **)
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   367
paulson@13163
   368
lemma diff_add_inverse: "(n#+m) #- n = natify(m)"
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   369
apply (subgoal_tac " (natify(n) #+ m) #- natify(n) = natify(m) ")
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   370
apply (rule_tac [2] n = "natify(n) " in nat_induct)
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   371
apply auto
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   372
done
paulson@13163
   373
paulson@13163
   374
lemma diff_add_inverse2: "(m#+n) #- n = natify(m)"
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   375
by (simp add: add_commute [of m] diff_add_inverse)
paulson@13163
   376
paulson@13163
   377
lemma diff_cancel: "(k#+m) #- (k#+n) = m #- n"
paulson@13163
   378
apply (subgoal_tac "(natify(k) #+ natify(m)) #- (natify(k) #+ natify(n)) =
paulson@13163
   379
                    natify(m) #- natify(n) ")
paulson@13163
   380
apply (rule_tac [2] n = "natify(k) " in nat_induct)
paulson@13163
   381
apply auto
paulson@13163
   382
done
paulson@13163
   383
paulson@13163
   384
lemma diff_cancel2: "(m#+k) #- (n#+k) = m #- n"
paulson@13163
   385
by (simp add: add_commute [of _ k] diff_cancel)
paulson@13163
   386
paulson@13163
   387
lemma diff_add_0: "n #- (n#+m) = 0"
paulson@13163
   388
apply (subgoal_tac "natify(n) #- (natify(n) #+ natify(m)) = 0")
paulson@13163
   389
apply (rule_tac [2] n = "natify(n) " in nat_induct)
paulson@13163
   390
apply auto
paulson@13163
   391
done
paulson@13163
   392
paulson@13361
   393
lemma pred_0 [simp]: "pred(0) = 0"
paulson@13361
   394
by (simp add: pred_def)
paulson@13361
   395
paulson@13361
   396
lemma eq_succ_imp_eq_m1: "[|i = succ(j); i\<in>nat|] ==> j = i #- 1 & j \<in>nat"
paulson@46820
   397
by simp
paulson@13361
   398
paulson@13361
   399
lemma pred_Un_distrib:
paulson@46820
   400
    "[|i\<in>nat; j\<in>nat|] ==> pred(i \<union> j) = pred(i) \<union> pred(j)"
paulson@46820
   401
apply (erule_tac n=i in natE, simp)
paulson@46820
   402
apply (erule_tac n=j in natE, simp)
paulson@13361
   403
apply (simp add:  succ_Un_distrib [symmetric])
paulson@13361
   404
done
paulson@13361
   405
paulson@13361
   406
lemma pred_type [TC,simp]:
paulson@13361
   407
    "i \<in> nat ==> pred(i) \<in> nat"
paulson@13361
   408
by (simp add: pred_def split: split_nat_case)
paulson@13361
   409
wenzelm@58860
   410
lemma nat_diff_pred: "[|i\<in>nat; j\<in>nat|] ==> i #- succ(j) = pred(i #- j)"
paulson@46820
   411
apply (rule_tac m=i and n=j in diff_induct)
paulson@13361
   412
apply (auto simp add: pred_def nat_imp_quasinat split: split_nat_case)
paulson@13361
   413
done
paulson@13361
   414
wenzelm@58860
   415
lemma diff_succ_eq_pred: "i #- succ(j) = pred(i #- j)"
paulson@13361
   416
apply (insert nat_diff_pred [of "natify(i)" "natify(j)"])
paulson@46820
   417
apply (simp add: natify_succ [symmetric])
paulson@13361
   418
done
paulson@13361
   419
paulson@13361
   420
lemma nat_diff_Un_distrib:
paulson@46820
   421
    "[|i\<in>nat; j\<in>nat; k\<in>nat|] ==> (i \<union> j) #- k = (i#-k) \<union> (j#-k)"
paulson@46820
   422
apply (rule_tac n=k in nat_induct)
paulson@46820
   423
apply (simp_all add: diff_succ_eq_pred pred_Un_distrib)
paulson@13361
   424
done
paulson@13361
   425
paulson@13361
   426
lemma diff_Un_distrib:
paulson@46820
   427
    "[|i\<in>nat; j\<in>nat|] ==> (i \<union> j) #- k = (i#-k) \<union> (j#-k)"
paulson@13361
   428
by (insert nat_diff_Un_distrib [of i j "natify(k)"], simp)
paulson@13361
   429
wenzelm@60770
   430
text\<open>We actually prove @{term "i #- j #- k = i #- (j #+ k)"}\<close>
paulson@13361
   431
lemma diff_diff_left [simplified]:
wenzelm@58860
   432
     "natify(i)#-natify(j)#-k = natify(i) #- (natify(j)#+k)"
paulson@13361
   433
by (rule_tac m="natify(i)" and n="natify(j)" in diff_induct, auto)
paulson@13361
   434
paulson@13163
   435
paulson@13163
   436
(** Lemmas for the CancelNumerals simproc **)
paulson@13163
   437
paulson@46821
   438
lemma eq_add_iff: "(u #+ m = u #+ n) \<longleftrightarrow> (0 #+ m = natify(n))"
paulson@13163
   439
apply auto
paulson@13163
   440
apply (blast dest: add_left_cancel_natify)
paulson@13163
   441
apply (simp add: add_def)
paulson@13163
   442
done
paulson@13163
   443
paulson@46821
   444
lemma less_add_iff: "(u #+ m < u #+ n) \<longleftrightarrow> (0 #+ m < natify(n))"
paulson@13163
   445
apply (auto simp add: add_lt_elim1_natify)
paulson@13163
   446
apply (drule add_lt_mono1)
paulson@13163
   447
apply (auto simp add: add_commute [of u])
paulson@13163
   448
done
paulson@13163
   449
paulson@13163
   450
lemma diff_add_eq: "((u #+ m) #- (u #+ n)) = ((0 #+ m) #- n)"
paulson@13163
   451
by (simp add: diff_cancel)
paulson@13163
   452
paulson@13163
   453
(*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
paulson@13163
   454
lemma eq_cong2: "u = u' ==> (t==u) == (t==u')"
paulson@13163
   455
by auto
paulson@13163
   456
paulson@46821
   457
lemma iff_cong2: "u \<longleftrightarrow> u' ==> (t==u) == (t==u')"
paulson@13163
   458
by auto
paulson@13163
   459
paulson@13163
   460
wenzelm@60770
   461
subsection\<open>Multiplication\<close>
paulson@13163
   462
paulson@13163
   463
lemma mult_0 [simp]: "0 #* m = 0"
paulson@13163
   464
by (simp add: mult_def)
paulson@13163
   465
paulson@13163
   466
lemma mult_succ [simp]: "succ(m) #* n = n #+ (m #* n)"
paulson@13163
   467
by (simp add: add_def mult_def natify_succ raw_mult_type)
paulson@13163
   468
paulson@13163
   469
(*right annihilation in product*)
paulson@13163
   470
lemma mult_0_right [simp]: "m #* 0 = 0"
paulson@13163
   471
apply (unfold mult_def)
paulson@13163
   472
apply (rule_tac n = "natify(m) " in nat_induct)
paulson@13163
   473
apply auto
paulson@13163
   474
done
paulson@13163
   475
paulson@13163
   476
(*right successor law for multiplication*)
paulson@13163
   477
lemma mult_succ_right [simp]: "m #* succ(n) = m #+ (m #* n)"
paulson@13163
   478
apply (subgoal_tac "natify(m) #* succ (natify(n)) =
paulson@13163
   479
                    natify(m) #+ (natify(m) #* natify(n))")
paulson@13163
   480
apply (simp (no_asm_use) add: natify_succ add_def mult_def)
paulson@13163
   481
apply (rule_tac n = "natify(m) " in nat_induct)
paulson@13163
   482
apply (simp_all add: add_ac)
paulson@13163
   483
done
paulson@13163
   484
paulson@13163
   485
lemma mult_1_natify [simp]: "1 #* n = natify(n)"
paulson@13163
   486
by auto
paulson@13163
   487
paulson@13163
   488
lemma mult_1_right_natify [simp]: "n #* 1 = natify(n)"
paulson@13163
   489
by auto
paulson@13163
   490
paulson@14060
   491
lemma mult_1: "n \<in> nat ==> 1 #* n = n"
paulson@13163
   492
by simp
paulson@13163
   493
paulson@14060
   494
lemma mult_1_right: "n \<in> nat ==> n #* 1 = n"
paulson@13163
   495
by simp
paulson@13163
   496
paulson@13163
   497
(*Commutative law for multiplication*)
paulson@13163
   498
lemma mult_commute: "m #* n = n #* m"
paulson@13163
   499
apply (subgoal_tac "natify(m) #* natify(n) = natify(n) #* natify(m) ")
paulson@13163
   500
apply (rule_tac [2] n = "natify(m) " in nat_induct)
paulson@13163
   501
apply auto
paulson@13163
   502
done
paulson@13163
   503
paulson@13163
   504
(*addition distributes over multiplication*)
paulson@13163
   505
lemma add_mult_distrib: "(m #+ n) #* k = (m #* k) #+ (n #* k)"
paulson@13163
   506
apply (subgoal_tac "(natify(m) #+ natify(n)) #* natify(k) =
paulson@13163
   507
                    (natify(m) #* natify(k)) #+ (natify(n) #* natify(k))")
paulson@13163
   508
apply (rule_tac [2] n = "natify(m) " in nat_induct)
paulson@13163
   509
apply (simp_all add: add_assoc [symmetric])
paulson@13163
   510
done
paulson@13163
   511
paulson@13163
   512
(*Distributive law on the left*)
paulson@13163
   513
lemma add_mult_distrib_left: "k #* (m #+ n) = (k #* m) #+ (k #* n)"
paulson@13163
   514
apply (subgoal_tac "natify(k) #* (natify(m) #+ natify(n)) =
paulson@13163
   515
                    (natify(k) #* natify(m)) #+ (natify(k) #* natify(n))")
paulson@13163
   516
apply (rule_tac [2] n = "natify(m) " in nat_induct)
paulson@13163
   517
apply (simp_all add: add_ac)
paulson@13163
   518
done
paulson@13163
   519
paulson@13163
   520
(*Associative law for multiplication*)
paulson@13163
   521
lemma mult_assoc: "(m #* n) #* k = m #* (n #* k)"
paulson@13163
   522
apply (subgoal_tac "(natify(m) #* natify(n)) #* natify(k) =
paulson@13163
   523
                    natify(m) #* (natify(n) #* natify(k))")
paulson@13163
   524
apply (rule_tac [2] n = "natify(m) " in nat_induct)
paulson@13163
   525
apply (simp_all add: add_mult_distrib)
paulson@13163
   526
done
paulson@13163
   527
paulson@13163
   528
(*for a/c rewriting*)
paulson@13163
   529
lemma mult_left_commute: "m #* (n #* k) = n #* (m #* k)"
paulson@13163
   530
apply (rule mult_commute [THEN trans])
paulson@13163
   531
apply (rule mult_assoc [THEN trans])
paulson@13163
   532
apply (rule mult_commute [THEN subst_context])
paulson@13163
   533
done
paulson@13163
   534
paulson@13163
   535
lemmas mult_ac = mult_assoc mult_commute mult_left_commute
paulson@13163
   536
paulson@13163
   537
paulson@13163
   538
lemma lt_succ_eq_0_disj:
paulson@14060
   539
     "[| m\<in>nat; n\<in>nat |]
paulson@46821
   540
      ==> (m < succ(n)) \<longleftrightarrow> (m = 0 | (\<exists>j\<in>nat. m = succ(j) & j < n))"
paulson@13163
   541
by (induct_tac "m", auto)
paulson@13163
   542
paulson@13163
   543
lemma less_diff_conv [rule_format]:
paulson@46821
   544
     "[| j\<in>nat; k\<in>nat |] ==> \<forall>i\<in>nat. (i < j #- k) \<longleftrightarrow> (i #+ k < j)"
paulson@13784
   545
by (erule_tac m = k in diff_induct, auto)
paulson@13163
   546
paulson@13163
   547
lemmas nat_typechecks = rec_type nat_0I nat_1I nat_succI Ord_nat
paulson@13163
   548
clasohm@0
   549
end