src/ZF/Arith.thy
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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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header{*Arithmetic Operators and Their Definitions*}
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theory Arith imports Univ begin
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text{*Proofs about elementary arithmetic: addition, multiplication, etc.*}
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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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notation (xsymbols)
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  mult  (infixr "#\<times>" 70)
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notation (HTML output)
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  mult  (infixr "#\<times>" 70)
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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{*@{text natify}, the Coercion to @{term nat}*}
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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{*Typing rules*}
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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{*Addition*}
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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{*Monotonicity of Addition*}
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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{*strict, in second argument*}
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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{*A [clumsy] way of lifting < monotonicity to @{text "\<le>"} monotonicity*}
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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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      and ford:    "!!i. i:k ==> Ord(f(i))"
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      and leij:    "i \<le> j"
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      and jink:    "j:k"
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  shows "f(i) \<le> f(j)"
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apply (insert leij jink)
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apply (blast intro!: leCI lt_mono ford elim!: leE)
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done
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text{*@{text "\<le>"} monotonicity, 1st argument*}
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lemma add_le_mono1: "[| i \<le> j; j\<in>nat |] ==> i#+k \<le> j#+k"
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apply (rule_tac f = "%j. j#+k" in Ord_lt_mono_imp_le_mono, typecheck)
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apply (blast intro: add_lt_mono1 add_type [THEN nat_into_Ord])+
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done
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text{*@{text "\<le>"} monotonicity, both arguments*}
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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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apply (rule add_le_mono1 [THEN le_trans], assumption+)
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apply (subst add_commute, subst add_commute, rule add_le_mono1, assumption+)
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done
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text{*Combinations of less-than and less-than-or-equals*}
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lemma add_lt_le_mono: "[| i<j; k\<le>l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
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apply (rule add_lt_mono1 [THEN lt_trans2], assumption+)
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apply (subst add_commute, subst add_commute, rule add_le_mono1, assumption+)
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   361
done
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   362
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lemma add_le_lt_mono: "[| i\<le>j; k<l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
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   364
by (subst add_commute, subst add_commute, erule add_lt_le_mono, assumption+)
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   365
c0c4af41fa3b Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
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text{*Less-than: in other words, strict in both arguments*}
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lemma add_lt_mono: "[| i<j; k<l; j\<in>nat; l\<in>nat |] ==> i#+k < j#+l"
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apply (rule add_lt_le_mono)
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apply (auto intro: leI)
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done
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(** Subtraction is the inverse of addition. **)
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lemma diff_add_inverse: "(n#+m) #- n = natify(m)"
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apply (subgoal_tac " (natify(n) #+ m) #- natify(n) = natify(m) ")
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apply (rule_tac [2] n = "natify(n) " in nat_induct)
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apply auto
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   378
done
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   379
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lemma diff_add_inverse2: "(m#+n) #- n = natify(m)"
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by (simp add: add_commute [of m] diff_add_inverse)
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lemma diff_cancel: "(k#+m) #- (k#+n) = m #- n"
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apply (subgoal_tac "(natify(k) #+ natify(m)) #- (natify(k) #+ natify(n)) =
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                    natify(m) #- natify(n) ")
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apply (rule_tac [2] n = "natify(k) " in nat_induct)
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   387
apply auto
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   388
done
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   389
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   390
lemma diff_cancel2: "(m#+k) #- (n#+k) = m #- n"
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   391
by (simp add: add_commute [of _ k] diff_cancel)
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   392
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   393
lemma diff_add_0: "n #- (n#+m) = 0"
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   394
apply (subgoal_tac "natify(n) #- (natify(n) #+ natify(m)) = 0")
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   395
apply (rule_tac [2] n = "natify(n) " in nat_induct)
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   396
apply auto
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   397
done
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   398
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lemma pred_0 [simp]: "pred(0) = 0"
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   400
by (simp add: pred_def)
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diff changeset
   401
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diff changeset
   402
lemma eq_succ_imp_eq_m1: "[|i = succ(j); i\<in>nat|] ==> j = i #- 1 & j \<in>nat"
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diff changeset
   403
by simp
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diff changeset
   404
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   405
lemma pred_Un_distrib:
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   406
    "[|i\<in>nat; j\<in>nat|] ==> pred(i \<union> j) = pred(i) \<union> pred(j)"
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diff changeset
   407
apply (erule_tac n=i in natE, simp)
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diff changeset
   408
apply (erule_tac n=j in natE, simp)
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diff changeset
   409
apply (simp add:  succ_Un_distrib [symmetric])
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diff changeset
   410
done
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diff changeset
   411
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   412
lemma pred_type [TC,simp]:
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diff changeset
   413
    "i \<in> nat ==> pred(i) \<in> nat"
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diff changeset
   414
by (simp add: pred_def split: split_nat_case)
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diff changeset
   415
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   416
lemma nat_diff_pred: "[|i\<in>nat; j\<in>nat|] ==> i #- succ(j) = pred(i #- j)";
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   417
apply (rule_tac m=i and n=j in diff_induct)
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   418
apply (auto simp add: pred_def nat_imp_quasinat split: split_nat_case)
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diff changeset
   419
done
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diff changeset
   420
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diff changeset
   421
lemma diff_succ_eq_pred: "i #- succ(j) = pred(i #- j)";
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   422
apply (insert nat_diff_pred [of "natify(i)" "natify(j)"])
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   423
apply (simp add: natify_succ [symmetric])
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   424
done
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parents: 13356
diff changeset
   425
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diff changeset
   426
lemma nat_diff_Un_distrib:
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   427
    "[|i\<in>nat; j\<in>nat; k\<in>nat|] ==> (i \<union> j) #- k = (i#-k) \<union> (j#-k)"
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diff changeset
   428
apply (rule_tac n=k in nat_induct)
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diff changeset
   429
apply (simp_all add: diff_succ_eq_pred pred_Un_distrib)
13361
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parents: 13356
diff changeset
   430
done
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paulson
parents: 13356
diff changeset
   431
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diff changeset
   432
lemma diff_Un_distrib:
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diff changeset
   433
    "[|i\<in>nat; j\<in>nat|] ==> (i \<union> j) #- k = (i#-k) \<union> (j#-k)"
13361
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parents: 13356
diff changeset
   434
by (insert nat_diff_Un_distrib [of i j "natify(k)"], simp)
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diff changeset
   435
5005d34425bb new lemmas
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diff changeset
   436
text{*We actually prove @{term "i #- j #- k = i #- (j #+ k)"}*}
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diff changeset
   437
lemma diff_diff_left [simplified]:
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diff changeset
   438
     "natify(i)#-natify(j)#-k = natify(i) #- (natify(j)#+k)";
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paulson
parents: 13356
diff changeset
   439
by (rule_tac m="natify(i)" and n="natify(j)" in diff_induct, auto)
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parents: 13356
diff changeset
   440
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   441
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   442
(** Lemmas for the CancelNumerals simproc **)
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diff changeset
   443
46821
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   444
lemma eq_add_iff: "(u #+ m = u #+ n) \<longleftrightarrow> (0 #+ m = natify(n))"
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diff changeset
   445
apply auto
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   446
apply (blast dest: add_left_cancel_natify)
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diff changeset
   447
apply (simp add: add_def)
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diff changeset
   448
done
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diff changeset
   449
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   450
lemma less_add_iff: "(u #+ m < u #+ n) \<longleftrightarrow> (0 #+ m < natify(n))"
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paulson
parents: 12114
diff changeset
   451
apply (auto simp add: add_lt_elim1_natify)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   452
apply (drule add_lt_mono1)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   453
apply (auto simp add: add_commute [of u])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   454
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   455
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   456
lemma diff_add_eq: "((u #+ m) #- (u #+ n)) = ((0 #+ m) #- n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   457
by (simp add: diff_cancel)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   458
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   459
(*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   460
lemma eq_cong2: "u = u' ==> (t==u) == (t==u')"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   461
by auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   462
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   463
lemma iff_cong2: "u \<longleftrightarrow> u' ==> (t==u) == (t==u')"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   464
by auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   465
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   466
13356
c9cfe1638bf2 improved presentation markup
paulson
parents: 13328
diff changeset
   467
subsection{*Multiplication*}
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   468
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   469
lemma mult_0 [simp]: "0 #* m = 0"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   470
by (simp add: mult_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   471
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   472
lemma mult_succ [simp]: "succ(m) #* n = n #+ (m #* n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   473
by (simp add: add_def mult_def natify_succ raw_mult_type)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   474
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   475
(*right annihilation in product*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   476
lemma mult_0_right [simp]: "m #* 0 = 0"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   477
apply (unfold mult_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   478
apply (rule_tac n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   479
apply auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   480
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   481
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   482
(*right successor law for multiplication*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   483
lemma mult_succ_right [simp]: "m #* succ(n) = m #+ (m #* n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   484
apply (subgoal_tac "natify(m) #* succ (natify(n)) =
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   485
                    natify(m) #+ (natify(m) #* natify(n))")
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   486
apply (simp (no_asm_use) add: natify_succ add_def mult_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   487
apply (rule_tac n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   488
apply (simp_all add: add_ac)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   489
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   490
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   491
lemma mult_1_natify [simp]: "1 #* n = natify(n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   492
by auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   493
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   494
lemma mult_1_right_natify [simp]: "n #* 1 = natify(n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   495
by auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   496
14060
c0c4af41fa3b Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents: 13784
diff changeset
   497
lemma mult_1: "n \<in> nat ==> 1 #* n = n"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   498
by simp
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   499
14060
c0c4af41fa3b Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents: 13784
diff changeset
   500
lemma mult_1_right: "n \<in> nat ==> n #* 1 = n"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   501
by simp
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   502
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   503
(*Commutative law for multiplication*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   504
lemma mult_commute: "m #* n = n #* m"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   505
apply (subgoal_tac "natify(m) #* natify(n) = natify(n) #* natify(m) ")
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   506
apply (rule_tac [2] n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   507
apply auto
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   508
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   509
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   510
(*addition distributes over multiplication*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   511
lemma add_mult_distrib: "(m #+ n) #* k = (m #* k) #+ (n #* k)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   512
apply (subgoal_tac "(natify(m) #+ natify(n)) #* natify(k) =
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   513
                    (natify(m) #* natify(k)) #+ (natify(n) #* natify(k))")
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   514
apply (rule_tac [2] n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   515
apply (simp_all add: add_assoc [symmetric])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   516
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   517
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   518
(*Distributive law on the left*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   519
lemma add_mult_distrib_left: "k #* (m #+ n) = (k #* m) #+ (k #* n)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   520
apply (subgoal_tac "natify(k) #* (natify(m) #+ natify(n)) =
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   521
                    (natify(k) #* natify(m)) #+ (natify(k) #* natify(n))")
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   522
apply (rule_tac [2] n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   523
apply (simp_all add: add_ac)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   524
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   525
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   526
(*Associative law for multiplication*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   527
lemma mult_assoc: "(m #* n) #* k = m #* (n #* k)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   528
apply (subgoal_tac "(natify(m) #* natify(n)) #* natify(k) =
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   529
                    natify(m) #* (natify(n) #* natify(k))")
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   530
apply (rule_tac [2] n = "natify(m) " in nat_induct)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   531
apply (simp_all add: add_mult_distrib)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   532
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   533
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   534
(*for a/c rewriting*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   535
lemma mult_left_commute: "m #* (n #* k) = n #* (m #* k)"
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   536
apply (rule mult_commute [THEN trans])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   537
apply (rule mult_assoc [THEN trans])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   538
apply (rule mult_commute [THEN subst_context])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   539
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   540
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   541
lemmas mult_ac = mult_assoc mult_commute mult_left_commute
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   542
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   543
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   544
lemma lt_succ_eq_0_disj:
14060
c0c4af41fa3b Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents: 13784
diff changeset
   545
     "[| m\<in>nat; n\<in>nat |]
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   546
      ==> (m < succ(n)) \<longleftrightarrow> (m = 0 | (\<exists>j\<in>nat. m = succ(j) & j < n))"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   547
by (induct_tac "m", auto)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   548
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   549
lemma less_diff_conv [rule_format]:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   550
     "[| j\<in>nat; k\<in>nat |] ==> \<forall>i\<in>nat. (i < j #- k) \<longleftrightarrow> (i #+ k < j)"
13784
b9f6154427a4 tidying (by script)
paulson
parents: 13361
diff changeset
   551
by (erule_tac m = k in diff_induct, auto)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   552
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   553
lemmas nat_typechecks = rec_type nat_0I nat_1I nat_succI Ord_nat
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 12114
diff changeset
   554
0
a5a9c433f639 Initial revision
clasohm
parents:
diff changeset
   555
end