# Theory Arith

```(*  Title:      ZF/Arith.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

(*"Difference" is subtraction of natural numbers.
There are no negative numbers; we have
m #- n = 0  iff  m<=n   and     m #- n = succ(k) iff m>n.
Also, rec(m, 0, λz w.z) is pred(m).
*)

section‹Arithmetic Operators and Their Definitions›

theory Arith imports Univ begin

definition
pred   :: "i⇒i"    (*inverse of succ*)  where
"pred(y) ≡ nat_case(0, λx. x, y)"

definition
natify :: "i⇒i"    (*coerces non-nats to nats*)  where
"natify ≡ Vrecursor(λf a. if a = succ(pred(a)) then succ(f`pred(a))
else 0)"

consts
raw_diff  :: "[i,i]⇒i"
raw_mult  :: "[i,i]⇒i"

primrec

primrec
raw_diff_0:     "raw_diff(m, 0) = m"
raw_diff_succ:  "raw_diff(m, succ(n)) =
nat_case(0, λx. x, raw_diff(m, n))"

primrec
"raw_mult(0, n) = 0"
"raw_mult(succ(m), n) = raw_add (n, raw_mult(m, n))"

definition
add :: "[i,i]⇒i"                    (infixl ‹#+› 65)  where
"m #+ n ≡ raw_add (natify(m), natify(n))"

definition
diff :: "[i,i]⇒i"                    (infixl ‹#-› 65)  where
"m #- n ≡ raw_diff (natify(m), natify(n))"

definition
mult :: "[i,i]⇒i"                    (infixl ‹#*› 70)  where
"m #* n ≡ raw_mult (natify(m), natify(n))"

definition
raw_div  :: "[i,i]⇒i"  where
"raw_div (m, n) ≡
transrec(m, λj f. if j<n | n=0 then 0 else succ(f`(j#-n)))"

definition
raw_mod  :: "[i,i]⇒i"  where
"raw_mod (m, n) ≡
transrec(m, λj f. if j<n | n=0 then j else f`(j#-n))"

definition
div  :: "[i,i]⇒i"                    (infixl ‹div› 70)  where
"m div n ≡ raw_div (natify(m), natify(n))"

definition
mod  :: "[i,i]⇒i"                    (infixl ‹mod› 70)  where
"m mod n ≡ raw_mod (natify(m), natify(n))"

declare rec_type [simp]
nat_0_le [simp]

lemma zero_lt_lemma: "⟦0<k; k ∈ nat⟧ ⟹ ∃j∈nat. k = succ(j)"
apply (erule rev_mp)
apply (induct_tac "k", auto)
done

(* @{term"⟦0 < k; k ∈ nat; ⋀j. ⟦j ∈ nat; k = succ(j)⟧ ⟹ Q⟧ ⟹ Q"} *)
lemmas zero_lt_natE = zero_lt_lemma [THEN bexE]

subsection‹‹natify›, the Coercion to \<^term>‹nat››

lemma pred_succ_eq [simp]: "pred(succ(y)) = y"
by (unfold pred_def, auto)

lemma natify_succ: "natify(succ(x)) = succ(natify(x))"
by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)

lemma natify_0 [simp]: "natify(0) = 0"
by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)

lemma natify_non_succ: "∀z. x ≠ succ(z) ⟹ natify(x) = 0"
by (rule natify_def [THEN def_Vrecursor, THEN trans], auto)

lemma natify_in_nat [iff,TC]: "natify(x) ∈ nat"
apply (rule_tac a=x in eps_induct)
apply (case_tac "∃z. x = succ(z)")
apply (auto simp add: natify_succ natify_non_succ)
done

lemma natify_ident [simp]: "n ∈ nat ⟹ natify(n) = n"
apply (induct_tac "n")
done

lemma natify_eqE: "⟦natify(x) = y;  x ∈ nat⟧ ⟹ x=y"
by auto

(*** Collapsing rules: to remove natify from arithmetic expressions ***)

lemma natify_idem [simp]: "natify(natify(x)) = natify(x)"
by simp

lemma add_natify1 [simp]: "natify(m) #+ n = m #+ n"

lemma add_natify2 [simp]: "m #+ natify(n) = m #+ n"

(** Multiplication **)

lemma mult_natify1 [simp]: "natify(m) #* n = m #* n"

lemma mult_natify2 [simp]: "m #* natify(n) = m #* n"

(** Difference **)

lemma diff_natify1 [simp]: "natify(m) #- n = m #- n"

lemma diff_natify2 [simp]: "m #- natify(n) = m #- n"

(** Remainder **)

lemma mod_natify1 [simp]: "natify(m) mod n = m mod n"

lemma mod_natify2 [simp]: "m mod natify(n) = m mod n"

(** Quotient **)

lemma div_natify1 [simp]: "natify(m) div n = m div n"

lemma div_natify2 [simp]: "m div natify(n) = m div n"

subsection‹Typing rules›

by (induct_tac "m", auto)

lemma add_type [iff,TC]: "m #+ n ∈ nat"

(** Multiplication **)

lemma raw_mult_type: "⟦m∈nat;  n∈nat⟧ ⟹ raw_mult (m, n) ∈ nat"
apply (induct_tac "m")
done

lemma mult_type [iff,TC]: "m #* n ∈ nat"

(** Difference **)

lemma raw_diff_type: "⟦m∈nat;  n∈nat⟧ ⟹ raw_diff (m, n) ∈ nat"
by (induct_tac "n", auto)

lemma diff_type [iff,TC]: "m #- n ∈ nat"

lemma diff_0_eq_0 [simp]: "0 #- n = 0"
unfolding diff_def
apply (rule natify_in_nat [THEN nat_induct], auto)
done

(*Must simplify BEFORE the induction: else we get a critical pair*)
lemma diff_succ_succ [simp]: "succ(m) #- succ(n) = m #- n"
apply (rule_tac x1 = n in natify_in_nat [THEN nat_induct], auto)
done

(*This defining property is no longer wanted*)
declare raw_diff_succ [simp del]

(*Natify has weakened this law, compared with the older approach*)
lemma diff_0 [simp]: "m #- 0 = natify(m)"

lemma diff_le_self: "m∈nat ⟹ (m #- n) ≤ m"
apply (subgoal_tac " (m #- natify (n)) ≤ m")
apply (rule_tac [2] m = m and n = "natify (n) " in diff_induct)
apply (erule_tac [6] leE)
done

(*Natify has weakened this law, compared with the older approach*)
lemma add_0_natify [simp]: "0 #+ m = natify(m)"

lemma add_succ [simp]: "succ(m) #+ n = succ(m #+ n)"

lemma add_0: "m ∈ nat ⟹ 0 #+ m = m"
by simp

lemma add_assoc: "(m #+ n) #+ k = m #+ (n #+ k)"
apply (subgoal_tac "(natify(m) #+ natify(n)) #+ natify(k) =
natify(m) #+ (natify(n) #+ natify(k))")
apply (rule_tac [2] n = "natify(m)" in nat_induct)
apply auto
done

(*The following two lemmas are used for add_commute and sometimes
elsewhere, since they are safe for rewriting.*)
lemma add_0_right_natify [simp]: "m #+ 0 = natify(m)"
apply (subgoal_tac "natify(m) #+ 0 = natify(m)")
apply (rule_tac [2] n = "natify(m)" in nat_induct)
apply auto
done

lemma add_succ_right [simp]: "m #+ succ(n) = succ(m #+ n)"
apply (rule_tac n = "natify(m) " in nat_induct)
done

lemma add_0_right: "m ∈ nat ⟹ m #+ 0 = m"
by auto

lemma add_commute: "m #+ n = n #+ m"
apply (subgoal_tac "natify(m) #+ natify(n) = natify(n) #+ natify(m) ")
apply (rule_tac [2] n = "natify(m) " in nat_induct)
apply auto
done

(*for a/c rewriting*)
done

(*Cancellation law on the left*)
apply (erule rev_mp)
apply (induct_tac "k", auto)
done

lemma add_left_cancel_natify: "k #+ m = k #+ n ⟹ natify(m) = natify(n)"
done

"⟦i = j;  i #+ m = j #+ n;  m∈nat;  n∈nat⟧ ⟹ m = n"

(*Thanks to Sten Agerholm*)
lemma add_le_elim1_natify: "k#+m ≤ k#+n ⟹ natify(m) ≤ natify(n)"
apply (rule_tac P = "natify(k) #+m ≤ natify(k) #+n" in rev_mp)
apply (rule_tac [2] n = "natify(k) " in nat_induct)
apply auto
done

lemma add_le_elim1: "⟦k#+m ≤ k#+n; m ∈ nat; n ∈ nat⟧ ⟹ m ≤ n"

lemma add_lt_elim1_natify: "k#+m < k#+n ⟹ natify(m) < natify(n)"
apply (rule_tac P = "natify(k) #+m < natify(k) #+n" in rev_mp)
apply (rule_tac [2] n = "natify(k) " in nat_induct)
apply auto
done

lemma add_lt_elim1: "⟦k#+m < k#+n; m ∈ nat; n ∈ nat⟧ ⟹ m < n"

lemma zero_less_add: "⟦n ∈ nat; m ∈ nat⟧ ⟹ 0 < m #+ n ⟷ (0<m | 0<n)"
by (induct_tac "n", auto)

(*strict, in 1st argument; proof is by rule induction on 'less than'.
Still need j∈nat, for consider j = omega.  Then we can have i<omega,
which is the same as i∈nat, but natify(j)=0, so the conclusion fails.*)
lemma add_lt_mono1: "⟦i<j; j∈nat⟧ ⟹ i#+k < j#+k"
apply (frule lt_nat_in_nat, assumption)
apply (erule succ_lt_induct)
done

text‹strict, in second argument›
lemma add_lt_mono2: "⟦i<j; j∈nat⟧ ⟹ k#+i < k#+j"

text‹A [clumsy] way of lifting < monotonicity to ‹≤› monotonicity›
lemma Ord_lt_mono_imp_le_mono:
assumes lt_mono: "⋀i j. ⟦i<j; j:k⟧ ⟹ f(i) < f(j)"
and ford:    "⋀i. i:k ⟹ Ord(f(i))"
and leij:    "i ≤ j"
and jink:    "j:k"
shows "f(i) ≤ f(j)"
apply (insert leij jink)
apply (blast intro!: leCI lt_mono ford elim!: leE)
done

text‹‹≤› monotonicity, 1st argument›
lemma add_le_mono1: "⟦i ≤ j; j∈nat⟧ ⟹ i#+k ≤ j#+k"
apply (rule_tac f = "λj. j#+k" in Ord_lt_mono_imp_le_mono, typecheck)
done

text‹‹≤› monotonicity, both arguments›
lemma add_le_mono: "⟦i ≤ j; k ≤ l; j∈nat; l∈nat⟧ ⟹ i#+k ≤ j#+l"
apply (rule add_le_mono1 [THEN le_trans], assumption+)
done

text‹Combinations of less-than and less-than-or-equals›

lemma add_lt_le_mono: "⟦i<j; k≤l; j∈nat; l∈nat⟧ ⟹ i#+k < j#+l"
apply (rule add_lt_mono1 [THEN lt_trans2], assumption+)
done

lemma add_le_lt_mono: "⟦i≤j; k<l; j∈nat; l∈nat⟧ ⟹ i#+k < j#+l"

text‹Less-than: in other words, strict in both arguments›
lemma add_lt_mono: "⟦i<j; k<l; j∈nat; l∈nat⟧ ⟹ i#+k < j#+l"
apply (auto intro: leI)
done

(** Subtraction is the inverse of addition. **)

lemma diff_add_inverse: "(n#+m) #- n = natify(m)"
apply (subgoal_tac " (natify(n) #+ m) #- natify(n) = natify(m) ")
apply (rule_tac [2] n = "natify(n) " in nat_induct)
apply auto
done

lemma diff_add_inverse2: "(m#+n) #- n = natify(m)"

lemma diff_cancel: "(k#+m) #- (k#+n) = m #- n"
apply (subgoal_tac "(natify(k) #+ natify(m)) #- (natify(k) #+ natify(n)) =
natify(m) #- natify(n) ")
apply (rule_tac [2] n = "natify(k) " in nat_induct)
apply auto
done

lemma diff_cancel2: "(m#+k) #- (n#+k) = m #- n"

lemma diff_add_0: "n #- (n#+m) = 0"
apply (subgoal_tac "natify(n) #- (natify(n) #+ natify(m)) = 0")
apply (rule_tac [2] n = "natify(n) " in nat_induct)
apply auto
done

lemma pred_0 [simp]: "pred(0) = 0"

lemma eq_succ_imp_eq_m1: "⟦i = succ(j); i∈nat⟧ ⟹ j = i #- 1 ∧ j ∈nat"
by simp

lemma pred_Un_distrib:
"⟦i∈nat; j∈nat⟧ ⟹ pred(i ∪ j) = pred(i) ∪ pred(j)"
apply (erule_tac n=i in natE, simp)
apply (erule_tac n=j in natE, simp)
done

lemma pred_type [TC,simp]:
"i ∈ nat ⟹ pred(i) ∈ nat"
by (simp add: pred_def split: split_nat_case)

lemma nat_diff_pred: "⟦i∈nat; j∈nat⟧ ⟹ i #- succ(j) = pred(i #- j)"
apply (rule_tac m=i and n=j in diff_induct)
apply (auto simp add: pred_def nat_imp_quasinat split: split_nat_case)
done

lemma diff_succ_eq_pred: "i #- succ(j) = pred(i #- j)"
apply (insert nat_diff_pred [of "natify(i)" "natify(j)"])
done

lemma nat_diff_Un_distrib:
"⟦i∈nat; j∈nat; k∈nat⟧ ⟹ (i ∪ j) #- k = (i#-k) ∪ (j#-k)"
apply (rule_tac n=k in nat_induct)
done

lemma diff_Un_distrib:
"⟦i∈nat; j∈nat⟧ ⟹ (i ∪ j) #- k = (i#-k) ∪ (j#-k)"
by (insert nat_diff_Un_distrib [of i j "natify(k)"], simp)

text‹We actually prove \<^term>‹i #- j #- k = i #- (j #+ k)››
lemma diff_diff_left [simplified]:
"natify(i)#-natify(j)#-k = natify(i) #- (natify(j)#+k)"
by (rule_tac m="natify(i)" and n="natify(j)" in diff_induct, auto)

(** Lemmas for the CancelNumerals simproc **)

lemma eq_add_iff: "(u #+ m = u #+ n) ⟷ (0 #+ m = natify(n))"
apply auto
done

lemma less_add_iff: "(u #+ m < u #+ n) ⟷ (0 #+ m < natify(n))"
done

lemma diff_add_eq: "((u #+ m) #- (u #+ n)) = ((0 #+ m) #- n)"

(*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
lemma eq_cong2: "u = u' ⟹ (t≡u) ≡ (t≡u')"
by auto

lemma iff_cong2: "u ⟷ u' ⟹ (t≡u) ≡ (t≡u')"
by auto

subsection‹Multiplication›

lemma mult_0 [simp]: "0 #* m = 0"

lemma mult_succ [simp]: "succ(m) #* n = n #+ (m #* n)"

(*right annihilation in product*)
lemma mult_0_right [simp]: "m #* 0 = 0"
unfolding mult_def
apply (rule_tac n = "natify(m) " in nat_induct)
apply auto
done

(*right successor law for multiplication*)
lemma mult_succ_right [simp]: "m #* succ(n) = m #+ (m #* n)"
apply (subgoal_tac "natify(m) #* succ (natify(n)) =
natify(m) #+ (natify(m) #* natify(n))")
apply (rule_tac n = "natify(m) " in nat_induct)
done

lemma mult_1_natify [simp]: "1 #* n = natify(n)"
by auto

lemma mult_1_right_natify [simp]: "n #* 1 = natify(n)"
by auto

lemma mult_1: "n ∈ nat ⟹ 1 #* n = n"
by simp

lemma mult_1_right: "n ∈ nat ⟹ n #* 1 = n"
by simp

(*Commutative law for multiplication*)
lemma mult_commute: "m #* n = n #* m"
apply (subgoal_tac "natify(m) #* natify(n) = natify(n) #* natify(m) ")
apply (rule_tac [2] n = "natify(m) " in nat_induct)
apply auto
done

lemma add_mult_distrib: "(m #+ n) #* k = (m #* k) #+ (n #* k)"
apply (subgoal_tac "(natify(m) #+ natify(n)) #* natify(k) =
(natify(m) #* natify(k)) #+ (natify(n) #* natify(k))")
apply (rule_tac [2] n = "natify(m) " in nat_induct)
done

(*Distributive law on the left*)
lemma add_mult_distrib_left: "k #* (m #+ n) = (k #* m) #+ (k #* n)"
apply (subgoal_tac "natify(k) #* (natify(m) #+ natify(n)) =
(natify(k) #* natify(m)) #+ (natify(k) #* natify(n))")
apply (rule_tac [2] n = "natify(m) " in nat_induct)
done

(*Associative law for multiplication*)
lemma mult_assoc: "(m #* n) #* k = m #* (n #* k)"
apply (subgoal_tac "(natify(m) #* natify(n)) #* natify(k) =
natify(m) #* (natify(n) #* natify(k))")
apply (rule_tac [2] n = "natify(m) " in nat_induct)
done

(*for a/c rewriting*)
lemma mult_left_commute: "m #* (n #* k) = n #* (m #* k)"
apply (rule mult_commute [THEN trans])
apply (rule mult_assoc [THEN trans])
apply (rule mult_commute [THEN subst_context])
done

lemmas mult_ac = mult_assoc mult_commute mult_left_commute

lemma lt_succ_eq_0_disj:
"⟦m∈nat; n∈nat⟧
⟹ (m < succ(n)) ⟷ (m = 0 | (∃j∈nat. m = succ(j) ∧ j < n))"
by (induct_tac "m", auto)

lemma less_diff_conv [rule_format]:
"⟦j∈nat; k∈nat⟧ ⟹ ∀i∈nat. (i < j #- k) ⟷ (i #+ k < j)"
by (erule_tac m = k in diff_induct, auto)

lemmas nat_typechecks = rec_type nat_0I nat_1I nat_succI Ord_nat

end
```