--- a/src/ZF/ArithSimp.thy Fri Jun 28 20:01:09 2002 +0200
+++ b/src/ZF/ArithSimp.thy Sat Jun 29 21:33:06 2002 +0200
@@ -8,4 +8,585 @@
theory ArithSimp = Arith
files "arith_data.ML":
+
+(*** Difference ***)
+
+lemma diff_self_eq_0: "m #- m = 0"
+apply (subgoal_tac "natify (m) #- natify (m) = 0")
+apply (rule_tac [2] natify_in_nat [THEN nat_induct], auto)
+done
+
+(**Addition is the inverse of subtraction**)
+
+(*We need m:nat even if we replace the RHS by natify(m), for consider e.g.
+ n=2, m=omega; then n + (m-n) = 2 + (0-2) = 2 ~= 0 = natify(m).*)
+lemma add_diff_inverse: "[| n le m; m:nat |] ==> n #+ (m#-n) = m"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (erule rev_mp)
+apply (rule_tac m = "m" and n = "n" in diff_induct, auto)
+done
+
+lemma add_diff_inverse2: "[| n le m; m:nat |] ==> (m#-n) #+ n = m"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (simp (no_asm_simp) add: add_commute add_diff_inverse)
+done
+
+(*Proof is IDENTICAL to that of add_diff_inverse*)
+lemma diff_succ: "[| n le m; m:nat |] ==> succ(m) #- n = succ(m#-n)"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (erule rev_mp)
+apply (rule_tac m = "m" and n = "n" in diff_induct)
+apply (simp_all (no_asm_simp))
+done
+
+lemma zero_less_diff [simp]:
+ "[| m: nat; n: nat |] ==> 0 < (n #- m) <-> m<n"
+apply (rule_tac m = "m" and n = "n" in diff_induct)
+apply (simp_all (no_asm_simp))
+done
+
+
+(** Difference distributes over multiplication **)
+
+lemma diff_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] m = "natify (m) " and n = "natify (n) " in diff_induct)
+apply (simp_all add: diff_cancel)
+done
+
+lemma diff_mult_distrib2: "k #* (m #- n) = (k #* m) #- (k #* n)"
+apply (simp (no_asm) add: mult_commute [of k] diff_mult_distrib)
+done
+
+
+(*** Remainder ***)
+
+(*We need m:nat even with natify*)
+lemma div_termination: "[| 0<n; n le m; m:nat |] ==> m #- n < m"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (erule rev_mp)
+apply (erule rev_mp)
+apply (rule_tac m = "m" and n = "n" in diff_induct)
+apply (simp_all (no_asm_simp) add: diff_le_self)
+done
+
+(*for mod and div*)
+lemmas div_rls =
+ nat_typechecks Ord_transrec_type apply_funtype
+ div_termination [THEN ltD]
+ nat_into_Ord not_lt_iff_le [THEN iffD1]
+
+lemma raw_mod_type: "[| m:nat; n:nat |] ==> raw_mod (m, n) : nat"
+apply (unfold raw_mod_def)
+apply (rule Ord_transrec_type)
+apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff])
+apply (blast intro: div_rls)
+done
+
+lemma mod_type [TC,iff]: "m mod n : nat"
+apply (unfold mod_def)
+apply (simp (no_asm) add: mod_def raw_mod_type)
+done
+
+
+(** Aribtrary definitions for division by zero. Useful to simplify
+ certain equations **)
+
+lemma DIVISION_BY_ZERO_DIV: "a div 0 = 0"
+apply (unfold div_def)
+apply (rule raw_div_def [THEN def_transrec, THEN trans])
+apply (simp (no_asm_simp))
+done (*NOT for adding to default simpset*)
+
+lemma DIVISION_BY_ZERO_MOD: "a mod 0 = natify(a)"
+apply (unfold mod_def)
+apply (rule raw_mod_def [THEN def_transrec, THEN trans])
+apply (simp (no_asm_simp))
+done (*NOT for adding to default simpset*)
+
+lemma raw_mod_less: "m<n ==> raw_mod (m,n) = m"
+apply (rule raw_mod_def [THEN def_transrec, THEN trans])
+apply (simp (no_asm_simp) add: div_termination [THEN ltD])
+done
+
+lemma mod_less [simp]: "[| m<n; n : nat |] ==> m mod n = m"
+apply (frule lt_nat_in_nat, assumption)
+apply (simp (no_asm_simp) add: mod_def raw_mod_less)
+done
+
+lemma raw_mod_geq:
+ "[| 0<n; n le m; m:nat |] ==> raw_mod (m, n) = raw_mod (m#-n, n)"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (rule raw_mod_def [THEN def_transrec, THEN trans])
+apply (simp add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2], blast)
+done
+
+
+lemma mod_geq: "[| n le m; m:nat |] ==> m mod n = (m#-n) mod n"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (case_tac "n=0")
+ apply (simp add: DIVISION_BY_ZERO_MOD)
+apply (simp add: mod_def raw_mod_geq nat_into_Ord [THEN Ord_0_lt_iff])
+done
+
+
+(*** Division ***)
+
+lemma raw_div_type: "[| m:nat; n:nat |] ==> raw_div (m, n) : nat"
+apply (unfold raw_div_def)
+apply (rule Ord_transrec_type)
+apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff])
+apply (blast intro: div_rls)
+done
+
+lemma div_type [TC,iff]: "m div n : nat"
+apply (unfold div_def)
+apply (simp (no_asm) add: div_def raw_div_type)
+done
+
+lemma raw_div_less: "m<n ==> raw_div (m,n) = 0"
+apply (rule raw_div_def [THEN def_transrec, THEN trans])
+apply (simp (no_asm_simp) add: div_termination [THEN ltD])
+done
+
+lemma div_less [simp]: "[| m<n; n : nat |] ==> m div n = 0"
+apply (frule lt_nat_in_nat, assumption)
+apply (simp (no_asm_simp) add: div_def raw_div_less)
+done
+
+lemma raw_div_geq: "[| 0<n; n le m; m:nat |] ==> raw_div(m,n) = succ(raw_div(m#-n, n))"
+apply (subgoal_tac "n ~= 0")
+prefer 2 apply blast
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (rule raw_div_def [THEN def_transrec, THEN trans])
+apply (simp add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2] )
+done
+
+lemma div_geq [simp]:
+ "[| 0<n; n le m; m:nat |] ==> m div n = succ ((m#-n) div n)"
+apply (frule lt_nat_in_nat, erule nat_succI)
+apply (simp (no_asm_simp) add: div_def raw_div_geq)
+done
+
+declare div_less [simp] div_geq [simp]
+
+
+(*A key result*)
+lemma mod_div_lemma: "[| m: nat; n: nat |] ==> (m div n)#*n #+ m mod n = m"
+apply (case_tac "n=0")
+ apply (simp add: DIVISION_BY_ZERO_MOD)
+apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff])
+apply (erule complete_induct)
+apply (case_tac "x<n")
+txt{*case x<n*}
+apply (simp (no_asm_simp))
+txt{*case n le x*}
+apply (simp add: not_lt_iff_le add_assoc mod_geq div_termination [THEN ltD] add_diff_inverse)
+done
+
+lemma mod_div_equality_natify: "(m div n)#*n #+ m mod n = natify(m)"
+apply (subgoal_tac " (natify (m) div natify (n))#*natify (n) #+ natify (m) mod natify (n) = natify (m) ")
+apply force
+apply (subst mod_div_lemma, auto)
+done
+
+lemma mod_div_equality: "m: nat ==> (m div n)#*n #+ m mod n = m"
+apply (simp (no_asm_simp) add: mod_div_equality_natify)
+done
+
+
+(*** Further facts about mod (mainly for mutilated chess board) ***)
+
+lemma mod_succ_lemma:
+ "[| 0<n; m:nat; n:nat |]
+ ==> succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))"
+apply (erule complete_induct)
+apply (case_tac "succ (x) <n")
+txt{* case succ(x) < n *}
+ apply (simp (no_asm_simp) add: nat_le_refl [THEN lt_trans] succ_neq_self)
+ apply (simp add: ltD [THEN mem_imp_not_eq])
+txt{* case n le succ(x) *}
+apply (simp add: mod_geq not_lt_iff_le)
+apply (erule leE)
+ apply (simp (no_asm_simp) add: mod_geq div_termination [THEN ltD] diff_succ)
+txt{*equality case*}
+apply (simp add: diff_self_eq_0)
+done
+
+lemma mod_succ:
+ "n:nat ==> succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))"
+apply (case_tac "n=0")
+ apply (simp (no_asm_simp) add: natify_succ DIVISION_BY_ZERO_MOD)
+apply (subgoal_tac "natify (succ (m)) mod n = (if succ (natify (m) mod n) = n then 0 else succ (natify (m) mod n))")
+ prefer 2
+ apply (subst natify_succ)
+ apply (rule mod_succ_lemma)
+ apply (auto simp del: natify_succ simp add: nat_into_Ord [THEN Ord_0_lt_iff])
+done
+
+lemma mod_less_divisor: "[| 0<n; n:nat |] ==> m mod n < n"
+apply (subgoal_tac "natify (m) mod n < n")
+apply (rule_tac [2] i = "natify (m) " in complete_induct)
+apply (case_tac [3] "x<n", auto)
+txt{* case n le x*}
+apply (simp add: mod_geq not_lt_iff_le div_termination [THEN ltD])
+done
+
+lemma mod_1_eq [simp]: "m mod 1 = 0"
+by (cut_tac n = "1" in mod_less_divisor, auto)
+
+lemma mod2_cases: "b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"
+apply (subgoal_tac "k mod 2: 2")
+ prefer 2 apply (simp add: mod_less_divisor [THEN ltD])
+apply (drule ltD, auto)
+done
+
+lemma mod2_succ_succ [simp]: "succ(succ(m)) mod 2 = m mod 2"
+apply (subgoal_tac "m mod 2: 2")
+ prefer 2 apply (simp add: mod_less_divisor [THEN ltD])
+apply (auto simp add: mod_succ)
+done
+
+lemma mod2_add_more [simp]: "(m#+m#+n) mod 2 = n mod 2"
+apply (subgoal_tac " (natify (m) #+natify (m) #+n) mod 2 = n mod 2")
+apply (rule_tac [2] n = "natify (m) " in nat_induct)
+apply auto
+done
+
+lemma mod2_add_self [simp]: "(m#+m) mod 2 = 0"
+by (cut_tac n = "0" in mod2_add_more, auto)
+
+
+(**** Additional theorems about "le" ****)
+
+lemma add_le_self: "m:nat ==> m le (m #+ n)"
+apply (simp (no_asm_simp))
+done
+
+lemma add_le_self2: "m:nat ==> m le (n #+ m)"
+apply (simp (no_asm_simp))
+done
+
+(*** Monotonicity of Multiplication ***)
+
+lemma mult_le_mono1: "[| i le j; j:nat |] ==> (i#*k) le (j#*k)"
+apply (subgoal_tac "natify (i) #*natify (k) le j#*natify (k) ")
+apply (frule_tac [2] lt_nat_in_nat)
+apply (rule_tac [3] n = "natify (k) " in nat_induct)
+apply (simp_all add: add_le_mono)
+done
+
+(* le monotonicity, BOTH arguments*)
+lemma mult_le_mono: "[| i le j; k le l; j:nat; l:nat |] ==> i#*k le j#*l"
+apply (rule mult_le_mono1 [THEN le_trans], assumption+)
+apply (subst mult_commute, subst mult_commute, rule mult_le_mono1, assumption+)
+done
+
+(*strict, in 1st argument; proof is by induction on k>0.
+ I can't see how to relax the typing conditions.*)
+lemma mult_lt_mono2: "[| i<j; 0<k; j:nat; k:nat |] ==> k#*i < k#*j"
+apply (erule zero_lt_natE)
+apply (frule_tac [2] lt_nat_in_nat)
+apply (simp_all (no_asm_simp))
+apply (induct_tac "x")
+apply (simp_all (no_asm_simp) add: add_lt_mono)
+done
+
+lemma mult_lt_mono1: "[| i<j; 0<k; j:nat; k:nat |] ==> i#*k < j#*k"
+apply (simp (no_asm_simp) add: mult_lt_mono2 mult_commute [of _ k])
+done
+
+lemma add_eq_0_iff [iff]: "m#+n = 0 <-> natify(m)=0 & natify(n)=0"
+apply (subgoal_tac "natify (m) #+ natify (n) = 0 <-> natify (m) =0 & natify (n) =0")
+apply (rule_tac [2] n = "natify (m) " in natE)
+ apply (rule_tac [4] n = "natify (n) " in natE)
+apply auto
+done
+
+lemma zero_lt_mult_iff [iff]: "0 < m#*n <-> 0 < natify(m) & 0 < natify(n)"
+apply (subgoal_tac "0 < natify (m) #*natify (n) <-> 0 < natify (m) & 0 < natify (n) ")
+apply (rule_tac [2] n = "natify (m) " in natE)
+ apply (rule_tac [4] n = "natify (n) " in natE)
+ apply (rule_tac [3] n = "natify (n) " in natE)
+apply auto
+done
+
+lemma mult_eq_1_iff [iff]: "m#*n = 1 <-> natify(m)=1 & natify(n)=1"
+apply (subgoal_tac "natify (m) #* natify (n) = 1 <-> natify (m) =1 & natify (n) =1")
+apply (rule_tac [2] n = "natify (m) " in natE)
+ apply (rule_tac [4] n = "natify (n) " in natE)
+apply auto
+done
+
+
+lemma mult_is_zero: "[|m: nat; n: nat|] ==> (m #* n = 0) <-> (m = 0 | n = 0)"
+apply auto
+apply (erule natE)
+apply (erule_tac [2] natE, auto)
+done
+
+lemma mult_is_zero_natify [iff]:
+ "(m #* n = 0) <-> (natify(m) = 0 | natify(n) = 0)"
+apply (cut_tac m = "natify (m) " and n = "natify (n) " in mult_is_zero)
+apply auto
+done
+
+
+(** Cancellation laws for common factors in comparisons **)
+
+lemma mult_less_cancel_lemma:
+ "[| k: nat; m: nat; n: nat |] ==> (m#*k < n#*k) <-> (0<k & m<n)"
+apply (safe intro!: mult_lt_mono1)
+apply (erule natE, auto)
+apply (rule not_le_iff_lt [THEN iffD1])
+apply (drule_tac [3] not_le_iff_lt [THEN [2] rev_iffD2])
+prefer 5 apply (blast intro: mult_le_mono1, auto)
+done
+
+lemma mult_less_cancel2 [simp]:
+ "(m#*k < n#*k) <-> (0 < natify(k) & natify(m) < natify(n))"
+apply (rule iff_trans)
+apply (rule_tac [2] mult_less_cancel_lemma, auto)
+done
+
+lemma mult_less_cancel1 [simp]:
+ "(k#*m < k#*n) <-> (0 < natify(k) & natify(m) < natify(n))"
+apply (simp (no_asm) add: mult_less_cancel2 mult_commute [of k])
+done
+
+lemma mult_le_cancel2 [simp]: "(m#*k le n#*k) <-> (0 < natify(k) --> natify(m) le natify(n))"
+apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
+apply auto
+done
+
+lemma mult_le_cancel1 [simp]: "(k#*m le k#*n) <-> (0 < natify(k) --> natify(m) le natify(n))"
+apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
+apply auto
+done
+
+lemma mult_le_cancel_le1: "k : nat ==> k #* m le k \<longleftrightarrow> (0 < k \<longrightarrow> natify(m) le 1)"
+by (cut_tac k = "k" and m = "m" and n = "1" in mult_le_cancel1, auto)
+
+lemma Ord_eq_iff_le: "[| Ord(m); Ord(n) |] ==> m=n <-> (m le n & n le m)"
+by (blast intro: le_anti_sym)
+
+lemma mult_cancel2_lemma:
+ "[| k: nat; m: nat; n: nat |] ==> (m#*k = n#*k) <-> (m=n | k=0)"
+apply (simp (no_asm_simp) add: Ord_eq_iff_le [of "m#*k"] Ord_eq_iff_le [of m])
+apply (auto simp add: Ord_0_lt_iff)
+done
+
+lemma mult_cancel2 [simp]:
+ "(m#*k = n#*k) <-> (natify(m) = natify(n) | natify(k) = 0)"
+apply (rule iff_trans)
+apply (rule_tac [2] mult_cancel2_lemma, auto)
+done
+
+lemma mult_cancel1 [simp]:
+ "(k#*m = k#*n) <-> (natify(m) = natify(n) | natify(k) = 0)"
+apply (simp (no_asm) add: mult_cancel2 mult_commute [of k])
+done
+
+
+(** Cancellation law for division **)
+
+lemma div_cancel_raw:
+ "[| 0<n; 0<k; k:nat; m:nat; n:nat |] ==> (k#*m) div (k#*n) = m div n"
+apply (erule_tac i = "m" in complete_induct)
+apply (case_tac "x<n")
+ apply (simp add: div_less zero_lt_mult_iff mult_lt_mono2)
+apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono]
+ div_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD])
+done
+
+lemma div_cancel:
+ "[| 0 < natify(n); 0 < natify(k) |] ==> (k#*m) div (k#*n) = m div n"
+apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)"
+ in div_cancel_raw)
+apply auto
+done
+
+
+(** Distributive law for remainder (mod) **)
+
+lemma mult_mod_distrib_raw:
+ "[| k:nat; m:nat; n:nat |] ==> (k#*m) mod (k#*n) = k #* (m mod n)"
+apply (case_tac "k=0")
+ apply (simp add: DIVISION_BY_ZERO_MOD)
+apply (case_tac "n=0")
+ apply (simp add: DIVISION_BY_ZERO_MOD)
+apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff])
+apply (erule_tac i = "m" in complete_induct)
+apply (case_tac "x<n")
+ apply (simp (no_asm_simp) add: mod_less zero_lt_mult_iff mult_lt_mono2)
+apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono]
+ mod_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD])
+done
+
+lemma mod_mult_distrib2: "k #* (m mod n) = (k#*m) mod (k#*n)"
+apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)"
+ in mult_mod_distrib_raw)
+apply auto
+done
+
+lemma mult_mod_distrib: "(m mod n) #* k = (m#*k) mod (n#*k)"
+apply (simp (no_asm) add: mult_commute mod_mult_distrib2)
+done
+
+lemma mod_add_self2_raw: "n \<in> nat ==> (m #+ n) mod n = m mod n"
+apply (subgoal_tac " (n #+ m) mod n = (n #+ m #- n) mod n")
+apply (simp add: add_commute)
+apply (subst mod_geq [symmetric], auto)
+done
+
+lemma mod_add_self2 [simp]: "(m #+ n) mod n = m mod n"
+apply (cut_tac n = "natify (n) " in mod_add_self2_raw)
+apply auto
+done
+
+lemma mod_add_self1 [simp]: "(n#+m) mod n = m mod n"
+apply (simp (no_asm_simp) add: add_commute mod_add_self2)
+done
+
+lemma mod_mult_self1_raw: "k \<in> nat ==> (m #+ k#*n) mod n = m mod n"
+apply (erule nat_induct)
+apply (simp_all (no_asm_simp) add: add_left_commute [of _ n])
+done
+
+lemma mod_mult_self1 [simp]: "(m #+ k#*n) mod n = m mod n"
+apply (cut_tac k = "natify (k) " in mod_mult_self1_raw)
+apply auto
+done
+
+lemma mod_mult_self2 [simp]: "(m #+ n#*k) mod n = m mod n"
+apply (simp (no_asm) add: mult_commute mod_mult_self1)
+done
+
+(*Lemma for gcd*)
+lemma mult_eq_self_implies_10: "m = m#*n ==> natify(n)=1 | m=0"
+apply (subgoal_tac "m: nat")
+ prefer 2
+ apply (erule ssubst)
+ apply simp
+apply (rule disjCI)
+apply (drule sym)
+apply (rule Ord_linear_lt [of "natify(n)" 1])
+apply simp_all
+ apply (subgoal_tac "m #* n = 0", simp)
+ apply (subst mult_natify2 [symmetric])
+ apply (simp del: mult_natify2)
+apply (drule nat_into_Ord [THEN Ord_0_lt, THEN [2] mult_lt_mono2], auto)
+done
+
+lemma less_imp_succ_add [rule_format]:
+ "[| m<n; n: nat |] ==> EX k: nat. n = succ(m#+k)"
+apply (frule lt_nat_in_nat, assumption)
+apply (erule rev_mp)
+apply (induct_tac "n")
+apply (simp_all (no_asm) add: le_iff)
+apply (blast elim!: leE intro!: add_0_right [symmetric] add_succ_right [symmetric])
+done
+
+lemma less_iff_succ_add:
+ "[| m: nat; n: nat |] ==> (m<n) <-> (EX k: nat. n = succ(m#+k))"
+by (auto intro: less_imp_succ_add)
+
+(* on nat *)
+
+lemma diff_is_0_lemma:
+ "[| m: nat; n: nat |] ==> m #- n = 0 <-> m le n"
+apply (rule_tac m = "m" and n = "n" in diff_induct, simp_all)
+done
+
+lemma diff_is_0_iff: "m #- n = 0 <-> natify(m) le natify(n)"
+by (simp add: diff_is_0_lemma [symmetric])
+
+lemma nat_lt_imp_diff_eq_0:
+ "[| a:nat; b:nat; a<b |] ==> a #- b = 0"
+by (simp add: diff_is_0_iff le_iff)
+
+lemma nat_diff_split:
+ "[| a:nat; b:nat |] ==>
+ (P(a #- b)) <-> ((a < b -->P(0)) & (ALL d:nat. a = b #+ d --> P(d)))"
+apply (case_tac "a < b")
+ apply (force simp add: nat_lt_imp_diff_eq_0)
+apply (rule iffI, simp_all)
+ apply clarify
+ apply (rotate_tac -1)
+ apply simp
+apply (drule_tac x="a#-b" in bspec)
+apply (simp_all add: Ordinal.not_lt_iff_le add_diff_inverse)
+done
+
+ML
+{*
+val diff_self_eq_0 = thm "diff_self_eq_0";
+val add_diff_inverse = thm "add_diff_inverse";
+val add_diff_inverse2 = thm "add_diff_inverse2";
+val diff_succ = thm "diff_succ";
+val zero_less_diff = thm "zero_less_diff";
+val diff_mult_distrib = thm "diff_mult_distrib";
+val diff_mult_distrib2 = thm "diff_mult_distrib2";
+val div_termination = thm "div_termination";
+val raw_mod_type = thm "raw_mod_type";
+val mod_type = thm "mod_type";
+val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
+val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
+val raw_mod_less = thm "raw_mod_less";
+val mod_less = thm "mod_less";
+val raw_mod_geq = thm "raw_mod_geq";
+val mod_geq = thm "mod_geq";
+val raw_div_type = thm "raw_div_type";
+val div_type = thm "div_type";
+val raw_div_less = thm "raw_div_less";
+val div_less = thm "div_less";
+val raw_div_geq = thm "raw_div_geq";
+val div_geq = thm "div_geq";
+val mod_div_equality_natify = thm "mod_div_equality_natify";
+val mod_div_equality = thm "mod_div_equality";
+val mod_succ = thm "mod_succ";
+val mod_less_divisor = thm "mod_less_divisor";
+val mod_1_eq = thm "mod_1_eq";
+val mod2_cases = thm "mod2_cases";
+val mod2_succ_succ = thm "mod2_succ_succ";
+val mod2_add_more = thm "mod2_add_more";
+val mod2_add_self = thm "mod2_add_self";
+val add_le_self = thm "add_le_self";
+val add_le_self2 = thm "add_le_self2";
+val mult_le_mono1 = thm "mult_le_mono1";
+val mult_le_mono = thm "mult_le_mono";
+val mult_lt_mono2 = thm "mult_lt_mono2";
+val mult_lt_mono1 = thm "mult_lt_mono1";
+val add_eq_0_iff = thm "add_eq_0_iff";
+val zero_lt_mult_iff = thm "zero_lt_mult_iff";
+val mult_eq_1_iff = thm "mult_eq_1_iff";
+val mult_is_zero = thm "mult_is_zero";
+val mult_is_zero_natify = thm "mult_is_zero_natify";
+val mult_less_cancel2 = thm "mult_less_cancel2";
+val mult_less_cancel1 = thm "mult_less_cancel1";
+val mult_le_cancel2 = thm "mult_le_cancel2";
+val mult_le_cancel1 = thm "mult_le_cancel1";
+val mult_le_cancel_le1 = thm "mult_le_cancel_le1";
+val Ord_eq_iff_le = thm "Ord_eq_iff_le";
+val mult_cancel2 = thm "mult_cancel2";
+val mult_cancel1 = thm "mult_cancel1";
+val div_cancel_raw = thm "div_cancel_raw";
+val div_cancel = thm "div_cancel";
+val mult_mod_distrib_raw = thm "mult_mod_distrib_raw";
+val mod_mult_distrib2 = thm "mod_mult_distrib2";
+val mult_mod_distrib = thm "mult_mod_distrib";
+val mod_add_self2_raw = thm "mod_add_self2_raw";
+val mod_add_self2 = thm "mod_add_self2";
+val mod_add_self1 = thm "mod_add_self1";
+val mod_mult_self1_raw = thm "mod_mult_self1_raw";
+val mod_mult_self1 = thm "mod_mult_self1";
+val mod_mult_self2 = thm "mod_mult_self2";
+val mult_eq_self_implies_10 = thm "mult_eq_self_implies_10";
+val less_imp_succ_add = thm "less_imp_succ_add";
+val less_iff_succ_add = thm "less_iff_succ_add";
+val diff_is_0_iff = thm "diff_is_0_iff";
+val nat_lt_imp_diff_eq_0 = thm "nat_lt_imp_diff_eq_0";
+val nat_diff_split = thm "nat_diff_split";
+*}
+
end