--- a/src/HOL/Word/Num_Lemmas.thy Tue Aug 28 19:45:45 2007 +0200
+++ b/src/HOL/Word/Num_Lemmas.thy Tue Aug 28 20:13:47 2007 +0200
@@ -7,12 +7,37 @@
theory Num_Lemmas imports Parity begin
-(* lemmas funpow_0 = funpow.simps(1) *)
+lemma contentsI: "y = {x} ==> contents y = x"
+ unfolding contents_def by auto
+
+lemma prod_case_split: "prod_case = split"
+ by (rule ext)+ auto
+
+lemmas split_split = prod.split [unfolded prod_case_split]
+lemmas split_split_asm = prod.split_asm [unfolded prod_case_split]
+lemmas "split.splits" = split_split split_split_asm
+
+lemmas funpow_0 = funpow.simps(1)
lemmas funpow_Suc = funpow.simps(2)
-(* used by BinGeneral.funpow_minus_simp *)
+
+lemma nonemptyE: "S ~= {} ==> (!!x. x : S ==> R) ==> R"
+ apply (erule contrapos_np)
+ apply (rule equals0I)
+ apply auto
+ done
lemma gt_or_eq_0: "0 < y \<or> 0 = (y::nat)" by auto
+constdefs
+ mod_alt :: "'a => 'a => 'a :: Divides.div"
+ "mod_alt n m == n mod m"
+
+ -- "alternative way of defining @{text bin_last}, @{text bin_rest}"
+ bin_rl :: "int => int * bit"
+ "bin_rl w == SOME (r, l). w = r BIT l"
+
+declare iszero_0 [iff]
+
lemmas xtr1 = xtrans(1)
lemmas xtr2 = xtrans(2)
lemmas xtr3 = xtrans(3)
@@ -22,7 +47,13 @@
lemmas xtr7 = xtrans(7)
lemmas xtr8 = xtrans(8)
-lemmas PlsMin_defs (*[intro!]*) =
+lemma Min_ne_Pls [iff]:
+ "Numeral.Min ~= Numeral.Pls"
+ unfolding Min_def Pls_def by auto
+
+lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric]
+
+lemmas PlsMin_defs [intro!] =
Pls_def Min_def Pls_def [symmetric] Min_def [symmetric]
lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI]
@@ -31,19 +62,36 @@
"False ==> number_of x = number_of y"
by auto
+lemmas nat_simps = diff_add_inverse2 diff_add_inverse
+lemmas nat_iffs = le_add1 le_add2
+
+lemma sum_imp_diff: "j = k + i ==> j - i = (k :: nat)"
+ by (clarsimp simp add: nat_simps)
+
lemma nobm1:
"0 < (number_of w :: nat) ==>
number_of w - (1 :: nat) = number_of (Numeral.pred w)"
apply (unfold nat_number_of_def One_nat_def nat_1 [symmetric] pred_def)
apply (simp add: number_of_eq nat_diff_distrib [symmetric])
done
-(* used in BinGeneral, BinOperations, BinBoolList *)
+
+lemma of_int_power:
+ "of_int (a ^ n) = (of_int a ^ n :: 'a :: {recpower, comm_ring_1})"
+ by (induct n) (auto simp add: power_Suc)
lemma zless2: "0 < (2 :: int)"
by auto
-lemmas zless2p [simp] = zless2 [THEN zero_less_power] (* keep *)
-lemmas zle2p [simp] = zless2p [THEN order_less_imp_le] (* keep *)
+lemmas zless2p [simp] = zless2 [THEN zero_less_power]
+lemmas zle2p [simp] = zless2p [THEN order_less_imp_le]
+
+lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
+lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
+
+-- "the inverse(s) of @{text number_of}"
+lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1"
+ using pos_mod_sign2 [of n] pos_mod_bound2 [of n]
+ unfolding mod_alt_def [symmetric] by auto
lemma emep1:
"even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1"
@@ -56,53 +104,192 @@
lemmas eme1p = emep1 [simplified add_commute]
+lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))"
+ by (simp add: le_diff_eq add_commute)
+
+lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))"
+ by (simp add: less_diff_eq add_commute)
+
lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))"
by (simp add: diff_le_eq add_commute)
-(* used by BinGeneral.sb_dec_lem' *)
+
+lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))"
+ by (simp add: diff_less_eq add_commute)
+
lemmas m1mod2k = zless2p [THEN zmod_minus1]
-(* used in WordArith *)
-
+lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2]
+lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified]
+lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified]
lemma p1mod22k:
"(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)"
by (simp add: p1mod22k' add_commute)
-(* used in BinOperations *)
+
+lemma z1pmod2:
+ "(2 * b + 1) mod 2 = (1::int)"
+ by (simp add: z1pmod2' add_commute)
+
+lemma z1pdiv2:
+ "(2 * b + 1) div 2 = (b::int)"
+ by (simp add: z1pdiv2' add_commute)
lemmas zdiv_le_dividend = xtr3 [OF zdiv_1 [symmetric] zdiv_mono2,
simplified int_one_le_iff_zero_less, simplified, standard]
-(* used in WordShift *)
+
+(** ways in which type Bin resembles a datatype **)
+
+lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c"
+ apply (unfold Bit_def)
+ apply (simp (no_asm_use) split: bit.split_asm)
+ apply simp_all
+ apply (drule_tac f=even in arg_cong, clarsimp)+
+ done
+
+lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard]
+
+lemma BIT_eq_iff [simp]:
+ "(u BIT b = v BIT c) = (u = v \<and> b = c)"
+ by (rule iffI) auto
+
+lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
+
+lemma less_Bits:
+ "(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)"
+ unfolding Bit_def by (auto split: bit.split)
+
+lemma le_Bits:
+ "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))"
+ unfolding Bit_def by (auto split: bit.split)
+
+lemma neB1E [elim!]:
+ assumes ne: "y \<noteq> bit.B1"
+ assumes y: "y = bit.B0 \<Longrightarrow> P"
+ shows "P"
+ apply (rule y)
+ apply (cases y rule: bit.exhaust, simp)
+ apply (simp add: ne)
+ done
+
+lemma bin_ex_rl: "EX w b. w BIT b = bin"
+ apply (unfold Bit_def)
+ apply (cases "even bin")
+ apply (clarsimp simp: even_equiv_def)
+ apply (auto simp: odd_equiv_def split: bit.split)
+ done
+
+lemma bin_exhaust:
+ assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
+ shows "Q"
+ apply (insert bin_ex_rl [of bin])
+ apply (erule exE)+
+ apply (rule Q)
+ apply force
+ done
+
+lemma bin_rl_char: "(bin_rl w = (r, l)) = (r BIT l = w)"
+ apply (unfold bin_rl_def)
+ apply safe
+ apply (cases w rule: bin_exhaust)
+ apply auto
+ done
+
+lemmas bin_rl_simps [THEN bin_rl_char [THEN iffD2], standard, simp] =
+ Pls_0_eq Min_1_eq refl
+
+lemma bin_abs_lem:
+ "bin = (w BIT b) ==> ~ bin = Numeral.Min --> ~ bin = Numeral.Pls -->
+ nat (abs w) < nat (abs bin)"
+ apply (clarsimp simp add: bin_rl_char)
+ apply (unfold Pls_def Min_def Bit_def)
+ apply (cases b)
+ apply (clarsimp, arith)
+ apply (clarsimp, arith)
+ done
+
+lemma bin_induct:
+ assumes PPls: "P Numeral.Pls"
+ and PMin: "P Numeral.Min"
+ and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
+ shows "P bin"
+ apply (rule_tac P=P and a=bin and f1="nat o abs"
+ in wf_measure [THEN wf_induct])
+ apply (simp add: measure_def inv_image_def)
+ apply (case_tac x rule: bin_exhaust)
+ apply (frule bin_abs_lem)
+ apply (auto simp add : PPls PMin PBit)
+ done
+
+lemma no_no [simp]: "number_of (number_of i) = i"
+ unfolding number_of_eq by simp
lemma Bit_B0:
"k BIT bit.B0 = k + k"
by (unfold Bit_def) simp
+lemma Bit_B1:
+ "k BIT bit.B1 = k + k + 1"
+ by (unfold Bit_def) simp
+
lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k"
by (rule trans, rule Bit_B0) simp
-(* used in BinOperations *)
+
+lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1"
+ by (rule trans, rule Bit_B1) simp
+
+lemma B_mod_2':
+ "X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0"
+ apply (simp (no_asm) only: Bit_B0 Bit_B1)
+ apply (simp add: z1pmod2)
+ done
+
+lemmas B1_mod_2 [simp] = B_mod_2' [OF refl, THEN conjunct1, standard]
+lemmas B0_mod_2 [simp] = B_mod_2' [OF refl, THEN conjunct2, standard]
+
+lemma axxbyy:
+ "a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==>
+ a = b & m = (n :: int)"
+ apply auto
+ apply (drule_tac f="%n. n mod 2" in arg_cong)
+ apply (clarsimp simp: z1pmod2)
+ apply (drule_tac f="%n. n mod 2" in arg_cong)
+ apply (clarsimp simp: z1pmod2)
+ done
+
+lemma axxmod2:
+ "(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)"
+ by simp (rule z1pmod2)
+
+lemma axxdiv2:
+ "(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)"
+ by simp (rule z1pdiv2)
+
+lemmas iszero_minus = trans [THEN trans,
+ OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard]
lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add_commute,
standard]
-(* used in WordArith *)
lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2], standard]
-(* used in WordShift *)
lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b"
by (simp add : zmod_zminus1_eq_if)
-(* used in BinGeneral *)
+
+lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c"
+ apply (unfold diff_int_def)
+ apply (rule trans [OF _ zmod_zadd1_eq [symmetric]])
+ apply (simp add: zmod_uminus zmod_zadd1_eq [symmetric])
+ done
lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c"
apply (unfold diff_int_def)
apply (rule trans [OF _ zmod_zadd_right_eq [symmetric]])
apply (simp add : zmod_uminus zmod_zadd_right_eq [symmetric])
done
-(* used in BinGeneral, WordGenLib *)
lemmas zmod_zsub_left_eq =
zmod_zadd_left_eq [where b = "- ?b", simplified diff_int_def [symmetric]]
-(* used in BinGeneral, WordGenLib *)
lemma zmod_zsub_self [simp]:
"((b :: int) - a) mod a = b mod a"
@@ -114,12 +301,10 @@
apply (subst zmod_zmult1_eq)
apply simp
done
-(* used in BinGeneral *)
lemmas rdmods [symmetric] = zmod_uminus [symmetric]
zmod_zsub_left_eq zmod_zsub_right_eq zmod_zadd_left_eq
zmod_zadd_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev
-(* used in WordArith, WordShift *)
lemma mod_plus_right:
"((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))"
@@ -128,12 +313,27 @@
apply arith
done
+lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)"
+ by (induct n) (simp_all add : mod_Suc)
+
+lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric],
+ THEN mod_plus_right [THEN iffD2], standard, simplified]
+
+lemmas push_mods' = zmod_zadd1_eq [standard]
+ zmod_zmult_distrib [standard] zmod_zsub_distrib [standard]
+ zmod_uminus [symmetric, standard]
+
+lemmas push_mods = push_mods' [THEN eq_reflection, standard]
+lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard]
+lemmas mod_simps =
+ zmod_zmult_self1 [THEN eq_reflection] zmod_zmult_self2 [THEN eq_reflection]
+ mod_mod_trivial [THEN eq_reflection]
+
lemma nat_mod_eq:
"!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)"
by (induct a) auto
lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
-(* used in WordArith, WordGenLib *)
lemma nat_mod_lem:
"(0 :: nat) < n ==> b < n = (b mod n = b)"
@@ -142,7 +342,6 @@
apply (erule subst)
apply (erule mod_less_divisor)
done
-(* used in WordArith *)
lemma mod_nat_add:
"(x :: nat) < z ==> y < z ==>
@@ -155,7 +354,10 @@
apply (rule nat_mod_eq')
apply arith
done
-(* used in WordArith, WordGenLib *)
+
+lemma mod_nat_sub:
+ "(x :: nat) < z ==> (x - y) mod z = x - y"
+ by (rule nat_mod_eq') arith
lemma int_mod_lem:
"(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
@@ -164,14 +366,12 @@
apply (erule_tac [!] subst)
apply auto
done
-(* used in WordDefinition, WordArith, WordShift *)
lemma int_mod_eq:
"(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b"
by clarsimp (rule mod_pos_pos_trivial)
lemmas int_mod_eq' = refl [THEN [3] int_mod_eq]
-(* used in WordDefinition, WordArith, WordShift, WordGenLib *)
lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a"
apply (cases "a < n")
@@ -195,15 +395,88 @@
"(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
(x + y) mod z = (if x + y < z then x + y else x + y - z)"
by (auto intro: int_mod_eq)
-(* used in WordArith, WordGenLib *)
lemma mod_sub_if_z:
"(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
(x - y) mod z = (if y <= x then x - y else x - y + z)"
by (auto intro: int_mod_eq)
-(* used in WordArith, WordGenLib *)
+
+lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric]
+lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
+
+(* already have this for naturals, div_mult_self1/2, but not for ints *)
+lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n"
+ apply (rule mcl)
+ prefer 2
+ apply (erule asm_rl)
+ apply (simp add: zmde ring_distribs)
+ apply (simp add: push_mods)
+ done
+
+(** Rep_Integ **)
+lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}"
+ unfolding equiv_def refl_def quotient_def Image_def by auto
+
+lemmas Rep_Integ_ne = Integ.Rep_Integ
+ [THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard]
+
+lemmas riq = Integ.Rep_Integ [simplified Integ_def]
+lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard]
+lemmas Rep_Integ_equiv = quotient_eq_iff
+ [OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard]
+lemmas Rep_Integ_same =
+ Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard]
+
+lemma RI_int: "(a, 0) : Rep_Integ (int a)"
+ unfolding int_def by auto
+
+lemmas RI_intrel [simp] = UNIV_I [THEN quotientI,
+ THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard]
+
+lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)"
+ apply (rule_tac z=x in eq_Abs_Integ)
+ apply (clarsimp simp: minus)
+ done
-lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
+lemma RI_add:
+ "(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==>
+ (a + c, b + d) : Rep_Integ (x + y)"
+ apply (rule_tac z=x in eq_Abs_Integ)
+ apply (rule_tac z=y in eq_Abs_Integ)
+ apply (clarsimp simp: add)
+ done
+
+lemma mem_same: "a : S ==> a = b ==> b : S"
+ by fast
+
+(* two alternative proofs of this *)
+lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)"
+ apply (unfold diff_def)
+ apply (rule mem_same)
+ apply (rule RI_minus RI_add RI_int)+
+ apply simp
+ done
+
+lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)"
+ apply safe
+ apply (rule Rep_Integ_same)
+ prefer 2
+ apply (erule asm_rl)
+ apply (rule RI_eq_diff')+
+ done
+
+lemma mod_power_lem:
+ "a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)"
+ apply clarsimp
+ apply safe
+ apply (simp add: zdvd_iff_zmod_eq_0 [symmetric])
+ apply (drule le_iff_add [THEN iffD1])
+ apply (force simp: zpower_zadd_distrib)
+ apply (rule mod_pos_pos_trivial)
+ apply (simp add: zero_le_power)
+ apply (rule power_strict_increasing)
+ apply auto
+ done
lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)"
by arith
@@ -215,14 +488,40 @@
lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm]
+lemma pl_pl_rels:
+ "a + b = c + d ==>
+ a >= c & b <= d | a <= c & b >= (d :: nat)"
+ apply (cut_tac n=a and m=c in nat_le_linear)
+ apply (safe dest!: le_iff_add [THEN iffD1])
+ apply arith+
+ done
+
+lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels]
+
+lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))"
+ by arith
+
+lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b"
+ by arith
+
+lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm]
+
lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)"
by arith
lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus]
+lemma nat_no_eq_iff:
+ "(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==>
+ (number_of b = (number_of c :: nat)) = (b = c)"
+ apply (unfold nat_number_of_def)
+ apply safe
+ apply (drule (2) eq_nat_nat_iff [THEN iffD1])
+ apply (simp add: number_of_eq)
+ done
+
lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right]
lemmas dtle = xtr3 [OF dme [symmetric] le_add1]
-(* used in WordShift *)
lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle]
lemma td_gal:
@@ -233,7 +532,6 @@
done
lemmas td_gal_lt = td_gal [simplified le_def, simplified]
-(* used in WordShift *)
lemma div_mult_le: "(a :: nat) div b * b <= a"
apply (cases b)
@@ -241,7 +539,6 @@
apply (rule order_refl [THEN th2])
apply auto
done
-(* used in WordArith *)
lemmas sdl = split_div_lemma [THEN iffD1, symmetric]
@@ -256,8 +553,22 @@
apply (rule_tac f="%n. n div f" in arg_cong)
apply (simp add : mult_ac)
done
-(* used in WordShift *)
+lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b"
+ apply (unfold dvd_def)
+ apply clarify
+ apply (case_tac k)
+ apply clarsimp
+ apply clarify
+ apply (cases "b > 0")
+ apply (drule mult_commute [THEN xtr1])
+ apply (frule (1) td_gal_lt [THEN iffD1])
+ apply (clarsimp simp: le_simps)
+ apply (rule mult_div_cancel [THEN [2] xtr4])
+ apply (rule mult_mono)
+ apply auto
+ done
+
lemma less_le_mult':
"w * c < b * c ==> 0 \<le> c ==> (w + 1) * c \<le> b * (c::int)"
apply (rule mult_right_mono)
@@ -267,7 +578,9 @@
done
lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified]
-(* used in WordArith *)
+
+lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
+ simplified left_diff_distrib, standard]
lemma lrlem':
assumes d: "(i::nat) \<le> j \<or> m < j'"
@@ -290,33 +603,20 @@
apply arith
apply simp
done
-(* used in BinBoolList *)
lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))"
by auto
-(* used in BinGeneral *)
lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i"
apply (induct i, clarsimp)
apply (cases j, clarsimp)
apply arith
done
-(* used in WordShift *)
-subsection "if simps"
-
-lemma if_Not_x: "(if p then ~ x else x) = (p = (~ x))"
- by auto
-
-lemma if_x_Not: "(if p then x else ~ x) = (p = x)"
- by auto
-
-lemma if_bool_simps:
- "If p True y = (p | y) & If p False y = (~p & y) &
- If p y True = (p --> y) & If p y False = (p & y)"
- by auto
-
-lemmas if_simps = if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps
-(* used in WordBitwise *)
+lemma nonneg_mod_div:
+ "0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b"
+ apply (cases "b = 0", clarsimp)
+ apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
+ done
end