author nipkow Tue Feb 17 18:48:17 2009 +0100 (2009-02-17) changeset 29948 cdf12a1cb963 parent 29947 0a51765d2084 child 29951 a70bc5190534
Cleaned up IntDiv and removed subsumed lemmas.
 src/HOL/Algebra/IntRing.thy file | annotate | diff | revisions src/HOL/Decision_Procs/cooper_tac.ML file | annotate | diff | revisions src/HOL/Decision_Procs/ferrack_tac.ML file | annotate | diff | revisions src/HOL/Decision_Procs/mir_tac.ML file | annotate | diff | revisions src/HOL/Divides.thy file | annotate | diff | revisions src/HOL/IntDiv.thy file | annotate | diff | revisions src/HOL/NumberTheory/Chinese.thy file | annotate | diff | revisions src/HOL/NumberTheory/IntPrimes.thy file | annotate | diff | revisions src/HOL/NumberTheory/Residues.thy file | annotate | diff | revisions src/HOL/Tools/Qelim/presburger.ML file | annotate | diff | revisions src/HOL/Word/Num_Lemmas.thy file | annotate | diff | revisions src/HOL/Word/WordGenLib.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Algebra/IntRing.thy	Mon Feb 16 19:35:52 2009 -0800
1.2 +++ b/src/HOL/Algebra/IntRing.thy	Tue Feb 17 18:48:17 2009 +0100
1.3 @@ -407,7 +407,7 @@
1.4
1.5    hence "b mod m = (x * m + a) mod m" by simp
1.6    also
1.7 -      have "\<dots> = ((x * m) mod m) + (a mod m)" by (simp add: zmod_zadd1_eq)
1.8 +      have "\<dots> = ((x * m) mod m) + (a mod m)" by (simp add: mod_add_eq)
1.9    also
1.10        have "\<dots> = a mod m" by simp
1.11    finally
```
```     2.1 --- a/src/HOL/Decision_Procs/cooper_tac.ML	Mon Feb 16 19:35:52 2009 -0800
2.2 +++ b/src/HOL/Decision_Procs/cooper_tac.ML	Tue Feb 17 18:48:17 2009 +0100
2.3 @@ -30,7 +30,7 @@
```
```     3.1 --- a/src/HOL/Decision_Procs/ferrack_tac.ML	Mon Feb 16 19:35:52 2009 -0800
3.2 +++ b/src/HOL/Decision_Procs/ferrack_tac.ML	Tue Feb 17 18:48:17 2009 +0100
3.3 @@ -34,7 +34,7 @@
```
```     4.1 --- a/src/HOL/Decision_Procs/mir_tac.ML	Mon Feb 16 19:35:52 2009 -0800
4.2 +++ b/src/HOL/Decision_Procs/mir_tac.ML	Tue Feb 17 18:48:17 2009 +0100
4.3 @@ -49,7 +49,7 @@
```
```     5.1 --- a/src/HOL/Divides.thy	Mon Feb 16 19:35:52 2009 -0800
5.2 +++ b/src/HOL/Divides.thy	Tue Feb 17 18:48:17 2009 +0100
5.3 @@ -173,7 +173,7 @@
5.4  qed
5.5
5.6  lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
5.7 -by (unfold dvd_def, auto)
5.8 +by (rule dvd_eq_mod_eq_0[THEN iffD1])
5.9
5.10  lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
5.11  by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
```
```     6.1 --- a/src/HOL/IntDiv.thy	Mon Feb 16 19:35:52 2009 -0800
6.2 +++ b/src/HOL/IntDiv.thy	Tue Feb 17 18:48:17 2009 +0100
6.3 @@ -451,9 +451,6 @@
6.4  lemma zmod_zero [simp]: "(0::int) mod b = 0"
6.5  by (simp add: mod_def divmod_def)
6.6
6.7 -lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
6.8 -by (simp add: div_def divmod_def)
6.9 -
6.10  lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
6.11  by (simp add: mod_def divmod_def)
6.12
6.13 @@ -729,18 +726,6 @@
6.14  apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
6.15  done
6.16
6.17 -lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
6.18 -apply (rule trans)
6.19 -apply (rule_tac s = "b*a mod c" in trans)
6.20 -apply (rule_tac [2] zmod_zmult1_eq)
6.22 -done
6.23 -
6.24 -lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
6.25 -apply (rule zmod_zmult1_eq' [THEN trans])
6.26 -apply (rule zmod_zmult1_eq)
6.27 -done
6.28 -
6.29  lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
6.31
6.32 @@ -749,11 +734,6 @@
6.33  apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
6.34  done
6.35
6.36 -lemma zmod_zmod_trivial: "(a mod b) mod b = a mod (b::int)"
6.37 -apply (case_tac "b = 0", simp)
6.38 -apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
6.39 -done
6.40 -
6.41  text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
6.42
6.44 @@ -768,11 +748,6 @@
6.45  apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
6.46  done
6.47
6.48 -lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
6.49 -apply (case_tac "c = 0", simp)
6.50 -apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_mod)
6.51 -done
6.52 -
6.53  instance int :: ring_div
6.54  proof
6.55    fix a b c :: int
6.56 @@ -971,7 +946,7 @@
6.57      P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
6.58  apply (rule iffI, clarify)
6.59   apply (erule_tac P="P ?x ?y" in rev_mp)
6.63   apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
6.64  txt{*converse direction*}
6.65 @@ -984,7 +959,7 @@
6.66      P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
6.67  apply (rule iffI, clarify)
6.68   apply (erule_tac P="P ?x ?y" in rev_mp)
6.72   apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
6.73  txt{*converse direction*}
6.74 @@ -1057,11 +1032,6 @@
6.75         simp)
6.76  done
6.77
6.78 -(*Not clear why this must be proved separately; probably number_of causes
6.79 -  simplification problems*)
6.80 -lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
6.81 -by auto
6.82 -
6.83  lemma zdiv_number_of_Bit0 [simp]:
6.84       "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =
6.85            number_of v div (number_of w :: int)"
6.86 @@ -1088,7 +1058,7 @@
6.87   apply (rule_tac [2] mult_left_mono)
6.89                        pos_mod_bound)
6.92  apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
6.93  apply (rule mod_pos_pos_trivial)
6.94  apply (auto simp add: mod_pos_pos_trivial ring_distribs)
6.95 @@ -1111,7 +1081,7 @@
6.96        (2::int) * (number_of v mod number_of w)"
6.97  apply (simp only: number_of_eq numeral_simps)
6.98  apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
6.101  done
6.102
6.103  lemma zmod_number_of_Bit1 [simp]:
6.104 @@ -1121,7 +1091,7 @@
6.105                  else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
6.106  apply (simp only: number_of_eq numeral_simps)
6.107  apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
6.110  done
6.111
6.112
6.113 @@ -1131,7 +1101,7 @@
6.114  apply (subgoal_tac "a div b \<le> -1", force)
6.115  apply (rule order_trans)
6.116  apply (rule_tac a' = "-1" in zdiv_mono1)
6.117 -apply (auto simp add: zdiv_minus1)
6.118 +apply (auto simp add: div_eq_minus1)
6.119  done
6.120
6.121  lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
6.122 @@ -1379,7 +1349,7 @@
6.123  apply (induct "y", auto)
6.124  apply (rule zmod_zmult1_eq [THEN trans])
6.125  apply (simp (no_asm_simp))
6.126 -apply (rule zmod_zmult_distrib [symmetric])
6.127 +apply (rule mod_mult_eq [symmetric])
6.128  done
6.129
6.130  lemma zdiv_int: "int (a div b) = (int a) div (int b)"
6.131 @@ -1420,7 +1390,7 @@
6.134    IntDiv.zmod_zmult1_eq     [symmetric]
6.135 -  IntDiv.zmod_zmult1_eq'    [symmetric]
6.136 +  mod_mult_left_eq          [symmetric]
6.137    IntDiv.zpower_zmod
6.138    zminus_zmod zdiff_zmod_left zdiff_zmod_right
6.139
```
```     7.1 --- a/src/HOL/NumberTheory/Chinese.thy	Mon Feb 16 19:35:52 2009 -0800
7.2 +++ b/src/HOL/NumberTheory/Chinese.thy	Tue Feb 17 18:48:17 2009 +0100
7.3 @@ -101,7 +101,7 @@
7.4    apply (induct l)
7.5     apply auto
7.6    apply (rule trans)
7.9    apply simp
7.11    done
7.12 @@ -237,10 +237,10 @@
7.13    apply (unfold lincong_sol_def)
7.14    apply safe
7.15      apply (tactic {* stac (thm "zcong_zmod") 3 *})
7.16 -    apply (tactic {* stac (thm "zmod_zmult_distrib") 3 *})
7.17 +    apply (tactic {* stac (thm "mod_mult_eq") 3 *})
7.18      apply (tactic {* stac (thm "zmod_zdvd_zmod") 3 *})
7.19        apply (tactic {* stac (thm "x_sol_lin") 5 *})
7.20 -        apply (tactic {* stac (thm "zmod_zmult_distrib" RS sym) 7 *})
7.21 +        apply (tactic {* stac (thm "mod_mult_eq" RS sym) 7 *})
7.22          apply (tactic {* stac (thm "zcong_zmod" RS sym) 7 *})
7.23          apply (subgoal_tac [7]
7.24            "0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
```
```     8.1 --- a/src/HOL/NumberTheory/IntPrimes.thy	Mon Feb 16 19:35:52 2009 -0800
8.2 +++ b/src/HOL/NumberTheory/IntPrimes.thy	Tue Feb 17 18:48:17 2009 +0100
8.3 @@ -88,7 +88,7 @@
8.4
8.5  lemma zgcd_zadd_zmult [simp]: "zgcd (m + n * k) n = zgcd m n"
8.6    apply (rule zgcd_eq [THEN trans])
8.9    apply (rule zgcd_eq [symmetric])
8.10    done
8.11
```
```     9.1 --- a/src/HOL/NumberTheory/Residues.thy	Mon Feb 16 19:35:52 2009 -0800
9.2 +++ b/src/HOL/NumberTheory/Residues.thy	Tue Feb 17 18:48:17 2009 +0100
9.3 @@ -53,7 +53,7 @@
9.4  lemma StandardRes_prop4: "2 < m
9.5       ==> [StandardRes m x * StandardRes m y = (x * y)] (mod m)"
9.6    by (auto simp add: StandardRes_def zcong_zmod_eq
9.7 -                     zmod_zmult_distrib [of x y m])
9.8 +                     mod_mult_eq [of x y m])
9.9
9.10  lemma StandardRes_lbound: "0 < p ==> 0 \<le> StandardRes p x"
9.11    by (auto simp add: StandardRes_def pos_mod_sign)
```
```    10.1 --- a/src/HOL/Tools/Qelim/presburger.ML	Mon Feb 16 19:35:52 2009 -0800
10.2 +++ b/src/HOL/Tools/Qelim/presburger.ML	Tue Feb 17 18:48:17 2009 +0100
10.3 @@ -124,7 +124,7 @@
10.4    @ map (symmetric o mk_meta_eq)
10.5      [@{thm "dvd_eq_mod_eq_0"}, @{thm "zdvd_iff_zmod_eq_0"}, @{thm "mod_add1_eq"},
10.10    @ [@{thm "mod_self"}, @{thm "zmod_self"}, @{thm "mod_by_0"},
10.11       @{thm "div_by_0"}, @{thm "DIVISION_BY_ZERO"} RS conjunct1,
```
```    11.1 --- a/src/HOL/Word/Num_Lemmas.thy	Mon Feb 16 19:35:52 2009 -0800
11.2 +++ b/src/HOL/Word/Num_Lemmas.thy	Tue Feb 17 18:48:17 2009 +0100
11.3 @@ -121,8 +121,8 @@
11.4
11.5  lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c"
11.6    apply (unfold diff_int_def)
11.7 -  apply (rule trans [OF _ zmod_zadd1_eq [symmetric]])
11.9 +  apply (rule trans [OF _ mod_add_eq [symmetric]])
11.11    done
11.12
11.13  lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c"
11.14 @@ -162,8 +162,8 @@
11.15  lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric],
11.16    THEN mod_plus_right [THEN iffD2], standard, simplified]
11.17
11.18 -lemmas push_mods' = zmod_zadd1_eq [standard]
11.19 -  zmod_zmult_distrib [standard] zmod_zsub_distrib [standard]
11.20 +lemmas push_mods' = mod_add_eq [standard]
11.21 +  mod_mult_eq [standard] zmod_zsub_distrib [standard]
11.22    zmod_uminus [symmetric, standard]
11.23
11.24  lemmas push_mods = push_mods' [THEN eq_reflection, standard]
```
```    12.1 --- a/src/HOL/Word/WordGenLib.thy	Mon Feb 16 19:35:52 2009 -0800
12.2 +++ b/src/HOL/Word/WordGenLib.thy	Tue Feb 17 18:48:17 2009 +0100
12.3 @@ -293,9 +293,9 @@
12.4    shows "(x + y) mod b = z' mod b'"
12.5  proof -
12.6    from 1 2[symmetric] 3[symmetric] have "(x + y) mod b = (x' mod b' + y' mod b') mod b'"