src/HOL/Integ/IntArith.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16417 9bc16273c2d4 child 17085 5b57f995a179 permissions -rw-r--r--
Constant "If" is now local
```     1 (*  Title:      HOL/Integ/IntArith.thy
```
```     2     ID:         \$Id\$
```
```     3     Authors:    Larry Paulson and Tobias Nipkow
```
```     4 *)
```
```     5
```
```     6 header {* Integer arithmetic *}
```
```     7
```
```     8 theory IntArith
```
```     9 imports Numeral
```
```    10 uses ("int_arith1.ML")
```
```    11 begin
```
```    12
```
```    13 text{*Duplicate: can't understand why it's necessary*}
```
```    14 declare numeral_0_eq_0 [simp]
```
```    15
```
```    16
```
```    17 subsection{*Instantiating Binary Arithmetic for the Integers*}
```
```    18
```
```    19 instance
```
```    20   int :: number ..
```
```    21
```
```    22 defs (overloaded)
```
```    23   int_number_of_def: "(number_of w :: int) == of_int (Rep_Bin w)"
```
```    24     --{*the type constraint is essential!*}
```
```    25
```
```    26 instance int :: number_ring
```
```    27 by (intro_classes, simp add: int_number_of_def)
```
```    28
```
```    29
```
```    30 subsection{*Inequality Reasoning for the Arithmetic Simproc*}
```
```    31
```
```    32 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
```
```    33 by simp
```
```    34
```
```    35 lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
```
```    36 by simp
```
```    37
```
```    38 lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
```
```    39 by simp
```
```    40
```
```    41 lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
```
```    42 by simp
```
```    43
```
```    44 text{*Theorem lists for the cancellation simprocs. The use of binary numerals
```
```    45 for 0 and 1 reduces the number of special cases.*}
```
```    46
```
```    47 lemmas add_0s = add_numeral_0 add_numeral_0_right
```
```    48 lemmas mult_1s = mult_numeral_1 mult_numeral_1_right
```
```    49                  mult_minus1 mult_minus1_right
```
```    50
```
```    51
```
```    52 subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
```
```    53
```
```    54 text{*Arithmetic computations are defined for binary literals, which leaves 0
```
```    55 and 1 as special cases. Addition already has rules for 0, but not 1.
```
```    56 Multiplication and unary minus already have rules for both 0 and 1.*}
```
```    57
```
```    58
```
```    59 lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
```
```    60 by simp
```
```    61
```
```    62
```
```    63 lemmas add_number_of_eq = number_of_add [symmetric]
```
```    64
```
```    65 text{*Allow 1 on either or both sides*}
```
```    66 lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
```
```    67 by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
```
```    68
```
```    69 lemmas add_special =
```
```    70     one_add_one_is_two
```
```    71     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
```
```    72     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
```
```    73
```
```    74 text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
```
```    75 lemmas diff_special =
```
```    76     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
```
```    77     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
```
```    78
```
```    79 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```    80 lemmas eq_special =
```
```    81     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
```
```    82     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
```
```    83     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
```
```    84     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
```
```    85
```
```    86 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```    87 lemmas less_special =
```
```    88   binop_eq [of "op <", OF less_number_of_eq_neg numeral_0_eq_0 refl, standard]
```
```    89   binop_eq [of "op <", OF less_number_of_eq_neg numeral_1_eq_1 refl, standard]
```
```    90   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_0_eq_0, standard]
```
```    91   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_1_eq_1, standard]
```
```    92
```
```    93 text{*Allow 0 or 1 on either side with a binary numeral on the other*}
```
```    94 lemmas le_special =
```
```    95     binop_eq [of "op \<le>", OF le_number_of_eq numeral_0_eq_0 refl, standard]
```
```    96     binop_eq [of "op \<le>", OF le_number_of_eq numeral_1_eq_1 refl, standard]
```
```    97     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_0_eq_0, standard]
```
```    98     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_1_eq_1, standard]
```
```    99
```
```   100 lemmas arith_special =
```
```   101        add_special diff_special eq_special less_special le_special
```
```   102
```
```   103
```
```   104 use "int_arith1.ML"
```
```   105 setup int_arith_setup
```
```   106
```
```   107
```
```   108 subsection{*Lemmas About Small Numerals*}
```
```   109
```
```   110 lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
```
```   111 proof -
```
```   112   have "(of_int -1 :: 'a) = of_int (- 1)" by simp
```
```   113   also have "... = - of_int 1" by (simp only: of_int_minus)
```
```   114   also have "... = -1" by simp
```
```   115   finally show ?thesis .
```
```   116 qed
```
```   117
```
```   118 lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
```
```   119 by (simp add: abs_if)
```
```   120
```
```   121 lemma abs_power_minus_one [simp]:
```
```   122      "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
```
```   123 by (simp add: power_abs)
```
```   124
```
```   125 lemma of_int_number_of_eq:
```
```   126      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
```
```   127 by (simp add: number_of_eq)
```
```   128
```
```   129 text{*Lemmas for specialist use, NOT as default simprules*}
```
```   130 lemma mult_2: "2 * z = (z+z::'a::number_ring)"
```
```   131 proof -
```
```   132   have "2*z = (1 + 1)*z" by simp
```
```   133   also have "... = z+z" by (simp add: left_distrib)
```
```   134   finally show ?thesis .
```
```   135 qed
```
```   136
```
```   137 lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
```
```   138 by (subst mult_commute, rule mult_2)
```
```   139
```
```   140
```
```   141 subsection{*More Inequality Reasoning*}
```
```   142
```
```   143 lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
```
```   144 by arith
```
```   145
```
```   146 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
```
```   147 by arith
```
```   148
```
```   149 lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
```
```   150 by arith
```
```   151
```
```   152 lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
```
```   153 by arith
```
```   154
```
```   155 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
```
```   156 by arith
```
```   157
```
```   158
```
```   159 subsection{*The Functions @{term nat} and @{term int}*}
```
```   160
```
```   161 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
```
```   162   @{term "w + - z"}*}
```
```   163 declare Zero_int_def [symmetric, simp]
```
```   164 declare One_int_def [symmetric, simp]
```
```   165
```
```   166 text{*cooper.ML refers to this theorem*}
```
```   167 lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
```
```   168
```
```   169 lemma nat_0: "nat 0 = 0"
```
```   170 by (simp add: nat_eq_iff)
```
```   171
```
```   172 lemma nat_1: "nat 1 = Suc 0"
```
```   173 by (subst nat_eq_iff, simp)
```
```   174
```
```   175 lemma nat_2: "nat 2 = Suc (Suc 0)"
```
```   176 by (subst nat_eq_iff, simp)
```
```   177
```
```   178 lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
```
```   179 apply (insert zless_nat_conj [of 1 z])
```
```   180 apply (auto simp add: nat_1)
```
```   181 done
```
```   182
```
```   183 text{*This simplifies expressions of the form @{term "int n = z"} where
```
```   184       z is an integer literal.*}
```
```   185 declare int_eq_iff [of _ "number_of v", standard, simp]
```
```   186
```
```   187 lemma split_nat [arith_split]:
```
```   188   "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
```
```   189   (is "?P = (?L & ?R)")
```
```   190 proof (cases "i < 0")
```
```   191   case True thus ?thesis by simp
```
```   192 next
```
```   193   case False
```
```   194   have "?P = ?L"
```
```   195   proof
```
```   196     assume ?P thus ?L using False by clarsimp
```
```   197   next
```
```   198     assume ?L thus ?P using False by simp
```
```   199   qed
```
```   200   with False show ?thesis by simp
```
```   201 qed
```
```   202
```
```   203
```
```   204 (*Analogous to zadd_int*)
```
```   205 lemma zdiff_int: "n \<le> m ==> int m - int n = int (m-n)"
```
```   206 by (induct m n rule: diff_induct, simp_all)
```
```   207
```
```   208 lemma nat_mult_distrib: "(0::int) \<le> z ==> nat (z*z') = nat z * nat z'"
```
```   209 apply (case_tac "0 \<le> z'")
```
```   210 apply (rule inj_int [THEN injD])
```
```   211 apply (simp add: int_mult zero_le_mult_iff)
```
```   212 apply (simp add: mult_le_0_iff)
```
```   213 done
```
```   214
```
```   215 lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
```
```   216 apply (rule trans)
```
```   217 apply (rule_tac [2] nat_mult_distrib, auto)
```
```   218 done
```
```   219
```
```   220 lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
```
```   221 apply (case_tac "z=0 | w=0")
```
```   222 apply (auto simp add: abs_if nat_mult_distrib [symmetric]
```
```   223                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
```
```   224 done
```
```   225
```
```   226
```
```   227 subsubsection "Induction principles for int"
```
```   228
```
```   229                      (* `set:int': dummy construction *)
```
```   230 theorem int_ge_induct[case_names base step,induct set:int]:
```
```   231   assumes ge: "k \<le> (i::int)" and
```
```   232         base: "P(k)" and
```
```   233         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   234   shows "P i"
```
```   235 proof -
```
```   236   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
```
```   237     proof (induct n)
```
```   238       case 0
```
```   239       hence "i = k" by arith
```
```   240       thus "P i" using base by simp
```
```   241     next
```
```   242       case (Suc n)
```
```   243       hence "n = nat((i - 1) - k)" by arith
```
```   244       moreover
```
```   245       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```   246       ultimately
```
```   247       have "P(i - 1)" by(rule Suc.hyps)
```
```   248       from step[OF ki1 this] show ?case by simp
```
```   249     qed
```
```   250   }
```
```   251   with ge show ?thesis by fast
```
```   252 qed
```
```   253
```
```   254                      (* `set:int': dummy construction *)
```
```   255 theorem int_gr_induct[case_names base step,induct set:int]:
```
```   256   assumes gr: "k < (i::int)" and
```
```   257         base: "P(k+1)" and
```
```   258         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   259   shows "P i"
```
```   260 apply(rule int_ge_induct[of "k + 1"])
```
```   261   using gr apply arith
```
```   262  apply(rule base)
```
```   263 apply (rule step, simp+)
```
```   264 done
```
```   265
```
```   266 theorem int_le_induct[consumes 1,case_names base step]:
```
```   267   assumes le: "i \<le> (k::int)" and
```
```   268         base: "P(k)" and
```
```   269         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   270   shows "P i"
```
```   271 proof -
```
```   272   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
```
```   273     proof (induct n)
```
```   274       case 0
```
```   275       hence "i = k" by arith
```
```   276       thus "P i" using base by simp
```
```   277     next
```
```   278       case (Suc n)
```
```   279       hence "n = nat(k - (i+1))" by arith
```
```   280       moreover
```
```   281       have ki1: "i + 1 \<le> k" using Suc.prems by arith
```
```   282       ultimately
```
```   283       have "P(i+1)" by(rule Suc.hyps)
```
```   284       from step[OF ki1 this] show ?case by simp
```
```   285     qed
```
```   286   }
```
```   287   with le show ?thesis by fast
```
```   288 qed
```
```   289
```
```   290 theorem int_less_induct [consumes 1,case_names base step]:
```
```   291   assumes less: "(i::int) < k" and
```
```   292         base: "P(k - 1)" and
```
```   293         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   294   shows "P i"
```
```   295 apply(rule int_le_induct[of _ "k - 1"])
```
```   296   using less apply arith
```
```   297  apply(rule base)
```
```   298 apply (rule step, simp+)
```
```   299 done
```
```   300
```
```   301 subsection{*Intermediate value theorems*}
```
```   302
```
```   303 lemma int_val_lemma:
```
```   304      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
```
```   305       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
```
```   306 apply (induct_tac "n", simp)
```
```   307 apply (intro strip)
```
```   308 apply (erule impE, simp)
```
```   309 apply (erule_tac x = n in allE, simp)
```
```   310 apply (case_tac "k = f (n+1) ")
```
```   311  apply force
```
```   312 apply (erule impE)
```
```   313  apply (simp add: abs_if split add: split_if_asm)
```
```   314 apply (blast intro: le_SucI)
```
```   315 done
```
```   316
```
```   317 lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
```
```   318
```
```   319 lemma nat_intermed_int_val:
```
```   320      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
```
```   321          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
```
```   322 apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
```
```   323        in int_val_lemma)
```
```   324 apply simp
```
```   325 apply (erule impE)
```
```   326  apply (intro strip)
```
```   327  apply (erule_tac x = "i+m" in allE, arith)
```
```   328 apply (erule exE)
```
```   329 apply (rule_tac x = "i+m" in exI, arith)
```
```   330 done
```
```   331
```
```   332
```
```   333 subsection{*Products and 1, by T. M. Rasmussen*}
```
```   334
```
```   335 lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
```
```   336 by arith
```
```   337
```
```   338 lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
```
```   339 apply (case_tac "\<bar>n\<bar>=1")
```
```   340 apply (simp add: abs_mult)
```
```   341 apply (rule ccontr)
```
```   342 apply (auto simp add: linorder_neq_iff abs_mult)
```
```   343 apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
```
```   344  prefer 2 apply arith
```
```   345 apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp)
```
```   346 apply (rule mult_mono, auto)
```
```   347 done
```
```   348
```
```   349 lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
```
```   350 by (insert abs_zmult_eq_1 [of m n], arith)
```
```   351
```
```   352 lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
```
```   353 apply (auto dest: pos_zmult_eq_1_iff_lemma)
```
```   354 apply (simp add: mult_commute [of m])
```
```   355 apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```   356 done
```
```   357
```
```   358 lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
```
```   359 apply (rule iffI)
```
```   360  apply (frule pos_zmult_eq_1_iff_lemma)
```
```   361  apply (simp add: mult_commute [of m])
```
```   362  apply (frule pos_zmult_eq_1_iff_lemma, auto)
```
```   363 done
```
```   364
```
```   365 ML
```
```   366 {*
```
```   367 val zle_diff1_eq = thm "zle_diff1_eq";
```
```   368 val zle_add1_eq_le = thm "zle_add1_eq_le";
```
```   369 val nonneg_eq_int = thm "nonneg_eq_int";
```
```   370 val abs_minus_one = thm "abs_minus_one";
```
```   371 val of_int_number_of_eq = thm"of_int_number_of_eq";
```
```   372 val nat_eq_iff = thm "nat_eq_iff";
```
```   373 val nat_eq_iff2 = thm "nat_eq_iff2";
```
```   374 val nat_less_iff = thm "nat_less_iff";
```
```   375 val int_eq_iff = thm "int_eq_iff";
```
```   376 val nat_0 = thm "nat_0";
```
```   377 val nat_1 = thm "nat_1";
```
```   378 val nat_2 = thm "nat_2";
```
```   379 val nat_less_eq_zless = thm "nat_less_eq_zless";
```
```   380 val nat_le_eq_zle = thm "nat_le_eq_zle";
```
```   381
```
```   382 val nat_intermed_int_val = thm "nat_intermed_int_val";
```
```   383 val pos_zmult_eq_1_iff = thm "pos_zmult_eq_1_iff";
```
```   384 val zmult_eq_1_iff = thm "zmult_eq_1_iff";
```
```   385 val nat_add_distrib = thm "nat_add_distrib";
```
```   386 val nat_diff_distrib = thm "nat_diff_distrib";
```
```   387 val nat_mult_distrib = thm "nat_mult_distrib";
```
```   388 val nat_mult_distrib_neg = thm "nat_mult_distrib_neg";
```
```   389 val nat_abs_mult_distrib = thm "nat_abs_mult_distrib";
```
```   390 *}
```
```   391
```
```   392 end
```