src/HOL/NatArith.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 17688 91d3604ec4b5 child 21238 c46bc715bdfd permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     1 (*  Title:      HOL/NatArith.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow and Markus Wenzel
```
```     4 *)
```
```     5
```
```     6 header {*Further Arithmetic Facts Concerning the Natural Numbers*}
```
```     7
```
```     8 theory NatArith
```
```     9 imports Nat
```
```    10 uses "arith_data.ML"
```
```    11 begin
```
```    12
```
```    13 setup arith_setup
```
```    14
```
```    15 text{*The following proofs may rely on the arithmetic proof procedures.*}
```
```    16
```
```    17 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
```
```    18   by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
```
```    19
```
```    20 lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
```
```    21 by (simp add: less_eq reflcl_trancl [symmetric]
```
```    22             del: reflcl_trancl, arith)
```
```    23
```
```    24 lemma nat_diff_split:
```
```    25     "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
```
```    26     -- {* elimination of @{text -} on @{text nat} *}
```
```    27   by (cases "a<b" rule: case_split)
```
```    28      (auto simp add: diff_is_0_eq [THEN iffD2])
```
```    29
```
```    30 lemma nat_diff_split_asm:
```
```    31     "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
```
```    32     -- {* elimination of @{text -} on @{text nat} in assumptions *}
```
```    33   by (simp split: nat_diff_split)
```
```    34
```
```    35 lemmas [arith_split] = nat_diff_split split_min split_max
```
```    36
```
```    37
```
```    38
```
```    39 lemma le_square: "m \<le> m*(m::nat)"
```
```    40 by (induct_tac "m", auto)
```
```    41
```
```    42 lemma le_cube: "(m::nat) \<le> m*(m*m)"
```
```    43 by (induct_tac "m", auto)
```
```    44
```
```    45
```
```    46 text{*Subtraction laws, mostly by Clemens Ballarin*}
```
```    47
```
```    48 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
```
```    49 by arith
```
```    50
```
```    51 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
```
```    52 by arith
```
```    53
```
```    54 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
```
```    55 by arith
```
```    56
```
```    57 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
```
```    58 by arith
```
```    59
```
```    60 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
```
```    61 by arith
```
```    62
```
```    63 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
```
```    64 by arith
```
```    65
```
```    66 (*Replaces the previous diff_less and le_diff_less, which had the stronger
```
```    67   second premise n\<le>m*)
```
```    68 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
```
```    69 by arith
```
```    70
```
```    71
```
```    72 (** Simplification of relational expressions involving subtraction **)
```
```    73
```
```    74 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
```
```    75 by (simp split add: nat_diff_split)
```
```    76
```
```    77 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
```
```    78 by (auto split add: nat_diff_split)
```
```    79
```
```    80 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
```
```    81 by (auto split add: nat_diff_split)
```
```    82
```
```    83 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
```
```    84 by (auto split add: nat_diff_split)
```
```    85
```
```    86
```
```    87 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
```
```    88
```
```    89 (* Monotonicity of subtraction in first argument *)
```
```    90 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
```
```    91 by (simp split add: nat_diff_split)
```
```    92
```
```    93 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
```
```    94 by (simp split add: nat_diff_split)
```
```    95
```
```    96 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
```
```    97 by (simp split add: nat_diff_split)
```
```    98
```
```    99 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
```
```   100 by (simp split add: nat_diff_split)
```
```   101
```
```   102 text{*Lemmas for ex/Factorization*}
```
```   103
```
```   104 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
```
```   105 by (case_tac "m", auto)
```
```   106
```
```   107 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
```
```   108 by (case_tac "m", auto)
```
```   109
```
```   110 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
```
```   111 by (case_tac "m", auto)
```
```   112
```
```   113
```
```   114 text{*Rewriting to pull differences out*}
```
```   115
```
```   116 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
```
```   117 by arith
```
```   118
```
```   119 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
```
```   120 by arith
```
```   121
```
```   122 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
```
```   123 by arith
```
```   124
```
```   125 (*The others are
```
```   126       i - j - k = i - (j + k),
```
```   127       k \<le> j ==> j - k + i = j + i - k,
```
```   128       k \<le> j ==> i + (j - k) = i + j - k *)
```
```   129 lemmas add_diff_assoc = diff_add_assoc [symmetric]
```
```   130 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
```
```   131 declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
```
```   132
```
```   133 text{*At present we prove no analogue of @{text not_less_Least} or @{text
```
```   134 Least_Suc}, since there appears to be no need.*}
```
```   135
```
```   136 ML
```
```   137 {*
```
```   138 val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
```
```   139 val nat_diff_split = thm "nat_diff_split";
```
```   140 val nat_diff_split_asm = thm "nat_diff_split_asm";
```
```   141 val le_square = thm "le_square";
```
```   142 val le_cube = thm "le_cube";
```
```   143 val diff_less_mono = thm "diff_less_mono";
```
```   144 val less_diff_conv = thm "less_diff_conv";
```
```   145 val le_diff_conv = thm "le_diff_conv";
```
```   146 val le_diff_conv2 = thm "le_diff_conv2";
```
```   147 val diff_diff_cancel = thm "diff_diff_cancel";
```
```   148 val le_add_diff = thm "le_add_diff";
```
```   149 val diff_less = thm "diff_less";
```
```   150 val diff_diff_eq = thm "diff_diff_eq";
```
```   151 val eq_diff_iff = thm "eq_diff_iff";
```
```   152 val less_diff_iff = thm "less_diff_iff";
```
```   153 val le_diff_iff = thm "le_diff_iff";
```
```   154 val diff_le_mono = thm "diff_le_mono";
```
```   155 val diff_le_mono2 = thm "diff_le_mono2";
```
```   156 val diff_less_mono2 = thm "diff_less_mono2";
```
```   157 val diffs0_imp_equal = thm "diffs0_imp_equal";
```
```   158 val one_less_mult = thm "one_less_mult";
```
```   159 val n_less_m_mult_n = thm "n_less_m_mult_n";
```
```   160 val n_less_n_mult_m = thm "n_less_n_mult_m";
```
```   161 val diff_diff_right = thm "diff_diff_right";
```
```   162 val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
```
```   163 val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
```
```   164 *}
```
```   165
```
```   166 subsection{*Embedding of the Naturals into any @{text
```
```   167 comm_semiring_1_cancel}: @{term of_nat}*}
```
```   168
```
```   169 consts of_nat :: "nat => 'a::comm_semiring_1_cancel"
```
```   170
```
```   171 primrec
```
```   172   of_nat_0:   "of_nat 0 = 0"
```
```   173   of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
```
```   174
```
```   175 lemma of_nat_1 [simp]: "of_nat 1 = 1"
```
```   176 by simp
```
```   177
```
```   178 lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
```
```   179 apply (induct m)
```
```   180 apply (simp_all add: add_ac)
```
```   181 done
```
```   182
```
```   183 lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
```
```   184 apply (induct m)
```
```   185 apply (simp_all add: mult_ac add_ac right_distrib)
```
```   186 done
```
```   187
```
```   188 lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
```
```   189 apply (induct m, simp_all)
```
```   190 apply (erule order_trans)
```
```   191 apply (rule less_add_one [THEN order_less_imp_le])
```
```   192 done
```
```   193
```
```   194 lemma less_imp_of_nat_less:
```
```   195      "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
```
```   196 apply (induct m n rule: diff_induct, simp_all)
```
```   197 apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
```
```   198 done
```
```   199
```
```   200 lemma of_nat_less_imp_less:
```
```   201      "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
```
```   202 apply (induct m n rule: diff_induct, simp_all)
```
```   203 apply (insert zero_le_imp_of_nat)
```
```   204 apply (force simp add: linorder_not_less [symmetric])
```
```   205 done
```
```   206
```
```   207 lemma of_nat_less_iff [simp]:
```
```   208      "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
```
```   209 by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
```
```   210
```
```   211 text{*Special cases where either operand is zero*}
```
```   212 lemmas of_nat_0_less_iff = of_nat_less_iff [of 0, simplified]
```
```   213 lemmas of_nat_less_0_iff = of_nat_less_iff [of _ 0, simplified]
```
```   214 declare of_nat_0_less_iff [simp]
```
```   215 declare of_nat_less_0_iff [simp]
```
```   216
```
```   217 lemma of_nat_le_iff [simp]:
```
```   218      "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
```
```   219 by (simp add: linorder_not_less [symmetric])
```
```   220
```
```   221 text{*Special cases where either operand is zero*}
```
```   222 lemmas of_nat_0_le_iff = of_nat_le_iff [of 0, simplified]
```
```   223 lemmas of_nat_le_0_iff = of_nat_le_iff [of _ 0, simplified]
```
```   224 declare of_nat_0_le_iff [simp]
```
```   225 declare of_nat_le_0_iff [simp]
```
```   226
```
```   227 text{*The ordering on the @{text comm_semiring_1_cancel} is necessary
```
```   228 to exclude the possibility of a finite field, which indeed wraps back to
```
```   229 zero.*}
```
```   230 lemma of_nat_eq_iff [simp]:
```
```   231      "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
```
```   232 by (simp add: order_eq_iff)
```
```   233
```
```   234 text{*Special cases where either operand is zero*}
```
```   235 lemmas of_nat_0_eq_iff = of_nat_eq_iff [of 0, simplified]
```
```   236 lemmas of_nat_eq_0_iff = of_nat_eq_iff [of _ 0, simplified]
```
```   237 declare of_nat_0_eq_iff [simp]
```
```   238 declare of_nat_eq_0_iff [simp]
```
```   239
```
```   240 lemma of_nat_diff [simp]:
```
```   241      "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::comm_ring_1)"
```
```   242 by (simp del: of_nat_add
```
```   243 	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
```
```   244
```
```   245
```
```   246 end
```