src/HOL/Groebner_Basis.thy
 author haftmann Thu May 06 23:11:57 2010 +0200 (2010-05-06) changeset 36720 41da7025e59c parent 36716 b09f3ad3208f child 36751 7f1da69cacb3 permissions -rw-r--r--
proper sublocales; no free-floating ML sections
```     1 (*  Title:      HOL/Groebner_Basis.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Semiring normalization and Groebner Bases *}
```
```     6
```
```     7 theory Groebner_Basis
```
```     8 imports Numeral_Simprocs Nat_Transfer
```
```     9 uses
```
```    10   "Tools/Groebner_Basis/normalizer.ML"
```
```    11   ("Tools/Groebner_Basis/groebner.ML")
```
```    12 begin
```
```    13
```
```    14 subsection {* Semiring normalization *}
```
```    15
```
```    16 setup Normalizer.setup
```
```    17
```
```    18 locale normalizing_semiring =
```
```    19   fixes add mul pwr r0 r1
```
```    20   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    21     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    22     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    23     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    24     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    25     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    26 begin
```
```    27
```
```    28 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    29 proof (induct p)
```
```    30   case 0
```
```    31   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    32 next
```
```    33   case Suc
```
```    34   from this [symmetric] show ?case
```
```    35     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    36 qed
```
```    37
```
```    38 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    39 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    40   fix q x y
```
```    41   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    42   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    43     by (simp add: mul_a)
```
```    44   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    45   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    46   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    47     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    48 qed
```
```    49
```
```    50 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    51 proof (induct p arbitrary: q)
```
```    52   case 0
```
```    53   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    54 next
```
```    55   case Suc
```
```    56   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    57 qed
```
```    58
```
```    59 lemma semiring_ops:
```
```    60   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    61     and "TERM r0" and "TERM r1" .
```
```    62
```
```    63 lemma semiring_rules:
```
```    64   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    65   "add (mul a m) m = mul (add a r1) m"
```
```    66   "add m (mul a m) = mul (add a r1) m"
```
```    67   "add m m = mul (add r1 r1) m"
```
```    68   "add r0 a = a"
```
```    69   "add a r0 = a"
```
```    70   "mul a b = mul b a"
```
```    71   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    72   "mul r0 a = r0"
```
```    73   "mul a r0 = r0"
```
```    74   "mul r1 a = a"
```
```    75   "mul a r1 = a"
```
```    76   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    77   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    78   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    79   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    80   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    81   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    82   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    83   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    84   "add (add a b) c = add a (add b c)"
```
```    85   "add a (add c d) = add c (add a d)"
```
```    86   "add (add a b) c = add (add a c) b"
```
```    87   "add a c = add c a"
```
```    88   "add a (add c d) = add (add a c) d"
```
```    89   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    90   "mul x (pwr x q) = pwr x (Suc q)"
```
```    91   "mul (pwr x q) x = pwr x (Suc q)"
```
```    92   "mul x x = pwr x 2"
```
```    93   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    94   "pwr (pwr x p) q = pwr x (p * q)"
```
```    95   "pwr x 0 = r1"
```
```    96   "pwr x 1 = x"
```
```    97   "mul x (add y z) = add (mul x y) (mul x z)"
```
```    98   "pwr x (Suc q) = mul x (pwr x q)"
```
```    99   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   100   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   101 proof -
```
```   102   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   103 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   104 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   105 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   106 next show "add r0 a = a" using add_0 by simp
```
```   107 next show "add a r0 = a" using add_0 add_c by simp
```
```   108 next show "mul a b = mul b a" using mul_c by simp
```
```   109 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   110 next show "mul r0 a = r0" using mul_0 by simp
```
```   111 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   112 next show "mul r1 a = a" using mul_1 by simp
```
```   113 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   114 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   115     using mul_c mul_a by simp
```
```   116 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   117     using mul_a by simp
```
```   118 next
```
```   119   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   120   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   121   finally
```
```   122   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   123     using mul_c by simp
```
```   124 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   125 next
```
```   126   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   127 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   128 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   129 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   130     using add_c add_a by simp
```
```   131 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   132 next show "add a (add c d) = add c (add a d)"
```
```   133     apply (simp add: add_a) by (simp only: add_c)
```
```   134 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   135 next show "add a c = add c a" by (rule add_c)
```
```   136 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   137 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   138 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   139 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   140 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   141 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   142 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   143 next show "pwr x 0 = r1" using pwr_0 .
```
```   144 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   145 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   146 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   147 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
```
```   148 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   149     by (simp add: nat_number' pwr_Suc mul_pwr)
```
```   150 qed
```
```   151
```
```   152
```
```   153 lemmas normalizing_semiring_axioms' =
```
```   154   normalizing_semiring_axioms [normalizer
```
```   155     semiring ops: semiring_ops
```
```   156     semiring rules: semiring_rules]
```
```   157
```
```   158 end
```
```   159
```
```   160 sublocale comm_semiring_1
```
```   161   < normalizing!: normalizing_semiring plus times power zero one
```
```   162 proof
```
```   163 qed (simp_all add: algebra_simps)
```
```   164
```
```   165 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
```
```   166
```
```   167 locale normalizing_ring = normalizing_semiring +
```
```   168   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   169     and neg :: "'a \<Rightarrow> 'a"
```
```   170   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   171     and sub_add: "sub x y = add x (neg y)"
```
```   172 begin
```
```   173
```
```   174 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   175
```
```   176 lemmas ring_rules = neg_mul sub_add
```
```   177
```
```   178 lemmas normalizing_ring_axioms' =
```
```   179   normalizing_ring_axioms [normalizer
```
```   180     semiring ops: semiring_ops
```
```   181     semiring rules: semiring_rules
```
```   182     ring ops: ring_ops
```
```   183     ring rules: ring_rules]
```
```   184
```
```   185 end
```
```   186
```
```   187 sublocale comm_ring_1
```
```   188   < normalizing!: normalizing_ring plus times power zero one minus uminus
```
```   189 proof
```
```   190 qed (simp_all add: diff_minus)
```
```   191
```
```   192 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
```
```   193
```
```   194 locale normalizing_field = normalizing_ring +
```
```   195   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   196     and inverse:: "'a \<Rightarrow> 'a"
```
```   197   assumes divide_inverse: "divide x y = mul x (inverse y)"
```
```   198      and inverse_divide: "inverse x = divide r1 x"
```
```   199 begin
```
```   200
```
```   201 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
```
```   202
```
```   203 lemmas field_rules = divide_inverse inverse_divide
```
```   204
```
```   205 lemmas normalizing_field_axioms' =
```
```   206   normalizing_field_axioms [normalizer
```
```   207     semiring ops: semiring_ops
```
```   208     semiring rules: semiring_rules
```
```   209     ring ops: ring_ops
```
```   210     ring rules: ring_rules
```
```   211     field ops: field_ops
```
```   212     field rules: field_rules]
```
```   213
```
```   214 end
```
```   215
```
```   216 locale normalizing_semiring_cancel = normalizing_semiring +
```
```   217   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   218   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   219     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   220 begin
```
```   221
```
```   222 lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   223 proof-
```
```   224   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   225   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   226     using add_mul_solve by blast
```
```   227   finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   228     by simp
```
```   229 qed
```
```   230
```
```   231 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   232   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   233 proof(clarify)
```
```   234   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   235     and eq: "add b (mul r c) = add b (mul r d)"
```
```   236   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   237   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   238     using mul_0 add_cancel by simp
```
```   239   thus "False" using add_mul_solve nz cnd by simp
```
```   240 qed
```
```   241
```
```   242 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
```
```   243 proof-
```
```   244   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
```
```   245   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
```
```   246 qed
```
```   247
```
```   248 declare normalizing_semiring_axioms' [normalizer del]
```
```   249
```
```   250 lemmas normalizing_semiring_cancel_axioms' =
```
```   251   normalizing_semiring_cancel_axioms [normalizer
```
```   252     semiring ops: semiring_ops
```
```   253     semiring rules: semiring_rules
```
```   254     idom rules: noteq_reduce add_scale_eq_noteq]
```
```   255
```
```   256 end
```
```   257
```
```   258 locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
```
```   259   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
```
```   260 begin
```
```   261
```
```   262 declare normalizing_ring_axioms' [normalizer del]
```
```   263
```
```   264 lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
```
```   265   semiring ops: semiring_ops
```
```   266   semiring rules: semiring_rules
```
```   267   ring ops: ring_ops
```
```   268   ring rules: ring_rules
```
```   269   idom rules: noteq_reduce add_scale_eq_noteq
```
```   270   ideal rules: subr0_iff add_r0_iff]
```
```   271
```
```   272 end
```
```   273
```
```   274 sublocale idom
```
```   275   < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
```
```   276 proof
```
```   277   fix w x y z
```
```   278   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```   279   proof
```
```   280     assume "w * y + x * z = w * z + x * y"
```
```   281     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
```
```   282     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```   283     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```   284     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
```
```   285     then show "w = x \<or> y = z" by auto
```
```   286   qed (auto simp add: add_ac)
```
```   287 qed (simp_all add: algebra_simps)
```
```   288
```
```   289 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
```
```   290
```
```   291 interpretation normalizing_nat!: normalizing_semiring_cancel
```
```   292   "op +" "op *" "op ^" "0::nat" "1"
```
```   293 proof (unfold_locales, simp add: algebra_simps)
```
```   294   fix w x y z ::"nat"
```
```   295   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   296     hence "y < z \<or> y > z" by arith
```
```   297     moreover {
```
```   298       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   299       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   300       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
```
```   301       hence "x*k = w*k" by simp
```
```   302       hence "w = x" using kp by simp }
```
```   303     moreover {
```
```   304       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   305       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   306       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
```
```   307       hence "w*k = x*k" by simp
```
```   308       hence "w = x" using kp by simp }
```
```   309     ultimately have "w=x" by blast }
```
```   310   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   311 qed
```
```   312
```
```   313 declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
```
```   314
```
```   315 locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
```
```   316 begin
```
```   317
```
```   318 declare normalizing_field_axioms' [normalizer del]
```
```   319
```
```   320 lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
```
```   321   semiring ops: semiring_ops
```
```   322   semiring rules: semiring_rules
```
```   323   ring ops: ring_ops
```
```   324   ring rules: ring_rules
```
```   325   field ops: field_ops
```
```   326   field rules: field_rules
```
```   327   idom rules: noteq_reduce add_scale_eq_noteq
```
```   328   ideal rules: subr0_iff add_r0_iff]
```
```   329
```
```   330 end
```
```   331
```
```   332 sublocale field
```
```   333   < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
```
```   334 proof
```
```   335 qed (simp_all add: divide_inverse)
```
```   336
```
```   337 declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
```
```   338
```
```   339
```
```   340 subsection {* Groebner Bases *}
```
```   341
```
```   342 lemmas bool_simps = simp_thms(1-34)
```
```   343
```
```   344 lemma dnf:
```
```   345     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
```
```   346     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
```
```   347   by blast+
```
```   348
```
```   349 lemmas weak_dnf_simps = dnf bool_simps
```
```   350
```
```   351 lemma nnf_simps:
```
```   352     "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
```
```   353     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
```
```   354   by blast+
```
```   355
```
```   356 lemma PFalse:
```
```   357     "P \<equiv> False \<Longrightarrow> \<not> P"
```
```   358     "\<not> P \<Longrightarrow> (P \<equiv> False)"
```
```   359   by auto
```
```   360
```
```   361 ML {*
```
```   362 structure Algebra_Simplification = Named_Thms(
```
```   363   val name = "algebra"
```
```   364   val description = "pre-simplification rules for algebraic methods"
```
```   365 )
```
```   366 *}
```
```   367
```
```   368 setup Algebra_Simplification.setup
```
```   369
```
```   370 declare dvd_def[algebra]
```
```   371 declare dvd_eq_mod_eq_0[symmetric, algebra]
```
```   372 declare mod_div_trivial[algebra]
```
```   373 declare mod_mod_trivial[algebra]
```
```   374 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
```
```   375 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
```
```   376 declare zmod_zdiv_equality[symmetric,algebra]
```
```   377 declare zdiv_zmod_equality[symmetric, algebra]
```
```   378 declare zdiv_zminus_zminus[algebra]
```
```   379 declare zmod_zminus_zminus[algebra]
```
```   380 declare zdiv_zminus2[algebra]
```
```   381 declare zmod_zminus2[algebra]
```
```   382 declare zdiv_zero[algebra]
```
```   383 declare zmod_zero[algebra]
```
```   384 declare mod_by_1[algebra]
```
```   385 declare div_by_1[algebra]
```
```   386 declare zmod_minus1_right[algebra]
```
```   387 declare zdiv_minus1_right[algebra]
```
```   388 declare mod_div_trivial[algebra]
```
```   389 declare mod_mod_trivial[algebra]
```
```   390 declare mod_mult_self2_is_0[algebra]
```
```   391 declare mod_mult_self1_is_0[algebra]
```
```   392 declare zmod_eq_0_iff[algebra]
```
```   393 declare dvd_0_left_iff[algebra]
```
```   394 declare zdvd1_eq[algebra]
```
```   395 declare zmod_eq_dvd_iff[algebra]
```
```   396 declare nat_mod_eq_iff[algebra]
```
```   397
```
```   398 use "Tools/Groebner_Basis/groebner.ML"
```
```   399
```
```   400 method_setup algebra = Groebner.algebra_method
```
```   401   "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
```
```   402
```
```   403 end
```