src/HOL/Library/Commutative_Ring.thy
 author haftmann Mon Jul 07 08:47:17 2008 +0200 (2008-07-07) changeset 27487 c8a6ce181805 parent 27368 9f90ac19e32b child 28952 15a4b2cf8c34 permissions -rw-r--r--
absolute imports of HOL/*.thy theories
```     1 (*  ID:         \$Id\$
```
```     2     Author:     Bernhard Haeupler
```
```     3
```
```     4 Proving equalities in commutative rings done "right" in Isabelle/HOL.
```
```     5 *)
```
```     6
```
```     7 header {* Proving equalities in commutative rings *}
```
```     8
```
```     9 theory Commutative_Ring
```
```    10 imports Plain "~~/src/HOL/List" Parity
```
```    11 uses ("comm_ring.ML")
```
```    12 begin
```
```    13
```
```    14 text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}
```
```    15
```
```    16 datatype 'a pol =
```
```    17     Pc 'a
```
```    18   | Pinj nat "'a pol"
```
```    19   | PX "'a pol" nat "'a pol"
```
```    20
```
```    21 datatype 'a polex =
```
```    22     Pol "'a pol"
```
```    23   | Add "'a polex" "'a polex"
```
```    24   | Sub "'a polex" "'a polex"
```
```    25   | Mul "'a polex" "'a polex"
```
```    26   | Pow "'a polex" nat
```
```    27   | Neg "'a polex"
```
```    28
```
```    29 text {* Interpretation functions for the shadow syntax. *}
```
```    30
```
```    31 fun
```
```    32   Ipol :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
```
```    33 where
```
```    34     "Ipol l (Pc c) = c"
```
```    35   | "Ipol l (Pinj i P) = Ipol (drop i l) P"
```
```    36   | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
```
```    37
```
```    38 fun
```
```    39   Ipolex :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
```
```    40 where
```
```    41     "Ipolex l (Pol P) = Ipol l P"
```
```    42   | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
```
```    43   | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
```
```    44   | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
```
```    45   | "Ipolex l (Pow p n) = Ipolex l p ^ n"
```
```    46   | "Ipolex l (Neg P) = - Ipolex l P"
```
```    47
```
```    48 text {* Create polynomial normalized polynomials given normalized inputs. *}
```
```    49
```
```    50 definition
```
```    51   mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
```
```    52   "mkPinj x P = (case P of
```
```    53     Pc c \<Rightarrow> Pc c |
```
```    54     Pinj y P \<Rightarrow> Pinj (x + y) P |
```
```    55     PX p1 y p2 \<Rightarrow> Pinj x P)"
```
```    56
```
```    57 definition
```
```    58   mkPX :: "'a::{comm_ring,recpower} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
```
```    59   "mkPX P i Q = (case P of
```
```    60     Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
```
```    61     Pinj j R \<Rightarrow> PX P i Q |
```
```    62     PX P2 i2 Q2 \<Rightarrow> (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"
```
```    63
```
```    64 text {* Defining the basic ring operations on normalized polynomials *}
```
```    65
```
```    66 function
```
```    67   add :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
```
```    68 where
```
```    69     "Pc a \<oplus> Pc b = Pc (a + b)"
```
```    70   | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
```
```    71   | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
```
```    72   | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
```
```    73   | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
```
```    74   | "Pinj x P \<oplus> Pinj y Q =
```
```    75       (if x = y then mkPinj x (P \<oplus> Q)
```
```    76        else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
```
```    77          else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
```
```    78   | "Pinj x P \<oplus> PX Q y R =
```
```    79       (if x = 0 then P \<oplus> PX Q y R
```
```    80        else (if x = 1 then PX Q y (R \<oplus> P)
```
```    81          else PX Q y (R \<oplus> Pinj (x - 1) P)))"
```
```    82   | "PX P x R \<oplus> Pinj y Q =
```
```    83       (if y = 0 then PX P x R \<oplus> Q
```
```    84        else (if y = 1 then PX P x (R \<oplus> Q)
```
```    85          else PX P x (R \<oplus> Pinj (y - 1) Q)))"
```
```    86   | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
```
```    87       (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
```
```    88        else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
```
```    89          else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
```
```    90 by pat_completeness auto
```
```    91 termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto
```
```    92
```
```    93 function
```
```    94   mul :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<otimes>" 70)
```
```    95 where
```
```    96     "Pc a \<otimes> Pc b = Pc (a * b)"
```
```    97   | "Pc c \<otimes> Pinj i P =
```
```    98       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
```
```    99   | "Pinj i P \<otimes> Pc c =
```
```   100       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
```
```   101   | "Pc c \<otimes> PX P i Q =
```
```   102       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
```
```   103   | "PX P i Q \<otimes> Pc c =
```
```   104       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
```
```   105   | "Pinj x P \<otimes> Pinj y Q =
```
```   106       (if x = y then mkPinj x (P \<otimes> Q) else
```
```   107          (if x > y then mkPinj y (Pinj (x-y) P \<otimes> Q)
```
```   108            else mkPinj x (Pinj (y - x) Q \<otimes> P)))"
```
```   109   | "Pinj x P \<otimes> PX Q y R =
```
```   110       (if x = 0 then P \<otimes> PX Q y R else
```
```   111          (if x = 1 then mkPX (Pinj x P \<otimes> Q) y (R \<otimes> P)
```
```   112            else mkPX (Pinj x P \<otimes> Q) y (R \<otimes> Pinj (x - 1) P)))"
```
```   113   | "PX P x R \<otimes> Pinj y Q =
```
```   114       (if y = 0 then PX P x R \<otimes> Q else
```
```   115          (if y = 1 then mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Q)
```
```   116            else mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Pinj (y - 1) Q)))"
```
```   117   | "PX P1 x P2 \<otimes> PX Q1 y Q2 =
```
```   118       mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2) \<oplus>
```
```   119         (mkPX (P1 \<otimes> mkPinj 1 Q2) x (Pc 0) \<oplus>
```
```   120           (mkPX (Q1 \<otimes> mkPinj 1 P2) y (Pc 0)))"
```
```   121 by pat_completeness auto
```
```   122 termination by (relation "measure (\<lambda>(x, y). size x + size y)")
```
```   123   (auto simp add: mkPinj_def split: pol.split)
```
```   124
```
```   125 text {* Negation*}
```
```   126 fun
```
```   127   neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   128 where
```
```   129     "neg (Pc c) = Pc (-c)"
```
```   130   | "neg (Pinj i P) = Pinj i (neg P)"
```
```   131   | "neg (PX P x Q) = PX (neg P) x (neg Q)"
```
```   132
```
```   133 text {* Substraction *}
```
```   134 definition
```
```   135   sub :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
```
```   136 where
```
```   137   "sub P Q = P \<oplus> neg Q"
```
```   138
```
```   139 text {* Square for Fast Exponentation *}
```
```   140 fun
```
```   141   sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   142 where
```
```   143     "sqr (Pc c) = Pc (c * c)"
```
```   144   | "sqr (Pinj i P) = mkPinj i (sqr P)"
```
```   145   | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<oplus>
```
```   146       mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"
```
```   147
```
```   148 text {* Fast Exponentation *}
```
```   149 fun
```
```   150   pow :: "nat \<Rightarrow> 'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   151 where
```
```   152     "pow 0 P = Pc 1"
```
```   153   | "pow n P = (if even n then pow (n div 2) (sqr P)
```
```   154        else P \<otimes> pow (n div 2) (sqr P))"
```
```   155
```
```   156 lemma pow_if:
```
```   157   "pow n P =
```
```   158    (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
```
```   159     else P \<otimes> pow (n div 2) (sqr P))"
```
```   160   by (cases n) simp_all
```
```   161
```
```   162
```
```   163 text {* Normalization of polynomial expressions *}
```
```   164
```
```   165 fun
```
```   166   norm :: "'a::{comm_ring,recpower} polex \<Rightarrow> 'a pol"
```
```   167 where
```
```   168     "norm (Pol P) = P"
```
```   169   | "norm (Add P Q) = norm P \<oplus> norm Q"
```
```   170   | "norm (Sub P Q) = norm P \<ominus> norm Q"
```
```   171   | "norm (Mul P Q) = norm P \<otimes> norm Q"
```
```   172   | "norm (Pow P n) = pow n (norm P)"
```
```   173   | "norm (Neg P) = neg (norm P)"
```
```   174
```
```   175 text {* mkPinj preserve semantics *}
```
```   176 lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
```
```   177   by (induct B) (auto simp add: mkPinj_def ring_simps)
```
```   178
```
```   179 text {* mkPX preserves semantics *}
```
```   180 lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
```
```   181   by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add ring_simps)
```
```   182
```
```   183 text {* Correctness theorems for the implemented operations *}
```
```   184
```
```   185 text {* Negation *}
```
```   186 lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
```
```   187   by (induct P arbitrary: l) auto
```
```   188
```
```   189 text {* Addition *}
```
```   190 lemma add_ci: "Ipol l (P \<oplus> Q) = Ipol l P + Ipol l Q"
```
```   191 proof (induct P Q arbitrary: l rule: add.induct)
```
```   192   case (6 x P y Q)
```
```   193   show ?case
```
```   194   proof (rule linorder_cases)
```
```   195     assume "x < y"
```
```   196     with 6 show ?case by (simp add: mkPinj_ci ring_simps)
```
```   197   next
```
```   198     assume "x = y"
```
```   199     with 6 show ?case by (simp add: mkPinj_ci)
```
```   200   next
```
```   201     assume "x > y"
```
```   202     with 6 show ?case by (simp add: mkPinj_ci ring_simps)
```
```   203   qed
```
```   204 next
```
```   205   case (7 x P Q y R)
```
```   206   have "x = 0 \<or> x = 1 \<or> x > 1" by arith
```
```   207   moreover
```
```   208   { assume "x = 0" with 7 have ?case by simp }
```
```   209   moreover
```
```   210   { assume "x = 1" with 7 have ?case by (simp add: ring_simps) }
```
```   211   moreover
```
```   212   { assume "x > 1" from 7 have ?case by (cases x) simp_all }
```
```   213   ultimately show ?case by blast
```
```   214 next
```
```   215   case (8 P x R y Q)
```
```   216   have "y = 0 \<or> y = 1 \<or> y > 1" by arith
```
```   217   moreover
```
```   218   { assume "y = 0" with 8 have ?case by simp }
```
```   219   moreover
```
```   220   { assume "y = 1" with 8 have ?case by simp }
```
```   221   moreover
```
```   222   { assume "y > 1" with 8 have ?case by simp }
```
```   223   ultimately show ?case by blast
```
```   224 next
```
```   225   case (9 P1 x P2 Q1 y Q2)
```
```   226   show ?case
```
```   227   proof (rule linorder_cases)
```
```   228     assume a: "x < y" hence "EX d. d + x = y" by arith
```
```   229     with 9 a show ?case by (auto simp add: mkPX_ci power_add ring_simps)
```
```   230   next
```
```   231     assume a: "y < x" hence "EX d. d + y = x" by arith
```
```   232     with 9 a show ?case by (auto simp add: power_add mkPX_ci ring_simps)
```
```   233   next
```
```   234     assume "x = y"
```
```   235     with 9 show ?case by (simp add: mkPX_ci ring_simps)
```
```   236   qed
```
```   237 qed (auto simp add: ring_simps)
```
```   238
```
```   239 text {* Multiplication *}
```
```   240 lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
```
```   241   by (induct P Q arbitrary: l rule: mul.induct)
```
```   242     (simp_all add: mkPX_ci mkPinj_ci ring_simps add_ci power_add)
```
```   243
```
```   244 text {* Substraction *}
```
```   245 lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
```
```   246   by (simp add: add_ci neg_ci sub_def)
```
```   247
```
```   248 text {* Square *}
```
```   249 lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
```
```   250   by (induct P arbitrary: ls)
```
```   251     (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_simps power_add)
```
```   252
```
```   253 text {* Power *}
```
```   254 lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
```
```   255   by (induct n) simp_all
```
```   256
```
```   257 lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
```
```   258 proof (induct n arbitrary: P rule: nat_less_induct)
```
```   259   case (1 k)
```
```   260   show ?case
```
```   261   proof (cases k)
```
```   262     case 0
```
```   263     then show ?thesis by simp
```
```   264   next
```
```   265     case (Suc l)
```
```   266     show ?thesis
```
```   267     proof cases
```
```   268       assume "even l"
```
```   269       then have "Suc l div 2 = l div 2"
```
```   270         by (simp add: nat_number even_nat_plus_one_div_two)
```
```   271       moreover
```
```   272       from Suc have "l < k" by simp
```
```   273       with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
```
```   274       moreover
```
```   275       note Suc `even l` even_nat_plus_one_div_two
```
```   276       ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
```
```   277     next
```
```   278       assume "odd l"
```
```   279       {
```
```   280         fix p
```
```   281         have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
```
```   282         proof (cases l)
```
```   283           case 0
```
```   284           with `odd l` show ?thesis by simp
```
```   285         next
```
```   286           case (Suc w)
```
```   287           with `odd l` have "even w" by simp
```
```   288           have two_times: "2 * (w div 2) = w"
```
```   289             by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
```
```   290           have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
```
```   291             by (simp add: power_Suc)
```
```   292 	  then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
```
```   293 	    by (simp add: numerals)
```
```   294           with Suc show ?thesis
```
```   295             by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci)
```
```   296         qed
```
```   297       } with 1 Suc `odd l` show ?thesis by simp
```
```   298     qed
```
```   299   qed
```
```   300 qed
```
```   301
```
```   302 text {* Normalization preserves semantics  *}
```
```   303 lemma norm_ci: "Ipolex l Pe = Ipol l (norm Pe)"
```
```   304   by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)
```
```   305
```
```   306 text {* Reflection lemma: Key to the (incomplete) decision procedure *}
```
```   307 lemma norm_eq:
```
```   308   assumes "norm P1 = norm P2"
```
```   309   shows "Ipolex l P1 = Ipolex l P2"
```
```   310 proof -
```
```   311   from prems have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
```
```   312   then show ?thesis by (simp only: norm_ci)
```
```   313 qed
```
```   314
```
```   315
```
```   316 use "comm_ring.ML"
```
```   317 setup CommRing.setup
```
```   318
```
```   319 end
```