src/HOL/Library/Commutative_Ring.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30663 0b6aff7451b2 child 31021 53642251a04f permissions -rw-r--r--
power operation defined generic
```     1 (*  Author:     Bernhard Haeupler
```
```     2
```
```     3 Proving equalities in commutative rings done "right" in Isabelle/HOL.
```
```     4 *)
```
```     5
```
```     6 header {* Proving equalities in commutative rings *}
```
```     7
```
```     8 theory Commutative_Ring
```
```     9 imports List Parity Main
```
```    10 uses ("comm_ring.ML")
```
```    11 begin
```
```    12
```
```    13 text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}
```
```    14
```
```    15 datatype 'a pol =
```
```    16     Pc 'a
```
```    17   | Pinj nat "'a pol"
```
```    18   | PX "'a pol" nat "'a pol"
```
```    19
```
```    20 datatype 'a polex =
```
```    21     Pol "'a pol"
```
```    22   | Add "'a polex" "'a polex"
```
```    23   | Sub "'a polex" "'a polex"
```
```    24   | Mul "'a polex" "'a polex"
```
```    25   | Pow "'a polex" nat
```
```    26   | Neg "'a polex"
```
```    27
```
```    28 text {* Interpretation functions for the shadow syntax. *}
```
```    29
```
```    30 fun
```
```    31   Ipol :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
```
```    32 where
```
```    33     "Ipol l (Pc c) = c"
```
```    34   | "Ipol l (Pinj i P) = Ipol (drop i l) P"
```
```    35   | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
```
```    36
```
```    37 fun
```
```    38   Ipolex :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
```
```    39 where
```
```    40     "Ipolex l (Pol P) = Ipol l P"
```
```    41   | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
```
```    42   | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
```
```    43   | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
```
```    44   | "Ipolex l (Pow p n) = Ipolex l p ^ n"
```
```    45   | "Ipolex l (Neg P) = - Ipolex l P"
```
```    46
```
```    47 text {* Create polynomial normalized polynomials given normalized inputs. *}
```
```    48
```
```    49 definition
```
```    50   mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
```
```    51   "mkPinj x P = (case P of
```
```    52     Pc c \<Rightarrow> Pc c |
```
```    53     Pinj y P \<Rightarrow> Pinj (x + y) P |
```
```    54     PX p1 y p2 \<Rightarrow> Pinj x P)"
```
```    55
```
```    56 definition
```
```    57   mkPX :: "'a::{comm_ring,recpower} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
```
```    58   "mkPX P i Q = (case P of
```
```    59     Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
```
```    60     Pinj j R \<Rightarrow> PX P i Q |
```
```    61     PX P2 i2 Q2 \<Rightarrow> (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"
```
```    62
```
```    63 text {* Defining the basic ring operations on normalized polynomials *}
```
```    64
```
```    65 function
```
```    66   add :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
```
```    67 where
```
```    68     "Pc a \<oplus> Pc b = Pc (a + b)"
```
```    69   | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
```
```    70   | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
```
```    71   | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
```
```    72   | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
```
```    73   | "Pinj x P \<oplus> Pinj y Q =
```
```    74       (if x = y then mkPinj x (P \<oplus> Q)
```
```    75        else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
```
```    76          else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
```
```    77   | "Pinj x P \<oplus> PX Q y R =
```
```    78       (if x = 0 then P \<oplus> PX Q y R
```
```    79        else (if x = 1 then PX Q y (R \<oplus> P)
```
```    80          else PX Q y (R \<oplus> Pinj (x - 1) P)))"
```
```    81   | "PX P x R \<oplus> Pinj y Q =
```
```    82       (if y = 0 then PX P x R \<oplus> Q
```
```    83        else (if y = 1 then PX P x (R \<oplus> Q)
```
```    84          else PX P x (R \<oplus> Pinj (y - 1) Q)))"
```
```    85   | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
```
```    86       (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
```
```    87        else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
```
```    88          else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
```
```    89 by pat_completeness auto
```
```    90 termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto
```
```    91
```
```    92 function
```
```    93   mul :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<otimes>" 70)
```
```    94 where
```
```    95     "Pc a \<otimes> Pc b = Pc (a * b)"
```
```    96   | "Pc c \<otimes> Pinj i P =
```
```    97       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
```
```    98   | "Pinj i P \<otimes> Pc c =
```
```    99       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
```
```   100   | "Pc c \<otimes> PX P i Q =
```
```   101       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
```
```   102   | "PX P i Q \<otimes> Pc c =
```
```   103       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
```
```   104   | "Pinj x P \<otimes> Pinj y Q =
```
```   105       (if x = y then mkPinj x (P \<otimes> Q) else
```
```   106          (if x > y then mkPinj y (Pinj (x-y) P \<otimes> Q)
```
```   107            else mkPinj x (Pinj (y - x) Q \<otimes> P)))"
```
```   108   | "Pinj x P \<otimes> PX Q y R =
```
```   109       (if x = 0 then P \<otimes> PX Q y R else
```
```   110          (if x = 1 then mkPX (Pinj x P \<otimes> Q) y (R \<otimes> P)
```
```   111            else mkPX (Pinj x P \<otimes> Q) y (R \<otimes> Pinj (x - 1) P)))"
```
```   112   | "PX P x R \<otimes> Pinj y Q =
```
```   113       (if y = 0 then PX P x R \<otimes> Q else
```
```   114          (if y = 1 then mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Q)
```
```   115            else mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Pinj (y - 1) Q)))"
```
```   116   | "PX P1 x P2 \<otimes> PX Q1 y Q2 =
```
```   117       mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2) \<oplus>
```
```   118         (mkPX (P1 \<otimes> mkPinj 1 Q2) x (Pc 0) \<oplus>
```
```   119           (mkPX (Q1 \<otimes> mkPinj 1 P2) y (Pc 0)))"
```
```   120 by pat_completeness auto
```
```   121 termination by (relation "measure (\<lambda>(x, y). size x + size y)")
```
```   122   (auto simp add: mkPinj_def split: pol.split)
```
```   123
```
```   124 text {* Negation*}
```
```   125 fun
```
```   126   neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   127 where
```
```   128     "neg (Pc c) = Pc (-c)"
```
```   129   | "neg (Pinj i P) = Pinj i (neg P)"
```
```   130   | "neg (PX P x Q) = PX (neg P) x (neg Q)"
```
```   131
```
```   132 text {* Substraction *}
```
```   133 definition
```
```   134   sub :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
```
```   135 where
```
```   136   "sub P Q = P \<oplus> neg Q"
```
```   137
```
```   138 text {* Square for Fast Exponentation *}
```
```   139 fun
```
```   140   sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   141 where
```
```   142     "sqr (Pc c) = Pc (c * c)"
```
```   143   | "sqr (Pinj i P) = mkPinj i (sqr P)"
```
```   144   | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<oplus>
```
```   145       mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"
```
```   146
```
```   147 text {* Fast Exponentation *}
```
```   148 fun
```
```   149   pow :: "nat \<Rightarrow> 'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
```
```   150 where
```
```   151     "pow 0 P = Pc 1"
```
```   152   | "pow n P = (if even n then pow (n div 2) (sqr P)
```
```   153        else P \<otimes> pow (n div 2) (sqr P))"
```
```   154
```
```   155 lemma pow_if:
```
```   156   "pow n P =
```
```   157    (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
```
```   158     else P \<otimes> pow (n div 2) (sqr P))"
```
```   159   by (cases n) simp_all
```
```   160
```
```   161
```
```   162 text {* Normalization of polynomial expressions *}
```
```   163
```
```   164 fun
```
```   165   norm :: "'a::{comm_ring,recpower} polex \<Rightarrow> 'a pol"
```
```   166 where
```
```   167     "norm (Pol P) = P"
```
```   168   | "norm (Add P Q) = norm P \<oplus> norm Q"
```
```   169   | "norm (Sub P Q) = norm P \<ominus> norm Q"
```
```   170   | "norm (Mul P Q) = norm P \<otimes> norm Q"
```
```   171   | "norm (Pow P n) = pow n (norm P)"
```
```   172   | "norm (Neg P) = neg (norm P)"
```
```   173
```
```   174 text {* mkPinj preserve semantics *}
```
```   175 lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
```
```   176   by (induct B) (auto simp add: mkPinj_def algebra_simps)
```
```   177
```
```   178 text {* mkPX preserves semantics *}
```
```   179 lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
```
```   180   by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add algebra_simps)
```
```   181
```
```   182 text {* Correctness theorems for the implemented operations *}
```
```   183
```
```   184 text {* Negation *}
```
```   185 lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
```
```   186   by (induct P arbitrary: l) auto
```
```   187
```
```   188 text {* Addition *}
```
```   189 lemma add_ci: "Ipol l (P \<oplus> Q) = Ipol l P + Ipol l Q"
```
```   190 proof (induct P Q arbitrary: l rule: add.induct)
```
```   191   case (6 x P y Q)
```
```   192   show ?case
```
```   193   proof (rule linorder_cases)
```
```   194     assume "x < y"
```
```   195     with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
```
```   196   next
```
```   197     assume "x = y"
```
```   198     with 6 show ?case by (simp add: mkPinj_ci)
```
```   199   next
```
```   200     assume "x > y"
```
```   201     with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
```
```   202   qed
```
```   203 next
```
```   204   case (7 x P Q y R)
```
```   205   have "x = 0 \<or> x = 1 \<or> x > 1" by arith
```
```   206   moreover
```
```   207   { assume "x = 0" with 7 have ?case by simp }
```
```   208   moreover
```
```   209   { assume "x = 1" with 7 have ?case by (simp add: algebra_simps) }
```
```   210   moreover
```
```   211   { assume "x > 1" from 7 have ?case by (cases x) simp_all }
```
```   212   ultimately show ?case by blast
```
```   213 next
```
```   214   case (8 P x R y Q)
```
```   215   have "y = 0 \<or> y = 1 \<or> y > 1" by arith
```
```   216   moreover
```
```   217   { assume "y = 0" with 8 have ?case by simp }
```
```   218   moreover
```
```   219   { assume "y = 1" with 8 have ?case by simp }
```
```   220   moreover
```
```   221   { assume "y > 1" with 8 have ?case by simp }
```
```   222   ultimately show ?case by blast
```
```   223 next
```
```   224   case (9 P1 x P2 Q1 y Q2)
```
```   225   show ?case
```
```   226   proof (rule linorder_cases)
```
```   227     assume a: "x < y" hence "EX d. d + x = y" by arith
```
```   228     with 9 a show ?case by (auto simp add: mkPX_ci power_add algebra_simps)
```
```   229   next
```
```   230     assume a: "y < x" hence "EX d. d + y = x" by arith
```
```   231     with 9 a show ?case by (auto simp add: power_add mkPX_ci algebra_simps)
```
```   232   next
```
```   233     assume "x = y"
```
```   234     with 9 show ?case by (simp add: mkPX_ci algebra_simps)
```
```   235   qed
```
```   236 qed (auto simp add: algebra_simps)
```
```   237
```
```   238 text {* Multiplication *}
```
```   239 lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
```
```   240   by (induct P Q arbitrary: l rule: mul.induct)
```
```   241     (simp_all add: mkPX_ci mkPinj_ci algebra_simps add_ci power_add)
```
```   242
```
```   243 text {* Substraction *}
```
```   244 lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
```
```   245   by (simp add: add_ci neg_ci sub_def)
```
```   246
```
```   247 text {* Square *}
```
```   248 lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
```
```   249   by (induct P arbitrary: ls)
```
```   250     (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci algebra_simps power_add)
```
```   251
```
```   252 text {* Power *}
```
```   253 lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
```
```   254   by (induct n) simp_all
```
```   255
```
```   256 lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
```
```   257 proof (induct n arbitrary: P rule: nat_less_induct)
```
```   258   case (1 k)
```
```   259   show ?case
```
```   260   proof (cases k)
```
```   261     case 0
```
```   262     then show ?thesis by simp
```
```   263   next
```
```   264     case (Suc l)
```
```   265     show ?thesis
```
```   266     proof cases
```
```   267       assume "even l"
```
```   268       then have "Suc l div 2 = l div 2"
```
```   269         by (simp add: nat_number even_nat_plus_one_div_two)
```
```   270       moreover
```
```   271       from Suc have "l < k" by simp
```
```   272       with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
```
```   273       moreover
```
```   274       note Suc `even l` even_nat_plus_one_div_two
```
```   275       ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
```
```   276     next
```
```   277       assume "odd l"
```
```   278       {
```
```   279         fix p
```
```   280         have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
```
```   281         proof (cases l)
```
```   282           case 0
```
```   283           with `odd l` show ?thesis by simp
```
```   284         next
```
```   285           case (Suc w)
```
```   286           with `odd l` have "even w" by simp
```
```   287           have two_times: "2 * (w div 2) = w"
```
```   288             by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
```
```   289           have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
```
```   290             by (simp add: power_Suc)
```
```   291 	  then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
```
```   292 	    by (simp add: numerals)
```
```   293           with Suc show ?thesis
```
```   294             by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci
```
```   295                      simp del: power_Suc)
```
```   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
```