src/HOL/Library/Commutative_Ring.thy
author wenzelm
Thu Jan 19 21:22:08 2006 +0100 (2006-01-19 ago)
changeset 18708 4b3dadb4fe33
parent 17516 45164074dad4
child 19736 d8d0f8f51d69
permissions -rw-r--r--
setup: theory -> theory;
     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 Main
    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 consts
    32   Ipol :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
    33   Ipolex :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
    34 
    35 primrec
    36   "Ipol l (Pc c) = c"
    37   "Ipol l (Pinj i P) = Ipol (drop i l) P"
    38   "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
    39 
    40 primrec
    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 constdefs
    51   mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol"
    52   "mkPinj x P \<equiv> (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 constdefs
    58   mkPX :: "'a::{comm_ring,recpower} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol"
    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 consts
    67   add :: "'a::{comm_ring,recpower} pol \<times> 'a pol \<Rightarrow> 'a pol"
    68   mul :: "'a::{comm_ring,recpower} pol \<times> 'a pol \<Rightarrow> 'a pol"
    69   neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
    70   sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
    71   pow :: "'a::{comm_ring,recpower} pol \<times> nat \<Rightarrow> 'a pol"
    72 
    73 text {* Addition *}
    74 recdef add "measure (\<lambda>(x, y). size x + size y)"
    75   "add (Pc a, Pc b) = Pc (a + b)"
    76   "add (Pc c, Pinj i P) = Pinj i (add (P, Pc c))"
    77   "add (Pinj i P, Pc c) = Pinj i (add (P, Pc c))"
    78   "add (Pc c, PX P i Q) = PX P i (add (Q, Pc c))"
    79   "add (PX P i Q, Pc c) = PX P i (add (Q, Pc c))"
    80   "add (Pinj x P, Pinj y Q) =
    81   (if x=y then mkPinj x (add (P, Q))
    82    else (if x>y then mkPinj y (add (Pinj (x-y) P, Q))
    83          else mkPinj x (add (Pinj (y-x) Q, P)) ))"
    84   "add (Pinj x P, PX Q y R) =
    85   (if x=0 then add(P, PX Q y R)
    86    else (if x=1 then PX Q y (add (R, P))
    87          else PX Q y (add (R, Pinj (x - 1) P))))"
    88   "add (PX P x R, Pinj y Q) =
    89   (if y=0 then add(PX P x R, Q)
    90    else (if y=1 then PX P x (add (R, Q))
    91          else PX P x (add (R, Pinj (y - 1) Q))))"
    92   "add (PX P1 x P2, PX Q1 y Q2) =
    93   (if x=y then mkPX (add (P1, Q1)) x (add (P2, Q2))
    94   else (if x>y then mkPX (add (PX P1 (x-y) (Pc 0), Q1)) y (add (P2,Q2))
    95         else mkPX (add (PX Q1 (y-x) (Pc 0), P1)) x (add (P2,Q2)) ))"
    96 
    97 text {* Multiplication *}
    98 recdef mul "measure (\<lambda>(x, y). size x + size y)"
    99   "mul (Pc a, Pc b) = Pc (a*b)"
   100   "mul (Pc c, Pinj i P) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))"
   101   "mul (Pinj i P, Pc c) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))"
   102   "mul (Pc c, PX P i Q) =
   103   (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))"
   104   "mul (PX P i Q, Pc c) =
   105   (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))"
   106   "mul (Pinj x P, Pinj y Q) =
   107   (if x=y then mkPinj x (mul (P, Q))
   108    else (if x>y then mkPinj y (mul (Pinj (x-y) P, Q))
   109          else mkPinj x (mul (Pinj (y-x) Q, P)) ))"
   110   "mul (Pinj x P, PX Q y R) =
   111   (if x=0 then mul(P, PX Q y R)
   112    else (if x=1 then mkPX (mul (Pinj x P, Q)) y (mul (R, P))
   113          else mkPX (mul (Pinj x P, Q)) y (mul (R, Pinj (x - 1) P))))"
   114   "mul (PX P x R, Pinj y Q) =
   115   (if y=0 then mul(PX P x R, Q)
   116    else (if y=1 then mkPX (mul (Pinj y Q, P)) x (mul (R, Q))
   117          else mkPX (mul (Pinj y Q, P)) x (mul (R, Pinj (y - 1) Q))))"
   118   "mul (PX P1 x P2, PX Q1 y Q2) =
   119   add (mkPX (mul (P1, Q1)) (x+y) (mul (P2, Q2)),
   120   add (mkPX (mul (P1, mkPinj 1 Q2)) x (Pc 0), mkPX (mul (Q1, mkPinj 1 P2)) y (Pc 0)) )"
   121 (hints simp add: mkPinj_def split: pol.split)
   122 
   123 text {* Negation*}
   124 primrec
   125   "neg (Pc c) = Pc (-c)"
   126   "neg (Pinj i P) = Pinj i (neg P)"
   127   "neg (PX P x Q) = PX (neg P) x (neg Q)"
   128 
   129 text {* Substraction *}
   130 constdefs
   131   sub :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol"
   132   "sub p q \<equiv> add (p, neg q)"
   133 
   134 text {* Square for Fast Exponentation *}
   135 primrec
   136   "sqr (Pc c) = Pc (c * c)"
   137   "sqr (Pinj i P) = mkPinj i (sqr P)"
   138   "sqr (PX A x B) = add (mkPX (sqr A) (x + x) (sqr B),
   139     mkPX (mul (mul (Pc (1 + 1), A), mkPinj 1 B)) x (Pc 0))"
   140 
   141 text {* Fast Exponentation *}
   142 lemma pow_wf:"odd n \<Longrightarrow> (n::nat) div 2 < n" by (cases n) auto
   143 recdef pow "measure (\<lambda>(x, y). y)"
   144   "pow (p, 0) = Pc 1"
   145   "pow (p, n) = (if even n then (pow (sqr p, n div 2)) else mul (p, pow (sqr p, n div 2)))"
   146 (hints simp add: pow_wf)
   147 
   148 lemma pow_if:
   149   "pow (p,n) =
   150    (if n = 0 then Pc 1 else if even n then pow (sqr p, n div 2)
   151     else mul (p, pow (sqr p, n div 2)))"
   152   by (cases n) simp_all
   153 
   154 (*
   155 lemma number_of_nat_B0: "(number_of (w BIT bit.B0) ::nat) = 2* (number_of w)"
   156 by simp
   157 
   158 lemma number_of_nat_even: "even (number_of (w BIT bit.B0)::nat)"
   159 by simp
   160 
   161 lemma pow_even : "pow (p, number_of(w BIT bit.B0)) = pow (sqr p, number_of w)"
   162   ( is "pow(?p,?n) = pow (_,?n2)")
   163 proof-
   164   have "even ?n" by simp
   165   hence "pow (p, ?n) = pow (sqr p, ?n div 2)"
   166     apply simp
   167     apply (cases "IntDef.neg (number_of w)")
   168     apply simp
   169     done
   170 *)
   171 
   172 text {* Normalization of polynomial expressions *}
   173 
   174 consts norm :: "'a::{comm_ring,recpower} polex \<Rightarrow> 'a pol"
   175 primrec
   176   "norm (Pol P) = P"
   177   "norm (Add P Q) = add (norm P, norm Q)"
   178   "norm (Sub p q) = sub (norm p) (norm q)"
   179   "norm (Mul P Q) = mul (norm P, norm Q)"
   180   "norm (Pow p n) = pow (norm p, n)"
   181   "norm (Neg P) = neg (norm P)"
   182 
   183 text {* mkPinj preserve semantics *}
   184 lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
   185   by (induct B) (auto simp add: mkPinj_def ring_eq_simps)
   186 
   187 text {* mkPX preserves semantics *}
   188 lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
   189   by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add ring_eq_simps)
   190 
   191 text {* Correctness theorems for the implemented operations *}
   192 
   193 text {* Negation *}
   194 lemma neg_ci: "\<And>l. Ipol l (neg P) = -(Ipol l P)"
   195   by (induct P) auto
   196 
   197 text {* Addition *}
   198 lemma add_ci: "\<And>l. Ipol l (add (P, Q)) = Ipol l P + Ipol l Q"
   199 proof (induct P Q rule: add.induct)
   200   case (6 x P y Q)
   201   show ?case
   202   proof (rule linorder_cases)
   203     assume "x < y"
   204     with 6 show ?case by (simp add: mkPinj_ci ring_eq_simps)
   205   next
   206     assume "x = y"
   207     with 6 show ?case by (simp add: mkPinj_ci)
   208   next
   209     assume "x > y"
   210     with 6 show ?case by (simp add: mkPinj_ci ring_eq_simps)
   211   qed
   212 next
   213   case (7 x P Q y R)
   214   have "x = 0 \<or> x = 1 \<or> x > 1" by arith
   215   moreover
   216   { assume "x = 0" with 7 have ?case by simp }
   217   moreover
   218   { assume "x = 1" with 7 have ?case by (simp add: ring_eq_simps) }
   219   moreover
   220   { assume "x > 1" from 7 have ?case by (cases x) simp_all }
   221   ultimately show ?case by blast
   222 next
   223   case (8 P x R y Q)
   224   have "y = 0 \<or> y = 1 \<or> y > 1" by arith
   225   moreover
   226   { assume "y = 0" with 8 have ?case by simp }
   227   moreover
   228   { assume "y = 1" with 8 have ?case by simp }
   229   moreover
   230   { assume "y > 1" with 8 have ?case by simp }
   231   ultimately show ?case by blast
   232 next
   233   case (9 P1 x P2 Q1 y Q2)
   234   show ?case
   235   proof (rule linorder_cases)
   236     assume a: "x < y" hence "EX d. d + x = y" by arith
   237     with 9 a show ?case by (auto simp add: mkPX_ci power_add ring_eq_simps)
   238   next
   239     assume a: "y < x" hence "EX d. d + y = x" by arith
   240     with 9 a show ?case by (auto simp add: power_add mkPX_ci ring_eq_simps)
   241   next
   242     assume "x = y"
   243     with 9 show ?case by (simp add: mkPX_ci ring_eq_simps)
   244   qed
   245 qed (auto simp add: ring_eq_simps)
   246 
   247 text {* Multiplication *}
   248 lemma mul_ci: "\<And>l. Ipol l (mul (P, Q)) = Ipol l P * Ipol l Q"
   249   by (induct P Q rule: mul.induct)
   250     (simp_all add: mkPX_ci mkPinj_ci ring_eq_simps add_ci power_add)
   251 
   252 text {* Substraction *}
   253 lemma sub_ci: "Ipol l (sub p q) = Ipol l p - Ipol l q"
   254   by (simp add: add_ci neg_ci sub_def)
   255 
   256 text {* Square *}
   257 lemma sqr_ci:"\<And>ls. Ipol ls (sqr p) = Ipol ls p * Ipol ls p"
   258   by (induct p) (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_eq_simps power_add)
   259 
   260 
   261 text {* Power *}
   262 lemma even_pow:"even n \<Longrightarrow> pow (p, n) = pow (sqr p, n div 2)" by (induct n) simp_all
   263 
   264 lemma pow_ci: "\<And>p. Ipol ls (pow (p, n)) = (Ipol ls p) ^ n"
   265 proof (induct n rule: nat_less_induct)
   266   case (1 k)
   267   have two:"2 = Suc (Suc 0)" by simp
   268   show ?case
   269   proof (cases k)
   270     case (Suc l)
   271     show ?thesis
   272     proof cases
   273       assume EL: "even l"
   274       have "Suc l div 2 = l div 2"
   275         by (simp add: nat_number even_nat_plus_one_div_two [OF EL])
   276       moreover
   277       from Suc have "l < k" by simp
   278       with 1 have "\<forall>p. Ipol ls (pow (p, l)) = Ipol ls p ^ l" by simp
   279       moreover
   280       note Suc EL even_nat_plus_one_div_two [OF EL]
   281       ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
   282     next
   283       assume OL: "odd l"
   284       with prems have "\<lbrakk>\<forall>m<Suc l. \<forall>p. Ipol ls (pow (p, m)) = Ipol ls p ^ m; k = Suc l; odd l\<rbrakk> \<Longrightarrow> \<forall>p. Ipol ls (sqr p) ^ (Suc l div 2) = Ipol ls p ^ Suc l"
   285       proof(cases l)
   286         case (Suc w)
   287         from prems have EW: "even w" by simp
   288         from two have two_times:"(2 * (w div 2))= w"
   289           by (simp only: even_nat_div_two_times_two[OF EW])
   290         have A: "\<And>p. (Ipol ls p * Ipol ls p) = (Ipol ls p) ^ (Suc (Suc 0))"
   291           by (simp add: power_Suc)
   292         from A two [symmetric] have "ALL p.(Ipol ls p * Ipol ls p) = (Ipol ls p) ^ 2"
   293           by simp
   294         with prems show ?thesis
   295           by (auto simp add: power_mult[symmetric, of _ 2 _] two_times mul_ci sqr_ci)
   296       qed simp
   297       with prems show ?thesis by simp
   298     qed
   299   next
   300     case 0
   301     then show ?thesis by simp
   302   qed
   303 qed
   304 
   305 text {* Normalization preserves semantics  *}
   306 lemma norm_ci:"Ipolex l Pe = Ipol l (norm Pe)"
   307   by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)
   308 
   309 text {* Reflection lemma: Key to the (incomplete) decision procedure *}
   310 lemma norm_eq:
   311   assumes eq: "norm P1  = norm P2"
   312   shows "Ipolex l P1 = Ipolex l P2"
   313 proof -
   314   from eq have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
   315   thus ?thesis by (simp only: norm_ci)
   316 qed
   317 
   318 
   319 text {* Code generation *}
   320 (*
   321 Does not work, since no generic ring operations implementation is there
   322 generate_code ("ring.ML") test = "norm"*)
   323 
   324 use "comm_ring.ML"
   325 setup CommRing.setup
   326 
   327 end