Theory Commutative_Ring

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theory Commutative_Ring
imports List Parity
uses comm_ring.ML
begin

(*  ID:         $Id: Commutative_Ring.thy,v 1.11 2007/12/10 10:24:12 haftmann Exp $
    Author:     Bernhard Haeupler

Proving equalities in commutative rings done "right" in Isabelle/HOL.
*)

header {* Proving equalities in commutative rings *}

theory Commutative_Ring
imports List Parity
uses ("comm_ring.ML")
begin

text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}

datatype 'a pol =
    Pc 'a
  | Pinj nat "'a pol"
  | PX "'a pol" nat "'a pol"

datatype 'a polex =
    Pol "'a pol"
  | Add "'a polex" "'a polex"
  | Sub "'a polex" "'a polex"
  | Mul "'a polex" "'a polex"
  | Pow "'a polex" nat
  | Neg "'a polex"

text {* Interpretation functions for the shadow syntax. *}

fun
  Ipol :: "'a::{comm_ring,recpower} list => 'a pol => 'a"
where
    "Ipol l (Pc c) = c"
  | "Ipol l (Pinj i P) = Ipol (drop i l) P"
  | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"

fun
  Ipolex :: "'a::{comm_ring,recpower} list => 'a polex => 'a"
where
    "Ipolex l (Pol P) = Ipol l P"
  | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
  | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
  | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
  | "Ipolex l (Pow p n) = Ipolex l p ^ n"
  | "Ipolex l (Neg P) = - Ipolex l P"

text {* Create polynomial normalized polynomials given normalized inputs. *}

definition
  mkPinj :: "nat => 'a pol => 'a pol" where
  "mkPinj x P = (case P of
    Pc c => Pc c |
    Pinj y P => Pinj (x + y) P |
    PX p1 y p2 => Pinj x P)"

definition
  mkPX :: "'a::{comm_ring,recpower} pol => nat => 'a pol => 'a pol" where
  "mkPX P i Q = (case P of
    Pc c => (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
    Pinj j R => PX P i Q |
    PX P2 i2 Q2 => (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"

text {* Defining the basic ring operations on normalized polynomials *}

function
  add :: "'a::{comm_ring,recpower} pol => 'a pol => 'a pol" (infixl "⊕" 65)
where
    "Pc a ⊕ Pc b = Pc (a + b)"
  | "Pc c ⊕ Pinj i P = Pinj i (P ⊕ Pc c)"
  | "Pinj i P ⊕ Pc c = Pinj i (P ⊕ Pc c)"
  | "Pc c ⊕ PX P i Q = PX P i (Q ⊕ Pc c)"
  | "PX P i Q ⊕ Pc c = PX P i (Q ⊕ Pc c)"
  | "Pinj x P ⊕ Pinj y Q =
      (if x = y then mkPinj x (P ⊕ Q)
       else (if x > y then mkPinj y (Pinj (x - y) P ⊕ Q)
         else mkPinj x (Pinj (y - x) Q ⊕ P)))"
  | "Pinj x P ⊕ PX Q y R =
      (if x = 0 then P ⊕ PX Q y R
       else (if x = 1 then PX Q y (R ⊕ P)
         else PX Q y (R ⊕ Pinj (x - 1) P)))"
  | "PX P x R ⊕ Pinj y Q =
      (if y = 0 then PX P x R ⊕ Q
       else (if y = 1 then PX P x (R ⊕ Q)
         else PX P x (R ⊕ Pinj (y - 1) Q)))"
  | "PX P1 x P2 ⊕ PX Q1 y Q2 =
      (if x = y then mkPX (P1 ⊕ Q1) x (P2 ⊕ Q2)
       else (if x > y then mkPX (PX P1 (x - y) (Pc 0) ⊕ Q1) y (P2 ⊕ Q2)
         else mkPX (PX Q1 (y-x) (Pc 0) ⊕ P1) x (P2 ⊕ Q2)))"
by pat_completeness auto
termination by (relation "measure (λ(x, y). size x + size y)") auto

function
  mul :: "'a::{comm_ring,recpower} pol => 'a pol => 'a pol" (infixl "⊗" 70)
where
    "Pc a ⊗ Pc b = Pc (a * b)"
  | "Pc c ⊗ Pinj i P =
      (if c = 0 then Pc 0 else mkPinj i (P ⊗ Pc c))"
  | "Pinj i P ⊗ Pc c =
      (if c = 0 then Pc 0 else mkPinj i (P ⊗ Pc c))"
  | "Pc c ⊗ PX P i Q =
      (if c = 0 then Pc 0 else mkPX (P ⊗ Pc c) i (Q ⊗ Pc c))"
  | "PX P i Q ⊗ Pc c =
      (if c = 0 then Pc 0 else mkPX (P ⊗ Pc c) i (Q ⊗ Pc c))"
  | "Pinj x P ⊗ Pinj y Q =
      (if x = y then mkPinj x (P ⊗ Q) else
         (if x > y then mkPinj y (Pinj (x-y) P ⊗ Q)
           else mkPinj x (Pinj (y - x) Q ⊗ P)))"
  | "Pinj x P ⊗ PX Q y R =
      (if x = 0 then P ⊗ PX Q y R else
         (if x = 1 then mkPX (Pinj x P ⊗ Q) y (R ⊗ P)
           else mkPX (Pinj x P ⊗ Q) y (R ⊗ Pinj (x - 1) P)))"
  | "PX P x R ⊗ Pinj y Q =
      (if y = 0 then PX P x R ⊗ Q else
         (if y = 1 then mkPX (Pinj y Q ⊗ P) x (R ⊗ Q)
           else mkPX (Pinj y Q ⊗ P) x (R ⊗ Pinj (y - 1) Q)))"
  | "PX P1 x P2 ⊗ PX Q1 y Q2 =
      mkPX (P1 ⊗ Q1) (x + y) (P2 ⊗ Q2) ⊕
        (mkPX (P1 ⊗ mkPinj 1 Q2) x (Pc 0) ⊕
          (mkPX (Q1 ⊗ mkPinj 1 P2) y (Pc 0)))"
by pat_completeness auto
termination by (relation "measure (λ(x, y). size x + size y)")
  (auto simp add: mkPinj_def split: pol.split)

text {* Negation*}
fun
  neg :: "'a::{comm_ring,recpower} pol => 'a pol"
where
    "neg (Pc c) = Pc (-c)"
  | "neg (Pinj i P) = Pinj i (neg P)"
  | "neg (PX P x Q) = PX (neg P) x (neg Q)"

text {* Substraction *}
definition
  sub :: "'a::{comm_ring,recpower} pol => 'a pol => 'a pol" (infixl "\<ominus>" 65)
where
  "sub P Q = P ⊕ neg Q"

text {* Square for Fast Exponentation *}
fun
  sqr :: "'a::{comm_ring,recpower} pol => 'a pol"
where
    "sqr (Pc c) = Pc (c * c)"
  | "sqr (Pinj i P) = mkPinj i (sqr P)"
  | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) ⊕
      mkPX (Pc (1 + 1) ⊗ A ⊗ mkPinj 1 B) x (Pc 0)"

text {* Fast Exponentation *}
fun
  pow :: "nat => 'a::{comm_ring,recpower} pol => 'a pol"
where
    "pow 0 P = Pc 1"
  | "pow n P = (if even n then pow (n div 2) (sqr P)
       else P ⊗ pow (n div 2) (sqr P))"
  
lemma pow_if:
  "pow n P =
   (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
    else P ⊗ pow (n div 2) (sqr P))"
  by (cases n) simp_all


text {* Normalization of polynomial expressions *}

fun
  norm :: "'a::{comm_ring,recpower} polex => 'a pol"
where
    "norm (Pol P) = P"
  | "norm (Add P Q) = norm P ⊕ norm Q"
  | "norm (Sub P Q) = norm P \<ominus> norm Q"
  | "norm (Mul P Q) = norm P ⊗ norm Q"
  | "norm (Pow P n) = pow n (norm P)"
  | "norm (Neg P) = neg (norm P)"

text {* mkPinj preserve semantics *}
lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
  by (induct B) (auto simp add: mkPinj_def ring_simps)

text {* mkPX preserves semantics *}
lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
  by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add ring_simps)

text {* Correctness theorems for the implemented operations *}

text {* Negation *}
lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
  by (induct P arbitrary: l) auto

text {* Addition *}
lemma add_ci: "Ipol l (P ⊕ Q) = Ipol l P + Ipol l Q"
proof (induct P Q arbitrary: l rule: add.induct)
  case (6 x P y Q)
  show ?case
  proof (rule linorder_cases)
    assume "x < y"
    with 6 show ?case by (simp add: mkPinj_ci ring_simps)
  next
    assume "x = y"
    with 6 show ?case by (simp add: mkPinj_ci)
  next
    assume "x > y"
    with 6 show ?case by (simp add: mkPinj_ci ring_simps)
  qed
next
  case (7 x P Q y R)
  have "x = 0 ∨ x = 1 ∨ x > 1" by arith
  moreover
  { assume "x = 0" with 7 have ?case by simp }
  moreover
  { assume "x = 1" with 7 have ?case by (simp add: ring_simps) }
  moreover
  { assume "x > 1" from 7 have ?case by (cases x) simp_all }
  ultimately show ?case by blast
next
  case (8 P x R y Q)
  have "y = 0 ∨ y = 1 ∨ y > 1" by arith
  moreover
  { assume "y = 0" with 8 have ?case by simp }
  moreover
  { assume "y = 1" with 8 have ?case by simp }
  moreover
  { assume "y > 1" with 8 have ?case by simp }
  ultimately show ?case by blast
next
  case (9 P1 x P2 Q1 y Q2)
  show ?case
  proof (rule linorder_cases)
    assume a: "x < y" hence "EX d. d + x = y" by arith
    with 9 a show ?case by (auto simp add: mkPX_ci power_add ring_simps)
  next
    assume a: "y < x" hence "EX d. d + y = x" by arith
    with 9 a show ?case by (auto simp add: power_add mkPX_ci ring_simps)
  next
    assume "x = y"
    with 9 show ?case by (simp add: mkPX_ci ring_simps)
  qed
qed (auto simp add: ring_simps)

text {* Multiplication *}
lemma mul_ci: "Ipol l (P ⊗ Q) = Ipol l P * Ipol l Q"
  by (induct P Q arbitrary: l rule: mul.induct)
    (simp_all add: mkPX_ci mkPinj_ci ring_simps add_ci power_add)

text {* Substraction *}
lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
  by (simp add: add_ci neg_ci sub_def)

text {* Square *}
lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
  by (induct P arbitrary: ls)
    (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_simps power_add)

text {* Power *}
lemma even_pow:"even n ==> pow n P = pow (n div 2) (sqr P)"
  by (induct n) simp_all

lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
proof (induct n arbitrary: P rule: nat_less_induct)
  case (1 k)
  show ?case
  proof (cases k)
    case 0
    then show ?thesis by simp
  next
    case (Suc l)
    show ?thesis
    proof cases
      assume "even l"
      then have "Suc l div 2 = l div 2"
        by (simp add: nat_number even_nat_plus_one_div_two)
      moreover
      from Suc have "l < k" by simp
      with 1 have "!!P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
      moreover
      note Suc `even l` even_nat_plus_one_div_two
      ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
    next
      assume "odd l"
      {
        fix p
        have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
        proof (cases l)
          case 0
          with `odd l` show ?thesis by simp
        next
          case (Suc w)
          with `odd l` have "even w" by simp
          have two_times: "2 * (w div 2) = w"
            by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
          have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
            by (simp add: power_Suc)
          then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
            by (simp add: numerals)
          with Suc show ?thesis
            by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci)
        qed
      } with 1 Suc `odd l` show ?thesis by simp
    qed
  qed
qed

text {* Normalization preserves semantics  *}
lemma norm_ci: "Ipolex l Pe = Ipol l (norm Pe)"
  by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)

text {* Reflection lemma: Key to the (incomplete) decision procedure *}
lemma norm_eq:
  assumes "norm P1 = norm P2"
  shows "Ipolex l P1 = Ipolex l P2"
proof -
  from prems have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
  then show ?thesis by (simp only: norm_ci)
qed


use "comm_ring.ML"
setup CommRing.setup

end

lemma pow_if:

  pow n P =
  (if n = 0 then Pc (1::'a)
   else if even n then pow (n div 2) (sqr P) else P  pow (n div 2) (sqr P))

lemma mkPinj_ci:

  Ipol l (mkPinj a B) = Ipol l (Pinj a B)

lemma mkPX_ci:

  Ipol l (mkPX A b C) = Ipol l (PX A b C)

lemma neg_ci:

  Ipol l (Commutative_Ring.neg P) = - Ipol l P

lemma add_ci:

  Ipol l (P  Q) = Ipol l P + Ipol l Q

lemma mul_ci:

  Ipol l (P  Q) = Ipol l P * Ipol l Q

lemma sub_ci:

  Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q

lemma sqr_ci:

  Ipol ls (sqr P) = Ipol ls P * Ipol ls P

lemma even_pow:

  even n ==> pow n P = pow (n div 2) (sqr P)

lemma pow_ci:

  Ipol ls (pow n P) = Ipol ls P ^ n

lemma norm_ci:

  Ipolex l Pe = Ipol l (norm Pe)

lemma norm_eq:

  norm P1.0 = norm P2.0 ==> Ipolex l P1.0 = Ipolex l P2.0