src/HOL/Library/Commutative_Ring.thy

added replace operation

(*  Author:     Bernhard Haeupler

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

header {* Proving equalities in commutative rings *}

theory Commutative_Ring
imports Plain "~~/src/HOL/List" "~~/src/HOL/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 \<Rightarrow> 'a pol \<Rightarrow> '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 \<Rightarrow> 'a polex \<Rightarrow> '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 \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
  "mkPinj x P = (case P of
    Pc c \<Rightarrow> Pc c |
    Pinj y P \<Rightarrow> Pinj (x + y) P |
    PX p1 y p2 \<Rightarrow> Pinj x P)"

definition
  mkPX :: "'a::{comm_ring,recpower} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
  "mkPX P i Q = (case P of
    Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
    Pinj j R \<Rightarrow> PX P i Q |
    PX P2 i2 Q2 \<Rightarrow> (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 \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
where
    "Pc a \<oplus> Pc b = Pc (a + b)"
  | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
  | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
  | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
  | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
  | "Pinj x P \<oplus> Pinj y Q =
      (if x = y then mkPinj x (P \<oplus> Q)
       else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
         else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
  | "Pinj x P \<oplus> PX Q y R =
      (if x = 0 then P \<oplus> PX Q y R
       else (if x = 1 then PX Q y (R \<oplus> P)
         else PX Q y (R \<oplus> Pinj (x - 1) P)))"
  | "PX P x R \<oplus> Pinj y Q =
      (if y = 0 then PX P x R \<oplus> Q
       else (if y = 1 then PX P x (R \<oplus> Q)
         else PX P x (R \<oplus> Pinj (y - 1) Q)))"
  | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
      (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
       else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
         else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
by pat_completeness auto
termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto

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

text {* Negation*}
fun
  neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> '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 \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
where
  "sub P Q = P \<oplus> neg Q"

text {* Square for Fast Exponentation *}
fun
  sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> '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) \<oplus>
      mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"

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


text {* Normalization of polynomial expressions *}

fun
  norm :: "'a::{comm_ring,recpower} polex \<Rightarrow> 'a pol"
where
    "norm (Pol P) = P"
  | "norm (Add P Q) = norm P \<oplus> norm Q"
  | "norm (Sub P Q) = norm P \<ominus> norm Q"
  | "norm (Mul P Q) = norm P \<otimes> 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 algebra_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 algebra_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 \<oplus> 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 algebra_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 algebra_simps)
  qed
next
  case (7 x P Q y R)
  have "x = 0 \<or> x = 1 \<or> 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: algebra_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 \<or> y = 1 \<or> 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 algebra_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 algebra_simps)
  next
    assume "x = y"
    with 9 show ?case by (simp add: mkPX_ci algebra_simps)
  qed
qed (auto simp add: algebra_simps)

text {* Multiplication *}
lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
  by (induct P Q arbitrary: l rule: mul.induct)
    (simp_all add: mkPX_ci mkPinj_ci algebra_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 algebra_simps power_add)

text {* Power *}
lemma even_pow:"even n \<Longrightarrow> 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 "\<And>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