(*  Author:     Bernhard Haeupler

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

 *)

 header {* Proving equalities in commutative rings *}

 theory Commutative_Ring

 imports Main Parity

 uses ("commutative_ring_tac.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. *}

 primrec

   Ipol :: "'a::{comm_ring_1} 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"

 primrec

   Ipolex :: "'a::{comm_ring_1} 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} 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} 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} 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*}

 primrec

   neg :: "'a::{comm_ring} 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} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)

 where

   "sub P Q = P \<oplus> neg Q"

 text {* Square for Fast Exponentation *}

 primrec

   sqr :: "'a::{comm_ring_1} 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_1} 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 *}

 primrec

   norm :: "'a::{comm_ring_1} 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: eval_nat_numeral 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

                      simp del: power_Suc)

         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 assms have "Ipol l (norm P1) = Ipol l (norm P2)" by simp

   then show ?thesis by (simp only: norm_ci)

 qed

 use "commutative_ring_tac.ML"

 method_setup comm_ring = {*

   Scan.succeed (SIMPLE_METHOD' o Commutative_Ring_Tac.tac)

 *} "reflective decision procedure for equalities over commutative rings"

 end