
wenzelm@17516: (*  ID:         $Id$
wenzelm@17516:     Author:     Bernhard Haeupler
wenzelm@17516: 
wenzelm@17516: Proving equalities in commutative rings done "right" in Isabelle/HOL.
wenzelm@17516: *)
wenzelm@17516: 
wenzelm@17516: header {* Proving equalities in commutative rings *}
wenzelm@17516: 
wenzelm@17516: theory Commutative_Ring
wenzelm@21256: imports Main Parity
wenzelm@17516: uses ("comm_ring.ML")
wenzelm@17516: begin
wenzelm@17516: 
wenzelm@17516: text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}
wenzelm@17516: 
wenzelm@17516: datatype 'a pol =
wenzelm@17516:     Pc 'a
wenzelm@17516:   | Pinj nat "'a pol"
wenzelm@17516:   | PX "'a pol" nat "'a pol"
wenzelm@17516: 
wenzelm@17516: datatype 'a polex =
wenzelm@20622:     Pol "'a pol"
wenzelm@17516:   | Add "'a polex" "'a polex"
wenzelm@17516:   | Sub "'a polex" "'a polex"
wenzelm@17516:   | Mul "'a polex" "'a polex"
wenzelm@17516:   | Pow "'a polex" nat
wenzelm@17516:   | Neg "'a polex"
wenzelm@17516: 
wenzelm@17516: text {* Interpretation functions for the shadow syntax. *}
wenzelm@17516: 
haftmann@22742: fun
wenzelm@17516:   Ipol :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
haftmann@22742: where
haftmann@22742:     "Ipol l (Pc c) = c"
haftmann@22742:   | "Ipol l (Pinj i P) = Ipol (drop i l) P"
haftmann@22742:   | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
wenzelm@17516: 
haftmann@22742: fun
haftmann@22742:   Ipolex :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
haftmann@22742: where
haftmann@22742:     "Ipolex l (Pol P) = Ipol l P"
haftmann@22742:   | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
haftmann@22742:   | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
haftmann@22742:   | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
haftmann@22742:   | "Ipolex l (Pow p n) = Ipolex l p ^ n"
haftmann@22742:   | "Ipolex l (Neg P) = - Ipolex l P"
wenzelm@17516: 
wenzelm@17516: text {* Create polynomial normalized polynomials given normalized inputs. *}
wenzelm@17516: 
wenzelm@19736: definition
wenzelm@21404:   mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
wenzelm@19736:   "mkPinj x P = (case P of
wenzelm@17516:     Pc c \<Rightarrow> Pc c |
wenzelm@17516:     Pinj y P \<Rightarrow> Pinj (x + y) P |
wenzelm@17516:     PX p1 y p2 \<Rightarrow> Pinj x P)"
wenzelm@17516: 
wenzelm@19736: definition
wenzelm@21404:   mkPX :: "'a::{comm_ring,recpower} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
wenzelm@19736:   "mkPX P i Q = (case P of
wenzelm@17516:     Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
wenzelm@17516:     Pinj j R \<Rightarrow> PX P i Q |
wenzelm@17516:     PX P2 i2 Q2 \<Rightarrow> (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"
wenzelm@17516: 
wenzelm@17516: text {* Defining the basic ring operations on normalized polynomials *}
wenzelm@17516: 
haftmann@22742: function
haftmann@22742:   add :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
haftmann@22742: where
haftmann@22742:     "Pc a \<oplus> Pc b = Pc (a + b)"
haftmann@22742:   | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
haftmann@22742:   | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
haftmann@22742:   | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
haftmann@22742:   | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
haftmann@22742:   | "Pinj x P \<oplus> Pinj y Q =
haftmann@22742:       (if x = y then mkPinj x (P \<oplus> Q)
haftmann@22742:        else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
haftmann@22742:          else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
haftmann@22742:   | "Pinj x P \<oplus> PX Q y R =
haftmann@22742:       (if x = 0 then P \<oplus> PX Q y R
haftmann@22742:        else (if x = 1 then PX Q y (R \<oplus> P)
haftmann@22742:          else PX Q y (R \<oplus> Pinj (x - 1) P)))"
haftmann@22742:   | "PX P x R \<oplus> Pinj y Q =
haftmann@22742:       (if y = 0 then PX P x R \<oplus> Q
haftmann@22742:        else (if y = 1 then PX P x (R \<oplus> Q)
haftmann@22742:          else PX P x (R \<oplus> Pinj (y - 1) Q)))"
haftmann@22742:   | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
haftmann@22742:       (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
haftmann@22742:        else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
haftmann@22742:          else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
haftmann@22742: by pat_completeness auto
haftmann@22742: termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto
wenzelm@17516: 
haftmann@22742: function
haftmann@22742:   mul :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<otimes>" 70)
haftmann@22742: where
haftmann@22742:     "Pc a \<otimes> Pc b = Pc (a * b)"
haftmann@22742:   | "Pc c \<otimes> Pinj i P =
haftmann@22742:       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
haftmann@22742:   | "Pinj i P \<otimes> Pc c =
haftmann@22742:       (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
haftmann@22742:   | "Pc c \<otimes> PX P i Q =
haftmann@22742:       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
haftmann@22742:   | "PX P i Q \<otimes> Pc c =
haftmann@22742:       (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
haftmann@22742:   | "Pinj x P \<otimes> Pinj y Q =
haftmann@22742:       (if x = y then mkPinj x (P \<otimes> Q) else
haftmann@22742:          (if x > y then mkPinj y (Pinj (x-y) P \<otimes> Q)
haftmann@22742:            else mkPinj x (Pinj (y - x) Q \<otimes> P)))"
haftmann@22742:   | "Pinj x P \<otimes> PX Q y R =
haftmann@22742:       (if x = 0 then P \<otimes> PX Q y R else
haftmann@22742:          (if x = 1 then mkPX (Pinj x P \<otimes> Q) y (R \<otimes> P)
haftmann@22742:            else mkPX (Pinj x P \<otimes> Q) y (R \<otimes> Pinj (x - 1) P)))"
haftmann@22742:   | "PX P x R \<otimes> Pinj y Q =
haftmann@22742:       (if y = 0 then PX P x R \<otimes> Q else
haftmann@22742:          (if y = 1 then mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Q)
haftmann@22742:            else mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Pinj (y - 1) Q)))"
haftmann@22742:   | "PX P1 x P2 \<otimes> PX Q1 y Q2 =
haftmann@22742:       mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2) \<oplus>
haftmann@22742:         (mkPX (P1 \<otimes> mkPinj 1 Q2) x (Pc 0) \<oplus>
haftmann@22742:           (mkPX (Q1 \<otimes> mkPinj 1 P2) y (Pc 0)))"
haftmann@22742: by pat_completeness auto
haftmann@22742: termination by (relation "measure (\<lambda>(x, y). size x + size y)")
haftmann@22742:   (auto simp add: mkPinj_def split: pol.split)
wenzelm@17516: 
wenzelm@17516: text {* Negation*}
haftmann@22742: fun
haftmann@22742:   neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
haftmann@22742: where
haftmann@22742:     "neg (Pc c) = Pc (-c)"
haftmann@22742:   | "neg (Pinj i P) = Pinj i (neg P)"
haftmann@22742:   | "neg (PX P x Q) = PX (neg P) x (neg Q)"
wenzelm@17516: 
wenzelm@17516: text {* Substraction *}
wenzelm@19736: definition
haftmann@22742:   sub :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
haftmann@22742: where
haftmann@22742:   "sub P Q = P \<oplus> neg Q"
wenzelm@17516: 
wenzelm@17516: text {* Square for Fast Exponentation *}
haftmann@22742: fun
haftmann@22742:   sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
haftmann@22742: where
haftmann@22742:     "sqr (Pc c) = Pc (c * c)"
haftmann@22742:   | "sqr (Pinj i P) = mkPinj i (sqr P)"
haftmann@22742:   | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<oplus>
haftmann@22742:       mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"
wenzelm@17516: 
wenzelm@17516: text {* Fast Exponentation *}
haftmann@22742: fun
haftmann@22742:   pow :: "nat \<Rightarrow> 'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
haftmann@22742: where
haftmann@22742:     "pow 0 P = Pc 1"
haftmann@22742:   | "pow n P = (if even n then pow (n div 2) (sqr P)
haftmann@22742:        else P \<otimes> pow (n div 2) (sqr P))"
haftmann@22742:   
wenzelm@17516: lemma pow_if:
haftmann@22742:   "pow n P =
haftmann@22742:    (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
haftmann@22742:     else P \<otimes> pow (n div 2) (sqr P))"
wenzelm@17516:   by (cases n) simp_all
wenzelm@17516: 
wenzelm@17516: 
wenzelm@17516: text {* Normalization of polynomial expressions *}
wenzelm@17516: 
haftmann@22742: fun
haftmann@22742:   norm :: "'a::{comm_ring,recpower} polex \<Rightarrow> 'a pol"
haftmann@22742: where
haftmann@22742:     "norm (Pol P) = P"
haftmann@22742:   | "norm (Add P Q) = norm P \<oplus> norm Q"
haftmann@22742:   | "norm (Sub P Q) = norm P \<ominus> norm Q"
haftmann@22742:   | "norm (Mul P Q) = norm P \<otimes> norm Q"
haftmann@22742:   | "norm (Pow P n) = pow n (norm P)"
haftmann@22742:   | "norm (Neg P) = neg (norm P)"
wenzelm@17516: 
wenzelm@17516: text {* mkPinj preserve semantics *}
wenzelm@17516: lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
wenzelm@17516:   by (induct B) (auto simp add: mkPinj_def ring_eq_simps)
wenzelm@17516: 
wenzelm@17516: text {* mkPX preserves semantics *}
wenzelm@17516: lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
wenzelm@17516:   by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add ring_eq_simps)
wenzelm@17516: 
wenzelm@17516: text {* Correctness theorems for the implemented operations *}
wenzelm@17516: 
wenzelm@17516: text {* Negation *}
wenzelm@20622: lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
wenzelm@20622:   by (induct P arbitrary: l) auto
wenzelm@17516: 
wenzelm@17516: text {* Addition *}
haftmann@22742: lemma add_ci: "Ipol l (P \<oplus> Q) = Ipol l P + Ipol l Q"
wenzelm@20622: proof (induct P Q arbitrary: l rule: add.induct)
wenzelm@17516:   case (6 x P y Q)
wenzelm@17516:   show ?case
wenzelm@17516:   proof (rule linorder_cases)
wenzelm@17516:     assume "x < y"
wenzelm@17516:     with 6 show ?case by (simp add: mkPinj_ci ring_eq_simps)
wenzelm@17516:   next
wenzelm@17516:     assume "x = y"
wenzelm@17516:     with 6 show ?case by (simp add: mkPinj_ci)
wenzelm@17516:   next
wenzelm@17516:     assume "x > y"
wenzelm@17516:     with 6 show ?case by (simp add: mkPinj_ci ring_eq_simps)
wenzelm@17516:   qed
wenzelm@17516: next
wenzelm@17516:   case (7 x P Q y R)
wenzelm@17516:   have "x = 0 \<or> x = 1 \<or> x > 1" by arith
wenzelm@17516:   moreover
wenzelm@17516:   { assume "x = 0" with 7 have ?case by simp }
wenzelm@17516:   moreover
wenzelm@17516:   { assume "x = 1" with 7 have ?case by (simp add: ring_eq_simps) }
wenzelm@17516:   moreover
wenzelm@17516:   { assume "x > 1" from 7 have ?case by (cases x) simp_all }
wenzelm@17516:   ultimately show ?case by blast
wenzelm@17516: next
wenzelm@17516:   case (8 P x R y Q)
wenzelm@17516:   have "y = 0 \<or> y = 1 \<or> y > 1" by arith
wenzelm@17516:   moreover
wenzelm@17516:   { assume "y = 0" with 8 have ?case by simp }
wenzelm@17516:   moreover
wenzelm@17516:   { assume "y = 1" with 8 have ?case by simp }
wenzelm@17516:   moreover
wenzelm@17516:   { assume "y > 1" with 8 have ?case by simp }
wenzelm@17516:   ultimately show ?case by blast
wenzelm@17516: next
wenzelm@17516:   case (9 P1 x P2 Q1 y Q2)
wenzelm@17516:   show ?case
wenzelm@17516:   proof (rule linorder_cases)
wenzelm@17516:     assume a: "x < y" hence "EX d. d + x = y" by arith
wenzelm@17516:     with 9 a show ?case by (auto simp add: mkPX_ci power_add ring_eq_simps)
wenzelm@17516:   next
wenzelm@17516:     assume a: "y < x" hence "EX d. d + y = x" by arith
wenzelm@17516:     with 9 a show ?case by (auto simp add: power_add mkPX_ci ring_eq_simps)
wenzelm@17516:   next
wenzelm@17516:     assume "x = y"
wenzelm@17516:     with 9 show ?case by (simp add: mkPX_ci ring_eq_simps)
wenzelm@17516:   qed
wenzelm@17516: qed (auto simp add: ring_eq_simps)
wenzelm@17516: 
wenzelm@17516: text {* Multiplication *}
haftmann@22742: lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
wenzelm@20622:   by (induct P Q arbitrary: l rule: mul.induct)
wenzelm@17516:     (simp_all add: mkPX_ci mkPinj_ci ring_eq_simps add_ci power_add)
wenzelm@17516: 
wenzelm@17516: text {* Substraction *}
haftmann@22742: lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
wenzelm@17516:   by (simp add: add_ci neg_ci sub_def)
wenzelm@17516: 
wenzelm@17516: text {* Square *}
haftmann@22742: lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
haftmann@22742:   by (induct P arbitrary: ls)
wenzelm@20622:     (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_eq_simps power_add)
wenzelm@17516: 
wenzelm@17516: text {* Power *}
haftmann@22742: lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
wenzelm@20622:   by (induct n) simp_all
wenzelm@17516: 
haftmann@22742: lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
haftmann@22742: proof (induct n arbitrary: P rule: nat_less_induct)
wenzelm@17516:   case (1 k)
wenzelm@17516:   show ?case
wenzelm@17516:   proof (cases k)
wenzelm@20622:     case 0
wenzelm@20622:     then show ?thesis by simp
wenzelm@20622:   next
wenzelm@17516:     case (Suc l)
wenzelm@17516:     show ?thesis
wenzelm@17516:     proof cases
wenzelm@20622:       assume "even l"
wenzelm@20622:       then have "Suc l div 2 = l div 2"
wenzelm@20622:         by (simp add: nat_number even_nat_plus_one_div_two)
wenzelm@17516:       moreover
wenzelm@17516:       from Suc have "l < k" by simp
haftmann@22742:       with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
wenzelm@17516:       moreover
wenzelm@20622:       note Suc `even l` even_nat_plus_one_div_two
wenzelm@17516:       ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
wenzelm@17516:     next
wenzelm@20622:       assume "odd l"
wenzelm@20622:       {
wenzelm@20622:         fix p
haftmann@22742:         have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
wenzelm@20622:         proof (cases l)
wenzelm@20622:           case 0
wenzelm@20622:           with `odd l` show ?thesis by simp
wenzelm@20622:         next
wenzelm@20622:           case (Suc w)
wenzelm@20622:           with `odd l` have "even w" by simp
wenzelm@20678:           have two_times: "2 * (w div 2) = w"
wenzelm@20678:             by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
haftmann@22742:           have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
wenzelm@20622:             by (simp add: power_Suc)
haftmann@22742: 	  then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
wenzelm@20678: 	    by (simp add: numerals)
wenzelm@20622:           with Suc show ?thesis
wenzelm@20678:             by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci)
wenzelm@20622:         qed
wenzelm@20622:       } with 1 Suc `odd l` show ?thesis by simp
wenzelm@17516:     qed
wenzelm@17516:   qed
wenzelm@17516: qed
wenzelm@17516: 
wenzelm@17516: text {* Normalization preserves semantics  *}
wenzelm@20622: lemma norm_ci: "Ipolex l Pe = Ipol l (norm Pe)"
wenzelm@17516:   by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)
wenzelm@17516: 
wenzelm@17516: text {* Reflection lemma: Key to the (incomplete) decision procedure *}
wenzelm@17516: lemma norm_eq:
wenzelm@20622:   assumes "norm P1 = norm P2"
wenzelm@17516:   shows "Ipolex l P1 = Ipolex l P2"
wenzelm@17516: proof -
wenzelm@20622:   from prems have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
wenzelm@20622:   then show ?thesis by (simp only: norm_ci)
wenzelm@17516: qed
wenzelm@17516: 
wenzelm@17516: 
wenzelm@17516: use "comm_ring.ML"
wenzelm@18708: setup CommRing.setup
wenzelm@17516: 
wenzelm@17516: end




