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theory Commutative_Ring(* ID: $Id: Commutative_Ring.thy,v 1.1 2005/09/20 12:10:29 wenzelm Exp $ Author: Bernhard Haeupler Proving equalities in commutative rings done "right" in Isabelle/HOL. *) header {* Proving equalities in commutative rings *} theory Commutative_Ring imports Main 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. *} consts Ipol :: "'a::{comm_ring,recpower} list => 'a pol => 'a" Ipolex :: "'a::{comm_ring,recpower} list => 'a polex => 'a" primrec "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 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. *} constdefs mkPinj :: "nat => 'a pol => 'a pol" "mkPinj x P ≡ (case P of Pc c => Pc c | Pinj y P => Pinj (x + y) P | PX p1 y p2 => Pinj x P)" constdefs mkPX :: "'a::{comm_ring,recpower} pol => nat => 'a pol => 'a pol" "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 *} consts add :: "'a::{comm_ring,recpower} pol × 'a pol => 'a pol" mul :: "'a::{comm_ring,recpower} pol × 'a pol => 'a pol" neg :: "'a::{comm_ring,recpower} pol => 'a pol" sqr :: "'a::{comm_ring,recpower} pol => 'a pol" pow :: "'a::{comm_ring,recpower} pol × nat => 'a pol" text {* Addition *} recdef add "measure (λ(x, y). size x + size y)" "add (Pc a, Pc b) = Pc (a + b)" "add (Pc c, Pinj i P) = Pinj i (add (P, Pc c))" "add (Pinj i P, Pc c) = Pinj i (add (P, Pc c))" "add (Pc c, PX P i Q) = PX P i (add (Q, Pc c))" "add (PX P i Q, Pc c) = PX P i (add (Q, Pc c))" "add (Pinj x P, Pinj y Q) = (if x=y then mkPinj x (add (P, Q)) else (if x>y then mkPinj y (add (Pinj (x-y) P, Q)) else mkPinj x (add (Pinj (y-x) Q, P)) ))" "add (Pinj x P, PX Q y R) = (if x=0 then add(P, PX Q y R) else (if x=1 then PX Q y (add (R, P)) else PX Q y (add (R, Pinj (x - 1) P))))" "add (PX P x R, Pinj y Q) = (if y=0 then add(PX P x R, Q) else (if y=1 then PX P x (add (R, Q)) else PX P x (add (R, Pinj (y - 1) Q))))" "add (PX P1 x P2, PX Q1 y Q2) = (if x=y then mkPX (add (P1, Q1)) x (add (P2, Q2)) else (if x>y then mkPX (add (PX P1 (x-y) (Pc 0), Q1)) y (add (P2,Q2)) else mkPX (add (PX Q1 (y-x) (Pc 0), P1)) x (add (P2,Q2)) ))" text {* Multiplication *} recdef mul "measure (λ(x, y). size x + size y)" "mul (Pc a, Pc b) = Pc (a*b)" "mul (Pc c, Pinj i P) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))" "mul (Pinj i P, Pc c) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))" "mul (Pc c, PX P i Q) = (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))" "mul (PX P i Q, Pc c) = (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))" "mul (Pinj x P, Pinj y Q) = (if x=y then mkPinj x (mul (P, Q)) else (if x>y then mkPinj y (mul (Pinj (x-y) P, Q)) else mkPinj x (mul (Pinj (y-x) Q, P)) ))" "mul (Pinj x P, PX Q y R) = (if x=0 then mul(P, PX Q y R) else (if x=1 then mkPX (mul (Pinj x P, Q)) y (mul (R, P)) else mkPX (mul (Pinj x P, Q)) y (mul (R, Pinj (x - 1) P))))" "mul (PX P x R, Pinj y Q) = (if y=0 then mul(PX P x R, Q) else (if y=1 then mkPX (mul (Pinj y Q, P)) x (mul (R, Q)) else mkPX (mul (Pinj y Q, P)) x (mul (R, Pinj (y - 1) Q))))" "mul (PX P1 x P2, PX Q1 y Q2) = add (mkPX (mul (P1, Q1)) (x+y) (mul (P2, Q2)), add (mkPX (mul (P1, mkPinj 1 Q2)) x (Pc 0), mkPX (mul (Q1, mkPinj 1 P2)) y (Pc 0)) )" (hints simp add: mkPinj_def split: pol.split) text {* Negation*} primrec "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 *} constdefs sub :: "'a::{comm_ring,recpower} pol => 'a pol => 'a pol" "sub p q ≡ add (p, neg q)" text {* Square for Fast Exponentation *} primrec "sqr (Pc c) = Pc (c * c)" "sqr (Pinj i P) = mkPinj i (sqr P)" "sqr (PX A x B) = add (mkPX (sqr A) (x + x) (sqr B), mkPX (mul (mul (Pc (1 + 1), A), mkPinj 1 B)) x (Pc 0))" text {* Fast Exponentation *} lemma pow_wf:"odd n ==> (n::nat) div 2 < n" by (cases n) auto recdef pow "measure (λ(x, y). y)" "pow (p, 0) = Pc 1" "pow (p, n) = (if even n then (pow (sqr p, n div 2)) else mul (p, pow (sqr p, n div 2)))" (hints simp add: pow_wf) lemma pow_if: "pow (p,n) = (if n = 0 then Pc 1 else if even n then pow (sqr p, n div 2) else mul (p, pow (sqr p, n div 2)))" by (cases n) simp_all (* lemma number_of_nat_B0: "(number_of (w BIT bit.B0) ::nat) = 2* (number_of w)" by simp lemma number_of_nat_even: "even (number_of (w BIT bit.B0)::nat)" by simp lemma pow_even : "pow (p, number_of(w BIT bit.B0)) = pow (sqr p, number_of w)" ( is "pow(?p,?n) = pow (_,?n2)") proof- have "even ?n" by simp hence "pow (p, ?n) = pow (sqr p, ?n div 2)" apply simp apply (cases "IntDef.neg (number_of w)") apply simp done *) text {* Normalization of polynomial expressions *} consts norm :: "'a::{comm_ring,recpower} polex => 'a pol" primrec "norm (Pol P) = P" "norm (Add P Q) = add (norm P, norm Q)" "norm (Sub p q) = sub (norm p) (norm q)" "norm (Mul P Q) = mul (norm P, norm Q)" "norm (Pow p n) = pow (norm p, n)" "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_eq_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_eq_simps) text {* Correctness theorems for the implemented operations *} text {* Negation *} lemma neg_ci: "!!l. Ipol l (neg P) = -(Ipol l P)" by (induct P) auto text {* Addition *} lemma add_ci: "!!l. Ipol l (add (P, Q)) = Ipol l P + Ipol l Q" proof (induct P Q 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_eq_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_eq_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_eq_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_eq_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_eq_simps) next assume "x = y" with 9 show ?case by (simp add: mkPX_ci ring_eq_simps) qed qed (auto simp add: ring_eq_simps) text {* Multiplication *} lemma mul_ci: "!!l. Ipol l (mul (P, Q)) = Ipol l P * Ipol l Q" by (induct P Q rule: mul.induct) (simp_all add: mkPX_ci mkPinj_ci ring_eq_simps add_ci power_add) text {* Substraction *} lemma sub_ci: "Ipol l (sub p q) = Ipol l p - Ipol l q" by (simp add: add_ci neg_ci sub_def) text {* Square *} lemma sqr_ci:"!!ls. Ipol ls (sqr p) = Ipol ls p * Ipol ls p" by (induct p) (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_eq_simps power_add) text {* Power *} lemma even_pow:"even n ==> pow (p, n) = pow (sqr p, n div 2)" by (induct n) simp_all lemma pow_ci: "!!p. Ipol ls (pow (p, n)) = (Ipol ls p) ^ n" proof (induct n rule: nat_less_induct) case (1 k) have two:"2 = Suc (Suc 0)" by simp show ?case proof (cases k) case (Suc l) show ?thesis proof cases assume EL: "even l" have "Suc l div 2 = l div 2" by (simp add: nat_number even_nat_plus_one_div_two [OF EL]) moreover from Suc have "l < k" by simp with 1 have "∀p. Ipol ls (pow (p, l)) = Ipol ls p ^ l" by simp moreover note Suc EL even_nat_plus_one_div_two [OF EL] ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow) next assume OL: "odd l" with prems have "[|∀m<Suc l. ∀p. Ipol ls (pow (p, m)) = Ipol ls p ^ m; k = Suc l; odd l|] ==> ∀p. Ipol ls (sqr p) ^ (Suc l div 2) = Ipol ls p ^ Suc l" proof(cases l) case (Suc w) from prems have EW: "even w" by simp from two have two_times:"(2 * (w div 2))= w" by (simp only: even_nat_div_two_times_two[OF EW]) have A: "!!p. (Ipol ls p * Ipol ls p) = (Ipol ls p) ^ (Suc (Suc 0))" by (simp add: power_Suc) from A two [symmetric] have "ALL p.(Ipol ls p * Ipol ls p) = (Ipol ls p) ^ 2" by simp with prems show ?thesis by (auto simp add: power_mult[symmetric, of _ 2 _] two_times mul_ci sqr_ci) qed simp with prems show ?thesis by simp qed next case 0 then show ?thesis by simp 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 eq: "norm P1 = norm P2" shows "Ipolex l P1 = Ipolex l P2" proof - from eq have "Ipol l (norm P1) = Ipol l (norm P2)" by simp thus ?thesis by (simp only: norm_ci) qed text {* Code generation *} (* Does not work, since no generic ring operations implementation is there generate_code ("ring.ML") test = "norm"*) use "comm_ring.ML" setup "CommRing.setup" end
lemma pow_wf:
odd n ==> n div 2 < n
lemma pow_if:
pow (p, n) = (if n = 0 then Pc (1::'a) else if even n then pow (sqr p, n div 2) else mul (p, pow (sqr p, n div 2)))
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 (add (P, Q)) = Ipol l P + Ipol l Q
lemma mul_ci:
Ipol l (mul (P, Q)) = Ipol l P * Ipol l Q
lemma sub_ci:
Ipol l (sub p 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 (p, n) = pow (sqr p, n div 2)
lemma pow_ci:
Ipol ls (pow (p, n)) = 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