(* ID: $Id$
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 ("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 \<Rightarrow> 'a pol \<Rightarrow> 'a"
Ipolex :: "'a::{comm_ring,recpower} list \<Rightarrow> 'a polex \<Rightarrow> '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. *}
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 *}
consts
add :: "'a::{comm_ring,recpower} pol \<times> 'a pol \<Rightarrow> 'a pol"
mul :: "'a::{comm_ring,recpower} pol \<times> 'a pol \<Rightarrow> 'a pol"
neg :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
sqr :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol"
pow :: "'a::{comm_ring,recpower} pol \<times> nat \<Rightarrow> 'a pol"
text {* Addition *}
recdef add "measure (\<lambda>(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 (\<lambda>(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 *}
definition
sub :: "'a::{comm_ring,recpower} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
"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 \<Longrightarrow> (n::nat) div 2 < n" by (cases n) auto
recdef pow "measure (\<lambda>(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 \<Rightarrow> '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: "Ipol l (neg P) = -(Ipol l P)"
by (induct P arbitrary: l) auto
text {* Addition *}
lemma add_ci: "Ipol l (add (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_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 \<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: 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 \<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 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: "Ipol l (mul (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_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: "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_eq_simps power_add)
text {* Power *}
lemma even_pow:"even n \<Longrightarrow> pow (p, n) = pow (sqr p, n div 2)"
by (induct n) simp_all
lemma pow_ci: "Ipol ls (pow (p, n)) = 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 (p, l)) = 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