--- a/src/HOL/Library/Commutative_Ring.thy Fri Oct 30 01:32:06 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,319 +0,0 @@
-(* Author: Bernhard Haeupler
-
-Proving equalities in commutative rings done "right" in Isabelle/HOL.
-*)
-
-header {* Proving equalities in commutative rings *}
-
-theory Commutative_Ring
-imports List Parity 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. *}
-
-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: 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
- 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 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