--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Commutative_Ring.thy Fri Oct 30 13:59:49 2009 +0100
@@ -0,0 +1,319 @@
+(* 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: 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 "commutative_ring_tac.ML"
+setup Commutative_Ring_Tac.setup
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy Fri Oct 30 13:59:49 2009 +0100
@@ -0,0 +1,391 @@
+(* Author: Bernhard Haeupler
+
+This theory is about of the relative completeness of method comm-ring
+method. As long as the reified atomic polynomials of type 'a pol are
+in normal form, the cring method is complete.
+*)
+
+header {* Proof of the relative completeness of method comm-ring *}
+
+theory Commutative_Ring_Complete
+imports Commutative_Ring
+begin
+
+text {* Formalization of normal form *}
+fun
+ isnorm :: "('a::{comm_ring}) pol \<Rightarrow> bool"
+where
+ "isnorm (Pc c) \<longleftrightarrow> True"
+ | "isnorm (Pinj i (Pc c)) \<longleftrightarrow> False"
+ | "isnorm (Pinj i (Pinj j Q)) \<longleftrightarrow> False"
+ | "isnorm (Pinj 0 P) \<longleftrightarrow> False"
+ | "isnorm (Pinj i (PX Q1 j Q2)) \<longleftrightarrow> isnorm (PX Q1 j Q2)"
+ | "isnorm (PX P 0 Q) \<longleftrightarrow> False"
+ | "isnorm (PX (Pc c) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm Q"
+ | "isnorm (PX (PX P1 j (Pc c)) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm (PX P1 j (Pc c)) \<and> isnorm Q"
+ | "isnorm (PX P i Q) \<longleftrightarrow> isnorm P \<and> isnorm Q"
+
+(* Some helpful lemmas *)
+lemma norm_Pinj_0_False:"isnorm (Pinj 0 P) = False"
+by(cases P, auto)
+
+lemma norm_PX_0_False:"isnorm (PX (Pc 0) i Q) = False"
+by(cases i, auto)
+
+lemma norm_Pinj:"isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
+by(cases i,simp add: norm_Pinj_0_False norm_PX_0_False,cases Q) auto
+
+lemma norm_PX2:"isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
+by(cases i, auto, cases P, auto, case_tac pol2, auto)
+
+lemma norm_PX1:"isnorm (PX P i Q) \<Longrightarrow> isnorm P"
+by(cases i, auto, cases P, auto, case_tac pol2, auto)
+
+lemma mkPinj_cn:"\<lbrakk>y~=0; isnorm Q\<rbrakk> \<Longrightarrow> isnorm (mkPinj y Q)"
+apply(auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
+apply(case_tac nat, auto simp add: norm_Pinj_0_False)
+by(case_tac pol, auto) (case_tac y, auto)
+
+lemma norm_PXtrans:
+ assumes A:"isnorm (PX P x Q)" and "isnorm Q2"
+ shows "isnorm (PX P x Q2)"
+proof(cases P)
+ case (PX p1 y p2) from prems show ?thesis by(cases x, auto, cases p2, auto)
+next
+ case Pc from prems show ?thesis by(cases x, auto)
+next
+ case Pinj from prems show ?thesis by(cases x, auto)
+qed
+
+lemma norm_PXtrans2: assumes A:"isnorm (PX P x Q)" and "isnorm Q2" shows "isnorm (PX P (Suc (n+x)) Q2)"
+proof(cases P)
+ case (PX p1 y p2)
+ from prems show ?thesis by(cases x, auto, cases p2, auto)
+next
+ case Pc
+ from prems show ?thesis by(cases x, auto)
+next
+ case Pinj
+ from prems show ?thesis by(cases x, auto)
+qed
+
+text {* mkPX conserves normalizedness (@{text "_cn"}) *}
+lemma mkPX_cn:
+ assumes "x \<noteq> 0" and "isnorm P" and "isnorm Q"
+ shows "isnorm (mkPX P x Q)"
+proof(cases P)
+ case (Pc c)
+ from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
+next
+ case (Pinj i Q)
+ from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
+next
+ case (PX P1 y P2)
+ from prems have Y0:"y>0" by(cases y, auto)
+ from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 y P2] norm_PX2[of P1 y P2])
+ with prems Y0 show ?thesis by (cases x, auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
+qed
+
+text {* add conserves normalizedness *}
+lemma add_cn:"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
+proof(induct P Q rule: add.induct)
+ case (2 c i P2) thus ?case by (cases P2, simp_all, cases i, simp_all)
+next
+ case (3 i P2 c) thus ?case by (cases P2, simp_all, cases i, simp_all)
+next
+ case (4 c P2 i Q2)
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+next
+ case (5 P2 i Q2 c)
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+next
+ case (6 x P2 y Q2)
+ from prems have Y0:"y>0" by (cases y, auto simp add: norm_Pinj_0_False)
+ from prems have X0:"x>0" by (cases x, auto simp add: norm_Pinj_0_False)
+ have "x < y \<or> x = y \<or> x > y" by arith
+ moreover
+ { assume "x<y" hence "EX d. y=d+x" by arith
+ then obtain d where "y=d+x"..
+ moreover
+ note prems X0
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
+ ultimately have ?case by (simp add: mkPinj_cn)}
+ moreover
+ { assume "x=y"
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ note prems Y0
+ moreover
+ ultimately have ?case by (simp add: mkPinj_cn) }
+ moreover
+ { assume "x>y" hence "EX d. x=d+y" by arith
+ then obtain d where "x=d+y"..
+ moreover
+ note prems Y0
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
+ ultimately have ?case by (simp add: mkPinj_cn)}
+ ultimately show ?case by blast
+next
+ case (7 x P2 Q2 y R)
+ have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+ moreover
+ { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+ moreover
+ { assume "x=1"
+ from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (R \<oplus> P2)" by simp
+ with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ moreover
+ { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+ then obtain d where X:"x=Suc (Suc d)" ..
+ from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+ with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
+ with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ ultimately show ?case by blast
+next
+ case (8 Q2 y R x P2)
+ have "x = 0 \<or> x = 1 \<or> x > 1" by arith
+ moreover
+ { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+ moreover
+ { assume "x=1"
+ from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (R \<oplus> P2)" by simp
+ with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ moreover
+ { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+ then obtain d where X:"x=Suc (Suc d)" ..
+ from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+ with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
+ with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ ultimately show ?case by blast
+next
+ case (9 P1 x P2 Q1 y Q2)
+ from prems have Y0:"y>0" by(cases y, auto)
+ from prems have X0:"x>0" by(cases x, auto)
+ from prems have NP1:"isnorm P1" and NP2:"isnorm P2" by (auto simp add: norm_PX1[of P1 _ P2] norm_PX2[of P1 _ P2])
+ from prems have NQ1:"isnorm Q1" and NQ2:"isnorm Q2" by (auto simp add: norm_PX1[of Q1 _ Q2] norm_PX2[of Q1 _ Q2])
+ have "y < x \<or> x = y \<or> x < y" by arith
+ moreover
+ {assume sm1:"y < x" hence "EX d. x=d+y" by arith
+ then obtain d where sm2:"x=d+y"..
+ note prems NQ1 NP1 NP2 NQ2 sm1 sm2
+ moreover
+ have "isnorm (PX P1 d (Pc 0))"
+ proof(cases P1)
+ case (PX p1 y p2)
+ with prems show ?thesis by(cases d, simp,cases p2, auto)
+ next case Pc from prems show ?thesis by(cases d, auto)
+ next case Pinj from prems show ?thesis by(cases d, auto)
+ qed
+ ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX P1 (x - y) (Pc 0) \<oplus> Q1)" by auto
+ with Y0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+ moreover
+ {assume "x=y"
+ from prems NP1 NP2 NQ1 NQ2 have "isnorm (P2 \<oplus> Q2)" "isnorm (P1 \<oplus> Q1)" by auto
+ with Y0 prems have ?case by (simp add: mkPX_cn) }
+ moreover
+ {assume sm1:"x<y" hence "EX d. y=d+x" by arith
+ then obtain d where sm2:"y=d+x"..
+ note prems NQ1 NP1 NP2 NQ2 sm1 sm2
+ moreover
+ have "isnorm (PX Q1 d (Pc 0))"
+ proof(cases Q1)
+ case (PX p1 y p2)
+ with prems show ?thesis by(cases d, simp,cases p2, auto)
+ next case Pc from prems show ?thesis by(cases d, auto)
+ next case Pinj from prems show ?thesis by(cases d, auto)
+ qed
+ ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX Q1 (y - x) (Pc 0) \<oplus> P1)" by auto
+ with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+ ultimately show ?case by blast
+qed simp
+
+text {* mul concerves normalizedness *}
+lemma mul_cn :"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
+proof(induct P Q rule: mul.induct)
+ case (2 c i P2) thus ?case
+ by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+next
+ case (3 i P2 c) thus ?case
+ by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+next
+ case (4 c P2 i Q2)
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with prems show ?case
+ by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
+next
+ case (5 P2 i Q2 c)
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with prems show ?case
+ by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
+next
+ case (6 x P2 y Q2)
+ have "x < y \<or> x = y \<or> x > y" by arith
+ moreover
+ { assume "x<y" hence "EX d. y=d+x" by arith
+ then obtain d where "y=d+x"..
+ moreover
+ note prems
+ moreover
+ from prems have "x>0" by (cases x, auto simp add: norm_Pinj_0_False)
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
+ ultimately have ?case by (simp add: mkPinj_cn)}
+ moreover
+ { assume "x=y"
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ with prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
+ moreover
+ note prems
+ moreover
+ ultimately have ?case by (simp add: mkPinj_cn) }
+ moreover
+ { assume "x>y" hence "EX d. x=d+y" by arith
+ then obtain d where "x=d+y"..
+ moreover
+ note prems
+ moreover
+ from prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
+ moreover
+ from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ moreover
+ with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
+ ultimately have ?case by (simp add: mkPinj_cn) }
+ ultimately show ?case by blast
+next
+ case (7 x P2 Q2 y R)
+ from prems have Y0:"y>0" by(cases y, auto)
+ have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+ moreover
+ { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+ moreover
+ { assume "x=1"
+ from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
+ with Y0 prems have ?case by (simp add: mkPX_cn)}
+ moreover
+ { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+ then obtain d where X:"x=Suc (Suc d)" ..
+ from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
+ moreover
+ from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+ moreover
+ from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
+ moreover
+ note prems
+ ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
+ with Y0 X have ?case by (simp add: mkPX_cn)}
+ ultimately show ?case by blast
+next
+ case (8 Q2 y R x P2)
+ from prems have Y0:"y>0" by(cases y, auto)
+ have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+ moreover
+ { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+ moreover
+ { assume "x=1"
+ from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
+ with Y0 prems have ?case by (simp add: mkPX_cn) }
+ moreover
+ { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+ then obtain d where X:"x=Suc (Suc d)" ..
+ from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
+ moreover
+ from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+ moreover
+ from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
+ moreover
+ note prems
+ ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
+ with Y0 X have ?case by (simp add: mkPX_cn) }
+ ultimately show ?case by blast
+next
+ case (9 P1 x P2 Q1 y Q2)
+ from prems have X0:"x>0" by(cases x, auto)
+ from prems have Y0:"y>0" by(cases y, auto)
+ note prems
+ moreover
+ from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+ moreover
+ from prems have "isnorm Q1" "isnorm Q2" by (auto simp add: norm_PX1[of Q1 y Q2] norm_PX2[of Q1 y Q2])
+ ultimately have "isnorm (P1 \<otimes> Q1)" "isnorm (P2 \<otimes> Q2)"
+ "isnorm (P1 \<otimes> mkPinj 1 Q2)" "isnorm (Q1 \<otimes> mkPinj 1 P2)"
+ by (auto simp add: mkPinj_cn)
+ with prems X0 Y0 have
+ "isnorm (mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2))"
+ "isnorm (mkPX (P1 \<otimes> mkPinj (Suc 0) Q2) x (Pc 0))"
+ "isnorm (mkPX (Q1 \<otimes> mkPinj (Suc 0) P2) y (Pc 0))"
+ by (auto simp add: mkPX_cn)
+ thus ?case by (simp add: add_cn)
+qed(simp)
+
+text {* neg conserves normalizedness *}
+lemma neg_cn: "isnorm P \<Longrightarrow> isnorm (neg P)"
+proof (induct P)
+ case (Pinj i P2)
+ from prems have "isnorm P2" by (simp add: norm_Pinj[of i P2])
+ with prems show ?case by(cases P2, auto, cases i, auto)
+next
+ case (PX P1 x P2)
+ from prems have "isnorm P2" "isnorm P1" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+ with prems show ?case
+ proof(cases P1)
+ case (PX p1 y p2)
+ with prems show ?thesis by(cases x, auto, cases p2, auto)
+ next
+ case Pinj
+ with prems show ?thesis by(cases x, auto)
+ qed(cases x, auto)
+qed(simp)
+
+text {* sub conserves normalizedness *}
+lemma sub_cn:"isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
+by (simp add: sub_def add_cn neg_cn)
+
+text {* sqr conserves normalizizedness *}
+lemma sqr_cn:"isnorm P \<Longrightarrow> isnorm (sqr P)"
+proof(induct P)
+ case (Pinj i Q)
+ from prems show ?case by(cases Q, auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
+next
+ case (PX P1 x P2)
+ from prems have "x+x~=0" "isnorm P2" "isnorm P1" by (cases x, auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+ with prems have
+ "isnorm (mkPX (Pc (1 + 1) \<otimes> P1 \<otimes> mkPinj (Suc 0) P2) x (Pc 0))"
+ and "isnorm (mkPX (sqr P1) (x + x) (sqr P2))"
+ by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
+ thus ?case by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
+qed simp
+
+text {* pow conserves normalizedness *}
+lemma pow_cn:"isnorm P \<Longrightarrow> isnorm (pow n P)"
+proof (induct n arbitrary: P rule: nat_less_induct)
+ case (1 k)
+ show ?case
+ proof (cases "k=0")
+ case False
+ then have K2:"k div 2 < k" by (cases k, auto)
+ from prems have "isnorm (sqr P)" by (simp add: sqr_cn)
+ with prems K2 show ?thesis
+ by (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
+ qed simp
+qed
+
+end
--- a/src/HOL/Decision_Procs/Decision_Procs.thy Fri Oct 30 01:32:06 2009 +0100
+++ b/src/HOL/Decision_Procs/Decision_Procs.thy Fri Oct 30 13:59:49 2009 +0100
@@ -1,7 +1,10 @@
-header {* Various decision procedures. typically involving reflection *}
+header {* Various decision procedures, typically involving reflection *}
theory Decision_Procs
-imports Cooper Ferrack MIR Approximation Dense_Linear_Order "ex/Approximation_Ex" "ex/Dense_Linear_Order_Ex" Parametric_Ferrante_Rackoff
+imports
+ Commutative_Ring Cooper Ferrack MIR Approximation Dense_Linear_Order Parametric_Ferrante_Rackoff
+ Commutative_Ring_Complete
+ "ex/Commutative_Ring_Ex" "ex/Approximation_Ex" "ex/Dense_Linear_Order_Ex"
begin
end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/commutative_ring_tac.ML Fri Oct 30 13:59:49 2009 +0100
@@ -0,0 +1,104 @@
+(* Author: Amine Chaieb
+
+Tactic for solving equalities over commutative rings.
+*)
+
+signature COMMUTATIVE_RING_TAC =
+sig
+ val tac: Proof.context -> int -> tactic
+ val setup: theory -> theory
+end
+
+structure Commutative_Ring_Tac: COMMUTATIVE_RING_TAC =
+struct
+
+(* Zero and One of the commutative ring *)
+fun cring_zero T = Const (@{const_name HOL.zero}, T);
+fun cring_one T = Const (@{const_name HOL.one}, T);
+
+(* reification functions *)
+(* add two polynom expressions *)
+fun polT t = Type (@{type_name Commutative_Ring.pol}, [t]);
+fun polexT t = Type (@{type_name Commutative_Ring.polex}, [t]);
+
+(* pol *)
+fun pol_Pc t = Const (@{const_name Commutative_Ring.pol.Pc}, t --> polT t);
+fun pol_Pinj t = Const (@{const_name Commutative_Ring.pol.Pinj}, HOLogic.natT --> polT t --> polT t);
+fun pol_PX t = Const (@{const_name Commutative_Ring.pol.PX}, polT t --> HOLogic.natT --> polT t --> polT t);
+
+(* polex *)
+fun polex_add t = Const (@{const_name Commutative_Ring.polex.Add}, polexT t --> polexT t --> polexT t);
+fun polex_sub t = Const (@{const_name Commutative_Ring.polex.Sub}, polexT t --> polexT t --> polexT t);
+fun polex_mul t = Const (@{const_name Commutative_Ring.polex.Mul}, polexT t --> polexT t --> polexT t);
+fun polex_neg t = Const (@{const_name Commutative_Ring.polex.Neg}, polexT t --> polexT t);
+fun polex_pol t = Const (@{const_name Commutative_Ring.polex.Pol}, polT t --> polexT t);
+fun polex_pow t = Const (@{const_name Commutative_Ring.polex.Pow}, polexT t --> HOLogic.natT --> polexT t);
+
+(* reification of polynoms : primitive cring expressions *)
+fun reif_pol T vs (t as Free _) =
+ let
+ val one = @{term "1::nat"};
+ val i = find_index (fn t' => t' = t) vs
+ in if i = 0
+ then pol_PX T $ (pol_Pc T $ cring_one T)
+ $ one $ (pol_Pc T $ cring_zero T)
+ else pol_Pinj T $ HOLogic.mk_nat i
+ $ (pol_PX T $ (pol_Pc T $ cring_one T)
+ $ one $ (pol_Pc T $ cring_zero T))
+ end
+ | reif_pol T vs t = pol_Pc T $ t;
+
+(* reification of polynom expressions *)
+fun reif_polex T vs (Const (@{const_name HOL.plus}, _) $ a $ b) =
+ polex_add T $ reif_polex T vs a $ reif_polex T vs b
+ | reif_polex T vs (Const (@{const_name HOL.minus}, _) $ a $ b) =
+ polex_sub T $ reif_polex T vs a $ reif_polex T vs b
+ | reif_polex T vs (Const (@{const_name HOL.times}, _) $ a $ b) =
+ polex_mul T $ reif_polex T vs a $ reif_polex T vs b
+ | reif_polex T vs (Const (@{const_name HOL.uminus}, _) $ a) =
+ polex_neg T $ reif_polex T vs a
+ | reif_polex T vs (Const (@{const_name Power.power}, _) $ a $ n) =
+ polex_pow T $ reif_polex T vs a $ n
+ | reif_polex T vs t = polex_pol T $ reif_pol T vs t;
+
+(* reification of the equation *)
+val cr_sort = @{sort "comm_ring_1"};
+
+fun reif_eq thy (eq as Const(@{const_name "op ="}, Type("fun", [T, _])) $ lhs $ rhs) =
+ if Sign.of_sort thy (T, cr_sort) then
+ let
+ val fs = OldTerm.term_frees eq;
+ val cvs = cterm_of thy (HOLogic.mk_list T fs);
+ val clhs = cterm_of thy (reif_polex T fs lhs);
+ val crhs = cterm_of thy (reif_polex T fs rhs);
+ val ca = ctyp_of thy T;
+ in (ca, cvs, clhs, crhs) end
+ else error ("reif_eq: not an equation over " ^ Syntax.string_of_sort_global thy cr_sort)
+ | reif_eq _ _ = error "reif_eq: not an equation";
+
+(* The cring tactic *)
+(* Attention: You have to make sure that no t^0 is in the goal!! *)
+(* Use simply rewriting t^0 = 1 *)
+val cring_simps =
+ [@{thm mkPX_def}, @{thm mkPinj_def}, @{thm sub_def}, @{thm power_add},
+ @{thm even_def}, @{thm pow_if}, sym OF [@{thm power_add}]];
+
+fun tac ctxt = SUBGOAL (fn (g, i) =>
+ let
+ val thy = ProofContext.theory_of ctxt;
+ val cring_ss = Simplifier.simpset_of ctxt (*FIXME really the full simpset!?*)
+ addsimps cring_simps;
+ val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g)
+ val norm_eq_th =
+ simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] @{thm norm_eq})
+ in
+ cut_rules_tac [norm_eq_th] i
+ THEN (simp_tac cring_ss i)
+ THEN (simp_tac cring_ss i)
+ end);
+
+val setup =
+ Method.setup @{binding comm_ring} (Scan.succeed (SIMPLE_METHOD' o tac))
+ "reflective decision procedure for equalities over commutative rings";
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/ex/Commutative_Ring_Ex.thy Fri Oct 30 13:59:49 2009 +0100
@@ -0,0 +1,48 @@
+(* Author: Bernhard Haeupler *)
+
+header {* Some examples demonstrating the comm-ring method *}
+
+theory Commutative_Ring_Ex
+imports Commutative_Ring
+begin
+
+lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
+by comm_ring
+
+lemma "((x::int) + y)^2 = x^2 + y^2 + 2*x*y"
+by comm_ring
+
+lemma "((x::int) + y)^3 = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
+by comm_ring
+
+lemma "((x::int) - y)^3 = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
+by comm_ring
+
+lemma "((x::int) - y)^2 = x^2 + y^2 - 2*x*y"
+by comm_ring
+
+lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
+by comm_ring
+
+lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
+by comm_ring
+
+lemma "(a::int)*b + a*c = a*(b+c)"
+by comm_ring
+
+lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
+by comm_ring
+
+lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
+by comm_ring
+
+lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
+by comm_ring
+
+lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
+by comm_ring
+
+lemma "(a::int)^10 - b^10 = (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9 )"
+by comm_ring
+
+end
--- a/src/HOL/IsaMakefile Fri Oct 30 01:32:06 2009 +0100
+++ b/src/HOL/IsaMakefile Fri Oct 30 13:59:49 2009 +0100
@@ -248,12 +248,12 @@
Groebner_Basis.thy \
Hilbert_Choice.thy \
Int.thy \
- IntDiv.thy \
List.thy \
Main.thy \
Map.thy \
Nat_Numeral.thy \
Nat_Transfer.thy \
+ Numeral_Simprocs.thy \
Presburger.thy \
Predicate_Compile.thy \
Quickcheck.thy \
@@ -382,7 +382,6 @@
Library/While_Combinator.thy Library/Product_ord.thy \
Library/Char_nat.thy Library/Char_ord.thy Library/Option_ord.thy \
Library/Sublist_Order.thy Library/List_lexord.thy \
- Library/Commutative_Ring.thy Library/comm_ring.ML \
Library/Coinductive_List.thy Library/AssocList.thy \
Library/Formal_Power_Series.thy Library/Binomial.thy \
Library/Eval_Witness.thy Library/Code_Char.thy \
@@ -785,6 +784,9 @@
$(LOG)/HOL-Decision_Procs.gz: $(OUT)/HOL \
Decision_Procs/Approximation.thy \
+ Decision_Procs/Commutative_Ring.thy \
+ Decision_Procs/Commutative_Ring_Complete.thy \
+ Decision_Procs/commutative_ring_tac.ML \
Decision_Procs/Cooper.thy \
Decision_Procs/cooper_tac.ML \
Decision_Procs/Dense_Linear_Order.thy \
@@ -795,6 +797,7 @@
Decision_Procs/Decision_Procs.thy \
Decision_Procs/ex/Dense_Linear_Order_Ex.thy \
Decision_Procs/ex/Approximation_Ex.thy \
+ Decision_Procs/ex/Commutative_Ring_Ex.thy \
Decision_Procs/ROOT.ML
@$(ISABELLE_TOOL) usedir $(OUT)/HOL Decision_Procs
@@ -937,7 +940,7 @@
HOL-ex: HOL $(LOG)/HOL-ex.gz
-$(LOG)/HOL-ex.gz: $(OUT)/HOL Library/Commutative_Ring.thy \
+$(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy \
Number_Theory/Primes.thy \
Tools/Predicate_Compile/predicate_compile_core.ML \
ex/Abstract_NAT.thy ex/Antiquote.thy ex/Arith_Examples.thy \
@@ -945,8 +948,8 @@
ex/Binary.thy ex/CTL.thy ex/Chinese.thy ex/Classical.thy \
ex/CodegenSML_Test.thy ex/Codegenerator_Candidates.thy \
ex/Codegenerator_Pretty.thy ex/Codegenerator_Pretty_Test.thy \
- ex/Codegenerator_Test.thy ex/Coherent.thy ex/Commutative_RingEx.thy \
- ex/Commutative_Ring_Complete.thy ex/Efficient_Nat_examples.thy \
+ ex/Codegenerator_Test.thy ex/Coherent.thy \
+ ex/Efficient_Nat_examples.thy \
ex/Eval_Examples.thy ex/Fundefs.thy ex/Groebner_Examples.thy \
ex/Guess.thy ex/HarmonicSeries.thy ex/Hebrew.thy \
ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy \
--- 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
--- a/src/HOL/Library/Library.thy Fri Oct 30 01:32:06 2009 +0100
+++ b/src/HOL/Library/Library.thy Fri Oct 30 13:59:49 2009 +0100
@@ -11,7 +11,6 @@
Code_Char_chr
Code_Integer
Coinductive_List
- Commutative_Ring
Continuity
ContNotDenum
Countable
--- a/src/HOL/Library/comm_ring.ML Fri Oct 30 01:32:06 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,109 +0,0 @@
-(* Author: Amine Chaieb
-
-Tactic for solving equalities over commutative rings.
-*)
-
-signature COMM_RING =
-sig
- val comm_ring_tac : Proof.context -> int -> tactic
- val setup : theory -> theory
-end
-
-structure CommRing: COMM_RING =
-struct
-
-(* The Cring exception for erronous uses of cring_tac *)
-exception CRing of string;
-
-(* Zero and One of the commutative ring *)
-fun cring_zero T = Const (@{const_name HOL.zero}, T);
-fun cring_one T = Const (@{const_name HOL.one}, T);
-
-(* reification functions *)
-(* add two polynom expressions *)
-fun polT t = Type ("Commutative_Ring.pol", [t]);
-fun polexT t = Type ("Commutative_Ring.polex", [t]);
-
-(* pol *)
-fun pol_Pc t = Const ("Commutative_Ring.pol.Pc", t --> polT t);
-fun pol_Pinj t = Const ("Commutative_Ring.pol.Pinj", HOLogic.natT --> polT t --> polT t);
-fun pol_PX t = Const ("Commutative_Ring.pol.PX", polT t --> HOLogic.natT --> polT t --> polT t);
-
-(* polex *)
-fun polex_add t = Const ("Commutative_Ring.polex.Add", polexT t --> polexT t --> polexT t);
-fun polex_sub t = Const ("Commutative_Ring.polex.Sub", polexT t --> polexT t --> polexT t);
-fun polex_mul t = Const ("Commutative_Ring.polex.Mul", polexT t --> polexT t --> polexT t);
-fun polex_neg t = Const ("Commutative_Ring.polex.Neg", polexT t --> polexT t);
-fun polex_pol t = Const ("Commutative_Ring.polex.Pol", polT t --> polexT t);
-fun polex_pow t = Const ("Commutative_Ring.polex.Pow", polexT t --> HOLogic.natT --> polexT t);
-
-(* reification of polynoms : primitive cring expressions *)
-fun reif_pol T vs (t as Free _) =
- let
- val one = @{term "1::nat"};
- val i = find_index (fn t' => t' = t) vs
- in if i = 0
- then pol_PX T $ (pol_Pc T $ cring_one T)
- $ one $ (pol_Pc T $ cring_zero T)
- else pol_Pinj T $ HOLogic.mk_nat i
- $ (pol_PX T $ (pol_Pc T $ cring_one T)
- $ one $ (pol_Pc T $ cring_zero T))
- end
- | reif_pol T vs t = pol_Pc T $ t;
-
-(* reification of polynom expressions *)
-fun reif_polex T vs (Const (@{const_name HOL.plus}, _) $ a $ b) =
- polex_add T $ reif_polex T vs a $ reif_polex T vs b
- | reif_polex T vs (Const (@{const_name HOL.minus}, _) $ a $ b) =
- polex_sub T $ reif_polex T vs a $ reif_polex T vs b
- | reif_polex T vs (Const (@{const_name HOL.times}, _) $ a $ b) =
- polex_mul T $ reif_polex T vs a $ reif_polex T vs b
- | reif_polex T vs (Const (@{const_name HOL.uminus}, _) $ a) =
- polex_neg T $ reif_polex T vs a
- | reif_polex T vs (Const (@{const_name Power.power}, _) $ a $ n) =
- polex_pow T $ reif_polex T vs a $ n
- | reif_polex T vs t = polex_pol T $ reif_pol T vs t;
-
-(* reification of the equation *)
-val cr_sort = @{sort "comm_ring_1"};
-
-fun reif_eq thy (eq as Const("op =", Type("fun", [T, _])) $ lhs $ rhs) =
- if Sign.of_sort thy (T, cr_sort) then
- let
- val fs = OldTerm.term_frees eq;
- val cvs = cterm_of thy (HOLogic.mk_list T fs);
- val clhs = cterm_of thy (reif_polex T fs lhs);
- val crhs = cterm_of thy (reif_polex T fs rhs);
- val ca = ctyp_of thy T;
- in (ca, cvs, clhs, crhs) end
- else raise CRing ("reif_eq: not an equation over " ^ Syntax.string_of_sort_global thy cr_sort)
- | reif_eq _ _ = raise CRing "reif_eq: not an equation";
-
-(* The cring tactic *)
-(* Attention: You have to make sure that no t^0 is in the goal!! *)
-(* Use simply rewriting t^0 = 1 *)
-val cring_simps =
- [@{thm mkPX_def}, @{thm mkPinj_def}, @{thm sub_def}, @{thm power_add},
- @{thm even_def}, @{thm pow_if}, sym OF [@{thm power_add}]];
-
-fun comm_ring_tac ctxt = SUBGOAL (fn (g, i) =>
- let
- val thy = ProofContext.theory_of ctxt;
- val cring_ss = Simplifier.simpset_of ctxt (*FIXME really the full simpset!?*)
- addsimps cring_simps;
- val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g)
- val norm_eq_th =
- simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] @{thm norm_eq})
- in
- cut_rules_tac [norm_eq_th] i
- THEN (simp_tac cring_ss i)
- THEN (simp_tac cring_ss i)
- end);
-
-val setup =
- Method.setup @{binding comm_ring} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
- "reflective decision procedure for equalities over commutative rings" #>
- Method.setup @{binding algebra} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
- "method for proving algebraic properties (same as comm_ring)";
-
-end;
--- a/src/HOL/ex/Codegenerator_Candidates.thy Fri Oct 30 01:32:06 2009 +0100
+++ b/src/HOL/ex/Codegenerator_Candidates.thy Fri Oct 30 13:59:49 2009 +0100
@@ -9,7 +9,6 @@
AssocList
Binomial
Fset
- Commutative_Ring
Enum
List_Prefix
Nat_Infinity
@@ -22,7 +21,7 @@
Tree
While_Combinator
Word
- "~~/src/HOL/ex/Commutative_Ring_Complete"
+ "~~/src/HOL/Decision_Procs/Commutative_Ring_Complete"
"~~/src/HOL/ex/Records"
begin
--- a/src/HOL/ex/Commutative_RingEx.thy Fri Oct 30 01:32:06 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,50 +0,0 @@
-(* ID: $Id$
- Author: Bernhard Haeupler
-*)
-
-header {* Some examples demonstrating the comm-ring method *}
-
-theory Commutative_RingEx
-imports Commutative_Ring
-begin
-
-lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
-by comm_ring
-
-lemma "((x::int) + y)^2 = x^2 + y^2 + 2*x*y"
-by comm_ring
-
-lemma "((x::int) + y)^3 = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
-by comm_ring
-
-lemma "((x::int) - y)^3 = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
-by comm_ring
-
-lemma "((x::int) - y)^2 = x^2 + y^2 - 2*x*y"
-by comm_ring
-
-lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
-by comm_ring
-
-lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
-by comm_ring
-
-lemma "(a::int)*b + a*c = a*(b+c)"
-by comm_ring
-
-lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
-by comm_ring
-
-lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
-by comm_ring
-
-lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
-by comm_ring
-
-lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
-by comm_ring
-
-lemma "(a::int)^10 - b^10 = (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9 )"
-by comm_ring
-
-end
--- a/src/HOL/ex/Commutative_Ring_Complete.thy Fri Oct 30 01:32:06 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,391 +0,0 @@
-(* Author: Bernhard Haeupler
-
-This theory is about of the relative completeness of method comm-ring
-method. As long as the reified atomic polynomials of type 'a pol are
-in normal form, the cring method is complete.
-*)
-
-header {* Proof of the relative completeness of method comm-ring *}
-
-theory Commutative_Ring_Complete
-imports Commutative_Ring
-begin
-
-text {* Formalization of normal form *}
-fun
- isnorm :: "('a::{comm_ring}) pol \<Rightarrow> bool"
-where
- "isnorm (Pc c) \<longleftrightarrow> True"
- | "isnorm (Pinj i (Pc c)) \<longleftrightarrow> False"
- | "isnorm (Pinj i (Pinj j Q)) \<longleftrightarrow> False"
- | "isnorm (Pinj 0 P) \<longleftrightarrow> False"
- | "isnorm (Pinj i (PX Q1 j Q2)) \<longleftrightarrow> isnorm (PX Q1 j Q2)"
- | "isnorm (PX P 0 Q) \<longleftrightarrow> False"
- | "isnorm (PX (Pc c) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm Q"
- | "isnorm (PX (PX P1 j (Pc c)) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm (PX P1 j (Pc c)) \<and> isnorm Q"
- | "isnorm (PX P i Q) \<longleftrightarrow> isnorm P \<and> isnorm Q"
-
-(* Some helpful lemmas *)
-lemma norm_Pinj_0_False:"isnorm (Pinj 0 P) = False"
-by(cases P, auto)
-
-lemma norm_PX_0_False:"isnorm (PX (Pc 0) i Q) = False"
-by(cases i, auto)
-
-lemma norm_Pinj:"isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
-by(cases i,simp add: norm_Pinj_0_False norm_PX_0_False,cases Q) auto
-
-lemma norm_PX2:"isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
-
-lemma norm_PX1:"isnorm (PX P i Q) \<Longrightarrow> isnorm P"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
-
-lemma mkPinj_cn:"\<lbrakk>y~=0; isnorm Q\<rbrakk> \<Longrightarrow> isnorm (mkPinj y Q)"
-apply(auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
-apply(case_tac nat, auto simp add: norm_Pinj_0_False)
-by(case_tac pol, auto) (case_tac y, auto)
-
-lemma norm_PXtrans:
- assumes A:"isnorm (PX P x Q)" and "isnorm Q2"
- shows "isnorm (PX P x Q2)"
-proof(cases P)
- case (PX p1 y p2) from prems show ?thesis by(cases x, auto, cases p2, auto)
-next
- case Pc from prems show ?thesis by(cases x, auto)
-next
- case Pinj from prems show ?thesis by(cases x, auto)
-qed
-
-lemma norm_PXtrans2: assumes A:"isnorm (PX P x Q)" and "isnorm Q2" shows "isnorm (PX P (Suc (n+x)) Q2)"
-proof(cases P)
- case (PX p1 y p2)
- from prems show ?thesis by(cases x, auto, cases p2, auto)
-next
- case Pc
- from prems show ?thesis by(cases x, auto)
-next
- case Pinj
- from prems show ?thesis by(cases x, auto)
-qed
-
-text {* mkPX conserves normalizedness (@{text "_cn"}) *}
-lemma mkPX_cn:
- assumes "x \<noteq> 0" and "isnorm P" and "isnorm Q"
- shows "isnorm (mkPX P x Q)"
-proof(cases P)
- case (Pc c)
- from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
-next
- case (Pinj i Q)
- from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
-next
- case (PX P1 y P2)
- from prems have Y0:"y>0" by(cases y, auto)
- from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 y P2] norm_PX2[of P1 y P2])
- with prems Y0 show ?thesis by (cases x, auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
-qed
-
-text {* add conserves normalizedness *}
-lemma add_cn:"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
-proof(induct P Q rule: add.induct)
- case (2 c i P2) thus ?case by (cases P2, simp_all, cases i, simp_all)
-next
- case (3 i P2 c) thus ?case by (cases P2, simp_all, cases i, simp_all)
-next
- case (4 c P2 i Q2)
- from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
-next
- case (5 P2 i Q2 c)
- from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
-next
- case (6 x P2 y Q2)
- from prems have Y0:"y>0" by (cases y, auto simp add: norm_Pinj_0_False)
- from prems have X0:"x>0" by (cases x, auto simp add: norm_Pinj_0_False)
- have "x < y \<or> x = y \<or> x > y" by arith
- moreover
- { assume "x<y" hence "EX d. y=d+x" by arith
- then obtain d where "y=d+x"..
- moreover
- note prems X0
- moreover
- from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
- ultimately have ?case by (simp add: mkPinj_cn)}
- moreover
- { assume "x=y"
- moreover
- from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- note prems Y0
- moreover
- ultimately have ?case by (simp add: mkPinj_cn) }
- moreover
- { assume "x>y" hence "EX d. x=d+y" by arith
- then obtain d where "x=d+y"..
- moreover
- note prems Y0
- moreover
- from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
- ultimately have ?case by (simp add: mkPinj_cn)}
- ultimately show ?case by blast
-next
- case (7 x P2 Q2 y R)
- have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
- moreover
- { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
- moreover
- { assume "x=1"
- from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (R \<oplus> P2)" by simp
- with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
- moreover
- { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
- then obtain d where X:"x=Suc (Suc d)" ..
- from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
- with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
- with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
- ultimately show ?case by blast
-next
- case (8 Q2 y R x P2)
- have "x = 0 \<or> x = 1 \<or> x > 1" by arith
- moreover
- { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
- moreover
- { assume "x=1"
- from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (R \<oplus> P2)" by simp
- with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
- moreover
- { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
- then obtain d where X:"x=Suc (Suc d)" ..
- from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
- with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
- with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
- ultimately show ?case by blast
-next
- case (9 P1 x P2 Q1 y Q2)
- from prems have Y0:"y>0" by(cases y, auto)
- from prems have X0:"x>0" by(cases x, auto)
- from prems have NP1:"isnorm P1" and NP2:"isnorm P2" by (auto simp add: norm_PX1[of P1 _ P2] norm_PX2[of P1 _ P2])
- from prems have NQ1:"isnorm Q1" and NQ2:"isnorm Q2" by (auto simp add: norm_PX1[of Q1 _ Q2] norm_PX2[of Q1 _ Q2])
- have "y < x \<or> x = y \<or> x < y" by arith
- moreover
- {assume sm1:"y < x" hence "EX d. x=d+y" by arith
- then obtain d where sm2:"x=d+y"..
- note prems NQ1 NP1 NP2 NQ2 sm1 sm2
- moreover
- have "isnorm (PX P1 d (Pc 0))"
- proof(cases P1)
- case (PX p1 y p2)
- with prems show ?thesis by(cases d, simp,cases p2, auto)
- next case Pc from prems show ?thesis by(cases d, auto)
- next case Pinj from prems show ?thesis by(cases d, auto)
- qed
- ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX P1 (x - y) (Pc 0) \<oplus> Q1)" by auto
- with Y0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
- moreover
- {assume "x=y"
- from prems NP1 NP2 NQ1 NQ2 have "isnorm (P2 \<oplus> Q2)" "isnorm (P1 \<oplus> Q1)" by auto
- with Y0 prems have ?case by (simp add: mkPX_cn) }
- moreover
- {assume sm1:"x<y" hence "EX d. y=d+x" by arith
- then obtain d where sm2:"y=d+x"..
- note prems NQ1 NP1 NP2 NQ2 sm1 sm2
- moreover
- have "isnorm (PX Q1 d (Pc 0))"
- proof(cases Q1)
- case (PX p1 y p2)
- with prems show ?thesis by(cases d, simp,cases p2, auto)
- next case Pc from prems show ?thesis by(cases d, auto)
- next case Pinj from prems show ?thesis by(cases d, auto)
- qed
- ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX Q1 (y - x) (Pc 0) \<oplus> P1)" by auto
- with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
- ultimately show ?case by blast
-qed simp
-
-text {* mul concerves normalizedness *}
-lemma mul_cn :"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
-proof(induct P Q rule: mul.induct)
- case (2 c i P2) thus ?case
- by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
-next
- case (3 i P2 c) thus ?case
- by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
-next
- case (4 c P2 i Q2)
- from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with prems show ?case
- by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
-next
- case (5 P2 i Q2 c)
- from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with prems show ?case
- by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
-next
- case (6 x P2 y Q2)
- have "x < y \<or> x = y \<or> x > y" by arith
- moreover
- { assume "x<y" hence "EX d. y=d+x" by arith
- then obtain d where "y=d+x"..
- moreover
- note prems
- moreover
- from prems have "x>0" by (cases x, auto simp add: norm_Pinj_0_False)
- moreover
- from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
- ultimately have ?case by (simp add: mkPinj_cn)}
- moreover
- { assume "x=y"
- moreover
- from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- with prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
- moreover
- note prems
- moreover
- ultimately have ?case by (simp add: mkPinj_cn) }
- moreover
- { assume "x>y" hence "EX d. x=d+y" by arith
- then obtain d where "x=d+y"..
- moreover
- note prems
- moreover
- from prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
- moreover
- from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
- moreover
- with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
- ultimately have ?case by (simp add: mkPinj_cn) }
- ultimately show ?case by blast
-next
- case (7 x P2 Q2 y R)
- from prems have Y0:"y>0" by(cases y, auto)
- have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
- moreover
- { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
- moreover
- { assume "x=1"
- from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
- with Y0 prems have ?case by (simp add: mkPX_cn)}
- moreover
- { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
- then obtain d where X:"x=Suc (Suc d)" ..
- from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
- moreover
- from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
- moreover
- from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
- moreover
- note prems
- ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
- with Y0 X have ?case by (simp add: mkPX_cn)}
- ultimately show ?case by blast
-next
- case (8 Q2 y R x P2)
- from prems have Y0:"y>0" by(cases y, auto)
- have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
- moreover
- { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
- moreover
- { assume "x=1"
- from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
- with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
- with Y0 prems have ?case by (simp add: mkPX_cn) }
- moreover
- { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
- then obtain d where X:"x=Suc (Suc d)" ..
- from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
- moreover
- from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
- moreover
- from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
- moreover
- note prems
- ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
- with Y0 X have ?case by (simp add: mkPX_cn) }
- ultimately show ?case by blast
-next
- case (9 P1 x P2 Q1 y Q2)
- from prems have X0:"x>0" by(cases x, auto)
- from prems have Y0:"y>0" by(cases y, auto)
- note prems
- moreover
- from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
- moreover
- from prems have "isnorm Q1" "isnorm Q2" by (auto simp add: norm_PX1[of Q1 y Q2] norm_PX2[of Q1 y Q2])
- ultimately have "isnorm (P1 \<otimes> Q1)" "isnorm (P2 \<otimes> Q2)"
- "isnorm (P1 \<otimes> mkPinj 1 Q2)" "isnorm (Q1 \<otimes> mkPinj 1 P2)"
- by (auto simp add: mkPinj_cn)
- with prems X0 Y0 have
- "isnorm (mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2))"
- "isnorm (mkPX (P1 \<otimes> mkPinj (Suc 0) Q2) x (Pc 0))"
- "isnorm (mkPX (Q1 \<otimes> mkPinj (Suc 0) P2) y (Pc 0))"
- by (auto simp add: mkPX_cn)
- thus ?case by (simp add: add_cn)
-qed(simp)
-
-text {* neg conserves normalizedness *}
-lemma neg_cn: "isnorm P \<Longrightarrow> isnorm (neg P)"
-proof (induct P)
- case (Pinj i P2)
- from prems have "isnorm P2" by (simp add: norm_Pinj[of i P2])
- with prems show ?case by(cases P2, auto, cases i, auto)
-next
- case (PX P1 x P2)
- from prems have "isnorm P2" "isnorm P1" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
- with prems show ?case
- proof(cases P1)
- case (PX p1 y p2)
- with prems show ?thesis by(cases x, auto, cases p2, auto)
- next
- case Pinj
- with prems show ?thesis by(cases x, auto)
- qed(cases x, auto)
-qed(simp)
-
-text {* sub conserves normalizedness *}
-lemma sub_cn:"isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
-by (simp add: sub_def add_cn neg_cn)
-
-text {* sqr conserves normalizizedness *}
-lemma sqr_cn:"isnorm P \<Longrightarrow> isnorm (sqr P)"
-proof(induct P)
- case (Pinj i Q)
- from prems show ?case by(cases Q, auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
-next
- case (PX P1 x P2)
- from prems have "x+x~=0" "isnorm P2" "isnorm P1" by (cases x, auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
- with prems have
- "isnorm (mkPX (Pc (1 + 1) \<otimes> P1 \<otimes> mkPinj (Suc 0) P2) x (Pc 0))"
- and "isnorm (mkPX (sqr P1) (x + x) (sqr P2))"
- by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
- thus ?case by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
-qed simp
-
-text {* pow conserves normalizedness *}
-lemma pow_cn:"isnorm P \<Longrightarrow> isnorm (pow n P)"
-proof (induct n arbitrary: P rule: nat_less_induct)
- case (1 k)
- show ?case
- proof (cases "k=0")
- case False
- then have K2:"k div 2 < k" by (cases k, auto)
- from prems have "isnorm (sqr P)" by (simp add: sqr_cn)
- with prems K2 show ?thesis
- by (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
- qed simp
-qed
-
-end
--- a/src/HOL/ex/ROOT.ML Fri Oct 30 01:32:06 2009 +0100
+++ b/src/HOL/ex/ROOT.ML Fri Oct 30 13:59:49 2009 +0100
@@ -45,8 +45,6 @@
"Groebner_Examples",
"MT",
"Unification",
- "Commutative_RingEx",
- "Commutative_Ring_Complete",
"Primrec",
"Tarski",
"Adder",