split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
(* Title: HOL/Semiring_Normalization.thy
Author: Amine Chaieb, TU Muenchen
*)
header {* Semiring normalization *}
theory Semiring_Normalization
imports Numeral_Simprocs Nat_Transfer
uses
"Tools/Groebner_Basis/normalizer.ML"
begin
setup Normalizer.setup
locale normalizing_semiring =
fixes add mul pwr r0 r1
assumes add_a:"(add x (add y z) = add (add x y) z)"
and add_c: "add x y = add y x" and add_0:"add r0 x = x"
and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0"
and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
begin
lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
proof (induct p)
case 0
then show ?case by (auto simp add: pwr_0 mul_1)
next
case Suc
from this [symmetric] show ?case
by (auto simp add: pwr_Suc mul_1 mul_a)
qed
lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
fix q x y
assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
by (simp add: mul_a)
also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
qed
lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
proof (induct p arbitrary: q)
case 0
show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
next
case Suc
thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
qed
lemma semiring_ops:
shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
and "TERM r0" and "TERM r1" .
lemma semiring_rules:
"add (mul a m) (mul b m) = mul (add a b) m"
"add (mul a m) m = mul (add a r1) m"
"add m (mul a m) = mul (add a r1) m"
"add m m = mul (add r1 r1) m"
"add r0 a = a"
"add a r0 = a"
"mul a b = mul b a"
"mul (add a b) c = add (mul a c) (mul b c)"
"mul r0 a = r0"
"mul a r0 = r0"
"mul r1 a = a"
"mul a r1 = a"
"mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
"mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
"mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
"mul (mul lx ly) rx = mul (mul lx rx) ly"
"mul (mul lx ly) rx = mul lx (mul ly rx)"
"mul lx (mul rx ry) = mul (mul lx rx) ry"
"mul lx (mul rx ry) = mul rx (mul lx ry)"
"add (add a b) (add c d) = add (add a c) (add b d)"
"add (add a b) c = add a (add b c)"
"add a (add c d) = add c (add a d)"
"add (add a b) c = add (add a c) b"
"add a c = add c a"
"add a (add c d) = add (add a c) d"
"mul (pwr x p) (pwr x q) = pwr x (p + q)"
"mul x (pwr x q) = pwr x (Suc q)"
"mul (pwr x q) x = pwr x (Suc q)"
"mul x x = pwr x 2"
"pwr (mul x y) q = mul (pwr x q) (pwr y q)"
"pwr (pwr x p) q = pwr x (p * q)"
"pwr x 0 = r1"
"pwr x 1 = x"
"mul x (add y z) = add (mul x y) (mul x z)"
"pwr x (Suc q) = mul x (pwr x q)"
"pwr x (2*n) = mul (pwr x n) (pwr x n)"
"pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
proof -
show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
next show "add r0 a = a" using add_0 by simp
next show "add a r0 = a" using add_0 add_c by simp
next show "mul a b = mul b a" using mul_c by simp
next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
next show "mul r0 a = r0" using mul_0 by simp
next show "mul a r0 = r0" using mul_0 mul_c by simp
next show "mul r1 a = a" using mul_1 by simp
next show "mul a r1 = a" using mul_1 mul_c by simp
next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
using mul_c mul_a by simp
next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
using mul_a by simp
next
have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
finally
show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
using mul_c by simp
next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
next
show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
next show "add (add a b) (add c d) = add (add a c) (add b d)"
using add_c add_a by simp
next show "add (add a b) c = add a (add b c)" using add_a by simp
next show "add a (add c d) = add c (add a d)"
apply (simp add: add_a) by (simp only: add_c)
next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
next show "add a c = add c a" by (rule add_c)
next show "add a (add c d) = add (add a c) d" using add_a by simp
next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
next show "pwr x 0 = r1" using pwr_0 .
next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
by (simp add: nat_number' pwr_Suc mul_pwr)
qed
lemmas normalizing_semiring_axioms' =
normalizing_semiring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules]
end
sublocale comm_semiring_1
< normalizing!: normalizing_semiring plus times power zero one
proof
qed (simp_all add: algebra_simps)
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
locale normalizing_ring = normalizing_semiring +
fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and neg :: "'a \<Rightarrow> 'a"
assumes neg_mul: "neg x = mul (neg r1) x"
and sub_add: "sub x y = add x (neg y)"
begin
lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
lemmas ring_rules = neg_mul sub_add
lemmas normalizing_ring_axioms' =
normalizing_ring_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules]
end
sublocale comm_ring_1
< normalizing!: normalizing_ring plus times power zero one minus uminus
proof
qed (simp_all add: diff_minus)
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
locale normalizing_field = normalizing_ring +
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and inverse:: "'a \<Rightarrow> 'a"
assumes divide_inverse: "divide x y = mul x (inverse y)"
and inverse_divide: "inverse x = divide r1 x"
begin
lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
lemmas field_rules = divide_inverse inverse_divide
lemmas normalizing_field_axioms' =
normalizing_field_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules
field ops: field_ops
field rules: field_rules]
end
locale normalizing_semiring_cancel = normalizing_semiring +
assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
and add_mul_solve: "add (mul w y) (mul x z) =
add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
begin
lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
proof-
have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
using add_mul_solve by blast
finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
by simp
qed
lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
\<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
proof(clarify)
assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
and eq: "add b (mul r c) = add b (mul r d)"
hence "mul r c = mul r d" using cnd add_cancel by simp
hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
using mul_0 add_cancel by simp
thus "False" using add_mul_solve nz cnd by simp
qed
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
proof-
have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
qed
declare normalizing_semiring_axioms' [normalizer del]
lemmas normalizing_semiring_cancel_axioms' =
normalizing_semiring_cancel_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
idom rules: noteq_reduce add_scale_eq_noteq]
end
locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
begin
declare normalizing_ring_axioms' [normalizer del]
lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: subr0_iff add_r0_iff]
end
sublocale idom
< normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
proof
fix w x y z
show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
proof
assume "w * y + x * z = w * z + x * y"
then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
then show "w = x \<or> y = z" by auto
qed (auto simp add: add_ac)
qed (simp_all add: algebra_simps)
declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
interpretation normalizing_nat!: normalizing_semiring_cancel
"op +" "op *" "op ^" "0::nat" "1"
proof (unfold_locales, simp add: algebra_simps)
fix w x y z ::"nat"
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
hence "y < z \<or> y > z" by arith
moreover {
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
then obtain k where kp: "k>0" and yz:"z = y + k" by blast
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
hence "x*k = w*k" by simp
hence "w = x" using kp by simp }
moreover {
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
then obtain k where kp: "k>0" and yz:"y = z + k" by blast
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
hence "w*k = x*k" by simp
hence "w = x" using kp by simp }
ultimately have "w=x" by blast }
thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
qed
declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
begin
declare normalizing_field_axioms' [normalizer del]
lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules
field ops: field_ops
field rules: field_rules
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: subr0_iff add_r0_iff]
end
sublocale field
< normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
proof
qed (simp_all add: divide_inverse)
declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
end