author | huffman |
Sun, 09 May 2010 17:47:43 -0700 | |
changeset 36777 | be5461582d0f |
parent 36756 | c1ae8a0b4265 |
child 36845 | d778c64fc35d |
permissions | -rw-r--r-- |
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(* Title: HOL/Semiring_Normalization.thy |
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Author: Amine Chaieb, TU Muenchen |
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*) |
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header {* Semiring normalization *} |
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theory Semiring_Normalization |
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imports Numeral_Simprocs Nat_Transfer |
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uses |
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"Tools/semiring_normalizer.ML" |
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begin |
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text {* FIXME prelude *} |
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class comm_semiring_1_cancel_norm (*FIXME name*) = comm_semiring_1_cancel + |
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assumes add_mult_solve: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" |
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sublocale idom < comm_semiring_1_cancel_norm |
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proof |
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fix w x y z |
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show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" |
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proof |
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assume "w * y + x * z = w * z + x * y" |
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then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps) |
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then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps) |
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then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps) |
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then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero) |
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then show "w = x \<or> y = z" by auto |
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qed (auto simp add: add_ac) |
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qed |
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instance nat :: comm_semiring_1_cancel_norm |
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proof |
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fix w x y z :: nat |
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{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" |
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hence "y < z \<or> y > z" by arith |
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moreover { |
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assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto) |
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then obtain k where kp: "k>0" and yz:"z = y + k" by blast |
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from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) |
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hence "x*k = w*k" by simp |
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hence "w = x" using kp by simp } |
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moreover { |
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assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto) |
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then obtain k where kp: "k>0" and yz:"y = z + k" by blast |
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from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) |
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hence "w*k = x*k" by simp |
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hence "w = x" using kp by simp } |
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ultimately have "w=x" by blast } |
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then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto |
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qed |
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setup Semiring_Normalizer.setup |
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locale normalizing_semiring = |
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fixes add mul pwr r0 r1 |
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assumes add_a:"(add x (add y z) = add (add x y) z)" |
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and add_c: "add x y = add y x" and add_0:"add r0 x = x" |
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and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x" |
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and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0" |
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and mul_d:"mul x (add y z) = add (mul x y) (mul x z)" |
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and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)" |
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begin |
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lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)" |
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proof (induct p) |
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case 0 |
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then show ?case by (auto simp add: pwr_0 mul_1) |
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next |
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case Suc |
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from this [symmetric] show ?case |
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by (auto simp add: pwr_Suc mul_1 mul_a) |
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qed |
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lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)" |
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proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1) |
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fix q x y |
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assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)" |
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have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))" |
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by (simp add: mul_a) |
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also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c) |
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also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a) |
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finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) = |
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mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c) |
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qed |
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lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)" |
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proof (induct p arbitrary: q) |
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case 0 |
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show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto |
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next |
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case Suc |
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thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc) |
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qed |
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lemma semiring_ops: |
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shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)" |
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and "TERM r0" and "TERM r1" . |
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lemma semiring_rules: |
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"add (mul a m) (mul b m) = mul (add a b) m" |
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"add (mul a m) m = mul (add a r1) m" |
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"add m (mul a m) = mul (add a r1) m" |
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"add m m = mul (add r1 r1) m" |
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"add r0 a = a" |
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"add a r0 = a" |
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"mul a b = mul b a" |
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"mul (add a b) c = add (mul a c) (mul b c)" |
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"mul r0 a = r0" |
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"mul a r0 = r0" |
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"mul r1 a = a" |
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"mul a r1 = a" |
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"mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" |
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"mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" |
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"mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" |
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"mul (mul lx ly) rx = mul (mul lx rx) ly" |
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"mul (mul lx ly) rx = mul lx (mul ly rx)" |
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"mul lx (mul rx ry) = mul (mul lx rx) ry" |
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"mul lx (mul rx ry) = mul rx (mul lx ry)" |
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"add (add a b) (add c d) = add (add a c) (add b d)" |
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"add (add a b) c = add a (add b c)" |
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"add a (add c d) = add c (add a d)" |
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"add (add a b) c = add (add a c) b" |
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"add a c = add c a" |
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"add a (add c d) = add (add a c) d" |
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"mul (pwr x p) (pwr x q) = pwr x (p + q)" |
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"mul x (pwr x q) = pwr x (Suc q)" |
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"mul (pwr x q) x = pwr x (Suc q)" |
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"mul x x = pwr x 2" |
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"pwr (mul x y) q = mul (pwr x q) (pwr y q)" |
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"pwr (pwr x p) q = pwr x (p * q)" |
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"pwr x 0 = r1" |
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"pwr x 1 = x" |
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"mul x (add y z) = add (mul x y) (mul x z)" |
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"pwr x (Suc q) = mul x (pwr x q)" |
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"pwr x (2*n) = mul (pwr x n) (pwr x n)" |
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"pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))" |
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proof - |
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show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp |
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next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp |
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next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp |
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next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp |
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next show "add r0 a = a" using add_0 by simp |
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next show "add a r0 = a" using add_0 add_c by simp |
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next show "mul a b = mul b a" using mul_c by simp |
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next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp |
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next show "mul r0 a = r0" using mul_0 by simp |
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next show "mul a r0 = r0" using mul_0 mul_c by simp |
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next show "mul r1 a = a" using mul_1 by simp |
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next show "mul a r1 = a" using mul_1 mul_c by simp |
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next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" |
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using mul_c mul_a by simp |
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next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" |
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using mul_a by simp |
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next |
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have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c) |
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also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp |
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finally |
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show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" |
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using mul_c by simp |
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next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp |
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next |
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show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a) |
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next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a ) |
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next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c) |
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next show "add (add a b) (add c d) = add (add a c) (add b d)" |
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using add_c add_a by simp |
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next show "add (add a b) c = add a (add b c)" using add_a by simp |
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next show "add a (add c d) = add c (add a d)" |
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apply (simp add: add_a) by (simp only: add_c) |
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next show "add (add a b) c = add (add a c) b" using add_a add_c by simp |
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next show "add a c = add c a" by (rule add_c) |
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next show "add a (add c d) = add (add a c) d" using add_a by simp |
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next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr) |
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next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp |
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next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp |
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next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) |
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next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul) |
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next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr) |
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next show "pwr x 0 = r1" using pwr_0 . |
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next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c) |
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next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp |
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next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp |
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next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr) |
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next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))" |
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by (simp add: nat_number' pwr_Suc mul_pwr) |
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qed |
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end |
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sublocale comm_semiring_1 |
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< normalizing!: normalizing_semiring plus times power zero one |
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proof |
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qed (simp_all add: algebra_simps) |
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lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms = |
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comm_semiring_1_axioms [normalizer |
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semiring ops: normalizing.semiring_ops |
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semiring rules: normalizing.semiring_rules] |
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declaration (in comm_semiring_1) |
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{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *} |
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locale normalizing_ring = normalizing_semiring + |
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fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
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and neg :: "'a \<Rightarrow> 'a" |
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assumes neg_mul: "neg x = mul (neg r1) x" |
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and sub_add: "sub x y = add x (neg y)" |
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begin |
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lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" . |
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lemmas ring_rules = neg_mul sub_add |
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end |
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sublocale comm_ring_1 |
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< normalizing!: normalizing_ring plus times power zero one minus uminus |
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proof |
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qed (simp_all add: diff_minus) |
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lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms = |
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comm_ring_1_axioms [normalizer |
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semiring ops: normalizing.semiring_ops |
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semiring rules: normalizing.semiring_rules |
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ring ops: normalizing.ring_ops |
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ring rules: normalizing.ring_rules] |
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declaration (in comm_ring_1) |
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{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *} |
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locale normalizing_semiring_cancel = normalizing_semiring + |
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assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z" |
234 |
and add_mul_solve: "add (mul w y) (mul x z) = |
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add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z" |
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begin |
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||
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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)" |
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proof- |
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have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp |
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also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)" |
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using add_mul_solve by blast |
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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)" |
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by simp |
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qed |
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lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk> |
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\<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)" |
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proof(clarify) |
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assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d" |
|
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and eq: "add b (mul r c) = add b (mul r d)" |
|
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hence "mul r c = mul r d" using cnd add_cancel by simp |
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hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)" |
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using mul_0 add_cancel by simp |
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thus "False" using add_mul_solve nz cnd by simp |
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qed |
|
257 |
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lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0" |
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proof- |
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have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel) |
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thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0) |
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qed |
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end |
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|
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sublocale comm_semiring_1_cancel_norm |
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< normalizing!: normalizing_semiring_cancel plus times power zero one |
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proof |
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qed (simp_all add: add_mult_solve) |
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270 |
|
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declare (in comm_semiring_1_cancel_norm) |
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normalizing_comm_semiring_1_axioms [normalizer del] |
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lemmas (in comm_semiring_1_cancel_norm) |
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normalizing_comm_semiring_1_cancel_norm_axioms = |
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comm_semiring_1_cancel_norm_axioms [normalizer |
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semiring ops: normalizing.semiring_ops |
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semiring rules: normalizing.semiring_rules |
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idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq] |
23252 | 280 |
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declaration (in comm_semiring_1_cancel_norm) |
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{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *} |
23252 | 283 |
|
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locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + |
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285 |
assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y" |
23252 | 286 |
|
36720 | 287 |
sublocale idom |
288 |
< normalizing!: normalizing_ring_cancel plus times power zero one minus uminus |
|
289 |
proof |
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qed simp |
23252 | 291 |
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declare (in idom) normalizing_comm_ring_1_axioms [normalizer del] |
23252 | 293 |
|
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lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer |
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semiring ops: normalizing.semiring_ops |
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semiring rules: normalizing.semiring_rules |
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ring ops: normalizing.ring_ops |
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ring rules: normalizing.ring_rules |
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idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq |
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300 |
ideal rules: normalizing.subr0_iff normalizing.add_r0_iff] |
23252 | 301 |
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declaration (in idom) |
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{* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *} |
23252 | 304 |
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locale normalizing_field = normalizing_ring_cancel + |
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306 |
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" |
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and inverse:: "'a \<Rightarrow> 'a" |
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308 |
assumes divide_inverse: "divide x y = mul x (inverse y)" |
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309 |
and inverse_divide: "inverse x = divide r1 x" |
23327 | 310 |
begin |
311 |
||
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312 |
lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" . |
23327 | 313 |
|
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314 |
lemmas field_rules = divide_inverse inverse_divide |
26314 | 315 |
|
23327 | 316 |
end |
317 |
||
36720 | 318 |
sublocale field |
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319 |
< normalizing!: normalizing_field plus times power zero one minus uminus divide inverse |
36720 | 320 |
proof |
321 |
qed (simp_all add: divide_inverse) |
|
28402 | 322 |
|
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lemmas (in field) normalizing_field_axioms = |
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324 |
field_axioms [normalizer |
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325 |
semiring ops: normalizing.semiring_ops |
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326 |
semiring rules: normalizing.semiring_rules |
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327 |
ring ops: normalizing.ring_ops |
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328 |
ring rules: normalizing.ring_rules |
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329 |
field ops: normalizing.field_ops |
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330 |
field rules: normalizing.field_rules |
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331 |
idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq |
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332 |
ideal rules: normalizing.subr0_iff normalizing.add_r0_iff] |
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333 |
|
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334 |
declaration (in field) |
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{* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *} |
28402 | 336 |
|
337 |
end |