author hoelzl Tue, 11 May 2010 19:21:39 +0200 changeset 36845 d778c64fc35d parent 36844 5f9385ecc1a7 child 36846 0f67561ed5a6
```--- a/src/HOL/Semiring_Normalization.thy	Tue May 11 19:21:05 2010 +0200
+++ b/src/HOL/Semiring_Normalization.thy	Tue May 11 19:21:39 2010 +0200
@@ -52,286 +52,155 @@

setup Semiring_Normalizer.setup

-locale normalizing_semiring =
-  fixes add mul pwr r0 r1
-    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)))"
-  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 (in comm_semiring_1) semiring_ops:
+  shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)"
+    and "TERM 0" and "TERM 1" .

-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 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)"
-  "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 "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 "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
-
-end
-
-sublocale comm_semiring_1
-  < normalizing!: normalizing_semiring plus times power zero one
-proof
+lemma (in comm_semiring_1) semiring_rules:
+  "(a * m) + (b * m) = (a + b) * m"
+  "(a * m) + m = (a + 1) * m"
+  "m + (a * m) = (a + 1) * m"
+  "m + m = (1 + 1) * m"
+  "0 + a = a"
+  "a + 0 = a"
+  "a * b = b * a"
+  "(a + b) * c = (a * c) + (b * c)"
+  "0 * a = 0"
+  "a * 0 = 0"
+  "1 * a = a"
+  "a * 1 = a"
+  "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
+  "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
+  "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
+  "(lx * ly) * rx = (lx * rx) * ly"
+  "(lx * ly) * rx = lx * (ly * rx)"
+  "lx * (rx * ry) = (lx * rx) * ry"
+  "lx * (rx * ry) = rx * (lx * ry)"
+  "(a + b) + (c + d) = (a + c) + (b + d)"
+  "(a + b) + c = a + (b + c)"
+  "a + (c + d) = c + (a + d)"
+  "(a + b) + c = (a + c) + b"
+  "a + c = c + a"
+  "a + (c + d) = (a + c) + d"
+  "(x ^ p) * (x ^ q) = x ^ (p + q)"
+  "x * (x ^ q) = x ^ (Suc q)"
+  "(x ^ q) * x = x ^ (Suc q)"
+  "x * x = x ^ 2"
+  "(x * y) ^ q = (x ^ q) * (y ^ q)"
+  "(x ^ p) ^ q = x ^ (p * q)"
+  "x ^ 0 = 1"
+  "x ^ 1 = x"
+  "x * (y + z) = (x * y) + (x * z)"
+  "x ^ (Suc q) = x * (x ^ q)"
+  "x ^ (2*n) = (x ^ n) * (x ^ n)"
+  "x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))"

lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
comm_semiring_1_axioms [normalizer
-    semiring ops: normalizing.semiring_ops
-    semiring rules: normalizing.semiring_rules]
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules]

declaration (in comm_semiring_1)
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_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"
-begin
-
-lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
+lemma (in comm_ring_1) ring_ops: shows "TERM (x- y)" and "TERM (- x)" .

-
-end
-
-sublocale comm_ring_1
-  < normalizing!: normalizing_ring plus times power zero one minus uminus
-proof
+lemma (in comm_ring_1) ring_rules:
+  "- x = (- 1) * x"
+  "x - y = x + (- y)"

lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
comm_ring_1_axioms [normalizer
-    semiring ops: normalizing.semiring_ops
-    semiring rules: normalizing.semiring_rules
-    ring ops: normalizing.ring_ops
-    ring rules: normalizing.ring_rules]
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules
+    ring ops: ring_ops
+    ring rules: ring_rules]

declaration (in comm_ring_1)
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}

-locale normalizing_semiring_cancel = normalizing_semiring +
-    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)"
+lemma (in comm_semiring_1_cancel_norm) noteq_reduce:
+  "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (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)"
-  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)"
+  also have "\<dots> \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)"
+  finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (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)"
+  "\<lbrakk>r \<noteq> 0 ; a = b \<and> c \<noteq> d\<rbrakk> \<Longrightarrow> a + (r * c) \<noteq> b + (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
+  assume nz: "r\<noteq> 0" and cnd: "c\<noteq>d"
+    and eq: "b + (r * c) = b + (r * d)"
+  have "(0 * d) + (r * c) = (0 * c) + (r * d)"
+    using add_imp_eq eq mult_zero_left by simp
+  thus "False" using add_mult_solve[of 0 d] nz cnd by simp
qed

-lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
+  "x = x + a \<longleftrightarrow> a = 0"
proof-
+  have "a = 0 \<longleftrightarrow> x + a = x + 0" using add_imp_eq[of x a 0] by auto
+  thus "x = x + a \<longleftrightarrow> a = 0" by (auto simp add: add_commute)
qed

-end
-
-sublocale comm_semiring_1_cancel_norm
-  < normalizing!: normalizing_semiring_cancel plus times power zero one
-proof
-
declare (in comm_semiring_1_cancel_norm)
normalizing_comm_semiring_1_axioms [normalizer del]

lemmas (in comm_semiring_1_cancel_norm)
normalizing_comm_semiring_1_cancel_norm_axioms =
comm_semiring_1_cancel_norm_axioms [normalizer
-    semiring ops: normalizing.semiring_ops
-    semiring rules: normalizing.semiring_rules
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules

declaration (in comm_semiring_1_cancel_norm)
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}

-locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
-  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
-
-sublocale idom
-  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
-proof
-qed simp
-
declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]

lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
-  semiring ops: normalizing.semiring_ops
-  semiring rules: normalizing.semiring_rules
-  ring ops: normalizing.ring_ops
-  ring rules: normalizing.ring_rules
+  semiring ops: semiring_ops
+  semiring rules: semiring_rules
+  ring ops: ring_ops
+  ring rules: ring_rules

declaration (in idom)
{* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}

-locale normalizing_field = normalizing_ring_cancel +
-  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)" .
+lemma (in field) field_ops:
+  shows "TERM (x / y)" and "TERM (inverse x)" .

-lemmas field_rules = divide_inverse inverse_divide
-
-end
-
-sublocale field
-  < normalizing!: normalizing_field plus times power zero one minus uminus divide inverse
-proof
+lemmas (in field) field_rules = divide_inverse inverse_eq_divide

lemmas (in field) normalizing_field_axioms =
field_axioms [normalizer
-    semiring ops: normalizing.semiring_ops
-    semiring rules: normalizing.semiring_rules
-    ring ops: normalizing.ring_ops
-    ring rules: normalizing.ring_rules
-    field ops: normalizing.field_ops
-    field rules: normalizing.field_rules
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules
+    ring ops: ring_ops
+    ring rules: ring_rules
+    field ops: field_ops
+    field rules: field_rules

declaration (in field)
{* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}

+hide_fact (open) normalizing_comm_semiring_1_axioms
+  normalizing_comm_semiring_1_cancel_norm_axioms semiring_ops semiring_rules
+
+hide_fact (open) normalizing_comm_ring_1_axioms
+  normalizing_idom_axioms ring_ops ring_rules
+
+hide_fact (open)  normalizing_field_axioms field_ops field_rules
+