--- a/src/HOL/Semiring_Normalization.thy Wed May 12 12:31:51 2010 +0200
+++ b/src/HOL/Semiring_Normalization.thy Wed May 12 12:31:52 2010 +0200
@@ -50,13 +50,18 @@
then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto
qed
+text {* semiring normalization proper *}
+
setup Semiring_Normalizer.setup
-lemma (in comm_semiring_1) semiring_ops:
+context comm_semiring_1
+begin
+
+lemma semiring_ops:
shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)"
and "TERM 0" and "TERM 1" .
-lemma (in comm_semiring_1) semiring_rules:
+lemma semiring_rules:
"(a * m) + (b * m) = (a + b) * m"
"(a * m) + m = (a + 1) * m"
"m + (a * m) = (a + 1) * m"
@@ -96,32 +101,42 @@
"x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))"
by (simp_all add: algebra_simps power_add power2_eq_square power_mult_distrib power_mult)
-lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
+lemmas normalizing_comm_semiring_1_axioms =
comm_semiring_1_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules]
-declaration (in comm_semiring_1)
+declaration
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
-lemma (in comm_ring_1) ring_ops: shows "TERM (x- y)" and "TERM (- x)" .
+end
-lemma (in comm_ring_1) ring_rules:
+context comm_ring_1
+begin
+
+lemma ring_ops: shows "TERM (x- y)" and "TERM (- x)" .
+
+lemma ring_rules:
"- x = (- 1) * x"
"x - y = x + (- y)"
by (simp_all add: diff_minus)
-lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
+lemmas normalizing_comm_ring_1_axioms =
comm_ring_1_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
ring rules: ring_rules]
-declaration (in comm_ring_1)
+declaration
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
-lemma (in comm_semiring_1_cancel_norm) noteq_reduce:
+end
+
+context comm_semiring_1_cancel_norm
+begin
+
+lemma 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
@@ -131,7 +146,7 @@
by simp
qed
-lemma (in comm_semiring_1_cancel_norm) add_scale_eq_noteq:
+lemma add_scale_eq_noteq:
"\<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> 0" and cnd: "c\<noteq>d"
@@ -141,29 +156,34 @@
thus "False" using add_mult_solve[of 0 d] nz cnd by simp
qed
-lemma (in comm_semiring_1_cancel_norm) add_0_iff:
+lemma add_0_iff:
"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
-declare (in comm_semiring_1_cancel_norm)
+declare
normalizing_comm_semiring_1_axioms [normalizer del]
-lemmas (in comm_semiring_1_cancel_norm)
+lemmas
normalizing_comm_semiring_1_cancel_norm_axioms =
comm_semiring_1_cancel_norm_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
idom rules: noteq_reduce add_scale_eq_noteq]
-declaration (in comm_semiring_1_cancel_norm)
+declaration
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
-declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]
+end
-lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
+context idom
+begin
+
+declare normalizing_comm_ring_1_axioms [normalizer del]
+
+lemmas normalizing_idom_axioms = idom_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
ring ops: ring_ops
@@ -171,15 +191,20 @@
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: right_minus_eq add_0_iff]
-declaration (in idom)
+declaration
{* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
-lemma (in field) field_ops:
+end
+
+context field
+begin
+
+lemma field_ops:
shows "TERM (x / y)" and "TERM (inverse x)" .
-lemmas (in field) field_rules = divide_inverse inverse_eq_divide
+lemmas field_rules = divide_inverse inverse_eq_divide
-lemmas (in field) normalizing_field_axioms =
+lemmas normalizing_field_axioms =
field_axioms [normalizer
semiring ops: semiring_ops
semiring rules: semiring_rules
@@ -190,16 +215,18 @@
idom rules: noteq_reduce add_scale_eq_noteq
ideal rules: right_minus_eq add_0_iff]
-declaration (in field)
+declaration
{* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
+end
+
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
+hide_fact (open) normalizing_field_axioms field_ops field_rules
hide_fact (open) add_scale_eq_noteq noteq_reduce