author  haftmann 
Wed, 12 May 2010 12:31:52 +0200  
changeset 36871  3763c349c8c1 
parent 36845  d778c64fc35d 
child 36872  6520ba1256a6 
permissions  rwrr 
36751
7f1da69cacb3
split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents:
36720
diff
changeset

1 
(* Title: HOL/Semiring_Normalization.thy 
23252  2 
Author: Amine Chaieb, TU Muenchen 
3 
*) 

4 

36751
7f1da69cacb3
split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents:
36720
diff
changeset

5 
header {* Semiring normalization *} 
28402  6 

36751
7f1da69cacb3
split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents:
36720
diff
changeset

7 
theory Semiring_Normalization 
36699
816da1023508
moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents:
36698
diff
changeset

8 
imports Numeral_Simprocs Nat_Transfer 
23252  9 
uses 
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36751
diff
changeset

10 
"Tools/semiring_normalizer.ML" 
23252  11 
begin 
12 

36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

13 
text {* FIXME prelude *} 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

14 

c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

15 
class comm_semiring_1_cancel_norm (*FIXME name*) = comm_semiring_1_cancel + 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

16 
assumes add_mult_solve: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

17 

c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

18 
sublocale idom < comm_semiring_1_cancel_norm 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

19 
proof 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

20 
fix w x y z 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

21 
show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

22 
proof 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

23 
assume "w * y + x * z = w * z + x * y" 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

24 
then have "w * y + x * z  w * z  x * y = 0" by (simp add: algebra_simps) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

25 
then have "w * (y  z)  x * (y  z) = 0" by (simp add: algebra_simps) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

26 
then have "(y  z) * (w  x) = 0" by (simp add: algebra_simps) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

27 
then have "y  z = 0 \<or> w  x = 0" by (rule divisors_zero) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

28 
then show "w = x \<or> y = z" by auto 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

29 
qed (auto simp add: add_ac) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

30 
qed 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

31 

c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

32 
instance nat :: comm_semiring_1_cancel_norm 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

33 
proof 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

34 
fix w x y z :: nat 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

35 
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z" 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

36 
hence "y < z \<or> y > z" by arith 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

37 
moreover { 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

38 
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z  y" in exI, auto) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

39 
then obtain k where kp: "k>0" and yz:"z = y + k" by blast 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

40 
from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

41 
hence "x*k = w*k" by simp 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

42 
hence "w = x" using kp by simp } 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

43 
moreover { 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

44 
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y  z" in exI, auto) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

45 
then obtain k where kp: "k>0" and yz:"y = z + k" by blast 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

46 
from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps) 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

47 
hence "w*k = x*k" by simp 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

48 
hence "w = x" using kp by simp } 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

49 
ultimately have "w=x" by blast } 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

50 
then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

51 
qed 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

52 

36871  53 
text {* semiring normalization proper *} 
54 

36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36751
diff
changeset

55 
setup Semiring_Normalizer.setup 
23252  56 

36871  57 
context comm_semiring_1 
58 
begin 

59 

60 
lemma semiring_ops: 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

61 
shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

62 
and "TERM 0" and "TERM 1" . 
23252  63 

36871  64 
lemma semiring_rules: 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

65 
"(a * m) + (b * m) = (a + b) * m" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

66 
"(a * m) + m = (a + 1) * m" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

67 
"m + (a * m) = (a + 1) * m" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

68 
"m + m = (1 + 1) * m" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

69 
"0 + a = a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

70 
"a + 0 = a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

71 
"a * b = b * a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

72 
"(a + b) * c = (a * c) + (b * c)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

73 
"0 * a = 0" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

74 
"a * 0 = 0" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

75 
"1 * a = a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

76 
"a * 1 = a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

77 
"(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

78 
"(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

79 
"(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

80 
"(lx * ly) * rx = (lx * rx) * ly" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

81 
"(lx * ly) * rx = lx * (ly * rx)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

82 
"lx * (rx * ry) = (lx * rx) * ry" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

83 
"lx * (rx * ry) = rx * (lx * ry)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

84 
"(a + b) + (c + d) = (a + c) + (b + d)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

85 
"(a + b) + c = a + (b + c)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

86 
"a + (c + d) = c + (a + d)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

87 
"(a + b) + c = (a + c) + b" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

88 
"a + c = c + a" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

89 
"a + (c + d) = (a + c) + d" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

90 
"(x ^ p) * (x ^ q) = x ^ (p + q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

91 
"x * (x ^ q) = x ^ (Suc q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

92 
"(x ^ q) * x = x ^ (Suc q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

93 
"x * x = x ^ 2" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

94 
"(x * y) ^ q = (x ^ q) * (y ^ q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

95 
"(x ^ p) ^ q = x ^ (p * q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

96 
"x ^ 0 = 1" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

97 
"x ^ 1 = x" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

98 
"x * (y + z) = (x * y) + (x * z)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

99 
"x ^ (Suc q) = x * (x ^ q)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

100 
"x ^ (2*n) = (x ^ n) * (x ^ n)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

101 
"x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

102 
by (simp_all add: algebra_simps power_add power2_eq_square power_mult_distrib power_mult) 
23252  103 

36871  104 
lemmas normalizing_comm_semiring_1_axioms = 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

105 
comm_semiring_1_axioms [normalizer 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

106 
semiring ops: semiring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

107 
semiring rules: semiring_rules] 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

108 

36871  109 
declaration 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

110 
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *} 
23573  111 

36871  112 
end 
23252  113 

36871  114 
context comm_ring_1 
115 
begin 

116 

117 
lemma ring_ops: shows "TERM (x y)" and "TERM ( x)" . 

118 

119 
lemma ring_rules: 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

120 
" x = ( 1) * x" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

121 
"x  y = x + ( y)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

122 
by (simp_all add: diff_minus) 
23252  123 

36871  124 
lemmas normalizing_comm_ring_1_axioms = 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

125 
comm_ring_1_axioms [normalizer 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

126 
semiring ops: semiring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

127 
semiring rules: semiring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

128 
ring ops: ring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

129 
ring rules: ring_rules] 
30866  130 

36871  131 
declaration 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

132 
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *} 
23327  133 

36871  134 
end 
135 

136 
context comm_semiring_1_cancel_norm 

137 
begin 

138 

139 
lemma noteq_reduce: 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

140 
"a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)" 
23252  141 
proof 
142 
have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

143 
also have "\<dots> \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

144 
using add_mult_solve by blast 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

145 
finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)" 
23252  146 
by simp 
147 
qed 

148 

36871  149 
lemma add_scale_eq_noteq: 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

150 
"\<lbrakk>r \<noteq> 0 ; a = b \<and> c \<noteq> d\<rbrakk> \<Longrightarrow> a + (r * c) \<noteq> b + (r * d)" 
23252  151 
proof(clarify) 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

152 
assume nz: "r\<noteq> 0" and cnd: "c\<noteq>d" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

153 
and eq: "b + (r * c) = b + (r * d)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

154 
have "(0 * d) + (r * c) = (0 * c) + (r * d)" 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

155 
using add_imp_eq eq mult_zero_left by simp 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

156 
thus "False" using add_mult_solve[of 0 d] nz cnd by simp 
23252  157 
qed 
158 

36871  159 
lemma add_0_iff: 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

160 
"x = x + a \<longleftrightarrow> a = 0" 
25250
b3a485b98963
(1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents:
23573
diff
changeset

161 
proof 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

162 
have "a = 0 \<longleftrightarrow> x + a = x + 0" using add_imp_eq[of x a 0] by auto 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

163 
thus "x = x + a \<longleftrightarrow> a = 0" by (auto simp add: add_commute) 
25250
b3a485b98963
(1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents:
23573
diff
changeset

164 
qed 
b3a485b98963
(1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents:
23573
diff
changeset

165 

36871  166 
declare 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

167 
normalizing_comm_semiring_1_axioms [normalizer del] 
23252  168 

36871  169 
lemmas 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

170 
normalizing_comm_semiring_1_cancel_norm_axioms = 
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

171 
comm_semiring_1_cancel_norm_axioms [normalizer 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

172 
semiring ops: semiring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

173 
semiring rules: semiring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

174 
idom rules: noteq_reduce add_scale_eq_noteq] 
23252  175 

36871  176 
declaration 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

177 
{* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *} 
23252  178 

36871  179 
end 
23252  180 

36871  181 
context idom 
182 
begin 

183 

184 
declare normalizing_comm_ring_1_axioms [normalizer del] 

185 

186 
lemmas normalizing_idom_axioms = idom_axioms [normalizer 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

187 
semiring ops: semiring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

188 
semiring rules: semiring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

189 
ring ops: ring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

190 
ring rules: ring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

191 
idom rules: noteq_reduce add_scale_eq_noteq 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

192 
ideal rules: right_minus_eq add_0_iff] 
23252  193 

36871  194 
declaration 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

195 
{* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *} 
23252  196 

36871  197 
end 
198 

199 
context field 

200 
begin 

201 

202 
lemma field_ops: 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

203 
shows "TERM (x / y)" and "TERM (inverse x)" . 
23327  204 

36871  205 
lemmas field_rules = divide_inverse inverse_eq_divide 
28402  206 

36871  207 
lemmas normalizing_field_axioms = 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

208 
field_axioms [normalizer 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

209 
semiring ops: semiring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

210 
semiring rules: semiring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

211 
ring ops: ring_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

212 
ring rules: ring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

213 
field ops: field_ops 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

214 
field rules: field_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

215 
idom rules: noteq_reduce add_scale_eq_noteq 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

216 
ideal rules: right_minus_eq add_0_iff] 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

217 

36871  218 
declaration 
36756
c1ae8a0b4265
moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents:
36753
diff
changeset

219 
{* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *} 
28402  220 

36871  221 
end 
222 

36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

223 
hide_fact (open) normalizing_comm_semiring_1_axioms 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

224 
normalizing_comm_semiring_1_cancel_norm_axioms semiring_ops semiring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

225 

d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

226 
hide_fact (open) normalizing_comm_ring_1_axioms 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

227 
normalizing_idom_axioms ring_ops ring_rules 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

228 

36871  229 
hide_fact (open) normalizing_field_axioms field_ops field_rules 
36845
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

230 

d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

231 
hide_fact (open) add_scale_eq_noteq noteq_reduce 
d778c64fc35d
Add rules directly to the corresponding class locales instead.
hoelzl
parents:
36756
diff
changeset

232 

28402  233 
end 