src/HOL/Semiring_Normalization.thy
 author haftmann Wed May 12 13:51:22 2010 +0200 (2010-05-12) changeset 36872 6520ba1256a6 parent 36871 3763c349c8c1 child 36873 112e613e8d0b permissions -rw-r--r--
tuned fact collection names and some proofs
1 (*  Title:      HOL/Semiring_Normalization.thy
2     Author:     Amine Chaieb, TU Muenchen
3 *)
5 header {* Semiring normalization *}
7 theory Semiring_Normalization
8 imports Numeral_Simprocs Nat_Transfer
9 uses
10   "Tools/semiring_normalizer.ML"
11 begin
13 text {* FIXME prelude *}
15 class comm_semiring_1_cancel_norm (*FIXME name*) = comm_semiring_1_cancel +
16   assumes add_mult_solve: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
18 sublocale idom < comm_semiring_1_cancel_norm
19 proof
20   fix w x y z
21   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
22   proof
23     assume "w * y + x * z = w * z + x * y"
24     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
25     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
26     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
27     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
28     then show "w = x \<or> y = z" by auto
30 qed
32 instance nat :: comm_semiring_1_cancel_norm
33 proof
34   fix w x y z :: nat
35   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
36     hence "y < z \<or> y > z" by arith
37     moreover {
38       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
39       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
40       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
41       hence "x*k = w*k" by simp
42       hence "w = x" using kp by simp }
43     moreover {
44       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
45       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
46       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
47       hence "w*k = x*k" by simp
48       hence "w = x" using kp by simp }
49     ultimately have "w=x" by blast }
50   then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto
51 qed
53 text {* semiring normalization proper *}
55 setup Semiring_Normalizer.setup
57 context comm_semiring_1
58 begin
60 lemma normalizing_semiring_ops:
61   shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)"
62     and "TERM 0" and "TERM 1" .
64 lemma normalizing_semiring_rules:
65   "(a * m) + (b * m) = (a + b) * m"
66   "(a * m) + m = (a + 1) * m"
67   "m + (a * m) = (a + 1) * m"
68   "m + m = (1 + 1) * m"
69   "0 + a = a"
70   "a + 0 = a"
71   "a * b = b * a"
72   "(a + b) * c = (a * c) + (b * c)"
73   "0 * a = 0"
74   "a * 0 = 0"
75   "1 * a = a"
76   "a * 1 = a"
77   "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
78   "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
79   "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
80   "(lx * ly) * rx = (lx * rx) * ly"
81   "(lx * ly) * rx = lx * (ly * rx)"
82   "lx * (rx * ry) = (lx * rx) * ry"
83   "lx * (rx * ry) = rx * (lx * ry)"
84   "(a + b) + (c + d) = (a + c) + (b + d)"
85   "(a + b) + c = a + (b + c)"
86   "a + (c + d) = c + (a + d)"
87   "(a + b) + c = (a + c) + b"
88   "a + c = c + a"
89   "a + (c + d) = (a + c) + d"
90   "(x ^ p) * (x ^ q) = x ^ (p + q)"
91   "x * (x ^ q) = x ^ (Suc q)"
92   "(x ^ q) * x = x ^ (Suc q)"
93   "x * x = x ^ 2"
94   "(x * y) ^ q = (x ^ q) * (y ^ q)"
95   "(x ^ p) ^ q = x ^ (p * q)"
96   "x ^ 0 = 1"
97   "x ^ 1 = x"
98   "x * (y + z) = (x * y) + (x * z)"
99   "x ^ (Suc q) = x * (x ^ q)"
100   "x ^ (2*n) = (x ^ n) * (x ^ n)"
101   "x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))"
102   by (simp_all add: algebra_simps power_add power2_eq_square power_mult_distrib power_mult)
104 lemmas normalizing_comm_semiring_1_axioms =
105   comm_semiring_1_axioms [normalizer
106     semiring ops: normalizing_semiring_ops
107     semiring rules: normalizing_semiring_rules]
109 declaration
110   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
112 end
114 context comm_ring_1
115 begin
117 lemma normalizing_ring_ops: shows "TERM (x- y)" and "TERM (- x)" .
119 lemma normalizing_ring_rules:
120   "- x = (- 1) * x"
121   "x - y = x + (- y)"
122   by (simp_all add: diff_minus)
124 lemmas normalizing_comm_ring_1_axioms =
125   comm_ring_1_axioms [normalizer
126     semiring ops: normalizing_semiring_ops
127     semiring rules: normalizing_semiring_rules
128     ring ops: normalizing_ring_ops
129     ring rules: normalizing_ring_rules]
131 declaration
132   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
134 end
136 context comm_semiring_1_cancel_norm
137 begin
139 lemma noteq_reduce:
140   "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> a * c + b * d \<noteq> a * d + b * c"
144   "r \<noteq> 0 \<Longrightarrow> a = b \<and> c \<noteq> d \<Longrightarrow> a + r * c \<noteq> b + r * d"
145 proof (rule notI)
146   assume nz: "r\<noteq> 0" and cnd: "a = b \<and> c\<noteq>d"
147     and eq: "a + (r * c) = b + (r * d)"
148   have "(0 * d) + (r * c) = (0 * c) + (r * d)"
149     using add_imp_eq eq mult_zero_left by (simp add: cnd)
150   then show False using add_mult_solve [of 0 d] nz cnd by simp
151 qed
154   "x = x + a \<longleftrightarrow> a = 0"
155   using add_imp_eq [of x a 0] by auto
157 declare
158   normalizing_comm_semiring_1_axioms [normalizer del]
160 lemmas
161   normalizing_comm_semiring_1_cancel_norm_axioms =
162   comm_semiring_1_cancel_norm_axioms [normalizer
163     semiring ops: normalizing_semiring_ops
164     semiring rules: normalizing_semiring_rules
165     idom rules: noteq_reduce add_scale_eq_noteq]
167 declaration
168   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
170 end
172 context idom
173 begin
175 declare normalizing_comm_ring_1_axioms [normalizer del]
177 lemmas normalizing_idom_axioms = idom_axioms [normalizer
178   semiring ops: normalizing_semiring_ops
179   semiring rules: normalizing_semiring_rules
180   ring ops: normalizing_ring_ops
181   ring rules: normalizing_ring_rules
182   idom rules: noteq_reduce add_scale_eq_noteq
183   ideal rules: right_minus_eq add_0_iff]
185 declaration
186   {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
188 end
190 context field
191 begin
193 lemma normalizing_field_ops:
194   shows "TERM (x / y)" and "TERM (inverse x)" .
196 lemmas normalizing_field_rules = divide_inverse inverse_eq_divide
198 lemmas normalizing_field_axioms =
199   field_axioms [normalizer
200     semiring ops: normalizing_semiring_ops
201     semiring rules: normalizing_semiring_rules
202     ring ops: normalizing_ring_ops
203     ring rules: normalizing_ring_rules
204     field ops: normalizing_field_ops
205     field rules: normalizing_field_rules
206     idom rules: noteq_reduce add_scale_eq_noteq
207     ideal rules: right_minus_eq add_0_iff]
209 declaration
210   {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
212 end
214 hide_fact (open) normalizing_comm_semiring_1_axioms
215   normalizing_comm_semiring_1_cancel_norm_axioms normalizing_semiring_ops normalizing_semiring_rules
217 hide_fact (open) normalizing_comm_ring_1_axioms
218   normalizing_idom_axioms normalizing_ring_ops normalizing_ring_rules
220 hide_fact (open) normalizing_field_axioms normalizing_field_ops normalizing_field_rules
222 hide_fact (open) add_scale_eq_noteq noteq_reduce
224 end