src/HOL/Semiring_Normalization.thy
 author haftmann Sat May 08 18:52:38 2010 +0200 (2010-05-08) changeset 36756 c1ae8a0b4265 parent 36753 5cf4e9128f22 child 36845 d778c64fc35d permissions -rw-r--r--
moved normalization proof tool infrastructure to canonical algebraic classes
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 setup Semiring_Normalizer.setup
55 locale normalizing_semiring =
56   fixes add mul pwr r0 r1
59     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
60     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
61     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
62     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
63 begin
65 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
66 proof (induct p)
67   case 0
68   then show ?case by (auto simp add: pwr_0 mul_1)
69 next
70   case Suc
71   from this [symmetric] show ?case
72     by (auto simp add: pwr_Suc mul_1 mul_a)
73 qed
75 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
76 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
77   fix q x y
78   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
79   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
80     by (simp add: mul_a)
81   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
82   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
83   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
84     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
85 qed
87 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
88 proof (induct p arbitrary: q)
89   case 0
90   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
91 next
92   case Suc
93   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
94 qed
96 lemma semiring_ops:
97   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
98     and "TERM r0" and "TERM r1" .
100 lemma semiring_rules:
101   "add (mul a m) (mul b m) = mul (add a b) m"
102   "add (mul a m) m = mul (add a r1) m"
103   "add m (mul a m) = mul (add a r1) m"
104   "add m m = mul (add r1 r1) m"
105   "add r0 a = a"
106   "add a r0 = a"
107   "mul a b = mul b a"
108   "mul (add a b) c = add (mul a c) (mul b c)"
109   "mul r0 a = r0"
110   "mul a r0 = r0"
111   "mul r1 a = a"
112   "mul a r1 = a"
113   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
114   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
115   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
116   "mul (mul lx ly) rx = mul (mul lx rx) ly"
117   "mul (mul lx ly) rx = mul lx (mul ly rx)"
118   "mul lx (mul rx ry) = mul (mul lx rx) ry"
119   "mul lx (mul rx ry) = mul rx (mul lx ry)"
124   "add a c = add c a"
126   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
127   "mul x (pwr x q) = pwr x (Suc q)"
128   "mul (pwr x q) x = pwr x (Suc q)"
129   "mul x x = pwr x 2"
130   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
131   "pwr (pwr x p) q = pwr x (p * q)"
132   "pwr x 0 = r1"
133   "pwr x 1 = x"
134   "mul x (add y z) = add (mul x y) (mul x z)"
135   "pwr x (Suc q) = mul x (pwr x q)"
136   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
137   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
138 proof -
139   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
140 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
141 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
142 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
143 next show "add r0 a = a" using add_0 by simp
144 next show "add a r0 = a" using add_0 add_c by simp
145 next show "mul a b = mul b a" using mul_c by simp
146 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
147 next show "mul r0 a = r0" using mul_0 by simp
148 next show "mul a r0 = r0" using mul_0 mul_c by simp
149 next show "mul r1 a = a" using mul_1 by simp
150 next show "mul a r1 = a" using mul_1 mul_c by simp
151 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
152     using mul_c mul_a by simp
153 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
154     using mul_a by simp
155 next
156   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
157   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
158   finally
159   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
160     using mul_c by simp
161 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
162 next
163   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
164 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
165 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
168 next show "add (add a b) c = add a (add b c)" using add_a by simp
169 next show "add a (add c d) = add c (add a d)"
172 next show "add a c = add c a" by (rule add_c)
173 next show "add a (add c d) = add (add a c) d" using add_a by simp
174 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
175 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
176 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
177 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
178 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
179 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
180 next show "pwr x 0 = r1" using pwr_0 .
181 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
182 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
183 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
184 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
185 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
186     by (simp add: nat_number' pwr_Suc mul_pwr)
187 qed
189 end
191 sublocale comm_semiring_1
192   < normalizing!: normalizing_semiring plus times power zero one
193 proof
194 qed (simp_all add: algebra_simps)
196 lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
197   comm_semiring_1_axioms [normalizer
198     semiring ops: normalizing.semiring_ops
199     semiring rules: normalizing.semiring_rules]
201 declaration (in comm_semiring_1)
202   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
204 locale normalizing_ring = normalizing_semiring +
205   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
206     and neg :: "'a \<Rightarrow> 'a"
207   assumes neg_mul: "neg x = mul (neg r1) x"
208     and sub_add: "sub x y = add x (neg y)"
209 begin
211 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
213 lemmas ring_rules = neg_mul sub_add
215 end
217 sublocale comm_ring_1
218   < normalizing!: normalizing_ring plus times power zero one minus uminus
219 proof
220 qed (simp_all add: diff_minus)
222 lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
223   comm_ring_1_axioms [normalizer
224     semiring ops: normalizing.semiring_ops
225     semiring rules: normalizing.semiring_rules
226     ring ops: normalizing.ring_ops
227     ring rules: normalizing.ring_rules]
229 declaration (in comm_ring_1)
230   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
232 locale normalizing_semiring_cancel = normalizing_semiring +
233   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) =
235     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
236 begin
238 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)"
239 proof-
240   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
241   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
242     using add_mul_solve by blast
243   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)"
244     by simp
245 qed
247 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
248   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
249 proof(clarify)
250   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
251     and eq: "add b (mul r c) = add b (mul r d)"
252   hence "mul r c = mul r d" using cnd add_cancel by simp
253   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
254     using mul_0 add_cancel by simp
255   thus "False" using add_mul_solve nz cnd by simp
256 qed
258 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
259 proof-
260   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
261   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
262 qed
264 end
266 sublocale comm_semiring_1_cancel_norm
267   < normalizing!: normalizing_semiring_cancel plus times power zero one
268 proof
271 declare (in comm_semiring_1_cancel_norm)
272   normalizing_comm_semiring_1_axioms [normalizer del]
274 lemmas (in comm_semiring_1_cancel_norm)
275   normalizing_comm_semiring_1_cancel_norm_axioms =
276   comm_semiring_1_cancel_norm_axioms [normalizer
277     semiring ops: normalizing.semiring_ops
278     semiring rules: normalizing.semiring_rules
279     idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq]
281 declaration (in comm_semiring_1_cancel_norm)
282   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
284 locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
285   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
287 sublocale idom
288   < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
289 proof
290 qed simp
292 declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]
294 lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
295   semiring ops: normalizing.semiring_ops
296   semiring rules: normalizing.semiring_rules
297   ring ops: normalizing.ring_ops
298   ring rules: normalizing.ring_rules
299   idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
300   ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
302 declaration (in idom)
303   {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
305 locale normalizing_field = normalizing_ring_cancel +
306   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
307     and inverse:: "'a \<Rightarrow> 'a"
308   assumes divide_inverse: "divide x y = mul x (inverse y)"
309      and inverse_divide: "inverse x = divide r1 x"
310 begin
312 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
314 lemmas field_rules = divide_inverse inverse_divide
316 end
318 sublocale field
319   < normalizing!: normalizing_field plus times power zero one minus uminus divide inverse
320 proof
321 qed (simp_all add: divide_inverse)
323 lemmas (in field) normalizing_field_axioms =
324   field_axioms [normalizer
325     semiring ops: normalizing.semiring_ops
326     semiring rules: normalizing.semiring_rules
327     ring ops: normalizing.ring_ops
328     ring rules: normalizing.ring_rules
329     field ops: normalizing.field_ops
330     field rules: normalizing.field_rules
331     idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
332     ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
334 declaration (in field)
335   {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
337 end