1(* Author: Various *)
2
3section \<open>Combination and Cancellation Simprocs for Numeral Expressions\<close>
4
5theory Numeral_Simprocs
6imports Divides
7begin
8
9ML_file "~~/src/Provers/Arith/assoc_fold.ML"
10ML_file "~~/src/Provers/Arith/cancel_numerals.ML"
11ML_file "~~/src/Provers/Arith/combine_numerals.ML"
12ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML"
13ML_file "~~/src/Provers/Arith/extract_common_term.ML"
14
15lemmas semiring_norm =
16  Let_def arith_simps diff_nat_numeral rel_simps
17  if_False if_True
18  add_0 add_Suc add_numeral_left
19  add_neg_numeral_left mult_numeral_left
20  numeral_One [symmetric] uminus_numeral_One [symmetric] Suc_eq_plus1
21  eq_numeral_iff_iszero not_iszero_Numeral1
22
23declare split_div [of _ _ "numeral k", arith_split] for k
24declare split_mod [of _ _ "numeral k", arith_split] for k
25
26text \<open>For \<open>combine_numerals\<close>\<close>
27
28lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
29by (simp add: add_mult_distrib)
30
31text \<open>For \<open>cancel_numerals\<close>\<close>
32
33lemma nat_diff_add_eq1:
34     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
35by (simp split: nat_diff_split add: add_mult_distrib)
36
37lemma nat_diff_add_eq2:
38     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
39by (simp split: nat_diff_split add: add_mult_distrib)
40
41lemma nat_eq_add_iff1:
42     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
43by (auto split: nat_diff_split simp add: add_mult_distrib)
44
45lemma nat_eq_add_iff2:
46     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
47by (auto split: nat_diff_split simp add: add_mult_distrib)
48
49lemma nat_less_add_iff1:
50     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
51by (auto split: nat_diff_split simp add: add_mult_distrib)
52
53lemma nat_less_add_iff2:
54     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
55by (auto split: nat_diff_split simp add: add_mult_distrib)
56
57lemma nat_le_add_iff1:
58     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
59by (auto split: nat_diff_split simp add: add_mult_distrib)
60
61lemma nat_le_add_iff2:
62     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
63by (auto split: nat_diff_split simp add: add_mult_distrib)
64
65text \<open>For \<open>cancel_numeral_factors\<close>\<close>
66
67lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
68by auto
69
70lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
71by auto
72
73lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
74by auto
75
76lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
77by auto
78
79lemma nat_mult_dvd_cancel_disj[simp]:
80  "(k*m) dvd (k*n) = (k=0 \<or> m dvd (n::nat))"
81by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1)
82
83lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
84by(auto)
85
86text \<open>For \<open>cancel_factor\<close>\<close>
87
88lemmas nat_mult_le_cancel_disj = mult_le_cancel1
89
90lemmas nat_mult_less_cancel_disj = mult_less_cancel1
91
92lemma nat_mult_eq_cancel_disj:
93  fixes k m n :: nat
94  shows "k * m = k * n \<longleftrightarrow> k = 0 \<or> m = n"
95  by auto
96
97lemma nat_mult_div_cancel_disj [simp]:
98  fixes k m n :: nat
99  shows "(k * m) div (k * n) = (if k = 0 then 0 else m div n)"
100  by (fact div_mult_mult1_if)
101
102lemma numeral_times_minus_swap:
103  fixes x:: "'a::comm_ring_1" shows  "numeral w * -x = x * - numeral w"
104  by (simp add: mult.commute)
105
106ML_file "Tools/numeral_simprocs.ML"
107
108simproc_setup semiring_assoc_fold
109  ("(a::'a::comm_semiring_1_cancel) * b") =
110  \<open>fn phi => Numeral_Simprocs.assoc_fold\<close>
111
112(* TODO: see whether the type class can be generalized further *)
113simproc_setup int_combine_numerals
114  ("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") =
115  \<open>fn phi => Numeral_Simprocs.combine_numerals\<close>
116
117simproc_setup field_combine_numerals
118  ("(i::'a::{field,ring_char_0}) + j"
119  |"(i::'a::{field,ring_char_0}) - j") =
120  \<open>fn phi => Numeral_Simprocs.field_combine_numerals\<close>
121
122simproc_setup inteq_cancel_numerals
123  ("(l::'a::comm_ring_1) + m = n"
124  |"(l::'a::comm_ring_1) = m + n"
125  |"(l::'a::comm_ring_1) - m = n"
126  |"(l::'a::comm_ring_1) = m - n"
127  |"(l::'a::comm_ring_1) * m = n"
128  |"(l::'a::comm_ring_1) = m * n"
129  |"- (l::'a::comm_ring_1) = m"
130  |"(l::'a::comm_ring_1) = - m") =
131  \<open>fn phi => Numeral_Simprocs.eq_cancel_numerals\<close>
132
133simproc_setup intless_cancel_numerals
134  ("(l::'a::linordered_idom) + m < n"
135  |"(l::'a::linordered_idom) < m + n"
136  |"(l::'a::linordered_idom) - m < n"
137  |"(l::'a::linordered_idom) < m - n"
138  |"(l::'a::linordered_idom) * m < n"
139  |"(l::'a::linordered_idom) < m * n"
140  |"- (l::'a::linordered_idom) < m"
141  |"(l::'a::linordered_idom) < - m") =
142  \<open>fn phi => Numeral_Simprocs.less_cancel_numerals\<close>
143
144simproc_setup intle_cancel_numerals
145  ("(l::'a::linordered_idom) + m \<le> n"
146  |"(l::'a::linordered_idom) \<le> m + n"
147  |"(l::'a::linordered_idom) - m \<le> n"
148  |"(l::'a::linordered_idom) \<le> m - n"
149  |"(l::'a::linordered_idom) * m \<le> n"
150  |"(l::'a::linordered_idom) \<le> m * n"
151  |"- (l::'a::linordered_idom) \<le> m"
152  |"(l::'a::linordered_idom) \<le> - m") =
153  \<open>fn phi => Numeral_Simprocs.le_cancel_numerals\<close>
154
155simproc_setup ring_eq_cancel_numeral_factor
156  ("(l::'a::{idom,ring_char_0}) * m = n"
157  |"(l::'a::{idom,ring_char_0}) = m * n") =
158  \<open>fn phi => Numeral_Simprocs.eq_cancel_numeral_factor\<close>
159
160simproc_setup ring_less_cancel_numeral_factor
161  ("(l::'a::linordered_idom) * m < n"
162  |"(l::'a::linordered_idom) < m * n") =
163  \<open>fn phi => Numeral_Simprocs.less_cancel_numeral_factor\<close>
164
165simproc_setup ring_le_cancel_numeral_factor
166  ("(l::'a::linordered_idom) * m <= n"
167  |"(l::'a::linordered_idom) <= m * n") =
168  \<open>fn phi => Numeral_Simprocs.le_cancel_numeral_factor\<close>
169
170(* TODO: remove comm_ring_1 constraint if possible *)
171simproc_setup int_div_cancel_numeral_factors
172  ("((l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) * m) div n"
173  |"(l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) div (m * n)") =
174  \<open>fn phi => Numeral_Simprocs.div_cancel_numeral_factor\<close>
175
176simproc_setup divide_cancel_numeral_factor
177  ("((l::'a::{field,ring_char_0}) * m) / n"
178  |"(l::'a::{field,ring_char_0}) / (m * n)"
179  |"((numeral v)::'a::{field,ring_char_0}) / (numeral w)") =
180  \<open>fn phi => Numeral_Simprocs.divide_cancel_numeral_factor\<close>
181
182simproc_setup ring_eq_cancel_factor
183  ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
184  \<open>fn phi => Numeral_Simprocs.eq_cancel_factor\<close>
185
186simproc_setup linordered_ring_le_cancel_factor
187  ("(l::'a::linordered_idom) * m <= n"
188  |"(l::'a::linordered_idom) <= m * n") =
189  \<open>fn phi => Numeral_Simprocs.le_cancel_factor\<close>
190
191simproc_setup linordered_ring_less_cancel_factor
192  ("(l::'a::linordered_idom) * m < n"
193  |"(l::'a::linordered_idom) < m * n") =
194  \<open>fn phi => Numeral_Simprocs.less_cancel_factor\<close>
195
196simproc_setup int_div_cancel_factor
197  ("((l::'a::euclidean_semiring_cancel) * m) div n"
198  |"(l::'a::euclidean_semiring_cancel) div (m * n)") =
199  \<open>fn phi => Numeral_Simprocs.div_cancel_factor\<close>
200
201simproc_setup int_mod_cancel_factor
202  ("((l::'a::euclidean_semiring_cancel) * m) mod n"
203  |"(l::'a::euclidean_semiring_cancel) mod (m * n)") =
204  \<open>fn phi => Numeral_Simprocs.mod_cancel_factor\<close>
205
206simproc_setup dvd_cancel_factor
207  ("((l::'a::idom) * m) dvd n"
208  |"(l::'a::idom) dvd (m * n)") =
209  \<open>fn phi => Numeral_Simprocs.dvd_cancel_factor\<close>
210
211simproc_setup divide_cancel_factor
212  ("((l::'a::field) * m) / n"
213  |"(l::'a::field) / (m * n)") =
214  \<open>fn phi => Numeral_Simprocs.divide_cancel_factor\<close>
215
216ML_file "Tools/nat_numeral_simprocs.ML"
217
218simproc_setup nat_combine_numerals
219  ("(i::nat) + j" | "Suc (i + j)") =
220  \<open>fn phi => Nat_Numeral_Simprocs.combine_numerals\<close>
221
222simproc_setup nateq_cancel_numerals
223  ("(l::nat) + m = n" | "(l::nat) = m + n" |
224   "(l::nat) * m = n" | "(l::nat) = m * n" |
225   "Suc m = n" | "m = Suc n") =
226  \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals\<close>
227
228simproc_setup natless_cancel_numerals
229  ("(l::nat) + m < n" | "(l::nat) < m + n" |
230   "(l::nat) * m < n" | "(l::nat) < m * n" |
231   "Suc m < n" | "m < Suc n") =
232  \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numerals\<close>
233
234simproc_setup natle_cancel_numerals
235  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" |
236   "(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" |
237   "Suc m \<le> n" | "m \<le> Suc n") =
238  \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numerals\<close>
239
240simproc_setup natdiff_cancel_numerals
241  ("((l::nat) + m) - n" | "(l::nat) - (m + n)" |
242   "(l::nat) * m - n" | "(l::nat) - m * n" |
243   "Suc m - n" | "m - Suc n") =
244  \<open>fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals\<close>
245
246simproc_setup nat_eq_cancel_numeral_factor
247  ("(l::nat) * m = n" | "(l::nat) = m * n") =
248  \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor\<close>
249
250simproc_setup nat_less_cancel_numeral_factor
251  ("(l::nat) * m < n" | "(l::nat) < m * n") =
252  \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor\<close>
253
254simproc_setup nat_le_cancel_numeral_factor
255  ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
256  \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor\<close>
257
258simproc_setup nat_div_cancel_numeral_factor
259  ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
260  \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor\<close>
261
262simproc_setup nat_dvd_cancel_numeral_factor
263  ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
264  \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor\<close>
265
266simproc_setup nat_eq_cancel_factor
267  ("(l::nat) * m = n" | "(l::nat) = m * n") =
268  \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_factor\<close>
269
270simproc_setup nat_less_cancel_factor
271  ("(l::nat) * m < n" | "(l::nat) < m * n") =
272  \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_factor\<close>
273
274simproc_setup nat_le_cancel_factor
275  ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
276  \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_factor\<close>
277
278simproc_setup nat_div_cancel_factor
279  ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
280  \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_factor\<close>
281
282simproc_setup nat_dvd_cancel_factor
283  ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
284  \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor\<close>
285
286declaration \<open>
287  K (Lin_Arith.add_simprocs
288      [@{simproc semiring_assoc_fold},
289       @{simproc int_combine_numerals},
290       @{simproc inteq_cancel_numerals},
291       @{simproc intless_cancel_numerals},
292       @{simproc intle_cancel_numerals},
293       @{simproc field_combine_numerals}]
294  #> Lin_Arith.add_simprocs
295      [@{simproc nat_combine_numerals},
296       @{simproc nateq_cancel_numerals},
297       @{simproc natless_cancel_numerals},
298       @{simproc natle_cancel_numerals},
299       @{simproc natdiff_cancel_numerals}])
300\<close>
301
302end
303