src/HOL/Numeral_Simprocs.thy
 author haftmann Tue Mar 31 21:54:32 2015 +0200 (2015-03-31) changeset 59867 58043346ca64 parent 59757 93d4169e07dc child 60758 d8d85a8172b5 permissions -rw-r--r--
given up separate type classes demanding `inverse 0 = 0`
```     1 (* Author: Various *)
```
```     2
```
```     3 section {* Combination and Cancellation Simprocs for Numeral Expressions *}
```
```     4
```
```     5 theory Numeral_Simprocs
```
```     6 imports Divides
```
```     7 begin
```
```     8
```
```     9 ML_file "~~/src/Provers/Arith/assoc_fold.ML"
```
```    10 ML_file "~~/src/Provers/Arith/cancel_numerals.ML"
```
```    11 ML_file "~~/src/Provers/Arith/combine_numerals.ML"
```
```    12 ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML"
```
```    13 ML_file "~~/src/Provers/Arith/extract_common_term.ML"
```
```    14
```
```    15 lemmas 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
```
```    23 declare split_div [of _ _ "numeral k", arith_split] for k
```
```    24 declare split_mod [of _ _ "numeral k", arith_split] for k
```
```    25
```
```    26 text {* For @{text combine_numerals} *}
```
```    27
```
```    28 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
```
```    29 by (simp add: add_mult_distrib)
```
```    30
```
```    31 text {* For @{text cancel_numerals} *}
```
```    32
```
```    33 lemma nat_diff_add_eq1:
```
```    34      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
```
```    35 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```    36
```
```    37 lemma nat_diff_add_eq2:
```
```    38      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
```
```    39 by (simp split add: nat_diff_split add: add_mult_distrib)
```
```    40
```
```    41 lemma nat_eq_add_iff1:
```
```    42      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
```
```    43 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    44
```
```    45 lemma nat_eq_add_iff2:
```
```    46      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
```
```    47 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    48
```
```    49 lemma nat_less_add_iff1:
```
```    50      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
```
```    51 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    52
```
```    53 lemma nat_less_add_iff2:
```
```    54      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
```
```    55 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    56
```
```    57 lemma nat_le_add_iff1:
```
```    58      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
```
```    59 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    60
```
```    61 lemma nat_le_add_iff2:
```
```    62      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
```
```    63 by (auto split add: nat_diff_split simp add: add_mult_distrib)
```
```    64
```
```    65 text {* For @{text cancel_numeral_factors} *}
```
```    66
```
```    67 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
```
```    68 by auto
```
```    69
```
```    70 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
```
```    71 by auto
```
```    72
```
```    73 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
```
```    74 by auto
```
```    75
```
```    76 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
```
```    77 by auto
```
```    78
```
```    79 lemma nat_mult_dvd_cancel_disj[simp]:
```
```    80   "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
```
```    81 by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1)
```
```    82
```
```    83 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
```
```    84 by(auto)
```
```    85
```
```    86 text {* For @{text cancel_factor} *}
```
```    87
```
```    88 lemmas nat_mult_le_cancel_disj = mult_le_cancel1
```
```    89
```
```    90 lemmas nat_mult_less_cancel_disj = mult_less_cancel1
```
```    91
```
```    92 lemma 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
```
```    97 lemma 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
```
```   102 lemma 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
```
```   106 ML_file "Tools/numeral_simprocs.ML"
```
```   107
```
```   108 simproc_setup semiring_assoc_fold
```
```   109   ("(a::'a::comm_semiring_1_cancel) * b") =
```
```   110   {* fn phi => Numeral_Simprocs.assoc_fold *}
```
```   111
```
```   112 (* TODO: see whether the type class can be generalized further *)
```
```   113 simproc_setup int_combine_numerals
```
```   114   ("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") =
```
```   115   {* fn phi => Numeral_Simprocs.combine_numerals *}
```
```   116
```
```   117 simproc_setup field_combine_numerals
```
```   118   ("(i::'a::{field,ring_char_0}) + j"
```
```   119   |"(i::'a::{field,ring_char_0}) - j") =
```
```   120   {* fn phi => Numeral_Simprocs.field_combine_numerals *}
```
```   121
```
```   122 simproc_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   {* fn phi => Numeral_Simprocs.eq_cancel_numerals *}
```
```   132
```
```   133 simproc_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   {* fn phi => Numeral_Simprocs.less_cancel_numerals *}
```
```   143
```
```   144 simproc_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   {* fn phi => Numeral_Simprocs.le_cancel_numerals *}
```
```   154
```
```   155 simproc_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   {* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *}
```
```   159
```
```   160 simproc_setup ring_less_cancel_numeral_factor
```
```   161   ("(l::'a::linordered_idom) * m < n"
```
```   162   |"(l::'a::linordered_idom) < m * n") =
```
```   163   {* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *}
```
```   164
```
```   165 simproc_setup ring_le_cancel_numeral_factor
```
```   166   ("(l::'a::linordered_idom) * m <= n"
```
```   167   |"(l::'a::linordered_idom) <= m * n") =
```
```   168   {* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *}
```
```   169
```
```   170 (* TODO: remove comm_ring_1 constraint if possible *)
```
```   171 simproc_setup int_div_cancel_numeral_factors
```
```   172   ("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n"
```
```   173   |"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") =
```
```   174   {* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *}
```
```   175
```
```   176 simproc_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   {* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *}
```
```   181
```
```   182 simproc_setup ring_eq_cancel_factor
```
```   183   ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
```
```   184   {* fn phi => Numeral_Simprocs.eq_cancel_factor *}
```
```   185
```
```   186 simproc_setup linordered_ring_le_cancel_factor
```
```   187   ("(l::'a::linordered_idom) * m <= n"
```
```   188   |"(l::'a::linordered_idom) <= m * n") =
```
```   189   {* fn phi => Numeral_Simprocs.le_cancel_factor *}
```
```   190
```
```   191 simproc_setup linordered_ring_less_cancel_factor
```
```   192   ("(l::'a::linordered_idom) * m < n"
```
```   193   |"(l::'a::linordered_idom) < m * n") =
```
```   194   {* fn phi => Numeral_Simprocs.less_cancel_factor *}
```
```   195
```
```   196 simproc_setup int_div_cancel_factor
```
```   197   ("((l::'a::semiring_div) * m) div n"
```
```   198   |"(l::'a::semiring_div) div (m * n)") =
```
```   199   {* fn phi => Numeral_Simprocs.div_cancel_factor *}
```
```   200
```
```   201 simproc_setup int_mod_cancel_factor
```
```   202   ("((l::'a::semiring_div) * m) mod n"
```
```   203   |"(l::'a::semiring_div) mod (m * n)") =
```
```   204   {* fn phi => Numeral_Simprocs.mod_cancel_factor *}
```
```   205
```
```   206 simproc_setup dvd_cancel_factor
```
```   207   ("((l::'a::idom) * m) dvd n"
```
```   208   |"(l::'a::idom) dvd (m * n)") =
```
```   209   {* fn phi => Numeral_Simprocs.dvd_cancel_factor *}
```
```   210
```
```   211 simproc_setup divide_cancel_factor
```
```   212   ("((l::'a::field) * m) / n"
```
```   213   |"(l::'a::field) / (m * n)") =
```
```   214   {* fn phi => Numeral_Simprocs.divide_cancel_factor *}
```
```   215
```
```   216 ML_file "Tools/nat_numeral_simprocs.ML"
```
```   217
```
```   218 simproc_setup nat_combine_numerals
```
```   219   ("(i::nat) + j" | "Suc (i + j)") =
```
```   220   {* fn phi => Nat_Numeral_Simprocs.combine_numerals *}
```
```   221
```
```   222 simproc_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   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *}
```
```   227
```
```   228 simproc_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   {* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *}
```
```   233
```
```   234 simproc_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   {* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *}
```
```   239
```
```   240 simproc_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   {* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *}
```
```   245
```
```   246 simproc_setup nat_eq_cancel_numeral_factor
```
```   247   ("(l::nat) * m = n" | "(l::nat) = m * n") =
```
```   248   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *}
```
```   249
```
```   250 simproc_setup nat_less_cancel_numeral_factor
```
```   251   ("(l::nat) * m < n" | "(l::nat) < m * n") =
```
```   252   {* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *}
```
```   253
```
```   254 simproc_setup nat_le_cancel_numeral_factor
```
```   255   ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
```
```   256   {* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *}
```
```   257
```
```   258 simproc_setup nat_div_cancel_numeral_factor
```
```   259   ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
```
```   260   {* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *}
```
```   261
```
```   262 simproc_setup nat_dvd_cancel_numeral_factor
```
```   263   ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
```
```   264   {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *}
```
```   265
```
```   266 simproc_setup nat_eq_cancel_factor
```
```   267   ("(l::nat) * m = n" | "(l::nat) = m * n") =
```
```   268   {* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *}
```
```   269
```
```   270 simproc_setup nat_less_cancel_factor
```
```   271   ("(l::nat) * m < n" | "(l::nat) < m * n") =
```
```   272   {* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *}
```
```   273
```
```   274 simproc_setup nat_le_cancel_factor
```
```   275   ("(l::nat) * m <= n" | "(l::nat) <= m * n") =
```
```   276   {* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *}
```
```   277
```
```   278 simproc_setup nat_div_cancel_factor
```
```   279   ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
```
```   280   {* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *}
```
```   281
```
```   282 simproc_setup nat_dvd_cancel_factor
```
```   283   ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
```
```   284   {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *}
```
```   285
```
```   286 declaration {*
```
```   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 *}
```
```   301
```
```   302 end
```