src/HOL/SMT_Examples/SMT_Examples.thy
 author kuncar Fri Dec 09 18:07:04 2011 +0100 (2011-12-09) changeset 45802 b16f976db515 parent 45393 13ab80eafd71 child 45972 deda685ba210 permissions -rw-r--r--
Quotient_Info stores only relation maps
```     1 (*  Title:      HOL/SMT_Examples/SMT_Examples.thy
```
```     2     Author:     Sascha Boehme, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Examples for the SMT binding *}
```
```     6
```
```     7 theory SMT_Examples
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
```
```    11 declare [[smt_oracle=false]]
```
```    12 declare [[smt_certificates="SMT_Examples.certs"]]
```
```    13 declare [[smt_fixed=true]]
```
```    14
```
```    15
```
```    16
```
```    17 section {* Propositional and first-order logic *}
```
```    18
```
```    19 lemma "True" by smt
```
```    20
```
```    21 lemma "p \<or> \<not>p" by smt
```
```    22
```
```    23 lemma "(p \<and> True) = p" by smt
```
```    24
```
```    25 lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt
```
```    26
```
```    27 lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)"
```
```    28   by smt
```
```    29
```
```    30 lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt
```
```    31
```
```    32 lemma "P=P=P=P=P=P=P=P=P=P" by smt
```
```    33
```
```    34 lemma
```
```    35   assumes "a | b | c | d"
```
```    36       and "e | f | (a & d)"
```
```    37       and "~(a | (c & ~c)) | b"
```
```    38       and "~(b & (x | ~x)) | c"
```
```    39       and "~(d | False) | c"
```
```    40       and "~(c | (~p & (p | (q & ~q))))"
```
```    41   shows False
```
```    42   using assms by smt
```
```    43
```
```    44 axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```    45   symm_f: "symm_f x y = symm_f y x"
```
```    46 lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f)
```
```    47
```
```    48 (*
```
```    49 Taken from ~~/src/HOL/ex/SAT_Examples.thy.
```
```    50 Translated from TPTP problem library: PUZ015-2.006.dimacs
```
```    51 *)
```
```    52 lemma
```
```    53   assumes "~x0"
```
```    54   and "~x30"
```
```    55   and "~x29"
```
```    56   and "~x59"
```
```    57   and "x1 | x31 | x0"
```
```    58   and "x2 | x32 | x1"
```
```    59   and "x3 | x33 | x2"
```
```    60   and "x4 | x34 | x3"
```
```    61   and "x35 | x4"
```
```    62   and "x5 | x36 | x30"
```
```    63   and "x6 | x37 | x5 | x31"
```
```    64   and "x7 | x38 | x6 | x32"
```
```    65   and "x8 | x39 | x7 | x33"
```
```    66   and "x9 | x40 | x8 | x34"
```
```    67   and "x41 | x9 | x35"
```
```    68   and "x10 | x42 | x36"
```
```    69   and "x11 | x43 | x10 | x37"
```
```    70   and "x12 | x44 | x11 | x38"
```
```    71   and "x13 | x45 | x12 | x39"
```
```    72   and "x14 | x46 | x13 | x40"
```
```    73   and "x47 | x14 | x41"
```
```    74   and "x15 | x48 | x42"
```
```    75   and "x16 | x49 | x15 | x43"
```
```    76   and "x17 | x50 | x16 | x44"
```
```    77   and "x18 | x51 | x17 | x45"
```
```    78   and "x19 | x52 | x18 | x46"
```
```    79   and "x53 | x19 | x47"
```
```    80   and "x20 | x54 | x48"
```
```    81   and "x21 | x55 | x20 | x49"
```
```    82   and "x22 | x56 | x21 | x50"
```
```    83   and "x23 | x57 | x22 | x51"
```
```    84   and "x24 | x58 | x23 | x52"
```
```    85   and "x59 | x24 | x53"
```
```    86   and "x25 | x54"
```
```    87   and "x26 | x25 | x55"
```
```    88   and "x27 | x26 | x56"
```
```    89   and "x28 | x27 | x57"
```
```    90   and "x29 | x28 | x58"
```
```    91   and "~x1 | ~x31"
```
```    92   and "~x1 | ~x0"
```
```    93   and "~x31 | ~x0"
```
```    94   and "~x2 | ~x32"
```
```    95   and "~x2 | ~x1"
```
```    96   and "~x32 | ~x1"
```
```    97   and "~x3 | ~x33"
```
```    98   and "~x3 | ~x2"
```
```    99   and "~x33 | ~x2"
```
```   100   and "~x4 | ~x34"
```
```   101   and "~x4 | ~x3"
```
```   102   and "~x34 | ~x3"
```
```   103   and "~x35 | ~x4"
```
```   104   and "~x5 | ~x36"
```
```   105   and "~x5 | ~x30"
```
```   106   and "~x36 | ~x30"
```
```   107   and "~x6 | ~x37"
```
```   108   and "~x6 | ~x5"
```
```   109   and "~x6 | ~x31"
```
```   110   and "~x37 | ~x5"
```
```   111   and "~x37 | ~x31"
```
```   112   and "~x5 | ~x31"
```
```   113   and "~x7 | ~x38"
```
```   114   and "~x7 | ~x6"
```
```   115   and "~x7 | ~x32"
```
```   116   and "~x38 | ~x6"
```
```   117   and "~x38 | ~x32"
```
```   118   and "~x6 | ~x32"
```
```   119   and "~x8 | ~x39"
```
```   120   and "~x8 | ~x7"
```
```   121   and "~x8 | ~x33"
```
```   122   and "~x39 | ~x7"
```
```   123   and "~x39 | ~x33"
```
```   124   and "~x7 | ~x33"
```
```   125   and "~x9 | ~x40"
```
```   126   and "~x9 | ~x8"
```
```   127   and "~x9 | ~x34"
```
```   128   and "~x40 | ~x8"
```
```   129   and "~x40 | ~x34"
```
```   130   and "~x8 | ~x34"
```
```   131   and "~x41 | ~x9"
```
```   132   and "~x41 | ~x35"
```
```   133   and "~x9 | ~x35"
```
```   134   and "~x10 | ~x42"
```
```   135   and "~x10 | ~x36"
```
```   136   and "~x42 | ~x36"
```
```   137   and "~x11 | ~x43"
```
```   138   and "~x11 | ~x10"
```
```   139   and "~x11 | ~x37"
```
```   140   and "~x43 | ~x10"
```
```   141   and "~x43 | ~x37"
```
```   142   and "~x10 | ~x37"
```
```   143   and "~x12 | ~x44"
```
```   144   and "~x12 | ~x11"
```
```   145   and "~x12 | ~x38"
```
```   146   and "~x44 | ~x11"
```
```   147   and "~x44 | ~x38"
```
```   148   and "~x11 | ~x38"
```
```   149   and "~x13 | ~x45"
```
```   150   and "~x13 | ~x12"
```
```   151   and "~x13 | ~x39"
```
```   152   and "~x45 | ~x12"
```
```   153   and "~x45 | ~x39"
```
```   154   and "~x12 | ~x39"
```
```   155   and "~x14 | ~x46"
```
```   156   and "~x14 | ~x13"
```
```   157   and "~x14 | ~x40"
```
```   158   and "~x46 | ~x13"
```
```   159   and "~x46 | ~x40"
```
```   160   and "~x13 | ~x40"
```
```   161   and "~x47 | ~x14"
```
```   162   and "~x47 | ~x41"
```
```   163   and "~x14 | ~x41"
```
```   164   and "~x15 | ~x48"
```
```   165   and "~x15 | ~x42"
```
```   166   and "~x48 | ~x42"
```
```   167   and "~x16 | ~x49"
```
```   168   and "~x16 | ~x15"
```
```   169   and "~x16 | ~x43"
```
```   170   and "~x49 | ~x15"
```
```   171   and "~x49 | ~x43"
```
```   172   and "~x15 | ~x43"
```
```   173   and "~x17 | ~x50"
```
```   174   and "~x17 | ~x16"
```
```   175   and "~x17 | ~x44"
```
```   176   and "~x50 | ~x16"
```
```   177   and "~x50 | ~x44"
```
```   178   and "~x16 | ~x44"
```
```   179   and "~x18 | ~x51"
```
```   180   and "~x18 | ~x17"
```
```   181   and "~x18 | ~x45"
```
```   182   and "~x51 | ~x17"
```
```   183   and "~x51 | ~x45"
```
```   184   and "~x17 | ~x45"
```
```   185   and "~x19 | ~x52"
```
```   186   and "~x19 | ~x18"
```
```   187   and "~x19 | ~x46"
```
```   188   and "~x52 | ~x18"
```
```   189   and "~x52 | ~x46"
```
```   190   and "~x18 | ~x46"
```
```   191   and "~x53 | ~x19"
```
```   192   and "~x53 | ~x47"
```
```   193   and "~x19 | ~x47"
```
```   194   and "~x20 | ~x54"
```
```   195   and "~x20 | ~x48"
```
```   196   and "~x54 | ~x48"
```
```   197   and "~x21 | ~x55"
```
```   198   and "~x21 | ~x20"
```
```   199   and "~x21 | ~x49"
```
```   200   and "~x55 | ~x20"
```
```   201   and "~x55 | ~x49"
```
```   202   and "~x20 | ~x49"
```
```   203   and "~x22 | ~x56"
```
```   204   and "~x22 | ~x21"
```
```   205   and "~x22 | ~x50"
```
```   206   and "~x56 | ~x21"
```
```   207   and "~x56 | ~x50"
```
```   208   and "~x21 | ~x50"
```
```   209   and "~x23 | ~x57"
```
```   210   and "~x23 | ~x22"
```
```   211   and "~x23 | ~x51"
```
```   212   and "~x57 | ~x22"
```
```   213   and "~x57 | ~x51"
```
```   214   and "~x22 | ~x51"
```
```   215   and "~x24 | ~x58"
```
```   216   and "~x24 | ~x23"
```
```   217   and "~x24 | ~x52"
```
```   218   and "~x58 | ~x23"
```
```   219   and "~x58 | ~x52"
```
```   220   and "~x23 | ~x52"
```
```   221   and "~x59 | ~x24"
```
```   222   and "~x59 | ~x53"
```
```   223   and "~x24 | ~x53"
```
```   224   and "~x25 | ~x54"
```
```   225   and "~x26 | ~x25"
```
```   226   and "~x26 | ~x55"
```
```   227   and "~x25 | ~x55"
```
```   228   and "~x27 | ~x26"
```
```   229   and "~x27 | ~x56"
```
```   230   and "~x26 | ~x56"
```
```   231   and "~x28 | ~x27"
```
```   232   and "~x28 | ~x57"
```
```   233   and "~x27 | ~x57"
```
```   234   and "~x29 | ~x28"
```
```   235   and "~x29 | ~x58"
```
```   236   and "~x28 | ~x58"
```
```   237   shows False
```
```   238   using assms by smt
```
```   239
```
```   240 lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
```
```   241   by smt
```
```   242
```
```   243 lemma
```
```   244   assumes "(\<forall>x y. P x y = x)"
```
```   245   shows "(\<exists>y. P x y) = P x c"
```
```   246   using assms by smt
```
```   247
```
```   248 lemma
```
```   249   assumes "(\<forall>x y. P x y = x)"
```
```   250   and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)"
```
```   251   shows "(EX y. P x y) = P x c"
```
```   252   using assms by smt
```
```   253
```
```   254 lemma
```
```   255   assumes "if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)"
```
```   256   shows "P x \<longrightarrow> P y"
```
```   257   using assms by smt
```
```   258
```
```   259
```
```   260 section {* Arithmetic *}
```
```   261
```
```   262 subsection {* Linear arithmetic over integers and reals *}
```
```   263
```
```   264 lemma "(3::int) = 3" by smt
```
```   265
```
```   266 lemma "(3::real) = 3" by smt
```
```   267
```
```   268 lemma "(3 :: int) + 1 = 4" by smt
```
```   269
```
```   270 lemma "x + (y + z) = y + (z + (x::int))" by smt
```
```   271
```
```   272 lemma "max (3::int) 8 > 5" by smt
```
```   273
```
```   274 lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt
```
```   275
```
```   276 lemma "P ((2::int) < 3) = P True" by smt
```
```   277
```
```   278 lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt
```
```   279
```
```   280 lemma
```
```   281   assumes "x \<ge> (3::int)" and "y = x + 4"
```
```   282   shows "y - x > 0"
```
```   283   using assms by smt
```
```   284
```
```   285 lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt
```
```   286
```
```   287 lemma
```
```   288   fixes x :: real
```
```   289   assumes "3 * x + 7 * a < 4" and "3 < 2 * x"
```
```   290   shows "a < 0"
```
```   291   using assms by smt
```
```   292
```
```   293 lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
```
```   294
```
```   295 lemma "distinct [x < (3::int), 3 \<le> x]" by smt
```
```   296
```
```   297 lemma
```
```   298   assumes "a > (0::int)"
```
```   299   shows "distinct [a, a * 2, a - a]"
```
```   300   using assms by smt
```
```   301
```
```   302 lemma "
```
```   303   (n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
```
```   304   (n = n' & n' < m) | (n = m & m < n') |
```
```   305   (n' < m & m < n) | (n' < m & m = n) |
```
```   306   (n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) |
```
```   307   (m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) |
```
```   308   (m = n & n < n') | (m = n' & n' < n) |
```
```   309   (n' = m & m = (n::int))"
```
```   310   by smt
```
```   311
```
```   312 text{*
```
```   313 The following example was taken from HOL/ex/PresburgerEx.thy, where it says:
```
```   314
```
```   315   This following theorem proves that all solutions to the
```
```   316   recurrence relation \$x_{i+2} = |x_{i+1}| - x_i\$ are periodic with
```
```   317   period 9.  The example was brought to our attention by John
```
```   318   Harrison. It does does not require Presburger arithmetic but merely
```
```   319   quantifier-free linear arithmetic and holds for the rationals as well.
```
```   320
```
```   321   Warning: it takes (in 2006) over 4.2 minutes!
```
```   322
```
```   323 There, it is proved by "arith". SMT is able to prove this within a fraction
```
```   324 of one second. With proof reconstruction, it takes about 13 seconds on a Core2
```
```   325 processor.
```
```   326 *}
```
```   327
```
```   328 lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3;
```
```   329          x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6;
```
```   330          x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk>
```
```   331  \<Longrightarrow> x1 = x10 & x2 = (x11::int)"
```
```   332   by smt
```
```   333
```
```   334
```
```   335 lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt
```
```   336
```
```   337 lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)" by smt
```
```   338
```
```   339 lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" by smt
```
```   340
```
```   341 lemma
```
```   342   assumes "x \<noteq> (0::real)"
```
```   343   shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x"
```
```   344   using assms by smt
```
```   345
```
```   346 lemma
```
```   347   assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"
```
```   348   shows "n mod 2 = 1 & m mod 2 = (1::int)"
```
```   349   using assms by smt
```
```   350
```
```   351
```
```   352
```
```   353 subsection {* Linear arithmetic with quantifiers *}
```
```   354
```
```   355 lemma "~ (\<exists>x::int. False)" by smt
```
```   356
```
```   357 lemma "~ (\<exists>x::real. False)" by smt
```
```   358
```
```   359 lemma "\<exists>x::int. 0 < x"
```
```   360   using [[smt_oracle=true]] (* no Z3 proof *)
```
```   361   by smt
```
```   362
```
```   363 lemma "\<exists>x::real. 0 < x"
```
```   364   using [[smt_oracle=true]] (* no Z3 proof *)
```
```   365   by smt
```
```   366
```
```   367 lemma "\<forall>x::int. \<exists>y. y > x"
```
```   368   using [[smt_oracle=true]] (* no Z3 proof *)
```
```   369   by smt
```
```   370
```
```   371 lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt
```
```   372
```
```   373 lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt
```
```   374
```
```   375 lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt
```
```   376
```
```   377 lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt
```
```   378
```
```   379 lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt
```
```   380
```
```   381 lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt
```
```   382
```
```   383 lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt
```
```   384
```
```   385 lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt
```
```   386
```
```   387 lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt
```
```   388
```
```   389 lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt
```
```   390
```
```   391 lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt
```
```   392
```
```   393 lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt
```
```   394
```
```   395 lemma "\<forall>x::int. SMT.trigger [[SMT.pat x]] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
```
```   396
```
```   397 lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by smt
```
```   398
```
```   399
```
```   400 subsection {* Non-linear arithmetic over integers and reals *}
```
```   401
```
```   402 lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
```
```   403   using [[smt_oracle=true]]
```
```   404   by smt
```
```   405
```
```   406 lemma  "(a::int) * (x + 1 + y) = a * x + a * (y + 1)"
```
```   407   by smt
```
```   408
```
```   409 lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)"
```
```   410   by smt
```
```   411
```
```   412 lemma
```
```   413   "(U::int) + (1 + p) * (b + e) + p * d =
```
```   414    U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)"
```
```   415   by smt
```
```   416
```
```   417 lemma [z3_rule]:
```
```   418   fixes x :: "int"
```
```   419   assumes "x * y \<le> 0" and "\<not> y \<le> 0" and "\<not> x \<le> 0"
```
```   420   shows False
```
```   421   using assms by (metis mult_le_0_iff)
```
```   422
```
```   423 lemma "x * y \<le> (0 :: int) \<Longrightarrow> x \<le> 0 \<or> y \<le> 0" by smt
```
```   424
```
```   425
```
```   426
```
```   427 subsection {* Linear arithmetic for natural numbers *}
```
```   428
```
```   429 lemma "2 * (x::nat) ~= 1" by smt
```
```   430
```
```   431 lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt
```
```   432
```
```   433 lemma "let x = (1::nat) + y in x - y > 0 * x" by smt
```
```   434
```
```   435 lemma
```
```   436   "let x = (1::nat) + y in
```
```   437    let P = (if x > 0 then True else False) in
```
```   438    False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"
```
```   439   by smt
```
```   440
```
```   441 lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt
```
```   442
```
```   443 lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt
```
```   444
```
```   445 definition prime_nat :: "nat \<Rightarrow> bool" where
```
```   446   "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
```
```   447 lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def)
```
```   448
```
```   449
```
```   450 section {* Pairs *}
```
```   451
```
```   452 lemma "fst (x, y) = a \<Longrightarrow> x = a"
```
```   453   using fst_conv
```
```   454   by smt
```
```   455
```
```   456 lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
```
```   457   using fst_conv snd_conv
```
```   458   by smt
```
```   459
```
```   460
```
```   461 section {* Higher-order problems and recursion *}
```
```   462
```
```   463 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i"
```
```   464   using fun_upd_same fun_upd_apply
```
```   465   by smt
```
```   466
```
```   467 lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
```
```   468   by smt
```
```   469
```
```   470 lemma "id 3 = 3 \<and> id True = True" by (smt id_def)
```
```   471
```
```   472 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i"
```
```   473   using fun_upd_same fun_upd_apply
```
```   474   by smt
```
```   475
```
```   476 lemma
```
```   477   "f (\<exists>x. g x) \<Longrightarrow> True"
```
```   478   "f (\<forall>x. g x) \<Longrightarrow> True"
```
```   479   by smt+
```
```   480
```
```   481 lemma True using let_rsp by smt
```
```   482
```
```   483 lemma "le = op \<le> \<Longrightarrow> le (3::int) 42" by smt
```
```   484
```
```   485 lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps)
```
```   486
```
```   487
```
```   488 lemma "(ALL x. P x) | ~ All P" by smt
```
```   489
```
```   490 fun dec_10 :: "nat \<Rightarrow> nat" where
```
```   491   "dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
```
```   492 lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
```
```   493
```
```   494
```
```   495 axiomatization
```
```   496   eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int"
```
```   497   where
```
```   498   eval_dioph_mod:
```
```   499   "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"
```
```   500   and
```
```   501   eval_dioph_div_mult:
```
```   502   "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n +
```
```   503    eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"
```
```   504 lemma
```
```   505   "(eval_dioph ks xs = l) =
```
```   506    (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and>
```
```   507     eval_dioph ks (map (\<lambda>x. x div 2) xs) =
```
```   508       (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"
```
```   509   using [[smt_oracle=true]] (*FIXME*)
```
```   510   by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
```
```   511
```
```   512
```
```   513 context complete_lattice
```
```   514 begin
```
```   515
```
```   516 lemma
```
```   517   assumes "Sup { a | i::bool . True } \<le> Sup { b | i::bool . True }"
```
```   518   and     "Sup { b | i::bool . True } \<le> Sup { a | i::bool . True }"
```
```   519   shows   "Sup { a | i::bool . True } \<le> Sup { a | i::bool . True }"
```
```   520   using assms by (smt order_trans)
```
```   521
```
```   522 end
```
```   523
```
```   524
```
```   525
```
```   526 section {* Monomorphization examples *}
```
```   527
```
```   528 definition Pred :: "'a \<Rightarrow> bool" where "Pred x = True"
```
```   529 lemma poly_Pred: "Pred x \<and> (Pred [x] \<or> \<not>Pred[x])" by (simp add: Pred_def)
```
```   530 lemma "Pred (1::int)" by (smt poly_Pred)
```
```   531
```
```   532 axiomatization g :: "'a \<Rightarrow> nat"
```
```   533 axiomatization where
```
```   534   g1: "g (Some x) = g [x]" and
```
```   535   g2: "g None = g []" and
```
```   536   g3: "g xs = length xs"
```
```   537 lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size)
```
```   538
```
```   539 end
```