src/HOL/Integ/Presburger.thy
author berghofe
Tue Mar 25 09:47:05 2003 +0100 (2003-03-25)
changeset 13876 68f4ed8311ac
child 14139 ca3dd7ed5ac5
permissions -rw-r--r--
New decision procedure for Presburger arithmetic.
     1 (*  Title:      HOL/Integ/Presburger.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 
     6 File containing necessary theorems for the proof
     7 generation for Cooper Algorithm  
     8 *)
     9 
    10 theory Presburger = NatSimprocs
    11 files
    12   ("cooper_dec.ML")
    13   ("cooper_proof.ML")
    14   ("qelim.ML")
    15   ("presburger.ML"):
    16 
    17 (* Theorem for unitifying the coeffitients of x in an existential formula*)
    18 
    19 theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
    20   apply (rule iffI)
    21   apply (erule exE)
    22   apply (rule_tac x = "l * x" in exI)
    23   apply simp
    24   apply (erule exE)
    25   apply (erule conjE)
    26   apply (erule dvdE)
    27   apply (rule_tac x = k in exI)
    28   apply simp
    29   done
    30 
    31 lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
    32 apply(unfold dvd_def)
    33 apply(rule iffI)
    34 apply(clarsimp)
    35 apply(rename_tac k)
    36 apply(rule_tac x = "-k" in exI)
    37 apply simp
    38 apply(clarsimp)
    39 apply(rename_tac k)
    40 apply(rule_tac x = "-k" in exI)
    41 apply simp
    42 done
    43 
    44 lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
    45 apply(unfold dvd_def)
    46 apply(rule iffI)
    47 apply(clarsimp)
    48 apply(rule_tac x = "-k" in exI)
    49 apply simp
    50 apply(clarsimp)
    51 apply(rule_tac x = "-k" in exI)
    52 apply simp
    53 done
    54 
    55 
    56 
    57 (*Theorems for the combination of proofs of the equality of P and P_m for integers x less than some integer z.*)
    58 
    59 theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    60   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    61   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    62   apply (erule exE)+
    63   apply (rule_tac x = "min z1 z2" in exI)
    64   apply simp
    65   done
    66 
    67 
    68 theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    69   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    70   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    71 
    72   apply (erule exE)+
    73   apply (rule_tac x = "min z1 z2" in exI)
    74   apply simp
    75   done
    76 
    77 
    78 (*Theorems for the combination of proofs of the equality of P and P_m for integers x greather than some integer z.*)
    79 
    80 theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    81   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    82   \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    83   apply (erule exE)+
    84   apply (rule_tac x = "max z1 z2" in exI)
    85   apply simp
    86   done
    87 
    88 
    89 theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    90   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    91   \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    92   apply (erule exE)+
    93   apply (rule_tac x = "max z1 z2" in exI)
    94   apply simp
    95   done
    96 (*=============================================================================*)
    97 (*Theorems for the combination of proofs of the modulo D property for P
    98 pluusinfinity*)
    99 (* FIXME : This is THE SAME theorem as for the minusinf version, but with +k.. instead of -k.. In the future replace these both with only one*)
   100 
   101 theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   102   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   103   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
   104   by simp
   105 
   106 
   107 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   108   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   109   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
   110   by simp
   111 
   112 (*=============================================================================*)
   113 (*This is one of the cases where the simplifed formula is prooved to habe some property
   114 (in relation to P_m) but we need to proove the property for the original formula (P_m)*)
   115 (*FIXME : This is exaclty the same thm as for minusinf.*)
   116 lemma pinf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
   117 by blast
   118 
   119 
   120 
   121 (*=============================================================================*)
   122 (*Theorems for the combination of proofs of the modulo D property for P
   123 minusinfinity*)
   124 
   125 theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   126   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   127   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
   128   by simp
   129 
   130 
   131 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   132   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   133   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
   134   by simp
   135 
   136 (*=============================================================================*)
   137 (*This is one of the cases where the simplifed formula is prooved to habe some property
   138 (in relation to P_m) but we need to proove the property for the original formula (P_m)*)
   139 
   140 lemma minf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
   141 by blast
   142 
   143 (*=============================================================================*)
   144 
   145 (*theorem needed for prooving at runtime divide properties using the arithmetic tatic
   146 (who knows only about modulo = 0)*)
   147 
   148 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
   149 by(simp add:dvd_def zmod_eq_0_iff)
   150 
   151 (*=============================================================================*)
   152 
   153 
   154 
   155 (*Theorems used for the combination of proof for the backwards direction of cooper's
   156 theorem. they rely exclusively on Predicate calculus.*)
   157 
   158 lemma not_ast_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P1(x) --> P1(x + d))
   159 ==>
   160 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   161 ==>
   162 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<or> P2(x)) --> (P1(x + d) \<or> P2(x + d))) "
   163 by blast
   164 
   165 
   166 
   167 lemma not_ast_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a- j)) --> P1(x) --> P1(x + d))
   168 ==>
   169 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   170 ==>
   171 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<and> P2(x)) --> (P1(x + d)
   172 \<and> P2(x + d))) "
   173 by blast
   174 
   175 lemma not_ast_p_Q_elim: "
   176 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
   177 ==> ( P = Q )
   178 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
   179 by blast
   180 (*=============================================================================*)
   181 
   182 
   183 (*Theorems used for the combination of proof for the backwards direction of cooper's
   184 theorem. they rely exclusively on Predicate calculus.*)
   185 
   186 lemma not_bst_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
   187 ==>
   188 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   189 ==>
   190 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<or> P2(x)) --> (P1(x - d)
   191 \<or> P2(x-d))) "
   192 by blast
   193 
   194 
   195 
   196 lemma not_bst_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
   197 ==>
   198 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   199 ==>
   200 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<and> P2(x)) --> (P1(x - d)
   201 \<and> P2(x-d))) "
   202 by blast
   203 
   204 lemma not_bst_p_Q_elim: "
   205 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) 
   206 ==> ( P = Q )
   207 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
   208 by blast
   209 (*=============================================================================*)
   210 (*The Theorem for the second proof step- about bset. it is trivial too. *)
   211 lemma bst_thm: " (EX (j::int) : {1..d}. EX (b::int) : B. P (b+j) )--> (EX x::int. P (x)) "
   212 by blast
   213 
   214 (*The Theorem for the second proof step- about aset. it is trivial too. *)
   215 lemma ast_thm: " (EX (j::int) : {1..d}. EX (a::int) : A. P (a - j) )--> (EX x::int. P (x)) "
   216 by blast
   217 
   218 
   219 (*=============================================================================*)
   220 (*This is the first direction of cooper's theorem*)
   221 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
   222 by blast
   223 
   224 (*=============================================================================*)
   225 (*The full cooper's theoorem in its equivalence Form- Given the premisses it is trivial
   226 too, it relies exclusively on prediacte calculus.*)
   227 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
   228 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
   229 by blast
   230 
   231 (*=============================================================================*)
   232 (*Some of the atomic theorems generated each time the atom does not depend on x, they
   233 are trivial.*)
   234 
   235 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
   236 by blast
   237 
   238 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
   239 by blast
   240 
   241 lemma not_bst_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> fm --> fm"
   242 by blast
   243 
   244 
   245 
   246 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
   247 by blast
   248 
   249 (* The next 2 thms are the same as the minusinf version*)
   250 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
   251 by blast
   252 
   253 lemma not_ast_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> fm --> fm"
   254 by blast
   255 
   256 
   257 (* Theorems to be deleted from simpset when proving simplified formulaes*)
   258 lemma P_eqtrue: "(P=True) = P"
   259   by rules
   260 
   261 lemma P_eqfalse: "(P=False) = (~P)"
   262   by rules
   263 
   264 (*=============================================================================*)
   265 
   266 (*Theorems for the generation of the bachwards direction of cooper's theorem*)
   267 (*These are the 6 interesting atomic cases which have to be proved relying on the
   268 properties of B-set ant the arithmetic and contradiction proofs*)
   269 
   270 lemma not_bst_p_lt: "0 < (d::int) ==>
   271  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ( 0 < -x + a) --> (0 < -(x - d) + a )"
   272 by arith
   273 
   274 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   275  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 < (x) + a) --> ( 0 < (x - d) + a)"
   276 apply clarsimp
   277 apply(rule ccontr)
   278 apply(drule_tac x = "x+a" in bspec)
   279 apply(simp add:atLeastAtMost_iff)
   280 apply(drule_tac x = "-a" in bspec)
   281 apply assumption
   282 apply(simp)
   283 done
   284 
   285 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
   286  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 = x + a) --> (0 = (x - d) + a )"
   287 apply clarsimp
   288 apply(subgoal_tac "x = -a")
   289  prefer 2 apply arith
   290 apply(drule_tac x = "1" in bspec)
   291 apply(simp add:atLeastAtMost_iff)
   292 apply(drule_tac x = "-a- 1" in bspec)
   293 apply assumption
   294 apply(simp)
   295 done
   296 
   297 
   298 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   299  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)"
   300 apply clarsimp
   301 apply(subgoal_tac "x = -a+d")
   302  prefer 2 apply arith
   303 apply(drule_tac x = "d" in bspec)
   304 apply(simp add:atLeastAtMost_iff)
   305 apply(drule_tac x = "-a" in bspec)
   306 apply assumption
   307 apply(simp)
   308 done
   309 
   310 
   311 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
   312  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )"
   313 apply(clarsimp simp add:dvd_def)
   314 apply(rename_tac m)
   315 apply(rule_tac x = "m - k" in exI)
   316 apply(simp add:int_distrib)
   317 done
   318 
   319 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
   320  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))"
   321 apply(clarsimp simp add:dvd_def)
   322 apply(rename_tac m)
   323 apply(erule_tac x = "m + k" in allE)
   324 apply(simp add:int_distrib)
   325 done
   326 
   327 
   328 
   329 (*Theorems for the generation of the bachwards direction of cooper's theorem*)
   330 (*These are the 6 interesting atomic cases which have to be proved relying on the
   331 properties of A-set ant the arithmetic and contradiction proofs*)
   332 
   333 lemma not_ast_p_gt: "0 < (d::int) ==>
   334  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ( 0 < x + t) --> (0 < (x + d) + t )"
   335 by arith
   336 
   337 
   338 lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
   339  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)"
   340   apply clarsimp
   341   apply (rule ccontr)
   342   apply (drule_tac x = "t-x" in bspec)
   343   apply simp
   344   apply (drule_tac x = "t" in bspec)
   345   apply assumption
   346   apply simp
   347   done
   348 
   349 lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
   350  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 = x + t) --> (0 = (x + d) + t )"
   351   apply clarsimp
   352   apply (drule_tac x="1" in bspec)
   353   apply simp
   354   apply (drule_tac x="- t + 1" in bspec)
   355   apply assumption
   356   apply(subgoal_tac "x = -t")
   357   prefer 2 apply arith
   358   apply simp
   359   done
   360 
   361 lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
   362  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(0 = x + t) --> ~(0 = (x + d) + t)"
   363   apply clarsimp
   364   apply (subgoal_tac "x = -t-d")
   365   prefer 2 apply arith
   366   apply (drule_tac x = "d" in bspec)
   367   apply simp
   368   apply (drule_tac x = "-t" in bspec)
   369   apply assumption
   370   apply simp
   371   done
   372 
   373 lemma not_ast_p_dvd: "(d1::int) dvd d ==>
   374  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> d1 dvd (x + t) --> d1 dvd ((x + d) + t )"
   375   apply(clarsimp simp add:dvd_def)
   376   apply(rename_tac m)
   377   apply(rule_tac x = "m + k" in exI)
   378   apply(simp add:int_distrib)
   379   done
   380 
   381 lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
   382  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(d1 dvd (x + t)) --> ~(d1 dvd ((x + d) + t ))"
   383   apply(clarsimp simp add:dvd_def)
   384   apply(rename_tac m)
   385   apply(erule_tac x = "m - k" in allE)
   386   apply(simp add:int_distrib)
   387   done
   388 
   389 
   390 
   391 (*=============================================================================*)
   392 (*These are the atomic cases for the proof generation for the modulo D property for P
   393 plusinfinity*)
   394 (*They are fully based on arithmetics*)
   395 
   396 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
   397  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
   398   apply(clarsimp simp add:dvd_def)
   399   apply(rule iffI)
   400   apply(clarsimp)
   401   apply(rename_tac n m)
   402   apply(rule_tac x = "m + n*k" in exI)
   403   apply(simp add:int_distrib)
   404   apply(clarsimp)
   405   apply(rename_tac n m)
   406   apply(rule_tac x = "m - n*k" in exI)
   407   apply(simp add:int_distrib zmult_ac)
   408   done
   409 
   410 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
   411  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
   412   apply(clarsimp simp add:dvd_def)
   413   apply(rule iffI)
   414   apply(clarsimp)
   415   apply(rename_tac n m)
   416   apply(erule_tac x = "m - n*k" in allE)
   417   apply(simp add:int_distrib zmult_ac)
   418   apply(clarsimp)
   419   apply(rename_tac n m)
   420   apply(erule_tac x = "m + n*k" in allE)
   421   apply(simp add:int_distrib zmult_ac)
   422   done
   423 
   424 (*=============================================================================*)
   425 (*These are the atomic cases for the proof generation for the equivalence of P and P
   426 plusinfinity for integers x greather than some integer z.*)
   427 (*They are fully based on arithmetics*)
   428 
   429 lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
   430   apply(rule_tac x = "-t" in exI)
   431   apply simp
   432   done
   433 
   434 lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
   435   apply(rule_tac x = "-t" in exI)
   436   apply simp
   437   done
   438 
   439 lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
   440   apply(rule_tac x = "-t" in exI)
   441   apply simp
   442   done
   443 
   444 lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
   445   apply(rule_tac x = "t" in exI)
   446   apply simp
   447   done
   448 
   449 lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
   450 by simp
   451 
   452 lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   453 by simp
   454 
   455 
   456 
   457 
   458 (*=============================================================================*)
   459 (*These are the atomic cases for the proof generation for the modulo D property for P
   460 minusinfinity*)
   461 (*They are fully based on arithmetics*)
   462 
   463 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
   464  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
   465 apply(clarsimp simp add:dvd_def)
   466 apply(rule iffI)
   467 apply(clarsimp)
   468 apply(rename_tac n m)
   469 apply(rule_tac x = "m - n*k" in exI)
   470 apply(simp add:int_distrib)
   471 apply(clarsimp)
   472 apply(rename_tac n m)
   473 apply(rule_tac x = "m + n*k" in exI)
   474 apply(simp add:int_distrib zmult_ac)
   475 done
   476 
   477 
   478 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
   479  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
   480 apply(clarsimp simp add:dvd_def)
   481 apply(rule iffI)
   482 apply(clarsimp)
   483 apply(rename_tac n m)
   484 apply(erule_tac x = "m + n*k" in allE)
   485 apply(simp add:int_distrib zmult_ac)
   486 apply(clarsimp)
   487 apply(rename_tac n m)
   488 apply(erule_tac x = "m - n*k" in allE)
   489 apply(simp add:int_distrib zmult_ac)
   490 done
   491 
   492 
   493 (*=============================================================================*)
   494 (*These are the atomic cases for the proof generation for the equivalence of P and P
   495 minusinfinity for integers x less than some integer z.*)
   496 (*They are fully based on arithmetics*)
   497 
   498 lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
   499 apply(rule_tac x = "-t" in exI)
   500 apply simp
   501 done
   502 
   503 lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
   504 apply(rule_tac x = "-t" in exI)
   505 apply simp
   506 done
   507 
   508 lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
   509 apply(rule_tac x = "-t" in exI)
   510 apply simp
   511 done
   512 
   513 
   514 lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
   515 apply(rule_tac x = "t" in exI)
   516 apply simp
   517 done
   518 
   519 lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
   520 by simp
   521 
   522 lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   523 by simp
   524 
   525 
   526 (*=============================================================================*)
   527 (*This Theorem combines whithnesses about P minusinfinity to schow one component of the
   528 equivalence proof for cooper's theorem*)
   529 
   530 (* FIXME: remove once they are part of the distribution *)
   531 theorem int_ge_induct[consumes 1,case_names base step]:
   532   assumes ge: "k \<le> (i::int)" and
   533         base: "P(k)" and
   534         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   535   shows "P i"
   536 proof -
   537   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
   538     proof (induct n)
   539       case 0
   540       hence "i = k" by arith
   541       thus "P i" using base by simp
   542     next
   543       case (Suc n)
   544       hence "n = nat((i - 1) - k)" by arith
   545       moreover
   546       have ki1: "k \<le> i - 1" using Suc.prems by arith
   547       ultimately
   548       have "P(i - 1)" by(rule Suc.hyps)
   549       from step[OF ki1 this] show ?case by simp
   550     qed
   551   }
   552   from this ge show ?thesis by fast
   553 qed
   554 
   555 theorem int_gr_induct[consumes 1,case_names base step]:
   556   assumes gr: "k < (i::int)" and
   557         base: "P(k+1)" and
   558         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   559   shows "P i"
   560 apply(rule int_ge_induct[of "k + 1"])
   561   using gr apply arith
   562  apply(rule base)
   563 apply(rule step)
   564  apply simp+
   565 done
   566 
   567 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   568 apply(induct rule: int_gr_induct)
   569  apply simp
   570  apply arith
   571 apply (simp add:int_distrib)
   572 apply arith
   573 done
   574 
   575 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   576 apply(induct rule: int_gr_induct)
   577  apply simp
   578  apply arith
   579 apply (simp add:int_distrib)
   580 apply arith
   581 done
   582 
   583 lemma  minusinfinity:
   584   assumes "0 < d" and
   585     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
   586     ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   587   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   588 proof
   589   assume eP1: "EX x. P1 x"
   590   then obtain x where P1: "P1 x" ..
   591   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   592   let ?w = "x - (abs(x-z)+1) * d"
   593   show "EX x. P x"
   594   proof
   595     have w: "?w < z" by(rule decr_lemma)
   596     have "P1 x = P1 ?w" using P1eqP1 by blast
   597     also have "\<dots> = P(?w)" using w P1eqP by blast
   598     finally show "P ?w" using P1 by blast
   599   qed
   600 qed
   601 
   602 (*=============================================================================*)
   603 (*This Theorem combines whithnesses about P minusinfinity to schow one component of the
   604 equivalence proof for cooper's theorem*)
   605 
   606 lemma plusinfinity:
   607   assumes "0 < d" and
   608     P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
   609     ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
   610   shows "(EX x::int. P1 x) --> (EX x::int. P x)"
   611 proof
   612   assume eP1: "EX x. P1 x"
   613   then obtain x where P1: "P1 x" ..
   614   from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
   615   let ?w = "x + (abs(x-z)+1) * d"
   616   show "EX x. P x"
   617   proof
   618     have w: "z < ?w" by(rule incr_lemma)
   619     have "P1 x = P1 ?w" using P1eqP1 by blast
   620     also have "\<dots> = P(?w)" using w P1eqP by blast
   621     finally show "P ?w" using P1 by blast
   622   qed
   623 qed
   624  
   625 
   626 
   627 (*=============================================================================*)
   628 (*Theorem for periodic function on discrete sets*)
   629 
   630 lemma minf_vee:
   631   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   632   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   633   (is "?LHS = ?RHS")
   634 proof
   635   assume ?LHS
   636   then obtain x where P: "P x" ..
   637   have "x mod d = x - (x div d)*d"
   638     by(simp add:zmod_zdiv_equality zmult_ac eq_zdiff_eq)
   639   hence Pmod: "P x = P(x mod d)" using modd by simp
   640   show ?RHS
   641   proof (cases)
   642     assume "x mod d = 0"
   643     hence "P 0" using P Pmod by simp
   644     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   645     ultimately have "P d" by simp
   646     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   647     ultimately show ?RHS ..
   648   next
   649     assume not0: "x mod d \<noteq> 0"
   650     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   651     moreover have "x mod d : {1..d}"
   652     proof -
   653       have "0 \<le> x mod d" by(rule pos_mod_sign)
   654       moreover have "x mod d < d" by(rule pos_mod_bound)
   655       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   656     qed
   657     ultimately show ?RHS ..
   658   qed
   659 next
   660   assume ?RHS thus ?LHS by blast
   661 qed
   662 
   663 (*=============================================================================*)
   664 (*Theorem for periodic function on discrete sets*)
   665 lemma pinf_vee:
   666   assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
   667   shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
   668   (is "?LHS = ?RHS")
   669 proof
   670   assume ?LHS
   671   then obtain x where P: "P x" ..
   672   have "x mod d = x + (-(x div d))*d"
   673     by(simp add:zmod_zdiv_equality zmult_ac eq_zdiff_eq)
   674   hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
   675   show ?RHS
   676   proof (cases)
   677     assume "x mod d = 0"
   678     hence "P 0" using P Pmod by simp
   679     moreover have "P 0 = P(0 + 1*d)" using modd by blast
   680     ultimately have "P d" by simp
   681     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   682     ultimately show ?RHS ..
   683   next
   684     assume not0: "x mod d \<noteq> 0"
   685     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   686     moreover have "x mod d : {1..d}"
   687     proof -
   688       have "0 \<le> x mod d" by(rule pos_mod_sign)
   689       moreover have "x mod d < d" by(rule pos_mod_bound)
   690       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   691     qed
   692     ultimately show ?RHS ..
   693   qed
   694 next
   695   assume ?RHS thus ?LHS by blast
   696 qed
   697 
   698 lemma decr_mult_lemma:
   699   assumes dpos: "(0::int) < d" and
   700           minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
   701           knneg: "0 <= k"
   702   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   703 using knneg
   704 proof (induct rule:int_ge_induct)
   705   case base thus ?case by simp
   706 next
   707   case (step i)
   708   show ?case
   709   proof
   710     fix x
   711     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   712     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
   713       using minus[THEN spec, of "x - i * d"]
   714       by (simp add:int_distrib zdiff_zdiff_eq[symmetric])
   715     ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
   716   qed
   717 qed
   718 
   719 lemma incr_mult_lemma:
   720   assumes dpos: "(0::int) < d" and
   721           plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
   722           knneg: "0 <= k"
   723   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   724 using knneg
   725 proof (induct rule:int_ge_induct)
   726   case base thus ?case by simp
   727 next
   728   case (step i)
   729   show ?case
   730   proof
   731     fix x
   732     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   733     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
   734       using plus[THEN spec, of "x + i * d"]
   735       by (simp add:int_distrib zadd_ac)
   736     ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
   737   qed
   738 qed
   739 
   740 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
   741 ==> (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)) --> (EX (x::int). P x) 
   742 ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
   743 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
   744 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
   745 apply(rule iffI)
   746 prefer 2
   747 apply(drule minusinfinity)
   748 apply assumption+
   749 apply(fastsimp)
   750 apply clarsimp
   751 apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
   752 apply(frule_tac x = x and z=z in decr_lemma)
   753 apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
   754 prefer 2
   755 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   756 prefer 2 apply arith
   757  apply fastsimp
   758 apply(drule (1) minf_vee)
   759 apply blast
   760 apply(blast dest:decr_mult_lemma)
   761 done
   762 
   763 (* Cooper Thm `, plus infinity version*)
   764 lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
   765 ==> (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)) --> (EX (x::int). P x) 
   766 ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) 
   767 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
   768 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
   769   apply(rule iffI)
   770   prefer 2
   771   apply(drule plusinfinity)
   772   apply assumption+
   773   apply(fastsimp)
   774   apply clarsimp
   775   apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
   776   apply(frule_tac x = x and z=z in incr_lemma)
   777   apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
   778   prefer 2
   779   apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   780   prefer 2 apply arith
   781   apply fastsimp
   782   apply(drule (1) pinf_vee)
   783   apply blast
   784   apply(blast dest:incr_mult_lemma)
   785   done
   786 
   787 
   788 (*=============================================================================*)
   789 
   790 (*Theorems for the quantifier elminination Functions.*)
   791 
   792 lemma qe_ex_conj: "(EX (x::int). A x) = R
   793 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
   794 		==> (EX (x::int). P x) = (Q & R)"
   795 by blast
   796 
   797 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
   798 		==> (EX (x::int). P x) = Q"
   799 by blast
   800 
   801 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
   802 by blast
   803 
   804 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
   805 by blast
   806 
   807 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
   808 by blast
   809 
   810 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
   811 by blast
   812 
   813 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
   814 by blast
   815 
   816 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
   817 by blast
   818 
   819 (* Theorems for proving NNF *)
   820 
   821 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
   822 by blast
   823 
   824 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
   825 by blast
   826 
   827 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
   828   by blast
   829 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
   830 by blast
   831 
   832 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
   833 by blast
   834 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
   835 by blast
   836 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
   837 by blast
   838 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
   839 by blast
   840 
   841 
   842 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   843   by simp
   844 
   845 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   846   by rules
   847 
   848 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
   849   by rules
   850 
   851 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
   852 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
   853 by blast
   854 
   855 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
   856 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
   857 by blast
   858 
   859 
   860 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   861 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   862 apply(fastsimp)
   863 done
   864 
   865 (* Theorems required for the adjustcoeffitienteq*)
   866 
   867 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
   868 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
   869 proof
   870   assume ?P
   871   thus ?Q
   872     apply(simp add:dvd_def)
   873     apply clarify
   874     apply(rename_tac d)
   875     apply(drule_tac f = "op * k" in arg_cong)
   876     apply(simp only:int_distrib)
   877     apply(rule_tac x = "d" in exI)
   878     apply(simp only:zmult_ac)
   879     done
   880 next
   881   assume ?Q
   882   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
   883   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib zmult_ac)
   884   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   885   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   886   thus ?P by(simp add:dvd_def)
   887 qed
   888 
   889 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
   890 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
   891 proof
   892   assume P: ?P
   893   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib zmult_ac)
   894 next
   895   assume ?Q
   896   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib zmult_ac)
   897   with gr0 have "0 < (c*n + t - m)" by(simp add:int_0_less_mult_iff)
   898   thus ?P by(simp)
   899 qed
   900 
   901 lemma ac_eq_eq : assumes not0: "0 ~= (k::int)" shows "((m::int) = (c*n+t)) = (k*m =((k*c)*n+(k*t)) )" (is "?P = ?Q")
   902 proof
   903   assume ?P
   904   thus ?Q
   905     apply(drule_tac f = "op * k" in arg_cong)
   906     apply(simp only:int_distrib)
   907     done
   908 next
   909   assume ?Q
   910   hence "m * k = (c*n + t) * k" by(simp add:int_distrib zmult_ac)
   911   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   912   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   913 qed
   914 
   915 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
   916 proof -
   917   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
   918   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib zmult_ac)
   919   also have "0<(-1*c)*n + (-t+1) = (0 < (k*(-1*c)*n) + (k*(-t+1)))" by(rule ac_lt_eq[of _ 0,OF gr0,simplified])
   920   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib zmult_ac)
   921   finally show ?thesis .
   922 qed
   923 
   924 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
   925 by arith
   926 
   927 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
   928 by simp
   929 
   930 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
   931 by simp
   932 
   933 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
   934 by simp
   935 
   936 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
   937 by simp
   938 
   939 (* Theorems for transforming predicates on nat to predicates on int*)
   940 
   941 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   942   by (simp split add: split_nat)
   943 
   944 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   945   apply (simp split add: split_nat)
   946   apply (rule iffI)
   947   apply (erule exE)
   948   apply (rule_tac x = "int x" in exI)
   949   apply simp
   950   apply (erule exE)
   951   apply (rule_tac x = "nat x" in exI)
   952   apply (erule conjE)
   953   apply (erule_tac x = "nat x" in allE)
   954   apply simp
   955   done
   956 
   957 theorem zdiff_int_split: "P (int (x - y)) =
   958   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   959   apply (case_tac "y \<le> x")
   960   apply (simp_all add: zdiff_int)
   961   done
   962 
   963 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
   964   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
   965     nat_0_le cong add: conj_cong)
   966   apply (rule iffI)
   967   apply rules
   968   apply (erule exE)
   969   apply (case_tac "x=0")
   970   apply (rule_tac x=0 in exI)
   971   apply simp
   972   apply (case_tac "0 \<le> k")
   973   apply rules
   974   apply (simp add: linorder_not_le)
   975   apply (drule zmult_zless_mono2_neg [OF iffD2 [OF zero_less_int_conv]])
   976   apply assumption
   977   apply (simp add: zmult_ac)
   978   done
   979 
   980 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
   981   by simp
   982 
   983 theorem number_of2: "(0::int) <= number_of bin.Pls" by simp
   984 
   985 theorem Suc_plus1: "Suc n = n + 1" by simp
   986 
   987 (* specific instances of congruence rules, to prevent simplifier from looping *)
   988 
   989 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
   990   by simp
   991 
   992 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<and> P) = (0 <= x \<and> P')"
   993   by simp
   994 
   995 use "cooper_dec.ML"
   996 use "cooper_proof.ML"
   997 use "qelim.ML"
   998 use "presburger.ML"
   999 
  1000 setup "Presburger.setup"
  1001 
  1002 end