src/HOL/Presburger.thy
author obua
Tue May 11 20:11:08 2004 +0200 (2004-05-11)
changeset 14738 83f1a514dcb4
parent 14577 dbb95b825244
child 14758 af3b71a46a1c
permissions -rw-r--r--
changes made due to new Ring_and_Field theory
     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 header {* Presburger Arithmetic: Cooper Algorithm *}
    11 
    12 theory Presburger = NatSimprocs + SetInterval
    13 files
    14   ("cooper_dec.ML")
    15   ("cooper_proof.ML")
    16   ("qelim.ML")
    17   ("presburger.ML"):
    18 
    19 text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
    20 
    21 theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
    22   apply (rule iffI)
    23   apply (erule exE)
    24   apply (rule_tac x = "l * x" in exI)
    25   apply simp
    26   apply (erule exE)
    27   apply (erule conjE)
    28   apply (erule dvdE)
    29   apply (rule_tac x = k in exI)
    30   apply simp
    31   done
    32 
    33 lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
    34 apply(unfold dvd_def)
    35 apply(rule iffI)
    36 apply(clarsimp)
    37 apply(rename_tac k)
    38 apply(rule_tac x = "-k" in exI)
    39 apply simp
    40 apply(clarsimp)
    41 apply(rename_tac k)
    42 apply(rule_tac x = "-k" in exI)
    43 apply simp
    44 done
    45 
    46 lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
    47 apply(unfold dvd_def)
    48 apply(rule iffI)
    49 apply(clarsimp)
    50 apply(rule_tac x = "-k" in exI)
    51 apply simp
    52 apply(clarsimp)
    53 apply(rule_tac x = "-k" in exI)
    54 apply simp
    55 done
    56 
    57 
    58 
    59 text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} less than some integer @{text z}.*}
    60 
    61 theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    62   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    63   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    64   apply (erule exE)+
    65   apply (rule_tac x = "min z1 z2" in exI)
    66   apply simp
    67   done
    68 
    69 
    70 theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    71   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    72   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    73 
    74   apply (erule exE)+
    75   apply (rule_tac x = "min z1 z2" in exI)
    76   apply simp
    77   done
    78 
    79 
    80 text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} greather than some integer @{text z}.*}
    81 
    82 theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    83   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    84   \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    85   apply (erule exE)+
    86   apply (rule_tac x = "max z1 z2" in exI)
    87   apply simp
    88   done
    89 
    90 
    91 theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    92   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    93   \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    94   apply (erule exE)+
    95   apply (rule_tac x = "max z1 z2" in exI)
    96   apply simp
    97   done
    98 
    99 text {*
   100   \medskip Theorems for the combination of proofs of the modulo @{text
   101   D} property for @{text "P plusinfinity"}
   102 
   103   FIXME: This is THE SAME theorem as for the @{text minusinf} version,
   104   but with @{text "+k.."} instead of @{text "-k.."} In the future
   105   replace these both with only one. *}
   106 
   107 theorem modd_pinf_conjI: "\<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 \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
   110   by simp
   111 
   112 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   113   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   114   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
   115   by simp
   116 
   117 text {*
   118   This is one of the cases where the simplifed formula is prooved to
   119   habe some property (in relation to @{text P_m}) but we need to prove
   120   the property for the original formula (@{text P_m})
   121 
   122   FIXME: This is exaclty the same thm as for @{text minusinf}. *}
   123 
   124 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)) "
   125   by blast
   126 
   127 
   128 text {*
   129   \medskip Theorems for the combination of proofs of the modulo @{text D}
   130   property for @{text "P minusinfinity"} *}
   131 
   132 theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   133   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   134   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
   135   by simp
   136 
   137 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   138   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   139   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
   140   by simp
   141 
   142 text {*
   143   This is one of the cases where the simplifed formula is prooved to
   144   have some property (in relation to @{text P_m}) but we need to
   145   prove the property for the original formula (@{text P_m}). *}
   146 
   147 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)) "
   148   by blast
   149 
   150 text {*
   151   Theorem needed for proving at runtime divide properties using the
   152   arithmetic tactic (which knows only about modulo = 0). *}
   153 
   154 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
   155   by(simp add:dvd_def zmod_eq_0_iff)
   156 
   157 text {*
   158   \medskip Theorems used for the combination of proof for the
   159   backwards direction of Cooper's Theorem. They rely exclusively on
   160   Predicate calculus.*}
   161 
   162 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))
   163 ==>
   164 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   165 ==>
   166 (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))) "
   167   by blast
   168 
   169 
   170 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))
   171 ==>
   172 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   173 ==>
   174 (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)
   175 \<and> P2(x + d))) "
   176   by blast
   177 
   178 lemma not_ast_p_Q_elim: "
   179 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
   180 ==> ( P = Q )
   181 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
   182   by blast
   183 
   184 text {*
   185   \medskip Theorems used for the combination of proof for the
   186   backwards direction of Cooper's Theorem. They rely exclusively on
   187   Predicate calculus.*}
   188 
   189 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))
   190 ==>
   191 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   192 ==>
   193 (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)
   194 \<or> P2(x-d))) "
   195   by blast
   196 
   197 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))
   198 ==>
   199 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   200 ==>
   201 (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)
   202 \<and> P2(x-d))) "
   203   by blast
   204 
   205 lemma not_bst_p_Q_elim: "
   206 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) 
   207 ==> ( P = Q )
   208 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
   209   by blast
   210 
   211 text {* \medskip This is the first direction of Cooper's Theorem. *}
   212 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
   213   by blast
   214 
   215 text {*
   216   \medskip The full Cooper's Theorem in its equivalence Form. Given
   217   the premises it is trivial too, it relies exclusively on prediacte calculus.*}
   218 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
   219 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
   220   by blast
   221 
   222 text {*
   223   \medskip Some of the atomic theorems generated each time the atom
   224   does not depend on @{text x}, they are trivial.*}
   225 
   226 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
   227   by blast
   228 
   229 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
   230   by blast
   231 
   232 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"
   233   by blast
   234 
   235 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
   236   by blast
   237 
   238 text {* The next two thms are the same as the @{text minusinf} version. *}
   239 
   240 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
   241   by blast
   242 
   243 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"
   244   by blast
   245 
   246 text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
   247 
   248 lemma P_eqtrue: "(P=True) = P"
   249   by rules
   250 
   251 lemma P_eqfalse: "(P=False) = (~P)"
   252   by rules
   253 
   254 text {*
   255   \medskip Theorems for the generation of the bachwards direction of
   256   Cooper's Theorem.
   257 
   258   These are the 6 interesting atomic cases which have to be proved relying on the
   259   properties of B-set and the arithmetic and contradiction proofs. *}
   260 
   261 lemma not_bst_p_lt: "0 < (d::int) ==>
   262  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 )"
   263   by arith
   264 
   265 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   266  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)"
   267 apply clarsimp
   268 apply(rule ccontr)
   269 apply(drule_tac x = "x+a" in bspec)
   270 apply(simp add:atLeastAtMost_iff)
   271 apply(drule_tac x = "-a" in bspec)
   272 apply assumption
   273 apply(simp)
   274 done
   275 
   276 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
   277  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 )"
   278 apply clarsimp
   279 apply(subgoal_tac "x = -a")
   280  prefer 2 apply arith
   281 apply(drule_tac x = "1" in bspec)
   282 apply(simp add:atLeastAtMost_iff)
   283 apply(drule_tac x = "-a- 1" in bspec)
   284 apply assumption
   285 apply(simp)
   286 done
   287 
   288 
   289 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   290  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)"
   291 apply clarsimp
   292 apply(subgoal_tac "x = -a+d")
   293  prefer 2 apply arith
   294 apply(drule_tac x = "d" in bspec)
   295 apply(simp add:atLeastAtMost_iff)
   296 apply(drule_tac x = "-a" in bspec)
   297 apply assumption
   298 apply(simp)
   299 done
   300 
   301 
   302 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
   303  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 )"
   304 apply(clarsimp simp add:dvd_def)
   305 apply(rename_tac m)
   306 apply(rule_tac x = "m - k" in exI)
   307 apply(simp add:int_distrib)
   308 done
   309 
   310 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
   311  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 ))"
   312 apply(clarsimp simp add:dvd_def)
   313 apply(rename_tac m)
   314 apply(erule_tac x = "m + k" in allE)
   315 apply(simp add:int_distrib)
   316 done
   317 
   318 text {*
   319   \medskip Theorems for the generation of the bachwards direction of
   320   Cooper's Theorem.
   321 
   322   These are the 6 interesting atomic cases which have to be proved
   323   relying on the properties of A-set ant the arithmetic and
   324   contradiction proofs. *}
   325 
   326 lemma not_ast_p_gt: "0 < (d::int) ==>
   327  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 )"
   328   by arith
   329 
   330 lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
   331  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)"
   332   apply clarsimp
   333   apply (rule ccontr)
   334   apply (drule_tac x = "t-x" in bspec)
   335   apply simp
   336   apply (drule_tac x = "t" in bspec)
   337   apply assumption
   338   apply simp
   339   done
   340 
   341 lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
   342  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 )"
   343   apply clarsimp
   344   apply (drule_tac x="1" in bspec)
   345   apply simp
   346   apply (drule_tac x="- t + 1" in bspec)
   347   apply assumption
   348   apply(subgoal_tac "x = -t")
   349   prefer 2 apply arith
   350   apply simp
   351   done
   352 
   353 lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
   354  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)"
   355   apply clarsimp
   356   apply (subgoal_tac "x = -t-d")
   357   prefer 2 apply arith
   358   apply (drule_tac x = "d" in bspec)
   359   apply simp
   360   apply (drule_tac x = "-t" in bspec)
   361   apply assumption
   362   apply simp
   363   done
   364 
   365 lemma not_ast_p_dvd: "(d1::int) dvd d ==>
   366  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 )"
   367   apply(clarsimp simp add:dvd_def)
   368   apply(rename_tac m)
   369   apply(rule_tac x = "m + k" in exI)
   370   apply(simp add:int_distrib)
   371   done
   372 
   373 lemma not_ast_p_ndvd: "(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(erule_tac x = "m - k" in allE)
   378   apply(simp add:int_distrib)
   379   done
   380 
   381 text {*
   382   \medskip These are the atomic cases for the proof generation for the
   383   modulo @{text D} property for @{text "P plusinfinity"}
   384 
   385   They are fully based on arithmetics. *}
   386 
   387 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
   388  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
   389   apply(clarsimp simp add:dvd_def)
   390   apply(rule iffI)
   391   apply(clarsimp)
   392   apply(rename_tac n m)
   393   apply(rule_tac x = "m + n*k" in exI)
   394   apply(simp add:int_distrib)
   395   apply(clarsimp)
   396   apply(rename_tac n m)
   397   apply(rule_tac x = "m - n*k" in exI)
   398   apply(simp add:int_distrib mult_ac)
   399   done
   400 
   401 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
   402  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
   403   apply(clarsimp simp add:dvd_def)
   404   apply(rule iffI)
   405   apply(clarsimp)
   406   apply(rename_tac n m)
   407   apply(erule_tac x = "m - n*k" in allE)
   408   apply(simp add:int_distrib mult_ac)
   409   apply(clarsimp)
   410   apply(rename_tac n m)
   411   apply(erule_tac x = "m + n*k" in allE)
   412   apply(simp add:int_distrib mult_ac)
   413   done
   414 
   415 text {*
   416   \medskip These are the atomic cases for the proof generation for the
   417   equivalence of @{text P} and @{text "P plusinfinity"} for integers
   418   @{text x} greater than some integer @{text z}.
   419 
   420   They are fully based on arithmetics. *}
   421 
   422 lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
   423   apply(rule_tac x = "-t" in exI)
   424   apply simp
   425   done
   426 
   427 lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
   428   apply(rule_tac x = "-t" in exI)
   429   apply simp
   430   done
   431 
   432 lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
   433   apply(rule_tac x = "-t" in exI)
   434   apply simp
   435   done
   436 
   437 lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
   438   apply(rule_tac x = "t" in exI)
   439   apply simp
   440   done
   441 
   442 lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
   443   by simp
   444 
   445 lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   446   by simp
   447 
   448 text {*
   449   \medskip These are the atomic cases for the proof generation for the
   450   modulo @{text D} property for @{text "P minusinfinity"}.
   451 
   452   They are fully based on arithmetics. *}
   453 
   454 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
   455  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
   456 apply(clarsimp simp add:dvd_def)
   457 apply(rule iffI)
   458 apply(clarsimp)
   459 apply(rename_tac n m)
   460 apply(rule_tac x = "m - n*k" in exI)
   461 apply(simp add:int_distrib)
   462 apply(clarsimp)
   463 apply(rename_tac n m)
   464 apply(rule_tac x = "m + n*k" in exI)
   465 apply(simp add:int_distrib mult_ac)
   466 done
   467 
   468 
   469 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
   470  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
   471 apply(clarsimp simp add:dvd_def)
   472 apply(rule iffI)
   473 apply(clarsimp)
   474 apply(rename_tac n m)
   475 apply(erule_tac x = "m + n*k" in allE)
   476 apply(simp add:int_distrib mult_ac)
   477 apply(clarsimp)
   478 apply(rename_tac n m)
   479 apply(erule_tac x = "m - n*k" in allE)
   480 apply(simp add:int_distrib mult_ac)
   481 done
   482 
   483 text {*
   484   \medskip These are the atomic cases for the proof generation for the
   485   equivalence of @{text P} and @{text "P minusinfinity"} for integers
   486   @{text x} less than some integer @{text z}.
   487 
   488   They are fully based on arithmetics. *}
   489 
   490 lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
   491 apply(rule_tac x = "-t" in exI)
   492 apply simp
   493 done
   494 
   495 lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
   496 apply(rule_tac x = "-t" in exI)
   497 apply simp
   498 done
   499 
   500 lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
   501 apply(rule_tac x = "-t" in exI)
   502 apply simp
   503 done
   504 
   505 
   506 lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
   507 apply(rule_tac x = "t" in exI)
   508 apply simp
   509 done
   510 
   511 lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
   512   by simp
   513 
   514 lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   515   by simp
   516 
   517 text {*
   518   \medskip This Theorem combines whithnesses about @{text "P
   519   minusinfinity"} to show one component of the equivalence proof for
   520   Cooper's Theorem.
   521 
   522   FIXME: remove once they are part of the distribution. *}
   523 
   524 theorem int_ge_induct[consumes 1,case_names base step]:
   525   assumes ge: "k \<le> (i::int)" and
   526         base: "P(k)" and
   527         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   528   shows "P i"
   529 proof -
   530   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
   531     proof (induct n)
   532       case 0
   533       hence "i = k" by arith
   534       thus "P i" using base by simp
   535     next
   536       case (Suc n)
   537       hence "n = nat((i - 1) - k)" by arith
   538       moreover
   539       have ki1: "k \<le> i - 1" using Suc.prems by arith
   540       ultimately
   541       have "P(i - 1)" by(rule Suc.hyps)
   542       from step[OF ki1 this] show ?case by simp
   543     qed
   544   }
   545   from this ge show ?thesis by fast
   546 qed
   547 
   548 theorem int_gr_induct[consumes 1,case_names base step]:
   549   assumes gr: "k < (i::int)" and
   550         base: "P(k+1)" and
   551         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   552   shows "P i"
   553 apply(rule int_ge_induct[of "k + 1"])
   554   using gr apply arith
   555  apply(rule base)
   556 apply(rule step)
   557  apply simp+
   558 done
   559 
   560 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   561 apply(induct rule: int_gr_induct)
   562  apply simp
   563  apply arith
   564 apply (simp add:int_distrib)
   565 apply arith
   566 done
   567 
   568 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   569 apply(induct rule: int_gr_induct)
   570  apply simp
   571  apply arith
   572 apply (simp add:int_distrib)
   573 apply arith
   574 done
   575 
   576 lemma  minusinfinity:
   577   assumes "0 < d" and
   578     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
   579     ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   580   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   581 proof
   582   assume eP1: "EX x. P1 x"
   583   then obtain x where P1: "P1 x" ..
   584   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   585   let ?w = "x - (abs(x-z)+1) * d"
   586   show "EX x. P x"
   587   proof
   588     have w: "?w < z" by(rule decr_lemma)
   589     have "P1 x = P1 ?w" using P1eqP1 by blast
   590     also have "\<dots> = P(?w)" using w P1eqP by blast
   591     finally show "P ?w" using P1 by blast
   592   qed
   593 qed
   594 
   595 text {*
   596   \medskip This Theorem combines whithnesses about @{text "P
   597   minusinfinity"} to show one component of the equivalence proof for
   598   Cooper's Theorem. *}
   599 
   600 lemma plusinfinity:
   601   assumes "0 < d" and
   602     P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
   603     ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
   604   shows "(EX x::int. P1 x) --> (EX x::int. P x)"
   605 proof
   606   assume eP1: "EX x. P1 x"
   607   then obtain x where P1: "P1 x" ..
   608   from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
   609   let ?w = "x + (abs(x-z)+1) * d"
   610   show "EX x. P x"
   611   proof
   612     have w: "z < ?w" by(rule incr_lemma)
   613     have "P1 x = P1 ?w" using P1eqP1 by blast
   614     also have "\<dots> = P(?w)" using w P1eqP by blast
   615     finally show "P ?w" using P1 by blast
   616   qed
   617 qed
   618  
   619 text {*
   620   \medskip Theorem for periodic function on discrete sets. *}
   621 
   622 lemma minf_vee:
   623   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   624   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   625   (is "?LHS = ?RHS")
   626 proof
   627   assume ?LHS
   628   then obtain x where P: "P x" ..
   629   have "x mod d = x - (x div d)*d"
   630     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   631   hence Pmod: "P x = P(x mod d)" using modd by simp
   632   show ?RHS
   633   proof (cases)
   634     assume "x mod d = 0"
   635     hence "P 0" using P Pmod by simp
   636     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   637     ultimately have "P d" by simp
   638     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   639     ultimately show ?RHS ..
   640   next
   641     assume not0: "x mod d \<noteq> 0"
   642     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   643     moreover have "x mod d : {1..d}"
   644     proof -
   645       have "0 \<le> x mod d" by(rule pos_mod_sign)
   646       moreover have "x mod d < d" by(rule pos_mod_bound)
   647       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   648     qed
   649     ultimately show ?RHS ..
   650   qed
   651 next
   652   assume ?RHS thus ?LHS by blast
   653 qed
   654 
   655 text {*
   656   \medskip Theorem for periodic function on discrete sets. *}
   657 
   658 lemma pinf_vee:
   659   assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
   660   shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
   661   (is "?LHS = ?RHS")
   662 proof
   663   assume ?LHS
   664   then obtain x where P: "P x" ..
   665   have "x mod d = x + (-(x div d))*d"
   666     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   667   hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
   668   show ?RHS
   669   proof (cases)
   670     assume "x mod d = 0"
   671     hence "P 0" using P Pmod by simp
   672     moreover have "P 0 = P(0 + 1*d)" using modd by blast
   673     ultimately have "P d" by simp
   674     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   675     ultimately show ?RHS ..
   676   next
   677     assume not0: "x mod d \<noteq> 0"
   678     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   679     moreover have "x mod d : {1..d}"
   680     proof -
   681       have "0 \<le> x mod d" by(rule pos_mod_sign)
   682       moreover have "x mod d < d" by(rule pos_mod_bound)
   683       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   684     qed
   685     ultimately show ?RHS ..
   686   qed
   687 next
   688   assume ?RHS thus ?LHS by blast
   689 qed
   690 
   691 lemma decr_mult_lemma:
   692   assumes dpos: "(0::int) < d" and
   693           minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
   694           knneg: "0 <= k"
   695   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   696 using knneg
   697 proof (induct rule:int_ge_induct)
   698   case base thus ?case by simp
   699 next
   700   case (step i)
   701   show ?case
   702   proof
   703     fix x
   704     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   705     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
   706       using minus[THEN spec, of "x - i * d"]
   707       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
   708     ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
   709   qed
   710 qed
   711 
   712 lemma incr_mult_lemma:
   713   assumes dpos: "(0::int) < d" and
   714           plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
   715           knneg: "0 <= k"
   716   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   717 using knneg
   718 proof (induct rule:int_ge_induct)
   719   case base thus ?case by simp
   720 next
   721   case (step i)
   722   show ?case
   723   proof
   724     fix x
   725     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   726     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
   727       using plus[THEN spec, of "x + i * d"]
   728       by (simp add:int_distrib zadd_ac)
   729     ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
   730   qed
   731 qed
   732 
   733 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
   734 ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
   735 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
   736 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
   737 apply(rule iffI)
   738 prefer 2
   739 apply(drule minusinfinity)
   740 apply assumption+
   741 apply(fastsimp)
   742 apply clarsimp
   743 apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
   744 apply(frule_tac x = x and z=z in decr_lemma)
   745 apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
   746 prefer 2
   747 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   748 prefer 2 apply arith
   749  apply fastsimp
   750 apply(drule (1) minf_vee)
   751 apply blast
   752 apply(blast dest:decr_mult_lemma)
   753 done
   754 
   755 text {* Cooper Theorem, plus infinity version. *}
   756 lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
   757 ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) 
   758 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
   759 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
   760   apply(rule iffI)
   761   prefer 2
   762   apply(drule plusinfinity)
   763   apply assumption+
   764   apply(fastsimp)
   765   apply clarsimp
   766   apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
   767   apply(frule_tac x = x and z=z in incr_lemma)
   768   apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
   769   prefer 2
   770   apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   771   prefer 2 apply arith
   772   apply fastsimp
   773   apply(drule (1) pinf_vee)
   774   apply blast
   775   apply(blast dest:incr_mult_lemma)
   776   done
   777 
   778 
   779 text {*
   780   \bigskip Theorems for the quantifier elminination Functions. *}
   781 
   782 lemma qe_ex_conj: "(EX (x::int). A x) = R
   783 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
   784 		==> (EX (x::int). P x) = (Q & R)"
   785 by blast
   786 
   787 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
   788 		==> (EX (x::int). P x) = Q"
   789 by blast
   790 
   791 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
   792 by blast
   793 
   794 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
   795 by blast
   796 
   797 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
   798 by blast
   799 
   800 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
   801 by blast
   802 
   803 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
   804 by blast
   805 
   806 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
   807 by blast
   808 
   809 text {* \bigskip Theorems for proving NNF *}
   810 
   811 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
   812 by blast
   813 
   814 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
   815 by blast
   816 
   817 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
   818   by blast
   819 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
   820 by blast
   821 
   822 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
   823 by blast
   824 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
   825 by blast
   826 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
   827 by blast
   828 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
   829 by blast
   830 
   831 
   832 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   833   by simp
   834 
   835 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   836   by rules
   837 
   838 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
   839   by rules
   840 
   841 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
   842 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
   843 by blast
   844 
   845 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
   846 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
   847 by blast
   848 
   849 
   850 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   851 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   852 apply(fastsimp)
   853 done
   854 
   855 text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
   856 
   857 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
   858 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
   859 proof
   860   assume ?P
   861   thus ?Q
   862     apply(simp add:dvd_def)
   863     apply clarify
   864     apply(rename_tac d)
   865     apply(drule_tac f = "op * k" in arg_cong)
   866     apply(simp only:int_distrib)
   867     apply(rule_tac x = "d" in exI)
   868     apply(simp only:mult_ac)
   869     done
   870 next
   871   assume ?Q
   872   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
   873   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
   874   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   875   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   876   thus ?P by(simp add:dvd_def)
   877 qed
   878 
   879 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
   880 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
   881 proof
   882   assume P: ?P
   883   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
   884 next
   885   assume ?Q
   886   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
   887   with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
   888   thus ?P by(simp)
   889 qed
   890 
   891 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")
   892 proof
   893   assume ?P
   894   thus ?Q
   895     apply(drule_tac f = "op * k" in arg_cong)
   896     apply(simp only:int_distrib)
   897     done
   898 next
   899   assume ?Q
   900   hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
   901   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   902   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   903 qed
   904 
   905 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
   906 proof -
   907   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
   908   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
   909   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])
   910   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
   911   finally show ?thesis .
   912 qed
   913 
   914 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
   915 by arith
   916 
   917 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
   918 by simp
   919 
   920 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
   921 by simp
   922 
   923 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
   924 by simp
   925 
   926 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
   927 by simp
   928 
   929 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
   930 
   931 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   932   by (simp split add: split_nat)
   933 
   934 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   935   apply (simp split add: split_nat)
   936   apply (rule iffI)
   937   apply (erule exE)
   938   apply (rule_tac x = "int x" in exI)
   939   apply simp
   940   apply (erule exE)
   941   apply (rule_tac x = "nat x" in exI)
   942   apply (erule conjE)
   943   apply (erule_tac x = "nat x" in allE)
   944   apply simp
   945   done
   946 
   947 theorem zdiff_int_split: "P (int (x - y)) =
   948   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   949   apply (case_tac "y \<le> x")
   950   apply (simp_all add: zdiff_int)
   951   done
   952 
   953 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
   954   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
   955     nat_0_le cong add: conj_cong)
   956   apply (rule iffI)
   957   apply rules
   958   apply (erule exE)
   959   apply (case_tac "x=0")
   960   apply (rule_tac x=0 in exI)
   961   apply simp
   962   apply (case_tac "0 \<le> k")
   963   apply rules
   964   apply (simp add: linorder_not_le)
   965   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
   966   apply assumption
   967   apply (simp add: mult_ac)
   968   done
   969 
   970 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
   971   by simp
   972 
   973 theorem number_of2: "(0::int) <= number_of bin.Pls" by simp
   974 
   975 theorem Suc_plus1: "Suc n = n + 1" by simp
   976 
   977 text {*
   978   \medskip Specific instances of congruence rules, to prevent
   979   simplifier from looping. *}
   980 
   981 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
   982   by simp
   983 
   984 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<and> P) = (0 <= x \<and> P')"
   985   by simp
   986 
   987 use "cooper_dec.ML"
   988 use "cooper_proof.ML"
   989 use "qelim.ML"
   990 use "presburger.ML"
   991 
   992 setup "Presburger.setup"
   993 
   994 end