src/HOL/Presburger.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 16417 9bc16273c2d4
child 16836 45a3dc4688bc
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/Integ/Presburger.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
     4 
     5 File containing necessary theorems for the proof
     6 generation for Cooper Algorithm  
     7 *)
     8 
     9 header {* Presburger Arithmetic: Cooper's Algorithm *}
    10 
    11 theory Presburger
    12 imports NatSimprocs SetInterval
    13 uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") ("presburger.ML")
    14 begin
    15 
    16 text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
    17 
    18 theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
    19   apply (rule iffI)
    20   apply (erule exE)
    21   apply (rule_tac x = "l * x" in exI)
    22   apply simp
    23   apply (erule exE)
    24   apply (erule conjE)
    25   apply (erule dvdE)
    26   apply (rule_tac x = k in exI)
    27   apply simp
    28   done
    29 
    30 lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
    31 apply(unfold dvd_def)
    32 apply(rule iffI)
    33 apply(clarsimp)
    34 apply(rename_tac k)
    35 apply(rule_tac x = "-k" in exI)
    36 apply simp
    37 apply(clarsimp)
    38 apply(rename_tac k)
    39 apply(rule_tac x = "-k" in exI)
    40 apply simp
    41 done
    42 
    43 lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
    44 apply(unfold dvd_def)
    45 apply(rule iffI)
    46 apply(clarsimp)
    47 apply(rule_tac x = "-k" in exI)
    48 apply simp
    49 apply(clarsimp)
    50 apply(rule_tac x = "-k" in exI)
    51 apply simp
    52 done
    53 
    54 
    55 
    56 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}.*}
    57 
    58 theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    59   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    60   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    61   apply (erule exE)+
    62   apply (rule_tac x = "min z1 z2" in exI)
    63   apply simp
    64   done
    65 
    66 
    67 theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    68   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    69   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    70 
    71   apply (erule exE)+
    72   apply (rule_tac x = "min z1 z2" in exI)
    73   apply simp
    74   done
    75 
    76 
    77 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}.*}
    78 
    79 theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    80   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    81   \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
    82   apply (erule exE)+
    83   apply (rule_tac x = "max z1 z2" in exI)
    84   apply simp
    85   done
    86 
    87 
    88 theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
    89   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
    90   \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
    91   apply (erule exE)+
    92   apply (rule_tac x = "max z1 z2" in exI)
    93   apply simp
    94   done
    95 
    96 text {*
    97   \medskip Theorems for the combination of proofs of the modulo @{text
    98   D} property for @{text "P plusinfinity"}
    99 
   100   FIXME: This is THE SAME theorem as for the @{text minusinf} version,
   101   but with @{text "+k.."} instead of @{text "-k.."} In the future
   102   replace these both with only one. *}
   103 
   104 theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   105   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   106   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
   107   by simp
   108 
   109 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   110   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   111   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
   112   by simp
   113 
   114 text {*
   115   This is one of the cases where the simplifed formula is prooved to
   116   habe some property (in relation to @{text P_m}) but we need to prove
   117   the property for the original formula (@{text P_m})
   118 
   119   FIXME: This is exaclty the same thm as for @{text minusinf}. *}
   120 
   121 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)) "
   122   by blast
   123 
   124 
   125 text {*
   126   \medskip Theorems for the combination of proofs of the modulo @{text D}
   127   property for @{text "P minusinfinity"} *}
   128 
   129 theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   130   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   131   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
   132   by simp
   133 
   134 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   135   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   136   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
   137   by simp
   138 
   139 text {*
   140   This is one of the cases where the simplifed formula is prooved to
   141   have some property (in relation to @{text P_m}) but we need to
   142   prove the property for the original formula (@{text P_m}). *}
   143 
   144 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)) "
   145   by blast
   146 
   147 text {*
   148   Theorem needed for proving at runtime divide properties using the
   149   arithmetic tactic (which knows only about modulo = 0). *}
   150 
   151 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
   152   by(simp add:dvd_def zmod_eq_0_iff)
   153 
   154 text {*
   155   \medskip Theorems used for the combination of proof for the
   156   backwards direction of Cooper's Theorem. They rely exclusively on
   157   Predicate calculus.*}
   158 
   159 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))
   160 ==>
   161 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   162 ==>
   163 (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))) "
   164   by blast
   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 text {*
   182   \medskip Theorems used for the combination of proof for the
   183   backwards direction of Cooper's Theorem. They rely exclusively on
   184   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 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))
   195 ==>
   196 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   197 ==>
   198 (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)
   199 \<and> P2(x-d))) "
   200   by blast
   201 
   202 lemma not_bst_p_Q_elim: "
   203 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) 
   204 ==> ( P = Q )
   205 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
   206   by blast
   207 
   208 text {* \medskip This is the first direction of Cooper's Theorem. *}
   209 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
   210   by blast
   211 
   212 text {*
   213   \medskip The full Cooper's Theorem in its equivalence Form. Given
   214   the premises it is trivial too, it relies exclusively on prediacte calculus.*}
   215 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
   216 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
   217   by blast
   218 
   219 text {*
   220   \medskip Some of the atomic theorems generated each time the atom
   221   does not depend on @{text x}, they are trivial.*}
   222 
   223 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
   224   by blast
   225 
   226 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
   227   by blast
   228 
   229 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"
   230   by blast
   231 
   232 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
   233   by blast
   234 
   235 text {* The next two thms are the same as the @{text minusinf} version. *}
   236 
   237 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
   238   by blast
   239 
   240 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"
   241   by blast
   242 
   243 text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
   244 
   245 lemma P_eqtrue: "(P=True) = P"
   246   by rules
   247 
   248 lemma P_eqfalse: "(P=False) = (~P)"
   249   by rules
   250 
   251 text {*
   252   \medskip Theorems for the generation of the bachwards direction of
   253   Cooper's Theorem.
   254 
   255   These are the 6 interesting atomic cases which have to be proved relying on the
   256   properties of B-set and the arithmetic and contradiction proofs. *}
   257 
   258 lemma not_bst_p_lt: "0 < (d::int) ==>
   259  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 )"
   260   by arith
   261 
   262 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   263  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)"
   264 apply clarsimp
   265 apply(rule ccontr)
   266 apply(drule_tac x = "x+a" in bspec)
   267 apply(simp add:atLeastAtMost_iff)
   268 apply(drule_tac x = "-a" in bspec)
   269 apply assumption
   270 apply(simp)
   271 done
   272 
   273 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
   274  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 )"
   275 apply clarsimp
   276 apply(subgoal_tac "x = -a")
   277  prefer 2 apply arith
   278 apply(drule_tac x = "1" in bspec)
   279 apply(simp add:atLeastAtMost_iff)
   280 apply(drule_tac x = "-a- 1" in bspec)
   281 apply assumption
   282 apply(simp)
   283 done
   284 
   285 
   286 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   287  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)"
   288 apply clarsimp
   289 apply(subgoal_tac "x = -a+d")
   290  prefer 2 apply arith
   291 apply(drule_tac x = "d" in bspec)
   292 apply(simp add:atLeastAtMost_iff)
   293 apply(drule_tac x = "-a" in bspec)
   294 apply assumption
   295 apply(simp)
   296 done
   297 
   298 
   299 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
   300  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 )"
   301 apply(clarsimp simp add:dvd_def)
   302 apply(rename_tac m)
   303 apply(rule_tac x = "m - k" in exI)
   304 apply(simp add:int_distrib)
   305 done
   306 
   307 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
   308  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 ))"
   309 apply(clarsimp simp add:dvd_def)
   310 apply(rename_tac m)
   311 apply(erule_tac x = "m + k" in allE)
   312 apply(simp add:int_distrib)
   313 done
   314 
   315 text {*
   316   \medskip Theorems for the generation of the bachwards direction of
   317   Cooper's Theorem.
   318 
   319   These are the 6 interesting atomic cases which have to be proved
   320   relying on the properties of A-set ant the arithmetic and
   321   contradiction proofs. *}
   322 
   323 lemma not_ast_p_gt: "0 < (d::int) ==>
   324  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 )"
   325   by arith
   326 
   327 lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
   328  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)"
   329   apply clarsimp
   330   apply (rule ccontr)
   331   apply (drule_tac x = "t-x" in bspec)
   332   apply simp
   333   apply (drule_tac x = "t" in bspec)
   334   apply assumption
   335   apply simp
   336   done
   337 
   338 lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<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 (drule_tac x="1" in bspec)
   342   apply simp
   343   apply (drule_tac x="- t + 1" in bspec)
   344   apply assumption
   345   apply(subgoal_tac "x = -t")
   346   prefer 2 apply arith
   347   apply simp
   348   done
   349 
   350 lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
   351  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)"
   352   apply clarsimp
   353   apply (subgoal_tac "x = -t-d")
   354   prefer 2 apply arith
   355   apply (drule_tac x = "d" in bspec)
   356   apply simp
   357   apply (drule_tac x = "-t" in bspec)
   358   apply assumption
   359   apply simp
   360   done
   361 
   362 lemma not_ast_p_dvd: "(d1::int) dvd d ==>
   363  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 )"
   364   apply(clarsimp simp add:dvd_def)
   365   apply(rename_tac m)
   366   apply(rule_tac x = "m + k" in exI)
   367   apply(simp add:int_distrib)
   368   done
   369 
   370 lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
   371  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 ))"
   372   apply(clarsimp simp add:dvd_def)
   373   apply(rename_tac m)
   374   apply(erule_tac x = "m - k" in allE)
   375   apply(simp add:int_distrib)
   376   done
   377 
   378 text {*
   379   \medskip These are the atomic cases for the proof generation for the
   380   modulo @{text D} property for @{text "P plusinfinity"}
   381 
   382   They are fully based on arithmetics. *}
   383 
   384 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
   385  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
   386   apply(clarsimp simp add:dvd_def)
   387   apply(rule iffI)
   388   apply(clarsimp)
   389   apply(rename_tac n m)
   390   apply(rule_tac x = "m + n*k" in exI)
   391   apply(simp add:int_distrib)
   392   apply(clarsimp)
   393   apply(rename_tac n m)
   394   apply(rule_tac x = "m - n*k" in exI)
   395   apply(simp add:int_distrib mult_ac)
   396   done
   397 
   398 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
   399  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
   400   apply(clarsimp simp add:dvd_def)
   401   apply(rule iffI)
   402   apply(clarsimp)
   403   apply(rename_tac n m)
   404   apply(erule_tac x = "m - n*k" in allE)
   405   apply(simp add:int_distrib mult_ac)
   406   apply(clarsimp)
   407   apply(rename_tac n m)
   408   apply(erule_tac x = "m + n*k" in allE)
   409   apply(simp add:int_distrib mult_ac)
   410   done
   411 
   412 text {*
   413   \medskip These are the atomic cases for the proof generation for the
   414   equivalence of @{text P} and @{text "P plusinfinity"} for integers
   415   @{text x} greater than some integer @{text z}.
   416 
   417   They are fully based on arithmetics. *}
   418 
   419 lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
   420   apply(rule_tac x = "-t" in exI)
   421   apply simp
   422   done
   423 
   424 lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
   425   apply(rule_tac x = "-t" in exI)
   426   apply simp
   427   done
   428 
   429 lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
   430   apply(rule_tac x = "-t" in exI)
   431   apply simp
   432   done
   433 
   434 lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
   435   apply(rule_tac x = "t" in exI)
   436   apply simp
   437   done
   438 
   439 lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
   440   by simp
   441 
   442 lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   443   by simp
   444 
   445 text {*
   446   \medskip These are the atomic cases for the proof generation for the
   447   modulo @{text D} property for @{text "P minusinfinity"}.
   448 
   449   They are fully based on arithmetics. *}
   450 
   451 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
   452  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
   453 apply(clarsimp simp add:dvd_def)
   454 apply(rule iffI)
   455 apply(clarsimp)
   456 apply(rename_tac n m)
   457 apply(rule_tac x = "m - n*k" in exI)
   458 apply(simp add:int_distrib)
   459 apply(clarsimp)
   460 apply(rename_tac n m)
   461 apply(rule_tac x = "m + n*k" in exI)
   462 apply(simp add:int_distrib mult_ac)
   463 done
   464 
   465 
   466 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
   467  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
   468 apply(clarsimp simp add:dvd_def)
   469 apply(rule iffI)
   470 apply(clarsimp)
   471 apply(rename_tac n m)
   472 apply(erule_tac x = "m + n*k" in allE)
   473 apply(simp add:int_distrib mult_ac)
   474 apply(clarsimp)
   475 apply(rename_tac n m)
   476 apply(erule_tac x = "m - n*k" in allE)
   477 apply(simp add:int_distrib mult_ac)
   478 done
   479 
   480 text {*
   481   \medskip These are the atomic cases for the proof generation for the
   482   equivalence of @{text P} and @{text "P minusinfinity"} for integers
   483   @{text x} less than some integer @{text z}.
   484 
   485   They are fully based on arithmetics. *}
   486 
   487 lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
   488 apply(rule_tac x = "-t" in exI)
   489 apply simp
   490 done
   491 
   492 lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
   493 apply(rule_tac x = "-t" in exI)
   494 apply simp
   495 done
   496 
   497 lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
   498 apply(rule_tac x = "-t" in exI)
   499 apply simp
   500 done
   501 
   502 
   503 lemma  len_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  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
   509   by simp
   510 
   511 lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   512   by simp
   513 
   514 text {*
   515   \medskip This Theorem combines whithnesses about @{text "P
   516   minusinfinity"} to show one component of the equivalence proof for
   517   Cooper's Theorem.
   518 
   519   FIXME: remove once they are part of the distribution. *}
   520 
   521 theorem int_ge_induct[consumes 1,case_names base step]:
   522   assumes ge: "k \<le> (i::int)" and
   523         base: "P(k)" and
   524         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   525   shows "P i"
   526 proof -
   527   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
   528     proof (induct n)
   529       case 0
   530       hence "i = k" by arith
   531       thus "P i" using base by simp
   532     next
   533       case (Suc n)
   534       hence "n = nat((i - 1) - k)" by arith
   535       moreover
   536       have ki1: "k \<le> i - 1" using Suc.prems by arith
   537       ultimately
   538       have "P(i - 1)" by(rule Suc.hyps)
   539       from step[OF ki1 this] show ?case by simp
   540     qed
   541   }
   542   from this ge show ?thesis by fast
   543 qed
   544 
   545 theorem int_gr_induct[consumes 1,case_names base step]:
   546   assumes gr: "k < (i::int)" and
   547         base: "P(k+1)" and
   548         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   549   shows "P i"
   550 apply(rule int_ge_induct[of "k + 1"])
   551   using gr apply arith
   552  apply(rule base)
   553 apply(rule step)
   554  apply simp+
   555 done
   556 
   557 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   558 apply(induct rule: int_gr_induct)
   559  apply simp
   560  apply arith
   561 apply (simp add:int_distrib)
   562 apply arith
   563 done
   564 
   565 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   566 apply(induct rule: int_gr_induct)
   567  apply simp
   568  apply arith
   569 apply (simp add:int_distrib)
   570 apply arith
   571 done
   572 
   573 lemma  minusinfinity:
   574   assumes "0 < d" and
   575     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
   576     ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   577   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   578 proof
   579   assume eP1: "EX x. P1 x"
   580   then obtain x where P1: "P1 x" ..
   581   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   582   let ?w = "x - (abs(x-z)+1) * d"
   583   show "EX x. P x"
   584   proof
   585     have w: "?w < z" by(rule decr_lemma)
   586     have "P1 x = P1 ?w" using P1eqP1 by blast
   587     also have "\<dots> = P(?w)" using w P1eqP by blast
   588     finally show "P ?w" using P1 by blast
   589   qed
   590 qed
   591 
   592 text {*
   593   \medskip This Theorem combines whithnesses about @{text "P
   594   minusinfinity"} to show one component of the equivalence proof for
   595   Cooper's Theorem. *}
   596 
   597 lemma plusinfinity:
   598   assumes "0 < d" and
   599     P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
   600     ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
   601   shows "(EX x::int. P1 x) --> (EX x::int. P x)"
   602 proof
   603   assume eP1: "EX x. P1 x"
   604   then obtain x where P1: "P1 x" ..
   605   from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
   606   let ?w = "x + (abs(x-z)+1) * d"
   607   show "EX x. P x"
   608   proof
   609     have w: "z < ?w" by(rule incr_lemma)
   610     have "P1 x = P1 ?w" using P1eqP1 by blast
   611     also have "\<dots> = P(?w)" using w P1eqP by blast
   612     finally show "P ?w" using P1 by blast
   613   qed
   614 qed
   615  
   616 text {*
   617   \medskip Theorem for periodic function on discrete sets. *}
   618 
   619 lemma minf_vee:
   620   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   621   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   622   (is "?LHS = ?RHS")
   623 proof
   624   assume ?LHS
   625   then obtain x where P: "P x" ..
   626   have "x mod d = x - (x div d)*d"
   627     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   628   hence Pmod: "P x = P(x mod d)" using modd by simp
   629   show ?RHS
   630   proof (cases)
   631     assume "x mod d = 0"
   632     hence "P 0" using P Pmod by simp
   633     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   634     ultimately have "P d" by simp
   635     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   636     ultimately show ?RHS ..
   637   next
   638     assume not0: "x mod d \<noteq> 0"
   639     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   640     moreover have "x mod d : {1..d}"
   641     proof -
   642       have "0 \<le> x mod d" by(rule pos_mod_sign)
   643       moreover have "x mod d < d" by(rule pos_mod_bound)
   644       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   645     qed
   646     ultimately show ?RHS ..
   647   qed
   648 next
   649   assume ?RHS thus ?LHS by blast
   650 qed
   651 
   652 text {*
   653   \medskip Theorem for periodic function on discrete sets. *}
   654 
   655 lemma pinf_vee:
   656   assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
   657   shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
   658   (is "?LHS = ?RHS")
   659 proof
   660   assume ?LHS
   661   then obtain x where P: "P x" ..
   662   have "x mod d = x + (-(x div d))*d"
   663     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   664   hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
   665   show ?RHS
   666   proof (cases)
   667     assume "x mod d = 0"
   668     hence "P 0" using P Pmod by simp
   669     moreover have "P 0 = P(0 + 1*d)" using modd by blast
   670     ultimately have "P d" by simp
   671     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   672     ultimately show ?RHS ..
   673   next
   674     assume not0: "x mod d \<noteq> 0"
   675     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   676     moreover have "x mod d : {1..d}"
   677     proof -
   678       have "0 \<le> x mod d" by(rule pos_mod_sign)
   679       moreover have "x mod d < d" by(rule pos_mod_bound)
   680       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   681     qed
   682     ultimately show ?RHS ..
   683   qed
   684 next
   685   assume ?RHS thus ?LHS by blast
   686 qed
   687 
   688 lemma decr_mult_lemma:
   689   assumes dpos: "(0::int) < d" and
   690           minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
   691           knneg: "0 <= k"
   692   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   693 using knneg
   694 proof (induct rule:int_ge_induct)
   695   case base thus ?case by simp
   696 next
   697   case (step i)
   698   show ?case
   699   proof
   700     fix x
   701     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   702     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
   703       using minus[THEN spec, of "x - i * d"]
   704       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
   705     ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
   706   qed
   707 qed
   708 
   709 lemma incr_mult_lemma:
   710   assumes dpos: "(0::int) < d" and
   711           plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
   712           knneg: "0 <= k"
   713   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   714 using knneg
   715 proof (induct rule:int_ge_induct)
   716   case base thus ?case by simp
   717 next
   718   case (step i)
   719   show ?case
   720   proof
   721     fix x
   722     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   723     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
   724       using plus[THEN spec, of "x + i * d"]
   725       by (simp add:int_distrib zadd_ac)
   726     ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
   727   qed
   728 qed
   729 
   730 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
   731 ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
   732 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
   733 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
   734 apply(rule iffI)
   735 prefer 2
   736 apply(drule minusinfinity)
   737 apply assumption+
   738 apply(fastsimp)
   739 apply clarsimp
   740 apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
   741 apply(frule_tac x = x and z=z in decr_lemma)
   742 apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
   743 prefer 2
   744 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   745 prefer 2 apply arith
   746  apply fastsimp
   747 apply(drule (1) minf_vee)
   748 apply blast
   749 apply(blast dest:decr_mult_lemma)
   750 done
   751 
   752 text {* Cooper Theorem, plus infinity version. *}
   753 lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
   754 ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) 
   755 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
   756 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
   757   apply(rule iffI)
   758   prefer 2
   759   apply(drule plusinfinity)
   760   apply assumption+
   761   apply(fastsimp)
   762   apply clarsimp
   763   apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
   764   apply(frule_tac x = x and z=z in incr_lemma)
   765   apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
   766   prefer 2
   767   apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   768   prefer 2 apply arith
   769   apply fastsimp
   770   apply(drule (1) pinf_vee)
   771   apply blast
   772   apply(blast dest:incr_mult_lemma)
   773   done
   774 
   775 
   776 text {*
   777   \bigskip Theorems for the quantifier elminination Functions. *}
   778 
   779 lemma qe_ex_conj: "(EX (x::int). A x) = R
   780 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
   781 		==> (EX (x::int). P x) = (Q & R)"
   782 by blast
   783 
   784 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
   785 		==> (EX (x::int). P x) = Q"
   786 by blast
   787 
   788 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
   789 by blast
   790 
   791 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
   792 by blast
   793 
   794 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
   795 by blast
   796 
   797 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
   798 by blast
   799 
   800 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
   801 by blast
   802 
   803 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
   804 by blast
   805 
   806 text {* \bigskip Theorems for proving NNF *}
   807 
   808 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
   809 by blast
   810 
   811 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
   812 by blast
   813 
   814 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
   815   by blast
   816 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
   817 by blast
   818 
   819 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
   820 by blast
   821 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
   822 by blast
   823 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
   824 by blast
   825 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
   826 by blast
   827 
   828 
   829 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   830   by simp
   831 
   832 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   833   by rules
   834 
   835 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
   836   by rules
   837 
   838 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
   839 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
   840 by blast
   841 
   842 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
   843 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
   844 by blast
   845 
   846 
   847 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   848 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   849 apply(fastsimp)
   850 done
   851 
   852 text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
   853 
   854 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
   855 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
   856 proof
   857   assume ?P
   858   thus ?Q
   859     apply(simp add:dvd_def)
   860     apply clarify
   861     apply(rename_tac d)
   862     apply(drule_tac f = "op * k" in arg_cong)
   863     apply(simp only:int_distrib)
   864     apply(rule_tac x = "d" in exI)
   865     apply(simp only:mult_ac)
   866     done
   867 next
   868   assume ?Q
   869   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
   870   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
   871   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   872   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   873   thus ?P by(simp add:dvd_def)
   874 qed
   875 
   876 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
   877 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
   878 proof
   879   assume P: ?P
   880   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
   881 next
   882   assume ?Q
   883   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
   884   with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
   885   thus ?P by(simp)
   886 qed
   887 
   888 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")
   889 proof
   890   assume ?P
   891   thus ?Q
   892     apply(drule_tac f = "op * k" in arg_cong)
   893     apply(simp only:int_distrib)
   894     done
   895 next
   896   assume ?Q
   897   hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
   898   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   899   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   900 qed
   901 
   902 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
   903 proof -
   904   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
   905   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
   906   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])
   907   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
   908   finally show ?thesis .
   909 qed
   910 
   911 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
   912 by arith
   913 
   914 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
   915 by simp
   916 
   917 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
   918 by simp
   919 
   920 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
   921 by simp
   922 
   923 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
   924 by simp
   925 
   926 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
   927 
   928 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   929   by (simp split add: split_nat)
   930 
   931 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   932   apply (simp split add: split_nat)
   933   apply (rule iffI)
   934   apply (erule exE)
   935   apply (rule_tac x = "int x" in exI)
   936   apply simp
   937   apply (erule exE)
   938   apply (rule_tac x = "nat x" in exI)
   939   apply (erule conjE)
   940   apply (erule_tac x = "nat x" in allE)
   941   apply simp
   942   done
   943 
   944 theorem zdiff_int_split: "P (int (x - y)) =
   945   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   946   apply (case_tac "y \<le> x")
   947   apply (simp_all add: zdiff_int)
   948   done
   949 
   950 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
   951   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
   952     nat_0_le cong add: conj_cong)
   953   apply (rule iffI)
   954   apply rules
   955   apply (erule exE)
   956   apply (case_tac "x=0")
   957   apply (rule_tac x=0 in exI)
   958   apply simp
   959   apply (case_tac "0 \<le> k")
   960   apply rules
   961   apply (simp add: linorder_not_le)
   962   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
   963   apply assumption
   964   apply (simp add: mult_ac)
   965   done
   966 
   967 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
   968   by simp
   969 
   970 theorem number_of2: "(0::int) <= Numeral0" by simp
   971 
   972 theorem Suc_plus1: "Suc n = n + 1" by simp
   973 
   974 text {*
   975   \medskip Specific instances of congruence rules, to prevent
   976   simplifier from looping. *}
   977 
   978 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
   979   by simp
   980 
   981 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
   982   by (simp cong: conj_cong)
   983 
   984 use "cooper_dec.ML"
   985 oracle
   986   presburger_oracle = CooperDec.mk_presburger_oracle
   987 
   988 use "cooper_proof.ML"
   989 use "qelim.ML"
   990 use "presburger.ML"
   991 
   992 setup "Presburger.setup"
   993 
   994 end