src/HOL/Integ/Presburger.thy
author chaieb
Fri Nov 18 07:13:58 2005 +0100 (2005-11-18)
changeset 18202 46af82efd311
parent 17589 58eeffd73be1
child 20051 859e7129961b
permissions -rw-r--r--
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
     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") 
    14 	("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
    15 begin
    16 
    17 text {* Theorem for unitifying the coeffitients of @{text 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 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}.*}
    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 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}.*}
    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 text {*
    98   \medskip Theorems for the combination of proofs of the modulo @{text
    99   D} property for @{text "P plusinfinity"}
   100 
   101   FIXME: This is THE SAME theorem as for the @{text minusinf} version,
   102   but with @{text "+k.."} instead of @{text "-k.."} In the future
   103   replace these both with only one. *}
   104 
   105 theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   106   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   107   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
   108   by simp
   109 
   110 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
   111   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
   112   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
   113   by simp
   114 
   115 text {*
   116   This is one of the cases where the simplifed formula is prooved to
   117   habe some property (in relation to @{text P_m}) but we need to prove
   118   the property for the original formula (@{text P_m})
   119 
   120   FIXME: This is exaclty the same thm as for @{text minusinf}. *}
   121 
   122 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)) "
   123   by blast
   124 
   125 
   126 text {*
   127   \medskip Theorems for the combination of proofs of the modulo @{text D}
   128   property for @{text "P minusinfinity"} *}
   129 
   130 theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   131   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   132   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
   133   by simp
   134 
   135 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
   136   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
   137   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
   138   by simp
   139 
   140 text {*
   141   This is one of the cases where the simplifed formula is prooved to
   142   have some property (in relation to @{text P_m}) but we need to
   143   prove the property for the original formula (@{text P_m}). *}
   144 
   145 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)) "
   146   by blast
   147 
   148 text {*
   149   Theorem needed for proving at runtime divide properties using the
   150   arithmetic tactic (which knows only about modulo = 0). *}
   151 
   152 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
   153   by(simp add:dvd_def zmod_eq_0_iff)
   154 
   155 text {*
   156   \medskip Theorems used for the combination of proof for the
   157   backwards direction of Cooper's Theorem. They rely exclusively on
   158   Predicate calculus.*}
   159 
   160 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))
   161 ==>
   162 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   163 ==>
   164 (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))) "
   165   by blast
   166 
   167 
   168 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))
   169 ==>
   170 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
   171 ==>
   172 (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)
   173 \<and> P2(x + d))) "
   174   by blast
   175 
   176 lemma not_ast_p_Q_elim: "
   177 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
   178 ==> ( P = Q )
   179 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
   180   by blast
   181 
   182 text {*
   183   \medskip Theorems used for the combination of proof for the
   184   backwards direction of Cooper's Theorem. They rely exclusively on
   185   Predicate calculus.*}
   186 
   187 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))
   188 ==>
   189 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   190 ==>
   191 (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)
   192 \<or> P2(x-d))) "
   193   by blast
   194 
   195 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))
   196 ==>
   197 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
   198 ==>
   199 (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)
   200 \<and> P2(x-d))) "
   201   by blast
   202 
   203 lemma not_bst_p_Q_elim: "
   204 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) 
   205 ==> ( P = Q )
   206 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
   207   by blast
   208 
   209 text {* \medskip This is the first direction of Cooper's Theorem. *}
   210 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
   211   by blast
   212 
   213 text {*
   214   \medskip The full Cooper's Theorem in its equivalence Form. Given
   215   the premises it is trivial too, it relies exclusively on prediacte calculus.*}
   216 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
   217 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
   218   by blast
   219 
   220 text {*
   221   \medskip Some of the atomic theorems generated each time the atom
   222   does not depend on @{text x}, they are trivial.*}
   223 
   224 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
   225   by blast
   226 
   227 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
   228   by blast
   229 
   230 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"
   231   by blast
   232 
   233 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
   234   by blast
   235 
   236 text {* The next two thms are the same as the @{text minusinf} version. *}
   237 
   238 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
   239   by blast
   240 
   241 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"
   242   by blast
   243 
   244 text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
   245 
   246 lemma P_eqtrue: "(P=True) = P"
   247   by iprover
   248 
   249 lemma P_eqfalse: "(P=False) = (~P)"
   250   by iprover
   251 
   252 text {*
   253   \medskip Theorems for the generation of the bachwards direction of
   254   Cooper's Theorem.
   255 
   256   These are the 6 interesting atomic cases which have to be proved relying on the
   257   properties of B-set and the arithmetic and contradiction proofs. *}
   258 
   259 lemma not_bst_p_lt: "0 < (d::int) ==>
   260  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 )"
   261   by arith
   262 
   263 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   264  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)"
   265 apply clarsimp
   266 apply(rule ccontr)
   267 apply(drule_tac x = "x+a" in bspec)
   268 apply(simp add:atLeastAtMost_iff)
   269 apply(drule_tac x = "-a" in bspec)
   270 apply assumption
   271 apply(simp)
   272 done
   273 
   274 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<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(subgoal_tac "x = -a")
   278  prefer 2 apply arith
   279 apply(drule_tac x = "1" in bspec)
   280 apply(simp add:atLeastAtMost_iff)
   281 apply(drule_tac x = "-a- 1" in bspec)
   282 apply assumption
   283 apply(simp)
   284 done
   285 
   286 
   287 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
   288  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)"
   289 apply clarsimp
   290 apply(subgoal_tac "x = -a+d")
   291  prefer 2 apply arith
   292 apply(drule_tac x = "d" in bspec)
   293 apply(simp add:atLeastAtMost_iff)
   294 apply(drule_tac x = "-a" in bspec)
   295 apply assumption
   296 apply(simp)
   297 done
   298 
   299 
   300 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
   301  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 )"
   302 apply(clarsimp simp add:dvd_def)
   303 apply(rename_tac m)
   304 apply(rule_tac x = "m - k" in exI)
   305 apply(simp add:int_distrib)
   306 done
   307 
   308 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
   309  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 ))"
   310 apply(clarsimp simp add:dvd_def)
   311 apply(rename_tac m)
   312 apply(erule_tac x = "m + k" in allE)
   313 apply(simp add:int_distrib)
   314 done
   315 
   316 text {*
   317   \medskip Theorems for the generation of the bachwards direction of
   318   Cooper's Theorem.
   319 
   320   These are the 6 interesting atomic cases which have to be proved
   321   relying on the properties of A-set ant the arithmetic and
   322   contradiction proofs. *}
   323 
   324 lemma not_ast_p_gt: "0 < (d::int) ==>
   325  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 )"
   326   by arith
   327 
   328 lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
   329  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)"
   330   apply clarsimp
   331   apply (rule ccontr)
   332   apply (drule_tac x = "t-x" in bspec)
   333   apply simp
   334   apply (drule_tac x = "t" in bspec)
   335   apply assumption
   336   apply simp
   337   done
   338 
   339 lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
   340  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 )"
   341   apply clarsimp
   342   apply (drule_tac x="1" in bspec)
   343   apply simp
   344   apply (drule_tac x="- t + 1" in bspec)
   345   apply assumption
   346   apply(subgoal_tac "x = -t")
   347   prefer 2 apply arith
   348   apply simp
   349   done
   350 
   351 lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
   352  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)"
   353   apply clarsimp
   354   apply (subgoal_tac "x = -t-d")
   355   prefer 2 apply arith
   356   apply (drule_tac x = "d" in bspec)
   357   apply simp
   358   apply (drule_tac x = "-t" in bspec)
   359   apply assumption
   360   apply simp
   361   done
   362 
   363 lemma not_ast_p_dvd: "(d1::int) dvd d ==>
   364  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 )"
   365   apply(clarsimp simp add:dvd_def)
   366   apply(rename_tac m)
   367   apply(rule_tac x = "m + k" in exI)
   368   apply(simp add:int_distrib)
   369   done
   370 
   371 lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
   372  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 ))"
   373   apply(clarsimp simp add:dvd_def)
   374   apply(rename_tac m)
   375   apply(erule_tac x = "m - k" in allE)
   376   apply(simp add:int_distrib)
   377   done
   378 
   379 text {*
   380   \medskip These are the atomic cases for the proof generation for the
   381   modulo @{text D} property for @{text "P plusinfinity"}
   382 
   383   They are fully based on arithmetics. *}
   384 
   385 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
   386  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
   387   apply(clarsimp simp add:dvd_def)
   388   apply(rule iffI)
   389   apply(clarsimp)
   390   apply(rename_tac n m)
   391   apply(rule_tac x = "m + n*k" in exI)
   392   apply(simp add:int_distrib)
   393   apply(clarsimp)
   394   apply(rename_tac n m)
   395   apply(rule_tac x = "m - n*k" in exI)
   396   apply(simp add:int_distrib mult_ac)
   397   done
   398 
   399 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
   400  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
   401   apply(clarsimp simp add:dvd_def)
   402   apply(rule iffI)
   403   apply(clarsimp)
   404   apply(rename_tac n m)
   405   apply(erule_tac x = "m - n*k" in allE)
   406   apply(simp add:int_distrib mult_ac)
   407   apply(clarsimp)
   408   apply(rename_tac n m)
   409   apply(erule_tac x = "m + n*k" in allE)
   410   apply(simp add:int_distrib mult_ac)
   411   done
   412 
   413 text {*
   414   \medskip These are the atomic cases for the proof generation for the
   415   equivalence of @{text P} and @{text "P plusinfinity"} for integers
   416   @{text x} greater than some integer @{text z}.
   417 
   418   They are fully based on arithmetics. *}
   419 
   420 lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
   421   apply(rule_tac x = "-t" in exI)
   422   apply simp
   423   done
   424 
   425 lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
   426   apply(rule_tac x = "-t" in exI)
   427   apply simp
   428   done
   429 
   430 lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
   431   apply(rule_tac x = "-t" in exI)
   432   apply simp
   433   done
   434 
   435 lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
   436   apply(rule_tac x = "t" in exI)
   437   apply simp
   438   done
   439 
   440 lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
   441   by simp
   442 
   443 lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   444   by simp
   445 
   446 text {*
   447   \medskip These are the atomic cases for the proof generation for the
   448   modulo @{text D} property for @{text "P minusinfinity"}.
   449 
   450   They are fully based on arithmetics. *}
   451 
   452 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
   453  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
   454 apply(clarsimp simp add:dvd_def)
   455 apply(rule iffI)
   456 apply(clarsimp)
   457 apply(rename_tac n m)
   458 apply(rule_tac x = "m - n*k" in exI)
   459 apply(simp add:int_distrib)
   460 apply(clarsimp)
   461 apply(rename_tac n m)
   462 apply(rule_tac x = "m + n*k" in exI)
   463 apply(simp add:int_distrib mult_ac)
   464 done
   465 
   466 
   467 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
   468  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
   469 apply(clarsimp simp add:dvd_def)
   470 apply(rule iffI)
   471 apply(clarsimp)
   472 apply(rename_tac n m)
   473 apply(erule_tac x = "m + n*k" in allE)
   474 apply(simp add:int_distrib mult_ac)
   475 apply(clarsimp)
   476 apply(rename_tac n m)
   477 apply(erule_tac x = "m - n*k" in allE)
   478 apply(simp add:int_distrib mult_ac)
   479 done
   480 
   481 text {*
   482   \medskip These are the atomic cases for the proof generation for the
   483   equivalence of @{text P} and @{text "P minusinfinity"} for integers
   484   @{text x} less than some integer @{text z}.
   485 
   486   They are fully based on arithmetics. *}
   487 
   488 lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
   489 apply(rule_tac x = "-t" in exI)
   490 apply simp
   491 done
   492 
   493 lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
   494 apply(rule_tac x = "-t" in exI)
   495 apply simp
   496 done
   497 
   498 lemma  le_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 
   504 lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
   505 apply(rule_tac x = "t" in exI)
   506 apply simp
   507 done
   508 
   509 lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
   510   by simp
   511 
   512 lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
   513   by simp
   514 
   515 text {*
   516   \medskip This Theorem combines whithnesses about @{text "P
   517   minusinfinity"} to show one component of the equivalence proof for
   518   Cooper's Theorem.
   519 
   520   FIXME: remove once they are part of the distribution. *}
   521 
   522 theorem int_ge_induct[consumes 1,case_names base step]:
   523   assumes ge: "k \<le> (i::int)" and
   524         base: "P(k)" and
   525         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   526   shows "P i"
   527 proof -
   528   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
   529     proof (induct n)
   530       case 0
   531       hence "i = k" by arith
   532       thus "P i" using base by simp
   533     next
   534       case (Suc n)
   535       hence "n = nat((i - 1) - k)" by arith
   536       moreover
   537       have ki1: "k \<le> i - 1" using Suc.prems by arith
   538       ultimately
   539       have "P(i - 1)" by(rule Suc.hyps)
   540       from step[OF ki1 this] show ?case by simp
   541     qed
   542   }
   543   from this ge show ?thesis by fast
   544 qed
   545 
   546 theorem int_gr_induct[consumes 1,case_names base step]:
   547   assumes gr: "k < (i::int)" and
   548         base: "P(k+1)" and
   549         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
   550   shows "P i"
   551 apply(rule int_ge_induct[of "k + 1"])
   552   using gr apply arith
   553  apply(rule base)
   554 apply(rule step)
   555  apply simp+
   556 done
   557 
   558 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   559 apply(induct rule: int_gr_induct)
   560  apply simp
   561  apply arith
   562 apply (simp add:int_distrib)
   563 apply arith
   564 done
   565 
   566 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   567 apply(induct rule: int_gr_induct)
   568  apply simp
   569  apply arith
   570 apply (simp add:int_distrib)
   571 apply arith
   572 done
   573 
   574 lemma  minusinfinity:
   575   assumes "0 < d" and
   576     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
   577     ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   578   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   579 proof
   580   assume eP1: "EX x. P1 x"
   581   then obtain x where P1: "P1 x" ..
   582   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   583   let ?w = "x - (abs(x-z)+1) * d"
   584   show "EX x. P x"
   585   proof
   586     have w: "?w < z" by(rule decr_lemma)
   587     have "P1 x = P1 ?w" using P1eqP1 by blast
   588     also have "\<dots> = P(?w)" using w P1eqP by blast
   589     finally show "P ?w" using P1 by blast
   590   qed
   591 qed
   592 
   593 text {*
   594   \medskip This Theorem combines whithnesses about @{text "P
   595   minusinfinity"} to show one component of the equivalence proof for
   596   Cooper's Theorem. *}
   597 
   598 lemma plusinfinity:
   599   assumes "0 < d" and
   600     P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
   601     ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
   602   shows "(EX x::int. P1 x) --> (EX x::int. P x)"
   603 proof
   604   assume eP1: "EX x. P1 x"
   605   then obtain x where P1: "P1 x" ..
   606   from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
   607   let ?w = "x + (abs(x-z)+1) * d"
   608   show "EX x. P x"
   609   proof
   610     have w: "z < ?w" by(rule incr_lemma)
   611     have "P1 x = P1 ?w" using P1eqP1 by blast
   612     also have "\<dots> = P(?w)" using w P1eqP by blast
   613     finally show "P ?w" using P1 by blast
   614   qed
   615 qed
   616  
   617 text {*
   618   \medskip Theorem for periodic function on discrete sets. *}
   619 
   620 lemma minf_vee:
   621   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   622   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   623   (is "?LHS = ?RHS")
   624 proof
   625   assume ?LHS
   626   then obtain x where P: "P x" ..
   627   have "x mod d = x - (x div d)*d"
   628     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   629   hence Pmod: "P x = P(x mod d)" using modd by simp
   630   show ?RHS
   631   proof (cases)
   632     assume "x mod d = 0"
   633     hence "P 0" using P Pmod by simp
   634     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   635     ultimately have "P d" by simp
   636     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   637     ultimately show ?RHS ..
   638   next
   639     assume not0: "x mod d \<noteq> 0"
   640     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   641     moreover have "x mod d : {1..d}"
   642     proof -
   643       have "0 \<le> x mod d" by(rule pos_mod_sign)
   644       moreover have "x mod d < d" by(rule pos_mod_bound)
   645       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   646     qed
   647     ultimately show ?RHS ..
   648   qed
   649 next
   650   assume ?RHS thus ?LHS by blast
   651 qed
   652 
   653 text {*
   654   \medskip Theorem for periodic function on discrete sets. *}
   655 
   656 lemma pinf_vee:
   657   assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
   658   shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
   659   (is "?LHS = ?RHS")
   660 proof
   661   assume ?LHS
   662   then obtain x where P: "P x" ..
   663   have "x mod d = x + (-(x div d))*d"
   664     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   665   hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
   666   show ?RHS
   667   proof (cases)
   668     assume "x mod d = 0"
   669     hence "P 0" using P Pmod by simp
   670     moreover have "P 0 = P(0 + 1*d)" using modd by blast
   671     ultimately have "P d" by simp
   672     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   673     ultimately show ?RHS ..
   674   next
   675     assume not0: "x mod d \<noteq> 0"
   676     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   677     moreover have "x mod d : {1..d}"
   678     proof -
   679       have "0 \<le> x mod d" by(rule pos_mod_sign)
   680       moreover have "x mod d < d" by(rule pos_mod_bound)
   681       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   682     qed
   683     ultimately show ?RHS ..
   684   qed
   685 next
   686   assume ?RHS thus ?LHS by blast
   687 qed
   688 
   689 lemma decr_mult_lemma:
   690   assumes dpos: "(0::int) < d" and
   691           minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
   692           knneg: "0 <= k"
   693   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   694 using knneg
   695 proof (induct rule:int_ge_induct)
   696   case base thus ?case by simp
   697 next
   698   case (step i)
   699   show ?case
   700   proof
   701     fix x
   702     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   703     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
   704       using minus[THEN spec, of "x - i * d"]
   705       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
   706     ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
   707   qed
   708 qed
   709 
   710 lemma incr_mult_lemma:
   711   assumes dpos: "(0::int) < d" and
   712           plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
   713           knneg: "0 <= k"
   714   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   715 using knneg
   716 proof (induct rule:int_ge_induct)
   717   case base thus ?case by simp
   718 next
   719   case (step i)
   720   show ?case
   721   proof
   722     fix x
   723     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   724     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
   725       using plus[THEN spec, of "x + i * d"]
   726       by (simp add:int_distrib zadd_ac)
   727     ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
   728   qed
   729 qed
   730 
   731 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
   732 ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
   733 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
   734 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
   735 apply(rule iffI)
   736 prefer 2
   737 apply(drule minusinfinity)
   738 apply assumption+
   739 apply(fastsimp)
   740 apply clarsimp
   741 apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
   742 apply(frule_tac x = x and z=z in decr_lemma)
   743 apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
   744 prefer 2
   745 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   746 prefer 2 apply arith
   747  apply fastsimp
   748 apply(drule (1) minf_vee)
   749 apply blast
   750 apply(blast dest:decr_mult_lemma)
   751 done
   752 
   753 text {* Cooper Theorem, plus infinity version. *}
   754 lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
   755 ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) 
   756 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
   757 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
   758   apply(rule iffI)
   759   prefer 2
   760   apply(drule plusinfinity)
   761   apply assumption+
   762   apply(fastsimp)
   763   apply clarsimp
   764   apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
   765   apply(frule_tac x = x and z=z in incr_lemma)
   766   apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
   767   prefer 2
   768   apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
   769   prefer 2 apply arith
   770   apply fastsimp
   771   apply(drule (1) pinf_vee)
   772   apply blast
   773   apply(blast dest:incr_mult_lemma)
   774   done
   775 
   776 
   777 text {*
   778   \bigskip Theorems for the quantifier elminination Functions. *}
   779 
   780 lemma qe_ex_conj: "(EX (x::int). A x) = R
   781 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
   782 		==> (EX (x::int). P x) = (Q & R)"
   783 by blast
   784 
   785 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
   786 		==> (EX (x::int). P x) = Q"
   787 by blast
   788 
   789 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
   790 by blast
   791 
   792 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
   793 by blast
   794 
   795 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
   796 by blast
   797 
   798 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
   799 by blast
   800 
   801 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
   802 by blast
   803 
   804 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
   805 by blast
   806 
   807 text {* \bigskip Theorems for proving NNF *}
   808 
   809 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
   810 by blast
   811 
   812 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
   813 by blast
   814 
   815 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
   816   by blast
   817 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
   818 by blast
   819 
   820 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
   821 by blast
   822 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
   823 by blast
   824 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
   825 by blast
   826 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
   827 by blast
   828 
   829 
   830 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   831   by simp
   832 
   833 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
   834   by iprover
   835 
   836 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
   837   by iprover
   838 
   839 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
   840 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
   841 by blast
   842 
   843 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
   844 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
   845 by blast
   846 
   847 
   848 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   849 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   850 apply(fastsimp)
   851 done
   852 
   853 text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
   854 
   855 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
   856 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
   857 proof
   858   assume ?P
   859   thus ?Q
   860     apply(simp add:dvd_def)
   861     apply clarify
   862     apply(rename_tac d)
   863     apply(drule_tac f = "op * k" in arg_cong)
   864     apply(simp only:int_distrib)
   865     apply(rule_tac x = "d" in exI)
   866     apply(simp only:mult_ac)
   867     done
   868 next
   869   assume ?Q
   870   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
   871   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
   872   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   873   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   874   thus ?P by(simp add:dvd_def)
   875 qed
   876 
   877 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
   878 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
   879 proof
   880   assume P: ?P
   881   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
   882 next
   883   assume ?Q
   884   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
   885   with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
   886   thus ?P by(simp)
   887 qed
   888 
   889 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")
   890 proof
   891   assume ?P
   892   thus ?Q
   893     apply(drule_tac f = "op * k" in arg_cong)
   894     apply(simp only:int_distrib)
   895     done
   896 next
   897   assume ?Q
   898   hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
   899   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
   900   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
   901 qed
   902 
   903 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
   904 proof -
   905   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
   906   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
   907   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])
   908   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
   909   finally show ?thesis .
   910 qed
   911 
   912 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
   913 by arith
   914 
   915 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
   916 by simp
   917 
   918 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
   919 by simp
   920 
   921 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
   922 by simp
   923 
   924 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
   925 by simp
   926 
   927 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
   928 
   929 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   930   by (simp split add: split_nat)
   931 
   932 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   933   apply (simp split add: split_nat)
   934   apply (rule iffI)
   935   apply (erule exE)
   936   apply (rule_tac x = "int x" in exI)
   937   apply simp
   938   apply (erule exE)
   939   apply (rule_tac x = "nat x" in exI)
   940   apply (erule conjE)
   941   apply (erule_tac x = "nat x" in allE)
   942   apply simp
   943   done
   944 
   945 theorem zdiff_int_split: "P (int (x - y)) =
   946   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   947   apply (case_tac "y \<le> x")
   948   apply (simp_all add: zdiff_int)
   949   done
   950 
   951 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
   952   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
   953     nat_0_le cong add: conj_cong)
   954   apply (rule iffI)
   955   apply iprover
   956   apply (erule exE)
   957   apply (case_tac "x=0")
   958   apply (rule_tac x=0 in exI)
   959   apply simp
   960   apply (case_tac "0 \<le> k")
   961   apply iprover
   962   apply (simp add: linorder_not_le)
   963   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
   964   apply assumption
   965   apply (simp add: mult_ac)
   966   done
   967 
   968 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
   969   by simp
   970 
   971 theorem number_of2: "(0::int) <= Numeral0" by simp
   972 
   973 theorem Suc_plus1: "Suc n = n + 1" by simp
   974 
   975 text {*
   976   \medskip Specific instances of congruence rules, to prevent
   977   simplifier from looping. *}
   978 
   979 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
   980   by simp
   981 
   982 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
   983   by (simp cong: conj_cong)
   984 
   985     (* Theorems used in presburger.ML for the computation simpset*)
   986     (* FIXME: They are present in Float.thy, so may be Float.thy should be lightened.*)
   987 
   988 lemma lift_bool: "x \<Longrightarrow> x=True"
   989   by simp
   990 
   991 lemma nlift_bool: "~x \<Longrightarrow> x=False"
   992   by simp
   993 
   994 lemma not_false_eq_true: "(~ False) = True" by simp
   995 
   996 lemma not_true_eq_false: "(~ True) = False" by simp
   997 
   998 
   999 lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
  1000   by simp
  1001 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" 
  1002   by (simp only: iszero_number_of_Pls)
  1003 
  1004 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
  1005   by simp
  1006 
  1007 lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
  1008   by simp
  1009 
  1010 lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)" 
  1011   by simp
  1012 
  1013 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
  1014   by simp
  1015 
  1016 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))" 
  1017   by simp
  1018 
  1019 lemma int_neg_number_of_Min: "neg (-1::int)"
  1020   by simp
  1021 
  1022 lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
  1023   by simp
  1024 
  1025 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (bin_add y (bin_minus x)))::int))"
  1026   by simp
  1027 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)"
  1028   by simp
  1029 
  1030 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))"
  1031   by simp
  1032 
  1033 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)"
  1034   by simp
  1035 
  1036 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)"
  1037   by simp
  1038 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
  1039   by simp
  1040 
  1041 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
  1042   by simp
  1043 
  1044 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
  1045   by simp
  1046 
  1047 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
  1048   by simp
  1049 
  1050 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
  1051   by simp
  1052 
  1053 lemma int_pow_1: "(a::int)^(Numeral1) = a"
  1054   by simp
  1055 
  1056 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
  1057   by simp
  1058 
  1059 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
  1060   by simp
  1061 
  1062 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
  1063   by simp
  1064 
  1065 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
  1066   by simp
  1067 
  1068 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
  1069   by simp
  1070 
  1071 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
  1072 proof -
  1073   have 1:"((-1)::nat) = 0"
  1074     by simp
  1075   show ?thesis by (simp add: 1)
  1076 qed
  1077 
  1078 use "cooper_dec.ML"
  1079 use "reflected_presburger.ML" 
  1080 use "reflected_cooper.ML"
  1081 oracle
  1082   presburger_oracle ("term") = ReflectedCooper.presburger_oracle
  1083 
  1084 use "cooper_proof.ML"
  1085 use "qelim.ML"
  1086 use "presburger.ML"
  1087 
  1088 setup "Presburger.setup"
  1089 
  1090 end