src/HOL/Presburger.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15013 34264f5e4691 child 15140 322485b816ac permissions -rw-r--r--
```     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 import NatSimprocs SetInterval
```
```    13 files ("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
```