src/HOL/Presburger.thy
 author paulson Mon Jan 12 16:51:45 2004 +0100 (2004-01-12) changeset 14353 79f9fbef9106 parent 14271 8ed6989228bb child 14378 69c4d5997669 permissions -rw-r--r--
Added lemmas to Ring_and_Field with slightly modified simplification rules

Deleted some little-used integer theorems, replacing them by the generic ones
in Ring_and_Field

Consolidated integer powers
```     1 (*  Title:      HOL/Integ/Presburger.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
```
```     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
```
```     5
```
```     6 File containing necessary theorems for the proof
```
```     7 generation for Cooper Algorithm
```
```     8 *)
```
```     9
```
```    10 theory Presburger = NatSimprocs
```
```    11 files
```
```    12   ("cooper_dec.ML")
```
```    13   ("cooper_proof.ML")
```
```    14   ("qelim.ML")
```
```    15   ("presburger.ML"):
```
```    16
```
```    17 (* Theorem for unitifying the coeffitients of x in an existential formula*)
```
```    18
```
```    19 theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
```
```    20   apply (rule iffI)
```
```    21   apply (erule exE)
```
```    22   apply (rule_tac x = "l * x" in exI)
```
```    23   apply simp
```
```    24   apply (erule exE)
```
```    25   apply (erule conjE)
```
```    26   apply (erule dvdE)
```
```    27   apply (rule_tac x = k in exI)
```
```    28   apply simp
```
```    29   done
```
```    30
```
```    31 lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
```
```    32 apply(unfold dvd_def)
```
```    33 apply(rule iffI)
```
```    34 apply(clarsimp)
```
```    35 apply(rename_tac k)
```
```    36 apply(rule_tac x = "-k" in exI)
```
```    37 apply simp
```
```    38 apply(clarsimp)
```
```    39 apply(rename_tac k)
```
```    40 apply(rule_tac x = "-k" in exI)
```
```    41 apply simp
```
```    42 done
```
```    43
```
```    44 lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
```
```    45 apply(unfold dvd_def)
```
```    46 apply(rule iffI)
```
```    47 apply(clarsimp)
```
```    48 apply(rule_tac x = "-k" in exI)
```
```    49 apply simp
```
```    50 apply(clarsimp)
```
```    51 apply(rule_tac x = "-k" in exI)
```
```    52 apply simp
```
```    53 done
```
```    54
```
```    55
```
```    56
```
```    57 (*Theorems for the combination of proofs of the equality of P and P_m for integers x less than some integer z.*)
```
```    58
```
```    59 theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    60   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    61   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
```
```    62   apply (erule exE)+
```
```    63   apply (rule_tac x = "min z1 z2" in exI)
```
```    64   apply simp
```
```    65   done
```
```    66
```
```    67
```
```    68 theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    69   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    70   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
```
```    71
```
```    72   apply (erule exE)+
```
```    73   apply (rule_tac x = "min z1 z2" in exI)
```
```    74   apply simp
```
```    75   done
```
```    76
```
```    77
```
```    78 (*Theorems for the combination of proofs of the equality of P and P_m for integers x greather than some integer z.*)
```
```    79
```
```    80 theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    81   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    82   \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
```
```    83   apply (erule exE)+
```
```    84   apply (rule_tac x = "max z1 z2" in exI)
```
```    85   apply simp
```
```    86   done
```
```    87
```
```    88
```
```    89 theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    90   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    91   \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
```
```    92   apply (erule exE)+
```
```    93   apply (rule_tac x = "max z1 z2" in exI)
```
```    94   apply simp
```
```    95   done
```
```    96 (*=============================================================================*)
```
```    97 (*Theorems for the combination of proofs of the modulo D property for P
```
```    98 pluusinfinity*)
```
```    99 (* FIXME : This is THE SAME theorem as for the minusinf version, but with +k.. instead of -k.. In the future replace these both with only one*)
```
```   100
```
```   101 theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
```
```   102   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
```
```   103   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
```
```   104   by simp
```
```   105
```
```   106
```
```   107 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
```
```   108   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
```
```   109   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
```
```   110   by simp
```
```   111
```
```   112 (*=============================================================================*)
```
```   113 (*This is one of the cases where the simplifed formula is prooved to habe some property
```
```   114 (in relation to P_m) but we need to proove the property for the original formula (P_m)*)
```
```   115 (*FIXME : This is exaclty the same thm as for minusinf.*)
```
```   116 lemma pinf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
```
```   117 by blast
```
```   118
```
```   119
```
```   120
```
```   121 (*=============================================================================*)
```
```   122 (*Theorems for the combination of proofs of the modulo D property for P
```
```   123 minusinfinity*)
```
```   124
```
```   125 theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
```
```   126   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
```
```   127   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
```
```   128   by simp
```
```   129
```
```   130
```
```   131 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
```
```   132   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
```
```   133   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
```
```   134   by simp
```
```   135
```
```   136 (*=============================================================================*)
```
```   137 (*This is one of the cases where the simplifed formula is prooved to habe some property
```
```   138 (in relation to P_m) but we need to proove the property for the original formula (P_m)*)
```
```   139
```
```   140 lemma minf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
```
```   141 by blast
```
```   142
```
```   143 (*=============================================================================*)
```
```   144
```
```   145 (*theorem needed for prooving at runtime divide properties using the arithmetic tatic
```
```   146 (who knows only about modulo = 0)*)
```
```   147
```
```   148 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
```
```   149 by(simp add:dvd_def zmod_eq_0_iff)
```
```   150
```
```   151 (*=============================================================================*)
```
```   152
```
```   153
```
```   154
```
```   155 (*Theorems used for the combination of proof for the backwards direction of cooper's
```
```   156 theorem. they rely exclusively on Predicate calculus.*)
```
```   157
```
```   158 lemma not_ast_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P1(x) --> P1(x + d))
```
```   159 ==>
```
```   160 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
```
```   161 ==>
```
```   162 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<or> P2(x)) --> (P1(x + d) \<or> P2(x + d))) "
```
```   163 by blast
```
```   164
```
```   165
```
```   166
```
```   167 lemma not_ast_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a- j)) --> P1(x) --> P1(x + d))
```
```   168 ==>
```
```   169 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
```
```   170 ==>
```
```   171 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<and> P2(x)) --> (P1(x + d)
```
```   172 \<and> P2(x + d))) "
```
```   173 by blast
```
```   174
```
```   175 lemma not_ast_p_Q_elim: "
```
```   176 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
```
```   177 ==> ( P = Q )
```
```   178 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
```
```   179 by blast
```
```   180 (*=============================================================================*)
```
```   181
```
```   182
```
```   183 (*Theorems used for the combination of proof for the backwards direction of cooper's
```
```   184 theorem. they rely exclusively on Predicate calculus.*)
```
```   185
```
```   186 lemma not_bst_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
```
```   187 ==>
```
```   188 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
```
```   189 ==>
```
```   190 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<or> P2(x)) --> (P1(x - d)
```
```   191 \<or> P2(x-d))) "
```
```   192 by blast
```
```   193
```
```   194
```
```   195
```
```   196 lemma not_bst_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
```
```   197 ==>
```
```   198 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
```
```   199 ==>
```
```   200 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<and> P2(x)) --> (P1(x - d)
```
```   201 \<and> P2(x-d))) "
```
```   202 by blast
```
```   203
```
```   204 lemma not_bst_p_Q_elim: "
```
```   205 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d))
```
```   206 ==> ( P = Q )
```
```   207 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
```
```   208 by blast
```
```   209 (*=============================================================================*)
```
```   210
```
```   211 (*This is the first direction of cooper's theorem*)
```
```   212 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
```
```   213 by blast
```
```   214
```
```   215 (*=============================================================================*)
```
```   216 (*The full cooper's theoorem in its equivalence Form- Given the premisses it is trivial
```
```   217 too, it relies exclusively on prediacte calculus.*)
```
```   218 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
```
```   219 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
```
```   220 by blast
```
```   221
```
```   222 (*=============================================================================*)
```
```   223 (*Some of the atomic theorems generated each time the atom does not depend on x, they
```
```   224 are trivial.*)
```
```   225
```
```   226 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
```
```   227 by blast
```
```   228
```
```   229 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
```
```   230 by blast
```
```   231
```
```   232 lemma not_bst_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> fm --> fm"
```
```   233 by blast
```
```   234
```
```   235
```
```   236
```
```   237 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
```
```   238 by blast
```
```   239
```
```   240 (* The next 2 thms are the same as the minusinf version*)
```
```   241 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
```
```   242 by blast
```
```   243
```
```   244 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"
```
```   245 by blast
```
```   246
```
```   247
```
```   248 (* Theorems to be deleted from simpset when proving simplified formulaes*)
```
```   249 lemma P_eqtrue: "(P=True) = P"
```
```   250   by rules
```
```   251
```
```   252 lemma P_eqfalse: "(P=False) = (~P)"
```
```   253   by rules
```
```   254
```
```   255 (*=============================================================================*)
```
```   256
```
```   257 (*Theorems for the generation of the bachwards direction of cooper's theorem*)
```
```   258 (*These are the 6 interesting atomic cases which have to be proved relying on the
```
```   259 properties of B-set ant the arithmetic and contradiction proofs*)
```
```   260
```
```   261 lemma not_bst_p_lt: "0 < (d::int) ==>
```
```   262  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ( 0 < -x + a) --> (0 < -(x - d) + a )"
```
```   263 by arith
```
```   264
```
```   265 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
```
```   266  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 < (x) + a) --> ( 0 < (x - d) + a)"
```
```   267 apply clarsimp
```
```   268 apply(rule ccontr)
```
```   269 apply(drule_tac x = "x+a" in bspec)
```
```   270 apply(simp add:atLeastAtMost_iff)
```
```   271 apply(drule_tac x = "-a" in bspec)
```
```   272 apply assumption
```
```   273 apply(simp)
```
```   274 done
```
```   275
```
```   276 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
```
```   277  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 = x + a) --> (0 = (x - d) + a )"
```
```   278 apply clarsimp
```
```   279 apply(subgoal_tac "x = -a")
```
```   280  prefer 2 apply arith
```
```   281 apply(drule_tac x = "1" in bspec)
```
```   282 apply(simp add:atLeastAtMost_iff)
```
```   283 apply(drule_tac x = "-a- 1" in bspec)
```
```   284 apply assumption
```
```   285 apply(simp)
```
```   286 done
```
```   287
```
```   288
```
```   289 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
```
```   290  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)"
```
```   291 apply clarsimp
```
```   292 apply(subgoal_tac "x = -a+d")
```
```   293  prefer 2 apply arith
```
```   294 apply(drule_tac x = "d" in bspec)
```
```   295 apply(simp add:atLeastAtMost_iff)
```
```   296 apply(drule_tac x = "-a" in bspec)
```
```   297 apply assumption
```
```   298 apply(simp)
```
```   299 done
```
```   300
```
```   301
```
```   302 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
```
```   303  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )"
```
```   304 apply(clarsimp simp add:dvd_def)
```
```   305 apply(rename_tac m)
```
```   306 apply(rule_tac x = "m - k" in exI)
```
```   307 apply(simp add:int_distrib)
```
```   308 done
```
```   309
```
```   310 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
```
```   311  ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))"
```
```   312 apply(clarsimp simp add:dvd_def)
```
```   313 apply(rename_tac m)
```
```   314 apply(erule_tac x = "m + k" in allE)
```
```   315 apply(simp add:int_distrib)
```
```   316 done
```
```   317
```
```   318
```
```   319
```
```   320 (*Theorems for the generation of the bachwards direction of cooper's theorem*)
```
```   321 (*These are the 6 interesting atomic cases which have to be proved relying on the
```
```   322 properties of A-set ant the arithmetic and 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
```
```   329 lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
```
```   330  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)"
```
```   331   apply clarsimp
```
```   332   apply (rule ccontr)
```
```   333   apply (drule_tac x = "t-x" in bspec)
```
```   334   apply simp
```
```   335   apply (drule_tac x = "t" in bspec)
```
```   336   apply assumption
```
```   337   apply simp
```
```   338   done
```
```   339
```
```   340 lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
```
```   341  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 )"
```
```   342   apply clarsimp
```
```   343   apply (drule_tac x="1" in bspec)
```
```   344   apply simp
```
```   345   apply (drule_tac x="- t + 1" in bspec)
```
```   346   apply assumption
```
```   347   apply(subgoal_tac "x = -t")
```
```   348   prefer 2 apply arith
```
```   349   apply simp
```
```   350   done
```
```   351
```
```   352 lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
```
```   353  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)"
```
```   354   apply clarsimp
```
```   355   apply (subgoal_tac "x = -t-d")
```
```   356   prefer 2 apply arith
```
```   357   apply (drule_tac x = "d" in bspec)
```
```   358   apply simp
```
```   359   apply (drule_tac x = "-t" in bspec)
```
```   360   apply assumption
```
```   361   apply simp
```
```   362   done
```
```   363
```
```   364 lemma not_ast_p_dvd: "(d1::int) dvd d ==>
```
```   365  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 )"
```
```   366   apply(clarsimp simp add:dvd_def)
```
```   367   apply(rename_tac m)
```
```   368   apply(rule_tac x = "m + k" in exI)
```
```   369   apply(simp add:int_distrib)
```
```   370   done
```
```   371
```
```   372 lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
```
```   373  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 ))"
```
```   374   apply(clarsimp simp add:dvd_def)
```
```   375   apply(rename_tac m)
```
```   376   apply(erule_tac x = "m - k" in allE)
```
```   377   apply(simp add:int_distrib)
```
```   378   done
```
```   379
```
```   380
```
```   381
```
```   382 (*=============================================================================*)
```
```   383 (*These are the atomic cases for the proof generation for the modulo D property for P
```
```   384 plusinfinity*)
```
```   385 (*They are fully based on arithmetics*)
```
```   386
```
```   387 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
```
```   388  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
```
```   389   apply(clarsimp simp add:dvd_def)
```
```   390   apply(rule iffI)
```
```   391   apply(clarsimp)
```
```   392   apply(rename_tac n m)
```
```   393   apply(rule_tac x = "m + n*k" in exI)
```
```   394   apply(simp add:int_distrib)
```
```   395   apply(clarsimp)
```
```   396   apply(rename_tac n m)
```
```   397   apply(rule_tac x = "m - n*k" in exI)
```
```   398   apply(simp add:int_distrib mult_ac)
```
```   399   done
```
```   400
```
```   401 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
```
```   402  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
```
```   403   apply(clarsimp simp add:dvd_def)
```
```   404   apply(rule iffI)
```
```   405   apply(clarsimp)
```
```   406   apply(rename_tac n m)
```
```   407   apply(erule_tac x = "m - n*k" in allE)
```
```   408   apply(simp add:int_distrib mult_ac)
```
```   409   apply(clarsimp)
```
```   410   apply(rename_tac n m)
```
```   411   apply(erule_tac x = "m + n*k" in allE)
```
```   412   apply(simp add:int_distrib mult_ac)
```
```   413   done
```
```   414
```
```   415 (*=============================================================================*)
```
```   416 (*These are the atomic cases for the proof generation for the equivalence of P and P
```
```   417 plusinfinity for integers x greather than some integer z.*)
```
```   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
```
```   447
```
```   448
```
```   449 (*=============================================================================*)
```
```   450 (*These are the atomic cases for the proof generation for the modulo D property for P
```
```   451 minusinfinity*)
```
```   452 (*They are fully based on arithmetics*)
```
```   453
```
```   454 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
```
```   455  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
```
```   456 apply(clarsimp simp add:dvd_def)
```
```   457 apply(rule iffI)
```
```   458 apply(clarsimp)
```
```   459 apply(rename_tac n m)
```
```   460 apply(rule_tac x = "m - n*k" in exI)
```
```   461 apply(simp add:int_distrib)
```
```   462 apply(clarsimp)
```
```   463 apply(rename_tac n m)
```
```   464 apply(rule_tac x = "m + n*k" in exI)
```
```   465 apply(simp add:int_distrib mult_ac)
```
```   466 done
```
```   467
```
```   468
```
```   469 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
```
```   470  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
```
```   471 apply(clarsimp simp add:dvd_def)
```
```   472 apply(rule iffI)
```
```   473 apply(clarsimp)
```
```   474 apply(rename_tac n m)
```
```   475 apply(erule_tac x = "m + n*k" in allE)
```
```   476 apply(simp add:int_distrib mult_ac)
```
```   477 apply(clarsimp)
```
```   478 apply(rename_tac n m)
```
```   479 apply(erule_tac x = "m - n*k" in allE)
```
```   480 apply(simp add:int_distrib mult_ac)
```
```   481 done
```
```   482
```
```   483
```
```   484 (*=============================================================================*)
```
```   485 (*These are the atomic cases for the proof generation for the equivalence of P and P
```
```   486 minusinfinity for integers x less than some integer z.*)
```
```   487 (*They are fully based on arithmetics*)
```
```   488
```
```   489 lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
```
```   490 apply(rule_tac x = "-t" in exI)
```
```   491 apply simp
```
```   492 done
```
```   493
```
```   494 lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
```
```   495 apply(rule_tac x = "-t" in exI)
```
```   496 apply simp
```
```   497 done
```
```   498
```
```   499 lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
```
```   500 apply(rule_tac x = "-t" in exI)
```
```   501 apply simp
```
```   502 done
```
```   503
```
```   504
```
```   505 lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
```
```   506 apply(rule_tac x = "t" in exI)
```
```   507 apply simp
```
```   508 done
```
```   509
```
```   510 lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
```
```   511 by simp
```
```   512
```
```   513 lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
```
```   514 by simp
```
```   515
```
```   516
```
```   517 (*=============================================================================*)
```
```   518 (*This Theorem combines whithnesses about P minusinfinity to schow one component of the
```
```   519 equivalence proof for cooper's theorem*)
```
```   520
```
```   521 (* FIXME: remove once they are part of the distribution *)
```
```   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 (*=============================================================================*)
```
```   594 (*This Theorem combines whithnesses about P minusinfinity to schow one component of the
```
```   595 equivalence proof for 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
```
```   617
```
```   618 (*=============================================================================*)
```
```   619 (*Theorem for periodic function on discrete sets*)
```
```   620
```
```   621 lemma minf_vee:
```
```   622   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
```
```   623   shows "(EX x. P x) = (EX j : {1..d}. P j)"
```
```   624   (is "?LHS = ?RHS")
```
```   625 proof
```
```   626   assume ?LHS
```
```   627   then obtain x where P: "P x" ..
```
```   628   have "x mod d = x - (x div d)*d"
```
```   629     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
```
```   630   hence Pmod: "P x = P(x mod d)" using modd by simp
```
```   631   show ?RHS
```
```   632   proof (cases)
```
```   633     assume "x mod d = 0"
```
```   634     hence "P 0" using P Pmod by simp
```
```   635     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
```
```   636     ultimately have "P d" by simp
```
```   637     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
```
```   638     ultimately show ?RHS ..
```
```   639   next
```
```   640     assume not0: "x mod d \<noteq> 0"
```
```   641     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
```
```   642     moreover have "x mod d : {1..d}"
```
```   643     proof -
```
```   644       have "0 \<le> x mod d" by(rule pos_mod_sign)
```
```   645       moreover have "x mod d < d" by(rule pos_mod_bound)
```
```   646       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
```
```   647     qed
```
```   648     ultimately show ?RHS ..
```
```   649   qed
```
```   650 next
```
```   651   assume ?RHS thus ?LHS by blast
```
```   652 qed
```
```   653
```
```   654 (*=============================================================================*)
```
```   655 (*Theorem for periodic function on discrete sets*)
```
```   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 Ring_and_Field.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 (* Cooper Thm `, 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 (*=============================================================================*)
```
```   778
```
```   779 (*Theorems for the quantifier elminination Functions.*)
```
```   780
```
```   781 lemma qe_ex_conj: "(EX (x::int). A x) = R
```
```   782 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
```
```   783 		==> (EX (x::int). P x) = (Q & R)"
```
```   784 by blast
```
```   785
```
```   786 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
```
```   787 		==> (EX (x::int). P x) = Q"
```
```   788 by blast
```
```   789
```
```   790 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
```
```   791 by blast
```
```   792
```
```   793 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
```
```   794 by blast
```
```   795
```
```   796 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
```
```   797 by blast
```
```   798
```
```   799 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
```
```   800 by blast
```
```   801
```
```   802 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
```
```   803 by blast
```
```   804
```
```   805 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
```
```   806 by blast
```
```   807
```
```   808 (* Theorems for proving NNF *)
```
```   809
```
```   810 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
```
```   811 by blast
```
```   812
```
```   813 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
```
```   814 by blast
```
```   815
```
```   816 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
```
```   817   by blast
```
```   818 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
```
```   819 by blast
```
```   820
```
```   821 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
```
```   822 by blast
```
```   823 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
```
```   824 by blast
```
```   825 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
```
```   826 by blast
```
```   827 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
```
```   828 by blast
```
```   829
```
```   830
```
```   831 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
```
```   832   by simp
```
```   833
```
```   834 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
```
```   835   by rules
```
```   836
```
```   837 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
```
```   838   by rules
```
```   839
```
```   840 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
```
```   841 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
```
```   842 by blast
```
```   843
```
```   844 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
```
```   845 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
```
```   846 by blast
```
```   847
```
```   848
```
```   849 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
```
```   850 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   851 apply(fastsimp)
```
```   852 done
```
```   853
```
```   854 (* Theorems required for the adjustcoeffitienteq*)
```
```   855
```
```   856 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
```
```   857 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
```
```   858 proof
```
```   859   assume ?P
```
```   860   thus ?Q
```
```   861     apply(simp add:dvd_def)
```
```   862     apply clarify
```
```   863     apply(rename_tac d)
```
```   864     apply(drule_tac f = "op * k" in arg_cong)
```
```   865     apply(simp only:int_distrib)
```
```   866     apply(rule_tac x = "d" in exI)
```
```   867     apply(simp only:mult_ac)
```
```   868     done
```
```   869 next
```
```   870   assume ?Q
```
```   871   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
```
```   872   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
```
```   873   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
```
```   874   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
```
```   875   thus ?P by(simp add:dvd_def)
```
```   876 qed
```
```   877
```
```   878 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
```
```   879 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
```
```   880 proof
```
```   881   assume P: ?P
```
```   882   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
```
```   883 next
```
```   884   assume ?Q
```
```   885   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
```
```   886   with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
```
```   887   thus ?P by(simp)
```
```   888 qed
```
```   889
```
```   890 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")
```
```   891 proof
```
```   892   assume ?P
```
```   893   thus ?Q
```
```   894     apply(drule_tac f = "op * k" in arg_cong)
```
```   895     apply(simp only:int_distrib)
```
```   896     done
```
```   897 next
```
```   898   assume ?Q
```
```   899   hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
```
```   900   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
```
```   901   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
```
```   902 qed
```
```   903
```
```   904 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
```
```   905 proof -
```
```   906   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
```
```   907   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
```
```   908   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])
```
```   909   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
```
```   910   finally show ?thesis .
```
```   911 qed
```
```   912
```
```   913 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
```
```   914 by arith
```
```   915
```
```   916 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
```
```   917 by simp
```
```   918
```
```   919 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
```
```   920 by simp
```
```   921
```
```   922 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
```
```   923 by simp
```
```   924
```
```   925 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
```
```   926 by simp
```
```   927
```
```   928 (* Theorems for transforming predicates on nat to predicates on int*)
```
```   929
```
```   930 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
```
```   931   by (simp split add: split_nat)
```
```   932
```
```   933 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
```
```   934   apply (simp split add: split_nat)
```
```   935   apply (rule iffI)
```
```   936   apply (erule exE)
```
```   937   apply (rule_tac x = "int x" in exI)
```
```   938   apply simp
```
```   939   apply (erule exE)
```
```   940   apply (rule_tac x = "nat x" in exI)
```
```   941   apply (erule conjE)
```
```   942   apply (erule_tac x = "nat x" in allE)
```
```   943   apply simp
```
```   944   done
```
```   945
```
```   946 theorem zdiff_int_split: "P (int (x - y)) =
```
```   947   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
```
```   948   apply (case_tac "y \<le> x")
```
```   949   apply (simp_all add: zdiff_int)
```
```   950   done
```
```   951
```
```   952 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
```
```   953   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
```
```   954     nat_0_le cong add: conj_cong)
```
```   955   apply (rule iffI)
```
```   956   apply rules
```
```   957   apply (erule exE)
```
```   958   apply (case_tac "x=0")
```
```   959   apply (rule_tac x=0 in exI)
```
```   960   apply simp
```
```   961   apply (case_tac "0 \<le> k")
```
```   962   apply rules
```
```   963   apply (simp add: linorder_not_le)
```
```   964   apply (drule zmult_zless_mono2_neg [OF iffD2 [OF zero_less_int_conv]])
```
```   965   apply assumption
```
```   966   apply (simp add: mult_ac)
```
```   967   done
```
```   968
```
```   969 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
```
```   970   by simp
```
```   971
```
```   972 theorem number_of2: "(0::int) <= number_of bin.Pls" by simp
```
```   973
```
```   974 theorem Suc_plus1: "Suc n = n + 1" by simp
```
```   975
```
```   976 (* specific instances of congruence rules, to prevent simplifier from looping *)
```
```   977
```
```   978 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
```
```   979   by simp
```
```   980
```
```   981 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<and> P) = (0 <= x \<and> P')"
```
```   982   by simp
```
```   983
```
```   984 use "cooper_dec.ML"
```
```   985 use "cooper_proof.ML"
```
```   986 use "qelim.ML"
```
```   987 use "presburger.ML"
```
```   988
```
```   989 setup "Presburger.setup"
```
```   990
```
```   991 end
```