src/HOL/Presburger.thy
 author haftmann Wed Nov 22 10:20:12 2006 +0100 (2006-11-22) changeset 21454 a1937c51ed88 parent 21046 fe1db2f991a7 child 22026 cc60e54aa7cb permissions -rw-r--r--
dropped eq const
```     1 (*  Title:      HOL/Integ/Presburger.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
```
```     4
```
```     5 File containing necessary theorems for the proof
```
```     6 generation for Cooper Algorithm
```
```     7 *)
```
```     8
```
```     9 header {* Presburger Arithmetic: Cooper's Algorithm *}
```
```    10
```
```    11 theory Presburger
```
```    12 imports NatSimprocs
```
```    13 uses
```
```    14   ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML")
```
```    15   ("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
```
```    16 begin
```
```    17
```
```    18 text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
```
```    19
```
```    20 theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
```
```    21   apply (rule iffI)
```
```    22   apply (erule exE)
```
```    23   apply (rule_tac x = "l * x" in exI)
```
```    24   apply simp
```
```    25   apply (erule exE)
```
```    26   apply (erule conjE)
```
```    27   apply (erule dvdE)
```
```    28   apply (rule_tac x = k in exI)
```
```    29   apply simp
```
```    30   done
```
```    31
```
```    32 lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
```
```    33 apply(unfold dvd_def)
```
```    34 apply(rule iffI)
```
```    35 apply(clarsimp)
```
```    36 apply(rename_tac k)
```
```    37 apply(rule_tac x = "-k" in exI)
```
```    38 apply simp
```
```    39 apply(clarsimp)
```
```    40 apply(rename_tac k)
```
```    41 apply(rule_tac x = "-k" in exI)
```
```    42 apply simp
```
```    43 done
```
```    44
```
```    45 lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
```
```    46 apply(unfold dvd_def)
```
```    47 apply(rule iffI)
```
```    48 apply(clarsimp)
```
```    49 apply(rule_tac x = "-k" in exI)
```
```    50 apply simp
```
```    51 apply(clarsimp)
```
```    52 apply(rule_tac x = "-k" in exI)
```
```    53 apply simp
```
```    54 done
```
```    55
```
```    56
```
```    57
```
```    58 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}.*}
```
```    59
```
```    60 theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    61   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    62   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
```
```    63   apply (erule exE)+
```
```    64   apply (rule_tac x = "min z1 z2" in exI)
```
```    65   apply simp
```
```    66   done
```
```    67
```
```    68
```
```    69 theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    70   \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    71   \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
```
```    72
```
```    73   apply (erule exE)+
```
```    74   apply (rule_tac x = "min z1 z2" in exI)
```
```    75   apply simp
```
```    76   done
```
```    77
```
```    78
```
```    79 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}.*}
```
```    80
```
```    81 theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    82   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    83   \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
```
```    84   apply (erule exE)+
```
```    85   apply (rule_tac x = "max z1 z2" in exI)
```
```    86   apply simp
```
```    87   done
```
```    88
```
```    89
```
```    90 theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
```
```    91   \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
```
```    92   \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
```
```    93   apply (erule exE)+
```
```    94   apply (rule_tac x = "max z1 z2" in exI)
```
```    95   apply simp
```
```    96   done
```
```    97
```
```    98 text {*
```
```    99   \medskip Theorems for the combination of proofs of the modulo @{text
```
```   100   D} property for @{text "P plusinfinity"}
```
```   101
```
```   102   FIXME: This is THE SAME theorem as for the @{text minusinf} version,
```
```   103   but with @{text "+k.."} instead of @{text "-k.."} In the future
```
```   104   replace these both with only one. *}
```
```   105
```
```   106 theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
```
```   107   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
```
```   108   \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
```
```   109   by simp
```
```   110
```
```   111 theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
```
```   112   \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
```
```   113   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
```
```   114   by simp
```
```   115
```
```   116 text {*
```
```   117   This is one of the cases where the simplifed formula is prooved to
```
```   118   habe some property (in relation to @{text P_m}) but we need to prove
```
```   119   the property for the original formula (@{text P_m})
```
```   120
```
```   121   FIXME: This is exaclty the same thm as for @{text minusinf}. *}
```
```   122
```
```   123 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)) "
```
```   124   by blast
```
```   125
```
```   126
```
```   127 text {*
```
```   128   \medskip Theorems for the combination of proofs of the modulo @{text D}
```
```   129   property for @{text "P minusinfinity"} *}
```
```   130
```
```   131 theorem modd_minf_conjI: "\<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 \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
```
```   134   by simp
```
```   135
```
```   136 theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
```
```   137   \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
```
```   138   \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
```
```   139   by simp
```
```   140
```
```   141 text {*
```
```   142   This is one of the cases where the simplifed formula is prooved to
```
```   143   have some property (in relation to @{text P_m}) but we need to
```
```   144   prove the property for the original formula (@{text P_m}). *}
```
```   145
```
```   146 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)) "
```
```   147   by blast
```
```   148
```
```   149 text {*
```
```   150   Theorem needed for proving at runtime divide properties using the
```
```   151   arithmetic tactic (which knows only about modulo = 0). *}
```
```   152
```
```   153 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
```
```   154   by(simp add:dvd_def zmod_eq_0_iff)
```
```   155
```
```   156 text {*
```
```   157   \medskip Theorems used for the combination of proof for the
```
```   158   backwards direction of Cooper's Theorem. They rely exclusively on
```
```   159   Predicate calculus.*}
```
```   160
```
```   161 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))
```
```   162 ==>
```
```   163 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
```
```   164 ==>
```
```   165 (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))) "
```
```   166   by blast
```
```   167
```
```   168
```
```   169 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))
```
```   170 ==>
```
```   171 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
```
```   172 ==>
```
```   173 (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)
```
```   174 \<and> P2(x + d))) "
```
```   175   by blast
```
```   176
```
```   177 lemma not_ast_p_Q_elim: "
```
```   178 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
```
```   179 ==> ( P = Q )
```
```   180 ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
```
```   181   by blast
```
```   182
```
```   183 text {*
```
```   184   \medskip Theorems used for the combination of proof for the
```
```   185   backwards direction of Cooper's Theorem. They rely exclusively on
```
```   186   Predicate calculus.*}
```
```   187
```
```   188 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))
```
```   189 ==>
```
```   190 (ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
```
```   191 ==>
```
```   192 (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)
```
```   193 \<or> P2(x-d))) "
```
```   194   by blast
```
```   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 text {* \medskip This is the first direction of Cooper's Theorem. *}
```
```   211 lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
```
```   212   by blast
```
```   213
```
```   214 text {*
```
```   215   \medskip The full Cooper's Theorem in its equivalence Form. Given
```
```   216   the premises it is trivial too, it relies exclusively on prediacte calculus.*}
```
```   217 lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
```
```   218 --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
```
```   219   by blast
```
```   220
```
```   221 text {*
```
```   222   \medskip Some of the atomic theorems generated each time the atom
```
```   223   does not depend on @{text x}, they are trivial.*}
```
```   224
```
```   225 lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
```
```   226   by blast
```
```   227
```
```   228 lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
```
```   229   by blast
```
```   230
```
```   231 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"
```
```   232   by blast
```
```   233
```
```   234 lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
```
```   235   by blast
```
```   236
```
```   237 text {* The next two thms are the same as the @{text minusinf} version. *}
```
```   238
```
```   239 lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
```
```   240   by blast
```
```   241
```
```   242 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"
```
```   243   by blast
```
```   244
```
```   245 text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
```
```   246
```
```   247 lemma P_eqtrue: "(P=True) = P"
```
```   248   by iprover
```
```   249
```
```   250 lemma P_eqfalse: "(P=False) = (~P)"
```
```   251   by iprover
```
```   252
```
```   253 text {*
```
```   254   \medskip Theorems for the generation of the bachwards direction of
```
```   255   Cooper's Theorem.
```
```   256
```
```   257   These are the 6 interesting atomic cases which have to be proved relying on the
```
```   258   properties of B-set and the arithmetic and contradiction proofs. *}
```
```   259
```
```   260 lemma not_bst_p_lt: "0 < (d::int) ==>
```
```   261  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 )"
```
```   262   by arith
```
```   263
```
```   264 lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
```
```   265  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)"
```
```   266 apply clarsimp
```
```   267 apply(rule ccontr)
```
```   268 apply(drule_tac x = "x+a" in bspec)
```
```   269 apply(simp add:atLeastAtMost_iff)
```
```   270 apply(drule_tac x = "-a" in bspec)
```
```   271 apply assumption
```
```   272 apply(simp)
```
```   273 done
```
```   274
```
```   275 lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
```
```   276  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 )"
```
```   277 apply clarsimp
```
```   278 apply(subgoal_tac "x = -a")
```
```   279  prefer 2 apply arith
```
```   280 apply(drule_tac x = "1" in bspec)
```
```   281 apply(simp add:atLeastAtMost_iff)
```
```   282 apply(drule_tac x = "-a- 1" in bspec)
```
```   283 apply assumption
```
```   284 apply(simp)
```
```   285 done
```
```   286
```
```   287
```
```   288 lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
```
```   289  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)"
```
```   290 apply clarsimp
```
```   291 apply(subgoal_tac "x = -a+d")
```
```   292  prefer 2 apply arith
```
```   293 apply(drule_tac x = "d" in bspec)
```
```   294 apply(simp add:atLeastAtMost_iff)
```
```   295 apply(drule_tac x = "-a" in bspec)
```
```   296 apply assumption
```
```   297 apply(simp)
```
```   298 done
```
```   299
```
```   300
```
```   301 lemma not_bst_p_dvd: "(d1::int) dvd d ==>
```
```   302  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 )"
```
```   303 apply(clarsimp simp add:dvd_def)
```
```   304 apply(rename_tac m)
```
```   305 apply(rule_tac x = "m - k" in exI)
```
```   306 apply(simp add:int_distrib)
```
```   307 done
```
```   308
```
```   309 lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
```
```   310  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 ))"
```
```   311 apply(clarsimp simp add:dvd_def)
```
```   312 apply(rename_tac m)
```
```   313 apply(erule_tac x = "m + k" in allE)
```
```   314 apply(simp add:int_distrib)
```
```   315 done
```
```   316
```
```   317 text {*
```
```   318   \medskip Theorems for the generation of the bachwards direction of
```
```   319   Cooper's Theorem.
```
```   320
```
```   321   These are the 6 interesting atomic cases which have to be proved
```
```   322   relying on the properties of A-set ant the arithmetic and
```
```   323   contradiction proofs. *}
```
```   324
```
```   325 lemma not_ast_p_gt: "0 < (d::int) ==>
```
```   326  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 )"
```
```   327   by arith
```
```   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 text {*
```
```   381   \medskip These are the atomic cases for the proof generation for the
```
```   382   modulo @{text D} property for @{text "P plusinfinity"}
```
```   383
```
```   384   They are fully based on arithmetics. *}
```
```   385
```
```   386 lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
```
```   387  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
```
```   388   apply(clarsimp simp add:dvd_def)
```
```   389   apply(rule iffI)
```
```   390   apply(clarsimp)
```
```   391   apply(rename_tac n m)
```
```   392   apply(rule_tac x = "m + n*k" in exI)
```
```   393   apply(simp add:int_distrib)
```
```   394   apply(clarsimp)
```
```   395   apply(rename_tac n m)
```
```   396   apply(rule_tac x = "m - n*k" in exI)
```
```   397   apply(simp add:int_distrib mult_ac)
```
```   398   done
```
```   399
```
```   400 lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
```
```   401  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
```
```   402   apply(clarsimp simp add:dvd_def)
```
```   403   apply(rule iffI)
```
```   404   apply(clarsimp)
```
```   405   apply(rename_tac n m)
```
```   406   apply(erule_tac x = "m - n*k" in allE)
```
```   407   apply(simp add:int_distrib mult_ac)
```
```   408   apply(clarsimp)
```
```   409   apply(rename_tac n m)
```
```   410   apply(erule_tac x = "m + n*k" in allE)
```
```   411   apply(simp add:int_distrib mult_ac)
```
```   412   done
```
```   413
```
```   414 text {*
```
```   415   \medskip These are the atomic cases for the proof generation for the
```
```   416   equivalence of @{text P} and @{text "P plusinfinity"} for integers
```
```   417   @{text x} greater than some integer @{text z}.
```
```   418
```
```   419   They are fully based on arithmetics. *}
```
```   420
```
```   421 lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
```
```   422   apply(rule_tac x = "-t" in exI)
```
```   423   apply simp
```
```   424   done
```
```   425
```
```   426 lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
```
```   427   apply(rule_tac x = "-t" in exI)
```
```   428   apply simp
```
```   429   done
```
```   430
```
```   431 lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
```
```   432   apply(rule_tac x = "-t" in exI)
```
```   433   apply simp
```
```   434   done
```
```   435
```
```   436 lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
```
```   437   apply(rule_tac x = "t" in exI)
```
```   438   apply simp
```
```   439   done
```
```   440
```
```   441 lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
```
```   442   by simp
```
```   443
```
```   444 lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
```
```   445   by simp
```
```   446
```
```   447 text {*
```
```   448   \medskip These are the atomic cases for the proof generation for the
```
```   449   modulo @{text D} property for @{text "P minusinfinity"}.
```
```   450
```
```   451   They are fully based on arithmetics. *}
```
```   452
```
```   453 lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
```
```   454  (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
```
```   455 apply(clarsimp simp add:dvd_def)
```
```   456 apply(rule iffI)
```
```   457 apply(clarsimp)
```
```   458 apply(rename_tac n m)
```
```   459 apply(rule_tac x = "m - n*k" in exI)
```
```   460 apply(simp add:int_distrib)
```
```   461 apply(clarsimp)
```
```   462 apply(rename_tac n m)
```
```   463 apply(rule_tac x = "m + n*k" in exI)
```
```   464 apply(simp add:int_distrib mult_ac)
```
```   465 done
```
```   466
```
```   467
```
```   468 lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
```
```   469  (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
```
```   470 apply(clarsimp simp add:dvd_def)
```
```   471 apply(rule iffI)
```
```   472 apply(clarsimp)
```
```   473 apply(rename_tac n m)
```
```   474 apply(erule_tac x = "m + n*k" in allE)
```
```   475 apply(simp add:int_distrib mult_ac)
```
```   476 apply(clarsimp)
```
```   477 apply(rename_tac n m)
```
```   478 apply(erule_tac x = "m - n*k" in allE)
```
```   479 apply(simp add:int_distrib mult_ac)
```
```   480 done
```
```   481
```
```   482 text {*
```
```   483   \medskip These are the atomic cases for the proof generation for the
```
```   484   equivalence of @{text P} and @{text "P minusinfinity"} for integers
```
```   485   @{text x} less than some integer @{text z}.
```
```   486
```
```   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 text {*
```
```   517   \medskip This Theorem combines whithnesses about @{text "P
```
```   518   minusinfinity"} to show one component of the equivalence proof for
```
```   519   Cooper's Theorem.
```
```   520
```
```   521   FIXME: remove once they are part of the distribution. *}
```
```   522
```
```   523 theorem int_ge_induct[consumes 1,case_names base step]:
```
```   524   assumes ge: "k \<le> (i::int)" and
```
```   525         base: "P(k)" and
```
```   526         step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   527   shows "P i"
```
```   528 proof -
```
```   529   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
```
```   530     proof (induct n)
```
```   531       case 0
```
```   532       hence "i = k" by arith
```
```   533       thus "P i" using base by simp
```
```   534     next
```
```   535       case (Suc n)
```
```   536       hence "n = nat((i - 1) - k)" by arith
```
```   537       moreover
```
```   538       have ki1: "k \<le> i - 1" using Suc.prems by arith
```
```   539       ultimately
```
```   540       have "P(i - 1)" by(rule Suc.hyps)
```
```   541       from step[OF ki1 this] show ?case by simp
```
```   542     qed
```
```   543   }
```
```   544   from this ge show ?thesis by fast
```
```   545 qed
```
```   546
```
```   547 theorem int_gr_induct[consumes 1,case_names base step]:
```
```   548   assumes gr: "k < (i::int)" and
```
```   549         base: "P(k+1)" and
```
```   550         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
```
```   551   shows "P i"
```
```   552 apply(rule int_ge_induct[of "k + 1"])
```
```   553   using gr apply arith
```
```   554  apply(rule base)
```
```   555 apply(rule step)
```
```   556  apply simp+
```
```   557 done
```
```   558
```
```   559 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
```
```   560 apply(induct rule: int_gr_induct)
```
```   561  apply simp
```
```   562 apply (simp add:int_distrib)
```
```   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 (simp add:int_distrib)
```
```   569 done
```
```   570
```
```   571 lemma  minusinfinity:
```
```   572   assumes "0 < d" and
```
```   573     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
```
```   574     ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
```
```   575   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
```
```   576 proof
```
```   577   assume eP1: "EX x. P1 x"
```
```   578   then obtain x where P1: "P1 x" ..
```
```   579   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
```
```   580   let ?w = "x - (abs(x-z)+1) * d"
```
```   581   show "EX x. P x"
```
```   582   proof
```
```   583     have w: "?w < z" by(rule decr_lemma)
```
```   584     have "P1 x = P1 ?w" using P1eqP1 by blast
```
```   585     also have "\<dots> = P(?w)" using w P1eqP by blast
```
```   586     finally show "P ?w" using P1 by blast
```
```   587   qed
```
```   588 qed
```
```   589
```
```   590 text {*
```
```   591   \medskip This Theorem combines whithnesses about @{text "P
```
```   592   minusinfinity"} to show one component of the equivalence proof for
```
```   593   Cooper's Theorem. *}
```
```   594
```
```   595 lemma plusinfinity:
```
```   596   assumes "0 < d" and
```
```   597     P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
```
```   598     ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
```
```   599   shows "(EX x::int. P1 x) --> (EX x::int. P x)"
```
```   600 proof
```
```   601   assume eP1: "EX x. P1 x"
```
```   602   then obtain x where P1: "P1 x" ..
```
```   603   from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
```
```   604   let ?w = "x + (abs(x-z)+1) * d"
```
```   605   show "EX x. P x"
```
```   606   proof
```
```   607     have w: "z < ?w" by(rule incr_lemma)
```
```   608     have "P1 x = P1 ?w" using P1eqP1 by blast
```
```   609     also have "\<dots> = P(?w)" using w P1eqP by blast
```
```   610     finally show "P ?w" using P1 by blast
```
```   611   qed
```
```   612 qed
```
```   613
```
```   614 text {*
```
```   615   \medskip Theorem for periodic function on discrete sets. *}
```
```   616
```
```   617 lemma minf_vee:
```
```   618   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
```
```   619   shows "(EX x. P x) = (EX j : {1..d}. P j)"
```
```   620   (is "?LHS = ?RHS")
```
```   621 proof
```
```   622   assume ?LHS
```
```   623   then obtain x where P: "P x" ..
```
```   624   have "x mod d = x - (x div d)*d"
```
```   625     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
```
```   626   hence Pmod: "P x = P(x mod d)" using modd by simp
```
```   627   show ?RHS
```
```   628   proof (cases)
```
```   629     assume "x mod d = 0"
```
```   630     hence "P 0" using P Pmod by simp
```
```   631     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
```
```   632     ultimately have "P d" by simp
```
```   633     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
```
```   634     ultimately show ?RHS ..
```
```   635   next
```
```   636     assume not0: "x mod d \<noteq> 0"
```
```   637     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
```
```   638     moreover have "x mod d : {1..d}"
```
```   639     proof -
```
```   640       have "0 \<le> x mod d" by(rule pos_mod_sign)
```
```   641       moreover have "x mod d < d" by(rule pos_mod_bound)
```
```   642       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
```
```   643     qed
```
```   644     ultimately show ?RHS ..
```
```   645   qed
```
```   646 next
```
```   647   assume ?RHS thus ?LHS by blast
```
```   648 qed
```
```   649
```
```   650 text {*
```
```   651   \medskip Theorem for periodic function on discrete sets. *}
```
```   652
```
```   653 lemma pinf_vee:
```
```   654   assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
```
```   655   shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
```
```   656   (is "?LHS = ?RHS")
```
```   657 proof
```
```   658   assume ?LHS
```
```   659   then obtain x where P: "P x" ..
```
```   660   have "x mod d = x + (-(x div d))*d"
```
```   661     by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
```
```   662   hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
```
```   663   show ?RHS
```
```   664   proof (cases)
```
```   665     assume "x mod d = 0"
```
```   666     hence "P 0" using P Pmod by simp
```
```   667     moreover have "P 0 = P(0 + 1*d)" using modd by blast
```
```   668     ultimately have "P d" by simp
```
```   669     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
```
```   670     ultimately show ?RHS ..
```
```   671   next
```
```   672     assume not0: "x mod d \<noteq> 0"
```
```   673     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
```
```   674     moreover have "x mod d : {1..d}"
```
```   675     proof -
```
```   676       have "0 \<le> x mod d" by(rule pos_mod_sign)
```
```   677       moreover have "x mod d < d" by(rule pos_mod_bound)
```
```   678       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
```
```   679     qed
```
```   680     ultimately show ?RHS ..
```
```   681   qed
```
```   682 next
```
```   683   assume ?RHS thus ?LHS by blast
```
```   684 qed
```
```   685
```
```   686 lemma decr_mult_lemma:
```
```   687   assumes dpos: "(0::int) < d" and
```
```   688           minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
```
```   689           knneg: "0 <= k"
```
```   690   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
```
```   691 using knneg
```
```   692 proof (induct rule:int_ge_induct)
```
```   693   case base thus ?case by simp
```
```   694 next
```
```   695   case (step i)
```
```   696   show ?case
```
```   697   proof
```
```   698     fix x
```
```   699     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
```
```   700     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
```
```   701       using minus[THEN spec, of "x - i * d"]
```
```   702       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
```
```   703     ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
```
```   704   qed
```
```   705 qed
```
```   706
```
```   707 lemma incr_mult_lemma:
```
```   708   assumes dpos: "(0::int) < d" and
```
```   709           plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
```
```   710           knneg: "0 <= k"
```
```   711   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
```
```   712 using knneg
```
```   713 proof (induct rule:int_ge_induct)
```
```   714   case base thus ?case by simp
```
```   715 next
```
```   716   case (step i)
```
```   717   show ?case
```
```   718   proof
```
```   719     fix x
```
```   720     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
```
```   721     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
```
```   722       using plus[THEN spec, of "x + i * d"]
```
```   723       by (simp add:int_distrib zadd_ac)
```
```   724     ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
```
```   725   qed
```
```   726 qed
```
```   727
```
```   728 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
```
```   729 ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
```
```   730 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
```
```   731 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
```
```   732 apply(rule iffI)
```
```   733 prefer 2
```
```   734 apply(drule minusinfinity)
```
```   735 apply assumption+
```
```   736 apply(fastsimp)
```
```   737 apply clarsimp
```
```   738 apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
```
```   739 apply(frule_tac x = x and z=z in decr_lemma)
```
```   740 apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
```
```   741 prefer 2
```
```   742 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
```
```   743 prefer 2 apply arith
```
```   744  apply fastsimp
```
```   745 apply(drule (1) minf_vee)
```
```   746 apply blast
```
```   747 apply(blast dest:decr_mult_lemma)
```
```   748 done
```
```   749
```
```   750 text {* Cooper Theorem, plus infinity version. *}
```
```   751 lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
```
```   752 ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D)
```
```   753 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
```
```   754 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
```
```   755   apply(rule iffI)
```
```   756   prefer 2
```
```   757   apply(drule plusinfinity)
```
```   758   apply assumption+
```
```   759   apply(fastsimp)
```
```   760   apply clarsimp
```
```   761   apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
```
```   762   apply(frule_tac x = x and z=z in incr_lemma)
```
```   763   apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
```
```   764   prefer 2
```
```   765   apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
```
```   766   prefer 2 apply arith
```
```   767   apply fastsimp
```
```   768   apply(drule (1) pinf_vee)
```
```   769   apply blast
```
```   770   apply(blast dest:incr_mult_lemma)
```
```   771   done
```
```   772
```
```   773
```
```   774 text {*
```
```   775   \bigskip Theorems for the quantifier elminination Functions. *}
```
```   776
```
```   777 lemma qe_ex_conj: "(EX (x::int). A x) = R
```
```   778 		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
```
```   779 		==> (EX (x::int). P x) = (Q & R)"
```
```   780 by blast
```
```   781
```
```   782 lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
```
```   783 		==> (EX (x::int). P x) = Q"
```
```   784 by blast
```
```   785
```
```   786 lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
```
```   787 by blast
```
```   788
```
```   789 lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
```
```   790 by blast
```
```   791
```
```   792 lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
```
```   793 by blast
```
```   794
```
```   795 lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
```
```   796 by blast
```
```   797
```
```   798 lemma qe_Not: "P = Q ==> (~P) = (~Q)"
```
```   799 by blast
```
```   800
```
```   801 lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
```
```   802 by blast
```
```   803
```
```   804 text {* \bigskip Theorems for proving NNF *}
```
```   805
```
```   806 lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
```
```   807 by blast
```
```   808
```
```   809 lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
```
```   810 by blast
```
```   811
```
```   812 lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
```
```   813   by blast
```
```   814 lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
```
```   815 by blast
```
```   816
```
```   817 lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
```
```   818 by blast
```
```   819 lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
```
```   820 by blast
```
```   821 lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
```
```   822 by blast
```
```   823 lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
```
```   824 by blast
```
```   825
```
```   826
```
```   827 lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
```
```   828   by simp
```
```   829
```
```   830 lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
```
```   831   by iprover
```
```   832
```
```   833 lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
```
```   834   by iprover
```
```   835
```
```   836 lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
```
```   837 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
```
```   838 by blast
```
```   839
```
```   840 lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
```
```   841 ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
```
```   842 by blast
```
```   843
```
```   844
```
```   845 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
```
```   846 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   847 apply(fastsimp)
```
```   848 done
```
```   849
```
```   850 text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
```
```   851
```
```   852 lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
```
```   853 shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
```
```   854 proof
```
```   855   assume ?P
```
```   856   thus ?Q
```
```   857     apply(simp add:dvd_def)
```
```   858     apply clarify
```
```   859     apply(rename_tac d)
```
```   860     apply(drule_tac f = "op * k" in arg_cong)
```
```   861     apply(simp only:int_distrib)
```
```   862     apply(rule_tac x = "d" in exI)
```
```   863     apply(simp only:mult_ac)
```
```   864     done
```
```   865 next
```
```   866   assume ?Q
```
```   867   then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
```
```   868   hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
```
```   869   hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
```
```   870   hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
```
```   871   thus ?P by(simp add:dvd_def)
```
```   872 qed
```
```   873
```
```   874 lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
```
```   875 shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
```
```   876 proof
```
```   877   assume P: ?P
```
```   878   show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
```
```   879 next
```
```   880   assume ?Q
```
```   881   hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
```
```   882   with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
```
```   883   thus ?P by(simp)
```
```   884 qed
```
```   885
```
```   886 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")
```
```   887 proof
```
```   888   assume ?P
```
```   889   thus ?Q
```
```   890     apply(drule_tac f = "op * k" in arg_cong)
```
```   891     apply(simp only:int_distrib)
```
```   892     done
```
```   893 next
```
```   894   assume ?Q
```
```   895   hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
```
```   896   hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
```
```   897   thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
```
```   898 qed
```
```   899
```
```   900 lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
```
```   901 proof -
```
```   902   have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
```
```   903   also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
```
```   904   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])
```
```   905   also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
```
```   906   finally show ?thesis .
```
```   907 qed
```
```   908
```
```   909 lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
```
```   910 by arith
```
```   911
```
```   912 lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
```
```   913 by simp
```
```   914
```
```   915 lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
```
```   916 by simp
```
```   917
```
```   918 lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
```
```   919 by simp
```
```   920
```
```   921 lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
```
```   922 by simp
```
```   923
```
```   924 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
```
```   925
```
```   926 theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
```
```   927   by (simp split add: split_nat)
```
```   928
```
```   929 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
```
```   930   apply (simp split add: split_nat)
```
```   931   apply (rule iffI)
```
```   932   apply (erule exE)
```
```   933   apply (rule_tac x = "int x" in exI)
```
```   934   apply simp
```
```   935   apply (erule exE)
```
```   936   apply (rule_tac x = "nat x" in exI)
```
```   937   apply (erule conjE)
```
```   938   apply (erule_tac x = "nat x" in allE)
```
```   939   apply simp
```
```   940   done
```
```   941
```
```   942 theorem zdiff_int_split: "P (int (x - y)) =
```
```   943   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
```
```   944   apply (case_tac "y \<le> x")
```
```   945   apply (simp_all add: zdiff_int)
```
```   946   done
```
```   947
```
```   948 theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
```
```   949   apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
```
```   950     nat_0_le cong add: conj_cong)
```
```   951   apply (rule iffI)
```
```   952   apply iprover
```
```   953   apply (erule exE)
```
```   954   apply (case_tac "x=0")
```
```   955   apply (rule_tac x=0 in exI)
```
```   956   apply simp
```
```   957   apply (case_tac "0 \<le> k")
```
```   958   apply iprover
```
```   959   apply (simp add: linorder_not_le)
```
```   960   apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
```
```   961   apply assumption
```
```   962   apply (simp add: mult_ac)
```
```   963   done
```
```   964
```
```   965 theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
```
```   966   by simp
```
```   967
```
```   968 theorem number_of2: "(0::int) <= Numeral0" by simp
```
```   969
```
```   970 theorem Suc_plus1: "Suc n = n + 1" by simp
```
```   971
```
```   972 text {*
```
```   973   \medskip Specific instances of congruence rules, to prevent
```
```   974   simplifier from looping. *}
```
```   975
```
```   976 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
```
```   977   by simp
```
```   978
```
```   979 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
```
```   980   by (simp cong: conj_cong)
```
```   981
```
```   982     (* Theorems used in presburger.ML for the computation simpset*)
```
```   983     (* FIXME: They are present in Float.thy, so may be Float.thy should be lightened.*)
```
```   984
```
```   985 lemma lift_bool: "x \<Longrightarrow> x=True"
```
```   986   by simp
```
```   987
```
```   988 lemma nlift_bool: "~x \<Longrightarrow> x=False"
```
```   989   by simp
```
```   990
```
```   991 lemma not_false_eq_true: "(~ False) = True" by simp
```
```   992
```
```   993 lemma not_true_eq_false: "(~ True) = False" by simp
```
```   994
```
```   995
```
```   996 lemma int_eq_number_of_eq:
```
```   997   "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
```
```   998   by simp
```
```   999 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
```
```  1000   by (simp only: iszero_number_of_Pls)
```
```  1001
```
```  1002 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
```
```  1003   by simp
```
```  1004
```
```  1005 lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
```
```  1006   by simp
```
```  1007
```
```  1008 lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
```
```  1009   by simp
```
```  1010
```
```  1011 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
```
```  1012   by simp
```
```  1013
```
```  1014 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
```
```  1015   by simp
```
```  1016
```
```  1017 lemma int_neg_number_of_Min: "neg (-1::int)"
```
```  1018   by simp
```
```  1019
```
```  1020 lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
```
```  1021   by simp
```
```  1022
```
```  1023 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
```
```  1024   by simp
```
```  1025 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
```
```  1026   by simp
```
```  1027
```
```  1028 lemma int_number_of_diff_sym:
```
```  1029   "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
```
```  1030   by simp
```
```  1031
```
```  1032 lemma int_number_of_mult_sym:
```
```  1033   "((number_of v)::int) * number_of w = number_of (v * w)"
```
```  1034   by simp
```
```  1035
```
```  1036 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
```
```  1037   by simp
```
```  1038 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
```
```  1039   by simp
```
```  1040
```
```  1041 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
```
```  1042   by simp
```
```  1043
```
```  1044 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
```
```  1045   by simp
```
```  1046
```
```  1047 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
```
```  1048   by simp
```
```  1049
```
```  1050 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
```
```  1051   by simp
```
```  1052
```
```  1053 lemma int_pow_1: "(a::int)^(Numeral1) = a"
```
```  1054   by simp
```
```  1055
```
```  1056 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
```
```  1057   by simp
```
```  1058
```
```  1059 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
```
```  1060   by simp
```
```  1061
```
```  1062 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
```
```  1063   by simp
```
```  1064
```
```  1065 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
```
```  1066   by simp
```
```  1067
```
```  1068 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
```
```  1069   by simp
```
```  1070
```
```  1071 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
```
```  1072 proof -
```
```  1073   have 1:"((-1)::nat) = 0"
```
```  1074     by simp
```
```  1075   show ?thesis by (simp add: 1)
```
```  1076 qed
```
```  1077
```
```  1078 use "cooper_dec.ML"
```
```  1079 use "reflected_presburger.ML"
```
```  1080 use "reflected_cooper.ML"
```
```  1081 oracle
```
```  1082   presburger_oracle ("term") = ReflectedCooper.presburger_oracle
```
```  1083
```
```  1084 use "cooper_proof.ML"
```
```  1085 use "qelim.ML"
```
```  1086 use "presburger.ML"
```
```  1087
```
```  1088 setup "Presburger.setup"
```
```  1089
```
```  1090 text {* Code generator setup *}
```
```  1091
```
```  1092 text {*
```
```  1093   Presburger arithmetic is necessary (at least convenient) to prove some
```
```  1094   of the following code lemmas on integer numerals.
```
```  1095 *}
```
```  1096
```
```  1097 lemma eq_number_of [code func]:
```
```  1098   "((number_of k)\<Colon>int) = number_of l \<longleftrightarrow> k = l"
```
```  1099   unfolding number_of_is_id ..
```
```  1100
```
```  1101 lemma less_eq_number_of [code func]:
```
```  1102   "((number_of k)\<Colon>int) <= number_of l \<longleftrightarrow> k <= l"
```
```  1103   unfolding number_of_is_id ..
```
```  1104
```
```  1105 lemma eq_Pls_Pls:
```
```  1106   "Numeral.Pls = Numeral.Pls" ..
```
```  1107
```
```  1108 lemma eq_Pls_Min:
```
```  1109   "Numeral.Pls \<noteq> Numeral.Min"
```
```  1110   unfolding Pls_def Min_def by auto
```
```  1111
```
```  1112 lemma eq_Pls_Bit0:
```
```  1113   "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
```
```  1114   unfolding Pls_def Bit_def bit.cases by auto
```
```  1115
```
```  1116 lemma eq_Pls_Bit1:
```
```  1117   "Numeral.Pls \<noteq> Numeral.Bit k bit.B1"
```
```  1118   unfolding Pls_def Bit_def bit.cases by arith
```
```  1119
```
```  1120 lemma eq_Min_Pls:
```
```  1121   "Numeral.Min \<noteq> Numeral.Pls"
```
```  1122   unfolding Pls_def Min_def by auto
```
```  1123
```
```  1124 lemma eq_Min_Min:
```
```  1125   "Numeral.Min = Numeral.Min" ..
```
```  1126
```
```  1127 lemma eq_Min_Bit0:
```
```  1128   "Numeral.Min \<noteq> Numeral.Bit k bit.B0"
```
```  1129   unfolding Min_def Bit_def bit.cases by arith
```
```  1130
```
```  1131 lemma eq_Min_Bit1:
```
```  1132   "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
```
```  1133   unfolding Min_def Bit_def bit.cases by auto
```
```  1134
```
```  1135 lemma eq_Bit0_Pls:
```
```  1136   "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
```
```  1137   unfolding Pls_def Bit_def bit.cases by auto
```
```  1138
```
```  1139 lemma eq_Bit1_Pls:
```
```  1140   "Numeral.Bit k bit.B1 \<noteq>  Numeral.Pls"
```
```  1141   unfolding Pls_def Bit_def bit.cases by arith
```
```  1142
```
```  1143 lemma eq_Bit0_Min:
```
```  1144   "Numeral.Bit k bit.B0 \<noteq> Numeral.Min"
```
```  1145   unfolding Min_def Bit_def bit.cases by arith
```
```  1146
```
```  1147 lemma eq_Bit1_Min:
```
```  1148   "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
```
```  1149   unfolding Min_def Bit_def bit.cases by auto
```
```  1150
```
```  1151 lemma eq_Bit_Bit:
```
```  1152   "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
```
```  1153     v1 = v2 \<and> k1 = k2"
```
```  1154   unfolding Bit_def
```
```  1155   apply (cases v1)
```
```  1156   apply (cases v2)
```
```  1157   apply auto
```
```  1158   apply arith
```
```  1159   apply (cases v2)
```
```  1160   apply auto
```
```  1161   apply arith
```
```  1162   apply (cases v2)
```
```  1163   apply auto
```
```  1164 done
```
```  1165
```
```  1166 lemmas eq_numeral_code [code func] =
```
```  1167   eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
```
```  1168   eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
```
```  1169   eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
```
```  1170
```
```  1171 lemma less_eq_Pls_Pls:
```
```  1172   "Numeral.Pls \<le> Numeral.Pls" ..
```
```  1173
```
```  1174 lemma less_eq_Pls_Min:
```
```  1175   "\<not> (Numeral.Pls \<le> Numeral.Min)"
```
```  1176   unfolding Pls_def Min_def by auto
```
```  1177
```
```  1178 lemma less_eq_Pls_Bit:
```
```  1179   "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
```
```  1180   unfolding Pls_def Bit_def by (cases v) auto
```
```  1181
```
```  1182 lemma less_eq_Min_Pls:
```
```  1183   "Numeral.Min \<le> Numeral.Pls"
```
```  1184   unfolding Pls_def Min_def by auto
```
```  1185
```
```  1186 lemma less_eq_Min_Min:
```
```  1187   "Numeral.Min \<le> Numeral.Min" ..
```
```  1188
```
```  1189 lemma less_eq_Min_Bit0:
```
```  1190   "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
```
```  1191   unfolding Min_def Bit_def by auto
```
```  1192
```
```  1193 lemma less_eq_Min_Bit1:
```
```  1194   "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
```
```  1195   unfolding Min_def Bit_def by auto
```
```  1196
```
```  1197 lemma less_eq_Bit0_Pls:
```
```  1198   "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
```
```  1199   unfolding Pls_def Bit_def by simp
```
```  1200
```
```  1201 lemma less_eq_Bit1_Pls:
```
```  1202   "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
```
```  1203   unfolding Pls_def Bit_def by auto
```
```  1204
```
```  1205 lemma less_eq_Bit_Min:
```
```  1206   "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
```
```  1207   unfolding Min_def Bit_def by (cases v) auto
```
```  1208
```
```  1209 lemma less_eq_Bit0_Bit:
```
```  1210   "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
```
```  1211   unfolding Min_def Bit_def bit.cases by (cases v) auto
```
```  1212
```
```  1213 lemma less_eq_Bit_Bit1:
```
```  1214   "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
```
```  1215   unfolding Min_def Bit_def bit.cases by (cases v) auto
```
```  1216
```
```  1217 lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
```
```  1218   less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
```
```  1219   less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1
```
```  1220
```
```  1221 end
```