author chaieb Mon, 11 Jun 2007 11:06:00 +0200 changeset 23314 6894137e854a parent 23313 af2847a95751 child 23315 df3a7e9ebadb
A new and cleaned up Theory for QE. for Presburger arithmetic
 src/HOL/Presburger.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Presburger.thy	Mon Jun 11 11:05:59 2007 +0200
+++ b/src/HOL/Presburger.thy	Mon Jun 11 11:06:00 2007 +0200
@@ -1,629 +1,191 @@
(*  Title:      HOL/Presburger.thy
ID:         \$Id\$
-    Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
+    Author:     Amine Chaieb, TU Muenchen
*)

-header {* Presburger Arithmetic: Cooper's Algorithm *}
-
theory Presburger
imports NatSimprocs SetInterval
-uses
-  ("Tools/Presburger/cooper_dec.ML")
-  ("Tools/Presburger/cooper_proof.ML")
-  ("Tools/Presburger/qelim.ML")
-  ("Tools/Presburger/reflected_presburger.ML")
-  ("Tools/Presburger/reflected_cooper.ML")
-  ("Tools/Presburger/presburger.ML")
+  uses "Tools/Presburger/cooper_data" "Tools/Presburger/qelim"
+       "Tools/Presburger/generated_cooper.ML"
+       ("Tools/Presburger/cooper.ML") ("Tools/Presburger/presburger.ML")
+
begin
-
-text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
-
-theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
-  apply (rule iffI)
-  apply (erule exE)
-  apply (rule_tac x = "l * x" in exI)
-  apply simp
-  apply (erule exE)
-  apply (erule conjE)
-  apply (erule dvdE)
-  apply (rule_tac x = k in exI)
-  apply simp
-  done
-
-lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
-apply(unfold dvd_def)
-apply(rule iffI)
-apply(clarsimp)
-apply(rename_tac k)
-apply(rule_tac x = "-k" in exI)
-apply simp
-apply(clarsimp)
-apply(rename_tac k)
-apply(rule_tac x = "-k" in exI)
-apply simp
-done
+setup {* Cooper_Data.setup*}

-lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
-apply(unfold dvd_def)
-apply(rule iffI)
-apply(clarsimp)
-apply(rule_tac x = "-k" in exI)
-apply simp
-apply(clarsimp)
-apply(rule_tac x = "-k" in exI)
-apply simp
-done
-
-
-
-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}.*}
-
-theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
-  \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
-  \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
-  apply (erule exE)+
-  apply (rule_tac x = "min z1 z2" in exI)
-  apply simp
-  done
-
-
-theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
-  \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
-  \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
-
-  apply (erule exE)+
-  apply (rule_tac x = "min z1 z2" in exI)
-  apply simp
-  done
-
-
-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}.*}
+section{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}

-theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
-  \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
-  \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
-  apply (erule exE)+
-  apply (rule_tac x = "max z1 z2" in exI)
-  apply simp
-  done
-
-
-theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
-  \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
-  \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
-  apply (erule exE)+
-  apply (rule_tac x = "max z1 z2" in exI)
-  apply simp
-  done
-
-text {*
-  \medskip Theorems for the combination of proofs of the modulo @{text
-  D} property for @{text "P plusinfinity"}
-
-  FIXME: This is THE SAME theorem as for the @{text minusinf} version,
-  but with @{text "+k.."} instead of @{text "-k.."} In the future
-  replace these both with only one. *}
-
-theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
-  \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
-  \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
-  by simp
-
-theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
-  \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
-  \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
-  by simp
-
-text {*
-  This is one of the cases where the simplifed formula is prooved to
-  habe some property (in relation to @{text P_m}) but we need to prove
-  the property for the original formula (@{text P_m})
-
-  FIXME: This is exaclty the same thm as for @{text minusinf}. *}
-
-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)) "
-  by blast
-
-
-text {*
-  \medskip Theorems for the combination of proofs of the modulo @{text D}
-  property for @{text "P minusinfinity"} *}
-
-theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
-  \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
-  \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
-  by simp
-
-theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
-  \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
-  \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
-  by simp
-
-text {*
-  This is one of the cases where the simplifed formula is prooved to
-  have some property (in relation to @{text P_m}) but we need to
-  prove the property for the original formula (@{text P_m}). *}
-
-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)) "
-  by blast
-
-text {*
-  Theorem needed for proving at runtime divide properties using the
-  arithmetic tactic (which knows only about modulo = 0). *}
-
-lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
+lemma minf:
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
+     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
+     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>x<z. F = F"
+  by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all

-text {*
-  \medskip Theorems used for the combination of proof for the
-  backwards direction of Cooper's Theorem. They rely exclusively on
-  Predicate calculus.*}
-
-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))
-==>
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
-==>
-(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))) "
-  by blast
-
-
-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))
-==>
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
-==>
-(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)
-\<and> P2(x + d))) "
-  by blast
-
-lemma not_ast_p_Q_elim: "
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
-==> ( P = Q )
-==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
-  by blast
-
-text {*
-  \medskip Theorems used for the combination of proof for the
-  backwards direction of Cooper's Theorem. They rely exclusively on
-  Predicate calculus.*}
-
-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))
-==>
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
-==>
-(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)
-\<or> P2(x-d))) "
-  by blast
-
-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))
-==>
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
-==>
-(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)
-\<and> P2(x-d))) "
-  by blast
-
-lemma not_bst_p_Q_elim: "
-(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d))
-==> ( P = Q )
-==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
-  by blast
-
-text {* \medskip This is the first direction of Cooper's Theorem. *}
-lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
-  by blast
-
-text {*
-  \medskip The full Cooper's Theorem in its equivalence Form. Given
-  the premises it is trivial too, it relies exclusively on prediacte calculus.*}
-lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
---> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
-  by blast
-
-text {*
-  \medskip Some of the atomic theorems generated each time the atom
-  does not depend on @{text x}, they are trivial.*}
-
-lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
-  by blast
-
-lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
-  by blast
+lemma pinf:
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
+     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
+     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>x>z. F = F"
+  by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all

-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"
-  by blast
-
-lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
-  by blast
-
-text {* The next two thms are the same as the @{text minusinf} version. *}
-
-lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
-  by blast
-
-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"
-  by blast
-
-text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
-
-lemma P_eqtrue: "(P=True) = P"
-  by iprover
-
-lemma P_eqfalse: "(P=False) = (~P)"
-  by iprover
-
-text {*
-  \medskip Theorems for the generation of the bachwards direction of
-  Cooper's Theorem.
-
-  These are the 6 interesting atomic cases which have to be proved relying on the
-  properties of B-set and the arithmetic and contradiction proofs. *}
-
-lemma not_bst_p_lt: "0 < (d::int) ==>
- 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 )"
-  by arith
-
-lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
- 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)"
-apply clarsimp
-apply(rule ccontr)
-apply(drule_tac x = "x+a" in bspec)
-apply(drule_tac x = "-a" in bspec)
-apply assumption
-apply(simp)
-done
-
-lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
- 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 )"
-apply clarsimp
-apply(subgoal_tac "x = -a")
- prefer 2 apply arith
-apply(drule_tac x = "1" in bspec)
-apply(drule_tac x = "-a- 1" in bspec)
-apply assumption
-apply(simp)
-done
-
-
-lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
- 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)"
-apply clarsimp
-apply(subgoal_tac "x = -a+d")
- prefer 2 apply arith
-apply(drule_tac x = "d" in bspec)
-apply(drule_tac x = "-a" in bspec)
-apply assumption
-apply(simp)
-done
-
-
-lemma not_bst_p_dvd: "(d1::int) dvd d ==>
- 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 )"
-apply(rename_tac m)
-apply(rule_tac x = "m - k" in exI)
-done
+lemma inf_period:
+  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
+    \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
+  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
+    \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
+  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
+  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
+  "\<forall>x k. F = F"
+by simp_all
+  (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
+    simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+

-lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
- 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 ))"
-apply(rename_tac m)
-apply(erule_tac x = "m + k" in allE)
-done
-
-text {*
-  \medskip Theorems for the generation of the bachwards direction of
-  Cooper's Theorem.
-
-  These are the 6 interesting atomic cases which have to be proved
-  relying on the properties of A-set ant the arithmetic and
-
-lemma not_ast_p_gt: "0 < (d::int) ==>
- 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 )"
-  by arith
-
-lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
- 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)"
-  apply clarsimp
-  apply (rule ccontr)
-  apply (drule_tac x = "t-x" in bspec)
-  apply simp
-  apply (drule_tac x = "t" in bspec)
-  apply assumption
-  apply simp
-  done
-
-lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
- 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 )"
-  apply clarsimp
-  apply (drule_tac x="1" in bspec)
-  apply simp
-  apply (drule_tac x="- t + 1" in bspec)
-  apply assumption
-  apply(subgoal_tac "x = -t")
-  prefer 2 apply arith
-  apply simp
-  done
-
-lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
- 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)"
-  apply clarsimp
-  apply (subgoal_tac "x = -t-d")
-  prefer 2 apply arith
-  apply (drule_tac x = "d" in bspec)
-  apply simp
-  apply (drule_tac x = "-t" in bspec)
-  apply assumption
-  apply simp
-  done
-
-lemma not_ast_p_dvd: "(d1::int) dvd d ==>
- 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 )"
-  apply(rename_tac m)
-  apply(rule_tac x = "m + k" in exI)
-  done
-
-lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
- 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 ))"
-  apply(rename_tac m)
-  apply(erule_tac x = "m - k" in allE)
-  done
-
-text {*
-  \medskip These are the atomic cases for the proof generation for the
-  modulo @{text D} property for @{text "P plusinfinity"}
-
-  They are fully based on arithmetics. *}
-
-lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
- (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
-  apply(rule iffI)
-  apply(clarsimp)
-  apply(rename_tac n m)
-  apply(rule_tac x = "m + n*k" in exI)
-  apply(clarsimp)
-  apply(rename_tac n m)
-  apply(rule_tac x = "m - n*k" in exI)
-  done
-
-lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
- (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
-  apply(rule iffI)
-  apply(clarsimp)
-  apply(rename_tac n m)
-  apply(erule_tac x = "m - n*k" in allE)
-  apply(clarsimp)
-  apply(rename_tac n m)
-  apply(erule_tac x = "m + n*k" in allE)
-  done
-
-text {*
-  \medskip These are the atomic cases for the proof generation for the
-  equivalence of @{text P} and @{text "P plusinfinity"} for integers
-  @{text x} greater than some integer @{text z}.
-
-  They are fully based on arithmetics. *}
-
-lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
-  apply(rule_tac x = "-t" in exI)
-  apply simp
-  done
-
-lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
-  apply(rule_tac x = "-t" in exI)
-  apply simp
-  done
-
-lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
-  apply(rule_tac x = "-t" in exI)
-  apply simp
-  done
-
-lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
-  apply(rule_tac x = "t" in exI)
-  apply simp
-  done
-
-lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
-  by simp
-
-lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
-  by simp
-
-text {*
-  \medskip These are the atomic cases for the proof generation for the
-  modulo @{text D} property for @{text "P minusinfinity"}.
-
-  They are fully based on arithmetics. *}
+section{* The A and B sets *}
+lemma bset:
+  "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
+  "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
+  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
+  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
+  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
+  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
+  "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
+  "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t))"
+  "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x - D) + t))"
+  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+  assume dp: "D > 0" and tB: "t - 1\<in> B"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
+    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
+    using dp tB by simp_all
+next
+  assume dp: "D > 0" and tB: "t \<in> B"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
+    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+    using dp tB by simp_all
+next
+  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
+next
+  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
+next
+  assume dp: "D > 0" and tB:"t \<in> B"
+  {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
+    hence "x -t \<le> D" and "1 \<le> x - t" by simp+
+      hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
+      with nob tB have "False" by simp}
+  thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
+next
+  assume dp: "D > 0" and tB:"t - 1\<in> B"
+  {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
+    hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
+      hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
+      with nob tB have "False" by simp}
+  thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
+next
+  assume d: "d dvd D"
+  {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
+      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
+  thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp
+next
+  assume d: "d dvd D"
+  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
+      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
+  thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
+qed blast

-lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
- (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
-apply(rule iffI)
-apply(clarsimp)
-apply(rename_tac n m)
-apply(rule_tac x = "m - n*k" in exI)
-apply(clarsimp)
-apply(rename_tac n m)
-apply(rule_tac x = "m + n*k" in exI)
-done
-
-
-lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
- (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
-apply(rule iffI)
-apply(clarsimp)
-apply(rename_tac n m)
-apply(erule_tac x = "m + n*k" in allE)
-apply(clarsimp)
-apply(rename_tac n m)
-apply(erule_tac x = "m - n*k" in allE)
-done
-
-text {*
-  \medskip These are the atomic cases for the proof generation for the
-  equivalence of @{text P} and @{text "P minusinfinity"} for integers
-  @{text x} less than some integer @{text z}.
-
-  They are fully based on arithmetics. *}
-
-lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
-apply(rule_tac x = "-t" in exI)
-apply simp
-done
-
-lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
-apply(rule_tac x = "-t" in exI)
-apply simp
-done
-
-lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
-apply(rule_tac x = "-t" in exI)
-apply simp
-done
-
-
-lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
-apply(rule_tac x = "t" in exI)
-apply simp
-done
-
-lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
-  by simp
-
-lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
-  by simp
-
-text {*
-  \medskip This Theorem combines whithnesses about @{text "P
-  minusinfinity"} to show one component of the equivalence proof for
-  Cooper's Theorem.
-
-  FIXME: remove once they are part of the distribution. *}
+lemma aset:
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
+  "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
+  "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
+  "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
+  "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
+  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
+  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
+  "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t))"
+  "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x + D) + t))"
+  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+  assume dp: "D > 0" and tA: "t + 1 \<in> A"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
+    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
+    using dp tA by simp_all
+next
+  assume dp: "D > 0" and tA: "t \<in> A"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
+    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+    using dp tA by simp_all
+next
+  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
+next
+  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
+next
+  assume dp: "D > 0" and tA:"t \<in> A"
+  {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
+    hence "t - x \<le> D" and "1 \<le> t - x" by simp+
+      hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps)
+      with nob tA have "False" by simp}
+  thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
+next
+  assume dp: "D > 0" and tA:"t + 1\<in> A"
+  {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
+    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
+      hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
+      with nob tA have "False" by simp}
+  thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
+next
+  assume d: "d dvd D"
+  {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
+      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
+  thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
+next
+  assume d: "d dvd D"
+  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
+      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
+  thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
+qed blast

-theorem int_ge_induct[consumes 1,case_names base step]:
-  assumes ge: "k \<le> (i::int)" and
-        base: "P(k)" and
-        step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
-  shows "P i"
-proof -
-  { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
-    proof (induct n)
-      case 0
-      hence "i = k" by arith
-      thus "P i" using base by simp
-    next
-      case (Suc n)
-      hence "n = nat((i - 1) - k)" by arith
-      moreover
-      have ki1: "k \<le> i - 1" using Suc.prems by arith
-      ultimately
-      have "P(i - 1)" by(rule Suc.hyps)
-      from step[OF ki1 this] show ?case by simp
-    qed
-  }
-  from this ge show ?thesis by fast
-qed
-
-theorem int_gr_induct[consumes 1,case_names base step]:
-  assumes gr: "k < (i::int)" and
-        base: "P(k+1)" and
-        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
-  shows "P i"
-apply(rule int_ge_induct[of "k + 1"])
-  using gr apply arith
- apply(rule base)
-apply(rule step)
- apply simp+
-done
-
-lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
-apply(induct rule: int_gr_induct)
- apply simp
-done
-
-lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
-apply(induct rule: int_gr_induct)
- apply simp
-done
+section{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}

-lemma  minusinfinity:
-  assumes "0 < d" and
-    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
-    ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
-  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
-proof
-  assume eP1: "EX x. P1 x"
-  then obtain x where P1: "P1 x" ..
-  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
-  let ?w = "x - (abs(x-z)+1) * d"
-  show "EX x. P x"
-  proof
-    have w: "?w < z" by(rule decr_lemma)
-    have "P1 x = P1 ?w" using P1eqP1 by blast
-    also have "\<dots> = P(?w)" using w P1eqP by blast
-    finally show "P ?w" using P1 by blast
-  qed
-qed
-
-text {*
-  \medskip This Theorem combines whithnesses about @{text "P
-  minusinfinity"} to show one component of the equivalence proof for
-  Cooper's Theorem. *}
-
-lemma plusinfinity:
-  assumes "0 < d" and
-    P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
-    ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
-  shows "(EX x::int. P1 x) --> (EX x::int. P x)"
-proof
-  assume eP1: "EX x. P1 x"
-  then obtain x where P1: "P1 x" ..
-  from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
-  let ?w = "x + (abs(x-z)+1) * d"
-  show "EX x. P x"
-  proof
-    have w: "z < ?w" by(rule incr_lemma)
-    have "P1 x = P1 ?w" using P1eqP1 by blast
-    also have "\<dots> = P(?w)" using w P1eqP by blast
-    finally show "P ?w" using P1 by blast
-  qed
-qed
-
-text {*
-  \medskip Theorem for periodic function on discrete sets. *}
-
-lemma minf_vee:
+subsection{* First some trivial facts about periodic sets or predicates *}
+lemma periodic_finite_ex:
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
shows "(EX x. P x) = (EX j : {1..d}. P j)"
(is "?LHS = ?RHS")
proof
assume ?LHS
then obtain x where P: "P x" ..
-  have "x mod d = x - (x div d)*d"
+  have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
hence Pmod: "P x = P(x mod d)" using modd by simp
show ?RHS
proof (cases)
@@ -644,425 +206,245 @@
qed
ultimately show ?RHS ..
qed
-next
-  assume ?RHS thus ?LHS by blast
-qed
+qed auto

-text {*
-  \medskip Theorem for periodic function on discrete sets. *}
+subsection{* The @{text "-\<infinity>"} Version*}
+
+lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"

-lemma pinf_vee:
-  assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
-  shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
-  (is "?LHS = ?RHS")
-proof
-  assume ?LHS
-  then obtain x where P: "P x" ..
-  have "x mod d = x + (-(x div d))*d"
-  hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
-  show ?RHS
-  proof (cases)
-    assume "x mod d = 0"
-    hence "P 0" using P Pmod by simp
-    moreover have "P 0 = P(0 + 1*d)" using modd by blast
-    ultimately have "P d" by simp
-    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
-    ultimately show ?RHS ..
-  next
-    assume not0: "x mod d \<noteq> 0"
-    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
-    moreover have "x mod d : {1..d}"
-    proof -
-      have "0 \<le> x mod d" by(rule pos_mod_sign)
-      moreover have "x mod d < d" by(rule pos_mod_bound)
-      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
-    qed
-    ultimately show ?RHS ..
-  qed
-next
-  assume ?RHS thus ?LHS by blast
+lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
+
+theorem int_induct[case_names base step1 step2]:
+  assumes
+  base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
+  step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
+  shows "P i"
+proof -
+  have "i \<le> k \<or> i\<ge> k" by arith
+  thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
qed

lemma decr_mult_lemma:
-  assumes dpos: "(0::int) < d" and
-          minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
-          knneg: "0 <= k"
+  assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
shows "ALL x. P x \<longrightarrow> P(x - k*d)"
using knneg
proof (induct rule:int_ge_induct)
case base thus ?case by simp
next
case (step i)
-  show ?case
-  proof
-    fix x
+  {fix x
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
-    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
-      using minus[THEN spec, of "x - i * d"]
+    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
-    ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
-  qed
+    ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
+  thus ?case ..
+qed
+
+lemma  minusinfinity:
+  assumes "0 < d" and
+    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
+  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
+proof
+  assume eP1: "EX x. P1 x"
+  then obtain x where P1: "P1 x" ..
+  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
+  let ?w = "x - (abs(x-z)+1) * d"
+  have w: "?w < z" by(rule decr_lemma)
+  have "P1 x = P1 ?w" using P1eqP1 by blast
+  also have "\<dots> = P(?w)" using w P1eqP by blast
+  finally have "P ?w" using P1 by blast
+  thus "EX x. P x" ..
+qed
+
+lemma cpmi:
+  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
+  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
+  and pd: "\<forall> x k. P' x = P' (x-k*D)"
+  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
+         (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L"  by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover
+ { fix x
+   assume P: "P x" and H: "\<not> ?R2"
+   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
+     hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
+     with nb P  have "P (y - D)" by auto }
+   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
+   with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
+   from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
+   let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
+   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+   from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
+   from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+   with periodic_finite_ex[OF dp pd]
+   have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed
+
+subsection {* The @{text "+\<infinity>"} Version*}
+
+lemma  plusinfinity:
+  assumes "(0::int) < d" and
+    P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
+  shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
+proof
+  assume eP1: "EX x. P' x"
+  then obtain x where P1: "P' x" ..
+  from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
+  let ?w' = "x + (abs(x-z)+1) * d"
+  let ?w = "x - (-(abs(x-z) + 1))*d"
+  have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
+  have w: "?w > z" by(simp only: ww', rule incr_lemma)
+  hence "P' x = P' ?w" using P1eqP1 by blast
+  also have "\<dots> = P(?w)" using w P1eqP by blast
+  finally have "P ?w" using P1 by blast
+  thus "EX x. P x" ..
qed

lemma incr_mult_lemma:
-  assumes dpos: "(0::int) < d" and
-          plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
-          knneg: "0 <= k"
+  assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
shows "ALL x. P x \<longrightarrow> P(x + k*d)"
using knneg
proof (induct rule:int_ge_induct)
case base thus ?case by simp
next
case (step i)
-  show ?case
-  proof
-    fix x
+  {fix x
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
-    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
-      using plus[THEN spec, of "x + i * d"]
+    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
-    ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
-  qed
+    ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
+  thus ?case ..
qed

-lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
-==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
-==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
-==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
-apply(rule iffI)
-prefer 2
-apply(drule minusinfinity)
-apply assumption+
-apply(fastsimp)
-apply clarsimp
-apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
-apply(frule_tac x = x and z=z in decr_lemma)
-apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
-prefer 2
-apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
-prefer 2 apply arith
- apply fastsimp
-apply(drule (1) minf_vee)
-apply blast
-apply(blast dest:decr_mult_lemma)
-done
-
-text {* Cooper Theorem, plus infinity version. *}
-lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
-==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D)
-==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
-==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
-  apply(rule iffI)
-  prefer 2
-  apply(drule plusinfinity)
-  apply assumption+
-  apply(fastsimp)
-  apply clarsimp
-  apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
-  apply(frule_tac x = x and z=z in incr_lemma)
-  apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
-  prefer 2
-  apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
-  prefer 2 apply arith
-  apply fastsimp
-  apply(drule (1) pinf_vee)
-  apply blast
-  apply(blast dest:incr_mult_lemma)
-  done
-
-
-text {*
-  \bigskip Theorems for the quantifier elminination Functions. *}
-
-lemma qe_ex_conj: "(EX (x::int). A x) = R
-		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
-		==> (EX (x::int). P x) = (Q & R)"
-by blast
-
-lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
-		==> (EX (x::int). P x) = Q"
-by blast
-
-lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
-by blast
-
-lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
-by blast
-
-lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
-by blast
-
-lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
-by blast
-
-lemma qe_Not: "P = Q ==> (~P) = (~Q)"
-by blast
-
-lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
-by blast
-
-text {* \bigskip Theorems for proving NNF *}
-
-lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
-by blast
-
-lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
-by blast
-
-lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
-  by blast
-lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
-by blast
-
-lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
-by blast
-lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
-by blast
-lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
-by blast
-lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
-by blast
-
-
-lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
-  by simp
-
-lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
-  by iprover
-
-lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
-  by iprover
-
-lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
-==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
-by blast
-
-lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
-==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
-by blast
-
+lemma cppi:
+  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
+  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
+  and pd: "\<forall> x k. P' x= P' (x-k*D)"
+  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L"  by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover
+ { fix x
+   assume P: "P x" and H: "\<not> ?R2"
+   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
+     hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
+     with nb P  have "P (y + D)" by auto }
+   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
+   with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
+   from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
+   let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
+   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+   from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
+   from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+   with periodic_finite_ex[OF dp pd]
+   have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed

lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
apply(fastsimp)
done

-text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
-
-lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
-shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
-proof
-  assume ?P
-  thus ?Q
-    apply clarify
-    apply(rename_tac d)
-    apply(drule_tac f = "op * k" in arg_cong)
-    apply(simp only:int_distrib)
-    apply(rule_tac x = "d" in exI)
-    apply(simp only:mult_ac)
-    done
-next
-  assume ?Q
-  then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
-  hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
-  hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
-  hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
-qed
-
-lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
-shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
-proof
-  assume P: ?P
-  show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
-next
-  assume ?Q
-  hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
-  with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
-  thus ?P by(simp)
-qed
+theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
+  apply (rule eq_reflection[symmetric])
+  apply (rule iffI)
+  defer
+  apply (erule exE)
+  apply (rule_tac x = "l * x" in exI)
+  apply (rule_tac x="x" in exI, simp)
+  apply (erule exE)
+  apply (erule conjE)
+  apply (erule dvdE)
+  apply (rule_tac x = k in exI)
+  apply simp
+  done

-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")
-proof
-  assume ?P
-  thus ?Q
-    apply(drule_tac f = "op * k" in arg_cong)
-    apply(simp only:int_distrib)
-    done
-next
-  assume ?Q
-  hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
-  hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
-  thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
-qed
+lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
+shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
+  using not0 by (simp add: dvd_def)

-lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
-proof -
-  have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
-  also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
-  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])
-  also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
-  finally show ?thesis .
-qed
-
-lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
-by arith
+lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))"
+by blast

-lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
-by simp
-
-lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
-by simp
-
-lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
-by simp
-
-lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
-by simp
-
+lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
+  by simp_all
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
-
-theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
+lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"

+lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
+  by (auto split add: split_nat)
+(rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)

-theorem zdiff_int_split: "P (int (x - y)) =
+lemma zdiff_int_split: "P (int (x - y)) =
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
-  apply (case_tac "y \<le> x")
+  by (case_tac "y \<le> x",simp_all add: zdiff_int)
+
+lemma zdvd_int: "(x dvd y) = (int x dvd int y)"
+  apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
+  apply (rule iffI)
+  apply iprover
+  apply (erule exE)
+  apply (case_tac "x=0")
+  apply (rule_tac x=0 in exI)
+  apply simp
+  apply (case_tac "0 \<le> k")
+  apply iprover
+  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
+  apply assumption
done

-
-theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
-  by simp
-
-theorem number_of2: "(0::int) <= Numeral0" by simp
-
-theorem Suc_plus1: "Suc n = n + 1" by simp
+lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
+lemma number_of2: "(0::int) <= Numeral0" by simp
+lemma Suc_plus1: "Suc n = n + 1" by simp

text {*
\medskip Specific instances of congruence rules, to prevent
simplifier from looping. *}

-theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
-  by simp
-
-theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
-  by (simp cong: conj_cong)
-
-    (* Theorems used in presburger.ML for the computation simpset*)
-    (* FIXME: They are present in Float.thy, so may be Float.thy should be lightened.*)
+theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp

-lemma lift_bool: "x \<Longrightarrow> x=True"
-  by simp
-
-lemma nlift_bool: "~x \<Longrightarrow> x=False"
-  by simp
-
-lemma not_false_eq_true: "(~ False) = True" by simp
-
-lemma not_true_eq_false: "(~ True) = False" by simp
-
-
+theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
+  by (simp cong: conj_cong)
lemma int_eq_number_of_eq:
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
by simp
-lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
-  by (simp only: iszero_number_of_Pls)

-lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
-  by simp
-
-lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
-  by simp
-
-lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
-  by simp
-
-lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
-  by simp
-
-lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
-  by simp
-
-lemma int_neg_number_of_Min: "neg (-1::int)"
-  by simp
-
-lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
-  by simp
-
-lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
-  by simp
-lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
-  by simp
-
-lemma int_number_of_diff_sym:
-  "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
-  by simp
-
-lemma int_number_of_mult_sym:
-  "((number_of v)::int) * number_of w = number_of (v * w)"
-  by simp
-
-lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
-  by simp
-  by simp

-  by simp
-
-lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
-  by simp
-
-lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
-  by simp
-
-lemma int_pow_0: "(a::int)^(Numeral0) = 1"
-  by simp
-
-lemma int_pow_1: "(a::int)^(Numeral1) = a"
-  by simp
-
-lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
-  by simp
-
-lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
-  by simp
+use "Tools/Presburger/cooper.ML"
+oracle linzqe_oracle ("term") = Coopereif.cooper_oracle

-lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
-  by simp
-
-lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
-  by simp
-
-lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
-  by simp
-
-lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
-proof -
-  have 1:"((-1)::nat) = 0"
-    by simp
-  show ?thesis by (simp add: 1)
-qed
-
-use "Tools/Presburger/cooper_dec.ML"
-use "Tools/Presburger/reflected_presburger.ML"
-use "Tools/Presburger/reflected_cooper.ML"
-oracle
-  presburger_oracle ("term") = ReflectedCooper.presburger_oracle
-
-use "Tools/Presburger/cooper_proof.ML"
-use "Tools/Presburger/qelim.ML"
use "Tools/Presburger/presburger.ML"

-setup "Presburger.setup"
-
+setup {*
+    (mk_arith_tactic "presburger" (fn i => fn st =>
+       (warning "Trying Presburger arithmetic ...";
+    Presburger.cooper_tac true ((ProofContext.init o theory_of_thm) st) i st)))
+  (* FIXME!!!!!!! get the right context!!*)
+*}
+method_setup presburger = {* Method.simple_args (Scan.optional (Args.\$\$\$ "elim" >> K false) true)
+  (fn q => fn ctxt =>  Method.SIMPLE_METHOD' (Presburger.cooper_tac q ctxt))*} ""
+(*
+method_setup presburger = {*
+  Method.ctxt_args (Method.SIMPLE_METHOD' o (Presburger.cooper_tac true))
+*} ""
+*)

subsection {* Code generator setup *}
-
text {*
Presburger arithmetic is convenient to prove some
of the following code lemmas on integer numerals:
@@ -1243,7 +625,6 @@
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
unfolding number_of_is_id ..

-
lemmas pred_succ_numeral_code [code func] =
arith_simps(5-12)

@@ -1277,4 +658,4 @@
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
less_number_of

-end
+end
\ No newline at end of file```