src/HOL/Integ/Presburger.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 16417 9bc16273c2d4
permissions -rw-r--r--
import -> imports
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(*  Title:      HOL/Integ/Presburger.thy
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    ID:         $Id$
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    Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
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File containing necessary theorems for the proof
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generation for Cooper Algorithm  
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*)
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header {* Presburger Arithmetic: Cooper's Algorithm *}
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theory Presburger
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imports NatSimprocs SetInterval
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files ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") ("presburger.ML")
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begin
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text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
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theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
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  apply (rule iffI)
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  apply (erule exE)
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  apply (rule_tac x = "l * x" in exI)
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  apply simp
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  apply (erule exE)
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  apply (erule conjE)
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  apply (erule dvdE)
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  apply (rule_tac x = k in exI)
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  apply simp
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  done
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lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
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apply(unfold dvd_def)
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apply(rule iffI)
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apply(clarsimp)
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apply(rename_tac k)
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apply(rule_tac x = "-k" in exI)
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apply simp
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apply(clarsimp)
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apply(rename_tac k)
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apply(rule_tac x = "-k" in exI)
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apply simp
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done
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lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
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apply(unfold dvd_def)
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apply(rule iffI)
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apply(clarsimp)
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apply(rule_tac x = "-k" in exI)
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apply simp
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apply(clarsimp)
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apply(rule_tac x = "-k" in exI)
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apply simp
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done
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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}.*}
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theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
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  \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
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  \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
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  apply (erule exE)+
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  apply (rule_tac x = "min z1 z2" in exI)
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  apply simp
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  done
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theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
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  \<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
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  \<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
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  apply (erule exE)+
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  apply (rule_tac x = "min z1 z2" in exI)
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  apply simp
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  done
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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}.*}
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theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
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  \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
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  \<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
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  apply (erule exE)+
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  apply (rule_tac x = "max z1 z2" in exI)
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  apply simp
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  done
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theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
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  \<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
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  \<exists>z::int. \<forall>x. z < x  \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
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  apply (erule exE)+
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  apply (rule_tac x = "max z1 z2" in exI)
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  apply simp
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  done
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text {*
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  \medskip Theorems for the combination of proofs of the modulo @{text
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  D} property for @{text "P plusinfinity"}
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  FIXME: This is THE SAME theorem as for the @{text minusinf} version,
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  but with @{text "+k.."} instead of @{text "-k.."} In the future
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  replace these both with only one. *}
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theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
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  \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
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  \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
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  by simp
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theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
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  \<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
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  \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
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  by simp
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text {*
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  This is one of the cases where the simplifed formula is prooved to
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  habe some property (in relation to @{text P_m}) but we need to prove
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  the property for the original formula (@{text P_m})
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  FIXME: This is exaclty the same thm as for @{text minusinf}. *}
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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)) "
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  by blast
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text {*
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  \medskip Theorems for the combination of proofs of the modulo @{text D}
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  property for @{text "P minusinfinity"} *}
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theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
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  \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
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  \<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
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  by simp
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theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
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  \<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
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  \<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
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  by simp
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text {*
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  This is one of the cases where the simplifed formula is prooved to
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  have some property (in relation to @{text P_m}) but we need to
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  prove the property for the original formula (@{text P_m}). *}
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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)) "
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  by blast
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text {*
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  Theorem needed for proving at runtime divide properties using the
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  arithmetic tactic (which knows only about modulo = 0). *}
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lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
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  by(simp add:dvd_def zmod_eq_0_iff)
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text {*
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  \medskip Theorems used for the combination of proof for the
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  backwards direction of Cooper's Theorem. They rely exclusively on
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  Predicate calculus.*}
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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))
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==>
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
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==>
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(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))) "
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  by blast
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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))
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==>
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
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==>
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(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)
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\<and> P2(x + d))) "
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  by blast
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lemma not_ast_p_Q_elim: "
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
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==> ( P = Q )
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==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
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  by blast
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text {*
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  \medskip Theorems used for the combination of proof for the
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  backwards direction of Cooper's Theorem. They rely exclusively on
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  Predicate calculus.*}
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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))
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==>
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
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==>
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(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)
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\<or> P2(x-d))) "
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  by blast
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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))
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==>
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
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==>
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(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)
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\<and> P2(x-d))) "
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  by blast
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lemma not_bst_p_Q_elim: "
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(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) 
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==> ( P = Q )
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==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
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  by blast
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text {* \medskip This is the first direction of Cooper's Theorem. *}
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lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
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  by blast
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text {*
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  \medskip The full Cooper's Theorem in its equivalence Form. Given
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  the premises it is trivial too, it relies exclusively on prediacte calculus.*}
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lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
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--> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
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  by blast
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text {*
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  \medskip Some of the atomic theorems generated each time the atom
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  does not depend on @{text x}, they are trivial.*}
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lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
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  by blast
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lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
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  by blast
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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"
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  by blast
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lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
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  by blast
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text {* The next two thms are the same as the @{text minusinf} version. *}
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lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
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  by blast
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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"
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  by blast
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text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
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lemma P_eqtrue: "(P=True) = P"
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  by rules
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lemma P_eqfalse: "(P=False) = (~P)"
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  by rules
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text {*
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  \medskip Theorems for the generation of the bachwards direction of
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  Cooper's Theorem.
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  These are the 6 interesting atomic cases which have to be proved relying on the
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  properties of B-set and the arithmetic and contradiction proofs. *}
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lemma not_bst_p_lt: "0 < (d::int) ==>
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 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 )"
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  by arith
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lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
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 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)"
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apply clarsimp
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apply(rule ccontr)
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apply(drule_tac x = "x+a" in bspec)
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apply(simp add:atLeastAtMost_iff)
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apply(drule_tac x = "-a" in bspec)
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apply assumption
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apply(simp)
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done
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lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
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 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 )"
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apply clarsimp
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apply(subgoal_tac "x = -a")
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 prefer 2 apply arith
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apply(drule_tac x = "1" in bspec)
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apply(simp add:atLeastAtMost_iff)
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apply(drule_tac x = "-a- 1" in bspec)
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apply assumption
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   282
apply(simp)
berghofe@13876
   283
done
berghofe@13876
   284
berghofe@13876
   285
berghofe@13876
   286
lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
berghofe@13876
   287
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)"
berghofe@13876
   288
apply clarsimp
berghofe@13876
   289
apply(subgoal_tac "x = -a+d")
berghofe@13876
   290
 prefer 2 apply arith
berghofe@13876
   291
apply(drule_tac x = "d" in bspec)
berghofe@13876
   292
apply(simp add:atLeastAtMost_iff)
berghofe@13876
   293
apply(drule_tac x = "-a" in bspec)
berghofe@13876
   294
apply assumption
berghofe@13876
   295
apply(simp)
berghofe@13876
   296
done
berghofe@13876
   297
berghofe@13876
   298
berghofe@13876
   299
lemma not_bst_p_dvd: "(d1::int) dvd d ==>
berghofe@13876
   300
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )"
berghofe@13876
   301
apply(clarsimp simp add:dvd_def)
berghofe@13876
   302
apply(rename_tac m)
berghofe@13876
   303
apply(rule_tac x = "m - k" in exI)
berghofe@13876
   304
apply(simp add:int_distrib)
berghofe@13876
   305
done
berghofe@13876
   306
berghofe@13876
   307
lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
berghofe@13876
   308
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))"
berghofe@13876
   309
apply(clarsimp simp add:dvd_def)
berghofe@13876
   310
apply(rename_tac m)
berghofe@13876
   311
apply(erule_tac x = "m + k" in allE)
berghofe@13876
   312
apply(simp add:int_distrib)
berghofe@13876
   313
done
berghofe@13876
   314
wenzelm@14577
   315
text {*
wenzelm@14577
   316
  \medskip Theorems for the generation of the bachwards direction of
wenzelm@14577
   317
  Cooper's Theorem.
berghofe@13876
   318
wenzelm@14577
   319
  These are the 6 interesting atomic cases which have to be proved
wenzelm@14577
   320
  relying on the properties of A-set ant the arithmetic and
wenzelm@14577
   321
  contradiction proofs. *}
berghofe@13876
   322
berghofe@13876
   323
lemma not_ast_p_gt: "0 < (d::int) ==>
berghofe@13876
   324
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ( 0 < x + t) --> (0 < (x + d) + t )"
wenzelm@14577
   325
  by arith
berghofe@13876
   326
berghofe@13876
   327
lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
berghofe@13876
   328
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)"
berghofe@13876
   329
  apply clarsimp
berghofe@13876
   330
  apply (rule ccontr)
berghofe@13876
   331
  apply (drule_tac x = "t-x" in bspec)
berghofe@13876
   332
  apply simp
berghofe@13876
   333
  apply (drule_tac x = "t" in bspec)
berghofe@13876
   334
  apply assumption
berghofe@13876
   335
  apply simp
berghofe@13876
   336
  done
berghofe@13876
   337
berghofe@13876
   338
lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
berghofe@13876
   339
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 = x + t) --> (0 = (x + d) + t )"
berghofe@13876
   340
  apply clarsimp
berghofe@13876
   341
  apply (drule_tac x="1" in bspec)
berghofe@13876
   342
  apply simp
berghofe@13876
   343
  apply (drule_tac x="- t + 1" in bspec)
berghofe@13876
   344
  apply assumption
berghofe@13876
   345
  apply(subgoal_tac "x = -t")
berghofe@13876
   346
  prefer 2 apply arith
berghofe@13876
   347
  apply simp
berghofe@13876
   348
  done
berghofe@13876
   349
berghofe@13876
   350
lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
berghofe@13876
   351
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(0 = x + t) --> ~(0 = (x + d) + t)"
berghofe@13876
   352
  apply clarsimp
berghofe@13876
   353
  apply (subgoal_tac "x = -t-d")
berghofe@13876
   354
  prefer 2 apply arith
berghofe@13876
   355
  apply (drule_tac x = "d" in bspec)
berghofe@13876
   356
  apply simp
berghofe@13876
   357
  apply (drule_tac x = "-t" in bspec)
berghofe@13876
   358
  apply assumption
berghofe@13876
   359
  apply simp
berghofe@13876
   360
  done
berghofe@13876
   361
berghofe@13876
   362
lemma not_ast_p_dvd: "(d1::int) dvd d ==>
berghofe@13876
   363
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> d1 dvd (x + t) --> d1 dvd ((x + d) + t )"
berghofe@13876
   364
  apply(clarsimp simp add:dvd_def)
berghofe@13876
   365
  apply(rename_tac m)
berghofe@13876
   366
  apply(rule_tac x = "m + k" in exI)
berghofe@13876
   367
  apply(simp add:int_distrib)
berghofe@13876
   368
  done
berghofe@13876
   369
berghofe@13876
   370
lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
berghofe@13876
   371
 ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(d1 dvd (x + t)) --> ~(d1 dvd ((x + d) + t ))"
berghofe@13876
   372
  apply(clarsimp simp add:dvd_def)
berghofe@13876
   373
  apply(rename_tac m)
berghofe@13876
   374
  apply(erule_tac x = "m - k" in allE)
berghofe@13876
   375
  apply(simp add:int_distrib)
berghofe@13876
   376
  done
berghofe@13876
   377
wenzelm@14577
   378
text {*
wenzelm@14577
   379
  \medskip These are the atomic cases for the proof generation for the
wenzelm@14577
   380
  modulo @{text D} property for @{text "P plusinfinity"}
berghofe@13876
   381
wenzelm@14577
   382
  They are fully based on arithmetics. *}
berghofe@13876
   383
berghofe@13876
   384
lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
berghofe@13876
   385
 (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
berghofe@13876
   386
  apply(clarsimp simp add:dvd_def)
berghofe@13876
   387
  apply(rule iffI)
berghofe@13876
   388
  apply(clarsimp)
berghofe@13876
   389
  apply(rename_tac n m)
berghofe@13876
   390
  apply(rule_tac x = "m + n*k" in exI)
berghofe@13876
   391
  apply(simp add:int_distrib)
berghofe@13876
   392
  apply(clarsimp)
berghofe@13876
   393
  apply(rename_tac n m)
berghofe@13876
   394
  apply(rule_tac x = "m - n*k" in exI)
paulson@14271
   395
  apply(simp add:int_distrib mult_ac)
berghofe@13876
   396
  done
berghofe@13876
   397
berghofe@13876
   398
lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
berghofe@13876
   399
 (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
berghofe@13876
   400
  apply(clarsimp simp add:dvd_def)
berghofe@13876
   401
  apply(rule iffI)
berghofe@13876
   402
  apply(clarsimp)
berghofe@13876
   403
  apply(rename_tac n m)
berghofe@13876
   404
  apply(erule_tac x = "m - n*k" in allE)
paulson@14271
   405
  apply(simp add:int_distrib mult_ac)
berghofe@13876
   406
  apply(clarsimp)
berghofe@13876
   407
  apply(rename_tac n m)
berghofe@13876
   408
  apply(erule_tac x = "m + n*k" in allE)
paulson@14271
   409
  apply(simp add:int_distrib mult_ac)
berghofe@13876
   410
  done
berghofe@13876
   411
wenzelm@14577
   412
text {*
wenzelm@14577
   413
  \medskip These are the atomic cases for the proof generation for the
wenzelm@14577
   414
  equivalence of @{text P} and @{text "P plusinfinity"} for integers
wenzelm@14577
   415
  @{text x} greater than some integer @{text z}.
wenzelm@14577
   416
wenzelm@14577
   417
  They are fully based on arithmetics. *}
berghofe@13876
   418
berghofe@13876
   419
lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
berghofe@13876
   420
  apply(rule_tac x = "-t" in exI)
berghofe@13876
   421
  apply simp
berghofe@13876
   422
  done
berghofe@13876
   423
berghofe@13876
   424
lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
berghofe@13876
   425
  apply(rule_tac x = "-t" in exI)
berghofe@13876
   426
  apply simp
berghofe@13876
   427
  done
berghofe@13876
   428
berghofe@13876
   429
lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
berghofe@13876
   430
  apply(rule_tac x = "-t" in exI)
berghofe@13876
   431
  apply simp
berghofe@13876
   432
  done
berghofe@13876
   433
berghofe@13876
   434
lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
berghofe@13876
   435
  apply(rule_tac x = "t" in exI)
berghofe@13876
   436
  apply simp
berghofe@13876
   437
  done
berghofe@13876
   438
berghofe@13876
   439
lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
wenzelm@14577
   440
  by simp
berghofe@13876
   441
berghofe@13876
   442
lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
wenzelm@14577
   443
  by simp
berghofe@13876
   444
wenzelm@14577
   445
text {*
wenzelm@14577
   446
  \medskip These are the atomic cases for the proof generation for the
wenzelm@14577
   447
  modulo @{text D} property for @{text "P minusinfinity"}.
wenzelm@14577
   448
wenzelm@14577
   449
  They are fully based on arithmetics. *}
berghofe@13876
   450
berghofe@13876
   451
lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
berghofe@13876
   452
 (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
berghofe@13876
   453
apply(clarsimp simp add:dvd_def)
berghofe@13876
   454
apply(rule iffI)
berghofe@13876
   455
apply(clarsimp)
berghofe@13876
   456
apply(rename_tac n m)
berghofe@13876
   457
apply(rule_tac x = "m - n*k" in exI)
berghofe@13876
   458
apply(simp add:int_distrib)
berghofe@13876
   459
apply(clarsimp)
berghofe@13876
   460
apply(rename_tac n m)
berghofe@13876
   461
apply(rule_tac x = "m + n*k" in exI)
paulson@14271
   462
apply(simp add:int_distrib mult_ac)
berghofe@13876
   463
done
berghofe@13876
   464
berghofe@13876
   465
berghofe@13876
   466
lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
berghofe@13876
   467
 (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
berghofe@13876
   468
apply(clarsimp simp add:dvd_def)
berghofe@13876
   469
apply(rule iffI)
berghofe@13876
   470
apply(clarsimp)
berghofe@13876
   471
apply(rename_tac n m)
berghofe@13876
   472
apply(erule_tac x = "m + n*k" in allE)
paulson@14271
   473
apply(simp add:int_distrib mult_ac)
berghofe@13876
   474
apply(clarsimp)
berghofe@13876
   475
apply(rename_tac n m)
berghofe@13876
   476
apply(erule_tac x = "m - n*k" in allE)
paulson@14271
   477
apply(simp add:int_distrib mult_ac)
berghofe@13876
   478
done
berghofe@13876
   479
wenzelm@14577
   480
text {*
wenzelm@14577
   481
  \medskip These are the atomic cases for the proof generation for the
wenzelm@14577
   482
  equivalence of @{text P} and @{text "P minusinfinity"} for integers
wenzelm@14577
   483
  @{text x} less than some integer @{text z}.
berghofe@13876
   484
wenzelm@14577
   485
  They are fully based on arithmetics. *}
berghofe@13876
   486
berghofe@13876
   487
lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
berghofe@13876
   488
apply(rule_tac x = "-t" in exI)
berghofe@13876
   489
apply simp
berghofe@13876
   490
done
berghofe@13876
   491
berghofe@13876
   492
lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
berghofe@13876
   493
apply(rule_tac x = "-t" in exI)
berghofe@13876
   494
apply simp
berghofe@13876
   495
done
berghofe@13876
   496
berghofe@13876
   497
lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
berghofe@13876
   498
apply(rule_tac x = "-t" in exI)
berghofe@13876
   499
apply simp
berghofe@13876
   500
done
berghofe@13876
   501
berghofe@13876
   502
berghofe@13876
   503
lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
berghofe@13876
   504
apply(rule_tac x = "t" in exI)
berghofe@13876
   505
apply simp
berghofe@13876
   506
done
berghofe@13876
   507
berghofe@13876
   508
lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
wenzelm@14577
   509
  by simp
berghofe@13876
   510
berghofe@13876
   511
lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
wenzelm@14577
   512
  by simp
berghofe@13876
   513
wenzelm@14577
   514
text {*
wenzelm@14577
   515
  \medskip This Theorem combines whithnesses about @{text "P
wenzelm@14577
   516
  minusinfinity"} to show one component of the equivalence proof for
wenzelm@14577
   517
  Cooper's Theorem.
berghofe@13876
   518
wenzelm@14577
   519
  FIXME: remove once they are part of the distribution. *}
wenzelm@14577
   520
berghofe@13876
   521
theorem int_ge_induct[consumes 1,case_names base step]:
berghofe@13876
   522
  assumes ge: "k \<le> (i::int)" and
berghofe@13876
   523
        base: "P(k)" and
berghofe@13876
   524
        step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
berghofe@13876
   525
  shows "P i"
berghofe@13876
   526
proof -
berghofe@13876
   527
  { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
berghofe@13876
   528
    proof (induct n)
berghofe@13876
   529
      case 0
berghofe@13876
   530
      hence "i = k" by arith
berghofe@13876
   531
      thus "P i" using base by simp
berghofe@13876
   532
    next
berghofe@13876
   533
      case (Suc n)
berghofe@13876
   534
      hence "n = nat((i - 1) - k)" by arith
berghofe@13876
   535
      moreover
berghofe@13876
   536
      have ki1: "k \<le> i - 1" using Suc.prems by arith
berghofe@13876
   537
      ultimately
berghofe@13876
   538
      have "P(i - 1)" by(rule Suc.hyps)
berghofe@13876
   539
      from step[OF ki1 this] show ?case by simp
berghofe@13876
   540
    qed
berghofe@13876
   541
  }
berghofe@13876
   542
  from this ge show ?thesis by fast
berghofe@13876
   543
qed
berghofe@13876
   544
berghofe@13876
   545
theorem int_gr_induct[consumes 1,case_names base step]:
berghofe@13876
   546
  assumes gr: "k < (i::int)" and
berghofe@13876
   547
        base: "P(k+1)" and
berghofe@13876
   548
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
berghofe@13876
   549
  shows "P i"
berghofe@13876
   550
apply(rule int_ge_induct[of "k + 1"])
berghofe@13876
   551
  using gr apply arith
berghofe@13876
   552
 apply(rule base)
berghofe@13876
   553
apply(rule step)
berghofe@13876
   554
 apply simp+
berghofe@13876
   555
done
berghofe@13876
   556
berghofe@13876
   557
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
berghofe@13876
   558
apply(induct rule: int_gr_induct)
berghofe@13876
   559
 apply simp
berghofe@13876
   560
 apply arith
berghofe@13876
   561
apply (simp add:int_distrib)
berghofe@13876
   562
apply arith
berghofe@13876
   563
done
berghofe@13876
   564
berghofe@13876
   565
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
berghofe@13876
   566
apply(induct rule: int_gr_induct)
berghofe@13876
   567
 apply simp
berghofe@13876
   568
 apply arith
berghofe@13876
   569
apply (simp add:int_distrib)
berghofe@13876
   570
apply arith
berghofe@13876
   571
done
berghofe@13876
   572
berghofe@13876
   573
lemma  minusinfinity:
berghofe@13876
   574
  assumes "0 < d" and
berghofe@13876
   575
    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
berghofe@13876
   576
    ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
berghofe@13876
   577
  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
berghofe@13876
   578
proof
berghofe@13876
   579
  assume eP1: "EX x. P1 x"
berghofe@13876
   580
  then obtain x where P1: "P1 x" ..
berghofe@13876
   581
  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
berghofe@13876
   582
  let ?w = "x - (abs(x-z)+1) * d"
berghofe@13876
   583
  show "EX x. P x"
berghofe@13876
   584
  proof
berghofe@13876
   585
    have w: "?w < z" by(rule decr_lemma)
berghofe@13876
   586
    have "P1 x = P1 ?w" using P1eqP1 by blast
berghofe@13876
   587
    also have "\<dots> = P(?w)" using w P1eqP by blast
berghofe@13876
   588
    finally show "P ?w" using P1 by blast
berghofe@13876
   589
  qed
berghofe@13876
   590
qed
berghofe@13876
   591
wenzelm@14577
   592
text {*
wenzelm@14577
   593
  \medskip This Theorem combines whithnesses about @{text "P
wenzelm@14577
   594
  minusinfinity"} to show one component of the equivalence proof for
wenzelm@14577
   595
  Cooper's Theorem. *}
berghofe@13876
   596
berghofe@13876
   597
lemma plusinfinity:
berghofe@13876
   598
  assumes "0 < d" and
berghofe@13876
   599
    P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
berghofe@13876
   600
    ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
berghofe@13876
   601
  shows "(EX x::int. P1 x) --> (EX x::int. P x)"
berghofe@13876
   602
proof
berghofe@13876
   603
  assume eP1: "EX x. P1 x"
berghofe@13876
   604
  then obtain x where P1: "P1 x" ..
berghofe@13876
   605
  from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
berghofe@13876
   606
  let ?w = "x + (abs(x-z)+1) * d"
berghofe@13876
   607
  show "EX x. P x"
berghofe@13876
   608
  proof
berghofe@13876
   609
    have w: "z < ?w" by(rule incr_lemma)
berghofe@13876
   610
    have "P1 x = P1 ?w" using P1eqP1 by blast
berghofe@13876
   611
    also have "\<dots> = P(?w)" using w P1eqP by blast
berghofe@13876
   612
    finally show "P ?w" using P1 by blast
berghofe@13876
   613
  qed
berghofe@13876
   614
qed
berghofe@13876
   615
 
wenzelm@14577
   616
text {*
wenzelm@14577
   617
  \medskip Theorem for periodic function on discrete sets. *}
berghofe@13876
   618
berghofe@13876
   619
lemma minf_vee:
berghofe@13876
   620
  assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
berghofe@13876
   621
  shows "(EX x. P x) = (EX j : {1..d}. P j)"
berghofe@13876
   622
  (is "?LHS = ?RHS")
berghofe@13876
   623
proof
berghofe@13876
   624
  assume ?LHS
berghofe@13876
   625
  then obtain x where P: "P x" ..
berghofe@13876
   626
  have "x mod d = x - (x div d)*d"
paulson@14271
   627
    by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
berghofe@13876
   628
  hence Pmod: "P x = P(x mod d)" using modd by simp
berghofe@13876
   629
  show ?RHS
berghofe@13876
   630
  proof (cases)
berghofe@13876
   631
    assume "x mod d = 0"
berghofe@13876
   632
    hence "P 0" using P Pmod by simp
berghofe@13876
   633
    moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
berghofe@13876
   634
    ultimately have "P d" by simp
berghofe@13876
   635
    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
berghofe@13876
   636
    ultimately show ?RHS ..
berghofe@13876
   637
  next
berghofe@13876
   638
    assume not0: "x mod d \<noteq> 0"
berghofe@13876
   639
    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
berghofe@13876
   640
    moreover have "x mod d : {1..d}"
berghofe@13876
   641
    proof -
berghofe@13876
   642
      have "0 \<le> x mod d" by(rule pos_mod_sign)
berghofe@13876
   643
      moreover have "x mod d < d" by(rule pos_mod_bound)
berghofe@13876
   644
      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
berghofe@13876
   645
    qed
berghofe@13876
   646
    ultimately show ?RHS ..
berghofe@13876
   647
  qed
berghofe@13876
   648
next
berghofe@13876
   649
  assume ?RHS thus ?LHS by blast
berghofe@13876
   650
qed
berghofe@13876
   651
wenzelm@14577
   652
text {*
wenzelm@14577
   653
  \medskip Theorem for periodic function on discrete sets. *}
wenzelm@14577
   654
berghofe@13876
   655
lemma pinf_vee:
berghofe@13876
   656
  assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
berghofe@13876
   657
  shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
berghofe@13876
   658
  (is "?LHS = ?RHS")
berghofe@13876
   659
proof
berghofe@13876
   660
  assume ?LHS
berghofe@13876
   661
  then obtain x where P: "P x" ..
berghofe@13876
   662
  have "x mod d = x + (-(x div d))*d"
paulson@14271
   663
    by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
berghofe@13876
   664
  hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
berghofe@13876
   665
  show ?RHS
berghofe@13876
   666
  proof (cases)
berghofe@13876
   667
    assume "x mod d = 0"
berghofe@13876
   668
    hence "P 0" using P Pmod by simp
berghofe@13876
   669
    moreover have "P 0 = P(0 + 1*d)" using modd by blast
berghofe@13876
   670
    ultimately have "P d" by simp
berghofe@13876
   671
    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
berghofe@13876
   672
    ultimately show ?RHS ..
berghofe@13876
   673
  next
berghofe@13876
   674
    assume not0: "x mod d \<noteq> 0"
berghofe@13876
   675
    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
berghofe@13876
   676
    moreover have "x mod d : {1..d}"
berghofe@13876
   677
    proof -
berghofe@13876
   678
      have "0 \<le> x mod d" by(rule pos_mod_sign)
berghofe@13876
   679
      moreover have "x mod d < d" by(rule pos_mod_bound)
berghofe@13876
   680
      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
berghofe@13876
   681
    qed
berghofe@13876
   682
    ultimately show ?RHS ..
berghofe@13876
   683
  qed
berghofe@13876
   684
next
berghofe@13876
   685
  assume ?RHS thus ?LHS by blast
berghofe@13876
   686
qed
berghofe@13876
   687
berghofe@13876
   688
lemma decr_mult_lemma:
berghofe@13876
   689
  assumes dpos: "(0::int) < d" and
berghofe@13876
   690
          minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
berghofe@13876
   691
          knneg: "0 <= k"
berghofe@13876
   692
  shows "ALL x. P x \<longrightarrow> P(x - k*d)"
berghofe@13876
   693
using knneg
berghofe@13876
   694
proof (induct rule:int_ge_induct)
berghofe@13876
   695
  case base thus ?case by simp
berghofe@13876
   696
next
berghofe@13876
   697
  case (step i)
berghofe@13876
   698
  show ?case
berghofe@13876
   699
  proof
berghofe@13876
   700
    fix x
berghofe@13876
   701
    have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
berghofe@13876
   702
    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
berghofe@13876
   703
      using minus[THEN spec, of "x - i * d"]
obua@14738
   704
      by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
berghofe@13876
   705
    ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
berghofe@13876
   706
  qed
berghofe@13876
   707
qed
berghofe@13876
   708
berghofe@13876
   709
lemma incr_mult_lemma:
berghofe@13876
   710
  assumes dpos: "(0::int) < d" and
berghofe@13876
   711
          plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
berghofe@13876
   712
          knneg: "0 <= k"
berghofe@13876
   713
  shows "ALL x. P x \<longrightarrow> P(x + k*d)"
berghofe@13876
   714
using knneg
berghofe@13876
   715
proof (induct rule:int_ge_induct)
berghofe@13876
   716
  case base thus ?case by simp
berghofe@13876
   717
next
berghofe@13876
   718
  case (step i)
berghofe@13876
   719
  show ?case
berghofe@13876
   720
  proof
berghofe@13876
   721
    fix x
berghofe@13876
   722
    have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
berghofe@13876
   723
    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
berghofe@13876
   724
      using plus[THEN spec, of "x + i * d"]
berghofe@13876
   725
      by (simp add:int_distrib zadd_ac)
berghofe@13876
   726
    ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
berghofe@13876
   727
  qed
berghofe@13876
   728
qed
berghofe@13876
   729
berghofe@13876
   730
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
berghofe@13876
   731
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
berghofe@13876
   732
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
berghofe@13876
   733
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
berghofe@13876
   734
apply(rule iffI)
berghofe@13876
   735
prefer 2
berghofe@13876
   736
apply(drule minusinfinity)
berghofe@13876
   737
apply assumption+
berghofe@13876
   738
apply(fastsimp)
berghofe@13876
   739
apply clarsimp
berghofe@13876
   740
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
berghofe@13876
   741
apply(frule_tac x = x and z=z in decr_lemma)
berghofe@13876
   742
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
berghofe@13876
   743
prefer 2
berghofe@13876
   744
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
berghofe@13876
   745
prefer 2 apply arith
berghofe@13876
   746
 apply fastsimp
berghofe@13876
   747
apply(drule (1) minf_vee)
berghofe@13876
   748
apply blast
berghofe@13876
   749
apply(blast dest:decr_mult_lemma)
berghofe@13876
   750
done
berghofe@13876
   751
wenzelm@14577
   752
text {* Cooper Theorem, plus infinity version. *}
berghofe@13876
   753
lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
berghofe@13876
   754
==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) 
berghofe@13876
   755
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
berghofe@13876
   756
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
berghofe@13876
   757
  apply(rule iffI)
berghofe@13876
   758
  prefer 2
berghofe@13876
   759
  apply(drule plusinfinity)
berghofe@13876
   760
  apply assumption+
berghofe@13876
   761
  apply(fastsimp)
berghofe@13876
   762
  apply clarsimp
berghofe@13876
   763
  apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
berghofe@13876
   764
  apply(frule_tac x = x and z=z in incr_lemma)
berghofe@13876
   765
  apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
berghofe@13876
   766
  prefer 2
berghofe@13876
   767
  apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
berghofe@13876
   768
  prefer 2 apply arith
berghofe@13876
   769
  apply fastsimp
berghofe@13876
   770
  apply(drule (1) pinf_vee)
berghofe@13876
   771
  apply blast
berghofe@13876
   772
  apply(blast dest:incr_mult_lemma)
berghofe@13876
   773
  done
berghofe@13876
   774
berghofe@13876
   775
wenzelm@14577
   776
text {*
wenzelm@14577
   777
  \bigskip Theorems for the quantifier elminination Functions. *}
berghofe@13876
   778
berghofe@13876
   779
lemma qe_ex_conj: "(EX (x::int). A x) = R
berghofe@13876
   780
		==> (EX (x::int). P x) = (Q & (EX x::int. A x))
berghofe@13876
   781
		==> (EX (x::int). P x) = (Q & R)"
berghofe@13876
   782
by blast
berghofe@13876
   783
berghofe@13876
   784
lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
berghofe@13876
   785
		==> (EX (x::int). P x) = Q"
berghofe@13876
   786
by blast
berghofe@13876
   787
berghofe@13876
   788
lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
berghofe@13876
   789
by blast
berghofe@13876
   790
berghofe@13876
   791
lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
berghofe@13876
   792
by blast
berghofe@13876
   793
berghofe@13876
   794
lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
berghofe@13876
   795
by blast
berghofe@13876
   796
berghofe@13876
   797
lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
berghofe@13876
   798
by blast
berghofe@13876
   799
berghofe@13876
   800
lemma qe_Not: "P = Q ==> (~P) = (~Q)"
berghofe@13876
   801
by blast
berghofe@13876
   802
berghofe@13876
   803
lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
berghofe@13876
   804
by blast
berghofe@13876
   805
wenzelm@14577
   806
text {* \bigskip Theorems for proving NNF *}
berghofe@13876
   807
berghofe@13876
   808
lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
berghofe@13876
   809
by blast
berghofe@13876
   810
berghofe@13876
   811
lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
berghofe@13876
   812
by blast
berghofe@13876
   813
berghofe@13876
   814
lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
berghofe@13876
   815
  by blast
berghofe@13876
   816
lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
berghofe@13876
   817
by blast
berghofe@13876
   818
berghofe@13876
   819
lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
berghofe@13876
   820
by blast
berghofe@13876
   821
lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
berghofe@13876
   822
by blast
berghofe@13876
   823
lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
berghofe@13876
   824
by blast
berghofe@13876
   825
lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
berghofe@13876
   826
by blast
berghofe@13876
   827
berghofe@13876
   828
berghofe@13876
   829
lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
berghofe@13876
   830
  by simp
berghofe@13876
   831
berghofe@13876
   832
lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
berghofe@13876
   833
  by rules
berghofe@13876
   834
berghofe@13876
   835
lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
berghofe@13876
   836
  by rules
berghofe@13876
   837
berghofe@13876
   838
lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
berghofe@13876
   839
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
berghofe@13876
   840
by blast
berghofe@13876
   841
berghofe@13876
   842
lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
berghofe@13876
   843
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
berghofe@13876
   844
by blast
berghofe@13876
   845
berghofe@13876
   846
berghofe@13876
   847
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
berghofe@13876
   848
apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
berghofe@13876
   849
apply(fastsimp)
berghofe@13876
   850
done
berghofe@13876
   851
wenzelm@14577
   852
text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
berghofe@13876
   853
berghofe@13876
   854
lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
berghofe@13876
   855
shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
berghofe@13876
   856
proof
berghofe@13876
   857
  assume ?P
berghofe@13876
   858
  thus ?Q
berghofe@13876
   859
    apply(simp add:dvd_def)
berghofe@13876
   860
    apply clarify
berghofe@13876
   861
    apply(rename_tac d)
berghofe@13876
   862
    apply(drule_tac f = "op * k" in arg_cong)
berghofe@13876
   863
    apply(simp only:int_distrib)
berghofe@13876
   864
    apply(rule_tac x = "d" in exI)
paulson@14271
   865
    apply(simp only:mult_ac)
berghofe@13876
   866
    done
berghofe@13876
   867
next
berghofe@13876
   868
  assume ?Q
berghofe@13876
   869
  then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
paulson@14271
   870
  hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
berghofe@13876
   871
  hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
berghofe@13876
   872
  hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
berghofe@13876
   873
  thus ?P by(simp add:dvd_def)
berghofe@13876
   874
qed
berghofe@13876
   875
berghofe@13876
   876
lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
berghofe@13876
   877
shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
berghofe@13876
   878
proof
berghofe@13876
   879
  assume P: ?P
paulson@14271
   880
  show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
berghofe@13876
   881
next
berghofe@13876
   882
  assume ?Q
paulson@14271
   883
  hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
paulson@14353
   884
  with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
berghofe@13876
   885
  thus ?P by(simp)
berghofe@13876
   886
qed
berghofe@13876
   887
berghofe@13876
   888
lemma ac_eq_eq : assumes not0: "0 ~= (k::int)" shows "((m::int) = (c*n+t)) = (k*m =((k*c)*n+(k*t)) )" (is "?P = ?Q")
berghofe@13876
   889
proof
berghofe@13876
   890
  assume ?P
berghofe@13876
   891
  thus ?Q
berghofe@13876
   892
    apply(drule_tac f = "op * k" in arg_cong)
berghofe@13876
   893
    apply(simp only:int_distrib)
berghofe@13876
   894
    done
berghofe@13876
   895
next
berghofe@13876
   896
  assume ?Q
paulson@14271
   897
  hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
berghofe@13876
   898
  hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
berghofe@13876
   899
  thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
berghofe@13876
   900
qed
berghofe@13876
   901
berghofe@13876
   902
lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
berghofe@13876
   903
proof -
berghofe@13876
   904
  have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
paulson@14271
   905
  also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
berghofe@13876
   906
  also have "0<(-1*c)*n + (-t+1) = (0 < (k*(-1*c)*n) + (k*(-t+1)))" by(rule ac_lt_eq[of _ 0,OF gr0,simplified])
paulson@14271
   907
  also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
berghofe@13876
   908
  finally show ?thesis .
berghofe@13876
   909
qed
berghofe@13876
   910
berghofe@13876
   911
lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
berghofe@13876
   912
by arith
berghofe@13876
   913
berghofe@13876
   914
lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
berghofe@13876
   915
by simp
berghofe@13876
   916
berghofe@13876
   917
lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
berghofe@13876
   918
by simp
berghofe@13876
   919
berghofe@13876
   920
lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
berghofe@13876
   921
by simp
berghofe@13876
   922
berghofe@13876
   923
lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
berghofe@13876
   924
by simp
berghofe@13876
   925
wenzelm@14577
   926
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
berghofe@13876
   927
berghofe@13876
   928
theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
berghofe@13876
   929
  by (simp split add: split_nat)
berghofe@13876
   930
berghofe@13876
   931
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
berghofe@13876
   932
  apply (simp split add: split_nat)
berghofe@13876
   933
  apply (rule iffI)
berghofe@13876
   934
  apply (erule exE)
berghofe@13876
   935
  apply (rule_tac x = "int x" in exI)
berghofe@13876
   936
  apply simp
berghofe@13876
   937
  apply (erule exE)
berghofe@13876
   938
  apply (rule_tac x = "nat x" in exI)
berghofe@13876
   939
  apply (erule conjE)
berghofe@13876
   940
  apply (erule_tac x = "nat x" in allE)
berghofe@13876
   941
  apply simp
berghofe@13876
   942
  done
berghofe@13876
   943
berghofe@13876
   944
theorem zdiff_int_split: "P (int (x - y)) =
berghofe@13876
   945
  ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
berghofe@13876
   946
  apply (case_tac "y \<le> x")
berghofe@13876
   947
  apply (simp_all add: zdiff_int)
berghofe@13876
   948
  done
berghofe@13876
   949
berghofe@13876
   950
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
berghofe@13876
   951
  apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
berghofe@13876
   952
    nat_0_le cong add: conj_cong)
berghofe@13876
   953
  apply (rule iffI)
berghofe@13876
   954
  apply rules
berghofe@13876
   955
  apply (erule exE)
berghofe@13876
   956
  apply (case_tac "x=0")
berghofe@13876
   957
  apply (rule_tac x=0 in exI)
berghofe@13876
   958
  apply simp
berghofe@13876
   959
  apply (case_tac "0 \<le> k")
berghofe@13876
   960
  apply rules
berghofe@13876
   961
  apply (simp add: linorder_not_le)
paulson@14378
   962
  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
berghofe@13876
   963
  apply assumption
paulson@14271
   964
  apply (simp add: mult_ac)
berghofe@13876
   965
  done
berghofe@13876
   966
berghofe@13876
   967
theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
berghofe@13876
   968
  by simp
berghofe@13876
   969
paulson@15013
   970
theorem number_of2: "(0::int) <= Numeral0" by simp
berghofe@13876
   971
berghofe@13876
   972
theorem Suc_plus1: "Suc n = n + 1" by simp
berghofe@13876
   973
wenzelm@14577
   974
text {*
wenzelm@14577
   975
  \medskip Specific instances of congruence rules, to prevent
wenzelm@14577
   976
  simplifier from looping. *}
berghofe@13876
   977
chaieb@14758
   978
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
berghofe@13876
   979
  by simp
berghofe@13876
   980
chaieb@14758
   981
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
chaieb@14758
   982
  by (simp cong: conj_cong)
berghofe@13876
   983
berghofe@13876
   984
use "cooper_dec.ML"
chaieb@14941
   985
oracle
chaieb@14941
   986
  presburger_oracle = CooperDec.mk_presburger_oracle
chaieb@14941
   987
berghofe@13876
   988
use "cooper_proof.ML"
berghofe@13876
   989
use "qelim.ML"
berghofe@13876
   990
use "presburger.ML"
berghofe@13876
   991
berghofe@13876
   992
setup "Presburger.setup"
berghofe@13876
   993
berghofe@13876
   994
end