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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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License: GPL (GNU GENERAL PUBLIC LICENSE)
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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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theory Presburger = NatSimprocs
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files
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("cooper_dec.ML")
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("cooper_proof.ML")
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("qelim.ML")
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("presburger.ML"):
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(* Theorem for unitifying the coeffitients of 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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(*Theorems for the combination of proofs of the equality of P and P_m for integers x less than some integer 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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(*Theorems for the combination of proofs of the equality of P and P_m for integers x greather than some integer 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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(*=============================================================================*)
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(*Theorems for the combination of proofs of the modulo D property for P
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pluusinfinity*)
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(* FIXME : This is THE SAME theorem as for the minusinf version, but with +k.. instead of -k.. In the future replace these both with only one*)
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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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(*=============================================================================*)
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(*This is one of the cases where the simplifed formula is prooved to habe some property
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(in relation to P_m) but we need to proove the property for the original formula (P_m)*)
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(*FIXME : This is exaclty the same thm as for 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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(*=============================================================================*)
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(*Theorems for the combination of proofs of the modulo D property for P
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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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(*=============================================================================*)
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(*This is one of the cases where the simplifed formula is prooved to habe some property
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(in relation to P_m) but we need to proove the property for the original formula (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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(*=============================================================================*)
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(*theorem needed for prooving at runtime divide properties using the arithmetic tatic
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(who 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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(*=============================================================================*)
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(*Theorems used for the combination of proof for the backwards direction of cooper's
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theorem. they rely exclusively on 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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(*=============================================================================*)
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(*Theorems used for the combination of proof for the backwards direction of cooper's
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theorem. they rely exclusively on 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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(*=============================================================================*)
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(*The Theorem for the second proof step- about bset. it is trivial too. *)
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lemma bst_thm: " (EX (j::int) : {1..d}. EX (b::int) : B. P (b+j) )--> (EX x::int. P (x)) "
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by blast
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(*The Theorem for the second proof step- about aset. it is trivial too. *)
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lemma ast_thm: " (EX (j::int) : {1..d}. EX (a::int) : A. P (a - j) )--> (EX x::int. P (x)) "
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by blast
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(*=============================================================================*)
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(*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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(*=============================================================================*)
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(*The full cooper's theoorem in its equivalence Form- Given the premisses it is trivial
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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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(*=============================================================================*)
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(*Some of the atomic theorems generated each time the atom does not depend on x, they
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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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(* The next 2 thms are the same as the 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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(* 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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(*=============================================================================*)
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(*Theorems for the generation of the bachwards direction of 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 ant 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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apply(simp)
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done
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lemma not_bst_p_ne: "\<lbrakk> 0 < d; (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(subgoal_tac "x = -a+d")
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prefer 2 apply arith
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apply(drule_tac x = "d" 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_dvd: "(d1::int) dvd d ==>
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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 )"
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apply(clarsimp simp add:dvd_def)
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apply(rename_tac m)
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apply(rule_tac x = "m - k" in exI)
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apply(simp add:int_distrib)
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done
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318 |
|
|
319 |
lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
|
|
320 |
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 ))"
|
|
321 |
apply(clarsimp simp add:dvd_def)
|
|
322 |
apply(rename_tac m)
|
|
323 |
apply(erule_tac x = "m + k" in allE)
|
|
324 |
apply(simp add:int_distrib)
|
|
325 |
done
|
|
326 |
|
|
327 |
|
|
328 |
|
|
329 |
(*Theorems for the generation of the bachwards direction of cooper's theorem*)
|
|
330 |
(*These are the 6 interesting atomic cases which have to be proved relying on the
|
|
331 |
properties of A-set ant the arithmetic and contradiction proofs*)
|
|
332 |
|
|
333 |
lemma not_ast_p_gt: "0 < (d::int) ==>
|
|
334 |
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 )"
|
|
335 |
by arith
|
|
336 |
|
|
337 |
|
|
338 |
lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
|
|
339 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)"
|
|
340 |
apply clarsimp
|
|
341 |
apply (rule ccontr)
|
|
342 |
apply (drule_tac x = "t-x" in bspec)
|
|
343 |
apply simp
|
|
344 |
apply (drule_tac x = "t" in bspec)
|
|
345 |
apply assumption
|
|
346 |
apply simp
|
|
347 |
done
|
|
348 |
|
|
349 |
lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
|
|
350 |
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 )"
|
|
351 |
apply clarsimp
|
|
352 |
apply (drule_tac x="1" in bspec)
|
|
353 |
apply simp
|
|
354 |
apply (drule_tac x="- t + 1" in bspec)
|
|
355 |
apply assumption
|
|
356 |
apply(subgoal_tac "x = -t")
|
|
357 |
prefer 2 apply arith
|
|
358 |
apply simp
|
|
359 |
done
|
|
360 |
|
|
361 |
lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
|
|
362 |
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)"
|
|
363 |
apply clarsimp
|
|
364 |
apply (subgoal_tac "x = -t-d")
|
|
365 |
prefer 2 apply arith
|
|
366 |
apply (drule_tac x = "d" in bspec)
|
|
367 |
apply simp
|
|
368 |
apply (drule_tac x = "-t" in bspec)
|
|
369 |
apply assumption
|
|
370 |
apply simp
|
|
371 |
done
|
|
372 |
|
|
373 |
lemma not_ast_p_dvd: "(d1::int) dvd d ==>
|
|
374 |
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 )"
|
|
375 |
apply(clarsimp simp add:dvd_def)
|
|
376 |
apply(rename_tac m)
|
|
377 |
apply(rule_tac x = "m + k" in exI)
|
|
378 |
apply(simp add:int_distrib)
|
|
379 |
done
|
|
380 |
|
|
381 |
lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
|
|
382 |
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 ))"
|
|
383 |
apply(clarsimp simp add:dvd_def)
|
|
384 |
apply(rename_tac m)
|
|
385 |
apply(erule_tac x = "m - k" in allE)
|
|
386 |
apply(simp add:int_distrib)
|
|
387 |
done
|
|
388 |
|
|
389 |
|
|
390 |
|
|
391 |
(*=============================================================================*)
|
|
392 |
(*These are the atomic cases for the proof generation for the modulo D property for P
|
|
393 |
plusinfinity*)
|
|
394 |
(*They are fully based on arithmetics*)
|
|
395 |
|
|
396 |
lemma dvd_modd_pinf: "((d::int) dvd d1) ==>
|
|
397 |
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
|
|
398 |
apply(clarsimp simp add:dvd_def)
|
|
399 |
apply(rule iffI)
|
|
400 |
apply(clarsimp)
|
|
401 |
apply(rename_tac n m)
|
|
402 |
apply(rule_tac x = "m + n*k" in exI)
|
|
403 |
apply(simp add:int_distrib)
|
|
404 |
apply(clarsimp)
|
|
405 |
apply(rename_tac n m)
|
|
406 |
apply(rule_tac x = "m - n*k" in exI)
|
|
407 |
apply(simp add:int_distrib zmult_ac)
|
|
408 |
done
|
|
409 |
|
|
410 |
lemma not_dvd_modd_pinf: "((d::int) dvd d1) ==>
|
|
411 |
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
|
|
412 |
apply(clarsimp simp add:dvd_def)
|
|
413 |
apply(rule iffI)
|
|
414 |
apply(clarsimp)
|
|
415 |
apply(rename_tac n m)
|
|
416 |
apply(erule_tac x = "m - n*k" in allE)
|
|
417 |
apply(simp add:int_distrib zmult_ac)
|
|
418 |
apply(clarsimp)
|
|
419 |
apply(rename_tac n m)
|
|
420 |
apply(erule_tac x = "m + n*k" in allE)
|
|
421 |
apply(simp add:int_distrib zmult_ac)
|
|
422 |
done
|
|
423 |
|
|
424 |
(*=============================================================================*)
|
|
425 |
(*These are the atomic cases for the proof generation for the equivalence of P and P
|
|
426 |
plusinfinity for integers x greather than some integer z.*)
|
|
427 |
(*They are fully based on arithmetics*)
|
|
428 |
|
|
429 |
lemma eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
|
|
430 |
apply(rule_tac x = "-t" in exI)
|
|
431 |
apply simp
|
|
432 |
done
|
|
433 |
|
|
434 |
lemma neq_eq_pinf: "EX z::int. ALL x. z < x --> ((~( 0 = x +t )) = True )"
|
|
435 |
apply(rule_tac x = "-t" in exI)
|
|
436 |
apply simp
|
|
437 |
done
|
|
438 |
|
|
439 |
lemma le_eq_pinf: "EX z::int. ALL x. z < x --> ( 0 < x +t = True )"
|
|
440 |
apply(rule_tac x = "-t" in exI)
|
|
441 |
apply simp
|
|
442 |
done
|
|
443 |
|
|
444 |
lemma len_eq_pinf: "EX z::int. ALL x. z < x --> (0 < -x +t = False )"
|
|
445 |
apply(rule_tac x = "t" in exI)
|
|
446 |
apply simp
|
|
447 |
done
|
|
448 |
|
|
449 |
lemma dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
|
|
450 |
by simp
|
|
451 |
|
|
452 |
lemma not_dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
|
|
453 |
by simp
|
|
454 |
|
|
455 |
|
|
456 |
|
|
457 |
|
|
458 |
(*=============================================================================*)
|
|
459 |
(*These are the atomic cases for the proof generation for the modulo D property for P
|
|
460 |
minusinfinity*)
|
|
461 |
(*They are fully based on arithmetics*)
|
|
462 |
|
|
463 |
lemma dvd_modd_minf: "((d::int) dvd d1) ==>
|
|
464 |
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
|
|
465 |
apply(clarsimp simp add:dvd_def)
|
|
466 |
apply(rule iffI)
|
|
467 |
apply(clarsimp)
|
|
468 |
apply(rename_tac n m)
|
|
469 |
apply(rule_tac x = "m - n*k" in exI)
|
|
470 |
apply(simp add:int_distrib)
|
|
471 |
apply(clarsimp)
|
|
472 |
apply(rename_tac n m)
|
|
473 |
apply(rule_tac x = "m + n*k" in exI)
|
|
474 |
apply(simp add:int_distrib zmult_ac)
|
|
475 |
done
|
|
476 |
|
|
477 |
|
|
478 |
lemma not_dvd_modd_minf: "((d::int) dvd d1) ==>
|
|
479 |
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
|
|
480 |
apply(clarsimp simp add:dvd_def)
|
|
481 |
apply(rule iffI)
|
|
482 |
apply(clarsimp)
|
|
483 |
apply(rename_tac n m)
|
|
484 |
apply(erule_tac x = "m + n*k" in allE)
|
|
485 |
apply(simp add:int_distrib zmult_ac)
|
|
486 |
apply(clarsimp)
|
|
487 |
apply(rename_tac n m)
|
|
488 |
apply(erule_tac x = "m - n*k" in allE)
|
|
489 |
apply(simp add:int_distrib zmult_ac)
|
|
490 |
done
|
|
491 |
|
|
492 |
|
|
493 |
(*=============================================================================*)
|
|
494 |
(*These are the atomic cases for the proof generation for the equivalence of P and P
|
|
495 |
minusinfinity for integers x less than some integer z.*)
|
|
496 |
(*They are fully based on arithmetics*)
|
|
497 |
|
|
498 |
lemma eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
|
|
499 |
apply(rule_tac x = "-t" in exI)
|
|
500 |
apply simp
|
|
501 |
done
|
|
502 |
|
|
503 |
lemma neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
|
|
504 |
apply(rule_tac x = "-t" in exI)
|
|
505 |
apply simp
|
|
506 |
done
|
|
507 |
|
|
508 |
lemma le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t = False )"
|
|
509 |
apply(rule_tac x = "-t" in exI)
|
|
510 |
apply simp
|
|
511 |
done
|
|
512 |
|
|
513 |
|
|
514 |
lemma len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t = True )"
|
|
515 |
apply(rule_tac x = "t" in exI)
|
|
516 |
apply simp
|
|
517 |
done
|
|
518 |
|
|
519 |
lemma dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
|
|
520 |
by simp
|
|
521 |
|
|
522 |
lemma not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
|
|
523 |
by simp
|
|
524 |
|
|
525 |
|
|
526 |
(*=============================================================================*)
|
|
527 |
(*This Theorem combines whithnesses about P minusinfinity to schow one component of the
|
|
528 |
equivalence proof for cooper's theorem*)
|
|
529 |
|
|
530 |
(* FIXME: remove once they are part of the distribution *)
|
|
531 |
theorem int_ge_induct[consumes 1,case_names base step]:
|
|
532 |
assumes ge: "k \<le> (i::int)" and
|
|
533 |
base: "P(k)" and
|
|
534 |
step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
|
|
535 |
shows "P i"
|
|
536 |
proof -
|
|
537 |
{ fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
|
|
538 |
proof (induct n)
|
|
539 |
case 0
|
|
540 |
hence "i = k" by arith
|
|
541 |
thus "P i" using base by simp
|
|
542 |
next
|
|
543 |
case (Suc n)
|
|
544 |
hence "n = nat((i - 1) - k)" by arith
|
|
545 |
moreover
|
|
546 |
have ki1: "k \<le> i - 1" using Suc.prems by arith
|
|
547 |
ultimately
|
|
548 |
have "P(i - 1)" by(rule Suc.hyps)
|
|
549 |
from step[OF ki1 this] show ?case by simp
|
|
550 |
qed
|
|
551 |
}
|
|
552 |
from this ge show ?thesis by fast
|
|
553 |
qed
|
|
554 |
|
|
555 |
theorem int_gr_induct[consumes 1,case_names base step]:
|
|
556 |
assumes gr: "k < (i::int)" and
|
|
557 |
base: "P(k+1)" and
|
|
558 |
step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
|
|
559 |
shows "P i"
|
|
560 |
apply(rule int_ge_induct[of "k + 1"])
|
|
561 |
using gr apply arith
|
|
562 |
apply(rule base)
|
|
563 |
apply(rule step)
|
|
564 |
apply simp+
|
|
565 |
done
|
|
566 |
|
|
567 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
|
|
568 |
apply(induct rule: int_gr_induct)
|
|
569 |
apply simp
|
|
570 |
apply arith
|
|
571 |
apply (simp add:int_distrib)
|
|
572 |
apply arith
|
|
573 |
done
|
|
574 |
|
|
575 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
|
|
576 |
apply(induct rule: int_gr_induct)
|
|
577 |
apply simp
|
|
578 |
apply arith
|
|
579 |
apply (simp add:int_distrib)
|
|
580 |
apply arith
|
|
581 |
done
|
|
582 |
|
|
583 |
lemma minusinfinity:
|
|
584 |
assumes "0 < d" and
|
|
585 |
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
|
|
586 |
ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
|
|
587 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
|
|
588 |
proof
|
|
589 |
assume eP1: "EX x. P1 x"
|
|
590 |
then obtain x where P1: "P1 x" ..
|
|
591 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
|
|
592 |
let ?w = "x - (abs(x-z)+1) * d"
|
|
593 |
show "EX x. P x"
|
|
594 |
proof
|
|
595 |
have w: "?w < z" by(rule decr_lemma)
|
|
596 |
have "P1 x = P1 ?w" using P1eqP1 by blast
|
|
597 |
also have "\<dots> = P(?w)" using w P1eqP by blast
|
|
598 |
finally show "P ?w" using P1 by blast
|
|
599 |
qed
|
|
600 |
qed
|
|
601 |
|
|
602 |
(*=============================================================================*)
|
|
603 |
(*This Theorem combines whithnesses about P minusinfinity to schow one component of the
|
|
604 |
equivalence proof for cooper's theorem*)
|
|
605 |
|
|
606 |
lemma plusinfinity:
|
|
607 |
assumes "0 < d" and
|
|
608 |
P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
|
|
609 |
ePeqP1: "EX z::int. ALL x. z < x --> (P x = P1 x)"
|
|
610 |
shows "(EX x::int. P1 x) --> (EX x::int. P x)"
|
|
611 |
proof
|
|
612 |
assume eP1: "EX x. P1 x"
|
|
613 |
then obtain x where P1: "P1 x" ..
|
|
614 |
from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
|
|
615 |
let ?w = "x + (abs(x-z)+1) * d"
|
|
616 |
show "EX x. P x"
|
|
617 |
proof
|
|
618 |
have w: "z < ?w" by(rule incr_lemma)
|
|
619 |
have "P1 x = P1 ?w" using P1eqP1 by blast
|
|
620 |
also have "\<dots> = P(?w)" using w P1eqP by blast
|
|
621 |
finally show "P ?w" using P1 by blast
|
|
622 |
qed
|
|
623 |
qed
|
|
624 |
|
|
625 |
|
|
626 |
|
|
627 |
(*=============================================================================*)
|
|
628 |
(*Theorem for periodic function on discrete sets*)
|
|
629 |
|
|
630 |
lemma minf_vee:
|
|
631 |
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
|
|
632 |
shows "(EX x. P x) = (EX j : {1..d}. P j)"
|
|
633 |
(is "?LHS = ?RHS")
|
|
634 |
proof
|
|
635 |
assume ?LHS
|
|
636 |
then obtain x where P: "P x" ..
|
|
637 |
have "x mod d = x - (x div d)*d"
|
|
638 |
by(simp add:zmod_zdiv_equality zmult_ac eq_zdiff_eq)
|
|
639 |
hence Pmod: "P x = P(x mod d)" using modd by simp
|
|
640 |
show ?RHS
|
|
641 |
proof (cases)
|
|
642 |
assume "x mod d = 0"
|
|
643 |
hence "P 0" using P Pmod by simp
|
|
644 |
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
|
|
645 |
ultimately have "P d" by simp
|
|
646 |
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
|
|
647 |
ultimately show ?RHS ..
|
|
648 |
next
|
|
649 |
assume not0: "x mod d \<noteq> 0"
|
|
650 |
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
|
|
651 |
moreover have "x mod d : {1..d}"
|
|
652 |
proof -
|
|
653 |
have "0 \<le> x mod d" by(rule pos_mod_sign)
|
|
654 |
moreover have "x mod d < d" by(rule pos_mod_bound)
|
|
655 |
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
|
|
656 |
qed
|
|
657 |
ultimately show ?RHS ..
|
|
658 |
qed
|
|
659 |
next
|
|
660 |
assume ?RHS thus ?LHS by blast
|
|
661 |
qed
|
|
662 |
|
|
663 |
(*=============================================================================*)
|
|
664 |
(*Theorem for periodic function on discrete sets*)
|
|
665 |
lemma pinf_vee:
|
|
666 |
assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
|
|
667 |
shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
|
|
668 |
(is "?LHS = ?RHS")
|
|
669 |
proof
|
|
670 |
assume ?LHS
|
|
671 |
then obtain x where P: "P x" ..
|
|
672 |
have "x mod d = x + (-(x div d))*d"
|
|
673 |
by(simp add:zmod_zdiv_equality zmult_ac eq_zdiff_eq)
|
|
674 |
hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
|
|
675 |
show ?RHS
|
|
676 |
proof (cases)
|
|
677 |
assume "x mod d = 0"
|
|
678 |
hence "P 0" using P Pmod by simp
|
|
679 |
moreover have "P 0 = P(0 + 1*d)" using modd by blast
|
|
680 |
ultimately have "P d" by simp
|
|
681 |
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
|
|
682 |
ultimately show ?RHS ..
|
|
683 |
next
|
|
684 |
assume not0: "x mod d \<noteq> 0"
|
|
685 |
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
|
|
686 |
moreover have "x mod d : {1..d}"
|
|
687 |
proof -
|
|
688 |
have "0 \<le> x mod d" by(rule pos_mod_sign)
|
|
689 |
moreover have "x mod d < d" by(rule pos_mod_bound)
|
|
690 |
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
|
|
691 |
qed
|
|
692 |
ultimately show ?RHS ..
|
|
693 |
qed
|
|
694 |
next
|
|
695 |
assume ?RHS thus ?LHS by blast
|
|
696 |
qed
|
|
697 |
|
|
698 |
lemma decr_mult_lemma:
|
|
699 |
assumes dpos: "(0::int) < d" and
|
|
700 |
minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
|
|
701 |
knneg: "0 <= k"
|
|
702 |
shows "ALL x. P x \<longrightarrow> P(x - k*d)"
|
|
703 |
using knneg
|
|
704 |
proof (induct rule:int_ge_induct)
|
|
705 |
case base thus ?case by simp
|
|
706 |
next
|
|
707 |
case (step i)
|
|
708 |
show ?case
|
|
709 |
proof
|
|
710 |
fix x
|
|
711 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
|
|
712 |
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
|
|
713 |
using minus[THEN spec, of "x - i * d"]
|
|
714 |
by (simp add:int_distrib zdiff_zdiff_eq[symmetric])
|
|
715 |
ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
|
|
716 |
qed
|
|
717 |
qed
|
|
718 |
|
|
719 |
lemma incr_mult_lemma:
|
|
720 |
assumes dpos: "(0::int) < d" and
|
|
721 |
plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
|
|
722 |
knneg: "0 <= k"
|
|
723 |
shows "ALL x. P x \<longrightarrow> P(x + k*d)"
|
|
724 |
using knneg
|
|
725 |
proof (induct rule:int_ge_induct)
|
|
726 |
case base thus ?case by simp
|
|
727 |
next
|
|
728 |
case (step i)
|
|
729 |
show ?case
|
|
730 |
proof
|
|
731 |
fix x
|
|
732 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
|
|
733 |
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
|
|
734 |
using plus[THEN spec, of "x + i * d"]
|
|
735 |
by (simp add:int_distrib zadd_ac)
|
|
736 |
ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
|
|
737 |
qed
|
|
738 |
qed
|
|
739 |
|
|
740 |
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
|
|
741 |
==> (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)) --> (EX (x::int). P x)
|
|
742 |
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
|
|
743 |
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
|
|
744 |
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
|
|
745 |
apply(rule iffI)
|
|
746 |
prefer 2
|
|
747 |
apply(drule minusinfinity)
|
|
748 |
apply assumption+
|
|
749 |
apply(fastsimp)
|
|
750 |
apply clarsimp
|
|
751 |
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
|
|
752 |
apply(frule_tac x = x and z=z in decr_lemma)
|
|
753 |
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
|
|
754 |
prefer 2
|
|
755 |
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
|
|
756 |
prefer 2 apply arith
|
|
757 |
apply fastsimp
|
|
758 |
apply(drule (1) minf_vee)
|
|
759 |
apply blast
|
|
760 |
apply(blast dest:decr_mult_lemma)
|
|
761 |
done
|
|
762 |
|
|
763 |
(* Cooper Thm `, plus infinity version*)
|
|
764 |
lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
|
|
765 |
==> (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)) --> (EX (x::int). P x)
|
|
766 |
==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D)
|
|
767 |
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
|
|
768 |
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
|
|
769 |
apply(rule iffI)
|
|
770 |
prefer 2
|
|
771 |
apply(drule plusinfinity)
|
|
772 |
apply assumption+
|
|
773 |
apply(fastsimp)
|
|
774 |
apply clarsimp
|
|
775 |
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
|
|
776 |
apply(frule_tac x = x and z=z in incr_lemma)
|
|
777 |
apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
|
|
778 |
prefer 2
|
|
779 |
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
|
|
780 |
prefer 2 apply arith
|
|
781 |
apply fastsimp
|
|
782 |
apply(drule (1) pinf_vee)
|
|
783 |
apply blast
|
|
784 |
apply(blast dest:incr_mult_lemma)
|
|
785 |
done
|
|
786 |
|
|
787 |
|
|
788 |
(*=============================================================================*)
|
|
789 |
|
|
790 |
(*Theorems for the quantifier elminination Functions.*)
|
|
791 |
|
|
792 |
lemma qe_ex_conj: "(EX (x::int). A x) = R
|
|
793 |
==> (EX (x::int). P x) = (Q & (EX x::int. A x))
|
|
794 |
==> (EX (x::int). P x) = (Q & R)"
|
|
795 |
by blast
|
|
796 |
|
|
797 |
lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
|
|
798 |
==> (EX (x::int). P x) = Q"
|
|
799 |
by blast
|
|
800 |
|
|
801 |
lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
|
|
802 |
by blast
|
|
803 |
|
|
804 |
lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
|
|
805 |
by blast
|
|
806 |
|
|
807 |
lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
|
|
808 |
by blast
|
|
809 |
|
|
810 |
lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
|
|
811 |
by blast
|
|
812 |
|
|
813 |
lemma qe_Not: "P = Q ==> (~P) = (~Q)"
|
|
814 |
by blast
|
|
815 |
|
|
816 |
lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
|
|
817 |
by blast
|
|
818 |
|
|
819 |
(* Theorems for proving NNF *)
|
|
820 |
|
|
821 |
lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
|
|
822 |
by blast
|
|
823 |
|
|
824 |
lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
|
|
825 |
by blast
|
|
826 |
|
|
827 |
lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
|
|
828 |
by blast
|
|
829 |
lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
|
|
830 |
by blast
|
|
831 |
|
|
832 |
lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
|
|
833 |
by blast
|
|
834 |
lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
|
|
835 |
by blast
|
|
836 |
lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
|
|
837 |
by blast
|
|
838 |
lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
|
|
839 |
by blast
|
|
840 |
|
|
841 |
|
|
842 |
lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
|
|
843 |
by simp
|
|
844 |
|
|
845 |
lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
|
|
846 |
by rules
|
|
847 |
|
|
848 |
lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
|
|
849 |
by rules
|
|
850 |
|
|
851 |
lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
|
|
852 |
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
|
|
853 |
by blast
|
|
854 |
|
|
855 |
lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
|
|
856 |
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
|
|
857 |
by blast
|
|
858 |
|
|
859 |
|
|
860 |
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
|
|
861 |
apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
|
|
862 |
apply(fastsimp)
|
|
863 |
done
|
|
864 |
|
|
865 |
(* Theorems required for the adjustcoeffitienteq*)
|
|
866 |
|
|
867 |
lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
|
|
868 |
shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
|
|
869 |
proof
|
|
870 |
assume ?P
|
|
871 |
thus ?Q
|
|
872 |
apply(simp add:dvd_def)
|
|
873 |
apply clarify
|
|
874 |
apply(rename_tac d)
|
|
875 |
apply(drule_tac f = "op * k" in arg_cong)
|
|
876 |
apply(simp only:int_distrib)
|
|
877 |
apply(rule_tac x = "d" in exI)
|
|
878 |
apply(simp only:zmult_ac)
|
|
879 |
done
|
|
880 |
next
|
|
881 |
assume ?Q
|
|
882 |
then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
|
|
883 |
hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib zmult_ac)
|
|
884 |
hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
|
|
885 |
hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
|
|
886 |
thus ?P by(simp add:dvd_def)
|
|
887 |
qed
|
|
888 |
|
|
889 |
lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
|
|
890 |
shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
|
|
891 |
proof
|
|
892 |
assume P: ?P
|
|
893 |
show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib zmult_ac)
|
|
894 |
next
|
|
895 |
assume ?Q
|
|
896 |
hence "0 < k*(c*n + t - m)" by(simp add: int_distrib zmult_ac)
|
|
897 |
with gr0 have "0 < (c*n + t - m)" by(simp add:int_0_less_mult_iff)
|
|
898 |
thus ?P by(simp)
|
|
899 |
qed
|
|
900 |
|
|
901 |
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")
|
|
902 |
proof
|
|
903 |
assume ?P
|
|
904 |
thus ?Q
|
|
905 |
apply(drule_tac f = "op * k" in arg_cong)
|
|
906 |
apply(simp only:int_distrib)
|
|
907 |
done
|
|
908 |
next
|
|
909 |
assume ?Q
|
|
910 |
hence "m * k = (c*n + t) * k" by(simp add:int_distrib zmult_ac)
|
|
911 |
hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
|
|
912 |
thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
|
|
913 |
qed
|
|
914 |
|
|
915 |
lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
|
|
916 |
proof -
|
|
917 |
have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
|
|
918 |
also have "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib zmult_ac)
|
|
919 |
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])
|
|
920 |
also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib zmult_ac)
|
|
921 |
finally show ?thesis .
|
|
922 |
qed
|
|
923 |
|
|
924 |
lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
|
|
925 |
by arith
|
|
926 |
|
|
927 |
lemma linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
|
|
928 |
by simp
|
|
929 |
|
|
930 |
lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
|
|
931 |
by simp
|
|
932 |
|
|
933 |
lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
|
|
934 |
by simp
|
|
935 |
|
|
936 |
lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
|
|
937 |
by simp
|
|
938 |
|
|
939 |
(* Theorems for transforming predicates on nat to predicates on int*)
|
|
940 |
|
|
941 |
theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
|
|
942 |
by (simp split add: split_nat)
|
|
943 |
|
|
944 |
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
|
|
945 |
apply (simp split add: split_nat)
|
|
946 |
apply (rule iffI)
|
|
947 |
apply (erule exE)
|
|
948 |
apply (rule_tac x = "int x" in exI)
|
|
949 |
apply simp
|
|
950 |
apply (erule exE)
|
|
951 |
apply (rule_tac x = "nat x" in exI)
|
|
952 |
apply (erule conjE)
|
|
953 |
apply (erule_tac x = "nat x" in allE)
|
|
954 |
apply simp
|
|
955 |
done
|
|
956 |
|
|
957 |
theorem zdiff_int_split: "P (int (x - y)) =
|
|
958 |
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
|
|
959 |
apply (case_tac "y \<le> x")
|
|
960 |
apply (simp_all add: zdiff_int)
|
|
961 |
done
|
|
962 |
|
|
963 |
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
|
|
964 |
apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
|
|
965 |
nat_0_le cong add: conj_cong)
|
|
966 |
apply (rule iffI)
|
|
967 |
apply rules
|
|
968 |
apply (erule exE)
|
|
969 |
apply (case_tac "x=0")
|
|
970 |
apply (rule_tac x=0 in exI)
|
|
971 |
apply simp
|
|
972 |
apply (case_tac "0 \<le> k")
|
|
973 |
apply rules
|
|
974 |
apply (simp add: linorder_not_le)
|
|
975 |
apply (drule zmult_zless_mono2_neg [OF iffD2 [OF zero_less_int_conv]])
|
|
976 |
apply assumption
|
|
977 |
apply (simp add: zmult_ac)
|
|
978 |
done
|
|
979 |
|
|
980 |
theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
|
|
981 |
by simp
|
|
982 |
|
|
983 |
theorem number_of2: "(0::int) <= number_of bin.Pls" by simp
|
|
984 |
|
|
985 |
theorem Suc_plus1: "Suc n = n + 1" by simp
|
|
986 |
|
|
987 |
(* specific instances of congruence rules, to prevent simplifier from looping *)
|
|
988 |
|
|
989 |
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
|
|
990 |
by simp
|
|
991 |
|
|
992 |
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::nat) \<and> P) = (0 <= x \<and> P')"
|
|
993 |
by simp
|
|
994 |
|
|
995 |
use "cooper_dec.ML"
|
|
996 |
use "cooper_proof.ML"
|
|
997 |
use "qelim.ML"
|
|
998 |
use "presburger.ML"
|
|
999 |
|
|
1000 |
setup "Presburger.setup"
|
|
1001 |
|
|
1002 |
end
|