# Theory Presburger

Up to index of Isabelle/HOL

theory Presburger
imports NatSimprocs SetInterval
uses (cooper_dec.ML) (cooper_proof.ML) (qelim.ML) (reflected_presburger.ML) (reflected_cooper.ML) (presburger.ML)
begin

    (*  Title:      HOL/Integ/Presburger.thy
        ID:         $Id: Presburger.thy,v 1.18 2005/09/22 21:56:32 nipkow Exp $
        Author:     Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen

    File containing necessary theorems for the proof
    generation for Cooper Algorithm
    *)

    header {* Presburger Arithmetic: Cooper's Algorithm *}

    theory Presburger
    imports NatSimprocs SetInterval
    uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML")
            ("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
    begin

    text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}

    theorem unity_coeff_ex: "(∃x::int. P (l * x)) = (∃x. l dvd (1*x+0) ∧ P x)"
      apply (rule iffI)
      apply (erule exE)
      apply (rule_tac x = "l * x" in exI)
      apply simp
      apply (erule exE)
      apply (erule conjE)
      apply (erule dvdE)
      apply (rule_tac x = k in exI)
      apply simp
      done

    lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
    apply(unfold dvd_def)
    apply(rule iffI)
    apply(clarsimp)
    apply(rename_tac k)
    apply(rule_tac x = "-k" in exI)
    apply simp
    apply(clarsimp)
    apply(rename_tac k)
    apply(rule_tac x = "-k" in exI)
    apply simp
    done

    lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
    apply(unfold dvd_def)
    apply(rule iffI)
    apply(clarsimp)
    apply(rule_tac x = "-k" in exI)
    apply simp
    apply(clarsimp)
    apply(rule_tac x = "-k" in exI)
    apply simp
    done

    text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} less than some integer @{text z}.*}

    theorem eq_minf_conjI: "∃z1::int. ∀x. x < z1 --> (A1 x = A2 x) ==>
      ∃z2::int. ∀x. x < z2 --> (B1 x = B2 x) ==>
      ∃z::int. ∀x. x < z --> ((A1 x ∧ B1 x) = (A2 x ∧ B2 x))"
      apply (erule exE)+
      apply (rule_tac x = "min z1 z2" in exI)
      apply simp
      done

    theorem eq_minf_disjI: "∃z1::int. ∀x. x < z1 --> (A1 x = A2 x) ==>
      ∃z2::int. ∀x. x < z2 --> (B1 x = B2 x) ==>
      ∃z::int. ∀x. x < z --> ((A1 x ∨ B1 x) = (A2 x ∨ B2 x))"

      apply (erule exE)+
      apply (rule_tac x = "min z1 z2" in exI)
      apply simp
      done

    text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} greather than some integer @{text z}.*}

    theorem eq_pinf_conjI: "∃z1::int. ∀x. z1 < x --> (A1 x = A2 x) ==>
      ∃z2::int. ∀x. z2 < x --> (B1 x = B2 x) ==>
      ∃z::int. ∀x. z < x --> ((A1 x ∧ B1 x) = (A2 x ∧ B2 x))"
      apply (erule exE)+
      apply (rule_tac x = "max z1 z2" in exI)
      apply simp
      done

    theorem eq_pinf_disjI: "∃z1::int. ∀x. z1 < x --> (A1 x = A2 x) ==>
      ∃z2::int. ∀x. z2 < x --> (B1 x = B2 x) ==>
      ∃z::int. ∀x. z < x  --> ((A1 x ∨ B1 x) = (A2 x ∨ B2 x))"
      apply (erule exE)+
      apply (rule_tac x = "max z1 z2" in exI)
      apply simp
      done

    text {*
      \medskip Theorems for the combination of proofs of the modulo @{text
      D} property for @{text "P plusinfinity"}

      FIXME: This is THE SAME theorem as for the @{text minusinf} version,
      but with @{text "+k.."} instead of @{text "-k.."} In the future
      replace these both with only one. *}

    theorem modd_pinf_conjI: "∀(x::int) k. A x = A (x+k*d) ==>
      ∀(x::int) k. B x = B (x+k*d) ==>
      ∀(x::int) (k::int). (A x ∧ B x) = (A (x+k*d) ∧ B (x+k*d))"
      by simp

    theorem modd_pinf_disjI: "∀(x::int) k. A x = A (x+k*d) ==>
      ∀(x::int) k. B x = B (x+k*d) ==>
      ∀(x::int) (k::int). (A x ∨ B x) = (A (x+k*d) ∨ B (x+k*d))"
      by simp

    text {*
      This is one of the cases where the simplifed formula is prooved to
      habe some property (in relation to @{text P_m}) but we need to prove
      the property for the original formula (@{text P_m})

      FIXME: This is exaclty the same thm as for @{text minusinf}. *}

    lemma pinf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
      by blast

    text {*
      \medskip Theorems for the combination of proofs of the modulo @{text D}
      property for @{text "P minusinfinity"} *}

    theorem modd_minf_conjI: "∀(x::int) k. A x = A (x-k*d) ==>
      ∀(x::int) k. B x = B (x-k*d) ==>
      ∀(x::int) (k::int). (A x ∧ B x) = (A (x-k*d) ∧ B (x-k*d))"
      by simp

    theorem modd_minf_disjI: "∀(x::int) k. A x = A (x-k*d) ==>
      ∀(x::int) k. B x = B (x-k*d) ==>
      ∀(x::int) (k::int). (A x ∨ B x) = (A (x-k*d) ∨ B (x-k*d))"
      by simp

    text {*
      This is one of the cases where the simplifed formula is prooved to
      have some property (in relation to @{text P_m}) but we need to
      prove the property for the original formula (@{text P_m}). *}

    lemma minf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x))  ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
      by blast

    text {*
      Theorem needed for proving at runtime divide properties using the
      arithmetic tactic (which knows only about modulo = 0). *}

    lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
      by(simp add:dvd_def zmod_eq_0_iff)

    text {*
      \medskip Theorems used for the combination of proof for the
      backwards direction of Cooper's Theorem. They rely exclusively on
      Predicate calculus.*}

    lemma not_ast_p_disjI: "(ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P1(x) --> P1(x + d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) ∨ P2(x)) --> (P1(x + d) ∨ P2(x + d))) "
      by blast

    lemma not_ast_p_conjI: "(ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a- j)) --> P1(x) --> P1(x + d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) ∧ P2(x)) --> (P1(x + d)
    ∧ P2(x + d))) "
      by blast

    lemma not_ast_p_Q_elim: "
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
    ==> ( P = Q )
    ==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
      by blast

    text {*
      \medskip Theorems used for the combination of proof for the
      backwards direction of Cooper's Theorem. They rely exclusively on
      Predicate calculus.*}

    lemma not_bst_p_disjI: "(ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) ∨ P2(x)) --> (P1(x - d)
    ∨ P2(x-d))) "
      by blast

    lemma not_bst_p_conjI: "(ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
    ==>
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) ∧ P2(x)) --> (P1(x - d)
    ∧ P2(x-d))) "
      by blast

    lemma not_bst_p_Q_elim: "
    (ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d))
    ==> ( P = Q )
    ==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
      by blast

    text {* \medskip This is the first direction of Cooper's Theorem. *}
    lemma cooper_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((R|Q) --> (EX x::int. P x )) "
      by blast

    text {*
      \medskip The full Cooper's Theorem in its equivalence Form. Given
      the premises it is trivial too, it relies exclusively on prediacte calculus.*}
    lemma cooper_eq_thm: "(R --> (EX x::int. P x))  ==> (Q -->(EX x::int.  P x )) ==> ((~Q)
    --> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
      by blast

    text {*
      \medskip Some of the atomic theorems generated each time the atom
      does not depend on @{text x}, they are trivial.*}

    lemma  fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
      by blast

    lemma  fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
      by blast

    lemma not_bst_p_fm: "ALL (x::int). Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> fm --> fm"
      by blast

    lemma  fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
      by blast

    text {* The next two thms are the same as the @{text minusinf} version. *}

    lemma  fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
      by blast

    lemma not_ast_p_fm: "ALL (x::int). Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> fm --> fm"
      by blast

    text {* Theorems to be deleted from simpset when proving simplified formulaes. *}

    lemma P_eqtrue: "(P=True) = P"
      by iprover

    lemma P_eqfalse: "(P=False) = (~P)"
      by iprover

    text {*
      \medskip Theorems for the generation of the bachwards direction of
      Cooper's Theorem.

      These are the 6 interesting atomic cases which have to be proved relying on the
      properties of B-set and the arithmetic and contradiction proofs. *}

    lemma not_bst_p_lt: "0 < (d::int) ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ( 0 < -x + a) --> (0 < -(x - d) + a )"
      by arith

    lemma not_bst_p_gt: "[| (g::int) ∈ B; g = -a |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 < (x) + a) --> ( 0 < (x - d) + a)"
    apply clarsimp
    apply(rule ccontr)
    apply(drule_tac x = "x+a" in bspec)
    apply(simp add:atLeastAtMost_iff)
    apply(drule_tac x = "-a" in bspec)
    apply assumption
    apply(simp)
    done

    lemma not_bst_p_eq: "[| 0 < d; (g::int) ∈ B; g = -a - 1 |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 = x + a) --> (0 = (x - d) + a )"
    apply clarsimp
    apply(subgoal_tac "x = -a")
     prefer 2 apply arith
    apply(drule_tac x = "1" in bspec)
    apply(simp add:atLeastAtMost_iff)
    apply(drule_tac x = "-a- 1" in bspec)
    apply assumption
    apply(simp)
    done

    lemma not_bst_p_ne: "[| 0 < d; (g::int) ∈ B; g = -a |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)"
    apply clarsimp
    apply(subgoal_tac "x = -a+d")
     prefer 2 apply arith
    apply(drule_tac x = "d" in bspec)
    apply(simp add:atLeastAtMost_iff)
    apply(drule_tac x = "-a" in bspec)
    apply assumption
    apply(simp)
    done

    lemma not_bst_p_dvd: "(d1::int) dvd d ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )"
    apply(clarsimp simp add:dvd_def)
    apply(rename_tac m)
    apply(rule_tac x = "m - k" in exI)
    apply(simp add:int_distrib)
    done

    lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))"
    apply(clarsimp simp add:dvd_def)
    apply(rename_tac m)
    apply(erule_tac x = "m + k" in allE)
    apply(simp add:int_distrib)
    done

    text {*
      \medskip Theorems for the generation of the bachwards direction of
      Cooper's Theorem.

      These are the 6 interesting atomic cases which have to be proved
      relying on the properties of A-set ant the arithmetic and
      contradiction proofs. *}

    lemma not_ast_p_gt: "0 < (d::int) ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ( 0 < x + t) --> (0 < (x + d) + t )"
      by arith

    lemma not_ast_p_lt: "[|0 < d ;(t::int) ∈ A |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)"
      apply clarsimp
      apply (rule ccontr)
      apply (drule_tac x = "t-x" in bspec)
      apply simp
      apply (drule_tac x = "t" in bspec)
      apply assumption
      apply simp
      done

    lemma not_ast_p_eq: "[| 0 < d; (g::int) ∈ A; g = -t + 1 |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 = x + t) --> (0 = (x + d) + t )"
      apply clarsimp
      apply (drule_tac x="1" in bspec)
      apply simp
      apply (drule_tac x="- t + 1" in bspec)
      apply assumption
      apply(subgoal_tac "x = -t")
      prefer 2 apply arith
      apply simp
      done

    lemma not_ast_p_ne: "[| 0 < d; (g::int) ∈ A; g = -t |] ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(0 = x + t) --> ~(0 = (x + d) + t)"
      apply clarsimp
      apply (subgoal_tac "x = -t-d")
      prefer 2 apply arith
      apply (drule_tac x = "d" in bspec)
      apply simp
      apply (drule_tac x = "-t" in bspec)
      apply assumption
      apply simp
      done

    lemma not_ast_p_dvd: "(d1::int) dvd d ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> d1 dvd (x + t) --> d1 dvd ((x + d) + t )"
      apply(clarsimp simp add:dvd_def)
      apply(rename_tac m)
      apply(rule_tac x = "m + k" in exI)
      apply(simp add:int_distrib)
      done

    lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
     ALL x. Q(x::int) ∧ ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(d1 dvd (x + t)) --> ~(d1 dvd ((x + d) + t ))"
      apply(clarsimp simp add:dvd_def)
      apply(rename_tac m)
      apply(erule_tac x = "m - k" in allE)
      apply(simp add:int_distrib)
      done

    text {*
      \medskip These are the atomic cases for the proof generation for the
      modulo @{text D} property for @{text "P plusinfinity"}

      They are fully based on arithmetics. *}

    lemma  dvd_modd_pinf: "((d::int) dvd d1) ==>
     (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
      apply(clarsimp simp add:dvd_def)
      apply(rule iffI)
      apply(clarsimp)
      apply(rename_tac n m)
      apply(rule_tac x = "m + n*k" in exI)
      apply(simp add:int_distrib)
      apply(clarsimp)
      apply(rename_tac n m)
      apply(rule_tac x = "m - n*k" in exI)
      apply(simp add:int_distrib mult_ac)
      done

    lemma  not_dvd_modd_pinf: "((d::int) dvd d1) ==>
     (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
      apply(clarsimp simp add:dvd_def)
      apply(rule iffI)
      apply(clarsimp)
      apply(rename_tac n m)
      apply(erule_tac x = "m - n*k" in allE)
      apply(simp add:int_distrib mult_ac)
      apply(clarsimp)
      apply(rename_tac n m)
      apply(erule_tac x = "m + n*k" in allE)
      apply(simp add:int_distrib mult_ac)
      done

    text {*
      \medskip These are the atomic cases for the proof generation for the
      equivalence of @{text P} and @{text "P plusinfinity"} for integers
      @{text x} greater than some integer @{text z}.

      They are fully based on arithmetics. *}

    lemma  eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
      apply(rule_tac x = "-t" in exI)
      apply simp
      done

    lemma  neq_eq_pinf: "EX z::int. ALL x.  z < x --> ((~( 0 = x +t )) = True )"
      apply(rule_tac x = "-t" in exI)
      apply simp
      done

    lemma  le_eq_pinf: "EX z::int. ALL x.  z < x --> ( 0 < x +t  = True )"
      apply(rule_tac x = "-t" in exI)
      apply simp
      done

    lemma  len_eq_pinf: "EX z::int. ALL x. z < x  --> (0 < -x +t  = False )"
      apply(rule_tac x = "t" in exI)
      apply simp
      done

    lemma  dvd_eq_pinf: "EX z::int. ALL x.  z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
      by simp

    lemma  not_dvd_eq_pinf: "EX z::int. ALL x. z < x  --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
      by simp

    text {*
      \medskip These are the atomic cases for the proof generation for the
      modulo @{text D} property for @{text "P minusinfinity"}.

      They are fully based on arithmetics. *}

    lemma  dvd_modd_minf: "((d::int) dvd d1) ==>
     (ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
    apply(clarsimp simp add:dvd_def)
    apply(rule iffI)
    apply(clarsimp)
    apply(rename_tac n m)
    apply(rule_tac x = "m - n*k" in exI)
    apply(simp add:int_distrib)
    apply(clarsimp)
    apply(rename_tac n m)
    apply(rule_tac x = "m + n*k" in exI)
    apply(simp add:int_distrib mult_ac)
    done

    lemma  not_dvd_modd_minf: "((d::int) dvd d1) ==>
     (ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
    apply(clarsimp simp add:dvd_def)
    apply(rule iffI)
    apply(clarsimp)
    apply(rename_tac n m)
    apply(erule_tac x = "m + n*k" in allE)
    apply(simp add:int_distrib mult_ac)
    apply(clarsimp)
    apply(rename_tac n m)
    apply(erule_tac x = "m - n*k" in allE)
    apply(simp add:int_distrib mult_ac)
    done

    text {*
      \medskip These are the atomic cases for the proof generation for the
      equivalence of @{text P} and @{text "P minusinfinity"} for integers
      @{text x} less than some integer @{text z}.

      They are fully based on arithmetics. *}

    lemma  eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
    apply(rule_tac x = "-t" in exI)
    apply simp
    done

    lemma  neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
    apply(rule_tac x = "-t" in exI)
    apply simp
    done

    lemma  le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t  = False )"
    apply(rule_tac x = "-t" in exI)
    apply simp
    done

    lemma  len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t  = True )"
    apply(rule_tac x = "t" in exI)
    apply simp
    done

    lemma  dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
      by simp

    lemma  not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
      by simp

    text {*
      \medskip This Theorem combines whithnesses about @{text "P
      minusinfinity"} to show one component of the equivalence proof for
      Cooper's Theorem.

      FIXME: remove once they are part of the distribution. *}

    theorem int_ge_induct[consumes 1,case_names base step]:
      assumes ge: "k ≤ (i::int)" and
            base: "P(k)" and
            step: "!!i. [|k ≤ i; P i|] ==> P(i+1)"
      shows "P i"
    proof -
      { fix n have "!!i::int. n = nat(i-k) ==> k <= i ==> P i"
        proof (induct n)
          case 0
          hence "i = k" by arith
          thus "P i" using base by simp
        next
          case (Suc n)
          hence "n = nat((i - 1) - k)" by arith
          moreover
          have ki1: "k ≤ i - 1" using Suc.prems by arith
          ultimately
          have "P(i - 1)" by(rule Suc.hyps)
          from step[OF ki1 this] show ?case by simp
        qed
      }
      from this ge show ?thesis by fast
    qed

    theorem int_gr_induct[consumes 1,case_names base step]:
      assumes gr: "k < (i::int)" and
            base: "P(k+1)" and
            step: "!!i. [|k < i; P i|] ==> P(i+1)"
      shows "P i"
    apply(rule int_ge_induct[of "k + 1"])
      using gr apply arith
     apply(rule base)
    apply(rule step)
     apply simp+
    done

    lemma decr_lemma: "0 < (d::int) ==> x - (abs(x-z)+1) * d < z"
    apply(induct rule: int_gr_induct)
     apply simp
     apply arith
    apply (simp add:int_distrib)
    apply arith
    done

    lemma incr_lemma: "0 < (d::int) ==> z < x + (abs(x-z)+1) * d"
    apply(induct rule: int_gr_induct)
     apply simp
     apply arith
    apply (simp add:int_distrib)
    apply arith
    done

    lemma  minusinfinity:
      assumes "0 < d" and
        P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
        ePeqP1: "EX z::int. ALL x. x < z --> (P x = P1 x)"
      shows "(EX x. P1 x) --> (EX x. P x)"
    proof
      assume eP1: "EX x. P1 x"
      then obtain x where P1: "P1 x" ..
      from ePeqP1 obtain z where P1eqP: "ALL x. x < z --> (P x = P1 x)" ..
      let ?w = "x - (abs(x-z)+1) * d"
      show "EX x. P x"
      proof
        have w: "?w < z" by(rule decr_lemma)
        have "P1 x = P1 ?w" using P1eqP1 by blast
        also have "… = P(?w)" using w P1eqP by blast
        finally show "P ?w" using P1 by blast
      qed
    qed

    text {*
      \medskip This Theorem combines whithnesses about @{text "P
      minusinfinity"} to show one component of the equivalence proof for
      Cooper's Theorem. *}

    lemma plusinfinity:
      assumes "0 < d" and
        P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
        ePeqP1: "EX z::int. ALL x. z < x  --> (P x = P1 x)"
      shows "(EX x::int. P1 x) --> (EX x::int. P x)"
    proof
      assume eP1: "EX x. P1 x"
      then obtain x where P1: "P1 x" ..
      from ePeqP1 obtain z where P1eqP: "ALL x. z < x --> (P x = P1 x)" ..
      let ?w = "x + (abs(x-z)+1) * d"
      show "EX x. P x"
      proof
        have w: "z < ?w" by(rule incr_lemma)
        have "P1 x = P1 ?w" using P1eqP1 by blast
        also have "… = P(?w)" using w P1eqP by blast
        finally show "P ?w" using P1 by blast
      qed
    qed

    text {*
      \medskip Theorem for periodic function on discrete sets. *}

    lemma minf_vee:
      assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
      shows "(EX x. P x) = (EX j : {1..d}. P j)"
      (is "?LHS = ?RHS")
    proof
      assume ?LHS
      then obtain x where P: "P x" ..
      have "x mod d = x - (x div d)*d"
        by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
      hence Pmod: "P x = P(x mod d)" using modd by simp
      show ?RHS
      proof (cases)
        assume "x mod d = 0"
        hence "P 0" using P Pmod by simp
        moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
        ultimately have "P d" by simp
        moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
        ultimately show ?RHS ..
      next
        assume not0: "x mod d ≠ 0"
        have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
        moreover have "x mod d : {1..d}"
        proof -
          have "0 ≤ x mod d" by(rule pos_mod_sign)
          moreover have "x mod d < d" by(rule pos_mod_bound)
          ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
        qed
        ultimately show ?RHS ..
      qed
    next
      assume ?RHS thus ?LHS by blast
    qed

    text {*
      \medskip Theorem for periodic function on discrete sets. *}

    lemma pinf_vee:
      assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
      shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
      (is "?LHS = ?RHS")
    proof
      assume ?LHS
      then obtain x where P: "P x" ..
      have "x mod d = x + (-(x div d))*d"
        by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
      hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
      show ?RHS
      proof (cases)
        assume "x mod d = 0"
        hence "P 0" using P Pmod by simp
        moreover have "P 0 = P(0 + 1*d)" using modd by blast
        ultimately have "P d" by simp
        moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
        ultimately show ?RHS ..
      next
        assume not0: "x mod d ≠ 0"
        have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
        moreover have "x mod d : {1..d}"
        proof -
          have "0 ≤ x mod d" by(rule pos_mod_sign)
          moreover have "x mod d < d" by(rule pos_mod_bound)
          ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
        qed
        ultimately show ?RHS ..
      qed
    next
      assume ?RHS thus ?LHS by blast
    qed

    lemma decr_mult_lemma:
      assumes dpos: "(0::int) < d" and
              minus: "ALL x::int. P x --> P(x - d)" and
              knneg: "0 <= k"
      shows "ALL x. P x --> P(x - k*d)"
    using knneg
    proof (induct rule:int_ge_induct)
      case base thus ?case by simp
    next
      case (step i)
      show ?case
      proof
        fix x
        have "P x --> P (x - i * d)" using step.hyps by blast
        also have "… --> P(x - (i + 1) * d)"
          using minus[THEN spec, of "x - i * d"]
          by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
        ultimately show "P x --> P(x - (i + 1) * d)" by blast
      qed
    qed

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

    lemma cpmi_eq: "0 < D ==> (EX z::int. ALL x. x < z --> (P x = P1 x))
    ==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
    ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
    ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
    apply(rule iffI)
    prefer 2
    apply(drule minusinfinity)
    apply assumption+
    apply(fastsimp)
    apply clarsimp
    apply(subgoal_tac "!!k. 0<=k ==> !x. P x --> P (x - k*D)")
    apply(frule_tac x = x and z=z in decr_lemma)
    apply(subgoal_tac "P1(x - (¦x - z¦ + 1) * D)")
    prefer 2
    apply(subgoal_tac "0 <= (¦x - z¦ + 1)")
    prefer 2 apply arith
     apply fastsimp
    apply(drule (1) minf_vee)
    apply blast
    apply(blast dest:decr_mult_lemma)
    done

    text {* Cooper Theorem, plus infinity version. *}
    lemma cppi_eq: "0 < D ==> (EX z::int. ALL x. z < x --> (P x = P1 x))
    ==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D)
    ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
    ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
      apply(rule iffI)
      prefer 2
      apply(drule plusinfinity)
      apply assumption+
      apply(fastsimp)
      apply clarsimp
      apply(subgoal_tac "!!k. 0<=k ==> !x. P x --> P (x + k*D)")
      apply(frule_tac x = x and z=z in incr_lemma)
      apply(subgoal_tac "P1(x + (¦x - z¦ + 1) * D)")
      prefer 2
      apply(subgoal_tac "0 <= (¦x - z¦ + 1)")
      prefer 2 apply arith
      apply fastsimp
      apply(drule (1) pinf_vee)
      apply blast
      apply(blast dest:incr_mult_lemma)
      done

    text {*
      \bigskip Theorems for the quantifier elminination Functions. *}

    lemma qe_ex_conj: "(EX (x::int). A x) = R
                    ==> (EX (x::int). P x) = (Q & (EX x::int. A x))
                    ==> (EX (x::int). P x) = (Q & R)"
    by blast

    lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
                    ==> (EX (x::int). P x) = Q"
    by blast

    lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
    by blast

    lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
    by blast

    lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
    by blast

    lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
    by blast

    lemma qe_Not: "P = Q ==> (~P) = (~Q)"
    by blast

    lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
    by blast

    text {* \bigskip Theorems for proving NNF *}

    lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
    by blast

    lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
    by blast

    lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
      by blast
    lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
    by blast

    lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
    by blast
    lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
    by blast
    lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
    by blast
    lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
    by blast

    lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
      by simp

    lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
      by iprover

    lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
      by iprover

    lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
    ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
    by blast

    lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
    ==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
    by blast

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

    text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}

    lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
    shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
    proof
      assume ?P
      thus ?Q
        apply(simp add:dvd_def)
        apply clarify
        apply(rename_tac d)
        apply(drule_tac f = "op * k" in arg_cong)
        apply(simp only:int_distrib)
        apply(rule_tac x = "d" in exI)
        apply(simp only:mult_ac)
        done
    next
      assume ?Q
      then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
      hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
      hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
      hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
      thus ?P by(simp add:dvd_def)
    qed

    lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
    shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
    proof
      assume P: ?P
      show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
    next
      assume ?Q
      hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
      with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
      thus ?P by(simp)
    qed

    lemma ac_eq_eq : assumes not0: "0 ~= (k::int)" shows "((m::int) = (c*n+t)) = (k*m =((k*c)*n+(k*t)) )" (is "?P = ?Q")
    proof
      assume ?P
      thus ?Q
        apply(drule_tac f = "op * k" in arg_cong)
        apply(simp only:int_distrib)
        done
    next
      assume ?Q
      hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
      hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
      thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
    qed

    lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
    proof -
      have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
      also have  "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
      also have "0<(-1*c)*n + (-t+1) = (0 < (k*(-1*c)*n) + (k*(-t+1)))" by(rule ac_lt_eq[of _ 0,OF gr0,simplified])
      also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
      finally show ?thesis .
    qed

    lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
    by arith

    lemma  linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
    by simp

    lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
    by simp

    lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
    by simp

    lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
    by simp

    text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}

    theorem all_nat: "(∀x::nat. P x) = (∀x::int. 0 <= x --> P (nat x))"
      by (simp split add: split_nat)

    theorem ex_nat: "(∃x::nat. P x) = (∃x::int. 0 <= x ∧ P (nat x))"
      apply (simp split add: split_nat)
      apply (rule iffI)
      apply (erule exE)
      apply (rule_tac x = "int x" in exI)
      apply simp
      apply (erule exE)
      apply (rule_tac x = "nat x" in exI)
      apply (erule conjE)
      apply (erule_tac x = "nat x" in allE)
      apply simp
      done

    theorem zdiff_int_split: "P (int (x - y)) =
      ((y ≤ x --> P (int x - int y)) ∧ (x < y --> P 0))"
      apply (case_tac "y ≤ x")
      apply (simp_all add: zdiff_int)
      done

    theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
      apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
        nat_0_le cong add: conj_cong)
      apply (rule iffI)
      apply iprover
      apply (erule exE)
      apply (case_tac "x=0")
      apply (rule_tac x=0 in exI)
      apply simp
      apply (case_tac "0 ≤ k")
      apply iprover
      apply (simp add: linorder_not_le)
      apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
      apply assumption
      apply (simp add: mult_ac)
      done

    theorem number_of1: "(0::int) <= number_of n ==> (0::int) <= number_of (n BIT b)"
      by simp

    theorem number_of2: "(0::int) <= Numeral0" by simp

    theorem Suc_plus1: "Suc n = n + 1" by simp

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

    theorem imp_le_cong: "(0 <= x ==> P = P') ==> (0 <= (x::int) --> P) = (0 <= x --> P')"
      by simp

    theorem conj_le_cong: "(0 <= x ==> P = P') ==> (0 <= (x::int) ∧ P) = (0 <= x ∧ P')"
      by (simp cong: conj_cong)

    use "cooper_dec.ML"
    use "reflected_presburger.ML"
    use "reflected_cooper.ML"
    oracle
      presburger_oracle ("term") = ReflectedCooper.presburger_oracle

    use "cooper_proof.ML"
    use "qelim.ML"
    use "presburger.ML"

    setup "Presburger.setup"

    end

theorem unity_coeff_ex:

      (∃x. P (l * x)) = (∃x. l dvd 1 * x + 0 ∧ P x)

lemma uminus_dvd_conv:

      (d dvd t) = (- d dvd t)

lemma uminus_dvd_conv':

      (d dvd t) = (d dvd - t)

theorem eq_minf_conjI:

      [| ∃z1. ∀x<z1. A1.0 x = A2.0 x; ∃z2. ∀x<z2. B1.0 x = B2.0 x |]
      ==> ∃z. ∀x<z. (A1.0 x ∧ B1.0 x) = (A2.0 x ∧ B2.0 x)

theorem eq_minf_disjI:

      [| ∃z1. ∀x<z1. A1.0 x = A2.0 x; ∃z2. ∀x<z2. B1.0 x = B2.0 x |]
      ==> ∃z. ∀x<z. (A1.0 x ∨ B1.0 x) = (A2.0 x ∨ B2.0 x)

theorem eq_pinf_conjI:

      [| ∃z1. ∀x. z1 < x --> A1.0 x = A2.0 x; ∃z2. ∀x. z2 < x --> B1.0 x = B2.0 x |]
      ==> ∃z. ∀x. z < x --> (A1.0 x ∧ B1.0 x) = (A2.0 x ∧ B2.0 x)

theorem eq_pinf_disjI:

      [| ∃z1. ∀x. z1 < x --> A1.0 x = A2.0 x; ∃z2. ∀x. z2 < x --> B1.0 x = B2.0 x |]
      ==> ∃z. ∀x. z < x --> (A1.0 x ∨ B1.0 x) = (A2.0 x ∨ B2.0 x)

theorem modd_pinf_conjI:

      [| ∀x k. A x = A (x + k * d); ∀x k. B x = B (x + k * d) |]
      ==> ∀x k. (A x ∧ B x) = (A (x + k * d) ∧ B (x + k * d))

theorem modd_pinf_disjI:

      [| ∀x k. A x = A (x + k * d); ∀x k. B x = B (x + k * d) |]
      ==> ∀x k. (A x ∨ B x) = (A (x + k * d) ∨ B (x + k * d))

lemma pinf_simp_eq:

      [| ∀x. P x = Q x; (∃x. P x) --> (∃x. F x) |] ==> (∃x. Q x) --> (∃x. F x)

theorem modd_minf_conjI:

      [| ∀x k. A x = A (x - k * d); ∀x k. B x = B (x - k * d) |]
      ==> ∀x k. (A x ∧ B x) = (A (x - k * d) ∧ B (x - k * d))

theorem modd_minf_disjI:

      [| ∀x k. A x = A (x - k * d); ∀x k. B x = B (x - k * d) |]
      ==> ∀x k. (A x ∨ B x) = (A (x - k * d) ∨ B (x - k * d))

lemma minf_simp_eq:

      [| ∀x. P x = Q x; (∃x. P x) --> (∃x. F x) |] ==> (∃x. Q x) --> (∃x. F x)

lemma zdvd_iff_zmod_eq_0:

      (m dvd n) = (n mod m = 0)

lemma not_ast_p_disjI:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> P1.0 x --> P1.0 (x + d);
         ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> P2.0 x --> P2.0 (x + d) |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) -->
              P1.0 x ∨ P2.0 x --> P1.0 (x + d) ∨ P2.0 (x + d)

lemma not_ast_p_conjI:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> P1.0 x --> P1.0 (x + d);
         ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> P2.0 x --> P2.0 (x + d) |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) -->
              P1.0 x ∧ P2.0 x --> P1.0 (x + d) ∧ P2.0 (x + d)

lemma not_ast_p_Q_elim:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> P x --> P (x + d); P = Q |]
      ==> ∀x. ¬ (∃j∈{1..d}. ∃a∈A. P (a - j)) --> P x --> P (x + d)

lemma not_bst_p_disjI:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> P1.0 x --> P1.0 (x - d);
         ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> P2.0 x --> P2.0 (x - d) |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) -->
              P1.0 x ∨ P2.0 x --> P1.0 (x - d) ∨ P2.0 (x - d)

lemma not_bst_p_conjI:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> P1.0 x --> P1.0 (x - d);
         ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> P2.0 x --> P2.0 (x - d) |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) -->
              P1.0 x ∧ P2.0 x --> P1.0 (x - d) ∧ P2.0 (x - d)

lemma not_bst_p_Q_elim:

      [| ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> P x --> P (x - d); P = Q |]
      ==> ∀x. ¬ (∃j∈{1..d}. ∃b∈B. P (b + j)) --> P x --> P (x - d)

lemma cooper_thm:

      [| R --> (∃x. P x); Q --> (∃x. P x) |] ==> R ∨ Q --> (∃x. P x)

lemma cooper_eq_thm:

      [| R --> (∃x. P x); Q --> (∃x. P x); ¬ Q --> (∃x. P x) --> R |]
      ==> (∃x. P x) = R ∨ Q

lemma fm_eq_minf:

      ∃z. ∀x<z. P = P

lemma fm_modd_minf:

      ∀x k. P = P

lemma not_bst_p_fm:

      ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> fm --> fm

lemma fm_eq_pinf:

      ∃z. ∀x. z < x --> P = P

lemma fm_modd_pinf:

      ∀x k. P = P

lemma not_ast_p_fm:

      ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> fm --> fm

lemma P_eqtrue:

      (P = True) = P

lemma P_eqfalse:

      (P = False) = (¬ P)

lemma not_bst_p_lt:

      0 < d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) -->
              0 < - x + a --> 0 < - (x - d) + a

lemma not_bst_p_gt:

      [| g ∈ B; g = - a |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> 0 < x + a --> 0 < x - d + a

lemma not_bst_p_eq:

      [| 0 < d; g ∈ B; g = - a - 1 |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> 0 = x + a --> 0 = x - d + a

lemma not_bst_p_ne:

      [| 0 < d; g ∈ B; g = - a |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) --> 0 ≠ x + a --> 0 ≠ x - d + a

lemma not_bst_p_dvd:

      d1.0 dvd d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) -->
              d1.0 dvd x + a --> d1.0 dvd x - d + a

lemma not_bst_p_ndvd:

      d1.0 dvd d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃b∈B. Q (b + j)) -->
              ¬ d1.0 dvd x + a --> ¬ d1.0 dvd x - d + a

lemma not_ast_p_gt:

      0 < d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> 0 < x + t --> 0 < x + d + t

lemma not_ast_p_lt:

      [| 0 < d; t ∈ A |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) -->
              0 < - x + t --> 0 < - (x + d) + t

lemma not_ast_p_eq:

      [| 0 < d; g ∈ A; g = - t + 1 |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> 0 = x + t --> 0 = x + d + t

lemma not_ast_p_ne:

      [| 0 < d; g ∈ A; g = - t |]
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) --> 0 ≠ x + t --> 0 ≠ x + d + t

lemma not_ast_p_dvd:

      d1.0 dvd d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) -->
              d1.0 dvd x + t --> d1.0 dvd x + d + t

lemma not_ast_p_ndvd:

      d1.0 dvd d
      ==> ∀x. Q x ∧ ¬ (∃j∈{1..d}. ∃a∈A. Q (a - j)) -->
              ¬ d1.0 dvd x + t --> ¬ d1.0 dvd x + d + t

lemma dvd_modd_pinf:

      d dvd d1.0 ==> ∀x k. (d dvd x + t) = (d dvd x + k * d1.0 + t)

lemma not_dvd_modd_pinf:

      d dvd d1.0 ==> ∀x k. (¬ d dvd x + t) = (¬ d dvd x + k * d1.0 + t)

lemma eq_eq_pinf:

      ∃z. ∀x. z < x --> (0 = x + t) = False

lemma neq_eq_pinf:

      ∃z. ∀x. z < x --> (0 ≠ x + t) = True

lemma le_eq_pinf:

      ∃z. ∀x. z < x --> (0 < x + t) = True

lemma len_eq_pinf:

      ∃z. ∀x. z < x --> (0 < - x + t) = False

lemma dvd_eq_pinf:

      ∃z. ∀x. z < x --> (d dvd x + t) = (d dvd x + t)

lemma not_dvd_eq_pinf:

      ∃z. ∀x. z < x --> (¬ d dvd x + t) = (¬ d dvd x + t)

lemma dvd_modd_minf:

      d dvd d1.0 ==> ∀x k. (d dvd x + t) = (d dvd x - k * d1.0 + t)

lemma not_dvd_modd_minf:

      d dvd d1.0 ==> ∀x k. (¬ d dvd x + t) = (¬ d dvd x - k * d1.0 + t)

lemma eq_eq_minf:

      ∃z. ∀x<z. (0 = x + t) = False

lemma neq_eq_minf:

      ∃z. ∀x<z. (0 ≠ x + t) = True

lemma le_eq_minf:

      ∃z. ∀x<z. (0 < x + t) = False

lemma len_eq_minf:

      ∃z. ∀x<z. (0 < - x + t) = True

lemma dvd_eq_minf:

      ∃z. ∀x<z. (d dvd x + t) = (d dvd x + t)

lemma not_dvd_eq_minf:

      ∃z. ∀x<z. (¬ d dvd x + t) = (¬ d dvd x + t)

theorem int_ge_induct:

      [| k ≤ i; P k; !!i. [| k ≤ i; P i |] ==> P (i + 1) |] ==> P i

theorem int_gr_induct:

      [| k < i; P (k + 1); !!i. [| k < i; P i |] ==> P (i + 1) |] ==> P i

lemma decr_lemma:

      0 < d ==> x - (¦x - z¦ + 1) * d < z

lemma incr_lemma:

      0 < d ==> z < x + (¦x - z¦ + 1) * d

lemma minusinfinity:

      [| 0 < d; ∀x k. P1.0 x = P1.0 (x - k * d); ∃z. ∀x<z. P x = P1.0 x |]
      ==> (∃x. P1.0 x) --> (∃x. P x)

lemma plusinfinity:

      [| 0 < d; ∀x k. P1.0 x = P1.0 (x + k * d); ∃z. ∀x. z < x --> P x = P1.0 x |]
      ==> (∃x. P1.0 x) --> (∃x. P x)

lemma minf_vee:

      [| 0 < d; ∀x k. P x = P (x - k * d) |] ==> (∃x. P x) = (∃j∈{1..d}. P j)

lemma pinf_vee:

      [| 0 < d; ∀x k. P x = P (x + k * d) |] ==> (∃x. P x) = (∃j∈{1..d}. P j)

lemma decr_mult_lemma:

      [| 0 < d; ∀x. P x --> P (x - d); 0 ≤ k |] ==> ∀x. P x --> P (x - k * d)

lemma incr_mult_lemma:

      [| 0 < d; ∀x. P x --> P (x + d); 0 ≤ k |] ==> ∀x. P x --> P (x + k * d)

lemma cpmi_eq:

      [| 0 < D; ∃z. ∀x<z. P x = P1.0 x;
         ∀x. ¬ (∃j∈{1..D}. ∃b∈B. P (b + j)) --> P x --> P (x - D);
         ∀x k. P1.0 x = P1.0 (x - k * D) |]
      ==> (∃x. P x) = ((∃j∈{1..D}. P1.0 j) ∨ (∃j∈{1..D}. ∃b∈B. P (b + j)))

lemma cppi_eq:

      [| 0 < D; ∃z. ∀x. z < x --> P x = P1.0 x;
         ∀x. ¬ (∃j∈{1..D}. ∃a∈A. P (a - j)) --> P x --> P (x + D);
         ∀x k. P1.0 x = P1.0 (x + k * D) |]
      ==> (∃x. P x) = ((∃j∈{1..D}. P1.0 j) ∨ (∃j∈{1..D}. ∃a∈A. P (a - j)))

lemma qe_ex_conj:

      [| (∃x. A x) = R; (∃x. P x) = (Q ∧ (∃x. A x)) |] ==> (∃x. P x) = (Q ∧ R)

lemma qe_ex_nconj:

      (∃x. P x) = (True ∧ Q) ==> (∃x. P x) = Q

lemma qe_conjI:

      [| P1.0 = P2.0; Q1.0 = Q2.0 |] ==> (P1.0 ∧ Q1.0) = (P2.0 ∧ Q2.0)

lemma qe_disjI:

      [| P1.0 = P2.0; Q1.0 = Q2.0 |] ==> (P1.0 ∨ Q1.0) = (P2.0 ∨ Q2.0)

lemma qe_impI:

      [| P1.0 = P2.0; Q1.0 = Q2.0 |] ==> (P1.0 --> Q1.0) = (P2.0 --> Q2.0)

lemma qe_eqI:

      [| P1.0 = P2.0; Q1.0 = Q2.0 |] ==> (P1.0 = Q1.0) = (P2.0 = Q2.0)

lemma qe_Not:

      P = Q ==> (¬ P) = (¬ Q)

lemma qe_ALL:

      (∃x. ¬ P x) = R ==> (∀x. P x) = (¬ R)

lemma nnf_im:

      [| (¬ P) = P1.0; Q = Q1.0 |] ==> (P --> Q) = (P1.0 ∨ Q1.0)

lemma nnf_eq:

      [| (P ∧ Q) = (P1.0 ∧ Q1.0); (¬ P ∧ ¬ Q) = (P2.0 ∧ Q2.0) |]
      ==> (P = Q) = (P1.0 ∧ Q1.0 ∨ P2.0 ∧ Q2.0)

lemma nnf_nn:

      P = Q ==> (¬ ¬ P) = Q

lemma nnf_ncj:

      [| (¬ P) = P1.0; (¬ Q) = Q1.0 |] ==> (¬ (P ∧ Q)) = (P1.0 ∨ Q1.0)

lemma nnf_ndj:

      [| (¬ P) = P1.0; (¬ Q) = Q1.0 |] ==> (¬ (P ∨ Q)) = (P1.0 ∧ Q1.0)

lemma nnf_nim:

      [| P = P1.0; (¬ Q) = Q1.0 |] ==> (¬ (P --> Q)) = (P1.0 ∧ Q1.0)

lemma nnf_neq:

      [| (P ∧ ¬ Q) = (P1.0 ∧ Q1.0); (¬ P ∧ Q) = (P2.0 ∧ Q2.0) |]
      ==> (P ≠ Q) = (P1.0 ∧ Q1.0 ∨ P2.0 ∧ Q2.0)

lemma nnf_sdj:

      [| (A ∧ ¬ B) = (A1.0 ∧ B1.0); (C ∧ ¬ D) = (C1.0 ∧ D1.0); A = (¬ C) |]
      ==> (¬ (A ∧ B ∨ C ∧ D)) = (A1.0 ∧ B1.0 ∨ C1.0 ∧ D1.0)

lemma qe_exI2:

      A = B ==> (∃x. A x) = (∃x. B x)

lemma qe_exI:

      (!!x. A x = B x) ==> (∃x. A x) = (∃x. B x)

lemma qe_ALLI:

      (!!x. A x = B x) ==> (∀x. A x) = (∀x. B x)

lemma cp_expand:

      (∃x. P x) = (∃j∈{1..d}. ∃b∈B. P1.0 j ∨ P (b + j))
      ==> (∃x. P x) = (∃j∈{1..d}. ∃b∈B. P1.0 j ∨ P (b + j))

lemma cppi_expand:

      (∃x. P x) = (∃j∈{1..d}. ∃a∈A. P1.0 j ∨ P (a - j))
      ==> (∃x. P x) = (∃j∈{1..d}. ∃a∈A. P1.0 j ∨ P (a - j))

lemma simp_from_to:

      {i..j} = (if j < i then {} else insert i {i + 1..j})

lemma ac_dvd_eq:

      0 ≠ k ==> (m dvd c * n + t) = (k * m dvd k * c * n + k * t)

lemma ac_lt_eq:

      0 < k ==> (m < c * n + t) = (k * m < k * c * n + k * t)

lemma ac_eq_eq:

      0 ≠ k ==> (m = c * n + t) = (k * m = k * c * n + k * t)

lemma ac_pi_eq:

      0 < k ==> (¬ 0 < c * n + t) = (0 < - k * c * n + (- k * t + k))

lemma binminus_uminus_conv:

      a - b = a + - b

lemma linearize_dvd:

      t = t1.0 ==> (d dvd t) = (d dvd t1.0)

lemma lf_lt:

      [| l = ll; r = lr |] ==> (l < r) = (ll < lr)

lemma lf_eq:

      [| l = ll; r = lr |] ==> (l = r) = (ll = lr)

lemma lf_dvd:

      [| l = ll; r = lr |] ==> (l dvd r) = (ll dvd lr)

theorem all_nat:

      (∀x. P x) = (∀x≥0. P (nat x))

theorem ex_nat:

      (∃x. P x) = (∃x≥0. P (nat x))

theorem zdiff_int_split:

      P (int (x - y)) = ((y ≤ x --> P (int x - int y)) ∧ (x < y --> P 0))

theorem zdvd_int:

      (x dvd y) = (int x dvd int y)

theorem number_of1:

      0 ≤ number_of n ==> 0 ≤ number_of (n BIT b)

theorem number_of2:

      0 ≤ Numeral0

theorem Suc_plus1:

      Suc n = n + 1

theorem imp_le_cong:

      (0 ≤ x ==> P = P') ==> (0 ≤ x --> P) = (0 ≤ x --> P')

theorem conj_le_cong:

      (0 ≤ x ==> P = P') ==> (0 ≤ x ∧ P) = (0 ≤ x ∧ P')