renamed NatSimprocs.thy to Arith_Tools.thy;
incorporated HOL/Presburger.thy into NatSimprocs.thy;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Arith_Tools.thy Thu Jun 21 17:28:50 2007 +0200
@@ -0,0 +1,1621 @@
+(* Title: HOL/Arith_Tools.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Author: Amine Chaieb, TU Muenchen
+*)
+
+header {* Setup of arithmetic tools *}
+
+theory Arith_Tools
+imports Groebner_Basis Dense_Linear_Order SetInterval
+uses
+ "~~/src/Provers/Arith/cancel_numeral_factor.ML"
+ "~~/src/Provers/Arith/extract_common_term.ML"
+ "int_factor_simprocs.ML"
+ "nat_simprocs.ML"
+ "Tools/Presburger/cooper_data.ML"
+ "Tools/Presburger/generated_cooper.ML"
+ ("Tools/Presburger/cooper.ML")
+ ("Tools/Presburger/presburger.ML")
+begin
+
+subsection {* Simprocs for the Naturals *}
+
+setup nat_simprocs_setup
+
+subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
+
+text{*Where K above is a literal*}
+
+lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
+by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
+
+text {*Now just instantiating @{text n} to @{text "number_of v"} does
+ the right simplification, but with some redundant inequality
+ tests.*}
+lemma neg_number_of_pred_iff_0:
+ "neg (number_of (Numeral.pred v)::int) = (number_of v = (0::nat))"
+apply (subgoal_tac "neg (number_of (Numeral.pred v)) = (number_of v < Suc 0) ")
+apply (simp only: less_Suc_eq_le le_0_eq)
+apply (subst less_number_of_Suc, simp)
+done
+
+text{*No longer required as a simprule because of the @{text inverse_fold}
+ simproc*}
+lemma Suc_diff_number_of:
+ "neg (number_of (uminus v)::int) ==>
+ Suc m - (number_of v) = m - (number_of (Numeral.pred v))"
+apply (subst Suc_diff_eq_diff_pred)
+apply simp
+apply (simp del: nat_numeral_1_eq_1)
+apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
+by (simp add: numerals split add: nat_diff_split)
+
+
+subsubsection{*For @{term nat_case} and @{term nat_rec}*}
+
+lemma nat_case_number_of [simp]:
+ "nat_case a f (number_of v) =
+ (let pv = number_of (Numeral.pred v) in
+ if neg pv then a else f (nat pv))"
+by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
+
+lemma nat_case_add_eq_if [simp]:
+ "nat_case a f ((number_of v) + n) =
+ (let pv = number_of (Numeral.pred v) in
+ if neg pv then nat_case a f n else f (nat pv + n))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: numeral_1_eq_Suc_0 [symmetric] Let_def
+ neg_imp_number_of_eq_0 neg_number_of_pred_iff_0)
+done
+
+lemma nat_rec_number_of [simp]:
+ "nat_rec a f (number_of v) =
+ (let pv = number_of (Numeral.pred v) in
+ if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
+apply (case_tac " (number_of v) ::nat")
+apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
+apply (simp split add: split_if_asm)
+done
+
+lemma nat_rec_add_eq_if [simp]:
+ "nat_rec a f (number_of v + n) =
+ (let pv = number_of (Numeral.pred v) in
+ if neg pv then nat_rec a f n
+ else f (nat pv + n) (nat_rec a f (nat pv + n)))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0
+ neg_number_of_pred_iff_0)
+done
+
+
+subsubsection{*Various Other Lemmas*}
+
+text {*Evens and Odds, for Mutilated Chess Board*}
+
+text{*Lemmas for specialist use, NOT as default simprules*}
+lemma nat_mult_2: "2 * z = (z+z::nat)"
+proof -
+ have "2*z = (1 + 1)*z" by simp
+ also have "... = z+z" by (simp add: left_distrib)
+ finally show ?thesis .
+qed
+
+lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
+by (subst mult_commute, rule nat_mult_2)
+
+text{*Case analysis on @{term "n<2"}*}
+lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
+by arith
+
+lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
+by arith
+
+lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
+by (simp add: nat_mult_2 [symmetric])
+
+lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
+apply (subgoal_tac "m mod 2 < 2")
+apply (erule less_2_cases [THEN disjE])
+apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
+done
+
+lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
+apply (subgoal_tac "m mod 2 < 2")
+apply (force simp del: mod_less_divisor, simp)
+done
+
+text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
+
+lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
+by simp
+
+lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
+by simp
+
+text{*Can be used to eliminate long strings of Sucs, but not by default*}
+lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
+by simp
+
+
+text{*These lemmas collapse some needless occurrences of Suc:
+ at least three Sucs, since two and fewer are rewritten back to Suc again!
+ We already have some rules to simplify operands smaller than 3.*}
+
+lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
+by (simp add: Suc3_eq_add_3)
+
+lemmas Suc_div_eq_add3_div_number_of =
+ Suc_div_eq_add3_div [of _ "number_of v", standard]
+declare Suc_div_eq_add3_div_number_of [simp]
+
+lemmas Suc_mod_eq_add3_mod_number_of =
+ Suc_mod_eq_add3_mod [of _ "number_of v", standard]
+declare Suc_mod_eq_add3_mod_number_of [simp]
+
+
+subsubsection{*Special Simplification for Constants*}
+
+text{*These belong here, late in the development of HOL, to prevent their
+interfering with proofs of abstract properties of instances of the function
+@{term number_of}*}
+
+text{*These distributive laws move literals inside sums and differences.*}
+lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
+declare left_distrib_number_of [simp]
+
+lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
+declare right_distrib_number_of [simp]
+
+
+lemmas left_diff_distrib_number_of =
+ left_diff_distrib [of _ _ "number_of v", standard]
+declare left_diff_distrib_number_of [simp]
+
+lemmas right_diff_distrib_number_of =
+ right_diff_distrib [of "number_of v", standard]
+declare right_diff_distrib_number_of [simp]
+
+
+text{*These are actually for fields, like real: but where else to put them?*}
+lemmas zero_less_divide_iff_number_of =
+ zero_less_divide_iff [of "number_of w", standard]
+declare zero_less_divide_iff_number_of [simp]
+
+lemmas divide_less_0_iff_number_of =
+ divide_less_0_iff [of "number_of w", standard]
+declare divide_less_0_iff_number_of [simp]
+
+lemmas zero_le_divide_iff_number_of =
+ zero_le_divide_iff [of "number_of w", standard]
+declare zero_le_divide_iff_number_of [simp]
+
+lemmas divide_le_0_iff_number_of =
+ divide_le_0_iff [of "number_of w", standard]
+declare divide_le_0_iff_number_of [simp]
+
+
+(****
+IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
+then these special-case declarations may be useful.
+
+text{*These simprules move numerals into numerators and denominators.*}
+lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
+by (simp add: times_divide_eq)
+
+lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
+by (simp add: times_divide_eq)
+
+lemmas times_divide_eq_right_number_of =
+ times_divide_eq_right [of "number_of w", standard]
+declare times_divide_eq_right_number_of [simp]
+
+lemmas times_divide_eq_right_number_of =
+ times_divide_eq_right [of _ _ "number_of w", standard]
+declare times_divide_eq_right_number_of [simp]
+
+lemmas times_divide_eq_left_number_of =
+ times_divide_eq_left [of _ "number_of w", standard]
+declare times_divide_eq_left_number_of [simp]
+
+lemmas times_divide_eq_left_number_of =
+ times_divide_eq_left [of _ _ "number_of w", standard]
+declare times_divide_eq_left_number_of [simp]
+
+****)
+
+text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks
+ strange, but then other simprocs simplify the quotient.*}
+
+lemmas inverse_eq_divide_number_of =
+ inverse_eq_divide [of "number_of w", standard]
+declare inverse_eq_divide_number_of [simp]
+
+
+text {*These laws simplify inequalities, moving unary minus from a term
+into the literal.*}
+lemmas less_minus_iff_number_of =
+ less_minus_iff [of "number_of v", standard]
+declare less_minus_iff_number_of [simp]
+
+lemmas le_minus_iff_number_of =
+ le_minus_iff [of "number_of v", standard]
+declare le_minus_iff_number_of [simp]
+
+lemmas equation_minus_iff_number_of =
+ equation_minus_iff [of "number_of v", standard]
+declare equation_minus_iff_number_of [simp]
+
+
+lemmas minus_less_iff_number_of =
+ minus_less_iff [of _ "number_of v", standard]
+declare minus_less_iff_number_of [simp]
+
+lemmas minus_le_iff_number_of =
+ minus_le_iff [of _ "number_of v", standard]
+declare minus_le_iff_number_of [simp]
+
+lemmas minus_equation_iff_number_of =
+ minus_equation_iff [of _ "number_of v", standard]
+declare minus_equation_iff_number_of [simp]
+
+
+text{*To Simplify Inequalities Where One Side is the Constant 1*}
+
+lemma less_minus_iff_1 [simp]:
+ fixes b::"'b::{ordered_idom,number_ring}"
+ shows "(1 < - b) = (b < -1)"
+by auto
+
+lemma le_minus_iff_1 [simp]:
+ fixes b::"'b::{ordered_idom,number_ring}"
+ shows "(1 \<le> - b) = (b \<le> -1)"
+by auto
+
+lemma equation_minus_iff_1 [simp]:
+ fixes b::"'b::number_ring"
+ shows "(1 = - b) = (b = -1)"
+by (subst equation_minus_iff, auto)
+
+lemma minus_less_iff_1 [simp]:
+ fixes a::"'b::{ordered_idom,number_ring}"
+ shows "(- a < 1) = (-1 < a)"
+by auto
+
+lemma minus_le_iff_1 [simp]:
+ fixes a::"'b::{ordered_idom,number_ring}"
+ shows "(- a \<le> 1) = (-1 \<le> a)"
+by auto
+
+lemma minus_equation_iff_1 [simp]:
+ fixes a::"'b::number_ring"
+ shows "(- a = 1) = (a = -1)"
+by (subst minus_equation_iff, auto)
+
+
+text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
+
+lemmas mult_less_cancel_left_number_of =
+ mult_less_cancel_left [of "number_of v", standard]
+declare mult_less_cancel_left_number_of [simp]
+
+lemmas mult_less_cancel_right_number_of =
+ mult_less_cancel_right [of _ "number_of v", standard]
+declare mult_less_cancel_right_number_of [simp]
+
+lemmas mult_le_cancel_left_number_of =
+ mult_le_cancel_left [of "number_of v", standard]
+declare mult_le_cancel_left_number_of [simp]
+
+lemmas mult_le_cancel_right_number_of =
+ mult_le_cancel_right [of _ "number_of v", standard]
+declare mult_le_cancel_right_number_of [simp]
+
+
+text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
+
+lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard]
+declare le_divide_eq_number_of [simp]
+
+lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard]
+declare divide_le_eq_number_of [simp]
+
+lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard]
+declare less_divide_eq_number_of [simp]
+
+lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard]
+declare divide_less_eq_number_of [simp]
+
+lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard]
+declare eq_divide_eq_number_of [simp]
+
+lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard]
+declare divide_eq_eq_number_of [simp]
+
+
+
+subsubsection{*Optional Simplification Rules Involving Constants*}
+
+text{*Simplify quotients that are compared with a literal constant.*}
+
+lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
+lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
+lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
+lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
+lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
+lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
+
+
+text{*Not good as automatic simprules because they cause case splits.*}
+lemmas divide_const_simps =
+ le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
+ divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
+ le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
+
+text{*Division By @{text "-1"}*}
+
+lemma divide_minus1 [simp]:
+ "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
+by simp
+
+lemma minus1_divide [simp]:
+ "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
+by (simp add: divide_inverse inverse_minus_eq)
+
+lemma half_gt_zero_iff:
+ "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
+by auto
+
+lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
+declare half_gt_zero [simp]
+
+(* The following lemma should appear in Divides.thy, but there the proof
+ doesn't work. *)
+
+lemma nat_dvd_not_less:
+ "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
+ by (unfold dvd_def) auto
+
+ML {*
+val divide_minus1 = @{thm divide_minus1};
+val minus1_divide = @{thm minus1_divide};
+*}
+
+
+subsection{* Groebner Bases for fields *}
+
+interpretation class_fieldgb:
+ fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
+
+lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
+lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
+ by simp
+lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
+ by simp
+lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y"
+ by simp
+lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y"
+ by simp
+
+lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
+
+lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
+ by (simp add: add_divide_distrib)
+lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
+ by (simp add: add_divide_distrib)
+
+declaration{*
+let
+ val zr = @{cpat "0"}
+ val zT = ctyp_of_term zr
+ val geq = @{cpat "op ="}
+ val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
+ val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
+ val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
+ val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
+
+ fun prove_nz ctxt =
+ let val ss = local_simpset_of ctxt
+ in fn T => fn t =>
+ let
+ val z = instantiate_cterm ([(zT,T)],[]) zr
+ val eq = instantiate_cterm ([(eqT,T)],[]) geq
+ val th = Simplifier.rewrite (ss addsimps simp_thms)
+ (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
+ (Thm.capply (Thm.capply eq t) z)))
+ in equal_elim (symmetric th) TrueI
+ end
+ end
+
+ fun proc ctxt phi ss ct =
+ let
+ val ((x,y),(w,z)) =
+ (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
+ val T = ctyp_of_term x
+ val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
+ val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
+ in SOME (implies_elim (implies_elim th y_nz) z_nz)
+ end
+ handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun proc2 ctxt phi ss ct =
+ let
+ val (l,r) = Thm.dest_binop ct
+ val T = ctyp_of_term l
+ in (case (term_of l, term_of r) of
+ (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
+ let val (x,y) = Thm.dest_binop l val z = r
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z]
+ val ynz = prove_nz ctxt T y
+ in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
+ end
+ | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
+ let val (x,y) = Thm.dest_binop r val z = l
+ val _ = map (HOLogic.dest_number o term_of) [x,y,z]
+ val ynz = prove_nz ctxt T y
+ in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
+ end
+ | _ => NONE)
+ end
+ handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
+ | is_number t = can HOLogic.dest_number t
+
+ val is_number = is_number o term_of
+
+ fun proc3 phi ss ct =
+ (case term_of ct of
+ Const(@{const_name "Orderings.less"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name "Orderings.less_eq"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+ let
+ val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name "Orderings.less"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const(@{const_name "Orderings.less_eq"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+ let
+ val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
+ val _ = map is_number [a,b,c]
+ val T = ctyp_of_term c
+ val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
+ in SOME (mk_meta_eq th) end
+ | _ => NONE)
+ handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
+
+fun add_frac_frac_simproc ctxt =
+ make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
+ name = "add_frac_frac_simproc",
+ proc = proc ctxt, identifier = []}
+
+fun add_frac_num_simproc ctxt =
+ make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
+ name = "add_frac_num_simproc",
+ proc = proc2 ctxt, identifier = []}
+
+val ord_frac_simproc =
+ make_simproc
+ {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
+ @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
+ @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
+ @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
+ @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
+ @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
+ name = "ord_frac_simproc", proc = proc3, identifier = []}
+
+val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
+ "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
+
+val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
+ "add_Suc", "add_number_of_left", "mult_number_of_left",
+ "Suc_eq_add_numeral_1"])@
+ (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
+ @ arith_simps@ nat_arith @ rel_simps
+val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
+ @{thm "divide_Numeral1"},
+ @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
+ @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
+ @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
+ @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
+ @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
+ @{thm "diff_def"}, @{thm "minus_divide_left"},
+ @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
+
+local
+open Conv
+in
+fun comp_conv ctxt = (Simplifier.rewrite
+(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
+ addsimps ths addsimps comp_arith addsimps simp_thms
+ addsimprocs field_cancel_numeral_factors
+ addsimprocs [add_frac_frac_simproc ctxt, add_frac_num_simproc ctxt,
+ ord_frac_simproc]
+ addcongs [@{thm "if_weak_cong"}]))
+then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
+ [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
+end
+
+fun numeral_is_const ct =
+ case term_of ct of
+ Const (@{const_name "HOL.divide"},_) $ a $ b =>
+ numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
+ | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
+ | t => can HOLogic.dest_number t
+
+fun dest_const ct = case term_of ct of
+ Const (@{const_name "HOL.divide"},_) $ a $ b=>
+ Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
+
+fun mk_const phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Normalizer.mk_cnumber cT a
+ else Thm.capply
+ (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+ (Normalizer.mk_cnumber cT a))
+ (Normalizer.mk_cnumber cT b)
+ end
+
+in
+ NormalizerData.funs @{thm class_fieldgb.axioms}
+ {is_const = K numeral_is_const,
+ dest_const = K dest_const,
+ mk_const = mk_const,
+ conv = K comp_conv}
+end
+
+*}
+
+
+subsection {* Ferrante and Rackoff algorithm over ordered fields *}
+
+lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
+proof-
+ assume H: "c < 0"
+ have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
+ also have "\<dots> = (0 < x)" by simp
+ finally show "(c*x < 0) == (x > 0)" by simp
+qed
+
+lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
+proof-
+ assume H: "c > 0"
+ hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
+ also have "\<dots> = (0 > x)" by simp
+ finally show "(c*x < 0) == (x < 0)" by simp
+qed
+
+lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
+proof-
+ assume H: "c < 0"
+ have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
+ also have "\<dots> = ((- 1/c)*t < x)" by simp
+ finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
+proof-
+ assume H: "c > 0"
+ have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
+ also have "\<dots> = ((- 1/c)*t > x)" by simp
+ finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
+qed
+
+lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
+ using less_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
+proof-
+ assume H: "c < 0"
+ have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
+ also have "\<dots> = (0 <= x)" by simp
+ finally show "(c*x <= 0) == (x >= 0)" by simp
+qed
+
+lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
+proof-
+ assume H: "c > 0"
+ hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
+ also have "\<dots> = (0 >= x)" by simp
+ finally show "(c*x <= 0) == (x <= 0)" by simp
+qed
+
+lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
+proof-
+ assume H: "c < 0"
+ have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
+ also have "\<dots> = ((- 1/c)*t <= x)" by simp
+ finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
+proof-
+ assume H: "c > 0"
+ have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
+ also have "\<dots> = ((- 1/c)*t >= x)" by simp
+ finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
+qed
+
+lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
+ using le_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
+lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
+proof-
+ assume H: "c \<noteq> 0"
+ have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
+ also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_eq_simps)
+ finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
+qed
+lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
+ using eq_diff_eq[where a= x and b=t and c=0] by simp
+
+
+interpretation class_ordered_field_dense_linear_order: dense_linear_order
+ ["op <=" "op <"
+ "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
+proof (unfold_locales,
+ simp_all only: ordered_field_no_ub ordered_field_no_lb,
+ auto simp add: linorder_not_le)
+ fix x y::'a assume lt: "x < y"
+ from less_half_sum[OF lt] show "x < (x + y) /2" by simp
+next
+ fix x y::'a assume lt: "x < y"
+ from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
+qed
+
+declaration{*
+let
+fun earlier [] x y = false
+ | earlier (h::t) x y =
+ if h aconvc y then false else if h aconvc x then true else earlier t x y;
+
+fun dest_frac ct = case term_of ct of
+ Const (@{const_name "HOL.divide"},_) $ a $ b=>
+ Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
+
+fun mk_frac phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Normalizer.mk_cnumber cT a
+ else Thm.capply
+ (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+ (Normalizer.mk_cnumber cT a))
+ (Normalizer.mk_cnumber cT b)
+ end
+
+fun whatis x ct = case term_of ct of
+ Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
+ if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
+ else ("Nox",[])
+| Const(@{const_name "HOL.plus"}, _)$y$_ =>
+ if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
+ else ("Nox",[])
+| Const(@{const_name "HOL.times"}, _)$_$y =>
+ if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
+ else ("Nox",[])
+| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
+
+fun xnormalize_conv ctxt [] ct = reflexive ct
+| xnormalize_conv ctxt (vs as (x::_)) ct =
+ case term_of ct of
+ Const(@{const_name "Orderings.less"},_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val cr = dest_frac c
+ val clt = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
+ (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val cr = dest_frac c
+ val clt = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+ (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
+ val rth = th
+ in rth end
+ | _ => reflexive ct)
+
+
+| Const(@{const_name "Orderings.less_eq"},_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
+ (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+ val cz = Thm.dest_arg ct
+ val neg = cr </ Rat.zero
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (if neg then Thm.capply (Thm.capply clt c) cz
+ else Thm.capply (Thm.capply clt cz) c))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+ (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
+ val rth = th
+ in rth end
+ | _ => reflexive ct)
+
+| Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
+ (case whatis x (Thm.dest_arg1 ct) of
+ ("c*x+t",[c,t]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val ceq = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val th = implies_elim
+ (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("x+t",[t]) =>
+ let
+ val T = ctyp_of_term x
+ val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
+ val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+ (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+ in rth end
+ | ("c*x",[c]) =>
+ let
+ val T = ctyp_of_term x
+ val cr = dest_frac c
+ val ceq = Thm.dest_fun2 ct
+ val cz = Thm.dest_arg ct
+ val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+ (Thm.capply @{cterm "Trueprop"}
+ (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+ val cth = equal_elim (symmetric cthp) TrueI
+ val rth = implies_elim
+ (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
+ in rth end
+ | _ => reflexive ct);
+
+local
+ val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
+ val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
+ val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
+in
+fun field_isolate_conv phi ctxt vs ct = case term_of ct of
+ Const(@{const_name "Orderings.less"},_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+| Const(@{const_name "Orderings.less_eq"},_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+
+| Const("op =",_)$a$b =>
+ let val (ca,cb) = Thm.dest_binop ct
+ val T = ctyp_of_term ca
+ val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
+ val nth = Conv.fconv_rule
+ (Conv.arg_conv (Conv.arg1_conv
+ (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+ val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+ in rth end
+| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
+| _ => reflexive ct
+end;
+
+fun classfield_whatis phi =
+ let
+ fun h x t =
+ case term_of t of
+ Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+ else Ferrante_Rackoff_Data.Nox
+ | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+ else Ferrante_Rackoff_Data.Nox
+ | Const(@{const_name "Orderings.less"},_)$y$z =>
+ if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+ else Ferrante_Rackoff_Data.Nox
+ | Const (@{const_name "Orderings.less_eq"},_)$y$z =>
+ if term_of x aconv y then Ferrante_Rackoff_Data.Le
+ else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+ else Ferrante_Rackoff_Data.Nox
+ | _ => Ferrante_Rackoff_Data.Nox
+ in h end;
+fun class_field_ss phi =
+ HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
+ addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
+
+in
+Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
+ {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
+end
+*}
+
+
+subsection {* Decision Procedure for Presburger Arithmetic *}
+
+setup CooperData.setup
+
+subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
+
+lemma minf:
+ "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
+ \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+ "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
+ \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
+ "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
+ "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
+ "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+ "\<exists>z.\<forall>x<z. F = F"
+ by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
+
+lemma pinf:
+ "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
+ \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+ "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
+ \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
+ "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
+ "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
+ "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+ "\<exists>z.\<forall>x>z. F = F"
+ by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
+
+lemma inf_period:
+ "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
+ \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
+ "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
+ \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
+ "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
+ "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
+ "\<forall>x k. F = F"
+by simp_all
+ (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
+ simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
+
+section{* The A and B sets *}
+lemma bset:
+ "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
+ "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
+ "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
+ "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
+ "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
+ "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
+ "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
+ "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
+ "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t))"
+ "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x - D) + t))"
+ "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+ assume dp: "D > 0" and tB: "t - 1\<in> B"
+ show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
+ apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
+ using dp tB by simp_all
+next
+ assume dp: "D > 0" and tB: "t \<in> B"
+ show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
+ apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+ using dp tB by simp_all
+next
+ assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
+next
+ assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
+next
+ assume dp: "D > 0" and tB:"t \<in> B"
+ {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
+ hence "x -t \<le> D" and "1 \<le> x - t" by simp+
+ hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
+ hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
+ with nob tB have "False" by simp}
+ thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
+next
+ assume dp: "D > 0" and tB:"t - 1\<in> B"
+ {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
+ hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
+ hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
+ hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
+ with nob tB have "False" by simp}
+ thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
+next
+ assume d: "d dvd D"
+ {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
+ by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
+ thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp
+next
+ assume d: "d dvd D"
+ {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
+ by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
+ thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
+qed blast
+
+lemma aset:
+ "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
+ "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
+ \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
+ "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
+ "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
+ "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
+ "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
+ "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
+ "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
+ "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t))"
+ "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x + D) + t))"
+ "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+ assume dp: "D > 0" and tA: "t + 1 \<in> A"
+ show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
+ apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
+ using dp tA by simp_all
+next
+ assume dp: "D > 0" and tA: "t \<in> A"
+ show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
+ apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+ using dp tA by simp_all
+next
+ assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
+next
+ assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
+next
+ assume dp: "D > 0" and tA:"t \<in> A"
+ {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
+ hence "t - x \<le> D" and "1 \<le> t - x" by simp+
+ hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
+ hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps)
+ with nob tA have "False" by simp}
+ thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
+next
+ assume dp: "D > 0" and tA:"t + 1\<in> A"
+ {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
+ hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
+ hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
+ hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
+ with nob tA have "False" by simp}
+ thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
+next
+ assume d: "d dvd D"
+ {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
+ by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
+ thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
+next
+ assume d: "d dvd D"
+ {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
+ by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
+ thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
+qed blast
+
+subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
+
+subsubsection{* First some trivial facts about periodic sets or predicates *}
+lemma periodic_finite_ex:
+ assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
+ shows "(EX x. P x) = (EX j : {1..d}. P j)"
+ (is "?LHS = ?RHS")
+proof
+ assume ?LHS
+ then obtain x where P: "P x" ..
+ have "x mod d = x - (x div d)*d" 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 \<noteq> 0"
+ have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
+ moreover have "x mod d : {1..d}"
+ proof -
+ from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
+ moreover from dpos 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
+qed auto
+
+subsubsection{* The @{text "-\<infinity>"} Version*}
+
+lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
+by(induct rule: int_gr_induct,simp_all add:int_distrib)
+
+lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
+by(induct rule: int_gr_induct, simp_all add:int_distrib)
+
+theorem int_induct[case_names base step1 step2]:
+ assumes
+ base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
+ step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
+ shows "P i"
+proof -
+ have "i \<le> k \<or> i\<ge> k" by arith
+ thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
+qed
+
+lemma decr_mult_lemma:
+ assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
+ shows "ALL x. P x \<longrightarrow> P(x - k*d)"
+using knneg
+proof (induct rule:int_ge_induct)
+ case base thus ?case by simp
+next
+ case (step i)
+ {fix x
+ have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
+ also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
+ by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
+ ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
+ thus ?case ..
+qed
+
+lemma minusinfinity:
+ assumes dpos: "0 < d" and
+ P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
+ shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
+proof
+ assume eP1: "EX x. P1 x"
+ then obtain x where P1: "P1 x" ..
+ from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
+ let ?w = "x - (abs(x-z)+1) * d"
+ from dpos have w: "?w < z" by(rule decr_lemma)
+ have "P1 x = P1 ?w" using P1eqP1 by blast
+ also have "\<dots> = P(?w)" using w P1eqP by blast
+ finally have "P ?w" using P1 by blast
+ thus "EX x. P x" ..
+qed
+
+lemma cpmi:
+ assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
+ and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
+ and pd: "\<forall> x k. P' x = P' (x-k*D)"
+ shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
+ (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L" by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover
+ { fix x
+ assume P: "P x" and H: "\<not> ?R2"
+ {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
+ hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
+ with nb P have "P (y - D)" by auto }
+ hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
+ with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
+ from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
+ let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
+ have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+ from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
+ from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+ with periodic_finite_ex[OF dp pd]
+ have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed
+
+subsubsection {* The @{text "+\<infinity>"} Version*}
+
+lemma plusinfinity:
+ assumes dpos: "(0::int) < d" and
+ P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
+ shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
+proof
+ assume eP1: "EX x. P' x"
+ then obtain x where P1: "P' x" ..
+ from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
+ let ?w' = "x + (abs(x-z)+1) * d"
+ let ?w = "x - (-(abs(x-z) + 1))*d"
+ have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
+ from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
+ hence "P' x = P' ?w" using P1eqP1 by blast
+ also have "\<dots> = P(?w)" using w P1eqP by blast
+ finally have "P ?w" using P1 by blast
+ thus "EX x. P x" ..
+qed
+
+lemma incr_mult_lemma:
+ assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
+ shows "ALL x. P x \<longrightarrow> P(x + k*d)"
+using knneg
+proof (induct rule:int_ge_induct)
+ case base thus ?case by simp
+next
+ case (step i)
+ {fix x
+ have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
+ also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
+ by (simp add:int_distrib zadd_ac)
+ ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
+ thus ?case ..
+qed
+
+lemma cppi:
+ assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
+ and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
+ and pd: "\<forall> x k. P' x= P' (x-k*D)"
+ shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L" by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover
+ { fix x
+ assume P: "P x" and H: "\<not> ?R2"
+ {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
+ hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
+ with nb P have "P (y + D)" by auto }
+ hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
+ with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
+ from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
+ let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
+ have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+ from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
+ from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+ with periodic_finite_ex[OF dp pd]
+ have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed
+
+lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
+apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
+apply(fastsimp)
+done
+
+theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
+ apply (rule eq_reflection[symmetric])
+ apply (rule iffI)
+ defer
+ apply (erule exE)
+ apply (rule_tac x = "l * x" in exI)
+ apply (simp add: dvd_def)
+ apply (rule_tac x="x" in exI, simp)
+ apply (erule exE)
+ apply (erule conjE)
+ apply (erule dvdE)
+ apply (rule_tac x = k in exI)
+ apply simp
+ done
+
+lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
+shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
+ using not0 by (simp add: dvd_def)
+
+lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
+ by simp_all
+text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
+lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
+ by (simp split add: split_nat)
+
+lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
+ apply (auto split add: split_nat)
+ apply (rule_tac x="int x" in exI, simp)
+ apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
+ done
+
+lemma zdiff_int_split: "P (int (x - y)) =
+ ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
+ by (case_tac "y \<le> x", simp_all add: zdiff_int)
+
+lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
+lemma number_of2: "(0::int) <= Numeral0" by simp
+lemma Suc_plus1: "Suc n = n + 1" by simp
+
+text {*
+ \medskip Specific instances of congruence rules, to prevent
+ simplifier from looping. *}
+
+theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
+
+theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
+ by (simp cong: conj_cong)
+lemma int_eq_number_of_eq:
+ "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
+ by simp
+
+lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
+unfolding dvd_eq_mod_eq_0[symmetric] ..
+
+lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
+unfolding zdvd_iff_zmod_eq_0[symmetric] ..
+declare mod_1[presburger]
+declare mod_0[presburger]
+declare zmod_1[presburger]
+declare zmod_zero[presburger]
+declare zmod_self[presburger]
+declare mod_self[presburger]
+declare DIVISION_BY_ZERO_MOD[presburger]
+declare nat_mod_div_trivial[presburger]
+declare div_mod_equality2[presburger]
+declare div_mod_equality[presburger]
+declare mod_div_equality2[presburger]
+declare mod_div_equality[presburger]
+declare mod_mult_self1[presburger]
+declare mod_mult_self2[presburger]
+declare zdiv_zmod_equality2[presburger]
+declare zdiv_zmod_equality[presburger]
+declare mod2_Suc_Suc[presburger]
+lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
+using IntDiv.DIVISION_BY_ZERO by blast+
+
+use "Tools/Presburger/cooper.ML"
+oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
+
+use "Tools/Presburger/presburger.ML"
+
+setup {*
+ arith_tactic_add
+ (mk_arith_tactic "presburger" (fn i => fn st =>
+ (warning "Trying Presburger arithmetic ...";
+ Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st)))
+ (* FIXME!!!!!!! get the right context!!*)
+*}
+
+method_setup presburger = {*
+let
+ fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+ fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
+ val addN = "add"
+ val delN = "del"
+ val elimN = "elim"
+ val any_keyword = keyword addN || keyword delN || simple_keyword elimN
+ val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+in
+ fn src => Method.syntax
+ ((Scan.optional (simple_keyword elimN >> K false) true) --
+ (Scan.optional (keyword addN |-- thms) []) --
+ (Scan.optional (keyword delN |-- thms) [])) src
+ #> (fn (((elim, add_ths), del_ths),ctxt) =>
+ Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
+end
+*} "Cooper's algorithm for Presburger arithmetic"
+
+lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+
+
+subsection {* Code generator setup *}
+
+text {*
+ Presburger arithmetic is convenient to prove some
+ of the following code lemmas on integer numerals:
+*}
+
+lemma eq_Pls_Pls:
+ "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
+
+lemma eq_Pls_Min:
+ "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma eq_Pls_Bit0:
+ "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
+ unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Pls_Bit1:
+ "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
+ unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Min_Pls:
+ "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma eq_Min_Min:
+ "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
+
+lemma eq_Min_Bit0:
+ "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
+ unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Min_Bit1:
+ "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
+ unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Bit0_Pls:
+ "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
+ unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Bit1_Pls:
+ "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
+ unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Bit0_Min:
+ "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
+ unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Bit1_Min:
+ "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
+ unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Bit_Bit:
+ "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
+ v1 = v2 \<and> k1 = k2"
+ unfolding Bit_def
+ apply (cases v1)
+ apply (cases v2)
+ apply auto
+ apply presburger
+ apply (cases v2)
+ apply auto
+ apply presburger
+ apply (cases v2)
+ apply auto
+ done
+
+lemma eq_number_of:
+ "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l"
+ unfolding number_of_is_id ..
+
+
+lemma less_eq_Pls_Pls:
+ "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
+
+lemma less_eq_Pls_Min:
+ "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_eq_Pls_Bit:
+ "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
+ unfolding Pls_def Bit_def by (cases v) auto
+
+lemma less_eq_Min_Pls:
+ "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_eq_Min_Min:
+ "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
+
+lemma less_eq_Min_Bit0:
+ "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
+ unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_eq_Min_Bit1:
+ "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
+ unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_eq_Bit0_Pls:
+ "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
+ unfolding Pls_def Bit_def by simp
+
+lemma less_eq_Bit1_Pls:
+ "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+ unfolding Pls_def Bit_def by auto
+
+lemma less_eq_Bit_Min:
+ "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
+ unfolding Numeral.Min_def Bit_def by (cases v) auto
+
+lemma less_eq_Bit0_Bit:
+ "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
+ unfolding Bit_def bit.cases by (cases v) auto
+
+lemma less_eq_Bit_Bit1:
+ "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+ unfolding Bit_def bit.cases by (cases v) auto
+
+lemma less_eq_Bit1_Bit0:
+ "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+ unfolding Bit_def by (auto split: bit.split)
+
+lemma less_eq_number_of:
+ "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
+ unfolding number_of_is_id ..
+
+
+lemma less_Pls_Pls:
+ "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp
+
+lemma less_Pls_Min:
+ "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_Pls_Bit0:
+ "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
+ unfolding Pls_def Bit_def by auto
+
+lemma less_Pls_Bit1:
+ "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
+ unfolding Pls_def Bit_def by auto
+
+lemma less_Min_Pls:
+ "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
+ unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_Min_Min:
+ "Numeral.Min < Numeral.Min \<longleftrightarrow> False" by simp
+
+lemma less_Min_Bit:
+ "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
+ unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit_Pls:
+ "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+ unfolding Pls_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit0_Min:
+ "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
+ unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_Bit1_Min:
+ "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
+ unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_Bit_Bit0:
+ "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+ unfolding Bit_def by (auto split: bit.split)
+
+lemma less_Bit1_Bit:
+ "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
+ unfolding Bit_def by (auto split: bit.split)
+
+lemma less_Bit0_Bit1:
+ "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+ unfolding Bit_def bit.cases by arith
+
+lemma less_number_of:
+ "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
+ unfolding number_of_is_id ..
+
+lemmas pred_succ_numeral_code [code func] =
+ arith_simps(5-12)
+
+lemmas plus_numeral_code [code func] =
+ arith_simps(13-17)
+ arith_simps(26-27)
+ arith_extra_simps(1) [where 'a = int]
+
+lemmas minus_numeral_code [code func] =
+ arith_simps(18-21)
+ arith_extra_simps(2) [where 'a = int]
+ arith_extra_simps(5) [where 'a = int]
+
+lemmas times_numeral_code [code func] =
+ arith_simps(22-25)
+ arith_extra_simps(4) [where 'a = int]
+
+lemmas eq_numeral_code [code func] =
+ eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
+ eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
+ eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
+ eq_number_of
+
+lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
+ less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
+ less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
+ less_eq_number_of
+
+lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
+ less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
+ less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
+ less_number_of
+
+end
--- a/src/HOL/IsaMakefile Thu Jun 21 15:42:15 2007 +0200
+++ b/src/HOL/IsaMakefile Thu Jun 21 17:28:50 2007 +0200
@@ -83,13 +83,13 @@
$(SRC)/Provers/quasi.ML $(SRC)/Provers/splitter.ML \
$(SRC)/Provers/trancl.ML $(SRC)/Tools/integer.ML \
$(SRC)/Tools/rat.ML Tools/TFL/casesplit.ML ATP_Linkup.thy \
- Accessible_Part.thy Datatype.thy Dense_Linear_Order.thy Divides.thy \
- Equiv_Relations.thy Extraction.thy Finite_Set.thy FixedPoint.thy \
- Fun.thy FunDef.thy HOL.thy Hilbert_Choice.thy Inductive.thy \
- IntArith.thy IntDef.thy IntDiv.thy Lattices.thy List.thy Main.thy \
- Map.thy Nat.ML Nat.thy NatBin.thy NatSimprocs.thy Numeral.thy \
- OrderedGroup.thy Orderings.thy Power.thy PreList.thy Predicate.thy \
- Presburger.thy Product_Type.thy ROOT.ML Recdef.thy \
+ Accessible_Part.thy Arith_Tools.thy Datatype.thy \
+ Dense_Linear_Order.thy Divides.thy Equiv_Relations.thy Extraction.thy \
+ Finite_Set.thy FixedPoint.thy Fun.thy FunDef.thy HOL.thy \
+ Hilbert_Choice.thy Inductive.thy IntArith.thy IntDef.thy IntDiv.thy \
+ Lattices.thy List.thy Main.thy Map.thy Nat.ML Nat.thy NatBin.thy \
+ Numeral.thy OrderedGroup.thy Orderings.thy Power.thy PreList.thy \
+ Predicate.thy Product_Type.thy ROOT.ML Recdef.thy \
Record.thy Refute.thy Relation.thy Relation_Power.thy \
Ring_and_Field.thy SAT.thy Set.thy SetInterval.thy Sum_Type.thy \
Tools/ATP/reduce_axiomsN.ML Tools/ATP/watcher.ML \
--- a/src/HOL/NatSimprocs.thy Thu Jun 21 15:42:15 2007 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,940 +0,0 @@
-(* Title: HOL/NatSimprocs.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Author: Amine Chaieb, TU Muenchen
-*)
-
-header {*Simprocs for the Naturals*}
-
-theory NatSimprocs
-imports Groebner_Basis Dense_Linear_Order
-uses
- "~~/src/Provers/Arith/cancel_numeral_factor.ML"
- "~~/src/Provers/Arith/extract_common_term.ML"
- "int_factor_simprocs.ML"
- "nat_simprocs.ML"
-begin
-
-setup nat_simprocs_setup
-
-subsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
-
-text{*Where K above is a literal*}
-
-lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
-by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
-
-text {*Now just instantiating @{text n} to @{text "number_of v"} does
- the right simplification, but with some redundant inequality
- tests.*}
-lemma neg_number_of_pred_iff_0:
- "neg (number_of (Numeral.pred v)::int) = (number_of v = (0::nat))"
-apply (subgoal_tac "neg (number_of (Numeral.pred v)) = (number_of v < Suc 0) ")
-apply (simp only: less_Suc_eq_le le_0_eq)
-apply (subst less_number_of_Suc, simp)
-done
-
-text{*No longer required as a simprule because of the @{text inverse_fold}
- simproc*}
-lemma Suc_diff_number_of:
- "neg (number_of (uminus v)::int) ==>
- Suc m - (number_of v) = m - (number_of (Numeral.pred v))"
-apply (subst Suc_diff_eq_diff_pred)
-apply simp
-apply (simp del: nat_numeral_1_eq_1)
-apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
-by (simp add: numerals split add: nat_diff_split)
-
-
-subsection{*For @{term nat_case} and @{term nat_rec}*}
-
-lemma nat_case_number_of [simp]:
- "nat_case a f (number_of v) =
- (let pv = number_of (Numeral.pred v) in
- if neg pv then a else f (nat pv))"
-by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
-
-lemma nat_case_add_eq_if [simp]:
- "nat_case a f ((number_of v) + n) =
- (let pv = number_of (Numeral.pred v) in
- if neg pv then nat_case a f n else f (nat pv + n))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: numeral_1_eq_Suc_0 [symmetric] Let_def
- neg_imp_number_of_eq_0 neg_number_of_pred_iff_0)
-done
-
-lemma nat_rec_number_of [simp]:
- "nat_rec a f (number_of v) =
- (let pv = number_of (Numeral.pred v) in
- if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
-apply (case_tac " (number_of v) ::nat")
-apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
-apply (simp split add: split_if_asm)
-done
-
-lemma nat_rec_add_eq_if [simp]:
- "nat_rec a f (number_of v + n) =
- (let pv = number_of (Numeral.pred v) in
- if neg pv then nat_rec a f n
- else f (nat pv + n) (nat_rec a f (nat pv + n)))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0
- neg_number_of_pred_iff_0)
-done
-
-
-subsection{*Various Other Lemmas*}
-
-subsubsection{*Evens and Odds, for Mutilated Chess Board*}
-
-text{*Lemmas for specialist use, NOT as default simprules*}
-lemma nat_mult_2: "2 * z = (z+z::nat)"
-proof -
- have "2*z = (1 + 1)*z" by simp
- also have "... = z+z" by (simp add: left_distrib)
- finally show ?thesis .
-qed
-
-lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
-by (subst mult_commute, rule nat_mult_2)
-
-text{*Case analysis on @{term "n<2"}*}
-lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
-by arith
-
-lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
-by arith
-
-lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
-by (simp add: nat_mult_2 [symmetric])
-
-lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
-apply (subgoal_tac "m mod 2 < 2")
-apply (erule less_2_cases [THEN disjE])
-apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
-done
-
-lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
-apply (subgoal_tac "m mod 2 < 2")
-apply (force simp del: mod_less_divisor, simp)
-done
-
-subsubsection{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
-
-lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
-by simp
-
-lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
-by simp
-
-text{*Can be used to eliminate long strings of Sucs, but not by default*}
-lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
-by simp
-
-
-text{*These lemmas collapse some needless occurrences of Suc:
- at least three Sucs, since two and fewer are rewritten back to Suc again!
- We already have some rules to simplify operands smaller than 3.*}
-
-lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
-by (simp add: Suc3_eq_add_3)
-
-lemmas Suc_div_eq_add3_div_number_of =
- Suc_div_eq_add3_div [of _ "number_of v", standard]
-declare Suc_div_eq_add3_div_number_of [simp]
-
-lemmas Suc_mod_eq_add3_mod_number_of =
- Suc_mod_eq_add3_mod [of _ "number_of v", standard]
-declare Suc_mod_eq_add3_mod_number_of [simp]
-
-
-
-subsection{*Special Simplification for Constants*}
-
-text{*These belong here, late in the development of HOL, to prevent their
-interfering with proofs of abstract properties of instances of the function
-@{term number_of}*}
-
-text{*These distributive laws move literals inside sums and differences.*}
-lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard]
-declare left_distrib_number_of [simp]
-
-lemmas right_distrib_number_of = right_distrib [of "number_of v", standard]
-declare right_distrib_number_of [simp]
-
-
-lemmas left_diff_distrib_number_of =
- left_diff_distrib [of _ _ "number_of v", standard]
-declare left_diff_distrib_number_of [simp]
-
-lemmas right_diff_distrib_number_of =
- right_diff_distrib [of "number_of v", standard]
-declare right_diff_distrib_number_of [simp]
-
-
-text{*These are actually for fields, like real: but where else to put them?*}
-lemmas zero_less_divide_iff_number_of =
- zero_less_divide_iff [of "number_of w", standard]
-declare zero_less_divide_iff_number_of [simp]
-
-lemmas divide_less_0_iff_number_of =
- divide_less_0_iff [of "number_of w", standard]
-declare divide_less_0_iff_number_of [simp]
-
-lemmas zero_le_divide_iff_number_of =
- zero_le_divide_iff [of "number_of w", standard]
-declare zero_le_divide_iff_number_of [simp]
-
-lemmas divide_le_0_iff_number_of =
- divide_le_0_iff [of "number_of w", standard]
-declare divide_le_0_iff_number_of [simp]
-
-
-(****
-IF times_divide_eq_right and times_divide_eq_left are removed as simprules,
-then these special-case declarations may be useful.
-
-text{*These simprules move numerals into numerators and denominators.*}
-lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)"
-by (simp add: times_divide_eq)
-
-lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)"
-by (simp add: times_divide_eq)
-
-lemmas times_divide_eq_right_number_of =
- times_divide_eq_right [of "number_of w", standard]
-declare times_divide_eq_right_number_of [simp]
-
-lemmas times_divide_eq_right_number_of =
- times_divide_eq_right [of _ _ "number_of w", standard]
-declare times_divide_eq_right_number_of [simp]
-
-lemmas times_divide_eq_left_number_of =
- times_divide_eq_left [of _ "number_of w", standard]
-declare times_divide_eq_left_number_of [simp]
-
-lemmas times_divide_eq_left_number_of =
- times_divide_eq_left [of _ _ "number_of w", standard]
-declare times_divide_eq_left_number_of [simp]
-
-****)
-
-text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks
- strange, but then other simprocs simplify the quotient.*}
-
-lemmas inverse_eq_divide_number_of =
- inverse_eq_divide [of "number_of w", standard]
-declare inverse_eq_divide_number_of [simp]
-
-
-subsubsection{*These laws simplify inequalities, moving unary minus from a term
-into the literal.*}
-lemmas less_minus_iff_number_of =
- less_minus_iff [of "number_of v", standard]
-declare less_minus_iff_number_of [simp]
-
-lemmas le_minus_iff_number_of =
- le_minus_iff [of "number_of v", standard]
-declare le_minus_iff_number_of [simp]
-
-lemmas equation_minus_iff_number_of =
- equation_minus_iff [of "number_of v", standard]
-declare equation_minus_iff_number_of [simp]
-
-
-lemmas minus_less_iff_number_of =
- minus_less_iff [of _ "number_of v", standard]
-declare minus_less_iff_number_of [simp]
-
-lemmas minus_le_iff_number_of =
- minus_le_iff [of _ "number_of v", standard]
-declare minus_le_iff_number_of [simp]
-
-lemmas minus_equation_iff_number_of =
- minus_equation_iff [of _ "number_of v", standard]
-declare minus_equation_iff_number_of [simp]
-
-
-subsubsection{*To Simplify Inequalities Where One Side is the Constant 1*}
-
-lemma less_minus_iff_1 [simp]:
- fixes b::"'b::{ordered_idom,number_ring}"
- shows "(1 < - b) = (b < -1)"
-by auto
-
-lemma le_minus_iff_1 [simp]:
- fixes b::"'b::{ordered_idom,number_ring}"
- shows "(1 \<le> - b) = (b \<le> -1)"
-by auto
-
-lemma equation_minus_iff_1 [simp]:
- fixes b::"'b::number_ring"
- shows "(1 = - b) = (b = -1)"
-by (subst equation_minus_iff, auto)
-
-lemma minus_less_iff_1 [simp]:
- fixes a::"'b::{ordered_idom,number_ring}"
- shows "(- a < 1) = (-1 < a)"
-by auto
-
-lemma minus_le_iff_1 [simp]:
- fixes a::"'b::{ordered_idom,number_ring}"
- shows "(- a \<le> 1) = (-1 \<le> a)"
-by auto
-
-lemma minus_equation_iff_1 [simp]:
- fixes a::"'b::number_ring"
- shows "(- a = 1) = (a = -1)"
-by (subst minus_equation_iff, auto)
-
-
-subsubsection {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
-
-lemmas mult_less_cancel_left_number_of =
- mult_less_cancel_left [of "number_of v", standard]
-declare mult_less_cancel_left_number_of [simp]
-
-lemmas mult_less_cancel_right_number_of =
- mult_less_cancel_right [of _ "number_of v", standard]
-declare mult_less_cancel_right_number_of [simp]
-
-lemmas mult_le_cancel_left_number_of =
- mult_le_cancel_left [of "number_of v", standard]
-declare mult_le_cancel_left_number_of [simp]
-
-lemmas mult_le_cancel_right_number_of =
- mult_le_cancel_right [of _ "number_of v", standard]
-declare mult_le_cancel_right_number_of [simp]
-
-
-subsubsection {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
-
-lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard]
-declare le_divide_eq_number_of [simp]
-
-lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard]
-declare divide_le_eq_number_of [simp]
-
-lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard]
-declare less_divide_eq_number_of [simp]
-
-lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard]
-declare divide_less_eq_number_of [simp]
-
-lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard]
-declare eq_divide_eq_number_of [simp]
-
-lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard]
-declare divide_eq_eq_number_of [simp]
-
-
-
-subsection{*Optional Simplification Rules Involving Constants*}
-
-text{*Simplify quotients that are compared with a literal constant.*}
-
-lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
-lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
-lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
-lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
-lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
-lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
-
-
-text{*Not good as automatic simprules because they cause case splits.*}
-lemmas divide_const_simps =
- le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
- divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
- le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
-
-subsubsection{*Division By @{text "-1"}*}
-
-lemma divide_minus1 [simp]:
- "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
-by simp
-
-lemma minus1_divide [simp]:
- "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)"
-by (simp add: divide_inverse inverse_minus_eq)
-
-lemma half_gt_zero_iff:
- "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))"
-by auto
-
-lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard]
-declare half_gt_zero [simp]
-
-(* The following lemma should appear in Divides.thy, but there the proof
- doesn't work. *)
-
-lemma nat_dvd_not_less:
- "[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)"
- by (unfold dvd_def) auto
-
-ML {*
-val divide_minus1 = @{thm divide_minus1};
-val minus1_divide = @{thm minus1_divide};
-*}
-
-
-section{* Installing Groebner Bases for Fields *}
-
-interpretation class_fieldgb:
- fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
-
-lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
-lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
- by simp
-lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
- by simp
-lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y"
- by simp
-lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y"
- by simp
-
-lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
-
-lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
- by (simp add: add_divide_distrib)
-lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
- by (simp add: add_divide_distrib)
-
-declaration{*
-let
- val zr = @{cpat "0"}
- val zT = ctyp_of_term zr
- val geq = @{cpat "op ="}
- val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
- val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
- val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
- val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
-
- fun prove_nz ctxt =
- let val ss = local_simpset_of ctxt
- in fn T => fn t =>
- let
- val z = instantiate_cterm ([(zT,T)],[]) zr
- val eq = instantiate_cterm ([(eqT,T)],[]) geq
- val th = Simplifier.rewrite (ss addsimps simp_thms)
- (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
- (Thm.capply (Thm.capply eq t) z)))
- in equal_elim (symmetric th) TrueI
- end
- end
-
- fun proc ctxt phi ss ct =
- let
- val ((x,y),(w,z)) =
- (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
- val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
- val T = ctyp_of_term x
- val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
- val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
- in SOME (implies_elim (implies_elim th y_nz) z_nz)
- end
- handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun proc2 ctxt phi ss ct =
- let
- val (l,r) = Thm.dest_binop ct
- val T = ctyp_of_term l
- in (case (term_of l, term_of r) of
- (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
- let val (x,y) = Thm.dest_binop l val z = r
- val _ = map (HOLogic.dest_number o term_of) [x,y,z]
- val ynz = prove_nz ctxt T y
- in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
- end
- | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
- let val (x,y) = Thm.dest_binop r val z = l
- val _ = map (HOLogic.dest_number o term_of) [x,y,z]
- val ynz = prove_nz ctxt T y
- in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
- end
- | _ => NONE)
- end
- handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
- | is_number t = can HOLogic.dest_number t
-
- val is_number = is_number o term_of
-
- fun proc3 phi ss ct =
- (case term_of ct of
- Const(@{const_name "Orderings.less"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name "Orderings.less_eq"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
- in SOME (mk_meta_eq th) end
- | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
- let
- val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name "Orderings.less"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
- in SOME (mk_meta_eq th) end
- | Const(@{const_name "Orderings.less_eq"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
- in SOME (mk_meta_eq th) end
- | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
- let
- val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
- val _ = map is_number [a,b,c]
- val T = ctyp_of_term c
- val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
- in SOME (mk_meta_eq th) end
- | _ => NONE)
- handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
-
-fun add_frac_frac_simproc ctxt =
- make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
- name = "add_frac_frac_simproc",
- proc = proc ctxt, identifier = []}
-
-fun add_frac_num_simproc ctxt =
- make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
- name = "add_frac_num_simproc",
- proc = proc2 ctxt, identifier = []}
-
-val ord_frac_simproc =
- make_simproc
- {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
- @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
- @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
- @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
- @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
- @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
- name = "ord_frac_simproc", proc = proc3, identifier = []}
-
-val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
- "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
-
-val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
- "add_Suc", "add_number_of_left", "mult_number_of_left",
- "Suc_eq_add_numeral_1"])@
- (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
- @ arith_simps@ nat_arith @ rel_simps
-val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
- @{thm "divide_Numeral1"},
- @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
- @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
- @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
- @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
- @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
- @{thm "diff_def"}, @{thm "minus_divide_left"},
- @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
-
-local
-open Conv
-in
-fun comp_conv ctxt = (Simplifier.rewrite
-(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
- addsimps ths addsimps comp_arith addsimps simp_thms
- addsimprocs field_cancel_numeral_factors
- addsimprocs [add_frac_frac_simproc ctxt, add_frac_num_simproc ctxt,
- ord_frac_simproc]
- addcongs [@{thm "if_weak_cong"}]))
-then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
- [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
-end
-
-fun numeral_is_const ct =
- case term_of ct of
- Const (@{const_name "HOL.divide"},_) $ a $ b =>
- numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
- | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
- | t => can HOLogic.dest_number t
-
-fun dest_const ct = case term_of ct of
- Const (@{const_name "HOL.divide"},_) $ a $ b=>
- Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
- | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
-
-fun mk_const phi cT x =
- let val (a, b) = Rat.quotient_of_rat x
- in if b = 1 then Normalizer.mk_cnumber cT a
- else Thm.capply
- (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
- (Normalizer.mk_cnumber cT a))
- (Normalizer.mk_cnumber cT b)
- end
-
-in
- NormalizerData.funs @{thm class_fieldgb.axioms}
- {is_const = K numeral_is_const,
- dest_const = K dest_const,
- mk_const = mk_const,
- conv = K comp_conv}
-end
-
-*}
-
-
-subsection {* Instantiation of quantifier elimination in dense linear order
- (Ferrante and Rackoff algorithm) to linear arithmetic over ordered fields.*}
-
-lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
-proof-
- assume H: "c < 0"
- have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
- also have "\<dots> = (0 < x)" by simp
- finally show "(c*x < 0) == (x > 0)" by simp
-qed
-
-lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
-proof-
- assume H: "c > 0"
- hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
- also have "\<dots> = (0 > x)" by simp
- finally show "(c*x < 0) == (x < 0)" by simp
-qed
-
-lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
-proof-
- assume H: "c < 0"
- have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
- also have "\<dots> = ((- 1/c)*t < x)" by simp
- finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
-qed
-
-lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
-proof-
- assume H: "c > 0"
- have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
- also have "\<dots> = ((- 1/c)*t > x)" by simp
- finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
-qed
-
-lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
- using less_diff_eq[where a= x and b=t and c=0] by simp
-
-lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
-proof-
- assume H: "c < 0"
- have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
- also have "\<dots> = (0 <= x)" by simp
- finally show "(c*x <= 0) == (x >= 0)" by simp
-qed
-
-lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
-proof-
- assume H: "c > 0"
- hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
- also have "\<dots> = (0 >= x)" by simp
- finally show "(c*x <= 0) == (x <= 0)" by simp
-qed
-
-lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
-proof-
- assume H: "c < 0"
- have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
- also have "\<dots> = ((- 1/c)*t <= x)" by simp
- finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
-qed
-
-lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
-proof-
- assume H: "c > 0"
- have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
- also have "\<dots> = ((- 1/c)*t >= x)" by simp
- finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
-qed
-
-lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
- using le_diff_eq[where a= x and b=t and c=0] by simp
-
-lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
-lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
-proof-
- assume H: "c \<noteq> 0"
- have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_eq_simps)
- finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
-qed
-lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
- using eq_diff_eq[where a= x and b=t and c=0] by simp
-
-
-interpretation class_ordered_field_dense_linear_order: dense_linear_order
- ["op <=" "op <"
- "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
-proof (unfold_locales,
- simp_all only: ordered_field_no_ub ordered_field_no_lb,
- auto simp add: linorder_not_le)
- fix x y::'a assume lt: "x < y"
- from less_half_sum[OF lt] show "x < (x + y) /2" by simp
-next
- fix x y::'a assume lt: "x < y"
- from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
-qed
-
-declaration{*
-let
-fun earlier [] x y = false
- | earlier (h::t) x y =
- if h aconvc y then false else if h aconvc x then true else earlier t x y;
-
-fun dest_frac ct = case term_of ct of
- Const (@{const_name "HOL.divide"},_) $ a $ b=>
- Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
- | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
-
-fun mk_frac phi cT x =
- let val (a, b) = Rat.quotient_of_rat x
- in if b = 1 then Normalizer.mk_cnumber cT a
- else Thm.capply
- (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
- (Normalizer.mk_cnumber cT a))
- (Normalizer.mk_cnumber cT b)
- end
-
-fun whatis x ct = case term_of ct of
- Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
- if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
- else ("Nox",[])
-| Const(@{const_name "HOL.plus"}, _)$y$_ =>
- if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
- else ("Nox",[])
-| Const(@{const_name "HOL.times"}, _)$_$y =>
- if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
- else ("Nox",[])
-| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
-
-fun xnormalize_conv ctxt [] ct = reflexive ct
-| xnormalize_conv ctxt (vs as (x::_)) ct =
- case term_of ct of
- Const(@{const_name "Orderings.less"},_)$_$Const(@{const_name "HOL.zero"},_) =>
- (case whatis x (Thm.dest_arg1 ct) of
- ("c*x+t",[c,t]) =>
- let
- val cr = dest_frac c
- val clt = Thm.dest_fun2 ct
- val cz = Thm.dest_arg ct
- val neg = cr </ Rat.zero
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (if neg then Thm.capply (Thm.capply clt c) cz
- else Thm.capply (Thm.capply clt cz) c))
- val cth = equal_elim (symmetric cthp) TrueI
- val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
- (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("x+t",[t]) =>
- let
- val T = ctyp_of_term x
- val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("c*x",[c]) =>
- let
- val cr = dest_frac c
- val clt = Thm.dest_fun2 ct
- val cz = Thm.dest_arg ct
- val neg = cr </ Rat.zero
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (if neg then Thm.capply (Thm.capply clt c) cz
- else Thm.capply (Thm.capply clt cz) c))
- val cth = equal_elim (symmetric cthp) TrueI
- val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
- (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
- val rth = th
- in rth end
- | _ => reflexive ct)
-
-
-| Const(@{const_name "Orderings.less_eq"},_)$_$Const(@{const_name "HOL.zero"},_) =>
- (case whatis x (Thm.dest_arg1 ct) of
- ("c*x+t",[c,t]) =>
- let
- val T = ctyp_of_term x
- val cr = dest_frac c
- val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
- val cz = Thm.dest_arg ct
- val neg = cr </ Rat.zero
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (if neg then Thm.capply (Thm.capply clt c) cz
- else Thm.capply (Thm.capply clt cz) c))
- val cth = equal_elim (symmetric cthp) TrueI
- val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
- (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("x+t",[t]) =>
- let
- val T = ctyp_of_term x
- val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("c*x",[c]) =>
- let
- val T = ctyp_of_term x
- val cr = dest_frac c
- val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
- val cz = Thm.dest_arg ct
- val neg = cr </ Rat.zero
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (if neg then Thm.capply (Thm.capply clt c) cz
- else Thm.capply (Thm.capply clt cz) c))
- val cth = equal_elim (symmetric cthp) TrueI
- val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
- (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
- val rth = th
- in rth end
- | _ => reflexive ct)
-
-| Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
- (case whatis x (Thm.dest_arg1 ct) of
- ("c*x+t",[c,t]) =>
- let
- val T = ctyp_of_term x
- val cr = dest_frac c
- val ceq = Thm.dest_fun2 ct
- val cz = Thm.dest_arg ct
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
- val cth = equal_elim (symmetric cthp) TrueI
- val th = implies_elim
- (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("x+t",[t]) =>
- let
- val T = ctyp_of_term x
- val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
- val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
- (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
- in rth end
- | ("c*x",[c]) =>
- let
- val T = ctyp_of_term x
- val cr = dest_frac c
- val ceq = Thm.dest_fun2 ct
- val cz = Thm.dest_arg ct
- val cthp = Simplifier.rewrite (local_simpset_of ctxt)
- (Thm.capply @{cterm "Trueprop"}
- (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
- val cth = equal_elim (symmetric cthp) TrueI
- val rth = implies_elim
- (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
- in rth end
- | _ => reflexive ct);
-
-local
- val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
- val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
- val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
-in
-fun field_isolate_conv phi ctxt vs ct = case term_of ct of
- Const(@{const_name "Orderings.less"},_)$a$b =>
- let val (ca,cb) = Thm.dest_binop ct
- val T = ctyp_of_term ca
- val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
- val nth = Conv.fconv_rule
- (Conv.arg_conv (Conv.arg1_conv
- (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
- val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
- in rth end
-| Const(@{const_name "Orderings.less_eq"},_)$a$b =>
- let val (ca,cb) = Thm.dest_binop ct
- val T = ctyp_of_term ca
- val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
- val nth = Conv.fconv_rule
- (Conv.arg_conv (Conv.arg1_conv
- (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
- val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
- in rth end
-
-| Const("op =",_)$a$b =>
- let val (ca,cb) = Thm.dest_binop ct
- val T = ctyp_of_term ca
- val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
- val nth = Conv.fconv_rule
- (Conv.arg_conv (Conv.arg1_conv
- (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
- val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
- in rth end
-| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
-| _ => reflexive ct
-end;
-
-fun classfield_whatis phi =
- let
- fun h x t =
- case term_of t of
- Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
- else Ferrante_Rackoff_Data.Nox
- | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
- else Ferrante_Rackoff_Data.Nox
- | Const(@{const_name "Orderings.less"},_)$y$z =>
- if term_of x aconv y then Ferrante_Rackoff_Data.Lt
- else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
- else Ferrante_Rackoff_Data.Nox
- | Const (@{const_name "Orderings.less_eq"},_)$y$z =>
- if term_of x aconv y then Ferrante_Rackoff_Data.Le
- else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
- else Ferrante_Rackoff_Data.Nox
- | _ => Ferrante_Rackoff_Data.Nox
- in h end;
-fun class_field_ss phi =
- HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
- addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
-
-in
-Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
- {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
-end
-*}
-
-end
--- a/src/HOL/PreList.thy Thu Jun 21 15:42:15 2007 +0200
+++ b/src/HOL/PreList.thy Thu Jun 21 17:28:50 2007 +0200
@@ -7,7 +7,7 @@
header {* A Basis for Building the Theory of Lists *}
theory PreList
-imports Wellfounded_Relations Presburger Relation_Power SAT
+imports Wellfounded_Relations Arith_Tools Relation_Power SAT
FunDef Recdef Extraction
begin
--- a/src/HOL/Presburger.thy Thu Jun 21 15:42:15 2007 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,692 +0,0 @@
-(* Title: HOL/Presburger.thy
- ID: $Id$
- Author: Amine Chaieb, TU Muenchen
-*)
-
-header {* Decision Procedure for Presburger Arithmetic *}
-
-theory Presburger
-imports NatSimprocs SetInterval
-uses
- "Tools/Presburger/cooper_data.ML"
- "Tools/Presburger/generated_cooper.ML"
- ("Tools/Presburger/cooper.ML")
- ("Tools/Presburger/presburger.ML")
-begin
-
-setup CooperData.setup
-
-
-subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
-
-lemma minf:
- "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
- \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
- "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
- \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
- "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
- "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
- "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
- "\<exists>z.\<forall>x<z. F = F"
- by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
-
-lemma pinf:
- "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
- \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
- "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
- \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
- "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
- "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
- "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
- "\<exists>z.\<forall>x>z. F = F"
- by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
-
-lemma inf_period:
- "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
- \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
- "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
- \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
- "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
- "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
- "\<forall>x k. F = F"
-by simp_all
- (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
- simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
-
-section{* The A and B sets *}
-lemma bset:
- "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
- "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
- "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
- "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
- "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
- "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
- "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
- "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
- "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t))"
- "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x - D) + t))"
- "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
-proof (blast, blast)
- assume dp: "D > 0" and tB: "t - 1\<in> B"
- show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
- apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
- using dp tB by simp_all
-next
- assume dp: "D > 0" and tB: "t \<in> B"
- show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
- apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
- using dp tB by simp_all
-next
- assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
-next
- assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
-next
- assume dp: "D > 0" and tB:"t \<in> B"
- {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
- hence "x -t \<le> D" and "1 \<le> x - t" by simp+
- hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
- with nob tB have "False" by simp}
- thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
-next
- assume dp: "D > 0" and tB:"t - 1\<in> B"
- {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
- hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
- hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
- with nob tB have "False" by simp}
- thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
-next
- assume d: "d dvd D"
- {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
- by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
- thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp
-next
- assume d: "d dvd D"
- {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
- by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
- thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
-qed blast
-
-lemma aset:
- "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
- "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
- \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
- "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
- "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
- "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
- "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
- "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
- "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
- "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t))"
- "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x + D) + t))"
- "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
-proof (blast, blast)
- assume dp: "D > 0" and tA: "t + 1 \<in> A"
- show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
- apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
- using dp tA by simp_all
-next
- assume dp: "D > 0" and tA: "t \<in> A"
- show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
- apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
- using dp tA by simp_all
-next
- assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
-next
- assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
-next
- assume dp: "D > 0" and tA:"t \<in> A"
- {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
- hence "t - x \<le> D" and "1 \<le> t - x" by simp+
- hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps)
- with nob tA have "False" by simp}
- thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
-next
- assume dp: "D > 0" and tA:"t + 1\<in> A"
- {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
- hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
- hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
- with nob tA have "False" by simp}
- thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
-next
- assume d: "d dvd D"
- {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
- by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
- thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
-next
- assume d: "d dvd D"
- {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
- by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
- thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
-qed blast
-
-subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
-
-subsubsection{* First some trivial facts about periodic sets or predicates *}
-lemma periodic_finite_ex:
- assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
- shows "(EX x. P x) = (EX j : {1..d}. P j)"
- (is "?LHS = ?RHS")
-proof
- assume ?LHS
- then obtain x where P: "P x" ..
- have "x mod d = x - (x div d)*d" 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 \<noteq> 0"
- have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
- moreover have "x mod d : {1..d}"
- proof -
- from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
- moreover from dpos 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
-qed auto
-
-subsubsection{* The @{text "-\<infinity>"} Version*}
-
-lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
-by(induct rule: int_gr_induct,simp_all add:int_distrib)
-
-lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
-by(induct rule: int_gr_induct, simp_all add:int_distrib)
-
-theorem int_induct[case_names base step1 step2]:
- assumes
- base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
- step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
- shows "P i"
-proof -
- have "i \<le> k \<or> i\<ge> k" by arith
- thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
-qed
-
-lemma decr_mult_lemma:
- assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
- shows "ALL x. P x \<longrightarrow> P(x - k*d)"
-using knneg
-proof (induct rule:int_ge_induct)
- case base thus ?case by simp
-next
- case (step i)
- {fix x
- have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
- also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
- by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
- ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
- thus ?case ..
-qed
-
-lemma minusinfinity:
- assumes dpos: "0 < d" and
- P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
- shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
-proof
- assume eP1: "EX x. P1 x"
- then obtain x where P1: "P1 x" ..
- from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
- let ?w = "x - (abs(x-z)+1) * d"
- from dpos have w: "?w < z" by(rule decr_lemma)
- have "P1 x = P1 ?w" using P1eqP1 by blast
- also have "\<dots> = P(?w)" using w P1eqP by blast
- finally have "P ?w" using P1 by blast
- thus "EX x. P x" ..
-qed
-
-lemma cpmi:
- assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
- and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
- and pd: "\<forall> x k. P' x = P' (x-k*D)"
- shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
- (is "?L = (?R1 \<or> ?R2)")
-proof-
- {assume "?R2" hence "?L" by blast}
- moreover
- {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
- moreover
- { fix x
- assume P: "P x" and H: "\<not> ?R2"
- {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
- hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
- with nb P have "P (y - D)" by auto }
- hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
- with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
- from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
- let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
- have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
- from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
- from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
- with periodic_finite_ex[OF dp pd]
- have "?R1" by blast}
- ultimately show ?thesis by blast
-qed
-
-subsubsection {* The @{text "+\<infinity>"} Version*}
-
-lemma plusinfinity:
- assumes dpos: "(0::int) < d" and
- P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
- shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
-proof
- assume eP1: "EX x. P' x"
- then obtain x where P1: "P' x" ..
- from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
- let ?w' = "x + (abs(x-z)+1) * d"
- let ?w = "x - (-(abs(x-z) + 1))*d"
- have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
- from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
- hence "P' x = P' ?w" using P1eqP1 by blast
- also have "\<dots> = P(?w)" using w P1eqP by blast
- finally have "P ?w" using P1 by blast
- thus "EX x. P x" ..
-qed
-
-lemma incr_mult_lemma:
- assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
- shows "ALL x. P x \<longrightarrow> P(x + k*d)"
-using knneg
-proof (induct rule:int_ge_induct)
- case base thus ?case by simp
-next
- case (step i)
- {fix x
- have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
- also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
- by (simp add:int_distrib zadd_ac)
- ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
- thus ?case ..
-qed
-
-lemma cppi:
- assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
- and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
- and pd: "\<forall> x k. P' x= P' (x-k*D)"
- shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
-proof-
- {assume "?R2" hence "?L" by blast}
- moreover
- {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
- moreover
- { fix x
- assume P: "P x" and H: "\<not> ?R2"
- {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
- hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
- with nb P have "P (y + D)" by auto }
- hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
- with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
- from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
- let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
- have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
- from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
- from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
- with periodic_finite_ex[OF dp pd]
- have "?R1" by blast}
- ultimately show ?thesis by blast
-qed
-
-lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
-apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
-apply(fastsimp)
-done
-
-theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
- apply (rule eq_reflection[symmetric])
- apply (rule iffI)
- defer
- apply (erule exE)
- apply (rule_tac x = "l * x" in exI)
- apply (simp add: dvd_def)
- apply (rule_tac x="x" in exI, simp)
- apply (erule exE)
- apply (erule conjE)
- apply (erule dvdE)
- apply (rule_tac x = k in exI)
- apply simp
- done
-
-lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
-shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
- using not0 by (simp add: dvd_def)
-
-lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
- by simp_all
-text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
-lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
- by (simp split add: split_nat)
-
-lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
- apply (auto split add: split_nat)
- apply (rule_tac x="int x" in exI, simp)
- apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
- done
-
-lemma zdiff_int_split: "P (int (x - y)) =
- ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
- by (case_tac "y \<le> x", simp_all add: zdiff_int)
-
-lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
-lemma number_of2: "(0::int) <= Numeral0" by simp
-lemma Suc_plus1: "Suc n = n + 1" by simp
-
-text {*
- \medskip Specific instances of congruence rules, to prevent
- simplifier from looping. *}
-
-theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
-
-theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
- by (simp cong: conj_cong)
-lemma int_eq_number_of_eq:
- "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
- by simp
-
-lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
-unfolding dvd_eq_mod_eq_0[symmetric] ..
-
-lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
-unfolding zdvd_iff_zmod_eq_0[symmetric] ..
-declare mod_1[presburger]
-declare mod_0[presburger]
-declare zmod_1[presburger]
-declare zmod_zero[presburger]
-declare zmod_self[presburger]
-declare mod_self[presburger]
-declare DIVISION_BY_ZERO_MOD[presburger]
-declare nat_mod_div_trivial[presburger]
-declare div_mod_equality2[presburger]
-declare div_mod_equality[presburger]
-declare mod_div_equality2[presburger]
-declare mod_div_equality[presburger]
-declare mod_mult_self1[presburger]
-declare mod_mult_self2[presburger]
-declare zdiv_zmod_equality2[presburger]
-declare zdiv_zmod_equality[presburger]
-declare mod2_Suc_Suc[presburger]
-lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
-using IntDiv.DIVISION_BY_ZERO by blast+
-
-use "Tools/Presburger/cooper.ML"
-oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
-
-use "Tools/Presburger/presburger.ML"
-
-setup {*
- arith_tactic_add
- (mk_arith_tactic "presburger" (fn i => fn st =>
- (warning "Trying Presburger arithmetic ...";
- Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st)))
- (* FIXME!!!!!!! get the right context!!*)
-*}
-
-method_setup presburger = {*
-let
- fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
- fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
- val addN = "add"
- val delN = "del"
- val elimN = "elim"
- val any_keyword = keyword addN || keyword delN || simple_keyword elimN
- val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-in
- fn src => Method.syntax
- ((Scan.optional (simple_keyword elimN >> K false) true) --
- (Scan.optional (keyword addN |-- thms) []) --
- (Scan.optional (keyword delN |-- thms) [])) src
- #> (fn (((elim, add_ths), del_ths),ctxt) =>
- Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
-end
-*} "Cooper's algorithm for Presburger arithmetic"
-
-lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-
-
-subsection {* Code generator setup *}
-
-text {*
- Presburger arithmetic is convenient to prove some
- of the following code lemmas on integer numerals:
-*}
-
-lemma eq_Pls_Pls:
- "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
-
-lemma eq_Pls_Min:
- "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma eq_Pls_Bit0:
- "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
- unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Pls_Bit1:
- "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
- unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Min_Pls:
- "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma eq_Min_Min:
- "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
-
-lemma eq_Min_Bit0:
- "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Min_Bit1:
- "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Bit0_Pls:
- "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
- unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Bit1_Pls:
- "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
- unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Bit0_Min:
- "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Bit1_Min:
- "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
- unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Bit_Bit:
- "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
- v1 = v2 \<and> k1 = k2"
- unfolding Bit_def
- apply (cases v1)
- apply (cases v2)
- apply auto
- apply presburger
- apply (cases v2)
- apply auto
- apply presburger
- apply (cases v2)
- apply auto
- done
-
-lemma eq_number_of:
- "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l"
- unfolding number_of_is_id ..
-
-
-lemma less_eq_Pls_Pls:
- "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Pls_Min:
- "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_eq_Pls_Bit:
- "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
- unfolding Pls_def Bit_def by (cases v) auto
-
-lemma less_eq_Min_Pls:
- "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_eq_Min_Min:
- "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Min_Bit0:
- "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
- unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_eq_Min_Bit1:
- "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
- unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_eq_Bit0_Pls:
- "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
- unfolding Pls_def Bit_def by simp
-
-lemma less_eq_Bit1_Pls:
- "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
- unfolding Pls_def Bit_def by auto
-
-lemma less_eq_Bit_Min:
- "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
- unfolding Numeral.Min_def Bit_def by (cases v) auto
-
-lemma less_eq_Bit0_Bit:
- "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit_def bit.cases by (cases v) auto
-
-lemma less_eq_Bit_Bit1:
- "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit_def bit.cases by (cases v) auto
-
-lemma less_eq_Bit1_Bit0:
- "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
- unfolding Bit_def by (auto split: bit.split)
-
-lemma less_eq_number_of:
- "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
- unfolding number_of_is_id ..
-
-
-lemma less_Pls_Pls:
- "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp
-
-lemma less_Pls_Min:
- "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_Pls_Bit0:
- "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
- unfolding Pls_def Bit_def by auto
-
-lemma less_Pls_Bit1:
- "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
- unfolding Pls_def Bit_def by auto
-
-lemma less_Min_Pls:
- "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
- unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_Min_Min:
- "Numeral.Min < Numeral.Min \<longleftrightarrow> False" by simp
-
-lemma less_Min_Bit:
- "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
- unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
-
-lemma less_Bit_Pls:
- "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
- unfolding Pls_def Bit_def by (auto split: bit.split)
-
-lemma less_Bit0_Min:
- "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
- unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_Bit1_Min:
- "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
- unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_Bit_Bit0:
- "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
- unfolding Bit_def by (auto split: bit.split)
-
-lemma less_Bit1_Bit:
- "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
- unfolding Bit_def by (auto split: bit.split)
-
-lemma less_Bit0_Bit1:
- "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
- unfolding Bit_def bit.cases by arith
-
-lemma less_number_of:
- "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
- unfolding number_of_is_id ..
-
-lemmas pred_succ_numeral_code [code func] =
- arith_simps(5-12)
-
-lemmas plus_numeral_code [code func] =
- arith_simps(13-17)
- arith_simps(26-27)
- arith_extra_simps(1) [where 'a = int]
-
-lemmas minus_numeral_code [code func] =
- arith_simps(18-21)
- arith_extra_simps(2) [where 'a = int]
- arith_extra_simps(5) [where 'a = int]
-
-lemmas times_numeral_code [code func] =
- arith_simps(22-25)
- arith_extra_simps(4) [where 'a = int]
-
-lemmas eq_numeral_code [code func] =
- eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
- eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
- eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
- eq_number_of
-
-lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
- less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
- less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
- less_eq_number_of
-
-lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
- less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
- less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
- less_number_of
-
-end
--- a/src/HOL/ex/PresburgerEx.thy Thu Jun 21 15:42:15 2007 +0200
+++ b/src/HOL/ex/PresburgerEx.thy Thu Jun 21 17:28:50 2007 +0200
@@ -5,7 +5,9 @@
header {* Some examples for Presburger Arithmetic *}
-theory PresburgerEx imports Presburger begin
+theory PresburgerEx
+imports Main
+begin
lemma "\<And>m n ja ia. \<lbrakk>\<not> m \<le> j; \<not> n \<le> i; e \<noteq> 0; Suc j \<le> ja\<rbrakk> \<Longrightarrow> \<exists>m. \<forall>ja ia. m \<le> ja \<longrightarrow> (if j = ja \<and> i = ia then e else 0) = 0" by presburger
lemma "(0::nat) < emBits mod 8 \<Longrightarrow> 8 + emBits div 8 * 8 - emBits = 8 - emBits mod 8"