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theory Renegar_Decision
imports "Renegar_Proofs"
"BKR_Decision"
begin
(* Note that there is significant overlap between Renegar and BKR in general, so there is some
similarity between this file and BKR_Decision.thy. However, there are also notable differences
as Renegar and BKR use different auxiliary polynomials in their decision procedures.
In general, the _R's on definition and lemma names in this file are to emphasize that we are
working with Renegar style.
*)
section "Algorithm"
(* The set of all rational sign vectors for qs wrt the set S
When S = UNIV, then this quantifies over all reals *)
definition consistent_sign_vectors_R::"real poly list \<Rightarrow> real set \<Rightarrow> rat list set"
where "consistent_sign_vectors_R qs S = (consistent_sign_vec qs) ` S"
primrec prod_list_var:: "real poly list \<Rightarrow> real poly"
where "prod_list_var [] = 1"
| "prod_list_var (h#T) = (if h = 0 then (prod_list_var T) else (h* prod_list_var T))"
primrec check_all_const_deg:: "real poly list \<Rightarrow> bool"
where "check_all_const_deg [] = True"
| "check_all_const_deg (h#T) = (if degree h = 0 then (check_all_const_deg T) else False)"
definition poly_f :: "real poly list \<Rightarrow> real poly"
where
"poly_f ps =
(if (check_all_const_deg ps = True) then [:0, 1:] else
(pderiv (prod_list_var ps)) * (prod_list_var ps)* ([:-(crb (prod_list_var ps)),1:]) * ([:(crb (prod_list_var ps)),1:]))"
definition find_consistent_signs_R :: "real poly list \<Rightarrow> rat list list"
where
"find_consistent_signs_R ps = find_consistent_signs_at_roots_R (poly_f ps) ps"
definition decide_universal_R :: "real poly fml \<Rightarrow> bool"
where [code]:
"decide_universal_R fml = (
let (fml_struct,polys) = convert fml;
conds = find_consistent_signs_R polys
in
list_all (lookup_sem fml_struct) conds
)"
definition decide_existential_R :: "real poly fml \<Rightarrow> bool"
where [code]:
"decide_existential_R fml = (
let (fml_struct,polys) = convert fml;
conds = find_consistent_signs_R polys
in
find (lookup_sem fml_struct) conds \<noteq> None
)"
subsection "Proofs"
definition roots_of_poly_f:: "real poly list \<Rightarrow> real set"
where "roots_of_poly_f qs = {x. poly (poly_f qs) x = 0}"
lemma prod_list_var_nonzero:
shows "prod_list_var qs \<noteq> 0"
proof (induct qs)
case Nil
then show ?case by auto
next
case (Cons a qs)
then show ?case by auto
qed
lemma q_dvd_prod_list_var_prop:
assumes "q \<in> set qs"
assumes "q \<noteq> 0"
shows "q dvd prod_list_var qs" using assms
proof (induct qs)
case Nil
then show ?case by auto
next
case (Cons a qs)
then have eo: "q = a \<or>q \<in> set qs" by auto
have c1: "q = a \<Longrightarrow> q dvd prod_list_var (a#qs)"
proof -
assume "q = a"
then have "prod_list_var (a#qs) = q*(prod_list_var qs)" using Cons.prems
unfolding prod_list_var_def by auto
then show ?thesis using prod_list_var_nonzero[of qs] by auto
qed
have c2: "q \<in> set qs \<longrightarrow> q dvd prod_list_var qs"
using Cons.prems Cons.hyps unfolding prod_list_var_def by auto
show ?case using eo c1 c2 by auto
qed
lemma check_all_const_deg_prop:
shows "check_all_const_deg l = True \<longleftrightarrow> (\<forall>p \<in> set(l). degree p = 0)"
proof (induct l)
case Nil
then show ?case by auto
next
case (Cons a l)
then show ?case by auto
qed
(* lemma prod_zero shows that the product of the polynomial list is 0 at x iff there is a polynomial
in the list that is 0 at x *)
lemma poly_f_nonzero:
fixes qs :: "real poly list"
shows "(poly_f qs) \<noteq> 0"
proof -
have eo: "(\<forall>p \<in> set qs. degree p = 0) \<or> (\<exists>p \<in> set qs. degree p > 0)"
by auto
have c1: "(\<forall>p \<in> set qs. degree p = 0) \<longrightarrow> (poly_f qs) \<noteq> 0"
unfolding poly_f_def using check_all_const_deg_prop by auto
have c2: "(\<exists>p \<in> set qs. degree p > 0) \<longrightarrow> (poly_f qs) \<noteq> 0"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume deg_q: "0 < degree q"
assume contrad: "poly_f qs = 0"
have nonconst: "check_all_const_deg qs = False" using deg_q check_all_const_deg_prop
q_in by auto
have h1: "prod_list_var qs \<noteq> 0" using prod_list_var_nonzero by auto
then have "degree (prod_list_var qs) > 0" using q_in deg_q h1
proof (induct qs)
case Nil
then show ?case by auto
next
case (Cons a qs)
have q_nonz: "q \<noteq> 0" using Cons.prems by auto
have q_ins: "q \<in> set (a # qs) " using Cons.prems by auto
then have "q = a \<or> q \<in> set qs" by auto
then have eo: " q = a \<or> List.member qs q" using in_set_member[of q qs]
by auto
have degq: "degree q > 0" using Cons.prems by auto
have h2: "(prod_list (a # qs)) = a* (prod_list qs)"
by auto
have isa: "q = a \<longrightarrow> 0 < degree (prod_list_var (a # qs))"
using h2 degree_mult_eq_0[where p = "q", where q = "prod_list_var qs"]
Cons.prems by auto
have inl: "List.member qs q \<longrightarrow> 0 < degree (prod_list_var (a # qs))"
proof -
have nonzprod: "prod_list_var (a # qs) \<noteq> 0" using prod_list_var_nonzero by auto
have "q dvd prod_list_var (a # qs)"
using q_dvd_prod_list_var_prop[where q = "q", where qs = "(a#qs)"] q_nonz q_ins
by auto
then show ?thesis using divides_degree[where p = "q", where q = "prod_list_var (a # qs)"] nonzprod degq
by auto
qed
then show ?case using eo isa by auto
qed
then have h2: "pderiv (prod_list_var qs) \<noteq> 0" using pderiv_eq_0_iff[where p = "prod_list_var qs"]
by auto
then have "pderiv (prod_list_var qs) * prod_list_var qs \<noteq> 0"
using prod_list_var_nonzero h2 by auto
then show "False" using contrad nonconst unfolding poly_f_def deg_q
by (smt (z3) mult_eq_0_iff pCons_eq_0_iff)
qed
show ?thesis using eo c1 c2 by auto
qed
lemma poly_f_roots_prop_1:
fixes qs:: "real poly list"
assumes non_const: "check_all_const_deg qs = False"
shows "\<forall>x1. \<forall>x2. ((x1 < x2 \<and> (\<exists>q1 \<in> set (qs). q1 \<noteq> 0 \<and> (poly q1 x1) = 0) \<and> (\<exists>q2\<in> set(qs). q2 \<noteq> 0 \<and> (poly q2 x2) = 0)) \<longrightarrow> (\<exists>q. x1 < q \<and> q < x2 \<and> poly (poly_f qs) q = 0))"
proof clarsimp
fix x1:: "real"
fix x2:: "real"
fix q1:: "real poly"
fix q2:: "real poly"
assume "x1 < x2"
assume q1_in: "q1 \<in> set qs"
assume q1_0: "poly q1 x1 = 0"
assume q1_nonz: "q1 \<noteq> 0"
assume q2_in: "q2 \<in> set qs"
assume q2_0: "poly q2 x2 = 0"
assume q2_nonz: "q2 \<noteq> 0"
have prod_z_x1: "poly (prod_list_var qs) x1 = 0" using q1_in q1_0
using q1_nonz q_dvd_prod_list_var_prop[of q1 qs] by auto
have prod_z_x2: "poly (prod_list_var qs) x2 = 0" using q2_in q2_0
using q2_nonz q_dvd_prod_list_var_prop[of q2 qs] by auto
have "\<exists>w>x1. w < x2 \<and> poly (pderiv (prod_list_var qs)) w = 0"
using Rolle_pderiv[where q = "prod_list_var qs"] prod_z_x1 prod_z_x2
using \<open>x1 < x2\<close> by blast
then obtain w where w_def: "w > x1 \<and>w < x2 \<and> poly (pderiv (prod_list_var qs)) w = 0"
by auto
then have "poly (poly_f qs) w = 0"
unfolding poly_f_def using non_const
by simp
then show "\<exists>q>x1. q < x2 \<and> poly (poly_f qs) q = 0"
using w_def by blast
qed
lemma main_step_aux1_R:
fixes qs:: "real poly list"
assumes non_const: "check_all_const_deg qs = True"
shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
proof -
have poly_f_is: "poly_f qs = [:0, 1:]" unfolding poly_f_def using assms
by auto
have same: "set (find_consistent_signs_at_roots_R [:0, 1:] qs) =
set (characterize_consistent_signs_at_roots [:0, 1:] qs)" using find_consistent_signs_at_roots_R[of "[:0, 1:]" qs]
by auto
have rech: "(sorted_list_of_set {x. poly ([:0, 1:]::real poly) x = 0}) = [0]" by auto
have alldeg0: "(\<forall>p \<in> set qs. degree p = 0)" using non_const check_all_const_deg_prop
by auto
then have allconst: "\<forall>p \<in> set qs. (\<exists>(k::real). p = [:k:])"
apply (auto)
by (meson degree_eq_zeroE)
then have allconstvar: "\<forall>p \<in> set qs. \<forall>(x::real). \<forall>(y::real). poly p x = poly p y"
by fastforce
have e1: "set (remdups (map (signs_at qs) [0])) \<subseteq>
consistent_sign_vectors_R qs UNIV"
unfolding signs_at_def squash_def consistent_sign_vectors_R_def consistent_sign_vec_def apply (simp)
by (smt (verit, best) class_ring.ring_simprules(2) comp_def image_iff length_map map_nth_eq_conv)
have e2: "consistent_sign_vectors_R qs UNIV \<subseteq> set (remdups (map (signs_at qs) [0])) "
unfolding signs_at_def squash_def consistent_sign_vectors_R_def consistent_sign_vec_def apply (simp)
using allconstvar
by (smt (verit, best) comp_apply image_iff insert_iff map_eq_conv subsetI)
have "set (remdups (map (signs_at qs) [0])) =
consistent_sign_vectors_R qs UNIV"
using e1 e2 by auto
then have "set (characterize_consistent_signs_at_roots [:0, 1:] qs) = consistent_sign_vectors_R qs UNIV"
unfolding characterize_consistent_signs_at_roots_def characterize_root_list_p_def
using rech by auto
then show ?thesis using same poly_f_is unfolding find_consistent_signs_R_def
by auto
qed
lemma sorted_list_lemma_var:
fixes l:: "real list"
fixes x:: "real"
assumes "length l > 1"
assumes strict_sort: "sorted_wrt (<) l"
assumes x_not_in: "\<not> (List.member l x)"
assumes lt_a: "x > (l ! 0)"
assumes b_lt: "x < (l ! (length l - 1))"
shows "(\<exists>n. n < length l - 1 \<and> x > l ! n \<and> x < l !(n+1))" using assms
proof (induct l)
case Nil
then show ?case by auto
next
case (Cons a l)
have len_gteq: "length l \<ge> 1" using Cons.prems(1)
by (metis One_nat_def Suc_eq_plus1 list.size(4) not_le not_less_eq)
have len_one: "length l = 1 \<Longrightarrow> (\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1))"
proof -
assume len_is: "length l = 1"
then have "x > (a#l) ! 0 \<and> x < (a#l) !1 " using Cons.prems(4) Cons.prems(5)
by auto
then show "(\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1))"
using len_is by auto
qed
have len_gt: "length l > 1 \<Longrightarrow> (\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1))"
proof -
assume len_gt_one: "length l > 1"
have eo: "x \<noteq> l ! 0" using Cons.prems(3) apply (auto)
by (metis One_nat_def Suc_lessD in_set_member len_gt_one member_rec(1) nth_mem)
have c1: "x < l ! 0 \<Longrightarrow> (\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1)) "
proof -
assume xlt: "x < l !0"
then have "x < (a#l) ! 1 "
by simp
then show ?thesis using Cons.prems(4) len_gt_one apply (auto)
using Cons.prems(4) Suc_lessD by blast
qed
have c2: "x > l ! 0 \<Longrightarrow> (\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1)) "
proof -
assume asm: "x > l ! 0"
have xlt_1: " x < l ! (length l - 1)"
using Cons.prems(5)
by (metis Cons.prems(1) One_nat_def add_diff_cancel_right' list.size(4) nth_Cons_pos zero_less_diff)
have ssl: "sorted_wrt (<) l " using Cons.prems(2)
using sorted_wrt.simps(2) by auto
have " \<not> List.member l x" using Cons.prems(3)
by (meson member_rec(1))
then have " \<exists>n<length l - 1. l ! n < x \<and> x < l ! (n + 1)"
using asm xlt_1 len_gt_one ssl Cons.hyps
by auto
then show ?thesis
by (metis One_nat_def Suc_eq_plus1 diff_Suc_1 less_diff_conv list.size(4) nth_Cons_Suc)
qed
show "(\<exists>n. n < length (a#l) - 1 \<and> x > (a#l) ! n \<and> x < (a#l) !(n+1))"
using eo c1 c2
by (meson linorder_neqE_linordered_idom)
qed
then show ?case
using len_gteq len_one len_gt
apply (auto)
by (metis One_nat_def less_numeral_extra(1) linorder_neqE_nat not_less nth_Cons_0)
qed
(* We want to show that our auxiliary polynomial has roots in all relevant intervals:
so it captures all of the zeros, and also it captures all of the points in between! *)
lemma all_sample_points_prop:
assumes is_not_const: "check_all_const_deg qs = False"
assumes s_is: "S = (characterize_root_list_p (pderiv (prod_list_var qs) * (prod_list_var qs) * ([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])))"(* properties about S*)
shows "consistent_sign_vectors_R qs UNIV = consistent_sign_vectors_R qs (set S)"
proof -
let ?zer_list = "sorted_list_of_set {(x::real). (\<exists>q \<in> set(qs). (q \<noteq> 0 \<and> poly q x = 0))} :: real list"
have strict_sorted_h: "sorted_wrt (<) ?zer_list" using sorted_sorted_list_of_set
strict_sorted_iff by auto
have poly_f_is: "poly_f qs = (pderiv (prod_list_var qs) * prod_list_var qs)* ([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])"
unfolding poly_f_def using is_not_const by auto
then have set_S_char: "set S = ({x. poly (poly_f qs) x = 0}::real set)"
using poly_roots_finite[of "poly_f qs"] set_sorted_list_of_set poly_f_nonzero[of qs]
using s_is unfolding characterize_root_list_p_def by auto
have difficult_direction: "consistent_sign_vectors_R qs UNIV \<subseteq> consistent_sign_vectors_R qs (set S)"
proof clarsimp
fix x
assume "x \<in> consistent_sign_vectors_R qs UNIV "
then have "\<exists>y. x = (consistent_sign_vec qs y)" unfolding consistent_sign_vectors_R_def by auto
then obtain y where y_prop: "x = consistent_sign_vec qs y" by auto
then have "\<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
proof -
have c1: "(\<exists>q \<in> (set qs). q \<noteq> 0 \<and> poly q y = 0) \<Longrightarrow> (\<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y)"
proof -
assume "(\<exists>q \<in> (set qs). q \<noteq> 0 \<and> poly q y = 0)"
then obtain q where "q \<in> (set qs) \<and> q \<noteq> 0 \<and> poly q y = 0" by auto
then have "poly (prod_list_var qs) y = 0"
using q_dvd_prod_list_var_prop[of q qs] by auto
then have "poly (pderiv (prod_list_var qs) * (prod_list_var qs)*([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])) y = 0"
by auto
then have "y \<in> (set S)"
using s_is unfolding characterize_root_list_p_def
proof -
have "y \<in> {r. poly (pderiv (prod_list_var qs) * (prod_list_var qs)*([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])) r = 0}"
using \<open>poly (pderiv (prod_list_var qs) * (prod_list_var qs)*([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])) y = 0\<close> by force
then show ?thesis
by (metis characterize_root_list_p_def is_not_const poly_f_def poly_f_nonzero poly_roots_finite s_is set_sorted_list_of_set)
qed
then show "\<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
by auto
qed
have len_gtz_prop: "length ?zer_list > 0 \<longrightarrow>
((\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<or>
(y < ?zer_list ! 0) \<or>
(y > ?zer_list ! (length ?zer_list - 1)) \<or>
(\<exists>k < (length ?zer_list - 1). y > ?zer_list ! k \<and> y < ?zer_list ! (k+1)))"
proof -
let ?c = "(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<or>
(y < ?zer_list ! 0) \<or>
(y > ?zer_list ! (length ?zer_list - 1)) \<or>
(\<exists>k < (length ?zer_list - 1). y > ?zer_list ! k \<and> y < ?zer_list ! (k+1))"
have lis1: "length ?zer_list = 1 \<Longrightarrow> ?c"
by auto
have h1: "\<not>(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<Longrightarrow> \<not> (List.member ?zer_list y)"
by (metis (no_types, lifting) in_set_conv_nth in_set_member)
have h2: "(length ?zer_list > 0 \<and> \<not>(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<and> \<not> (y < ?zer_list ! 0)) \<Longrightarrow> y > ?zer_list ! 0"
by auto
have h3: "(length ?zer_list > 1 \<and> \<not>(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<and> \<not> (y > ?zer_list ! (length ?zer_list - 1))) \<Longrightarrow>
y < ?zer_list ! (length ?zer_list - 1)"
apply (auto)
by (smt (z3) diff_Suc_less gr_implies_not0 not_gr_zero)
have "length ?zer_list > 1 \<and> \<not>(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<and> \<not> (y < ?zer_list ! 0) \<and> \<not> (y > ?zer_list ! (length ?zer_list - 1))
\<Longrightarrow> (\<exists>k < (length ?zer_list - 1). y > ?zer_list ! k \<and> y < ?zer_list ! (k+1))"
using h1 h2 h3 strict_sorted_h sorted_list_lemma_var[of ?zer_list y]
using One_nat_def Suc_lessD by presburger
then have lgt1: "length ?zer_list > 1 \<Longrightarrow> ?c"
by auto
then show ?thesis using lis1 lgt1
by (smt (z3) diff_is_0_eq' not_less)
qed
have neg_crb_in: "(- crb (prod_list_var qs)) \<in> set S"
using set_S_char poly_f_is by auto
have pos_crb_in: "(crb (prod_list_var qs)) \<in> set S"
using set_S_char poly_f_is by auto
have set_S_nonempty: "set S \<noteq> {}" using neg_crb_in by auto
have finset: "finite {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
proof -
have "\<forall>q \<in> set qs. q\<noteq> 0 \<longrightarrow> finite {x. poly q x = 0} "
using poly_roots_finite by auto
then show ?thesis by auto
qed
have c2: "\<not>(\<exists>q \<in> (set qs). q \<noteq> 0 \<and> poly q y = 0) \<Longrightarrow> \<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
proof -
assume "\<not>(\<exists>q \<in> (set qs). q \<noteq> 0 \<and> poly q y = 0)"
have c_c1: "length ?zer_list = 0 \<Longrightarrow> \<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
proof -
assume "length ?zer_list = 0"
then have "\<forall>q \<in> set (qs). \<forall> (x:: real). \<forall>(y::real). squash (poly q x) = squash (poly q y)"
proof clarsimp
fix q x y
assume czer: "card {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0} = 0"
assume qin: "q \<in> set qs"
have fin_means_empty: "{x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0} = {}"
using finset czer
by auto
have qzer: "q = 0 \<Longrightarrow> squash (poly q x) = squash (poly q y)" by auto
have qnonz: "q \<noteq> 0 \<Longrightarrow> squash (poly q x) = squash (poly q y)"
proof -
assume qnonz: "q \<noteq> 0"
then have noroots: "{x. poly q x = 0} = {}" using qin finset
using Collect_empty_eq fin_means_empty by auto
have nonzsq1: "squash (poly q x) \<noteq> 0" using fin_means_empty qnonz czer qin
unfolding squash_def by auto
then have eo: "(poly q x) > 0 \<or> (poly q x) < 0" unfolding squash_def
apply (auto)
by presburger
have eo1: "poly q x > 0 \<Longrightarrow> poly q y > 0"
using noroots poly_IVT_pos[of y x q] poly_IVT_neg[of x y q]
apply (auto)
by (metis linorder_neqE_linordered_idom)
have eo2: "poly q x < 0 \<Longrightarrow> poly q y < 0"
using noroots poly_IVT_pos[of x y q] poly_IVT_neg[of y x q]
apply (auto) by (metis linorder_neqE_linordered_idom)
then show "squash (poly q x) = squash (poly q y)"
using eo eo1 eo2 unfolding squash_def by auto
qed
show "squash (poly q x) = squash (poly q y)"
using qzer qnonz
by blast
qed
then have "\<forall>q \<in> set (qs). squash (poly q y) = squash (poly q (- crb (prod_list_var qs)))"
by auto
then show "\<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
using neg_crb_in unfolding consistent_sign_vec_def squash_def
apply (auto)
by (metis (no_types, opaque_lifting) antisym_conv3 class_field.neg_1_not_0 equal_neg_zero less_irrefl of_int_minus)
qed
have c_c2: "length ?zer_list > 0 \<Longrightarrow> \<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
proof -
assume lengt: "length ?zer_list > 0"
let ?t = " \<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
have sg1: "(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w) \<Longrightarrow> ?t"
proof -
assume "(\<exists>w. w < length ?zer_list \<and> y = ?zer_list ! w)"
then obtain w where w_prop: "w < length ?zer_list \<and> y = ?zer_list ! w" by auto
then have " y \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using finset set_sorted_list_of_set[of "{x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"]
by (smt (verit, best) nth_mem)
then have "y \<in> {x. poly (poly_f qs) x = 0}" using poly_f_is
using \<open>\<not> (\<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q y = 0)\<close> by blast
then show ?thesis using set_S_char
by blast
qed
have sg2: "(y < ?zer_list ! 0) \<Longrightarrow> ?t"
proof -
assume ylt: "y < ?zer_list ! 0"
have ynonzat_some_qs: "\<forall>q \<in> (set qs). q \<noteq> 0 \<longrightarrow> poly q y \<noteq> 0"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume qnonz: "q \<noteq> 0"
assume "poly q y = 0"
then have "y \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list y"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "y \<ge> ?zer_list ! 0" using strict_sorted_h
using \<open>\<not> (\<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q y = 0)\<close> \<open>poly q y = 0\<close> q_in qnonz by blast
then show "False" using ylt
by auto
qed
let ?ncrb = "(- crb (prod_list_var qs))"
have "\<forall>x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}. poly (prod_list_var qs) x = 0"
using q_dvd_prod_list_var_prop
by fastforce
then have "poly (prod_list_var qs) (sorted_list_of_set {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0} ! 0) = 0"
using finset set_sorted_list_of_set
by (metis (no_types, lifting) lengt nth_mem)
then have ncrblt: "?ncrb < ?zer_list ! 0" using prod_list_var_nonzero crb_lem_neg[of "prod_list_var qs" "?zer_list ! 0"]
by auto
have qzerh: "\<forall>q \<in> (set qs). q = 0 \<longrightarrow> squash (poly q ?ncrb) = squash (poly q y)"
by auto
have "\<forall>q \<in> (set qs). q \<noteq> 0 \<longrightarrow> squash (poly q ?ncrb) = squash (poly q y)"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume qnonz: "q \<noteq> 0"
have nonzylt:"\<not>(\<exists>x \<le> y. poly q x = 0)"
proof clarsimp
fix x
assume xlt: "x \<le> y"
assume "poly q x = 0"
then have "x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list x"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "x \<ge> ?zer_list ! 0" using strict_sorted_h
by (metis (no_types, lifting) gr_implies_not0 in_set_conv_nth in_set_member not_less sorted_iff_nth_mono sorted_list_of_set(2))
then show "False" using xlt ylt
by auto
qed
have nonzncrb:"\<not>(\<exists>x \<le> (real_of_int ?ncrb). poly q x = 0)"
proof clarsimp
fix x
assume xlt: "x \<le> - real_of_int (crb (prod_list_var qs))"
assume "poly q x = 0"
then have "x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list x"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "x \<ge> ?zer_list ! 0" using strict_sorted_h
by (metis (no_types, lifting) gr_implies_not0 in_set_conv_nth in_set_member not_less sorted_iff_nth_mono sorted_list_of_set(2))
then show "False" using xlt ncrblt
by auto
qed
have c1: " (poly q ?ncrb) > 0 \<Longrightarrow> (poly q y) > 0"
proof -
assume qncrbgt: "(poly q ?ncrb) > 0"
then have eq: "?ncrb = y \<Longrightarrow> poly q y > 0 " by auto
have gt: " ?ncrb > y \<Longrightarrow> poly q y > 0" using qncrbgt qnonz poly_IVT_pos[of y ?ncrb q] poly_IVT_neg[of ?ncrb y q] nonzncrb nonzylt
apply (auto)
by (meson less_eq_real_def linorder_neqE_linordered_idom)
have lt: "?ncrb < y \<Longrightarrow> poly q y > 0" using qncrbgt
using qnonz poly_IVT_pos[of y ?ncrb q] poly_IVT_neg[of ?ncrb y q] nonzncrb nonzylt
apply (auto)
by (meson less_eq_real_def linorder_neqE_linordered_idom)
then show ?thesis using eq gt lt apply (auto)
by (meson linorder_neqE_linordered_idom)
qed
have c2: "(poly q ?ncrb) < 0 \<Longrightarrow> (poly q y) < 0"
using poly_IVT_pos[of ?ncrb y q] poly_IVT_neg[of y ?ncrb q] nonzncrb nonzylt
apply (auto)
by (metis less_eq_real_def linorder_neqE_linordered_idom)
have eo: "(poly q ?ncrb) > 0 \<or> (poly q ?ncrb) < 0"
using nonzncrb
by auto
then show "squash (poly q (- real_of_int (crb (prod_list_var qs)))) = squash (poly q y)"
using c1 c2
by (smt (verit, ccfv_SIG) of_int_minus squash_def)
qed
then have "\<forall>q \<in> (set qs). squash (poly q ?ncrb) = squash (poly q y)"
using qzerh by auto
then have "consistent_sign_vec qs ?ncrb = consistent_sign_vec qs y"
unfolding consistent_sign_vec_def squash_def
by (smt (z3) map_eq_conv)
then show ?thesis using neg_crb_in by auto
qed
have sg3: " (y > ?zer_list ! (length ?zer_list - 1)) \<Longrightarrow> ?t"
proof -
assume ygt: "y > ?zer_list ! (length ?zer_list - 1)"
have ynonzat_some_qs: "\<forall>q \<in> (set qs). q \<noteq> 0 \<longrightarrow> poly q y \<noteq> 0"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume qnonz: "q \<noteq> 0"
assume "poly q y = 0"
then have "y \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list y"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "y \<le> ?zer_list ! (length ?zer_list - 1)" using strict_sorted_h
using \<open>\<not> (\<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q y = 0)\<close> \<open>poly q y = 0\<close> q_in qnonz by blast
then show "False" using ygt
by auto
qed
let ?crb = "crb (prod_list_var qs)"
have "\<forall>x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}. poly (prod_list_var qs) x = 0"
using q_dvd_prod_list_var_prop
by fastforce
then have "poly (prod_list_var qs) (sorted_list_of_set {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0} ! 0) = 0"
using finset set_sorted_list_of_set
by (metis (no_types, lifting) lengt nth_mem)
then have crbgt: "?crb > ?zer_list ! (length ?zer_list - 1)" using prod_list_var_nonzero crb_lem_pos[of "prod_list_var qs" "?zer_list ! (length ?zer_list - 1)"]
by (metis (no_types, lifting) \<open>\<forall>x\<in>{x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}. poly (prod_list_var qs) x = 0\<close> diff_less finset lengt less_numeral_extra(1) nth_mem set_sorted_list_of_set)
have qzerh: "\<forall>q \<in> (set qs). q = 0 \<longrightarrow> squash (poly q ?crb) = squash (poly q y)"
by auto
have "\<forall>q \<in> (set qs). q \<noteq> 0 \<longrightarrow> squash (poly q ?crb) = squash (poly q y)"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume qnonz: "q \<noteq> 0"
have nonzylt:"\<not>(\<exists>x \<ge> y. poly q x = 0)"
proof clarsimp
fix x
assume xgt: "x \<ge> y"
assume "poly q x = 0"
then have "x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list x"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "x \<le> ?zer_list ! (length ?zer_list - 1)" using strict_sorted_h
by (metis (no_types, lifting) One_nat_def Suc_leI Suc_pred diff_Suc_less in_set_conv_nth in_set_member lengt not_less sorted_iff_nth_mono sorted_list_of_set(2))
then show "False" using xgt ygt
by auto
qed
have nonzcrb:"\<not>(\<exists>x \<ge> (real_of_int ?crb). poly q x = 0)"
proof clarsimp
fix x
assume xgt: "x \<ge> real_of_int (crb (prod_list_var qs))"
assume "poly q x = 0"
then have "x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}"
using q_in qnonz by auto
then have "List.member ?zer_list x"
by (smt (verit, best) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have "x \<le> ?zer_list ! (length ?zer_list - 1)" using strict_sorted_h
by (meson \<open>\<forall>x\<in>{x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}. poly (prod_list_var qs) x = 0\<close> \<open>x \<in> {x. \<exists>q\<in>set qs. q \<noteq> 0 \<and> poly q x = 0}\<close> crb_lem_pos not_less prod_list_var_nonzero xgt)
then show "False" using xgt crbgt
by auto
qed
have c1: " (poly q ?crb) > 0 \<Longrightarrow> (poly q y) > 0"
proof -
assume qcrbgt: "(poly q ?crb) > 0"
then have eq: "?crb = y \<Longrightarrow> poly q y > 0 " by auto
have gt: " ?crb > y \<Longrightarrow> poly q y > 0" using qcrbgt qnonz poly_IVT_pos[of y ?crb q] poly_IVT_neg[of ?crb y q] nonzcrb nonzylt
apply (auto)
by (meson less_eq_real_def linorder_neqE_linordered_idom)
have lt: "?crb < y \<Longrightarrow> poly q y > 0" using qcrbgt
using qnonz poly_IVT_pos[of y ?crb q] poly_IVT_neg[of ?crb y q] nonzcrb nonzylt
apply (auto)
by (meson less_eq_real_def linorder_neqE_linordered_idom)
then show ?thesis using eq gt lt apply (auto)
by (meson linorder_neqE_linordered_idom)
qed
have c2: "(poly q ?crb) < 0 \<Longrightarrow> (poly q y) < 0"
using poly_IVT_pos[of ?crb y q] poly_IVT_neg[of y ?crb q] nonzcrb nonzylt
apply (auto)
by (metis less_eq_real_def linorder_neqE_linordered_idom)
have eo: "(poly q ?crb) > 0 \<or> (poly q ?crb) < 0"
using nonzcrb
by auto
then show "squash (poly q (real_of_int (crb (prod_list_var qs)))) = squash (poly q y)"
using c1 c2
by (smt (verit, ccfv_SIG) of_int_minus squash_def)
qed
then have "\<forall>q \<in> (set qs). squash (poly q ?crb) = squash (poly q y)"
using qzerh by auto
then have "consistent_sign_vec qs ?crb = consistent_sign_vec qs y"
unfolding consistent_sign_vec_def squash_def
by (smt (z3) map_eq_conv)
then show ?thesis using pos_crb_in by auto
qed
have sg4: " (\<exists>k < (length ?zer_list - 1). y > ?zer_list ! k \<and> y < ?zer_list ! (k+1)) \<Longrightarrow> ?t"
proof -
assume " (\<exists>k < (length ?zer_list - 1). y > ?zer_list ! k \<and> y < ?zer_list ! (k+1))"
then obtain k where k_prop: "k < (length ?zer_list - 1) \<and> y > ?zer_list ! k \<and> y < ?zer_list ! (k+1)"
by auto
have ltk: "(?zer_list ! k) < (?zer_list ! (k+1)) "
using strict_sorted_h
using k_prop by linarith
have q1e: "(\<exists>q1\<in>set qs. q1 \<noteq> 0 \<and> poly q1 (?zer_list ! k) = 0)"
by (smt (z3) One_nat_def Suc_lessD add.right_neutral add_Suc_right finset k_prop less_diff_conv mem_Collect_eq nth_mem set_sorted_list_of_set)
have q2e: "(\<exists>q2\<in>set qs. q2 \<noteq> 0 \<and> poly q2 (?zer_list ! (k + 1)) = 0)"
by (smt (verit, del_insts) finset k_prop less_diff_conv mem_Collect_eq nth_mem set_sorted_list_of_set)
then have "(\<exists>q>(?zer_list ! k). q < (?zer_list ! (k + 1)) \<and> poly (poly_f qs) q = 0)"
using poly_f_roots_prop_1[of qs] q1e q2e ltk is_not_const
by auto
then have "\<exists>s \<in> set S. s > ?zer_list ! k \<and> s < ?zer_list ! (k+1)"
using poly_f_is
by (smt (z3) k_prop mem_Collect_eq set_S_char)
then obtain s where s_prop: "s \<in> set S \<and> s > ?zer_list ! k \<and> s < ?zer_list ! (k+1)" by auto
have qnon: "\<forall>q \<in> set qs. q\<noteq> 0 \<longrightarrow> squash (poly q s) = squash (poly q y)"
proof clarsimp
fix q
assume q_in: "q \<in> set qs"
assume qnonz: "q \<noteq> 0"
have sgt: "s > y \<Longrightarrow> squash (poly q s) = squash (poly q y)"
proof -
assume "s > y"
then have "\<nexists>x. List.member ?zer_list x \<and> y \<le> x \<and> x \<le> s"
using sorted_list_lemma[of y s k ?zer_list] k_prop strict_sorted_h s_prop y_prop
using less_diff_conv by blast
then have nox: "\<nexists>x. poly q x = 0 \<and> y \<le> x \<and> x \<le> s" using q_in qnonz
by (metis (mono_tags, lifting) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have c1: "poly q s \<noteq> 0" using s_prop q_in qnonz
by (metis (mono_tags, lifting) \<open>y < s\<close> less_eq_real_def )
have c2: "poly q s > 0 \<Longrightarrow> poly q y > 0"
using poly_IVT_pos poly_IVT_neg nox
by (meson \<open>y < s\<close> less_eq_real_def linorder_neqE_linordered_idom)
have c3: "poly q s < 0 \<Longrightarrow> poly q y < 0" using poly_IVT_pos poly_IVT_neg nox
by (meson \<open>y < s\<close> less_eq_real_def linorder_neqE_linordered_idom)
show ?thesis using c1 c2 c3 unfolding squash_def
by auto
qed
have slt: "s < y \<Longrightarrow> squash (poly q s) = squash (poly q y)"
proof -
assume slt: "s < y"
then have "\<nexists>x. List.member ?zer_list x \<and> s \<le> x \<and> x \<le> y"
using sorted_list_lemma[of s y k ?zer_list] k_prop strict_sorted_h s_prop y_prop
using less_diff_conv by blast
then have nox: "\<nexists>x. poly q x = 0 \<and> s \<le> x \<and> x \<le> y" using q_in qnonz
by (metis (mono_tags, lifting) finset in_set_member mem_Collect_eq set_sorted_list_of_set)
then have c1: "poly q s \<noteq> 0" using s_prop q_in qnonz
by (metis (mono_tags, lifting) \<open>s < y\<close> less_eq_real_def )
have c2: "poly q s > 0 \<Longrightarrow> poly q y > 0"
using poly_IVT_pos poly_IVT_neg nox
by (meson \<open>s < y\<close> less_eq_real_def linorder_neqE_linordered_idom)
have c3: "poly q s < 0 \<Longrightarrow> poly q y < 0" using poly_IVT_pos poly_IVT_neg nox
by (meson \<open>s < y\<close> less_eq_real_def linorder_neqE_linordered_idom)
show ?thesis using c1 c2 c3 unfolding squash_def
by auto
qed
have "s = y \<Longrightarrow> squash (poly q s) = squash (poly q y)"
by auto
then show "squash (poly q s) = squash (poly q y)"
using sgt slt
by (meson linorder_neqE_linordered_idom)
qed
have "\<forall>q \<in> set qs. q= 0 \<longrightarrow> squash (poly q s) = squash (poly q y)" by auto
then have "\<forall>q \<in> set qs. squash (poly q s) = squash (poly q y)"
using qnon
by fastforce
then show ?thesis
using s_prop unfolding squash_def consistent_sign_vec_def apply (auto)
by (metis (no_types, opaque_lifting) class_field.neg_1_not_0 equal_neg_zero less_irrefl linorder_neqE_linordered_idom)
qed
show ?thesis
using lengt sg1 sg2 sg3 sg4 len_gtz_prop is_not_const
by fastforce
qed
show "\<exists> k \<in> (set S). consistent_sign_vec qs k = consistent_sign_vec qs y"
using c_c1 c_c2 by auto
qed
show ?thesis
using c1 c2 by auto
qed
then show "x \<in> consistent_sign_vectors_R qs (set S)"
using y_prop unfolding consistent_sign_vectors_R_def
by (metis imageI)
qed
have easy_direction: "consistent_sign_vectors_R qs (set S) \<subseteq> consistent_sign_vectors_R qs UNIV "
using consistent_sign_vectors_R_def by auto
then show ?thesis using difficult_direction easy_direction by auto
qed
lemma main_step_aux2_R:
fixes qs:: "real poly list"
assumes is_not_const: "check_all_const_deg qs = False"
shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
proof -
have poly_f_is: "poly_f qs = (pderiv (prod_list_var qs)) * (prod_list_var qs)* ([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])"
using is_not_const unfolding poly_f_def by auto
let ?p = "(pderiv (prod_list_var qs)) * (prod_list_var qs)* ([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:])"
let ?S = "characterize_root_list_p (pderiv (prod_list_var qs) * (prod_list_var qs) * ([:-(crb (prod_list_var qs)),1:]) * ([:(crb (prod_list_var qs)),1:]))"
have "set (remdups
(map (signs_at qs) ?S))
= consistent_sign_vectors_R qs (set ?S)"
unfolding signs_at_def squash_def consistent_sign_vectors_R_def consistent_sign_vec_def
by (smt (verit, best) comp_apply map_eq_conv set_map set_remdups)
then have "set (characterize_consistent_signs_at_roots ?p qs) = consistent_sign_vectors_R qs UNIV"
unfolding characterize_consistent_signs_at_roots_def using assms all_sample_points_prop[of qs]
by auto
then show ?thesis
unfolding find_consistent_signs_R_def using find_consistent_signs_at_roots_R poly_f_is poly_f_nonzero[of qs]
by auto
qed
lemma main_step_R:
fixes qs:: "real poly list"
shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
using main_step_aux1_R main_step_aux2_R by auto
(* The universal and existential decision procedure for real polys are easy
if we know the consistent sign vectors *)
lemma consistent_sign_vec_semantics_R:
assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length ls"
shows "lookup_sem fml (map (\<lambda>p. poly p x) ls) = lookup_sem fml (consistent_sign_vec ls x)"
using assms apply (induction)
by (auto simp add: consistent_sign_vec_def)
lemma universal_lookup_sem_R:
assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"
assumes "set signs = consistent_sign_vectors_R qs UNIV"
shows "(\<forall>x::real. lookup_sem fml (map (\<lambda>p. poly p x) qs)) \<longleftrightarrow>
list_all (lookup_sem fml) signs"
using assms(2) unfolding consistent_sign_vectors_R_def list_all_iff
by (simp add: assms(1) consistent_sign_vec_semantics_R)
lemma existential_lookup_sem_R:
assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"
assumes "set signs = consistent_sign_vectors_R qs UNIV"
shows "(\<exists>x::real. lookup_sem fml (map (\<lambda>p. poly p x) qs)) \<longleftrightarrow>
find (lookup_sem fml) signs \<noteq> None"
using assms(2) unfolding consistent_sign_vectors_R_def find_None_iff
by (simp add: assms(1) consistent_sign_vec_semantics_R)
lemma decide_univ_lem_helper_R:
fixes fml:: "real poly fml"
assumes "(fml_struct,polys) = convert fml"
shows "(\<forall>x::real. lookup_sem fml_struct (map (\<lambda>p. poly p x) polys)) \<longleftrightarrow> (decide_universal_R fml)"
using assms universal_lookup_sem_R main_step_R unfolding decide_universal_R_def apply (auto)
apply (metis assms convert_closed fst_conv snd_conv)
by (metis (full_types) assms convert_closed fst_conv snd_conv)
lemma decide_exis_lem_helper_R:
fixes fml:: "real poly fml"
assumes "(fml_struct,polys) = convert fml"
shows "(\<exists>x::real. lookup_sem fml_struct (map (\<lambda>p. poly p x) polys)) \<longleftrightarrow> (decide_existential_R fml)"
using assms existential_lookup_sem_R main_step_R unfolding decide_existential_R_def apply (auto)
apply (metis assms convert_closed fst_conv snd_conv)
by (metis (full_types) assms convert_closed fst_conv snd_conv)
lemma convert_semantics_lem_R:
assumes "\<And>p. p \<in> set (poly_list fml) \<Longrightarrow>
ls ! (index_of ps p) = poly p x"
shows "real_sem fml x = lookup_sem (map_fml (index_of ps) fml) ls"
using assms apply (induct fml)
by auto
lemma convert_semantics_R:
shows "real_sem fml x = lookup_sem (fst (convert fml)) (map (\<lambda>p. poly p x) (snd (convert fml)))"
unfolding convert_def Let_def apply simp
apply (intro convert_semantics_lem_R)
by (simp add: index_of_lookup(1) index_of_lookup(2))
(* Main result *)
theorem decision_procedure_R:
shows "(\<forall>x::real. real_sem fml x) \<longleftrightarrow> (decide_universal_R fml)"
"\<exists>x::real. real_sem fml x \<longleftrightarrow> (decide_existential_R fml)"
using convert_semantics_lem_R decide_univ_lem_helper_R apply (auto)
apply (simp add: convert_semantics_R)
apply (metis convert_def convert_semantics_R fst_conv snd_conv)
using convert_semantics_lem_R
by (metis convert_def convert_semantics_R decide_exis_lem_helper_R fst_conv snd_conv)
end
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