imports Reflected_Multivariate_Polynomial Dense_Linear_Order DP_Library Code_Target_Numeral Old_Recdef

(* Title: HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy Author: Amine Chaieb *) section ‹A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008› theory Parametric_Ferrante_Rackoff imports Reflected_Multivariate_Polynomial Dense_Linear_Order DP_Library "HOL-Library.Code_Target_Numeral" "HOL-Library.Old_Recdef" begin subsection ‹Terms› datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm | Neg tm | Sub tm tm | CNP nat poly tm text ‹A size for poly to make inductive proofs simpler.› primrec tmsize :: "tm ⇒ nat" where "tmsize (CP c) = polysize c" | "tmsize (Bound n) = 1" | "tmsize (Neg a) = 1 + tmsize a" | "tmsize (Add a b) = 1 + tmsize a + tmsize b" | "tmsize (Sub a b) = 3 + tmsize a + tmsize b" | "tmsize (Mul c a) = 1 + polysize c + tmsize a" | "tmsize (CNP n c a) = 3 + polysize c + tmsize a " text ‹Semantics of terms tm.› primrec Itm :: "'a::{field_char_0,field} list ⇒ 'a list ⇒ tm ⇒ 'a" where "Itm vs bs (CP c) = (Ipoly vs c)" | "Itm vs bs (Bound n) = bs!n" | "Itm vs bs (Neg a) = -(Itm vs bs a)" | "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b" | "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b" | "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a" | "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t" fun allpolys :: "(poly ⇒ bool) ⇒ tm ⇒ bool" where "allpolys P (CP c) = P c" | "allpolys P (CNP n c p) = (P c ∧ allpolys P p)" | "allpolys P (Mul c p) = (P c ∧ allpolys P p)" | "allpolys P (Neg p) = allpolys P p" | "allpolys P (Add p q) = (allpolys P p ∧ allpolys P q)" | "allpolys P (Sub p q) = (allpolys P p ∧ allpolys P q)" | "allpolys P p = True" primrec tmboundslt :: "nat ⇒ tm ⇒ bool" where "tmboundslt n (CP c) = True" | "tmboundslt n (Bound m) = (m < n)" | "tmboundslt n (CNP m c a) = (m < n ∧ tmboundslt n a)" | "tmboundslt n (Neg a) = tmboundslt n a" | "tmboundslt n (Add a b) = (tmboundslt n a ∧ tmboundslt n b)" | "tmboundslt n (Sub a b) = (tmboundslt n a ∧ tmboundslt n b)" | "tmboundslt n (Mul i a) = tmboundslt n a" primrec tmbound0 :: "tm ⇒ bool" ― ‹a tm is INDEPENDENT of Bound 0› where "tmbound0 (CP c) = True" | "tmbound0 (Bound n) = (n>0)" | "tmbound0 (CNP n c a) = (n≠0 ∧ tmbound0 a)" | "tmbound0 (Neg a) = tmbound0 a" | "tmbound0 (Add a b) = (tmbound0 a ∧ tmbound0 b)" | "tmbound0 (Sub a b) = (tmbound0 a ∧ tmbound0 b)" | "tmbound0 (Mul i a) = tmbound0 a" lemma tmbound0_I: assumes nb: "tmbound0 a" shows "Itm vs (b#bs) a = Itm vs (b'#bs) a" using nb by (induct a rule: tm.induct) auto primrec tmbound :: "nat ⇒ tm ⇒ bool" ― ‹a tm is INDEPENDENT of Bound n› where "tmbound n (CP c) = True" | "tmbound n (Bound m) = (n ≠ m)" | "tmbound n (CNP m c a) = (n≠m ∧ tmbound n a)" | "tmbound n (Neg a) = tmbound n a" | "tmbound n (Add a b) = (tmbound n a ∧ tmbound n b)" | "tmbound n (Sub a b) = (tmbound n a ∧ tmbound n b)" | "tmbound n (Mul i a) = tmbound n a" lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t) auto lemma tmbound_I: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n ≤ length bs" shows "Itm vs (bs[n:=x]) t = Itm vs bs t" using nb le bnd by (induct t rule: tm.induct) auto fun decrtm0 :: "tm ⇒ tm" where "decrtm0 (Bound n) = Bound (n - 1)" | "decrtm0 (Neg a) = Neg (decrtm0 a)" | "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)" | "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)" | "decrtm0 (Mul c a) = Mul c (decrtm0 a)" | "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)" | "decrtm0 a = a" fun incrtm0 :: "tm ⇒ tm" where "incrtm0 (Bound n) = Bound (n + 1)" | "incrtm0 (Neg a) = Neg (incrtm0 a)" | "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)" | "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)" | "incrtm0 (Mul c a) = Mul c (incrtm0 a)" | "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)" | "incrtm0 a = a" lemma decrtm0: assumes nb: "tmbound0 t" shows "Itm vs (x # bs) t = Itm vs bs (decrtm0 t)" using nb by (induct t rule: decrtm0.induct) simp_all lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t" by (induct t rule: decrtm0.induct) simp_all primrec decrtm :: "nat ⇒ tm ⇒ tm" where "decrtm m (CP c) = (CP c)" | "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))" | "decrtm m (Neg a) = Neg (decrtm m a)" | "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)" | "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)" | "decrtm m (Mul c a) = Mul c (decrtm m a)" | "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))" primrec removen :: "nat ⇒ 'a list ⇒ 'a list" where "removen n [] = []" | "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))" lemma removen_same: "n ≥ length xs ⟹ removen n xs = xs" by (induct xs arbitrary: n) auto lemma nth_length_exceeds: "n ≥ length xs ⟹ xs!n = []!(n - length xs)" by (induct xs arbitrary: n) auto lemma removen_length: "length (removen n xs) = (if n ≥ length xs then length xs else length xs - 1)" by (induct xs arbitrary: n, auto) lemma removen_nth: "(removen n xs)!m = (if n ≥ length xs then xs!m else if m < n then xs!m else if m ≤ length xs then xs!(Suc m) else []!(m - (length xs - 1)))" proof (induct xs arbitrary: n m) case Nil then show ?case by simp next case (Cons x xs) let ?l = "length (x # xs)" consider "n ≥ ?l" | "n < ?l" by arith then show ?case proof cases case 1 with removen_same[OF this] show ?thesis by simp next case nl: 2 consider "m < n" | "m ≥ n" by arith then show ?thesis proof cases case 1 then show ?thesis using Cons by (cases m) auto next case 2 consider "m ≤ ?l" | "m > ?l" by arith then show ?thesis proof cases case 1 then show ?thesis using Cons by (cases m) auto next case ml: 2 have th: "length (removen n (x # xs)) = length xs" using removen_length[where n = n and xs= "x # xs"] nl by simp with ml have "m ≥ length (removen n (x # xs))" by auto from th nth_length_exceeds[OF this] have "(removen n (x # xs))!m = [] ! (m - length xs)" by auto then have "(removen n (x # xs))!m = [] ! (m - (length (x # xs) - 1))" by auto then show ?thesis using ml nl by auto qed qed qed qed lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" and nle: "m ≤ length bs" shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t" using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth) primrec tmsubst0:: "tm ⇒ tm ⇒ tm" where "tmsubst0 t (CP c) = CP c" | "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)" | "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))" | "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)" | "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)" | "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" | "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)" lemma tmsubst0: "Itm vs (x # bs) (tmsubst0 t a) = Itm vs (Itm vs (x # bs) t # bs) a" by (induct a rule: tm.induct) auto lemma tmsubst0_nb: "tmbound0 t ⟹ tmbound0 (tmsubst0 t a)" by (induct a rule: tm.induct) auto primrec tmsubst:: "nat ⇒ tm ⇒ tm ⇒ tm" where "tmsubst n t (CP c) = CP c" | "tmsubst n t (Bound m) = (if n=m then t else Bound m)" | "tmsubst n t (CNP m c a) = (if n = m then Add (Mul c t) (tmsubst n t a) else CNP m c (tmsubst n t a))" | "tmsubst n t (Neg a) = Neg (tmsubst n t a)" | "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)" | "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" | "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)" lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n ≤ length bs" shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a" using nb nlt by (induct a rule: tm.induct) auto lemma tmsubst_nb0: assumes tnb: "tmbound0 t" shows "tmbound0 (tmsubst 0 t a)" using tnb by (induct a rule: tm.induct) auto lemma tmsubst_nb: assumes tnb: "tmbound m t" shows "tmbound m (tmsubst m t a)" using tnb by (induct a rule: tm.induct) auto lemma incrtm0_tmbound: "tmbound n t ⟹ tmbound (Suc n) (incrtm0 t)" by (induct t) auto text ‹Simplification.› consts tmadd:: "tm × tm ⇒ tm" recdef tmadd "measure (λ(t,s). size t + size s)" "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) = (if n1 = n2 then let c = c1 +⇩_{p}c2 in if c = 0⇩_{p}then tmadd(r1,r2) else CNP n1 c (tmadd (r1, r2)) else if n1 ≤ n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) else (CNP n2 c2 (tmadd (CNP n1 c1 r1, r2))))" "tmadd (CNP n1 c1 r1, t) = CNP n1 c1 (tmadd (r1, t))" "tmadd (t, CNP n2 c2 r2) = CNP n2 c2 (tmadd (t, r2))" "tmadd (CP b1, CP b2) = CP (b1 +⇩_{p}b2)" "tmadd (a, b) = Add a b" lemma tmadd[simp]: "Itm vs bs (tmadd (t, s)) = Itm vs bs (Add t s)" apply (induct t s rule: tmadd.induct) apply (simp_all add: Let_def) apply (case_tac "c1 +⇩_{p}c2 = 0⇩_{p}") apply (case_tac "n1 ≤ n2") apply simp_all apply (case_tac "n1 = n2") apply (simp_all add: field_simps) apply (simp only: distrib_left[symmetric]) apply (auto simp del: polyadd simp add: polyadd[symmetric]) done lemma tmadd_nb0[simp]: "tmbound0 t ⟹ tmbound0 s ⟹ tmbound0 (tmadd (t, s))" by (induct t s rule: tmadd.induct) (auto simp add: Let_def) lemma tmadd_nb[simp]: "tmbound n t ⟹ tmbound n s ⟹ tmbound n (tmadd (t, s))" by (induct t s rule: tmadd.induct) (auto simp add: Let_def) lemma tmadd_blt[simp]: "tmboundslt n t ⟹ tmboundslt n s ⟹ tmboundslt n (tmadd (t, s))" by (induct t s rule: tmadd.induct) (auto simp add: Let_def) lemma tmadd_allpolys_npoly[simp]: "allpolys isnpoly t ⟹ allpolys isnpoly s ⟹ allpolys isnpoly (tmadd(t, s))" by (induct t s rule: tmadd.induct) (simp_all add: Let_def polyadd_norm) fun tmmul:: "tm ⇒ poly ⇒ tm" where "tmmul (CP j) = (λi. CP (i *⇩_{p}j))" | "tmmul (CNP n c a) = (λi. CNP n (i *⇩_{p}c) (tmmul a i))" | "tmmul t = (λi. Mul i t)" lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)" by (induct t arbitrary: i rule: tmmul.induct) (simp_all add: field_simps) lemma tmmul_nb0[simp]: "tmbound0 t ⟹ tmbound0 (tmmul t i)" by (induct t arbitrary: i rule: tmmul.induct) auto lemma tmmul_nb[simp]: "tmbound n t ⟹ tmbound n (tmmul t i)" by (induct t arbitrary: n rule: tmmul.induct) auto lemma tmmul_blt[simp]: "tmboundslt n t ⟹ tmboundslt n (tmmul t i)" by (induct t arbitrary: i rule: tmmul.induct) (auto simp add: Let_def) lemma tmmul_allpolys_npoly[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "allpolys isnpoly t ⟹ isnpoly c ⟹ allpolys isnpoly (tmmul t c)" by (induct t rule: tmmul.induct) (simp_all add: Let_def polymul_norm) definition tmneg :: "tm ⇒ tm" where "tmneg t ≡ tmmul t (C (- 1,1))" definition tmsub :: "tm ⇒ tm ⇒ tm" where "tmsub s t ≡ (if s = t then CP 0⇩_{p}else tmadd (s, tmneg t))" lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)" using tmneg_def[of t] by simp lemma tmneg_nb0[simp]: "tmbound0 t ⟹ tmbound0 (tmneg t)" using tmneg_def by simp lemma tmneg_nb[simp]: "tmbound n t ⟹ tmbound n (tmneg t)" using tmneg_def by simp lemma tmneg_blt[simp]: "tmboundslt n t ⟹ tmboundslt n (tmneg t)" using tmneg_def by simp lemma [simp]: "isnpoly (C (-1, 1))" unfolding isnpoly_def by simp lemma tmneg_allpolys_npoly[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "allpolys isnpoly t ⟹ allpolys isnpoly (tmneg t)" unfolding tmneg_def by auto lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)" using tmsub_def by simp lemma tmsub_nb0[simp]: "tmbound0 t ⟹ tmbound0 s ⟹ tmbound0 (tmsub t s)" using tmsub_def by simp lemma tmsub_nb[simp]: "tmbound n t ⟹ tmbound n s ⟹ tmbound n (tmsub t s)" using tmsub_def by simp lemma tmsub_blt[simp]: "tmboundslt n t ⟹ tmboundslt n s ⟹ tmboundslt n (tmsub t s)" using tmsub_def by simp lemma tmsub_allpolys_npoly[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "allpolys isnpoly t ⟹ allpolys isnpoly s ⟹ allpolys isnpoly (tmsub t s)" unfolding tmsub_def by (simp add: isnpoly_def) fun simptm :: "tm ⇒ tm" where "simptm (CP j) = CP (polynate j)" | "simptm (Bound n) = CNP n (1)⇩_{p}(CP 0⇩_{p})" | "simptm (Neg t) = tmneg (simptm t)" | "simptm (Add t s) = tmadd (simptm t,simptm s)" | "simptm (Sub t s) = tmsub (simptm t) (simptm s)" | "simptm (Mul i t) = (let i' = polynate i in if i' = 0⇩_{p}then CP 0⇩_{p}else tmmul (simptm t) i')" | "simptm (CNP n c t) = (let c' = polynate c in if c' = 0⇩_{p}then simptm t else tmadd (CNP n c' (CP 0⇩_{p}), simptm t))" lemma polynate_stupid: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "polynate t = 0⇩_{p}⟹ Ipoly bs t = (0::'a)" apply (subst polynate[symmetric]) apply simp done lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t" by (induct t rule: simptm.induct) (auto simp add: Let_def polynate_stupid) lemma simptm_tmbound0[simp]: "tmbound0 t ⟹ tmbound0 (simptm t)" by (induct t rule: simptm.induct) (auto simp add: Let_def) lemma simptm_nb[simp]: "tmbound n t ⟹ tmbound n (simptm t)" by (induct t rule: simptm.induct) (auto simp add: Let_def) lemma simptm_nlt[simp]: "tmboundslt n t ⟹ tmboundslt n (simptm t)" by (induct t rule: simptm.induct) (auto simp add: Let_def) lemma [simp]: "isnpoly 0⇩_{p}" and [simp]: "isnpoly (C (1, 1))" by (simp_all add: isnpoly_def) lemma simptm_allpolys_npoly[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "allpolys isnpoly (simptm p)" by (induct p rule: simptm.induct) (auto simp add: Let_def) declare let_cong[fundef_cong del] fun split0 :: "tm ⇒ poly × tm" where "split0 (Bound 0) = ((1)⇩_{p}, CP 0⇩_{p})" | "split0 (CNP 0 c t) = (let (c', t') = split0 t in (c +⇩_{p}c', t'))" | "split0 (Neg t) = (let (c, t') = split0 t in (~⇩_{p}c, Neg t'))" | "split0 (CNP n c t) = (let (c', t') = split0 t in (c', CNP n c t'))" | "split0 (Add s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 +⇩_{p}c2, Add s' t'))" | "split0 (Sub s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 -⇩_{p}c2, Sub s' t'))" | "split0 (Mul c t) = (let (c', t') = split0 t in (c *⇩_{p}c', Mul c t'))" | "split0 t = (0⇩_{p}, t)" declare let_cong[fundef_cong] lemma split0_stupid[simp]: "∃x y. (x, y) = split0 p" apply (rule exI[where x="fst (split0 p)"]) apply (rule exI[where x="snd (split0 p)"]) apply simp done lemma split0: "tmbound 0 (snd (split0 t)) ∧ Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t" apply (induct t rule: split0.induct) apply simp apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def mult.assoc distrib_left[symmetric]) apply (simp add: Let_def split_def field_simps) apply (simp add: Let_def split_def field_simps) done lemma split0_ci: "split0 t = (c',t') ⟹ Itm vs bs t = Itm vs bs (CNP 0 c' t')" proof - fix c' t' assume "split0 t = (c', t')" then have "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp qed lemma split0_nb0: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "split0 t = (c',t') ⟹ tmbound 0 t'" proof - fix c' t' assume "split0 t = (c', t')" then have "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp qed lemma split0_nb0'[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "tmbound0 (snd (split0 t))" using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff) lemma split0_nb: assumes nb: "tmbound n t" shows "tmbound n (snd (split0 t))" using nb by (induct t rule: split0.induct) (auto simp add: Let_def split_def) lemma split0_blt: assumes nb: "tmboundslt n t" shows "tmboundslt n (snd (split0 t))" using nb by (induct t rule: split0.induct) (auto simp add: Let_def split_def) lemma tmbound_split0: "tmbound 0 t ⟹ Ipoly vs (fst (split0 t)) = 0" by (induct t rule: split0.induct) (auto simp add: Let_def split_def) lemma tmboundslt_split0: "tmboundslt n t ⟹ Ipoly vs (fst (split0 t)) = 0 ∨ n > 0" by (induct t rule: split0.induct) (auto simp add: Let_def split_def) lemma tmboundslt0_split0: "tmboundslt 0 t ⟹ Ipoly vs (fst (split0 t)) = 0" by (induct t rule: split0.induct) (auto simp add: Let_def split_def) lemma allpolys_split0: "allpolys isnpoly p ⟹ allpolys isnpoly (snd (split0 p))" by (induct p rule: split0.induct) (auto simp add: isnpoly_def Let_def split_def) lemma isnpoly_fst_split0: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "allpolys isnpoly p ⟹ isnpoly (fst (split0 p))" by (induct p rule: split0.induct) (auto simp add: polyadd_norm polysub_norm polyneg_norm polymul_norm Let_def split_def) subsection ‹Formulae› datatype fm = T| F| Le tm | Lt tm | Eq tm | NEq tm| NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm text ‹A size for fm.› fun fmsize :: "fm ⇒ nat" where "fmsize (NOT p) = 1 + fmsize p" | "fmsize (And p q) = 1 + fmsize p + fmsize q" | "fmsize (Or p q) = 1 + fmsize p + fmsize q" | "fmsize (Imp p q) = 3 + fmsize p + fmsize q" | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" | "fmsize (E p) = 1 + fmsize p" | "fmsize (A p) = 4+ fmsize p" | "fmsize p = 1" lemma fmsize_pos[termination_simp]: "fmsize p > 0" by (induct p rule: fmsize.induct) simp_all text ‹Semantics of formulae (fm).› primrec Ifm ::"'a::linordered_field list ⇒ 'a list ⇒ fm ⇒ bool" where "Ifm vs bs T = True" | "Ifm vs bs F = False" | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)" | "Ifm vs bs (Le a) = (Itm vs bs a ≤ 0)" | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)" | "Ifm vs bs (NEq a) = (Itm vs bs a ≠ 0)" | "Ifm vs bs (NOT p) = (¬ (Ifm vs bs p))" | "Ifm vs bs (And p q) = (Ifm vs bs p ∧ Ifm vs bs q)" | "Ifm vs bs (Or p q) = (Ifm vs bs p ∨ Ifm vs bs q)" | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) ⟶ (Ifm vs bs q))" | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)" | "Ifm vs bs (E p) = (∃x. Ifm vs (x#bs) p)" | "Ifm vs bs (A p) = (∀x. Ifm vs (x#bs) p)" fun not:: "fm ⇒ fm" where "not (NOT (NOT p)) = not p" | "not (NOT p) = p" | "not T = F" | "not F = T" | "not (Lt t) = Le (tmneg t)" | "not (Le t) = Lt (tmneg t)" | "not (Eq t) = NEq t" | "not (NEq t) = Eq t" | "not p = NOT p" lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)" by (induct p rule: not.induct) auto definition conj :: "fm ⇒ fm ⇒ fm" where "conj p q ≡ (if p = F ∨ q = F then F else if p = T then q else if q = T then p else if p = q then p else And p q)" lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)" by (cases "p=F ∨ q=F", simp_all add: conj_def) (cases p, simp_all) definition disj :: "fm ⇒ fm ⇒ fm" where "disj p q ≡ (if (p = T ∨ q = T) then T else if p = F then q else if q = F then p else if p = q then p else Or p q)" lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)" by (cases "p = T ∨ q = T", simp_all add: disj_def) (cases p, simp_all) definition imp :: "fm ⇒ fm ⇒ fm" where "imp p q ≡ (if p = F ∨ q = T ∨ p = q then T else if p = T then q else if q = F then not p else Imp p q)" lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)" by (cases "p = F ∨ q = T") (simp_all add: imp_def) definition iff :: "fm ⇒ fm ⇒ fm" where "iff p q ≡ (if p = q then T else if p = NOT q ∨ NOT p = q then F else if p = F then not q else if q = F then not p else if p = T then q else if q = T then p else Iff p q)" lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)" by (unfold iff_def, cases "p = q", simp, cases "p = NOT q", simp) (cases "NOT p= q", auto) text ‹Quantifier freeness.› fun qfree:: "fm ⇒ bool" where "qfree (E p) = False" | "qfree (A p) = False" | "qfree (NOT p) = qfree p" | "qfree (And p q) = (qfree p ∧ qfree q)" | "qfree (Or p q) = (qfree p ∧ qfree q)" | "qfree (Imp p q) = (qfree p ∧ qfree q)" | "qfree (Iff p q) = (qfree p ∧ qfree q)" | "qfree p = True" text ‹Boundedness and substitution.› primrec boundslt :: "nat ⇒ fm ⇒ bool" where "boundslt n T = True" | "boundslt n F = True" | "boundslt n (Lt t) = tmboundslt n t" | "boundslt n (Le t) = tmboundslt n t" | "boundslt n (Eq t) = tmboundslt n t" | "boundslt n (NEq t) = tmboundslt n t" | "boundslt n (NOT p) = boundslt n p" | "boundslt n (And p q) = (boundslt n p ∧ boundslt n q)" | "boundslt n (Or p q) = (boundslt n p ∧ boundslt n q)" | "boundslt n (Imp p q) = ((boundslt n p) ∧ (boundslt n q))" | "boundslt n (Iff p q) = (boundslt n p ∧ boundslt n q)" | "boundslt n (E p) = boundslt (Suc n) p" | "boundslt n (A p) = boundslt (Suc n) p" fun bound0:: "fm ⇒ bool" ― ‹a Formula is independent of Bound 0› where "bound0 T = True" | "bound0 F = True" | "bound0 (Lt a) = tmbound0 a" | "bound0 (Le a) = tmbound0 a" | "bound0 (Eq a) = tmbound0 a" | "bound0 (NEq a) = tmbound0 a" | "bound0 (NOT p) = bound0 p" | "bound0 (And p q) = (bound0 p ∧ bound0 q)" | "bound0 (Or p q) = (bound0 p ∧ bound0 q)" | "bound0 (Imp p q) = ((bound0 p) ∧ (bound0 q))" | "bound0 (Iff p q) = (bound0 p ∧ bound0 q)" | "bound0 p = False" lemma bound0_I: assumes bp: "bound0 p" shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p" using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"] by (induct p rule: bound0.induct) auto primrec bound:: "nat ⇒ fm ⇒ bool" ― ‹a Formula is independent of Bound n› where "bound m T = True" | "bound m F = True" | "bound m (Lt t) = tmbound m t" | "bound m (Le t) = tmbound m t" | "bound m (Eq t) = tmbound m t" | "bound m (NEq t) = tmbound m t" | "bound m (NOT p) = bound m p" | "bound m (And p q) = (bound m p ∧ bound m q)" | "bound m (Or p q) = (bound m p ∧ bound m q)" | "bound m (Imp p q) = ((bound m p) ∧ (bound m q))" | "bound m (Iff p q) = (bound m p ∧ bound m q)" | "bound m (E p) = bound (Suc m) p" | "bound m (A p) = bound (Suc m) p" lemma bound_I: assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n ≤ length bs" shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p" using bnd nb le tmbound_I[where bs=bs and vs = vs] proof (induct p arbitrary: bs n rule: fm.induct) case (E p bs n) have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" for y proof - from E have bnd: "boundslt (length (y#bs)) p" and nb: "bound (Suc n) p" and le: "Suc n ≤ length (y#bs)" by simp+ from E.hyps[OF bnd nb le tmbound_I] show ?thesis . qed then show ?case by simp next case (A p bs n) have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" for y proof - from A have bnd: "boundslt (length (y#bs)) p" and nb: "bound (Suc n) p" and le: "Suc n ≤ length (y#bs)" by simp_all from A.hyps[OF bnd nb le tmbound_I] show ?thesis . qed then show ?case by simp qed auto fun decr0 :: "fm ⇒ fm" where "decr0 (Lt a) = Lt (decrtm0 a)" | "decr0 (Le a) = Le (decrtm0 a)" | "decr0 (Eq a) = Eq (decrtm0 a)" | "decr0 (NEq a) = NEq (decrtm0 a)" | "decr0 (NOT p) = NOT (decr0 p)" | "decr0 (And p q) = conj (decr0 p) (decr0 q)" | "decr0 (Or p q) = disj (decr0 p) (decr0 q)" | "decr0 (Imp p q) = imp (decr0 p) (decr0 q)" | "decr0 (Iff p q) = iff (decr0 p) (decr0 q)" | "decr0 p = p" lemma decr0: assumes nb: "bound0 p" shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)" using nb by (induct p rule: decr0.induct) (simp_all add: decrtm0) primrec decr :: "nat ⇒ fm ⇒ fm" where "decr m T = T" | "decr m F = F" | "decr m (Lt t) = (Lt (decrtm m t))" | "decr m (Le t) = (Le (decrtm m t))" | "decr m (Eq t) = (Eq (decrtm m t))" | "decr m (NEq t) = (NEq (decrtm m t))" | "decr m (NOT p) = NOT (decr m p)" | "decr m (And p q) = conj (decr m p) (decr m q)" | "decr m (Or p q) = disj (decr m p) (decr m q)" | "decr m (Imp p q) = imp (decr m p) (decr m q)" | "decr m (Iff p q) = iff (decr m p) (decr m q)" | "decr m (E p) = E (decr (Suc m) p)" | "decr m (A p) = A (decr (Suc m) p)" lemma decr: assumes bnd: "boundslt (length bs) p" and nb: "bound m p" and nle: "m < length bs" shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p" using bnd nb nle proof (induct p arbitrary: bs m rule: fm.induct) case (E p bs m) have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" for x proof - from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" and nle: "Suc m < length (x#bs)" by auto from E(1)[OF bnd nb nle] show ?thesis . qed then show ?case by auto next case (A p bs m) have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" for x proof - from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" and nle: "Suc m < length (x#bs)" by auto from A(1)[OF bnd nb nle] show ?thesis . qed then show ?case by auto qed (auto simp add: decrtm removen_nth) primrec subst0 :: "tm ⇒ fm ⇒ fm" where "subst0 t T = T" | "subst0 t F = F" | "subst0 t (Lt a) = Lt (tmsubst0 t a)" | "subst0 t (Le a) = Le (tmsubst0 t a)" | "subst0 t (Eq a) = Eq (tmsubst0 t a)" | "subst0 t (NEq a) = NEq (tmsubst0 t a)" | "subst0 t (NOT p) = NOT (subst0 t p)" | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" | "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" | "subst0 t (E p) = E p" | "subst0 t (A p) = A p" lemma subst0: assumes qf: "qfree p" shows "Ifm vs (x # bs) (subst0 t p) = Ifm vs ((Itm vs (x # bs) t) # bs) p" using qf tmsubst0[where x="x" and bs="bs" and t="t"] by (induct p rule: fm.induct) auto lemma subst0_nb: assumes bp: "tmbound0 t" and qf: "qfree p" shows "bound0 (subst0 t p)" using qf tmsubst0_nb[OF bp] bp by (induct p rule: fm.induct) auto primrec subst:: "nat ⇒ tm ⇒ fm ⇒ fm" where "subst n t T = T" | "subst n t F = F" | "subst n t (Lt a) = Lt (tmsubst n t a)" | "subst n t (Le a) = Le (tmsubst n t a)" | "subst n t (Eq a) = Eq (tmsubst n t a)" | "subst n t (NEq a) = NEq (tmsubst n t a)" | "subst n t (NOT p) = NOT (subst n t p)" | "subst n t (And p q) = And (subst n t p) (subst n t q)" | "subst n t (Or p q) = Or (subst n t p) (subst n t q)" | "subst n t (Imp p q) = Imp (subst n t p) (subst n t q)" | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)" | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)" | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)" lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n ≤ length bs" shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p" using nb nlm proof (induct p arbitrary: bs n t rule: fm.induct) case (E p bs n) have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p" for x proof - from E have bn: "boundslt (length (x#bs)) p" by simp from E have nlm: "Suc n ≤ length (x#bs)" by simp from E(1)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp then show ?thesis by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) qed then show ?case by simp next case (A p bs n) have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p" for x proof - from A have bn: "boundslt (length (x#bs)) p" by simp from A have nlm: "Suc n ≤ length (x#bs)" by simp from A(1)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp then show ?thesis by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) qed then show ?case by simp qed (auto simp add: tmsubst) lemma subst_nb: assumes tnb: "tmbound m t" shows "bound m (subst m t p)" using tnb tmsubst_nb incrtm0_tmbound by (induct p arbitrary: m t rule: fm.induct) auto lemma not_qf[simp]: "qfree p ⟹ qfree (not p)" by (induct p rule: not.induct) auto lemma not_bn0[simp]: "bound0 p ⟹ bound0 (not p)" by (induct p rule: not.induct) auto lemma not_nb[simp]: "bound n p ⟹ bound n (not p)" by (induct p rule: not.induct) auto lemma not_blt[simp]: "boundslt n p ⟹ boundslt n (not p)" by (induct p rule: not.induct) auto lemma conj_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (conj p q)" using conj_def by auto lemma conj_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (conj p q)" using conj_def by auto lemma conj_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (conj p q)" using conj_def by auto lemma conj_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (conj p q)" using conj_def by auto lemma disj_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (disj p q)" using disj_def by auto lemma disj_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (disj p q)" using disj_def by auto lemma disj_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (disj p q)" using disj_def by auto lemma disj_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (disj p q)" using disj_def by auto lemma imp_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (imp p q)" using imp_def by (cases "p = F ∨ q = T") (simp_all add: imp_def) lemma imp_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (imp p q)" using imp_def by (cases "p = F ∨ q = T ∨ p = q") (simp_all add: imp_def) lemma imp_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (imp p q)" using imp_def by (cases "p = F ∨ q = T ∨ p = q") (simp_all add: imp_def) lemma imp_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (imp p q)" using imp_def by auto lemma iff_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (iff p q)" unfolding iff_def by (cases "p = q") auto lemma iff_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (iff p q)" using iff_def unfolding iff_def by (cases "p = q") auto lemma iff_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (iff p q)" using iff_def unfolding iff_def by (cases "p = q") auto lemma iff_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (iff p q)" using iff_def by auto lemma decr0_qf: "bound0 p ⟹ qfree (decr0 p)" by (induct p) simp_all fun isatom :: "fm ⇒ bool" ― ‹test for atomicity› where "isatom T = True" | "isatom F = True" | "isatom (Lt a) = True" | "isatom (Le a) = True" | "isatom (Eq a) = True" | "isatom (NEq a) = True" | "isatom p = False" lemma bound0_qf: "bound0 p ⟹ qfree p" by (induct p) simp_all definition djf :: "('a ⇒ fm) ⇒ 'a ⇒ fm ⇒ fm" where "djf f p q ≡ (if q = T then T else if q = F then f p else (let fp = f p in case fp of T ⇒ T | F ⇒ q | _ ⇒ Or (f p) q))" definition evaldjf :: "('a ⇒ fm) ⇒ 'a list ⇒ fm" where "evaldjf f ps ≡ foldr (djf f) ps F" lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)" apply (cases "q = T") apply (simp add: djf_def) apply (cases "q = F") apply (simp add: djf_def) apply (cases "f p") apply (simp_all add: Let_def djf_def) done lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) ⟷ (∃p ∈ set ps. Ifm vs bs (f p))" by (induct ps) (simp_all add: evaldjf_def djf_Or) lemma evaldjf_bound0: assumes nb: "∀x∈ set xs. bound0 (f x)" shows "bound0 (evaldjf f xs)" using nb apply (induct xs) apply (auto simp add: evaldjf_def djf_def Let_def) apply (case_tac "f a") apply auto done lemma evaldjf_qf: assumes nb: "∀x∈ set xs. qfree (f x)" shows "qfree (evaldjf f xs)" using nb apply (induct xs) apply (auto simp add: evaldjf_def djf_def Let_def) apply (case_tac "f a") apply auto done fun disjuncts :: "fm ⇒ fm list" where "disjuncts (Or p q) = disjuncts p @ disjuncts q" | "disjuncts F = []" | "disjuncts p = [p]" lemma disjuncts: "(∃q ∈ set (disjuncts p). Ifm vs bs q) = Ifm vs bs p" by (induct p rule: disjuncts.induct) auto lemma disjuncts_nb: "bound0 p ⟹ ∀q ∈ set (disjuncts p). bound0 q" proof - assume nb: "bound0 p" then have "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct) auto then show ?thesis by (simp only: list_all_iff) qed lemma disjuncts_qf: "qfree p ⟹ ∀q ∈ set (disjuncts p). qfree q" proof - assume qf: "qfree p" then have "list_all qfree (disjuncts p)" by (induct p rule: disjuncts.induct) auto then show ?thesis by (simp only: list_all_iff) qed definition DJ :: "(fm ⇒ fm) ⇒ fm ⇒ fm" where "DJ f p ≡ evaldjf f (disjuncts p)" lemma DJ: assumes fdj: "∀p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))" and fF: "f F = F" shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)" proof - have "Ifm vs bs (DJ f p) = (∃q ∈ set (disjuncts p). Ifm vs bs (f q))" by (simp add: DJ_def evaldjf_ex) also have "… = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct) auto finally show ?thesis . qed lemma DJ_qf: assumes fqf: "∀p. qfree p ⟶ qfree (f p)" shows "∀p. qfree p ⟶ qfree (DJ f p)" proof clarify fix p assume qf: "qfree p" have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) from disjuncts_qf[OF qf] have "∀q∈ set (disjuncts p). qfree q" . with fqf have th':"∀q∈ set (disjuncts p). qfree (f q)" by blast from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp qed lemma DJ_qe: assumes qe: "∀bs p. qfree p ⟶ qfree (qe p) ∧ (Ifm vs bs (qe p) = Ifm vs bs (E p))" shows "∀bs p. qfree p ⟶ qfree (DJ qe p) ∧ (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))" proof clarify fix p :: fm and bs assume qf: "qfree p" from qe have qth: "∀p. qfree p ⟶ qfree (qe p)" by blast from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto have "Ifm vs bs (DJ qe p) ⟷ (∃q∈ set (disjuncts p). Ifm vs bs (qe q))" by (simp add: DJ_def evaldjf_ex) also have "… = (∃q ∈ set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto also have "… = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct) auto finally show "qfree (DJ qe p) ∧ Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast qed fun conjuncts :: "fm ⇒ fm list" where "conjuncts (And p q) = conjuncts p @ conjuncts q" | "conjuncts T = []" | "conjuncts p = [p]" definition list_conj :: "fm list ⇒ fm" where "list_conj ps ≡ foldr conj ps T" definition CJNB :: "(fm ⇒ fm) ⇒ fm ⇒ fm" where "CJNB f p ≡ (let cjs = conjuncts p; (yes, no) = partition bound0 cjs in conj (decr0 (list_conj yes)) (f (list_conj no)))" lemma conjuncts_qf: "qfree p ⟹ ∀q ∈ set (conjuncts p). qfree q" proof - assume qf: "qfree p" then have "list_all qfree (conjuncts p)" by (induct p rule: conjuncts.induct) auto then show ?thesis by (simp only: list_all_iff) qed lemma conjuncts: "(∀q∈ set (conjuncts p). Ifm vs bs q) = Ifm vs bs p" by (induct p rule: conjuncts.induct) auto lemma conjuncts_nb: "bound0 p ⟹ ∀q∈ set (conjuncts p). bound0 q" proof - assume nb: "bound0 p" then have "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct) auto then show ?thesis by (simp only: list_all_iff) qed fun islin :: "fm ⇒ bool" where "islin (And p q) = (islin p ∧ islin q ∧ p ≠ T ∧ p ≠ F ∧ q ≠ T ∧ q ≠ F)" | "islin (Or p q) = (islin p ∧ islin q ∧ p ≠ T ∧ p ≠ F ∧ q ≠ T ∧ q ≠ F)" | "islin (Eq (CNP 0 c s)) = (isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s)" | "islin (NEq (CNP 0 c s)) = (isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s)" | "islin (Lt (CNP 0 c s)) = (isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s)" | "islin (Le (CNP 0 c s)) = (isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s)" | "islin (NOT p) = False" | "islin (Imp p q) = False" | "islin (Iff p q) = False" | "islin p = bound0 p" lemma islin_stupid: assumes nb: "tmbound0 p" shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)" using nb by (cases p, auto, rename_tac nat a b, case_tac nat, auto)+ definition "lt p = (case p of CP (C c) ⇒ if 0>⇩_{N}c then T else F| _ ⇒ Lt p)" definition "le p = (case p of CP (C c) ⇒ if 0≥⇩_{N}c then T else F | _ ⇒ Le p)" definition "eq p = (case p of CP (C c) ⇒ if c = 0⇩_{N}then T else F | _ ⇒ Eq p)" definition "neq p = not (eq p)" lemma lt: "allpolys isnpoly p ⟹ Ifm vs bs (lt p) = Ifm vs bs (Lt p)" apply (simp add: lt_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply (simp_all add: isnpoly_def) done lemma le: "allpolys isnpoly p ⟹ Ifm vs bs (le p) = Ifm vs bs (Le p)" apply (simp add: le_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply (simp_all add: isnpoly_def) done lemma eq: "allpolys isnpoly p ⟹ Ifm vs bs (eq p) = Ifm vs bs (Eq p)" apply (simp add: eq_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply (simp_all add: isnpoly_def) done lemma neq: "allpolys isnpoly p ⟹ Ifm vs bs (neq p) = Ifm vs bs (NEq p)" by (simp add: neq_def eq) lemma lt_lin: "tmbound0 p ⟹ islin (lt p)" apply (simp add: lt_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply simp_all apply (rename_tac nat a b, case_tac nat) apply simp_all done lemma le_lin: "tmbound0 p ⟹ islin (le p)" apply (simp add: le_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply simp_all apply (rename_tac nat a b, case_tac nat) apply simp_all done lemma eq_lin: "tmbound0 p ⟹ islin (eq p)" apply (simp add: eq_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply simp_all apply (rename_tac nat a b, case_tac nat) apply simp_all done lemma neq_lin: "tmbound0 p ⟹ islin (neq p)" apply (simp add: neq_def eq_def) apply (cases p) apply simp_all apply (rename_tac poly, case_tac poly) apply simp_all apply (rename_tac nat a b, case_tac nat) apply simp_all done definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0⇩_{p}then lt s else Lt (CNP 0 c s))" definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0⇩_{p}then le s else Le (CNP 0 c s))" definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0⇩_{p}then eq s else Eq (CNP 0 c s))" definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0⇩_{p}then neq s else NEq (CNP 0 c s))" lemma simplt_islin[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "islin (simplt t)" unfolding simplt_def using split0_nb0' by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly]) lemma simple_islin[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "islin (simple t)" unfolding simple_def using split0_nb0' by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin) lemma simpeq_islin[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "islin (simpeq t)" unfolding simpeq_def using split0_nb0' by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin) lemma simpneq_islin[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "islin (simpneq t)" unfolding simpneq_def using split0_nb0' by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin) lemma really_stupid: "¬ (∀c1 s'. (c1, s') ≠ split0 s)" by (cases "split0 s") auto lemma split0_npoly: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" and n: "allpolys isnpoly t" shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))" using n by (induct t rule: split0.induct) (auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid) lemma simplt[simp]: "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)" proof - have n: "allpolys isnpoly (simptm t)" by simp let ?t = "simptm t" show ?thesis proof (cases "fst (split0 ?t) = 0⇩_{p}") case True then show ?thesis using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs] by (simp add: simplt_def Let_def split_def lt) next case False then show ?thesis using split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def) qed qed lemma simple[simp]: "Ifm vs bs (simple t) = Ifm vs bs (Le t)" proof - have n: "allpolys isnpoly (simptm t)" by simp let ?t = "simptm t" show ?thesis proof (cases "fst (split0 ?t) = 0⇩_{p}") case True then show ?thesis using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs] by (simp add: simple_def Let_def split_def le) next case False then show ?thesis using split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def) qed qed lemma simpeq[simp]: "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)" proof - have n: "allpolys isnpoly (simptm t)" by simp let ?t = "simptm t" show ?thesis proof (cases "fst (split0 ?t) = 0⇩_{p}") case True then show ?thesis using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs] by (simp add: simpeq_def Let_def split_def) next case False then show ?thesis using split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def) qed qed lemma simpneq[simp]: "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)" proof - have n: "allpolys isnpoly (simptm t)" by simp let ?t = "simptm t" show ?thesis proof (cases "fst (split0 ?t) = 0⇩_{p}") case True then show ?thesis using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs] by (simp add: simpneq_def Let_def split_def) next case False then show ?thesis using split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def) qed qed lemma lt_nb: "tmbound0 t ⟹ bound0 (lt t)" apply (simp add: lt_def) apply (cases t) apply auto apply (rename_tac poly, case_tac poly) apply auto done lemma le_nb: "tmbound0 t ⟹ bound0 (le t)" apply (simp add: le_def) apply (cases t) apply auto apply (rename_tac poly, case_tac poly) apply auto done lemma eq_nb: "tmbound0 t ⟹ bound0 (eq t)" apply (simp add: eq_def) apply (cases t) apply auto apply (rename_tac poly, case_tac poly) apply auto done lemma neq_nb: "tmbound0 t ⟹ bound0 (neq t)" apply (simp add: neq_def eq_def) apply (cases t) apply auto apply (rename_tac poly, case_tac poly) apply auto done lemma simplt_nb[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "tmbound0 t ⟹ bound0 (simplt t)" proof (simp add: simplt_def Let_def split_def) assume nb: "tmbound0 t" then have nb': "tmbound0 (simptm t)" by simp let ?c = "fst (split0 (simptm t))" from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] have th: "∀bs. Ipoly bs ?c = Ipoly bs 0⇩_{p}" by auto from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] have ths: "isnpolyh ?c 0" "isnpolyh 0⇩_{p}0" by (simp_all add: isnpoly_def) from iffD1[OF isnpolyh_unique[OF ths] th] have "fst (split0 (simptm t)) = 0⇩_{p}" . then show "(fst (split0 (simptm t)) = 0⇩_{p}⟶ bound0 (lt (snd (split0 (simptm t))))) ∧ fst (split0 (simptm t)) = 0⇩_{p}" by (simp add: simplt_def Let_def split_def lt_nb) qed lemma simple_nb[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "tmbound0 t ⟹ bound0 (simple t)" proof(simp add: simple_def Let_def split_def) assume nb: "tmbound0 t" then have nb': "tmbound0 (simptm t)" by simp let ?c = "fst (split0 (simptm t))" from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] have th: "∀bs. Ipoly bs ?c = Ipoly bs 0⇩_{p}" by auto from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] have ths: "isnpolyh ?c 0" "isnpolyh 0⇩_{p}0" by (simp_all add: isnpoly_def) from iffD1[OF isnpolyh_unique[OF ths] th] have "fst (split0 (simptm t)) = 0⇩_{p}" . then show "(fst (split0 (simptm t)) = 0⇩_{p}⟶ bound0 (le (snd (split0 (simptm t))))) ∧ fst (split0 (simptm t)) = 0⇩_{p}" by (simp add: simplt_def Let_def split_def le_nb) qed lemma simpeq_nb[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "tmbound0 t ⟹ bound0 (simpeq t)" proof (simp add: simpeq_def Let_def split_def) assume nb: "tmbound0 t" then have nb': "tmbound0 (simptm t)" by simp let ?c = "fst (split0 (simptm t))" from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] have th: "∀bs. Ipoly bs ?c = Ipoly bs 0⇩_{p}" by auto from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] have ths: "isnpolyh ?c 0" "isnpolyh 0⇩_{p}0" by (simp_all add: isnpoly_def) from iffD1[OF isnpolyh_unique[OF ths] th] have "fst (split0 (simptm t)) = 0⇩_{p}" . then show "(fst (split0 (simptm t)) = 0⇩_{p}⟶ bound0 (eq (snd (split0 (simptm t))))) ∧ fst (split0 (simptm t)) = 0⇩_{p}" by (simp add: simpeq_def Let_def split_def eq_nb) qed lemma simpneq_nb[simp]: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "tmbound0 t ⟹ bound0 (simpneq t)" proof (simp add: simpneq_def Let_def split_def) assume nb: "tmbound0 t" then have nb': "tmbound0 (simptm t)" by simp let ?c = "fst (split0 (simptm t))" from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]] have th: "∀bs. Ipoly bs ?c = Ipoly bs 0⇩_{p}" by auto from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]] have ths: "isnpolyh ?c 0" "isnpolyh 0⇩_{p}0" by (simp_all add: isnpoly_def) from iffD1[OF isnpolyh_unique[OF ths] th] have "fst (split0 (simptm t)) = 0⇩_{p}" . then show "(fst (split0 (simptm t)) = 0⇩_{p}⟶ bound0 (neq (snd (split0 (simptm t))))) ∧ fst (split0 (simptm t)) = 0⇩_{p}" by (simp add: simpneq_def Let_def split_def neq_nb) qed fun conjs :: "fm ⇒ fm list" where "conjs (And p q) = conjs p @ conjs q" | "conjs T = []" | "conjs p = [p]" lemma conjs_ci: "(∀q ∈ set (conjs p). Ifm vs bs q) = Ifm vs bs p" by (induct p rule: conjs.induct) auto definition list_disj :: "fm list ⇒ fm" where "list_disj ps ≡ foldr disj ps F" lemma list_conj: "Ifm vs bs (list_conj ps) = (∀p∈ set ps. Ifm vs bs p)" by (induct ps) (auto simp add: list_conj_def) lemma list_conj_qf: " ∀p∈ set ps. qfree p ⟹ qfree (list_conj ps)" by (induct ps) (auto simp add: list_conj_def) lemma list_disj: "Ifm vs bs (list_disj ps) = (∃p∈ set ps. Ifm vs bs p)" by (induct ps) (auto simp add: list_disj_def) lemma conj_boundslt: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (conj p q)" unfolding conj_def by auto lemma conjs_nb: "bound n p ⟹ ∀q∈ set (conjs p). bound n q" apply (induct p rule: conjs.induct) apply (unfold conjs.simps) apply (unfold set_append) apply (unfold ball_Un) apply (unfold bound.simps) apply auto done lemma conjs_boundslt: "boundslt n p ⟹ ∀q∈ set (conjs p). boundslt n q" apply (induct p rule: conjs.induct) apply (unfold conjs.simps) apply (unfold set_append) apply (unfold ball_Un) apply (unfold boundslt.simps) apply blast apply simp_all done lemma list_conj_boundslt: " ∀p∈ set ps. boundslt n p ⟹ boundslt n (list_conj ps)" unfolding list_conj_def by (induct ps) auto lemma list_conj_nb: assumes bnd: "∀p∈ set ps. bound n p" shows "bound n (list_conj ps)" using bnd unfolding list_conj_def by (induct ps) auto lemma list_conj_nb': "∀p∈set ps. bound0 p ⟹ bound0 (list_conj ps)" unfolding list_conj_def by (induct ps) auto lemma CJNB_qe: assumes qe: "∀bs p. qfree p ⟶ qfree (qe p) ∧ (Ifm vs bs (qe p) = Ifm vs bs (E p))" shows "∀bs p. qfree p ⟶ qfree (CJNB qe p) ∧ (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))" proof clarify fix bs p assume qfp: "qfree p" let ?cjs = "conjuncts p" let ?yes = "fst (partition bound0 ?cjs)" let ?no = "snd (partition bound0 ?cjs)" let ?cno = "list_conj ?no" let ?cyes = "list_conj ?yes" have part: "partition bound0 ?cjs = (?yes,?no)" by simp from partition_P[OF part] have "∀q∈ set ?yes. bound0 q" by blast then have yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') then have yes_qf: "qfree (decr0 ?cyes)" by (simp add: decr0_qf) from conjuncts_qf[OF qfp] partition_set[OF part] have " ∀q∈ set ?no. qfree q" by auto then have no_qf: "qfree ?cno" by (simp add: list_conj_qf) with qe have cno_qf:"qfree (qe ?cno)" and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+ from cno_qf yes_qf have qf: "qfree (CJNB qe p)" by (simp add: CJNB_def Let_def split_def) have "Ifm vs bs p = ((Ifm vs bs ?cyes) ∧ (Ifm vs bs ?cno))" for bs proof - from conjuncts have "Ifm vs bs p = (∀q∈ set ?cjs. Ifm vs bs q)" by blast also have "… = ((∀q∈ set ?yes. Ifm vs bs q) ∧ (∀q∈ set ?no. Ifm vs bs q))" using partition_set[OF part] by auto finally show ?thesis using list_conj[of vs bs] by simp qed then have "Ifm vs bs (E p) = (∃x. (Ifm vs (x#bs) ?cyes) ∧ (Ifm vs (x#bs) ?cno))" by simp also fix y have "… = (∃x. (Ifm vs (y#bs) ?cyes) ∧ (Ifm vs (x#bs) ?cno))" using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast also have "… = (Ifm vs bs (decr0 ?cyes) ∧ Ifm vs bs (E ?cno))" by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv) also have "… = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))" using qe[rule_format, OF no_qf] by auto finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" by (simp add: Let_def CJNB_def split_def) with qf show "qfree (CJNB qe p) ∧ Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast qed consts simpfm :: "fm ⇒ fm" recdef simpfm "measure fmsize" "simpfm (Lt t) = simplt (simptm t)" "simpfm (Le t) = simple (simptm t)" "simpfm (Eq t) = simpeq(simptm t)" "simpfm (NEq t) = simpneq(simptm t)" "simpfm (And p q) = conj (simpfm p) (simpfm q)" "simpfm (Or p q) = disj (simpfm p) (simpfm q)" "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)" "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))" "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))" "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))" "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))" "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))" "simpfm (NOT (Eq t)) = simpneq t" "simpfm (NOT (NEq t)) = simpeq t" "simpfm (NOT (Le t)) = simplt (Neg t)" "simpfm (NOT (Lt t)) = simple (Neg t)" "simpfm (NOT (NOT p)) = simpfm p" "simpfm (NOT T) = F" "simpfm (NOT F) = T" "simpfm p = p" lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p" by (induct p arbitrary: bs rule: simpfm.induct) auto lemma simpfm_bound0: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "bound0 p ⟹ bound0 (simpfm p)" by (induct p rule: simpfm.induct) auto lemma lt_qf[simp]: "qfree (lt t)" apply (cases t) apply (auto simp add: lt_def) apply (rename_tac poly, case_tac poly) apply auto done lemma le_qf[simp]: "qfree (le t)" apply (cases t) apply (auto simp add: le_def) apply (rename_tac poly, case_tac poly) apply auto done lemma eq_qf[simp]: "qfree (eq t)" apply (cases t) apply (auto simp add: eq_def) apply (rename_tac poly, case_tac poly) apply auto done lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def) lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def) lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def) lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def) lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def) lemma simpfm_qf[simp]: "qfree p ⟹ qfree (simpfm p)" by (induct p rule: simpfm.induct) auto lemma disj_lin: "islin p ⟹ islin q ⟹ islin (disj p q)" by (simp add: disj_def) lemma conj_lin: "islin p ⟹ islin q ⟹ islin (conj p q)" by (simp add: conj_def) lemma assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "qfree p ⟹ islin (simpfm p)" by (induct p rule: simpfm.induct) (simp_all add: conj_lin disj_lin) consts prep :: "fm ⇒ fm" recdef prep "measure fmsize" "prep (E T) = T" "prep (E F) = F" "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))" "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))" "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))" "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" "prep (E p) = E (prep p)" "prep (A (And p q)) = conj (prep (A p)) (prep (A q))" "prep (A p) = prep (NOT (E (NOT p)))" "prep (NOT (NOT p)) = prep p" "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))" "prep (NOT (A p)) = prep (E (NOT p))" "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))" "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))" "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))" "prep (NOT p) = not (prep p)" "prep (Or p q) = disj (prep p) (prep q)" "prep (And p q) = conj (prep p) (prep q)" "prep (Imp p q) = prep (Or (NOT p) q)" "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))" "prep p = p" (hints simp add: fmsize_pos) lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p" by (induct p arbitrary: bs rule: prep.induct) auto text ‹Generic quantifier elimination.› function (sequential) qelim :: "fm ⇒ (fm ⇒ fm) ⇒ fm" where "qelim (E p) = (λqe. DJ (CJNB qe) (qelim p qe))" | "qelim (A p) = (λqe. not (qe ((qelim (NOT p) qe))))" | "qelim (NOT p) = (λqe. not (qelim p qe))" | "qelim (And p q) = (λqe. conj (qelim p qe) (qelim q qe))" | "qelim (Or p q) = (λqe. disj (qelim p qe) (qelim q qe))" | "qelim (Imp p q) = (λqe. imp (qelim p qe) (qelim q qe))" | "qelim (Iff p q) = (λqe. iff (qelim p qe) (qelim q qe))" | "qelim p = (λy. simpfm p)" by pat_completeness simp_all termination by (relation "measure fmsize") auto lemma qelim: assumes qe_inv: "∀bs p. qfree p ⟶ qfree (qe p) ∧ (Ifm vs bs (qe p) = Ifm vs bs (E p))" shows "⋀ bs. qfree (qelim p qe) ∧ (Ifm vs bs (qelim p qe) = Ifm vs bs p)" using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] by (induct p rule: qelim.induct) auto subsection ‹Core Procedure› fun minusinf:: "fm ⇒ fm" ― ‹Virtual substitution of -∞› where "minusinf (And p q) = conj (minusinf p) (minusinf q)" | "minusinf (Or p q) = disj (minusinf p) (minusinf q)" | "minusinf (Eq (CNP 0 c e)) = conj (eq (CP c)) (eq e)" | "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))" | "minusinf (Lt (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~⇩_{p}c)))" | "minusinf (Le (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~⇩_{p}c)))" | "minusinf p = p" fun plusinf:: "fm ⇒ fm" ― ‹Virtual substitution of +∞› where "plusinf (And p q) = conj (plusinf p) (plusinf q)" | "plusinf (Or p q) = disj (plusinf p) (plusinf q)" | "plusinf (Eq (CNP 0 c e)) = conj (eq (CP c)) (eq e)" | "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))" | "plusinf (Lt (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))" | "plusinf (Le (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))" | "plusinf p = p" lemma minusinf_inf: assumes lp: "islin p" shows "∃z. ∀x < z. Ifm vs (x#bs) (minusinf p) ⟷ Ifm vs (x#bs) p" using lp proof (induct p rule: minusinf.induct) case 1 then show ?case apply auto apply (rule_tac x="min z za" in exI) apply auto done next case 2 then show ?case apply auto apply (rule_tac x="min z za" in exI) apply auto done next case (3 c e) then have nbe: "tmbound0 e" by simp from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto next case c: 2 have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x < - ?e" using pos_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x > - ?e" using neg_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto qed then show ?thesis by auto qed next case (4 c e) then have nbe: "tmbound0 e" by simp from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x < - ?e" using pos_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x > - ?e" using neg_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto qed then show ?thesis by auto qed next case (5 c e) then have nbe: "tmbound0 e" by simp from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all then have nc': "allpolys isnpoly (CP (~⇩_{p}c))" by (simp add: polyneg_norm) note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x < - ?e" using pos_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x > - ?e" using neg_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] c by auto qed then show ?thesis by auto qed next case (6 c e) then have nbe: "tmbound0 e" by simp from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all then have nc': "allpolys isnpoly (CP (~⇩_{p}c))" by (simp add: polyneg_norm) note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x < - ?e" using pos_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))" if "x < -?e / ?c" for x proof - from that have "?c * x > - ?e" using neg_less_divide_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto qed qed auto lemma plusinf_inf: assumes lp: "islin p" shows "∃z. ∀x > z. Ifm vs (x#bs) (plusinf p) ⟷ Ifm vs (x#bs) p" using lp proof (induct p rule: plusinf.induct) case 1 then show ?case apply auto apply (rule_tac x="max z za" in exI) apply auto done next case 2 then show ?case apply auto apply (rule_tac x="max z za" in exI) apply auto done next case (3 c e) then have nbe: "tmbound0 e" by simp from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto next case c: 2 have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x > - ?e" using pos_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x < - ?e" using neg_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto qed then show ?thesis by auto qed next case (4 c e) then have nbe: "tmbound0 e" by simp from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x > - ?e" using pos_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x < - ?e" using neg_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto qed then show ?thesis by auto qed next case (5 c e) then have nbe: "tmbound0 e" by simp from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all then have nc': "allpolys isnpoly (CP (~⇩_{p}c))" by (simp add: polyneg_norm) note eqs = lt[OF nc(1), where ?'a = 'a] lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x > - ?e" using pos_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x < - ?e" using neg_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] c by auto qed then show ?thesis by auto qed next case (6 c e) then have nbe: "tmbound0 e" by simp from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all then have nc': "allpolys isnpoly (CP (~⇩_{p}c))" by (simp add: polyneg_norm) note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a] let ?c = "Ipoly vs c" fix y let ?e = "Itm vs (y#bs) e" consider "?c = 0" | "?c > 0" | "?c < 0" by arith then show ?case proof cases case 1 then show ?thesis using eqs by auto next case c: 2 have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x > - ?e" using pos_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e > 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto next case c: 3 have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))" if "x > -?e / ?c" for x proof - from that have "?c * x < - ?e" using neg_divide_less_eq[OF c, where a="x" and b="-?e"] by (simp add: mult.commute) then have "?c * x + ?e < 0" by simp then show ?thesis using tmbound0_I[OF nbe, where b="y" and b'="x"] c eqs by auto qed then show ?thesis by auto qed qed auto lemma minusinf_nb: "islin p ⟹ bound0 (minusinf p)" by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb) lemma plusinf_nb: "islin p ⟹ bound0 (plusinf p)" by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb) lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)" shows "∃x. Ifm vs (x#bs) p" proof - from bound0_I [OF minusinf_nb[OF lp], where bs ="bs"] ex have th: "∀x. Ifm vs (x#bs) (minusinf p)" by auto from minusinf_inf[OF lp, where bs="bs"] obtain z where z: "∀x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp moreover have "z - 1 < z" by simp ultimately show ?thesis using z by auto qed lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)" shows "∃x. Ifm vs (x#bs) p" proof - from bound0_I [OF plusinf_nb[OF lp], where bs ="bs"] ex have th: "∀x. Ifm vs (x#bs) (plusinf p)" by auto from plusinf_inf[OF lp, where bs="bs"] obtain z where z: "∀x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp moreover have "z + 1 > z" by simp ultimately show ?thesis using z by auto qed fun uset :: "fm ⇒ (poly × tm) list" where "uset (And p q) = uset p @ uset q" | "uset (Or p q) = uset p @ uset q" | "uset (Eq (CNP 0 a e)) = [(a, e)]" | "uset (Le (CNP 0 a e)) = [(a, e)]" | "uset (Lt (CNP 0 a e)) = [(a, e)]" | "uset (NEq (CNP 0 a e)) = [(a, e)]" | "uset p = []" lemma uset_l: assumes lp: "islin p" shows "∀(c,s) ∈ set (uset p). isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s" using lp by (induct p rule: uset.induct) auto lemma minusinf_uset0: assumes lp: "islin p" and nmi: "¬ (Ifm vs (x#bs) (minusinf p))" and ex: "Ifm vs (x#bs) p" (is "?I x p") shows "∃(c, s) ∈ set (uset p). x ≥ - Itm vs (x#bs) s / Ipoly vs c" proof - have "∃(c, s) ∈ set (uset p). Ipoly vs c < 0 ∧ Ipoly vs c * x ≤ - Itm vs (x#bs) s ∨ Ipoly vs c > 0 ∧ Ipoly vs c * x ≥ - Itm vs (x#bs) s" using lp nmi ex apply (induct p rule: minusinf.induct) apply (auto simp add: eq le lt polyneg_norm) apply (auto simp add: linorder_not_less order_le_less) done then obtain c s where csU: "(c, s) ∈ set (uset p)" and x: "(Ipoly vs c < 0 ∧ Ipoly vs c * x ≤ - Itm vs (x#bs) s) ∨ (Ipoly vs c > 0 ∧ Ipoly vs c * x ≥ - Itm vs (x#bs) s)" by blast then have "x ≥ (- Itm vs (x#bs) s) / Ipoly vs c" using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x] by (auto simp add: mult.commute) then show ?thesis using csU by auto qed lemma minusinf_uset: assumes lp: "islin p" and nmi: "¬ (Ifm vs (a#bs) (minusinf p))" and ex: "Ifm vs (x#bs) p" (is "?I x p") shows "∃(c,s) ∈ set (uset p). x ≥ - Itm vs (a#bs) s / Ipoly vs c" proof - from nmi have nmi': "¬ Ifm vs (x#bs) (minusinf p)" by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a]) from minusinf_uset0[OF lp nmi' ex] obtain c s where csU: "(c,s) ∈ set (uset p)" and th: "x ≥ - Itm vs (x#bs) s / Ipoly vs c" by blast from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto qed lemma plusinf_uset0: assumes lp: "islin p" and nmi: "¬ (Ifm vs (x#bs) (plusinf p))" and ex: "Ifm vs (x#bs) p" (is "?I x p") shows "∃(c, s) ∈ set (uset p). x ≤ - Itm vs (x#bs) s / Ipoly vs c" proof - have "∃(c, s) ∈ set (uset p). Ipoly vs c < 0 ∧ Ipoly vs c * x ≥ - Itm vs (x#bs) s ∨ Ipoly vs c > 0 ∧ Ipoly vs c * x ≤ - Itm vs (x#bs) s" using lp nmi ex apply (induct p rule: minusinf.induct) apply (auto simp add: eq le lt polyneg_norm) apply (auto simp add: linorder_not_less order_le_less) done then obtain c s where csU: "(c, s) ∈ set (uset p)" and x: "Ipoly vs c < 0 ∧ Ipoly vs c * x ≥ - Itm vs (x#bs) s ∨ Ipoly vs c > 0 ∧ Ipoly vs c * x ≤ - Itm vs (x#bs) s" by blast then have "x ≤ (- Itm vs (x#bs) s) / Ipoly vs c" using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"] by (auto simp add: mult.commute) then show ?thesis using csU by auto qed lemma plusinf_uset: assumes lp: "islin p" and nmi: "¬ (Ifm vs (a#bs) (plusinf p))" and ex: "Ifm vs (x#bs) p" (is "?I x p") shows "∃(c,s) ∈ set (uset p). x ≤ - Itm vs (a#bs) s / Ipoly vs c" proof - from nmi have nmi': "¬ (Ifm vs (x#bs) (plusinf p))" by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a]) from plusinf_uset0[OF lp nmi' ex] obtain c s where csU: "(c,s) ∈ set (uset p)" and th: "x ≤ - Itm vs (x#bs) s / Ipoly vs c" by blast from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto qed lemma lin_dense: assumes lp: "islin p" and noS: "∀t. l < t ∧ t< u ⟶ t ∉ (λ(c,t). - Itm vs (x#bs) t / Ipoly vs c) ` set (uset p)" (is "∀t. _ ∧ _ ⟶ t ∉ (λ(c,t). - ?Nt x t / ?N c) ` ?U p") and lx: "l < x" and xu: "x < u" and px: "Ifm vs (x # bs) p" and ly: "l < y" and yu: "y < u" shows "Ifm vs (y#bs) p" using lp px noS proof (induct p rule: islin.induct) case (5 c s) from "5.prems" have lin: "isnpoly c" "c ≠ 0⇩_{p}" "tmbound0 s" "allpolys isnpoly s" and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))" and noS: "∀t. l < t ∧ t < u ⟶ t ≠ - Itm vs (x # bs) s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp_all from ly yu noS have yne: "y ≠ - ?Nt x s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp then have ycs: "y < - ?Nt x s / ?N c ∨ y > -?Nt x s / ?N c" by auto consider "?N c = 0" | "?N c > 0" | "?N c < 0" by arith then show ?case proof cases case 1 then show ?thesis using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) next case N: 2 from px pos_less_divide_eq[OF N, where a="x" and b="-?Nt x s"] have px': "x < - ?Nt x s / ?N c" by (auto simp add: not_less field_simps) from ycs show ?thesis proof assume y: "y < - ?Nt x s / ?N c" then have "y * ?N c < - ?Nt x s" by (simp add: pos_less_divide_eq[OF N, where a="y" and b="-?Nt x s", symmetric]) then have "?N c * y + ?Nt x s < 0" by (simp add: field_simps) then show ?thesis using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp next assume y: "y > -?Nt x s / ?N c" with yu have eu: "u > - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≤ l" by (cases "- ?Nt x s / ?N c > l") auto with lx px' have False by simp then show ?thesis .. qed next case N: 3 from px neg_divide_less_eq[OF N, where a="x" and b="-?Nt x s"] have px': "x > - ?Nt x s / ?N c" by (auto simp add: not_less field_simps) from ycs show ?thesis proof assume y: "y > - ?Nt x s / ?N c" then have "y * ?N c < - ?Nt x s" by (simp add: neg_divide_less_eq[OF N, where a="y" and b="-?Nt x s", symmetric]) then have "?N c * y + ?Nt x s < 0" by (simp add: field_simps) then show ?thesis using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp next assume y: "y < -?Nt x s / ?N c" with ly have eu: "l < - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≥ u" by (cases "- ?Nt x s / ?N c < u") auto with xu px' have False by simp then show ?thesis .. qed qed next case (6 c s) from "6.prems" have lin: "isnpoly c" "c ≠ 0⇩_{p}" "tmbound0 s" "allpolys isnpoly s" and px: "Ifm vs (x # bs) (Le (CNP 0 c s))" and noS: "∀t. l < t ∧ t < u ⟶ t ≠ - Itm vs (x # bs) s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp_all from ly yu noS have yne: "y ≠ - ?Nt x s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp then have ycs: "y < - ?Nt x s / ?N c ∨ y > -?Nt x s / ?N c" by auto have ccs: "?N c = 0 ∨ ?N c < 0 ∨ ?N c > 0" by dlo consider "?N c = 0" | "?N c > 0" | "?N c < 0" by arith then show ?case proof cases case 1 then show ?thesis using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) next case N: 2 from px pos_le_divide_eq[OF N, where a="x" and b="-?Nt x s"] have px': "x ≤ - ?Nt x s / ?N c" by (simp add: not_less field_simps) from ycs show ?thesis proof assume y: "y < - ?Nt x s / ?N c" then have "y * ?N c < - ?Nt x s" by (simp add: pos_less_divide_eq[OF N, where a="y" and b="-?Nt x s", symmetric]) then have "?N c * y + ?Nt x s < 0" by (simp add: field_simps) then show ?thesis using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp next assume y: "y > -?Nt x s / ?N c" with yu have eu: "u > - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≤ l" by (cases "- ?Nt x s / ?N c > l") auto with lx px' have False by simp then show ?thesis .. qed next case N: 3 from px neg_divide_le_eq[OF N, where a="x" and b="-?Nt x s"] have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps) from ycs show ?thesis proof assume y: "y > - ?Nt x s / ?N c" then have "y * ?N c < - ?Nt x s" by (simp add: neg_divide_less_eq[OF N, where a="y" and b="-?Nt x s", symmetric]) then have "?N c * y + ?Nt x s < 0" by (simp add: field_simps) then show ?thesis using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp next assume y: "y < -?Nt x s / ?N c" with ly have eu: "l < - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≥ u" by (cases "- ?Nt x s / ?N c < u") auto with xu px' have False by simp then show ?thesis .. qed qed next case (3 c s) from "3.prems" have lin: "isnpoly c" "c ≠ 0⇩_{p}" "tmbound0 s" "allpolys isnpoly s" and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))" and noS: "∀t. l < t ∧ t < u ⟶ t ≠ - Itm vs (x # bs) s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp_all from ly yu noS have yne: "y ≠ - ?Nt x s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp then have ycs: "y < - ?Nt x s / ?N c ∨ y > -?Nt x s / ?N c" by auto consider "?N c = 0" | "?N c < 0" | "?N c > 0" by arith then show ?case proof cases case 1 then show ?thesis using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) next case 2 then have cnz: "?N c ≠ 0" by simp from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps) from ycs show ?thesis proof assume y: "y < -?Nt x s / ?N c" with ly have eu: "l < - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≥ u" by (cases "- ?Nt x s / ?N c < u") auto with xu px' have False by simp then show ?thesis .. next assume y: "y > -?Nt x s / ?N c" with yu have eu: "u > - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≤ l" by (cases "- ?Nt x s / ?N c > l") auto with lx px' have False by simp then show ?thesis .. qed next case 3 then have cnz: "?N c ≠ 0" by simp from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps) from ycs show ?thesis proof assume y: "y < -?Nt x s / ?N c" with ly have eu: "l < - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≥ u" by (cases "- ?Nt x s / ?N c < u") auto with xu px' have False by simp then show ?thesis .. next assume y: "y > -?Nt x s / ?N c" with yu have eu: "u > - ?Nt x s / ?N c" by auto with noS ly yu have th: "- ?Nt x s / ?N c ≤ l" by (cases "- ?Nt x s / ?N c > l") auto with lx px' have False by simp then show ?thesis .. qed qed next case (4 c s) from "4.prems" have lin: "isnpoly c" "c ≠ 0⇩_{p}" "tmbound0 s" "allpolys isnpoly s" and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))" and noS: "∀t. l < t ∧ t < u ⟶ t ≠ - Itm vs (x # bs) s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp_all from ly yu noS have yne: "y ≠ - ?Nt x s / ⦇c⦈⇩_{p}⇗^{vs⇖}" by simp then have ycs: "y < - ?Nt x s / ?N c ∨ y > -?Nt x s / ?N c" by auto show ?case proof (cases "?N c = 0") case True then show ?thesis using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]) next case False with yne eq_divide_eq[of "y" "- ?Nt x s" "?N c"] show ?thesis by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) qed qed (auto simp add: tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"]) lemma inf_uset: assumes lp: "islin p" and nmi: "¬ (Ifm vs (x#bs) (minusinf p))" (is "¬ (Ifm vs (x#bs) (?M p))") and npi: "¬ (Ifm vs (x#bs) (plusinf p))" (is "¬ (Ifm vs (x#bs) (?P p))") and ex: "∃x. Ifm vs (x#bs) p" (is "∃x. ?I x p") shows "∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). ?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2) p" proof - let ?Nt = "λx t. Itm vs (x#bs) t" let ?N = "Ipoly vs" let ?U = "set (uset p)" from ex obtain a where pa: "?I a p" by blast from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi have nmi': "¬ (?I a (?M p))" by simp from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi have npi': "¬ (?I a (?P p))" by simp have "∃(c,t) ∈ set (uset p). ∃(d,s) ∈ set (uset p). ?I ((- ?Nt a t/?N c + - ?Nt a s /?N d) / 2) p" proof - let ?M = "(λ(c,t). - ?Nt a t / ?N c) ` ?U" have fM: "finite ?M" by auto from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] have "∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). a ≤ - ?Nt x t / ?N c ∧ a ≥ - ?Nt x s / ?N d" by blast then obtain "c" "t" "d" "s" where ctU: "(c,t) ∈ ?U" and dsU: "(d,s) ∈ ?U" and xs1: "a ≤ - ?Nt x s / ?N d" and tx1: "a ≥ - ?Nt x t / ?N c" by blast from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a ≤ - ?Nt a s / ?N d" and tx: "a ≥ - ?Nt a t / ?N c" by auto from ctU have Mne: "?M ≠ {}" by auto then have Une: "?U ≠ {}" by simp let ?l = "Min ?M" let ?u = "Max ?M" have linM: "?l ∈ ?M" using fM Mne by simp have uinM: "?u ∈ ?M" using fM Mne by simp have ctM: "- ?Nt a t / ?N c ∈ ?M" using ctU by auto have dsM: "- ?Nt a s / ?N d ∈ ?M" using dsU by auto have lM: "∀t∈ ?M. ?l ≤ t" using Mne fM by auto have Mu: "∀t∈ ?M. t ≤ ?u" using Mne fM by auto have "?l ≤ - ?Nt a t / ?N c" using ctM Mne by simp then have lx: "?l ≤ a" using tx by simp have "- ?Nt a s / ?N d ≤ ?u" using dsM Mne by simp then have xu: "a ≤ ?u" using xs by simp from finite_set_intervals2[where P="λx. ?I x p",OF pa lx xu linM uinM fM lM Mu] consider u where "u ∈ ?M" "?I u p" | t1 t2 where "t1 ∈ ?M" "t2∈ ?M" "∀y. t1 < y ∧ y < t2 ⟶ y ∉ ?M" "t1 < a" "a < t2" "?I a p" by blast then show ?thesis proof cases case 1 then have "∃(nu,tu) ∈ ?U. u = - ?Nt a tu / ?N nu" by auto then obtain tu nu where tuU: "(nu, tu) ∈ ?U" and tuu: "u = - ?Nt a tu / ?N nu" by blast have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / 2) p" using ‹?I u p› tuu by simp with tuU show ?thesis by blast next case 2 have "∃(t1n, t1u) ∈ ?U. t1 = - ?Nt a t1u / ?N t1n" using ‹t1 ∈ ?M› by auto then obtain t1u t1n where t1uU: "(t1n, t1u) ∈ ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast have "∃(t2n, t2u) ∈ ?U. t2 = - ?Nt a t2u / ?N t2n" using ‹t2 ∈ ?M› by auto then obtain t2u t2n where t2uU: "(t2n, t2u) ∈ ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast have "t1 < t2" using ‹t1 < a› ‹a < t2› by simp let ?u = "(t1 + t2) / 2" have "t1 < ?u" using less_half_sum [OF ‹t1 < t2›] by auto have "?u < t2" using gt_half_sum [OF ‹t1 < t2›] by auto have "?I ?u p" using lp ‹∀y. t1 < y ∧ y < t2 ⟶ y ∉ ?M› ‹t1 < a› ‹a < t2› ‹?I a p› ‹t1 < ?u› ‹?u < t2› by (rule lin_dense) with t1uU t2uU t1u t2u show ?thesis by blast qed qed then obtain l n s m where lnU: "(n, l) ∈ ?U" and smU:"(m,s) ∈ ?U" and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / 2) p" by blast from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / 2) p" by simp with lnU smU show ?thesis by auto qed section ‹The Ferrante - Rackoff Theorem› theorem fr_eq: assumes lp: "islin p" shows "(∃x. Ifm vs (x#bs) p) ⟷ (Ifm vs (x#bs) (minusinf p) ∨ Ifm vs (x#bs) (plusinf p) ∨ (∃(n, t) ∈ set (uset p). ∃(m, s) ∈ set (uset p). Ifm vs (((- Itm vs (x#bs) t / Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) / 2)#bs) p))" (is "(∃x. ?I x p) ⟷ ?M ∨ ?P ∨ ?F" is "?E ⟷ ?D") proof show ?D if ?E proof - consider "?M ∨ ?P" | "¬ ?M" "¬ ?P" by blast then show ?thesis proof cases case 1 then show ?thesis by blast next case 2 from inf_uset[OF lp this] have ?F using ‹?E› by blast then show ?thesis by blast qed qed show ?E if ?D proof - from that consider ?M | ?P | ?F by blast then show ?thesis proof cases case 1 from minusinf_ex[OF lp this] show ?thesis . next case 2 from plusinf_ex[OF lp this] show ?thesis . next case 3 then show ?thesis by blast qed qed qed section ‹First implementation : Naive by encoding all case splits locally› definition "msubsteq c t d s a r = evaldjf (case_prod conj) [(let cd = c *⇩_{p}d in (NEq (CP cd), Eq (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (Eq (CP c)) (Eq (CP d)), Eq r)]" lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s" shows "bound0 (msubsteq c t d s a r)" proof - have th: "∀x ∈ set [(let cd = c *⇩_{p}d in (NEq (CP cd), Eq (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (Eq (CP c)) (Eq (CP d)), Eq r)]. bound0 (case_prod conj x)" using lp by (simp add: Let_def t s) from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def) qed lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))" shows "Ifm vs (x#bs) (msubsteq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2)#bs) (Eq (CNP 0 a r))" (is "?lhs = ?rhs") proof - let ?Nt = "λx t. Itm vs (x#bs) t" let ?N = "λp. Ipoly vs p" let ?c = "?N c" let ?d = "?N d" let ?t = "?Nt x t" let ?s = "?Nt x s" let ?a = "?N a" let ?r = "?Nt x r" from lp have lin:"isnpoly a" "a ≠ 0⇩_{p}" "tmbound0 r" "allpolys isnpoly r" by simp_all note r = tmbound0_I[OF lin(3), of vs _ bs x] consider "?c = 0" "?d = 0" | "?c = 0" "?d ≠ 0" | "?c ≠ 0" "?d = 0" | "?c ≠ 0" "?d ≠ 0" by blast then show ?thesis proof cases case 1 then show ?thesis by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex) next case cd: 2 then have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" by simp have "?rhs = Ifm vs (-?s / (2*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (-?s / (2*?d)) + ?r = 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * ⦇d⦈⇩_{p}⇗^{vs⇖}))"]) also have "… ⟷ 2 * ?d * (?a * (-?s / (2*?d)) + ?r) = 0" using cd(2) mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp also have "… ⟷ (- ?a * ?s) * (2*?d / (2*?d)) + 2 * ?d * ?r= 0" by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left) also have "… ⟷ - (?a * ?s) + 2*?d*?r = 0" using cd(2) by simp finally show ?thesis using cd by (simp add: r[of "- (Itm vs (x # bs) s / (2 * ⦇d⦈⇩_{p}⇗^{vs⇖}))"] msubsteq_def Let_def evaldjf_ex) next case cd: 3 from cd(2) have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2 * ?c)" by simp have "?rhs = Ifm vs (-?t / (2*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (-?t / (2*?c)) + ?r = 0" by (simp add: r[of "- (?t/ (2 * ?c))"]) also have "… ⟷ 2 * ?c * (?a * (-?t / (2 * ?c)) + ?r) = 0" using cd(1) mult_cancel_left[of "2 * ?c" "(?a * (-?t / (2 * ?c)) + ?r)" 0] by simp also have "… ⟷ (?a * -?t)* (2 * ?c) / (2 * ?c) + 2 * ?c * ?r= 0" by (simp add: field_simps distrib_left[of "2 * ?c"] del: distrib_left) also have "… ⟷ - (?a * ?t) + 2 * ?c * ?r = 0" using cd(1) by simp finally show ?thesis using cd by (simp add: r[of "- (?t/ (2 * ?c))"] msubsteq_def Let_def evaldjf_ex) next case cd: 4 then have cd2: "?c * ?d * 2 ≠ 0" by simp from add_frac_eq[OF cd, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r = 0" by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) = 0" using cd mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2 * ?c * ?d)) + ?r" 0] by simp also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )) + 2 * ?c * ?d * ?r = 0" using nonzero_mult_div_cancel_left [OF cd2] cd by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps) qed qed definition "msubstneq c t d s a r = evaldjf (case_prod conj) [(let cd = c *⇩_{p}d in (NEq (CP cd), NEq (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (Eq (CP c)) (Eq (CP d)), NEq r)]" lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s" shows "bound0 (msubstneq c t d s a r)" proof - have th: "∀x∈ set [(let cd = c *⇩_{p}d in (NEq (CP cd), NEq (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (Eq (CP c)) (Eq (CP d)), NEq r)]. bound0 (case_prod conj x)" using lp by (simp add: Let_def t s) from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def) qed lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))" shows "Ifm vs (x#bs) (msubstneq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (NEq (CNP 0 a r))" (is "?lhs = ?rhs") proof - let ?Nt = "λx t. Itm vs (x#bs) t" let ?N = "λp. Ipoly vs p" let ?c = "?N c" let ?d = "?N d" let ?t = "?Nt x t" let ?s = "?Nt x s" let ?a = "?N a" let ?r = "?Nt x r" from lp have lin:"isnpoly a" "a ≠ 0⇩_{p}" "tmbound0 r" "allpolys isnpoly r" by simp_all note r = tmbound0_I[OF lin(3), of vs _ bs x] consider "?c = 0" "?d = 0" | "?c = 0" "?d ≠ 0" | "?c ≠ 0" "?d = 0" | "?c ≠ 0" "?d ≠ 0" by blast then show ?thesis proof cases case 1 then show ?thesis by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex) next case cd: 2 from cd(1) have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2 * ?d)" by simp have "?rhs = Ifm vs (-?s / (2*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (-?s / (2*?d)) + ?r ≠ 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * ⦇d⦈⇩_{p}⇗^{vs⇖}))"]) also have "… ⟷ 2*?d * (?a * (-?s / (2*?d)) + ?r) ≠ 0" using cd(2) mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp also have "… ⟷ (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r≠ 0" by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left) also have "… ⟷ - (?a * ?s) + 2*?d*?r ≠ 0" using cd(2) by simp finally show ?thesis using cd by (simp add: r[of "- (Itm vs (x # bs) s / (2 * ⦇d⦈⇩_{p}⇗^{vs⇖}))"] msubstneq_def Let_def evaldjf_ex) next case cd: 3 from cd(2) have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" by simp have "?rhs = Ifm vs (-?t / (2*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (-?t / (2*?c)) + ?r ≠ 0" by (simp add: r[of "- (?t/ (2 * ?c))"]) also have "… ⟷ 2*?c * (?a * (-?t / (2*?c)) + ?r) ≠ 0" using cd(1) mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp also have "… ⟷ (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r ≠ 0" by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left) also have "… ⟷ - (?a * ?t) + 2*?c*?r ≠ 0" using cd(1) by simp finally show ?thesis using cd by (simp add: r[of "- (?t/ (2*?c))"] msubstneq_def Let_def evaldjf_ex) next case cd: 4 then have cd2: "?c * ?d * 2 ≠ 0" by simp from add_frac_eq[OF cd, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r ≠ 0" by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) ≠ 0" using cd mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r ≠ 0" using nonzero_mult_div_cancel_left[OF cd2] cd by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps) qed qed definition "msubstlt c t d s a r = evaldjf (case_prod conj) [(let cd = c *⇩_{p}d in (lt (CP (~⇩_{p}cd)), Lt (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (let cd = c *⇩_{p}d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (lt (CP (~⇩_{p}c))) (Eq (CP d)), Lt (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP (~⇩_{p}d))) (Eq (CP c)), Lt (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (Eq (CP c)) (Eq (CP d)), Lt r)]" lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s" shows "bound0 (msubstlt c t d s a r)" proof - have th: "∀x∈ set [(let cd = c *⇩_{p}d in (lt (CP (~⇩_{p}cd)), Lt (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (let cd = c *⇩_{p}d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (lt (CP (~⇩_{p}c))) (Eq (CP d)), Lt (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP (~⇩_{p}d))) (Eq (CP c)), Lt (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (Eq (CP c)) (Eq (CP d)), Lt r)]. bound0 (case_prod conj x)" using lp by (simp add: Let_def t s lt_nb) from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def) qed lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))" shows "Ifm vs (x#bs) (msubstlt c t d s a r) ⟷ Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Lt (CNP 0 a r))" (is "?lhs = ?rhs") proof - let ?Nt = "λx t. Itm vs (x#bs) t" let ?N = "λp. Ipoly vs p" let ?c = "?N c" let ?d = "?N d" let ?t = "?Nt x t" let ?s = "?Nt x s" let ?a = "?N a" let ?r = "?Nt x r" from lp have lin:"isnpoly a" "a ≠ 0⇩_{p}" "tmbound0 r" "allpolys isnpoly r" by simp_all note r = tmbound0_I[OF lin(3), of vs _ bs x] consider "?c = 0" "?d = 0" | "?c * ?d > 0" | "?c * ?d < 0" | "?c > 0" "?d = 0" | "?c < 0" "?d = 0" | "?c = 0" "?d > 0" | "?c = 0" "?d < 0" by atomize_elim auto then show ?thesis proof cases case 1 then show ?thesis using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex) next case cd: 2 then have cd2: "2 * ?c * ?d > 0" by simp from cd have c: "?c ≠ 0" and d: "?d ≠ 0" by auto from cd2 have cd2': "¬ 2 * ?c * ?d < 0" by simp from add_frac_eq[OF c d, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0" by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) < 0" using cd2 cd2' mult_less_cancel_left_disj[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r < 0" using nonzero_mult_div_cancel_left[of "2*?c*?d"] c d by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd c d nc nd cd2' by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) next case cd: 3 then have cd2: "2 * ?c * ?d < 0" by (simp add: mult_less_0_iff field_simps) from cd have c: "?c ≠ 0" and d: "?d ≠ 0" by auto from add_frac_eq[OF c d, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s) / (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)) + ?r < 0" by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)) + ?r) > 0" using cd2 order_less_not_sym[OF cd2] mult_less_cancel_left_disj[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp also have "… ⟷ ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r < 0" using nonzero_mult_div_cancel_left[of "2 * ?c * ?d"] c d by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd c d nc nd by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) next case cd: 4 from cd(1) have c'': "2 * ?c > 0" by (simp add: zero_less_mult_iff) from cd(1) have c': "2 * ?c ≠ 0" by simp from cd(2) have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?t / (2 * ?c) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?t / (2 * ?c))+ ?r < 0" by (simp add: r[of "- (?t / (2 * ?c))"]) also have "… ⟷ 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) < 0" using cd(1) mult_less_cancel_left_disj[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp also have "… ⟷ - ?a * ?t + 2 * ?c * ?r < 0" using nonzero_mult_div_cancel_left[OF c'] ‹?c > 0› by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) finally show ?thesis using cd nc nd by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) next case cd: 5 from cd(1) have c': "2 * ?c ≠ 0" by simp from cd(1) have c'': "2 * ?c < 0" by (simp add: mult_less_0_iff) from cd(2) have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?t / (2*?c))+ ?r < 0" by (simp add: r[of "- (?t / (2*?c))"]) also have "… ⟷ 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) > 0" using cd(1) order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_less_cancel_left_disj[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp also have "… ⟷ ?a*?t - 2*?c *?r < 0" using nonzero_mult_div_cancel_left[OF c'] cd(1) order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd nc nd by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) next case cd: 6 from cd(2) have d'': "2 * ?d > 0" by (simp add: zero_less_mult_iff) from cd(2) have d': "2 * ?d ≠ 0" by simp from cd(1) have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?s / (2 * ?d))+ ?r < 0" by (simp add: r[of "- (?s / (2 * ?d))"]) also have "… ⟷ 2 * ?d * (?a * (- ?s / (2 * ?d))+ ?r) < 0" using cd(2) mult_less_cancel_left_disj[of "2 * ?d" "?a * (- ?s / (2 * ?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp also have "… ⟷ - ?a * ?s+ 2 * ?d * ?r < 0" using nonzero_mult_div_cancel_left[OF d'] cd(2) by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) finally show ?thesis using cd nc nd by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) next case cd: 7 from cd(2) have d': "2 * ?d ≠ 0" by simp from cd(2) have d'': "2 * ?d < 0" by (simp add: mult_less_0_iff) from cd(1) have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?s / (2 * ?d)) + ?r < 0" by (simp add: r[of "- (?s / (2 * ?d))"]) also have "… ⟷ 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) > 0" using cd(2) order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_less_cancel_left_disj[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp also have "… ⟷ ?a * ?s - 2 * ?d * ?r < 0" using nonzero_mult_div_cancel_left[OF d'] cd(2) order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally show ?thesis using cd nc nd by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) qed qed definition "msubstle c t d s a r = evaldjf (case_prod conj) [(let cd = c *⇩_{p}d in (lt (CP (~⇩_{p}cd)), Le (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (let cd = c *⇩_{p}d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (lt (CP (~⇩_{p}c))) (Eq (CP d)), Le (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP c)) (Eq (CP d)), Le (Sub (Mul a t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP (~⇩_{p}d))) (Eq (CP c)), Le (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (lt (CP d)) (Eq (CP c)), Le (Sub (Mul a s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (Eq (CP c)) (Eq (CP d)), Le r)]" lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s" shows "bound0 (msubstle c t d s a r)" proof - have th: "∀x∈ set [(let cd = c *⇩_{p}d in (lt (CP (~⇩_{p}cd)), Le (Add (Mul (~⇩_{p}a) (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (let cd = c *⇩_{p}d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)⇩_{p}*⇩_{p}cd) r)))), (conj (lt (CP (~⇩_{p}c))) (Eq (CP d)) , Le (Add (Mul (~⇩_{p}a) t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul ((2)⇩_{p}*⇩_{p}c) r))), (conj (lt (CP (~⇩_{p}d))) (Eq (CP c)) , Le (Add (Mul (~⇩_{p}a) s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul ((2)⇩_{p}*⇩_{p}d) r))), (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (case_prod conj x)" using lp by (simp add: Let_def t s lt_nb) from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def) qed lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))" shows "Ifm vs (x#bs) (msubstle c t d s a r) ⟷ Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Le (CNP 0 a r))" (is "?lhs = ?rhs") proof - let ?Nt = "λx t. Itm vs (x#bs) t" let ?N = "λp. Ipoly vs p" let ?c = "?N c" let ?d = "?N d" let ?t = "?Nt x t" let ?s = "?Nt x s" let ?a = "?N a" let ?r = "?Nt x r" from lp have lin:"isnpoly a" "a ≠ 0⇩_{p}" "tmbound0 r" "allpolys isnpoly r" by simp_all note r = tmbound0_I[OF lin(3), of vs _ bs x] have cd_cs: "?c * ?d < 0 ∨ ?c * ?d > 0 ∨ (?c = 0 ∧ ?d = 0) ∨ (?c = 0 ∧ ?d < 0) ∨ (?c = 0 ∧ ?d > 0) ∨ (?c < 0 ∧ ?d = 0) ∨ (?c > 0 ∧ ?d = 0)" by auto moreover { assume c: "?c = 0" and d: "?d = 0" then have ?thesis using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex) } moreover { assume dc: "?c * ?d > 0" from dc have dc': "2 * ?c * ?d > 0" by simp then have c: "?c ≠ 0" and d: "?d ≠ 0" by auto from dc' have dc'': "¬ 2 * ?c * ?d < 0" by simp from add_frac_eq[OF c d, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r ≤ 0" by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) ≤ 0" using dc' dc'' mult_le_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r ≤ 0" using nonzero_mult_div_cancel_left[of "2*?c*?d"] c d by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally have ?thesis using dc c d nc nd dc' by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } moreover { assume dc: "?c * ?d < 0" from dc have dc': "2 * ?c * ?d < 0" by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos) then have c: "?c ≠ 0" and d: "?d ≠ 0" by auto from add_frac_eq[OF c d, of "- ?t" "- ?s"] have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r ≤ 0" by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"]) also have "… ⟷ (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) ≥ 0" using dc' order_less_not_sym[OF dc'] mult_le_cancel_left[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp also have "… ⟷ ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r ≤ 0" using nonzero_mult_div_cancel_left[of "2 * ?c * ?d"] c d by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally have ?thesis using dc c d nc nd by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } moreover { assume c: "?c > 0" and d: "?d = 0" from c have c'': "2 * ?c > 0" by (simp add: zero_less_mult_iff) from c have c': "2 * ?c ≠ 0" by simp from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?t / (2 * ?c))+ ?r ≤ 0" by (simp add: r[of "- (?t / (2 * ?c))"]) also have "… ⟷ 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) ≤ 0" using c mult_le_cancel_left[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp also have "… ⟷ - ?a*?t+ 2*?c *?r ≤ 0" using nonzero_mult_div_cancel_left[OF c'] c by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) finally have ?thesis using c d nc nd by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } moreover { assume c: "?c < 0" and d: "?d = 0" then have c': "2 * ?c ≠ 0" by simp from c have c'': "2 * ?c < 0" by (simp add: mult_less_0_iff) from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?t / (2*?c))+ ?r ≤ 0" by (simp add: r[of "- (?t / (2*?c))"]) also have "… ⟷ 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) ≥ 0" using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_le_cancel_left[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp also have "… ⟷ ?a * ?t - 2 * ?c * ?r ≤ 0" using nonzero_mult_div_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally have ?thesis using c d nc nd by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } moreover { assume c: "?c = 0" and d: "?d > 0" from d have d'': "2 * ?d > 0" by (simp add: zero_less_mult_iff) from d have d': "2 * ?d ≠ 0" by simp from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?s / (2 * ?d) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a * (- ?s / (2 * ?d))+ ?r ≤ 0" by (simp add: r[of "- (?s / (2*?d))"]) also have "… ⟷ 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) ≤ 0" using d mult_le_cancel_left[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp also have "… ⟷ - ?a * ?s + 2 * ?d * ?r ≤ 0" using nonzero_mult_div_cancel_left[OF d'] d by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right) finally have ?thesis using c d nc nd by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } moreover { assume c: "?c = 0" and d: "?d < 0" then have d': "2 * ?d ≠ 0" by simp from d have d'': "2 * ?d < 0" by (simp add: mult_less_0_iff) from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)" by (simp add: field_simps) have "?rhs ⟷ Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" by (simp only: th) also have "… ⟷ ?a* (- ?s / (2*?d))+ ?r ≤ 0" by (simp add: r[of "- (?s / (2*?d))"]) also have "… ⟷ 2*?d * (?a* (- ?s / (2*?d))+ ?r) ≥ 0" using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_le_cancel_left[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp also have "… ⟷ ?a * ?s - 2 * ?d * ?r ≤ 0" using nonzero_mult_div_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' by (simp add: algebra_simps diff_divide_distrib del: distrib_right) finally have ?thesis using c d nc nd by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) } ultimately show ?thesis by blast qed fun msubst :: "fm ⇒ (poly × tm) × (poly × tm) ⇒ fm" where "msubst (And p q) ((c, t), (d, s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))" | "msubst (Or p q) ((c, t), (d, s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))" | "msubst (Eq (CNP 0 a r)) ((c, t), (d, s)) = msubsteq c t d s a r" | "msubst (NEq (CNP 0 a r)) ((c, t), (d, s)) = msubstneq c t d s a r" | "msubst (Lt (CNP 0 a r)) ((c, t), (d, s)) = msubstlt c t d s a r" | "msubst (Le (CNP 0 a r)) ((c, t), (d, s)) = msubstle c t d s a r" | "msubst p ((c, t), (d, s)) = p" lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d" shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) = Ifm vs (((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) p" using lp by (induct p rule: islin.induct) (auto simp add: tmbound0_I [where b = "(- (Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}) - (Itm vs (x # bs) s / ⦇d⦈⇩_{p}⇗^{vs⇖})) / 2" and b' = x and bs = bs and vs = vs] msubsteq msubstneq msubstlt [OF nc nd] msubstle [OF nc nd]) lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s" shows "bound0 (msubst p ((c,t),(d,s)))" using lp t s by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb) lemma fr_eq_msubst: assumes lp: "islin p" shows "(∃x. Ifm vs (x#bs) p) ⟷ (Ifm vs (x#bs) (minusinf p) ∨ Ifm vs (x#bs) (plusinf p) ∨ (∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). Ifm vs (x#bs) (msubst p ((c, t), (d, s)))))" (is "(∃x. ?I x p) = (?M ∨ ?P ∨ ?F)" is "?E = ?D") proof - from uset_l[OF lp] have th: "∀(c, s)∈set (uset p). isnpoly c ∧ tmbound0 s" by blast { fix c t d s assume ctU: "(c, t) ∈set (uset p)" and dsU: "(d,s) ∈set (uset p)" and pts: "Ifm vs ((- Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}+ - Itm vs (x # bs) s / ⦇d⦈⇩_{p}⇗^{vs⇖}) / 2 # bs) p" from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all from msubst_I[OF lp norm, of vs x bs t s] pts have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" .. } moreover { fix c t d s assume ctU: "(c, t) ∈ set (uset p)" and dsU: "(d,s) ∈set (uset p)" and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))" from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all from msubst_I[OF lp norm, of vs x bs t s] pts have "Ifm vs ((- Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}+ - Itm vs (x # bs) s / ⦇d⦈⇩_{p}⇗^{vs⇖}) / 2 # bs) p" .. } ultimately have th': "(∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). Ifm vs ((- Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}+ - Itm vs (x # bs) s / ⦇d⦈⇩_{p}⇗^{vs⇖}) / 2 # bs) p) ⟷ ?F" by blast from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis . qed lemma simpfm_lin: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" shows "qfree p ⟹ islin (simpfm p)" by (induct p rule: simpfm.induct) (auto simp add: conj_lin disj_lin) definition "ferrack p ≡ let q = simpfm p; mp = minusinf q; pp = plusinf q in if (mp = T ∨ pp = T) then T else (let U = alluopairs (remdups (uset q)) in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))" lemma ferrack: assumes qf: "qfree p" shows "qfree (ferrack p) ∧ Ifm vs bs (ferrack p) = Ifm vs bs (E p)" (is "_ ∧ ?rhs = ?lhs") proof - let ?I = "λx p. Ifm vs (x#bs) p" let ?N = "λt. Ipoly vs t" let ?Nt = "λx t. Itm vs (x#bs) t" let ?q = "simpfm p" let ?U = "remdups(uset ?q)" let ?Up = "alluopairs ?U" let ?mp = "minusinf ?q" let ?pp = "plusinf ?q" fix x let ?I = "λp. Ifm vs (x#bs) p" from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" . from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" . from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" . from uset_l[OF lq] have U_l: "∀(c, s)∈set ?U. isnpoly c ∧ c ≠ 0⇩_{p}∧ tmbound0 s ∧ allpolys isnpoly s" by simp { fix c t d s assume ctU: "(c, t) ∈ set ?U" and dsU: "(d,s) ∈ set ?U" from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto from msubst_I[OF lq norm, of vs x bs t s] msubst_I[OF lq norm(2,1), of vs x bs s t] have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps) } then have th0: "∀x ∈ set ?U. ∀y ∈ set ?U. ?I (msubst ?q (x, y)) ⟷ ?I (msubst ?q (y, x))" by auto { fix x assume xUp: "x ∈ set ?Up" then obtain c t d s where ctU: "(c, t) ∈ set ?U" and dsU: "(d,s) ∈ set ?U" and x: "x = ((c, t),(d, s))" using alluopairs_set1[of ?U] by auto from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] have nbs: "tmbound0 t" "tmbound0 s" by simp_all from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp } with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"] have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast with mp_nb pp_nb have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by simp from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def) have "?lhs ⟷ (∃x. Ifm vs (x#bs) ?q)" by simp also have "… ⟷ ?I ?mp ∨ ?I ?pp ∨ (∃(c, t)∈set ?U. ∃(d, s)∈set ?U. ?I (msubst (simpfm p) ((c, t), d, s)))" using fr_eq_msubst[OF lq, of vs bs x] by simp also have "… ⟷ ?I ?mp ∨ ?I ?pp ∨ (∃(x, y) ∈ set ?Up. ?I ((simpfm ∘ msubst ?q) (x, y)))" using alluopairs_bex[OF th0] by simp also have "… ⟷ ?I ?mp ∨ ?I ?pp ∨ ?I (evaldjf (simpfm ∘ msubst ?q) ?Up)" by (simp add: evaldjf_ex) also have "… ⟷ ?I (disj ?mp (disj ?pp (evaldjf (simpfm ∘ msubst ?q) ?Up)))" by simp also have "… ⟷ ?rhs" using decr0[OF th1, of vs x bs] apply (simp add: ferrack_def Let_def) apply (cases "?mp = T ∨ ?pp = T") apply auto done finally show ?thesis using thqf by blast qed definition "frpar p = simpfm (qelim p ferrack)" lemma frpar: "qfree (frpar p) ∧ (Ifm vs bs (frpar p) ⟷ Ifm vs bs p)" proof - from ferrack have th: "∀bs p. qfree p ⟶ qfree (ferrack p) ∧ Ifm vs bs (ferrack p) = Ifm vs bs (E p)" by blast from qelim[OF th, of p bs] show ?thesis unfolding frpar_def by auto qed section ‹Second implemenation: Case splits not local› lemma fr_eq2: assumes lp: "islin p" shows "(∃x. Ifm vs (x#bs) p) ⟷ (Ifm vs (x#bs) (minusinf p) ∨ Ifm vs (x#bs) (plusinf p) ∨ Ifm vs (0#bs) p ∨ (∃(n, t) ∈ set (uset p). Ipoly vs n ≠ 0 ∧ Ifm vs ((- Itm vs (x#bs) t / (Ipoly vs n * 2))#bs) p) ∨ (∃(n, t) ∈ set (uset p). ∃(m, s) ∈ set (uset p). Ipoly vs n ≠ 0 ∧ Ipoly vs m ≠ 0 ∧ Ifm vs (((- Itm vs (x#bs) t / Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /2)#bs) p))" (is "(∃x. ?I x p) = (?M ∨ ?P ∨ ?Z ∨ ?U ∨ ?F)" is "?E = ?D") proof assume px: "∃x. ?I x p" have "?M ∨ ?P ∨ (¬ ?M ∧ ¬ ?P)" by blast moreover { assume "?M ∨ ?P" then have "?D" by blast } moreover { assume nmi: "¬ ?M" and npi: "¬ ?P" from inf_uset[OF lp nmi npi, OF px] obtain c t d s where ct: "(c, t) ∈ set (uset p)" "(d, s) ∈ set (uset p)" "?I ((- Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}+ - Itm vs (x # bs) s / ⦇d⦈⇩_{p}⇗^{vs⇖}) / (1 + 1)) p" by auto let ?c = "⦇c⦈⇩_{p}⇗^{vs⇖}" let ?d = "⦇d⦈⇩_{p}⇗^{vs⇖}" let ?s = "Itm vs (x # bs) s" let ?t = "Itm vs (x # bs) t" have eq2: "⋀(x::'a). x + x = 2 * x" by (simp add: field_simps) { assume "?c = 0 ∧ ?d = 0" with ct have ?D by simp } moreover { assume z: "?c = 0" "?d ≠ 0" from z have ?D using ct by auto } moreover { assume z: "?c ≠ 0" "?d = 0" with ct have ?D by auto } moreover { assume z: "?c ≠ 0" "?d ≠ 0" from z have ?F using ct apply - apply (rule bexI[where x = "(c,t)"]) apply simp_all apply (rule bexI[where x = "(d,s)"]) apply simp_all done then have ?D by blast } ultimately have ?D by auto } ultimately show "?D" by blast next assume "?D" moreover { assume m: "?M" from minusinf_ex[OF lp m] have "?E" . } moreover { assume p: "?P" from plusinf_ex[OF lp p] have "?E" . } moreover { assume f:"?F" then have "?E" by blast } ultimately show "?E" by blast qed definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))" definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))" definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))" definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))" definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))" lemma msubsteq2: assumes nz: "Ipoly vs c ≠ 0" and l: "islin (Eq (CNP 0 a b))" shows "Ifm vs (x#bs) (msubsteq2 c t a b) = Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Eq (CNP 0 a b))" using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}", symmetric] by (simp add: msubsteq2_def field_simps) lemma msubstltpos: assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))" shows "Ifm vs (x#bs) (msubstltpos c t a b) = Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))" using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}", symmetric] by (simp add: msubstltpos_def field_simps) lemma msubstlepos: assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))" shows "Ifm vs (x#bs) (msubstlepos c t a b) = Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Le (CNP 0 a b))" using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}", symmetric] by (simp add: msubstlepos_def field_simps) lemma msubstltneg: assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))" shows "Ifm vs (x#bs) (msubstltneg c t a b) = Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))" using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}", symmetric] by (simp add: msubstltneg_def field_simps del: minus_add_distrib) lemma msubstleneg: assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))" shows "Ifm vs (x#bs) (msubstleneg c t a b) = Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Le (CNP 0 a b))" using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}", symmetric] by (simp add: msubstleneg_def field_simps del: minus_add_distrib) fun msubstpos :: "fm ⇒ poly ⇒ tm ⇒ fm" where "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)" | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)" | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r" | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)" | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r" | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r" | "msubstpos p c t = p" lemma msubstpos_I: assumes lp: "islin p" and pos: "Ipoly vs c > 0" shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" using lp pos by (induct p rule: islin.induct) (auto simp add: msubsteq2 msubstltpos[OF pos] msubstlepos[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}" bs x] field_simps) fun msubstneg :: "fm ⇒ poly ⇒ tm ⇒ fm" where "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)" | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)" | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r" | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)" | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r" | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r" | "msubstneg p c t = p" lemma msubstneg_I: assumes lp: "islin p" and pos: "Ipoly vs c < 0" shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" using lp pos by (induct p rule: islin.induct) (auto simp add: msubsteq2 msubstltneg[OF pos] msubstleneg[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / ⦇c⦈⇩_{p}⇗^{vs⇖}" bs x] field_simps) definition "msubst2 p c t = disj (conj (lt (CP (polyneg c))) (simpfm (msubstpos p c t))) (conj (lt (CP c)) (simpfm (msubstneg p c t)))" lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c ≠ 0" shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t / Ipoly vs c #bs) p" proof - let ?c = "Ipoly vs c" from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~⇩_{p}c))" by (simp_all add: polyneg_norm) from nz have "?c > 0 ∨ ?c < 0" by arith moreover { assume c: "?c < 0" from c msubstneg_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"] have ?thesis by (auto simp add: msubst2_def) } moreover { assume c: "?c > 0" from c msubstpos_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"] have ?thesis by (auto simp add: msubst2_def) } ultimately show ?thesis by blast qed lemma msubsteq2_nb: "tmbound0 t ⟹ islin (Eq (CNP 0 a r)) ⟹ bound0 (msubsteq2 c t a r)" by (simp add: msubsteq2_def) lemma msubstltpos_nb: "tmbound0 t ⟹ islin (Lt (CNP 0 a r)) ⟹ bound0 (msubstltpos c t a r)" by (simp add: msubstltpos_def) lemma msubstltneg_nb: "tmbound0 t ⟹ islin (Lt (CNP 0 a r)) ⟹ bound0 (msubstltneg c t a r)" by (simp add: msubstltneg_def) lemma msubstlepos_nb: "tmbound0 t ⟹ islin (Le (CNP 0 a r)) ⟹ bound0 (msubstlepos c t a r)" by (simp add: msubstlepos_def) lemma msubstleneg_nb: "tmbound0 t ⟹ islin (Le (CNP 0 a r)) ⟹ bound0 (msubstleneg c t a r)" by (simp add: msubstleneg_def) lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t" shows "bound0 (msubstpos p c t)" using lp tnb by (induct p c t rule: msubstpos.induct) (auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb) lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" and lp: "islin p" and tnb: "tmbound0 t" shows "bound0 (msubstneg p c t)" using lp tnb by (induct p c t rule: msubstneg.induct) (auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb) lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0,field})" and lp: "islin p" and tnb: "tmbound0 t" shows "bound0 (msubst2 p c t)" using lp tnb by (simp add: msubst2_def msubstneg_nb msubstpos_nb lt_nb simpfm_bound0) lemma mult_minus2_left: "-2 * (x::'a::comm_ring_1) = - (2 * x)" by simp lemma mult_minus2_right: "(x::'a::comm_ring_1) * -2 = - (x * 2)" by simp lemma islin_qf: "islin p ⟹ qfree p" by (induct p rule: islin.induct) (auto simp add: bound0_qf) lemma fr_eq_msubst2: assumes lp: "islin p" shows "(∃x. Ifm vs (x#bs) p) ⟷ ((Ifm vs (x#bs) (minusinf p)) ∨ (Ifm vs (x#bs) (plusinf p)) ∨ Ifm vs (x#bs) (subst0 (CP 0⇩_{p}) p) ∨ (∃(n, t) ∈ set (uset p). Ifm vs (x# bs) (msubst2 p (n *⇩_{p}(C (-2,1))) t)) ∨ (∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *⇩_{p}c*⇩_{p}d) (Add (Mul d t) (Mul c s)))))" (is "(∃x. ?I x p) = (?M ∨ ?P ∨ ?Pz ∨ ?PU ∨ ?F)" is "?E = ?D") proof - from uset_l[OF lp] have th: "∀(c, s)∈set (uset p). isnpoly c ∧ tmbound0 s" by blast let ?I = "λp. Ifm vs (x#bs) p" have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def) note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0⇩_{p}", simplified] have eq1: "(∃(n, t) ∈ set (uset p). ?I (msubst2 p (n *⇩_{p}(C (-2,1))) t)) ⟷ (∃(n, t) ∈ set (uset p). ⦇n⦈⇩_{p}⇗^{vs⇖}≠ 0 ∧ Ifm vs (- Itm vs (x # bs) t / (⦇n⦈⇩_{p}⇗^{vs⇖}* 2) # bs) p)" proof - { fix n t assume H: "(n, t) ∈ set (uset p)" "?I(msubst2 p (n *⇩_{p}C (-2, 1)) t)" from H(1) th have "isnpoly n" by blast then have nn: "isnpoly (n *⇩_{p}(C (-2,1)))" by (simp_all add: polymul_norm n2) have nn': "allpolys isnpoly (CP (~⇩_{p}(n *⇩_{p}C (-2, 1))))" by (simp add: polyneg_norm nn) then have nn2: "⦇n *⇩_{p}(C (-2,1)) ⦈⇩_{p}⇗^{vs⇖}≠ 0" "⦇n ⦈⇩_{p}⇗^{vs⇖}≠ 0" using H(2) nn' nn by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff) from msubst2[OF lp nn nn2(1), of x bs t] have "⦇n⦈⇩_{p}⇗^{vs⇖}≠ 0 ∧ Ifm vs (- Itm vs (x # bs) t / (⦇n⦈⇩_{p}⇗^{vs⇖}* 2) # bs) p" using H(2) nn2 by (simp add: mult_minus2_right) } moreover { fix n t assume H: "(n, t) ∈ set (uset p)" "⦇n⦈⇩_{p}⇗^{vs⇖}≠ 0" "Ifm vs (- Itm vs (x # bs) t / (⦇n⦈⇩_{p}⇗^{vs⇖}* 2) # bs) p" from H(1) th have "isnpoly n" by blast then have nn: "isnpoly (n *⇩_{p}(C (-2,1)))" "⦇n *⇩_{p}(C (-2,1)) ⦈⇩_{p}⇗^{vs⇖}≠ 0" using H(2) by (simp_all add: polymul_norm n2) from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *⇩_{p}(C (-2,1))) t)" using H(2,3) by (simp add: mult_minus2_right) } ultimately show ?thesis by blast qed have eq2: "(∃(c, t) ∈ set (uset p). ∃(d, s) ∈ set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *⇩_{p}c*⇩_{p}d) (Add (Mul d t) (Mul c s)))) ⟷ (∃(n, t)∈set (uset p). ∃(m, s)∈set (uset p). ⦇n⦈⇩_{p}⇗^{vs⇖}≠ 0 ∧ ⦇m⦈⇩_{p}⇗^{vs⇖}≠ 0 ∧ Ifm vs ((- Itm vs (x # bs) t / ⦇n⦈⇩_{p}⇗^{vs⇖}+ - Itm vs (x # bs) s / ⦇m⦈⇩_{p}⇗^{vs⇖}) / 2 # bs) p)" proof - { fix c t d s assume H: "(c,t) ∈ set (uset p)" "(d,s) ∈ set (uset p)" "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *⇩_{p}c*⇩_{p}d) (Add (Mul d t) (Mul c s)))" from H(1,2) th have "isnpoly c" "isnpoly d" by blast+